thesis_esch2.

thesis_esch2.
Adaptive Multiscale Finite Element
Method for Subsurface Flow Simulation
Adaptive Multiscale Finite Element
Method for Subsurface Flow Simulation
PROEFSCHRIFT
ter verkrijging van de graad van doctor
aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus, prof.ir. K.C.A.M. Luyben,
voorzitter van het College voor Promoties,
in het openbaar te verdedigen op maandag 15 november 2010 om 15:00 uur
door Johannes Maria VAN ESCH
civiel ingenieur
geboren te Heemstede
Dit proefschrift is goedgekeurd door de promotor:
Prof.dr.ir. F.B.J. Barends
Samenstelling promotie-commissie:
Rector Magnificus,
Technische Universiteit
Prof.dr.ir. F.B.J. Barends,
Technische Universiteit
Prof.dr. J. Bruining,
Technische Universiteit
Prof.dr. M.A. Hicks,
Technische Universiteit
Prof.dr.ir. T.N. Olsthoorn, Technische Universiteit
Prof.dr. R.J. Schotting,
Universiteit Utrecht
Prof.dr.-ing. P.A. Vermeer, Universität Stuttgart
Dr.ir. J.A.M. Teunissen,
Deltares
Delft, voorzitter
Delft, promotor
Delft
Delft
Delft
Printed by:
Wöhrmann Print Service
P.O. Box 92
7202 CZ Zutphen
The Netherlands
Telephone: +31 575 58 53 00
E-mail: [email protected]
ISBN 978-90-8570-608-3
c 2010 by J.M. van Esch
All rights reserved. No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or mechanical, including
photocopying, recording or by any information storage and retrieval system, without
written consent of the publisher.
Printed in The Netherlands
Summary
Natural geological formations generally show multiscale structural and functional heterogeneity evolving over many orders of magnitude in space and time. In subsurface
hydrological simulations the geological model focuses on the structural hierarchy of
physical sub units and the flow model addresses the functional hierarchy of the flow
process. Flow quantities like pressure, flux and dissipation relate to each other by
constitutive relations and structural sub-unit parameters like porosity and hydraulic
permeability. Hydraulic permeability includes the (steady state) intrinsic permeability
of the solid phase and the (time dependent) relative permeability. The permeability is
the dominant parameter and a highly heterogeneous parameter affecting the groundwater flow at field scale.
Laboratory experiments provide measurements of the sub-unit parameters on a
fine scale. If laboratory measurements are treated stochastically within the geological
model, then the structural model of the subsurface should be built on the same scale
as these indicating measurements. Fully resolved flow simulations on the field scale
are however intractable and a new adaptive multiscale technique has therefore been
developed. Though constitutive relations may change at different scales, Darcy’s law
is supposed to remain valid on both laboratory scale and field scale.
Nowadays upscaling methods are applied, which aim to propagate information over
this hierarchy of scales both functionally and structurally from the fine scale to the
coarse scale and not vise versa, by computing effective or equivalent material behavior.
Permeability is not an additive parameter so it is not possible to calculate an equivalent
coarse scale permeability as a simple average of fine scale measurements. A flow criterion or a criterion about energy dissipation often defines an equivalent permeability.
Only if the scale of variation is much smaller than the coarse observation scale then the
equivalent permeability matches the effective permeability. This effective permeability
is a constant second order tensor variable on the coarse scale, whereas the equivalent
permeability is non-unique and depends on the boundary conditions of the sample domain. The effective permeability holds for discrete hierarchical systems where scales
can be decomposed.
The newly developed adaptive multiscale technique extends the original two-level
multiscale finite element method to a hierarchy of scales. Multiscale finite element
methods capture the fine scale behavior on a coarse mesh by multiscale basis functions. The weights of the multiscale basis functions follow from local flow simulation.
The procedure removes fine scale nodes from the subdomain, but introduces errors at
the subdomain boundaries. The two-level method forms a class of subdomain decomv
vi
Summary
position techniques. It can be shown that a sequential implementation of the method
is not faster than an optimal solver like the full multigrid solver, however the method
is suitable for parallel implementation.
The proposed method is based on a conformal nodal finite element formulation over
simplex elements. The conformal finite element method obtains mass conservation on a
nodal basis, and does not preserve continuity of flux over the inter-element boundaries.
A mesh refinement criterion detects zones in which large errors occur over the element
edges, and an adaptive refinement procedure enriches the mesh locally to correct the
error in the velocity field. Multiscale basis functions follow from solving local flow
problems over patches of simplex elements. Linear boundary conditions close oversampled subdomains and reduce the effect of the imposed boundary condition on the
patch. This procedure obtains more accurate coarse scale behavior then a procedure
that operates on the patch directly. However, over-sampled local flow problems introduce discontinuities in the basis functions and introduce new nodal connectivities on
the coarser scale. For this reason closure of the local flow problems by dimensionally reduced flow problems is preferred. A second refinement criterion compares oversampled
and non-oversampled function values. Pressure-dissipation averaging approximates the
multiscale coarse grid operator and supports a functional adaptive formulation. The
multiscale averaging procedure computes equivalent permeability tensor components,
and reproduces the sparse matrix structure on the coarse scale. The loading cases for
the local problems follow from a summation of multiscale basis functions. The multiscale basis functions extrapolate the coarse scale solution to the fine scale. On this
scale discontinuities in the velocity field are detected and compared to the refinement
criterion. The computed equivalent permeability is used in the framework of a geometric multigrid solver to compute the coarse scale operators. The hierarchy of multiscale
basis functions, which relate the pressure on each coarse level to the next fine level,
generates the intergrid transfer operators.
The proposed approach provides a robust and efficient algorithm, based on the concept of the multiscale finite element method, for simulating partly saturated subsurface
flow and fully coupled solute transport and heat transport through hierarchical heterogeneous formations. The multiscale finite element formulation produces numerically
homogenized discrete flow equations, and upscales the permeability. The adaptive
formulation obtains locally refined velocity fields, which support accurate transport
computations. The performance of the method is illustrated by a set of realistic case
studies.
John van Esch.
Contents
Summary
v
1 Introduction
1
2 Mathematical model
2.1 Balance equations . . . . . . . . . . .
2.1.1 Mass balance equation . . . .
2.1.2 Momentum balance equation .
2.1.3 Energy balance equation . . .
2.2 Constitutive relations . . . . . . . . .
2.2.1 Porosity . . . . . . . . . . . .
2.2.2 Saturation . . . . . . . . . . .
2.2.3 Permeability . . . . . . . . . .
2.2.4 Density . . . . . . . . . . . . .
2.2.5 Viscosity . . . . . . . . . . . .
2.3 Problem definition . . . . . . . . . . .
2.3.1 System of differential equations
2.3.2 Initial conditions . . . . . . . .
2.3.3 Boundary conditions . . . . .
3 Numerical model
3.1 Basic equations . . . . . . . . . . .
3.1.1 Flow equation . . . . . . . .
3.1.2 Solute transport equation . .
3.1.3 Heat transport equation . .
3.2 Problem definition . . . . . . . . . .
3.2.1 System of algebraic equations
3.2.2 Initial conditions . . . . . . .
3.2.3 Boundary conditions . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. .
. .
.
.
.
.
.
.
. .
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
9
10
10
12
15
17
18
20
22
23
24
24
24
28
28
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
33
34
34
40
45
49
50
58
59
4 Adaptive multiscale method
61
4.1 Adaptive formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.1 Discrete adaptivity . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1.2 Functional adaptivity . . . . . . . . . . . . . . . . . . . . . . . . 70
vii
viii
Contents
4.2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5 Model verification
5.1 Saturated and unsaturated flow . . .
5.1.1 Lamb and Whitman’s problem
5.1.2 Forsyth et al’s problem . . . .
5.2 Solute transport and heat transport .
5.2.1 Henry’s problem . . . . . . . .
5.2.2 Elder’s heat problem . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
89
. 90
. 90
. 94
. 98
. 99
. 101
6 Flow simulation
6.1 Marine system . . . . .
6.1.1 Geological model
6.1.2 Flow model . . .
6.2 Fluvial system . . . . .
6.2.1 Geological model
6.2.2 Flow model . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4.3
Multiscale formulation . . . . .
4.2.1 Multiscale basis functions
4.2.2 Coarse mesh equations . .
Multigrid formulation . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
74
75
81
82
107
108
108
109
114
115
117
7 Conclusions
121
A Continuum mechanics
123
B Finite element method
137
C Multigrid method
153
Notations
173
Bibliography
177
Samenvatting
187
Chapter 1
Introduction
Natural earth materials are of irregular and complex nature [55, 31]. In general soils
are not homogeneous, nor uniformly random but may contain multiple, nested, natural length and time scales or even continuous evolving scales [29]. The multiscale
heterogeneity evolves over many orders of magnitude in space and time. As for many
multiscale problems two types of heterogeneity are often considered: structural heterogeneity and functional heterogeneity. Structural heterogeneity deals with the hierarchy of physical sub units and functional heterogeneity deals with the hierarchy of
processes. Both structural and functional hierarchies are mostly strongly connected
and may be discrete or continuous. The discrete type incorporates a finite number of
nested structural subunits or subprocesses. The number of subunits and subprocesses
goes to infinity without a clear cut decomposition for the continuous type. Periodic
porous media for instance show a discrete hierarchy and fractal porous media show a
continuous hierarchy.
In subsurface hydrological simulations, the geological model focuses on the structural hierarchy of physical subunits and the flow model addresses the functional hierarchy of the flow process. Flow quantities like pressure, flux and dissipation relate
to each other by constitutive relations and structural subunit parameters like porosity
and hydraulic permeability. Porosity generally varies over a single order of magnitude.
The hydraulic permeability however may vary over eight orders of magnitude due to
variations in soil structure and soil type or in the degree of saturation. The scale of
variability of the permeability may be a fraction of a meter, and time variations may
take place over a period of an hour. Figure 1.1 presents two single scale periodic permeability fields. Both single scale fields contribute (by adding the logarithmic values
attached to their subunits) to the multiscale periodic field shown in the most right
picture.
In heterogeneous fields the window of observation ld and the resolution of the mead
surement (l/n) may alter the measured value of the field variable. Here, l is length,
d corresponds to the dimension and n denotes the number of pixels in one dimension.
An apparatus may for instance consider a pixel to be totally void if the void space
intersects the pixel area, and totally solid if not. Under these assumptions the porosity
is the weighted sum of the pixel areas labeled void divided by the observation area.
1
2
1. Introduction
field 1
field 2
1.0E-03
field 3
1.0E+00
Figure 1.1: Periodic permeability fields.
The void space of the porous medium is fractal if the porosity goes to zero as the
number of pixels nd goes to infinity. If the porosity goes to one then the solid space is
generally fractal. Figure 1.2 presents a fractal field. The resolution has been increased
from the left to the right (low porosity values prevail). If not fractal then the porosity
field 1
field 2
1.0E-02
field 3
1.0E+00
Figure 1.2: Fractal porosity fields.
goes monotonically to some constant average if the number of observation pixels goes
to infinity. For a common focal point and a fixed resolution the window of observation
may also be changed. The porosity as a function of observation area may show one or
more plateaux, if it possesses a discrete hierarchy and the porosity has one or more natural scales. These asymptotic limits correspond to representative elementary volumes
(REV). If there are no plateaus then the heterogeneity is said to evolve with respect to
the observation area of the instrument. Figure 1.3 presents the state averages of the
periodic fields of Figure 1.1 and the fractal fields of Figure 1.2. A porous medium may
possess both fractal characteristics and natural scales.
Heterogeneity may also be observed in different sampling locations and different
sampling frequencies. This may be measured explicitly by multiple experiments over a
field with a constant window of observation and resolution. Still, if the samples show
the same results, heterogeneity may be implicitly present due to scale constants of the
experimental equipment or the choice of the sample locations. Only if the correlation
scale is much less than the size of the flow domain, statistical stationarity prevails.
3
1.0
0.8
0.8
mean state (-)
mean state (-)
periodic field data
1.0
0.6
0.4
0.2
fractal field data
0.6
0.4
0.2
0.0
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
sampling area (-)
0.2
0.4
0.6
0.8
1.0
sampling area (-)
field 1
field 2
field 3
Figure 1.3: State average for periodic and fractal fields.
For these conditions, a statistical framework of porous medium variables can be built
by random space functions. Then the actual porous medium corresponds to a single
realization of the random space functions ensemble. The space average is often captured by the assemble average under the ergodic hypothesis [33]. As measuring devices
that provide a fully resolved deterministic geologic description of the subsurface are
not available, the field has to be based on a set of measurements at different locations.
Geo-statistical methods then generate the geological model on field scale. Laboratory
experiments provide measurements of the subunit parameters at a fine scale. If laboratory measurements are treated stochastically within the geological model, then the
structural model of the subsurface should be built on the same scale as these supporting measurements. Geo-statistical methods generate flow problems with strongly
discontinuous coefficients. Figure 1.4 gives an example of the intrinsic permeability
variation on field scale. At the present, fully resolved flow simulations are computationally intractable at this resolution, and approximated flow problems are solved
instead using upscaling techniques or more general multiscale methods. Solving the
upscaled flow problems still requires an efficient solution technique like multigrid or
adaptive gridding.
Well established commercially available codes for solving partly saturated flow problems coupled to solute transport and heat transport in the subsoil include: Sutra [118],
Hst3D [78], Feflow [39] and Plaxflow [20]. All of them simulate both steady state and
transient flow in a three-dimensional domain. None of these codes however, are capable of dealing with heterogeneous permeability fields which show a large contrast, and
most of them operate on fixed meshes or grids.
Upscaling techniques
The development of upscaling techniques, particularly in petroleum reservoir engineering, has been motivated by geo-statistical reservoir description algorithms [6, 8, 42,
43, 64, 96, 97, 23]. Nowadays, upscaling methods are applied that aim to propagate
information over the hierarchy of scales both functionally and structurally from fine
4
1. Introduction
1.0E-18
1.0E-10
Figure 1.4: Intrinsic permeability field (m2 ) for layer 36 of the tenth SPE comparative
solution project [26].
scale to coarse scale and not vise versa. In subsurface flow simulations the permeability
is the dominant parameter and a highly heterogeneous parameter affecting the subsurface flow at field scale. Permeability is not an additive parameter so it is not possible
to calculate an equivalent coarse scale permeability as a simple average of fine scale
measurements. A complete equivalence between the heterogeneous medium and the
artificial homogeneous medium is impossible. Therefore a flow criterion or a criterion
on the energy dissipation often defines an equivalent permeability. Only if the scale
of variation is much smaller than the coarse scale then the equivalent permeability
matches the effective permeability. The effective permeability is a constant second
order tensor variable on the coarse scale, and applies to a medium that is statistically
homogeneous on the large scale [101]. Effective permeability only holds for discrete
hierarchical systems where scales can be decomposed. On a finite-size block the concept of statistical homogeneity does not hold if the observed block is too small. Here
the upscaled permeability or block permeability is the equivalent permeability of this
finite-size volume [101]. The upscaled permeability is non-unique and depends on the
boundary conditions of the sample domain.
An overview of scaling techniques can be found in literature [30, 101]. Power averaging, renormalization and homogenization will be mentioned here. Power averaging
generalizes simple averages like arithmetic mean, geometric mean and harmonic mean.
A more advanced averaging technique is renormalization. This technique locally computes a scalar permeability coefficient by treating the medium as a resistance network
[76, 77, 54]. The method of homogenization assumes a periodic structure of the permeability coefficients [67, 94]. The effective permeability then follows from a local flow
computation on a domain subjected to periodic boundary conditions. Numerical homogenization techniques either apply local scaling [129, 42, 7] or global scaling [16, 65].
In addition global-local scaling techniques [24, 56, 86, 44] first compute local averages
5
and then replace them by global flow computation averages, if necessary, to obtain
a predefined degree of accuracy. Zijl and Trykozko [129] proposed a local numerical
volume averaging technique, which calculates equivalent permeability tensors. They
pointed out that their pressure-dissipation approach has mathematical advantages,
such as the possibility to use it in combination with multigrid methods.
For all these scaling techniques, the permeability coefficient needs to be resolved
on the finest grid. If the field is under-resolved then numerical homogenization results
depend on the size of the finest grid as the discretization on the finest grid introduces
some kind of averaging already. For numerical homogenization techniques over-resolved
computations increase the accuracy of the predicted equivalent permeability.
Multiscale finite element method
Multiscale finite element methods are designed to capture the large scale behavior of
the solution without resolving the fine scale features [81, 68, 69, 24, 73, 124, 2, 63, 3,
22, 27, 46, 63, 75, 80, 87, 91, 74]. These methods capture the small scale behavior
within each element by multiscale finite element basis functions. The modified basis
functions need to be compatible, which means that they need to be C 0 -continuous
across the element boundaries and they should satisfy the homogeneous flow equation
(no source term) for each coarse scale element. Element wise local boundary conditions
mostly follow from a dimensionally reduced problem, as linear boundary conditions
often produce less accurate algorithms. The small-scale information is transferred to
the large-scale by integration over the small-scale elements and constructing a largescale element matrix. This way, the method reduces the size of the computation. For a
two-dimensional problem the operation count is about twice that of a conventional finite
element method. However, the construction of the basis functions is fully decoupled
between elements and is naturally adapted to parallel computing. The method is
not restricted to assumptions on the media, such as scale separation and periodicity.
Moreover, the number of scales involved is irrelevant to the computational cost. In its
original form, the two-level method can be seen as a domain decomposition technique
[98]. The procedure removes fine scale nodes from the subdomain, but introduces errors
at the subdomain boundaries. These errors are not corrected iteratively as in standard
domain decomposition methods.
Multiscale finite elements do not calculate effective permeability tensors explicitly,
but implicitly capture the fine-scale permeability distribution. MacLachlan [92] related
multigrid upscaling of governing equations to multiscale finite element discretization.
He obtained multiscale basis function weights recursively by directional lumping and
variational coarsening.
Multigrid solver
Multigrid solvers are able to obtain optimal convergence properties but have to be adjusted for problems with strongly discontinuous coefficient fields [5, 37, 38, 28, 34, 103,
102, 120, 45]. Multigrid methods are mostly used in combination with finite difference
techniques, although the variational finite element framework applies to multigrid more
naturally [25, 90]. The methods apply a series of coarser grids to accelerate convergence
6
1. Introduction
of basic relaxation methods. For problems with locally highly varying or discontinuous
coefficients functions, the performance and robustness of a multigrid solver depends
highly on the choice of the coarse grid operators. In the best case, computational work
and memory requirements scale linearly with the number of unknowns. There are two
classes of multigrid methods: geometric multigrid and algebraic multigrid. Geometric
multigrid operates on a hierarchy of grids, whereas algebraic multigrid generates a hierarchy of algebraic equations. Both types apply a discretization of the flow problem on
the finest grid. The discretization of the flow equation can be interpreted as an averaging or filtering in itself. In the geometric framework, homogenization methods provide
equivalent material behavior on a coarse grid. Flow equations are then discretized on
coarser grids to provide coarse grid operators. In the algebraic framework variational
coarsening provides a discretization on successive levels. Matrix dependent interpolation supports variational coarsening, and aims to provide coarse grid operators that
provide good convergence rates for the multigrid algorithm. As an alternative, Schur
complements of the approximated fine-scale matrix with respect to the coarser scale
can be used. Variational coarsening implicitly generates multiscale basis functions.
Multigrid upscaling of the permeability field was used by Griebel [59] and Knapek
[82]. They applied variational coarsening to calculate coarse-scale equations and extracted an equivalent permeability tensor by interpreting the stencil of the multigrid
homogenized operator as a linear combination of the finite difference stencils associated
with second order derivatives. As Griebel and Knapek pointed out, matrix dependent
prolongation and Schur complement approximations lead to energy dependent averaging procedures and to averaged equations. Moulton [95] was able to obtain the
equivalent permeability tensor directly. All authors apply a lumping procedure to calculate the intergrid transfer operators used in the variational coarsening, which was
proposed by Dendy [38].
Adaptive formulation
Adaptive grid refinement concentrates the computational work on localized phenomena
like sinks and concentration fronts [1, 11, 17, 49, 58, 93, 109, 126, 36, 35]. Figure 1.5
shows the flexibility of the finite element method with this respect. The refinement
technique proposed in this thesis divides a coarse line element in two fine line elements, a triangular element splits into four child elements and a tetrahedral element
generates eight child elements. Successive refinement of a line element over five levels generates 16 subelements out of a single line element at the base level. A single
triangular element subdivides over five levels into 256 subelements and a tetrahedral
parent element generates 4096 child elements on the fine level. These number show
that an adaptive grid refinement technique that concentrates the computational work
on localized phenomena reduces the computation effort to a large extent, especially in
three dimensions.
Scope of the work
The aim of this work is to provide a fast and robust algorithm to simulate groundwater
flow, solute transport and heat transport through highly heterogeneous porous media.
7
Figure 1.5: A finite element mesh with refinements.
Flow and transport simulations will be carried out on a geo-statistically generated
geological model at field scale supported by laboratory scale measurements. The flow
domain may either be saturated or (partly) unsaturated. As a result of changes in
solute mass fraction and temperature the density and viscosity of the fluid may vary in
time. The hydraulic permeability tensor depends on the intrinsic permeability tensor,
the relative permeability, the viscosity of the fluid, the density of the fluid and the
gravitation constant. The algorithm should be able to capture both the steady state
and time-dependent fine scale character of the hydraulic permeability tensor. Though
constitutive relations may change for different scales, Darcy’s law is supposed to remain
valid on both laboratory scale and field scale. The algorithm had to be based on the
nodal conforming finite element method because this method is widely used to solve
geotechnical problems as it is able to capture complex geometries. The algorithm
should support parallel computing and adaptive grid refinement.
Outline of the thesis
The outline of this thesis follows the modeling process. First the physical behavior of
the subsurface system is captured by a mathematical model. This model translates to
a numerical model which has to be solved. The behavior of the numerical model will
be verified for a number of known partly homogeneous benchmark problems. Finally,
the model will be applied to resolve two problems that focus on flow and transport
through a highly heterogeneous subsurface.
Chapter 2 presents the mathematical model as a set of coupled partial differential
equations. Continuum mechanics at microscopic level produces microscopic balance
equations and the method of volume averaging derives the macroscopic balance equations for mass, momentum and energy. Equations of state relate the unknowns in the
balance equations to primary unknowns. Both empirical and more fundamental constitutive relations for porosity, degree of saturation, permeability, fluid density and fluid
viscosity are presented. The resulting partial differential equations are subjected to
boundary conditions and initial conditions for the fully coupled subsurface flow, solute
8
1. Introduction
transport and heat transport problem.
Chapter 3 gives the numerical model as a set of algebraic equations. The conformal
finite element method first discretizes the partial differential equations of the mathematical model in space. The finite element method applies to the basic equations
for flow, solute transport and heat transport separately. An implicit time integration
scheme integrates the ordinary differential equation of the problems. Nonlinearities in
the equations are resolved sequentially by Picard iterations. Finally the fully coupled
resulting sets of linearized algebraic equations are presented, together with a procedure
for generation of initial conditions and handling of linear and geometrically nonlinear
boundary conditions.
Chapter 4 presents the newly developed adaptive multiscale finite element method
and its derived multigrid solver. The adaptive formulation and the multiscale formulation will be explained separately. Discrete adaptivity formulates mesh refinement,
and functional adaptivity facilitates coarsening of the mesh. Multiscale basis functions
extrapolate the coarse scale results to the fine scale and construct intergrid transfer
operators. The coarse mesh equations apply to the adaptively refined mesh and the
coarse grid operators apply to the sequence of coarse grids, which support the multigrid
solver.
Chapter 5 covers the model verification part on piecewise homogeneous domains.
The first section focuses on saturated and unsaturated flow and considers stable saturated flow though a dam and potentially instable unsaturated furrow infiltration. The
second section focuses on solute transport and heat transport and covers stable density
driven solute transport for a salt tongue that intrudes a fresh water domain and unstable convective heat transport in a domain heated from below. Adaptive results are
compared with results of standard fine scale conforming finite element calculations.
Chapter 6 includes two applications of the adaptive multiscale finite element method
on highly heterogeneous domains. The first section includes a simplified geothermal
storage process in a shallow marine system. The second application focuses on geothermal energy production in a deep fluvial system.
Chapter 7 contains the conclusions.
Chapter 2
Mathematical model
The subsurface is considered to be a porous medium [12, 116, 51, 33, 9, 52]. A porous
medium may consists of a single solid phase and multiple fluid phases. Phases are material subdomains that are separated from each other by phase interfaces. Each phase
may consist of a number of miscible chemical species. In the present approach the fluid
phases are considered to be immiscible. The mathematical model of this multi-phase
multi-species system, is based on fundamental physical principles: mass conservation,
momentum conservation, and energy conservation [85, 108]. The balance laws are firstly
formulated on a microscopic level by an Eulerian or spatial description. A spatial averaging procedure transforms the microscopic balance equations over representative
elementary volumes to the macroscopic level [67, 123]. At the macroscopic scale, the
phases are described as overlapping continua, where macroscopic physical laws and
conservation principles apply. Corresponding constitutive relations express unknowns
in the governing balance equations on the macroscale in measurable parameters. These
relations often have an empirical nature. Equations of state give fundamental thermodynamic relationships. This thesis focuses on a three-phase system, which consists of
a solid phase (soil skeleton), a liquid phase (groundwater) and a gas phase (air). The
model is restricted to single phase flow as the gas-phase is considered to be stagnant
and the solid phase is assumed to deform relatively slowly.
Section 2.1 presents the macroscopic balance equations for conservation of mass,
conservation of momentum and conservation of energy. Balance equations apply to the
solid phase, the liquid phase and the liquid phase components. Section 2.2 gives the
constitutive equations that relate unknowns in the balance equation like porosity, saturation, permeability, density and viscosity to a set of independent primary unknowns;
pressure, mass fraction and temperature. Section 2.3 states the mathematical problem
which needs to be solved subject to properly defined initial conditions and boundary
conditions. Characteristic numbers describe the flow regime and support the dimensional analysis. Throughout this section, the coordinate free tensor notation will be
used, which is outlined in appendix A. This appendix also introduces the concept of
continuum mechanics and the method of volume averaging.
9
10
2.1
2. Mathematical model
Balance equations
Macroscopic balance equations for conservation of mass, conservation of momentum
and conservation of energy form the basis of the mathematical flow and transport
model. The momentum conservation equation for the liquid phase supports the constitutive law of Darcy. The momentum conservation equation for a constituent of the
liquid phase supports an Fickian type equation for non-convective species flux. The
macroscopic energy balance equation for the complete system assumes thermal equilibrium between all phases.
Section 2.1.1 presents the mass balance equations, which prescribe mass conservation of the solid phase, mass conservation of the fluid phase and mass conservation of
chemical constituents of the liquid phase. Section 2.1.2 gives the momentum balance
equations, which involve momentum conservation of the liquid phase and momentum
conservation of the liquid phase chemical components. Energy conservation for the
total system gives the energy balance equation presented by section 2.1.3.
2.1.1
Mass balance equation
The microscopic equation for conservation of mass is given by equation (A.19) and
the macroscopic mass conservation equation for a chemical species in the fluid phase
is given by equation (A.35). This equation holds for all chemical components of all
phases of the porous medium. Diersch and Kolditz [41] presented the macroscopic
mass balance equation for a multi-phase, multi-species system as
∂
α
α
(φα ρα ωkα ) + ∇· (φα ρα ωkα v α ) + ∇·j α
(2.1)
k = Mk + Rk ,
∂t
where α is the phase indicator and k relates the chemical component or miscible species
of the phase. For the porous medium under investigation α ∈ (l, g, s) where l denotes
the liquid phase, g indicates the gas phase, and s is the solid phase of the porous
medium. In the left hand side of equation (2.1) t [t] denotes time, φα [−] is the volume
fraction of the α-phase, ρα [ml−3] is the phase density, ωkα [−] is the mass fraction of
species k in the α-phase, j [ml−2t−1 ] is the non-convective species flux vector due to
dispersion or diffusion, and v [lt−1 ] is the phase velocity vector. In the right hand side
Rα [ml−3t−1 ] is the chemical reaction rate term and M α [ml−3 t−1 ] denotes the bulk
solute mass source term for internal and external transfer. The mass source term will
have a negative value if mass is extracted from the porous medium. The sum of the
mass fraction over all species in the phase equals one, and the sum of the non-convective
terms and chemical rate terms equals zero, according to
α
α
n
X
α
α
φ = 1,
nk
X
k
α
ωkα
= 1,
n
X
α
α
jα
k
= 0,
n
X
Rα
k = 0.
(2.2)
α
Using these expressions, the mass balance equation for each phase can be written as
∂
(φα ρα ) + ∇· (φα ρα v α ) = M α .
∂t
P
By definition the total mass source term reads k Mkα = M α .
(2.3)
11
2.1. Balance equations
Solid phase mass balance equation
The mass balance for the solid phase follows from the macroscopic mass balance equation (2.3) and the definition of the solid phase volume fraction given by equation (A.30)
as
∂
((1 − n)ρs ) + ∇· ((1 − n)ρs v s ) = 0,
(2.4)
∂t
where the source term for the solid phase has not been included, so M s = 0. For incompressible solid grains, the density of the solid phase ρs is constant and the equation
simplifies to
∂n
− (1 − n)∇·v s + v s ·∇n = 0.
(2.5)
∂t
An indicative value for the solid phase density of sand is 2.65 · 103 kg/m3 . In general,
porous media deform slowly and v s ·∇n ≈ 0, so the mass balance equation for the solid
phase becomes
∂n
= (1 − n) ∇·v s .
(2.6)
∂t
Here ∇·v s is the volume strain rate of the solid matrix.
Liquid phase mass balance equation
The liquid phase mass balance equation follows from equation (2.3) and the definition
of the liquid phase volume fraction given by equation (A.31), and reads
∂
nSρl + ∇· nSρl v l = M l .
∂t
(2.7)
Saturation reads S = V l /V p , where V l is the volume of the liquid phase and V p is
the total pore volume. According to the definition of the filtration velocity given by
equation (A.37) and the definition of the liquid phase volume fraction presented by
equation (A.31), the macroscopic velocity vector of the liquid phase reads
vl =
ql
+ vs,
nS
(2.8)
where q l is the relative (Darcy) bulk velocity of the liquid phase [lt−1 ]. Rewriting
the mass balance equation in terms of the filtration velocity and expanding the time
derivative term yields
Sρl
∂n
∂S
∂ρl
+ nρl
+ nS
+ ∇· ρl q l + nSρl ∇·v s + v s ·∇ nSρl = M l .
∂t
∂t
∂t
(2.9)
Using the definition for the divergence of the solid phase macroscopic velocity given by
equation (2.6), the liquid phase mass balance equation reads
Sρl ∂n
∂S
∂ρl
+ nρl
+ nS
+ ∇· ρl q l = M l ,
1 − n ∂t
∂t
∂t
(2.10)
where for slowly
deformable media and slightly compressible fluids it was assumed that
v s ·∇ nSρl ≈ 0.
12
2. Mathematical model
Liquid phase component mass balance equation
The macroscopic mass balance equation for a single species k in the liquid phase follows
from equation (2.1) and the definition of the liquid phase volume fraction given by
equation (A.31). The liquid phase component mass balance equation then reads
∂
nSρl ωkl + ∇· nSρl ωkl v l + ∇·j lk = Mkl + Rlk .
∂t
(2.11)
Expanding the time derivative term and inserting the expression for the macroscopic
liquid phase velocity presented by equation (2.8) gives
∂n
∂S
∂ωl
∂ρl
+ nρl ωkl
+ nSρl k + nSωkl
+ ∇· ρl ωkl q l
∂t
∂t
∂t
∂t
+ nSρl ωkl ∇·v s + v s ·∇ nSρl ωkl + ∇·j lk = Mkl + Rlk . (2.12)
For slowly deformable media and slightly compressible fluids v s ·∇ nSρl ωkl ≈ 0. Inserting equation (2.6) which gives an expression for the macroscopic velocity, the solute
mass balance equation yields
Sρl ωkl
Sρl ωkl ∂n
∂S
∂ωl
∂ρl
+ nρl ωkl
+ nSρl k + nSωkl
+ ∇· ρl ωkl q l
1 − n ∂t
∂t
∂t
∂t
+ ∇·j lk = Mkl + Rlk . (2.13)
This form is known as the divergent form of the transport equation. The convective
form follows from extracting ωkl times the liquid phase mass balance equation (2.10)
from the divergent transport equation (2.13). The convective form reads
nSρl
∂ωkl
+ ρl q l ·∇ωkl + ∇·j lk = Mkl − M l ωk + Rlk ,
∂t
(2.14)
where M l [ml−3t−1 ] denotes the liquid source term, and Mkl [ml−3t−1 ] expresses the
solute mass entering the porous medium.
2.1.2
Momentum balance equation
The microscopic equation for conservation of momentum is given by equation (A.20).
Diersch and Kolditz [41] presented the general macroscopic momentum balance equation for a phase as
∂
(φα ρα v α ) + ∇· (φα ρα v α v α ) − ∇·σ α = φα ρα g + φα f α ,
∂t
(2.15)
where φα ρα g [lt−2 ] represents an external body force vector due to gravity only as
ionic attractions are neglected, f α [ml−2t−2 ] denotes an internal drag vector or momentum exchange vector, and σ α [ml−1t−2 ] denotes the viscous stress tensor if a fluid
is considered or the Cauchy stress tensor for the case of a solid phase.
13
2.1. Balance equations
For a fluid in motion, the stress tensor is decomposed in two parts and is written
as
D
σ α = −φα pα I + φα (σ α ) .
(2.16)
D
The deviatoric part of the stress tensor (σ α ) depends on the rate of deformation,
expressed by ∇v α , the velocity gradient and the dynamic viscosity of the fluid µα
[ml−1t−1 ]. The pressure p [ml−1t−2 ] does not depend on the rate of deformation. A
constitutive equation expresses the inter-facial drag term of momentum exchange for
a fluid phase in measurable flow quantities. For linear drag the Darcy term is imposed
as
−1
f α = −µα (K α ) ·q α ,
(2.17)
−1
where (K α ) denotes the inverse of the intrinsic permeability tensor K [l2] and q α
[lt−1 ] is the volumetric flux density. This constitutive relation can be extended with
a quadratic drag term (to model turbulent flow), known as the Forchheimer term, to
hold
f α = −µα (K α )−1 ·q α − Cf q α µα (K α )−1 ·q α .
(2.18)
Here Cf [l−1 t] is the Forchheimer coefficient, and q α [lt−1 ] is the magnitude of the
volumetric flux density q α .
Leijnse [88] extracted an expression for the non-convective flux from the momentum
equation of a constituent in the fluid phase. In this section the resulting phenomenological law for the non-convective species fluxes j α
k will be given. This Fickian type
equation reads
α α
α
jα
(2.19)
k = −ρ D ·∇ωk ,
where Dα [l2 t−1 ] is the second rank hydrodynamic dispersion tensor. According to
De Josselin de Jong [32] and Scheidegger [105] hydrodynamic dispersion consists of
molecular diffusion and mechanical dispersion. For an isotropic porous medium the
hydrodynamic dispersion tensor reads
Dα = φα Dcα T + φα Dα
m,
α
Dα
m = αt v I + (αl − αt )
vαvα
,
vα
(2.20)
where vα = kv α k2 [lt−1 ] is the magnitude of the phase flux vector, vv is the dyadicproduct of the velocity vector v, Dc [l2 t−1 ] denotes the coefficient of molecular diffusion,
and T [−] is the tortuosity tensor. The mechanical dispersion is written as Dm [l2 t−1 ],
I [−] denotes the unit tensor, αl [l] is the longitudinal dispersivity and αt [l] is the
transverse dispersivity. Both transverse and longitudinal dispersivity are length scale
dependent due to the multiscale nature of the subsurface. Dispersivity parameters
apply to an equivalent homogeneous porous medium.
Leijnse, Hassanizadeh and Schotting [88, 61, 62, 106] proposed a nonlinear extension
that captures large concentration gradients. This implicit formulation of the nonconvective species mass flux reads
α α
α α
α
jα
k + Ch jk j k = −ρ D ·∇ωk ,
(2.21)
−2 −1
where jkα = kj α
t ] is the magnitude of the species mass flux vector and CH
k k2 [ml
−1 2
[m l t] is known as the non-Fickian high-concentration dispersion coefficient.
14
2. Mathematical model
Liquid phase momentum balance equation
The liquid phase momentum balance equation results from equation (2.15) and the
expression for the volume fraction of the liquid phase given by equation (A.31). Conservation of momentum for the liquid phase is expressed by
∂
nSρl v l + ∇· nSρl v l v l − ∇·σ l = nSρl g + nSf l .
(2.22)
∂t
∂
Disregarding the inertial terms ∂t
nSρl v l for slow movement of the fluid, excluding
the compression and deformation work, by ∇· ρl v l v l ≈ 0, inserting the expression for
the viscous stress tensor given by equation (2.16) and the internal drag term expressed
by equation (2.18) gives
D
− σ l ∇ (nS)
−1
−1
= nSρl g − nSµl K l
·q l − Cf nSq l µl K l
·q l . (2.23)
nS∇pl + pl ∇ (nS) − nS∇· σ l
D
D
Replacing ∇· σ l
by the Brinkman term µl ∇2 q l where ∇2 = ∇·∇, the momentum
equation for the liquid phase reads
or
−1
−1
∇pl − µl ∇2 q l = ρl g − µl K l
·q l − Cf q l µl K l
·q l ,
q l + Cf q l q l = −
Kl
· ∇pl − ρl g + K l ·∇2 q l .
l
µ
(2.24)
(2.25)
This equation was derived by Leijnse [88] for saturated flow conditions. For unsaturated
flow the term ∇ (nS) is small. Without the Forchheimer and Brinkman effects the fluid
momentum equation is then given by
ql = −
Kl
· ∇pl − ρl g .
l
µ
(2.26)
Here q l [lt−2 ] is the volumetric flux density, K [l2 ] is the second rank (intrinsic) permeability tensor of the porous medium, µ [ml−1t−1 ] denotes the dynamic viscosity of the
liquid, p [ml−1t−2 ] denotes the pressure, and g [lt−2 ] is the gravitational body force
vector. Equation (2.26) is known as Darcy’s law. This empirical equation will be used
for saturated soils. A phenomenologic law will extend Darcy’s law for unsaturated flow
conditions.
Liquid phase component momentum balance equation
The Fickian type equation (2.19) for the non-convective species flux in the liquid phase
reads
j lk = −ρl Dl ·∇ωkl .
(2.27)
15
2.1. Balance equations
The hydrodynamic dispersion tensor for the liquid phase reads
Dl = nSDcα T + Dlm ,
Dlm = αt q l I + (αl − αt )
ql ql
.
ql
(2.28)
Here equation (2.8) is approximated by v l ≈ q l /nS, excluding solid matrix velocity
effects.
2.1.3
Energy balance equation
The local microscopic equation for conservation of energy was given by equation (A.21).
Diersch and Kolditz [41] presented the phase related macroscopic energy balance equation as
vα vα
vα vα
∂
α α
α
α α α
α
φ ρ Eh +
+ ∇· φ ρ v
Eh +
∂t
2
2
α
α α
α α
+ ∇·j h − ∇· (σ ·v ) = φ ρ g·v α + φα f α ·v α + H α , (2.29)
where j h [mt−3 ] expresses the non-convective (heat) energy flux, Eh [l2 t−2 ] denotes
the internal (thermal) energy, vα [lt−1 ] is the magnitude of the macroscopic phase
velocity vector v α , and H α [ml−1 t−3 ] accounts for the energy source, which captures
internal and external heat supply. For subsurface flow kinetic energy effects are usually
disregarded, so ∂ (vα vα ) /∂t ≈ 0, ∇· (vα vα ) ≈ 0, ∇· (σ α ·v α ) ≈ 0, φα ρα g·v α ≈ 0, and
φα f α ·v α ≈ 0. These assumptions reduce the macroscopic energy balance equation to
∂
(φα ρα Ehα ) + ∇· (φα ρα v α Ehα ) + ∇·j hφ = H α .
∂t
(2.30)
The thermodynamic state function for internal energy can be formulated as
!
nα
k
α
X
∂p
α
α
α
α
α
dEhα = T α
−
p
dν
+
c
dT
+
µα
(2.31)
k dωk ,
∂T α ν α ,ωα
k=1
k
α
α
−1 3
µα
k
2 −2
where ν = 1/ρ [m l ] is the specific volume,
[l t ] denotes the chemical potential of species k and cα [l2t−2 T−1 ] is the specific heat capacity. The chemical
potential depends on pressure, mass fraction of the constituents and temperature as
α
α
α
α
µα
k = µk (p , ωk , T ). The specific heat capacity can be written as
cα =
∂Ehα
∂T α
.
(2.32)
α
ν α ,ωk
If pressure and chemical effects are negligible then the internal energy only depends on
temperature. The equation of state for this case can be written as
dEhα = cα dT α .
(2.33)
For a constant heat capacity, equation (2.33) is reformulated as
Ehα = cα (T − T0α ) .
(2.34)
16
2. Mathematical model
A phenomenological law for the non-convective heat energy flux [48] generalizes the
corresponding linear Fourier law as
α
α
jα
h = −H ·∇T ,
α
−3
(2.35)
−1
where H [mlt T ] is the hydrodynamic thermo-dispersivity tensor and T [T] denotes temperature. Like the hydrodynamic dispersivity tensor, the hydrodynamic
thermo-dispersivity tensor consists of a conductive part and a mechanical part. For the
solid phase the mechanical part vanishes, and the hydrodynamic dispersivity tensor is
expressed as
H l = nSλl I + nScl ρl D lm ,
H s = (1 − n)λs I,
(2.36)
where λl and λs [mlt−3T−1 ] denote heat conductivity of the fluid phase and the solid
phase respectively.
Combined liquid and solid phase energy balance equation
For local thermal equilibrium conditions T = T α over all phases, a single energy balance
equation replaces the energy balance equations for each phase. Usually the thermal
capacity and conductivity of the gas phase is small with respect to the solid and
liquid phase so a combined liquid and solid phase energy balance equation captures
the conservation of energy in the porous medium. The first term of equation (2.30) and
the definition of the internal energy given by equation (2.34) gives the time dependent
term in the solid-fluid system as
∂
∂
∂
∂
nSρl Ehl +
((1 − n)ρs Ehs ) =
nScl ρl T +
((1 − n)cs ρs T ) .
(2.37)
∂t
∂t
∂t
∂t
The second term of the energy balance equation (2.30) includes the liquid phase velocity
given by equation (2.8), and specifies the divergence form in the solid-fluid system as
∇· nSρl v l Ehl + ∇· ((1 − n)ρs v s Ehs ) = ∇· cl ρl T q l + nST cl ρl ∇·v s
+ (1 − n)T cs ρs ∇·v s + v s ·∇ nST cl ρl + v s ·∇ ((1 − n)T cs ρs v s ) . (2.38)
For slowly deforming media v s ·∇ cl nST ρl ≈ 0 and v s ·∇ ((1 − n)cs T ρs v s ) ≈ 0.
Using these restriction and substituting the expression for the divergence of the macroscopic solid phase velocity given by equation (2.6), the second term of the combined
energy balance equation reads
∇· nSρl v l Ehl + ∇· ((1 − n)ρs v s Ehs )
nST cl ρl
∂n
∂n
∇ρs
+ cs ρs T
. (2.39)
= ∇· ρl cl T q l +
1−n
∂t
∂t
Assuming the specific heat capacity cs and cl and the solid phase density ρs to be constant, the combined macroscopic liquid phase and solid phase energy balance equation
can be written as
∂T
cl ST ρl ∂n
∂S
∂ρl
+ cl nT ρl
+ cl nST
+ cl nSρl + (1 − n)cs ρs
1 − n ∂t
∂t
∂t
∂t
l
l l l
+ ∇· ρ c q T + ∇·j e + ∇·j se = H l . (2.40)
17
2.2. Constitutive relations
This is the divergent form of the combined energy balance equation, also known as the
heat transport equation. External heat supply to the liquid phase was introduced by
H l . Subtracting cl ρl times the mass balance equation of the liquid phase, given by
equation (2.10), produces the convective form of the energy conservation equation for
the porous system. The convective form of the energy conservation equation reads
nSρl cl + (1 − n) ρs cs
∂T
+ ρl cl q l ·∇T + ∇·j le + ∇·j se = H l − M l cl T,
∂t
(2.41)
where the non-convective heat energy fluxes are given by
j le = −H l ·∇T,
j se = −H s ·∇T.
(2.42)
These two fluxes combine to an average flux, which reads j e = j le + j se , as the flux
between the two constituents is neglected. This averaged flux reads j e = −H·∇T .
The hydrodynamic thermo-dispersion tensors for the liquid and the solid phase are
expressed as
H = H l + H s,
H l = nSλl I + cl ρl Dlm ,
H s = (1 − n)λs I.
(2.43)
Equation (2.41) expressed energy conservation for the porous medium and equation (2.42) formulated the non-convective heat flux. An indicative value for the specific
heat capacity of sandstone is about 8.4 · 102 J/kgK at 293 K and the thermal conductivity is about 3.5 J/msK at 293 K. The specific heat capacity of pure water is approximately 4.2 · 103 J/kgK at 293 K and its thermal conductivity is about 0.6 J/msK
at 293 K.
2.2
Constitutive relations
A number of previously introduced solid material parameters like solid phase density,
specific heat capacity, and thermal conductivity are assumed to be constant in time but
may vary in space. The specific heat capacity and the thermal conductivity of the liquid
phase are considered to be constants in time and space. Longitudinal and transverse
dispersivity are assumed to be constant in time but have to capture the hierarchical
structure of the heterogeneous subsurface. In most practical applications their values
depend on the problem size. In this thesis their value relates to the fine element size
as heterogeneous fields are considered explicitly. Molecular diffusion is often neglected
as its value is small relative to mechanical dispersion. Constitutive relations for the
remaining variables formulate material behavior in terms of measurable quantities.
Here fluid pressure, solute mass fraction, and porous medium temperature are chosen
as measurable independent variables.
Section 2.2.1 gives an expression for the porosity as a function of fluid pressure.
The section presents the equilibrium equation for the solid matrix which results from
the momentum equation for the solid matrix. If this equation is decoupled from the
flow equation, then it expresses porosity for the case of dissipating water pressures.
Empirical relations give a functional dependency for the degree of saturation on the
capillary pore pressure and give a relation between relative permeability and saturation.
18
2. Mathematical model
Section 2.2.2 and section 2.2.3 present the empirical Van Genuchten relations that
express saturation and relative permeability as functions of pore pressure. Fluid density
and fluid viscosity relate to pressure, solute mass fraction and temperature and their
empirical relations will be discussed in section 2.2.4 and section 2.2.5.
2.2.1
Porosity
Biot consolidation theory couples the equilibrium equation for the solid skeleton to the
storage equation of the fluid. The equilibrium equation follows from the momentum
equation of the solid phase, neglects the acceleration terms, and adopts kinematic and
constitutive relations. For the solid, kinematic relations express the deformation in
terms of strains. Constitutive relations for the material behavior relate stresses to
strains. The storage equation follows from the mass balance equation of the fluid and
Darcy’s law. This subsection proposes a decoupling of both these equations, which
gives a relation between porosity and fluid pressure. Concentration and temperature
effects are assumed to be negligible and compression work of the skeleton only is taken
into account. An elastic material is considered for which the small deformation theory
or linear deformation theory applies, as displacements and displacement gradients are
small compared to unity.
According to the mass balance equation for the solid phase given by equation (2.6)
the rate of change of porosity is equal to the divergence of the macroscopic solid phase
velocity. Only the volume of the pores changes as the volume of the solids is considered
constant. The relation between solid phase velocity and solid phase displacements can
be written as
∂
∂
∂εv
v s = us ,
∇·v s =
(∇·us ) =
.
(2.44)
∂t
∂t
∂t
Substituting this expression into equation (2.6) gives
∂εv
∂n
= (1 − n)
,
∂t
∂t
(2.45)
where volume strain εv [−] is the first invariant of the strain tensor ε [−] and reads
εv = tr (ε) = ∇·us .
(2.46)
Strains are considered to be positive for extension. The volume strain depends on the
stresses in the porous medium.
The macroscopic momentum balance equation (2.15) without acceleration terms
resembles the macroscopic static equilibrium of a continuum [116, 19], written as
∇·σ + γ = 0,
(2.47)
where σ [ml−1t−2 ] represents the Cauchy (total) stress tensor and γ [ml−2t−2 ] denotes the body force vector. Stresses are positive for tensile stress and negative for
compression. For a partly saturated soil the weight of the soil is given by γ =
nSρl g + (1 − n) ρs g. The balance of moments requires the Cauchy stress tensor to
be symmetrical σ = σ T . External forces on the boundary are written as τ = σ·n,
19
2.2. Constitutive relations
where τ [mt−2] is the boundary traction vector and n [l] is the normal vector pointing
out of the domain. Terzaghi’s principle states that the total stress sums the liquid
pressure in the pores and the effective stress in the soil skeleton. Terzaghi’s principle
is formulated as
σ = σ 0 − pl I,
(2.48)
where σ 0 [ml−1 t−2 ] denotes the effective stress and I [−] is the identity tensor. A
minus sign applies on the pressure as it is a compressive stress and both effective and
total stress are positive for tensile stress. The (effective) stress-strain relationship of
an isotropic linear elastic material can be written as
σ 0 = λl tr (ε) I + 2µl ε,
(2.49)
where λl [ml−1t−2 ] and µl [ml−1t−2 ] are called the Lamé constants.
Substitution of the constitutive relation given by equation (2.49) into equation (2.47),
which states the static equilibrium of a continuum, gives
γ + 2µl ∇ε + λl ∇tr(ε) − ∇p = 0.
(2.50)
For small deformations the kinematic relation expresses the linear strain tensor in terms
of displacements. This relation is written as
1
T
ε=
(∇us ) + (∇us ) ,
(2.51)
2
T
where (∇us ) is the conjugate of dyad (∇us ) and ε represents the symmetric part
of the dyad (∇us ). The equilibrium equation for the solid matrix follows from equation (2.50) and equation (2.51) as
γ + (µl + λl ) ∇∇·us + µl ∇2 us − ∇p = 0.
(2.52)
The expression for the volumetric strain, given by equation (2.46), reformulates equation (2.52) as
γ + (µl + λl ) ∇εv + µl ∇2 us − ∇p = 0.
(2.53)
This equation already relates the strain tensor to the pore water pressure.
For an isotropic material the volume strain is often assumed to be a function of
isotropic effective stress. Verruijt [116] stated that in many cases consolidation takes
place while the loading of the soil remains constant and it may well be assumed that
changes in the total stress will be small. If a soil layer of large extent is loaded by a
constant surface load, then the horizontal deformations are negligible and the vertical
total stress remains almost constant. For a linear elastic material the relation between
vertical effective stress and vertical strain then reads
0
σzz
= (λl + 2µl ) εzz .
(2.54)
The total vertical stress is decomposed into the effective stress and the pore pressure
0
according to σzz = σzz
− p. As the total stress is assumed to be constant in time, the
variation of volumetric strain in time is expressed as
∂p
∂εv
= αp ,
∂t
∂t
(2.55)
20
2. Mathematical model
where αp [m−1 lt2 ] denotes the compressibility of the solid skeleton, and αp = 1/(λl + 2µl )
holds. Deformation of the solid skeleton is taken into account by equation (2.45). Using the result of equation (2.55) the change in porosity can be related to the change in
pore pressure as
1 ∂n
∂p
= αp ,
(2.56)
1 − n ∂t
∂t
For small time steps this equation can be expressed as dn = αp (1 − n) dp, and for the
porosity the following relation holds
n = 1 − (1 − n0 ) exp{−αp (p − p0 )},
(2.57)
where n0 [−] is the reference porosity at reference pressure p0 [ml−1t−2 ]. For small
deformations its linearized form reads
n = n0 + αp (1 − n0 ) (p − p0 ) .
(2.58)
Freeze and Cherry [52] propose αp = 1 · 10−10 ms2 /kg for sound bedrock and αp =
1 · 10−7 ms2 /kg for clay. It must be noted that these expressions only hold for the
one-dimensional dissipation of pore-water pressures, and do not capture pore pressure
generation by mechanical or hydraulical loading on the subsurface.
2.2.2
Saturation
Constitutive equations for the degree of saturation of a fluid in a porous medium are
based on the macroscopic capillary pressure. The capillary pressure denotes the difference of the non-wetting and the wetting phase pressure. For liquid-gas flow systems
the liquid phase is the wetting phase and the gas phase is the non-wetting phase. The
macroscopic capillary pressure pc [ml−1t−2 ] reads
pc = pg − pl .
(2.59)
If the gas phase is stagnant, the pressure in the gas phase is constant and equals the
atmospheric pressure if the phase is continuous. Then, pc = −pl holds for unsaturated
conditions and pc = 0 applies for saturated conditions. In general
the constitutive
relation for two immiscible phases reads pc = pc S l , T l , T g , ωkl , ωkg . Van Genuchten
and Brooks-Corey simplified this by an empirical relation pc = pc (S), where S denotes
the saturation of the wetting phase [117, 9].
Reversely, Van Genuchten-Mualem express the saturation as a functional relation
of wetting phase pressure according to S = S(p). The Van Genuchten model is written
as
g −g
S = Sr + (Ss − Sr ) [1 + |ga ψ| n ] m if ψ < ψa
,
(2.60)
S = Ss
for ψ ≥ ψa
where ψ [l] denotes the pressure head ψ = pl /ρl g, ψa [l] is the air-entry pressure head,
which is constraint by ψa ≤ 0. Sr [−] is the minimal saturation and Ss [−] denotes
the maximum degree of saturation. The minimal saturation deviates from zero due
to chemically attached water or entrapped water pockets. The maximum degree of
21
2.2. Constitutive relations
saturation is less than one mainly as a result of entrapped air. The Van Genuchten
relation counts two empirical shape factors that have to be measured in the laboratory:
gn [−] and ga [l−1 ]. For convenience a third shape factor (Mualem assumption) was
introduced as gm = (gn − 1)/gn [−].
Table 2.1: Staringreeks Van Genuchten shape factors.
soil type
O1 sand
O8 sandy clay
O11 clay
O14 loam
O16 peat
Sr
-
n
-
Kd
m/d
ga
1/m
gl
-
gn
-
0.03
0.00
0.00
0.03
0.00
0.36
0.47
0.42
0.38
0.89
0.1522
0.0908
0.0138
0.0151
0.0107
2.24
1.36
1.91
0.30
1.03
0.000
-0.803
-1.384
-0.292
-1.411
2.286
1.342
1.152
1.728
1.376
Figure 2.1 shows the functional saturation relation for five Dutch soil types, which
are reported in the Staring series [111], and table 2.1 shows their parameters. The
relative permeability relates to the saturation of the soil and will be discussed in the
next section. Instead of saturation the Staring series use water content θ [−], which
relates to saturation and porosity as θ = nS.
1.0
relative permeability
saturation
1.0E+0
1.0E-1
rel permeability (-)
saturation (-)
0.8
0.6
0.4
0.2
1.0E-2
1.0E-3
1.0E-4
1.0E-5
1.0E-6
1.0E-7
0.0
-1.0E+6
1.0E-8
-7.5E+5
-5.0E+5
-2.5E+5
0.0E+0
-1.0E+6 -8.0E+5 -6.0E+5 -4.0E+5 -2.0E+5 0.0E+0
pressure (Pa)
pressure (Pa)
O1
O8
O11
O14
O16
Figure 2.1: Saturation and relative permeability for Staring series materials.
The Staring series do not provide a value for the porosity and here the porous
medium is assumed to be fully saturated for Ss = 1. Under this assumption the
porosity follows from the maximum water content n = θs and the residual saturation
follows from Sr = θr /n. For time dependent problems the pressure derivative of the
saturation may be used. This derivative reads
dS
ga (gn − 1)(Ss − Sr )(ga ψ)(gn −1)
=
.
g (2g −1)/gn
dp
[1 + (ga ψp ) n ] n
(2.61)
22
2. Mathematical model
Alternatively Brooks-Corey parametric model is given by
−b
S = Sr + (Ss − Sr ) |bα ψ| n if ψ < −b−1
α
,
S = Ss
for ψ ≥ −b−1
α
(2.62)
where bα [l−1 ] is a curve fitting parameter, bn [−] denotes the pore distribution index
bn ≥ 1.
Many materials show a non-unique relationship for the saturation and wetting phase
pressure. This hysteresis is caused by the variation in pore diameters, differences in
radii of advancing and receding menisci, air entrapment etc. Van Dam [111] proposes
a main wetting curve and a main drying curve. The main drying curve describes the
gradual desorption of an initially saturated sample, and the main wetting curve follows
from an gradual adsorption experiment of an initially dry sample. Partially drying
and partially wetting follow drying scanning curves and wetting scanning curves. In
general, the degree of saturation for a given pressure will be higher on the main drying
curve than the saturation on the main wetting curve. Figure 2.1 shows the main drying
curves for saturation as a function of pore pressure. The hysteresis can be captured by
an alternative relationship, but it remains typically empirical [84, 114].
2.2.3
Permeability
For saturated conditions, the permeability of the porous medium is expressed by the
intrinsic permeability tensor. The intrinsic permeability tensor is a second order tensor.
If the working lines of K·n coincide, then n is called an eigenvector of K, the corresponding eigenvalues λi follow from (K − λI) ·n = 0. The eigenvalues correspond
to principal permeabilities and the eigenvectors capture the layering of the porous
medium.
If the porous medium is partly saturated then the intrinsic permeability is scaled
by the relative permeability, which is often given as a function of saturation. The
permeability of the porous medium for a fluid phase (α ∈ l, g) is thus given by
K α = krα (S α ) K,
(2.63)
where the relative permeability is denoted by kr [−] and its value holds 0 < kr ≤ 1.
Mualem-Van Genuchten and Brooks-Corey proposed empirical relations for the relative
permeability [117, 9]. The Van Genuchten relation reads
gl
kr = (Se )
h
gm i2
1 − 1 − Se 1/gm
,
Se =
S − Sr
.
Ss − Sr
(2.64)
The empirical shape factor gl [−] is often set to 0.5. Figure 2.1 presents the material
behavior for five Dutch soils: sand O1, sandy clay O8, clay O11, loam O14 and peat
O16. For these materials the Van Genuchten shape factors and the Darcy permeability
were given in table 2.1. The Darcy permeability tensor K d [lt−1 ] is derived from the
intrinsic permeability tensor K and reads
Kd =
Kρg
,
µ
(2.65)
23
2.2. Constitutive relations
where g denotes the magnitude of the gravity vector. The Darcy permeability tensor
expresses a bulk property of the porous medium, as the intrinsic permeability tensor
depends on the solid phase, and density and dynamic viscosity are fluid properties.
The Darcy permeability will not be used in this thesis since viscosity and density vary
in time and space.
Brooks-Corey’s parametric model expresses the relative permeability as
kr = Sebκ ,
Se =
S − Sr
,
Ss − Sr
(2.66)
where bκ = 2/bn + bl + 2 [−] is a curve fitting parameter and bl [−] is known as the
pore connectivity parameter.
2.2.4
Density
This subsection considers the density of a binary fluid, composed of pure water and
dissolved salt in the range from pure water to saturated brine. The general equation
of state for the density of this liquid phase reads
α
dρ =
1 ∂ρα
ρα ∂pα
α
α
ρ dp +
1 ∂ρα
ρα ∂ωkα
α
ρ
dωkα
+
1 ∂ρα
ρα ∂T α
ρα dT α
= βp ρα dpα + βω ρα dωkα − βT ρα dT α , (2.67)
where βp [m−1 lt2 ] denotes the liquid phase compressibility, βωk [−] is the volumetric
solute coefficient, and βT [T−1 ] is the thermal expansion coefficient. For constant
compressibility and expansion coefficients the liquid density can be written as
ρl = ρl0 exp{βp (p − p0 ) + βω (ω − ω0 ) − βT (T − T0 )}.
(2.68)
Here p0 [ml−1t−2 ] denotes the reference fluid pressure, ω0 [−] is the reference salt
concentration, and T0 [T] is the reference temperature. At these reference values the
density of the fluid, expressed as ρl0 = ρl0 (p0 , ω0 , T0 ) [ml−3], is determined. A first
order Taylor expansion gives a linear approximation, which reads
ρl = ρl0 [1 + βp (p − p0 ) + βω (ω − ω0 ) − βT (T − T0 )] .
(2.69)
The density of the fluid depends primary upon the fluid solute concentration and its
temperature. In the range of 293 K to 333 K the thermal expansion coefficient βT is
about 3.75 · 10−4 1/K for pure water (ω0 = 0). The thermal expansion coefficient for
fresh water in the range of 273 K to 373 K varies from −0.68·10−4 1/K to 7.5·10−4 1/K
and is zero at 277 K. The solute coefficient βω is approximately 0.7 for pure water and
salt mixtures, at 293 K, ρ0 equals 998.2 kg/m3. The density of seawater (ω = 0.024)
is about 1022 kg/m3, for a saturated brine (ω = 0.302) the density can be as high as
1300 kg/m3 . The weak dependency of density with pressure follows from the measured
compressibility coefficient: for pure water at 293 K βp is about 4.47 · 10−10 ms2 /kg.
24
2.2.5
2. Mathematical model
Viscosity
The dynamic viscosity of a binary fluid of pure water and dissolved salt constituents,
can be expressed empirically [15] as
µl = µl0 γ0 + γ1 ω + γ2 ω2 + γ3 ω3 ,
(2.70)
where µl0 [ml−1t−1 ] is the reference viscosity and γi [−] are polynomial constants.
Lever and Jackson [89] determined the coefficients for dissolves salt (NaCl) as γ0 = 1,
γ1 = 1.85, γ2 = −4.1 and γ3 = 44.5. According to this relation the viscosity for
seawater is about 1.04 · 10−3 kg/ms and for saturated brine it equals 2.39 · 10−3 kg/ms.
The reference viscosity of pure water at 293 K equals 1.0 · 10−3 kg/ms.
The dynamic viscosity of pure water, as a function of temperature, is given by
µl = 243.18 · 10−7 10247.8/T −140 .
(2.71)
According to this equation the viscosity of pure water at 293 K equals 1.01·10−3 kg/ms,
and at 333 K the viscosity is 4.68 · 10−3 kg/ms.
2.3
Problem definition
This section poses the mathematical model, subjected to initial and boundary conditions, which needs to be solved in order to obtain the solution for a specific subsurface
flow problem. The mathematical model is composed out of the previously given balance
equations and the constitutive relations, and forms a set of coupled nonlinear partial
differential equations formulated in terms of independent primary variables. Physically
admissible initial conditions and boundary conditions pose coupled constraints.
Section 2.3.1 states the system of equations composed by the flow equation, the mass
transport equation and the heat transport equation, and selects the primary variables:
fluid pressure, solute mass fraction, and porous medium temperature. These basic
equations are coupled by expressions for the convective flux and dispersive fluxes, and
the constitutive relations for porosity, saturation, permeability, density and viscosity.
The section also presents the Rayleigh number that characterize the flow regime in the
case of density driven flow. Section 2.3.2 gives a set of initial conditions and section 2.3.3
gathers boundary conditions for the coupled set of equations. This section considers
flow of a binary liquid, and drops the liquid phase index and the chemical species index.
2.3.1
System of differential equations
The flow equation follows from the mass balance equation of the liquid phase given by
equation (2.10) and Darcy’s law presented by equation (2.26) as
∂p
kr K
∂ρS
+ Sραp
− ∇· ρ
· (∇p − ρg) = ρ̃Q.
(2.72)
n
∂t
∂t
µ
The pressure in the liquid phase p is the primary unknown in the flow equation. In this
formulation the compressibility of the solid skeleton αp captures the time dependent
2.3. Problem definition
25
variation of porosity according to equation (2.56). The dynamic viscosity µ, expressed
by equation (2.71), of the water phase has a functional relation with temperature. The
density ρ of the water phase, according to equation (2.69), depends mainly on the mass
fraction of the constituents (fresh water and dissolved salt). The porosity n, according
to equation (2.58), depends on the fluid pressure p only. The intrinsic permeability K
of the solid skeleton is assumed to be constant in time. The Van Genuchten model poses
a unique relationship of saturation S with pore pressure by equation (2.60), and relates
relative permeability kr to pore pressure, according to equation (2.64). The volumetric
source term Q in the right hand side is predefined. Alternatively a prescribed mass
source M = ρQ can be applied. The density of the fluid ρ̃ is prescribed for positive
values of the source term (injection into the subsurface). The gravitational acceleration
g is constant.
The solute transport equation follows from the mass transport equation for a water
phase component, as given by equation (2.14). If the non-convective flux is replaced by
the Fickian type dispersion equation, given by equation (2.27) and chemical reactions
are neglected, then the convective form of the solute transport equation reads
∂ω
+ ρq·∇ω − ∇·ρD·∇ω = ρ̃ (ω̃ − ω) Q.
(2.73)
∂t
The mass fraction ω of the chemical species is the primary unknown in this equation.
The Darcy flux q couples the flow equation to the mass transport equation via the
convective term and the dispersive term in the solute transport equation, since the
hydrodynamic dispersion D incorporates the Darcy flux. Darcy’s law is given by
equation (2.26), the hydrodynamic dispersion follows from equation (2.28). The right
hand side cancels out for a negative value of the volumetric source (extraction out of
the subsurface) where ω̃ = ω holds.
The heat transport equation follows from the combined liquid and solid phase energy
balance equation (2.41). If the non-convective heat flux is captured by equation (2.42),
then the heat transport equation reads
∂T
(cnSρ + (1 − n) cs ρs )
(2.74)
+ cρq·∇T − ∇·H·∇T = cρ̃ T̃ − T Q,
∂t
where temperature T is the primary variable, which applies to the solid skeleton and
the groundwater. The heat capacity of the water phase c is assumed to be constant.
The density of the solid phase ρs and the specific heat capacity of the solid matrix cs
are assumed to be constant in time but may vary in space. The Darcy flux q couples the
flow equation to the heat transport equation. The hydrodynamic thermal dispersion
H incorporates the Darcy flux according to equation (2.43). For a positive value of the
volumetric source term, the temperature of the injected water is predefined as T̃ = T
holds. For a negative volumetric source term the right hand side of the heat transport
equation becomes zero.
nSρ
Characteristic numbers
Heat and solute concentration differences may introduce potentially unstable situations
if a lower density fluid lies underneath a higher density fluid [41, 121, 66]. In a subcritical conductive system, characterized by low Rayleigh numbers, density changes
26
2. Mathematical model
only with depth. Flow occurs when the critical Rayleigh number is exceeded. If
Ra > 4π 2 , the conduction state in a porous medium heated from below becomes
unstable [66]. In this first critical state there are three solutions: two stable steady
convection solutions and one unstable conduction solution. The convection solutions
show convection cells, their flow direction is opposite to each other. The rotation
direction depends on the initial disturbance. In bifurcation theory this is called a pitch
fork bifurcation. The steady convection solutions become unstable if the Rayleigh
number increases further. At the second critical Rayleigh number the flow regime
changes again. At this point the steady convective regime becomes unstable and there
is a transition to a fluctuating convective regime. At this point a Hopf bifurcation
occurs and the steady rotating state switches to an oscillatory rotating state, for which
the rotating velocity changes in time. Other critical numbers may be observed for even
higher Rayleigh numbers.
Characteristic Rayleigh numbers will be derived for a simplified set of equations.
At first, flow in a fully saturated non-deforming porous medium in the absence of
source terms will be considered. Secondly the Oberbeck-Boussinesq approximation
will be imposed. This approximation simplifies the flow equation and both transport
equations by neglecting density changes, except for the buoyancy term ρg in Darcy’s
law. The flow equation (2.72) then follows from the mass balance equation for the fluid
and Darcy’s law as
K
∇·q = 0,
q = − · (∇p − ρg) .
(2.75)
µ
The solute transport equation (2.73) simplifies to
∂ω
+ q·∇ω − n∇·D·∇ω = 0,
D = DI.
(2.76)
∂t
This expression imposes Fick’s law, which replaces the hydrodynamic dispersion by a
constant diffusion term. The heat transport equation (2.74), supplemented by Fick’s
law for thermal conductivity reads
n
(cρ)
e
∂T
e
+ cρq·∇T − (cρ) ∇·H·∇T = 0,
∂t
H = HI,
(2.77)
where (cρ)e = (ncρ + (1 − n) cs ρs ), λe = nλl + (1 − n)λs , and H = λe / (cρ)e .
The Rayleigh numbers Ra, which characterize the onset of natural convection, will
be derived on a two-dimension Cartesian coordinate system. First the stream function
Ψ will be introduced. The definition of the stream function in a two-dimensional
Cartesian domain follows from qx = −∂ψ/∂z and qz = ∂ψ/∂x and dψ = qx dz − qz dx.
Continuity of mass according to equation (2.75) gives
∂2Ψ
∂2Ψ
+
= 0.
(2.78)
∂z∂x ∂x∂z
This relation proves that the stream function is a single-valued function [32]. Darcy’s
law according to equation (2.75), restated for the two-dimensional case in a homogeneous domain, reads
−
µ
∂p
qx = − ,
K
∂x
µ
∂p
qz = −
+ ρg.
K
∂z
(2.79)
27
2.3. Problem definition
Pressure is a physical scalar which has one value in every field point [32], and the
following relation holds
∂2p
∂2p
−
+
= 0.
(2.80)
∂x∂z
∂z∂x
Differentiating Darcy’s flux components and subtracting them gives
µ ∂qx ∂qz
∂ρg
−
=
.
(2.81)
K ∂z
∂x
∂x
Reformulated in stream functions, this relations reads
gK ∂ρ ∂ω
∂ρ ∂T
∂2Ψ ∂2Ψ
+
=−
+
.
∂x2
∂z 2
µ
∂ω ∂x ∂T ∂x
(2.82)
The linear constitutive relation (2.69) gives an expression for the density derivatives,
according to ∂ρ/∂ω = ρ0 βω and ∂ρ/∂T = ρ0 βT . The Rayleigh numbers will be formulated for a coupled flow-solute transport and a coupled flow-heat transport problem
separately as the Fickian diffusion term varies per problem. The Rayleigh numbers
follow from a dimensionless formulation [66]. Spatial dimensions are transformed using a typical length unit L. The dimensionless coordinates follow from x̌ = x/L and
ž = z/L. The time scale for the solute transport equation transforms to ť = tD/L2 , and
the dimensionless Darcy velocity reads q̌ = qL/(nD). The solute transport equation
then reads
∂ ω̌ ∂ 2 ω̌ ∂ 2 ω̌
∂ ω̌
∂ ω̌
+ q̌z
−
− 2,
(2.83)
= q̌x
∂ x̌
∂ ž
∂ x̌2
∂ ž
∂ ť
where the mass fraction is normalized by ω̌ = (ω − ωmin )/(ωmax − ωmin ). The stream
function follows from Ψ̌ = Ψ/(nD), and the solute transport part of equation (2.82)
reads
∂ 2 Ψ̌ ∂ 2 Ψ̌
gρ0 βω KL ∂ ω̌
+
=−
,
(2.84)
2
2
∂ x̌
∂ ž
nµD ∂ x̌
where the right hand side of the original equation was multiplied by L/(nD) in order
to obtain the dimensionless formulation. The solute Rayleigh number follows from the
right hand side, and reads
g∆ρKL
.
(2.85)
Raω =
nµD
Here ∆ρ is the maximum density change due to the mass fraction variation.
The time scale for the heat transport equation transforms to ť = tH/L2 , and the
dimensionless Darcy velocity reads q̌ = qL(ρc)/H(ρc)e . The heat transport equation
then reads
∂ Ť
∂ Ť
∂ Ť
∂ 2 Ť
∂ 2 Ť
= q̌x
+ q̌z
−
−
,
(2.86)
∂ x̌
∂ ž
∂ x̌2
∂ ž 2
∂ ť
where the temperature is normalized by Ť = (T − Tmin )/(Tmax − Tmin ). The dimensionless stream function for this problem follows from Ψ̌ = (ρc)/(ρc)e ·Ψ/H, the heat
transport part of equation (2.82) reads
∂ 2 Ψ̌ ∂ 2 Ψ̌
gρcρ0 βT KL ∂ Ť
+
=
,
2
2
∂ x̌
∂ ž
µλe
∂ x̌
(2.87)
28
2. Mathematical model
where the right hand side of the original equation was multiplied by (ρcL)/H(ρc)e ,
in order to obtain the dimensionless formulation. The thermal Rayleigh number for a
linear expression of the density state reads
RaT =
gρc∆ρKL
,
µλe
(2.88)
where ∆ρ is the change in density due to the variation in temperature.
2.3.2
Initial conditions
Initial conditions specify the state of the system at the start of the simulation. These
conditions may be specified by the distribution of the primary variables in space as
p = p0
∩
ω = ω0
∩
T = T0
on Ω,
(2.89)
where p0 = p(x, t0 ) is the initially prescribed pressure field, ω0 = ω(x, t0 ) denotes
the prescribed mass fraction field, T 0 = T (x, t0 ) is the initial temperature field, and
Ω denotes the flow domain. Alternatively, a set of derived parameter fields, like a
velocity field, a density field or a viscosity field could specify the starting conditions.
In general, the fields of primary variables or derived variables do not construct an
equilibrium condition. Deviations from the equilibrium condition will dissipate during
the transient computation. An initial state at equilibrium simplifies the analysis of
simulation results. A set of boundary conditions that construct the initial state can
be used to compute the initial situation at equilibrium. For calculation speed it is
preferable to start with a guess of the initial state of the system that does not vary too
much from the equilibrium situation.
2.3.3
Boundary conditions
This subsection proposes a set of physically admissible boundary conditions. Continuity
of total mass flux, solute mass flux and heat energy flux requires that the fluxes on
both sides of the boundary should be the same. This condition is formulated as
ρq·n|− = ρq·n|+
∩
∩
(ρωq − ρD·∇ω) ·n|− = (ρωq − ρD·∇ω) ·n|+
(cρqT − H·∇T ) ·n|− = (cρqT − H ·∇T ) ·n|+
on Γ, (2.90)
where n is the outward normal, |− indicates the flux inside the domain and |+ denotes
the flux outside the flow domain over the boundary Γ. For an in-flow boundary condition from a well mixed reservoir (outside the porous medium) the diffusive fluxes just
outside the flow domain can be neglected (ρD·∇ω) ·n|+ ≈ 0 and (H·∇T ) ·n|+ ≈ 0.
Equation (2.90) applies directly to the divergent form of the transport equations.
A more convenient formulation of the boundary conditions for the convective form of
the transport equations reads
− ρq·n = ρ̃q n
∩
(ρD·∇ω) ·n = ρ̃q n (ω̃ − ω)
∩
(H·∇T ) ·n = cρ̃q n T̃ − T
on Γ. (2.91)
29
2.3. Problem definition
This condition replaces the convective flux outside the porous medium by the product
of a prescribed volumetric influx q n and a variable liquid phase density ρ̃, which equals
the density of the liquid phase within the medium ρ if outflow occurs or is set to a
prescribed density ρ if inflow takes place. The convective flux reads ρq·n|+ = −ρ̃q n .
According to equation (2.91), the advective species mass flux equals ρq n ω for flow
into the medium given by positive values of q n . For outflow the advective mass flux
depends on the density and mass fraction inside the porous medium and reads −ρq n ω.
The diffusive flux for this condition is set to zero, as ω̃ = ω. The advective heat flux
reads cρq n T for inflow and cρq n T for outflow. The average porous medium diffusive
heat flux is set to zero for outflow conditions.
For no-flow boundary conditions the total mass flux, the solute mass flux, and the
heat energy flux across the boundary are zero. If the convective flux is set to zero then
the diffusive solute mass flux and diffusive energy flux are zero as well. This boundary
condition reads
ρq·n = 0
∩
(ρD·∇ω) ·n = 0
∩
(H·∇T ) ·n = 0
on
Γ1 .
(2.92)
These boundary conditions for the flow equation and for the transport equations are
known as a Neumann condition.
To be consistent with the source term Q in the governing equations (2.72), (2.73)
and (2.74), the prescribed boundary flux q n will be positive for flow into the porous
medium and negative for flow out of the porous medium. For q n > 0 the boundary
condition reads
− ρq·n = ρq n
∩
(ρD·∇ω) ·n = ρq n (ω − ω)
on Γ2 , (2.93)
(H·∇T ) ·n = −cρ T − T q·n
on Γ3 , (2.94)
∩
(H·∇T ) ·n = cρq n T − T
where ρq n denotes the prescribed total mass flux, ρq n ω is the prescribed salt mass
fraction flux, and cρq n T denotes the pre-defined heat energy flux. As n points outside
the flow domain, the in-product q·n specifies outflow. The no-flow condition given by
equation (2.92) forms a special case of this inflow condition. The constraints on the
transport equations are mixed type conditions, also called Robin boundary conditions.
If a Dirichlet condition is imposed on the flow equation (prescribed pressure), then
the corresponding flux can be calculated. If q·n < 0, inflow occurs and the combined
conditions read
p=p
∩
(ρD·∇ω) ·n = −ρ (ω − ω) q·n
∩
where p denotes the prescribed boundary pressure.
If the (negative) prescribed volumetric flux q n specifies outflow, then its value is
given by the mass fraction inside the domain. This also holds for the temperature. If
q n < 0 holds then the outflow boundary condition reads
−ρq·n = ρq n
∩
(ρD·∇ω) ·n = 0
∩
(H·∇T ) ·n = 0
on Γ4 .
(2.95)
30
2. Mathematical model
This condition follows from equation (2.91) with ω̃ = ω and T̃ = T . The no-flow
condition given by equation (2.92) also forms a special case of this outflow condition.
If a Dirichlet condition is imposed on the flow equation and q·n > 0 applies then
the outflow condition is reformulated as
p=p
∩
(ρD·∇ω) ·n = 0
∩
(H·∇T ) ·n = 0
on Γ5 .
(2.96)
For a boundary that separates two parts of the porous medium the mass-fraction
and the temperature should be the same on both parts of the boundary. Inflow and
outflow boundary conditions now read
−ρq·n = ρq n
∩
ω=ω
∩
T =T
on
Γ6 .
(2.97)
Alternatively for a given pressure at the internal boundary the condition is reformulated as
p=p ∩ ω=ω ∩ T =T
on Γ7 .
(2.98)
The previous boundary conditions impose either a pressure or a total mass flux on
the flow equation. A seepage condition or free-outflow condition imposes a geometrically non-linear boundary conditions. According to this condition, the atmospheric
pressure is imposed as long as outflow takes place and a no-flow condition holds if the
pore pressure at the boundary (inside the flow domain) is negative. This condition is
written as
p=0
if
ρq·n ≥ 0
.
(2.99)
ρq·n = 0
for
p<0
The seepage condition prevents water from flowing into the domain. If water is
available by precipitation, the seepage condition is generalized as the ponding condition.
For this condition the prescribed influx at the boundary will be positive (q n > 0) and
water is allowed to flow into the domain. The rate at which inflow takes place depends
on the capacity of the soil. If the pressure at the boundary exceeds the ponding
pressure (pp > 0) then infiltration will take place at a rate lower than the precipitation
rate. Under this condition the ponding pressure prescribes the infiltration rate. The
ponding pressure corresponds to the depth of small depressions in the terrain. If these
depressions are filled, overland flow will occur. The ponding condition generalizes
the seepage condition, by replacing the atmospheric pressure by the ponding pressure
which seems realistic for outflow conditions as well, because depressions can get filled
by out-flowing water.
if
−ρq·n ≤ ρq n
p = pp
.
(2.100)
−ρq·n = ρq n
for
p < pp
The evaporation (of bare soil) boundary condition (q n < 0) also generalizes the
seepage boundary condition. The evaporation flux is limited by an evaporation pressure
(pe < 0) which follows from the humidity of air. This boundary condition is written as
p = pe
if
ρq·n ≤ −ρq n
.
(2.101)
ρq·n = −ρq n
for
p > pe
2.3. Problem definition
31
The evaporation boundary condition limits the outflux to the flux that follows from
the prescribed evaporation pressure. During evaporation, the saturation of the soil and
its relative permeability decreases, and the soil becomes less capable to support the
outflux. The modeling of transpiration by vegetation-roots follows the same approach,
the outflux however is simulated by a volumetric source term. Compared to the bare
soil situation where the flux is imposed on the boundary, the permeability of the soil
will decrease less and can support the imposed outflux longer.
Seepage conditions, ponding conditions and evaporation conditions are generalized
as

