Silje Kjonaas Teveldal (main adviser), 2014.

Silje Kjonaas Teveldal (main adviser), 2014.
Dynamic Capillary Effects in the Simulation of
Flow and Transport in Porous Media:
A New Linearisation Method
Master’s Thesis in Applied and Computational Mathematics
Silje Kjønaas Teveldal
Department of Mathematics
University of Bergen
December 25, 2014
ii
Acknowledgements
First and foremost I would like to extend my everlasting gratitude towards my
supervisor Florin A. Radu. Thank you for your excellent guidance and advice, as
well as your never-ending patience when I come to you for help. I greatly appreciate
the opportunities to travel as part of my degree thanks to your dedication and
initiative.
A special thanks to Florian Doster for his input and helpful advice, as well as the
rest of the people at the Institute of Petroleum Engineering at Heriot-Watt University in Edinburgh. Their generous hospitality made my stay both educational
and memorable.
Thanks to all my friends, especially the ones I have made along the way, for making
my time at university the best time of my life yet. Also, I am forever grateful for
my wonderful family and boyfriend for standing by me and always lending their
support.
Last but not least, I would like to pay tribute to all the fantastic teachers and
lecturers I have had over the years whose inspiration has been invaluable in my
choice to pursue a degree in mathematics.
Silje Kjønaas Teveldal,
December 2014
iii
Abstract
In this thesis mathematical models with and without dynamic capillary effects are
developed to model water flow and solute transport through a porous medium.
The system of equations are discretised using the finite volume method TPFA
in space and the backward Euler method in time. To solve the nonlinear systems appearing at each time step numerically, robust linearisation methods are
proposed. These methods do not involve the computation of derivatives. The
methods are analysed and have been shown to be linearly convergent and robust.
Moreover, the convergence was shown to be independent of mesh size. The influence that the dynamic effects have on flow and transport is studied numerically.
Additional numerical experiments were conducted to study the convergence of the
linearisation schemes. The numerical results are shown to be in correspondence
with the theoretical results.
v
Contents
Acknowledgements
iii
Abstract
v
1 Introduction
1
2 Mathematical Modelling of Porous Media Flow
2.1 Flow in Porous Media . . . . . . . . . . . . . . . .
2.1.1 Physical Properties . . . . . . . . . . . . . .
2.1.2 Fluid Properties . . . . . . . . . . . . . . . .
2.1.3 Darcy’s Law . . . . . . . . . . . . . . . . . .
2.1.4 Mass Conservation . . . . . . . . . . . . . .
2.1.5 Diffusion and Transport Equations . . . . .
2.2 Two-Phase Flow Model . . . . . . . . . . . . . . . .
2.2.1 Two-Phase Flow . . . . . . . . . . . . . . .
2.2.2 Capillary Pressure . . . . . . . . . . . . . .
2.2.3 Richards’ Equation . . . . . . . . . . . . . .
2.2.4 Parameterisations . . . . . . . . . . . . . . .
2.3 Non-Standard Models . . . . . . . . . . . . . . . . .
2.3.1 Dynamic Capillary Pressure and Hysteresis .
2.3.2 Extension of the Standard Model . . . . . .
2.4 The Mathematical Model . . . . . . . . . . . . . . .
2.4.1 Simplifications of the Mathematical Model .
2.4.2 Representative Equations . . . . . . . . . .
3 Numerical Methods
3.1 Grid . . . . . . . . . . . . . . .
3.2 Discretisation in Space . . . . .
3.2.1 Finite Difference Method
3.2.2 TPFA . . . . . . . . . .
3.2.3 Boundary Conditions . .
3.3 Discretisation in Time . . . . .
3.4 Linearisation . . . . . . . . . .
3.4.1 Linearisation Richards .
vii
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31
35
36
37
Contents
3.5
3.6
3.4.2 Linearisation Dynamic Capillary Pressure . . . . . .
Discretisation in Space and Time . . . . . . . . . . . . . . .
3.5.1 Convection-Diffusion Equation . . . . . . . . . . . . .
3.5.2 Standard Richards’ Equation . . . . . . . . . . . . .
3.5.3 Richards’ Equation with Dynamic Capillary Pressure
Implementation . . . . . . . . . . . . . . . . . . . . . . . . .
4 Numerical Results
4.1 Convergence for an Academical Example . . . . . . . . .
4.2 Numerical Simulations . . . . . . . . . . . . . . . . . . .
4.2.1 Numerical Solutions of Flow and Transport . . .
4.2.2 Convergence History of the Linearisation Schemes
5 Conclusion
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55
58
59
69
75
Chapter 1
Introduction
A classic case of gravity-driven flow in porous media is infiltration of water through
soil. An important application connected to this is the pollution of groundwater.
Organic compounds deposited on the ground surface or being used at i.e. chemical
factories, enter the soil and through infiltration of water containing the dissolved
substances these contaminants can reach the groundwater. Stability of the flow
field is a key question for the infiltration, as the formation of what is known as
preferential flow paths can create large consequences on the transport of contaminants to ground and surface waters. It has been shown through experiments that
the instability is related to a phenomena in one dimensional infiltrations called
saturation or pressure overshoot [9].
Through experimental evidence presented in [18] it was suggested that hysteresis
and dynamic effects in the capillary pressure relationship represent an important
role and have the potential of describing phenomena of saturation overshoot and
preferential flow paths. In order to evaluate the safety connected to a contaminated site, a reliable prediction of the water movement and solute transport is
of key importance. So to model the water flow the Richards equation will be
used. However, when this equation is based on the static relation for the capillary pressure, given by pc = pa − pw , it is unable to predict unstable infiltrations.
Thus, mathematical models including dynamic or non-equilibrium effects will be
considered.
Robust and flexible numerical methods are needed to successfully handle these
conditions. To recognise the most optimal numerical methods, a set of conditions
the methods should satisfy are introduced. Such conditions include the principle
of mass conservation as well as the method’s ability to yield explicit expressions
for the fluid flux inside the medium at a lowest possible CPU time. The finite
1
2
Chapter 1. Introduction
volume method TPFA satisfies these conditions, while the classical methods such
as finite difference (FD) and finite element method (FEM) are not optimal with
regards to these conditions [19].
In this thesis the TPFA method will be used for the spatial discretisation and
the backward Euler method for the temporal discretisation. On each time step
a nonlinear system of equations arises. In order to solve this system numerically
a linearisation scheme is needed. A common method for solving such systems is
Newton’s method. However, two main concerns regarding the Newton method is
that the Jacobian of the system has to be assembled, as well as the fact that the
convergence of the method is not guaranteed when the initial guess is not “close
enough” to the solution, implying a restriction on the time step.
Therefore, to treat the nonlinearities of the PDEs in the mathematical models,
two linearisation schemes will be proposed and analysed. The first scheme applies
to the standard Richards equation and is based on the works presented in [37]
and [36]. It is in this thesis applied for the TPFA method for the first time. The
second linearisation scheme applies to Richards’ equation with dynamic capillarity.
It continues the works in [36, 37]. The scheme is new. The linearisation schemes
do not involve the computation of derivatives and will be shown to be very robust.
Moreover, the convergence of the schemes is independent of the mesh diameter,
this being an important advantage when compared to other linearisation methods.
The outline of this thesis is set up in the following way. In Chapter 2 representative mathematical models will be developed to describe flow and transport in
porous media. Chapter 3 gives an introduction to the numerical methods used
to solve the system of equations. The equations are discretised using the TPFA
and backward Euler method. Additionally, to treat the nonlinearities of the model
equations, two numerical linearisation methods are proposed and the convergence
analysis is presented. Further, numerical simulations are conducted and the numerical results are then presented in Chapter 4. Here, the flow and transport
profiles as well as the convergence of the linearisation schemes will be evaluated
numerically. The conclusion is given in Chapter 5.
Chapter 2
Mathematical Modelling of
Porous Media Flow
This chapter is devoted to giving an overview of the equations describing flow in
porous media. Through discussion of the physical properties of the fluids and the
porous medium, the background for the mathematical model will be provided and
the governing equations used to model contaminant transport will be presented.
Further, to account for possible dynamic effects related to the phenomena of saturation overshoot and the origination of preferential flow paths, certain extensions
of the standard model will be given. To conclude the chapter, some assumptions
in order to simplify the model will be presented and a summary of the main equations throughout the chapter is made. The aim of this chapter is thus to present
and explain the equations needed to construct the representative mathematical
models which are used in the numerical methods discussed later in the thesis.
2.1
Flow in Porous Media
Several important properties of the fluid and the porous medium has to be taken
into consideration when deriving the mathematical model. These properties, as
well as the main equations for describing flow in porous media, being Darcy’s law
and equations of mass conservation, will be presented in the following.
3
4
Chapter 2. Mathematical Modelling of Porous Media Flow
2.1.1
Physical Properties of a Porous Medium
The focus of this thesis includes the transport of contaminants through soil, which
can be defined as a porous medium. However, soil is merely one example in a
vast group of porous materials and domains ranging from lungs and kidneys to
groundwater aquifers and oil reservoirs. A common factor in all these examples
is that part of the domain is occupied by the solid matrix, while the remaining
part known as the void space consists of pores. The pores in a porous medium
are occupied either by a single fluid phase, or by multiple fluid phases, e.g., gas,
water and oil, with each phase occupying a distinct portion of the void space [1].
Flow pathways exist within the pore space of the material, often consisting of
a complex structure of both interconnected and isolated pores. The fine scale of
these flow paths cannot reasonably be resolved, instead averages over length scales
more convenient are defined. The scale of choice is referred to as a representative
elementary volume (REV) [2]. Then to one mathematical point in space, within
the porous medium, the properties of the REV surrounding this point is associated,
see Figure 2.1.
The length scale of the REV typically range from one centimetre to a few tens of
centimetres, and is large enough to allow for meaningful averages of the void space
and solid matrix to be defined and laboratory measurements to be made [2]. This
is known as the continuum approach and is recognized for its ability in preserving
heterogeneities in the medium even though an exact small-scale representation of
the pores is not obtained [14].
REV
Solid matrix
x
Porous medium
Void space
Figure 2.1: The representative elementary volume (REV) of point x in space
of a porous medium.
Chapter 2. Mathematical Modelling of Porous Media Flow
5
Let Ω denote the REV. By expressing the void space volume as Ωv and the solid
matrix volume as Ωs , so that
Ω = Ωv + Ωs ,
(2.1)
the porosity can be defined as
φ=
Ωv
.
Ω
(2.2)
In other words, the porosity is the ratio between the volume of the voids in the
REV and the total volume of the REV.
Below the ground surface the domain is commonly divided into a saturated zone
and an unsaturated zone. The two zones are separated by the water table, where
the pressure head is equal to the atmospheric pressure [13]. In the saturated zone
all available pores are filled with water, while in the unsaturated zone this is not
the case. Hence, there are two phases present in the void space of the unsaturated
zone. These are water and air. When dealing with two-phase flow, the need to
indicate the fraction of the pore space occupied by each fluid develops. Denoting
Ωw and Ωa as the volume occupied by water and air respectively, the fraction of
water, also known as water saturation, is given by
Sw =
Ωw
,
Ωv
(2.3)
Sa =
Ωa
.
Ωv
(2.4)
and the fraction of air by
Since the sum of the volumes occupied by water and air equals the void space
volume
Ωv = Ωw + Ωa ,
(2.5)
the sum of the water saturation and the air fraction equals one
Sw + Sa = 1.
(2.6)
From this, the water content θ can be defined as the volume of water divided by
the total volume of the REV. It is given by the porosity φ, multiplied with the
water saturation Sw
Ωw Ωv
Ωw
θ w = Sw φ =
=
.
(2.7)
Ωv Ω
Ω
6
2.1.2
Chapter 2. Mathematical Modelling of Porous Media Flow
Fluid Properties
Viscosity of a fluid, µ, and density of a fluid, ρ, are two important properties
in the modelling of flow in porous media. The viscosity is a measure of internal
friction within a phase, and describes the phases’ resistance to flow [14]. Meaning,
the higher the value - the slower the flow.
The density is defined as the ratio between mass and volume of a fluid
ρ=
Mass of fluid
.
Volume of fluid
(2.8)
For a given temperature, T , the density of a fluid is normally dependent on the
pressure, p, applied to the fluid. Therefore, in practice, a fluid is usually compressible.
2.1.3
Darcy’s Law
First published by Henry Darcy in 1856, Darcy’s law is one of the most important building blocks for the description of flow in porous media [2]. The basis of
the relation was formed by the study of empirical experiments related to water
treatment and the design of sand filters (detailed explanation of the experiments
in [2] p.17-19). It is worth noting, that due to friction between the phase and the
wall of the pore being a dominating factor for flow in pores, the hydrodynamic
flow equations, e.g. the Navier-Stokes equation, cannot be used to model flow in
a porous medium [3].
From his experiments, Darcy found that the volumetric flow rate q is proportional
to the cross-sectional area A, the difference in hydraulic head h, and inversely
proportional to the distance between the measurement points l [2]. Giving,
q∼
A(h2 − h1 )
.
l
(2.9)
By including the hydraulic conductivity κ, and dividing by the area A, eq. (2.9)
can be expressed as the volumetric flux u of water through the column
u≡
q
κ(h2 − h1 )
=
.
A
l
The relation was later derived mathematically [2].
(2.10)
Chapter 2. Mathematical Modelling of Porous Media Flow
Introducing h =
equation yields
p
ρg
+ z and k =
u=−
κµ
,
ρg
7
the extension of Darcy’s law into a differential
k
κ
∇(p + ρgz) = − (∇p − ρg),
ρg
µ
(2.11)
where k is the permeability, µ the fluid viscosity, p the pressure, ρ the fluid density, g is the gravitational acceleration and z the height against the gravitational
direction. The gradient of the hydraulic head represents the fluids ability to flow
at a given spatial point in the porous medium.
The hydraulic head, h, is found by examining the state of water in the porous
medium, which is described by its energy [13]. From elementary physics, recall
that
Energy = Kinetic Energy + Potential Energy.
(2.12)
Assuming the flow to be a so-called laminar flow, meaning the flow of water
being sufficiently slow, the kinetic energy may be neglected. Additionally, by
disregarding the influence on the flow by all other factors than the pressure and
gravitational forces acting on the fluid, the potential energy at a given spatial
point in the porous medium may be written as
Potential Energy = Pressure Potential + Gravitational Potential.
(2.13)
The potential energy of a fluid in a porous medium is often called the hydraulic
potential [13]. By inserting the formulas for the potentials into eq. (2.13), the
following equation is obtained
mgh = pV + mgz.
(2.14)
Here m is the mass of the fluid, V the volume, p is the pressure on the fluid at
the spatial point being considered and z the elevation from a reference level called
datum. Manipulations of eq. (2.14) gives the formula describing hydraulic head
h=
p
pV
+z =
+ z.
mg
ρg
(2.15)
The second equality holds from the fact that ρ = m
, and the minus sign in Darcy’s
V
law is added since a fluid in a porous medium flows from higher values to lower
values of hydraulic head.
8
Chapter 2. Mathematical Modelling of Porous Media Flow
The permeability k is another important property of the porous medium [2]. It
measures the ability of the porous medium to transmit fluid, and is an average
property of the medium [4]. Meaning, a porous medium with a large permeability
has a higher ability to transmit fluid through its pore space than a porous medium
with a small permeability. However, if the permeability is sufficiently close to zero
the porous medium is so-called impermeable, which means that it does not transmit
fluid through its pores [13].
An interesting aspect of flow in porous media is that the porous medium may
allow a fluid to flow more easily in one direction than another [2]. Therefore, the
concepts of anisotropic and isotropic material and homogeneous and heterogeneous
material are important to derive for further understanding of the permeability.
If the material making up the solid matrix of a porous medium is anisotropic,
the permeability changes value depending on the direction being considered. If
there are no directional differences in the permeability, the material is said to be
isotropic. When the permeability changes as a function of spatial location, the
material is referred to as heterogeneous. Conversely, when a material is spatially
uniform, it is called homogeneous [2]. For homogeneous and isotropic media, the
permeability k is a constant scalar [3]. The medium considered in this thesis is
assumed to be an isotropic, homogeneous medium.
Although there are two phases present in unsaturated soil, the interest in this
thesis lies in modelling the flow of the water phase. The presence of the air phase
does however influence the flow of water, seeing as the air phase occupies some of
the pore space reducing the set of pores through which the water phase is able to
flow. This results in an increase in difficulty in the fluid flow, which is reflected
in a lower value of the apparent permeability [2]. To account for this reduction,
the absolute permeability k in eq. (2.11) is multiplied with a relative permeability
kr , which is a function of volumetric occupancy of the fluids [2]. The relative
permeability of the water phase depends on the water saturation
kr = kr (Sw ),
(2.16)
whereas the absolute permeability k is a material parameter depending on the
medium. The effective permeability for the water phase is given by
ke = kr (Sw )k,
(2.17)
which is a reduced permeability due to the presence of an air phase in addition to
the water phase.
Chapter 2. Mathematical Modelling of Porous Media Flow
9
Darcy’s law for the water phase is then given by,
uw = −
kr (Sw )k
(∇pw − ρw g).
µw
(2.18)
Here µw denotes water viscosity, ρw the water density and pw the water pressure.
2.1.4
Mass Conservation
The mathematical statement of the principle of conservation of mass is another
important building block when modelling flow in porous media. The basis for
the equation of mass conservation is formed by the statement that the change of
mass of a particular substance within a volume has to be equal to the amount of
mass created inside the volume, minus the mass that leaves the volume through
its boundaries [2]. In the following, the mass conservation equation will be derived
in a similar fashion as in [5] and [2].
n
Ω
∂Ω
Figure 2.2: A domain Ω with boundary ∂Ω and outward unit normal n
First, introducing an arbitrary volume Ω, with boundary ∂Ω and outward unit
normal n, see Figure 2.2. The mass per total volume of a species is given by the
porosity φ times its density ρ. Then, the time variation of the total mass in Ω is
given by
Z
∂
φρ dV.
(2.19)
∂t Ω
Using Leibniz integral rule [6] this expression becomes
Z
Ω
∂
(φρ) dV.
∂t
(2.20)
For this derivative not to equal zero, there has to be a source or sink inside Ω or
a flux through the boundary ∂Ω. Both will be considered in the following.
10
Chapter 2. Mathematical Modelling of Porous Media Flow
The net flux over the boundary ∂Ω is given by
Z
(ρu) · n dS,
(2.21)
∂Ω
