A mixed-dimensional finite volume method for Rainer Helmig

A mixed-dimensional finite volume method for Rainer Helmig
A mixed-dimensional finite volume method for
two-phase flow in fractured porous media
Volker Reichenberger Hartmut Jakobs Peter Bastian
Rainer Helmig
IWR, Universität Heidelberg, Im Neuenheimer Feld 368, 69120 Heidelberg
IWS, Universität Stuttgart, Pfaffenwaldring 61, 70569 Stuttgart
Abstract
We present a vertex-centered finite volume method for the fully coupled, fully implicit discretization of two-phase flow in fractured porous media. Fractures are discretely modeled as lower dimensional elements. The method works on unstructured,
locally refined grids and on parallel computers with distributed memory. An implicit
time discretization is employed and the nonlinear systems of equations are solved
with a parallel Newton-multigrid method. Results from two-dimensional and threedimensional simulations are presented.
Key words: Multiphase flow; Numerical reservoir simulation; Fractured reservoir;
Mixed-dimensional Finite Volume Method; Multigrid Methods.
1
Introduction
Fractures are present in many subsurface systems and can strongly influence
or even dominate fluid flow and transport behavior. Because of the importance
of fractured systems for secondary recovery techniques, waste disposal safety
analysis and remediation techniques they have been investigated since the
1960s [1,2] and still receive considerable attention. The difficulties which arise
for conceptual and numerical modeling stem from the strongly heterogeneous
and anisotropic nature of the fracture-matrix system. For multiphase flow
Email addresses: [email protected] (Volker
Reichenberger), [email protected] (Hartmut Jakobs),
[email protected] (Peter Bastian),
[email protected] (Rainer Helmig).
Preprint submitted to Elsevier Science
9 August 2005
problems the behavior is much more complex than in the case of a single
phase, since in the presence of two phases fractures may act as barriers or
main flow paths, depending on saturation. Finally, uncertainties associated
with field problems introduce difficulties into the simulation process.
Fractures appear on different length scales. We consider subsurface systems
where the location and properties of the dominant fractures are known or
can be estimated to good agreement with the site, with the length of these
fractures on the order of the size of the domain under investigation. Special
consideration is given to a combined treatment of the behavior of the fluid
phases in the fractures and the rock matrix, which should model accurately
the interchange processes between fractures and rock matrix. This suggests
a coupled approach where the fluid flow is described by a set of equations
which are valid in both parts of the domain and which doesn’t introduce
artificial exchange terms between these parts. Instead the model incorporates
a physically meaningful description of the fluid phase behavior at the media
discontinuities based on the accurate treatment of the capillary pressure. We
apply the extended interface conditions derived in [3], which were originally
developed for media discontinuities like lenses and layers and which are used
for fractures here. This addresses the above mentioned issue of heterogeneities.
As for the problem of the anisotropic nature of fractures, we do not approach
this issue from the modeling side but instead seek an appropriate discretization method. A discretization of a fractured medium domain with volumetric
elements in the fractures requires a mesh which resolves the geometry of the
problem. Due to the small fracture widths the resulting mesh will either consist of a very large number of elements or the mesh will contain elements
with a very large aspect ratio. These elements are known to introduce difficulties into the solution behavior of iterative solvers. Our approach is to
apply a mixed-dimensional finite volume discretization method which realizes
fractures as lower-dimensional elements. We use one-dimensional elements for
fractures in two-dimensional domains and two-dimensional fracture elements
in three-dimensional domains. These elements can be used in an extension of
the finite volume method of [4]. This method is suited for unstructured grids
and adaptive grid refinement as well as for efficient parallelization by domain
decomposition. The nonlinear systems arising from the implicit discretization
are solved by an inexact Newton method which uses multigrid for the solution
of the linearized systems.
The presented method is implemented in software. The features of this software can be summarized as follows.
• Domains in 2D and 3D with unstructured, hybrid grids and adaptive grid
refinement.
• Fully coupled mass-conserving vertex centered finite volume discretization.
2
• Physically correct treatment of capillary pressure at fracture–matrix interfaces by extended interface conditions.
• Phases can be compressible or incompressible.
• Fully implicit time discretization with selectable time stepping schemes.
• Efficient solution of the linearized systems with multigrid methods.
• Parallelization for distributed memory computers.
To the best knowledge of the authors, this combination of features in software
is unique. For more details we refer to [5].
The organization of this paper is as follows. In the next section we present the
model equations which are valid in fractures and rock matrix, and we explain
the interface conditions at the boundaries between fractures and rock matrix. Then in section 3 we present the finite volume method for the elements
of mixed dimension and include some notes on implementation, parallelization and pre- and post-processing. We explain the time discretization and
the nonlinear and linear solution methods. In section 4 we present results.
In two dimensions a comparison of the mixed-dimensional and a standard
finite volume method shows the adequacy and limitations of the method. In
three dimensions, we show an application in a domain with a complex fracture
network.
2
Two-phase flow equations
We consider an isothermal system of two phases, the wetting phase w (water
in our context) and the nonwetting phase n. The phases are considered to
consist of a single component each, i. e. we will not consider phase transitions.
The non-wetting phase can be any non-aqueous phase liquid (napl) like oil or
gas—the underlying equations are the same. Compressible phases are included
in the model and the software implementation, which is crucial for water-gas
systems. Because of the smaller density and viscosity differences the treatment
of water-oil systems is numerically easier. We only present water-gas systems
to prove the capability of the method. If a gas phase is referred to explicitly,
we denote it with g instead of n.
Central to the derivation of the governing equations is the existence of a
representative elementary volume (REV) for the fracture and the matrix. Since
we treat fractures discretely, we assume that an REV exists for the rock matrix
as well as for the fractures. See [6] for an overview of current questions arising
in fracture flow modelling.
3
2.1 Conservation of mass and Darcy’s law
We consider the two-phase system in a porous medium which fills the domain
Ω ⊂ Rd , d = 2, 3. We use the set of equations also employed in [4] and refer to
[7] and [8] for a more thorough description. The governing equations are the
conservation of mass for each phase α = w, g:
∂(Φρα Sα )
+ ∇ · (ρα vα ) = ρα qα ,
∂t
(1)
where w denotes the water phase and n denotes the napl phase. We assume
that the effective porosity Φ only depends on position, Φ = Φ(x). Sα = Sα (x, t)
is the saturation of phase α and the two phases fill the pore space,
Sw + Sn = 1.
(2)
ρα = ρα (x, t) is the density of phase α. For incompressible fluids ρα is constant,
otherwise an equation of state for ideal gases is employed, pα = ρα RT , with
the phase pressure pα , an individual gas constant R and the temperature T .
We assume isothermal conditions. qα = qα (x, t) is the source term of phase
α and vα is the Darcy velocity of phase α. The Darcy velocity is given by a
multiphase extension of Darcy’s law for each phase α = w, n,
vα = −
krα K
(∇pα − ρα g).
µα
(3)
The Darcy velocity only depends on the macroscopic phase pressure pα . K is
the tensor of absolute permeability, krα is the relative permeability of phase α
(see next section 2.2). µα = µα (pα ) is the dynamic viscosity, g is the gravitation
vector (0, 0, −g), g = 9.81[m/s2 ].