p = pp
if
ρq·n ≥ −ρq n ∩ q n ≥ 0

for
pe < p < pp
.
(2.102)
−ρq·n = ρ̃q n

p = pe
if
ρq·n ≤ −ρq n ∩ q n < 0
According to this expression the pressure at the boundary of the flow domain is bounded
between a positive pressure at ponding conditions pp and a negative pressure during
evapotranspiration pe . Ponding conditions take place if the potential influx exceeds
the influx capacity of the soil. This condition is enforced if the outflux found for
the imposed pressure (which will have a negative value during infiltration) exceeds
the negative value of the potential influx q n . If the ponding condition applies, the
pressure reads p = pp , and equation (2.94) holds for liquid phase mass fraction and
porous medium temperature. If the maximum evaporation condition, given by p = pe ,
holds then equation (2.96) is imposed. If the flux is prescribed by −ρq·n = ρq n
then equation (2.93) applies for inflow and equation (2.95) holds for outflow. For
embankments a free water surface could generate an extra condition on the pressure.
If (part of) the boundary submerges the conditions of equation (2.102) are overruled,
and infiltration or exfiltration takes place at a pre-described pressure.
32
2. Mathematical model
Chapter 3
Numerical model
Analytical solutions may exist for a set of simplified flow and transport equations
posed on specific homogeneous geometries and closed by linear boundary conditions.
In general however, due to the nonlinearities and the coupling of the equations no
analytical solutions can be found and one has to rely on approximate numerical solution methods like the finite difference method, the finite volume method or the finite
element method [71, 72, 125, 13] to solve the equations. For geotechnical simulations
the finite element method is preferred, as this method captures complex geometries
more flexibly [116, 20, 19]. Finite element formulations are based either on a weighted
residual approach or a variational approach. The weighted residual approach applies
to a wider range of applications than the variational approach. However, if an equivalent minimization problem exists, both approaches produce the same set of equations.
For transient boundary value problems, the method of lines transforms the partial differential equations into (nonlinear) algebraic equations. According to this method a
semi-discretization in space first translates the partial differential equations into ordinary differential equations. Next a time integration scheme produces the final set of
algebraic equations. Nonlinear solvers, based on Picard techniques or Newton-Raphson
techniques, resolve the nonlinearities. The final set of linearized equations then can
be solved by iterative solvers like conjugate gradient methods or multigrid methods.
In this thesis the Galerkin weighted residual method discretizes the partial differential equations in space. A weighted finite difference method integrates the resulting
ordinary differential equations in time and provides the set of algebraic equations.
Appendix B outlines the finite element method. Section 3.1 presents the numerical
formulation for each of the basic equations separately. Section 3.2 gives the overall
numerical problem. The problem includes the resulting set of algebraic equations for
the flow problem, the solute transport problem and the heat transport problem into a
single set of algebraic equations and poses initial conditions and physically admissible
boundary conditions. Throughout this chapter the summation notation convention is
applied and results are presented for Cartesian coordinates.
33
34
3.1
3. Numerical model
Basic equations
For all basic processes, a partial differential equation poses the corresponding problem. This partial differential equation needs to be supplemented by a specification of
material behavior, a statement of initial conditions, and a definition of boundary conditions. The method of lines first discretizes the differential equations in space. The
spatial discretization procedure includes the weak formulation of the problem and the
application of Green’s theorem. The spatial discretization generates a local set of ordinary differential equations. Gaussian quadrature replaces integration by summation.
The method of lines then integrates the set of local ordinary differential equations in
time. A finite difference time integration scheme obtains the global the set of algebraic
equations.
Section 3.2.1 presents the numerical formulation of the flow equation, section 3.2.2
presents the algebraic solute transport equation, and section 3.2.3 discretizes the heat
transport equation. The basic equations share their primary variables: pressure, mass
fraction and temperature.
3.1.1
Flow equation
The primary unknown in the flow equations is the pore pressure p = p(x, t), which is
a function of the independent ordinates space and time. The trial function p̂ approximates the pressure field as
p(x, t) ≈ p̂(x, t) = pb (t)Nb (x),
b = 1, . . . , nn .
(3.1)
Here Nb denotes the basis function or shape function associated with node b that
interpolates the discrete nodal values of the pressure pb in space. The number of nodes
is given by nn .
Problem definition
The differential equation, which describes flow of a single fluid in a porous medium
given by equation (2.72) reformulated in Cartesian coordinates reads
∂ρS
∂p
∂ ρkr Kij ∂p
n
+ αp ρS
=
− ρgj
+ ρ̃Q
on Ω.
(3.2)
∂t
∂t
∂xi
µ
∂xj
The compressibility of the solid skeleton αp = αp (x), and the components of the
intrinsic permeability tensor Kij = Kij (x) are solid phase properties. Both properties
are assumed to be constant in time but may vary in space. Expressions for porosity
n, saturation S, relative permeability kr , fluid density and fluid viscosity will be given
T
at the end of this section. The gravitational acceleration vector [g] = [0, 0, −9.81] is
constant and its components are given by gi . The volumetric source term Q = Q(x, t)
[t−1 ] is predefined, and applies over the flow domain. The liquid phase density in
the right hand side ρ̃ = ρ is predefined for positive values of the volumetric source
(injection into the subsurface). For negative values of the volumetric source the fluid
phase density ρ̃ = ρ depends on the density in the porous medium (extraction from
35
3.1. Basic equations
the subsurface). An injection well located at a nodal position xa can be expressed in
terms of the volumetric source term as Qδ = Q (δ(x − xa ), t) [lnd t−1 ], where δ denotes
the Dirac delta function.
Initial conditions are given by a primary variable field, the pressure field p0 = p(x, 0)
on the flow domain reads
p = p0
on Ω.
(3.3)
Alternatively a saturation field could be given on parts of the flow domain. The functional relation only holds for the unsaturated part of the domain. However, for the
saturated part of the domain, no unique relation with the pore pressure exists and for
low saturation, the determination of the pressure becomes less accurate.
Three types of boundary conditions complete the flow equation. Dirichlet or firsttype boundary conditions prescribe the pressure on parts of the boundary. Dirichlet
boundary conditions can be expressed by
p=p
on
Γp1 ,
(3.4)
Γp1
where
is the part of the boundary where these conditions apply.
Von Neumann or second-type boundary conditions prescribe the derivative of the
pressure or flux on the boundary. These second-type boundary conditions read
ρkKij ∂p
− ρgj ni = ρ̃q n
on Γp2 ,
(3.5)
µ
∂xj
here ρ̃q n denotes the convective mass flux over the boundary, and q n captures the
volumetric part. If q n > 0 then flow takes place into the flow domain. The mass flux
follows from ρ̃q n = ρq n . Here ρ is the known density of the in-flowing fluid. If outflow
occurs then the density reads ρ̃ = ρ.
Robin or third-type boundary conditions relate a flux to a pressure, and can be
written as
ρkKij ∂p
− ρgj ni = r(p − p)
on Γp3 .
(3.6)
µ
∂xj
This type of boundary condition simulates the behavior of a drain, for example. Flow
out of the drain occurs if the pressure at the boundary (in the flow domain) is lower
than the pressure in the drain p > p. Outflow (into the drain) occurs in the opposite
case. For this case the drain must be able to support the flux. In equation (3.6), r
denotes an empirical resistance factor.
Spatial discretization
A weak form of equation (3.2) follows from integration over the flow domain and
multiplication by weighting functions Wa , which are attached to nodes. The first weak
form is given by
Z
Z
∂
∂ p̂
Wa n (Sρ) dΩ +
Wa αpρS dΩ
∂t
∂t
e
e
Ω
ZΩ
Z
∂ ρkKij ∂ p̂
=
Wa
− ρgj
dΩ +
Wa ρ̃QdΩ. (3.7)
∂xi
µ
∂xj
Ωe
Ωe
36
3. Numerical model
This finite element formulation of the flow equation is given for a single element, where
a = 1, . . . , nen and nen is the number of nodes in the element. Application of Green’s
theorem to the first term on the right hand side of equation (3.7) gives
∂
Wa n (Sρ) dΩ +
∂t
e
Ω
Z
Z
∂ p̂
ρkKij ∂Wa ∂ p̂
Wa αpρS dΩ = −
− ρgj dΩ
∂t
µ
∂xi ∂xj
Ωe
Ωe
Z
Z
ρkKij ∂ p̂
+
Wa
− ρgj ni dΓ +
Wa ρ̃QdΩ. (3.8)
µ
∂xj
e
Γ
Ωe
Z
Green’s theorem reduces the order of integration, which makes this second form more
suitable for further elaboration. Essential boundary conditions, which prescribe the
primary unknown at the boundary are taken into account by a modification of the
weighting function according to Wa = pa Wa . Substitution of the interpolation functions Nb which span the trial functions for the pressure, and replacing the weighting
functions by the basis functions Na according to Galerkin’s method, results
Z
Z
d
dp̂
Na αp ρS dΩ
Na n (Sρ) dΩ +
dt
dt
e
Ωe
Z Ω
Z
ρkKij ∂Na
ρkKij ∂Na ∂Nb
=−
pb dΩ +
ρgj dΩ
µ
∂x
∂x
µ ∂xi
e
e
i
j
Ω
Ω
Z
Z
Z
+
Na Nb (ρ̃q n )b dΓ +
Na Nb r (p − p)b dΓ +
Na Nb ρ̃Q b dΩ, (3.9)
Γe2
Γe3
Ωe
where second-type and third-type boundary conditions are incorporated. Prescribed
boundary fluxes and volumetric fluxes are given at the nodal points. Their values are
integrated over the element boundaries and over the elements, respectively.
Integration in time
The finite difference method translates the set of ordinary differential equations into a
set of algebraic equations. The time derivative term in the flow equation is approximated tangentially as
pn+1
d
(Sρ)|
− (Sρ)|
(Sρ) ≈
dt
p̂n+1 − p̂n
pn
ω n+1
dp̂
(Sρ)|
− (Sρ)|
+
dt
ω̂n+1 − ω̂n
+
ωn
(Sρ)|
dω̂
dt
T n+1
T̂ n+1
− (Sρ)|
−
T̂ n
Tn
dT̂
, (3.10)
dt
where p̂n+1 denotes the pressure trail function at the new time step and p̂n is the
pressure trail function at the old time step. The denoted product of density and
pn+1
pn+1 ,ω n+θ ,T n+θ
saturation (Sρ)|
is short for (Sρ)|
and its value is calculated for the
pressure at the new time step; the mass fraction and the temperature are obtained for
an intermediate time step n + θ where 0 ≤ θ ≤ 1. The same short notation was used
ω n+1
for (Sρ)|
where the mass fraction is calculated for the new time step and pressure
and temperature follow from an intermediate time step.
37
3.1. Basic equations
A global set of ordinary differential equations follows from the assemblage of the
local results obtained by the spatial discretization step. The global set is written as
Mab
dpb
= Sab pb + Fa ,
dt
(3.11)
where the capacity matrix components Mab [lnd −2 t2 ] are calculated as
!
Z
pn+1
pn
ne
(Sρ)|
− (Sρ)|
Mab = A
Na Nb n
+ αp ρS dΩ,
pn+1 − pn
e=1 Ωe
(3.12)
and approximated by
Mab ≈
ne
e
wm
e=1
A
pn+1
"
Na Nb
(Sρ)|
− (Sρ)|
n
pn+1 − pn
pn
+ αp ρS
!
#e
|J|
.
(3.13)
m
Numerical integration approximates the integral by a summation over the integration
points m, wm are the integration weights in an isoparametric local element and |J| is
the determinant of the Jacobian matrix which translates the local element position to
the global element position. The number of elements is denoted by ne . The components
of the conductivity matrix Sab [lnd −2 t] follow from
Z
Z
ne
ne
ρkr Kij ∂Na ∂Nb
Sab = − A
dΩ − A
Na Nb rdΓ,
(3.14)
µ
∂xi ∂xj
e=1 Ωe
e=1 Γe3
or
ne
Sab ≈ − A
e=1
e
wm
e
nb3
ρkr Kij ∂Na ∂ξk ∂Nb ∂ξl
e
|J|
− A wm (rNa Nb |J|)m .
µ
∂ξk ∂xi ∂ξl ∂xj
e=1
m
(3.15)
Here nb3 expresses the number of nodes on the boundary of the element where thirdtype boundary conditions apply. Part of the time derivative term is transferred into
the force vector. The components of the force vector Fa [mlnd −3 t−1 ] follow from
Z
Z
ne
ne
ρkr Kij ∂Na
Fa = A
Na Nb (ρ̃q n )b dΓ
ρgj dΩ + A
µ
∂xi
e=1 Ωe
e=1 Γe2
Z
Z
ne
ne
+A
Na Nb rpb dΓ + A
Na Nb ρ̃Q b dΩ
e=1
−
ne
A
e=1
Γe3
Z
Ωe
e=1
Ωe
1
ω n+1 ,T n+1
ω n ,T n
Na Nb n (Sρ)|
− (Sρ)|
dΩ, (3.16)
∆t
b
local-global mapping and numerical integration gives
e
nb2
ne
ρkr Kij ∂Na ∂ξk
e
e
ρgj |J|
+ A wm (Na Nb |J|)m (ρ̃q n )b
Fa ≈ A wm
µ
∂ξk ∂xi
e=1
e=1
m
nb3
e
+ A wm (rNa Nb |J|)m pb + ρ̃Qδ a
e=1
−
ne
e
1 h
A wm ∆t
e=1
ie
n+1
n+1
n
n
nNa (Sρ)| ω ,T
− (Sρ)| ω ,T |J| . (3.17)
m
38
3. Numerical model
Here nb2 and nb3 denote the element nodes on second-type and third-type element
boundary respectively. Volumetric sources are assumed to be collocated in the nodes
of the finite element grid, and simulate point sources. Boundary fluxes are integrated
over the element boundary.
The θ-weighted time integration procedure provides a first order representation
of the time derivative. The linear integration scheme transfers the set of ordinary
differential equations to a set of algebraic equations, and rewrites equation (3.11) as
n+θ
Mab
pn+1
− pnb
n+1 n+1
n n
b
= θSab
pb + (1 − θ)Sab
pb + θFan+1 + (1 − θ)Fan .
∆t
(3.18)
n+θ
n+1
Both Mab
, Sab
and Fan+1 depend on the yet unknown solution at the new time step
n + 1. A successive substitution solution procedure resolves the nonlinearities by
n+1,r
n+θ
Mab
− θ∆tSab
pbn+1,r+1
n,r n
n+θ
= Mab
+ (1 − θ)∆tSab
pb + θ∆tFan+1,r + (1 − θ)∆tFan , (3.19)
where r denotes the current iteration and r + 1 expresses the new (nonlinear) iteration.
Algebraic equations
For a fully-implicit (θ = 1) time integration scheme equation (3.19) can be written as
n+1,r
n+1,r
Mab
− ∆tSab
pbn+1,r+1 = Mab pnb + ∆tFan+1,r .
(3.20)
The final set of linearized algebraic equations, which approximates the flow equation
then reads
(3.21)
Apab pb = Bap .
Here pb is short for pbn+1,r+1 . The components of the stiffness matrix Apab [lnd −2 t2 ] read
Apab
=
ne
e
wm
e=1
A
e
(Na Nb Nc |J|)m
pn+1
n
n+1
(Sρ)|
− (Sρ)|
pn+1 − pn
pn
+ αp (ρS)
n+1
!
c
e n+1
∂Na ∂ξk ∂Nb ∂ξl
ρk
e
+ ∆t A wm Kij Nc
|J|
∂ξ
∂x
∂ξ
∂x
µ c
k
i
l
j
e=1
m
ne
nb3
e
+ ∆t A wm r n+1 Na Nb |J| m . (3.22)
e=1
Row-sum lumping improves the condition of the matrix and generates an explicit formulation for the pressure when an Euler forward time integration scheme is adopted.
Replacing Nb by δab produces a diagonal capacity matrix. However, the accuracy of
the solution decreases by lumping the matrix. For this reason the conductivity matrix
is not altered by a lumping procedure. The components of the force vector Bap [mlnd −3 ]
39
3.1. Basic equations
read
!
pn+1
pn
(Sρ)|
− (Sρ)|
+ αp ρS pnb
= A
n
pn+1 − pn
e=1
c
e n+1
ne
n
b2
∂Na ∂ξk
ρk
n+1
e
e
+ ∆t A wm
ρgj |J|
+ ∆t A wm (Na Nb |J|)m (ρ̃q n )b
Kij Nc
∂ξ
∂x
µ
k
i
e=1
e=1
m
c
nb3
e n+1
n+1
n+1
+ ∆t A wm r
Na Nb |J| m pb + ∆t ρ̃Qδ a
Bap
ne
e
wm
(Na Nb Nc
e
|J|)m
n+1
e=1
−
ne
A
e=1
e
wm
(Na Nb
n+1
n+1
n
n
e
|J|)m nn+1 (Sρ)ω ,T
− nn+1 (Sρ)ω ,T
. (3.23)
b
According to this equation porosity, saturation, relative permeability, fluid density and
fluid viscosity need to be calculated at the nodes. Only their products are interpolated
over the elements. This approach reduces the order of the functions on the element and
the algorithm does not require higher order integration schemes. It reduces the number
of integration points and saves computational time needed to set up the equations.
The nodal point porosity (n)a follows from equation (2.58) as
(n)a = n0 + αp (1 − n0 ) (pa − p0 ) ,
(3.24)
where n0 and p0 are constant over the element. Because adjacent elements may have different material parameters the porosity field shows discontinuities over element boundaries which coincide with material domain boundaries. This is also the case for synthetic fields, where the porosity is attached to the elements rather than the nodes. The
porosity in integration point m follows from interpolation nm = Nam na .
The degree of saturation at the nodal points Sa follows from equation (2.60) as
−g
(S)a = Sr + (Ss − Sr ) [1 + |ga ψ|gn ] m if ψ < 0
,
(3.25)
(S)a = Ss
for ψ ≥ 0
where ψ = |pa /(ρg)|. The empirical shape factors ga , gn and gm = (gn − 1)/gn , the
minimal degree of saturation Sr and the maximum degree of saturation Ss are constant
over the element. The saturation in integration point m follows from interpolation
Sm = Nam Sa . Although the pressure field is continuous, the saturation field shows
discontinuities for (partly) homogeneous situations.
The nodal values of the relative permeability (kr )a according to equation (2.64)
follows from the degree of saturation as
(
gm i2
h
1/g
if ψ < 0 ,
(kr )a = (Se )gl 1 − 1 − Se m
(3.26)
(kr )a = 1
for ψ ≥ 0
where Se = (Sa − Sr )/(Ss − Sr ). The shape factor gl is constant over the element,
the relative permeability in the integration points follows from interpolation (kr )m =
Nam (kr )a . Like the saturation field the relative permeability field may show discontinuities.
40
3. Numerical model
The nodal point density (ρ)a follows from equation (2.69) as
(ρ)a = ρ0 [1 + βp (pa − p0 ) + βω (ωa − ω0 ) − βT (Ta − T0 )] .
(3.27)
The reference values ρ0 , p0 , ω0 , and T0 are constant, as well as the coefficients βp , βω
and βT . The density at the integration point follows from ρm = Nam ρa . The density
field is C0 continuous for linear interpolation functions.
The dynamic viscosity of the fluid phase at the nodal points (µ)a , follows from the
nodal point temperature, and is written according to equation (2.71) as
(µ)a = 2.4318 · 10−5 10247.8/Ta −140.
(3.28)
Multiplying the nodal values by the interpolation functions gives a continuous viscosity
field. At the integration points the dynamic viscosity reads µm = Nam µa .
3.1.2
Solute transport equation
The mass fraction of the solute ω is the primary unknown in the solute transport
equation. The trial function of the mass fraction field ω̂ approximates the mass fraction
field as
ω(x, t) ≈ ω̂(x, t) = ωb (t)Nb (x),
b = 1, . . . , nn .
(3.29)
The finite element method discretizes the spatial domain and the nodal values of the
mass fraction vary in time. The finite element basis functions Nb interpolate the nodal
values over the flow domain. In this expression nn denotes the number of finite element
nodes.
Problem definition
The solute transport equation (2.73) written in index notation on a Cartesian field
reads
∂ω
∂ω
∂
∂ω
nSρ
on Ω,
(3.30)
= −ρqi
+
ρDij
+ ρ̃ (ω̃ − ω) Q
∂t
∂xi
∂xi
∂xj
where qi denotes the components of the volumetric Darcy flux vector, and Dij expresses
the components of the hydrodynamic dispersion tensor. The volumetric source term
Q = Q(x, t) is positive for flow into the domain. A local positive value simulates
an infiltration well and a negative value simulates an abstraction well. The injected
solute mass is predefined as ρ̃ω̃Q = ρωQ for a positive value of the volumetric source
(injection). For a negative value of the volumetric source the abstracted mass fraction
follows from ω̃ = ω, and the density of the fluid reads ρ̃ = ρ. So, for negative values of
the volumetric source the third term on the right hand side disappears.
The volumetric convective flux follows from Darcy’s equation (2.26) as
kr Kij ∂p
qi = −
− ρgj .
(3.31)
µ
∂xj
41
3.1. Basic equations
The coefficients in this equation were previously explained. The hydrodynamic dispersion tensor coefficients follow from equation (2.28), and reads
Dij = nSDc δij + (αl − αt )
qi qj
+ αt qδij ,
q
(3.32)
where q is the magnitude of the Darcy flux. The longitudinal dispersivity αl = αl (x)
and transverse dispersivity αt = αt (x) are assumed to be time independent, and vary
in space only. The molecular diffusivity Dc is assumed to be constant and includes the
micro turtuosity of the soil structure.
The initial condition for the solute transport equation is given by a mass fraction
field, expressed as
ω = ω0
on Ω,
(3.33)
where Ω denotes the flow domain, Γ will express the boundary of the domain.
First-type boundary conditions or Dirichlet boundary conditions specify the mass
fraction on parts of the flow domain boundary as
ω=ω
on
Γω
1.
(3.34)
Robin or third-type boundary conditions express the mass flux into the flow domain
as
∂ω
ρDij
ni = ρ̃q n (ω̃ − ω)
on Γω
(3.35)
2.
∂xj
Here q n specifies an influx. The constraint poses a Robin condition for flow into the
domain and reduces to a Von Neumann condition, which sets the dispersive flux to
zero, for outflow.
Spatial discretization
A weak formulation of the solute transport equation (3.30) over a single element reads
∂ ω̂
Wa nSρ
dΩ = −
∂t
e
Ω
∂ ω̂
dΩ
∂xi
Z
Z
∂ ω̂
∂
+
Wa
ρDij
dΩ +
Wa ρ̃ (ω̃ − ω̂) QdΩ. (3.36)
∂xi
∂xj
Ωe
Ωe
Z
Z
Wa ρqi
Ωe
The size of this local set of equations equals the number of nodes on the element. The
weight function Wa is associated with node a of element e. Application of Green’s
theorem to the third term of equation (3.36) gives
Z
Ωe
Wa nSρ
∂ ω̂
dΩ = −
∂t
Z
∂ ω̂
∂Wa ∂ ω̂
Wa ρqi
dΩ −
ρDij
dΩ
∂xi
∂xi ∂xj
Ωe
Ωe
Z
Z
∂ ω̂
+
Wa ρDij
ni dΓ +
Wa ρ̃ (ω̃ − ω̂) QdΩ. (3.37)
∂xj
Γe
Ωe
Z
42
3. Numerical model
Here the order of integration has been reduced. Substitution of the interpolation
function for the mass fraction trial function and replacing the weight functions by
the basis functions according to Galerkin’s principle gives
Z
Z
Z
dω̂
∂Nb
∂Na ∂Nb
Na nSρ dΩ = −
ρqi Na
ωb dΩ −
ρDij
ωb dΩ
dt
∂x
∂xi ∂xj
i
Ωe
Ωe
Ωe
Z
Z
+
Na Nb (ρ̃ (ω̃ − ω) q n )b dΓ +
Na Nb ρ̃ (ω̃ − ω) Q b dΩ, (3.38)
Γe2
Ωe
where the weak boundary condition disappears for a prescribed outflow flux.
Integration in time
The diffusive form of the solute transport equation gives rise to the following time
dependent terms
pn+1
(Sρω̂n+θ )
− (Sρω̂n+θ )
d
(Sρω̂) ≈
dt
p̂n+1 − p̂n
ω n+1
pn
dp̂
dt
T n+1
n
(Sρω̂n+1 )
− (Sρω̂n )|ω dω̂
(Sρω̂n+θ )
− (Sρω̂n+θ )
+
+
ω̂n+1 − ω̂n
dt
T̂ n+1 − T̂ n
Tn
dT̂
. (3.39)
dt
Multiplication of the semi-discrete flow equation (3.10) by ω̂n+θ , and subtracting this
expression from the diffusive form gives the convective formulation of the transport
equation. The time dependent term for this formulation reads
dω̂
(Sρ)|
≈
(Sρ)
dt
ω n+1
ω̂n+1 − (Sρ)|
ω̂n+1 − ω̂n
ωn
ω̂n dω̂
dt
−
(Sρ)|
ω n+1
ω̂n+θ − (Sρ)|
ω̂n+1 − ω̂n
ωn
ω̂n+θ dω̂
. (3.40)
dt
According to this expression it follows that for θ = 1 the mass term should be evaluated
as (Sρ)n , and for θ = 0 the mass term should hold (Sρ)n+1 . For this reason, secants in
the flow equation should be evaluated in the point (p0 , ω0 , T 0 ) if the convective form is
integrated implicitly, and in the point (p1 , ω1 , T 1 ) if the convective form is integrated
explicitly. It is noted that the secant procedure aims to find the change in (Sρ) over
a time step for the flow equation. The convective form applies its actual value, which
should be calculated at the end of the time step for an implicit formulation and at the
start of the time step for an explicit form.
A global set of ordinary differential equations follows from the assemble of local sets
of ordinary differential equations and can be written as
Mab
dωb
= Sab ωb + Fa .
dt
The components of the capacity matrix Mab [mlnd −3 ] read
Z
ne
Mab = A
nSρNa Nb dΩ,
e=1
Ωe
(3.41)
(3.42)
43
3.1. Basic equations
and its approximation
Mab ≈
ne
e
e
A wm (nSρNa Nb |J|)m .
e=1
(3.43)
This expression approximates the integrals by numerical integration. The capacity matrix has a consistent form. The components of the conductivity matrix Sab [mlnd −3 t−1 ]
are given by
Z
Z
ne
ne
∂Nb
∂Na ∂Nb
Sab = − A
dΩ − A
dΩ
ρqi Na
ρDij
∂x
∂xi ∂xj
e
e
i
e=1 Ω
e=1 Ω
Z
Z
ne
ne
−A
Na Nb (ρq n )b dΓ − A
Na Nb ρQ b dΩ. (3.44)
e=1
Γe2
e=1
Ωe
Numerical integration and local-global mapping obtains
e
e
ne
ne
∂Nb ∂ξk
∂Na ∂ξk ∂Nb ∂ξl
e
e
|J|
− A wm ρDij
|J|
Sab ≈ − A wm ρqi Na
∂ξk ∂xi
∂ξk ∂xi ∂ξl ∂xj
e=1
e=1
m
m
nb2
e
e
− A wm
(Na Nb |J|)m (ρq n )b − ρQδ a . (3.45)
e=1
Here nb2 denotes the number of nodes on an inflow boundary. The inverse of the
Jacobian matrix multiplies the derivatives of the basis functions on the mapped local
element in order to obtain the derivatives on the global element. The determinant of
the Jacobian matrix |J| maps the length, surface area or volume of the local element to
the global element. The prescribed volumetric flux Nb Qb is replaced by a point source
Qa , which simulates wells. The force vector components Fa [mlnd−3 t−1 ] follow from
Z
Z
ne
ne
Na Nb (ρωq n )b dΓ + A
Na Nb ρωQ b dΩ,
Fa = A
(3.46)
e=1
Γe2
e=1
Ωe
which is approximated by
Fa ≈
nb2
e
e
A wm (Na Nb |J|)m (ρωqn )b +
e=1
ρωQδ
a
.
(3.47)
The boundary flux term is only defined for boundary segments where q n > 0 holds.
Linear θ-weighted time integration transfers the set of ordinary differential equations to a set of algebraic equations. This procedure modifies equation (3.41) to
ωbn+1 − ωbn
n+1 n+1
n n
= θSab
ωb + (1 − θ)Sab
ωb + θFan+1 + (1 − θ)Fan .
(3.48)
∆t
Implicit formulations need updated computation of the capacity matrix components
n+θ
n+1
Mab
, the conductivity matrix components Sab
and the force vector entries Fan+1 .
The components follow from the yet unknown values of the primary variables. A Picard
iteration scheme resolves the nonlinearities by
n+1,r
n+θ
Mab
− θ∆tSab
ωbn+1,r+1
n,r n
n+θ
= Mab
+ (1 − θ)∆tSab
ωb + θ∆tFan+1,r + (1 − θ)∆tFan . (3.49)
n+θ
Mab
44
3. Numerical model
Here r denotes the current nonlinear iteration and r + 1 is the new iteration step. At
each step a linearized set of equations needs to be solved.
Algebraic equations
For an explicit integration scheme (θ = 0) the nonlinearities in the set of equations
disappear. For pure convective flow the Galerkin explicit time integration schemes are
unconditionally unstable [128]. For this reason, an implicit time integration scheme is
preferred. The set of algebraic equations reads
n+1,r
n+1,r
n n
Mab
− ∆tSab
ωbn+1 = Mab
ωb + ∆tFan+1,r .
(3.50)
This set of equations can be rewritten as
ω
Aω
ab ωb = Ba ,
where ωb is short for
Aω
ab
ωbn+1 .
The stiffness matrix
Aω
ab
(3.51)
nd −3
[ml
] reads
e
ne
ne
∂Nb ∂ξk
e
n+1
e
e
n+1
= A wm (Na Nb Nc |J|)m (nSρ) c + ∆t A wm (ρqi )
Na
|J|
∂ξk ∂xi
e=1
e=1
m
e
ne
∂N
∂ξ
∂N
∂ξ
k
b
l
a
e
|J|
+ ∆t A wm
(ρDij )n+1
∂ξk ∂xi ∂ξl ∂xj
e=1
m
nb2
n+1
e
n+1
e
+ ∆t A wm
(Na Nb |J|)m (ρq n )b + ∆t ρQδ a , (3.52)
e=1
and the force vector
Baω =
ne
Baω
[ml
nd−3
] is given by
e
n+1
A wme (Na Nb Nc |J|)m ωbn (nSρ)c
e=1
nb2
e
n+1
e
(Na Nb |J|)m (ρq n ω)b
+ ∆t A wm
e=1
+ ∆t ρQδ ω
n+1
a
. (3.53)
Here, second-type boundaries only contribute, if the flux prescribes flow into the flow
domain.
The Darcy (mass) flux components in an integration point can be found by
ρkr Kij ∂Na ∂ξk
x
pa − Na (ρgj )a
.
(3.54)
(ρqi )m = −
µ
∂ξk ∂xj
m
This expression interpolates the pressure with a lower order interpolation function than
the density scaled gravitational vector. As a result, the velocity distribution will not be
consistent over the element. For the convective term the velocity will be integrated over
the element and the error in the approximation will cancel out. But this is not the case
for the dispersive term and Voss and Souza [118] proposed a second order reduction
scheme for the interpolation of the gravity term. They first wrote the gravity vector
in local coordinates for node a as
∂xj x
ξ
(ρgi )a =
ρg
.
(3.55)
∂ξi j a
45
3.1. Basic equations
Secondly, they interpolated the nodal vectors over the interior of the elements by the
scaled derivative of the interpolation function. This interpolation reads
ρgiξ = Nai ρgiξ ,
(3.56)
a
where no summation over i takes place. The modified interpolation function in the ith
direction are found, according to
Nai = r i
∂Na
,
∂ξi
r i = 1/
X ∂Na
,
∂ξi
a
(3.57)
where r i forces the sum of the reduced interpolation functions to be one, on each
location. The global gravity force vector components follow from the local components
by multiplying them by the Jacobian matrix. This procedure reads
ρgix =
∂ξj ξ
ρg .
∂xi j
(3.58)
A consistent velocity field finally follows from
∂ξk ∂Na
ρkr
pa
(ρqi )m = − Kij Nb
∂xj ∂ξk m
µ b
∂ξk k
ρkr
+ Kij Nb
Na
ρgkξ
, (3.59)
∂xj
µ b
a
m
where no summation over k in the second term takes place.
The mass equivalent of the hydrodynamic dispersion Dij in integration point m
follows from
qi qj
(ρDij )m = (ρnS)m Dc δij + (αl − αt )Na
ρa + (αt qNa )m ρa δij ,
(3.60)
q m
where the molecular dispersion Dc is constant, the longitudinal dispersivity αl and
transverse dispersivity αt are constant over the element. The components of the Darcy
flux vector (qi )m and its magnitude (q)m follow from equation (3.59).
3.1.3
Heat transport equation
Temperature T is the primary unknown in the heat transport equation. The finite
element formulation approximates the temperature by a trial function T̂ . The nodal
values of the trial functions Tb vary in time, finite element basis functions interpolate
these nodal values in space, as
T (x, t) ≈ T̂ (x, t) = Tb (t)Nb (x).
b = 1, . . . , nn ,
(3.61)
Here Nb denotes the basis function or shape function associated with node b, and nn
is the number of nodes in the finite element mesh.
46
3. Numerical model
Problem definition
The heat transport equation (2.74), reformulated in index notation on Cartesian coordinates reads
nSρc
∂T
∂T
∂T
+ (1 − n) ρs cs
= −ρcqi
∂t
∂t
∂xi
∂
∂T
+
Hij
+ cρ̃ T̃ − T Q
∂xi
∂xj
on Ω, (3.62)
where c denotes the specific heat capacity of the fluid phase, cs = cs (x) is the specific
heat capacity of the solid phase, and Hij = Hij (x, t) denotes the hydrodynamic thermal
dispersion tensor components. If the volumetric source term specifies outflow then
T̃ = T and the source term disappears from the equation. Thermal energy still leaves
the system at these points at a rate cρT Q.
The hydrodynamic thermal dispersion Hij includes the mechanical part of the hydrodynamic dispersion tensor, and conductivity of the partly saturated soil. The coefficients of the hydrodynamic thermal dispersion tensor follow from equation (2.43),
and are written as
qi qj
s
Hij = nSλδij + (1 − n) λij + cρ (αl − αt )
+ δij αt q .
(3.63)
q
The thermal components of the soil skeleton λsij = λsij (x) vary in space, but are assumed
to be constant in time. The water phase conductivity λ is assumed to be a constant
scalar variable.
The initial condition is given by an initial temperature distribution in space, written
as
T = T0
on Ω.
(3.64)
A temperature distribution at equilibrium follows from a boundary value problem
where the conditions on the boundary Γ of the flow domain Ω need to be specified.
First-type boundary conditions or Dirichlet boundary conditions prescribe the temperature on parts of the flow boundary, according to
T =T
on
ΓT1 .
(3.65)
Robin or third-type boundary conditions specify the derivative of the temperature over
the boundary, which corresponds to an energy flux. This condition reads
Hij
∂T
ni = cρ̃q n T̃ − T
∂xj
on
ΓT2 ,
(3.66)
where the volumetric boundary flux q n is positive for inflow. For this condition the
temperature of the fluid reads T̃ = T , and the fluid density is prescribed as ρ̃ = ρ. If
the boundary flux has a negative value then the dispersive flux over the boundary is
set to zero.
47
3.1. Basic equations
Spatial discretization
Multiplication of the heat transport equation (3.62) by a weighting function and integration over a single element gives
Z
Wa nSρc
Ωe
∂ Tˆ
dΩ +
∂t
Z
+
Z
Z
∂ T̂
∂ T̂
dΩ = −
Wa ρcqi
dΩ
∂t
∂xi
Ωe
!
Z
∂ T̂
Hij
dΩ +
Wa cρ̃ T̃ − T̂ QdΩ, (3.67)
∂xj
Ωe
Wa (1 − n) ρs cs
Ωe
Wa
Ωe
∂
∂xi
where a = 1, . . . , nne and nne denotes the number of nodal points in the element. This
number equals the number of equations in the local set. Green’s theorem on the fourth
term of equation (3.67) reduces the order of integration. The result is written as
∂ Tˆ
dΩ +
∂t
Z
∂ T̂
∂ T̂
dΩ = −
Wa ρcqi
dΩ
∂t
∂x
e
e
e
i
Ω
Ω
Ω
Z
Z
Z
∂Wa ∂ T̂
∂ T̂
dΩ +
Wa Hij
ni dΓ +
Wa cρ̃ T̃ − T̂ QdΩ. (3.68)
−
Hij
∂xi ∂xj
∂xj
Γe
Ωe
Ωe
Z
Wa nSρc
Z
Wa (1 − n) ρs cs
Substitution of the interpolation functions for the temperature trial function and replacing the weighting functions by the basis functions according to Galerkin’s method,
gives the local set of equations, which can be expressed as
Z
Ωe
Z
dT̂
dT̂
dΩ +
Na (1 − n) ρs cs
dΩ
dt
dt
e
Ω
Z
Z
∂Nb
∂Na ∂Nb
Tb dΩ −
Tb dΩ
=−
Na ρcqi
Hij
∂xi
∂xi ∂xj
Ωe
Ωe
Z
Z
h i
h i
+
Na Nb cρ̃ T̃ − T q n dΓ +
Na Nb cρ̃ T̃ − T Q dΩ. (3.69)
Na nSρc
b
Γe2
Ωe
b
In this expression, the boundary integral was replaced by the weak definition. Essential (first type) boundary conditions are included by a modification of the weighting
functions.
Integration in time
The global set of ordinary differential equations is written as
Mab
dTb
= Sab Tb + Fa.
dt
(3.70)
The components of the consistent form of the capacity matrix Mab [mlnd−1 t−2 T−1 ] are
given by
Z
Mab =
ne
A
e=1
Na Nb (nSρc + (1 − n) ρs cs ) dΩ,
Ωe
(3.71)
48
3. Numerical model
Numerical integration over local elements gives
Mab ≈
ne
e
s s
A wm [Na Nb (nSρc + (1 − n) ρ c
e=1
) |J|]em .
(3.72)
Here numerical integration approximates the integral over the element. The local matrices are assembled in the global capacity matrix. The components of the conductivity
matrix Sab [mlnd −1 t−3 T−1 ] follow from
Z
Z
ne
ne
∂Nb
∂Na ∂Nb
Sab = − A
ρcqi Na
dΩ − A
Hij
dΩ
∂x
∂xi ∂xj
e
e
i
e=1 Ω
e=1 Ω
Z
Z
ne
ne
−A
Na Nb (cρ̃q n )b dΓ − A
Na Nb cρ̃Q b dΩ, (3.73)
e=1
Γe2
e=1
Ωe
which is approximated by
e
e
ne
ne
∂Nb ∂ξk
∂Na ∂ξk ∂Nb ∂ξl
e
e
|J|
− A wm Hij
|J|
Sab ≈ − A wm cρqi Na
∂ξk ∂xi
∂ξk ∂xi ∂ξl ∂xj
e=1
e=1
m
m
nb2
e
e
− A wm c (Na Nb |J|)m (ρq n )b − cρQδ a . (3.74)
e=1
Here nb2 denotes the number of boundary nodes on the second-type inflow boundary.
If outflow is prescribed then the term is excluded from the formulation. The volumetric
source is replaced by a point source Qa located in node a, which is included only if the
source has a positive value and energy is flowing into the domain. The components of
the force vector Fa [mlnd −1 t−3 ] read
Z
Z
ne
ne
Fa = A
Na Nb cρq n T b dΓ + A
Na Nb cρQT b dΩ,
(3.75)
e=1
Γe2
or
Fa ≈
nb2
e=1
e
e
A wm c (Na Nb |J|)m
e=1
ρq n T
Ωe
b
+ cρQδ T
a
.
(3.76)
Both terms only apply for inflow conditions (q n > 0 and Q > 0); for outflow conditions
the terms become zero.
Linear θ-weighted time integration translates equation (3.70) into
n+θ
Mab
Tbn+1 − Tbn
n+1 n+1
n n
= θSab
Tb
+ (1 − θ)Sab
Tb + θFan+1 + (1 − θ)Fan ,
∆t
(3.77)
n+θ
where 0 < θ < 1. In general, components of the capacity matrix Mab
, the conducn+1
n+1
tivity matrix Sab , and the force vector Fa
follow from the yet unknown values of
the primary variables. Successive substitution or Picard iteration resolves the nonlinearities by
n+1,r
n+θ
Mab
− θ∆tSab
Tbn+1,r+1
n,r
nn+θ
+ (1 − θ)∆tSab
= Mab
Tbn + θ∆tFan+1,r + (1 − θ)∆tFan , (3.78)
where r denotes the current iteration step and r + 1 is the new step.
49
3.2. Problem definition
Algebraic equations
The implicit formulation for the heat transport equation reads
n+1,r
n+1,r
n+1,r n
Mab
− ∆tSab
Tbn+1 = Mab
Tb + ∆tFan+1,r .
(3.79)
The corresponding set of linear algebraic equation reads
ATab Tb = BaT ,
(3.80)
where Tbn+1 is short for Tb . The stiffness matrix components ATab [mlnd−1 t−2 T−1 ] read
ATab =
ne
e
n+1
A wme (Na Nb Nc |J|)m (nSρc + (1 − n) ρs cs )c
e=1
e
e
ne
ne
∂Nb ∂ξk
n+1
n+1 ∂Na ∂ξk ∂Nb ∂ξl
e
e
+ ∆t A wm
c (ρqi )
Na
|J|
+ ∆t A wm
Hij
|J|
∂ξk ∂xi
∂ξk ∂xi ∂ξl ∂xj
e=1
e=1
m
m
nb2
n+1
e
n+1
e
+ ∆t A wm c (Na Nb |J|)m (ρq n )b + c∆t ρQδ a , (3.81)
e=1
and the components of the right hand side force vector BaT [mlnd −1 t−2 ] read
BaT =
ne
e
e
n
A wm (Na Nb Nc |J|)m Tb
e=1
n+1
(nSρc + (1 − n) ρs cs )c
nb2
e
e
+ ∆t A wm
c (Na Nb |J|)m ρq n T
e=1
n+1
b
+ c∆t ρQδ T
n+1
a
. (3.82)
The solution of this set of equation is found sequentially.
In order to set up the set of algebraic equations, the components of hydrodynamic
thermal dispersion tensor Hij need to be calculated in the integration points m. The
components of the tensor follow from
(Hij )m = (nS)m λδij + (1 − n) λsij
m
qi qj
+ δij Na αt q
ρa , (3.83)
+ c Na (αl − αt )
q
m
where the liquid phase specific heat capacity c and the conductivity λ are constant.
The thermal conductivity coefficients for the soil skeleton λsij are constant in time and
constant over the element.
3.2
Problem definition
The finite element method transforms the set of partial differential equations into a set
of algebraic equations. The number of algebraic equations equals the number of nodes
in the mesh times the degrees of freedom per node. Here three degrees of freedom for
each node apply: pressure, mass fraction and temperature. The set of equations can
50
3. Numerical model
either be solved in its coupled full-size form or in a decomposed form by decoupling
the equations per degree of freedom.
Section 3.2.1 presents the system of algebraic equations and the solution strategy
for the partly linearized set of coupled algebraic equations. Section 3.2.2 proposes a
procedure to include initial conditions and section 3.2.3 formulates set of physically
admissible boundary conditions that combine the conditions given for each equation
separately.
3.2.1
System of algebraic equations
The coupled solution procedure requires the formulation of the constitutive relation for
fluid density and fluid viscosity as a function of all primary unknowns. Introduced into
the set of equations this formulation produces coupling terms in the global conductivity
matrix. The capacity matrix in the flow equation already contained coupling terms
through the functional dependency of relations on mass fraction and temperature.
The decoupled solution procedure accounts for the coupling terms by the force vector.
Generally, the Darcy velocity and dispersivity are included in the set of transport
equations. For this reason both coupled and decoupled equations have to be solved
iteratively. In this thesis, the decoupled procedure is preferred as it saves computer
memory and is able to deal with more complex constitutive relations.
At each time step, a linearized symmetric flow equation and two non-symmetric
transport equations need to be solved, because an implicit time integration procedure
is used for all equations. Coupling terms and nonlinearities are resolved sequentially.
The amount of work depends on the nonlinearities present in each of the equations, on
the coupling terms that have to be accounted for, and on the required time step size
relative to the simulated process.
Stability requirements may reduce the allowable time step size. The time step
size is also restricted to the range of the nonlinear solver and the mesh refinement
strategy. The stability analysis of the set of coupled nonlinear equations is not easily
performed, and only strongly simplified stability checks can be carried out. In this
thesis, an implicit time integration formulation is proposed, which is less restrictive
on the time step size. A Picard iteration scheme is preferred over a Newton-Raphson
scheme, because the range of convergence of the first scheme is wider [100].
The proposed (decoupled) sequential procedure is expressed by
Apab (pn , pr , ωr , T r ) pr+1
= Bap (pn , pr , ωn , ωr , T n , T r ) ,
b
r+1
Aω
, ωr , T r ωbr+1 = Baω pr+1 , ωn , ωr , T r ,
ab p
ATab pr+1 , ωr+1 , T r Tbr+1 = BaT pr+1 , ωr+1 , T n , T r .
(3.84)
Here n indicates the current time step, r is short for the current iteration step at
the new time step n + 1 and r + 1 denotes the new iteration step at new time step
n + 1. The sequential procedure first solves the pressure equation, posed by the first
line in equation (3.84). This line presents the set of algebraic equations that poses
the flow problem. Both the left hand side matrix and the right hand side vector are
based on primary unknowns evaluated on the current and the new time step. At
51
3.2. Problem definition
the new time step these unknowns are approximated by their values at the current
nonlinear approximation at this time step or their values at the previous time step at
the start of the nonlinear iteration. The expression for the system matrix coefficients
was given by equation (3.22). The right hand side vector coefficients were expressed by
equation (3.23). The Darcy velocity field follows from the updated pressure field, and
will be imposed into the transport equations. The second line in equation (3.84) states
the solute transport equation. System matrix coefficients and right hand side vector
coefficients followed from equations (3.52) and (3.53). The updated mass fraction field
is used in the heat transport computation. The third line in equation (3.84) gives the
heat transport equation, for which the coefficients of the system matrix were given by
equation (3.81) and the right hand side vector coefficients followed from equation (3.82).
The sequential procedure is repeated until the solution of pressure, mass fraction and
temperature stabilize or successive changes in the nodal values are within a prescribed
tolerance. The algorithm switches to the next time level after it achieves convergence.
Implicit formulations of linear convection-diffusion problems are found to be unconditionally stable. This does not hold for the nonlinear flow problem. For nonlinear
problems the time step size is restricted by the range of convergence of the nonlinear
solver. The number of Picard iterations per time step provides a heuristic basis for
an automatic time step algorithm. The proposed time step algorithm considers four
situations. If the number of iterations needed to achieve convergence in a time step
is less than a lower limit value (nr < 5), then the size of the next time step may be
doubled according to ∆tn+1 = 2.0 ∆tn . Dissipation of an infiltration front may involve
this situation. If on the other hand the number of nonlinear iterations exceeds an
upper limit value, (nr > 10), then the size of the next time step needs to be reduced
by ∆tn+1 = 0.5 ∆tn . If the number of nonlinear iterations exceeds a second upper
limit value, (nr > 20), then the size of the current time step will be reduced once more
according to ∆tn = 0.5 ∆tn . This situation may occur if boundary conditions change
or infiltration fronts propagate into a new material while the time step is relatively
large.
Stability and accuracy
This section considers the stability and accuracy of a one-dimensional convectiondiffusion test problem, written as
C
∂u
∂u
∂2u
+U
− D 2 = 0,
∂t
∂x
∂x
on
Ω.
(3.85)
This expression does not include source terms and simplifies the previously discussed
flow and transport equations. The flow equation (3.2) reduces to this form if fluid
density, fluid viscosity, intrinsic permeability of the soil structure and saturation and
relative permeability of the subsoil do not vary in time and space. For this case the
coefficients read C = αp ρS and D = ρkr Kij /µ. The solute transport equation (3.30)
complies to the test problem if Fick’s law for diffusion holds and density changes are not
taken into account. The convection-diffusion coefficients then read C = nSρ, U = ρqi
and D = ρDij . The test problem simplifies the heat transport equation (3.62) for
C = (nSρc + (1 − n)ρs cs ), U = ρcqi and D = ρHij . The thermal conductivity of
52
3. Numerical model
the fluid phase and the solid phase both contribute to the constant diffusion coefficient. Hydrodynamic dispersion is disregarded for the stability and accuracy analysis.
The unknown u in the test problem, either specifies pressure p, mass fraction ω or
temperature T .
Spatial discretization and integration in time lead to algebraic equations (3.18),
(3.48) and (3.77). These algebraic equations simplify the test problem to
n+θ
Mab
un+1
− unb
n+1 n+1
n n
b
= θSab
ub + (1 − θ)Sab
ub .
∆t
(3.86)
In this section the convective term and the diffusive term will be considered separately.
The consistent-mass capacity matrix according to equations (3.13), (3.43) and (3.72)
reads
Z
ne
CNa Nb dΩ.
(3.87)
Mab = A
e=1
Ωe
D
The diffusive part of the conductivity matrix Sab
according to equation (3.15), (3.45)
and (3.74) reads
Z
ne
∂Na ∂Nb
D
Sab = − A
D
dΩ.
(3.88)
∂x ∂x
e
e=1 Ω
U
The convective part of the conductivity matrix Sab
follows from the numerical derivation of transport equations (3.45) and (3.74) as
Z
ne
∂Nb
U
Sab
= −A
U Na
dΩ.
(3.89)
∂x
e=1 Ωe
The nodal equation for the test problem on a uniform grid reads
∆xC n
∆xC n+1
uj−1 + 4un+1
uj−1 + 4unj + unj+1
+ un+1
j
j+1 −
6
6
∆tU
∆tU
n+1
n+1
+θ
−uj−1 + uj+1 + (1 − θ)
−unj−1 + unj+1
2
2
D∆t
∆tD
n+1
n+1
n+1
+θ
−uj−1 + 2uj − uj+1 + (1 − θ)
−unj−1 + 2unj − unj+1 = 0. (3.90)
∆x
∆x
The finite difference form follows from this equation by mass lumping and multiplication by 1/∆x∆t. This form reads
C n+1
C n
U
U
n+1
uj −
uj + θ
−un+1
−unj−1 + unj+1
j−1 + uj+1 + (1 − θ)
∆t
∆t
2∆x
2∆x
D
D
n+1
n+1
n+1
+θ
−uj−1 + 2uj − uj+1 + (1 − θ)
−unj−1 + 2unj − unj+1 = 0. (3.91)
2
2
∆x
∆x
Both finite element and finite difference formulation solve a modified equation, which
follows from a Taylor series expansion with reference point at (j∆x, nθ∆t). The expansion of the discrete unknown un+1
in time gives
j
un+1
≈ un+θ
+ (1 − θ)∆t
j
j
∂u
∂t
+ (1 − θ)2
un+θ
j
∆t2 ∂ 2 u
2 ∂t2
un+θ
j
+ O ∆t3 .
(3.92)
53
3.2. Problem definition
The expansion of unj+1 in space and time reads
unj+1 ≈ un+θ
+ ∆x
j
∂u
∂x
+ θ∆x∆t
− θ∆t
un+θ
j
∂2u
∂x∂t
∂u
∂t
+ θ2
ujn+θ
+
un+θ
j
∆x2 ∂ 2 u
2 ∂x2
∆t2 ∂ 2 u
2 ∂t2
un+θ
j
un+θ
j
+ O ∆x3 + O ∆t3 . (3.93)
The modified equation for the consistent-mass discretization with reference point at
(j∆x, nθ∆t) reads
C
∂u
∂2 u
C∆t ∂ 2 u C∆x2 ∂ 3 u
∂u
+U
− D 2 = − (1 − 2θ)
−
∂t
∂x
∂x
2 ∂t2
6 ∂t∂x2
2 3
2 3
2 4
C∆t ∂ u U ∆x ∂ u D∆x ∂ u
C∆t2 ∂ 3 u
−
−
+
+ θ(1 − θ)
3
3
4
6 ∂t
6 ∂x
12 ∂x
2 ∂t3
2
3
2
U ∆t ∂ u
D∆t ∂ 4 u
− θ (1 − θ)
+
θ
(1
−
θ)
+ . . . (3.94)
2 ∂t2 ∂x
3 ∂t2 ∂x2
The right hand side presents the truncation errors, which are in the order of O(∆t, ∆x2)
for the fully-implicit and explicit form, and correspond to O(∆t2 , ∆x2) for θ = 21 .
The first order derivative term in time introduces numerical diffusion [119]. The time
derivative converts into spatial derivatives by
∂2u
∂
∂u
∂2 u
U 2 ∂ 2 u 2U D ∂ 3 u D2 ∂ 4 u
U
−D 2 =
−
+
.
(3.95)
C 2 =−
∂t
∂t
∂x
∂x
C ∂x2
C ∂x3
C ∂x4
The numerical diffusion, introduced by integration in time, reads
Dnum = θ −
1
2
U 2 ∆t
.
C
(3.96)
This artificial diffusion contributes to the diffusion D in equation (3.85). The actual
velocity and actual diffusion in this equation hold U/C and D/C. For an explicit
scheme the numerical diffusion equals Dnum = − 12 U 2 ∆t/C, whereas for a fully implicit
scheme this term reads Dnum = 21 U 2 ∆t/C. For stability the total diffusion should be
positive, D + Dnum > 0. A positive value of the numerical diffusion stabilizes the
method and adds diffusion to the system. The stability requirement poses a restriction
on the time step for the explicit formulation according to
∆t < 2CD/U 2 .
(3.97)
For a time weighting factor 21 ≤ θ ≤ 1 the total diffusion is positive independent of
the time step. For accuracy D Dnum in order to obtain accurate solutions. The
approximate equation for the lumped-mass form gives
C
∂u
∂u
∂p2
C∆t ∂ 2 u U ∆x2 ∂ 3 u D∆x2 ∂ 4 u
+U
− D 2 = − (1 − 2θ)
−
+
+ . . . (3.98)
∂t
∂x
∂ x
2 ∂t2
6 ∂x3
12 ∂x4
54
3. Numerical model
For θ = 0 this form corresponds to the forward in time central in space finite difference
approximation, a backward in time central in space finite difference formulation follows
from θ = 1.
Alternatively Von Neumann analysis considers the stability in terms of Fourier
series. The initial condition has to be developed into a Fourier series of cosine functions
with various wave lengths L. The initial condition reads u(x, 0) = u0 cos(kx), where
k = 2π/L will be represented by the real part of
u(x, 0) = u0 exp (ikx) .
(3.99)
In a grid point the initial value reads
uj = u0 exp (ijξ) ,
(3.100)
where ξ = k∆x = 2π∆x/L. The smallest grid size is ∆x = 0 and the largest grid size
is ∆x = L/2 as a wave can only be represented by at least two grid points. For this
reason 0 ≤ ξ ≤ π holds. In the adjacent grid point the discrete value of the unknown
equals
unj+1 = u0 exp (iξ) exp (ijξ) .
(3.101)
For the next time step the grid point value reads
un+1
= u0 ρ exp (ijξ) ,
j
(3.102)
where ρ denotes the amplification factor. The modulus of the amplification factor |ρ|
multiplies the amplitude of the initial signal. The requirement for stability therefore
states |ρ| ≤ 1. The phase shift follows from the argument of the amplification factor,
which will be written as arg(ρ). The relative celerity ratio is the ratio of arguments of
the amplification factor at different time steps and expresses the ratio of phase angles
[128] by arg(ρ)L/2πU ∆t. Both amplitude ratio and relative celerity ratio should be
close to unity for accuracy. The discrete grid point expressions entered in the nodal
consistent-mass equation (3.90) give the expression for the amplification factor for the
test problem. Using {exp(iξ)+exp(−iξ)}/2 = cos ξ and {exp(iξ)−exp(−iξ)}/2i = sin ξ
the expression for the amplification factor reads
ρ=
(cos ξ + 2) + 3(1 − θ)λ (cos ξ − 1) − 3σi (1 − θ) sin ξ
,
(cos ξ + 2) − 3θλ (cos ξ − 1) + 3σiθ sin ξ
(3.103)
where λ = 2D∆t/C∆x2, and σ = U ∆t/C∆x. Figure 3.1 show the amplitude ratio
and relative celerity ratio for a number of combinations of λ and σ.
The amplification factor for the lumped-mass formulation, given by equation (3.91),
reads
1 + (1 − θ)λ (cos ξ − 1) − σ (1 − θ) i sin ξ
ρ=
.
(3.104)
1 − θλ (cos ξ − 1) + σθi sin ξ
According to these expressions both methods are unconditionally stable for
The explicit mass-lumped method is stable for
σ 2 ≤ λ ≤ 1.
1
2
≤ θ ≤ 1.
(3.105)
55
3.2. Problem definition
θ = 0.0
1.00
0.00
1.0E+0
θ = 0.0
2.00
amplitude ratio
relative celerity
2.00
1.00
0.00
1.0E+1
1.0E+2
1.0E+0
L/dx
θ = 0.5
1.00
0.00
1.0E+0
1.00
0.00
1.0E+1
1.0E+2
1.0E+0
L/dx
θ = 1.0
1.0E+2
θ = 1.0
2.00
1.00
0.00
1.0E+0
1.0E+1
L/dx
amplitude ratio
relative celerity
2.00
1.0E+2
θ = 0.5
2.00
amplitude ratio
relative celerity
2.00
1.0E+1
L/dx
1.00
0.00
1.0E+1
1.0E+2
1.0E+0
L/dx
1.0E+1
1.0E+2
L/dx
1
2
3
4
Figure 3.1: Amplification factors for consistent-mass formulation 1: λ = σ = 0.1, 2:
λ = σ = 0.5, 3: λ = σ = 1.0, 4: λ = σ = 2.0.
This condition states two restrictions on the maximal time step. The first condition
λ ≤ 1 states ∆t ≤ ∆t1 = C∆x2 /2D. The second condition σ 2 ≤ λ poses ∆t ≤ ∆t2 =
2CD/U 2 . This condition was given already by equation (3.97). If the condition of the
56
3. Numerical model
θ = 0.0
1.00
0.00
1.0E+0
θ = 0.0
2.00
amplitude ratio
relative celerity
2.00
1.00
0.00
1.0E+1
1.0E+2
1.0E+0
L/dx
θ = 0.5
1.00
0.00
1.0E+0
1.00
0.00
1.0E+1
1.0E+2
1.0E+0
L/dx
θ = 1.0
1.0E+2
θ = 1.0
2.00
1.00
0.00
1.0E+0
1.0E+1
L/dx
amplitude ratio
relative celerity
2.00
1.0E+2
θ = 0.5
2.00
amplitude ratio
relative celerity
2.00
1.0E+1
L/dx
1.00
0.00
1.0E+1
1.0E+2
1.0E+0
L/dx
1.0E+1
1.0E+2
L/dx
1
2
3
4
Figure 3.2: Amplification factors for lumped-mass formulation 1: λ = σ = 0.1, 2:
λ = σ = 0.5, 3: λ = σ = 1.0, 4: λ = σ = 2.0.
cell Reynolds number or cell Peclet number given by Pe = |U ∆x/2D| ≤ 1 holds, then
the first condition is more restrictive as the Peclet number follows from ∆t1 ≤ ∆t2 .
Equation (3.105) states that the Courant-Friedrichs-Lewy condition, expressed by the
57
3.2. Problem definition
Courant number Cr = |U ∆t/C∆x| is necessary but not sufficient as λ ≤ 1 and Cr = σ.
For pure convection the explicit scheme is unconditionally unstable. For pure diffusion
only the first time step criterion holds. An alternative formulation of equation (3.105)
reads
Cr2 ≤ Cr/Pe ≤ 1.
(3.106)
If the spatial discretization follows from the cell Peclet number, then the Courant
number should be smaller than the Peclet number if the Peclet number is less than
unity. The Courant number should be less then the reciprocal value of the Peclet
number if the Peclet number exceeds one. The time step size subsequently follows
from the Courant number. In two-dimensional or three-dimensional problems the time
step size depends on the cell with maximum velocity. In general, its position will vary
in time.
For the flow problem the first stability criterion specifies λ by
λp =
2kr Kmax ∆t
,
µαpS∆x2min
(3.107)
where Kmax denotes the maximum eigenvalue of the intrinsic permeability tensor and
∆xmin is approximated by the diameter of the incircle of the element. For an explicit
formulation this number has to hold λ < 1. The cell Peclet number for the solute
transport equation reads
∆xmax
Peω =
,
(3.108)
2αl
and the Courant number is given by
Crω =
|q|∆t
,
nS∆xmin
(3.109)
where the ratio of dispersivity tensor Dij to Darcy’s flux qi equals the longitudinal
dispersivity αl , ∆xmax denotes the diameter of the circumcircle. For the heat transport
equation the Peclet number is given by
PeT =
ρc|q|∆xmax
.
2 (nSλl + (1 − n) λsl + cραl q)
(3.110)
The Courant number for this equation reads
CrT =
ρc|q|∆t
.
(nSρc + (1 − n)ρs cs ) ∆xmin
(3.111)
The stability limits given above have to be satisfied for the explicit formulation of the
linearized basic equations. The conditions can also give some guidance for solving the
non-linear flow and transport equation implicitly. Although the implicit time integration provided an unconditional stable method for solving the test problem, the time
step is limited by the range of convergence of the nonlinear solver for more advanced
problems.
The accuracy of the solution depends on the time step size and the size of the
elements. On non-uniform grids the accuracy of the spatial discretization also depends
58
3. Numerical model
on the ratio of adjacent element sizes, as lower order Taylor terms do not cancel out.
Comparing computational results with results obtained on a globally refined mesh
and decreased time step size provides a more rigorous indication on the accuracy of
the solution. Numerical diffusion produced by time integration, can effectively be
reduced by decreasing the time step size. Wiggles also relate to the accuracy of the
discretization method [119]. These short waves, L ≈ 2∆x, may occur in both steadystate and transient convection-diffusion problems if the Peclet condition is not satisfied.
The convective form of the transport equations sets the diffusive flux over the outflow
boundary to zero. This prevents the boundary condition to trigger wiggles. Still
permeability contrasts in the domain may generate regions with steep concentration
gradients, which are able to trigger wiggles. Local refinement reduces the Peclet number
and suppresses wiggles.
3.2.2
Initial conditions
Initial conditions are given by nodal values of the primary variables: pressure pa ,
mass fraction ωa and temperature Ta . Basis functions interpolate the nodal values as
p = pa Na , ω = ωa Na and T = Ta Na and span three continuous parameter fields. If
these fields are specified separately, generally they do not comply with an equilibrium
condition.
A better way to state the initial condition is to perform a steady state calculation
for the coupled flow and transport problem, given a proper set of boundary conditions.
The steady state solution can be obtained from the set of governing algebraic equations
by setting the time derivatives to zero. Alternatively, a time step algorithm can be used
as an iterative scheme to derive the steady state situation. The second approach gives
best results in combination with adaptive mesh refinement. For both schemes the
starting vectors for the primary variables need still to be specified. For this a constant
mass fraction field specifies the mass fraction vector as
ω = ω0
on Ω,
(3.112)
where ω0 could be specified by the reference mass fraction ω0 , which is part of the
expression for the fluid density. The reference temperature then specifies the starting
temperature field as
T = T0
on Ω,
(3.113)
and a fully saturated hydrostatic pressure field specifies the initial pressure distribution.
This field is given by
p = ρg (zmax − z)
on Ω,
(3.114)
where z is pointing in the opposite direction of the gravity vector, and zmax is larger
than the length of the domain in z-direction to ensure a saturated situation in the
entire flow domain. If the fluid density relates linearly with solute mass fraction and
temperature, the density can be approximated by the reference density ρ0 , as it shows a
weak relation to the pore pressure. In general, the density is calculated from the solute
mass fraction field and the initial temperature field. Nonlinearities in the starting
vector are disregarded. This situation suppresses nonlinearities associated with partly
59
3.2. Problem definition
unsaturated domains. Numerical tests show that the solution of a de-watering problem
is easier obtained than the solution of a re-wetting problem.
3.2.3
Boundary conditions
No-flow boundary conditions of equation (2.92) pose natural conditions on the boundary segment Γ1 where they apply. The volumetric boundary flux reads q n = 0 and the
condition combines the previously defined condition as
T
Γ1 = Γp2 ∩ Γω
2 ∩ Γ2 .
(3.115)
Inflow boundary conditions may either be prescribed by a predefined volumetric
boundary flux, given by equation (2.93), on the segment Γ2 or may result from a given
pressure for equation (2.93) at the boundary Γ3 . The prescribed boundary flux q n > 0
is imposed by a combination of
T
Γ2 = Γp2 ∩ Γω
2 ∩ Γ2 .
(3.116)
Inflow boundary conditions which result from a prescribed pressure p combine
T
Γ3 = Γp1 ∩ Γω
2 ∩ Γ2 .
(3.117)
Outflow boundary conditions are also obtained by either a prescribed boundary
flux, given by equation (2.95), or by a prescribed pressure as equation (2.96) on the
boundary elements where they apply. Prescribed boundary fluxes on Γ4 for which
(q n < 0) combine the conditions on the basic equations as
T
Γ4 = Γp2 ∩ Γω
2 ∩ Γ2 .
(3.118)
Outflow boundary conditions on Γ5 generated by a predefined pore water pressure p,
gather the separately formulated conditions according to
T
Γ5 = Γp1 ∩ Γω
2 ∩ Γ2 .
(3.119)
Here the third-type boundary conditions for the transport equations do not contribute
to the formulated set of equations as the boundary integrals only apply to second-type
inflow boundary conditions.
Artificial conditions on boundaries that separate parts of the porous medium do
not distinguish between inflow and outflow. Both inflow and outflow conditions are
based on a predefined boundary flux q n , given by equation (2.97). The conditions are
set as
T
Γ6 = Γp2 ∩ Γω
(3.120)
1 ∩ Γ1 .
Artificial inflow and outflow conditions which include a prescribed pressure condition
p, according to equation (2.98) are given by
T
Γ7 = Γp1 ∩ Γω
1 ∩ Γ1 .
(3.121)
The definition of geometrical nonlinear conditions depends on the imposed boundary flux. Seepage conditions of equation (2.99) specify q n = 0, precipitation conditions
60
3. Numerical model
given by equation (2.100) impose q n > 0 and evaporation conditions of equation (2.101)
propose q n < 0. Precipitation conditions and evaporation conditions both generalize
the seepage condition. The boundary conditions impose either Γp1 or Γp2 conditions.
Seepage conditions impose natural Γp2 conditions on the boundary segments if unsaturated pore pressures are calculated. If imposing these conditions results in saturated
pore pressures then the boundary type changes to a Γp1 . These conditions should not
generate an in-flux. The seepage boundary condition is written as
r
pr+1 = 0
if (ρqi ) ni ≥ 0
,
(3.122)
r+1
(ρqi )
ni = 0
for
pr < 0
where r + 1 denotes the next nonlinear iteration and r is the current iteration. First
the condition of all seepage boundary nodes has to be checked and in the next iteration
only the node with the largest error in the condition will be modified. Alternatively
the node with the smallest elevation could be updated. In general, bad convergence
behavior results from updating all nodes at the same iteration step.
Precipitation boundaries replace the atmospheric zero pressure condition by a ponding depth pressure p = pp and replace the condition on the flux by qn = q n corresponding to the precipitation flux. The predefined flux has a positive value and the
precipitation condition reads
r
pr+1 = pp
if
− (ρqi ) ni ≤ ρq n
.
(3.123)
for
pr < pp
− (ρqi )r+1 ni = ρq n
Evaporation conditions also generalize seepage conditions. The evaporation condition replaces the atmospheric pressure limit by a pressure based on the humidity of air
p = pe and replace the condition on the flux by a potential evaporation flux qn = q n .
r
pr+1 = pe
if
(ρqi ) ni ≤ −ρq n
.
(3.124)
r+1
(ρqi )
ni = −ρq n
for
pr > pe
The relative permeability of a drying top soil will decrease and after a while the soil
will not be able to produce the potential outflux. The actual flux then follows from
the pressure condition.
A rising surface water level overrules the geometrical nonlinear conditions if the
pressure is higher than the ponding depth pressure. A functional relation of the water
level in time is imposed by
pn = p(t).
(3.125)
If this condition applies, inflow or outflow could occur.
Chapter 4
Adaptive multiscale method
Adaptive grid refinement aims to reach fine-scale accuracy on partially refined grids. It
reduces the size of the system of algebraic equations, and cuts down the computational
costs for solving the set of equations. Compared to a fine-scale discretization, the
number of integration points is less, which saves the computational work in setting
up the system of equations. Coarsening the mesh on a heterogeneous field requires
the computation of equivalent field behavior. Multiscale finite element methods bring
fine scale information to the coarse scale by calculating multiscale basis functions [24,
2, 63, 68, 69, 73, 80]. The finite element procedure then constructs the algebraic
coarse scale equations by using these functions. Subsequently the global flow problem
is solved at the coarse mesh. The multiscale basis functions extrapolate the coarse
scale solution to the fine scale. In order to obtain the multiscale basis functions the
domain is decomposed in coarse scale elements, and flow problems are solved locally.
The local problems can be closed in their subdomains by the solution of dimensionally
reduced flow problems. This closure introduces errors in the global solution if flow
on the coarse scale is not tangential to the domain boundary. Unlike other domain
decomposition techniques, this error is not corrected iteratively [98]. The proposed
adaptive multiscale finite element method corrects these errors by refining the mesh
locally in zones of interest. The method is attractive when applying parallel computing.
The formulation facilitates a multigrid solver. Intergrid transfer operators interpolate
fine-scale information to the coarse scale and extrapolate coarse scale information back
to the fine scale. Coarse grid operators apply to the coarsened meshes.
Section 4.1 presents the adaptive formulation. Discrete adaptivity refines the finite
element mesh locally on homogeneous and piecewise homogeneous fields. Functional
adaptivity supports coarsening of the mesh on homogeneous fields. Section 4.2 presents
the multiscale formulation by introducing multiscale basis functions and constructing
the coarse mesh discretization. Section 4.3 outlines the multigrid formulation. The
multiscale formulations will be illustrated on a number of periodic fields, further elaborated in appendix B. Appendix C introduces the multigrid method that will be used
to solve the algebraic equations, both globally and locally.
61
62
4.1
4. Adaptive multiscale method
Adaptive formulation
Enrichment of finite element meshes may either be predefined, or may depend on
calculated results. Static refinement increases the accuracy for instance around wells
and near tips of sheet pile walls or resolves the resolution of a highly permeable streak.
For static refinement the refined parts of the mesh do not change in time. Dynamic
refinement is able to capture for instance concentration fronts. For dynamic mesh
refinement the refined areas do change in time. Adaptive refinement requires the
postulation of a set of refinement criteria. Figure 4.1 illustrates an adaptive mesh
refinement procedure on four finite element levels.
Figure 4.1: Mesh refinement.
Section 4.1.1 focuses on the adaptive formulation, and presents a nested hierarchy
of finite element meshes. It introduces variational coarsening, which merges fine scale
elements into a single coarse scale element, and regularization, which obtains a gradual
refinement. At the end of the section a refinement criterion is postulated. Section 4.1.2
focuses on functional adaptivity. It introduces a numerical homogenization algorithm
called pressure-dissipation averaging for which local boundary conditions have to be
proposed. Section 4.3 outlines the multigrid solver, which applies the multiscale formulation to speed up its convergence behavior.
4.1.1
Discrete adaptivity
In order to facilitate the multiscale algorithm, the adapted mesh is composed of parts
of a nested hierarchy of meshes. Figure 4.2 shows a mesh hierarchy with uniform mesh
refinement. The proposed adaptive formulation use simplex elements, which are more
flexible in dealing with complex geometries. Simplex elements, like triangular and
tetrahedral elements construct conforming meshes without hanging grid points more
generally, which is beneficial for the accuracy of an adaptive method [112, 113].
•..............................................................................•.......
...
.... ...
...
....
...
...
....
...
...
...
.
.
.
...
.
...
..
.
.
...
.
...
..
.
...
.
.
...
..
.
...
.
...
.
...
...
.
...
.
.
...
.
.
.
...
... .....
...
... ....
..
.
. .
.............................................................................
•
•
•..............................................•.............................................•.......
... ....
.. .
...
....
... ...
...
..
..
....
....
...
.... .... .......
...
...
.
.
.
.
.
.
..
.
.
...
.
.
......................................................................................
.
.
.
.
.
...
.
...
.. ... .....
.
.
...
...
.
... .. ......
...
...
....
... ....
.
.
.
....
..
....
.
... ......
.... ....
....
.... ..
... .....
...
.................................................................................
•
•
•
•
•
•
•.............................•.............................•..............................•............................•.......
... .... .. .... .. ... .. .... ..
.. . .
. ..
.
.
•..............................•....................................•................................•...................................•.........
... ... .. ... .. ... .. ... ..
......
...... ...
.
.
.......
.
.
•..........................•..............................•...............................•...........................•.........
... .... .. ... .. ... .. .... ..
. ..
...
.
.
•..............................•....................................•...............................•..................................•........
... .... .. .... .. .... .. .... ..
.
.
...
.
..
•....................•......................•.......................•...................•......
Figure 4.2: Constant mesh hierarchy.
63
4.1. Adaptive formulation
..
......
...........
......
... ......
....
....
.... ...
... .....
.... ......
.... ...... ....
...
.... ....
.... ....
...
...
...
........
...
...
.........
.... . ....
... ....... .... ......
..
... ... ....
... ..... ... ...... ...... ... ...... ....
.
......
.... .....
...
.
........
........................................................................................
..
...
.... ..............
....... ..
....... ...
........
...
.......
....
...
.................
..........
........ ............. ...
...
...
...
... .......................... ....
...
...
....... ..
............
..
...
........
.. ....... ....
... .............
... ...............
.............. ...
. ...............................................................................
1
•
?
?
K1
K2
•
2
1
•
?
K1
•
3
?
K2
•
2
Figure 4.3: Standard basis functions for line elements.
Element hierarchies
Figure 4.3 illustrates the hierarchical formulation of finite element basis functions. The
left picture shows the layout of a test problem on the domain x ∈ [0, 1] and standard
basis functions for a line element, the right picture shows the standard basis functions
for its child elements.
The standard basis functions for the test problem, on the coarse grid, in global
coordinates, read
N1k−1 = 1 − x,
N2k−1 = x.
(4.1)
Standard basis functions on the fine-grid for the first element are given by
N1k = 1 − 2x,
N3k = 2x,
(4.2)
N3k = 2 − 2x.
(4.3)
and for the second element they read
N2k = 2x − 1,
Standard coarse-scale basis functions Nak−1 for a line element can be expressed as
the sum of its child elements basis functions Nak by
N1k−1 = N1k + 12 N3k ,
N2k−1 = N2k + 12 N3k .
(4.4)
A generalized formulation in index notation reads
k
Nak−1 = Pba
Nbk .
The standard interpolation matrix for line elements P k then reads