where u is the volumetric flux vector and n denotes the outward unit normal to
the surface ∂Ω. Defining the source density Q, the total production or destruction
is given by
Z
Q dV.
(2.22)
Ω
Collecting the previous terms (eqs. (2.20) to (2.22)), the mass conservation equation on integral form is described by
Z
Z
Z
∂
(φρ) dV +
(ρu) · n dS =
Q dV.
(2.23)
∂Ω
Ω
Ω ∂t
To obtain the more general form of the mass conservation equation, the divergence
theorem [6] is applied to the boundary integral, u is assumed to be sufficiently
smooth and it is acknowledged that eq. (2.23) holds for any arbitrary closed volume
Ω. Then, the differential form of the mass conservation equation is obtained
∂
(φρ) + ∇ · (ρu) = Q.
∂t
(2.24)
As in the previous section covering Darcy’s law, the mass conservation equation
can be expressed for the water phase. Water being an immiscible fluid as it does
not mix with air, the mass of the water phase is a conserved quantity, satisfying
∂(ρw φw Sw )
+ ∇ · (ρw uw ) = Qw .
∂t
(2.25)
Using that θw = φw Sw the equation for the conservation of mass for the water
phase is described by the following equation,
∂ρw θw
+ ∇ · (ρw uw ) = Qw .
∂t
uw is the volumetric flux obtained from Darcy’s law.
(2.26)
Chapter 2. Mathematical Modelling of Porous Media Flow
2.1.5
11
Diffusion and Transport Equations
The point of interest in this thesis lies not only in the overall fluid phase, but
rather in the movement of one or more of the components that make up the phase
in question, namely the water phase. More specifically, the transport of dissolved
contaminants. Let c = c(x, t) define the concentration of a component. The
conservation equation for the component within the fluid phase can then be given
as
∂c
+ ∇ · J = Q,
(2.27)
∂t
where J denotes the flux over the boundary ∂Ω of the domain Ω. Flux is a quantity
defined on a “per area, per time” basis [2]. Q still denotes any sources or sinks.
A dissolved component may be transported by the means of advective transport or
molecular diffusion. Advective transport is referred to as the transport by the bulk
flow of the fluid phase, whereas for diffusion the fluid is at rest and the molecules
move from areas of high concentration to areas of low concentration by random
movements of the dissolved particles. Experimental evidence leads to the following
law
J(1) = −D∇c
(2.28)
describing diffusion. Where D is the diffusion coefficient, also called molecular
diffusivity [5]. Equation (2.28) is known as Fick’s first law [7].
Inserting eq. (2.28) into eq. (2.27), the diffusion equation is obtained
∂c
− ∇ · (D∇c) = Q.
∂t
(2.29)
For a fluid in motion, convection of the particles takes place. This being described
by
J(2) = uc.
(2.30)
Here, u is the velocity of the fluid found by Darcy’s law (eq. (2.11)). By taking both
transport and diffusive processes into account, the convection-diffusion equation
is obtained
∂c
− ∇ · (D∇c − uc) = Q.
(2.31)
∂t
The relative strength between the two processes in eq. (2.31) is measured by the
Péclet number [5]. One process may dominate the other, in which case the dominated process may be ignored and only the dominating process considered. I.e.,
12
Chapter 2. Mathematical Modelling of Porous Media Flow
if convection dominates, diffusion can be ignored, and the transport equation is
considered
∂c
+ ∇ · (uc) = Q.
(2.32)
∂t
The diffusion equation, eq. (2.29), is a second order parabolic partial differential
equation (PDE), while the transport equation, eq. (2.32), is a first order hyperbolic
PDE [8]. Adaptive discretisation techniques will be necessary due to the fact that
the different nature of the two processes has to be reflected in the model [5]. In
this thesis, examples containing both diffusion and transport will be considered,
hence the convection-diffusion equation will be applied.
As stated at the start of this section, transport of dissolved contaminants in the
water phase, i.e. organic solvents, are of interest. By denoting a component within
the fluid phase by subscript i, the concentration of the component is defined as
the ratio of the mass of component i to the total mass of the fluid phase [2],[5].
The subscript is omitted in the following, due to the fact that only one component
is assumed dissolved. Applying an averaging procedure, recalling that the water
content is given by θw := Sw φ and using a phenomenological description for the
diffusive mass flux [5], the resulting differential equation for the water phase reads
∂(θw c)
− ∇ · (θw D∇c − uw c) = Q.
∂t
(2.33)
If the production rate Q is independent of c, eq. (2.33) is linear [5]. This is assumed
true in later chapters. Equation (2.33) models transport through diffusion and
convection of a dissolved substance.
2.2
Two-Phase Flow Model
When modelling flow through a porous medium such as soil, different considerations are necessary when encountering the saturated and the unsaturated zone.
Recalling that the saturated zone is completely filled with the water phase, this
relates to single-phase flow. However, in order to model flow through the unsaturated zone the need to develop the concept of two-phase flow emerges. This need
originates from the fact that both a water and an air phase is present in this zone.
To construct the two-phase flow model, which in this thesis is the simplified model
known as the Richards equation, some important properties of two-phase flow has
to be considered.
Chapter 2. Mathematical Modelling of Porous Media Flow
2.2.1
13
Two-Phase Flow
Modelling of two-phase flow in porous media, concerns the simultaneous flow of
two fluid phases within a porous medium. Again, the two fluid phases in question
being water and air. One of the phases is referred to as the wetting phase, while
the other is referred to as the nonwetting phase. This is defined in such a way that
the fluid which is preferentially attracted by the solid is called the wetting fluid,
while the other fluid is referred to as the nonwetting fluid [2]. The contact angle
is defined as the angle between the fluid-fluid interface and the solid, and is used
to determine whether or not a fluid is a wetting or a nonwetting fluid. The fluid
on the side of the interface with an angle less than 90° with respect to the solid
surface is the wetting fluid [2].
In section 2.1.1, the saturation (Sα ) for each fluid phase was defined. Here, α
denotes either the wetting phase (α = ω) or the nonwetting phase (α = n). For
the porous medium soil, with water and air as the two phases present, water is the
wetting fluid and air the nonwetting fluid. Hence, α = w, a is chosen to represent
the water and air phase respectively.
2.2.2
Capillary Pressure
An important role in describing two-phase flow in a porous medium, is played
by the existence of fluid-fluid interfaces at the pore scale. This allows the two
fluids to coexist in the pore space. From the fact that these interfaces can support
nonzero stresses, different pressures can exist on either side of the interphase [2].
Hence, each phase usually has a different pressure. The difference between the
phase pressures is defined as the capillary pressure, denoted by pc , and is defined
by
pa − pw = pc (Sw ).
(2.34)
Here, pa is the pressure of the air phase, while pw represents the pressure of the
water phase. The capillary pressure can be measured as a function of the water
saturation Sw , and is a hysteretic function [9] which will be defined in section 2.3.1.
14
2.2.3
Chapter 2. Mathematical Modelling of Porous Media Flow
Richards’ Equation
From a two-phase flow system, a simplified representation often used to describe
water movement in unsaturated soils can be derived. This is the Richards equation.
The applications of this equation are valid for a two-phase porous medium where
the two phases are water and air. The domain of interest being the shallow soil
zone whose top boundary corresponds to the land surface [2].
Some important properties allow for this simplification, the biggest and most important one being the assumption that the air phase is at constant pressure everywhere in the soil [13]. This assumption is based on other simplifications, which
include rapid flow of air, meaning that air movements are driven by small pressure
gradients, and that the domain is interconnected and connected to the exterior
atmosphere [2].
Considering these simplifications yields an approach for deriving Richards’ equation. The simple two-phase flow model for immiscible fluids is represented by the
following set of equations
Qα
∂θα
+ ∇ · (uα ) =
= fα ,
∂t
ρα
kr,α k
(∇pα − ρα g),
uα = −
µα
(2.35)
(2.36)
Sw + Sa = 1,
(2.37)
pa − pw = pc (Sw ).
(2.38)
Recall, θα is the water content and uα the volumetric flux, with α = w, a, representing the water and air phase respectively. Equation (2.35) is the mass conservation equation and holds because the density ρα is set to be constant. The second
equation is the Darcy law, while eq. (2.37) is the relation given by eq. (2.6), see
section 2.1.1. Equation (2.38) is the capillary pressure.
Since the air pressure is assumed to be equal to the atmospheric pressure everywhere, i.e. pa = 0, this means that one of the primary unknowns is eliminated
and one of the equations, this being the air-phase equation, can be eliminated [2].
The capillary pressure pc is now equal to the negative of the water pressure pw ,
pc (Sw ) = −pw .
(2.39)
The capillary pressure is positive because the water pressure is less than the atmospheric pressure in the unsaturated zone [2]. By expressing Darcy’s law with
Chapter 2. Mathematical Modelling of Porous Media Flow
15
respect to the pressure head, Ψw = ρpwwg , and the height against the gravitational
direction z,
uw = −K(θw (Ψw ))∇(Ψw + z),
(2.40)
where K(θw (Ψw )) is the hydraulic conductivity and both K and θ are given functions of Ψ. This can be inserted into eq. (2.35) and hence give rise to what is
defined as the Richards’ equation,
∂θw (Ψw )
− ∇ · [K(θw (Ψw ))∇(Ψw + z)] = fw .
∂t
(2.41)
θw (Ψw ) is obtained by inverting the relation given by eq. (2.39). Equation (2.41)
is a nonlinear PDE consisting of the second derivative with respect to space and
first derivative with respect to time. Thus, it is recognized as a parabolic PDE.
2.2.4
Parameterisations
As stated in section 2.2.3, the hydraulic conductivity K, and water content θ are
known functions of the pressure head Ψ. Based on experimental results, different
functional relationships have been proposed for describing the dependency between
K, θ and Ψ [38]. From this point on, the van Genuchten-Mualem parameterisation
is applied. It is given by
For Ψ ≤ 0
n−1
n
1
,
θ(Ψ) = θR + (θS − θR )
1 + (−αΨ)n
2
n−1
n
1
n
K(θ(Ψ)) = KS θ(Ψ) 2 1 − 1 − θ(Ψ) n−1
h
i
1−n 2
1 − (−αΨ)n−1 [1 + (−αΨ)n ] n
= KS
.
n−1
[1 + (−αΨ)n ] 2n
(2.42)
(2.43)
For Ψ > 0
θ = θS ,
(2.44)
K = KS .
(2.45)
Here, θR is the residual water content, θS the saturated water content, KS is the
saturated hydraulic conductivity and α and n are van Genuchten curve fitting
parameters [10]. θR , θS , α, n, KS are material specific constants. Equations (2.42)
16
Chapter 2. Mathematical Modelling of Porous Media Flow
and (2.43) are valid for the unsaturated soil zone, while eqs. (2.44) and (2.45)
holds for the saturated zone.
In this thesis the focus is on the strictly unsaturated flow regime, i.e. Ψ < 0,
θ0 > 0 and K > 0. From [38], it is worth noting that in the present setting the
Richards equation degenerates whenever Ψ → −∞, implying that both θ0 (Ψ) and
K(θ(Ψ)) are approaching 0, or situated in the fully saturated regime (Ψ ≥ 0), when
θ0 (Ψ) = 0. The regions of degeneracy depend on the saturation of the medium;
therefore these regions are not known a priori and may vary in space and time.
2.3
Non-Standard Models
The model derived in section 2.2 is based on the validity of eq. (2.34) (and
eq. (2.39)). This includes an equilibrium assumption, which is not necessarily
true. There is experimental evidence (Hassanizadeh et al. [18, 39]) that dynamic
effects and hysteresis are playing an important role and therefore eq. (2.34) is not
valid in this form. These effects also have the potential of describing phenomena
such as saturation overshooting or finger formation, see D.A.Dicarlo [9], which is
not the case for Richards’ equation based on the static relation eq. (2.34).
In the following some mathematical models which include non-equilibrium effects
and/or hysteretic effects will be presented. Such models are referred to as nonstandard models.
2.3.1
Dynamic Capillary Pressure and Hysteresis
To present the principle of hysteresis, an experiment from [2] p. 77-79 will be
reproduced. Assuming a sample of porous medium with pores filled with the
wetting fluid, see Figure 2.3. There is a left reservoir of wetting fluid and a right
reservoir with nonwetting fluid. Further assuming that the pressure in the two
reservoirs are controllable, the top and bottom of the sample are impermeable and
the influence of gravity can be neglected.
By increasing the pressure in the nonwetting fluid, it is possible to measure the
amount of wetting fluid displaced. When equilibrium is reached, a data point
relating the capillary pressure and the saturation is produced. The experiment
can be repeated with varying pressure differences to collect several data points.
Plotting these data points will give a typical capillary pressure - saturation curve,
see Figure 2.4.
Chapter 2. Mathematical Modelling of Porous Media Flow
17
Wetting fluid
Impermeable
Wetting fluid
reservoir
PorousMedium
Nonwetting fluid
reservoir
Impermeable
Figure 2.3: Sample of porous medium with left reservoir filled with wetting
fluid and right reservoir filled with nonwetting fluid.
From this, four points can be made. First, the process where nonwetting fluid displaces wetting fluid is referred to as drainage, while when wetting fluid displaces
nonwetting it is called imbibition. Second, the residual saturation values (Sαres )
are clearly represented, which in this case means that the soil is never completely
dry. Thirdly, there is an obvious difference between the curves for drainage and
imbibition, meaning that the relation between the capillary pressure and the saturation depends on the history. Therefore, it is not enough to know the saturation
at one point to determine the capillary pressure, but it is also important to know if
the saturation is increasing (imbibition) or decreasing (drainage). Such behaviour
is called hysteresis, or the process is said to be hysteretic. The primary drainage
curve includes full saturation (Sw = 1) as one of its end points, relating to no
nonwetting fluid to start with. Main drainage and main imbibition curves are
curves that begin at the residual saturation points of the other fluid. Lastly, the
curves that begin at points between the two residual saturations are referred to as
scanning curves. The fourth point is that a finite capillary pressure is required before any drainage displacement begins. The capillary pressure required to initiate
displacement of the wetting fluid is called the entry pressure. Capillary exclusion
is the phenomenon whereby nonwetting fluid is unable to enter particular spatial
regions that are filled with wetting fluid due to failure to reach this entry pressure.
Thus an important feature of the macroscopic capillary pressure-saturation curve
is its hysteretic behaviour observed when reversing the flow direction, e.g. from
drainage to imbibition [16]. The standard relationship assumed between capillary
pressure and saturation, see eq. (2.38), is empirical in nature, and as such lacks
a firm theoretical foundation [39]. In fact, the relationship pn − pω = pc is valid
only under static condition. Under dynamic conditions, pn − pω depends on the
18
Chapter 2. Mathematical Modelling of Porous Media Flow
Capillary Pressure
Scanning Curves
Main Drainage
Primary Drainage
Main Imbibition
0 0
res
(1-snres)
sw
1
Wetting Fluid Saturation
Figure 2.4: Typical form for a capillary pressure-saturation curve. From
Nordbotten et al. [2].
flow velocity, which at larger time scales manifest itself as a change in saturation
with time [16].
Hassanizadeh and Gray (1990) [18] suggested that the hysteretic behaviour of the
capillary pressure is related to the configuration of interfaces, since fluid pressures vary spatially within each flowing phase, macroscale (or average) pressure
values will be different from pressure values at the interface [16]. Based on thermodynamic considerations, they concluded that the hysteretic behaviour of the
capillary pressure-saturation relationship can be modelled by including the specific interfacial area in the formulation [16]. In other words, they advocated a
dynamic capillary pressure, where the capillary pressure depends not only on the
saturation and saturation direction but also the rate of saturation change [9]. For
the remainder of this thesis, the focus will be on the effects of including dynamic
capillary pressure in the model, while the effects of hysteresis are left unexplored.
2.3.2
Extension of the Standard Model
In order to include non-equilibrium effects in the model, an extension will be
added to the Richards’ equation. Most continuum models that are proposed are
extensions of Richards’ equation as derivation of this only requires the Darcy
law and conservation of mass, in addition to the fact that the Richards equation
works in almost all cases [9]. The extension is related to the concept of dynamic
capillary pressure and yields additional terms of the θ − Ψ relationship. In the
Chapter 2. Mathematical Modelling of Porous Media Flow
19
literature there are several different extension models available, a collection of
which is presented in [16], p. 10-11. For the further study of these phenomena
in this thesis, an equation based on thermodynamic considerations suggested by
Hassanizadeh and Gray (1990) [18] and Kalaydjian (1992) [17] is applied [16]. The
equation relates the difference in the phase pressure pa − pw to a capillary pressure
pc by
∂θw
.
(2.46)
pa − pw = pc (θw ) − τ (θw )
∂t
Here, τ (θw ) ≥ 0 is a non-equilibrium coefficient. In this formulation, pc is an
intrinsic property of the porous medium-fluids system, whereas the fluid pressure
difference pa − pw is dependent on flow dynamics (and thus initial and boundary
conditions) [16]. The pressure difference between the air and water phase is equal
to a capillary pressure only under static condition. Equation (2.46) suggests that
at any given point in time where equilibrium is disturbed, the saturation will
change to reestablish the equilibrium condition, and the coefficient τ (θw ) controls
this process [16].
Recall from section 2.2.3, that since the pressure of the air phase is assumed equal
to the atmospheric pressure everywhere, there is no need to solve for pa . Also
utilising that pw can be set to be equal to the pressure head Ψw , eq. (2.46) takes
the form
∂θw
Ψw = −pc (θw ) + τ (θw )
,
(2.47)
∂t
with pc being the equilibrium (or static) capillary pressure.
The time derivative of θw in the Richards’ equation (eq. (2.41)) is replaced by the
rearrangement of eq. (2.47), given by
∂θw
1
1
=
Ψw +
pc (θw ).
∂t
τ (θw )
τ (θw )
(2.48)
The static capillary pressure pc (θw ) is given by the van Genuchten-Mualem parameterisation
n1
− n−1
n
θw −θR
−1
θS −θR
pc (θw ) =
.
(2.49)
α
Equation (2.49) is obtained by inverting the relation eq. (2.42)
20
Chapter 2. Mathematical Modelling of Porous Media Flow
2.4
The Mathematical Model
As a final note to the chapter, a summary of the equations derived in the previous
sections is presented. Thus establishing the representative mathematical models
used to describe porous media flow. The models will found the basis for the
numerical analysis developed in chapter 3.
2.4.1
Simplifications of the Mathematical Model
The following assumptions are set to hold true for all problems considered in
subsequent chapters:
ˆ the density ρ is constant,
ˆ the porosity φ is constant,
ˆ the temperature T is constant,
ˆ dissolved components do not influence the flow.
The first assumption is utilised in section 2.2.3, eq. (2.35), to yield the described
Richards’ equation (eq. (2.41)). From assuming the temperature and porosity to
be constant the fluids are said to be incompressible, see section 2.1.2. The final
assumption is necessary to ensure that no additional relations or equations are
needed to describe the flow of water containing dissolved components.
2.4.2
Representative Equations
Summing up the equations for the flow in a porous medium introduced in sections 2.1 to 2.3, yields the following mathematical models.
Standard Models
The Richards equation without any additional relations or extensions, as well as
the convection-diffusion equation are defined as the standard models. Assuming
Chapter 2. Mathematical Modelling of Porous Media Flow
21
all equations models flow for the water phase, the subscript w is omitted for the
remainder of the thesis, and the model for the Richards’ equation becomes