These equations are valid in the matrix and in the fracture if the flow is
laminar in both regions. If the fracture is open, the local cubic law derived
from the Hele-Shaw analog (see [9]) can be employed to define the absolute
permeability for the flow of a single incompressible fluid phase. The absolute
permeability is then K = b2 /12 where b is the distance of the two parallel
plates. Extensions to the local cubic law which incorporate fracture surface
roughness can be found in [10].
2.2 Constitutive relations
The set of equations (1) needs to be completed with constitutive relations. We
use the Brooks-Corey capillary pressure function [11]
1
−λ
pc (Sw ) = pd Se
4
(4)
which employs the effective saturation Se of the wetting phase,
Se =
Sw − Swr
1 − Swr
Swr ≤ Sw ≤ 1.
(5)
pd is the entry pressure, Swr is the residual water saturation, and λ is related
to the pore size distribution: Materials with small variations in pore size have
a large λ value while materials with large variations in pore sizes have small
λ values. Usually λ is in the range 0.2 ≤ λ ≤ 3.
The relative permeability functions after Brooks-Corey are
2+3λ
λ
krw (Sw ) = Se
,
(6)
2+λ
λ
krn (Sw ) = (1 − Se )2 (1 − Se
).
(7)
Across the interface between the wetting phase and the non-wetting phase
a jump discontinuity occurs in the pressure, because the pressure pn in the
non-wetting phase is larger than the pressure pw in the wetting phase. This
jump is the capillary pressure pc ,
pc = pn − pw ≥ 0.
(8)
The model for the flow of a wetting fluid phase w and a non-wetting fluid
phase n in a porous medium is now described by inserting (3) into (1)
krw
∂(Φρw Sw )
K(∇pw − ρw g)) = ρw qw in Ω,
− ∇ · (ρw
∂t
µw
krn
∂(Φρn Sn )
− ∇ · (ρn
K(∇pn − ρn g)) = ρn qn in Ω,
∂t
µn
(9a)
(9b)
This model will be used throughout the rest of this paper. The coupling of
saturation and pressure by
Sw + Sn = 1,
and
pn − pw = pc ,
(10)
makes only two out of the four variables pw , pn , Sw , and Sn independent variables.
The model has to be complemented by appropriate boundary conditions and
initial conditions which have to be chosen consistent with (10). We delay the
formulation of the boundary conditions until section 3.1, where an appropriate
choice of unknowns is made based on the constitutive relationships (10).
5
fracture
matrix
pc
f
*
m
Fig. 1. Capillary pressure curves in a rock matrix and a fracture according to the
Brooks-Corey model.
2.3 Interface conditions at Media Discontinuities
The governing equations for two-phase fluid flow in porous media are only valid
if the media properties are subject to slow and smooth variation. At media
discontinuities with sharp changes in properties like permeability or porosity it
is necessary to introduce interface conditions which model the correct physical
behavior. We adapt the approach of van Duijn, Molenaar and de Neef [3] for
the treatment of media discontinuities to the case of fractured media here.
The significant influence of the capillary pressure on the fluid flow especially
at media discontinuities has been shown in laboratory experiments ([12,13]).
These results indicate that it is especially important to capture the effects of
the capillary forces in the description of the interface conditions since they are
responsible for trapping and pooling at media discontinuities.
The partial differential equations for two-phase flow are of second order in
space. An interface condition at an inner boundary does therefore have to
consist of two conditions. The first condition is fluid conservation, so we require
continuity of the flux of both phases across the interface.
For the derivation of the second condition we consider two parts of the domain,
a fracture Ωf and the matrix Ωm . (The derivation is the same if we consider
lenses or layers of different materials.) We assume a mobile wetting phase in
both matrix and fracture, hence we require that pw is continuous across the
fracture-matrix interface Γ. The absolute permeabilities in their respective
domains are

K f (x) if x ∈ Ωf ,
(11)
K(x) =
K m (x) if x ∈ Ωm .
Accordingly, the porosity Φ depends on the domain as well as the capillary
pressure function pc (Sw ) and the relative permeability functions krα . The capillary pressure functions pc (Sw ) are shaped like the functions in figure 1.
Two assumptions are essential for our framework: we do not consider blocking
fractures (e. g. fractures filled with clay) and assume that
6
• the absolute permeability in the matrix is smaller than the absolute permeability in the fractures, K m (x) < K f (y) for all x, y ∈ Ω, and
• the capillary pressure function values in the matrix are larger than the
capillary pressure function value in the fractures for the same saturation,
f
pm
c (S) > pc (S). This implies that the entry pressure of the rock matrix is
larger than the entry pressure of the fractures.
For the Brooks-Corey capillary pressure relation, the entry pressure is positive,
and there is a saturation Sw⋆ (see figure 1) such that continuity of capillary pressure can only be achieved if Swf ≤ Sw⋆ . In [3] it is shown for a one-dimensional
problem that for Swf > Sw⋆ saturation Swm must be 1 in that case. Thus we
arrive at the following condition for the saturation at the media interface:
Swm

1
=
−1
pfc (Swf )
(pm
c )
if Swf > Sw⋆ ,
else.
(12)
Physically, if the non-wetting phase is not present in the matrix (i. e. Swm = 1
and Snm = 0), then pm
n on the matrix side is undefined and consequently also
capillary pressure on the matrix side is undefined. The interface condition (12)
is called extended capillary pressure condition and a major part of the paper is
concerned with the incorporation of this condition into the numerical scheme.
2.4 Summary of model assumptions and classification of the model
The model equations of the previous sections only differentiate in terms of
material properties between fractures and matrix. This is different than in the
classical approach to fracture modeling by double porosity models [1,2], where
different equations are valid in the different regions and the coupling between
the domains is handled by the introduction of exchange terms. Different exchange terms have been proposed ([14]), but the displeasing introduction of an
additional modeling parameter remains. Other extensions are multi-porosity
models which extend the model by differentiating between two or more matrix species with different permeabilities, or the dual-permeability models of
[15,16], which considers the rock matrix not only as a storage term but allow for
transmissivity in the matrix. A mathematical analysis of the double-porosity
model was given in [17], [18], [19]. The derivation can be found for singlephase fluids in [20] and for two-phase flow in [21]. Computational aspects are
considered in [22].
Double-porosity and double permeability models fall into the class of equivalence models and have mostly been employed for problems with periodic
fracture networks. An alternative approach is the modeling of fractures with
lower dimensional elements (sometimes called shell elements). This concept
7
has been applied for the numerical simulation of fracture networks which neglect the influence of the matrix altogether and only model fluid flow in the
fracture network. Apart from this approach, which is only justified for rocks
with very small matrix conductivity, there have been finite element models
which employ elements of different dimensionality. The term discrete model
can be found in the literature for both approaches; to distinguish both the
former is also called discrete fracture network model. The combination of discrete fracture network models with a continuum model is also known as the
hybrid model. [23] was one of the earliest papers on numerical simulation of water flow in fractured porous media. It contains two finite element models, one
with two-dimensional elements for rock matrix and fractures and one model
with one-dimensional elements for the fracture network which does not take
the rock matrix into account. [24] formulated a finite element model in three
space dimensions with tetrahedral elements in the rock matrix and triangular
elements in the fractures; a similar approach is presented in [25] and in [26].
The approach of element types of different dimension is also pursued in [27,28].
The investigation of unsaturated flow received less attention in the past. [29]
modeled unsaturated flow with a discrete fracture approach. [30] considered
solute transport in a fractured porous medium with discrete fractures modeled
by one-dimensional equations and the matrix modeled by two-dimensional
equations, coupled by exchange terms. In [31] mixed-dimensional elements
were employed for two-phase flow; see also [32].