1
0
k
P = 0
1 .
1/2 1/2
(4.5)
(4.6)
The coarsening procedure will be given for the one-dimensional test problem illustrated by figure 4.3. This problem considers steady state saturated groundwater flow
with constant fluid properties on the flow domain x ∈ [0, 1]. The gravity vector points
in opposite x-direction. The permeability shows a discontinuity for x = 0.5 and has a
value K1 for x ∈ [0, 0.5] and a value K2 for x = [0.5, 1]. The problem will be solved
by two line elements on the fine scale, where the fine-scale node is positioned in the
middle of both coarse-scale nodes.
64
4. Adaptive multiscale method
Matrix equations representing steady state groundwater flow at a coarse scale k − 1
k−1 k−1
written as Lab
ub = fak−1 , can be derived by using basis functions or their derivatives
at the coarse level. The consistent (coarse-grid) approach applies a weighted summation
over fine-scale integration points, and computes matrix components as
k−1
Lab
=
nk−1
e
A
e=1
Z
ρkr K ∂Nak−1 ∂Nbk−1
dΩ,
µ
∂x
∂x
Ωe
(4.7)
approximated by
k−1
Lab
≈
nk−1
e
e
A w̃m
e=1
ρkr K ∂Nak−1 ∂ξ ∂Nbk−1 ∂ξ
|J|
µ
∂ξ ∂x ∂ξ ∂x
!e
,
(4.8)
m
where ρ denotes the mass density of the fluid [ml−3], kr represents the relative permeability of the porous medium [−], K is the intrinsic permeability of the solid matrix [l2 ]
and µ expresses the dynamic viscosity of the fluid [ml−1t−1 ]. The weights at fine-scale
e
integration points m are given by w̃m
. The points are located at fine grid positions.
The superscript e denotes the element number and nek−1 indicates the number of elements that compose the coarse-scale mesh. Inserting the basis functions given by
equation (4.1) gives a coarse-grid matrix represented by
k−1 ρkr K 1 + K 2 −K 1 − K 2
L
=
,
(4.9)
2µ −K 1 − K 2 K 1 + K 2
where K 1 is the intrinsic permeability measured in the first integration points, which
is located in the first fine-scale element and K 2 is attached to the second fine-scale
element. It’s value is obtained in the second integration point.
The right hand side of the discrete flow equation is written as
fak−1 =
or
fak−1 ≈
nk−1
e
A
e=1
nk−1
e
A
e=1
e
wm
Z
Ωe
ρkr K ∂Nak−1
ρgdΩ
µ
∂x
ρk r K ∂Nak−1 ∂ξ
ρg|J|
µ
∂ξ ∂x
(4.10)
e
,
(4.11)
m
where g denotes the gravitational force [lt−2 ], g has a negative value as the gravity
vector points in the opposite direction to the local coordinate vector. Coarse-scale
basis functions given by equation (4.1) yield
k−1 ρkr −K 1 − K 2
f
= ρg
.
(4.12)
2µ K 1 + K 2
The coarse-grid matrix and vector could alternatively be constructed by assembling
the fine-scale matrix over the fine-grid element domains that contribute to the coarse
parent element using the standard interpolation matrix given by equation (4.6) in a
65
4.1. Adaptive formulation
variational (fine-grid) approach. This formulation expresses the coarse scale matrix
components as
Z
nk−1
e
ρkr K k ∂Nck k ∂Ndk
k−1
Lab
= A
Pca
P
dΩ,
(4.13)
µ
∂x db ∂x
e=1 Ωe
approximated by
k−1
Lab
≈
nk
e
e
wm
e=1
A
ρkr K k ∂Nck ∂ξ k ∂Ndk ∂ξ
Pca
P
|J|
µ
∂ξ ∂x db ∂ξ ∂x
e
,
(4.14)
m
and the right hand side vector components as
fak−1 =
or
fak−1
≈
nk−1
e
nk−1
e
A
e=1
A
e=1
Z
ρkr K k ∂Nck
Pca
ρgdΩ,
µ
∂x
e
wm
ρk r K k ∂Nck ∂ξ
Pca
ρg|J|
µ
∂ξ ∂x
Ωe
(4.15)
e
.
(4.16)
m
k
The components of the interpolation matrix Pab
construct the matrix P k . The
k
weights of the fine-scale integration points are given by wm
= 1, as summation over
a single integration point (nm = 1) provides the exact value of the integral for constant porous media properties. Fine-scale basis functions given by equations (4.2) and
(4.3) and standard interpolation given by equation (4.6) also produce the coarse-grid
matrix of equation (4.9) and coarse-grid vector of equation (4.12). Inconsistent results will generally be obtained if the integration points in the coarse-grid elements do
not coincide with the integration points of a single fine-grid element for the case of a
non-homogeneous field function K, which varies over the fine scale elements.
In this thesis, the coarsening process of a multi-dimensional problem is carried out
sequentially. The procedure facilitates two fine scale elements to merge into a single
coarse element. For the triangular child elements of figure 4.4 this procedure is carried
3 ..
3 ..
•.............
6 ?
•
?
1
•
•
?
•
5
•
•............
... ....
...
...
....
...
....
...
....
...
....
...
....
.. ..
...
.... ......
...
.
.
.
....
...
...
.
.
.
....
...
..
.
.
....
.
... ...
....
... .....
....
.. ....
..
........................................................................
6 ?
◦
?
4
?
3 ..
•.............
... ....
....
...
...
...
....
...
....
...
.
..........................................
.....
...
.... .. ...
.
...
.
.
. .. ....
...
... .... ......
...
...
....
.
.
...
....
..... .......
...
....
....
.. .....
..
..........................................................................
2
1
•
•
?
◦
4
?
5
•
6
2
1
... ....
...
...
....
...
....
...
....
...
....
...
....
...
....
....
...
....
...
....
...
....
...
....
...
....
...
.
.......................................................................
◦
•
?
?
◦
?
◦
4
?
5
•
2
Figure 4.4: Multistep coarsening of triangular elements.
out in two steps. In the first step, element (6, 4, 3) and element (1, 4, 6) are mapped and
generate the new element (1, 4, 3). Element (5, 2, 4) and element (1, 5, 4) produce the
new element (1, 2, 4). In the second step element (1, 4, 3) and element (1, 2, 4) merge
into the coarse-grid element (1, 2, 3). Figure 4.5 depicts the hierarchy of parent and
child elements. Variational coarsening merges basis functions of fine scale elements into
coarse scale basis functions of their parent element, and removes hanging grid nodes.
66
4. Adaptive multiscale method
(1, 2, 3)
c-level
.........
.......... ...................
..........
...........
..........
..........
...........
...........
..........
.
(1, 4, 3)
i-level
(1, 2, 4)
..
...... .........
.....
.....
.....
......
......
.....
......
.
(6, 4, 3)
f-level
...
..... ..........
.....
.....
.....
.....
......
.....
..
.....
(1, 4, 6)
(5, 2, 4)
(1, 5, 4)
Figure 4.5: Triangular element hierarchy.
3 ..
3 ..
•.............
•.............
... ....
...
...
....
...
....
...
...
...
...
....
...
...
......
.... ......
...
...
.
2
.
...
.
.
.. ....
..
.
...
.
.. ........
.
... ......
..
.
..
.. ......
... ....
.
.
.
.
.
.
.
.
. ...
........
..............................................................................
•
1
4
+
N
•
•
2
3 ..
•.............
N
... .....
4
... . .....
... .. ......
... .. . .....
... .. .. ......
... .. .. . .....
... . .. .. ..... ....
... .. .. .........
.. .....
....
... .. . ......
...
...
... .. ................
....
... .......
...
... .......................
..
....
..
........
........................................................................
•
1
=
4
•
•
N + 12 N4
... .....
... .. .....
2
... .. ......
... .. .. ....
... . . ......
... .. .. .. ....
... .. .. .. .......
... .. .. .. .. .....
... .. .. .. .. .......
... . . . . . ....
... .. .. .. .. .. .......
... .. .. .. .. .. .. .....
.. . . . . . . . ....
...............................................................................
2
1
•
•
2
Figure 4.6: Merged fine element basis functions.
Figure 4.6 illustrates the merging procedure over two triangular elements. The coarse
scale triangular basis functions in global coordinates read
N1k−1 = 1 − x − y,
N2k−1 = x,
N3k−1 = y.
(4.17)
Fine scale basis functions over element (1,2,4) are written as
N1k = 1 − x − y,
N2k = x,
N3k = 0,
N4k = y,
(4.18)
N4k = 1 − x.
(4.19)
and fine scale basis functions over element (1,3,4) read
N1k = 0,
N3k = −1 + x + y,
N2k = 1 − y,
Linear interpolation reproduces standard basis functions on the coarse scale, which can
be written as
N1k−1 = N1k ,
N2k−1 = N2k + 12 N4k ,
4 ...................................................................... 3
•.............
•
....
... ....
...
....
...
...
....
...
.
...
...
...
.
.
...
...
....
...
.
...
....
...
...
...
....
...
...
...
...
...
....
...
...
...
....
...
... ....
...
.
.
... ...
...
.
.
......................................................................
•.............
9
•
•
•
...
..
.... ..
...
... ....
... ..
...
....
.... ....
...
...
....
....
...
.
.
.
.
....
.
.
... ... ...
...
.... .. ...
...
...
... .. ....
..
.................................................................................
...
... ... .....
.
....
.
... ... .....
....
.
...
.
....
...
....
..
....
...
...
.... ....
... ......
...
.... ..
... .....
...
..................................................................................
•
4
4
3
5
3
•
1
1
•
2
(4.20)
4 ......................................8................................. 3
1
1
N3k−1 = N3k + 12 N4k .
1
•
•
7
2
1
•
2
6
•
2
Figure 4.7: Triangular parent and child element patches.
In order to obtain equivalent material behavior via the computation of multiscale
basis functions, as will be explained in the next section, the standard basis functions
67
4.1. Adaptive formulation
domain
N1
N2
N3
0
N4
1
Figure 4.8: Linear basis functions over a span of two triangular elements.
over a patch of elements will be formulated. Figure 4.7 shows a patch of triangular
elements, which forms a quadrilateral domain. Coarse scale basis functions over the
patch of triangular elements follow from the fine scale basis functions as
N1k−1 = N1k + 12 N6k + 21 N9k ,
N2k−1 = N2k + 12 N6k + 21 N7k + 12 N5k ,
N3k−1 = N3k + 12 N7k + 21 N8k + 12 N5k ,
N4k−1 = N4k + 12 N8k + 21 N9k .
(4.21)
Figure 4.8 presents linear basis functions on two triangular elements, which constitute to the patch, graphically.
Refinement procedure
Figure 4.9 depicts a mesh hierarchy with local mesh refinement. The finest grid in
this figure shows non-conforming elements and hanging nodes indicated by the symbol
. The shape functions across the non-conforming element boundaries are in general
not continuous. For this case continuous shape functions are obtained by merging of
elements. For more general situations, adaptive refinement may still result in non•...............................................................................•........
...
... ...
...
....
...
...
...
..
....
...
.
.
.
...
...
..
.
.
.
...
...
..
.
.
.
...
...
..
.
.
.
...
...
..
.
.
.
...
....
...
.
...
..
.
... .......
...
... .....
....
......
......................................................................
•
•
•
•
•
•.............................•.............................•..............................................•........
... .... .. .... ..
.
.
.... ..
•..............................•............................................. ............ ........
... .... .. .... .. ....
..
.
.
.
•...................................................•....................................................•..........
•
•
•
•
•..............................................•..............................................•........
... ....
.
..
...
.... ....
...
...
....
...
...
....
.... ... ......
...
.... .. ....
...
...
.... ... ....
..
..................................................................................
.. ... ....
.
...
.
....
.. .. ....
.
.
....
.
.
....
....
.
....
...
....
.
....
.... ...
.
... .......
.
.... ...
.... ......
.....
.
....................................................................................
...
....
... .... .......
...
....
....
.
..
...
....
.... ...
...
... .......
.... ...
.
.
.
.
.
.........................................................................................
•
•
Figure 4.9: Adaptive mesh hierarchy.
conforming meshes if hanging grid points occur on multiple grid levels. To circumvent
this a regularization procedure is proposed here. Figure 4.10 illustrates this procedure.
The upper four pictures of figure 4.10 show the process of adaptive refinement for a
triangular element mesh without regularization. The first refinement step generates
four child elements on level 2 for the lower left element. The next step refines one level
2 element and the last step refines two level 3 elements. The mesh on the finest level
68
4. Adaptive multiscale method
consists of 14 elements. In the final configuration, one of the level 2 elements is attached
to two level 4 elements and one level 3 element. On the internal boundary hanging
nodes are present. The four pictures at the bottom of figure 4.10 show the hierarchy of
meshes when adopting mesh regularization. The first refinement step already produces
a hanging grid node that can not be removed by merging level 2 elements. Here, the
adjacent coarse element needs to be refined as well. However, the four child elements
of this element can merge into two transition elements. The second refinement step
generates four level 3 elements, two of them could form a transition element making
renormalization unnecessary. The last refinement step introduces 16 level 4 elements
and eight of them merge into 4 transition elements. The most right transition elements
are connected to the hanging grid node, which was generated in the second refinement
step, and now the adjacent level 2 elements have to be refined. This requirement also
introduces four extra level 3 elements and two merged level 3 elements.
...................................................................
...
... ....
...
... .....
...
...
...
....
...
...
.
...
...
...
....
....
.....
.....
....
...
...
....
...
...
....
.
....
....
.... .....
...
.... ..
...
.
................................................................
..................................................................
...
... .....
...
... .....
....
...
...
....
...
...
....
...
...
.
..........................................
.....
..
.
.
.
.
.
...
.
.
.. ... ......
.
...
.....
.
.
.
....
....
.... ......
.... ....
.... ...
... .....
...
.
......
........
.
..............................................................
...................................................................
...
... ....
...
... .....
....
...
...
...
...
...
.
....
...
...
.
.
......................................
.....
. .. ...
..
.
.
...
.
.. ... .....
.
...
.....
.
.
...
.
.
....
.... ............................
... .....
... .... ... ..... ...
.... ..
........ .. .......
.
.
................................................................
..................................................................
...
... ....
...
... ......
....
...
...
....
...
...
....
...
...
.
.........................................
.....
..
. .. ....
.
.
...
.
.. ... ....
.
.
...
.....
.
.
. .....
.... ............................
.... ....
... ......................... ...
.... ...
......
..............................
................................................................
..................................................................
... .....
...
.. .....
..
...
...
....
...
...
....
....
...
...
....
....
....
....
...
...
....
...
....
...
....
...
.
.... ....
...
.... ..
...
.
.. ..
..................................................................
...................................................................
.. .
... .....
.... ...
.. ....
.... ....
...
....
...
....
...
.
...
.
.
.
...
...
....
... .....
............................................
....
.
.
.
.
...
...
.
.
.
.... ... ......
...
...
.
.
.. ... ....
...
.
.
.
.
.... ...
.
... .....
.
.
.
... ....
.
... .....
.
.
.
.. .
.
........................................................................
...................................................................
..
... .....
.... ...
.. ....
.... ...
....
...
...
...
...
.
.
...
.
.
....
...
....
.. ......
.........................................................................
....
.
..
.
.
.
........
.
.
.
... .... ...... ... ...... ...... ....
... ............................................ ...
... ............ ... ............ ...
... ..... ... ..... ... .... ... ..... ...
. .
.
.
................................................................................
....................................................................
..
... ....
.... ..
.. .....
.... ....
...
....
...
....
.
...
....
.
.
....
...
...
....
. ...
..............................................................................
... ...
.
.
.
... ....
.
.
.
... ..... ...... ... ...... ...... .....
........................................................... ...
........ .............. ....... ............ ...
... ............................... ... ..... ... ..... ...
.. . . .. . .
.
.......................................................................................
Figure 4.10: Adaptive mesh refinement without and with regularization.
The regularization algorithm obtains its result by setting a counter for each node
equal to the finest level of the elements that share this node. Comparing the element
levels with their nodal counters produces flags for elements that need to be refined. If
the difference between levels and counters is larger than one, then the element needs to
be refined and the level for these elements has to be increased by one. This procedure
is repeated until no element level has to be increased. After identifying all elements
that need to be refined, the final mesh can be constructed, without the need to repeat
the regularization.
Refinement criteria
Conformal finite elements obtain nodal based mass continuity. The pressure gradient
tangential to the element boundary is continuous, whereas the gradient perpendicular
to the boundary is discontinuous and continuity of flux over the element boundary
is not preserved. The refinement criterion proposed here, detects this discontinuity
and mesh enrichment corrects the accuracy of the flow field locally. This procedure
improves the accuracy of the transport simulation, which is based on the velocity field.
Figure 4.11 shows the Darcy flux vector in a single integration point and its components
69
4.1. Adaptive formulation
perpendicular and tangential to the element boundaries in the flow points.
3 ..
3 ..
•............
•............
... ....
...
...
....
...
....
...
....
...
....
...
.... ....
...
..............
...
....... ......
...
.......
....
...
....
...
....
....
...
...
...
.......................................................................
?
1
•
•
2
... ....
...
...
....
.
...
....
......
...
......
....
.
...
.
......... ........................... ............
.. . ......... ....
.... ....
... ....
..
...
....
....
....
..
....
....
....
...
....
........
..
..........................................................................
.
...........................
1
?
?
•
?
•
2
Figure 4.11: Darcy flux in integration point and flow points.
The expression for the Darcy mass flux components, postulated in global coordinates, and obtained in the flow points reads
ρkr Kij ∂Nc ∂ξ
qi =
pc − ρgi .
(4.22)
µ
∂ξ ∂xi
The flow points are located on the edges of two-dimensional elements and on the faces
of three-dimensional elements. The magnitude of the volumetric flux over the element
boundaries reads
q n = q i ni ,
(4.23)
where ni is the outward pointing normal vector on the element edge. The difference
in magnitude of the flux perpendicular to the element boundary, denoted by ∆qn,
computes the refinement criterion. This criterion reads
∆qn > c1 q l ,
∆qn = |qn1 − qn2 |,
(4.24)
where q l expresses the mean perpendicular flux over the global flow field.
Figure 4.12 illustrates the refinement procedure on a checkerboard cell. The cell
is closed for flow on the horizontal boundary parts of the cell and prescribed constant
pressure boundary conditions apply on the vertical boundary parts. The mesh refinement concentrates on the singular point in the center of the cell. The upper plots of
figure 4.12 show three levels of full mesh refinement and one adaptively refined mesh.
The lower plots relate to the upper pictures and show the pressure fields, which were
obtained by three levels of fixed mesh refinement and one adaptive mesh refinement.
The refinement procedure reaches fine scale accuracy on an adaptively refined mesh.
The accuracy of a numerical computation on a piecewise homogeneous field also
depends on the resolution of the structure. Figure 4.13 poses the structural multiscale
problem for diagonally layered cells. Adaptivity captures the geometry of the structures more accurately [115]. The upper plots of figure 4.13 illustrate the capability
of structural adaptivity. The first three pictures resolve the structure by full mesh
refinement, the last picture applies adaptive refinement and obtains the finest scale
resolution on a partly refined grid. Adaptive refinement improves the accuracy of the
pressure field as can be seen in the related pictures at the bottom in the figure. The
boundary conditions prescribe a constant pressure at the vertical boundaries and a
no-flow condition at the horizontal boundaries of the cell.
70
4. Adaptive multiscale method
level 1
level 2
level 3
adaptive
level 1
level 2
level 3
adaptive
0
1
Figure 4.12: Discrete adaptivity on a checkerboard cell.
4.1.2
Functional adaptivity
Homogenization techniques are used to reduce the fine scale information by calculating
equivalent material behavior on a coarse scale. Only if the scale of variation is much
smaller than the coarse observation scale then the equivalent parameter is a constant
variable on the sampling domain. In general, the equivalent parameter depends on the
scale of the averaging domain and in the case of permeability, for instance, on closure of
the domain. If the equivalent parameters are computed numerically, then their values
also depend on the discretization of the flow problem. Refining the mesh increases
the accuracy of the solution, and reduces the effect of local boundary conditions. Homogenization techniques obtain equivalent permeabilities either by solving local flow
problems or by solving the global flow problem a limited number of times. Alternatively, the local prediction could be updated during the actual global flow simulation.
Variational averaging
For the one-dimensional test case given in section 4.1.1 the effective permeability hKi
could be sampled in the coarse-scale integration point. Elaboration gives
Z
nk−1
e
ρkr hKi ∂Nak−1 ∂Nbk−1
k−1
Lab
= A
dΩ,
(4.25)
µ
∂x
∂x
e=1 Ωe
approximated by
k−1
Lab
≈
nk−1
e
e
A wm
e=1
ρkr hKi ∂Nak−1 ∂ξ ∂Nbk−1 ∂ξ
|J|
µ
∂ξ ∂x ∂ξ ∂x
!e
,
(4.26)
m
k−1
where wm
= 1 for a single integration point. The coarse-scale matrix then reads
k−1 ρkr hKi − hKi
.
(4.27)
L
=
µ − hKi hKi
71
4.1. Adaptive formulation
level 1
level 2
level 3
adaptive
level 1
level 2
level 3
adaptive
0
1
Figure 4.13: Functional adaptivity for diagonal layered cells.
The right hand side of the discrete flow equation is written as
fak−1 =
or
fak−1 ≈
nk−1
e
nk−1
e
A
e=1
A
e=1
Z
ρkr hKi ∂Nak−1
ρgdΩ,
µ
∂x
e
wm
ρk r hKi ∂Nak−1 ∂ξ
ρg|J|
µ
∂ξ ∂x
Ωe
(4.28)
e
.
(4.29)
m
The coarse scale vector reads
k−1 ρkr − hKi
f
= ρg
.
2µ hKi
(4.30)
Equations (4.9) and (4.12) are retrieved, if the arithmetic averaged (equivalent) permeability hKi = (K12 +K22 )/2 is applied. This equivalent permeability does not match the
effective permeability given by the harmonic mean hKi = (2K12 K22 )/(K12 +K22 ). For an
arithmetic average, high conductivities dominate the coarse-grid behavior. The proposed adaptive formulation applies a more advanced numerical homogenization technique called pressure-dissipation averaging [129].
Pressure-dissipation averaging
The pressure-dissipation averaging approach presented in appendix A combines conservation of driving force and conservation of energy dissipation. The set of algebraic
equations for this approach, in Cartesian coordinates, reads
−∂p(a)k
−∂p(b)k
∂p(a)k (b)k
K̃ij = −
q
,
(4.31)
∂xi
∂xj
∂xi i
72
4. Adaptive multiscale method
G(1)
domain
G(2)
G(3)
0
G(4)
1
Figure 4.14: Linear boundary conditions on homogeneous (an)isotropic field.
where both (a) and (b) indicate loading cases. The flow problem on the patch could
be closed by linear boundary conditions.
The linear loading cases read
G(1) = 1 − x,
G(2) = 1 − y,
(4.32)
G(4) =
(4.33)
or alternatively
G(3) = 1 −
1
2
(x + y) ,
1
2
(1 + x − y) .
Figure 4.14 shows the pressure fields, which result from loading by linear boundary
conditions on a homogeneous (an)isotropic cell. Two orthogonal loading cases give a
set of four algebraic equations and solve a two-dimensional problem. The number of unknowns equals the square of the spatial dimension. For symmetric permeability tensors
the number of unknowns reduces to three in two dimensions and six in three spatial dimensions. If off-diagonal components are disregarded, the number of unknowns equals
the number of spatial dimensions nd . Better known algorithms like pressure-flux averaging (appendix A) operate on a single loading case. This formulation produces nd
equations, which equals the number of diagonal tensor components. However, loading
in x-direction produces flow in x-direction only, so Kxx can be calculated, and Kyy
remains unknown. For both procedures nd (preferably orthogonal) loading cases are
required, to produce a non-singular matrix-vector equation. The set of algebraic equations, which follows from discretizing the mathematical pressure-dissipation problem
is written as
(ab)
(4.34)
Φij K̃ij = φ(ab) .
The fine-grid vector components are written as
e
nk
e
ρkr ∂Nck ∂ξk ∂Ndk ∂ξl (b)k
(ab)
e
(a)k
φ
= A wm pc Kij
p |J|
.
µ ∂ξk ∂xi ∂ξl ∂xj d
e=1
m
(4.35)
The coarse-scale matrix components are given by
(ab)
Φij
=
nk−1
e
A
e=1
e
wm
∂Nck−1 ∂ξk ∂Ndk−1 ∂ξl (b)k−1
p
|J|
µ ∂ξk ∂xi ∂ξl ∂xj d
ρkr
p(a)k−1
c
!e
.
(4.36)
m
Load vector components p(a)k−1 and p(b)k−1 are given for the coarse scale and will be
prolongated to the fine scale in the next section. The full matrix-vector form of the set
73
4.1. Adaptive formulation
domain
G(1)
G(2)
G(3)
G(4)
domain
G(1)
G(2)
G(3)
G(4)
0
1
Figure 4.15: Linear boundary conditions on a horizontally layered cell.
of equations in two dimensions reads
 (11)
(11)
(11)
Φxx
Φxy
Φyx
 (12)
(12)
(12)
Φxy
Φyx
Φxx
 (21)
(21)
(21)
Φxx
Φxy
Φyx
(22)
(22)
(22)
Φxx
Φxy
Φyx

  (11)
(11) 
Φyy
K̃xx
φ
(12)  
φ(12)
Φyy  K̃xy 
=

(21) .
(21) 
Φyy  K̃yx  φ 
(22)
φ(22)
K̃yy
Φyy
(4.37)
This expression produces a symmetric permeability tensor. An explicit expression
follows from loading in x-direction and y-direction.
The set of equations will be solved for local element patches. Figure 4.7 presented
the two-dimensional scaling patch. Fine-grid permeability tensors apply to the finegrid patches (1, 6, 5, 9), (2, 7, 5, 6), (3, 8, 5, 7) and (4, 9, 5, 8), and equivalent anisotropic
permeability coefficients are calculated for the coarse-grid patch (1, 2, 3, 4).
The pressure fields that result from loading by linear boundary conditions on a
horizontally layered cell are presented by figure 4.15. Fine scale permeability tensors
are considered to be isotropic. The intrinsic permeability of the low permeability
material equals k = 10−2 and its value for the high permeability material is k = 1. The
effective permeability components for this periodic cell read kxx = 5.05 · 10−1, kxy = 0,
kyx = 0, and kyy = 1.98 · 10−2 . The upper plots of the figure show fully-resolved finite
element mesh solutions, and the lower plots give over-resolved solutions. The pictures
show that mesh refinement increases the accuracy of the solution. Pressure-dissipation
averaging extracts inaccurate permeability components from the fully-resolved pressure
fields (kxx = 5.05 · 10−1, kxy = −2.62 · 10−17, kyx = 2.78 · 10−17, and kyy = 3.84 · 10−1)
and a bit more accurate results from the over-resolved pressure fields (kxx = 5.05·10−1,
kxy = −8.06 · 10−17, kyx = 6.93 · 10−16, and kyy = 2.77 · 10−1 ). For both cases the
inaccuracy increases for larger fine-scale permeability contrasts. Off-diagonal terms
show a small numerical error, as they deviate from zero.
Figure 4.16 presents the pressure fields, which results from loading by linear essential
boundary conditions on a horizontal layered field composed out of 4 × 4 periodic cells.
74
4. Adaptive multiscale method
global
G(1)
G(2)
G(3)
G(4)
local
G(1)
G(2)
G(3)
G(4)
0
1
Figure 4.16: Global-local boundary conditions on a horizontally layered cell.
The upper pictures of this figure show the dissipation of the boundary effects. Still
pressure-dissipation averaging obtains inaccurate results (kxx = 5.05 · 10−1 , kxy =
1.64·10−16, kyx = 3.64·10−16, and kyy = 1.11·10−1). The solution of the global problem
could however be used to generate boundary conditions for a local problem. The
pressure fields for this global-local procedure are given in the lower plots of figure 4.16.
Pressure-dissipation averaging now computes accurate permeability tensor components
(kxx = 5.05 · 10−1 , kxy = 2.55 · 10−3 , kyx = 2.55 · 10−3 , and kyy = 1.99 · 10−2 ).
For this case global pressure results could directly be imposed on the local problem,
because the mesh size of the local problem equals the mesh size of the global problem.
The boundary conditions which were extracted from the global computation on this
horizontal layered cell, generate a local flow field that nearly corresponds to the flow
field for periodic boundary conditions on the periodic cell. For the horizontally layered
cell these boundary conditions match oscillating boundary conditions, which follow
from closure by a dimensionally reduced flow problem.
4.2
Multiscale formulation
The proposed multiscale formulation computes basis functions by solving local flow
problems over patches of simplex elements. Only if the patch aligns with the small
scale structure then the periodic boundary conditions match with the solution of dimensionally reduced flow computations. Single simplex elements do not meet this
condition. This restriction becomes more severe in a hierarchical setting, where the
averaging domain might only cover a small number of fine scale elements. The original
two-level formulation for the multiscale basis functions, which relates the pressure on
each level to the fine scale pressure, is used to interpolate the coarse scale results and
to obtain estimates for the fine scale fluxes perpendicular to the element boundaries,
which are used for the refinement criterion.
75
4.2. Multiscale formulation
•..............
... ....
.
•................................•...................
... ... .. ...
.
.......
.
•............................•................................•................
... .... .. .... .. ....
.
..
...
...........
•................................•...................................•...............................•
.
... ... .. .... .. .... .... ......
..
.
.........
.
.......
.
........................................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
• • • • •
•..............
... ....
.
◦........ ..◦..........
...
....
.
...
◦..... ◦ ..◦..........
....
...
.
◦........ ◦ ◦ ..◦..........
...
...
..
..
•...............◦.................◦..................◦.................•....
Figure 4.17: Direct multiscale basis function formulation.
Section 4.2.1 presents the formulation of the multiscale basis functions. Section 4.2.2
outlines the construction of the coarse grid operator.
4.2.1
Multiscale basis functions
Figure 4.17 illustrates the coarsening procedure according to the two-level multiscale
finite element approach. The expression for the multiscale basis function reads
Nac = Pca Ncf ,
(4.38)
where Nac denotes the coarse scale basis function attached to node a, and Ncf expresses
the fine scale basis function attached to node c. In multiscale finite element literature,
the matrix coefficients Pca are called basis function weights. In the previous sections
the coefficients had constant values and where part of the interpolation matrix. Here
their values will depend on the solution of local problems.
Variational formulation
Despite consistency of both formulations given by equations (4.8), (4.11) and equations
(4.14), (4.16) for the one-dimensional test problem in the previous section, it can be
shown that the consistent coarse-grid approach is not accurate, when a volume average
of the permeability is adopted. The accuracy of the variational fine-grid formulation
can be improved by using multiscale basis functions, where the coefficients of the
interpolation matrix follow from a local flow analysis, and therefore better account
for permeability variations. For a line element depicted in figure 4.18 the multiscale
basis functions read
N1k−1 = N1k + P31 N3k ,
N2k−1 = N2k + P32N3k ,
k
k
where the weights P31
and P32
need to be calculated.
......
....
..........
.... ...
.... .......
... .....
...
...
.... .
.... ...... ....
.... ...... .......
........
........
..
...
.
...
...
.......
.........
.
.... .. ...
.... ...... ..... ......
... .... ....... ....
.
.
.. .... .. ....
.
.... ..
.... ......
..
.
.........
.
.
.. .
................................................................................................
.........
...........
.............. ....
... ......
................
....
.........
...
....
.....
..... .......
.
..
..
.
.. ..
...
...
....
................
.
.
.
.
.
.
....
.
....
....
.
.............. ...
.. ....... ....
..
..............
..
.... .....
....
....
................................................................................................
1
•
?
K1
◦
3
?
K2
•
2
1
•
?
K1
•
3
?
K2
•
2
Figure 4.18: Multiscale basis functions for line elements.
(4.39)
76
4. Adaptive multiscale method
Disregarding gravity on the test problem illustrated by figure 4.3, the continuity of
flux over the fine-scale node gives K 1 (p1 − p3 ) = K 2 (p3 − p2 ), where K 1 is the permeability of the left element and K 2 is the permeability of the right element. Pressures
are given as nodal values. From this expression the pressure in the fine-scale node reads
p3 =
K1
K2
p
+
p2 .
1
K1 + K2
K1 + K2
(4.40)
The multiscale basis functions that ensure continuity of flow can now be written as
N1k−1 = N1k +
K1
N k,
K1 + K2 3
and the interpolation matrix reads

k 
Pab = 
1
0
K1
K1 + K2
N2k−1 = N2k +
K2
N k,
K1 + K2 3

0

1
.
K2
K1 + K2
(4.41)
(4.42)
Figure 4.18 shows the modified basis functions graphically. Using this interpolation
matrix, variational coarsening now obtains an accurate coarse scale stiffness matrix
from the fine scale stiffness matrix. The fine-scale matrix coefficients read
Z
nk
e
ρkr K ∂Nak ∂Nbk
Lkab = A
dΩ,
(4.43)
µ
∂x ∂x
e=1 Ωe
or
Lkab ≈
nk
e
e
A wm
e=1
ρkr K ∂Nak ∂ξ ∂Nbk ∂ξ
|J|
µ
∂ξ ∂x ∂ξ ∂x
e
.
(4.44)
m
For elements with length 0.5 the fine-scale stiffness matrix can be written as
 1

0
−K 1
k 2ρkr K
 0
L =
K2
−K 2  .
µ
1
2
−K
−K
K1 + K2
(4.45)
Variational coarsening generates the coarse-scale matrix, which can be expressed as
k−1
Lab
=
nk−1
e
A
e=1
Z
Ωe
ρkr K k ∂Nck k ∂Ndk
Pca
P
dΩ,
µ
∂x db ∂x
(4.46)
approximated by
k−1
Lab
≈
nk
e
k e
A Pca wm
e=1
ρkr K ∂Nck ∂ξ ∂Ndk ∂ξ
|J|
µ
∂ξ ∂x ∂ξ ∂x
e
k
Pdb
.
(4.47)
m
Inserting expression (4.42) for the interpolation matrix gives
1 2
k−1 ρkr
2
K K
−K 1 K 2
L
=
.
µ K 1 + K 2 −K 1 K 2 K 1 K 2
(4.48)
77
4.2. Multiscale formulation
The right hand side of the discrete fine scale flow equation is written as
fak
=
nk
e
Z
ρkr K ∂Nak
ρgdΩ,
µ
∂x
ρkr K ∂Nak ∂ξ
ρg|J|
µ
∂ξ ∂x
A
e=1
Ωe
approximated by
fak ≈
nk
e
e
A wm
e=1
(4.49)
e
.
(4.50)
m
Inserting the fine-scale basis functions gives the fine-scale vector expressed by


−K 1
k
ρkr 
K2  .
f = ρg
µ
1
K − K2
(4.51)
Variational coarsening produces the coarse-grid vector as
fak
or
fak ≈
=
nk−1
e
A
e=1
nk
e
Z
Ωe
k e
A Pbawm
e=1
ρkr K k ∂Nbk
Pba
ρgdΩ,
µ
∂x
ρkr K ∂Nbk ∂ξ
ρg|J|
µ
∂ξ ∂x
(4.52)
e
.
(4.53)
m
Inserting the fine-scale basis functions and the expression for the interpolation matrix,
according to equation (4.42), gives
k−1 ρkr
2ρg
−K 1 K 2
f
=
.
(4.54)
µ K1 + K2 K1K2
This result is also obtained from a homogeneous coarse-scale formulation and an effective harmonic mean permeability hKi = 2(K 1 K 2 )/(K 1 +K 2 ) sampled in a coarse-scale
integration point. For a harmonic average, low conductivities dominate the coarse-grid
average. This example indicates the importance of the way to take conductivity extremes into account.
In conclusion, the multiscale formulation, with interpolation weights given by the
prolongation coefficients of equation (4.42), captures fine-scale behavior of the solution
on a coarse mesh. Solving the coarse-scale problem will give the exact flow rates over the
coarse-scale elements. The coarse-scale nodal values of the pressure are obtained with
fine-scale accuracy. The fine-scale variation of the pressure is found by interpolation of
the coarse-scale nodal values with the interpolation operator given by equation (4.42).
This conclusion only holds for the one-dimensional case as will be explained next.
The proposed multiscale basis functions on a quadrilateral patch of triangular elements, follow from bi-linear boundary conditions. For a unit square domain these
conditions coincide with the definition of the quadrilateral basis functions and read
B1 = (1 − x) · (1 − y),
B2 = x · (1 − y),
B3 = x · y,
B4 = (1 − x) · y.
(4.55)
Figure 4.19 shows multiscale basis functions on a patch of eight elements generated
78
4. Adaptive multiscale method
domain
N11
N21
N31
N41
domain
N1
N2
N3
N4
0
1
Figure 4.19: Multiscale basis functions on a patch of two triangular parent elements.
out of two triangular parent elements. Standard coarse scale basis functions over the
patch of quadrilateral elements follow from the fine scale basis functions as
N1k−1 = N1k + 12 N6k + 12 N9k + 14 N5k ,
N2k−1 = N2k + 12 N6k + 12 N7k + 14 N5k ,
N3k−1 = N3k + 12 N7k + 12 N8k + 14 N5k ,
N4k−1 = N4k + 12 N8k + 12 N9k + 14 N5k .
(4.56)
This formulation restores symmetry on the patch. The span of basis functions N1 and
N3 increases compared with the standard formulation, which was depicted by figure 4.8.
Sampling domain
The values of the multiscale weights often strongly depend on the closure of the local problem and the discretization size of this problem. These effects become more
dominant when the averaging volume is small compared to the scale of parameter variation. Oversampling reduces the boundary effects by posing the boundary conditions
on a sampling domain, which extends the patch of simplex elements as depicted by
figure 4.20. The overlapping domains are closed by either linear or oscillating essential
boundary conditions. Multiscale basis functions, which apply to the patch, now follow
from the oversampled multiscale basis functions as
Na = cab Nb∗ ,
(4.57)
where Nb∗ denote the base functions over the sampling domain, Na is the base function
over the element patch and cab are the coefficients, which follow from Na (xb ) = δab .
79
4.2. Multiscale formulation
well conditioned
∗
4∗ ..........................9
5∗
7∗
3∗
................................................................................................
•.....
•
•
•
•
8
•
3
•
8∗
•
7
•
5∗
..
....
...
..
...
...
∗ .....
....
...
..
................................................................
..
.
.
.
.
...
...
... ....
...
.. ...
.
.
...
....
... ....
.
...
.. ...
.
.
...
..
... ....
.
.
... .... .....
.
...
...
...
.
.... .. ...
∗ ...
...
..
.... ... ....
....
.
.
...
...
.
.
.
.
.
..
.
.
.
...
........................................................................
...
...
...
.
.. ... .....
.
.
.
...
.
.
...
...
.
.. ... .....
.
.
.
.
...
.
...
...
....
.
.
..
.
.
.
.
...
.
.
.
.... .
...
... ...
.... ...
..
...
... .....
.....
...
......................................................................
...
∗ ....
...
...
...
...
...
...
...
...
...
...
..........................................................................................................................
8
•
4
•
•
5
•
9
•
•
•
•
6 •
1∗
•
1
•
9∗
5
•
6
5∗
•
•6∗
2
•
•
7∗
2∗
.......
...
...
...
..
...
.
......
δ
ill conditioned
∗
4∗ ..........................9
5∗
7∗
3∗
....................................................................................................
•.....
•
•
•
•
.. ... ... .... ... ... .... ... ... .... ....
..................................................................................................................................................................................
...
...............................................................................................................................................
...
.
...........................................................................................................
..................................................................................................................................................................................
∗ .....
...
........................................................................
..................................................................................................................................................
..
.
.
...
... .....
...
.. ......................
...
... ....
.... ............................
...
... .............................................................
....
...
...
... .... .......
.
...
..........................................................................
.... .. ....
∗ .....
...
... ... ...
...................................................
...
..
. .
.
..................................................................................................................................
...
.
.....................................
.
.
...
...
.. ... .....
.
.
...................................................
.
...
...
.. ... ....
.
.
......................................
.
...
....
...
.
..
.
.
.
.
.
.
.
... .............................................................
...
... .....
.
.
.
.
..
.
.
.... ....................................................
... ...
.
.
...
.
.
.
.......................................................................................
∗ ....
..
...
...
...
...
...
...
...
...
...
..........................................................................................................................
8
•
4
•
•
5
•
9
•
•
•
•
6 •
1∗
•
1
•
9∗
8
5
•
3
•
8∗
•
7
•
5∗
•
6
•
•
5∗
2
7∗
.......
...
...
...
...
...
......
δ
•6∗
•
2∗
Figure 4.20: Oversampling domain.
domain
N1∗
N2∗
N3∗
N4∗
subdomain
N1
N2
N3
N4
0
1
Figure 4.21: Multiscale basis functions on a horizontally layered field.
As an example
 ∗
N1 (x1 )
N1∗ (x2 )
 ∗
N1 (x3 )
N1∗ (x4 )
N2∗ (x1 )
N2∗ (x2 )
N2∗ (x3 )
N2∗ (x4 )
N3∗ (x1 )
N3∗ (x2 )
N3∗ (x3 )
N3∗ (x4 )
   
N4∗ (x1 ) c11
1
c12  0
N4∗ (x2 )
  =  ,
N4∗ (x3 ) c13  0
N4∗ (x4 ) c14
0
(4.58)
gives N1 = c11 N1∗ + c12 N2∗ + c13 N3∗ + c14 N4∗ .
Figure 4.21 presents multiscale basis functions, which follow from linear closure of a
horizontally layered domain. This domain over samples the subdomain that captures a
single periodic cell. The bottom row presents the extracted multiscale basis functions
on the reduced domain. Results are given for a permeability contrast of 102 .
Reduction of basis functions from a domain to a subdomain might pose an illconditioned set of equations. The picture right in figure 4.20 shows an example for
this. Here the coarse scale node 3, is isolated from the subdomain by a low permeable
80
4. Adaptive multiscale method
domain
N11
N21
N31
N41
domain
N11
N21
N31
N41
0
1
Figure 4.22: Multiscale basis functions for a horizontal layered cell.
zone. For this situation the components of the third column of the matrix given by
equation (4.58) become small and c13 is undetermined. For this reason a second approach will be considered. This approach collapses the mesh over the boundary zone
and generates dimensionally reduced flow problems. Figure 4.20 also illustrates the
second approach that generates conforming basis functions by first computing oscillating boundary conditions. The algebraic form of the reduction problem associated with
the bottom boundary of the patch can be written as
   

Q1
p1
a11
a15 a16
a19






p
a
a
a
a
22
25
26
27
  2   Q2 

a51 a52 a55 a56 a57 a59  p5  Q5 
  =  .

(4.59)
  p6   Q 6 
a61 a62 a65 a66
   


  p7   Q 7 
a72 a75
a77
Q9
p9
a91
a95
a99
The lumped expression for the pressure at node 6 then follows from
(a61 + a51 ) p1 + (a62 + a52 ) p2 + (a65 + a66 + a55 + a56 ) p6 = 0.
(4.60)
The lumping process is illustrated by figure C.8. The connectivities to nodes on the
opposite side of the boundary generalize this expression as
(a61 + a51 + a5∗ 1 ) p1 + (a62 + a52 + a5∗ 2 ) p2
+ (a65 + a65∗ + a66 + a55 + a5∗ 5∗ + a56 + a5∗ 6 ) p6 = 0. (4.61)
Figure 4.22 presents fully-resolved and over-resolved multiscale basis functions for
a horizontally layered cell. These functions follow from a local flow computation with
oscillating boundary conditions.
Appendix B presents multiscale basis functions for four periodic fields. The appendix includes results for the horizontally layered field (figure B.12), the checkerboard
81
4.2. Multiscale formulation
field (figure B.13), the domain with inclusions (figure B.14), and the diagonal layered
field (figure B.15). The permeability contrast for all domains equals 102 and results
are obtained for an over-resolved finite element discretization.
4.2.2
Coarse mesh equations
Oversampling obtains best results for the computation of equivalent permeability. However the multiscale basis functions are no longer continuous over their domain boundaries and the span increases, if these basis functions are applied to obtain the coarse
grid operator. The method sometimes even produces basis function weights significantly larger than one on interpolation points for ill-conditioned situations. Sampling
directly over simplex element enforces the compatibility conditions. Using a patch of
elements already can be seen as oversampling, as the node in the center of the patch
is connected to all coarse grid nodes. Pressure-dissipation averaging prevents new connectivities to occur. Alternatively a dual mesh can be constructed. The stiffness matrix
then follows as the sum of the algebraic equations derived on the original mesh and the
dual mesh. This stiffness matrix also includes the extra connectivities. The dual mesh
corrects mesh dependency and the sum of both stiffness matrices is able to incorporate
non-diagonal terms in the permeability tensor at all nodes. The costs of this procedure
is relatively low as the dual meshes need not be constructed on the finest level.
Pressure-dissipation formulation
Variational coarsening generates the coarse-scale matrix, given by
Z
nk−1
e
ρkr Kij k ∂Nck k ∂Ndk
k−1
Lab
= A
Pca
P
dΩ,
µ
∂xi db ∂xj
e=1 Ωe
(4.62)
approximated by
k−1
Lab
≈
nk
e
A
e=1
e
k
wm
Pca
ρkr Kij ∂Nck ∂ξk ∂Ndk ∂ξl
|J|
µ
∂ξk ∂xi ∂ξl ∂xj
e
The right hand side of the discrete flow equation is written as
Z
nk
e
ρkr Kij
∂Nck
k−1
fa = A
Pca
ρgj dΩ,
µ
∂xi
e=1 Ωe
or
fak−1 ≈
nk
e
e
A wm Pca
e=1
ρkr Kij ∂Nck ∂ξk
ρgj |J|
µ
∂ξk ∂xi
k
Pdb
.
(4.63)
m
(4.64)
e
.
(4.65)
m
Evaluating the coarse scale operator over a subdomain of fine scale elements is computationally expensive. Pressure-dissipation averaging obtains the same coarse grid
operator if the procedure applies to all fine scale elements, which form the coarse scale
element. Multiscale basis functions modify expression (4.35) as
e
k
k
nk−1
e
(b)k
(ab)
e
(a)k ρkr ∂Nc ∂ξk ∂Nd ∂ξl
Pdb pb |J|
φij = A wm Pca pa
.
(4.66)
µ ∂ξk ∂xi ∂ξl ∂xj
e=1
m
82
4. Adaptive multiscale method
domain
G(1)
G(2)
0
G(3)
G(4)
1
Figure 4.23: Oversampled solution on a horizontally layered field.
The linear loading case for the patch given by figure 4.7, can be constructed out of
coarse scale basis functions. These conditions reads
G(1) = N1k−1 + N4k−1 ,
G(2) = N1k−1 + N2k−1,
(4.67)
or alternatively
G(3) = N1k−1 + 12 N2k−1 + 12 N4k−1 ,
G(4) = N2k−1 + 12 N1k−1 + 12 N3k−1 .
(4.68)
Figure 4.23 shows the reconstruction of the loading cases by adding the multiscale
basis functions for the horizontally layered periodic cell. Pressure-dissipation averaging
extracts accurate equivalent permeability tensor component for this case (kxx = 5.05 ·
10−1 , kxy = −8.31 · 10−16 , kyx = −1.76 · 10−15 , and kyy = 1.98 · 10−2 ).
Appendix A includes the reconstruction of the loading cases for the horizontally
layered field (figure A.2), the checkerboard field (figure A.3), the domain with inclusions (figure A.4), and the diagonally layered field (figure A.5). Basis functions where
obtained on overlapping (oversampled) subdomains and non-overlapping (oscillating)
subdomains. All results are compared with periodic loading of the periodic cell in accompanying pictures. The appendix also presents the results for the equivalent permeability in table A.1, table A.2, table A.3, and table A.4. These tables show that a fully
resolved horizontally layered cell and a fully resolved domain with a single inclusion already provide accurate equivalent permeability tensor components. The checkerboard
field requires an over-resolved finite element mesh, which captures the singular points.
A fine mesh captures the structure of the diagonally layered field more accurately, and
improves the prediction of the equivalent permeability components. Measuring the
difference of equivalent permeability predictions provides a second refinement criterion
|tr(K̃)ov − tr(K̃)os | > c2 · tr(K̃)os ,
(4.69)
where tr(K̃)ov denotes the trace of the equivalent permeability found by oversampling
the subdomain. The first invariant tr(K̃)os follows from imposing oscillating boundary
conditions. The trace of the tensor matrix sums the diagonal components of the matrix.
4.3
Multigrid formulation
Domain decomposition methods like multigrid algorithms can be used as complete iterative algorithms to solve sets of equations directly or may be used as preconditioners
83
4.3. Multigrid formulation
for Krylov subspace methods like the conjugate gradient method. Domain decomposition algorithms can be divided in two classes. Overlapping domain algorithms
refer to Schwarz methods and include multigrid methods. Non-overlapping domain
algorithms refer to Schur complement methods and provide substructuring techniques.
Additive Schwarz methods solve the system of algebraic equations on overlapping subdomains and do not use updated values on the subdomain boundaries, which follow
from computations on neighboring subdomains. These methods are suitable for parallel implementation and relate to Jacobi solvers. Multiplicative Schwarz methods use
updated boundary information as soon as it becomes available. These methods relate
to Gauss-Seidel algorithms and need coloring for parallel computation. The convergence rate for the additive Schwarz method is lower than the rate for the multiplicative
Schwarz method. In general convergence is poor if there is no overlap domain (δ = 0)
in figure 4.20. Multiscale finite element methods can be classified as non-iterative domain decomposition methods. In classical multigrid, local iterative solvers like Jacobi
or Gauss-Seidel remove the high frequency components of the error. Multilevel overlapping domain decomposition applies Schwarz smoothers. Jacobi multigrid follows as
a multilevel additive Schwarz method on matching grids and subdomains with minimal overlap as depicted in figure 4.24. This figure shows two overlapping subdomains
that consist of the support of the basis function associated with nodes 6 and 7. The
overlap consists of 2 elements (3,6,7) and (6,7,11). Multiplicative Schwarz domain
decomposition gives Gauss-Seidel multigrid on domains with minimal overlap.
12
9 ..................................10
11
.......................................................................
5
1
•.............
•
•
•
•
•
•
•
...
...
.........
...
..
.... .. ....
... ....
...
.... ... ......
....
...
...
....
...
.
....
....
.
.
.
...
...
.
.
....
.
.... ... ....
.
...
.
.
.... ....
.... ... .....
...
.... ..
....
.... .. ....
..................................................................................................................
.
...
....
...
..........
.... ..
...
.... ... ....
...
... ..
.... ... .......
... ...
....
....
.
....
....
....
.
.
.
.
.
...
.
....
.
.
.
... .. ...
... .......
... .. ...
....
...
.... .. ....
..........
..
..
.................................................................................................................
6
2
•
•
7
3
8
•
•
4
Figure 4.24: Minimal overlapping subdomains.
Appendix C presents the linear multigrid algorithm and generalizes this algorithm
to the non-linear Full Approximation Storage (FAS) algorithm. The multigrid method
presented in this section extends the two-level multiscale finite element method to a
multilevel algorithm. The proposed geometric multigrid method is a physically based
technique needed in an adaptive formulation. It operates on a predefined grid hierarchy
like the one shown by Figure 4.1. The section considers the matrix properties of the
fine scale operator. The construction of the coarse scale operator and intergrid transfer
operators will be outlined and procedures for measuring the convergence behavior of
the algorithm will be given.
Fine grid operator
Multigrid methods are well established for solving self-adjoint boundary value problems
on fields of coefficients that show mild variations over the domain. The self-adjoint
operator for these problems corresponds to its adjoint, or equivalently the operator
84
4. Adaptive multiscale method
matrix is a Hermitian matrix. This matrix is its own conjugate transpose, and if the
Hermitian matrix is real then the matrix is symmetric. Matrices of real discretized selfadjoint boundary value problems are symmetric, sparse, and weakly diagonal dominant.
For weakly diagonal dominant matrices the diagonal element is at least as large as the
sum of the absolute values of the off-diagonal elements. It can be shown that if a matrix
is symmetric and weakly diagonal dominant then the matrix is positive definite. For
a positive definite matrix the following conditions apply: for all u 6= 0 the relation
uT Au > 0 holds and A = AT . An M-matrix forms a special class, this matrix is
symmetric and positive definite with non-positive off-diagonal terms.
The fine scale operator matrix of the flow problem can be expressed by a finite
difference stencil. The finite difference formulation follows from a uniform finite element
mesh. Two configurations of the center nodal have to be investigated. Figure 4.25
shows these configurations. Components of the conductivity matrix which follow from
•.............................................•..............................................•........
... ....
.
.. ..
....
...
....
.... ...
....
.
....
...
....
.... .... ......
...
..
.... .. ....
...
.
.
.
.
..................................................................................
.........
.
.
....
....
.
... ... .....
...
.
.
...
.... ... ......
....
...
....
.
....
.
.
...
.
.
.... ....
..
... .......
.
.
... ...
.... .
..........
.....
.......................................................................
I •
•
•
•
•
•
•
...............................................................................
...
.... .. ....
....
.... .... .......
....
...
....
....
....
....
....
.....
..
.... ....
... .......
..
.... ..
.
... .....
.
......................................................................................
... .....
...
... ...
... .....
...
.... ..
....
... ...
....
....
....
....
.
.
...
.
....
.
.... .. ...
...
.... .. ....
...
.... .. ....
....
..
...........................................................................
•
•
• II
•
Figure 4.25: Uniform triangular mesh configurations.
the finite element discretization of the diffusive term read
Z
ne
∂Na
∂Nb
Sab = A
Kij
dΩ
∂xj
e=1 Ωe ∂xi
(4.70)
For configuration I the finite difference stencil reads
ShI

0 0
= Kxx −1 2
0 0



0
0 −1 0
−1 + Kyy 0
2 0
0
0 −1 0



1 0 −1
1
Kyx 
Kxy 
0 0
0+
0
+
2
2
−1 0
1
−1

0 −1
0
0  . (4.71)
0
1
In two dimensions the conductivity matrix is independent of the size of the uniform
element distribution. This stencil captures the off-diagonal terms in the permeability
matrix. However for boundary edge nodes configuration II applies. The finite difference
stencil for these nodes reads