∂t θ(Ψ) − ∇ · [K(Ψ)∇(Ψ + z)] = f,




θ(Ψ), K(Ψ),
in Ω,
in Ω,
(2.50)


Ψ(t, x)|t=0 = Ψ0 (x),
in Ω,




Ψ(t, x) = ΨD or nT K∇(Ψ + z) = qN1 , on ∂Ω,
with θ(Ψ) and K(Ψ) given by the van Genuchten-Mualem parameterisation in
section 2.2.4. Initial and boundary conditions are needed to ensure uniqueness
of the solution, given by Ψ0 (x) and ΨD , qN1 respectively. Here ΨD represents
Dirichlet boundary conditions and qN1 are Neumann boundary conditions (see
section 3.2.3). Ω is a domain in space consisting of a porous medium with ∂Ω as
its boundary.
The model for the convection-diffusion equation becomes


∂t (θc) − ∇ · (θD∇c − uc) = Q,




u = −K(Ψ)∇(Ψ + z),
in Ω,
in Ω,
(2.51)


c(t, x)|t=0 = c0 (x),
in Ω,




c(t, x) = cD or nT (θD∇c − uc) = qN2 , on ∂Ω.
The convection-diffusion equation is coupled with the Richards equation, so that
the volumetric flux, u, originally resulting from the Darcy law (eq. (2.40)), and the
water content θ are obtained from the computations of Richards’ equation. As in
the case of the model for the Richards equation, initial and boundary conditions
are included to ensure uniqueness of the solution, where cD and qN2 are Dirichlet
and Neumann conditions respectively.
Non-Standard Model
The non-standard model is given by the Richards equation with the extension
for the dynamic capillary pressure, defined in section 2.3.2, included. The model
becomes
22
Chapter 2. Mathematical Modelling of Porous Media Flow


∂t θ − ∇ · [K(θ)∇(Ψ + z)] = f,





Ψ = −pc (θ) + τ (θ)∂t θ(Ψ),


K(θ), pc (θ), τ (θ)




Ψ(t, x)|t=0 = Ψ0 (x),





Ψ(t, x) = ΨD or nT K∇(Ψ + z) = qN1 ,
in Ω,
in Ω,
in Ω,
(2.52)
in Ω,
on ∂Ω.
In this, the time derivative in Richards’ equation is replaced by the expression given
by the second equation, resulting in a pseudo-parabolic equation, where dynamic
effects are included in the capillary pressure [22]. The capillary pressure pc (θ) is
described in eq. (2.49), and τ (θ) is given by some function. K(θ) is given by the
first relation in eq. (2.43), and the initial and boundary conditions will be defined
in the same manner as in the standard model, (2.50). The Convection-Diffusion
equation, given by (2.51), is also coupled with this model. The case when τ = 0
corresponds to the standard model.
Chapter 3
Numerical Methods
In the previous chapter, a coupled set of partial differential equations was obtained.
These sets of equations will be solved numerically in one spatial dimension (1D),
using finite difference methods and a finite volume method known as the twopoint flux approximation scheme or simply TPFA [12]. This chapter will present
the theoretical background for the discretisation of the equations, also including
the fully discretised schemes in space and time. The Richards equation and the
equation giving the non-standard extension are both nonlinear, and thus need to
be linearised in order to be solved numerically. In section 3.4 robust linearisation
schemes are presented for Richards’ equation with and without dynamic capillary
pressure. The scheme for the Richards equation with dynamic capillarity is new,
whereas the one for the standard Richards’ equation is the one in [37, 40] but for
a TPFA discretisation. The schemes will be shown to be robust and linearly convergent, in addition to have certain advantages compared to the more widespread
and commonly used linearisation methods. Lastly, a short comment about the
implementation of the numerical schemes is made.
3.1
Grid
One of the first steps of implementing methods for solving a mathematical problem numerically, requires a ’geometric discretisation’ of the domain Ω [5]. The
discretisation is often constructed by placing grid points throughout the domain
and connecting these points using nonintersecting, straight lines [19]. In two dimensions (2D) the grid points now make up the corners of the grid cells.
23
24
Chapter 3. Numerical Methods
Figure 3.1:
Point-distributed
grid
Figure 3.2: Cell-centered grid
For a function f (x) defined on the domain of the grid, the grid points are used to
provide a discretised representation of the function. Let xi , i = 1, 2, ..., N denote
the grid points of a grid. A discretised representation of f is then given by
f = [f (x1 ), ...., f (xN )]T .
(3.1)
There are in principle two main types of grids, when applied results in different
discretised representations of the function f . These are point-distributed grids
and cell-centred grids. When the grid points are placed at the corners of the cells
it is known as a point-distributed grid, while a cell-centred grid has its grid points
in the centres of the cell [19], see Figures 3.1 and 3.2. For a cell-centred grid
the grid must be generated before the cell-centred points can be determined, even
for one-dimensional grids. In general, it is not possible to place a random set of
points throughout the domain, before creating a grid around the points, so that
they make up the cell-centres [19].
The TPFA scheme is a cell-centred finite-volume method [12]. Thus, the onedimensional interval in space is discretised using a cell-centred grid. The spacing
is assumed to be equidistant. First, the interval [0, L] is divided into N equal
cells. The walls of the cells are given by xi+ 1 = ih for i = 1, 2, . . . , N , where
2
h = xi+ 1 − xi− 1 = L/N . Then x 1 = 0 and xN + 1 = L denotes the boundaries of
2
2
2
2
the domain, see Figure 3.3.
0
L
x
x1
2
x3
2
xi− 1
2
xi+ 1
2
xN − 1
2
xN + 1
2
Figure 3.3: Space interval divided into N equidistant cells.
After the space interval is divided into equidistant cells, the cell-centred points
are defined and denoted by the grid points x1 , x2 , . . . , xN , see Figure 3.4. Since
the grid is equidistant, the distance between neighbouring grid points equals the
Chapter 3. Numerical Methods
0
25
x1
xi
xN
L
x
x1
2
x3
2
xi− 1
2
xi+ 1
2
xN − 1
2
xN + 1
2
Figure 3.4: Cell-centred discretisation in space.
length of each cell, ∆x = xi+1 − xi = h. Then x1 = h/2 and xi = ih − h/2 for
i = 0, . . . , N . The rightmost grid point is xN = N h − h/2 = L − h/2.
Equivalently to the discretisation in space, there is also a discretisation in time,
see Figure 3.5. The time interval [0, T ], spans from initial time, t0 = 0, to final
time, tm = T . As in the case with the spatial grid points, the time steps are
assumed equidistant, given by ∆t = tj+1 − tj = T /m, for j = 0, 1, . . . , m.
0
T
t
t0
t1
tj
tj+1
tm−1
tm
Figure 3.5: Equidistant time discretisation.
3.2
Discretisation in Space
In order to solve a PDE by the means of a numerical method, the equation must
be discretised. The PDEs in the mathematical models from section 2.4.2 are dependent on space and time, and thus need to be discretised with respect to both
variables. The main focus in this section will however be on the discretisation
in space and on boundary conditions. There are several ways to perform spatial
discretisation of a PDE. The method chosen in this thesis is, as mentioned previously the TPFA scheme, which is recognised as a control volume method (CVM)
also referred to as a finite-volume method [12, 19]. CVMs are a class of numerical
methods used to apply spatial discretisation to PDEs. They are popular methods
due to the fact that they satisfy the physical principle of mass conservation, in
addition to being fairly easy to formulate for complex grids. For 1D, the TPFA is
equivalent to the finite difference method for a cell-centred grid.
26
3.2.1
Chapter 3. Numerical Methods
Finite Difference Method
One of many different approaches to solving PDEs numerically, are finite difference
methods. It is an elementary discretisation method, in which the derivatives in a
differential equation are replaced by finite difference approximations at a discrete
set of points in space or time. The resulting set of equations, can then be solved by
algebraic methods [20]. For further study of the finite difference method, besides
what will be presented in the following, see [8, 19, 20, 21].
To establish appropriate finite difference approximations of derivatives, Taylor
series are applied. Recall from calculus [6], that for f (x ± h) = f (xi±1 ),
f (xi±1 ) = f (x) ± f 0 (xi )h + f 00 (xi )
h3
h2
± f 000 (xi ) + . . .
2!
3!
(3.2)
Using the + series, the forward difference approximation for the first derivative is
attained
f (xi+1 ) − f (xi )
f 0 (xi ) =
+ O(h),
(3.3)
h
while the − series results in the backward difference approximation for the first
derivative
f (xi ) − f (xi−1 )
f 0 (xi ) =
+ O(h).
(3.4)
h
These approximations are first order accurate, given by the term O(h), also referred
to as the truncation error. To attain a better approximation, the negative sign
in eq. (3.2) can be subtracted from the positive sign, to get the centred difference
approximation
f (xi+1 ) − f (xi−1 )
f 0 (xi ) =
+ O(h2 ).
(3.5)
2h
This approximation is second order accurate (O(h2 )).
For the second order derivative, a centred finite difference approximation is achieved
by adding the two series in eq. (3.2), to give
f 00 (xi ) =
f (xi+1 ) − 2f (xi ) + f (xi−1 )
+ O(h2 ).
h2
(3.6)
As with the centred difference, this approximation yields a truncation error of order
h2 , meaning they are more accurate approximations than eqs. (3.3) and (3.4). This
is as expected from geometric considerations.
Chapter 3. Numerical Methods
27
By applying eq. (3.6) to the ordinary differential equation (ODE)
− (Kux )x = f,
(3.7)
with K constant, the approximation for the second derivative becomes
− Kuxx ≈ K
−u(xi+1 ) + 2u(xi ) − u(xi−1 )
.
h2
(3.8)
This can be expressed by the linear system matrix system
Au = b,
where
(3.9)

2 −1

−1 2 −1


K

... ... ...
A= 2
h 



−1 2 −1

−1 2
0
0







,





(3.10)
and

f (x0 )
f (x1 )
..
.







,

b=



 f (xN ) 
f (xN +1 )
(3.11)
for i = 1, . . . , N .
3.2.2
Two-Point Flux Approximation
Unlike finite difference methods where partial derivatives are replaced by divided
differences, see section 3.2.1, finite volume methods have a more physical motivation and are derived from conservation of (physical) quantities over cell volumes
[12]. One finite volume method is the TPFA scheme, which is undoubtedly one
of the simplest discretisation techniques for elliptic equations. However, it is still
widely used for simulation purposes due to its simplicity and the method yielding
explicit expressions for the fluxes and harmonic averaging of the permeability, see
[19]. It holds when the flux is equal on both sides of the cell wall. The theory
28
Chapter 3. Numerical Methods
described in this section is based on the lecture notes by I.Aavatsmark [19] and obtained through personal communication with F.A.Radu [13] while he was lecturing
the course Flow in Porous Media (MAT254).
As in section 3.2.1, an ordinary differential equation (ODE) is given by
− (Kux )x = f,
(3.12)
where K = K(x) denotes the permeability and f some source-term. The index
denotes the derivative with respect to x. The TPFA is based on the integral
formulation of the problem given by
Z
Z
q · n dS =
f dx.
(3.13)
∂Ωi
Ωi
Applying the divergence theorem [6] generates the left hand side of eq. (3.13),
with q = −Kux . The domain of the ODE is discretised by a cell-centred, onedimensional grid, see Figure 3.6. The grid points denoted by xi for i = 0, . . . , N
are thus located at the centres of the cells, and the cell walls are given by xi− 1
2
and xi+ 1 .
2
∆xi−1
∆xi− 3
2
∆xi−1
∆xi+1
∆xi
∆xi
∆xi− 1
∆xi+ 1
2
∆xi+1
2
∆xi+ 3
2
Figure 3.6: Cell-centred grid with grid points xi and cell walls xi+ 1 .
2
Integrating eq. (3.13) over the ith cell, from xi− 1 to xi+ 1 , gives
2
Z
qi+ 1 − qi− 1 =
2
2
2
xi+ 1
2
f (x)dx.
(3.14)
xi− 1
2
To obtain an expression for qi+ 1 as a function of u, it first requires a rewriting of
2
the relation q = −Kux to get
ux = −
q
.
K(x)
(3.15)
By integrating eq. (3.15) from xi to xi+1 , the relation,
Z
xi+1
ui+1 − ui = −qi+ 1
2
xi
1
dx,
K(x)
(3.16)
Chapter 3. Numerical Methods
29
is obtained. This relation can be manipulated to give the desired expression for
qi+ 1 , given by
2
ui+1 − ui
qi+ 1 = − R xi+1 1
.
(3.17)
2
dx
K(x)
xi
It now remains to derive an expression for the integral
Z
xi+1
xi
1
dx.
K(x)
(3.18)
For a cell-centred grid, K(x) is assumed constant on each cell, denoted by the
values at the grid points so that Ki = K(xi ). From Figure 3.6, ∆xi = xi+ 1 − xi− 1 ,
2
2
represents the distance between the walls of the cell. Since xi and xi+1 are grid
points of two neighbouring cells, the integral is approximated by taking the average
over the two cells involved, given by
Z
xi+1
xi
1
1
dx =
K(x)
2
∆xi+1 ∆xi
+
Ki+1
Ki
.
(3.19)
The approximation in eq. (3.19) holds for non-equidistant, cell-centred grids. In
this thesis however, equidistant, cell-centred grids with ∆xi = h are considered,
which leads to the expression
Z
xi+1
xi
h
1
dx =
K(x)
2
1
1
+
Ki+1 Ki
.
(3.20)
Inserting eq. (3.20) into eq. (3.17) gives
ui+1 − ui
.
qi+ 1 = − 2
h
1
1
+
2 Ki+1
Ki
(3.21)
Finally, by inserting the equations for q for an equidistant, cell-centred grid,
eq. (3.14) becomes
u − ui−1
i
−
h
1
1
+ Ki−1
2 Ki
u − ui
i+1
=
h
1
1
+ Ki
2 Ki+1
Z
xi+ 1
2
f (x)dx.
(3.22)
xi− 1
2
Equation (3.22) can be expressed as
ai (ui − ui−1 ) − ai+1 (ui+1 − ui ) = bi ,
(3.23)
30
Chapter 3. Numerical Methods
by denoting
ai =
1
h
2
Z
xi+ 1
1
Ki
2
bi =
,
(3.24)
f (x)dx ≈ hfi ,
(3.25)
+
1
Ki−1
xi− 1
2
where fi is defined as
1
fi =
h
Z
xi+ 1
2
f (x)dx,
(3.26)
xi− 1
2
and the second relation in eq. (3.25) arise from applying the midpoint rule [24].
Rearranging eq. (3.23) leads to the system of equations
− ai ui−1 + (ai + ai+1 )ui − ai+1 ui+1 = bi ,
(3.27)
for i = 1, . . . , n. Hence, being a system of n equations. The unknowns ui can be
collected in the vector u = [u0 , . . . , un+1 ]T . That is, there are n + 2 unknowns
in n equations. In order to achieve a unique solution, additional boundary conditions are needed and will be considered in the following section, see section 3.2.3.
The system of equations in eq. (3.27) can be represented by the following matrix
representation
Au = b,
(3.28)
where the sparse, tridiagonal coefficient matrix A is given by


−a1 a1 + a2
−a2


−a2
a2 + a3 −a3










..
..
..
A=

.
.
.