We consider the lack of an exchange term between fractures and rock matrix
in the discrete model to be an important conceptual advantage of our method.
The assumptions which are imposed on the model can be summarized as
follows.
(1) We assume that fracture width is orders of magnitude smaller than the
fracture length. Fractures are of essentially planar geometry, with an associated aperture in each point.
(2) We assume that the multiphase fluid flow equations (9) are valid in the
rock matrix and the fractures, i. e. the multiphase extension of Darcy’s
law is valid. This implies that we assume a laminar flow regime in both
domains and that an rev can be found for fracture and matrix. Multicomponent and non-isothermal behavior of the fluids is not considered.
(3) The absolute permeability of the fractures is larger than the absolute
permeability of the rock matrix. Fractures may be open or filled, but we
do not consider blocking fractures.
(4) Relative permeability functions and capillary pressure functions exist for
fractures and matrix. The capillary pressure function is assumed to be
strictly monotone decreasing, and we assume that the capillary pressure
functions for rock and matrix do not intersect.
8
(5) The wetting phase exists and is mobile in fractures and rock matrix.
3
Discretization with a vertex centered finite volume method
The character of multi-phase flow problems in porous media with discretely
modeled fractures brings along with it some requirements for the discretization
method. The complex geometry of fractured media can only be represented
with unstructured grids. Consequently, the numerical scheme has to work on
unstructured grids. Special attention has to be given to the fact that solutions
of the multiphase flow equation (9) can exhibit sharp fronts. This suggests a
locally mass conservative method since methods which are not locally mass
conservative can fail to predict the correct location of shocks or sharp fronts
(see [33,34]). Finite volume methods are suited for unstructured grids and are
locally mass conservative. Properties of the finite volume method are analyzed
in [35,36].
For the time discretization we have to consider that for parabolic equations
explicit methods are only stable if the time step ∆t is on the order O(h2 ) with
h being the mesh size. This leads to very small time step sizes if the mesh is
sufficiently fine. In order to be able to apply the method to a wide range of
applications with large time steps, we choose an implicit time discretization
method. The implicit time discretization generates large nonlinear systems
of equations. They can be solved by inexact Newton methods. The linearized
systems in the Newton method can be solved efficiently by multigrid methods,
possibly accelerated by a Krylov-subspace method such as GMRES or BiCGSTAB.
A popular method in the context of reservoir modeling is the IMPES scheme
(implicit pressure, explicit saturation), which decouples the equations into an
elliptic water phase pressure equation (which is solved implicitly) and a NAPL
phase saturation equation (which is solved explicitly). The fully coupled, fully
implicit approach has the advantage of being more robust and applicable to a
wider range of problems. It also avoids the possibly very small time step sizes
of the explicit saturation step. The stability of the fully coupled fully implicit
scheme allows for larger time steps, although they should not be taken too
large for accuracy reasons.
To compute a numerical solution for the two-phase flow equations, first an
adequate formulation for the problem has to be found. This is done in section 3.1. Then the finite volume method is presented in sections 3.2 to 3.6.
The description of the discretization method considers the domains Ωm and Ωf
separately and then explains how a coupled treatment within the framework
of conforming methods is possible. The concluding sections give an overview of
9
the time discretization, the Newton method, the multigrid method and related
implementational issues.
3.1 The phase pressure-saturation formulation
As already pointed out, only two out of the four variables pw , pn , Sw , and Sn
in the multiphase flow equations (9) can be chosen as independent variables.
We choose the substitutions
Sw = 1 − S n ,
pn = pw + pc (1 − Sn )
(13)
to obtain the (pw , Sn )-formulation. Other choices are possible, see [8]. Despite
the choice of Sn as a primary unknown, we sometimes write Sw in the following if it facilitates reading. Of course a similar exchange of pw and pn is
not possible, because the choice of pw as an independent variable has larger
implications. Formulations based on pw assume that the water phase exists everywhere in the domain. Because we consider problems in initially fully watersaturated domains with a residual water saturation (i. e. the water phase is
never completely replaced by gas) this is the appropriate choice.
The equations now read
krw
∂(Φ̺w (Sw ))
− ∇ · (̺w
K(∇pw − ̺w g))
−̺w qw = 0,
∂t
µw
∂(Φ̺n (Sn ))
krg
− ∇ · (̺n K(∇pw + ∇pc (Sw ) − ̺n g)) −̺n qn = 0,
∂t
µn
(14a)
(14b)
where we used Sw for 1 − Sn . We consider these equations in (0, T ) × Ω.
Ω ⊂ Rd , (d = 2, 3) is a domain with polygonal or polyhedral boundary for
d = 2 and d = 3 respectively. The equations are complemented with initial
conditions and boundary conditions of Neumann or Dirichlet type on the
respective boundaries Γαn and Γαd
pw (x, 0) = pw0(x)
pw (x, t) = pwd(x, t) on Γwd
̺w vw · n = φw (x, t) on Γwn
Sn (x, 0) = Sn0 (x) ∀x ∈ Ω ,
Sn (x, t) = Snd (x, t) on Γnd ,
̺n vn · n = φn (x, t) on Γnn .
(15a)
(15b)
(15c)
If both phases are incompressible no initial condition for pw is required. Γpwd
should have positive measure to determine pw uniquely.
10
In the following we assume the dependencies
g = constant
qα = qα (x, t)
pc = pc (x, Sw )
krα = krα (x, Sα )
̺α = ̺α (pα )
µα = µα (pα )
Φ = Φ(x)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
The influence of fractures on the fluid flow is included through the dependency
of the quantities in equation (22) on the position, i. e. the values are different
depending on whether they are evaluated in a fracture or in the rock matrix.
3.2 The geometry of the problem
In the previous section the domain was only specified with respect to the shape
of its boundary and without the description of fractures. We now explain the
assumptions on the fracture network which are essential for the discretization
method. In the following superscript m denotes entities in the volumetric rock
matrix and superscript f denotes entities in the fracture network.
Let Ω ⊂ Rd be a polygonal or polyhedral domain for d = 2 or d = 3, respectively. The domain contains a nonempty set of fractures {f1 , . . . , fF }. Each
fracture fi is a (d − 1)-dimensional object—i. e. we identify each fracture with
its middle surface—and each fracture fi has a width δi associated with it,
which may be variable in the fracture. For simplicity we assume the fractures
to have a planar geometry: in a two-dimensional domain the fractures are line
segments and in a three-dimensional domain we assume polygonal shape of
the fractures (although circular or elliptic shapes can also be treated by the
method, as well as non-planar shapes). The union of the fractures constitutes
the fracture network
f
Ω =
F
[
i=1
fi ⊂ Ω.
(23)
The domain of the rock matrix Ωm is the whole domain,
Ωm = Ω.
(24)
This means that the domains of the fracture network and the rock matrix
overlap.
11
Fig. 2. Example domain with fractures and mesh resolving the fracture network
geometry.
Fig. 3. Mesh, dual grid and fracture elements/volumes.