0 0
0
0 −1 0
II 2 0 .
Sh = Kxx −1 2 −1 + Kyy 0
(4.72)
0 0
0
0 −1 0
Because of the absence of the diagonal connectivities this stencil does not capture the
effects of off-diagonal permeability terms. Configuration II produces an M-matrix,
85
4.3. Multigrid formulation
configuration I does not produce such a matrix if off-diagonal tensor components are
considered.
Variable coefficient problems and variable mesh size problems generate anisotropic
algebraic problems, and weak connections in one direction. Local smoothers reduce
the error in the direction of the strong connection but do not reduce the high frequency error in the direction of the weak connection. As a result multigrid solvers
show poor performance as convergence factors degrade. Semicoarsening, block or line
relaxation improves the convergence rate of the multigrid solvers on anisotropic problems. Semicoarsening applies point relaxation and coarsens the mesh in the direction
of the high connectivity because multigrid convergence can only be expected in this
direction. Semicoarsening does not use weakly connected neighbors for intergrid transfer operations. Line or block relaxation solves the strongly coupled unknowns directly
and applies sequential standard coarsening. Variable-coefficient problems require more
sophisticated multigrid methods if the amplitude of coefficient varies strongly over the
field and the frequency variation is high.
Intergrid transfer operator
Multiscale basis functions provide intergrid transfer operators. The intergrid transfer
operators are based on a hierarchical formulation of the multiscale basis functions.
Figure 4.26 illustrates the sequential procedure. The sequential expression for the
•..............
... ....
.
•................................•.................
... ... .. ....
.
......
•.............................•.................................•................
... .... .. .... .. ....
. .
.
•.............................•......................................•...............................•.................
... .... . ... . .... . ...
.
.
..
...........
......
.
.
•................•.................•..................•......................•..
•..............
... ....
.
◦........ ..◦..........
...
...
..
..
•.....................◦..........................•................
. .. ....
...
.
.
.
.
.
◦........ .......◦.... ◦........ .◦...........
... ....
...
..
..
.......
.
.
•...............◦.................•................◦....................•..
•..............
... ....
.
◦........ ..◦..........
...
...
..
..
◦...... ◦ .◦...........
...
....
.
◦........ ◦ ◦ .◦...........
...
....
...
.
•...............◦................◦.................◦...................•...
Figure 4.26: Sequential multicale basis functions formulation.
multiscale basis function reads
k
Nak−1 = Pca
Nck ,
k−1 k−1
Nak−2 = Pca
Nc ,
(4.73)
where k indicates the mesh level. Prolongation and restriction operators
k transfer information over the hierarchy of grids. The prolongation operator Pk−1
, which interpolates coarse scale results to the fine scale,
includes
the
components
of
the intergrid
transfer operator. The restriction matrix Rkk−1 injects the fine scale information to
the coarse scale. The matrix follows as the (weighted) transpose of the prolongation
matrix, as a variational formulation will be applied.
Coarse grid operator
The multiscale homogenization formulation provides equivalent coarse-scale behavior. Coarse grid operators follow from pressure-dissipation homogenization on nonoversampled subdomains, and multiscale basis function loading conditions. Coarse-grid
discretization produces coarse-grid operators. Hierarchical scaling, implicit in algebraic
86
4. Adaptive multiscale method
multigrid and explicit in geometric multigrid, produces less accurate equivalent permeability tensors due to the sequential closure of local flow problems. More accurate
results are obtained if the fine scale information is brought to each scale. The method
approximates variational coarsening, and provides an alternative for more complex
problems. The coarse-grid operator and coarse-grid force vector then follow from the
fine-scale operator and intergrid transfer operators as
k−1 k−1 k k k−1 k−1 k L
= Rk
L Pk−1 ,
f
= Rk
f .
(4.74)
This expression is known as the Galerkin approximation. Variational coarsening
uses the adjoint of the prolongation operator as a restriction operator
k−1 k T
Rk
= Pk−1 .
(4.75)
In the Galerkin formulation the restriction matrix is not weighted by the sum of its
rows.
Full approximation storage algorithm
Non-linear algebraic equations can be solved by the full approximation storage algorithm (FAS). Here a non-linear Gauss-Seidel relaxation is mostly used to smooth
the non-linear algebraic equations. FAS modifies the coarse scale equations with a
τ -correction (appendix C). This correction improves the accuracy of the coarse scale
solution. If the system matrix is a linear operator, then the FAS scheme reduces to the
linear correction scheme. Adaptive methods use estimates of the discretization error as
a criterion for mesh refinement. In a multilevel setting the discretization errors can be
estimated on several levels and the error on the finest level can be extrapolation. The
FAS scheme provides the approximation on each level, which can be used directly. The
proposed multigrid algorithm is based on the FAS scheme and applies Gauss-Seidel iterations to smooth the high frequency modes of the error. A V-cycle schedule performs
the multigrid iterations. The algorithm applies non-weighted restriction of the defect,
weighted restriction of the primary variable and multiscale basis function interpolation
of the unknown pressure.
Multigrid is an iterative method and needs a stopping criterion. This criterion will
be based on the algebraic error. The final algebraic error has to be below the level of
discretization error. The algebraic error is the difference between the solution of the
algebraic equation and the solution of the exact discrete solution. The discretization
error compares the solution of the algebraic equations to the solution of the continuous
problem given by differential equations in discrete points. The maximum or infinity
norm and Eulidean or 2-norm can be used to measure the error of the solution. Here
the discrete L2 norm of a function will be used. This Eulerian norm reads
s
Z
ne
2
kek2 = A
(Na ũa − Na ua ) dΩ,
(4.76)
e=1
Ωe
where ũa denotes a fine scale reference solution. A discrete error norm which shows a
decrease of a factor four, on a uniform two-dimensional mesh and standard coarsening,
87
4.3. Multigrid formulation
Table 4.1: Error norm checkerboard problem.
cycle
kek12
kek22
kek32
1
2.03 · 10+0
1.06 · 10+0
2
+0
2.03 · 10
+0
3
4
2.03 · 10
5
6
2.03 · 10
7
8
kek42
7.20 · 10−1
6.10 · 10−1
1.05 · 10
−1
4.37 · 10
4.41 · 10−2
2.03 · 10+0
1.05 · 10+0
4.29 · 10−1
3.31 · 10−3
+0
+0
1.05 · 10
−1
4.28 · 10
2.49 · 10−4
2.03 · 10+0
1.05 · 10+0
4.28 · 10−1
1.87 · 10−5
+0
+0
1.05 · 10
−1
4.28 · 10
1.40 · 10−6
2.03 · 10+0
1.05 · 10+0
4.28 · 10−1
1.05 · 10−7
+0
+0
4.28 · 10
7.34 · 10−9
2.03 · 10
1.05 · 10
−1
Table 4.2: Residual norm checkerboard problem.
cycle
krk12
krk22
1
8.19 · 10−07
1.76 · 10−03
9.94 · 10−03
1.14 · 10−02
2
1.02 · 10
4.72 · 10
−04
3.26 · 10
8.74 · 10−04
3
1.90 · 10−16
1.31 · 10−08
1.09 · 10−05
6.64 · 10−05
4
5.25 · 10
3.71 · 10
−07
3.67 · 10
5.00 · 10−06
5
5.25 · 10−16
1.07 · 10−13
1.24 · 10−08
3.76 · 10−07
6
5.25 · 10
1.07 · 10
−10
4.15 · 10
2.83 · 10−08
7
5.25 · 10−16
1.07 · 10−13
1.40 · 10−11
2.13 · 10−09
8
5.25 · 10
1.07 · 10
1.40 · 10
1.60 · 10−10
−12
−16
−16
−16
krk32
−06
−11
−13
−13
krk42
−11
as the resolution doubles indicates second order discretization. The residual norm is
given by
s
Z
krk2 =
ne
A
e=1
2
(Na ra ) dΩ.
(4.77)
Ωe
If the ratio of the residual norm kr r+1 k2 /kr r k2 approaches a nearly constant ratio
after a number of cycles then this ratio estimates the asymptotic convergence factor
of the algorithm. A sharp increase at the end of the cyclic iterations indicates that
the algebraic approximation is accurate to near machine precision. The smoothing
rate of standard relaxation schemes indicates the convergence factor for reducing the
oscillatory modes of the discrete error. The overall convergence factor for a multigrid
scheme should be small and independent of the discretization size. For homogeneous
problems convergence factors of 0.05 can be obtained, for variable coefficient problems
convergence coefficients in the 0.2-0.3 range can be expected.
Table 4.1 presents
the error norm per cycle for the checkerboard problem already shown by figure 4.12.
88
4. Adaptive multiscale method
Results were obtained for a hierarchy of finite element meshes consisting of 2, 8, 32,
128, 512 and 2048 triangular elements. The error norm is presented for computations
with meshes that consist of 25, 81, 289 and 1089 nodes at the finest level. Gauss-Seidel
iterations smooth the error on each level by two pre-smoothing and two post-smoothing
operations. All results are compared with the final result of the 1089 node computation.
The error norm reduces linearly for fixed refinement. Table 4.2 presents the residual
norm for the checkerboard problem per cycle. The convergence factor is 0.03 for the
third computation and equals 0.07 for the fourth analysis.
Chapter 5
Model verification
For partly saturated porous media the permeability depends on the saturation of the
medium by the fluid phase. The saturation itself depends on the pore-water pressure
[71]. A constitutive model captures this behavior and introduces non-linearities into
the flow equation. Stable flow conditions occur, if the unsaturated zone is on top of the
saturated zone, as in phreatic surface problems. Depending on the constitutive equations, infiltration problems may cause instable flow conditions. Small perturbations in
the permeability field or porosity field for instance may trigger these instabilities and
the infiltration fronts then show fingering. Transport of a solute is coupled to the flow
of the fluid for which the solute contributes as a single component. The fluid velocity
is present in the convective term and scales the hydrodynamic dispersivity in the diffusive term of the transport equation. Fully coupled flow and solute transport equations
capture density driven flow. Here, the fluid density, present in the flow equation, depends on the solute mass fraction. For density driven flow stable flow situations occur
if the fluid density decreases with elevation height. Depending on the hydrodynamic
dispersion, instable situation may prevail if a denser fluid overlays a less dense fluid.
For this situation convective cells may develop and even for the case without any diffusion, mixing takes place by convection. Heat transport gives rise to fully coupled
flow and transport equations, as the fluid density and fluid viscosity both depend on
pressure and temperature. Unstable flow conditions may occur in a porous medium
that is heated from below.
This chapter considers the implementation of the saturated and unsaturated flow
model, coupled with solute transport and heat transport. The accuracy of the proposed
adaptive refinement formulation will be verified for a number of well known benchmark
problems by comparing its results with fine grid results. The efficiency of the algorithm
follows from the number of degrees of freedom that provide the level of accuracy. For
this the L2 Euclidean norm will be used throughout the section. Section 5.1 verifies
the modeling of saturated and unsaturated flow under stable and instable conditions.
Section 5.2 verifies the modeling of solute and heat transport for similar conditions.
89
90
5.1
5. Model verification
Saturated and unsaturated flow
This section verifies the adaptive multilevel formulation for saturated and unsaturated
flow. The constitutive equations are given by either an artificial model or the Van
Genuchten material model. The problems that will be investigated, impose prescribed
pressure boundary conditions, prescribed flux conditions and seepage conditions. The
seepage conditions pose a prescribed pressure, if outflow occurs, and impose closed
boundary conditions, if the pressure becomes negative under unsaturated conditions.
Section 5.1.1 presents the Lamb and Whitman problem [39, 51, 19]. The problem
involves stable saturated flow though an earth dam, but is solved in this section in
a partly saturated domain. The modified simulation focuses on the position of the
phreatic line and length of the internal seepage face along the material interface. Section 5.1.2 presents Forsyth’s problem. This problem involves infiltration in a large
caisson filled with four different porous materials [40]. The numerical simulation focuses on the infiltration process and on the effect of material contrasts.
5.1.1
Lamb and Whitman’s problem
Lamb and Whitman considered saturated flow through a cross section of an earth
dam resting on an impervious foundation [51]. A rock toe in the dam prevents water
from eroding the construction. For this reason, the toe is constructed out of a coarser
material. The rock toe is more permeable than the base material of the dam, which
reduces the water pressure distribution in the dam. Water pressures will have a large
impact on the stability of the dam and need to be considered in the design of the
construction. Lamb and Whitman constructed a flow net to simulate the steady state
seepage though the dam. An iterative procedure had to be used because the flow
domain is closed by a phreatic line, and the position of this line is not known in
advance. The pressure head is zero along the phreatic line, so here the total head
equals the elevation head.
Figure 5.1 shows the layout of Lamb and Whitman’s earth dam problem. The
................................................
.·
·..·....·... . . . . . . . . . . . ...........
........ . . . . . . . . . . . . . . . . ...........
.·
·
.
.
.
.
·
.
. . . ....
.. . . . . . . . .
·...·.....·.... . . . . . . . . . . . . . . . . . . . . . . . . .............
...... . . . . . . . . . . . . . . . ..........
.·
.
·
.
.
.
·.. . . . . . . . . . . . . . . . . . ..
........................................
..
.....
....
...
....
..
...
.. . . . . . . . . . . . . . . . . . . . ......
.
.
.
....
.
......
..... . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
...
.
.
.
.
.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .........
.
.
.
.
.
.... . . . . . . . . . . . . . . . . . . . . . . . . . .......
....
.
.
.
.
.
...
. . . . . . . . . ...... .......
..... . . . . . . . . . . . . . . .
.
.
.
.
.....
...
... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......
.
..
.
.
.
.
.
..
.. . . .........
.
...
........ ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ .
. . .... ......
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ..
........................................
.....
y···
···x
··
···
·
···
zone 1
zone 2
18 m
5m
9m
8.9 m
3.6 m
9m
......................................................................................................................................................................................................................................................................................
Figure 5.1: Lamb and Whitman’s problem definition.
base of the dam is 41 m, its height is 12.5 m. The base material of the dam has a
permeability of 13.13 m/d, the permeability of the rock toe is set to 13.13 · 108 m/d.
On the upstream side the water is at reservoir level of 12 m. At the downstream side,
the water is at soil surface level. Here the total head equals zero. The simulation will
follow the benchmark problem outlined in the Plaxflow manual [19].
5.1. Saturated and unsaturated flow
91
The steady state situation is obtained by a transient computation. The simulation
starts with a fully unsaturated cross section, and then computes the filling of the dam
after a gradual rise of river water level. The numerical simulation follows the physical
process and provides time dependent water pressure fields, which can be used in a
decoupled stability assessment. The model is also able to compute water pressures
in the unsaturated zone. In general, these negative pressures will stiffen the soil.
The influence of flow through the unsaturated zone is reduced however by imposing a
constitutive model which provides a sharp transition zone between the saturated and
unsaturated domain. This makes the comparison to the original solution easier. The
simulation time needed to reach the steady state condition is about 15 days. The
hierarchy of finite element meshes consists of 30, 99, 357, 1353 nodes, and 48, 176, 672,
2624 elements. On the most coarse level the size of the elements ranges between 3.816
m up till 5.184 m in x-direction and 2.65 m till 3.6 m in y-direction. The simulation
period of 1.5 · 106 s was divided in 150 steps of 1.0 · 104 s each.
Initial conditions specify a hydrostatic fully unsaturated field. These conditions are
given by
p0 = −γ · y,
(5.1)
where γ equals 104 Pa. The pressure at the top of the dam then equals −1.25 · 105 Pa
at the beginning of the simulation.
Pressure conditions apply to the inner and the outer slope of the dam, and simulate
the reservoir and polder influence. Boundary conditions on the outer slope specify water
loading, and read

if 0 < t ≤ t1 ∩ p ≥ 0
 p = −γ · y + 0.125 · t
q x nx + q y ny = 0
for 0 < t ≤ t1 ∩ p < 0 .
(5.2)

p = −γ · y + γh
if
t > t1
At t1 = 106 s the water level of the reservoir equals 12.5 m, which corresponds to
the construction height h of the dam. The boundary condition at this outer slope
generalizes the seepage condition. Outflow of groundwater only occurs for saturated
conditions, on the unsaturated part of the boundary outflow of water is restricted. A
prescribed pressure condition holds for the submerged part of the boundary. Boundary
conditions on the inland slope specify either a no-flow condition or prescribe a pressure
condition. No-flow conditions apply on unsaturated parts of the boundary and pressure
conditions hold for parts of the boundary where outflow takes place. The position of the
separated boundary parts is not known in advance and will be calculated automatically.
These geometrically nonlinear boundary conditions read
p=0
if
q x nx + q y ny ≥ 0
.
(5.3)
q x nx + q y ny = 0
for
p<0
At the bottom of the model no-flow conditions apply. The no-flow conditions are given
by qy = 0 .
Table 5.1 lists the material parameters and their values. The transition from satu-
92
5. Model verification
Table 5.1: Material parameters for Lamb and Whitman’s problem.
parameter
value
permeability soil κ1
permeability rock κ2
porosity soil and rock n1 , n2
density water ρ
viscosity water ν
compressibility skeleton αp
1.52 · 10−11 m2
1.52 · 10−7 m2
0.35
103 kg/m3
10−3 Pas
0 m2 /N
rated to unsaturated condition is obtained by a

Ss

ψ
S
+
· (Ss − Sr )
S=
s
|ψ ∗ |

Sr
and
kr =



1
4ψ
|ψ ∗ |
10
10−4
linearized material model, given by
if
ψ>0
for ψ∗ ≤ ψ ≤ 0 ,
if
ψ < ψ∗
if
(5.4)
ψ>0
for ψ ≤ ψ ≤ 0 ,
if
ψ < ψ∗
∗
(5.5)
where the pressure head reads ψ = p/ρg and ψ∗ denotes a minimum threshold value,
which relates to the height of a coarse grid element. Simulations were carried out for
a residual saturation Sr of 0.1, a maximum saturation Ss of 1.0, and a threshold value
ψ∗ of 0.5 m. The relative permeability varies over four orders of magnitude, and the
permeability contrast over the rock and soil material equals 108 . The gravity constant
is 10.0 m/s2 .
Figure 5.2 presents the pressure field for a number of time steps. The pictures only
show the saturated part of the flow domain. Although a relatively sharp transition zone
is enforced, groundwater flow occurs above the phreatic surface as well. This situation
was not captured by Lamb and Whitman, but occurs in real physical systems. The
pictures show the time dependent filling process of the dam. The position of the
phreatic line at the left side of the flow domain coincides with the reservoir level at all
time. Water infiltrates into the dam under stable flow conditions and reaches the toe
of the embankment in about 4.6 days. Due to the large permeability contrast of the
rock fill in the toe of the dam and the soil in the rest of the dam only a small pressure
gradient supports the water flux out of the soil through the rock toe and the saturated
zone in the toe is hardly visible. This situation generates an internal seepage face,
which is most clearly present in the steady state situation. Due to the flow in a small
zone above the phreatic level, the pressure will not be zero at the material interface
and the solution does not match the saturated model solution exactly. However, in
the physical system the pressure at the material interface will be less than zero as
well. The actual values depend on the unsaturated soil behavior. Figure 5.2 compares
results found by the adaptive multilevel method with results obtained by a fine mesh
93
5.1. Saturated and unsaturated flow
adaptive multilevel
fine single level
t = 2 · 105 s
t = 4 · 105 s
t = 6 · 105 s
t = 8 · 105 s
t = 1 · 106 s
t = 1.2 · 106 s
0.00E+00
1.25E+05
Figure 5.2: Pressure (Pa) fields for Lamb and Whitman’s problem.
computation. This mesh applies level four elements over the entire flow domain. The
level four elements are shown in parts of the adaptively refined mesh. Results on the
adaptive mesh compare well with results on the fine mesh. Mesh refinement corrects
the error in the continuity of flux over the element boundaries. For this simulation the
refinement concentrates on the phreatic line.
94
5. Model verification
The computational behavior for the Lamb and Whitman problem is presented by
figure 5.3. This figure shows the number of nodes and the related pressure field error
as a function of time. The error is calculated as
s
Z
kek2 =
ne
A
e=1
2
(Na ũa − Na ua ) dΩ.
(5.6)
Ωe
where ũa denotes a fine scale reference solution. Scaling the error by kũk2 could reduce
the dimension of the parameter.
kek2 error
degrees of freedom
1.5E+3
8.0E+4
enorm (-)
nodes (-)
6.0E+4
1.0E+3
5.0E+2
4.0E+4
2.0E+4
0.0E+0
0.0E+0
0.0E+0
5.0E+5
1.0E+6
1.5E+6
0.0E+0
time (s)
level1
5.0E+5
1.0E+6
1.5E+6
time (s)
level2
level3
level4
adaptive
Figure 5.3: Computational results for Lamb and Whitman’s problem.
The adaptive mesh solution reduces the number of nodes from 1353 on level 4 to
about 357 on level 3. Still the error norm of the adaptive algorithm is well below the
norm for a level 3 computation over the entire simulation. The enhanced accuracy
of the adaptive algorithm at 5 · 105 s requires the generation of a limited number of
additional elements. These elements are positioned mainly along the phreatic line. As
the length of the phreatic line decreases later on, the number of additional elements is
reduced as well.
5.1.2
Forsyth et al’s problem
Forsyth et al. simulated the infiltration process into a large caisson in a two dimensional
cross section [50]. The horizontal dimension of the caisson is 8 m and vertical dimension
equals 6.5 m. The problem shows a gradual change in permeability over four materials,
which are present in the caisson. A zone of 2 × 1 m, with a permeability of 41.52 m/d
is included in a larger zone of 8 × 5.6 m with a relatively low permeability of 4.15 m/d.
On top of this zone rests a layer of 0.5 m with a permeability of 4.70 m/d. The layer
on top of this layer is 0.4 m thick and has a permeability of 7.91 m/d. Figure 5.4 shows
the layout of the problem posed by Forsyth et al [50].
The simulation, presented here, follows the benchmark problem outlined in the Feflow manual [40]. The manual considered two initial field conditions. These conditions
pose a fully unsaturated non-equilibrium starting condition. The infiltration process
95
5.1. Saturated and unsaturated flow
2.25
m
................................
.. .. .. .. ..
......... ......... ........ ......... .........
........................................................................................................................................................................................................................ .
......... . . . . . . . . . . . . . . . . . . . ... ....
................................................................................................................................................................................................................................... ...
...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ...
..... . . . . . . . . . . . . . . . . . . . . . . . . . . ... ...
....................................................................................................................................................................................................................................................... ...
........ ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
....... ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
....... ... .... ................................................................... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
....... ..... ...... ...... ..... ...... .... ..... .... ...... . ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ...
....... . . . .... . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . .... ...
....... ... .... ..... ... .... ... ... ... ..... . ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
....... ..... ...... ...... .... .... ... ... ... ...... . ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ...
....... . . . . .......................................................... . . . . . . . . . . . . . . . . . . . . . . . . .... ...
....... ..... ...... .... ..... ...... .... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ....... ....
....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ..
....... ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
........ ..... ...... .... ..... ...... .... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ...
....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ...
....... ..... ...... .... ..... ...... .... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ...
....... ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ...
....... ..... ...... .... ..... ...... .... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ...
....... ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
....... ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ...
........ ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ..... ....
..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ..
........ ..... ...... .... ..... ...... .... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ...
........ ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ...... ....
...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ..
....... ..... ...... .... ..... ...... .... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ..... .... ...... ...
........... ... .... ... ... .... ... ... ... .... ... ... .... ... ... .... ... ... .... ... ... .... ... ... ...... ....
..................................................................................................................................................................................................................................................... ..
...................................................................................................................................................................................................................
··································
zone 1
zone 2
zone 4
zone 3
y
x
1m
2m
......
..
....
......
.....
.....
.....
..
......
....
.......
.....
.....
...
...
....
..
...
...
....
..
...
...
...
...
...
...
...
...
...
.......
..
0.4 m
0.5 m
0.6 m
1m
4m
5m
..................................................................................................................................................................................................................
Figure 5.4: Forsyth et al.’s problem definition.
was simulated over 30 days. A time step of 10−3 days, and a spatial discretization by
90 × 21 quadrilateral elements was selected. Saturation results were compared for a
mesh of 28,917 nodes which was obtained by splitting each quadrilateral element into
two triangular elements and then refining the triangular elements twice.
This section selects the most extreme initial case for comparison. The hierarchy of
finite element meshes consists of 99, 357, 1353, 5265 nodes, and 162, 644, 2568, 10,256
elements. At the coarsest level, the size of the elements range from 0.5 m up till 1.0 m
in x-direction and 0.4 m till 1.0 m in y-direction. The simulation period of 2.592 · 106
s was divided in 300 steps of 8640 s each.
Initial dry conditions pose a constant pressure on the flow domain, and read
p0 = −106 Pa.
(5.7)
These conditions do not state an equilibrium situation, so the result of the time dependent simulation will be influenced by the starting conditions.
An infiltration boundary condition applies to the left most 2.25 m on the top of the
model, according to
qn = 0.02 m/d.
(5.8)
Feflow sets the remaining part of the boundaries as impervious. This condition is simply expressed by qx = 0 on the vertical boundary parts, and qy = 0 on the horizontal
boundary parts. The proposed boundary conditions do not state a well-posed boundary value problem, because the actual value of the pressure solution depends on the
initial field only. A boundary value problem is well posed if it satisfies the compatibility
condition and it has a unique solution. Well-posed problems are solvable. Compatibility requires that the sum of all source terms is zero. Additional Dirichlet conditions
enforce a unique solution. The simulation presented here applies alternative boundary
conditions at the top of the domain, which read p = −106 Pa.
96
5. Model verification
relative permeability
saturation
1.0
1.0E+0
1.0E-1
rel. permeability (-)
saturation (-)
0.8
0.6
0.4
0.2
1.0E-2
1.0E-3
1.0E-4
1.0E-5
0.0
1.0E-6
-1.0E+5
-7.5E+4
-5.0E+4
-2.5E+4
0.0E+0
-1.0E+5
-7.5E+4
pressure (Pa)
-5.0E+4
-2.5E+4
0.0E+0
pressure (Pa)
zone 1
zone 2
zone 3
zone 4
Figure 5.5: Material behavior for Forsyth et al.’s problem.
The density of the water ρ is 1000 kg/m3, and its viscosity ν is 10−3 Pa s. The
gravity constant is set to 10.0 m/s2 . Figure 5.5 shows Van Genuchten material behavior
for Forsyth et al’s problem graphically. Table 5.2 includes the coefficients for these Van
Genuchten model functions.
Table 5.2: Van Genuchten shape factors for Forsyth et al’s problem.
zone
Sr
-
n
-
K
m2
ga
1/m
gl
-
gn
-
1
2
3
4
0.2771
0.2806
0.2643
0.2643
0.368
0.351
0.325
0.325
9.153 · 10−12
5.445 · 10−12
4.805 · 10−12
4.805 · 10−11
3.34
3.63
3.45
3.45
0.5
0.5
0.5
0.5
1.982
1.632
1.573
1.573
Figure 5.6 shows the saturation fields for a number of time steps, and compares the
results of the adaptive computation to the fine mesh calculation. The saturation fields
are derived from continuous pressure fields. The saturation fields show discontinuities
over the material boundaries due to the difference in unsaturated behavior for the
individual soil materials. Adaptive grid results compare well with fine scale results,
although the adaptive algorithm reduces the number of elements considerably. Grid
refinement concentrates on zones with a sharp saturation gradients and zones with
curved streamlines. Both zones were detected by considering the error in continuity of
flux over the element boundaries. The adaptive simulation starts with a very coarse
mesh. After one day of infiltration the infiltration front reaches the second layer.
During this first period infiltration is almost vertical. Next water infiltrates into the
first layer in horizontal direction as well because of the permeability contrast with the
second layer. The infiltration speed is lower in the second layer mainly due to the
97
5.1. Saturated and unsaturated flow
adaptive multilevel
fine single level
t=1d
t = 10 d
t = 20 d
t = 30 d
0.4
0.8
Figure 5.6: Saturation (-) fields for Forsyth’s problem.
98
5. Model verification
lower value of intrinsic permeability of this layer. The degree of saturation in the top
layer is less as a results of the difference in saturation curves, which apply to both
layers. After 10 days of infiltration the infiltration front extends to the bottom level of
the included zone. Saturation contours are continuous over zone 3 and 4, because the
unsaturated behavior of the materials that fill these zones is the same. The intrinsic
permeability of zone 4 is a factor 10 higher than the intrinsic permeability of zone 3.
During the infiltration process the embedded zone drains its surrounding and transports
the incoming water faster than the material next to it. As a result bypass flow occurs
that infiltrates zone 3 at the bottom of zone 4 after some time.
kek2 error
5.0E+4
4.4E+3
4.0E+4
err norm (-)
nodes (-)
degrees of freedom
5.5E+3
3.3E+3
2.2E+3
1.1E+3
3.0E+4
2.0E+4
1.0E+4
0.0E+0
0.0E+0
0.0E+0
8.6E+5
1.7E+6
2.6E+6
0.0E+0
time (s)
level1
8.6E+5
1.7E+6
2.6E+6
time (s)
level2
level3
level4
adaptive
Figure 5.7: Computational results for Forsyth et al.’s problem.
Computational results for the modified Forsyth et al’s problem are shown in figure 5.7. The accuracy of the pressure field was calculated by equation (5.6). In the
first period of the simulation, up to one day, the number of nodes in the adaptive
mesh simulation is well below level 3 refinement of the mesh. This gives less accurate
results. From that moment on the number of nodes in the adaptive mesh corresponds
approximately to the level 3 mesh, which was constructed out of 1352 nodes and 2568
elements. The infiltration front however, was captured by level 4 elements, improving
the accuracy of the adaptive mesh solution. A full level 4 mesh would require 5265
nodes and 10256 elements.
5.2
Solute transport and heat transport
This section deals with solute transport and heat transport and covers the verification of the adaptive formulation for two well-known fully coupled flow and transport
benchmarks: Henry’s problem and Elder’s problem. These cases involve saturated
groundwater flow through homogeneous domains. Henry adds mass transport to the
problems, and Elder couples heat transport to the flow problem. Both problems consider density changes and density driven flow. The problems are closed by prescribed
fluid flux conditions or prescribed pressure conditions. The boundary conditions either
generate flow into the domain or flow out of the domain. For the pressure condition this
99
5.2. Solute transport and heat transport
..........................................................................................................................................................................................................................................................................................................................................................................................................
.................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
..................... . . . . . . . . . . . . . . . . . . ...
...................... . . . . . . . . . . . . . . . . . . .....
.................. . . . . . . . . . . . . . . . . . . ....
................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .....
.................... . . . . . . . . . . . . . . . . . . ....
..................... . . . . . . . . . . . . . . . . . . ....
................... . . . . . . . . . . . . . . . . . . .....
.................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
..................... . . . . . . . . . . . . . . . . . . ....
.................... . . . . . . . . . . . . . . . . . . .....
................... . . . . . . . . . . . . . . . . . . ....
................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
....................... . . . . . . . . . . . . . . . . . . .....
...........................................................................................................................................................................................................................................................................................................................................................................................................
y
··
··
··
··
··
··
··
··
··
·
x
...
......
..
...
...
...
...
....
..
....
...
...
...
...
...
...
...
..
.......
..
1m
2m
..................................................................................................................................................................................................
Figure 5.8: Definition of Henry’s problem
depends on the pressure in the domain. Flow into the domain constitutes a prescribed
solute mass flux or a prescribed energy flux. A weak boundary condition simulates the
solute transport or heat transport out of the domain, by imposing a zero diffusive flux.
Section 5.2.1 presents Henry’s problem. This problem involves salt water intrusion
into a domain in the opposite direction of the natural fresh water flow [4, 21, 39, 53, 66,
83, 79, 90, 107]. Flow conditions for this problem are stable as salt water under lays
fresh water. The problem verifies the accuracy and efficiency of mesh refinement for this
type of density driven flow, and focuses on geometrically nonlinear boundary conditions
that result from the prescribed pressure conditions, which apply on the outflow side
of the domain. Section 5.2.2 presents Elder’s problem [47, 4, 83, 107, 121, 39, 79, 66].
This problem involves potentially unstable heat flow and considers a domain which is
heated from below. This benchmark focuses on flow that is purely driven by density
differences and generates natural convection. Special attention will be paid to the effect
of the discretization of the problem on instabilities.
5.2.1
Henry’s problem
Henry’s problem involves seawater intrusion into a confined layer. At the inland side
fresh water flows into the domain, and salt water intrudes the domain at the opposite
side. Due to the difference in density, fresh water flows over the salt water. Henry
derived an analytical expression for the stream function and calculated the concentration field in the form of a Fourier series. The resulting algebraic equations, which
determine the coefficients of the Fourier series were solved numerically. Henry obtained
a dimensionless solution for the steady state solution in the cross section.
The simulation presented here, follows the benchmark problem outlined in the Sutra
manual [118]. In this manual Voss proposed dimensions for the problem that make a
comparison with other models possible. He also extended the original problem to a
transient problem, which finally obtains the steady state situation. The total simulation
time is 100 minutes, for which the steady state can essentially be reached. The cross
section has a length of 2.0 m and a height of 1.0 m. The Sutra manual proposed a
mesh that consists of 20 × 10 elements of size 0.1 × 0.1 m. A uniform time step of
1.0 minute was selected for the discretization in time. Figure 5.8 shows the layout of
Henry’s problem.
100
5. Model verification
The Feflow manual proposes a fine scale elements size of 0.0125 × 0.0125 m. Based
on this information, the hierarchy of finite element meshes constructed here, consists of
66, 231, 861, 3321 nodes, and 105, 410, 1620, 6440 elements. At the coarsest level the
size of the elements equals 0.2 m in both directions. The simulation period of 6.0 · 103
s was divided in 100 steps of 60 s each.
The initial state of the system is obtained by a steady state computation based
on freshwater inflow, zero concentration on both vertical boundaries, and specified
pressures at the sea side. This pressure condition already corresponds to enforcing a
hydrostatic seawater pressure distribution as will be shown below.
No-flow boundary conditions apply to the top and bottom of the domain. Fixedflow conditions apply to the left vertical boundary. Here a mass flux of 6.6 · 10−2 kg/s
holds and the salt mass fraction is set to zero. This condition poses the inflow of fresh
water into the domain, and is written as
qx = 6.6 · 10−5 m/s
∩
ω = 0.
(5.9)
At the right vertical boundary, a hydrostatic pressure is prescribed. Water that enters
the domain through the lower part of the boundary, has the density of seawater. The
diffusive influx jn = −D∂ω/∂x of the water that leaves the domain at the upper part
of the boundary is set to zero. The extend of both boundary parts varies throughout
the simulation. This geometrically nonlinear boundary condition reads
(
0
if qx > 0
p = ρs g(h − y) ∩ jn =
,
(5.10)
(ω − ω) qx if qx ≤ 0
where ω = 3.571 · 10−2 .
Table 5.3 lists the material parameters and their values. The density of the binary
fluid depends on the mass fraction of the solute. The constitutive relation for the
density is simplified to
ρ = ρ0 + βω (ω − ω0 ) .
(5.11)
Henry’s problem specifies a velocity-independent diffusion. This Fickian dispersion is
incorporated into the model by setting the molecular diffusion to 18.8571 · 10−6 m2 /s.
The total dispersion coefficient then equals 6.6 · 10−6 m/s2 , as the porosity is set to
0.35. In physical systems a velocity based dispersion describes the mixing process more
accurately, and the transition zone between salt and fresh water can be relatively thin.
The gravity constant is 9.8 m/s2 .
Figure 5.10 presents mass fraction fields for a number of time steps. Initially the
domain is filled with fresh water only. During the simulation salt water intrudes into
the domain. The greater density of the salt water drives the intrusion. At the bottom
of the flow domain the saltwater flux points in the opposite direction of the fresh water
flux at the top of the domain. As a result a single eddy will evolve. At the top of the
eddy mixing occurs. Both adaptive mesh results and fine grid results compare well,
and the final result at 1.6 hours corresponds to the steady state result presented in the
Sutra manual.
Figure 5.9 shows the number of nodes of the adaptively refined mesh and compares
them to the number of nodes on the uniform meshes. The corresponding accuracy of
101
5.2. Solute transport and heat transport
Table 5.3: Material parameters for Henry’s problem.
parameter
value
permeability skeleton κ
density fresh-water ρ0
density sea-water ρs
density variability coefficient βω
viscosity water ν
porosity skeleton n
diffusion coefficient Dc
longitudinal dispersivities αl , αt
compressibility skeleton αp
1.020408 · 10−9 m2
1000 kg/m3
1024.99 kg/m3
700 kg/m3
10−3 Pas
0.35
1.88571 · 10−5 m2 /s
0m
0 m2 /N
kek2 error
1.0E-3
3.0E+3
7.5E-4
enorm (-)
nodes (-)
degrees of freedom
4.0E+3
2.0E+3
1.0E+3
5.0E-4
2.5E-4
0.0E+0
0.0E+0
0.0E+0
2.0E+3
4.0E+3
6.0E+3
0.0E+0
time (s)
level1
2.0E+3
4.0E+3
6.0E+3
time (s)
level2
level3
level4
adaptive
Figure 5.9: Computational results for Henry’s problem.
the solution, now expressed in mass fraction according to equation (5.6), is also given
as a function of time. The figure shows that the number of nodes in the adaptive
mesh computation approximately corresponds to the number of nodes in the level 3
computation. The accuracy of both level 3 calculation and adaptive mesh computation
only slightly differs from the level 4 computation. Aiming at level 3 accuracy with an
adaptive mesh build out a number of nodes that corresponds to the level 2 mesh could
improve the efficiency of the adaptive method. Figure 5.10 shows a large part of the
domain that contains level 1 elements. This points out that adaptive refinement of
local phenomena is especially efficient.
5.2.2
Elder’s heat problem
Elder’s heat problem considers transient thermal convection in a confined cross-sectional
rectangular region. Part of the bottom boundary holds a fixed temperature greater
102
5. Model verification
adaptive multilevel
fine single level
t = 0.33 h
t = 0.66 h
t=1h
t = 1.33 h
t = 1.66 h
3.6E-03
3.6E-02
Figure 5.10: Mass fraction (-) fields for Henry’s problem.
103
5.2. Solute transport and heat transport
than the initial uniform temperature in the domain. The conducted heat into the pore
water decreases the density and induces circulating flow. Elder used a finite difference
technique to solve the set of equations, which was formulated in terms of dimensionless
variables.
The simulation that is presented here, follows the benchmark problem outlined in
the Hst3d manual [79]. Figure 5.11 shows the layout of the modified Elder’s problem.
The cross section is 4 m long and 1 m high. The heat source is present along the middle
2 m of the bottom boundary. The box is closed by impermeable walls on the remaining
horizontal and vertical boundaries. Kippe [79] proposed a uniformly discretization of
•........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................•....
... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ...
.... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ....
.. ...
.. .
... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ....
.... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ....
.. ...
.. .
... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ....
.... .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ....
.. ...
.. .
... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... ....
......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ....
.......
.. .
... ... . .... . . . . . . . ... . .... ...... .. ... ........ ..... ...... ..... .. ..... ........ ... .. ... .. .... . .. . . . . . . .... ....
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
y
x
1m
2m
..
......
....
..
.....
..
...
...
...
...
...
...
...
.....
..
1m
1m
....................................................................................................................................................................................................................................................................................................
Figure 5.11: Definition of Elder’s thermal problem.
the domain with nodal spacing of 0.091 m in the horizontal direction and 0.040 m in
the vertical direction, and uniform time steps of 596 s over a period of 1.1925 · 105 s.
This period corresponds to a dimensionless time of 0.1, used by Elder.
The hierarchy of finite element meshes consists of 66, 231, 861, 3321 nodes, and
105, 410, 1620, 6440 elements. At the coarsest level, the size of the elements equals 0.2
m in x-direction and 0.2 m in y-direction. The simulation period of 1.192 · 105 s was
divided by 2000 steps of 59.6 s each.
Initial conditions pose a hydrostatic pressure and a uniform temperature on the
flow domain. This initial condition reads
p0 = ρ0 g(h − y)
∩
T 0 = 283 K.
(5.12)
The boundary condition at the top of the domain reads
qy = 0
∩
T = 283 K.
(5.13)
A pressure of 0 Pa at the upper corners of the region is proposed in order to generate
a well posed problem. At the corner points of this boundary, the condition is replaced
by
p = 0 ∩ T = 283 K.
(5.14)
In the middle part of the boundary at the bottom, the following conditions hold
qy = 0
∩
T = 293 K.
(5.15)
104
5. Model verification
Table 5.4: Material parameters for Elder’s problem.
parameter
value
thermal conductivity solid phase λs
thermal conductivity fluid phase λ
thermal heat capacity solid phase cρs
specific heat capacity fluid phase c
thermal expansion coefficient fluid phase βT
reference density fluid phase ρ0
reference temperature T0
density solid phase ρs
permeability skeleton κ
porosity skeleton n
viscosity fluid phase ν
molecular diffusion coefficient Dc
dispersivities porous medium αl , αt
compressibility skeleton αp
2.0 W/mK
0.6 W/mK
2.0 · 106 J/m3 K
4182 J/kgK
2 · 10−4 1/K
1000 kg/m3
293 K
2650 kg/m3
9.084 · 10−9 m2
0.1
10−3 Pas
0 m2 /s
0m
0 m2 /N
At the remaining part of the bottom boundary, and on the vertical boundaries the
conditions read
qy = 0
∩
∂T
= 0,
∂y
qx = 0
∩
∂T
= 0.
∂x
(5.16)
Table 5.4 presents the material parameters for Elder’s problem and their values.
The fluid density varies between 1000 kg/m3 and 998 kg/m3 . The constitutive relation
for the density is simplified by
ρ = ρ0 − βT (T − T0 ) .
(5.17)
The compressibility of water and solids is disregarded. The gravitational constant
equals 9.807 m/s2 .
Figure 5.12 presents the computed temperature fields for an adaptively refined mesh
and the corresponding fields for a uniform fine mesh. Both series only show half of
the original flow domain, and the results will be discussed for this reduced domain.
In the first stage of the simulation two eddies emerge on both sides above the heater
end, which are driven by the rising fluid. On the fine grid two small eddies become
visible at 3.3 h. These eddies are positioned next to the larger eddies. A third small
eddy becomes visible at 6.6 h at the right vertical side of the domain, which models
symmetry of the original flow domain. The adaptive mesh shows different results. Here
convective transport of heat triggers an eddy at the right side of the domain at an early
stage. At 6.6 h the system contains five eddies of about equal strength. For both fine
scale computation and adaptive mesh computation the eddies shift to the right later
105
5.2. Solute transport and heat transport
adaptive multilevel
fine single level
t = 3.3 h
t = 6.6 h
t = 16.6 h
t = 24.8 h
t = 33.1 h
285
293
Figure 5.12: Temperature (K) fields for Elder’s problem.
106
5. Model verification
on and form a system that contains three eddies at 16.6 h. Finally the eddies melt into
one roll and reach a stable steady state situation. In the center part of the original
domain the fluid flow is upward, at the ends downward flow occurs. The figure shows
a large mesh dependency of the results in the development stage.
kek2 error
degrees of freedom
4.0E+3
2.6
2.2
1.9
err norm (-)
nodes (-)
3.0E+3
2.0E+3
1.5
1.1
0.7
1.0E+3
0.4
0.0E+0
0.0E+0
0.0
3.0E+4
6.0E+4
8.9E+4
1.2E+5
0.0E+0
3.0E+4
time (s)
level1
6.0E+4
8.9E+4
1.2E+5
time (s)
level2
level3
level4
adaptive
Figure 5.13: Computational results for Elder’s problem.
Figure 5.13 presents the adaptively generated number of nodes and the accuracy
of the temperature field as a function of time. Although the number of nodes is above
the number of nodes generated by a level 3 discretization, the accuracy of the adaptive
mesh computation does not improve much during the early development stage. The
reason is that the adaptively refined mesh triggers a different convective flow pattern.
At the end of the computation the adaptive mesh computation gives more accurate
results than the level 3 calculation although the number of nodes is about the same.
Elder’s problem poses a complicated verification test, as the size of the mesh influences the simulated transport mechanism. Adding a local refinement criterion on
the projected error of the mass fraction might resolve this problem. At this stage
the flow problem has been solved on a hierarchy of adaptively refined finite element