−an−1 an−1 + an
−an


−an
an + an+1 −an+1
0
(3.29)
0
As a final mark, it is worth noting that the TPFA scheme only yields consistent flux
approximations for K-orthogonal grids [14, 23]. A grid is said to be K-orthogonal
if and only if the flux across all edges can be approximated to a two-point flux in
a consistent way [19]. For further reading on the properties of K-orthogonal grids
see [19] p. 144-146.
Chapter 3. Numerical Methods
3.2.3
31
Boundary Conditions
PDEs has to be supplemented by initial and boundary conditions in order to specify a particular situation where a unique solution is expected, see section 2.4.2.
Boundary conditions are specifications on the boundary of the domain, ∂Ω [5].
For a one-dimensional domain, or interval, the boundary consists of two separate
boundary points located at the left and right side of the interval, see Figure 3.4.
Two of the principle types of boundary conditions will be presented in the following, these are Dirichlet and Neumann boundary conditions. Depending on the
problem, it can in some cases be convenient to apply the same boundary condition at the left and right boundary, while in others a combination of boundary
conditions are more fitting.
Dirichlet Boundary Conditions
For a system of PDEs, the Dirichlet boundary condition is given by the function
value of the unknown at the boundaries of the domain. To provide an outline of
Dirichlet boundary conditions, the stationary, one-dimensional problem eq. (3.12),
where u = u(x) is the unknown, will be considered on the domain [0, L]. The
Dirichlet boundary conditions are then given by
u(0) = u0 ,
(3.30)
u(L) = uL .
(3.31)
and
Since the grid applied in the discretisation of the domain is assumed to be cellcentred, Figure 3.4, the discretised form of the boundary conditions become
u 1 = u0 ,
(3.32)
un+ 1 = uL ,
(3.33)
2
and
2
For a point-distributed grid, the boundary points will coincide with the grid points
at the left- and rightmost endpoints, making the construction of Dirichlet boundary conditions straight forward, as u is defined at the grid points. This is not
the case however for cell-centred grids, as is evident from eqs. (3.32) and (3.33).
Alternative methods for handling the boundary conditions are thus needed. In
32
Chapter 3. Numerical Methods
x0 0 x1
xn L xn+1
xi
x
x− 1
2
x1
x3
2
2
xi− 1
xi+ 1
2
2
xn− 1
xn+ 1
2
2
xn+ 3
2
Figure 3.7: Cell-centred grid with ghost cells included at the boundaries of
the interval.
x1
x0 = 0
xi
xn
xn+1 = L
x
x1
2
x3
2
xi− 1
2
xi+ 1
2
xn− 1
2
xn+ 1
2
Figure 3.8: Cell-centred grid with adjusted half cells near the boundaries of
the interval.
[21], two ways to construct Dirichlet boundary conditions for cell-centred grids are
presented.
The first method consists of including a ghost cell at each boundary as shown in
Figure 3.7. It is then assumed that the Dirichlet boundary condition is prescribed
at the centre of the ghost cell and the usual approach is followed to derive the
difference equation on the first and last cell. Two additional grid points, x0 and
xn+1 , are thus created, and the Dirichlet boundary conditions can be defined as
u0 = u0 ,
(3.34)
un+1 = uL .
(3.35)
and
It is noted in [21] that prescribing the Dirichlet boundary condition at the centre of
the ghost cell instead of the boundary is a first order approximation, and will not
be adequate if the solution is strongly dependent on the distance away from the
boundary condition. The second method is to include a half cell at the endpoints
of the interval as done in Figure 3.8. In this case, no equation is derived on the
1st cell, but rather derived on the 2nd cell and when it reaches the 1st cell it takes
on the Dirichlet boundary condition [21].
When applying the method of adding ghost cells to the ends of an interval discretised using a cell-centred grid, the Dirichlet boundary condition is more or less
handled the same way as for a point-distributed or vertex centred grid. The difference is where the grid points, xi , are defined and consequently in what points
Chapter 3. Numerical Methods
33
u is computed. This can be demonstrated by using the discretisation of eq. (3.12)
given by eq. (3.27) in section 3.2.2. Thus, the system of equations looks like
−a1 u0 + (a1 + a2 )u1 − a2 u2 = b1
−a2 u1 + (a2 + a3 )u2 − a3 u3 = b2
..
.
−an−1 un−2 + (an−1 + an )un−1 − an un = bn−1
−an un−1 + (an + an+1 )un − an+1 un+1 = bn
The Dirichlet boundary conditions are given by u0 = u0 and un+1 = uL (eqs. (3.34)
and (3.35)), and the terms involving the boundary conditions can be moved to the
right hand side of the system, giving
(a1 + a2 )u1 − a2 u2 = b1 + a1 u0
−a2 u1 + (a2 + a3 )u2 − a3 u3 = b2
..
.
−an−1 un−2 + (an−1 + an )un−1 − an un = bn−1
−an un−1 + (an + an+1 )un = bn + an+1 uL
This can be expressed as the system
Au = b,
(3.36)
for u = [u1 , . . . , un ]T , where

a1 + a2
−a2

a2 + a3 −a3
 −a2




..
..
..
A=
.
.
.




−an−1 an−1 + an
−an

−an
an + an+1
0
0













(3.37)
and

b 1 + a1 u 0
b2
..
.







.
b=






bn−1
bn + an+1 uL
(3.38)
34
Chapter 3. Numerical Methods
Neumann Boundary Conditions
For the case when Neumann boundary conditions are imposed, the values that the
derivative of the function is to take on the boundary are specified. That is, the
Neumann boundary conditions for a one-dimensional domain is given by
− Kux (0) = uα ,
(3.39)
− Kux (L) = uβ
(3.40)
and
Recalling that x 1 = 0 and xn+ 1 = L, see Figure 3.4. Again, demonstrating by
2
2
the discretisation of eq. (3.12), and using the fact that the flux q is given by
q = −Ki (x)ux , the derivative at the cell wall, xi+ 1 , is expressed by qi+ 1 . Thus,
2
2
the boundary conditions are given by
q 1 = uα ,
(3.41)
qn+ 1 = uβ .
(3.42)
qi+ 1 − qi− 1 = bi ,
(3.43)
2
and
2
From eqs. (3.14) and (3.25),
2
2
for i = 1, . . . , n. The first equation, when i = 1, becomes
q 3 − q 1 = b1 ,
2
2
(3.44)
where q 1 = uα is given by the boundary condition, eq. (3.41), and eq. (3.44) can
2
be expressed as
q 3 = b1 + u α .
(3.45)
2
Similarly for the last equation, when i = n, eq. (3.41) becomes
qn− 1 = bn + uβ .
2
(3.46)
From observation of the above equations, it is worth noting that when discretising a
system of equations using a cell-centred grid like the one in Figure 3.4, Neumann
boundary conditions are the natural choice. This is due to the flux boundary
condition being located at the boundary, making considerations of ghost cells or
half-cells unnecessary.
Chapter 3. Numerical Methods
35
Lastly, recalling eqs. (3.21) and (3.24), where
qi+ 1 = −ai+1 (ui+1 − ui ),
(3.47)
2
the first and last equation become
a2 u1 − a2 u2 = b1 + uα
(3.48)
−an un−1 + an un = bn + uβ ,
(3.49)
while the other equations remain unchanged. As in the case with Dirichlet boundary conditions, the set of equations can be displayed with the matrix representation
Au = b,
where
(3.50)

a2
−a2

−a2 a2 + a3 −a3




..
..
..
A=
.
.
.




−an−1 an−1 + an −an

−an
an
0
0













(3.51)
and

b1 + u α


 b2 
 . 
. 
b=
 . 


 bn−1 
bn + u β

(3.52)
for u = [u1 , . . . , un ]T .
3.3
Discretisation in Time
To present the time discretisation methods, the ODE
u0 (t) = F (t, u(t)),
(3.53)
will be considered. From the forward difference approximation in eq. (3.3), the
resulting numerical scheme discretised with respect to time is known as the forward
36
Chapter 3. Numerical Methods
Euler scheme, or the explicit Euler’s method [25, 26], given by
un+1 = un + ∆tF (tn , un ).
(3.54)
If the backwards difference approximation in eq. (3.4) is applied instead, the resulting scheme will now be the backward Euler scheme, also known as the implicit
Euler’s method [25, 26],
un+1 = un + ∆tF (tn+1 , un+1 ).
(3.55)
For both relations, n = 1, . . . , T denotes the time steps, ∆t denotes the uniform
length of each time step and un represents the numerical approximation of the
exact solution u at time tn = n∆t.
What separates the two methods is the input of the function F , characterising the
method as an implicit or explicit method. In [24] a method is called explicit if
un+1 can be computed in terms of the previous values uk , k ≤ n, while it is said
to be implicit if un+1 depends implicitly on itself through F .
The Euler methods of eqs. (3.54) and (3.55) are Taylor methods of order one [25],
meaning they are both first order methods [26]. The stability conditions of the two
methods do however differ [24, 25]. While the backward Euler method satisfies
the absolute stability property,
|un | −→ 0 as tn −→ +∞,
(3.56)
where u is the solution of u0 (t) = λt with λ < 0, for any value of ∆t. This is
not the case for the forward Euler method, where the method is stable only for
certain values of ∆t. For a more detailed study of the stability conditions of the
two methods, see [24] p. 480 or [25] p. 609-610. An implicit method generally
requires more work than an explicit one, in particular if F is a nonlinear function
with respect to u, in which case a linearisation technique must be applied to solve
eq. (3.55), see section 3.4. If F is a linear function of u, the equation is simply
rearranged and solved for un+1 .
3.4
Linearisation of Nonlinear Equations
The mathematical models presented in this thesis consist of coupled, nonlinear
PDEs. This makes the design and implementation of efficient numerical schemes
a challenging task. To discretise the equations in space, a locally conservative
Chapter 3. Numerical Methods
37
finite volume [19, 28, 29, 30, 31] discretisation, see section 3.2.2, is applied due to
its alleviation of several stability issues. Furthermore, a fully implicit temporal
discretisation is implemented because of its stability properties for all time-scales,
see section 3.3.
The spatial and temporal discretisations thus lead to a large system of nonlinear
equations for each time step. Common methods for solving such systems are the
fix point method [24], Picard’s method [5] or Newton’s method [5, 32, 33, 34, 35].
The two former are linearly convergent while the latter is quadratically convergent
[27]. Newton’s method is a powerful tool when applied to systems arising from
discretisation of parabolic equations, but the quadratic convergence does however
come at a price of only local convergence in solution space. Two main concerns
regarding Newton’s method is that the Jacobian matrix of the system needs to
be assembled, as well as the fact that the convergence of the algorithm is not
guaranteed when the initial guess is not “close enough”, which implies a restriction
on the time step [27].
To treat the nonlinearities of the PDEs in the mathematical models of section 2.4.2,
two linearisation schemes are proposed and analysed. The first scheme applies to
the standard Richards equation (see section 2.4.2) and is based on the works of
M.Slodicka [37] and Radu et al. [36]. It is in this thesis applied for the TPFA
method for the first time. See Radu et al. [27] for a similar scheme for two-phase
flow. The second linearisation scheme applies to Richards’ equation with dynamic
capillarity (see section 2.4.2). It continues the works in [36, 37]. The scheme
is new. A similar linearisation method was proposed by Fan et al. [22], with
saturation as the primary variable.
The linearisation schemes do not involve the computation of derivatives and are
shown to be very robust (see sections 3.4.1 and 3.4.2). Moreover, the convergence
of the schemes is independent of the mesh diameter, this being an important
advantage when compared to other linearisation methods.
3.4.1
Linearisation Scheme for the Richards Equation
The linearisation scheme for the Richards equation is based on the schemes presented in M.Slodicka [37], Radu et al. [36] and F.A.Radu [40]. See [37] for a
Galerkin FE based scheme and [36, 40] for mixed finite element (MFEM). Similarly to the proposed linearisation scheme derived for the finite volume method,
multi-point flux approximation (MPFA) [12, 19], in the recently published report
by Radu et al. [27], the scheme is in the following applied to the TPFA method.
38
Chapter 3. Numerical Methods
Throughout this thesis the following assumptions on the data and the solution of
the continuous problem are considered.
(A1) θ(·) is monotone increasing and Lipschitz continuous.
(A2) pc (·) is monotone decreasing and Lipschitz continuous.
(A3) K is constant, K > 0.
(A4) τ is constant, τ > 0.
A fully implicit temporal discretisation of Richards’ equation (eq. (3.117)), by the
backward Euler method, leads to the nonlinear system
θn+1 + ∆tAn+1 Ψn+1 = ∆tf n+1 + θn ,
(3.57)
where θn = θ(Ψn ), θn+1 = θ(Ψn+1 ) with Ψn , Ψn+1 piecewise constants on the cells,
and An+1 is the system matrix given by