3.3 The finite volume grids and the dual grids
The discretization method requires a mesh for Ωm and Ωf . For the volumetric mesh we consider a subdivision Ehm of Ωm into K m elements, Ehm =
S m
′
m
m
m
m
m
{Ωm
e Ωe = Ω and Ωe ∩ Ωe′ = ∅ for e 6= e . Ωe is the open
1 , . . . , ΩK } with
subdomain covered by the element with index e. h denotes the diameter of the
largest element. The subdivision has to resolve the geometry of the fractures,
comparable to domains with inner boundaries. Figure 2 shows an example for
m
a two-dimensional mesh. The volumetric elements Ωm
e of Eh are triangles or
quadrilaterals in two dimensions and tetrahedra, pyramids, prisms, or hexahedra in three dimensions. Hybrid grids, i. e. grids of mixed element type are
admissible, but we require that Ehm is a triangulation: no vertex of an element
lies in the interior of a side of another element.
The volumetric elements are complemented with lower dimensional elements
on the fractures which are line elements for two-dimensional problems and triangles or quadrilaterals for three-dimensional problems. The fractures appear
as inner boundaries to the domain where material properties change, see also
[37]. The fracture elements constitute a mesh Ehf = {Ωf1 , . . . , ΩfK f } which is
conforming with the volumetric mesh, i. e. each Ωfe is an element face or face
for the two-dimensional and three-dimensional case, respectively.
The vertex centered finite volume method requires the construction of a socalled secondary mesh. For the volumetric mesh it is constructed by connecting
12
element e
m
γe,i,i
′
bm
i
γ
γef′′ ,i,i′
xi
xi′
bm
i′
fracture
element e′
Fig. 4. Notation for control volumes
element barycenters with edge midpoints as shown in figure 3 in two dimensions. In three dimensions, first the element barycenters are connected to face
barycenters and then these are connected with edge midpoints. Vertices of the
grid are denoted by vi and their corresponding coordinate vector by xi . By
construction each control volume contains exactly one vertex, and the conm
m
m
trol volume containing vertex vi is denoted by bm
i , thus Bh = {b1 , . . . , bN m }.
For two-dimensional fractures, the dual grid is generated in a similar manner.
One-dimensional elements are simply divided into parts of equal length. This
construction results in a conforming dual mesh for volumetric and fracture elements. The fracture dual mesh is denoted Bhf = {bf1 , . . . , bfN f }. Fracture and
f
matrix control volumes are related via bfi = bm
i ∩Ω .
m
m
The interface between control volume bm
i and bi′ inside of element Ωe is denoted by
m
m
m
m
γe,i,i
′ = Ωe ∩ ∂bi ∩ ∂bi′
m
m m
m
for Ωm
e ∈ Eh , bi , bi′ ∈ Bh
(25)
and from that the internal skeleton of the volumetric dual grid is denoted by
m
m
m
m m
m
Γm
int = {γe,i,i′ | Ωe ∈ Eh , bi , bi′ ∈ Bh , (xi , xi′ ) is an edge}.
(26)
Note that that the condition that (xi , xi′ ) is an edge of the (primal) grid
m
ensures that γe,i,i
′ is a (d − 1)-dimensional object.
For the fracture dual grid we define
f
f
f
f
γe,i,i
′ = Ωe ∩ ∂bi ∩ ∂bi′
for Ωfe ∈ Ehf , bfi , bfi′ ∈ Bhf
(27)
and the union of all internal control volume intersections (which are points
for one-dimensional fractures and edges for two-dimensional fractures) of the
fracture dual grid is denoted by
f
f
f f
f
f
Γfint = {γe,i,i
′ | Ωe ∈ Eh , bi , bi′ ∈ Bh , (xi , xi′ ) is an edge}.
13
(28)
f
f
In the two-dimensional case, γe,i,i
′ is already determined uniquely by e (or bi
and bfi′ ), but in the three-dimensional case all three arguments are needed.
The external skeleton is the union of the element faces on the domain boundary, defined for the volumetric dual mesh and the fracture dual mesh,
m
m
m
m
m
m
m
m
Γm
ext = {γe,i | γe,i = ∂Ωe ∩ ∂bi ∩ ∂Ω for Ωe ∈ Eh , bi ∈ Bh },
(29)
f
f
Γfext = {γe,i
| γe,i
= ∂Ωfe ∩ ∂bfi ∩ ∂Ω for Ωfe ∈ Ehf , bfi ∈ Bhf }.
(30)
With each element of the skeleton we associate a fixed unit normal n. For
f
f
m
γ m ∈ Γm
∈ Γm
ext and γ ∈ Γext we choose the outward unit normal. For γ
int
the sign of n is chosen arbitrarily, but fixed—a possible choice is to let n point
from the element with the larger index to the element with the lower index.
For γ f ∈ Γfint the normal vector is perpendicular to γ f , lies within the (planar)
fracture element and the sign is again choosen arbitrarily, but fixed.
For any function f defined on Ωm or Ωf , which may be discontinuous on Γm
int
f
or
x
∈
γ
∈
Γ
or Γfint , respectively, we define the jump of f at x ∈ γ ∈ Γm
int
int
to be
[v](x) = lim v(x + ǫn) − lim v(x − ǫn),
(31)
ǫ→0+
ǫ→0+
where n is the normal to γ.
3.4 Approximation spaces and weak formulation
For the discretization we introduce the standard conforming, piecewise linear
finite element spaces in the matrix and fracture domain
m
Vhm = {v ∈ C 0 (Ωm ) | v is linear on Ωm
e ∈ Eh },
(32)
Vhf = {v ∈ C 0 (Ωf ) | v is linear on Ωfe ∈ Ehf },
(33)
and the non-conforming test spaces (based on the secondary mesh):
m
Whm = {w ∈ L2 (Ωm ) | w is constant on each bm
i ∈ Bh },
Whf
2
f
= {w ∈ L (Ω ) | w is constant on each
bfi
∈
(34)
Bhf }.
(35)
The spaces Vhm and Vhf are equipped with the standard nodal basis
m
f
f
f
f
f N
m
m N
Vhm ∋ ϕm
i (xj ) = δij , Vh ∋ ϕi (xj ) = δij , Φh = {ϕi }i=1 , Φh = {ϕi }i=1 ,
whereas for Whm and Whf we take the basis functions that are the characteristic
functions of the control volumes:
m
f
f
f
f
f N
m
m N
ψim = χ(bm
i ), ψi = χ(bi ), Ψh = {ψi }i=1 , Ψh = {ψi }i=1 .
14
Fig. 5. Basis functions for volumetric elements and fracture elements.
Figure 5 shows two basis functions for Vhm and Vhf . Depending on the element
type (e.g. for quadrilaterals), the term linear has to be replaced by multi-linear
in (32),(33) as well as in the following.
In what follows, we will only describe the treatment of homogeneous Dirichlet
boundary conditions to avoid cumbersome notation. The subspaces for homogeneous Dirichlet boundary conditions for fracture, matrix and both phases
α = n, w are
τ
Vhα0
= {v ∈ Vhτ | v|Γαd = 0},
τ
Whα0
= {w ∈ Whτ | w|Γαd = 0},
(36)
(37)
with τ = m, f . Inhomogeneous Dirichlet boundary conditions can be treated
as in [7]. In the case of mixed Dirichlet and Neumann boundary conditions
it is necessary to employ separate function spaces for water pressure and gas
saturation, which adhere to the respective boundary conditions. These spaces
can depend on time, a feature which is not explicitly mentioned. This is only
τ
τ
a notational convenience and at all times Vhα0
and Whα0
should be thought of
τ
τ
as Vhα0 (t) and Whα0 (t).