meshes using the proposed multigrid solver. The transport equation has been solved by
a preconditioned bi-conjugate gradient solver on a single level. Solving the transport
equation sequentially over at least two levels facilitates extrapolation to the finest level
where the refinement criterion could be postulated.
Chapter 6
Flow simulation
The tenth SPE (Society of Petroleum Engineers) comparative solution project aimed
to compare upgridding and upscaling techniques for two problems [26, 56, 86]. The first
problem involved flow through a two-dimensional vertical cross section, for which the
geological model was captured by two-thousand cells. The model domain is 2500×50 ft,
and a fine grid covered the model by uniform sized grid blocks of 100 × 20 ft. Upscaling
techniques should generate equivalent permeability for a specified uniform upgridded
model of 25 coarse grid cells and for a non-uniform adaptive grid model of 100 cells
at maximum. The second problem involved a three-dimension waterflood problem
of over 1.1-million cells. For this number of cells results could still be compared to
fully resolved computations. The model domain is 1200 × 2200 × 170 ft. The top 70
ft represents the near-shore Tarbert formation, and the bottom 100 ft represents the
fluvial Upper Ness formation. For this model the fine-scale cell size is 20 × 10 × 2 ft.
Porosity and permeability are constant over the fine scale cells. The porosity field is
strongly correlated to the permeability field. The quotient of vertical and horizontal
permeability is 0.3 in highly permeable channels and 0.001 in the background formation.
Both formations show large permeability variations of 8 till 12 orders of magnitude.
Participants to the project produced upscaled permeabilities on grids of 30 × 55 cells
up till 5 × 5 grid cells. The three-dimensional reservoir is produced using a water drive
from a single well in the center of the model and four producers in the corners of the
model. Wells are completely vertical throughout the model. Prescribed bottom hole
pressure hold for all wells. The process was simulated over 2000 days of production.
Production curves and average field pressure were compared.
This chapter includes two applications of the adaptive multiscale finite element
method on these highly heterogeneous domains. Instead of oil production, energy
storage and heat production will be considered here. Section 6.1 considers geothermal
storage in the shallow marine formation. Section 6.2 focuses on geothermal energy
production in a deep fluvial system. For both applications the geological model and
the corresponding flow model will be presented. Adaptive multiscale results will be
compared with fully resolved solutions.
107
108
6.1
6. Flow simulation
Marine system
The top 70 ft of the second SPE model represents the near-shore Tarbert formation.
The distribution of porosity and permeability was given for 60 × 220 × 35 cells. The
parameter values are constant for each cell and discontinuous over the cell boundaries.
Here the problem is reduced to 60 × 220 × 32 cells, because this fits a hierarchy of three
levels. The geological model now includes 422,400 cells. The size of the problem was
set to 366 × 671 × 20 m, and the size of the cells equals 6.1 × 3.05 × 0.625 m. The
aspect ratio of the elements is 9.76, which is relatively large. The flow model splits each
quadrilateral cell into 6 tetrahedral elements and generates 2,534,400 elements in total.
The number of nodes equals 444,873, and as flow and transport will be considered, two
systems with 444,873 degrees of freedom apply. The coarsened mesh at level 2 includes
52,800 cells, and the level-one mesh contains 6,600 cells. For this mesh, the degrees of
freedom reduce to 16,128.
This section presents flow simulation results for a two-dimensional area: the top
layer of the formation. The geological model is restricted to 60 × 220 cells on the finest
level. The number of elements in the hierarchy of finite element meshes is 13,200, 3,300
and 825, the number of nodes equals 13,481, 3,441, and 896.
6.1.1
Geological model
Figure 6.1 shows the porosity field for the geological model of the Tarbert formation,
and Figure 6.2 presents the distribution of the porosity and the permeability for the
670.56
502.92
51.74
44.14
365.60
274.20
36.54
182.80
91.40
0.0
335.28
167.64
28.94
0.00 0.00
21.34
0.5
Figure 6.1: Tarbert porosity (-) field.
top layer of the marine system. Porosity varies between 0.05 and 0.45, permeability
fluctuates between 3.03 · 10−17 m2 and 4.65 · 10−11 m2 . Both parameters fit into a
single distribution.
109
6.1. Marine system
porosity
permeability
5.2
4.5E-01
3.4E-01
frequency (-)
frequency (-)
4.2
3.1
2.1
2.3E-01
1.1E-01
1.0
0.0
0.0E+00
0.0
0.1
0.2
porosity (-)
0.3
0.4
0.5
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
permeability (m2)
Figure 6.2: Material property distribution Tarbert formation top layer.
Figure 6.3 presents the porosity distribution and the permeability distribution over
the top layer. The figures show the small scale distribution on level 3, and its upscaled
values at level 2 and level 1. The pictures show that as the scale decreases, the resolution becomes less. The pictures at the bottom presents adaptive scaling results. The
refinement is based on the flow problem imposed. Here, the error in volumetric flux
over the element boundaries, as discussed in chapter 4, detects the regions on which
the mesh has to be refined. The pictures show a low permeable zone at the left side
that stretches from the bottom to the top of the flow domain. This zone is present at
all scales.
6.1.2
Flow model
The flow model simulates a simplified heat storage system. In the summer period, a
source injects heated water into the system. During this period a sink produces water
at subsurface temperature for cooling purposes. In winter time the system is reversed.
During this period the energy stored in the system is recaptured and cooled down water
is injected.
The simulation considers the top layer of 607 × 366 meter of the marine system.
The fine scale permeability distribution was first multiplied by a factor of ten in order to
represent an aquifer, then the porosity was set to 0.35. Natural conditions for this layer
impose a pressure of 9 · 105 Pa, as the layer is positioned 90 meter below the phreatic
surface, and a temperature of 13o C. Artificial conditions apply on the boundaries of
the flow model. In a top view, the conditions at the vertical boundaries prescribe a
constant pressure of 9 · 105 Pa. Along the horizontal sides of the flow domain no-flow
conditions apply. The simulation introduces a hot well at the point 240 - 240 m and a
cold well at 480 - 96 m. The injector well and producer well operate independently as
their distance is 280 m. The hot well injects water of 24o C in the summer period. The
pressure at the bottom of the well is 1.1 · 106 Pa (20 m above the natural head) and the
infiltration rate equals 1.7 m3 /h per meter filter length. Hot water is produced in the
winter at the same rate by lowering the head 20 m below the natural head, imposing
a pressure of 7 · 105 Pa. In the winter period the cold well injects water of 8o C into
110
6. Flow simulation
porosity fields
permeability fields
scale 1
scale 2
scale 3
adaptive scaling
5.0E-02
5.0E-01
1.0E-17
1.0E-09
Figure 6.3: Porosity (-) and permeability (m2 ) fields, heat storage system.
111
6.1. Marine system
the aquifer at a rate of 1.7 m3 /h per meter filter length, at a pressure of 1 · 106 Pa.
During the summer period the cold well produces water at the same rate by lowering
the natural head by 10 m, which sets the producer pressure to 8 · 105 Pa. A total filter
length of 30 m obtains realistic flow rates of 51 m3 /h. The total simulation time is 365
days, divided in 4 seasons of 91.25 days each, and discretized by 400 time steps. The
(level 3) finite element mesh was constructed out of 13,481 nodes and 26,400 elements.
Table 6.1 presents the material parameters for the flow simulation and their values.
The fluid density is practically constant, as the temperature varies between 281 K and
297 K. The constitutive relation for the density is simplified by
ρ = ρ0 − βT (T − T0 ) .
(6.1)
Density effects also play a minor role as a horizontal section has been studied here.
The compressibility of water and solid matrix is neglected.
Table 6.1: Material parameters for the flow simulation.
parameter
value
thermal conductivity solid phase λs
thermal conductivity fluid phase λ
thermal heat capacity solid phase cρs
specific heat capacity fluid phase c
thermal expansion coefficient fluid phase βT
reference density fluid phase ρ0
reference temperature T0
density solid phase ρs
viscosity fluid phase ν
molecular diffusion coefficient Dc
dispersivities porous medium αl, αt
compressibility skeleton αp
1.7 W/mK
0.6 W/mK
2.324 · 106 J/m3 K
4182 J/kgK
2 · 10−4 1/K
1200 kg/m3
293 K
2800 kg/m3
10−3 Pas
0 m2 /s
3m
0 m2 /N
Figure 6.4 presents the pressure fields and the velocity fields for the top layer.
The coarse scale pressure field already compares well with the pressure field at the fine
scale. The velocity field at the coarse scale is not captured that well at the coarse scale.
This will have a large impact on the results of transport simulations. The adaptive
refinement algorithm is able to correct the velocity field locally. The algorithm restores
the accuracy near the injector well and producer well, where flow velocities are relatively
large and flow lines are curved.
The heat storage problem was represented by a fully coupled set of algebraic equations presented in chapter 3 although non-linear effects are small. The flow criterion
constructs an adapted mesh, which consists of 2,902 nodes and 5,202 elements. The
size of the sets of equations have been reduced by a factor of 4.642, the number of
coefficients in the sparce matrix reduce linearly. The pressure field was resolved by
112
6. Flow simulation
pressure fields
velocity fields
scale 1
scale 2
scale 3
adaptive scaling
7.0E+05
1.1E+06
Figure 6.4: Pressure (Pa) and velocity fields, heat storage system.
113
6.1. Marine system
adaptive multiscale
fine single scale
t = 45.6 days
t = 91.3 days
t = 228.1 days
t = 274.8 days
281
297
Figure 6.5: Temperature (K) fields, heat storage system.
114
6. Flow simulation
the multigrid solver outlined in chapter 4. The first step requires 13 cycles with 2
Gauss Seidel pre-smoothing operations and 2 post-smoothing operations. The convergence factor equals 0.22. A single V-cycle resolves the pressure field in the next
steps because boundary conditions do not change, storage is neglected, density effects
are small, and the mesh does not change. The temperature field was resolved by a
pre-conditioned bi-conjugate gradient solver. The computational work for setting up
the systems of equations has been reduced by a factor of 5.07, because the velocity
field did not change during the simulation and there was no need in constructing a
new data structure for the system matrix and the right hand side vector. The overall
computation time for the fully refined (level 3) simulation has been reduced by adaptive refinement by a factor 4.72. Posing an additional refinement criterion on local
temperature change modifies the mesh per time step and requires the construction of
a new system matrix. This reduces the overall computation time factor to 4.17.
Figure 6.5 shows a sequence of temperature fields, and compares results on the
adaptively refined mesh to results on a (level 3) fine mesh. The upper pictures at
45.6 days and 91.3 days show a temperature increase during the summer period. The
pictures at 228.1 days and 274.8 days show a decrease in temperature during the winter
period. The results show that the convective heat flux is relatively low compared to
the diffusive heat flux by conduction and hydrodynamic dispersion. The temperature
distributions obtained by the adaptive mesh computation compare well with the results
of the fine mesh calculation.
Simulations on the Tarbert formation showed that the multiscale finite element
method based on an oversampled computation of basis functions sometimes produced
basis function weight significantly larger than one or smaller than zero on interpolation points. This was due to the ill-conditioned local problems, which were discussed
in chapter 4. Imposing oscillatory conditions on the local flow problem bounds the
values of the basis functions between zero and one and basis functions remain C 0 continuous over the element boundaries. The continuity property of basis functions on
non-overlapping domains supplies unique intergrid transfer operators for the multigrid
solver. The proposed multigrid solver obtains good results for this application.
6.2
Fluvial system
The bottom 100 ft of the SPE model represents the fluvial Upper Ness formation. The
distribution of porosity and permeability was given by 60×220×50 constant cell values.
Here the problem is reduced to 60 × 220 × 48 cells, to fit a hierarchy of three levels.
The geological model then includes 633,600 cells and its size meets 366 × 671 × 30
m. The flow model generates 3,801,600 tetrahedral elements. The number of nodes
equals 660,569. Two unknown values apply to each node. Coarsening the fine mesh
produces a mesh of 79,200 cells at the first coarse level. This mesh merges to a 9,900
cell mesh on the next coarse level. The degrees of freedom reduces to 23,296 on the
most coarse level. This section presents results for the top layer of the Upper Ness
formation. The geological model is restricted to 60 × 220 cells on the finest level. The
number of elements in the hierarchy of finite element meshes is 13,200, 3,300, and 825,
the number of nodes equals 13,481, 3,441, and 896.
115
6.2. Fluvial system
6.2.1
Geological model
Figure 6.6 shows the porosity field for the fully resolved fluvial Upper-Ness formation,
and Figure 6.7 presents the distribution of the porosity and the permeability for the top
layer of this system. Porosity varies between 0.05 and 0.41, and permeability ranges
from 2.16 · 10−17 m2 up to 1.9 · 10−10 m2 . For the fluvial system the permeability distribution can not be given by a single distribution. High conductivity streaks introduce
heterogeneity at a larger scale than the variation in the background. The permeability
distribution reveals this multiscale nature. The material behavior corresponds to the
behavior given by Table 6.1 for the Tarbert heat storage problem.
670
503
21
16
335
11
366
274
183
168
5
91
0
0
0
0.0
0.5
Figure 6.6: Upper-Ness porosity (-) field.
permeability
0.6
8.5
0.5
frequency (-)
frequency (-)
porosity
10.6
6.4
4.3
2.1
0.4
0.2
0.1
0.0
0.0
0.0
0.1
0.2
porosity (-)
0.3
0.4
0.5
1.0E-18
1.0E-16
1.0E-14
1.0E-12
1.0E-10
permeability (m2)
Figure 6.7: Material property distribution layer 36.
Figure 6.8 presents the porosity and permeability variation over the top layer. Fine
scale parameter values are given on the third scale and upscaled values are given on
116
6. Flow simulation
porosity fields
permeability fields
scale 1
scale 2
scale 3
adaptive scaling
5.0E-02
5.0E-01
1.0E-18
1.0E-10
Figure 6.8: Porosity (-) and permeability (m2 ) fields, thermal energy system.
6.2. Fluvial system
117
the second scale and first scale. Fixed uniform scaling shows that at coarser scales
the high permeability streak is interrupted at a number of locations. Adaptive scaling
captures the high conductivity streak.
6.2.2
Flow model
The flow model simulates a simplified geothermal energy production system. The
producer well extracts hot water from the system, which can be used for heating of green
houses for instance. The source well injects cooled down water into the subsurface.
The simulation considers the top layer of the fluvial system with a horizontal dimension of 670 × 366 m. The natural state of the system prescribes a constant pressure of
2 · 107 Pa, as the layer is positioned at a depth of 2 km, and a constant temperature of
60o C. The natural heat gradient was supposed to be 3o C per 100 m. The simulation
introduces an injector well at horizontal coordinates 84 - 336 m, and a producer well
at the coordinates 588 - 48 m. The distance between producer well and injector well
is 580 m, which is relatively small. For this setup the cold front can be expected after
a relatively short time period in the producer well. The temperature of the produced
water will be low if conduction is not able to restore the temperature in the convective
front. At that state the system becomes less effective or even fails. Injection and production are simulated by constant pressure wells. Injection raises the head by 200 m,
giving a bottom hole pressure of 2.2 · 107 Pa, and extraction lowers the head by 200 m
to a corresponding pressure of 1.8 · 107 Pa. The injected water has a temperature of 10o
C. On the horizontal sides (in plane view) of the model no-flow boundary conditions
apply. On the vertical ends a prescribed pressure condition of 2 · 107 Pa, holds. The
fine scale mesh consists of 13,481 nodes and 26,400 elements. The total simulation time
of 16 year was discretized by 1000 finite difference time steps.
Figure 6.9 shows pressure fields and velocity fields for three fixed grids and one
adaptively refined grid. The pressure field at the coarse scale compares well with the
fine scale (level 3) pressure field. The velocity fields follow from the derivative of the
pressure and are calculated less accurately for this reason. Velocity vectors decrease
in size at the coarse scale because of low permeable disturbance in the high permeable
streak. The proposed refinement indicator detects adjacent elements for which the
normal flux on shared element edges show a large discontinuity, and exceeds the refinement criterion. Adaptivity corrects this result locally and restores the preferential
flow pattern.
Figure 6.10 show the evolution of the cold front. The simulation shows that the
system will be operational for more then 16 years despite the short distance between
injector and producer well. However, production rates are very low due to the heterogeneous structure of the subsurface. The production rate for the layer under investigation
is 0.07 m3 /h per meter filter length. In practice a filter of 60 meter is often used, which
intersects many of these layers. Thermal energy systems become economically feasible
for production rates of 100 to 150 m3 /h and operational periods of 90 years. Figure 6.10
compares temperature fields obtained by a computation on the fine (level 3) scale to
fields on the adaptively refined mesh for a number of time steps. The results compare
well.
118
6. Flow simulation
pressure fields
velocity fields
scale 1
scale 2
scale 3
adaptive scaling
1.8E+07
2.2E+07
Figure 6.9: Pressure (Pa) and velocity fields, thermal energy system.
119
6.2. Fluvial system
adaptive multiscale
fine single scale
t = 4 years
t = 8 years
t = 12 years
t = 16 years
283
333
Figure 6.10: Temperature (K) fields, thermal energy system.
120
6. Flow simulation
The coupled flow and transport problem was solved by the fully coupled system of
equations presented in chapter 3 although non-linear effects are small. This approach
requires setting up a system of algebraic equations for each non-linear iteration, which
apply to all time steps, and solving the linearized sets of equations. The adapted
mesh consists of 3,912 nodes and 6,882 elements. The size of the sets of equations
have been reduced by a factor of 3.452 , the number of coefficients in the sparce matrix
reduce linearly by a factor of 3.45. The pressure field was resolved by the multigrid
solver outlined in chapter 4. The first step requires 17 cycles with 2 Gauss Seidel
pre-smoothing operations and 2 post-smoothing operations. The convergence factor
equals 0.31. A single V-cycle resolves the pressure field in the next steps because
boundary conditions do not change, storage is neglected and density effects are small.
The temperature field was resolved by a pre-conditioned bi-conjugate gradient solver.
The computational work for setting up the systems of equations has been reduced by
a factor of 3.84, because the velocity field did not change during the simulation and
there was no need in constructing a new data structure for the system matrix and the
right hand side vector. The overall computation time for the fully refined (level 3)
simulation has been reduced by the newly developed adaptive multiscale finite element
method by a factor 3.26. This application shows that simulations with an adaptively
refined mesh are computationally efficient. This especially holds for three dimensional
problems. However, their efficiency in reducing the number of unknowns, depends on
the complexity of the structure. The transport simulation shows that the proposed
refinement criterion effectively captures high permeability streaks.
Chapter 7
Conclusions
Simulations with an adaptively refined mesh are computationally efficient, especially for
solving three-dimensional problems. Adaptivity reduces the number of finite element
nodes on smooth parts of the domain. Setting up the coarsened system of equations
requires less operations, the equations that need to be solved require less computer
memory and the equations can be solved faster. However, adaptive techniques modify
the structure of the algebraic equations in time, which produces computational overhead. Flow simulations show that a heuristic threshold on the velocity change over the
flow domain reduces the update frequency, and restores the computational efficiency.
In this thesis meshes are composed out of simplex elements. Without any measurements taken, adaptivity generates hanging grid nodes and non-conforming meshes. The
proposed procedure efficiently restores conformality of meshes by a regularization algorithm and an element merging procedure. The mesh regularization algorithm prevents
the generation of more than one hanging grid point on a non-refined element. Remaining hanging nodes are removed by merging elements, for which a sequential coarsening
approach was introduced. The applied conformal finite element method obtains nodal
based mass continuity, whereas mass fluxes over the internal element boundaries are
discontinuous in general. The proposed refinement indicator detects adjacent elements
for which the normal flux on the shared element edge shows a large discontinuity, and
exceeds the refinement criterion. Adaptivity improves the accuracy of the velocity field
by refining the mesh locally. A hierarchy of finite element meshes supports the adaptive
formulation. Multiscale basis functions extrapolate the coarse scale results over this
hierarchy of scales. The refinement indicator operates on the finest scale. The proposed
refinement criterion effectively captures complex flow patterns like high permeability
streaks.
The proposed multiscale formulation captures the coarse scale behavior by modified
basis functions. The functions follow from the solution of local flow problems over
patches of simplex elements. Only if the patch aligns with the small scale structure, its
boundaries coincide with the principal directions of the equivalent permeability tensor
on the patch. Single simplex elements do not meet this condition.
A number of periodic test problems show that oversampling the patch domain
obtains best results for the computation of the equivalent coarse scale behavior. As
121
122
7. Conclusions
a consequence however, the multiscale basis functions are no longer continuous over
the domain boundaries and their span increases, if these basis functions are applied
to obtain the coarse grid operator. Simulations on a geostatistically generated field
show that the method sometimes even produces basis function weights substantially
larger than one or smaller than zero on interpolation points. Sampling directly over the
patch of simplex element, and closure by conditions that follow from a dimensionally
reduced flow problems, enforces the compatibility conditions. For this reason closure of
the local flow problems by dimensionally reduced flow problems is preferred. A second
refinement criterion compares oversampled and non-oversampled function values.
Using a patch of elements can already be seen as oversampling, as the node in
the center of the patch is connected to all coarse grid nodes. Pressure-dissipation
averaging approximates the multiscale coarse-mesh operator. The solutions of the
local flow problems follow as combinations of multiscale basis functions. The explicit
computation of the equivalent permeability tensor reduces the number of integration
points needed to compute the coarse scale operator in time. The procedure does not
introduce new connectivities on the coarse scale. Moreover the procedure supports
parallel computing.
The computation of a full equivalent permeability tensor reduces the process dependency of the variable. However, the equivalent permeability is a non-unique parameter
and depends on the boundary conditions posed on its local domain. Standard multiscale finite element methods interpolate coarse scale results obtained on the global
domain to the fine scale, and can not use these global results on the boundaries of the
local domains. Domain decomposition techniques correct the solution of the local problem iteratively. Alternatively local-global scaling updates the equivalent permeability
on subdomains with updated global information. In this thesis local refinement of the
mesh corrects the pre-scaled results. The accuracy of the equivalent permeability is
improved by refining the local mesh and relating the equivalent behavior on all scales
to the finest grid permeability directly.
Applications of the newly developed adaptive multiscale finite element method show
more accurate solutions of transport problems posed on highly heterogeneous and channelized system than solutions found on uniformly coarsened grids when compared to
fine grid results.
A hierarchical formulation of coarse grid operators, as proposed by standard multigrid methods, provides less accurate equivalent material behavior compared to a direct
multiscale approach. The sequential closure of local flow problems by directional lumping causes these inaccuracies. The proposed multigrid solver obtains good results, in
the sense of operation counts, for the simplified heat storage and energy production
applications. The intergrid transfer operators follow from multiscale weight functions
obtained on non-overlapping subdomains. The computation of the coarse grid operator applies the pressure-dissipation averaging result also found on non-overlapping
subdomains.
Appendix A
Continuum mechanics
Continuum theory aims to describe relationships between gross phenomena and averages the structure of the material at a smaller scale [85, 108]. The material of a
continuum is supposed to be indefinitely divisible. Continuum mechanics applies to
multiscale systems. For multiphase systems the method of volume averaging can be
used to derive continuum equations [67, 123]. The technique applies spatial smoothing
to translate microscopic equations, which are valid within a particular phase, to macroscopic equations, which are valid throughout the continuum. Homogenization theory
assumes that macroscopic equations remain valid on the mezoscale and produces equivalent material behavior on this scale. In a porous medium, Navier-Stokes transport
equations, obtained by continuum theory on the microscale, are valid in the pores.
The equations are subjected to boundary conditions at the solid-liquid interface. The
method of volume averaging gives a local volume averaged transport equation which
corresponds to Darcy’s law, if inertial effects are disregarded. Homogenization theory
assumes Darcy’s law to remain valid on the mezoscale and computes an equivalent
permeability on this scale.
This appendix discusses the concept of continuum mechanics. Tensor calculus first
presents a coordinate free notation. The microscopic formulation for transport problems presents microscopic balance equations. The macroscopic formulation uses volume averaging and translates the equations to the macroscopic scale. The mezoscopic
formulation applies homogenization [129] and gives three requirements for upscaling.
These requirements produce three approaches for upscaling.
Tensor calculus
Tensor analysis will be used for the mathematical description of the mechanical and
thermal behavior of the porous medium continuum. It applies a coordinate free vector
and tensor notation.
The lowest order tensor is a zero order tensor or scalar. A first order tensor or
vector has a length and a direction along a working line. Two vectors are identical if
they have parallel working lines, are pointing in the same direction and have the same
length. The vector u can be written as a unique combination of components related
123
124
A. Continuum mechanics
to three independent base vectors {e1 , e2 , e3 }. In index notation, the vector can then
T
be expressed as u = ui ei . In matrix-vector notation the vector reads u = [u] [e],
T
or alternatively u = [e] [u]. The components of the vector follow from the inverse
expression as dot-product of the vector and the coordinate base vectors. These components read ui = u·ei . In matrix-vector notation the column of vector components is
given by [u] = u· [e], or [u] = [e] ·u. The scalar product, dot-product or in-product of
two vectors given by u·v = ui vj ei ·ej reads
u·v = kuk2 kvk2 cosϑ,
(A.1)
where ϑ is the enclosed smallest angle and k.k2 denotes the L2 -norm which gives the
length of a vector. With respect to a Cartesian vector base ei ·ej = δij and u·v = ui vi .
Here δij is the Kronecker delta function: δij = 1 for i = j and δij = 0 for i 6= j.
The dyadic-product, tensor-product or open product operator of two vectors given
by uv = ui vj ei ej provides a linear mapping. The mapping of a vector w onto r is
given by
uv·w = r.
(A.2)
T
The dyad vu forms the conjugate or transpose of uv and will be written as (uv) . A
second-order tensor also transforms any vector into another vector. This transformation
or linear mapping is written as
A·u = v.
(A.3)
With respect to a vector base of three independent vectors {e1 , e2 , e3 }, the tensor is
given in index notation as A = Aij ei ej . In three spatial dimensions it expresses a
sum of 9 dyadic-products. The matrix-vector notation reads A = [e]T [A] [e]. Here
[e] is a column of vectors and [A] is the matrix containing the tensor coefficients.
The inverse relation gives the matrix of the tensor with respect to the vector base as
T
Aij = ei ·A·ej . In matrix-vector notation this expression reads [A] = [e] ·A· [e] . In
general the dot-product can be written as A·u = ei Aij ej ·ek uk . For a Cartesian vector
base A·u = Aij uj ei since ej ·ek = δjk . A symmetric tensor is specified by A = AT .
With respect to an orthonormal vector base, its components satisfy Aij = Aji . The
components of an antisymmetric tensor A = −AT , then satisfy Aij = −Aji .
The dot-product of two tensors A·B = ei Aij ej ·ek Bkl el follows from
(A·B) ·u = A· (B·u) .
(A.4)
This expression should hold for all u. For a Cartesian vector base the dot-product of
two tensors reads A·B = ei Aik Bkj ej . The scalar product or double dot-product of
two tensors will be written as
A:B = tr (A·B) ,
(A.5)
where tr(.) gives the first invariant or trace of a tensor specified as tr (A) = ei ·A·ei =
Aii . The scalar product of two tensors for a Cartesian vector base reads A:B = Aij Bji .
The inverse of a regular tensor A will be written as A−1 . If det (A) 6= 0, then
A·A−1 = I,
(A.6)
125
where I = δij ei ej denotes the unit tensor. For an orthogonal tensor AT ·A = I holds.
The deviatoric part of A will be indicated by AD , which is given as
AD = A − 31 tr (A) I.
(A.7)
The gradient operator ∇ will be introduced for a differentiable quantity A as
∂A
= A (x + dx, t) − A (x, t) ≡ dx·∇A.
(A.8)
∂x
For Cartesian coordinates the gradient operator can be written as ∇ = ∂/∂xi ei . Here
the Laplace operator ∇2 reads ∂ 2 /∂x2i ei . The gradient operator produces a vector
∇a if it works on a scalar a, and produces a dyadic-product ∇v if it operates on a
vector field v. For a Cartesian vector base ∇a = ∂a/∂xi ei and ∇v = ∂vj /∂xi ei ej .
The divergence of the velocity field ∇·v is the scalar field, given by the trace of the
dyadic-product as
∇·v = tr (∇v) .
(A.9)
dA = dx·
For a Cartesian vector base ∇·v = ∂vj /∂xi ei ·ej = ∂vi /∂xi .
The divergence of a tensor field ∇·A is a vector field, such that for any vector v
the following relation holds
(∇·A) ·v ≡ ∇· (A·v) − tr AT ·∇v .
(A.10)
For a Cartesian vector base ∇·A = ∂Aij /∂xi ej .
The material time derivative DA/Dt of a differentiable quantity A expresses the
rate of change of a material particle. For a small time step dt this material derivative
reads
DA
dt = A (x0 , t + dt) − A (x0 , t) .
(A.11)
Dt
Here the quantity A can be a scalar field, a vector field or a tensor field. The path
line or motion of every particle is given by x = x(x0 , t) with x0 = x(x0 , t0 ), denoting the original position of the material point. The displacement field is given
by u = x(x0 , t) − x, and the differential quantity of a material point is written as
A(x0 , t) = A(x0 (x, t), t) = A(x, t). The velocity of the particle v is given by the material time derivative of the position vector v = Dx/Dt. The spatial time derivative
of a differentiable quantity ∂A/∂t reads
∂A
dt = A (x, t + dt) − A (x, t) .
(A.12)
∂t
The relation between the material time derivative and the spatial derivative thus reads
DA
∂A
=
+ v·∇A,
Dt
∂t
(A.13)
where v·∇A is the convective derivative. For a tensor-function f = f(A) the time
derivative of the function reads
∂f
∂f ∂A
=
:
.
(A.14)
∂t
∂A ∂t
This expression gives the rate of change at a fixed point in space.
126
A. Continuum mechanics
Microscopic formulation
Continuum mechanics at the microscopic level assumes that physical properties like
density and pressure are continuously distributed in space and there exist an infinitesimally small material point as the material is regarded as indefinitely divisible. The
microscopic level considers one phase only, however a number of miscible chemical
species may define the material of this phase. The mass density ρα [ml−3] of the
mixture of the α-phase is given by
α
α
ρ =
nk
X
ρα
k,
(A.15)
k=1
−3
where ρα
k [ml ] is the mass of the constituent k per unit volume of phase α, and
α
nk [−] is the number of miscible chemical species that compose the phase. The mass
density of the phase mixture in general also depends on the pressure p [ml−1t−2 ] and
temperature T [K] in the phase. The mass fraction or mass-based concentration of
species k in the α-phase denoted by ωkα [−] is given as
ωkα =
ρα
k
.
ρα
(A.16)
Conservation of mass requires that the sum of the species mass fractions is one, or
Pnαk α
k=1 ωk = 1.
Global conservation laws or global balance equations apply to a material volume for
which the surface moves with the same speed as the material points. The Lagrangian
or material description of the continuum is associated with global conservation laws.
A spatial volume or control volume is not bounded by material points. If this volume
is fixed in space with respect to a reference frame, it is called a fixed control volume
or fixed partial volume and associated to the Eulerian or spatial description of the
continuum. Local conservation laws can be derived from global conservation laws.
Local balance equations are valid in every point of the volume.
The formulation of conservation equations will be based on Reynolds’s transport
theorem, which uses extensive and intensive quantities. Extensive quantities like mass
M , momentum M v and energy E are proportional to the amount of material under
consideration. Derived intensive quantities are mass per unit of mass (unity), momentum per unit of mass (velocity v), and energy per unit of mass (specific energy Ee ).
Pressure p, temperature T and density ρ are intensive quantities as well.
The transport theorem relates an extensive scalar or vector quantity Ψ(t) to the
corresponding continuous and differentiable intensive quantity ψ(x, t). For a given
mass M at time t the transport theorem is formulated as
Z
Z
Ψm (t) =
ψ(x, t)dM =
ρ(x, t)ψ(x, t)dV,
(A.17)
M (t)
Vm (t)
where dM denotes differential mass and dV is a differential volume, ρ is the mass
density, and Vm is the material volume. The rate of change of the extensive quantity
127
over the material volume reads
Z
Z
dΨm
Dψ
∂ρψ
=
+ ∇· (ρψv) dV =
ρ
dV,
dt
∂t
Vm (t)
Vm (t) Dt
(A.18)
where the results of the local equation for conservation of mass is used already. The
microscopic equation for conservation of extensive mass (Ψ = M ) follows according to
the transport equation for a unit intensive quantity (ψ = 1). The local equation, which
is valid for every arbitrary volume, for conservation of mass reads
Dρ
+ ρ∇·v = 0.
Dt
(A.19)
Conservation of momentum follows from the transport equation, where the extensive quantity is momentum (Ψ = M v) and the intensive quantity is momentum per
unit mass (ψ = v). The second conservation equation reads
ρ
Dv
− ∇·σ − ρg = 0,
Dt
(A.20)
where σ [ml−1t−2 ] is the (Cauchy) stress tensor and g [lt−2 ] denotes the gravitational
body force. The global equation for conservation of moment of momentum states
that the rate of change of moment of momentum of a material volume is equal to the
resulting moments applied by an external force. The local equation for conservation
of moment of momentum simply states that the stress tensor should be equal to its
transpose, formulated as σ = σ T .
The first law of thermodynamics states that the rate of change of internal kinetic
energy of a material volume is equal to the mechanical power performed by external
loads and the supplied heat per unit of time. This first law on a microscopic level is
written as
DEe
− ρr − σ:Λ + ∇q e = 0,
(A.21)
ρ
Dt
where Ee [−] denotes the specific internal energy, r [−] is the specific radiation density, and q e [−] denotes the heat flux vector at the surface. The rate of deformation tensor Λ [t−1 ] denotes the symmetric part of the velocity gradient vector
Λ = 12 (∇v)T + (∇v) .
As an example, the constitutive relations for a Newtonian fluid will be given and
the local momentum balance equation will be derived for an incompressible fluid. For a
Newtonian fluid, or linear viscous fluid the fluid stress associated with motion depends
linearly on the instantaneous value of the rate of deformation.
Water and air are unable to sustain shear stresses without continuous deformation (notion of fluidity). Incompressible fluids like water undergo neglectable density
changes under a wide range of loads, whereas compressible fluids like air show large
density changes. For fluid substances the total stress tensor is decomposed in a viscous
stress part and a pressure part. The constitutive relation between the stress tensor,
pressure and rate of deformation reads
σ = −pI + λtr (Λ) I + 2µΛ.
(A.22)
128
A. Continuum mechanics
Here µ is called the first coefficient of viscosity or simply viscosity, λ+ 23 µ relates the viscous mean normal stress to the rate of change of volume and is known as the coefficient
of bulk viscosity. Using this relation in the momentum balance equation (A.20) gives
the Navier-Stokes equation for an incompressible fluid where trΛ = 0. This relation
reads
∂v
ρ
= ρg − ∇p − ρv· (∇v) + µ∇·∇v.
(A.23)
∂t
For flow through (unfractured) porous media the acceleration term ρv· (∇v) generally
vanishes as velocities are relatively small.
Macroscopic formulation
This subsection deals with continuum mechanics of the multi-phase porous medium
system. The porous medium under investigation consists of a solid phase (the soil
skeleton) and two immiscible fluid phases: a liquid phase (water) and a gas phase
(air). Porous medium phases will be denoted by the superscript α. For the multiphase system, the solid structure is indicated by α = s, the liquid phase is identified
by α = l and the gas phase is denoted by α = g. Each phase consists of miscible
chemical components called phase species. A species will be indicated by a subscript k.
Species that pass through a phase interface are regarded as separate, phase-pertinent
constituents. According to this definition, the total number of chemical species in the
system nk equals the sum of the species per phase nα
k and is written as
α
nk =
n
X
nα
k.
(A.24)
α=1
At the macroscopic scale the phases are described as overlapping continua. An
averaging technique produces the macroscopic equations and uses generalized functions
to identify the portions of space. An intensive quantity of a constituent of the liquid
phase, ψkl for instance, is multiplied by a generalized function which has a value one
in the liquid phase and a value zero in the other phases. Leijnse [88] introduces three
averaging operators for an intensive quantity: the volume averaging operator hψk iv , the
α
intrinsic volume averaging operator hψk iv and the intrinsic mass averaging operator
α
hψk im . The definition of the volume averaging operator reads
Z
1
hψk iv =
ψk dV,
(A.25)
δV δV
where δV is the volume of the representative elementary volume. The intrinsic volume
average operator averages the quantity over the α-phase only, as
Z
1
α
hψk iv =
ψk dV,
(A.26)
δV α δV α
Here δV α denotes the volume of the α-phase within the representative elementary
volume. The intrinsic mass averaging operator weights the intensive quantity by the
129
density of the α-phase and reads
hψk iα
m =
1
α
hρk iv δV α
Z
ρk ψk dV.
(A.27)
δV α
The relation between a volume averaged quantity and an intrinsic volume averaged
quantity for a solid-fluid system is written as
hψk iv =
δV s
δV f
hψk isv +
hψk ifv .
δV
δV
(A.28)
f
If the constituent exists in the fluid phase only, then hψk iv = n hψk iv , where the volume
fraction n [−] denotes (volumetric) porosity. In general the volume fraction φα [−] of
an α-phase can be expressed as
δV α
,
(A.29)
φα =
δV
P α
where α φ = 1. For a solid-fluid system the following relation holds
φs = 1 − n,
φf = n.
(A.30)
If the liquid phase and the gas phase form the fluid phase, then the volume fraction of
the fluid phase reads φf = φl + φg . The part of the pores that is filled with the liquid
phase is denoted by the saturation S l , and S g specifies the part of the pores filled with
the gas phase. For the saturation the following relations hold S l + S g = 1, 0 ≤ S l ≤ 1,
0 ≤ S g ≤ 1. The volume fraction of the liquid phase and the gas phase can now be
written as
φl = nS,
φg = n (1 − S) ,
(A.31)
where S = S l [−] denotes the degree of saturation, which equals the volume of void
space occupied by the liquid relative to the volume of the void space.
The reformulated Eulerian description of the microscopic mass balance equation (A.19)
for a species in the fluid phase reads
∂ρfk
+ ∇· ρfk v fk = 0.
(A.32)
∂t
The macroscopic mass balance equation follows from the microscopic mass balance
equation as
Z ∂ f
1
f
f
f
φ hρk iv + ∇· φf hρk iv hv k im +
ρk v fk − v fs ·nfs = 0, (A.33)
∂t
δV Γ
where Γ denotes the boundary of the elementary volume δV , v fs denotes the velocity of the fluid-solid interface, and nfs is the outward pointing normal vector. The
microscopic mass density of a phase is formed by the sum of its constituent densities
Pnαk α
Pnαk
α
α
as ρα = k=1
ρk . The macroscopic mass density reads hρiv = k=1
hρk iv . In the
formulation of the macroscopic mass balance equation, the mass averaged velocity of
a fluid was used. This velocity reads
Pnαk
nα
f
f
k
X
f
f
f
k=1 hρk iv hv k im
hvim =
=
hωk im hv k im ,
(A.34)
Pnαk
f
hρ
i
k v
k=1
k=1
130
A. Continuum mechanics
α
α
where the macroscopic mass fraction follows from hωk iα
m = hρk iv /hρiv , the sum of all
Pnαk
α
constituents within a phase reads
k=1 hωk im = 1 and inversely the concentration
α
α
α
α
ck ≡ hρk iv = hωk im hρiv . In contrast with the volume averaged velocity, the mass
averaged velocity is a physically meaningful quantity. In the absence of mass exchange
at the interface the velocity of the fluid-solid interface will be zero, and the macroscopic
mass balance equation (A.33) can then be written as
∂ f f f
φ ρ ωk + ∇· φf ρf ωkf v f + ∇·j fk = 0,
∂t
(A.35)
f
where the macroscopic fluid density ρf = hρk iv and the macroscopic mass averaged
f
velocity of the fluid phase reads v f = hvim . The non-convective flux reads
j fk = φf ωk ρ v k − v f .
(A.36)
f
Here the intrinsic mass average velocities, of the constituent is written as v k = hv k im .
Instead of the fluid velocity, the bulk velocity for the fluid phase or filtration velocity
q f is often used in computational subsurface flow formulations. The bulk velocity
expresses the fluid velocity v f relative to the macroscopic solid phase velocity v s as
(A.37)
q f = φf v f − v s .
Mezoscopic formulation
The sum of the mass fraction over all species in the liquid phase equals one, and the
sum of the non-convective terms equals zero, according to
α
nk
X
α
ωkα
= 1,
k
n
X
jα
k = 0.
(A.38)
α
Using these expressions, the steady state macroscopic mass balance equation for the
fluid reads
∇· (ρq) = 0,
(A.39)
where the index of the fluid has been dropped. Darcy’s law gives an empirical relation
for the bulk velocity on the macroscale, which reads
q=−
kr
K (∇p − ρg) .
µ
(A.40)
Here kr denotes the relative permeability, µ indicates the dynamic viscosity of the fluid,
and K is the intrinsic permeability of the porous medium. Darcy’s law holds for the
macroscale and follows from the Navier-Stokes equation on the microscale. The flow
equation results from the macroscopic mass balance equation and Darcy’s law as
ρkr
∇·
K· (∇p − ρg) = 0.
(A.41)
µ
131
This equation holds on the macroscopic scale and is supposed to hold on the mezoscale
as well. Homogenization techniques scale macro-scale permeabilities to mezo-scale permeabilities for the flow problem. Coarse-scale permeabilities mutually relate spatially
averaged pressure, flux and dissipation. Their values are derived from the spatially averaged continuity equation, Darcy’s law and the dissipation equation. Zijl and Trykozko
[129] proposed three requirements for upscaling: conservation of driving force, volumetric conservation of mass continuity, and conservation of energy dissipation per unit
volume. A combination of these requirements produces three approaches for upscaling:
pressure-flux averaging, pressure-dissipation averaging and flux-dissipation averaging.
Only for periodic porous media these approaches yield the same coarse-scale permeability.
Numerical homogenization introduces an averaging operator. The spatial volumeaveraging operator h.i, reads
Z
1
hfi =
fdΩ,
(A.42)
V Ω
R
where V = Ω 1dΩ denotes the volume of subdomain Ω.
The first homogenization requirement gives an expression for the upscaled pressure,
and considers conservation of driving force by pressure differences only. This expression
follows from the requirement on the averaged coarse-scale pressure that should match
the volume averaged fine-scale pressure distribution as pk−1 = pk . The superscript
k denotes the fine scale and k − 1 denotes the coarse scale. As ∇ pk = ∇pk ,
conservation of driving force reads
−∇pk−1 = −∇pk .
(A.43)
The second requirement expresses the upscaled flux components as fine-scale flux
components, by considering (volumetric) conservation of mass continuity. This requirement reads
(A.44)
∇·q k−1 = ∇·q k .
The second requirement states that the continuity equation for the upscaled flux equals
the steady state continuity for the fine-scale flux. The upscaled flux components
q k−1 ·ei should match the spatially averaged flux components q k ·ei . The averaged
flux components follow from Darcy’s law as q k ·ei = −K·∇pk ·ei , where p equals
pressure minus the hydrostatic pressure, and the mobility is assumed to be constant.
The third requirement considers conservation of energy dissipation per unit volume
−∇pk−1 ·q k−1 = −∇pk ·q k .
(A.45)
According to this condition the dissipation equation on the coarse scale matches the
averaged dissipation equation on the fine scale. The dissipation equation expresses
the rate of irreversible conversion of mechanical energy to internal energy within the
upscaling domain (per unit volume).
In general three requirements over-specify the upscaling problem, and only two out
of three requirements can be satisfied simultaneously. The first averaging approach,
called pressure-flux averaging, combines the first and second requirement as
D
D
E
E
∇p(a)k K̃ = q (a)k ,
(A.46)
132
A. Continuum mechanics
3 ........................................................................ 4
•.............
•
..
..
... ....
....
....
...
.
...
.
...
....
...
...
...
....
...
...
...
...
...
....
...
...
...
....
...
...
...
...
.
...
.
....
...
...
....
...
.
... ...
...
.
.
........
...
.
.....................................................................
3 ........................................9.................................. 4
•.............
1
7
•
•
•
•
4
4
3
5
3
•
1
1
1
•
...
...
...
.... ...
... ....
... ..
...
....
...
.... ....
.
.
.
.
.
.
.
...
....
.
..
....
... .... ......
...
... .. ..
...
..
....................................................................................
.
.
.
.
.
...
...
.
.
.. ... .....
.
..
.
...
.
.
...
.... .... ......
...
...
....
...
.
.
.
..
.
...
.
.... ...
.... .......
....
... ...
... .....
...
..
.............................................................................
2
1
•
•
8
2
1
•
2
6
•
2
Figure A.1: Triangular parent and child element patches.
where a = (1 . . . nd ) indicates the loading case, and K̃ denotes the equivalent permeability tensor.
The second averaging approach combines the first requirement and third requirement and is named pressure-dissipation averaging. The set of algebraic equations for
this approach reads
D
E
D
E D
E
−∇p(a)k ·K̃· −∇p(b)k = −∇p(a)k ·q (b)k ,
(A.47)
here both (a) and (b) indicate loading cases.
Flux-dissipation averaging forms the third averaging approach, and combines the
second and the third requirement as
E
D
E D
E
D
−1
(A.48)
q (a)k ·K̃ · q (b)k = −∇p(a)k ·q (b)k .
All three expressions present a set of four algebraic equations in two dimensions, and
give nine equation in a three-dimensional space.
Homogenization theory applies to (structurally and functionally) periodic fields.
For these fields analytical expressions of the effective coarse scale permeability are
available. This makes the fields convenient for comparison [59, 82, 129]. Numerical homogenization techniques approximate the effective behavior of these fields by imposing
periodic boundary conditions. Periodic boundary conditions for the patch of elements
depicted by figure A.1 follow from
p2 = p1 + ∆px
p8 = p7 + ∆px,
p3 = p1 + ∆py p9 = p6 + ∆py ,
p4 = p1 + ∆px + ∆py .
(A.49)
These relations reduce the set of equations, which followed from the finite element
133
discretization of the steady state flow problem, as