2 −1

−1 2 −1


K

.. .. ..
= 2
.
.
.
h 



−1 2 −1

−1 2
0
An+1
0







,





(3.58)
for the 1-dimensional case and K constant. For a detailed explanation of how
eq. (3.57) is derived see section 3.5.2.
Instead of applying any of the standard approaches to solving eq. (3.57) the linearisation method from Radu et al. [27, 36], M.Slodicka [37] and F.A.Radu [40]
is considered. The linearisation is applied in combination with the TPFA method
for the first time. Let
d
LΨ = supΨ
θ(Ψ).
(3.59)
dΨ
d
In practice, any constant LΨ ≥ supψ dψ
θ(ψ) will ensure convergence of the scheme,
as it will be shown in Theorem 3.1. Then, iterate j + 1 is obtained by solving the
following system of equations
LΨ (Ψn+1,j+1 − Ψn+1,j ) + θ(Ψn+1,j ) + ∆tAn+1 Ψn+1,j+1 = ∆tf n+1 + θ(Ψn ), (3.60)
Chapter 3. Numerical Methods
39
where Ψn+1,0 = Ψn . Note that eq. (3.60) is a linear elliptic system. To evaluate
the convergence of the linear system the error at iteration step j + 1 is introduced
en+1,j+1
= Ψn+1,j+1 − Ψn+1 .
Ψ
(3.61)
In order to show the convergence of the scheme, it will be proven that
||en+1,j+1
|| −→ 0 when j −→ ∞,
Ψ
where ||·|| is the notation for the discrete L2 norm, ||en+1,j+1
||2 :=
Ψ
(3.62)
P
mi |Ψn+1,j+1
−
i
i
Ψn+1
|2 . Here mi represents the length of each subinterval i of the 1-dimensional
i
domain, mi = h for an equidistant grid. The discrete L2 scalar product is defined
P
by hu, vi := mi ui vi .
i
Theorem 3.1. Assuming (A1),(A3) and that the time step ∆t is sufficiently small,
and that the finite volume method satisfies (3.66), the linearisation scheme (3.60)
is (at least) linearly convergent.
Proof. Subtracting eq. (3.57) from eq. (3.60) results in
LΨ (Ψn+1,j+1 − Ψn+1,j ) + θ(Ψn+1,j ) − θ(Ψn+1 ) + ∆tAn+1 (Ψn+1,j+1 − Ψn+1 ) = 0,
(3.63)
which can be expressed as
LΨ (en+1,j+1
− en+1,j
) + θ(Ψn+1,j ) − θ(Ψn+1 ) + ∆tAn+1 (en+1,j+1
) = 0.
Ψ
Ψ
Ψ
(3.64)
Then multiplying with en+1,j+1
(the multiplication is done element wise, and
Ψ
summed up after weighing by the cell mass mi ) to obtain
LΨ hen+1,j+1
− en+1,j
, en+1,j+1
i + h(θ(Ψn+1,j ) − θ(Ψn+1 )), en+1,j+1
i
Ψ
Ψ
Ψ
Ψ
+ ∆thAn+1 en+1,j+1
, en+1,j+1
i = 0.
Ψ
Ψ
(3.65)
The following inequality is needed for the finite volume method, which can be
easily verified (see [27]):
hAn+1 en+1,j+1
, en+1,j+1
i ≥ a||en+1,j+1
||2
Ψ
Ψ
Ψ
(3.66)
where a not depending on ∆t. The relation
hu − v, ui =
|u|2 |u − v|2 |v|2
+
−
2
2
2
(3.67)
40
Chapter 3. Numerical Methods
is also utilised. Combining eqs. (3.65) to (3.67), and performing some algebraic
manipulations leads to
LΨ
LΨ n+1,j+1
+ a∆t ||en+1,j+1
||2 +
||eΨ
− en+1,j
||2 + h(θ(Ψn+1,j ) − θ(Ψn+1 )), en+1,j+1
i
Ψ
Ψ
Ψ
2
2
LΨ n+1,j 2
≤
||eΨ || + h(θ(Ψn+1,j ) − θ(Ψn+1 )), en+1,j+1
− en+1,j
i.
Ψ
Ψ
2
(3.68)
By using the monotonicity of θ, its Lipschitz continuity and the Young inequality,
i.e. |uv| ≤ 2 |u|2 + 21 |v|2 for all > 0, this further becomes
LΨ n+1,j+1
1
LΨ
+ a∆t ||en+1,j+1
||2 +
||eΨ
− en+1,j
||2 + ||θ(Ψn+1,j ) − θ(Ψn+1 )||2
Ψ
Ψ
2
2
Lθ
1
LΨ n+1,j 2
||eΨ || + ||θ(Ψn+1,j ) − θ(Ψn+1 )||2 + ||en+1,j+1
− en+1,j
||2 ,
≤
Ψ
2
2
2 Ψ
(3.69)
Lθ
1
1
with the condition on Lθ : Lθ ≥ 2LΨ . From LΨ > 2 and letting = LΨ , this gives
LΨ n+1,j 2
LΨ
+ a∆t ||en+1,j+1
||2 ≤
||e
|| .
Ψ
2
2 Ψ
(3.70)
Finally,
||en+1,j+1
||2 ≤
Ψ
LΨ
||en+1,j ||2 .
(LΨ + 2∆ta) Ψ
(3.71)
Equation (3.71) is clearly a contraction [24], hence proving the convergence as
||en+1,j+1
|| −→ 0 when j −→ ∞.
Ψ
3.4.2
Linearisation Scheme for Richards’ Equation with
Dynamic Capillary Pressure
The linearisation scheme for Richards’ equation with dynamic capillary pressure
about to be presented, is a new scheme continuing the works of M.Slodicka [37],
Radu et al. [27] and F.A.Radu [40]. A similar scheme is given by Fan et al. [22],
where they solve for saturation and not pressure as in this thesis.
A fully implicit temporal discretisation of Richards’ equation with dynamic capillary pressure (eqs. (3.136) and (3.137)), by the backward Euler method, leads to
the nonlinear system
θn+1 + ∆tAn+1 Ψn+1 = ∆tf n+1 + θn ,
(3.72)
∆tΨn+1 = −∆tpc (θn+1 ) + τ θn+1 − τ θn ,
(3.73)
Chapter 3. Numerical Methods
41
where An+1 is the system matrix defined as (3.58). For a detailed explanation of
how eqs. (3.72) and (3.73) is derived see section 3.5.3. Note that this system is
coupled. The new linearisation method is considered when solving the system of
equations. Iterate j + 1 is obtained by solving
LΨ (Ψn+1,j+1 − Ψn+1,j ) + θn+1,j+1 + ∆tAn+1 Ψn+1,j+1 = ∆tf n+1 + θn ,
(3.74)
∆tΨn+1,j+1 = −∆tpc (θn+1,j ) + τ θn+1,j+1 − τ θn + Lθ (θn+1,j+1 − θn+1,j ).
(3.75)
and
LΨ , Lθ are positive constants to be specified at a later time. The errors at iteration
step j + 1 are introduced to evaluate the convergence of the linear system
en+1,j+1
= Ψn+1,j+1 − Ψn+1 ,
Ψ
(3.76)
en+1,j+1
= θn+1,j+1 − θn+1 .
θ
(3.77)
Again, the convergence will be proved by showing that
||en+1,j+1
||, ||en+1,j+1
|| −→ 0 when j −→ ∞,
Ψ
θ
(3.78)
where || · || is the notation for the discrete L2 norm, defined in the same manner
as in section 3.4.1.
Theorem 3.2. Assuming (A1)-(A4) and that the time step ∆t is sufficiently
small, and that the finite volume method satisfies (3.66), the linearisation scheme
(3.74) and (3.75) is (at least) linearly convergent for any LΨ > 0 and Lθ ≥ ∆tLpc .
Proof. Subtracting eqs. (3.72) and (3.73) from eqs. (3.74) and (3.75) results in
LΨ (en+1,j+1
− en+1,j
) + en+1,j+1
+ ∆tAn+1 (en+1,j+1
) = 0,
Ψ
Ψ
Ψ
θ
(3.79)
and
− en+1,j
) − ∆t(pc (θn+1,j ) − pc (θn+1 )) + τ en+1,j+1
= ∆ten+1,j+1
. (3.80)
Lθ (en+1,j+1
Ψ
θ
θ
θ
42
Chapter 3. Numerical Methods
Multiplying with ∆ten+1,j+1
in eq. (3.79) and en+1,j+1
in eq. (3.80) and then adding
Ψ
θ
the results gives
∆tLΨ hen+1,j+1
− en+1,j
, en+1,j+1
i + ∆then+1,j+1
, en+1,j+1
i
Ψ
Ψ
Ψ
Ψ
θ
+ ∆t2 hAn+1 en+1,j+1
, en+1,j+1
i + Lθ hen+1,j+1
− en+1,j
, en+1,j+1
i
Ψ
Ψ
θ
θ
θ
− ∆thpc (θn+1,j ) − pc (θn+1 ), en+1,j+1
i + τ ||en+1,j+1
||2 = ∆then+1,j+1
, en+1,j+1
i.
Ψ
θ
θ
θ
(3.81)
Applying (3.66) and (3.67) yields
∆t
∆t
n+1,j 2
LΨ ||en+1,j+1
||2 +
LΨ ||en+1,j+1
− eΨ
|| + ∆t2 a||en+1,j+1
||2
Ψ
Ψ
Ψ
2
2
Lθ
Lθ
+ ||eθn+1,j+1 ||2 + ||en+1,j+1
− en+1,j
||2 − ∆thpc (θn+1,j ) − pc (θn+1 ), en+1,j
i
θ
θ
2
2 θ
∆t
Lθ
||2
||2 ≤
LΨ ||en+1,j
||2 + ||en+1,j
+ τ ||en+1,j+1
Ψ
θ
2
2 θ
+ ∆thpc (θn+1,j ) − pc (θn+1 ), en+1,j+1
− en+1,j
i.
θ
θ
(3.82)
Using the fact that pc is monotone decreasing and Lipschitz continuous
− hpc (θn+1,j ) − pc (θn+1 ), en+1,j
i≥
θ
1
||pc (θn+1,j ) − pc (θn+1 )||2 ,
Lpc
(3.83)
and applying Young’s inequality with = Lpc , gives
∆tLΨ n+1,j+1 2 ∆tLΨ n+1,j+1
||eΨ
|| +
||eΨ
− en+1,j
||2 + ∆t2 a||en+1,i+1
||2
Ψ
Ψ
2
2
Lθ
∆t
+
+ τ ||en+1,j+1
||2 +
||pc (θn+1,j ) − pc (θn+1 )||2
θ
2
Lpc
Lθ
∆tLΨ n+1,j 2 Lθ n+1,j 2
+ ||en+1,j+1
− en+1,j
||eΨ || + ||eθ
||
||2 ≤
θ
θ
2
2
2
∆t
∆tLpc n+1,j+1
||pc (θn+1,j ) − pc (θn+1 )||2 +
||eθ
− en+1,j
+
||2 ,
θ
2Lpc
2
(3.84)
which in turn becomes
∆tLΨ n+1,j+1 2 ∆tLΨ n+1,j+1
||2
||eΨ
|| +
||eΨ
− en+1,j
||2 + ∆t2 a||en+1,j+1
Ψ
Ψ
2
2
Lθ
∆t
n+1,j+1 2
+
+ τ ||eθ
|| +
||pc (θn+1,j ) − pc (θn+1 )||2
(3.85)
2
2Lpc
Lθ ∆tLpc
∆tLΨ n+1,j 2 Lθ n+1,j 2
+
−
||en+1,j+1
− en+1,j
||2 ≤
||eΨ || + ||eθ
|| ,
θ
θ
2
2
2
2
Chapter 3. Numerical Methods
43
with the condition on Lθ : Lθ ≥ ∆tLpc . Finally, multiplying with 2 and leaving
out two positive terms leads to
(∆tLΨ + 2∆t2 a)||en+1,j+1
||2 + (Lθ + 2τ )||en+1,j+1
||2 ≤ ∆tLΨ ||en+1,j
||2 + Lθ ||en+1,j
||2 .
Ψ
Ψ
θ
θ
(3.86)
In order to show the convergence eq. (3.86) will be expressed on the form
γT n+1,j+1 ≤ γ1 T n+1,j , with γ ≥ γ1 .
(3.87)
||2 ,
T n+1,j+1 = ∆tLΨ ||en+1,j+1
||2 + Lθ ||en+1,j+1
Ψ
θ
(3.88)
By letting
eq. (3.86) becomes
T n+1,j+1 + 2∆t2 a||en+1,j+1
||2 + 2τ ||en+1,j+1
||2 ≤ T n+1,j .
Ψ
θ
(3.89)
The next step consists of defining
2∆t2 a
2τ
(∆tLΨ ||en+1,j+1
||2 ) + (Lθ ||en+1,j+1
||2 ) ≥ min
Ψ
θ
∆tLΨ
Lθ
2∆ta 2τ
+
LΨ
Lθ
T n+1,j+1 ,
(3.90)
and inserting this into eq. (3.89) which gives
min
2∆ta 2τ
+
LΨ
Lθ
and further
T n+1,j+1 ≤ + 1 T n+1,j+1 ≤ T n+1,j ,
1
min
2∆ta
LΨ
+
2τ
Lθ
+1
T n+1,j .
(3.91)
(3.92)
Equation (3.92) is a contraction, hence the system converges linearly for any
LΨ > 0 and Lθ ≥ ∆tLpc , as ||en+1,j+1
||, ||en+1,j+1
|| −→ 0 when j −→ ∞.
Ψ
θ
3.5
Discretisation of the System of Equations in
Space and Time
To this point, the focus of the spatial discretisations has been on the second
order in space ODE, eq. (3.12). However, the main equations to be evaluated in
this thesis, given by the mathematical models (2.50)-(2.52) in section 2.4.2, are
44
Chapter 3. Numerical Methods
PDEs that are second order in space and first order in time. This section will
thus present the spatial and temporal discretisation of the convection-diffusion
equation, the Richards equation and the equations including dynamic capillarity.
The two latter being nonlinear equations undergoing additional considerations
with the linearisation schemes presented in section 3.4.
3.5.1
Convection-Diffusion Equation
The convection-diffusion equation from (2.51) is on the form
(θc)t + (−θDc cx + q(x, t)c)x = Q(x, t),
(3.93)
where q(x, t) is the water flux, Q(x, t) is the source or sink term, and c=c(x,t) the
concentration of the species, see section 2.1.5. Dc is the diffusion coefficient, and
θ is some function
θ = θ(x, t),
(3.94)
which models the water content in the porous medium. θ and q are obtained by
solving the flow equation numerically.
In this case F in the Euler implicit method, eq. (3.54), becomes
F (t, c(t)) = Q(x, t) + (θDc cx )x − (q(x, t)c)x ,
(3.95)
and the approximation of the time derivative is given by the backward difference,
eq. (3.4)
θn+1 cn+1 − θn cn
,
(3.96)
(θ(x, t)c)t =
∆t
where θn+1 = θ(x, tn+1 ). Thus, the implicit scheme for time step n + 1 is
n+1 n+1
c )x = ∆tQn+1 + θn cn .
θn+1 cn+1 − ∆t (θn+1 Dc cn+1
x )x − (q
(3.97)
To ease the notation going into the spatial discretisation, a variable D is denoted
as
D(x, t) = θn+1 Dc ,
(3.98)
and
dn+1 = −Dcn+1
x .
(3.99)
θn+1 cn+1 + ∆t(dn+1
+ (q n+1 cn+1 )x ) = ∆tQn+1 + θn cn ,
x
(3.100)
Then eq. (3.93) becomes
Chapter 3. Numerical Methods
45
from the linearity of the derivative.
Applying the cell-centred grid, see Figure 3.4, and integrating over each cell,
[xi− 1 , xi+ 1 ], results in
2
2
Z
xi+ 1
2
xi− 1
Z2 x
∆t
n+1 n+1
n+1 n+1
θn+1 cn+1 dx + ∆t(dn+1
− dn+1
+ qi+
− qi−
)=
1 c
1 c
i+ 1
i− 1
i+ 1
i− 1
2
i+ 1
2
Z
n+1
Q
2
2
2
2
(3.101)
xi+ 1
2
dx +
xi− 1
2
n n
θ c dx,
xi− 1
2
2
where e.g. di+ 1 = d(xi+ 1 , tn+1 , u(xi+ 1 , tn+1 )) and qi+ 1 = q(xi+ 1 , tn+1 ).
2
2
2
2
2
To approximate the integrals, the midpoint rule is applied the same way as in
eq. (3.25). Given that xi is the midpoint in the interval [xi− 1 , xi+ 1 ] and the length
2
2
of the interval is denoted by h, the integrals in eq. (3.101) are replaced by
xi+ 1
Z
2
Qn+1 dx ≈ ∆thQn+1
,
i
(3.102)
θn+1 cn+1 dx ≈ hθin+1 cn+1
i
(3.103)
∆t
xi− 1
2
Z
xi+ 1
2
xi− 1
2
and
Z
xi+ 1
2
θn cn dx ≈ hθin cni ,
(3.104)
xi− 1
2
which leads to
n+1 n+1
n+1 n+1
hθin+1 cn+1
+ ∆t(dn+1
− dn+1
+ qi+
− qi−
) = ∆thQn+1
+ hθin cni . (3.105)
1 c
1 c
i
i
i+ 1
i− 1
i+ 1
i− 1
2
2
2
2
2
2
Since the applied grid is cell-centred, the function values are defined at the centres
of the cells, leaving the values at the walls of the cells to be determined. These
values are approximated by the mean of the function values of the two neighbouring
cells, yielding
cn+1
+ cn+1
i
i+1
,
(3.106)
cn+1
≈
i+ 12
2
for i = 0, . . . , N . Also recalling from eq. (3.21) that
qi+ 1 = −ai+1 (ui+1 − ui ),
2
(3.107)
46
Chapter 3. Numerical Methods
where ai is given by eq. (3.24), for i = 0, . . . , N . Inserting eq. (3.106) gives
n+1 n+1
n+1 n+1
n+1
qi+
− qi−
≈ qi+
1 c
1 c
1
i+ 1
i− 1
2
2
2
2
2
n+1
n+1
cn+1
+ cn+1
i
i+1
n+1 ci−1 + ci
− qi−
,
1
2
2
2
(3.108)
for i = 1, . . . , N . Rearranging eq. (3.108) leads to
1 n+1 n+1
n+1 n+1
n+1 n+1
n+1
n+1 n+1
n+1 n+1
(qi+ 1 ci+1 + (qi+
qi+
− qi−
1 − q
1 ci−1 ).
1 c
1 − q
1 c
1 ≈
1 )ci
i+ 2
i− 2 i− 2
i−
2
2
2
2
2
2
(3.109)
To find an expression for dn+1
, the procedure defined in section 3.2.2 is followed.
i+ 1
2
= −dn+1 /D for an equidistant grid with h denoting the length of
Integrating cn+1
x
each cell, leads to
cn+1
i+1
−
h
−dn+1
i+ 21 2
1
1
+
Di Di+1
cn+1
i
=
dn+1
i+ 12
n+1
cn+1
i+1 − ci
.
=− h
1
1
+
2 Di
Di+1
and
,
(3.110)
(3.111)
This means that
n+1
− cn+1
cn+1
cn+1
i
i−1
i+1 − ci
n+1
,
+
−
d
=
−
dn+1
i− 21
i+ 12
1
1
1
1
h
h
+
+
2 Di
Di+1
2 Di
Di−1
(3.112)
and by further letting
si =
1
h
2
1
Di
+
1
,
(3.113)
Di−1
eq. (3.112) simplifies to
n+1
dn+1
− dn+1
= −si+1 (cn+1
) + si (cn+1
− cn+1
i+1 − ci
i
i−1 ).
i+ 1
i− 1
2
(3.114)
2
Inserting eqs. (3.109) and (3.114) into eq. (3.105) yields
n+1
n+1
+ ∆tsi (cn+1
− cn+1
)+
hθin+1 cn+1
i
i
i−1 ) − ∆tsi+1 (ci+1 − ci
∆t n+1 n+1
n+1
n+1 n+1
n+1 n+1
n+1
+ (qi+
)ci − qi−
+ hθin cni .
(q 1 c
1 − q
1 ci−1 ) = ∆thQi
i− 21
2
2
2 i+ 2 i+1
(3.115)
Chapter 3. Numerical Methods
47
Collecting the terms and dividing by h finally leads to
∆t
θin+1 cn+1
+
i
h
∆t
−
h
si +
−si+1 +
n+1
qi−
1
2
2
n+1
qi+
1
2
2
!
∆t
cn+1
i+1 +
h
si + si+1 +
n+1
n+1
qi+
1 − q
i− 1
2
2
2
!
cn+1
i
!
n+1
cn+1
+ θin cni .
i−1 = ∆tQi
(3.116)
This is a linear system discretised with respect to both space and time, which is
implemented and solved numerically for cn+1
, i = 1, . . . , N , n = 0, . . . , T .
i
3.5.2
Standard Richards’ Equation
The standard Richards equation is given by
(θ(Ψ))t − (K(Ψ)(Ψ(x, t) + x)x )x = f (x, t),
(3.117)
where Ψ(x, t) is the pressure head, f (x, t) is the source or sink term divided by
the density ρ, (see eq. (2.35)), and x denotes the height against the reference level
(datum). θ(Ψ) and K(Ψ) are nonlinear functions given by some parameterisation. Nonlinear equations, as is the case for the Richards equation, need to be
linearised in order to be solved by the means of numerical methods. The linearisation scheme presented in section 3.4 is therefore applied, in order to compute
the linear, discretised system needed for it to be solved numerically.
Starting with the temporal discretisation the backward Euler scheme (eq. (3.55))
is applied, where
F (t, Ψ(t)) = f (x, t) + (K(Ψ)(Ψ(x, t) + x)x )x ,
(3.118)
which leads to
n+1
θ(Ψn+1 ) − θ(Ψn ) − ∆t(K(Ψn+1 )Ψn+1
))x = ∆tf n+1 ,
x )x − ∆t(K(Ψ
(3.119)
as the implicit scheme for time step n + 1, where Ψn+1 = Ψ(x, tn+1 ). The standard approach to solving this system is to apply Newton’s method. However, as
mentioned in section 3.4, this method has several drawbacks. Instead applying
48
Chapter 3. Numerical Methods
the robust linearisation method given, yields
LΨ (Ψn+1,j+1 − Ψn+1,j ) + θ(Ψn+1,j ) − θ(Ψn ) − ∆t(K(Ψn+1,j )Ψn+1,j+1
)x
x
− ∆tKx (Ψn+1,j ) = ∆tf n+1 ,
(3.120)
for obtaining iteration step j +1, where LΨ is some positive constant. Rearranging
eq. (3.120) so that the known values at previous time and iteration steps are located
on the right hand side of the equation, leads to
LΨ Ψn+1,j+1 − ∆t(K(Ψn+1,j )Ψn+1,j+1
)x = ∆tf n+1 + LΨ Ψn+1,j − θ(Ψn+1,j )
x
+ θ(Ψn ) + ∆tKx (Ψn+1,j ),
(3.121)
concluding the temporal discretisation and the iterative approach.
The first step in the spatial discretisation is to define the flux as
q = −K(Ψn+1,j )Ψn+1,j+1
,
x
(3.122)
then eq. (3.120) becomes
LΨ Ψn+1,j+1 + ∆tqx = ∆tf n+1 + LΨ Ψn+1,j − θ(Ψn+1,j ) + θ(Ψn ) + ∆tKx (Ψn+1,j ).
(3.123)
Applying the cell-centred grid, see fig. 3.4, and integrating over each cell, [xi− 1 , xi+ 1 ],
2
2
results in
Z x 1
Z x 1
Z x 1
i+ 2
i+ 2
i+ 2
n+1,j+1
n+1
LΨ
Ψ
dx + ∆t(qi+ 1 − qi− 1 ) = ∆t
f
+ LΨ
Ψn+1,j dx
2
xi− 1
Z
−
2
xi+ 1
2
2
xi− 1
2
θ(Ψn+1,j ) dx +
xi− 1
xi+ 1
Z
2
2
θ(Ψn ) dx + ∆t(Ki+ 1 (Ψn+1,j ) − Ki− 1 (Ψn+1,j )).
xi− 1
2
xi− 1
2
2
2
(3.124)
As in section 3.5.1, eqs. (3.103) to (3.105), the integrals of eq. (3.124) are approximated by the midpoint rule. Inserting these approximations into eq. (3.124),
recalling that the grid is equidistant with h = ∆xi , gives
LΨ hΨn+1,j+1
+ ∆t(qi+ 1 − qi− 1 ) = ∆thfin+1 + LΨ hΨn+1,j
− hθi (Ψn+1,j ) + hθi (Ψn )
i
i
2
n+1,j
+ ∆t(Ki+ 1 (Ψ
2
2
) − Ki− 1 (Ψn+1,j )),
2
(3.125)
n+1,j
where θi (Ψ
)=
θ(Ψn+1,j
).
i
Chapter 3. Numerical Methods
49
To determine the values at the walls of the cells, the mean of the function values
of the two neighbouring cells is computed so that
Ki+ 1 (Ψn+1,j ) ≈
2
Ki (Ψn+1,j ) + Ki+1 (Ψn+1,j )
2
(3.126)
Ki−1 (Ψn+1,j ) + Ki (Ψn+1,j )
,
2
(3.127)
for i = 0, . . . , N , and
Ki− 1 (Ψn+1,j ) ≈
2
for i = 1, . . . , N . This gives
∆t(Ki+ 1 (Ψn+1,j ) − Ki− 1 (Ψn+1,j ))
2
2
Ki (Ψn+1,j ) + Ki+1 (Ψn+1,j ) Ki−1 (Ψn+1,j ) + Ki (Ψn+1,j )
≈ ∆t
−
2
2
(3.128)
for i = 1, . . . , N . Subtracting the common terms yield
∆t(Ki+ 1 (Ψn+1,j ) − Ki− 1 (Ψn+1,j )) ≈
2
2
∆t
(Ki+1 (Ψn+1,j ) − Ki−1 (Ψn+1,j )). (3.129)
2
The expression for qi+ 1 is found by applying the TPFA scheme, see section 3.2.2.
2
=
−q/K(Ψn+1,j ) for an equidistant grid with h denoting the
Integrating Ψn+1,j+1
x
length of each cell, leads to
Ψn+1,j+1
i+1
−
Ψn+1,j+1
i
and
qi+ 1
2
h
= −qi+ 1
2 2
1
1
+
Ki Ki+1
,
Ψn+1,j+1
− Ψn+1,j+1
i+1
i
.
=−
h
1
1
+
2 Ki
Ki+1
(3.130)
(3.131)
This means that
qi+ 1 − qi− 1 = −
2
2
Ψn+1,j+1
− Ψn+1,j+1
Ψn+1,j+1 − Ψn+1,j+1
i+1
i
i−1
+ i 1
1
h
1
h
1
+
+
2 Ki
Ki+1
2 Ki
Ki−1
(3.132)
and by further letting
ai =
1
h
2
1
Ki
+
1
Ki−1
from eq. (3.24), eq. (3.132) simplifies to
− Ψn+1,j+1
) + ai (Ψn+1,j+1
− Ψn+1,j+1
).
qi+ 1 − qi− 1 = −ai+1 (Ψn+1,j+1
i+1
i
i
i−1
2
2
(3.133)
50
Chapter 3. Numerical Methods
Inserting eqs. (3.129) and (3.133) into eq. (3.125) yields
LΨ hΨn+1,j+1
+ ∆tai (Ψn+1,j+1
− Ψn+1,j+1
) − ∆tai+1 (Ψn+1,j+1
− Ψn+1,j+1
) = ∆thfin+1
i
i
i−1
i+1
i
∆t
(Ki+1 (Ψn+1,j ) − Ki−1 (Ψn+1,j )).
+ LΨ hΨn+1,j
− hθi (Ψn+1,j ) + hθi (Ψn ) +
i
2
(3.134)
Collecting the terms and dividing by h finally leads to
∆t
∆t
∆t
ai Ψn+1,j+1
+
(ai + ai+1 )Ψn+1,j+1
−
ai+1 Ψn+1,j+1
= ∆tfin+1
i−1
i
i+1
h
h
h
∆t
− θi (Ψn+1,j ) + θi (Ψn ) +
(Ki+1 (Ψn+1,j ) − Ki−1 (Ψn+1,j )).
2h
(3.135)
LΨ Ψn+1,j+1
−
i
+ LΨ Ψn+1,j
i
The original nonlinear system is now a linear system discretised with respect to
space and time, and is implemented and solved numerically for Ψn+1,j+1
, i=1,. . . ,N,
i
n = 1, . . . , T , j = 1, . . . , J.
3.5.3
Richards’ Equation with Dynamic Capillary Pressure
In the case of dynamic capillary pressure, two equations are considered. These are
θt − (K(θ)(Ψ(x, t) + x)x )x = f (x, t),
(3.136)
and
∂θ
,
(3.137)
∂t
where the first equation is the Richards equation considered in section 3.5.2, and
the second is the extension accommodating dynamic capillary pressure. pc (θ) is the
capillary pressure and τ (θ) = τ some constant. Both equations are nonlinear and
need to be linearised with the method described in section 3.4. Equations (3.136)
and (3.137) are coupled, meaning one cannot be solved without the other.
Ψ(x, t) = −pc (θ) + τ (θ)
The implicit numerical schemes when applying the backward Euler method (eq. (3.55)),
are
n+1
θn+1 − θn − ∆t(K(θn+1 )Ψn+1
))x = ∆tf n+1
(3.138)
x )x − ∆t(K(θ
from eq. (3.119), and
∆tΨn+1 = −∆tpc (θn+1 ) + τ (θn+1 − θn ).
(3.139)
Chapter 3. Numerical Methods
51
Again, the robust iterative scheme for Richards’ equation is previously given by
eq. (3.120) in section 3.5.2
LΨ (Ψn+1,j+1 − Ψn+1,j ) + θn+1,j+1 − θn − ∆t(K(θn+1,j )Ψn+1,j+1
)x
x
− ∆tKx (θn+1,j ) = ∆tf n+1 ,
(3.140)
while the iterative scheme for the dynamic capillary pressure becomes
∆tΨn+1,j+1 = −∆tpc (θn+1,j ) + τ θn+1,j+1 − τ θn + Lθ (θn+1,j+1 − θn+1,j ), (3.141)
when applying the linearisation method. LΨ and Lθ are positive constants.
Remark. In the case of a non-constant τ , i.e. τ = τ (θ), the iterative scheme
(3.141) above becomes
∆tΨn+1,j+1 = −∆tpc (θn+1,j )+τ (θn+1,j )θn+1,j+1 −τ (θn+1,j )θn +Lθ (θn+1,j+1 −θn+1,j ).
(3.142)
This scheme was also implemented and results are presented in section 4.2.
In order to solve the coupled system, eq. (3.141) is first solved with respect to
θn+1,j+1 , before being inserted into eq. (3.140). Rearranging eq. (3.141) gives
θn+1,j+1 =
1
(∆tΨn+1,j+1 + ∆tpc (θn+1,j ) + τ θn + Lθ θn+1,j ),
(τ + Lθ )
(3.143)
and by inserting the relation in eq. (3.140), it follows that
LΨ (Ψn+1,j+1 − Ψn+1,j )
1
+
(∆tΨn+1,j+1 + ∆tpc (θn+1,j ) + τ θn + Lθ θn+1,j )
(τ + Lθ )
(3.144)
− θn − ∆t(K(θn+1,j )Ψn+1,j+1
)x − ∆tKx (θn+1,j ) = ∆tf n+1 .
x
Moving the known terms from the previous time and iteration steps to the right
hand side og the equation, leads to
∆t
Ψn+1,j+1 − ∆t(K(θn+1,j )Ψn+1,j+1
)x
x
(τ + Lθ )
∆t
τ
= ∆tf n+1 + LΨ Ψn+1,j −
pc (θn+1,j ) −
θn
(τ + Lθ )
(τ + Lθ )
Lθ
−
θn+1,j + θn + ∆tKx (θn+1,j ).
(τ + Lθ )
LΨ Ψn+1,j+1 +
(3.145)
52
Chapter 3. Numerical Methods
Integrating each term over each cell [xi− 1 , xi+ 1 ], inserting eqs. (3.129) and (3.133)
2
2
from section 3.5.2 and approximating the remaining integrals using the midpoint
rule, see eqs. (3.102) to (3.104), yields
LΨ hΨn+1,j+1
+
i
∆th
Ψn+1,j+1 + ∆tai (Ψn+1,j+1
− Ψn+1,j+1
)
i
i−1
(τ + Lθ ) i
− ∆tai+1 (Ψn+1,j+1
− Ψn+1,j+1
) = ∆thfin+1 + LΨ hΨn+1,j
i+1
i
i
τ
h
L
h
∆th
θ
pc (θn+1,j ) −
θn −
θn+1,j + hθin
−
(τ + Lθ ) i
(τ + Lθ ) i
(τ + Lθ ) i
∆t
+
(Ki+1 (θn+1,j ) − Ki−1 (θn+1,j )).
2
(3.146)
Finally, collecting the terms and dividing by h, gives
LΨ +
∆t
(τ + Lθ )
∆t
∆t
ai Ψn+1,j+1
+
(ai + ai+1 )Ψn+1,j+1
i−1
i
h
h
∆t
∆t
−
ai+1 Ψn+1,j+1
= ∆tfin+1 + LΨ Ψn+1,j
−
pc (θn+1,j )
i+1
i
h
(τ + Lθ ) i
∆t
τ
Lθ
θin+1,j +
(Ki+1 (θn+1,j ) − Ki−1 (θn+1,j ))
+ 1−
θin −
(τ + Lθ )
(τ + Lθ )
2h
(3.147)
Ψn+1,j+1
−
i
Equation (3.147) represents the linear, discretised system for the case with Richards’
equation and dynamic capillary pressure. This system is implemented and solved
numerically for Ψn+1,j+1
, i=1,. . . ,N, n = 1, . . . , T , j = 1, . . . , J.
i
3.6
Implementation
The numerical computations based on the discretised system of equations derived in sections 3.5.1 to 3.5.3 are performed using the software MATLAB. The
discretised equations are implemented using functions where values of all known
variables, as well as initial and boundary conditions, are defined. Approximated
solutions of the unknown variables are computed at time step n+1 as time evolves.
The approximations will improve for each time step in the case of convergence of
the system.
As well as being evaluated at each time step, the solutions of the linearised
Richards’ equation, with and without dynamic capillary pressure are evaluated
at each iteration step until some stopping criteria is met. There are two possible
types of stopping criteria [24]. One based on evaluating the residual and the other
Chapter 3. Numerical Methods
53
is based on the increment. In both cases a fixed tolerance on the approximated solution, ε, is defined. When considering the residual approach, the iterative process
terminates at the first step i where ||f (Ψn+1,i )|| < ε. While when evaluating the
increment, the iterative process terminates when ||Ψn+1,i+1 − Ψn+1,i || < ε. The
latter is applied in the numerical schemes constructed for this thesis. In the possible event that the method does not converge, a security measure to prevent an
indefinite continuation of the iterative process is included. That is, after a maximum number of iteration steps set high enough to ensure a good approximation
of the solution in the case of convergence, the method is terminated.
The linear system at each time step can be solved by the means of a direct solver
or an iterative one (e.g. Jacobi or Gauss-Seidel, [24, 25]). In this work a direct
solver is utilised. The problem being one dimensional gives rise to relatively small
linear systems, making the use of a direct solver convenient.
Finally, the solutions of Ψ (pressure), θ (water content) and c (concentration) as
well as the convergence history of the linearisation schemes, are presented graphically at certain time and grid steps.
Chapter 4
Numerical Results
This chapter is devoted to presenting the results attained from numerical simulations constructed on the basis of the theoretical framework and discretised models
described in preceding chapters. The first section gives the results of a numerical
convergence test, where a constructed analytical solution was applied to verify
the convergence of the numerical method. Section 4.2 contains the numerical results produced when the developed numerical schemes are applied to the transport
problem. Comparing the results with and without dynamic capillary pressure is
the main focus. Also, in order to confirm the statements made about the linearisation schemes in section 3.4, the convergence history displaying the number of
iterations for different grid lengths h are presented. From this, it will be shown
that the linearisation schemes are at least linearly convergent, robust and indeed
independent of mesh size.
4.1
Convergence of the Numerical Solution for
an Academical Example
The purpose of this section is to show that the numerical solution of the system
converges. To do so a numerical test is presented, based the approach described
in [38], section 3. The numerical simulations are performed by choosing a simple
analytical solution that satisfies the given initial and boundary conditions and by
adjusting the functions so that this chosen solution corresponds to the exact solution of the equation. The numerical solutions are then compared to the analytical
solution by computing the L2 error.
55
56
Chapter 4. Numerical Results
The convergence test is performed on (2.50) with gravitation and a source term
∂t θ(Ψ) − ∂x (K(Ψ)∂x (Ψ + x)) = f (x, t) in (0, T ) × (0, 1),
(4.1)
with final time T = 1. For the numerical test LΨ = 0.1 and the coefficient functions
θ(Ψ), K(Ψ) are set to
1 − Ψ2
,
2
K(Ψ) = 1 − Ψ2 ,
θ(Ψ) =
(4.2)
(4.3)
The initial condition is given by
Ψ|t=0 = 0,
(4.4)
and Neumann boundary conditions are applied at both end points of the interval
q = K(x = 0)t,
(4.5)
q = −K(x = 1)t.
(4.6)
Note that tests with homogeneous Dirichlet boundary conditions (Ψ|x=0 = Ψ|x=1 =
0) were also conducted, giving the same convergence results.
Constructing an analytical solution for Ψ(x, t) that satisfies the given initial and
boundary conditions yields
Ψana (x, t) = −tx(1 − x).
(4.7)
To find the source term, eqs. (4.2) and (4.3) are inserted into eq. (4.1) and the
partial derivatives are calculated from
f (x, t) =
1 − Ψ2ana (x, t)
2
− ((1 − Ψ2ana (x, t))(Ψanax (x, t) + 1))x ,
(4.8)
t
which becomes
f (x, t) = −tx2 (1−x)2 +2(t3 x(5x3 −10x2 +6x−1)−t)+(2t2 x(2x2 −3x+1)), (4.9)
when Ψana is defined by eq. (4.7). Inserting eq. (4.9) into eq. (4.1) gives the
equation to be solved in the numerical convergence test.
Solving the equation given by eqs. (4.1) and (4.9) numerically, yields numerical
solutions which can be compared to the analytical solution of eq. (4.7). The
Chapter 4. Numerical Results
57
procedure for solving Richards’ equation by the means of a numerical method is
described in section 3.5.2. The Neumann boundary conditions are treated as in
section 3.2.3.
To compare the numerical solutions to the analytical one, the L2 norm is computed
as a measure of the difference between the two. Recalling that the L2 norm
[5, 21, 24, 25] is defined as
Z
2
||Ψ||2 =
|Ψ| dx
12
,
(4.10)
for a function Ψ. The error is computed as
E = ||Ψana (x, T ) − Ψnum (x, T )||2 ,
(4.11)
where Ψana (x, T ) and Ψnum (x, T ) is the analytical and numerical solution respectively. t = T for t ∈ [0, 1] represents the final time, thus T = 1. The squared error
is given by
2
E = ||Ψana (x, T ) −
Ψnum (x, T )||22
Z
=
1
|Ψana (x, T ) − Ψnum (x, T )|2 dx,
(4.12)
0
for x ∈ [0, 1]. To solve the equations numerically a cell-centered grid is applied to the interval [0, 1], dividing it into subintervals or cells with midpoints
xi , i = 0, . . . , n, see chapter 3. Integrating over each cell [xi− 1 , xi+ 1 ] gives
2
2
E =
n Z
X
i=1
xi+ 1
2
|Ψana (x, T ) − Ψnum (x, T )|2 dx,
2
(4.13)
xi− 1
2
which is becomes
2
E '
n
X
h|Ψana (xi , T ) − Ψnum (xi , T )|2 ,
(4.14)
i=1
when the integrals are approximated using the midpoint rule with h representing
the uniform length of each cell.
The computations are performed on a uniform grid with h = ∆t = 0.1 and halving
the grid size (h) and time step (∆t) successively. Figure 4.1 show the solutions
for h = 0.1, 0.05, 0.025, 0.0125, 0.0625. The numerical solutions for pressure Ψ are
compared to the analytical solution marked with blue stars, and it is clear that the
numerical solution converges to the analytical one, due to the decreasing difference
in the solutions for decreasing grid and time steps.
58
Chapter 4. Numerical Results
Solutions of the Pressure
0
−0.05
−0.1
−0.15
−0.2
−0.25
0
0.2
0.4
0.6
0.8
1
Figure 4.1: Numerical solutions for h = 0.1, 0.05, 0.025, 0.0125, 0.0625 compared to the analytical solution marked by stars.
According to Theorem 1 in [38], E ≤ C(∆t + h) with C constant not depending
on the discretisation parameters ∆t and h. The results are presented in table 4.1.
As predicted in Theorem 1, the reduction of the error E is of factor 2, which means
a convergence of O(∆t + h) or O(∆t2 + h2 ) for E2 .
Table 4.1: Errors E for different h
Time step (∆t)
0.1
0.05
0.025
0.0125
0.00625
4.2
Step length (h)
0.1
0.05
0.025
0.0125
0.00625
Error (E)
0.018378609390458
0.009022002647100
0.004505633495526
0.002254288598985
0.001127719602277
Reduction
2.037087563520698
2.002382718447396
1.998694176759216
1.998979705977481
Numerical Simulations
Having established the convergence of the method, it can be used to perform
numerical simulations for the transport problem with and without including dynamic capillary pressure. The numerical results of these simulations are presented
in section 4.2.1. To evaluate the convergence of the linearisation methods, the
convergence history of the schemes are presented in section 4.2.2.
Chapter 4. Numerical Results
4.2.1
59
Numerical Solutions of Flow and Transport
The equations that are considered in the numerical simulations are defined in
(2.50)-(2.52), and are restated below.
(θ(Ψ))t − (K(θ(Ψ))(Ψ + x)x )x = f (x, t),
is again the standard Richards equation, where θ(Ψ), K(θ(Ψ))) are assumed to be
constitutive relations given by the van Genuchten-Mualem parameterisation, see
section 2.2.4,
For Ψ ≤ 0
n−1
n
1
,
θ(Ψ) = θR + (θS − θR )
1 + (−αΨ)n
h
i
1−n 2
n−1
n n
1 − (−αΨ)
[1 + (−αΨ) ]
K(θ(Ψ)) = KS
.
n−1
[1 + (−αΨ)n ] 2n
For Ψ > 0
θ = θS ,
K = KS .
Unless otherwise stated, the van Genuchten-Mualem formula is used to compute
the hydraulic conductivity K. Referred to as non-constant K(θ).
To include dynamic capillarity, recall that the extension
τ (θ)∂t θ(Ψ) = Ψ + pc (θ),
replaces the partial derivative of θ with respect to t in Richards’ equation. For the
first case considered, τ (θ) = τ is some constant not depending on θ. Also recall
that for the case with dynamic capillary pressure θ 6= θ(Ψ) (and K = K(θ)).
To model the transport of the dissolved substance, the convection-diffusion equation given by
(θc)t + (−θDc cx + q(x, t)c)x = Q(x, t),
is coupled to the flow equations.
The spatial domain range from x ∈ [0, 1], where x = 1 corresponds to the ground
surface. On this interval, the initial condition when t = 0 and boundary conditions
60
Chapter 4. Numerical Results
for x = 0 and x = 1 are defined. For the standard and non-standard Richards’
equation, the boundary conditions are set to be Dirichlet for the left boundary
and Neumann for the right boundary, so that
Ψ(0, t) = −1,
q(1, t) = −3 × 10−3 .
For the transport problem, a homogeneous initial condition,
c(x, 0) = 0,
is given and the boundary conditions are defined as Dirichlet conditions for both
boundaries,
c(0, t) = 0,
c(1, t) = 1.
The known variables in the van Genuchten-Mualem parameterisation are defined
by the residual water content θR = 0.026 and the saturated water content θS =
0.420. Further, α = 0.95, n = 1.9 and the saturated hydraulic conductivity
KS = 2 × 10−2 . These values are based on realistic values found in the PhD thesis
of E.Schneid [43]. Note that the magnitude of the variables corresponds to values
found in i.e. Hassanizadeh et al. [39] and Fučı́k et al. [42], when converting from
seconds into days. The latter being the time scale in this thesis. The diffusion
coefficient Dc = 10−1 .
In the following, two examples are considered when computing the numerical results. Boundary conditions, the initial condition for the convection-diffusion equation and known variables remain the same, while the initial condition for the
Richards equation is adjusted.
Example I
The initial condition for the pressure Ψ is given by
Ψ(x, 0) = −x − 1,
in order to model the water flow in the strictly unsaturated region. This is to
ensure the validity of assumption (A2) in section 3.4.1. For the case when both
the saturated and the unsaturated region is included in the model, the given pc (θ)
is not Lipschitz continuous.
Chapter 4. Numerical Results
61
When computing the standard Richards equation LΨ = 0.07, while in the case with
dynamic capillary pressure LΨ = 0.001, Lθ = 1 and τ = 20. With the parameters
and conditions defined above, simulations are performed with step size h = 0.005
and time step ∆t = 0.01.
The numerical solutions for the pressure Ψ and water content θ with and without
dynamic capillary pressure for constant K = KS , are given in Figure 4.2. In
Figure 4.3 solutions of the same variables but for non-constant K(θ) are presented.
The solutions are shown for t = 0.01 (T1 ), t = 0.5 (Tim ) and t = 1 (Tend ).
Water content without dynamic capillary pressure
0.32
−1.1
0.31
−1.2
0.3
−1.3
0.29
Water content
Pressure
Pressure without dynamic capillary pressure
−1
−1.4
−1.5
−1.6
−1.7
0.27
0.26
0.25
T
−1.8
0.2
1
Tim
0.23
Tend
0
T
0.24
1
Tim
−1.9
−2
0.28
0.4
0.6
0.8
0.22
1
Tend
0
0.2
0.4
x
0.6
0.8
1
x
Pressure with dynamic capillary pressure
Water content with dynamic capillary pressure
−0.95
0.32
0.31
−1
0.3
−1.05
Water content
Pressure
0.29
−1.1
−1.15
−1.2
0.28
0.27
0.26
0.25
−1.25
T1
0
0.2
Tim
0.23
Tend
−1.35
T1
0.24
Tim
−1.3
0.4
0.6
x
0.8
1
0.22
Tend
0
0.2
0.4
0.6
0.8
1
x
Figure 4.2: Pressure and water content profiles for the standard Richards
equation (top), and with the influence of dynamic capillary pressure (bottom).
Computations are done with constant K = KS , τ = 20 at time T1 = 0.01,
Tim = 0.5, Tend = 1.
Nonhomogeneous Neumann boundary conditions, as defined on the right boundary
of the domain (the assumed ground surface), means there is a flow or flux across
the boundary. In other words, the pressure is increased at the boundary which is
the same as water being pumped into the domain. The nonhomogeneous Dirichlet
condition gives the value of the solution at the left boundary. The given value is a
62
Chapter 4. Numerical Results
Water content without dynamic capillary pressure
0.32
−1.1
0.31
−1.2
0.3
−1.3
0.29
Water content
Pressure
Pressure without dynamic capillary pressure
−1
−1.4
−1.5
−1.6
−1.7
0.27
0.26
0.25
T
−1.8
Tim
0.2
1
Tim
0.23
Tend
0
T
0.24
1
−1.9
−2
0.28
0.4
0.6
0.8
0.22
1
Tend
0
0.2
0.4
x
Pressure with dynamic capillary pressure
0.32
−1
0.31
1
0.3
−1.1
0.29
Water content
−1.2
Pressure
0.8
Water content with dynamic capillary pressure
−0.9
−1.3
−1.4
−1.5
0.28
0.27
0.26
0.25
−1.6
T1
T1
0.24
T
T
im
−1.7
−1.8
0.6
x
0
0.2
im
0.23
Tend
0.4
0.6
x
0.8
1
0.22
Tend
0
0.2
0.4
0.6
0.8
1
x
Figure 4.3: Pressure and water content profiles for the standard Richards
equation (top), and with the influence of dynamic capillary pressure (bottom).
Computations are done with non-constant K(θ) and constant τ = 20 at time
T1 = 0.01, Tim = 0.5, Tend = 1.
result of the chosen initial condition which as mentioned previously, is a measure
to ensure Lipschitz continuous pc (θ).
As time evolves there is an increase in pressure and water content throughout the
domain. The observed increase is larger in the case with constant K = KS than
in the case where the van Genuchten-Mualem parameterisation for the hydraulic
conductivity is applied. This is true for both the pressure and the water content
profiles. As will be evident shortly, this effects the transport of the dissolved
substance. The dynamic effects are obvious in both cases (constant and nonconstant K), and a clear difference between the standard and non-standard profiles
for pressure and water content is seen.
In Figure 4.4 numerical solutions of Richards’ equation with dynamic capillary
pressure for varying values of τ (θ) are presented.
Chapter 4. Numerical Results
63
Pressure at T for different tau, constant K
Water content at T for different tau, constant K
1
1
−1
0.32
−1.1
0.31
−1.2
0.3
−1.3
0.29
tau=0
tau=20
Water content
Pressure
tau=1−theta2
−1.4
−1.5
−1.6
−1.7
0.28
0.27
0.26
0.25
tau=0
tau=20
−1.8
0.24
tau=1−theta2
−1.9
−2
tau=102e−7.7theta
0.23
tau=102e−7.7theta
0
0.2
0.4
0.6
0.8
0.22
1
0
0.2
0.4
0.6
0.8
1
x
x
Pressure at Tim for different tau, constant K
Water content at Tim for different tau, constant K
−0.9
0.32
tau=0
tau=20
−1
tau=0
tau=20
0.31
2
tau=1−theta2
tau=1−theta
tau=102e−7.7theta
tau=102e−7.7theta
0.3
Water content
Pressure
−1.1
−1.2
−1.3
0.29
0.28
0.27
0.26
−1.4
0.25
−1.5
−1.6
0.24
0
0.2
0.4
0.6
0.8
0.23
1
0
0.2
0.4
0.6
0.8
1
x
x
Pressure at Tend for different tau, constant K
Water content at Tend for different tau, constant K
−1
0.33
−1.05
0.32
tau=0
tau=20
tau=1−theta2
−1.1
Water content
Pressure
−1.15
−1.2
−1.25
−1.3
−1.35
tau=0
tau=20
−1.4
tau=1−theta
−1.45
0.3
0.29
0.28
0.27
0.26
2
−1.5
tau=102e−7.7theta
0.31
0.25
tau=102e−7.7theta
0
0.2
0.4
0.6
x
0.8
1
0.24
0
0.2
0.4
0.6
0.8
1
x
Figure 4.4: Pressure and water content profiles for different values of τ ,
τ = 0, 20, 1 − θ2 , 102 × e−7.7θ . Computations are done with constant K = KS
at time T1 = 0.01 (top), Tim = 0.5 (middle), Tend = 1 (bottom).
64
Chapter 4. Numerical Results
Pressure at T for different tau, non−const. K
1
Water content at T for different tau, non−const. K
1
−1
0.32
−1.1
0.31
−1.2
0.3
−1.3
0.29
tau=0
tau=20
Water content
Pressure
tau=1−theta2
−1.4
−1.5
−1.6
−1.7
0.28
0.27
0.26
0.25
tau=0
tau=20
−1.8
0.24
tau=1−theta2
−1.9
−2
tau=102e−7.7theta
0.23
tau=102e−7.7theta
0
0.2
0.4
0.6
0.8
0.22
1
0
0.2
0.4
0.6
0.8
1
x
x
Pressure at Tim for different tau, non−const. K
Water content at Tim for different tau, non−const. K
−1
0.32
−1.1
0.31
tau=0
tau=20
tau=1−theta2
−1.2
Water content
Pressure
−1.3
−1.4
−1.5
−1.6
−1.7
−1.8
−1.9
−2
tau=102e−7.7theta
0.3
0.29
0.28
0.27
0.26
tau=0
tau=20
0.25
tau=1−theta2
0.24
tau=102e−7.7theta
0
0.2
0.4
0.6
0.8
0.23
1
0
0.2
0.4
0.6
0.8
1
x
x
Pressure at Tend for different tau, non−const. K
Water content at Tend for different tau, non−const. K
−0.6
0.32
tau=0
tau=20
0.31
−0.8
tau=1−theta2
tau=102e−7.7theta
0.3
Water content
Pressure
−1
−1.2
−1.4
0.29
0.28
0.27
0.26
−1.6
tau=0
tau=20
−1.8
0.25
2
tau=1−theta
0.24
tau=102e−7.7theta
−2
0
0.2
0.4
0.6
x
0.8
1
0.23
0
0.2
0.4
0.6
0.8
1
x
Figure 4.5: Pressure and water content profiles for different values of τ ,
τ = 0, 20, 1 − θ2 , 102 × e−7.7θ . Computations are done with non-constant K(θ)
at time T1 = 0.01 (top), Tim = 0.5 (middle), Tend = 1 (bottom).
Chapter 4. Numerical Results
65
The solutions for pressure and water content for constant K = KS are given at
T1 = 0.01, Tim = 0.5 and Tend = 1 with τ = 0, 20, 1 − θ2 and 102 × e−7.7θ . The
computations are performed with step size h, time step ∆t and values of LΨ and
Lθ as defined previously.
Figure 4.5 presents similar solutions as the ones displayed in Figure 4.4, but for
non-constant K(θ). The different nonlinear functions of the dynamic effect coefficient τ (θ) are based on experimentally determined models found in [42]. Recall
that τ (θ) = 0 corresponds to the standard Richards equation not including any
dynamic effects. By comparing the solutions of τ = 0 in Figures 4.4 and 4.5 to
the solutions in Figures 4.2 and 4.3 at each time, it is evident that this is indeed
the case. For all values of τ , the increase as time evolves in pressure and water
content are larger for constant than for non-constant hydraulic conductivity, K.
As mentioned previously this effects the transport of the dissolved component.
The transport takes place through convection and diffusion of the substance, so
by coupling the convection-diffusion equation to the flow equations, the numerical
results shown in Figure 4.6 are obtained. The left figure displays the transport
both in connection with the standard Richards equation and dynamic capillarity
for constant K = KS , while the figure on the right displays it for non-constant
K(θ). The solutions are also for these given at time T1 = 0.01, Tim = 0.5 and
Tend = 1, τ = 20.
Transport, non−constant K
1
0.9
0.9
0.8
0.8
0.7
0.7
Concentration
Concentration
Transport, constant K
1
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
x
0.8
1
0
0
0.2
0.4
0.6
0.8
1
x
Figure 4.6: Transport profiles for standard Richards equation (blue solid line)
and with dynamic capillary pressure (green solid line). The computations are
done with constant K = KS (left) and non-constant K(θ) (right) at time T1 =
0.01, Tim = 0.5, Tend = 1, τ = 20.
66
Chapter 4. Numerical Results
For the case with constant K the dynamic effects are clear, portraying different
transport profiles for the standard and non-standard model. In other words showing the effects of including dynamic capillary pressure in the model. However,
when K is non-constant, the dynamic effects are hardly noticeable. This is mainly
caused by the small change in water content, θ, for the non-constant case.
Recall from section 2.4.2 that coupling the convection-diffusion equation to the
flow equations means that water content, θ, and volumetric flux, q, are obtained
from the computations of Richards’ equation. Assuming that the flux undergo
little change during the simulations, the main influence on the transport is clearly
caused by the water content.
The change in θ when comparing the standard and the non-standard model is
small for the case with non-constant K (see figs. 4.3 and 4.5), thus accounting
for the observed similarities in the transport profiles for the standard and nonstandard model for non-constant K(θ). In a similar fashion, the bigger difference
in θ for constant K accounts for the observed dynamic effects in this case.
Example II
The initial condition for the pressure Ψ is given by
Ψ(x, 0) = −1,
and is therefore a more physical example than the previous one, resulting in a
constant pressure and water content profile at the initial time.
As in example I, numerical results are computed by implementing certain values of
τ , defined above, in the scheme modelling dynamic effects. Recall that LΨ = 0.001
and Lθ = 1. Together with the parameters and conditions defined at the start of
the section, simulations are performed with step size h = 0.005 and time step
∆t = 0.01.
Numerical solutions of pressure Ψ and water content θ for the different τ are given
in Figures 4.7 and 4.8. Figure 4.7 show the solutions computed with constant
K = KS and Figure 4.8 show the ones computed with non-constant K(θ). The
solutions are given at T1 = 0.01, Tim = 0.5 and Tend = 1. Again, as time evolves
there is an increase in pressure and water content throughout the domain due to
the flux across the left boundary. This is the case for all values of τ . The observed
increase is higher for the case with constant K than non-constant K. However,
when comparing these results to those in example I (figs. 4.4 and 4.5) the increase
is less for constant K and slightly higher for non-constant K.
Chapter 4. Numerical Results
67
Pressure at T for different tau, constant K
Water content at T for different tau, constant K
1
1
−0.9
0.3172
tau=0
tau=20
tau=0
tau=20
2
tau=1−theta
−0.92
tau=1−theta2
0.317
tau=102e−7.7theta
tau=102e−7.7theta
Water content
Pressure
−0.94
−0.96
−0.98
−1
−1.02
0.3168
0.3166
0.3164
0.3162
0
0.2
0.4
0.6
0.8
0.316
1
0
0.2
0.4
x
Pressure at Tim for different tau, constant K
tau=0
tau=20
tau=0
tau=20
0.325
2
tau=1−theta2
tau=1−theta
−0.92
0.324
tau=102e−7.7theta
−0.93
tau=102e−7.7theta
0.323
Water content
Pressure
1
0.326
−0.91
−0.94
−0.95
−0.96
0.322
0.321
0.32
−0.97
0.319
−0.98
0.318
−0.99
0.317
0
0.2
0.4
0.6
0.8
0.316
1
0
0.2
0.4
0.6
0.8
1
x
x
Pressure at Tend for different tau, constant K
Water content at Tend for different tau, constant K
−0.9
0.33
tau=0
tau=20
−0.91
tau=0
tau=20
0.328
2
tau=1−theta
−0.92
2
tau=1−theta
tau=102e−7.7theta
tau=102e−7.7theta
0.326
Water content
−0.93
Pressure
0.8
Water content at Tim for different tau, constant K
−0.9
−1
0.6
x
−0.94
−0.95
−0.96
−0.97
0.324
0.322
0.32
−0.98
0.318
−0.99
−1
0
0.2
0.4
0.6
x
0.8
1
0.316
0
0.2
0.4
0.6
0.8
1
x
Figure 4.7: Pressure and water content profiles for different values of τ both
constant and non-linear, τ = 0, 20, 1 − θ2 , 102 × e−7.7θ . Computations are done
with constant K = KS at time T1 = 0.01 (top), Tim = 0.5 (middle), Tend = 1
(bottom).
68
Chapter 4. Numerical Results
Pressure at T for different tau, non−const. K
Water content at T for different tau, non−const. K
1
1
−0.65
0.321
tau=0
tau=20
−0.7
tau=0
tau=20
0.3205
2
tau=1−theta2
tau=1−theta
0.32
tau=102e−7.7theta
−0.75
tau=102e−7.7theta
Water content
Pressure
0.3195
−0.8
−0.85
−0.9
0.319
0.3185
0.318
0.3175
−0.95
0.317
−1
−1.05
0.3165
0
0.2
0.4
0.6
0.8
0.316
1
0.2
0.4
0.6
0.8
1
x
Pressure at Tim for different tau, non−const. K
Water content at Tim for different tau, non−const. K
−0.6
0.345
tau=0
tau=20
−0.65
tau=0
tau=20
2
tau=1−theta
tau=1−theta2
0.34
tau=102e−7.7theta
−0.7
tau=102e−7.7theta
Water content
−0.75
Pressure
0
x
−0.8
−0.85
−0.9
0.335
0.33
0.325
−0.95
0.32
−1
−1.05
0
0.2
0.4
0.6
0.8
0.315
1
0.2
0.4
0.6
0.8
1
x
Pressure at Tend for different tau, non−const. K
Water content at Tend for different tau, non−const. K
−0.6
0.35
tau=0
tau=20
−0.65
tau=0
tau=20
0.345
2
tau=1−theta
tau=102e−7.7theta
Water content
0.34
−0.75
−0.8
−0.85
0.335
0.33
0.325
−0.9
0.32
−0.95
−1
2
tau=1−theta
tau=102e−7.7theta
−0.7
Pressure
0
x
0
0.2
0.4
0.6
x
0.8
1
0.315
0
0.2
0.4
0.6
0.8
1
x
Figure 4.8: Pressure and water content profiles for different values of τ both
constant and non-linear, τ = 0, 20, 1 − θ2 , 102 × e−7.7θ . Computations are done
with non-constant K(θ) at time T1 = 0.01 (top), Tim = 0.5 (middle), Tend = 1
(bottom).
Chapter 4. Numerical Results
69
In Figure 4.9 the transport profiles for the standard Richards equation as well as
the non-standard equation, including dynamic capillarity, are given. The solutions
for constant K = KS are shown in the figure on the left, while the solutions for
the non-constant K(θ) are given by the figure on the right. As in Figure 4.6 the
solutions are given at T1 = 0.01, Tim = 0.5 and Tend = 1, τ = 20.
Neither profile show a clear effect of including dynamic capillary pressure. This
is manily due to the small change in water content θ (see Figures 4.7 and 4.8) as
suggested previously.
Transport, non−constant K
1
0.9
0.9
0.8
0.8
0.7
0.7
Concentration
Concentration
Transport, constant K
1
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
x
0.8
1
0
0
0.2
0.4
0.6
0.8
1
x
Figure 4.9: Transport profiles for standard Richards equation (blue solid line)
and with dynamic capillary pressure (green solid line). The computations are
done with constant K = KS (left) and non-constant K(θ) (right) at time T1 =
0.01, Tim = 0.5, Tend = 1, τ = 20.
4.2.2
Convergence History of the Linearisation Schemes
In this section numerical simulations for the academical example, section 4.1,
and example I, section 4.2.1, are performed. However, the focus now lies on the
convergence of the linearisation schemes introduced in section 3.4.
For all simulations a constant time step of 0.1 is used, and a tolerance in residual
error of 10−4 for the non-linear system is given. A one dimensional grid with
varying mesh size h = 0.02, 0.008 and 0.005 is considered.
Figure 4.10 illustrates the convergence of the academical example. It is clear that
the linear error reduction factor is independent of mesh size, with four iterations
needed to reach the tolerance, for any value of h.
70
Chapter 4. Numerical Results
0
10
−1
Error (pressure)
10
−2
10
−3
10
−4
10
1
2
3
4
5
Iteration number
Figure 4.10: Convergence history of relative pressure error at T1 = 0.1 for
various mesh sizes for the academical example. Illustrated by h = 0.02 (green
solid line), h = 0.008 (blue dashed line) and h = 0.005 (red triangles). With
constant ∆t = 0.1.
In Figure 4.11 the convergence for the standard Richards equation for constant
K = KS and non-constant K(θ) is presented. For K constant no more than nine
iterations are needed. This is depicted by the figure on the right. In the case
with non-constant K, see the figure to the left, the negligible difference between
h = 0.02 and h = 0.008, 0.005 results in a total of six iterations. Hence, the
linearisation scheme’s independence of mesh size is obvious when applied to the
standard Richards equation.
0
0
10
10
−1
−1
10
Error (pressure)
Error (pressure)
10
−2
10
−3
−3
10
10
−4
10
−2
10
−4
2
4
6
Iteration number
8
10
10
1
2
3
4
5
6
Iteration number
Figure 4.11: Convergence history of relative pressure error at T1 = 0.1 for
various mesh sizes for the standard Richards equation. Illustrated by h = 0.02
(green solid line), h = 0.008 (blue dashed line) and h = 0.005 (red triangles).
Computations are done with constant K = KS (left) and non-constant K(θ)
(right), ∆t = 0.1.
Chapter 4. Numerical Results
71
Remark. In section 3.4.1 the convergence of the linearisation scheme for the
standard Richards equation was given by the contraction
f (·) =
L
≤ 1,
L+α
where L is some Lipschitz constant and α ∝ ∆ta with a not depending on ∆t
(see (3.66), section 3.4.1). From this relation some restrictions on the choice of
L, α arises, illustrated by taking the limits of the relation as limL→∞ f (L) and
limα→0 f (α):
L
= 1,
lim f (L) = lim
L→∞
L→∞ L + α
L
lim f (α) = lim
= 1.
α→0
α→0 L + α
In other words, the convergence will be very poor (or nonexisting) for L too big
and/or a, ∆t too small, implying a mild restriction on the time step size.
1
1
0.9
0.9
0.8
0.8
0.7
0.7
L/(L+alpha)
L/(L+alpha)
Figure 4.12 illustrates the rate at which the relations approaches 1. The figure
on the left shows the rate approaching 1 when L increases and α = 0.01, 0.1, 1,
while the one on the right shows the rate approaching 1 when α decreases and
L = 0.1, 1, 10.
0.6
0.5
0.6
0.5
0.4
0.4
0.3
0.3
0.2
0.2
alpha = 0.01
alpha = 0.1
alpha = 1
0.1
0
L = 0.1
L=1
L = 10
0
20
40
60
L
80
100
0.1
0
0
20
40
60
80
100
alpha
L
Figure 4.12: Rate of function f = L+α
approaching 1 for L increasing (left)
and α decreasing (right). Computed with α = 0.01 (green line), α = 0.1 (blue
line) and α = 1 (red line) for increasing L. Computed with L = 0.1 (green line),
L = 1 (blue line) and L = 10 (red line) for decreasing α.
72
Chapter 4. Numerical Results
0
0
10
10
−1
−1
10
Error (pressure)
Error (pressure)
10
−2
10
−3
−3
10
10
−4
10
−2
10
−4
1
2
3
4
5
6
7
8
10
9
2
4
Iteration number
0
10
12
3.5
4
10
−1
−1
10
Error (water content)
10
Error (water content)
8
0
10
−2
10
−3
−2
10
−3
10
10
−4
10
6
Iteration number
−4
1
2
3
Iteration number
4
5
10
1
1.5
2
2.5
3
Iteration number
Figure 4.13: Convergence history of relative pressure error (top) and relative
water content error (bottom) with dynamic capillary pressure, at T1 = 0.1
for various mesh sizes. Illustrated by h = 0.02 (green solid line), h = 0.008
(blue dashed line) and h = 0.005 (red triangles). Computations are done with
constant K = KS (left) and non-constant K(θ) (right), ∆t = 0.1 and τ = 20.
As in the case for the academical example and the standard Richards equation,
the convergence of the linearisation scheme for Richards’ equation with dynamic
capillarity, where τ = 20, is independent of h, see Figure 4.13. This is evident with
regards to pressure as well as the water content. An increase in the number of
iterations needed is observed for the pressure solution between constant K = KS
and non-constant K(θ), while a slight decrease is observed for the water content.
In Figure 4.14 the convergence with nonlinear τ = 1 − θ2 is shown. With just
over four iterations needed for the pressure and just over three for the water
content for all values of h, this is also independent of mesh size. The convergence
results presented so far in this section, verifies the robustness and generality of the
iterative schemes as well as the suitability of the finite volume discretisation.
Chapter 4. Numerical Results
73
0
0
10
10
−1
−1
10
Error (pressure)
Error (pressure)
10
−2
10
−3
−3
10
10
−4
10
−2
10
−4
1
2
3
4
10
5
1
2
Iteration number
0
10
5
0
−1
−1
10
Error (water content)
Error (water content)
4
10
10
−2
10
−3
−2
10
−3
10
10
−4
10
3
Iteration number
−4
1
1.5
2
2.5
3
Iteration number
3.5
4
10
1
1.5
2
2.5
3
3.5
4
Iteration number
Figure 4.14: Convergence history of relative pressure error (top) and relative
water content error (bottom) with dynamic capillary pressure, at T1 = 0.1
for various mesh sizes. Illustrated by h = 0.02 (green solid line), h = 0.008
(blue dashed line) and h = 0.005 (red triangles). Computations are done with
constant K = KS (left) and non-constant K(θ) (right), ∆t = 0.1 and τ = 1−θ2 .
In order to further explore the robustness and generality of the linearisation
schemes, the convergence history for different values of the dynamic effect coefficient τ is computed, see Figure 4.15. The computations were performed with
h = 0.005. Throughout these numerical simulations, the error reduction factor is
independent of grid size and no more than six iterations was needed.
74
Chapter 4. Numerical Results
0
0
10
10
tau = 0
tau = 20
2
theta
1−theta2
−1
10
−2
10
−3
102e−7.7theta
−2
10
−3
10
10
−4
10
1−theta2
−1
10
102e−7.7theta
Error (pressure)
Error (pressure)
tau = 0
tau = 20
2
theta
−4
1
2
3
4
5
10
6
1
2
Iteration number
3
4
5
0
10
tau = 0
tau = 20
tau = 0
tau = 20
2
2
theta
theta
1−theta2
−1
10
1−theta2
−1
10
102e−7.7theta
Error (water content)
Error (water content)
7
0
10
−2
10
−3
102e−7.7theta
−2
10
−3
10
10
−4
10
6
Iteration number
−4
1
2
3
Iteration number
4
5
10
1
1.5
2
2.5
3
3.5
Iteration number
Figure 4.15: Convergence history of relative pressure error (top) and relative water content error (bottom), at T1 = 0.1 for various values of τ .
τ = 0, 20, θ2 , 1−θ2 , 102 ×e−7.7θ . Computations are done with constant K = KS
(left) and non-constant K(θ) (right), ∆t = 0.1 and h = 0.005.
4
Chapter 5
Conclusion
In this thesis dynamic capillary effects on numerical simulations of flow and transport in porous media have been considered. To do so, mathematical models to
model flow and transport were constructed, given by the Richards equation and the
convection-diffusion equation. In order to account for dynamic effects related to
phenomena such as saturation overshoot and formation of preferential flow paths,
an extension describing dynamic capillary pressure was included in the standard
model, hence developing what was referred to as a non-standard model. The aim
was thus to evaluate the influence of dynamic effects on the flow and transport of
a dissolved substance.
Richard’s equation admits in general no analytical solutions, hence numerical solutions have to be considered. In this thesis the discretisation in time was performed
by using the backward Euler method and in space the cell-centered finite volume
method TPFA. To solve the nonlinear systems appearing at each time step, robust
linearisation methods were proposed. The scheme for the Richards equation with
dynamic capillarity is new. The schemes were analysed to prove the convergence
of the methods. All simulations were conducted using the Matlab implementation
environment and the numerical simulations supported the evidence put forward
by the theoretical analysis. The linearisation schemes are very robust, shown to be
linearly convergent and independent of mesh diameter, which is an argument for
the efficiency of the schemes. Another advantage of the presented schemes is that
they do not involve the calculations of derivatives. The numerical scheme for the
standard Richards equation, based on the works of Slodicka (2002) and Pop and
Radu (2004) [37, 40], and the new scheme including the dynamic capillarity are
relatively simple to implement and are valuable alternatives to Picard or Newton
methods.
75
76
Chapter 5. Conclusion
The numerical simulations of the flow and transport models were performed on
two examples with different initial conditions. Numerical solutions based on the
standard and the non-standard Richards equation were presented. In terms of the
flow, the dynamic effect was evident and a clear difference between the solutions
of the standard and non-standard case was seen. This was true for both the
pressure, Ψ, and water content, θ. As well as computing numerical solutions to
problems matching the theoretical analysis where constant K, τ > 0 were assumed,
numerical simulations were performed on problems containing non-constant K, τ ,
implying that similar characteristics also holds for such problems. This is however
in the need of further exploration and a proper theoretical analysis should be
established. Regarding the transport problem, dynamic effects were observed in
the case with constant K = KS for Example I (see section 4.2.1). For the remaining
transport computations, the chosen values and conditions resulted in a change in
θ not significant enough to portray dynamic effects.
If time had allowed it, the model would be extended to include hysteresis as well
as dynamic capillarity in an effort to obtain saturation overshoot profiles and
further investigating the influence on the transport. Applying the implemented
schemes to more realistic examples over longer time spans would hopefully result
in solutions showing the true dynamic effects and underline the importance of
including non-standard aspects in the model.
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