3.5 Discontinuous saturation spaces and projections
The phase saturations Sn and Sw are discontinuous at interfaces between media with different properties as well as at all vertices vi ∈ Ωf , because these
vertices are shared by the rock matrix and the fracture network. A discontinuous saturation can not be represented by the standard conforming finite
element spaces Vhm and Vhf so instead we choose discontinuous saturation
spaces
m
Shm = {v ∈ L2 (Ωm ) | v|Ωm
linear for Ωm
e ∈ Eh },
e
Shf = {v ∈ L2 (Ωf ) | v|Ωfe linear for Ωfe ∈ Ehf }.
(38)
(39)
f
By means of the mappings Πm
h and Πh ,
m
m
Πm
h : V h → Sh ,
Πfh
:
Vhf
15
→
Shf ,
(40)
(41)
it is possible to formulate the discretization by the conforming finite element
functions from the previous section, but to employ the correct discontinuous saturation function wherever appropriate. These mappings employ the
extended interface conditions from section 2.3.
Shm is only continuous within elements so we define the mapping for a given
x ∈ Ωm
e . The values of the function
sh = Πm
h vh
vh ∈ Vhm , sh ∈ Shm
(42)
on Ωm
e are uniquely determined by values in the corners of the element via
sh (x) =
X
Sie ϕm
i (x)
(43)
i∈V m (e)
where V m (e) is the set of indices of the corner vertices of Ωm
e . Assuming that
vh is the finite element function representing non-wetting phase saturation the
nodal values Sie in element Ωm
e are found by



v(xi )
Sie = 0


1−S
if pc (xe , 1 − v(xi )) = pcmin (xi ),
if pcmin(xi ) < pc (xe , 1),
else, with S from pc (xe , S) = pcmin (xi ).
(44)
Here we employ the minimal capillary pressure function pcmin(x) with nodal
values defined as
pcmin (xi ) =
min pc (xe , 1 − v(xi ))
Ωe ∈E(xi )
(45)
and we denote the barycenter of element e as xe . E(x) is the set of all elements
which contain x in their closure:
f
f
f
m
m
E(x) = {Ωm
e ∈ Eh | x ∈ Ωe } ∪ {Ωe ∈ Eh | x ∈ Ωe }.
(46)
For fractures the same construction is employed, only that the fracture basis
functions are used instead of matrix basis functions,
sh (x ∈ Ωfe ) =
X
i∈V f (e)
Sie ϕfi (x) for sh ∈ Shf , vh ∈ Vhf .
(47)
Here, V f (e) denotes the indices of the vertices of fracture element Ωfe .
So far, Vhm and Vhf have been treated as seperate spaces with seperate unknowns in the vertices of the grid. The projection Λh , to be defined now,
16
rock matrix
fracture
rock matrix
vh ∈ Vhm
Πm
h vh
Λh vh
Fig. 6. A function vh ∈ Vh and the mapping into the saturation spaces, Πm
h vh and
f
Πh Λh vh .
connects these two spaces. Consider
Λh : Vhm → Vhf
vhm 7→ vhf
with vhm (x) = vhf (x) for all x ∈ Ωf
,
(48)
i.e. due to the conformity of matrix and fracture mesh we define Λh vh to be
the finite element function on the fracture mesh whose nodal values coincide
with a given finite element function on the matrix mesh. Figure 6 shows a
function vh ∈ Vh and illustrates the mappings Πm
h and Λh .
Similarly, we need also a projection Ξh : Whm → Whf that connects the test
spaces. This projection is defined as follows:
Ξh (whm ) = whf ⇔ whf (xi ) = whm (xi ) ∀bfi ∈ Bhf , xi center of bfi .
(49)
3.6 Weak formulation
The weak formulation of equations (14) for the rock matrix is found by multiplying with the test functions and integration by parts. We employ the fol17
lowing forms in the formulation of the weak form for the rock matrix:
m
m
m
mm
w (pwh , Snh , wwh ) :=
X
Z
X
Z
m
m
Φ̺w (1 − Snh
)wwh
dx,
bm ∈Bhm bm
m
m
m
am
w (pwh , Snh , wwh )
:=
m
̺w vwm · n [wwh
] ds
γ m ∈Γm
int γ m
Z
X
+
γ m ∈Γm
ext ∩Γwn γ m
m
m
m
mm
n (pwh , Snh , wnh ) :=
X
Z
X
Z
(50a)
m
φw wwh
ds −
X
(50b)
Z
m
̺w qw wwh
dx,
bm ∈Bhm bm
m m
Φ̺n Snh
wnh dx,
(50c)
bm ∈Bhm bm
m
m
m
am
n (pwh , Snh , wnh ) :=
m
̺n vnm · n [wnh
] ds
γ m ∈Γm
int γ m
+
Z
X
m
φn wnh
γ m ∈Γm
ext ∩Γnn γ m
ds −
X
(50d)
Z
m
̺n qn wnh
dx
bm ∈Bhm bm
m
m
m
m
m
m
with pm
wh ∈ Vh , Snh ∈ Sh from (38) and wwh , wnh ∈ Wh . The terms in
m
m
mα are called accumulation term and the terms in aα are called internal
flux term, boundary flux term and source and sink term, respectively. For
the numerical evaluation of the accumulation term we employ a midpoint
rule, which corresponds to the mass lumping approach in the finite element
method. The Darcy velocities in the interior flux terms are evaluated with an
upwind scheme. For the water phase this is for a given control volume face
m
γe,i,i
′
Z
̺w vwm
m
γe,i,i
′
· n [wh ] ds =
Z
m
γe,i,i
′
̺w λ⋆wγ m ′ ṽwm · n [wh ] ds
e,i,i
(51)
with the upwind evaluation of the mobility
λ⋆wγ m
e,i,i′
m
γe,i,i
′
= (1 − β)λwh (x
)+β·
and the directional part of the velocity
m
γe,i,i
′
ṽw = −K(xe )(∇p(x

λ
if ṽwm · n ≥ 0
λwh (xi′ ) else
wh (xi )
m
γe,i,i
′
) − ̺w (x
)g)
(52)
(53)
γm
m
x e,i,i′ is the barycenter of γe,i,i
′ and xi is the grid vertex inside control volume
bm
.
The
source
and
sink
terms
and the boundary flux terms are evaluated
i
by the midpoint rule. The analogous evaluation scheme is employed for the
non-wetting phase saturation. The parameter β controls the upwinding strategy. For β = 1 full upwinding is achieved, while β = 0 results in a central
differencing scheme. We employ a fixed β, but adaptive choices depending on
the local Peclet number are possible ([36]).
The corresponding forms mfα (·, ·, ·) and afα (·, ·, ·) for the fractures are derived
18
by replacing superscript m with f :
f
f
mfw (pfwh , Snh
, wwh
)
:=
X Z
f
f
Φ̺w (1 − Snh
)wwh
δ dx,
bf ∈Bhf bf
f
f
afw (pfwh , Snh
, wwh
)
:=
X Z
f
̺w vwf · n [wwh
]δ ds
γ f ∈Γfint γ f
Z
X
+
f
φw wwh
δ
γ f ∈Γfext ∩Γwn γ f
f
f
mfn (pfwh , Snh
, wnh
) :=
X Z
(54a)
ds −
X Z
(54b)
f
̺w qw wwh
δ dx,
bf ∈Bhf bf
f
f
Φ̺n Snh
wnh
δ dx,
(54c)
bf ∈Bhf bf
f
f
afn (pfwh , Snh
, wnh
)
:=
X Z
γ f ∈Γfint γ f
+
X
Z
γ f ∈Γfext ∩Γnn γ f
f
̺n vnf · n [wnh
]δ ds
f
φn wnh
δ ds −
X Z
(54d)
f
̺n qn wnh
δ dx.
bf ∈Bhf bf
Here function δ(x) denotes the width of the fracture. In the evaluation of
these terms the lower dimension of the integrals has to be taken into account
by using appropriate integral transformations and in the evaluation of the
directional velocity.