a11
a15
a16
a17

a22
a25
a26
a28 



 
a
a
a
a37 
33
35
39

 p1


a
a
a
a
44
45
49
48

 
a51 + a52 + a53 + a54 a55 a56 + a59 a57 + a58  p5 

 p6 

a61 + a62
a65
a66
a67 + a68 

 p7

a71 + a73
a75 a76 + a79
a77 



a82 + a84
a85 a86 + a89
a88 
a93 + a94
a95
a99
a97 + a98
  

Q1
Q2   −a22 − a28

  

Q3  
−a33 − a39 
  

Q4   −a44 − a48
x
−a44 − a49 
  

 + −a52 − a54 − a58 −a53 − a54 − a59  ∆py . (A.50)
=
Q
5
  
 ∆p
Q6   −a62 − a68

  

Q7  
−a73 − a79 
  

Q8  −a82 − a84 − a88
−a84 − a89 
Q9
−a94 − a98
−a93 − a94 − a99
Continuity of flow suggests
Q1 + Q2 + Q3 + Q4 = 0,
Q6 + Q9 = 0,
Q7 + Q8 = 0 Q5 = 0.
(A.51)
Finally, these conditions reduce the set of algebraic equations to four, from which the
fine scale pressure unknowns p5 , p6 , and p7 can be calculated given a coarse scale
pressure p1 . The remaining pressures result from equation (A.49).
In this section numerical homogenization by pressure-dissipation averaging will be
applied for a number of periodic fields. Multiscale basis functions reconstruct the
pressure fields of local flow problems. The basis functions follow either from an oversampling procedure or from closure by oscillating conditions of the subdomain. Appendix B includes the multiscale basis functions for all fields. The pressure fields will
be compared with fields that follow from closure by periodic loading conditions.
Figure A.2, A.3, A.4 and A.5 show the local flow fields for the horizontally layered
cell, the checker board cell, the cell with an inclusion and the diagonally layered cell.
The upper row of plots show the reconstructed flow fields that follow from summing
basis functions obtained by the oversampling procedure. Here the sample domain was
extended by a zone with a thickness of a complete periodic cell. The middle row of
plots shows a reconstruction by basis functions that follows from the oscillating closure
procedure. Results from both procedures can be compared with local flow fields that
follow from loading with periodic boundary conditions. These results are given by
the lower plots. The effective permeability follows from homogenization theory and
the equivalent permeability follows from pressure-dissipation averaging. Equivalent
permeability is obtained on a number of discretization scales, and gathered in Table A.1,
A.2, A.3, and A.4.
134
A. Continuum mechanics
Table A.1: Equivalent permeability for the horizontally layered cell.
nodes
kxx
5.05 · 10
−1
kxy
kyx
kyy
0
0
1.98 · 10−2
periodic
25
81
−1
5.05 · 10
5.05 · 10−1
−16
−9.71 · 10
−8.31 · 10−16
−15
−1.09 · 10
−1.76 · 10−15
1.98 · 10
1.98 · 10−2
oversampled
25
81
5.05 · 10−1
5.05 · 10−1
−1.11 · 10−15
−1.59 · 10−15
−1.11 · 10−15
−1.53 · 10−15
1.98 · 10−2
1.98 · 10−2
oscillating
−2
Table A.2: Equivalent permeability for the checkerboard cell.
nodes
kxx
1.00 · 10
−1
kxy
kyx
kyy
0
0
1.00 · 10−1
periodic
25
81
−1
4.07 · 10
2.99 · 10−1
−4
−7.57 · 10
−3.83 · 10−4
−4
−7.57 · 10
−3.83 · 10−4
3.85 · 10
2.99 · 10−1
oversampled
25
81
4.34 · 10−1
3.87 · 10−1
−9.90 · 10−3
−3.12 · 10−2
−9.90 · 10−3
−3.12 · 10−2
4.34 · 10−1
3.87 · 10−1
oscillating
−1
Table A.3: Equivalent permeability for the domain with single inclusion.
nodes
kxx
kxy
kyx
kyy
5.92 · 10−1
0
0
5.92 · 10−1
periodic
25
81
−1
6.31 · 10
5.92 · 10−1
−16
−4.92 · 10
−7.78 · 10−16
−16
−4.36 · 10
−2.21 · 10−15
6.31 · 10
5.92 · 10−1
oversampled
25
81
6.42 · 10−1
6.10 · 10−1
−6.45 · 10−16
−1.05 · 10−15
−6.70 · 10−16
−8.87 · 10−16
6.42 · 10−1
6.10 · 10−1
oscillating
−1
Table A.4: Equivalent permeability for the diagonally layered cell.
nodes
kxx
kxy
kyx
kyy
2.62 · 10
2.42 · 10
2.42 · 10
2.62 · 10−1
periodic
25
81
−1
2.72 · 10
2.65 · 10−1
−1
2.34 · 10
2.39 · 10−1
−1
2.34 · 10
2.39 · 10−1
2.72 · 10
2.65 · 10−1
oversampled
25
81
5.82 · 10−1
4.68 · 10−1
5.44 · 10−1
4.42 · 10−1
5.44 · 10−1
4.42 · 10−1
5.82 · 10−1
4.68 · 10−1
oscillating
−1
−1
−1
−1
135
oversampled
G(1)
G(2)
G(3)
G(4)
oscillating
G(1)
G(2)
G(3)
G(4)
periodic
G(1)
G(2)
G(3)
G(4)
-0.1
1.1
Figure A.2: Pressure distribution on a horizontally layered field.
oversampled
G(1)
G(2)
G(3)
G(4)
oscillating
G(1)
G(2)
G(3)
G(4)
periodic
G(1)
G(2)
G(3)
G(4)
-0.1
1.1
Figure A.3: Pressure distributions on a checkerboard field.
136
A. Continuum mechanics
oversampled
G(1)
G(2)
G(3)
G(4)
oscillating
G(1)
G(2)
G(3)
G(4)
periodic
G(1)
G(2)
G(3)
G(4)
-0.1
1.1
Figure A.4: Pressure distribution on a domain with inclusions.
oversampled
G(1)
G(2)
G(3)
G(4)
oscillating
G(1)
G(2)
G(3)
G(4)
periodic
G(1)
G(2)
G(3)
G(4)
-0.1
1.1
Figure A.5: Pressure distribution on a diagonally layered field.
Appendix B
Finite element method
The finite element method is a numerical method for solving differential equations
[10, 70, 127, 128, 14]. The method applies the concept of piecewise approximations
and interconnected elements in a discretization step. The elements may have simple
triangular shapes or tetrahedral shapes. Elements are connected through nodes. For
the case of linear elements, the nodes are located in the corners of the elements. Compatibility conditions need to be satisfied when connecting element boundaries. After
discretization, local matrix expressions are developed, which relate the nodal variables
of each element. Next, the element matrices are assembled to form the global set
of equations. Boundary conditions are incorporated into the resulting global matrixvector equation. Finally, the global set of equations has to be solved.
This appendix introduces the finite element method, which applies to the weak
formulation for a differential equation. Numerical integration in space and integration
in time are briefly discussed, followed by coordinate transformation for vector and
tensor properties. Basis functions and weighting functions are given for a linear element
in global coordinates. Isoparametric elements present the basis functions on local
coordinates and provide a convenient mapping to global coordinates. Next, the element
hierarchy will be presented for simplex elements. This hierarchy will be used for the
construction of multiscale basis functions.
Finite element formulation
A general partial differential problem introduces the finite element method. This problem can be written as
L (u) = f
on Ω,
(B.1)
where Ω is the domain, and u = u(x, t) denotes the exact solution of the unknown,
which can be a function of space x and time t. For nonlinear problems the operator
L and the force vector f may be functions of space and time but may also depend
on the unknown u. The finite element method applies a trial function û(x, t) that
approximates the unknown function u. In this thesis the trial functions will be written
as
û(x, t) = ua (t)Na (x),
a = 1, . . . , nn ,
(B.2)
137
138
B. Finite element method
where Na are linear independent basis functions or shape functions, which do not depend on time. The number of independent basis functions is expressed by nn . The basis
functions are specified over the entire solution domain and need to satisfy the essential
boundary conditions. In general, the trial function will not satisfy equation (B.1) and
a residual r can be calculated as
r = L (û) − f.
(B.3)
The method of weighted residuals calculates the trial function coefficients ua such
that the residual r is minimized in some sense. Hence, a weighted integral of the
residual is formed over the entire solution domain, and a linear combination of linearly
independent trial functions is considered. The weighted residual integral is then set to
zero according to
Z
Z
Wa rdΩ =
Wa (L(û) − f) dΩ = 0,
a = 1, . . . , nn .
(B.4)
Ω
Ω
The residual is orthogonalized to nn linear independent weight functions Wa . The
Galerkin weighted residual method chooses the weight functions identical to the basis
functions. The basis functions are defined piecewise in each element, which restricts
integration over these non-overlapping subdomains. Basis functions or shape functions
are in fact interpolation functions over an element. The interpolation functions are
associated with a single node of the element. The conformal finite element method
specifies compatibility conditions, which require that interpolation functions need to
be C0 -continuous over the element boundaries and the following condition holds
Na (xb ) = δab ,
(B.5)
where δab denotes the Dirac delta function, which has a value 1 if a = b and 0 if a 6= b.
The global coordinate of node b is written as xb . The global set of algebraic equations
then follows from the assembly of local sets. This process is written as
Z
Z
ne
Wae (L(Nb ub ) − f)e dΩ,
(B.6)
Wa (L(Nb ub ) − f) dΩ = A
Ω
e=1
where ne is the number of elements, and
Ωe
ne
A denotes the assemblage operator.
e=1
This
expression presents the global set of simultaneous equations for a, where b = 1, . . . , nn .
Weak formulation
An essential part of the finite element method is the weak formulation of a mathematical
problem. This weak formulation will be discussed in more detail for an elliptic test
problem. Hughes [70] posed this problem as
∂2u
= −f,
∂x2
−
∂u(0)
= h,
∂x
u(1) = g,
(B.7)
where f is a smooth scalar-valued function defined on the closed unit interval [0, 1]
and f(x) is a real function in [0, 1]. The interval [0, 1] is the domain of the problem,
139
which will be indicated by Ω and < is the range. The strong, or classical form of the
boundary-value problem is stated as: given f : Ω → < and g and h, find u : Ω → <,
such that equation (B.7) is satisfied.
At first two classes of functions need to be characterized in order to define the
weak or variational form of the strong formulation. The first class is composed of trial
solutions û, which need to satisfy û(1) = g. The derivatives of these trial solutions are
required to be square integrable, so
Z
1
0
∂ û
∂x
2
dΩ < ∞.
(B.8)
H 1 -functions from a Hilbert-space satisfy this condition. If two functions û and w are
square integrable themselves, then their in-product exists, so
Z 1
ûwdΩ < ∞.
(B.9)
0
The set of trial functions U now consists of all functions which have square integrable
derivatives and have a value g on x = 1. This selection is written as
U = û|û ∈ H 1 , û(1) = g ,
(B.10)
where û is a member of the set. The collection of weighting functions has a homogeneous
boundary condition for x = 1. This collection W will be written as
W = w|w ∈ H 1 , w(1) = 0 .
(B.11)
A weak formulation of the strong form given by equation (B.7) is found by multiplying both sides with a smooth (interpolation) function w and by integration over the
domain. This weak form reads
Z 1
Z 1
∂2u
w 2 dx = −
wfdx.
(B.12)
∂x
0
0
Application of Green’s theorem reduces the order of differentiation and gives a lower
order weak form, written as
Z 1
Z 1
∂u ∂w
dx =
wfdx + w(0)h.
(B.13)
0 ∂x ∂x
0
The weak formulation is stated as: given f : Ω → <, g and h, find u : Ω → <, such that
equation (B.13) is satisfied. This formulation is often called the virtual work equation
or virtual displacement equation in mechanics. It should be noted that equation (B.12)
and equation (B.13) are both weak forms of equation (B.7). However, equation (B.13) is
preferred in finite element formulations because the required order of the interpolation
functions is lower.
For non-homogeneous natural boundary conditions Green’s theorem introduces a
boundary integral. For the case of a multi-dimensional problem Green’s theorem
140
B. Finite element method
rewrites the integral over the weighted divergence of a flux field with components
vi in two steps. Integration by parts in the first step gives
Z
Z
Z
∂vi
∂
∂w
w
dΩ =
(wvi ) dΩ −
vi
dΩ.
(B.14)
∂xi
∂xi
Ω
Ω ∂xi
Ω
The second step applies Gauss theorem on the first term in the right hand side, so that
equation (B.14) becomes
Z
Z
Z
∂vi
∂w
w
dΩ = −
vi
dΩ + wvi ni dΓ,
(B.15)
∂xi
∂xi
Ω
Ω
Γ
where ni are the components of a vector on the boundary of the domain, pointing
outside.
Coordinate transformations
Coordinate transformations facilitate the finite element formulation of multidimensional partial differential equations. The mapping of a two-dimensional vector g given
in global coordinates (gx , gy ) into its local expression (gξ , gη ) can be given by
gξ
cos α sin α gx
=
,
(B.16)
gη
− sin α cos α gy
where α is known as an Eulerian angle. According to figure B.1 cos α = dx/dξ = dy/dη
and sin α = dy/dξ = −dx/dη. More general, the Jacobian matrix transforms the
η ...
y
.
...
......
.......
...
..
..
..
...
..
...
....
..
...
..
...
..
....
...
..
...
...
..
..
..
...........
... ....
......... .
... ...
.........
.. ...
.........
.. ..
........
.
.
.
.
.
.
.
.
.. ..
..
.. ..
.........
.... ................
.................................................................................................
ξ
α
x
Figure B.1: Coordinate transformation.
coordinates from the global system (x, y) to the local system (ξ, η) as
∂x
 ∂ξ

[J] ≡ 
 ∂x
∂η


∂y
∂ξ 
cos α sin α

.
=
− sin α cos α
∂y 
∂η
(B.17)
141
The back transformation is achieved by the inverse of the Jacobian matrix. This inverse
matrix follows from


∂ξ ∂η
 ∂x ∂x  cos α − sin α


−1
[J] ≡ 
.
(B.18)
=
sin α cos α
 ∂ξ ∂η 
∂y ∂y
By definition the matrix product of both gives the unit matrix. The vector transformation from the local system (ξ, η) to the global system (x, y) and backward reads
ξ −1 T x
g = J
[g ] ,
T
[gx ] = [J]
ξ
g .
(B.19)
In index notations the relationship between the vector components in global coordinates
gix into its local coordinate components giξ is expressed by
giξ = gjx
∂ξi
,
∂xj
gix = gjξ
∂xi
.
∂ξj
(B.20)
The conductivity tensor K has orthotropic properties. The principal conductivities
are given along their principal directions. These directions are turned against the global
Cartesian coordinate system by rigid body rotation. If the conductivity matrix is a
diagonal matrix, then its components contain the conductivities along the principal
directions, and a rotation matrix maps this matrix into global Cartesian coordinates.
The transformation between coordinates (x, y) and (ξ, η) may be written as
Kξη Kξη
cos α sin α Kxx Kxy cos α − sin α
=
,
(B.21)
Kηξ Kηη
− sin α cos α Kyx Kyy sin α cos α
or more general by
ξ −1 T
[K x ] J −1 ,
K = J
T
[K x ] = [J]
ξ
K [J] .
(B.22)
In index notations the relations between the tensor components in local rotated coorξ
x
dinates Kij
and global coordinates Kij
read
ξ
x
Kij
= Kkl
∂ξi ∂ξj
,
∂xk ∂xl
ξ
x
Kij
= Kkl
∂xi ∂xj
.
∂ξk ∂ξl
(B.23)
Two-dimensional anisotropy rotates the principal axes into the global Cartesian coordinates by a given angle α between the first principal axis of the permeability tensor
and the first Cartesian direction.
Isoparametric elements
Isoparametric elements support a convenient method to obtain the integrals over the
elements by mapping the global element to a local element. Figure B.2 illustrates the
mapping of a triangular element from global to local coordinates. The transformation
142
B. Finite element method
ξ
12 .......
...
......
•.............................
...........
..
............
..
...........
..
...........
..
...........
..
............
..
..
.
..
...
..
...
..
...
.
..
.
.
..
.
.
.
..
.
..
...
..
...
..
...
..
...
..
.
.
.
..
..
..
...
..
...
..
...
..
...
..
.
.
.
.. ..
.. ...
.....
.
∗
y
...
......
....
...
...
...
...
...
...
....
...........................................................
∗
∗
•
•
•.............
11
... ....
...
...
....
...
....
....
...
....
...
....
...
....
...
....
...
....
....
...
....
...
....
...
....
...
...
.................................................................................................
.
∗
. .
.... ....
.....................................
.... ...
. .
∗
x
3 ........
∗
∗
∗
•
•
1
2
η
10
Figure B.2: Triangular elements, global and local coordinates.
from local coordinates ξ of an isoparametric element to global coordinates x with
vector component xi of the actual element is obtained by xi (ξ) = xai Na (ξ). Na (ξ)
denotes the basis function value at location ξ and subscript a is the node number. This
expression will be written for short as
xi = xai Na .
(B.24)
As an example, the shape functions for a triangular element with local nodal coordinates (ξ, η)1 = (0, 0), (ξ, η)2 = (1, 0) and (ξ, η)3 = (0, 1) read
N1 = 1 − ξ − η,
N2 = ξ,
N3 = η.
(B.25)
According to equation (B.24) the global coordinates (x, y) of a point inside the local
element with local coordinates (ξ, η) follow explicitly from
x
x
x − x1 x3 − x1 ξ
+ 1 .
(B.26)
= 2
y1
y
y2 − y1 y3 − y1 η
The inverse form obtains the local coordinates from the global coordinates. For this
mapping a set of equations needs to be solved.
To obtain global derivatives of the shape functions, the Jacobian matrix and its
inverse have to be calculated. The components Jij of the Jacobian matrix follow from
Jij ≡
∂xj
∂Na
=
xaj .
∂ξi
∂ξi
(B.27)
For a linear triangular element the Jacobian matrix is invariant over the element. This
is for example not the case for a bi-linear quadrilateral element. Components of the
inverse of the Jacobian matrix are expressed as
−1
Jij
≡
∂ξj
.
∂xi
(B.28)
−1
By definition Jik Jkj
= δij where δij is the Dirac delta. For a one-dimensional element
the Jacobian is a scalar J and its inverse is written as 1/J. In multiple dimensions the
143
Jacobian inverse can be found by Cramer’s rule. The global derivatives of the shape
function now read
∂Na
∂ξj ∂Na
−1 ∂Na
=
= Jij
.
(B.29)
∂xi
∂xi ∂ξj
∂ξj
For spatial integration the absolute value of the determinant of the Jacobian matrix
|J| ≡ |det [J]| is needed. For a two-dimensional case the Jacobian determinant reads
∂x1 ∂x2
∂x2 ∂x1
−
.
∂ξ1 ∂ξ2
∂ξ1 ∂ξ2
|J| =
(B.30)
Basis functions
This subsection introduces basis functions on a single level. Figure B.3 presents a linear
and a quadratic line element and their nodal locations. The local shape functions for
1
•...................................................................•..
2
1
•................................•...................................•..
3
2
Figure B.3: Line element, linear and quadratic.
a line element with local nodal coordinates (ξ)1 = (0) and (ξ)2 = (1) are given by
N1 = 1 − ξ,
N2 = ξ.
(B.31)
For a quadratic element with nodal coordinates (ξ)1 = (0.0), (ξ)2 = (1.0) and (ξ)3 =
(0.5), the shape functions read
N1 = 2ξ 2 − 3ξ + 1,
N2 = 2ξ 2 − ξ,
N3 = −4ξ 2 + 4ξ.
(B.32)
The left picture of figure B.4 shows a linear three-node triangular element and
the picture in the right displays a six node quadratic triangular element. The nodal
3 ..
3 ..
•............
•............
... ....
....
...
....
...
....
...
....
...
....
..
...
...
....
...
....
...
...
....
...
....
...
....
...
....
...
..........................................................................
1
•
•
6
2
1
... ....
....
...
....
...
...
...
....
...
....
....
..
...
...
....
...
....
...
...
...
....
...
....
...
....
..
...
..........................................................................
•
•
•
•
4
5
•
2
Figure B.4: Linear and quadratic triangular elements.
point coordinates of the iso-parametric linear element are (ξ, η)1 = (0.0, 0.0), (ξ, η)2 =
(1.0, 0.0) and (ξ, η)3 = (0.0, 1.0). Associated to these nodes the linear shape functions
read
N1 = 1 − ξ − η,
N2 = ξ,
N3 = η.
(B.33)
144
B. Finite element method
The coordinates of the nodes added by the quadratic element are (ξ, η)4 = (0.5, 0.5),
(ξ, η)5 = (0.5, 0.0) and (ξ, η)6 = (0.0, 0.5), and the quadratic shape functions read
N1 = (1 − 2ξ − 2η) (1 − ξ − η) ,
N4 = 4ξη,
N2 = ξ (2ξ − 1) ,
N5 = 4ξ (1 − ξ − η) ,
N3 = η (2η − 1) ,
N6 = 4η (1 − ξ − η) .
(B.34)
Finally figure B.5 presents the linear and quadratic tetrahedral elements. The
ζ
ζ
...
........
...
...
......
..... .......
.
. .. .....
.
.
....
.. ..
.....
.. ....
....
...
....
..
....
...
...
.....
..
.
....
..
..
.
.
....
..
....
...... ....
.
.....
..
...... ..
....
..... ... ..
.
.
....
.
.... .. ...... ....
....
.
.....
.
.
.
.
.
... .... ....
.
.
....
.
.
.... .......................................
.
....
...........
..... ....
...................
.
.
.
.
.
..... ..
.
.
.
.
.
.
.
.
.
.
.
.
.
....... ............................
..........
4
•
•
η
•
3
•
2
•
...
.......
.....
.
......
..... .......
.
. ... .....
.
.
.....
... ....
....
....
... ...
.....
...
....
.....
..
.....
.
.
...
..
....
.
....
...
...... ....
.
.....
...... ...
.....
....
....... .... .. .
....
.. .... .. ....
....
....
.... .... ...
....
...
. ..... .... ......... ...................
.....
.
.....
...
....................
..... ..
...................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..... ..
.....
....................................
10
η
ξ
3
4
•
•
•
7
•
•
1
•
8
•
1
6
9
•
•
5
ξ
2
Figure B.5: Linear and quadratic tetrahedral element.
nodal point coordinates of the linear tetrahedral element are given by (ξ, η, ζ)1 =
(0.0, 0.0, 0.0), (ξ, η, ζ)2 = (1.0, 0.0, 0.0), (ξ, η, ζ)3 = (0.0, 1.0, 0.0) and (ξ, η, ζ)4 = (0.0, 0.0, 1.0).
The shape functions then read
N1 = 1 − ξ − η − ζ,
N2 = ξ,
N3 = η,
N4 = ζ.
(B.35)
The coordinates of the extra nodes introduced by the quadratic element are (ξ, η, ζ)5 =
(0.5, 0.0, 0.0), (ξ, η, ζ)6 = (0.5, 0.5, 0.0), (ξ, η, ζ)7 = (0.0, 0.5, 0.0), (ξ, η, ζ)8 = (0.0, 0.0, 0.5),
(ξ, η, ζ)9 = (0.5, 0.0, 0.5) and (ξ, η, ζ)10 = (0.0, 0.5, 0.5). The shape functions for the
quadratic tetrahedral element are given by
N1 = (1 − 2ξ − 2η − 2ζ) (1 − ξ − η − ζ) , N2 = ξ (2ξ − 1) , N3 = η (2η − 1) ,
N4 = ζ (2ζ − 1) , N5 = 4ξ (1 − ξ − η − ζ) , N6 = 4ξη, N7 = 4η (1 − ξ − η − ζ) ,
N8 = 4ζ (1 − ξ − η − ζ) , N9 = 4ξζ, N10 = 4ηζ.
(B.36)
Multilevel basis functions
The single level basis functions, which were presented in the previous subsection, extend
to multilevel basis function. Figure B.6 shows a one-dimensional parent element and its
two child elements. Node 3 which is present in the refined element case can be used in
the definition of the modified basis function of the parent element. Figure B.7 depicts
the element hierarchy. The coarse scale element denoted by its nodes (1, 2) is divided
into two fine scale elements (3, 2) and (1, 3). The definition of the basis functions
for the fine scale element in coarse scale coordinates outlines the modification of the
basis functions. The basis functions for the coarse scale line element with local nodal
coordinates (ξ)1 = (0) and (ξ)2 = (1) were already given by
N1k−1 = 1 − ξ,
N2k−1 = ξ.
(B.37)
145
1
•.................................◦..................................•..
3
2
1
•................................•...................................•..
3
2
Figure B.6: Parent line element and child elements.
(1, 2)
..
........ ..............
........
........
........
.......
........
........
.
.......
(3, 2)
(1, 3)
Figure B.7: Line element hierarchy.
Here k −1 denotes the coarse scale. For the first fine scale element with nodal locations
(ξ)1 = (0.0) and (ξ)3 = (0.5), the basis functions read
N1k = 1 − 2ξ,
N3k = 2ξ.
(B.38)
For the second fine scale element with nodal locations (ξ)3 = (0.5) and (ξ)2 = (1.0),
the basis functions read
N2k = 2ξ − 1,
N3k = 2 − 2ξ.
(B.39)
Standard coarse scale basis functions Nak−1 can now be written as the sum of fine scale
basis functions Nak . For the linear line element this expression reads
N1k−1 = N1k + 12 N3k ,
N2k−1 = N2k + 12 N3k .
(B.40)
k−1 T
In matrix-vector notation this expression is given by N1
= [P ] N1k . The interpolation matrix [P ] reads
1.0 0.0 0.5
T
[P ] =
.
(B.41)
0.0 1.0 0.5
The interpolation matrix displays the fine scale basis function weights. The multiscale
finite element method aims at a modification of these weight such that it obtains fine
scale accuracy on a coarse scale discretization. The multilevel formulation of basis
functions also applies to higher-order elements and multidimensional elements.
The left picture of figure B.8 presents one triangular parent elements. The rightmost
picture presents four child elements which are derived from an intermediate refinement
phase for which two elements where generated. The hierarchy of elements is also presented by figure B.9. The coarse scale element (1, 2, 3) denoted by its nodes, is first
divided into element (1, 4, 3) and (1, 2, 4). The first of them generates (6, 4, 3) and
(1, 4, 6) and the second intermediate element generates (5, 2, 4) and (1, 5, 4). Standard
coarse scale linear triangular elements basis functions follow from their fine scale counterparts as
N1k−1 = N1k + 21 N5k + 12 N6k ,
N2k−1 = N2k + 12 N4k + 12 N5k ,
N3k−1 = N3k + 12 N4k + 12 N6k .
(B.42)
146
B. Finite element method
3 ..
6
1
3 ..
3 ..
•.............
◦
◦
•
◦
6
4
•
5
•............
•............
... ....
....
...
...
...
....
...
...
...
....
...
...
....
...
...
...
....
...
...
....
...
...
...
....
...
...
...
...
........................................................................
2
1
... ....
....
...
...
...
...
...
....
...
....
....
...
...
...
.... ......
...
....
....
.
...
.
..
....
.
.
...
.
....
... .......
....
... ....
....
.. .....
..
..........................................................................
◦
•
•
◦
4
•
5
6
1
2
... ....
....
...
....
...
....
...
....
...
...
...............................................
....
.. .. ..
.... ... .......
..
.
.
.
....
....
....
....
....
... .......
...
....
... ....
....
...
..
..........
...
......................................................................
•
•
•
•
4
•
5
2
Figure B.8: Triangular parent and two-step child elements.
(1, 2, 3)
c-level
........
.......... ....................
..........
...........
...........
..........
..........
..........
.
...........
(1, 4, 3)
i-level
(1, 2, 4)
..........
..... .........
.....
......
......
......
...
......
...............
......
......
......
.....
.....
......
.
.
.
.
.
.
.
(6, 4, 3)
f-level
(1, 4, 6)
(5, 2, 4)
(1, 5, 4)
Figure B.9: Triangular element hierarchy.
Alternatively the coarse scale basis functions can be found by weighting the fine scale
basis functions with the coefficients of the interpolation matrix. This matrix is given
by


1.0 0.0 0.0 0.0 0.5 0.5
[P ]T = 0.0 1.0 0.0 0.5 0.5 0.0 .
(B.43)
0.0 0.0 1.0 0.5 0.0 0.5
Figure B.10 shows the refinement process of a tetrahedral parent element. In the
8
1
.... .... .......
...
.
...
... ........
..
....
...
...
....
...
....
...
...
.....
...
...
....
...
....
...
.....
...
..
....
...
....
.....
...
...
..
........
... ...... ....... ...
...
...
.
.
.
.
.
.
.
.
.
.
....
...
.....
..
...... ....... ..
...
..
............
...
..
............
... ....
............
............ ... ..
..............
.
.
◦
◦
◦
•
9
7
5
◦
•
◦
◦
•
2
6
8
3
1
.... .... ......
..
...
...
... ........
..
....
...
...
.....
...
....
...
...
.....
...
...
....
..........
...
... .............. ........
.
.
..
.
............ ....
... ...
.
.................
.
.....
...
..
...
.
.
..
.
.
.
...
.... .......
... ......
.. ...... ... ....
.
... ..... . ....... .......... ....
..
...
............. ......
.
...
............
..
...........
... ....
............
............ ..... ...
.............
.
◦
◦
•
•
5
◦
10
9
7
•
◦
◦
•
2
6
4 ........ ......................
4 ........ .....................
4 ........ .....................
10
•...........
•............
•............
•...........
4 ........ ......................
8
3
1
.... .... ......
... .....
...
....
...
..
.....
...
...
...
.......
...
... . ..... ........
....
.........
...
.....
....
...
...
. ................
..
........... ............................................
.
................
.....
... ....
......
.
.
.
..
.
.
.
... ..... ....... ........
.
... ........
.
.
.
.. .. ..
... ....... . ....... .......... .... ....... ....
.
.
.................................. ....... ....... ............ .............
...........
...
.
............
............ .... ....
............ .. ..
......
.
•
◦
•
•
9
7
5
◦
10
•
◦
•
•
2
6
8
3
1
.... .... ......
...
.
...
... ........
..
...
...
...
. .... ..............
....... ....... ...... ...... ........
....
... ................... ........... ..
.....
............. ..
....
...
. .................
..
......... ...................................................
.
.................
.
.
...
....... ... . ...... .......
.
.
....
..
. .
. .......
.. ........
.... ............ ................... ....
... ....... . ....... ... ..... ........ ............. ..
.............................. ......... ....... ........... ................
.
............
........... ... . ....... ........ ....
.
............
............ .... ...
.............
.
•
•
•
•
9
7
5
10
•
•
•
•
•
3
6
2
Figure B.10: Tetrahedral parent and child elements.
first step the coarse element (1, 2, 3, 4) generates two intermediate elements (1, 9, 3, 4)
and (1, 2, 3, 9). The second step splits these elements into (1, 9, 10, 4), (1, 9, 3, 10),
(5, 2, 3, 9), and (1, 5, 3, 9). Finally the third step constructs the elements on the fine
scale: (8, 9, 10, 4), (1, 9, 10, 8), (7, 9, 3, 10), (1, 9, 7, 10), (5, 6, 3, 9), (5, 2, 6, 9), (7, 5, 3, 9)
and (1, 5, 7, 9). Figure B.11 illustrates this process graphically. Standard coarse scale
147
(1, 2, 3, 4)
.................................
....................
.....................
....................
....................
.....................
....................
....................
....................
....................
..
(1, 9, 3, 4)
(1, 2, 3, 9)
.......
.......... ....................
...........
...........
..........
..........
..........
...........
...........
.
(1, 9,...10, 4)
(1, 9,..3,
10)
.
..... ........
.....
.....
.....
.....
.....
.....
....
.
.... .........
.....
.....
.....
.....
.....
.....
.....
.......
.......... ....................
...........
...........
..........
..........
..........
...........
...........
.
(5, 2,....3, 9)
..... .........
.....
.....
....
.....
.....
.....
.....
(1, 5,...3, 9)
.... .........
.....
.....
.....
.....
.....
.....
.....
(8, 9, 10, 4)(1, 9, 10, 8)(7, 9, 3, 10)(1, 9, 7, 10) (5, 6, 3, 9) (5, 2, 6, 9) (7, 5, 3, 9) (1, 5, 7, 9)
Figure B.11: Tetraheral element hierarchy.
linear tetrahedral elements basis functions follow from their fine scale counterparts as
N1k−1 = N1k + 21 N5k + 12 N7k + 12 N8k ,
N2k−1 = N2k + 12 N5k + 12 N6k + 12 N9k ,
k
N3k−1 = N3k + 12 N6k + 12 N7k + 12 N10
,
k
N4k−1 = N4k + 12 N8k + 12 N9k + 12 N10
.
(B.44)
The interpolation matrix which relates the fine scale elements to the coarse scale parent
reads


1.0 0.0 0.0 0.0 0.5 0.0 0.5 0.5 0.0 0.0
0.0 1.0 0.0 0.0 0.5 0.5 0.0 0.0 0.5 0.0

[P ]T = 
(B.45)
0.0 0.0 1.0 0.0 0.0 0.5 0.5 0.0 0.0 0.5 .
0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.5 0.5 0.5
Integration in space
Integral functions need to be evaluated in the finite element formulation of partial
differential equations. Integration of a spatial function f(x) over the two-dimensional
element area Ωe can be written as
Z
Z Z
Z Z
f(x)dΩ =
f(x, y)dxdy =
f(x(ξ, η), y(ξ, η))|J|dξdη,
(B.46)
Ωe
x
y
ξ
η
where the Jacobian determinant supports a coordinate transformation. Numerical
integration approximates this integral. Throughout this thesis numerical integration is
carried out by Gaussian quadrature. In two directions the numerical approximation to
the previous equation reads
Z Z
f(x(ξ, η), y(ξ, η))|J|dξdη
ξ
η
≈
n
m1 n
m2
X
X
i=1 j=1
wi wj f(x(ξ ij , η ij ), y(ξ ij , η ij ))|J|ij = wm (f|J|)m , (B.47)
148
B. Finite element method
where nm1 is the number of integration points in ξ direction and nm2 is the number
of integration points in η direction. In the short form nm = nm1 × nm2 is the total
number of integration points, and wm is the weight of integration point m. Both the
function and the Jacobian determinant are evaluated at the local point m. For simplex
elements, the Jacobian matrix is constant over the element.
Table B.1 presents the interpolation points, their position and weights for a general
line element. According to this table, a single integration point calculates the integral
of a linear function over the line element exactly. Two integration points obtain cubic
accuracy. The sum of the weights equals one for all integration schemes, as the sum
Table B.1: BAR2 element, quadrature point information.
point
ξ
w
order
1
0.50000000
1.0000000
linear
1
2
0.21113249
0.78886751
0.5000000
0.5000000
cubic
corresponds to the length of the isoparametric element.
Table B.2 assembles the integration points for the triangular element and Table B.3 gathers the locations of the quadrature points in a tetrahedral element and
their weights. The integration points weights for triangular elements sum to 0.5 which
Table B.2: TRI3 element, quadrature point information.
point
ξ
η
w
order
1
0.33333333
0.33333333
0.50000000
linear
1
2
3
4
0.33333333
0.20000000
0.60000000
0.20000000
0.33333333
0.20000000
0.20000000
0.60000000
-0.28125000
0.26041667
0.26041667
0.26041667
cubic
corresponds to the area of the local element. For tetrahedral elements the sum equals
0.167, this corresponds to the volume of the local element.
For linear accuracy the integration error reads R = O(h2 ), for cubic accuracy the
error reads R = O(h4 ). Here h is the corresponding length of the mapped global
element.
149
Table B.3: TET4 element, quadrature point information.
point
ξ
η
ζ
w
order
1
0.25000000
0.25000000
0.25000000
0.16666667
linear
1
2
3
4
5
0.25000000
0.16666667
0.50000000
0.16666667
0.16666667
0.25000000
0.16666667
0.16666667
0.50000000
0.16666667
0.25000000
0.16666667
0.16666667
0.16666667
0.50000000
-0.13333333
0.07500000
0.07500000
0.07500000
0.07500000
cubic
Integration in time
The finite element discretization in space generally results in a (non-linear) set of
ordinary differential equations, which can be expressed as
Mab
dub
= Sab ub + Fa ,
dt
(B.48)
where ub denote the components of the unknown, Mab are the components of the
capacity matrix, Sab denotes the conductivity matrix components and Fa are the force
vector components. Finite difference time integration schemes translate the set of
(nonlinear) ordinary differential equations into a set of (nonlinear) algebraic equations.
The result of a linear time integration scheme, can be written as
n+θ
Mab
un+1
− unb
n+1 n+1
n n
b
= θSab
ub + (1 − θ)Sab
ub + θFan+1 + (1 − θ)Fan .
∆t
(B.49)
In this expression n + 1 denotes the next time level, n is the current time level and θ is
n+θ
a time-weighing factor for which 0 ≤ θ ≤ 1. The capacity matrix Mab
is obtained by
a weighted average of its value in the current and the next time step. The conductivity
matrix and force vector are evaluated separately at the current and next time step.
Explicit time integration follows for θ = 0, fully implicit time integration for θ = 1,
and second order accuracy time integration is obtained for θ = 21 .
Large sets of equations are often integrated explicitly and the matrix is diagonalized
by a process called mass lumping. The lumped capacity matrix approximates the
consistent mass matrix as
Z
Z
CNa Nb dΩ ≈
δab CNa dΩ.
(B.50)
Ωe
Ωe
This procedure results in an explicit expression of the unknown approximation even
for nonlinear ordinary differential equations.
If the capacity matrix depends on the unknown in time, and an implicit scheme
was chosen (θ > 0), then the application of the chain rule gives
dMab dub
= Sab ub + Fa .
du dt
(B.51)
150
B. Finite element method
The partial derivative of the capacity matrix coefficients are obtained either tangentially
to the point in time n + θ or follow from their values at the current and new time step.
The tangential form is written as
∂Mab
∂ub
n+θ
u1b − u0b
n+1 n+1
n n
= θSab
ub + (1 − θ)Sab
ub + θFan+1 + (1 − θ)Fan .
∆t
(B.52)
The secant form is expressed as
Mab un+1
− Mab (unb ) dub
b
n+1 n+1
n n
= θSab
ub +(1 −θ)Sab
ub +θFan+1 +(1 −θ)Fan . (B.53)
dt
un+1
− unb
b
In addition to numerical analysis, numerical experiments can show which form is preferred.
Multiscale basis functions
Multiscale basis functions extend multilevel basis functions. Figures B.12, B.13, B.14
and B.15 present multiscale basis functions for a number of periodic fields. The functions capture the fine scale behavior of horizontally layered fields, checkerboard fields,
domains with inclusions and diagonally layered fields on a patch of elements. Multiscale basis functions were obtained by two techniques. The first approach computes
multiscale basis functions on an extended local domain. The domain is closed by linear
boundary conditions, and includes a subdomain of elements that captures a single periodic cell. The second approach closes the local flow problems by oscillating boundary
conditions, which follow from dimensionally reduced flow problems. The permeability
contrast for all periodic problems equals 102 and the finite element mesh over-resolves
the structure. For each periodic case the upper plots give the basis functions on the
oversampled domain. The plots below restrict these basis functions to a subdomain
according to the oversampling approach. The lower plots give the multiscale basis
functions that follow from closure of the local flow problems by oscillating boundary
conditions.
151
linear
N1∗
N2∗
N3∗
N4∗
oversampled
N1
N2
N3
N4
oscillating
N1
N2
N3
N4
0
1
Figure B.12: Multiscale basis functions on a horizontally layered field.
linear
N1∗
N2∗
N3∗
N4∗
oversampled
N1
N2
N3
N4
oscillating
N1
N2
N3
N4
0
1
Figure B.13: Multiscale basis functions on a checkerboard field.
152
B. Finite element method
linear
N1∗
N2∗
N3∗
N4∗
oversampled
N1
N2
N3
N4
oscillating
N1
N2
N3
N4
0
1
Figure B.14: Multiscale basis functions on a field with inclusions.
linear
N1∗
N2∗
N3∗
N4∗
oversampled
N1
N2
N3
N4
oscillating
N1
N2
N3
N4
0
1
Figure B.15: Multiscale basis functions on a diagonally layered field.
Appendix C
Multigrid method
Numerical solutions for sets of algebraic equations are either obtained by direct numerical solvers or by iterative numerical solvers [104, 57]. Gauss-Jordan elimination provides a classical direct solution procedure. Basic iterative solvers are Jacobi’s method
and the Gauss-Seidel method. Conjugate gradient solvers and multigrid solvers are
classified as more advanced iterative solvers, which aim to solve a set of nu equations
in O(nu ) operations [60, 110, 122, 18]. Multigrid solvers improve basic iterative solvers,
and accelerate the relaxation of smooth components of the fine-grid residual by solving
the residuals in the unknowns at a coarser grid.
This appendix introduces a linear multigrid algorithm by a two-grid correction
scheme. The nonlinear multigrid algorithm then generalizes the linear concept. Special techniques must be applied to restore the efficiency of the multigrid solver for
problems with highly heterogeneous field parameters. A number of algebraic multigrid
techniques will be given, which construct an (approximate) Schur complement with
respect to the coarse grid. The Schur complement provides a separation of scales, as
coarse-grid unknowns are related to other coarse-grid unknowns only. Gauss-Jordan
elimination introduces the Schur complement, which provides the exact coarse-scale
operator and optimal interpolation operator. However, a global system of equations
needs to be inverted to obtain the Schur complement in multi-dimensional cases. The
construction of an approximated matrix overcomes this problem. For quadrilateral
elements a directional lumping procedure is often applied.
Multigrid algorithms
Basic iterative solvers like Gauss-Seidel shows poor convergence behavior as the number
of iterations increase. The Gauss-Seidel iteration scheme for a lexicographic ordered
mesh, presented by the picture left in figure C.1, can be written as
ur+1
=−
i
nu
i−1
1 X
1 X
1
Lij ur+1
−
Lij urj +
fi ,
j
Lii
Lii
Lii
j=1
(C.1)
j=i+1
where nu is the number of unknowns, and summation only takes place over j, and
i = 1, . . . , nu . The algorithm takes updated values of the unknowns into account,
153
154
C. Multigrid method
as soon as they become available. In matrix-vector form this equation reads [u]1 =
−1
0
[RG] [u] + LD
+ [f], where [RG] is the iteration matrix and LD denotes the
diagonal matrix. For the Gauss-Seidel method the iteration matrix reads
−1 U LD − LL
L .
(C.2)
L
Here [LU ] is the upper matrix and L is the lower matrix. The performance of the
[RG ] =
11
13
12
14
•..............................................•..............................................•...............................................•...............................................•.......
... ....
.. . ..
.. ..
...
...
...
... .... ......
...
.... ...
....
..
...
....
....
...
...
.... ...
....
....
....
.... ... ......
...
.... ... .....
.
.
.
.
.
...
.
.
... . ..
...
. . ..
.
..........................................................................................................................................................................
..
.
.
.
.
.
..
.
.
...........
.
.. .........
.
.
.
.
....
.
.
.
...
.
.... .. ....
.... .. ....
....
....
.....
... .... ......
.... .... ......
.
.
.
...
..
.
.
.
....
....
.
..
.
.
..
.
.
.
... ......
.
.
.
.
.
... ...
... .. ...
...
.... ...
.... .. ....
.... ......
.....
.
...................................................................................................................................................................
•
•
6
1
•
•
7
2
•
•
8
3
•
•
9
4
•
•
15
10
5
6
7
14
15
•..............................................•...............................................•...............................................•..............................................•........
... ....
... . ..
.. ..
...
...
....
...
... .... ......
.... ...
..
...
...
....
...
...
...
.... ....
....
....
....
.... .... ......
...
.... .. .....
.
.
.
.
...
.
.
... . ..
...
.... .. .....
.. .. ..
.
..
...
..................................................................................................................................................................
.
. ... ....
.. ... .....
.
.
.
.
....
.
.
.
....
.
.
.. ... .....
... ... ......
.
.
.
....
.
.
....
.
....
.
....
..
...
....
..
....
....
....
....
....
....
.
.
.
.
.
... ......
.
.
.
.
.
.
.... ....
... .. ....
...
... .. ...
.
.... .......
.....
.
. .
........................................................................................................................................................................
•
•
11
1
•
•
4
9
•
•
12
2
•
•
5
10
•
•
8
13
3
Figure C.1: Node ordering schemes.
Gauss-Seidel method depends on the mesh ordering. A red-black ordering of the mesh
points improves the convergence speed of the Gauss-Seidel method. The picture right
in figure C.1 presents this alternative node ordering. Jacobi’s method follows from the
Gauss-Seidel method if no updates are used for the unknowns, by this Jacobi’s method
becomes mesh independent. Both Jacobi’s method and Gauss-Seidel method take into
account only the direct neighboring nodes of the unknown ui , and therefore they are
called local operations.
The effectiveness of the scheme in reducing the error dependents on the spatial
frequency of the error components. For basic iterative methods, the convergence behavior deteriorates as iterations continue. The reason for this becomes clear if a discrete
Fourier analysis of the error is elaborated. The relaxation process is very effective on
error components whose periods are small, but is ineffective in smoothing the lowfrequency components of the error. For solving a discrete boundary value problem on
a mesh with characteristic size h, periods of a few h are called high-frequency errors,
and errors with periods greater than about 10h are called low frequency errors. Multigrid methods use a sequence of coarser meshes to speed up their convergence. In a few
iterations basic relaxation methods reduce the high frequency errors on the fine mesh.
The remaining error is then transfered to a coarser grid where it can be reduced more
effectively, as low frequency error components on a fine mesh become high frequency
components on a coarse mesh.
Linear multigrid algorithm
The basic idea of the linear multigrid algorithm will be introduced on two grid levels,
with uniform grid sizes h and H, and standard coarsening H = 2h. The continuous
unknown will be written as u and its discrete approximation on the grids is denoted as
uh and uH . As the multigrid method is an iterative technique, the sequential solution
at iteration r will be written as urh and urH . An iteration from the fine level to the
coarsest level and back to the fine level, called cycle, produces the next iterate.
155
The continuous boundary value problem that resembles a linear flow problem will
be written symbolically as
L(u) = f
on Ω,
(C.3)
where L is the continuous linear elliptic differential operator and f is the source function. On a given uniform grid Ωh with mesh size h, its discrete form is written as
Lh (uh ) = fh
on Ωh ,
(C.4)
with linear elliptic operator matrix Lh , source vector fh , and discrete approximation to
the continuous equation uh . The jet unknown exact solution uh to the set of algebraic
equations will be approximated by urh .
The error in the approximate solution or correction erh reads
erh = uh − urh .
(C.5)
The defect correction scheme improves the approximate solution urh at level h in iteration r. The defect or residual drh follows from
drh = fh − Lh (urh ).
(C.6)
Since Lh is linear, Lh (urh ) = Lh (uh ) − Lh (erh ) holds and therefore the error satisfies
Lh (erh ) = drh .
(C.7)
This equation is known as the defect equation. The approximate solution for the error,
written as êrh , can be found from
L̂h (êrh ) = drh ,
(C.8)
where L̂h is an approximated operator. Multigrid techniques use the approximation of
the operator on the next coarse level LH to calculate the error at that level as
LH (erH ) = drH .
(C.9)
This coarse-scale equation has a smaller dimension and is easier to solve than the finescale problem. To obtain a solution for the error erH on the coarse-scale, first the defect
of the fine grid has to be transfered to the coarse grid. The defect drH on the coarse
grid is found from an intergrid transfer operation which is written as
drH = R(drh ),
(C.10)
here R denotes the restriction operator. This operator is also known as fine-to-coarse
operator, injection operator or extrapolation operator. After solving the defect equation at the coarse grid, the solution of erH has to be transferred back to the fine grid.
The corresponding error êrh at the fine scale is found by interpolation as
êrh = P(erH ),
(C.11)
156
C. Multigrid method
where P denotes the prolongation operator, which is also known as coarse-to-fine operator or interpolation operator. Finally the corrected value of the unknown at the fine
mesh nodes, is found as
ur+1
= urh + êrh .
(C.12)
h
A two-level multigrid iteration, or cycle, results from the two-grid correction step
if it is preceded by a pre-smoothing step on the fine level and completed by a postsmoothing operation on the fine grid. Figure C.2 presents a two grid cycle graphically.
pre-smoothing •.....
restriction
• post-smoothing
..
..
..
..
..
...
..
..
.
.
... ...
... .......
.. ..
... ...
.. ..
.. ..
.. ...
.. ...
......... ...
.
. ... ... ..
.. ...
....
prolongation
◦ coarse grid solution
Figure C.2: Two-grid cycle.
Equation C.13 specifies the call of a single cycle of the linear two-grid
method
LT G(.), which produces a new fine-scale approximation of the unknowns ur+1
given
h
an initial guess of the unknowns [vh ] = [urh ]. Here ν1 accounts for the number of
pre-smoothing operations and ν2 denotes the number of post-smoothing operations.
[uh ] = LT G([vh ] , [Lh ] , [fh ] , ν1 , ν2)
(C.13)
• relax ν1 times on [Lh ] [uh ] = [fh ] with initial guess [uh ] = [vh ]
• compute the fine-grid defect [dh ] = [fh ] − [Lh ] [uh ]
• restrict the fine-scale defect to the coarse grid [dH ] = RH
h [dh ]
• solve the defect equation [LH ] [eH ] = [dH ]
h
• prolongate the coarse-grid correction to the fine grid [êh ] = PH
[eH ]
• correct the fine-scale approximation [vh ] = [uh ] + [êh ]
• relax ν2 times on [Lh ] [uh ] = [fh ] with initial guess [uh ] = [vh ]
0
1
0
Given an initial
2 guess uh 1 the
first cycle produces uh = LT G( uh , . . .), the second
cycle gives uh = LT G( uh , . . .) and so on. The two-grid method is easily extended to
a three-grid method if the defect equation is solved by a recursive call of the two-grid
method. General multigrid methods solve a nested sequence of algebraic equations,
which is labeled by k = 1, . . . , kf . Here k = 1 indicates the coarsest grid and kf
indicates the finest grid. Equation (C.14) generalizes the two-grid method, and gives
a recursive formulation
algorithm LM
of the
linear
multigrid
k G(.) for
a single cycle.
k
Relaxing ν1 times on Lk uk = f k with
initial
guess
u
=
v
has now been
k k k
replaced by the expression uk = LS(
v
,
L
,
f
,
ν
,
ν
).
1 2
On three grid levels, the call u3 = LM G(3, . . .) on the finest grid for k = 3 carries
out a pre-smoothing operation on u3 . It then proceeds to a coarser scale in the
downward stroke, and calls the linear multigrid operator u2 = LM G(2, . . .) before
157
it proceeds to a finer scale along the upward stroke.
This recursive call sets k = 2
and performs a pre-smoothing operation on u2 , then solves the linear equation at
the coarsest grid u1 = LS(1, . . .), and performs a post-smoothing operation on u2 .
The algorithm
leaves the recursive call and proceeds with a post-smoothing operation
on u3 .
uk = LM G(k, vk , Lk , f k , ν1 , ν2)
(C.14)
• downward stroke
• pre-smoothing uk = LS( vk , Lk , f k , ν1)
• compute defect dk = f k − Lk uk
• restrict defect dk−1 = Rkk−1 dk
• introduce vk−1 = [0]
• solve coarse grid equation
• if k > 2 : uk−1 = LM G(k − 1, vk−1 , Lk−1 , dk−1 , ν1 , ν2 )
• if k = 2 : uk−1 = LS( vk−1 , Lk−1 , dk−1 , )
• upward stroke
k k−1
u
• prolongate correction êk = Pk−1
k k k • compute new approximation w = u + ê
• post-smoothing uk = LS( w k , Lk , f k , ν2)
The discrete differential operator at the finest level follows from a finite
element
discretization of the flow equation. Its matrix representation is given by Lkf . The
right hand side vector is not included in the operator and written separately as f kf .
Equation (C.14) provides coarse-grid right-hand side vectors. At the coarser levels
k = 1, . . . , kf−1 however, coarse-grid operators have to be calculated and intergrid
transfer operators have to be quantified.
k The matrix representation of the discrete
prolongation operator is denoted by Pk−1
as it interpolates the result from grid k − 1
to grid k. The matrix representation of the restriction operator is given by Rkk−1 as
it transfers the results from level k to k − 1. The extra computational effort of the
multigrid method over basic iterative methods is limited. For a n-dimensional problem
the relaxation work and memory requirement decrease by a factor 1/2n each time the
grid spacing is doubled. Accordingly, for a three-dimensional problem the memory
requirement only increases with about 15% as 1 + 1/8 + 1/64 + . . . ≈ 1.143.
Non-linear multigrid algorithm
The nonlinear multigrid algorithm known as the Full Approximation Storage (FAS)
algorithm [99] extends the linear multigrid algorithm. Press et al. [99] presented
two complementary viewpoints for the relation between coarse and fine grids for the
FAS algorithm. The first viewpoint poses that coarse grids are used to accelerate
158
C. Multigrid method
the convergence of the smooth components of the fine-grid residuals. The second
viewpoint states that fine grids are used to compute correction terms to the coarsegrid equations. The correction terms project fine-grid accuracy to the coarse grid. The
FAS algorithm provides a stopping criterion and an approximate solution at each grid
level. If the algorithm applies nonlinear smoothing operators, it is capable to solve
nonlinear equations. The nonlinear algorithm generalizes the linear algorithm.
The nonlinear continuous boundary value problem is written symbolically as
N (u) = f
on
Ωh .
(C.15)
The discrete boundary value problem on a uniform grid h reads
N h (uh ) = fh
on Ωh .
(C.16)
The error in the discrete equation will be written as
eh = u − uh .
(C.17)
The two-grid correction scheme uses the approximate solution urh of iteration r at the
fine level h. Following the first viewpoint, the error in the approximate solution erh
reads
erh = uh − urh ,
(C.18)
where uh is the unknown exact discrete solution at mesh h. The discrete equation can
now be written as
N h (urh + erh ) = fh .
(C.19)
The defect equation follows from
N h (urh + erh ) − N h (urh ) = drh ,
(C.20)
where drh = fh − N h (urh ). The approximate solution on a coarse level is found by
solving
N H (urH ) = N H (R (urh )) + R (drh ) .
(C.21)
Now the coarse-grid correction reads
erH = urH + R (urh ) .
(C.22)
The coarse-grid correction modifies the solution at the fine scale according to
ur+1
= urh + P (erH ) .
h
(C.23)
This process is repeated to obtain the next iterate ur+2
h , and so on. A stopping criterion
could be based on the defect norm, written as
kdh k ≤ .
A reasonable choice of can be found from the second viewpoint.
(C.24)
159
This dual view point considers local truncation errors τh that can be regarded as
corrections to the discrete source terms fh
N h (u) = fh + τh ,
(C.25)
where u is the exact solution of the continuum equation. The relative truncation error
that corrects the discrete equation at grid size h to the continuous equation now reads
τh = N h (u) − fh . The truncation error defined on the coarse-grid relative to the fine
grid is written as
τH = N H (R (uh )) − R (N h (uh )) .
(C.26)
Here uh is the unknown exact discrete solution on the fine scale. The relative truncation
error τH corrects the coarse-scale source term fH and makes the solution of the coarsegrid equation equal to the solution of the fine-grid equation. The corrected coarse-scale
equations are expressed by
N H (uH ) = fH + τH .
(C.27)
An approximation of τH follows from the approximate solution on the fine scale urh and
reads
r
τH
= N H (R (urh )) − R (N h (urh )) .
(C.28)
Substitution of equation (C.28) into equation (C.27) gives
N H (urH ) = N H (R (urh )) + [fH − R (N h (urh ))] ,
(C.29)
where the last two terms form the defect dH . If the coarse-grid force vector is obtained
by reduction as fH = R (fh ), then equation (C.29) gives equation (C.21). For a typical
case of second-order accurate differencing Press et all. [99] found τ ' 1/3τhr and hence
a good criterion is
1
= kτhr k.
(C.30)
3
This choice of prevents iterating beyond the point when the remaining error is dominated by the local truncation error.
Equation (C.31) gathers the two-grid nonlinear multigrid scheme. The nonlinear
two-grid algorithm N T G(.) produces a new fine-scale approximation [uh ], given an
initial guess of the unknowns [vh ]. It carries out ν1 pre-smoothing operations and
ν2 post-smoothing operations. The matrix representation of the nonlinear operator
N h is written as [Nh ]. The nonlinear two-grid algorithm N T G(.) restricts the finegrid approximation and the fine-grid defect. It solves the coarse-grid problem and
obtains coarse-grid approximations. Finally it computes the fine-grid correction as the
difference between the restricted fine-grid approximation and the calculated coarse-grid
approximation. The linear two-grid algorithm LT G(.) restricted the fine-grid defect
only. LT G(.) computed the coarse-grid correction by solving the defect equation. It
did not calculate coarse-grid approximations; coarse-grid corrections were calculated
160
C. Multigrid method
instead.
[uh ] = N T G([vh ] , [Nh ] , [fh ] , ν1, ν2 )
(C.31)
• relax ν1 times on [Nh ] [uh ] = [fh ] with initial guess [uh ] = [vh ]
• restrict the fine-grid approximation to the coarse grid [vH ] = RH
h [uh ]
• compute the fine-grid defect [dh ] = [fh ] − [Nh ] [uh ]
• restrict the fine-grid defect to the coarse grid [dH ] = RH
h [dh ]
• solve the coarse-grid problem [NH ] [uH ] = [NH ] [vH ] + [dH ]
• compute the coarse-grid correction [eH ] = [uH ] − [vH ]
h
• prolongate the coarse-grid correction to the fine grid [êh ] = PH
[eH ]
• correct the fine-scale approximation [vh ] = [uh ] + [êh ]
• relax ν2 times on [Nh ] [uh ] = [fh ] with initial guess [uh ] = [vh ]
The multigrid generalization is presented by equation (C.32). This equation presents
the recursive formulation of the nonlinear multigrid iterator N M G(.) for a single Vcycle, depicted by figure C.2, based on the FAS algorithm. The nested sequence of
algebraic equations is labeled k = 1, . . . , kf where 1 indicates the coarsest grid and kf
is the finest grid.
In general the nonlinear multigrid algorithm uses a nonlinear smoother N S(.) to
smooth high frequency errors on a fine mesh, whereas the linear multigrid algorithm
used a linear smoother LS(.). The FAS algorithm presented by equation (C.32) needs
one extra set of vectors to store the right hand side. This was not needed for the
LM G(.) algorithm given by equation (C.14). For a linear operator the nonlinear algorithm resembles the linear algorithm.
The formulation of the linear multigrid iterator LM G(.) and the nonlinear multigrid iterator N M G(.) was presented for a V-cycle schedule. Both methods can easily
be extended to alternative schedules, which are computationally more effective occasionally. Figure C.3 presents the V-cycle next to the W-cycle and the F-cycle. All
cycles should be carried out in sequence until a stopping criterion is met.
level
level
level
level
4
3
2
1
V-cycle
W-cycle
F-cycle
•......
•..
...
..
..
..
..
•.....
•
.
.
...
..
•...... ...•....
.. ..
◦...
•......
•...
..
..
...
•.......
•
•
..
......
.
.
.
.. ...
...
...
•...... ...•........ ...•..... •........ ...•........ ...•....
.... .....
.... ....
◦.. ◦..
◦.. ◦..
.
•......
•
..
..
..
..
..
•......
•
.....
•
.
.
.
. .
..
..
.. ..
•....... ....•....... ...•.... •........ ...•.....
... .. .....
.....
◦.. ◦..
◦..
Figure C.3: Four level cycle schemes.
The efficiency of multigrid is improved by a procedure called ’nested iteration’ for
certain applications. The full multigrid algorithm or nested iteration procedure forms
an extension of the linear multigrid algorithm. The full multigrid algorithm starts on
the coarsest mesh, and needs a discrete force vector as well as a discrete coarse-grid
161
operator. A F-cycle scheme, without pre-smoothing operations on the first decent
supplemented by zero initial solution, produces a full multigrid.
Multigrid operators
The coarse-grid operator for the model problem with homogeneous field functions,
simply follow from a finite element discretization on the coarse level. However, for
multi-dimensional problems the intergrid transfer operations already pose a problem
as fine scale unknowns relate to coarse scale unknowns and other fine scale unknows.
For this reason a concept of separation of variables needs to be posed.
k
u = N M G(k, vk , N k , f k , ν1, ν2 )
(C.32)
• downward stroke
• pre-smoothing uk = N S( vk , N k , f k , ν1 )
• compute defect dk = f k − N k uk
• restrict defect dk−1 = Rkk−1 dk
• restrict approximation vk−1 = Rkk−1 uk
• compute right hand side gk−1 = N k−1 vk−1 + sk+1 dk−1
• solve coarse-grid equation
• if k > 2 : uk−1 = N M G(k − 1, vk−1 , N k−1 , gk−1 , ν1 , ν2)
• if k = 2 : uk−1 = N S( vk−1 , N k−1 , gk−1 , )
• upward stroke
• compute correction ek−1 = uk−1 − vk−1
k k−1 e
• prolongate correction êk = Pk−1
1 • compute new approximation w k = uk + k−1 êk
s
• post-smoothing uk = N S( w k , N k , f k , ν2 )
Gauss-Jordan elimination and the construction of the Schur complement introduce
algebraic multigrid. The solution of a set of algebraic equations with two unknowns
can easily be found by Gauss-Jordan elimination. The set of equations is written as
L11 L12
u1
f1
=
.
(C.33)
L21 L22
u2
f2
Subsequently multiplying the second row by m1 = L12 /L22 , and subtracting the result
from the first row, restates the set of equations as
L11 − L21 L12 /L22
0
u1
f1 − m1 f2
=
.
(C.34)
L21
L22
u2
f2
162
C. Multigrid method
Multiplying the first row by m2 = (L21 L22 ) / (L11 L22 − L12 L21 ) and subtracting the
result from the second row gives
L11 − L21 L12 /L22
0
u1
f1 − m1 f2
=
.
(C.35)
0
L22
u2
f2 − m2 (f1 − m1 f2 )
This direct solution introduces the concept of separation of variables, but it becomes
inefficient for large sets of equations.
Schur complement
Figure C.4 presents a coarse mesh with nodes belonging to Ωk−1 and its refined mesh
with nodes belonging to Ωk . In a multigrid setting, coarse meshes are available at
grid levels k = 1, . . . , kf − 1. At the fine mesh, a global coarse-fine node ordering (c-f
ordering) is proposed. The set of equations which follows from a discretization at the
fine mesh will now be rewritten as two separate sets: one set for the fine nodes also
present in the coarse mesh (c-nodes) and one set for the fine-grid nodes that do not
appear in the coarse grid (f-nodes). For the fine grid shown in figure C.4, nodal point
4
5
•...............................................................................•..................................................................................•......
6
•
1
•
2
•
4
14
15
5
•..............................................•...............................................•..............................................•..............................................•......
.
... ....
.
.. .. ..
.. ..
....
....
...
....
.... ... ......
.... ...
....
....
.
.
....
...
...
...
...
.... .... ......
.... .... ......
...
...
...
.... .. ....
.... .. ....
.
...
.
.
.
.
.
.
.
...................................................................................................................................................................
.
........
...........
.
..
.
.
....
.
.
.
...
... ... ......
... ... .....
..
.
...
.
.
...
.
.
.
.... .... ......
.... .... ......
....
....
...
...
.
...
...
.
.
.
.
.
.
.
...
.
.
.
.
...
.
.
...
.... .. ....
.
.
.
.
.
.
.
.
... ......
....
.... .. ...
.... ....
..........
......
....
.........
....
..............................................................................................................................................
...
.. ..
... ..
...
.... ....
.... ....
...
....
....
..
..
...
...
....
...
...
....
...
.
.
.
...
.
.
.
.
.
...
.
..
...
.
...
.
.
.
.
.
.
...
..
...
..
.
...
.
.
.
.
.
...
.
...
..
..
.
.
.
.
.
.
...
.
.
...
.
..
..
.
.
.
.
.
.
.
.
...
...
.
.
.
.
.
.
.
.
.
.
.
...
.
.
...
.
.... ......
... .......
...
. ....
.. ....
.
...
.................................................................................................................................................
•
3
•
9
1
•
•
10
7
•
•
11
2
•
•
12
8
•
•
6
13
3
Figure C.4: Coarse mesh and c-f ordered fine mesh.
1, . . . , 6 are c-nodes belonging to Ωk−1 ∩ Ωk , and nodal points 7, . . . , 15 are f-nodes
belonging to Ωk \Ωk−1 .
For a discrete problem Lkab ukb = fak on a grid Ωk , the linear set of equations can be
partitioned in coarse-grid entries belonging to Ωk−1 and fine-grid entries that do not
belong to the coarse-grid part Ωk \Ωk−1 as
" 12 # " 1 # " 1 #
L11
L
u
f
21 22 2 = 2 ,
(C.36)
L
L
u
f
where u1 collects the unknowns in c-points and u2 gathers the f-points unknowns.
The block matrix L11 collects c-point to c-point connectivities and the block matrix
22 L
contains f-point to f-point connectivities. The optimal matrix-depended prolongation
12 arises
from
the exact block Gaussian elimination of the outer diagonal blocks
L
and L21 which collect f-node to c-node connectivities and vice versa. This
163
elimination process reads
"
[I]
[0]
−1 # " 11 L
− L12 L22
21 L
[I]
#
12 # "
[I]
[0]
L
22 −1 21 L
− L22
L
[I]
"
#
21 −1
L11 − L12 L22
L
[0]
=
22 . (C.37)
[0]
L
Equation (C.37) presents coarse-grid interactions by the matrix
k−1 11 12 22 −1 21 L
= L − L
L
L .
(C.38)
This matrix forms the Schur complement of Lk with respect to Ωk−1 . For a system
of two unknowns the scalar form was found by Gauss-Jordan elimination Lk−1 =
Lk11 − Lk21 Lk12 /Lk22 .
Alternatively, the same result can be found by variational coarsening. The Galerkin
condition poses the first variational property and reads
k−1 k−1 k k L
= Rk
L Pk−1 ,
(C.39)
with a restriction operator Rkk−1 given by
k−1 h
−1 i
Rk
= [I] − L12 L22
,
k and a prolongation operator Pk−1
expressed as
k
Pk−1
=
[I]
−1 21 − L22
L
.
(C.40)
(C.41)
For symmetric systems the second variational property holds, which states Rkk−1 ≡
k T
Pk−1 . Variational coarsening then also produces the Schur complement.
For a one-dimensional problem, L22 is diagonal and its inverse can easily be found.
In general however, this matrix is not a diagonal matrix, and a set of equations has
to be solved in order to obtain the coarse-grid matrix and intergrid transfer operator
matrices. For this general case, the coarse-grid operator as well as the prolongation
operator and restriction operator loose their sparse nature. To illustrate this, figure C.5
presents a single
triangular
parent
element and its four child elements. The fine-scale
equations read Lk uk = f k , and the discrete fine-scale operator is given by
 k
L11
 0

k  0
L =
Lk
 41
Lk51
Lk61
0
Lk22
0
Lk42
Lk52
0
0
0
Lk33
Lk43
0
Lk63
Lk14
Lk24
Lk34
Lk44
Lk54
Lk64
Lk15
Lk25
0
Lk45
Lk55
0

Lk16
0 

Lk36 
.
Lk46 

0 
Lk66
(C.42)
164
C. Multigrid method
3 ..
3 ..
•.............
•.............
6
1
... ....
....
...
....
...
....
...
....
...
....
...
....
...
...
....
...
...
...
....
...
...
....
...
...
...
....
...
........................................................................
◦
◦
•
◦
4
5
•
6
2
1
... ....
....
...
...
...
....
...
...
..
...
........................................
.
...
.........
...
... ... .....
.
..
.... .... ......
...
....
...
.
.
.
.
...
... .....
....
....
... .....
...
...
...
.......
.
.
.....................................................................
•
•
•
•
4
5
•
2
Figure C.5: Triangular parent and child elements.
The set of equations is decomposed as
"
12 # " 1 # " 1 #
L11
L
u
f
21 22 2 = 2 ,
L
L
u
f
(C.43)
where
 k

L44 Lk45 Lk46
0 .
(C.44)
L22 = Lk54 Lk55
k
L64
0 Lk66
−1
The inverse of this block matrix [A] = L22
has a full structure, and the expression of
its components gets complicated. The sub structuring technique builds super elements.
Approximate Schur complement
To overcome the problem of inverting the matrix with f-node interactions, approximations of the discrete differential operator are used to obtain the coarse-grid operator.
The general approach is to replace the set of equations by a modified set of equations,
for which the inverse matrix can easily be obtained. The coarse-grid operator and intergrid transfer operator then follow from the Schur complement of the approximated
fine-grid operator. It can be shown that the prolongation operator produces an optimal interpolation for the altered discretization, in the sense that variational coarsening
of the approximated coarse-grid operator with this prolongation operator obtains the
same result. However, the coarse-grid matrix does not in general maintain its sparse
structure.
h i h i
The modified set of algebraic equations L̃k uk = f˜k , which follows from the
approximation of the fine-grid discretization, will be written in block matrix-vector
form as
" 11 12 # " # " 1 #
f
L
L
u1
i
i h
h
2 = h 2 i .
(C.45)
22
21
L̃
u
f˜
L̃
For the approximation, firstly the computational costs for calculating the inverse
i
i−1 h
h
i
h
L̃21 should approximate
of L̃22 should be low. Secondly, the product L̃22
22 −1 21 L
L . Thirdly, the interpolation matrix has to be a local operator.
165
The Schur complement of the approximated matrix reads
i−1 h
i
h
Lk−1 = L11 − L12 L̃22
L̃21 .
(C.46)
ih ih
i
h
k
, where
This results follows from variational coarsening as Lk−1 = R̃kk−1 L̃k P̃k−1
the restriction matrix reads
i−1 k−1
h
Rk
= [I] − L12 L̃22
,
(C.47)
and the prolongation matrix is formulated as
"
#
i
h
[I]
k
h
i−1 h
i .
P̃k−1
=
− L̃22
L̃21
(C.48)
h k i
In general, the Galerkin approximation, where Rkk−1 ≡ P̃k−1
, does not obtain the
same approximated Schur complement. Two modifications will be given to illustrate
the approximation procedure for the model problem associated to figure C.5.
Reusken [103] presented the first modification. The equation for fine-grid node 5
reads
L51 u1 + L52 u2 + L54 u4 + L55 u5 = f5 .
(C.49)
A linear interpolation for the unknown in fine-grid point 4: u4 = (u2 + u3 )/2 then
replaces the expression for node 5 by
L54
L54
L51 u1 + L52 +
u2 +
u3 + L55 u5 = f5 .
(C.50)
2
2
Repeating the same procedure for unknowns
u4 and u6 produces an approximated
h
i
22
f-node to f-node connectivity matrix L̃ , which reads
 k
L44
h
i
22

L̃
= 0
0
0
Lk55
0

0
0 .
Lk66
(C.51)
The approximated f-node to c-node connectivity matrix is given by
h

k
L41
i 

L̃21 = 


L45
Lk
+
+ 46
2
2
Lk51
Lk61
Lk
+ 45
2
Lk54
k
L52 +
2
Lk64
2
Lk42
k
Lk46
+
2 


Lk54
 .
2 k 
L 
Lk63 + 64
2
Lk43
(C.52)
Now, the approximate Schur complement as well as the prolongation and restriction
matrices can be calculated straight forwardly.
166
C. Multigrid method
Wagner et al. [120] suggested the second modification. They proposed not to take a
linear interpolation for the f-grid contributions but to take a matrix depended interpolation. The unknown in point 4 was pre-calculated as u4 = (L42 u2 +L43 u3 )/(L42 +L43 ).
Using this expression, the discrete equation for point 5 becomes
L54 L42
L54
L54 L43
L51 u1 + L52 +
u2 +
u3 + L55 u4 = f5 −
f4 .
(C.53)
L42 + L43
L42 + L43
L44
The coarse-scale operator follows from equation (C.46) and the prolongation operator
is computed by equation (C.48). This equation produces an interpolation which is
not dimensionally reduced. Wagner et al. applied their black box multigrid procedure
successfully, solved an elliptic problem (steady state flow equation), a Poisson problem
(transient flow equation) and a ’hyperbolic’ problem (convection dominated transport
equation). However, a disadvantage of both Wagner’s and Reusken’s technique is the
fact that both introduce connectivities on the coarse scale, which are not present at a
direct finite element coarse-scale computation.
Variational coarsening over quadrilateral elements
Dendy [38] proposed directional lumping, based on a nine point finite difference molecule
for regular grids. A directional lumping procedure reduces the interpolation dimension
and generates a more local prolongation matrix. Variational coarsening uses this prolongation matrix, and produces a coarse-grid operator, which is based on the fine-grid
discretization only.
Figure C.6 presents a coarse mesh and a fine mesh with coarse - intermediate fine node ordering (c-i-f ordering). C-nodes denote nodes on the fine grid that are also
4
5
•........................................................................•..........................................................................•.......
6
•
1
•
2
•
4
5
14
15
•......................................•........................................•.......................................•.......................................•.......
.
.
.
...
.
....
....
....
...
....
...
...
...
...
...
...
...
...
...
...
.
.
.
...
.
.
.
.
....................................................................................................................................................
....
....
....
....
....
.
.
.
...
.
.
.
...
...
....
....
....
...
...
...
...
....
...
...
...
...
...
...
...
...
...
...
..
...............................................................................................................................................
...
.
.
...
....
....
...
...
...
...
...
...
...
.
.
...
.
..
.
.
...
...
.
.
.
...
...
.
.
.
...
...
...
....
...
...
...
...
...
...
...
...
..
.
.
..............................................................................................................................................
•
3
•
11
1
•
•
7
9
•
•
12
2
•
•
8
10
•
•
6
13
3
Figure C.6: Coarse mesh and c-i-f ordered fine mesh.
present in the coarse-grid Ωk−1 ∩ Ωk , (node 1, . . . , 6). The points belonging only to the
fine grid Ωk \Ωk−1 , will be called either f-nodes (node 9, . . . , 15) or i-nodes (node 7 and
8). For a quadrilateral mesh, the i-nodes are interpolated from f-nodes and c-nodes,
only when the f-node values are known. The approximation aims to relate f-nodes to
c-nodes only. For this reason the fine-scale equation has to be approximated by
1