We are now in a position to formulate the time continuous discrete problem:
m
m
Find pwh (t) ∈ Vhw0
and Snh (t) ∈ Vhn0
such that
∂ m
f
f
mw (pwh , Πm
h Snh , wwh ) + mw (Λh pwh , Πh Λh Snh , Ξh wwh )
∂t
f
m
f
+ am
w (pwh , Πh Snh , wwh ) + aw (Λh pwh , Πh Λh Snh , Ξh wwh ) = 0 (55)
and
∂ m
f
f
mn (pwh , Πm
h Snh , wnh ) + mn (Λh pwh , Πh Λh Snh , Ξh wnh )
∂t
f
m
f
+ am
n (pwh , Πh Snh , wnh ) + an (Λh pwh , Πh Λh Snh , Ξh wnh ) = 0 (56)
m
m
holds for all test functions wwh ∈ Whw0
, wnh ∈ Whn0
and 0 < t < T . Note that
all forms are evaluated with respect to the discontinuous saturation Πm
h Snh
which is obtained from the continuous saturation. The fractures do not hold
seperate degrees of freedom but are coupled to the matrix via the projection
Λh .
19
3.7 Time discretization
The traditional approach to the numerical solution of time-dependent partial
differential equations is by the method of lines. First, a spatial discretization
is applied to the problem (i. e. the finite volume method from the previous
section), which leads to a system of ordinary differential equations. This system is then solved by a time differencing scheme which can be chosen from the
wide range of available methods, see [38,39]. The arising system of ordinary
differential equations is stiff and should be treated by implicit methods.
We divide the time interval (0, T ) into discrete time steps
0 = t0 , . . . , tM = T
of variable or fixed size and employ superscript n notation for functions (e. g.
pwh ) and coefficient vectors (e. g. Snh ) denoting values at time step tn :
pwh (tn ) = pnwh and Snh (tn ) = Snnh .
We employ the standard (multi-)linear nodal finite element basis; this introduces a unique relationship between the discrete functions pwh , Snh and their
coefficient vectors pw and Sn .
The application of the finite volume discretization scheme leads to the semidiscretization
∂
Mw (pw (t), Sn (t)) + Aw (pw (t), Sn (t)) = 0,
∂t
∂
Mn (pw (t), Sn (t)) + An (pw (t), Sn (t)) = 0,
∂t
(57)
(58)
where M corresponds to m and A corresponds to a. The system can be written
as



 
∂pw (t)
Mww Mwn   ∂t  Aw (pw , Sn )
(59)


=0
+
∂Sn (t)
Mnw Mnn
A
(p
,
S
)
n
w
n
∂t
with the submatrices
(Mαw )ij =
∂Mαw,i
∂pw,j
(Mαg )ij =
∂Mαg,i
.
∂Sn,j
(60)
This results in a system of differential algebraic equations (DAE) of index 1
in implicit form. The matrix M,

M=


Mww Mwn 
Mnw Mnn
20
,
(61)
is singular in the incompressible case. An analysis for the incompressible case
shows that a discrete form of the elliptic equation has to be satisfied. This is
called the implicit constraint. For this reason explicit methods cannot be used
for the fully coupled system. One backward Euler step guarantees the validity
of the implicit constraint. Time steps computed with other choices than θ = 1
in the one step θ method below do not fulfill this property, but they leave
the implicit constraint fulfilled if it is satisfied in the preceding time step. For
this reason, we always employ one backward Euler step as the first time step,
regardless of the time differencing scheme of the subsequent steps.
The time step scheme in the one step θ notation reads as follows. For n =
0, 1, . . . , M − 1 find pnw , Snn such that for α = w, n
n
n
Mn+1
+ Mnα + ∆tn θ(An+1
α
α ) + ∆t (1 − θ)(Aα ) = 0
(62)
For θ = 1 this yields the backward (or implicit) Euler scheme, θ = 1/2 yields
the Crank-Nicholson scheme. The implicit Euler scheme is first-order accurate
and has very good stability (strongly A-stable, [38]), the Crank-Nicholson
scheme is second order accurate but has weaker damping properties which
may cause stability problems (only A-stable).
Closely related is the fractional-step-θ scheme. It consists of three sub-steps
tn → tn+α → tn+1−α → tn+1 , where each sub-step k is a one-step-θ step with
θk and ∆tk chosen as
√
θ1 = 2 − 2
√
θ2 = 2 − 1
√
θ3 = 2 − 2
√
∆t1 = (1 − 2/2)∆t = α∆t,
√
∆t2 = ( 2 − 1)∆t = (1 − α)∆t,
√
∆t3 = (1 − 2/2)∆t = α∆t
The θi can be chosen different than above as long as θ1 = θ3 = θ ∈ (1/2, 1],
and θ2 √
= 1 − θ holds. The fractional-step-θ scheme is of second order for
α = 1− 2/2 and strongly A-stable for any θ ∈ (1/2, 1]. The scheme possesses,
other than the Crank-Nicholson scheme, the full smoothing property in case
of rough initial data. Note that the sub-stepping does not result in higher
computational cost since the step size ∆t can be chosen three times larger
than for the single-step-θ scheme.
Eq. (62) results in a large system of nonlinear algebraic equations. This system
is solved using Newton’s method and the arising linear systems are solved with
a geometric multigrid method. For details of the globalization strategy, the
multigrid method and the parallelization we refer to [4].
21
4
1
3
2
Fig. 7. Four domain features which are difficult for automatic mesh generation (left)
and possible elements of good quality (right side of illustration on the right) and
poor quality (left side of illustration on the right) for parallel and almost parallel
fractures (right).
3.8 Implementation issues
The simulator is implemented based on the numerical software toolbox UG
[40]. The large number of features (adaptivity on hybrid grids in 3D, parallelization) could otherwise not be achieved.
The implementation of finite volume codes often uses a loop over all elements
and calculates the contribution of the dual grid skeleton from inside each
element. The implementation of the presented method for rock matrix and
fractures can be done based on a volumetric element code without the need to
introduce the notion of lower-dimensional elements into the code, if fractures
are represented as inner boundaries and if each element calculates not only the
f
contributions from Γm
int , but also from Γint . Calculation of the contribution to
the stiffness matrix and the defect from Ωfe is done at both neighboring volume
elements, where each contributes half of the value. This approach is advantageous, because it stays within the element-wise implementation paradigm and
doesn’t require data communication if the method is implemented on parallel
computers with a domain decomposition approach.
Numerical simulations of multi-phase flow in fractured porous media rely on
accurate knowledge about the material properties and domain geometry. In
fractured systems the exact location of the fractures is often not known, but
a good approximation of the fracture network is crucial for the simulation
process. In these situations a fracture generator can be employed to generate fracture networks based on prescribed geological data. An example for a
fractured domain generator is FRAC3D [41], which was used to generate the
example presented in section 4.2.
The automatic generation of meshes has persisted as one of the most challenging tasks of the simulation process. The mesh generation for fractured
porous media has to treat four configurations which are especially difficult.
They are depicted in figure 7 for a two-dimensional domain, but the even
22
more severe difficulties for three-dimensional domains can be traced back to
these fundamental configurations:
(1) Fracture intersections, where one fracture end protrudes only slightly
from the intersection. A fine mesh is required in the vicinity of this region,
which should coarsen rapidly outside this region. In three-dimensional
domains, the protruding fracture end can have a “difficult” shape, e. g.
a very flat triangle.
(2) Almost meeting fractures, where one fracture ends in close proximity to
the other. The mesh should also possess a fine resolution only near this
region.
(3) Parallel fractures very close to each other should be meshed with quadrilaterals in two space dimensions and with hexahedrons or prisms in three
space dimensions, because they can cause the generation of elements with
large inner angles.
(4) Fractures intersecting at a very small angle can lead to the generation of
elements with very large angles.
In both latter cases, automatic mesh generation is especially susceptible if
the grid vertices are placed by a penalty functional which aims at an even
distribution of vertices. In the case of parallel or almost parallel fractures this
necessarily leads to large angles, where two vertices on the two fractures, placed
close to each other, would produce preferable elements. This is illustrated on
the right in figure 7. Even if the quadrilateral elements on the right side are
divided into triangles, their largest angles are still close to 90◦ , while the largest
angles of the elements on the left side are close to 180◦ . Grids for the examples
in the following section were created with the mesh generator ART [42].
Finally, the visualization of fractured porous media requires rendering of values
on hyperplanes and separate data sets for saturation values in fractures and
matrix. Visualizations in the following section were done with OpenDX [43].
4
Examples
Both following examples are taken from [44].
4.1 Validation of the model
The first numerical experiment is chosen to assess the difference between a
lower-dimensional fracture approach and a fracture with volumetric elements.
The setup includes one vertical fracture inside a domain of 1[m] × 1[m]; a
23
0.5 m
b
1m
0.2 m
1m
Fig. 8. Sketch of the domain for the vertical water-gas flow example and Sg and uw
at t = 70 s.
sketch is displayed in figure 8. The fracture is located along the line from
(0.5m, 0.2m) to (0.5m, 0.8m). We consider inflow of a compressible gas phase
at the south boundary.
The parameters of the simulation are artificially chosen but give a representative picture of fracture-matrix interaction. The parameters are
pn
[kg/m3 ]
84149.6
µg = 1.65 · 10−5 [Pa s]
Φm = 0.1
K m = 10−12
m
Swr
=0
m
Sgr = 0
ρw = 1000 [kg/m3 ]
µw
Φf
Kf
f
Swr
f
Sgr
ρg =
= 10−3 [Pa s]
= 0.3
= 10−8
=0
=0
λf = 2
λf = 1000 [Pa]
λm = 2
λm = 2000 [Pa]
The fracture width b is chosen as 0.005 [m]. Boundary conditions are Sg =
pw = 0 on the north boundary, and Neumann boundary conditions elsewhere.
At the south boundary the value of the Neumann boundary condition for the
saturation is −2.5 · 10−5 [kg/m2 ], all remaining Neumann boundary condition
values are 0.
Without the fracture, the problem would be quasi one-dimensional, but a look
at the flow field reveals how the influence of the fracture affects the solution
behavior (figure 8, right). We choose a vertical fracture in order to exclude as
much grid dependent phenomena as possible from the experiment.
Two different coarse grids are employed. The first grid employs the mixed dimensional finite volume method and models the fracture by one-dimensional
elements. The second grid resolves the fracture with two-dimensional elements,
24
Fig. 9. Comparison of the gas saturation along the vertical line at times t = 74s
and t = 100s for b = 0.005 on level l = 4.
using only quadrilateral elements. The number of elements and nodes in the
grids for each refinement level is given in table 1. For the simulation we employ
the backward Euler scheme with fixed time step size. The nonlinear equations
are solved by the inexact Newton method with line search. The linear systems are solved with the V(2,2)-cycle multigrid method with ILU smoothing,
accelerated by Bi-CGSTAB.
We compared the discretization schemes by plotting the value of Sg at different
time steps along the line from (0.5, 0) to (0.5, 1). The time steps are chosen as
t = 74 s, 100 s. At t = 74 s gas has entered the fracture and at t = 100 s the
gas has reached the fracture end and has penetrated the matrix. In figure 9
we display the shape of the saturation curve for the two discretizations at
t = 74 s and t = 100 s. This is an interesting comparison, because it compares
the mixed-dimensional model with a “trusted one”, the model with only twodimensional elements.
The difference in the location of the gas front arises from the different behavone-dimensional
Level l
two-dimensional
∆t [s]
#E
#N
#E
#N
2
2
153
128
221
192
3
1
561
512
825
768
4
0.5
2145
2048
3185
3072
5
0.25
8385
8192
12513
12288
6
0.5
33153
32768
7
0.25
131841
131072
Table 1
Time step ∆t, number of elements #E and nodes #N for grids of the vertical gas
flow problem.
25
Number of nonlinear iterations
Number of linear iterations
1600
number of nonlinear iterations
number of linear iterations
16000
1D
2D
14000
12000
10000
8000
6000
4000
2000
1D
2D
1400
1200
1000
800
600
400
200
0
0
0
50
100
150
200
250
300
350
400
0
50
100
150
200
250
300
350
400
timestep
timestep
Fig. 10. Linear and nonlinear iterations for vertical infiltration problem with
b = 0.005 on level l = 4.
ior of the two models at the fracture ends where the pressure exhibits large
gradients. A grid refinement study did not positively reveal that both models converge towards the same solution. Note also that the mixed-dimensional
approach requires h ≫ b and thus asymptotic convergence has no meaning.
Both approaches can only be compared for a limited range of values for the
mesh size h. Different values of the fracture width b lead to similar results,
therefore we restricted the presentation to one representative value.
Note that the total mass in the system is the same in both realizations. A contradicting impression could arise because the saturation curve resembles a onedimensional model problem and seems to reveal that in the lower-dimensional
case less mass is present. This is not the case—only the mass in the fracture is
different. Apparently, the fracture geometry has a notable influence on the solution. Triangular or rounded shapes of the fracture ends lead to yet different
solutions. The difference are small enough to be of relatively small significance
when compared to other uncertainties associated with the modeling and simulation process.
We compared the results also for a domain where the fracture extends up to
the south domain boundary and gas flows directly into the fracture. In this
case, no differences between the saturation curves is visible. This implies that
in the case of direct inflow of gas into the fractures, the differences are smaller
than in the model example.
The differences in computation time are quite remarkable. In figure 10 we plot
the number of linear multigrid cycles necessary for the simulation with 400
time steps. The systems in the lower-dimensional realization are easier to solve
and require fewer iterations than the two-dimensional realization. The curves
also reveal that the system solution only starts to get demanding when the
gas has reached the fracture and the nonlinearities in the constitutive relations
from the discontinuous material properties make the systems more difficult to
solve.
26
Fig. 11. Domain with eight fractures and the initial grid.
4.2 3D example
We demonstrate the capability of the numerical simulator with an example
of a complex fracture network. The enclosing domain for the gas-water flow
simulation has a size of 12×12×18 [m] and contains an interconnected fracture
network with eight fractures. The domain and the initial coarse grid is shown
in figure 11. Two fractures are connected to the south domain boundary. It is
through these fractures that the compressible gas phase enters the system.
The parameters of the simulation are (some of them taken from [45])
ρw
µw
Φf
Kf
f
Swr
f
Sgr
= 1000 [kg/m3 ]
= 10−3 [Pa s]
= 0.084
= 2.41 · 10−12
= 0.04
=0
ρg
µg
Φm
Km
m
Swr
m
Sgr
= pg /84149.6 [kg/m3 ]
= 1.65 · 10−5 [Pa s]
= 0.114
= 3.86 · 10−15
= 0.18
=0
λf = 1.13
λm = 0.684
pfd = 3186.78 [Pa]
pm
d = 15559 [Pa]
We prescribe a hydrostatic pressure at the boundaries with pw = 9810 at the
north boundary. The boundary condition for the saturation is Sg = 0, except
for the south boundary with Neumann boundary conditions φ = 0 except for
the lines where the fractures intersect the domain boundary. There we set
φ = −2.4 · 10−4 .
We consider two cases of grid refinement. In the case of uniform refinement 37
million elements exist on level 5; this grid does not fit into the memory of a
single processor computer. The computation was carried out on the HELICS
27
Fig. 12. Local grid refinement of the fractured domain.
cluster (HEidelberg LInux Cluster System) consisting of AMD Athlon 1.3 GHz
processors connected by a Myrinet 2000 interconnect. An alternative configuration is reached by local refinement. After one step of uniform refinement
we refine only elements which contain a fracture node. The resulting grid is
then suitable for typical workstation computers; the sequential results were
obtained on a 1.8 GHz PowerMac G5 with 2 GB DDR SDRAM main memory. The exact number of elements and nodes is shown in the following table.
The series of locally refined grids is depicted in figure 12.
uniform refinement
Level l
local refinement
#E
#N
#E
#N
0
1.143
300
1.143
300
1
9.144
1.943
9.144
1.943
2
73.152
13.833
52.782
9.584
3
585.216
104.033
224.747
40.413
4
4.681.728
806.145
848.863 152.319
5
37.453.824 6.345.473 3.199.899 573.632
28
Fig. 13. Saturation in fractures and rock matrix. Saturation in the matrix corresponds to the isosurface for Sg = 0.01.
P
#P
T [s]
∆t [s]
nnl
nlin
steps
TD [s]
TA [s]
TL [s]
T [s]
16
4.5
0.125
291
1934
36
17.9
40.53
9.44
35260
256
4.5
0.0625
575
4204
72
9.19
19.77
5.01
37714
Table 2
Results from the parallel computation. All times are measured in seconds. Note that
one step of the time stepping scheme consists of three one step θ sub-steps.
The simulation uses the finite volume scheme with full upwinding and the
fractional-step-θ scheme. In each Newton step a defect reduction of 10−5
is prescribed, and a linear defect reduction of 10−5. The multigrid method
uses a V(2,2)-cycle with symmetric Gauß-Seidel smoothing. The smoothing
is damped with a factor 0.8. The time step size changes depending on the
number of required nonlinear iterations.
Figure 13 shows the gas saturation in the fracture network and the matrix
for three time steps obtained with the locally refined mesh. The gas fills the
fracture rapidly and enters the rock matrix when the entry pressure is reached.
The fluid velocity in the fractures depends on their orientation and is fastest
for the almost vertical fractures. At fracture intersections, the gas distribution
is also influenced by the fracture orientation and most gas enters into fractures
with vertical or almost vertical orientation.
The parallel simulation was carried out on 16 and 256 nodes with the uniformly
refined grids. With this configuration, the results in table 2 are obtained.
We show the number of processors #P , the simulated time T , the time step
size ∆t, the number of nonlinear solution steps nnl , the number of linear
solution steps nlin , the number of time steps, the average execution times
for nonlinear defect calculation TD , assembling of the Jacobian TA and one
P
linear cycle TL and the total computing time T for obtaining the solution
(nnl · (TD + TA ) + nlin · TL ). Note that for one refinement step the number of
29
elements increases by a factor of eight and that the time step size is halved.
P
This implies that for perfect speedup, the total execution time T would be
the same for both configurations. Instead, the run time is larger by a factor of
1.07. The additional runtime is explained by the longer time for each multigrid
solution cycle and the larger number of nonlinear and linear steps necessary in
the refined computation. Note that the fractional θ scheme has been used in
the simulations, which consists of three one step θ steps. This means that when
calculating the average number of Newton steps in one time step, nnl /(steps),
one should divide the number by 3. Thus the method needs less than three
Newton step on average in each one step θ step.
5
Conclusion
We presented a fully coupled, fully implicit, mixed-dimensional vertex centered finite volume method for the discretization of two-phase flow problems
on unstructured hybrid grids. The simulator uses a multigrid method for the
fast solution of the linearized systems and it’s solution is accelerated by using a
parallel implementation. A comparison for a model problem in 1D showed that
the differences between the mixed-dimensional discretization and a fully volumetric discretization are small. The mixed-dimensional discretization yields
systems which are far easier to solve than those from the fully volumetric
discretization approach. In 3D we showed the simulation of compressible gas
flow in a complex fracture network. Through local grid refinement, a suitably
fine mesh can be employed for the fractures, while the problem still fits into
memory of a typical desktop computer.
An extension to the case of multiphase, multicomponent flow would be very
interesting for long-term nuclear waste repository safety analysis. The flexibility of the simulator also allows for the simulation of problems from oil reservoir
engineering.
Acknowledgements
Financial support for this work was provided by the BMBF (Federal Ministry
for education and research, Bundesministerium für Bildung und Forschung)
under contract FKZ 02E 9370.
Symbols
30
α
phase n or w
ρ
density
[kg/m3 ]
Φ
effective porosity
[-]
p
pressure
[Pa]
λ
Brooks-Corey parameter
[−]
φ
Value of Neumann boundary condition
[kg/m2 ]
µ
viscosity
[-]
θ
time discretization parameter
∆t
time step size
Bhτ
Dual grid, τ = m, f
Ehτ
Mesh, τ = m, f
J
Jacobian matrix
K
absolute permeability
Sα
saturation of phase α
Sαr
residual saturation of phase α
Vhτ
conforming finite element space, τ = m, f
Whτ
finite volume test space, τ = m, f
bτi
box volume (dual grid), τ = m, f
d
space dimension (2 or 3)
e
finite element index
g
gravity vector
[m/s2 ]
pα
phase pressure, α = n, w
[Pa]
pc
capillary pressure
[Pa]
pd
entry pressure
[Pa]
n
normal
[-]
vα
Darcy velocity of phase α
[m/s]
v
basis function
w
test function
x
position in Rd
[m2 ]
31
Ω
Domain
Ωf
Domain of volumetric fractures
Ωm
Domain of rock matrix
Ωτe
Domain of finite element e, τ = m, f
Λh
Projection into fracture space
Πh
Mapping into saturation space
Γτint
interior grid skeleton, τ = m, f
Γτext
interior grid skeleton, τ = m, f
Subscript symbols
g
gas phase related quantity
n
non-wetting phase related quantity
w
wetting phase related quantity
Superscript symbols
f
A fracture related quantity
m
A rock matrix related quantity
n
time step
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