11 12 13  
1
 
L
L
L
f
u
 " 21 # " 22 23 #   " 2 #   " 2 # 
L
L
f
L

,

u
(C.54)
=
h i

h
i
h
i



3
3
33
31
u
f˜
L̃
0
L̃
167
where the subvector u1 collects unknowns at the c-nodes, u2 contains unknowns at
the i-nodes, and u3 collects unknowns at the f-nodes. In this approximation f-node
to i-nodes connectivities, expressed by the block matrix L32 , are canceled out. The
Schur complement of the approximated matrix reads
" 22 23 #−1 " 21 #
L
L
k−1 11 12 13 hL i
h
i
L
= L −
L
L
.
(C.55)
33
L̃31
0
L̃
Alternatively, the Schur complement can be obtained by variational coarsening that
incorporates the Galerkin approximation. The lumping procedure will be explained for
the mesh of quadrilateral elements shown in figure C.7. Here c-nodes 1, . . . , 4, f-nodes
4, . . . , 8 and i-node 5 are the nodes on the fine mesh. The equation for f-nodes laying
4 ........................................8............................... 3
4 ..........................................8.............................. 3
9
7
9
2
1
1
•.....
◦
◦
◦
•
◦
•
..
...
...
...
...
...
...
...
...
...
....
...
..
...
...
...
...
...
...
...
...
...
...
...
...
..
.......................................................................
5
6
◦
•
•.....
•
•
•
•
•
•
..
..
...
...
...
...
...
...
...
...
...
...
...
...
...
..
.
.
............................................................................
.
....
...
...
.
..
.
...
....
....
...
....
...
...
...
...
...
...
...
..
........................................................................
5
6
•
•
7
2
Figure C.7: Quadrilateral parent and child elements.
on a horizontal boundary is altered by vertical lumping. This will be demonstrated for
node 6 in figure C.7. The algebraic equation for u6 reads
L61 u1 + L62 u2 + L65 u5 + L66 u6 + L67 u7 + L69 u9 = f6 ,
(C.56)
and the lumped expression is given by
(L65 + L66 )u6 + (L61 + L69 )u1 + (L62 + L67 )u2 = f6 .
(C.57)
In the same way horizontal lumping is applied to modify the equations for f-nodes
laying on a vertical boundary. The approximated f-node to f-node matrix reads
 k

L66 + Lk65
0
0
0
h
i 

0
Lk77 + Lk75
0
0
.
L̃33 = 
(C.58)
k
k


0
0
L88 + L85
0
k
k
0
0
0
L99 + L95
The f-node to c-node matrix is approximated by
 k
L + Lk69 Lk62 + Lk67
0
h
i  61
k
k
k
0
L
+
L
L
+
Lk78
31
72
76
73
L̃
=
k

0
0
L83 + Lk87
k
k
L91 + L96
0
0

0

0

k
k .
L84 + L89
Lk94 + Lk98
(C.59)
The f-node to i-node matrix is replaced by the zero matrix. Matrix dependent prolongation operator coefficients for node 6 are found to be P61 = (L61 + L69 )/(L66 + L65 )
168
C. Multigrid method
and P62 = (L62 + L67 )/(L66 + L65 ). In the next step, f-node expressions are used
to calculate the expression for i-node 5. The contribution of L56 u6 is replaced by
L56 (L61 + L69 )/(L66 + L65 )u1 + L56 (L62 + L67 )/(L66 + L65 )u2 and so on. The expression for node 5 is finally given by
L91 + L96
L61 + L69
+ L56
u1
L51 + L59
L99 + L95
L66 + L65
L62 + L67
L72 + L76
+ L52 + L56
+ L57
u2
L66 + L65
L77 + L75
L83 + L87
L73 + L78
+ L58
u3
+ L53 + L57
L77 + L75
L88 + L85
L84 + L89
L94 + L98
+ L54 + L58
+ L59
u4 + L55 u5 = f5 . (C.60)
L88 + L85
L99 + L95
From this expression the prolongation operator coefficients are derived. Fine-grid unknowns follow from coarse-grid unknowns only, and interpolation does not have to be
done in a prescribed order.
Variational coarsening over triangular elements
This subsection proposes a new variational coarsening procedure over triangular elements. The procedure generalizes directional lumping over simplex elements, by adding
basis functions and conserving the flux over an inner element boundary. Figure C.8
presents isolines of the basis functions N2 and N3 , and the newly constructed basis
function Ñ1 , which follows from Ñ1 = N1 + N4 . These basis functions reduce the dimensions of the flow problem by directional lumping. The dimensionally reduced flow
3 ..
3 ..
•.............
•.............
•
1
•
4
•
N
•
2
3 ..
•.............
N
... ......
3
.... .....
....
.....
... ... ......
.
. ..
.......
.. ......
...
..
......
... ....
.... .......
... ..
....
....
.
...
...
.....
.
.
....
.
.... ......
....
... .....
....
..
........
.....................................................................
... ....
...
...
....
...
....
...
....
...
....
...
...
....
...
.... ....
...
2
.... .........
.
...
.
...
...
.
.. .......
.
..
. .
... ......
..
.. .......
... ....
.
.
..
..
.. ..
.. .....
.................................................................................
1
•
... ......
.... .....
....
1
.....
... ... ......
..
.
.. ...
.......
.. ........
...
.
... ...
... ....
... ...
... ..
.... ...........
...
.....
....
..
...
.... ......
..
.. .......
..
..
... .....
...
..
..
.
.
.. ....
.......
.........................................................................
4
•
•
2
1
•
4
N + N4
•
2
Figure C.8: Lumped fine element basis functions.
h i problem L̂k ûk = [0] relates the pressure in node 4 (or 1) to the pressures in nodes
2 and 3. The associated connectivity matrix reads

 k
L̂11 L̂k12 L̂k13
h i
(C.61)
L̃k = L̂k21 L̂k22 L̂k23  ,
k
k
k
L̂31 L̂32 L̂33
169
where the components in the first row, are given by
Z
kr ρ ∂N1k
∂N4k
∂N1k
∂N4k
k−1
L̂11
=
Kij
+
+
dΩ
µ
∂xi
∂xi
∂xj
∂xj
Ω
Z
∂N4k ∂N2k
kr ρ ∂N1k
k−1
+
dΩ
L̂12 =
Kij
µ
∂xi
∂xi
∂xj
Ω
Z
kr ρ ∂N1k
∂N4k ∂N3k
k−1
+
dΩ,
L̂13
=
Kij
µ
∂xi
∂xi
∂xj
Ω
second row components read
k−1
L̂21
k−1
L̂22
k−1
L̂23
kr ρ ∂N2k ∂N1k
∂N4k
=
Kij
+
dΩ
µ ∂xi
∂xj
∂xj
Ω
Z
kr ρ ∂N2k ∂N2k
dΩ
=
Kij
µ ∂xi ∂xj
Ω
Z
kr ρ ∂N2k ∂N3k
=
Kij
dΩ,
µ ∂xi ∂xj
Ω
Z
and third row components are written as
Z
kr ρ ∂N3k ∂N1k
∂N4k
k−1
+
dΩ
L̂31
=
Kij
µ ∂xi
∂xj
∂xj
Ω
Z
kr ρ ∂N3k ∂N2k
k−1
dΩ
L̂32
=
Kij
µ ∂xi ∂xj
Ω
Z
kr ρ ∂N3k ∂N3k
k−1
L̂33
=
Kij
dΩ.
µ ∂xi ∂xj
Ω
(C.62)
Interpolation weights follow from the first row of the set of algebraic equations as
k−1
k−1
k
P42
= L̂12
/L̂11
,
k−1
k−1
k
P43
= L̂13
/L̂11
,
(C.63)
or in components of the original matrix as
k
P42
=
Lk11
Lk12 + Lk42
,
+ Lk41 + Lk14 + Lk44
k
P43
=
Lk11
Lk13 + Lk43
.
+ Lk41 + Lk14 + Lk44
(C.64)
k
For a homogeneous element standard interpolation weights are found as P42
= 1/2
k
and P43 = 1/2. In general, the modified basis function Ñ1 gives an expression for
the continuity of flow across the line which divides the coarse element into two child
elements. Variational coarsening produces the coarse scale conductivity matrices as
k−1 k−1 k k L
= Rk
L Pk−1 .
(C.65)
A multistep refinement procedure splits a single triangular coarse element into four
child elements. For this reason the coarsening procedure is carried out in two steps.
170
C. Multigrid method
3 ..
3 ..
•.............
K
6 ?
4
4
.......
•
•
..•
.... .. ...
... .... .......
.
.
.
? K K ...... ... ...... K
...
1.....
2
..
.. ? .... ? .......
....
.
.
•
•...................................•.....................................•...
1
3 ..
•.............
... ....
....
...
....
...
.... 4
...
....
...
........................................
..
...
...
..
....
...
...
....
...
.
.
.
3
.... .......
... .....
...
1
5
2
•.............
... ....
...
...
....
...
....
...
....
...
....
...
...
...
.......
...
.... ......
....
...
....
.
.
...
...
....
....
... ......
....
... ...
....
.
. .
.............................................................................
6 ?
◦
?
1
•
•
?
◦
4
?
5
•
6
2
1
... ....
...
...
....
...
....
...
....
...
....
...
....
....
...
....
...
....
...
....
...
....
...
....
...
....
...
.
.......................................................................
◦
•
?
?
◦
?
◦
4
?
5
•
2
Figure C.9: Multistep coarsening of triangular elements.
Figure C.9 illustrates this sequential coarsening procedure. In the first step, element
(6, 4, 3) and element (1, 4, 6) are mapped on the intermediate element (1, 4, 3). Element
(5, 2, 4) and (1, 5, 4) produce the second intermediate element (1, 2, 4). In the next step,
element (1, 4, 3) and (1, 2, 4) merge into the coarse-grid element (1, 2, 3).
The assemble of element (6, 4, 3) and (1, 4, 6) produce a local matrix for which the
column and row numbers correspond to 1, 3, 4, 6. Variational coarsening maps this
conductivity matrix to a new conductivity matrix with global numbering 1, 3, 4, which
corresponds to the intermediate element (1, 4, 3). This variational coarsening step reads
k−1
L11
k−1
L
21
k−1
L31

k−1
L12
k−1
L22
k−1
L32

1
= 0
0
k−1 
L13
k−1 
L23
k−1
L33
lc
0 0
1 0
0 1
 k

L11
R14
 0
R24 
Lk31
0 lc
Lk41
0
Lk22
Lk32
Lk42
Lk13
Lk23
Lk33
Lk43
 
Lk14
1
 0
Lk24 
 
Lk34   0
Lk44 lc P41
0
1
0
P42

0
0
 . (C.66)
1
0 lc
If Kxx = Kyy and Kxy = Kyx = 0, then the conductivity matrix, which relates to
element (1, 3, 4), and follows from variational coarsening and R16 = P61 = K3 /(K3 +
K4 ), R36 = P63 = K4 /(K3 + K4 ), reads


K
0
−K3
k∗ 3,4
ρkr  3
0
K4
−K4  .
(C.67)
L
=
2µ
−K3 −K4 K3 + K4
These findings will be interpreted by constructing the coarse grid matrix for equivalent
permeabilities. The coarse grid analogy for element (1, 3, 4) reads


k∗3,4 ρkr hKxx i + hKyy i hKxx i − hKyy i −2 hKxx i
hKxx i − hKyy i hKxx i + hKyy i −2 hKxx i
L
=
4µ
−2 hKxx i
−2 hKxx i
4 hKxx i
 v

v
v
v
v
K + Kyx −Kxy
+ Kyx
−2Kyx
ρkr  xy
v
v
v
v
v 
+
Kxy
− Kyx
−Kxy
− Kyx
2Kyx
. (C.68)
4µ
v
v
−2Kxy
2Kxy
0
For this case the averaged permeability coefficients read hKxxi = hKyy i = (K3 + K4 )/2
and hKxy i = hKyx i = (K3 − K4 )/2.
171
For element (1, 2, 4), the conductivity matrix follows from variational coarsening
and R15 = P51 = K1 /(K1 + K2 ), R25 = P52 = K2 /(K1 + K2 ). This matrix reads


K1
0
−K1
k∗ 1,2
ρkr 
L
=
0
K2
−K2  .
(C.69)
2µ
−K1 −K2 K1 + K2
The coarse grid analogy for element (1, 2, 4) reads


hKxx i + hKyy i − hKxx i + hKyy i −2 hKyy i
k∗1,2 ρkr
− hKxx i + hKyy i hKxx i + hKyy i −2 hKyy i
L
=
4µ
−2 hKyy i
−2 hKyy i
4 hKyy i
 v

v
v
v
v
K + Kyx
Kxy − Kyx
−2Kxy
ρkr  xyv
v
v
v
v 
−Kxy + Kyx
−Kxy
− Kyx
2Kyx
+
. (C.70)
4µ
v
v
−2Kyx
2Kyx
0
Here the averaged components read hKxx i = hKyy i = (K1 + K2 )/2 and hKxy i =
hKyx i = (K1 − K2 )/2. Components of the conductivity matrix for element (1, 2, 3)
follow from both intermediate conductivity matrices, variational coarsening and setting
R24 = P42 = K2 /(K2 + K4 ), R34 = P43 = K4 /(K2 + K4 ).
The lumping process is repeated over the second step, where element (1, 4, 3) and
element (1, 2, 4) merge into the coarse-grid element (1, 2, 3). The equivalent coefficients
finally read
K2 K2 (K1 + K3 )
K2 K4
+
,
2 (K2 + K4 ) 2 (K2 + K4 ) (K2 + K4 )
K2 K4
K4 K4 (K1 + K3 )
=
+
,
2 (K2 + K4 ) 2 (K2 + K4 ) (K2 + K4 )
K2 K4
K2 K4 (K1 + K3 )
=−
+
.
2 (K2 + K4 ) 2 (K2 + K4 ) (K2 + K4 )
v
Kxx
=
v
Kyy
v
v
Kxy
= Kyx
(C.71)
The method implicitly provides a symmetric permeability tensor and symmetric
coarse scale matrix, but reduces the flow dimension in three directions. Due to the
sequential setting this has a large effect on the resulting equivalent permeability as
will be illustrated by the next test problem. Figure C.10 shows a model problem for
which the hierarchical method fails. For element (1,2,4) the equivalent permeability
v
Kyy
follows from equation (C.71), and reads
v1
Kyy
=
Ka (Ka Kb + 3Ka Ka )
.
2 (Ka + Kb ) (Ka + Kb )
(C.72)
Whereas for element (1,3,4) this reads
v2
Kyy
=
Ka (Ka Kb + 3Kb Kb )
.
2 (Ka + Kb ) (Ka + Kb )
(C.73)
v1
v1
v1
v2
v1
The interpolation weights follow from P42 = Kyy
/(Kyy
+ Kyy
) and P43 = Kyy
/(Kyy
+
v1
Kyy ). Only if the vertical permeability of element (1, 2, 4) and element (1, 3, 4) have
172
C. Multigrid method
Kb
K
a
K
2 •....................
.... ......
... . . ....
... . . ....
... . . . . . ......
....
...
....
...
....
....
....
....
...
.
....
..
......
....
.
..
.
.......................................................................
... . . ...... . . . ....
.
.
.... . . . . . .... . . . .......
.... . . . . .......
... . . . . . . . ......
.
..
....
...
....
....
....
....
.
.
.
...... .....
.... .....
...
9◦
4◦
b
Ka
7◦
3•
◦6
◦8
◦5
•1
Kb
K
a
K
2 •....................
... .....
.. . ......
... . . ....
... . . . .....
.... . . ..........
... ....
.... ......
... ....... .......
....
....
... .........
......
... ...... ........
.....
.......
.
................................................................
... ..... . . . ... . . . . . . ......
.
..
.
.
.
.... . ................. . . . . . ........
... . ....... ..... . . .....
.
.... ..... . . ........ .......
.....
..
....
....
...
...
....
... ......
... ........
........
9◦
4•
b
Ka
7◦
3•
◦6
◦8
◦5
•1
Kb
K
a
K
b
Ka
2 •...................
..... ......
... . . ...
.. . ... .....
.. . ..... . ...
.................................................
...
.... ..... ... ......
. ... .. ...
....
.............................. ......
....
........ .......
....
....
.
........ .
..................................................................
...... ... . ... . . .... ....
...... ........ . . . ...... . . . .......
.... . ...... ............................ ......
.. . . . .... ... . ....
.... . . ....... ..................
........................................
... ....
...
......
....
..
.
...... ......
.. .....
....
9◦
•6
4•
◦8
7◦
•5
•1
3•
Figure C.10: Multi-step test problem.
the same value, then P42 = P43 = 0.5, as one would expect from a reduced flow
computation over the element boundary. For this reason the proposed variational
coarsening procedure over triangular elements has been rejected, and patches of simplex
elements were proposed in this thesis.
Notations
Quantities
a, A
v
A
I
va
Aab
[b]
[A]ab A
δab
scalar
vector
second order tensor
second order unit tensor
Cartesian vector components
Cartesian tensor components
column
matrix
block matrix
Kronecker delta
Operations
L
N
∆
∇
∂/∂t
D/Dt
h.i
|a|
u·v
u∗v
uv
k.k
AD
AT
A−1
A:B
tr(A)
det(A)
[A]T
linear operator
nonlinear operator
increment
gradient operator
partial time derivative
material time derivative
volume average operator
absolute value of a scalar
in-product or dot-product of two vectors
outer-product of two vectors
dyadic-product or tensor-product of two vectors
vector norm or tensor norm
deviatoric part of a second-order tensor
conjugation or transpose of a second-order tensor
inverse of a second-order tensor
scalar-product or double-dot product of two second-order tensors
trace or first invariant of a second-order tensor
determinant or third invariant of a second-order tensor
transpose of a matrix
173
174
[A]−1
Notations
inversion of a matrix
Subscripts and Superscripts
a, b
(c)
e
g
i, j
k
l
n
m
r
s
x
α
ξ
matrix index
loading case
element
gas phase
space index
scale index
liquid or fluid phase
time step
gauss point
iteration step
solid phase
global coordinate
phase indicator
local coordinate
Integers
nb
nd
ne
nen
nk
nn
nu
nα
nα
k
number of boundary elements
flow domain dimension
number of elements
number of element nodes
number of species
number of nodes
number of unknowns
number of phases
number of species per phase
Units
[l]
[m]
[t]
[T]
length
mass
time
temperature
Symbols
[Ap ]
[Aω ]
[AT ]
bl
bn
flow stiffness matrix [lnd −2 t2 ]
solute stiffness matrix [mlnd−3 ]
heat stiffness matrix [mlnd −1 t−2 T−1 ]
pore-connectivity parameter [−]
pore size distribution index [−]
175
bα
bκ
[B p ]
[B ω ]
[B T ]
c
Cf
Ch
Cr
Dc
D
Dm
Eh
fk
[F p]
[F ω ]
[F T ]
ga
gl , gm , gn
g
H
H
j
j
jh
kr
M
[M p]
[M ω ]
[M T ]
n
n
p
pc
pe
pp
Pe
q
qn
q
Q
r
R
Ra
S
Brooks-Corey curve-fitting parameter [l−1]
Brooks-Corey curve-fitting parameter [−]
flow force vector [mlnd−3 ]
solute force vector [mlnd −3 ]
heat force vector [mlnd −1 t−2 ]
specific heat capacity [l2 t−2 T−1 ]
Forchheimer coefficient [l−1 t]
non-Fickian high concentration dispersion coefficient [m−1 l2 t]
Courant number [−]
coefficient of molecular dispersion [l2t−1 ]
hydrodynamic dispersion tensor [l2t−1 ]
mechanical dispersion tensor [l2t−1 ]
internal thermal energy density [l2 t−2 ]
internal drag vector or momentum exchange [ml−2t−2 ]
flow force vector [mlnd−3 t−1 ]
solute force vector [mlnd −3 t−1 ]
heat force vector [mlnd −1 t−3 ]
Genuchten shape factor [l−1 ]
Genuchten shape factors [−]
gravitational body force vector [lt−2]
energy source term for heat supply [ml−1t−3 ]
hydrodynamic thermo-dispersion tensor [mlt−3T−1 ]
magnitude of the non-convective mass flux vector [ml−2t−1 ]
non-convective (dispersive and diffusive) flux vector [ml−2 t−1 ]
non-convective heat energy flux [mt−3 ]
relative permeability [−]
mass source term [ml−3t−1 ]
flow compressibility matrix [lnd −2 t2 ]
solute compressibility matrix [mlnd−3 ]
heat compressibility matrix [mlnd−1 t−2 T−1 ]
porosity [−]
unit normal vector [−]
pressure [ml−1t−2 ]
capillary pressure [ml−1t−2 ]
evaporation threshold pressure [ml−1t−2 ]
ponding threshold pressure [ml−1t−2 ]
Peclet number [−]
magnitude of the volumetric flux density [lt−1]
boundary source flux [lt−1 ]
volumetric flux density [lt−1]
bulk volumetric solute source term [t−1 ]
resistance term [l−1 t]
chemical reaction rate term [ml−3t−1 ]
Rayleigh number [−]
saturation of the liquid phase in the void space [−]
176
Sr
Ss
[S p ]
[S ω ]
[S T ]
t
T
T
u
v
x
w
x, y, z
αl
αp
αt
βpl
βωl
k
βT l
Γ
γ
ε
εv
θ
K
Kd
λ
λl
µ
µk
µl
ν
ξ, η, ζ
ρ
σk
σ
σ0
τ
[Υ]
φ
ψ
ψa
ω
Ω
Notations
minimal saturation [−]
maximum saturated [−]
flow conductivity matrix [lnd −2 t]
solute conductivity matrix [mlnd −3 t−1 ]
heat conductivity matrix [mlnd −1 t−3 T−1 ]
time [t]
tortuosity tensor [−]
temperature [T]
displacement vector [l]
velocity vector [lt−1 ]
spatial coordinate vector [l]
Gauss weight [−]
global coordinates [l]
longitudinal dispersivity [l]
compressibility of the solid skeleton [m−1lt2 ]
transverse dispersivity [l]
liquid phase compressibility [m−1 lt2 ]
volumetric solute expansion coefficient [−]
thermal expansion coefficient [T−1 ]
domain boundary [lnd −1 ]
volumetric body force [ml−2t−2 ]
linear strain tensor [−]
volumetric strain [−]
time weighing factor [−]
intrinsic permeability tensor [l2 ]
hydraulic conductivity tensor [lt−1 ]
thermal conductivity [mlt−3T−1 ]
Lamé constant [ml−1t−2 ]
dynamic viscosity [ml−1t−1 ]
chemical potential of species k [l2t−2 ]
Lamé constant [ml−1t−2 ]
specific volume [m−1l3 ]
local coordinates [l]
density [ml−3 ]
viscous stress tensor [ml−1t−2 ]
Cauchy (total) stress tensor [ml−1t−2 ]
effective stress tensor[ml−1 t−2 ]
boundary traction tensor [mt−2 ]
pressure-dissipation matrix [m2 l−4 t−3 ]
volume fraction [−]
pressure head [l]
air-entry pressure head [l]
mass fraction [−]
domain [lnd ]
Bibliography
[1] J. E. Aarnes, V. L. Hauge, and Y. Efendiev. Coarsening of three-dimensional
structured and unstructured grids for subsurface flow. Advances in Water Resources, 30(11):2177–2193, November 2007.
[2] J.E. Aarnes, V. Kippe, and K.-A. Lie. Mixed multiscale finite elements and
streamline methods for reservoir simulation of large geo-models. Advances in
Water Resources, 28:257–271, 2005.
[3] J.E. Aarnes, S. Krogstad, and K.-A. Lie. Multiscale mixed/mimetic methods on
corner-point grids. Computational Geosciences, 12(3):297–315, September 2008.
[4] PH. Ackerer, A. Younes, and R. Mose. Modeling variable density flow and solute
transport in porous medium: 1. numerical model and verification. Transport in
Porous Media, 35:345–373, 1999.
[5] R.E. Alcouffe, A. Brandt, J.E. Dendy, Jr., and J.W. Painter. The multigrid
method for the diffusion equation with strongly discontinuous coefficients. SIAM
Journal on Scientific Computing, 2(4):430–453, 1981.
[6] T. Arbogast. The implementation of a locally conservative numerical subgrid
upscaling scheme for two-phase Darcy flow. Computational Geosciences, 6:453–
481, 2002.
[7] T. Arbogast and S.L. Bryant. Numerical subgrid upscaling for waterflood simulations. In 16th SPE Symposium on Reservoir Simulation, page SPE66375,
Houston, Texas, 2001. Society of Petroleum Engineers.
[8] J.-L. Auriault and J. Lewandowska. Upscaling: Cell symmetries and scale separation. Transport in Porous Media, 43:473–485, 2001.
[9] K. Aziz and A. Settari. Petroleum Reservoir Simulation. Blitzprint Ltd., Calgary,
Alberta, 2002.
[10] K.-J. Bathe. Finite Element Procedures. Prentice-Hall, 1996.
[11] M. Bause and P. Knabner. Computation of variably saturated flow by adaptive
mixed hybrid finite element methods. Advances in Water Resources, 27:565–581,
2004.
177
178
Bibliography
[12] J. Bear. Dynamics of Fluids in Porous Media. Dover Publications, 1988.
[13] J. Bear and A. Verruijt. Modeling Groundwater Flow and Pollution, volume 2
of Theory and Applications of Transport in Porous Media. D. Reidel Publishing
Company, Dordrecht, The Netherlands, 1987.
[14] T. Belytschko, W.K. Liu, and B. Moran. Nonlinear Finite Elements for Continua
and Structures. Wiley, USA, 2008.
[15] R.B. Bird, W.E. Stewart, and E.N. Lightfoot. Transport Phenomena. John Wiley
and Sons, 1960.
[16] Ø. Bø e. Analysis of an upscaling method based on conservation of dissipation.
Transport in Porous Media, 17:77–86, 1994.
[17] A.G.L. Borthwick, R.D. Marchant, and G.J.M. Copeland. Adaptive hierarchical
grid model of water-born pollutant dispersion. Advances in Water Resources,
23:849–865, 2000.
[18] W.L. Briggs, H. van Emden, and S.F. McCormick. A Multigrid Tutorial. SIAM,
Society for Industrial and Applied Mathematics, USA, 2000.
[19] R.B.J. Brinkgreve. Plaxis, Finite Element Code for Soil and Rock Analyses.
Plaxis B.V., Delft, The Netherlands, 8th edition, 2002.
[20] R.B.J. Brinkgreve, R. Al-Khoury, and J.M. van Esch. Plaxflow. Plaxis B.V.,
Delft, The Netherlands, 1th edition, 2003.
[21] M.A. Bùes and C. Oltean. Numerical simulations for saltwater intrusion by the
mixed hybrid finite element method and discontinuous finite element method.
Transport in Porous Media, 40:171–200, 2000.
[22] F. Chen and L. Ren. Application of the finite difference heterogeneous multiscale
method to the Richards’ equation. Water Resources Research, 44:W07413, July
2008.
[23] T. Chen, M.G. Gerritsen, J.V. Lambers, and L.J. Durlofsky. Global variable
compact multipoint methods for accurate upscaling with full-tensor effects. Computational Geosciences, 14(1), January 2010.
[24] Z. Chen and T.Y. Hou. A mixed multi scale finite element method for elliptic
problems with oscillating coefficients. Mathematics of Computation, 72(242):541–
576, 2002.
[25] H.-P. Cheng, G.-T. Yeh, J. Xu, M.-H Li, and R. Carsel. A study of incorporating
the multigrid method into the three-dimensional finite element discretization: A
modular setting and application. International Journal for Numerical Methods
in Engineering, 41:499–526, 1998.
Bibliography
179
[26] M.A. Christie and M.J. Blunt. Tenth spe comparative solution project: A comparison of upscaling techniques. SPE Reservoir Evaluation & Engineering, pages
308–317, August 2001.
[27] J. Chu, Y. Efendiev, V. Ginting, and T.Y. Hou. Flow based oversampling
technique for multiscale finite element methods. Advances in Water Resources,
31(4):599–608, April 2008.
[28] C.R. Cole and H.P. Foote. Multigrid Methods for Solving Multiscale Transport
Problems, pages 273–303. Volume 10 of Cushman [29], 1990.
[29] J.H. Cushman. Dynamics of Fluids in Hierarchical Porous Media. Academic
Press, 1990.
[30] J.H. Cushman, L.S. Bennethum, and B.X. Hu. A primer on upscaling tools for
porous media. Advances in Water Resources, 25:1043–1067, 2002.
[31] G. Dagan. Flow and Transport in Porous Formations. Springer, 1989.
[32] G. de Josselin de Jong. Longitudinal and transverse diffusion in granular deposits.
Transactions, American Geophysical Union, 39:67–74, 1958.
[33] G. de Marsily. Quantitative Hydrogeology, Groundwater Hydrology for Engineers.
Academic Press, 1986.
[34] P.M. de Zeeuw. Matrix-dependent prolongations and restrictions in a blackbox
multigrid solver. Journal of Computational Mathematics, 33:1–27, 1990.
[35] L. Demkowicz. Computing with hp-Adaptive Finite Elements, Frontiers: Three
Dimensional Elliptic and Maxwell Problems, volume 2. Chapman & Hall/CRC,
Boca Raton, USA, 2007.
[36] L. Demkowicz. Computing with hp-Adaptive Finite Elements, One and Two
Dimensional Elliptic and Maxwell Problems, volume 1. Chapman & Hall/CRC,
Boca Raton, USA, 2007.
[37] J.E. Dendy, Jr. Black box multigrid. Journal of Computational Physics, 48:366–
386, 1982.
[38] J.E. Dendy, Jr. Black box multigrid for non-symmetric problems. Applied Mathematics and Computation, 13:261–283, 1983.
[39] H.-J.G. Diersch. Feflow, Finite Element Subsurface Flow & Transport Simulation
System, Users Manual. Wasy Institute for Water Resources Planning and System
Research, Berlin, Germany, 2005.
[40] H.-J.G. Diersch. Feflow, Finite Element Subsurface Flow & Transport Simulation
System, White papers. Wasy Institute for Water Resources Planning and System
Research, Berlin, Germany, 2005.
180
Bibliography
[41] H.-J.G Diersch and O. Kolditz. Variable-density flow and transport in porous
media: Approaches and challenges. Advances in Water Resources, 25:899–944,
2002.
[42] L.J. Durlofsky. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resources Research, 27(5):699–708,
1991.
[43] L.J. Durlofsky. Representation of grid block permeability in coarse scale models
of randomly heterogeneous porous media. Water Resources Research, 28(7):1791–
1800, 1992.
[44] L.J. Durlofsky, Y. Efendiev, and V. Ginting. An adaptive local/global multiscale
finite volume element method for two-phase flow simulations. Advances in Water
Resources, 30(3):576–588, March 2007.
[45] M.G. Edwards and C.F. Roger. Multigrid and renormalization for reservoir simulation. In P.W. Hemker and P. Wesseling, editors, Multigrid methods IV, pages
189–200, Amsterdam, 1994. Birkhäuser Verlag.
[46] Y. Efendiev, T. Hou, and T. Strinopoulos. Multiscale simulations of porous media
flows in flow-based coordinate system. Computational Geosciences, 12(3):257–
272, September 2008.
[47] J.W. Elder. Numerical experiments with free convection in a vertical slot. Journal
of Fluid Mechanics, 24(4):823–843, 1966.
[48] H.I. Ene and D. Poliševski. Thermal Flow in Porous Media, volume 1 of Theory
and Applications of Transport in Porous Media. D. Reidel Publishing Company,
1987.
[49] M.W. Farthing and C.T. Miller. A comparison of high-resolution, finite-volume,
adaptive-stencil schemes for simulating advective-dispersive transport. Advances
in Water Resources, 24:29–48, 2001.
[50] P.A. Forsyth, Y.S. Wu, and K. Pruess. Robust numerical methods for saturatedunsaturated flow with dry initial conditions in heterogeneous media. Advances
in Water Resources, 18:25–38, 1995.
[51] D.G. Fredlund and H. Rahardjo. Soil Mechanics for Unsaturated Soils. John
Wiley & Sons, 1993.
[52] R.A. Freeze and J.A. Cherry. Groundwater. Prentice Hall, 1979.
[53] G. Galeati, G. Gambolati, and S.P. Neuman. Coupled and partially coupled
Euler-Lagrangian model of freshwater-seawater mixing. Water Resources Research, 28(1):149–165, 1992.
[54] Y. Gautier and B. Noetinger. Preferential flow-paths detection for heterogeneous
reservoirs using a new renormalization technique. In P.R. Thomassen, editor, 4th
European Conference on the Mathematics of Oil Recovery, 1992.
Bibliography
181
[55] L.W. Gelhar. Stochastic Subsurface Hydrology. Prentice-Hall, 1993.
[56] M. Gerritsen and J.V. Lambers. Integration of local/global upscaling and grid
adaptivity for simulation of subsurface flow in heterogeneous formations. Computational Geosciences, 12(2), June 2008.
[57] G.H. Golub and C.F. van Loan. Matrix Computations. John Hopkins University
Press, London, United Kingdom, 3rd edition, 1996.
[58] H. Gotovac, V. Cvetkovic, and R. Andricevic. Adaptive fup multi-resolution
approach to flow and advective transport in highly heterogeneous porous media: Methodology, accuracy and convergence. Advances in Water Resources,
32(6):779–968, June 2009.
[59] M. Griebel and S. Knapek. A multigrid-homogenization method. 13th GAMMSeminar Kiel on Numerical Treatment of Multi Scale Problems, pages 187–202,
1997.
[60] W. Hackbush. Multi-Grid Methods and Applications, volume 4 of Springer Series
in Computational Mathematics. Springer, 2003.
[61] S.M. Hassanizadeh. On the transient non-Fickian dispersion theory. Transport
in Porous Media, 23:107–124, 1996.
[62] S.M. Hassanizadeh and A. Leijnse. A non-linear theory of high-concentrationgradient dispersion in porous media. Advances in Water Resources, 18(4):203–
215, 1995.
[63] X. He and L. Ren. Finite volume multiscale finite element method for solving
the groundwater flow problems in heterogeneous porous media. Water Resources
Research, 41:W10417, 2005.
[64] K.J. Hersvik and M.S. Espedal. Adaptive hierarchical upscaling of flow in heterogeneous reservoirs based on an a posteriori error estimate. Computational
Geosciences, 2:311–336, 1998.
[65] L. Holden and B.F. Nielsen. Global upscaling of permeability in heterogeneous
reservoirs; the output least squares (OLS) method. Transport in Porous Media,
40:115–143, 2000.
[66] E. Holzbecher. Modeling Density-Driven Flow in Porous Media. Springer, 1998.
[67] U. Hornung. Homogenization and Porous Media. Springer, 1997.
[68] T.Y. Hou and X.-H. Wu. A multiscale finite element method for elliptic problems
in composite materials and porous media. Journal of Computational Physics,
134:169–189, 1997.
[69] T.Y. Hou, X.-H. Wu, and Z. Cai. Convergence of the multiscale finite element
method for elliptic problems with rapidly oscillating coefficients. Mathematics of
Computation, 68(227):913–943, 1999.
182
Bibliography
[70] T.J.R. Hughes. The Finite Element Method, Linear Static and Dynamic Finite
Element Analysis. Dover Publications, Mineola, New York, USA, 2000.
[71] P.S. Huyakorn and G.F. Pinder. Computational Methods in Subsurface Flow.
Academic Press, 1983.
[72] J. Istok. Groundwater Modeling by the Finite Element Method, volume 13 of
Water Resources Monograph. American Geophysical Union, 1990.
[73] P. Jenny, S.H. Lee, and H.A. Tchelepi. Multi-scale finite-volume method for elliptic problems in subsurface flow simulation. Journal of Computational Physics,
187:47–67, 2003.
[74] L. Jiang, Y. Efendiev, and I. Mishev. Mixed multiscale finite element methods
using approximate global information based on partial upscaling. Computational
Geosciences, 14(2), March 2010.
[75] R. Juanes and F.X. Dub. A locally conservative variational multiscale method for
the simulation of porous media flow with multiscale source terms. Computational
Geosciences, 12(3):273–295, September 2008.
[76] P.R. King. The use of renormalization for calculating effective permeability.
Transport in Porous Media, 4:37–58, 1989.
[77] P.R. King. Upscaling permeability: Error analysis for renormalization. Transport
in Porous Media, 23:337–354, 1996.
[78] K.L. Kipp, Jr. HST3D, A Computer Code for Simulation of Heat and Solute
Transport in Three-Dimensional Ground-Water Flow Systems. U.S. Geological
Survey, Denver, Colorado, USA, 1987.
[79] K.L. Kipp, Jr. Guide to the revisited Heat and Solute Transport Simulator: Hst3d
- Version 2. U.S. Geological Survey, Denver, Colorado, USA, 1997.
[80] V. Kippe, J.E. Aarnes, and K.-A. Lie. Multiscale finite-element methods for elliptic problems in porous media flow. In Proceedings of the 16th International
Conference on Computational Methods in Water Resources, Copenhagen, Denmark, 2006.
[81] V. Kippe, J.E. Aarnes, and K.-A. Lie. A comparison of multiscale methods for
elliptic problems in porous media flow. Computational Geosciences, 12(3):377–
398, September 2008.
[82] S. Knapek. Matrix-dependent multigrid homogenization for diffusion problems.
SIAM Journal on Scientific Computing, 20(2):515–533, 1998.
[83] O. Kolditz, R. Ratke, H.-J.G. Diersch, and W. Zielke. Coupled groundwater
flow and transport: 1. verification of variable density flow and transport models.
Advances in Water Resources, 21(1):27–46, 1998.
Bibliography
183
[84] J.B. Kool and J.C. Parker. Development and evaluation of closed-form expressions for hysteretic soil hydraulic properties. Water Resources Research,
23(1):105–114, 1987.
[85] W.M. Lai, D. Rubin, and E. Krempl. Introduction to Continuum Mechanics.
Pergamon Press, 3rd edition, 1993.
[86] J.V. Lambers, M.G. Gerritsen, and B.T. Mallison. Accurate local upscaling
with variable compact multipoint transmissibility calculations. Computational
Geosciences, 12(3), September 2008.
[87] S.H. Lee, C. Wolfsteiner, and H.A. Tchelepi. Multiscale finite-volume formulation
for multiphase flow in porous media: Black oil formulation of compressible, threephase flow with gravity. Computational Geosciences, 12(3):351–366, September
2008.
[88] A. Leijnse. Three-Dimensional Modeling of Coupled Flow and Transport in
Porous Media. PhD thesis, University of Notre Dame, Notre Dame, Indiana,
USA, 1992.
[89] D.A. Lever and C.P. Jackson. On the equations for the flow of concentrated salt
solution through a porous medium. Harwell Laboratory report AERE-R 11765,
1985.
[90] M.-H. Li, H.-P. Cheng, and G.-T. Yeh. Solving 3d subsurface flow and transport
with adaptive multigrid. Journal of Hydrological Engineering, pages 74–81, 2000.
[91] I. Lunati and P. Jenny. Multiscale finite-volume method for density-driven flow
in porous media. Computational Geosciences, 12(3):337–350, September 2008.
[92] S.P. MacLachlan and J.D. Moulton. Multilevel upscaling through variational
coarsening. Water Resources Research, 42:W02418, 2006.
[93] S. Mehl and M.C. Hill. Three-dimensional local grid refinement for block-centered
finite-difference groundwater models using iteratively coupled shared nodes: a
new method of interpolation and analysis of errors. Advances in Water resources,
27:899–912, 2004.
[94] C.C. Mei. Method of homogenization applied to dispersion in porous media.
Transport in Porous Media, 9:261–274, 1992.
[95] J.D. Moulton, J.E. Dendy, Jr., and J.M. Hyman. The black box multigrid numerical homogenization algorithm. Journal of Computational Physics, 142:80–108,
1998.
[96] B.F. Nielsen and A. Tvieto. An upscaling method for one-phase flow in heterogeneous reservoirs. a weighted output least squares (WOLS) approach. Computational Geosciences, 2:93–123, 1998.
184
Bibliography
[97] S. Nilsen and M.S. Espedal. Wavelet upscaling based of piecewise bilinear approximation on the permeability field. Transport in Porous Media, 23:125–134,
1996.
[98] J. M. Nordbotten and P. E. Bjórstad. On the relationship between the multiscale
finite-volume method and domain decomposition preconditioners. Computational
Geosciences, 12(3):367–376, September 2008.
[99] W.H. Press, W.T. Vetterling, S.A. Teukolsky, and B.P. Flannery. Numerical
recipes in C, The Art of Scientific Computing. Cambridge University Press, New
York, USA, 2nd edition, 2002.
[100] M. Putti and C. Paniconi. Picard and Newton linearization for the coupled model
of saltwater intrusion in aquifers. Advances in Water Resources, 18(3):159–170,
1995.
[101] Ph. Renard and G. de Marsily. Calculating equivalent permeability: A review.
Advances in Water Resources, 20(5–6):253–278, 1997.
[102] A. Reusken. Multigrid with matrix-dependent transfer operators for a singular
perturbation problem. Computing, 50:199–211, 1993.
[103] A. Reusken. Multigrid with matrix-dependent transfer operators for convectiondiffusion problems. In Proceedings of the Fourth European Multigrid Conference,
Brinkhauser, Basel, 1994. INSM.
[104] Y. Saad. Iterative Methods for Sparse Linear Systems. Society for Industrial and
Applied Mathematics, 2nd edition, 2000.
[105] A.E. Scheidegger. General theory of dispersion in porous media. Journal of
Geophysical Research, 66(10):3273–3278, 1961.
[106] R.J. Schotting, H. Moser, and S.M. Hassanizadeh. High-concentration-gradient
dispersion in porous media: Experiments, analysis and approximations. Advances
in Water Resources, 22(7):665–680, 1999.
[107] M.J. Simpson and T.P. Clement. Theoretical analysis of the worthiness of Henry
and Elder problems as benchmarks of density-dependent groundwater flow models. Advances in Water Resources, 26:17–31, 2003.
[108] C.W.M. Sitters. Continuum Mechanics. Delft University of Technology, Delft,
The Netherlands, 1996.
[109] J. Trangenstein. Multi-scale iterative techniques and adaptive mesh refinement
for flow in porous media. Advances in Water Resources, 25:1175–1213, 2002.
[110] U. Trottenberg, C. Oosterlee, and A. Schüller. Multigrid. Academic Press, 2001.
[111] J.C. van Dam. Field-Scale Water Flow and Solute Transport, SWAP Model
Concepts, Parameter Estimation and Case Studies. PhD thesis, Wageningen
Universiteit, Wageningen, The Netherlands, 2000.
Bibliography
185
[112] J.M. van Esch. Adaptive grid modeling of unsaturated groundwater flow. In
F.B.J. Barends and P.M.P.C. Steijger, editors, Learned and Applied Soil Mechanics Out of Delft, pages 9–14, Delft, The Netherlands, 2002. Delft University
of Technology, A.A. Balkema.
[113] J.M. van Esch. Adaptive multigrid modeling of density dependent groundwater
flow. In R.H. Boekelman et al., editor, Proceedings 17th Salt Water Intrusion
Meeting, pages 82–92, Delft, The Netherlands, 2002. Delft University of Technology, Hydrology & Ecology Section, Faculty of Civil Engineering and Geosciences.
[114] J.M. van Esch, M.A. Van, R.F.A. Hendriks, and J.J.H. van den Akker. Geohydrologic design procedure for peat dykes under drying conditions. In G.N.
Pande and S. Pietruszczak, editors, Numerical Models in Geomechanics (Numog
X). Taylor & Francis Group, 2007.
[115] J.M. van Esch, A.F. van Tol, H.R. Havinga, A.M.W. Duijvenstijn, B.J. Schat,
and J.C.W.M. de Wit. Functional analyses of jet grout bodies based on Monte
Carlo simulations. In G. Barla and M. Barla, editors, Proceedings of the 11th
International Conference on Computer Methods and Advances in Geomechanics,
pages 643–650, Torino, Italy, 2005. Department of Structural and Geotechnical
Engineering Politecnico di Torino, Patron Editore.
[116] A. Verruijt. Computational Geomechanics, volume 7 of Theory and Applications
of Transport in Porous Media. Kluwer Academic Publishers, 1995.
[117] T. Vogel, M.Th. van Genuchten, and M. Cislerova. Effect of the shape of the
soil hydraulic functions near saturation on variably-saturated flow predictions.
Advances in Water Resources, 24:133–144, 2001.
[118] C.I. Voss and A.M. Provost. SUTRA, A Model for Saturated-Unsaturated,
Variable-Density Ground-Water Flow with Solute or Energy Transport. U.S. Geological Survey, Reston, Virginia, USA, 2003.
[119] C.B. Vreugdenhil. Computational Hydraulics. Springer, 1989.
[120] C. Wagner, W. Kinzelbach, and G. Wittum. Schur-complement multigrid, a
robust method for groundwater flow and transport problems. Numerische Mathematik, 75:523–545, 1997.
[121] D. Weatherill, C.T. Simmons, C.I. Voss, and N.I. Robinson. Testing densitydependent groundwater flow models: Two-dimensional steady state unstable convection in infinite, finite and inclined porous layers. Advances in Water Resources,
27:547–562, 2004.
[122] P. Wesseling. An Introduction to Multigrid Methods. Edwards, Philadelphia,
USA, 2004.
[123] S. Whitaker. The Method of Volume Averaging, volume 13 of Theory and Applications of Transport in Porous Media. Kluwer Academic Publishers, 1999.
186
Bibliography
[124] S. Ye, Y. Xue, and C. Xie. Application of the multiscale finite element method
to flow in heterogeneous porous media. Water Resources Research, 40:W09202,
2004.
[125] G.-T. Yeh. Computational Subsurface Hydrology, Fluid Flows. Kluwer Academic
Publishers, 1999.
[126] G.T. Yeh. A Lagrangian-Eulerian method with zoomable hidden fine-mesh approach to solving advection-dispersion equations. Water Resources Research,
26(6):1133–1144, 1990.
[127] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, Basic Formulation and Linear Problems, volume 1. McGraw-Hill Book Company, 4th edition,
1989.
[128] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, Solid and Fluid
Mechanics Dynamics and Non-Linearity, volume 2. McGraw-Hill Book Company,
4th edition, 1989.
[129] W. Zijl and A. Trykozko. Numerical homogenization of the absolute permeability
using the conformal-nodal and mixed-hybrid finite element method. Transport
in Porous Media, 44:32–62, 2001.
Samenvatting
Natuurlijke geologische formaties vertonen veelal een structurele en functionele heterogeniteit over vele ordes van grootte in de ruimte en de tijd. In geohydrologische
simulaties wordt de structurele hiërarchie van fysische subeenheden met een geologisch model beschreven en de functionele hiërarchie van het stromingsproces met een
stromingsmodel. Stromingseigenschappen zoals druk, flux en dissipatie kunnen aan
elkaar gerelateerd worden met constitutieve relaties en parameters van de subeenheden zoals porositeit en hydraulische doorlatendheid. Componenten van de hydraulische
doorlatendheid zijn de (stationaire) intrinsieke doorlatendheid en de (tijdsafhankelijke)
relatieve doorlatendheid. De hydraulische doorlatendheid is de meest dominante en de
meest heterogene parameter die de stroming van grondwater op veldschaal beı̈nvloed.
Laboratorium experimenten geven metingen van de parameters van subeenheden
op een fijne schaal. Als laboratorium metingen stochastisch worden gebruikt in het geologische model dan moet het structurele model van de ondergrond op de zelfde schaal
als die van de metingen worden opgesteld. Stromingsberekeningen met deze resolutie
op veldschaal zijn onhanteerbaar en daarom is hier een nieuwe adaptive multiscale techniek ontwikkeld. Terwijl constitutieve relaties kunnen veranderen bij overgangen naar
verschillende schalen, wordt Darcy’s wet geldig verondersteld op de laboratoriumschaal
en de veldschaal.
Tegenwoordig worden opschalingsmethoden toegepast die, zowel functioneel als
structureel, informatie overdragen over de hiërachie van schalen. Dit gebeurt van
de fijne schaal naar de grove schaal en niet vise versa, met effectieve of equivalente
parameters. De doorlatendheid is geen additieve parameter en het is dus niet mogelijk
om een equivalente doorlatendheid op de grove schaal te berekenen als een eenvoudig
gemiddelde van metingen op een fijne schaal. De equivalente doorlatendheid wordt
vaak bepaald op basis van een stromingscriterium of een energiedissipatie criterium.
De equivalente doorlatendheid komt alleen overeen met de effectieve doorlatendheid als
de variatieschaal veel kleiner is dan de schaal waarop de doorlatendheid wordt bepaald.
Deze effectieve doorlatendheid is een constante tweede orde tensor parameter. De equivalente doorlatendheid is echter niet uniek en is afhankelijk van de randvoorwaarden
van het monster. Effectieve doorlatendheid geldt alleen voor discrete hiërarchische
systemen waarvan de schalen kunnen worden gescheiden.
De nieuw ontwikkelde multischaal techniek breidt de oorspronkelijke multischaal
eindige elementen methode op twee niveaus uit naar een hiërarchie van schalen. Met
multischaal eindige elementen methoden wordt het fijne schaal gedrag door middel
van multischaal basis functies vastgelegd op de grove schaal. De gewichten van de
187
188
Samenvatting
multischaal basis functies volgen uit locale stromingsberekeningen. De procedure verwijdert alle fijne schaal knopen uit het subdomein waardoor echter fouten worden
geı̈ntroduceerd aan de randen van het subdomein. De op deze wijze geformuleerde
methode vormt een klasse van subdomein decompositie technieken. Het kan worden
aangetoond dat een sequentiële implementatie niet sneller is dan een optimale solver
zoals de multigrid methode. De methode leent zich echter goed voor een parallelle
implementatie.
De voorgestelde methode is gebaseerd op een conforme eindige elementen formulering voor simplex elementen. De conforme eindige elementen methode formuleert
behoud van massa per knoop, continuı̈teit van flux over de element randen wordt niet
behouden. Zones waarin grote fouten in de snelheden over de elementranden optreden
worden met een mesh verfijningscriterium gedetecteerd, waarna een adaptieve verfijningsprocedure het mesh lokaal verrijkt zodat de fout wordt gecorrigeerd. Multiscale
basisfuncties volgen uit de oplossing van locale stromingsproblemen over patchen van
simplex elementen. Lineaire randvoorwaarden worden opgelegd op de randen van uitgebreide subdomeinen, waarmee randeffecten op de patch worden gereduceerd. Met
deze procedure wordt het gedrag op de grove schaal nauwkeuriger bepaald. Uitgebreide
locale stromingsproblemen introduceren echter discontinuı̈teten in de basisfuncties en
genereren nieuwe relaties tussen knopen op de grovere schaal. Daarom wordt de voorkeur gegeven aan het oplossen van het lokale problem met randvoorwaarden die volgen
uit een dimensioneel gereduceerd probleem. Een tweede verfijningscriterium vergelijkt
functiewaarden op patches met waarden op uitgebreide subdomeinen. Druk-dissipatie
middeling benadert de multiscale operator op het grove grid en ondersteunt een functionele adaptieve formulering. De multiscale middelingsmethode berekent equivalente
doorlatendheidstensor componenten en reproduceert de ijle matrix op de grove schaal.
De belastingsgevallen voor de locale problemen volgen uit het sommeren van multiscale basis functies. De multiscale basis functies extrapoleren de oplossing van de grove
schaal naar de fijne schaal. Op deze schaal worden discontinuı̈teiten in het snelheidsveld gedetecteerd en vergeleken met het eerste criterium. De berekende equivalente
doorlatendheid wordt gebruikt in het raamwerk van een geometrische multigrid methode bij het berekenen van grove schaal operatoren. Met de hiërachie van multiscale
basis functies, die de druk op ieder grof niveau relateert aan het volgende fijne niveau,
worden de intergrid transfer operatoren gegenereerd.
De voorgestelde aanpak geeft een robuust en efficiënt algoritme, gebaseerd op de
multiscale eindige elementen methode. Met het algoritme kan stroming door gedeeltelijk verzadigde hiërachische heterogene formaties, en het daaraan volledig gekoppeld
transport van opgeloste stoffen en warmte, worden gesimuleerd. Met de multiscale eindige elementen formulering worden numeriek gehomogeniseerde discrete stromingsvergelijkingen gegenereerd en worden lokaal verfijnde snelheidsvelden bepaald, die nauwkeurige transportberekeningen ondersteunen. De prestatie van het algoritme wordt
met een aantal realistische cases geı̈llustreerd.
John van Esch.
Curriculum vitae
John van Esch was born on the 10th of June 1966 in Heemstede.
In 1985 he started his Civil Engineering study at Hogeschool Haarlem. He was a
practical trainee for a period of one year, firstly at water company ’Gemeentewaterleidingen Amsterdam’ and secondly at geotechnical consultancy ’Tjaden Grondmechanica’. He obtained a B.Eng. degree in 1990. In 1989 he became a student at the faculty
of Civil Engineering of Delft University of Technology. He specialized in geo-technology
and hydrology, graduated (cum laude) in 1992 obtaining a MSc degree. During the
following years, he participated in a number of advanced courses at Stanford University, Wageningen University, IHE Institute for Water Education, Wessex Institute of
Technology, and Universitat Politecnia de Catalunya.
He started to work as a research engineer in the field of subsurface hydrology at
Grondmechanica Delft, Department of Mathematics and Informatics in 1992. In this
position, he collaborated in a research program on the assessment of time dependent
stability behavior of embankments, and developed numerical tools for operating geohydraulic control systems, for analyzing time series from well measurements, and for
analyzing retaining wall constructions.
In 1995, he became a research assistant at the Geotechnical Laboratory of Delft
University of Technology where he carried out research on a part time basis, for
a period of five years. During this period he focused on computational subsurface
flow. At GeoDelft, he switched to the Department of Strategic Research. Next to
geo-hydrological model applications and geo-statistical analysis, he participated in
a EU-project on the interaction of subsurface hydrology and stability of landslides.
He co-worked with Plaxis for about two year on the development of the unsaturated
groundwater flow simulator Plaxflow.
In the period of 2005 till 2007, he worked at GeoDelft’s Department of Soil structures. He participated in a research program on the development of a geo-hydrologic
design procedure for peat dikes under drying and wetting conditions, and developed
an algorithm for functional analyses of jetgrout bodies.
In 2008, he became a research engineer in the field of computational groundwater
mechanics at the Deltares Software Center. He co-works with Universität Stuttgart
on the development of a consolidation module, based on the material point method.
In 2009 he switched to Deltares’ foundation and subsurface structure unit and studied
two-phase dynamics.
189
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement