Mathematical and numerical analysis for non-equilibrium two

Mathematical and numerical analysis for non-equilibrium
two phase flow models in porous media
Cao, X.
Published: 22/03/2016
Document Version
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences
between the submitted version and the official published version of record. People interested in the research are advised to contact the
author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
Citation for published version (APA):
Cao, X. (2016). Mathematical and numerical analysis for non-equilibrium two phase flow models in porous media
Eindhoven: Technische Universiteit Eindhoven
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners
and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.
• You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal ?
Take down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Download date: 29. Jul. 2017
Mathematical and Numerical
Analysis for Non-Equilibrium
Two Phase Flow Models in
Porous Media
Xiulei Cao
A catalogue record is available from the Eindhoven University of Technology Library
ISBN: 978-90-386-4025-9
Copyright © 2016 by X. Cao, Eindhoven, The Netherlands.
All rights are reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author.
Mathematical and Numerical Analysis for
Non-Equilibrium Two Phase Flow Models in
Porous Media
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de
Technische Universiteit Eindhoven, op gezag van de
rector magnificus prof.dr.ir. F.P.T. Baaijens, voor een
commissie aangewezen door het College voor
Promoties, in het openbaar te verdedigen
op dinsdag 22 maart 2016 om 16.00 uur
door
Xiulei Cao
geboren te Jilin, China
Dit proefschrift is goedgekeurd door de promotoren en de samenstelling van
de promotiecommissie is als volgt:
voorzitter:
1e promotor:
2e promotor:
leden:
prof.dr. J. de Vlieg
prof.dr. I.S. Pop (Universiteit Hasselt)
prof.dr.ir. C.J. van Duijn
prof.dr.-ing. R. Helmig (Universität Stuttgart)
prof.dr. B. Schweizer (Technische Universität Dortmund)
prof.dr. B. Wohlmuth (Technische Universität München)
prof.dr.ir. E.H. van Brummelen
prof.dr.ir. B. Koren
Het onderzoek of ontwerp dat in dit proefschrift wordt beschreven is uitgevoerd in overeenstemming met de TU/e Gedragscode Wetenschapsbeoefening.
Abstract
Flow models in porous media are encountered in many real life applications of highest
societal relevance. Typically, such models are built on so-called equilibrium assumptions,
where besides physical laws like mass balance or the Darcy’s law, algebraic relationships
are assumed between model unknowns. In this thesis we consider non-equilibrium models,
where dynamic effects and hysteresis are taken into account in the capillary pressure saturation relationship.
Such models provide solutions that are ruled out by equilibrium models: saturation
overshoot, or the development of fingers during infiltration in homogeneous media. Such
effects have been observed experimentally. The focus here is on the mathematical and
numerical analysis for such models.
We start with simplified, scalar model of pseudo-parabolic type which incorporates
dynamic capillary effects. For this, we prove the uniqueness of a weak solution.
Then we consider the full two-phase porous media flow model, again incorporating
dynamic capillary effects. In this case, we prove the existence of a solution for cases that
include degeneracy (the vanishing of highest order terms).
Furthermore, for this model a multipoint flux approximation method is proposed.
We give a rigorous convergence proof for both saturation and fluxes. The numerical
experiments show a robust convergence of the method even for irregular grids.
Next we consider heterogeneous media consisting of homogeneous blocks. For this
case, we derive the conditions to be imposed for the coupling of the models defined in
each block at the separating interface.
For the two-phase flow model involving both dynamic capillarity and hysteresis, we
prove again the uniqueness of a solution. The final chapter is a numerical approximation:
the moisture transport in concrete during wetting-drying cycles.
Keywords: Non-equilibrium, dynamic effects, hysteresis, degeneracy, multipoint flux approximation.
MSC 2010: 35K65, 65J05, 65N08, 65N12, 65Z05, 76S05.
i
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Uniqueness of a weak solution for the scalar
2.1 Introduction . . . . . . . . . . . . . . . .
2.2 Uniqueness of the weak solution . . . . .
2.3 Conclusions . . . . . . . . . . . . . . . .
model
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . .
3 Existence of weak solutions to the two-phase model
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Assumptions and known results . . . . . . . . . . . . . . . . .
3.3 Existence, uniqueness of solutions in the regularized case . . .
3.3.1 The weak solution concept . . . . . . . . . . . . . . .
3.3.2 The time discretization . . . . . . . . . . . . . . . . .
3.3.3 A priori estimates . . . . . . . . . . . . . . . . . . . .
3.3.4 Existence of weak solutions to the regularized problem
3.3.5 Uniqueness of the weak solution for Problem Pδ . . . .
3.4 Existence of weak solutions for Problem P . . . . . . . . . . .
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Finite volume scheme
4.1 Introduction . . . . . . . . .
4.2 The weak solution . . . . . .
4.3 The finite volume scheme . .
4.3.1 Meshes and notations
4.3.2 The scheme . . . . .
4.3.3 A priori estimates and
4.4 Convergence of the scheme .
4.4.1 Compactness results .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
. . . . . . . . .
existence of the
. . . . . . . . .
. . . . . . . . .
iii
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
fully discrete
. . . . . . .
. . . . . . .
1
1
2
7
9
9
12
16
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
17
17
19
22
22
24
28
33
35
36
52
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
solution
. . . . .
. . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
53
53
54
56
56
57
59
65
65
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
iv
CONTENTS
4.5
4.4.2 Convergence results . . . . . . . . . . . . . . . . . . . . . . . . .
Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Two
5.1
5.2
5.3
phase flow in heterogeneous porous media
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . .
Conditions at the interface . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Constant saturation at the coarse side of the interface . . .
5.3.2 Non-constant saturation at the coarse side of the interface .
5.3.3 Comparison of extended pressure conditions with static case
5.4 Numerical schemes and examples . . . . . . . . . . . . . . . . . . .
5.4.1 Linear numerical scheme . . . . . . . . . . . . . . . . . . .
5.4.2 Fully implicit scheme . . . . . . . . . . . . . . . . . . . . .
5.4.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . .
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Uniqueness of a solution for the hysteresis
6.1 Introduction . . . . . . . . . . . . . . .
6.2 Uniqueness of the weak solution . . . .
6.3 Conclusions . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
70
74
79
79
80
83
85
89
97
99
99
102
105
108
model
111
. . . . . . . . . . . . . . . . . . 111
. . . . . . . . . . . . . . . . . . 113
. . . . . . . . . . . . . . . . . . 121
7 Moisture transport in concrete
7.1 Mathematical model . . . . . . . . . . . . . . . .
7.2 Approach . . . . . . . . . . . . . . . . . . . . . .
7.2.1 Standard model: one diffusion coefficient
7.2.2 Hysteretic model . . . . . . . . . . . . .
7.2.3 Numerical results . . . . . . . . . . . . .
7.3 Experiment . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
123
123
125
125
126
128
128
Summary
131
Curriculum Vitae
133
List of publications
135
Acknowledgments
137
Bibliography
139
Chapter 1
Introduction
1.1
Motivation
Many processes in the real life are involving flow in porous media: oil recovery, biological
systems, technological applications (batteries, catalytic converters), or underground waste
disposal (CO2 sequestration). Understanding such processes require the development and
analysis of appropriate mathematical models, completed by numerical simulations.
Single and multi-phase porous media flow models are commonly written in terms of
quantities like phase saturation, velocity and pressures. Besides balance equations (mass
conservation), the mathematical models are including functional relationships relating
the quantities of interest. Traditionally, these constitutive equations are obtained under
equilibrium assumptions: for example, for a given medium and at a given phase saturation,
the capillary pressure is fixed. This assumes a static distribution of phases in the pores
and disregards the history of the system (hysteresis).
However, the relations measured under infiltration and drainage are different, and
experiments have proven limitations of the standard approaches. For instance, the experiments reported in the work of DiCarlo [38] show that in homogeneous media, the
saturation profiles may become non-monotonic during infiltration (see Figure 1.1). In the
two dimensional case, F. Rezanezhad and the coauthors [93] have given an experimental
setup and results presenting a fingering behavior (see Figure 1.2).
1
2
CHAPTER 1. INTRODUCTION
Figure 1.1 Water saturation overshoot for different fluxes at the inlet taken from [38]. Observe
the non-monotonic saturation profiles encountered at large fluxes q.
(a)
(b)
Figure 1.2 The setup (a) and flow path (b) in the experiments reported in [93]. Observe the
fingers appearing in the right.
Such results are ruled out by classical models, assuming equilibrium relationships.
Specifically, for the experiments as reported in [38], standard models would predict monotonic saturation profiles but can not explain the fingering effect. This testifies including
non-equilibrium effects in the mathematical models.
1.2
Mathematical models
As discussed, the mathematical models which describe two phase flow in porous media consist of the mass conservation law, the Darcy’s law and constitutive relationships
between the difference of phase pressures and the phase saturation (the proportion of
the void space that is occupied by that phase in a reference elementary volume). Under
equilibrium conditions, denoting the non-wetting and wetting phase pressures by pn and
1.2. MATHEMATICAL MODELS
3
pw , the capillary pressure, defined as the phase pressure difference, is assumed to be a
monotone function of the wetting saturation Sw :
(1.1)
pn − pw = pc (Sw ).
The function pc is obtained experimentally.
With Swr and Snr being so-called irreducible wetting phase saturation and residual
non-wetting phase saturation (an amount of wetting/non-wetting phase that can not be
removed from the porous media under standard drainage or imbibition), one defines the
effective water saturation Se as
Se =
Sw − Swr
.
1 − Swr − Snr
(1.2)
In [22], Brooks and Corey proposed the relation
1
−λ
pc (Se ) = Pb Se
(1.3)
,
here Pb and λ are characteristic constants of the porous medium. λ is a number which
characterizes the pore-size distribution. In [58], Van Genuchten proposed
pc (Se ) =
n1
1 − m1
Se − 1 ,
a
(1.4)
Pn-Pw, [kPa]
Water saturation [-]
where a, m and n are given parameters. For both models above, it is clear that the
capillary pressure is a decreasing function of Sw . However, experimental results have
invalided this assumption. For example, the breakthrough curves of water saturation and
pressure differences in Figure 1.3 are reported in [17]. According to these curves, we
can easily conclude that the difference of phase pressures is not a monotone function of
saturation any more.
time [s]
time [s]
Figure 1.3 The water saturation (left) and the phase pressure difference (right) in the
experiment reported in [17]. Observe the non-monotone dependence.
Alternatively to (1.3) and (1.4), in [60] Hassanizadeh and Gray proposed a model that
4
CHAPTER 1. INTRODUCTION
includes dynamic effects. Play-type hysteresis is included later in [13], leading to
pn − pw ∈ pc (Sw ) − γsign(∂t Sw ) − τ ∂t Sw .
(1.5)
Here γ, τ ≥ 0 are two parameters or even functions depending on Sw . The multi-valued
sign function is defined by


ξ > 0,

1
sign(ξ) =
[−1, 1]


−1
ξ = 0,
ξ < 0.
In [17], the τ (Sw ) is proposed by the experimental results (see Figure 1.4), while CampsRoach and the co-authors introduced it in [26] (see Figure 1.5).
Figure 1.4 The parameter function τ (Sw ), obtained experimentally in [17].
Figure 1.5 The function τ (Sw ), measured experimentally in [26].
1.2. MATHEMATICAL MODELS
5
By the mass balance equation for each of the fluid phases, one has (see [12, 61])
φ
∂Sα
+ ∇ · qα = fα
∂t
(1.6)
(α = w, n).
Here φ, qα (α = w, n) denote the porosity [−], the volumetric velocity of the α phase
1
[m
s ], fα [ s ] is the source of the phase α. For simplicity, the source is assumed to be 0
here. The volumetric velocity qα is deduced from the Darcy’s law as
qα = −
k̄
krα (Sα )∇Ψα ,
µα
(1.7)
where k̄ [m2 ] is the absolute permeability of the porous medium, Ψα [kg · m−1 · s−2 ] the
phase potential, µα [kg · m−1 · s−1 ] the viscosity and krα the relative permeability of the
α phase. Normally, krα is assumed to be a function of Sw . In [24], Burdine proposed as
2+3λ
λ
krw = Se
(1.8)
,
for wetting phase and
2+λ
(1.9)
krn = (1 − Se )2 (1 − Se λ ),
for non-wetting phase. Additionally, Ψw and Ψn are given by
Ψw = pw + ρw gz,
(1.10)
Ψn = pn + ρn gz,
with ρα (α = w, n), g denoting density [kg · m−3 ] and the gravitational acceleration
[m · s−2 ]. Further, z is the vertical coordinator, being position in the upward direction.
For two phase porous media flows, one has
(1.11)
Sw + Sn = 1.
Combining the equations (1.6)-(1.7) and (1.10)-(1.11), together with (1.5), one obtains
the following system in terms of the unknown triple (Sw , pw , pn ):
φ∂t Sw − ∇ · (
k̄
k̄
krw (Sw )∇pw ) − ∇ · ( krw ρw gz) = 0,
µw
µw
(1.12)
k̄
k̄
krn (Sw )∇pn ) − ∇ · ( krn ρn gz) = 0,
µn
µn
(1.13)
−φ∂t Sw − ∇ · (
(1.14)
pn − pw ∈ pc (Sw ) − γsign(∂t Sw ) − τ ∂t Sw .
Assuming k̄ constant, we derive the system to a dimensionless form by introducing the
reference quantities T , L, Pr and defining
t :=
t
,
T
x :=
x
,
L
pn :=
pn
,
Pr
pw :=
pw
,
Pr
pc :=
pc
,
Pr
(1.15)
6
CHAPTER 1. INTRODUCTION
and
kw :=
k̄Pr T krw
,
L2 φ µw
kn :=
k̄Pr T krn
.
L2 φ µn
(1.16)
With the dimensionless quantities
ρw gzL
→
−
g1 :=
,
Pr
ρn gzL
→
−
g2 :=
,
Pr
γ :=
γ
,
T Pr
τ :=
τ
,
T Pr
(1.17)
we obtain the following system:
−
∂t Sw − ∇ · (kw (Sw )∇pw ) − ∇ · (kw →
g1 ) = 0,
(1.18)
−
−∂t Sw − ∇ · (kn (Sw )∇pn ) − ∇ · (kn →
g2 ) = 0,
(1.19)
pn − pw ∈ pc (Sw ) − γsign(∂t Sw ) − τ ∂t Sw .
(1.20)
Observe that kw , kn or pc may vanish, or become ∞ whenever Sw = 0 or Sw = 1. Such
cases we call degenerate. If instead, m, M > 0 exist such that m ≤ kw , kn , pc ≤ M , we
speak about non-degenerate cases.
(1.18) - (1.20) can be written in a different form. Summing (1.18) and (1.19) gives
−∇ · Q = 0,
(1.21)
−
−
with Q = kw (Sw )∇pw + kw (Sw )→
g1 + kn (Sw )∇pn + kn (Sw )→
g2 denoting the total flow.
Then, one can replace (1.18) or (1.19) by (1.21). In particular cases e.g. in one spatial
dimension, (1.21) implies Q = q (constant w.r.t spatial parameter). Then, the system
(1.18) - (1.20) reduces to
kw kw kn ∂Sw
−
−
+∇· q
+ ∇ · (→
g2 − →
g1 )
∂t
kw + kn
kw + kn
k k
w n
∇(pn − pw ) ,
= −∇ ·
kw + kn
pn − pw ∈ pc (Sw ) − γsign(∂t Sw ) − τ ∂t Sw .
(1.22)
(1.23)
Moreover, in the absence of hysteretic effects (γ = 0), one obtains the pseudo-parabolic
equation so-called scalar model:
kw kw kn ∂Sw
−
−
+∇· q
+ ∇ · (→
g2 − →
g1 )
∂t
kw + kn
kw + kn
k k
k k
∂Sw w n
w n
= −∇ ·
∇pc (Sw ) + ∇ ·
∇(τ
) .
kw + kn
kw + kn
∂t
(1.24)
The existence of weak solutions for this model has been proved in [77]. The result includes
degenerate cases i.e. when kw or kn may become 0. In [53], one obtains the existence
and uniqueness of a weak solution for a linear highest order term model. The equivalence
of different formulations is showed in [52]. Traveling wave solutions are analyzed in [40].
1.3. OUTLINE OF THE THESIS
7
Numerical schemes are discussed in [62, 63].
Returning to the original system (1.18) - (1.20), we distinguish types of non-equilibrium
models:
A) τ := τ (Sw ), γ = 0
A result in this thesis is the existence of weak solutions, including degenerate cases,
when kw or kn may vanish.
B) τ = 0, γ 6= 0
This is a so called play-type hysteresis model, yielding two different pn − pw curves
(see Figure 1.6), an imbibition curve and drainage one: pn − pw = pc (Sw ) − γ, pn − pw =
pc (Sw ) + γ. Here γ may be a function of Sw . Existence and stability analysis for such
models are obtained in [10, 96].
Figure 1.6 Two capillary pressure - saturation curves including hysteresis.
C) τ 6= 0, γ 6= 0
This is the full model including dynamic effects and play-type hysteresis. [68] and [71]
deal with the existence of weak solutions for this model in non-degenerate cases. In [96],
Schweizer extended the result to degenerate cases. Numerical schemes are discussed
in [68,71,92]. In this thesis, we will show the uniqueness of a weak solution to this model.
1.3
Outline of the thesis
This thesis is structured as follows:
In Chapter 2, without degeneracy, we provide the uniqueness of a weak solution for
the scalar model. We first transform the scalar model into an equivalent form, then prove
the uniqueness of a weak solution to the equivalent system which leads to the uniqueness
of a weak solution for the original model. To do so, we use a Green function approach.
In Chapter 3, we analyze a degenerate elliptic-parabolic system with the dynamic
effects τ = τ (Sw ) in the phase-pressure difference and without hysteresis. We first
transform the system (1.12) - (1.14) with γ = 0 into a two-equation system as in [35].
8
CHAPTER 1. INTRODUCTION
Then, by regularizing the coefficients, the existence and uniqueness of weak solutions
in the non-degenerate case is obtained. Further, the coefficients are assumed to have
a certain structure which has physical meaning when the system becomes degenerate.
Finally, we let the regularization parameter go to zero to show the existence of weak
solutions under degenerate case.
In Chapter 4, still with γ = 0 and τ constant, we present a finite volume method for
the simulation of the solution. The method is based on a multi-point flux approximation.
An energy estimate is derived for the numerical solution, and compactness arguments
are used to prove the convergence to the weak solution as the mesh size tends to zero.
Finally, we present some numerical results to confirm the convergence proved rigorously.
Chapter 5 discusses the two phase flow in heterogeneous media. The dynamic effects are taken into account in phase pressure difference. We consider a one-dimensional
heterogeneous case, with two adjacent homogeneous blocks separated by an interface.
The absolute permeability is assumed constant, but different in each block. This may
lead to the entrapment of the non-wetting phase (say oil) when flowing from the coarse
material into the fine material. We derive the interface conditions coupling the models
in each homogeneous block. In doing so, the interface is approximated by a thin porous
layer, and its thickness is then passed to zero. Such results have been obtained earlier for
standard models, based on the equilibrium relationship between the capillary pressure and
the saturation. Then, oil is trapped until its saturation on the coarse material side of the
interface exceeds an entry value. In the non-equilibrium case, the situation is different.
Due to the dynamic effects, oil may still flow into the fine material even after the saturation drops under the entry point, and this flow may continue for a certain amount of
time that is proportional to the non-equilibrium effects. This suggests that operating in a
dynamic regime reduces the account of oil trapped at interfaces, leading to an enhanced
oil recovery. Finally, we present some numerical results supporting the theoretical findings.
In Chapter 6, both the dynamic effects and hysteresis are included in the capillary
pressure, which is the full model of (1.12) - (1.14). The existence of weak solutions for
this model has been proved in [68]. In this chapter, we define an auxiliary elliptic system
to show the uniqueness of a weak solution for this problem.
In Chapter 7, we consider a practical situation related to the durability analysis of
concrete. Specifically, moisture transport in concrete is discussed. Moisture transport in
marine environment, where drying and wetting cycles occur, leads chloride to penetrate
into reinforced concrete structures. When chloride reaches the rebars, corrosion can
appear leading to a decrease of the service life time of the structures. To predict this
life time, one needs to understand the moisture transport. This process is modeled in
various ways. For example, in [8, 37, 75, 84, 98] moisture transport is described in terms
of a single diffusion coefficient. In [74, 101], two different diffusion coefficients are used
for drying and wetting. In this chapter we compare the two approaches for alternating
drying/wetting cycles and present a numerical scheme. In addition to the modeling, an
experimental set-up is proposed to validate the models.
Chapter 2
Uniqueness of a weak solution
for the scalar model
2.1
Introduction
In this chapter, we consider the scalar model which is proposed in porous media under
non-equilibrium conditions:
∂t u + ∇ · F(u) + ∇ · (H(u)∇(pc (u) − τ ∂t u)) = 0.
(2.1)
Here the two-phase flow model includes dynamic effects in the capillary pressure. If under
equilibrium conditions, the difference of the pressures in the two phases (wetting and
non-wetting) is a decreasing function in terms of the (say) wetting phase saturation u
pn − pw = pc (u).
(2.2)
Existence and uniqueness as such models are analyzed in [35,69]. However, the equilibrium
assumption does not hold in several real life applications, like paper drying. Further,
experimental evidence for the non-validity of the equilibrium assumption are provided e.g.
in [17, 38, 64]). In this case, dynamic effects have to be included. An example in this
sense is the model proposed in [60]:
pn − pw = pc (u) − τ ∂t u.
(2.3)
Here τ > 0 is a parameter accounting for the dynamic effects.
Inspired by this, we consider here a simplified model obtained from the mass conservation and the Darcy’s law for each phase (see [12, 61]), assuming that the medium is fully
This chapter is a collaborative work with I.S. Pop and it has been published in Applied Mathematics
Letters, 46(2015): 25-30.
9
10CHAPTER 2. UNIQUENESS OF A WEAK SOLUTION FOR THE SCALAR MODEL
saturated by the two phases. Generally, this leads to a system of two equations, a parabolic one for the wetting phase saturation, and an elliptic one for the total flux. Here we
assume the total flux known, and focus on the mass balance for the wetting phase. With
the phase pressure difference in (2.3), the model reads as (2.1) (see [28,77]). It is defined
in a bounded and connected domain Ω in Rd (d ≥ 1) with a given time interval (0, T ].
Further, by Ω̄ we mean the closure of Ω, and by ∂Ω its boundary. In the above, F and
H denote the water fractional flow function and the capillary induced diffusion function,
and pc is the equilibrium capillary pressure (see (2.2) and (2.3)). These are non-linear
functions defined for the physically relevant interval u ∈ [0, 1]. For the mathematical
analysis, we extend H, pc and F continuously to the entire R. Throughout this chapter,
we assume
A1: H: R −→ R is Lipschitz continuous, and a h0 exists such that
0 < h0 ≤ H(u) if 0 < u < 1, and H(u) = h0 otherwise.
(2.4)
0
A2: pc ∈ C 1 (R) is a decreasing function, and mp , Mp exist such that 0 < mp ≤ |pc (u)| ≤
Mp < +∞, for all u ∈ R.
A3: F: R −→ Rd is Lipschitz continuous.
A4: The functions F and H are bounded, |F(u)| + |H(u)| ≤ M < +∞, for all u ∈ R.
To complete the model, the initial and boundary conditions are given
u(·, 0) = u0 ,
and u = 0,
at ∂Ω.
(2.5)
For the initial data, we assume
A5: u0 ∈ C 0,α (Ω̄) for some α ∈ (0, 1], u0 = 0 at ∂Ω and u0 ∈ [0, 1] in Ω.
Further, the domain Ω has a smooth boundary:
A6: Ω is a C 1,α domain.
The motivation for (2.1) is the two phase flow in porous media. Generally, such
models are not only non-linear, but may also degenerate. Whenever one of the phase is
not present. More precise, functional relationships between quantities: the permeability,
pressure, and saturation lead to cases when, e.g. H(u) = 0 for u = 0 or u = 1 (see
[12, 34, 82]). Then the highest order term on (2.1) is vanishing, and the equation is not
parabolic anymore. This makes the analysis and numerical simulation of such problems
complicated. A usual technique to overcome the difficulties related to degeneracy is
regularization. For example, one can approximate non-linearities by regular permeabilities,
which are bounded away from 0 and ∞. The uniqueness result obtained here works in
the regularized cases, and was still an open issue even for regular cases.
2.1. INTRODUCTION
11
In this chapter we prove the uniqueness of a weak solution to (2.1) with the given
initial and boundary conditions. This solution solves
Problem P: Find u ∈ W 1,2 (0, T ; W01,2 (Ω)), such that u(·, 0) = u0 , ∇∂t u ∈ L2 (0, T ;
L2 (Ω)d ) and
Z
T
Z
T
Z
Z
T
Z
F(u) · ∇φdxdt
∂t uφdxdt −
0
Z
Ω
0
Ω
Z
T
Z
H(u)∇∂t u · ∇φdxdt = 0,
H(u)∇pc (u) · ∇φdxdt + τ
−
0
Ω
0
(2.6)
Ω
for any φ ∈ L2 (0, T ; W01,2 (Ω)).
Note the non-linearity appearing in the highest order term, ∇ · (H(u)∇∂t u). For the
linear case, when only ∆∂t u is appearing, existence and uniqueness results are obtained
in [53]. Also we refer to [28, 77, 78] for the existence of weak solutions to the nonlinear
Problem P. However, the uniqueness in the case considered here is still an open issue.
To overcome the difficulty related to the non-linearity appearing in the highest order
term, we use an additional unknown p (see [14, 52]). Closest to our work are the recent
results in [14]. There, the uniqueness is obtained for a similar, even degenerate model,
but in the absence of convective terms, and more important, when the non-linearities
appear only under all derivatives e.g. ∂t ∆ψ(u). Apart from some particular cases (i.e.
H ≡ constant), the model considered here involves the term ∇ · (H(u)∇∂t u), which
cannot be transformed to the form in [14] . Furthermore, the uniqueness in [14] is proved
for an alternative formulation when (2.1) is written as a system. In the non-degenerate
case, the equivalence between (2.1) and its reformulation as a system (like (2.7)-(2.8))
is proved in [52], while in the degenerate case, it is still open. Thus, uniqueness results
in [14], whenever the model considered here matches the frame work there, gives also
uniqueness for (2.1). However, this is only for particular cases, as mentioned. Besides, in
this chapter, we provide an alternative uniqueness proof, based on Green functions.
To avoid the confusion between u given by Problem P and the solution pair of the
following system, we denote the saturation latter by s. With this, (2.1) can be rewritten
formally as the system
∂t s + ∇ · F(s) + ∇ · (H(s)∇p) = 0,
(2.7)
p = pc (s) − τ ∂t s.
(2.8)
For the sake of presentation we finally assume
A7: pc (0) = 0.
Remark 2.1.1. A7 can be avoided: if pc (0) 6= 0, one defines p = pc (s) − pc (0) − τ ∂t s
in (2.8).
Finally, we mention that throughout this chapter, C denotes a generic constant. For this
system, a weak solution is a pair (s, p) solving
12CHAPTER 2. UNIQUENESS OF A WEAK SOLUTION FOR THE SCALAR MODEL
Problem Ps : Find s ∈ W 1,2 (0, T ; L2 (Ω)) and p ∈ L2 (0, T ; W01,2 (Ω)), such that s(·, 0) =
u0 and
(∂t s, φ) − (F(s), ∇φ) − (H(s)∇p, ∇φ) = 0,
(2.9)
(p, ψ) = (pc (s), ψ) − τ (∂t s, ψ),
(2.10)
for any φ ∈ L2 (0, T ; W01,2 (Ω)), ψ ∈ L2 (0, T ; L2 (Ω)). Following [52], the Problems P and
Ps are equivalent. Specifically, under the assumptions A1-A4, if (s, p) is a solution to
Problem Ps , then u = s solves Problem P. Conversely, if u solves Problem P, then (s, p)
with s = u and p defined in (2.8) solves Problem Ps . This equivalence will be exploited
below.
2.2
Uniqueness of the weak solution
In this section we prove the uniqueness of weak solutions for Problem Ps . The equivalence
result also provides uniqueness for Problem P. The first step is to obtain the essential
boundedness of ∇p:
Lemma 2.2.1. The solution component p of Problem Ps satisfies ∇p ∈ L∞ ((0, T ] × Ω).
Proof. As shown in [28], if u solves Problem P, one has u ∈ L∞ (0, T ; W01,2 (Ω)), ∂t u ∈
L∞ (0, T ; W01,2 (Ω)). In view of the equivalent result, from (2.10) one has p ∈ L∞ (0, T ;
W01,2 (Ω)). Further, for almost every t, one can use (2.8) to eliminate ∂t s in (2.7)
∇ · (H(s)∇p) +
1
1
p = pc (s) − ∇ · F(s).
τ
τ
(2.11)
Note that given s ∈ L∞ (0, T ; W01,2 (Ω)), (2.11) is a linear elliptic equation in p with the
1
right hand side pc (s) − ∇ · F(s) ∈ L2 (Ω). Then as in [35, 70], one obtains
τ
kpkL∞ (0,T ;C 0,β (Ω̄)) ≤ C,
0 < β ≤ 1.
(2.12)
Then for almost every t ∈ (0, T ] and for almost every x, y ∈ Ω, x 6= y, by using (2.8), we
obtain
p(t, x) − p(t, y)
pc (u)(x) − pc (u)(y)
u(t, x) − u(t, y)
=
− τ ∂t
.
(2.13)
|x − y|β
|x − y|β
|x − y|β
Since p ∈ C 0,β (Ω̄), this implies
sup
x,y∈Ω,x6=y
|p(t,x)−p(t,y)|
|x−y|β
≤ C.
Define w : (0, T ] × Ω2 → R:
w=
s(t, x) − s(t, y)
.
|x − y|β
(2.14)
2.2. UNIQUENESS OF THE WEAK SOLUTION
13
Clearly, a ξ exists such that
0
τ ∂t w − pc (ξ)w =
p(t, x) − p(t, y)
.
|x − y|β
(2.15)
Multiplying w in the two sides of (2.15), and integrating from 0 to t̃ (t̃ is arbitrary in
(0, T ]) give us
τ 2
w −
2
t̃
Z
0
pc (ξ)w2 dt =
Z
0
0
t̃
2
p(t, x) − p(t, y)
τ
wdt +
w(0, ·) .
β
|x − y|
2
(2.16)
According to A2 and A5, by using the Cauchy - Schwarz inequality, one has
u0 (x) − u0 (y)
|x − y|β
Z t̃
w2 dt.
≤C+
τ w2 ≤ τ
2
Z t̃ +
0
p(t, x) − p(t, y)
|x − y|β
2
Z
+
t̃
w2 dt
0
(2.17)
0
By Gronwall’s inequality we obtain
(2.18)
w2 ≤ C,
for any time t̃ ∈ (0, T ]. This implies
|s(t, x) − s(t, y)|
≤ C,
|x − y|β
for almost every x, y ∈ Ω, for every t.
(2.19)
Let Ωc be the subset of Ω, where (2.19) holds everywhere. Clearly, m(Ω\Ωc ) = 0. For
any x ∈ Ω\Ωc , we can find a sequence {xn }n∈N ∈ Ωc converging to x, and define
(2.20)
s(t, x) = lim s(t, xn ).
n→+∞
xn ∈Ωc
According to (2.19), s(t, x) does not depend on the choice of {xn }n∈N . Then, one has
s ∈ C 0,β (Ω̄) (see [48]). Finally, by Theorem 8.33 and Corollary 8.35 in Chapter 8 in [59],
we get
|p|1,β ≤ C(|p|0 + |pc (s)|0 + |F(s)|0,β ),
(2.21)
implying ∇p ∈ L∞ ((0, T ] × Ω).
To show the uniqueness of system (2.7) - (2.8): let g ∈ L2 (Ω) and define Gg as the
solution of the elliptic problem (see [53]):
−τ ∇(H(u)∇Gg ) + Gg = g,
in Ω,
(2.22)
with Gg = 0 at the boundary ∂Ω. Here u is a solution of Problem P. It is easy to show
Gg ∈ W01,2 (Ω)
and kGg kW 1,2 (Ω) ≤ CkgkL2 (Ω) .
(2.23)
14CHAPTER 2. UNIQUENESS OF A WEAK SOLUTION FOR THE SCALAR MODEL
We have the following result:
Theorem 2.2.1. Under the assumptions A1-A7, Problem P has a unique solution u.
Proof. As discussed above, the uniqueness of a solution to Problem P follows directly from
the uniqueness to Problem Ps . Assume now Problem Ps has two solutions, (u, pu ), (v, pv ),
then one has
(∂t (u − v), φ) − (H(u)∇pu − H(v)∇pv , ∇φ) − (F(u) − F(v), ∇φ) = 0,
(pu − pv , ψ) − (pc (u) − pc (v), ψ) + τ (∂t (u − v), ψ) = 0,
(2.24)
(2.25)
for any φ ∈ L2 (0, T ; W01,2 (Ω)), ψ ∈ L2 (0, T ; L2 (Ω)).
Further, we rewrite the above system as follows
(∂t (u − v), φ) − (F(u) − F(v), ∇φ)
−(H(u)(∇pu − ∇pv ), ∇φ) − ((H(u) − H(v))∇pv , ∇φ) = 0,
(2.26)
(pu − pv , ψ) − (pc (u) − pc (v), ψ) + τ (∂t (u − v), ψ) = 0.
(2.27)
Taking g = u − v into (2.22), one gets Gu−v ∈ W01,2 (Ω) and
(Gu−v , λ) + τ (H(u)∇Gu−v , ∇λ) = (u − v, λ),
(2.28)
for any λ ∈ W01,2 (Ω). Further, we have
kGu−v kW 1,2 (Ω) ≤ Cku − vkL2 (Ω) .
(2.29)
Setting φ = Gu−v in (2.26) gives
(∂t (u − v), Gu−v ) − (H(u)(∇pu − ∇pv ), ∇Gu−v )
−((H(u) − H(v))∇pv , ∇Gu−v ) − (F(u) − F(v), ∇Gu−v ) = 0,
(2.30)
and choose λ = pu − pv in (2.28), we find that
1
1
−(H(u)∇(pu − pv ), ∇Gu−v ) = − (u − v, pu − pv ) + (Gu−v , pu − pv ).
τ
τ
(2.31)
Substituting (2.31) into (2.30) leads to
1
1
(u − v, pu − pv ) + (Gu−v , pu − pv )
τ
τ
−((H(u) − H(v))∇pv , ∇Gu−v ) − (F(u) − F(v), ∇Gu−v ) = 0.
(∂t (u − v), Gu−v ) −
(2.32)
Taking the test function ψ = Gu−v in (2.27), we obtain
(∂t (u − v), Gu−v ) +
1
1
(Gu−v , pu − pv ) = (pc (u) − pc (v), Gu−v ).
τ
τ
(2.33)
2.2. UNIQUENESS OF THE WEAK SOLUTION
15
Further, substituting (2.33) into (2.32) gives
1
1
(pc (u) − pc (v), Gu−v ) − (u − v, pu − pv )
τ
τ
−((H(u) − H(v))∇pv , ∇Gu−v ) − (F(u) − F(v), ∇Gu−v ) = 0.
(2.34)
Setting ψ = u − v in (2.27) gives
1
1
− (u − v, pu − pv ) = (∂t (u − v), u − v) − (pc (u) − pc (v), u − v).
τ
τ
(2.35)
Combining (2.34) and (2.35) to eliminate − τ1 (u−v, pu −pv ) and integrating the resulting
equation in time over (0, t), with t ∈ (0, T ] arbitrary lead to
Z
Z t
1 t
(∂t (u − v), u − v)dz −
(u − v, pc (u) − pc (v))dz −
(F(u) − F(v), ∇Gu−v )dz
τ 0
0
0
Z
Z t
1 t
(Gu−v , pc (u) − pc (v))dz +
=−
((H(u) − H(v))∇pv , ∇Gu−v )dz.
(2.36)
τ 0
0
Z
t
We proceed by estimating each term called T1 , T2 , T3 , T4 , T5 . For T1 , since u(0) = v(0) =
u0 , one has
Z tZ
1
(2.37)
∂t (u − v)(u − v)dxdz = k(u − v)(t)k2L2 (Ω) .
2
0
Ω
For T2 , by A2, one obtains
1
−
τ
Z tZ
(u − v)(pc (u) − pc (v))dxdz ≥ 0.
0
(2.38)
Ω
Similarly, since F is Lipschitz continuous, and according to (2.29), we get the estimates
for T3 and T4 :
Z Z
1 t
(F(u) − F(v))∇Gu−v dxdz| ≤
τ 0 Ω
Z Z
1 t
|
Gu−v (pc (u) − pc (v))dxdz| ≤
τ 0 Ω
|
Z
C t
ku − vk2L2 (Ω) dz,
τ 0
Z
C t
ku − vk2L2 (Ω) dz.
τ 0
(2.39)
(2.40)
Finally, for the last term T5 , by using the Cauchy - Schwarz inequality, A1 and ∇p ∈
L∞ ((0, T ] × Ω), we obtain
Z tZ
Z tZ
(H(u) − H(v))∇pv · ∇Gu−v dxdz ≤ C
0
Ω
|(H(u) − H(v)) · ∇Gu−v |dxdz
0
Z
≤C
0
Ω
t
ku − vk2L2 (Ω) dz.
16CHAPTER 2. UNIQUENESS OF A WEAK SOLUTION FOR THE SCALAR MODEL
Summarizing the above leads to
k(u − v)(·, t)k2L2 (Ω) ≤ C
Z
t
ku − vk2L2 (Ω) dz.
(2.41)
0
Since t is arbitrary, the Gronwall lemma shows that
k(u − v)(·, t)k2L2 (Ω) = 0,
for all t,
(2.42)
implying the uniqueness for Problem Ps and therefore for Problem P as well.
2.3
Conclusions
In this chapter, we have proved the uniqueness of a weak solution to a pseudo-parabolic
equation modeling two-phase flow in porous media, and including dynamic effects in the
capillary pressure. The major difficulty is in the non-linearity of the third order derivative
term. The proof uses the equivalence of the original problem with a mixed form of it. By
this, the third order derivative term is avoided.
Chapter 3
Existence of weak solutions to
the two-phase model
3.1
Introduction
We analyze the existence and, where appropriate, uniqueness of a weak solution to the
elliptic - parabolic system:
∂t u + ∇ · (kn (u)∇p) − ∆θ(u) = 0,
(3.1)
∇ · (k(u)∇p) + ∇ · (kw (u)∇(τ (u)∂t u)) = 0,
(3.2)
with k = kw + kn . The equations hold in Q := (0, TM ] × Ω. Here Ω is a bounded
domain in Rd (d = 1, 2, 3), having Lipschtiz continuous boundary, and TM > 0 is a given
maximal time. The unknowns are u and p. The work is motivated by two-phase flow in
porous media (e.g. oil and water).
The system (3.1) - (3.2) models two phase flow in porous media, with dynamic effects
in the phase pressure difference. It is obtained by including the Darcy’s law for both phases
in the mass conservation laws. With w, n being indices for the wetting, respectively, nonwetting phase, the mass conservation equations are (see [12, 61]):
φ
∂sα
+ ∇ · qα = 0,
∂t
α = w, n.
(3.3)
The coefficient φ represents the porosity of the porous medium, while sα and qα denote
the saturation and the volumetric velocity of the α phase. The volumetric velocity qα is
This chapter is a collaborative work with I.S. Pop and it has been published in Journal of Differential
Equations, 260 (2016): 2418-2456.
17
18CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
deduced from the Darcy’s law as
qα = −
k̄
krα (sα )∇pα ,
µα
α = w, n,
(3.4)
where k̄ is the absolute permeability of the porous medium, pα the pressure, µα the
viscosity and krα the relative permeability of the α phase. The specific function of krα is
assumed to be known. Substituting (3.4) in (3.3) gives
φ
k̄krα
∂sα
−∇·(
∇pα ) = 0,
∂t
µα
α = w, n.
(3.5)
We assume that only two phases are present
(3.6)
sw + sn = 1.
To complete the model, one commonly assumes a relationship between the phase pressure
difference and sw . Under equilibrium assumption, this is
pn − pw = pc (sw ),
with a given function pc = pc (·). Experimental results [38] have, however, proved the
limitation of this assumption. Alternatively, in [60] the following relation is proposed:
pn − pw = pc (sw ) − τ (sw )
∂sw
.
∂t
(3.7)
The damping function τ as well as the function pc , which represents the capillary pressure
under equilibrium condition, are assumed to be known. Summing the two equations from
(3.5) and making use of (3.6) gives:
∇ · F = 0,
where F =
k̄krw
µw ∇pw
+
k̄krn
µn ∇pn
denotes the total flow.
Introducing the normalized relative permeabilities
kα :=
krα
,
µα
α = w, n,
and with k = kw + kn , we also define the fractional flow function
fw (sw ) :=
kw
.
k
(3.8)
3.2. ASSUMPTIONS AND KNOWN RESULTS
19
Then, we follow [5, 33], and define the global pressure p
Z
sw
p = pn −
0
fw (z)pc (z)dz,
(3.9)
CD
which leads to the following expression of the water pressure
Z sw
0
pw = p +
fw (z)pc (z)dz − pc (sw ) + τ (sw )∂t sw .
(3.10)
CD
Here CD ∈ (0, 1) is a constant that will be used as the boundary value of water saturation.
Furthermore, define the complementary pressure θ as the integral (Kirchhoff) transformation
Z sw
0
kw kn
θ(sw ) = −
(z)pc (z)dz.
(3.11)
k
CD
Then from (3.5) for the wetting phase, by using (3.6) and (3.9) gives
φ∂t sw + ∇ · (k̄kn ∇p) − ∇ · (k̄∇θ(sw )) = 0.
(3.12)
∇ · (k̄k∇p) + ∇ · k̄kw ∇(τ ∂t sw ) = 0.
(3.13)
Finally, (3.8) becomes
The system (3.12)-(3.13) is in dimensional form. Taking Lr , Tr , and Pr as characteristic
values for the length, time, global pressure, respectively, scaling the space variable x with
Lr , the time t with Tr and the pressure p with Pr , and assuming
Tr
φ
=
,
2
Lr
k̄Pr
we obtain the following system:
∂t u + ∇ · (kn ∇p) − ∆θ(u) = 0,
(3.14)
∇ · (k∇p) + ∇ · (kw ∇(τ ∂t u)) = 0,
(3.15)
with u = sw .
3.2
Assumptions and known results
For the non-linearities appearing in the model, we refer to [12, 61, 82]. The special assumptions are given below. Here we mention the typical choices of permeability in the
literature ( [22, 24])
kw (u) = uα ,
kn (u) = (1 − u)β ,
with α, β > 1,
20CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
and
0
−pc (u) = u−λ ,
λ > 1.
These are defined for u ∈ [0, 1] (the physical range).
The model (3.14) - (3.15) is completed by the initial condition
u(0, ·) = u0
in Ω,
(3.16)
and the boundary conditions
u = CD , p = 0
at ∂Ω,
for all
t > 0,
(3.17)
where u0 is a given function and the constant CD satisfies 0 < CD < 1.
Remark 3.2.1. To avoid an excess of technicalities, the pressure boundary condition in
(3.17) is assumed to be constant. Other types of boundary conditions, as Neumann can
be considered. Similarly, the boundary values of u may be non-constant, but should be
bounded away from 0 and 1.
In this chapter we assume
A1: The functions kw , kn [0, 1] −→ [0, 1] are C 1 , kw is an increasing function, and kn is
decreasing. pc ∈ C 1 ((0, 1], R+ ) is a decreasing function. τ ∈ C 1 ([0, 1), R+ ) satisfies
the following:
i: There exists τ0 > 0 such that τ ≥ τ0 for all u ∈ [0, 1).
ii: There exist u∗ ∈ [0, 1) , ω > 0 and Cτ > 0 such that τ (u) = Cτ (1 − u)−ω for
all u ∈ [u∗ , 1). The restriction on ω > 0 will be given in assumption A5 below.
0
Furthermore, we assume −pc kw ≤ C < +∞.
These functions (see also [12]) are defined on (0, 1), which is the physically relevant range.
For the analysis, we extend them to R in the following way:
a: kw (u) = kw (0), if u ≤ 0, kw (u) = kw (1), if u ≥ 1.
b: kn (u) = kn (0), if u ≤ 0, kn (u) = kn (1), if u ≥ 1.
c:
1
p0c (u)
=
1
,
p0c (0)
if u ≤ 0,
1
p0c (u)
=
1
,
p0c (1)
if u ≥ 1.
d: 1/τ (u) = 1/τ (0), if u ≤ 0, 1/τ (u) = 1/τ (1), if u ≥ 1.
0
Remark 3.2.2. By A1, pc and τ may become unbounded. Then an extension by continuity is not meaningful. However, this extension makes sense for the reciprocals p10 ,
1
τ.
c
3.2. ASSUMPTIONS AND KNOWN RESULTS
21
0
Remark 3.2.3. The assumptions on kw , kn and pc are commonly used functions in porous
media literature. For the behavior of τ , we refer to [26,94]. The experimental results there
show that τ is bounded away from 0. Moreover, the plots close to full water saturation
suggest a rapid increase of τ close to u = 1. Furthermore, such a profile is obtained in [18]
by means of upscaling techniques. This motivated considering the present framework for
the existence proof. At the same time, we mention that other types of τ −curves have
also been proposed in e.g. [56, 57]. The existence proof in such cases remains an open
research question.
In the proofs below, we will use the functions (see [28, 77]) G, Γ : R → R ∪ {±∞}
Z
u
G(u) =
CD
τk
(z)dz,
kw kn
Z
u
G(z)dz,
Γ(u) =
(3.18)
CD
and the function T (u) : R → R ∪ {±∞}
Z
T (u) =
u
(3.19)
τ (z)dz + CT ,
0
where CT is a constant specified below. Clearly, Γ is a convex function satisfying Γ(0) =
G(0) = 0, implying that
Γ(u) ≥ 0,
for all u ∈ R.
(3.20)
A2: The initial condition u0 satisfies u0 ∈ W01,2 (Ω) + CD ,
W 1,2 (Ω).
R
Ω
Γ(u0 ) < +∞, T (u0 ) ∈
The problem considered here is an extension of scalar, two-phase flow models studied
in [28], while only one pseudo-parabolic equation is considered in the unknown u. The
model in [28] can be derived from the present one assuming that the total flow is known
(see [5, 52]). After non-dimensionlization and some algebraic manipulation one gets
∂t u + ∇ · F − ∆θ(u) − τ ∇ · (
kw kn
∇∂t u) = 0.
k
(3.21)
The existence and uniqueness of the weak solution have been shown in [52] in the linear
case of the third order derivative term. In the nonlinear and non-degenerate case of the
third order derivative term, we refer to [16] which proved the existence of a solution
and uniqueness in one and two dimensional cases. When the coefficients also depend on
the time derivative of the unknown, the existence of weak solutions has been analyzed
in [95]. The present work is closer to [77], where a degenerate pseudo parabolic equation
modeling one-phase flow is considered. The existence is proved based on regularization.
For uniqueness but in the non-degenerate case, we refer to [14, 30]. Here we build on
similar ideas in [77] to prove for the existence for the two-phase model.
Under equilibrium assumptions (τ = 0), such models are analyzed e.g. in [6, 35,
69]. For non-equilibrium, two-phase flow models are analyzed in [68], where existence
22CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
is obtained for non-degenerate cases, but including hysteresis effects, and in [32], where
uniqueness is proved again in non-degenerate cases.
In what follows, we analyze first the regularized model, and prove the existence and
uniqueness of a solution, as well as a priori energy estimates. This is achieved by Rothe’s
method [65]. Finally, we pass the regularization parameter to 0 to obtain the existence of
weak solutions in the degenerate case. Note that uniqueness remains open.
3.3
Existence, uniqueness of solutions in the regularized
case
3.3.1
The weak solution concept
We use the standard spaces L2 (Ω), W 1,2 (Ω) and W01,2 (Ω) in the theory of partial differential equation. By (·, ·) we mean the scalar product and k·k stands for the corresponding
norm in L2 (Ω), or where needed, in (L2 (Ω))d . Furthermore, L2 (0, TM ; X) denotes the
Bochner space of X -valued functions. Let us now define the set
V := W01,2 (Ω) + CD .
Inspired by [77], a weak solution to the model (3.14)-(3.17) solves the following
∗
Problem P: Find u ∈ L2 (0, TM ; V ), such that ∂t u ∈ L2 (Q) and p ∈ L2 (0, TM ; W01,r (Ω))
√
(for some particular r∗ ∈ (1, 2)), such that u(0, ·) = u0 , kn ∇p ∈ L2 (0, TM ; L2 (Ω)d ),
√
kw (∇p + ∇∂t T (u)) ∈ L2 (0, TM ; L2 (Ω)d ), and
for any φ, ψ ∈
(∂t u, φ) − (kn ∇p, ∇φ) + (∇θ(u), ∇φ) = 0,
(3.22)
(k∇p, ∇ψ) + (kw ∇∂t T (u), ∇ψ) = 0,
(3.23)
L2 (0, TM ; W01,2 (Ω)).
Before dealing with the degenerate case, we consider first the regularized case. To
this aim, we define:
a’: kwδ (u) = kw (u + δ), if 0 ≤ u ≤ 1 − δ, kwδ (u) = kw (δ), if u < 0, kwδ (u) = kw (1),
if u > 1 − δ.
b’: knδ (u) = kn (u − δ), if δ ≤ u ≤ 1, knδ (u) = kn (0), if u < δ, knδ (u) = kn (1 − δ),
if u > 1.
0
0
0
0
c’: −pcδ (u) = −pc (u + δ), if 0 ≤ u ≤ 1 − δ, −pcδ (u) = −pc 0 (δ), if v < 0, −pcδ (u) =
0
−pc (1), if u > 1 − δ.
3.3. EXISTENCE, UNIQUENESS OF SOLUTIONS IN THE REGULARIZED CASE 23
d’:


τ (u),


 τ (u∗ ),
τδ (u) :=

τ (u − δ),



τ (1 − δ),
if
if
if
if
u < u∗ ,
u ∈ [u∗ , u∗ + δ],
u ∈ [u∗ + δ, 1),
u ≥ 1.
Furthermore, we let
u
Z
Gδ (u) =
CD
kδ (u) = kwδ (u) + knδ (u),
Z
τδ kδ
(z)dz,
kwδ knδ
u
Gδ (z)dz,
Γδ (u) =
(3.24)
CD
Z u
0
kwδ knδ
kwδ
, θδ (u) = −
(z)pcδ (z)dz, (3.25)
kδ
kδ
CD
Z u
Tδ (u) =
τδ (z)dz + CT δ .
(3.26)
fwδ =
0
Here CT δ is a constant which will be specified in Section 3.4.
We now consider the regularized approximation of (3.14) - (3.15)
∂t uδ + ∇ · (knδ ∇pδ ) − ∆θδ (uδ ) = 0,
(3.27)
∇ · (kδ ∇pδ ) + ∇ · (kwδ ∇∂t Tδ (uδ )) = 0,
(3.28)
with the following initial and boundary conditions
uδ (0, ·) = u0
in Ω,
uδ = CD , pδ = 0 at ∂Ω,
for all
t > 0.
We start by proving the existence of a solution for the following
Problem Pδ : Find uδ ∈ L2 (0, TM ; V ) and pδ ∈ L2 (0, TM ; W01,2 (Ω)), such that ∂t uδ ∈
L2 (Q), uδ (0, ·) = u0 , ∇∂t Tδ (uδ ) ∈ L2 (0, TM ; L2 (Ω)d ) and
Z
TM
Z
(∂t uδ , φ)dt −
0
TM
Z
(knδ ∇pδ , ∇φ)dt +
0
Z
TM
(∇θδ (uδ ), ∇φ)dt = 0,
(3.29)
0
Z
(kδ ∇pδ , ∇ψ)dt +
0
TM
TM
(kwδ ∇∂t Tδ (uδ ), ∇ψ)dt = 0,
(3.30)
0
for any φ, ψ ∈ L2 (0, TM ; W01,2 (Ω)).
The proof is based on the method of Rothe [65]. In what follows we use the elementary
inequality
1 2 σ 2
a + b ,
for any a, b ∈ R, σ > 0.
(3.31)
ab ≤
2σ
2
24CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
3.3.2
The time discretization
With N ∈ N, let h = TM /N and consider the Euler implicit discretization of (3.29)(3.30).
Problem Pnδ : Given un−1
∈ V (n = 1, 2...N ), find unδ ∈ V and pnδ ∈ W01,2 (Ω), such
δ
that
un − un−1
δ
( δ
, φ) − (knδ (unδ )∇pnδ , ∇φ) + (∇θδ (unδ ), ∇φ) = 0,
(3.32)
h
(kδ (unδ )∇pnδ , ∇ψ) + (kwδ (unδ )∇
Tδ (unδ ) − Tδ (un−1
)
δ
, ∇ψ) = 0,
h
(3.33)
for any φ, ψ ∈ W01,2 (Ω). Here (·, ·) means L2 inner product.
Lemma 3.3.1. Under the assumptions A1, A2, Problem Pδn has a solution.
Proof. We start with a finite dimensional approximation (Galerkin), for which we prove
the existence of a solution to Problem Pnδ . To this aim, we use Lemma 1.4 (p. 164
in [100]). Then we pass to the limit for the discrete solution, and use compactness to
show the existence of a solution for Problem Pnδ .
∞
Let {vm }m=1 be the countable basis of the separable space W01,2 (Ω). Set Wm =
span{v }(k = 1, ..., m). Then, given α1 , ..., αm ∈ R, β1 , ..., βm ∈ R, define v =
Pm m
Pm
i=1 αi vi ∈ Wm , w =
i=1 βi vi ∈ Wm , and set ς = v + CD . A finite dimensional
solution (ς, w) ∈ {Wm + CD } × {Wm } of (3.32)-(3.33) satisfies
(
ς − un−1
δ
, φ) − (knδ (ς)∇w, ∇φ) + (∇θδ (ς), ∇φ) = 0,
h
(kδ (ς)∇w, ∇ψ) + (kwδ (ς)∇
Tδ (ς) − Tδ (un−1
)
δ
, ∇ψ) = 0,
h
(3.34)
(3.35)
for any φ, ψ ∈ Wm .
To avoid an excess of notations in the first part, we do not use any different indices
for the solution pair (ς, w) in the finite dimensional case. For i = 1, ..., m, define
β̃i = (kδ (ς)∇w, ∇vi ) +
1
(kwδ (ς)∇(Tδ (ς) − Tδ (uδn−1 )), ∇vi ), (i = 1, 2, ..., m). (3.36)
h
Note that if (ς, w) is a solution pair to (3.34)-(3.35), one has β̃i = 0 for all i. By
Pm
w = i=1 βi vi , one clearly has
m
X
i=1
βi β̃i = (kδ (ς)∇w, ∇w) +
1
(kwδ (ς)∇(Tδ (ς) − Tδ (un−1
)), ∇w)
δ
h
p
2 1
1
= kδ (ς)∇w + (τδ (ς)kwδ (ς)∇ς, ∇w) − (τδ (un−1
)kwδ (ς)∇un−1
, ∇w).
δ
δ
h
h
(3.37)
3.3. EXISTENCE, UNIQUENESS OF SOLUTIONS IN THE REGULARIZED CASE 25
Further, we define g : R → R by
Z
ς
g(ς) =
0
τδ kwδ
(z)dz.
knδ
By A1, there exist 0 < M0 ≤ M1 < ∞ possibly depending on δ such that
g(ς)ς ≥ M0 ς 2 ,
|g(ς)| ≤ M1 |ς|.
(3.38)
As above, for i = 1, ..., m, we define
li =
(g(ς), vi )
2
kvi k
which immediately implies
g(ς) =
(i = 1, 2, ..., m),
m
X
li vi .
(3.39)
i=1
Define
ς − un−1
vi
1
1
δ
l˜i = (
, ) − (knδ (ς)∇w, ∇vi ) + (∇θδ (ς), ∇vi ),
(3.40)
h
h
h
h
as in above , if (ς, w) is a solution pair, then ˜li = 0 for all i. Similarly, since ∇g(ς) =
τδ kwδ
(ς)∇ς, we get
knδ
m
X
1
1
τδ kwδ
li ˜li = 2 (ς − un−1
, g(ς)) − (knδ (ς)∇w,
(ς)∇ς)
δ
h
h
knδ
i=1
+
1
τδ kwδ
(∇θδ (ς),
(ς)∇ς).
h
knδ
(3.41)
Adding (3.37) and (3.41) yields
m
X
i=1
βi β̃i +
m
X
p
2
1
li ˜li = kδ (ς)∇w + 2 (ς − un−1
, g(ς))
δ
h
i=1
+
1
τδ kwδ
1
(∇θδ (ς),
(ς)∇ς) − (τδ (un−1
)kwδ (ς)∇un−1
, ∇w).
δ
δ
h
knδ
h
(3.42)
The existence of a solution is provided if the sum on the left in (3.42) becomes positive
for (l1 , ..., lm ) and (β1 , ..., βm ) sufficiently large. To prove this, denote the terms on the
26CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
right by T1 , T2 , T3 , T4 . Note that T1 is positive. For T2 , we use (3.38) to obtain
1
1
1
(ς − un−1
, g(ς)) = 2 (ς, g(ς)) − 2 (un−1
, g(ς))
δ
δ
h2
h
h
2 2 M0
M0
1 M
2
2
≥ 2 · kςk − 2 · 1 uδn−1 − 2 · kςk
h
2h M0
2h
2
M0
1 M2 2
.
= 2 · kςk − 2 · 1 un−1
δ
2h
2h M0
(3.43)
Recalling (3.25), T3 becomes
1
τδ kwδ
1 kwδ · knδ 0
τδ kwδ
(∇θδ (ς),
(ς)∇ς) = (−
pcδ (ς)∇ς,
∇ς)
h
knδ
h
kδ
knδ
q
2
1 kwδ
0
.
√
= −τ
p
(ς)∇ς
δ
cδ
h k
δ
For T4 , one gets
2
1 p
τ (un−1 )k (ς)
2
δ δ
wδ
n−1 p
∇uδ − kδ (ς)∇w .
2
kδ (ς)
(3.44)
Using (3.43)-(3.44), the Poincaré and Cauchy-Swarchz inequalities lead to
1
1
− (τδ (un−1
)kwδ (ς)∇un−1
, ∇w) ≥ − 2
δ
δ
h
2h
m
X
i=1
βi β̃i +
m
X
M2
M0
M3
2
2
2
li ˜li ≥
k∇wk + 2 kςk +
k∇ςk
2
2h
h
i=1
2
n−1
1 1 M12 τδ (uδ )kwδ (ς)
un−1 2
p
− 2
−
∇un−1
δ
δ
2
2h
2h M0
kδ (ς)
M0
M2
M3
2
2
2
kwk + 2 kςk +
k∇ςk
2
2h
h
2
n−1
2
1 1 M2 τδ (uδ )kwδ (ς)
n−1 .
p
− 2
∇uδ − 2 1 un−1
δ
2h
2h M0
kδ (ς)
≥C(Ω)
(3.45)
Note that M2 is independent of δ, and M3 may depend on δ, but they all not depend
on the dimension of the space W
! that (3.36) and (3.40) define a mapping
!m . We know
˜
l
l
ζm : R2m → R2m by ζm
=
, which is continuous by (A1). As klk +
β̃
β
kβk (e.g. in the standard Euclidean norm) becomes large enough, ζm is strictly positive,
and Lemma 1.4 in [100] (p. 164) guarantees that ζm has a zero, that means there exists
a solution to the discrete system. We now pass to the limit as m → ∞. Denoting by
(um , pm ) ∈ {Wm + CD } × {Wm } the finite-dimensional solution (ς, w) obtained above,
3.3. EXISTENCE, UNIQUENESS OF SOLUTIONS IN THE REGULARIZED CASE 27
from (3.45), we get
M2
M3
M0
2
2
2
C(Ω)
k∇pm k + 2 kum k +
k∇um k
2
2h
h
2
n−1
2
1 M2 1 τδ (uδ )kwδ (um )
n−1 p
∇uδ + 2 1 un−1
≤ 2
δ
2h 2h M0
kδ (um )
≤C,
where C may depend on δ, but not on the dimension m. This means that um is uniformly
bounded in V , pm is uniformly bounded in W01,2 (Ω), so we can find ũ ∈ V, p̃ ∈ W01,2 (Ω),
such that, um * ũ weakly in V and pm * p̃ weakly in W01,2 (Ω). The compact embedding
of W 1,2 (Ω) into L2 (Ω) gives um → ũ and pm → p̃ strongly in L2 (Ω).
We show that the limit pair (ũ, p̃) ∈ (V × W01,2 (Ω)) is a solution of (3.32) and (3.33).
By the boundedness of knδ , the sequence knδ (um )∇pm is bounded uniformly in L2 (Ω)
(w.r.t. m), so it has a weak limit χ. We identify this limit as knδ (ũ)∇p̃ implying that
(knδ (um )∇pm , ∇φ) −→ (knδ (ũ)∇p̃, ∇φ), for any φ ∈ W01,2 (Ω).
To do so, we choose the test function φ ∈ C0∞ (Ω), clearly, one yields
(knδ (um )∇pm , ∇φ) = (∇pm , knδ (ũ)∇φ) + (∇pm , (knδ (um ) − knδ (ũ)∇φ)),
and since knδ (ũ) ∈ L∞ (Ω), one has
(∇pm , kδ (ũ)∇φ) −→ (∇p̃, knδ (ũ)∇φ).
We now show that the limit of (∇pm , (knδ (um )−knδ (ũ))∇φ) is 0. Since ∇pm is bounded
uniformly in (L2 (Ω))d , one has
|(∇pm , (knδ (um ) − knδ (ũ)∇φ))| ≤ k∇pm k · k(knδ (um ) − knδ (ũ))∇φk
Z
21
.
≤C
(knδ (um ) − knδ (ũ))2 |∇φ|2 dx
Ω
Because knδ is bounded, we get
|(knδ (um ) − knδ (ũ))∇φ| ≤ C|∇φ|
in Ω.
Further, since um −→ ũ strongly in L2 (Ω), in the view of the continuity of knδ we have
knδ (um ) −→ knδ (ũ) a.e.
28CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
Then, by the Dominated Convergence Theorem
Z
2
2
12
(knδ (um ) − knδ (ũ)) |∇φ| dx
−→ 0.
Ω
Therefore, for any φ ∈ C0∞ (Ω), since knδ (um )∇pm → χ (weakly in W 1,2 (Ω)), due to the
density of C0∞ (Ω) in W 1,2 (Ω), this allows identifying χ = knδ (ũ)∇p̃ for any φ ∈ W01,2 (Ω).
Similarly, we can also prove that
(kδ (um )∇pm , ∇φ) −→ (kδ (ũ)∇p̃, ∇φ),
(τδ (un−1
)kδ (um )∇un−1 , ∇φ) −→ (τδ (un−1
)kδ (ũ)∇un−1 , ∇φ),
δ
δ
(τδ (um )kwδ (um )∇um , ∇φ) −→ (τδ (ũ)kwδ (ũ)∇ũ, ∇φ),
(θδ (um )∇um , ∇φ) −→ (θδ (ũ)∇ũ, ∇φ),
for any φ ∈ W01,2 (Ω).
From now on a solution pair to Problem Pnδ is denoted by (unδ , pnδ ).
3.3.3
A priori estimates
Having established the existence for the time discrete problems, we now prove the existence
of a solution to Problem Pδ . To achieve this, we use the elementary result.
Lemma 3.3.2. Let k ∈ 1, ..., N , m ≥ 1. Given two sets of real vectors ak , bk ∈ Rm (k =
1, ..., N ), one has
N
X
N
(ak − ak−1 , ak ) =
k=1
X
1 N 2
(|a | − |a0 |2 +
|ak − ak−1 |2 ).
2
k=1
We have the following:
Lemma 3.3.3. A C > 0, not depending on h exists such that, for any N ∗ ∈ 1, 2, ..., N ,
one has
Z
2
∗
1
N∗ (u
)
Γδ (uN
)dx
+
∇T
≤ C,
δ
δ
δ
2
Ω
2
N∗
N ∗ q
X
X
∇(Tδ (unδ ) − Tδ (un−1 ))2 + h
−p0 τδ (un )∇unδ ≤ C,
δ
cδ
δ
n=1
n=1
N∗
∗ 2 1 X
N ∗ 2
N 2
k∇unδ k ≤ C.
uδ + ∇uδ + h
2
n=1
Proof. Taking φ =
Z
un
δ
CD
τδ kδ
(z)dz in (3.32) and ψ =
kwδ knδ
Z
un
δ
CD
(3.46)
τδ
(z)dz in (3.33) (both
kwδ
3.3. EXISTENCE, UNIQUENESS OF SOLUTIONS IN THE REGULARIZED CASE 29
being in W01,2 (Ω)) gives
(
unδ − un−1
δ
,
h
Z
un
δ
CD
τδ kδ
τδ kδ
(z)dz) − (knδ (unδ )∇pnδ ,
(un )∇unδ )
kwδ knδ
kwδ knδ δ
τδ kδ
+ (∇θδ (unδ ),
(un )∇unδ ) = 0,
kwδ knδ δ
(3.47)
and
)
Tδ (unδ ) − Tδ (un−1
1
τδ n
δ
,
(uδ )∇unδ ) + (kwδ (unδ )∇
∇Tδ (unδ )) = 0.
kwδ
h
kwδ (unδ )
(3.48)
With Gδ defined in (3.24), in view of its monotonicity we have
(kδ (unδ )∇pnδ ,
(unδ − un−1
) · Gδ (unδ ) ≥ Γδ (unδ ) − Γδ (un−1
).
δ
δ
Adding (3.47) and (3.48) and using (3.49) leads to
Z
τδ kδ
1
(Γδ (unδ ) − Γδ (un−1
))dx + (∇θδ (unδ ),
(un )∇unδ )
δ
h Ω
kwδ knδ δ
1
)), ∇Tδ (unδ )) ≤ 0.
+ (∇(Tδ (unδ ) − Tδ (un−1
δ
h
(3.49)
(3.50)
Summing up (3.50) for n = 1 to N ∗ gives
∗
Z
∗
(Γδ (uN
δ )
0
− Γδ (u ))dx + h
Ω
N
X
(∇θδ (unδ ),
n=1
τδ kδ
(un )∇unδ )
kwδ knδ δ
∗
+
N
X
(∇(Tδ (unδ ) − Tδ (uδn−1 )), ∇Tδ (unδ )) ≤ 0.
(3.51)
n=1
0
Since θδ (unδ ) = −
knδ kwδ n
0
(uδ ) · pcδ (unδ ), one has
kδ
∗
h
N
X
(∇θ(unδ ),
n=1
2
N ∗ q
X
τδ kδ
0
n
n
n
n
(uδ )∇uδ ) = h
−pcδ τδ (uδ )∇uδ .
kwδ knδ
n=1
According to Lemma 3.3.2, we get
∗
N
X
n=1
∗
(∇(Tδ (unδ )
−
Tδ (un−1
)), ∇Tδ (unδ ))
δ
N
1 X
∇(Tδ (unδ ) − Tδ (un−1 ))2
=
δ
2 n=1
∗ 2
1
1
∇Tδ (u0 )2 ,
+ ∇Tδ (uN
δ ) −
2
2
30CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
which implies
Z
∗
Γδ (uN
δ )
Ω
2
N ∗ q
X
0
n
n
+h
−pcδ τδ (uδ )∇uδ n=1
N∗
2
2
1 X
1
N∗ ∇(Tδ (unδ ) − Tδ (un−1 ))2 + 1 +
(u
)
≤ Γδ (u0 ) + ∇Tδ (u0 ) .
∇T
δ
δ
δ
2 n=1
2
2
Recalling (3.20) , (A1) and (A2), uniformly w.r. t. δ, it holds that
Z
u0
Z Z
0
Z
u
Γδ (u )dx =
Ω
CD
Ω
u0
Z Z
CD
Z
v
=
CD
Ω
u0
Z Z
CD
Z
v
≤
CD
Ω
Z
CD
τδ kδ
dzdvdx
kwδ knδ
Z Z u0 Z v
τδ
τδ
(z)dzdvdx +
(z)dzdvdx
kwδ
k
Ω CD CD nδ
Z Z u0 Z v
τ
τ
(z)dzdvdx +
(z)dzdvdx
kw
k
Ω CD CD n
0
Γ(u )dx < ∞.
=
Ω
Furthermore, we also have
Z
Z
Z
Z
|∇Tδ (u0 )|2 dx =
τδ2 |∇u0 |2 dx ≤
τ 2 |∇u0 |2 dx =
|∇T (u0 )|2 dx ≤ C.
Ω
Ω
Ω
Ω
Therefore, we obtain
Z
∗
Γδ (uN
δ )dx +
Ω
and
∗ 2
1
∇Tδ (uN
δ ) ≤ C,
2
2
N∗
N ∗ q
X
1 X
∇(Tδ (unδ ) − Tδ (un−1 ))2 + h
−p0 τδ (un )∇unδ ≤ C.
δ
cδ
δ
2 n=1
n=1
By the definition of τδ , this immediately gives
N∗
X
1
N ∗ 2
2
+
h
k∇unδ k ≤ C.
∇uδ 2
n=1
To complete the proof, we use the Poincaré’s inequality
∗
∗
N N∗ ≤ kuN
uδ 2
δ − CD kL2 (Ω) + kCD kL2 (Ω) ≤ C(Ω) ∇uδ L (Ω)
L2 (Ω)
+ C ≤ C.
3.3. EXISTENCE, UNIQUENESS OF SOLUTIONS IN THE REGULARIZED CASE 31
Lemma 3.3.4. For any N ∗ ∈ 1, 2, ..., N, we have
∗
∗
N
N
X
X
∇(Tδ (unδ ) − Tδ (un−1 ))2 +
k∇(unδ − un−1
)k2 ≤ Ch,
δ
δ
n=1
n=1
∗
∗
N
N
X
X
n
Tδ (unδ ) − Tδ (un−1 )2 +
uδ − un−1 2 ≤ Ch,
δ
δ
n=1
n=1
h
N∗
X
2
k∇pnδ k ≤ C,
(3.52)
n=1
where C is independent of h.
Proof. Testing in both (3.32) and (3.33) with h(Tδ (unδ )−Tδ (uδn−1 )), adding the resulting
gives
(unδ − un−1
, Tδ (unδ ) − Tδ (un−1
)) + h(kwδ (unδ )∇pnδ , ∇(Tδ (unδ ) − Tδ (un−1
)))
δ
δ
δ
2
q
n−1
n−1 n
n
n
n
+
kwδ (uδ )∇(Tδ (uδ ) − Tδ (uδ )) + h(∇θδ (uδ ), ∇(Tδ (uδ ) − Tδ (uδ ))) = 0.
(3.53)
With ψ = hpnδ in (3.33), we have
q
2
n
n
h kδ (uδ )∇pδ = −(kwδ (unδ )∇(Tδ (unδ ) − Tδ (un−1
)), ∇pnδ )
δ
k (un )
q
wδ δ
n−1 n
n
n
≤ p
∇(Tδ (uδ ) − Tδ (uδ )) · kδ (uδ )∇pδ ,
kδ (unδ )
(3.54)
giving
−h(kwδ (unδ )∇pnδ , ∇(Tδ (unδ )
−
Tδ (un−1
)))
δ
2
k (un )
wδ δ
n−1 n
≤ p
∇(T
(u
)
−
T
(u
))
.
δ
δ
δ
δ
kδ (unδ )
Then, (3.53) becomes
(unδ
−
uδn−1 , Tδ (unδ )
−
Tδ (un−1
))
δ
2
q
n−1 n
n
+ kwδ (uδ )∇(Tδ (uδ ) − Tδ (uδ ))
2
k (un )
wδ
n−1 n
δ
p
+h(∇θδ (unδ ), ∇(Tδ (unδ ) − Tδ (un−1
)))
≤
∇(T
(u
)
−
T
(u
))
. (3.55)
δ
δ
δ
δ
δ
kδ (unδ )
32CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
In view of the definition kδ , one has
2
q
2 k (un )
wδ δ
n−1 n
kwδ (un )∇(Tδ (unδ ) − Tδ (un−1 )) − p
∇(T
(u
)
−
T
(u
))
δ
δ
δ
δ
δ
δ
k (un )
δ
δ
r
2
k k
wδ nδ
n−1 n
n
=
(uδ )∇(Tδ (uδ ) − Tδ (uδ )) .
kδ
Further, since
r
2
1
kwδ knδ n
n−1
n−1 n
n
n
h (∇θ(uδ ), ∇(Tδ (uδ ) − Tδ (uδ ))) ≤ (uδ )∇(Tδ (uδ ) − Tδ (uδ ))
2
kδ
r
2
h2 kwδ knδ n 0
(uδ )pcδ (unδ )∇unδ ,
+
2 kδ
(3.55) yields
r
2
1
kwδ knδ n
n−1 n
−
−
+ (uδ )∇(Tδ (uδ ) − Tδ (uδ ))
2
kδ
r
2
r
2
q
k
0
h2
h2 kwδ knδ n 0
0
nδ
n
n
n
n
n
|kwδ pcδ |∞ (uδ )pcδ (uδ )∇uδ ≤
(uδ ) −pcδ τδ (uδ )∇uδ .
≤
kδ τδ
2 kδ
2
(unδ
un−1
, Tδ (unδ )
δ
Tδ (un−1
))
δ
(3.56)
By (3.46) and (A1) and (A2), we have
2
(unδ − un−1
, Tδ (unδ ) − Tδ (un−1
)) + ∇(Tδ (unδ ) − Tδ (un−1
)) ≤ Ch2 .
δ
δ
δ
(3.57)
This leads to
∗
N
X
∗
(unδ
−
un−1
, Tδ (unδ )
δ
−
Tδ (un−1
))
δ
N
X
∇(Tδ (unδ ) − Tδ (un−1 ))2 ≤ Ch.
+
δ
n=1
n=1
By the definition of T , a ξ exists, such that
p
2
(unδ − uδn−1 , Tδ (unδ ) − Tδ (un−1
)) = τδ (ξ)(unδ − un−1
) ≤ Ch2 .
δ
δ
Similarly, a ξ˜ exists, such that
2
1
n−1 n−1
n−1
n
n
n
(uδ − uδ , Tδ (uδ ) − Tδ (uδ )) = q
(Tδ (uδ ) − Tδ (uδ )) ≤ Ch2 .
˜
τδ (ξ)
3.3. EXISTENCE, UNIQUENESS OF SOLUTIONS IN THE REGULARIZED CASE 33
Then we obtain
∗
∗
N
N
X
X
n
Tδ (unδ ) − Tδ (un−1 )2 +
uδ − un−1 2 ≤ Ch.
δ
δ
n=1
n=1
Furthermore, according to (A1), (3.46) and (3.57), one also has
∗
N
X
k∇(unδ − un−1
)k2 ≤ Ch.
δ
n=1
Finally, (3.52) follows from (3.54) and (3.57).
3.3.4
Existence of weak solutions to the regularized problem
To show the existence of a solution to Problem Pδ , we consider linear interpolation in
time:
t − tn−1
TδN (t) = Tδ (un−1
)+
(Tδ (unδ ) − Tδ (un−1
)),
δ
δ
h
UδN (t) = un−1
+
δ
t − tn−1 n
(uδ − un−1
),
δ
h
and piecewise constant functions in time
T̄δN (t) = Tδ (unδ ), ŪδN (t) = unδ , P̄δN (t) = pnδ ,
for t ∈ (tn−1 , tn ], n = 1, 2, ..., N. Clearly, T̄δN ∈ L2 (Q), ŪδN ∈ L2 (0, TM ; V ), P̄δN (t) ∈
L2 (0, TM ; W01,2 (Ω)).
We have the following result:
Theorem 3.3.1. Under the assumptions A1 and A2, Problem Pδ has a solution (uδ , pδ ).
Proof. According to the priori estimates in Lemma 3.3.3 and Lemma 3.3.4, we have
Z
0
TM
2
kTδN (t)kL2 (Ω)
N Z
X
tn
2
t − tn−1
n−1
n−1 n
dt =
(Tδ (uδ ) − Tδ (uδ ))
Tδ (uδ ) +
dt
h
n−1
L2 (Ω)
n=1 t
N Z tn X
Tδ (unδ ) − Tδ (un−1 )2 2
Tδ (un−1 )2 2
≤2
+
dt
δ
δ
L (Ω)
L (Ω)
n=1
tn−1
≤ C,
Z
TM
0
Z
0
TM
2
k∇TδN (t)kL2 (Ω) dt ≤ C,
2
k∂t UδN (t)kL2 (Ω) dt =
N
1 X
unδ − un−1 2 2
≤ C,
δ
L (Ω)
h n=1
34CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
TM
Z
2
k∂t TδN (t)kL2 (Ω) dt =
0
N
1 X
Tδ (unδ ) − Tδ (un−1 )2 2
≤ C,
δ
L (Ω)
h n=1
and
Z
TM
2
k∂t ∇TδN (t)kL2 (Ω) dt =
0
2
1
∇(Tδ (unδ ) − Tδ (un−1 ))
δ
2 dt
tn−1 h
L (Ω)
n=1
=
tn
N Z
X
N
1 X
∇(Tδ (unδ ) − Tδ (un−1 ))2 2
δ
L (Ω)
h n=1
≤ C.
In the same way, one obtains similar estimates
Z
TM
0
Z
+
0
TM
TM
Z
2
kUδN (t)kL2 (Ω) dt +
Z
2
k∂t UδN (t)kL2 (Ω) dt +
2
k∇UδN (t)kL2 (Ω) dt
0
TM
2
k∂t ∇UδN (t)kL2 (Ω) dt ≤ C.
0
Therefore, the sequences {TδN }N ∈N and {UδN }N ∈N are uniformly bounded in W 1,2 (0, TM ;
W 1,2 (Ω)), so there exist two sub-sequences (still denoted by {TδN } and {UδN } ) which
converge weakly to some Tδ∗ ∈ W 1,2 (0, TM ; W 1,2 (Ω)) and uδ ∈ W 1,2 (0, TM ; V ).
For any φ, ψ ∈ L2 (0, TM ; W01,2 (Ω)), (3.32)-(3.33) give
Z
TM
Z
Z
TM
Z
0
Ω
Z
TM
Z
0
Ω
Z
Z
∇θδ (ŪδN )∇φdxdt = 0,
0
TM
Z
Ω
Ω
Z
kδ (ŪδN )∇P̄δN ∇ψdxdt +
0
TM
knδ (ŪδN )∇P̄δN ∇φdxdt+
∂t UδN φdxdt−
kwδ (ŪδN )∇∂t TδN ∇ψdxdt = 0.
0
(3.58)
(3.59)
Ω
Clearly, TδN (t) → Tδ∗ strongly in L2 (Q). By Lemma 3.2 in [73], it follows that T̄δN (t) →
Tδ∗ strongly in L2 (Q) as well and a similar conclusion can be drawn for ŪδN and uδ . By the
continuity of kδ , kwδ , knδ , we also have knδ (ŪδN ) → knδ (uδ ), kwδ (ŪδN ) → kwδ (uδ ), and
kδ (ŪδN ) → kδ (uδ ). Now we show that ∂t ∇Tδ∗ = ∂t ∇Tδ (uδ ). Since T̄δN (t) = Tδ (unδ ) =
Tδ (ŪδN ) converges to Tδ∗ , by the definition of Tδ , we also find that Tδ (ŪδN ) → Tδ (uδ ),
then we have Tδ∗ = Tδ (uδ ).
Similarly, for P̄δN (t), we have a pδ such that
∇P̄δN * ∇pδ
weakly in L2 (0, T ; L2 (Ω)d ).
As in the proof of Lemma 3.3.1, we get
knδ (ŪδN )∇P̄δN * knδ (uδ )∇pδ
weakly in L2 (0, TM ; L2 (Ω)d ),
kδ (ŪδN )∇P̄δN * kδ (uδ )∇pδ
weakly in L2 (0, TM ; L2 (Ω)d ),
3.3. EXISTENCE, UNIQUENESS OF SOLUTIONS IN THE REGULARIZED CASE 35
kwδ (ŪδN )∇∂t TδN * kwδ (uδ )∇∂t Tδ (uδ )
0
0
θδ (ŪδN )∇ŪN * θδ (uδ )∇uδ
weakly in L2 (0, TM ; L2 (Ω)d ),
weakly in L2 (0, TM ; L2 (Ω)d ).
Combining the results, we obtain that (uδ , pδ ) is the solution pair of Problem Pδ .
3.3.5
Uniqueness of the weak solution for Problem Pδ
After having obtained the existence of a weak solution, we show its uniqueness. To do
so, we consider the following system:
∂t s − ∇ · (kwδ (s)∇pw ) = 0,
(3.60)
−∂t s − ∇ · (knδ (s)∇pn ) = 0,
(3.61)
pn − pw = pcδ (s) − pcδ (CD ) − ∂t Tδ (s).
(3.62)
Formally, this is equivalent to (3.27) - (3.28), in the sense that uδ = s and the global
pressure p is given by (3.9) or (3.10), often finding pw , pn in (3.60) - (3.62). Clearly, the
initial and boundary conditions should be compatible with the original ones:
s(0, ·) = u0
pw = pn = 0,
in Ω,
at ∂Ω, for all t > 0.
Furthermore, to prove the uniqueness, we also assume
A3: Ω is a C 0, domain for some 0 < ≤ 1.
A4: u0 ∈ C 0, (Ω̄).
A weak solution of (3.60)-(3.62) solves
Problem Pe : Given s(0, ·) = u0 , find pw ∈ L2 (0, TM ; W01,2 (Ω)), pn ∈ L2 (0, TM ; W01,2 (Ω))
and s ∈ L2 (0, TM ; L2 (Ω)), such that
(∂t s, φ) + (kwδ (s)∇pw , ∇φ) = 0,
(3.63)
−(∂t s, ψ) − (knδ (s)∇pn , ∇ψ) = 0,
(3.64)
(pn − pw , ρ) = (pcδ (s) − pcδ (CD ), ρ) − (∂t Tδ (s), ρ),
(3.65)
for all φ, ψ ∈ L2 (0, T ; W01,2 (Ω)) and ρ ∈ L2 (0, T ; L2 (Ω)).
The equivalence above can be made more precisely.
Lemma 3.3.5. Problem Pe and Problem Pδ are equivalent. Specifically, (uδ , pδ ) is a
solution to Problem Pδ if and only if (s, pw , pn ) solves Problem Pe , with s = uδ , pw =
Ru
Ru
0
0
pδ + CDδ fwδ (z)pcδ (z)dz −pc (uδ )+pcδ (CD )+τ (uδ )∂t uδ , pn = pδ + CDδ fwδ (z)pcδ (z)dz.
Proof. The proof follows the ideas in [52], see [88] for the underlying ideas.
36CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
Theorem 3.3.2. If assumptions A1-A4 are satisfied, then there exists at most one solution (uδ , pδ ) for Problem Pδ .
Proof. The uniqueness of a solution for Problem Pe is proved in [32]. Since Problems Pe
and Pδ are equivalent, this immediately concludes the proof.
3.4
Existence of weak solutions for Problem P
Below we obtain the existence result for the degenerate case. More precise, the nonlinear
functions kw , kn , pc should behave as stated below.
A5: There exist α > 0, β > 0, λ > 0, ω > 0, and for different dimensions, we assume
the following
if d = 3: α ≥ λ > α/3 + 10/3, ω > 5/2, e := ω + β > 5,
if d = 2 or d = 1: α ≥ λ > 4, ω > 2, e > 4.
With this, for some constants Cw , Cp , Cn , Cτ > 0 there hold:
lim kw (u) · u−α = Cw ,
u&0
0
(3.66)
lim pc (u) · uλ = −Cp ,
(3.67)
lim kn (u) · (1 − u)−β = Cn ,
(3.68)
lim τ (u) · (1 − u)ω = Cτ .
(3.69)
u&0
u%1
u%1
To avoid unnecessary technical complications, we restrict the proof to the cases kw =
0
u , kn = (1 − u)β , pc (u) = −uλ , and τ (u) = (1 − u)−ω . We note that this type of
behavior is commonly encountered in the porous media literature [12, 22, 61, 82]. Less
standard is the function τ . The structure considered here agrees with the one proposed
in [17, 18, 26, 94]. Then the regularized τ becomes


if u < δ,
 1,
−ω
τδ (u) =
(1 + δ − u) , if δ ≤ u < 1,

 δ −ω ,
if u ≥ 1.
α
More general cases, in particular for τ , but still respecting assumption A1, A2 can be
considered as well. Then all integrals appearing below should be decomposed into two
parts, one integral up to u∗ , and the second from u∗ to u. To avoid an excess of technical
calculation we restrict the proof to the case given above. However, in the general case
the existence proof would rely on the same ideas.
0
Since in (A1) we assume that the product kw · pc is uniformly bounded in R, this
implies α ≥ λ. According to the definition of τ and its regularization, one has T (u) and
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
Tδ (u) as follows:


 u+
1
ω−1 ,
1
ω−1 (1
1−ω
37
if u < 0,
, if 0 ≤ u < 1,
if u ≥ 1,
− u)

 +∞,

1

if u < δ,
 u + ω−1 − δ,
1
1−ω
Tδ (u) =
, if δ ≤ u < 1,
ω−1 (1 + δ − u)

 u−1 + 1
1
,
if u ≥ 1.
δω
ω−1 δ ω−1
T (u) =
Remark 3.4.1. Choosing CT = 1/(ω − 1) in (3.19) and CT δ = 1/(ω − 1) − δ in (3.26)
1
1
gives simple expressions for T (u) = ω−1
(1 − u)1−ω and Tδ (u) = ω−1
(1 + δ − u)1−ω
(u ∈ [δ, 1)). Nevertheless, the choice of CT and CT δ has no importance for the proof.
Here we define the characteristic function
(
1, if v ∈ (a, b),
χ(a,b) =
0, if v ∈
/ (a, b).
Then we have the following results
Lemma 3.4.1. Let suppose the hypotheses (A1), (A2) and (A5). Then there exists a
constant C > 0 independent of δ, such that for the first component of the weak solution
pair (uδ , pδ ) to Problem Pδ , one has
k(|uδ | + δ)2−α kL∞ (0,TM ;L1 (Ω)) ≤ C,
(3.70)
k(|1 − uδ | + δ)2−e kL∞ (0,TM ;L1 (Ω)) ≤ C,
(3.71)
k∇Tδ (uδ )kL∞ (0,TM ;L2 (Ω)) + k∇uδ kL∞ (0,TM ;L2 (Ω)) ≤ C,
q
−p0 τδ ∇uδ ≤ C.
cδ
(3.72)
(3.73)
L2 (0,TM ;L2 (Ω))
Proof. With t ∈ (0, TM ], taking the test function φ = χ(0,t)
Z uδ
τδ
and ψ = χ(0,t)
(z)dz in (3.30) gives
CD kwδ
Z
t
Z
uδ
(∂t uδ ,
0
CD
τδ kδ
(z)dz)dt −
kwδ knδ
Z
0
t
Z
0
t
τδ kδ
(∇pδ ,
∇uδ )dt +
kwδ
τδ kδ
(∇pδ ,
∇uδ )dt +
kwδ
Z
t
Z
uδ
CD
τδ kδ
(z)dz in (3.29)
kwδ knδ
2
Z t q
−p0 τδ ∇uδ dt = 0,
cδ
0
(3.74)
(∇∂t Tδ (uδ ), ∇Tδ (uδ ))dt = 0.
0
(3.75)
38CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
Adding (3.74) and (3.75) gives
2
Z
Z t q
τδ kδ
1 t d
0
∂t uδ
(z)dzdxdt+
−pcδ τδ ∇uδ dt+
k∇Tδ (uδ )k2 dt = 0.
k
k
2
dt
0
Ω
CD wδ nδ
0
0
(3.76)
By the definition of kδ , one has
Z
Z
Z
Z uδ
Z uδ
Z uδ
τδ kδ
τδ
τδ
∂t uδ
(z)dzdx =
∂t uδ
(z)dzdx +
∂t uδ
(z)dzdx.
k
k
k
k
Ω
CD wδ nδ
Ω
CD wδ
Ω
CD nδ
Z tZ
Z
uδ
The following identities hold a.e.
Z uδ
Z uδ
Z uδ
τδ
τδ
τδ
∂t uδ
(z)dz = ∂t uδ
(z)dz −
(z)dz ,
z
CD kwδ
CD kwδ
CD kwδ
Z
uδ
∂t uδ
CD
Z uδ
Z uδ
τδ
τδ
τδ
(z)dz = ∂t (uδ − 1)
(z)dz −
(z)dz .
(z − 1)
knδ
knδ
1
CD knδ
As in [77], we define the functions Ewδ , Enδ : R → R
Z
y
v
τδ
(CD + δ)2−α
(CD + δ)1−α
dzdv +
+
CD
(1 − α)(2 − α)
α−1
CD kwδ
Z y
(CD + δ)1−α
(CD + δ)2−α
τδ
τδ z
+
CD ,
dz −
dz +
kwδ
(1 − α)(2 − α)
α−1
CD kwδ
Z
Ewδ (y) =
CD
y
Z
=y
Z
CD
yZ v
Z
y
τδ
δ 2−e
dzdv +
(1 − e)(2 − e)
1
CD knδ
Z y
Z y
τδ
(z − 1)τδ
δ 2−e
=(y − 1)
dz −
dz +
,
knδ
(1 − e)(2 − e)
CD knδ
1
Enδ (y) =
and
Z
v
Ẽwδ (y) =
CD
CD
1
kwδ
(z)dzdv +
(CD + δ)2−α
(CD + δ)1−α
+
CD .
(1 − α)(2 − α)
α−1
(3.77)
The choice of these terms is justified by the following calculations. Recalling (A5), for
0 ≤ uδ ≤ 1 − δ we have
Z uδ Z v
1
(uδ + δ)2−α
(CD + δ)1−α
(z)dzdv =
+
uδ
(1 − α)(2 − α)
α−1
CD CD kwδ
−
(CD + δ)2−α
(CD + δ)1−α
−
CD .
(1 − α)(2 − α)
α−1
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
39
Similarly, for uδ < 0, one has
Z
uδ
Z
v
1
kwδ
CD
CD
(z)dzdv =
δ −α 2 (CD + δ)1−α − δ 1−α
δ 2−α
uδ +
uδ +
2
α−1
(1 − α)(2 − α)
−
(CD + δ)2−α
(CD + δ)1−α
−
CD ,
(1 − α)(2 − α)
α−1
and for uδ > 1 − δ, we get
Z
uδ
Z
CD
v
CD
1
kwδ
(z)dzdv =
(CD + δ)1−α − 1
1
(uδ − (1 − δ))2
+
uδ +
2
α−1
(1 − α)(2 − α)
+
1−δ
(CD + δ)2−α
(CD + δ)1−α
−
−
CD .
α − 1 (1 − α)(2 − α)
α−1
Note that the calculations above hold for the choice kw (u) = u−α . If, instead, kw behaves
like in (3.66), there the expressions on the right in the above are dominating terms, the
reminders being regular w.r.t. δ.
In this way Ẽwδ (uδ ) becomes
Ẽwδ (uδ ) =
(CD +δ)1−α −δ 1−α
δ −α 2
δ 2−α
uδ + (1−α)(2−α)
,
2 uδ +
α−1
(uδ +δ)2−α
(CD +δ)1−α
uδ ,
(1−α)(2−α) +
α−1
(CD +δ)1−α −1
(uδ −(1−δ))2
1
+
uδ + (1−α)(2−α)
2
α−1







for uδ < 0,
for 0 ≤ uδ ≤ 1 − δ,
+
1−δ
α−1 ,
for uδ > 1 − δ.
We note that
Ẽwδ (uδ ) ≥
0
Ẽwδ
(uδ )
:=







δ −α 2
δ 2−α
2 uδ + (1−α)(2−α) ,
2−α
(uδ +δ)
(1−α)(2−α) ,
(uδ −(1−δ))2
1
+ (1−α)(2−α)
,
2
for uδ < 0,
for 0 ≤ uδ ≤ 1 − δ,
for uδ > 1 − δ,
0
whereas Ẽwδ
(uδ ) ≥ C(|uδ | + δ)2−α (C > 0 independent of δ).
In the same way, for Enδ , since τ1δ · knδ = (1 − z + δ)e (see (A5)), for δ ≤ uδ ≤ 1,
one has
Z uδ Z v
τδ
(1 − uδ + δ)2−e
δ 2−e
dzdv =
−
(1 − e)(2 − e)
(1 − e)(2 − e)
1
CD knδ
+
(1 − CD + δ)1−e
(1 − uδ ).
e−1
Further, for uδ < δ, one gets
Z
1
uδ
Z
v
CD
1
1−δ
τδ
(uδ − δ)2
dzdv =
+
+
knδ
2
(1 − e)(2 − e) e − 1
+
(1 + δ − CD )1−e − 1
δ 2−e
(1 − uδ ) −
,
e−1
(1 − e)(2 − e)
40CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
and for uδ > 1, we have
Z uδ Z v
τδ
δ −e
δ 1−e
(1 + δ − CD )1−e
dzdv =
(uδ − 1)2 + (
−
)(uδ − 1).
2
e−1
e−1
CD knδ
1
Then Enδ (v) rewrites
Enδ (uδ ) =







(uδ −δ)2
(1+δ−CD )1−e −1
1
+ (1−e)(2−e)
+ 1−δ
(1 − uδ ),
2
e−1 +
e−1
(1−uδ +δ)2−e
(1−CD +δ)1−e
(1 − uδ ),
(1−e)(2−e) +
e−1
δ −e (uδ −1)2
δ 1−e −(1+δ−CD )1−e
δ 2−e
+(
)(uδ − 1) + (1−e)(2−e)
,
2
e−1
Enδ (uδ ) ≥
0
Enδ
(uδ )




:=
(uδ −δ)2
1
+ (1−e)(2−e)
+ 1−δ
2
e−1 ,
2−e
(1−uδ +δ)
(1−e)(2−e) ,
δ 2−e
δ −e
2
2 (uδ − 1) + (1−e)(2−e) ,



for uδ < δ,
for δ ≤ uδ ≤ 1,
for uδ > 1,
for uδ < δ,
for δ ≤ uδ ≤ 1,
(3.78)
for uδ > 1,
and
(3.79)
0
Enδ
(uδ ) ≥ C(|1 − uδ | + δ)2−e ,
with C > 0 independent of δ.
Z
Substitute Ewδ + Enδ into (3.76) instead of
uδ
Z
∂t uδ
CD
Ω
τδ
(z)dzdx, then we have
kwδ knδ
t q
1
0
Enδ (uδ (t))dx +
k −pcδ τδ ∇uδ k2 dt + k∇Tδ (uδ (t))k2
2
Ω
Z
Z0
1
0
0
=
Ewδ (u )dx +
(3.80)
Enδ (u )dx + k∇Tδ (u0 )k2 .
2
Ω
Ω
Z
Z
Ewδ (uδ (t))dx+
Ω
Z
As the proof of Lemma 3.3.3, one has
u0
Z
Γδ (u0 ) =
Z
CD
v
CD
τδ
(z)dzdv +
kwδ
Z
u0
CD
Z
v
CD
τδ
(z)dzdv ≤ C,
knδ
and
Z
0
Z
2
τδ2 |∇u0 |2 dx
|∇Tδ (u )| dx =
Ω
Ω
Z
2
≤
0 2
Z
τ |∇u | dx =
Ω
|∇T (u0 )|2 dx ≤ C.
Ω
These lead to
Z
0
Z Z
u0
Z
v
Ewδ (u )dx =
Ω
Ω
≤ C.
CD
CD
τδ
dzdvdx +
kwδ
Z
Ω
(CD + δ)2−α
(CD + δ)1−α CD
+
dx
(1 − α)(2 − α)
(α − 1)
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
41
Similarly, we also have
Z
Z Z
0
u0
v
Z
Enδ (u )dx =
Ω
CD
Ω
Z Z
u0
CD
v
Z
=
CD
Ω
Z
+
Ω
CD
τδ
dx −
knδ
Z Z
τδ
dx +
knδ
Z
Ω
1
Z
CD
v
CD
τδ
dx +
knδ
Z
Ω
δ 2−e
dx
(1 − e)(2 − e)
2−e
Ω
(1 − CD + δ)
dx
(1 − e)(2 − e)
(1 − CD + δ)1−e (1 − CD )
dx
(e − 1)
≤C.
Observe that Ewδ is convex, allowing a minimum at uδ = CD , then we have Ewδ ≥
(CD +δ)2−α
(CD +δ)1−α
CD > 0. And Enδ is also positive obtained from (3.78) and
(1−α)(2−α) +
α−1
(3.79). These give the estimates from (3.80):
Z
Z
Ewδ (uδ ) +
Enδ (uδ ) ≤ C.
Ω
Ω
Since τδ is far away from 0, then one has
Z
Ẽwδ (uδ ) ≤ C.
Ω
These lead to
Z
TM
q
0
k −pcδ τδ ∇uδ k2 dt ≤ C,
(3.81)
(|uδ | + δ)2−α dx ≤ C,
(3.82)
(|1 − uδ | + δ)2−e dx ≤ C,
(3.83)
0
Z
Ω
Z
Ω
k∇Tδ (uδ )k2 + k∇uδ k2 ≤ C.
Lemma 3.4.2. Under the assumptions (A1), (A2) and (A5), there exists a constant
C > 0 independent of δ, such that the weak solution pair (uδ , pδ ) of Problem Pδ satisfies:
Z
TM
Z
2
TM
k∂t uδ k dt +
0
kp
0
Z
TM
r
k
+
0
Z
0
TM
Z
p
2
k knδ ∇pδ k +
0
TM
1
τδ (uδ )
∂t Tδ (uδ )k2 dt
kwδ knδ
∇∂t Tδ (uδ )k2 dt ≤ C,
kδ
(3.84)
p
k kwδ ∇(pδ + ∂t Tδ (uδ ))k2 ≤ C.
(3.85)
42CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
Proof. Testing by ∂t Tδ (uδ ) both in (3.29) and (3.30), adding the resulting gives
Z
TM
Z
TM
(kwδ ∇pδ , ∇∂t Tδ (uδ ))dt
(∂t uδ , ∂t Tδ (uδ ))dt +
0
0
Z
TM
TM
Z
k
(∇θδ (uδ ), ∇∂t Tδ (uδ ))dt +
+
p
kwδ ∇∂t Tδ (uδ )k2 dt = 0.
(3.86)
0
0
Further, taking ψ = pδ in (3.30) gives
2
p
kwδ
p
k
∇p
·
,
kδ ∇pδ = −(kwδ ∇∂t Tδ (uδ ), ∇pδ ) ≤ √ ∇∂t Tδ (uδ )
δ
δ
kδ
implying
2
kwδ
−(kwδ ∇pδ , ∇∂t Tδ (uδ )) ≤ √ ∇∂t Tδ (uδ )
.
kδ
Then (3.86) becomes
TM
Z
Z
TM
p
2
kwδ ∇∂t Tδ (uδ ) dt
0
0
2
Z TM
Z TM kwδ
+
(∇θδ (uδ ), ∇∂t Tδ (uδ ))dt ≤
√k ∇∂t Tδ (uδ ) dt.
δ
0
0
(∂t uδ , ∂t Tδ (uδ ))dt +
(3.87)
Further, one has
2
2 r k k
p
2 kwδ
wδ
nδ
√
=
∇∂
T
(u
)
∇∂
T
(u
)
kwδ ∇∂t Tδ (uδ ) − t δ δ t δ δ ,
k
kδ
δ
and
r
2
1
kwδ knδ
|(∇θ(uδ ), ∇∂t Tδ (uδ ))| ≤ ∇∂t Tδ (uδ )
2
kδ
r
2
0
1
kwδ knδ
(−pcδ )∇uδ .
+ 2
kδ
(3.88)
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
43
Then (3.87) leads to
r
2
Z
1 TM kwδ knδ
(∂t uδ , ∂t Tδ (uδ ))dt +
∇∂t Tδ (uδ ) dt
2
k
δ
0
0
r
2
Z
1 TM kwδ knδ 0
≤
pcδ ∇uδ dt
−
2 0 kδ
2
r
Z
k q
0
1 TM
0
nδ
≤
|kwδ pcδ |∞ −pcδ τδ ∇uδ dt.
kδ τδ
2 0
Z
TM
(3.89)
q
2
0
0
By using pcδ τδ ∇uδ ≤ C, |kwδ pcδ |∞ ≤ C, and since by (A1), τ1δ is bounded, we
have
r
2
Z TM
Z
1 TM kwδ knδ
(∂t uδ , ∂t Tδ (uδ ))dt +
∇∂t Tδ (uδ ) dt ≤ C.
(3.90)
2 0 kδ
0
Then, by (3.88), this particularly implies
TM
Z
|(∇θ(uδ ), ∇∂t Tδ (uδ ))| ≤ C.
(3.91)
0
Clearly,
Z
TM
√
2
k τδ ∂t uδ k dt =
Z
0
0
TM
2
Z
1
√ ∂t Tδ (uδ ) dt =
τδ
TM
(∂t uδ , ∂t Tδ (uδ ))dt ≤ C.
0
Testing again (3.29) with φ = ∂t Tδ (uδ ), we have
Z
TM
Z
TM
(knδ ∇pδ , ∇∂t Tδ (uδ )) dt
(∂t uδ , ∂t Tδ (uδ ))dt−
0
0
Z
TM
(∇θδ (uδ ), ∇∂t Tδ (uδ )) dt = 0.
+
(3.92)
0
Choosing now ψ = pδ + ∂t Tδ (uδ ) in (3.30) gives
Z
TM
Z
TM
(knδ ∇pδ , ∇pδ )dt +
0
(knδ ∇pδ , ∇∂t Tδ (uδ ))dt
0
Z
TM
k
+
0
p
kwδ ∇(pδ + ∂t Tδ (uδ ))k2 dt = 0.
(3.93)
44CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
Adding the equations (3.92) and (3.93), and using (3.91), we find
Z
TM
√
k τδ ∂t uδ k2 dt+
Z
0
TM
TM
Z
p
k knδ ∇pδ k2 dt +
0
p
k kwδ ∇(pδ + ∂t Tδ (uδ ))k2 dt
0
Z
TM
(3.94)
|(∇θδ (uδ ), ∇∂t Tδ (uδ ))|dt ≤ C,
≤
0
which concludes the proof.
Furthermore, by (A1), from (3.73), one gets
TM
Z
0
Z
q
0
2
k −pcδ (uδ )∇uδ k dxdt +
TM
p
k τδ (uδ )∇uδ k2 dxdt ≤ C.
(3.95)
0
With the notation:
[uδ ]1−δ
= max{0, min{1 − δ, uδ }},
0
since ∇
Z
uδ p
−pcδ 0 (z)dz =
p
−pcδ 0 (uδ )∇uδ ∈ L2 (Q), recalling that (A5) implies λ > 2
0
for all d, and
Z
0
[uδ ]1−δ
0
(z + δ)−λ/2 dz =
1
(([uδ ]1−δ
+ δ)1−λ/2 − δ 1−λ/2 ),
0
1 − λ/2
we have
∇([uδ ]1−δ
+ δ)1−λ/2 ∈ L2 (Q),
0
and by (3.95) it is bounded uniformly w.r.t. δ. Further, since the trace of [uδ ]1−δ
+ δ on
0
1−λ/2 1−λ/2
1−δ
∂Ω is CD +δ, applying the Poincaré’s inequality for [uδ ]0 +δ
− CD +δ
,
one immediately obtains that
k([uδ ]1−δ
+ δ)1−λ/2 k2L2 (0,TM ;W 1,2 (Ω)) ≤ C,
0
for some δ-independent C. By Sobolev Embedding Theorem, one obtains
([uδ ]1−δ
+ δ)1−λ/2 ∈ L2 (0, TM ; C(Ω̄)),
0
([uδ ]1−δ
+ δ)1−λ/2 ∈ L2 (0, TM ; Lr (Ω)),
0
if d = 1,
for any r ∈ (1, +∞),
2d
([uδ ]1−δ
+ δ)1−λ/2 ∈ L2 (0, TM ; L d−2 (Ω)),
0
if d = 2,
if d > 2,
and the respective norms are bounded uniformly w.r.t. δ.
Similarly, for
[uδ ]1δ := max{δ, min{1, uδ }},
one has
(1 − [uδ ]1δ + δ)1−ω/2 ∈ L2 (0, TM ; C(Ω̄)),
if d = 1,
(3.96)
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
45
for any r ∈ (1, +∞),
(1 − [uδ ]1δ + δ)1−ω/2 ∈ L2 (0, TM ; Lr (Ω)),
(1 − [uδ ]1δ + δ)1−ω/2 ∈ L2 (0, TM ; L
2d
d−2
if d = 2,
if d > 2.
(Ω)),
(3.97)
−γw
Lemma 3.4.3. For γw , γn , γτ chosen appropriately, the functions kwδ ([uδ ]1−δ
)
,
0
−γn
γτ
knδ ([uδ ]1δ )
, and τδ ([uδ ]1δ )
are in L1 (Q) and have uniformly bounded norms
w.r.t. δ.
Proof. We detail the proof for knδ and τδ , the arguments for kwδ being identical to those
in [77]. To do so, we consider the cases d = 1, 2, and 3 separately (d > 3 being similar
5ω
10
to d = 3). We start with the case d = 3, and choose γn = β1 ( 2β
3 + 3 − 3 ). By (A5),
3
one gets γn > 1. Applying Hölder inequality, for p = 3, q = 2 , one gets for a.e. t:
Z
(1 −
[uδ ]1δ
+ δ)
−γn β
Ω
Z
dxdt =
(1 −
+ δ)
dx dt
Ω
Z
1
(2−e)·2/3
·
(1 − [uδ ]δ + δ)
dx
[uδ ]1δ
(1−ω/2)2
Ω
1/3
(1 − [uδ ]1δ + δ)(1−ω/2)6 dx
dt
Z
≤
Ω
Z
·
(1 −
[uδ ]1δ
+ δ)
2−e
2/3
dx
.
Ω
Due to (3.71) and (3.97), we have (1 − [uδ ]1δ + δ)2−e ∈ L∞ (0, TM ; L1 (Ω)) and (1 −
[uδ ]1δ + δ)1−ω/2 ∈ L2 (0, TM ; L6 (Ω)), and the norms are bounded uniformly w.r.t. δ.
This implies:
Z
TM
Z
(1 −
0
[uδ ]1δ
+ δ)
−γn β
Z
TM
Z
dxdt ≤
(1 −
Ω
0
(1−ω/2)6
+ δ)
1/3
dx
dt
Ω
·
[uδ ]1δ
max
0≤t≤TM
Z
(1 −
[uδ ]1δ
2−e
+ δ)
2/3
dx
Ω
≤C.
With γτ =
1 2β
ω( 3
+
5ω
3
−
10
3 ),
γτ
the estimate for τδ ([uδ ]1δ )
follows similarly.
For d = 2, we choose any r > max{(2α − 4)/λ, (e − 2)/(ω − 2), (2e − 4)/(e − 4)} and
define
1
2
4
γn = −
4 − − ω − e(1 − ) ,
β
r
r
respectively,
1
γτ = −
ω
4
2
4 − − ω − e(1 − ) ,
r
r
46CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
and we apply Hölder inequality for p =
Z
and q =
r
2
(1 − [uδ ]1δ + δ)−γn β dxdt ≤
to obtain for a.e. t
2/r
dt
(1 − [uδ ]1δ + δ)(1−ω/2)r dx
Z
Ω
r
r−2
Ω
Z
·
(1 −
[uδ ]1δ
2−e
+ δ)
r−2/r
,
dx
Ω
and
Z
(1 − [uδ ]1δ + δ)−γτ ω dxdt ≤
2/r
(1 − [uδ ]1δ + δ)(1−ω/2)r dx
dt
Z
Ω
Ω
Z
·
(1 −
[uδ ]1δ
2−e
+ δ)
r−2/r
dx
.
Ω
Then, the proof continues as before.
Finally, for d = 1, we take γn = (ω + e − 4)/β and γτ = (ω + e − 4)/ω. Similarly, by
assumption (A5), we have γn , γτ > 1. Then, use (3.71), (3.96) to estimate for a.e. t
Z
Z
1
−γn β
(1 − [uδ ]δ + δ)
dx = (1 − [uδ ]1δ + δ)(1−ω/2)2 (1 − [uδ ]1δ + δ)2−e dx
Ω
Ω
2
≤ max(1 − [uδ ]1δ + δ)1−ω/2
Ω̄
Z
· (1 − [uδ ]1δ + δ)2−e dx ∈ L1 (0, TM ),
Ω
and
Z
Ω
(1 − [uδ ]1δ + δ)−γτ ω dx =
Z
(1 − [uδ ]1δ + δ)(1−ω/2)2 (1 − [uδ ]1δ + δ)2−e dx
Ω
2
≤ max(1 − [uδ ]1δ + δ)1−ω/2
Ω̄
Z
· (1 − [uδ ]1δ + δ)2−e dx ∈ L1 (0, TM ),
Ω
and the proof follows again as before.
Now we obtain further estimates for (uδ , pδ ).
Lemma 3.4.4. Let d = 1, 2, or 3 and assume (A1), (A2) and (A5). There exist
r1 , r2 , r3 ∈ (1, 2) and C > 0 independent of δ, such that the weak solution pair (uδ , pδ )
satisfies for all δ > 0
k∂t Tδ (uδ )kLr1 (Q) + k∇pδ kLr2 (Q) + k∇(pδ + ∂t Tδ (uδ ))kLr3 (Q) ≤ C.
Proof. The proof uses the estimates in Lemma 3.4.3 and distinguishes as before three
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
47
cases, d = 1, 2, and 3. We start with the latter.
Case 1: d =3
By (3.84), one has √1τδ ∂t Tδ (uδ ) ∈ L2 (Q). So here we show for any r1 ∈ (1, 2), one
has ∂t Tδ (uδ ) ∈ Lr1 (Q). Moreover, for appropriately chosen r1 , the corresponding norm
is bounded uniformly w.r.t. δ. To see this, we apply Hölder inequality to get
TM
Z
Z
Ω
0
TM
Z
Z
1
r /2
| √ ∂t Tδ (uδ )|r1 τδ 1 dxdt
τδ
Ω
0
!r1 /2
Z TM Z
1
≤
|∂t Tδ (uδ )|2 dxdt
0
Ω τδ
!1−r1 /2
Z
Z
|∂t Tδ (uδ )|r1 dxdt =
TM
r /(2−r1 )
·
0
Ω
τδ 1
dxdt
.
The first integral on the right hand side is bounded by (3.84), the second we recall
r1
5ω
10
= γτ = ω1 ( 2β
Lemma 3.4.3 and choose r1 such that 2−r
3 + 3 − 3 ), which, by (A5),
1
r1 = 2β+5ω−10)
3β+4ω−5 , satisfies r1 ∈ (1, 2).
Similarly, one also has the estimate
Z
TM
Z
TM
Z
r2
Z
|∇pδ | dxdt =
0
Ω
0
Ω
TM
Z
p
−r /2
| knδ ∇pδ |r2 knδ 2 dxdt
!r2 /2
Z
2
≤
knδ |∇pδ | dxdt
0
Ω
TM
Z
!1−r2 /2
Z
·
0
We obtain
r2 =
Ω
−r /(2−r2 )
knδ 2
dxdt
.
2(3ω + 2e − 10)
,
(5e − 10)
and following (A5), one has r2 ∈ (1, 2).
Similarly, one has
Z
0
TM
Z
|∇(pδ + ∂t Tδ (uδ ))|r3 dxdt =
TM
Z
Ω
Z
0
p
p
| kwδ ∇(pδ + ∂t Tδ (uδ ))|r3 ( kwδ )−r3 /2 dxdt
Ω
Z
TM
!r3 /2
Z
2
≤
kwδ |∇∂t (pδ + Tδ (uδ ))| dxdt
0
Ω
Z
TM
Z
·
(kwδ )
0
≤C,
!1−r3 /2
−r3 /(2−r3 )
Ω
dxdt
48CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
then for this case, we have
r3 =
2(3λ + 2α − 10)
∈ (1, 2),
3λ + 5α − 10
when α > 5 and λ > 10/3 + α/3, this implies ∇(pδ + ∂t Tδ (uδ )) ∈ Lr3 (Q).
Case 2: d =2
Similarly, for two dimensional case, we show for any r1 ∈ (1, 2), one has ∂t Tδ (uδ ) ∈
Lr1 (Q) and the corresponding norm is bounded uniformly w.r.t. δ. Since one has
Z
TM
Z
0
Z
|∂t Tδ (uδ )|r1 dxdt ≤
Ω
TM
0
Z
!r1 /2
1
|∂t Tδ (uδ )|2 dxdt
Ω τδ
!1−r1 /2
Z
r1 /(2−r1 )
τδ
dxdt
.
Z
TM
·
0
Ω
Then, for any r > 2(e − 2)/(2ω + e − 4), we solve
r1 =
2(ω + e − 2(e − 2)/r − 4)
∈ (1, 2),
(2ω + e − 2(e − 2)/r − 4)
which implies ∂t Tδ (uδ ) ∈ Lr1 (Q).
Using the same way, we also get ∇pδ ∈ Lr2 (Q) for
r2 =
2(ω + e − 2(e − 2)/r − 4)
,
(ω + e + β − 2(e − 2)/r − 4)
and ∇(pδ + ∂t Tδ (uδ )) ∈ Lr3 (Q) for
r3 =
2(4 − λ − α + 2(α − 2)/r)
.
(4 − λ − 2α + 2(α − 2)/r)
Case 3: d =1
The proof follows as before, we have ∂t Tδ (uδ ) ∈ Lr1 (Q) for
r1 =
2(ω + e − 4)
∈ (1, 2),
(2ω + e − 4)
∇pδ ∈ Lr2 (Q), for
r2 =
2(ω + e − 4)
∈ (1, 2),
(ω + e + β − 4)
and furthermore, ∇(pδ + ∂t Tδ (uδ )) ∈ Lr3 (Q) for
r3 =
2(4 − λ − α)
∈ (1, 2).
(4 − λ − 2α)
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
49
Then we have the estimates
k∂t Tδ (uδ )kLr1 (Q) ≤ C,
k∇pδ kLr2 (Q) ≤ C,
with r1 ∈ (1, 2),
with r2 ∈ (1, 2),
with r3 ∈ (1, 2).
k∇(pδ + ∂t Tδ (uδ ))kLr3 (Q) ≤ C,
(3.98)
(3.99)
(3.100)
With r∗ = min{r1 , r2 , r3 }, by Lemmas 3.4.1, 3.4.2 and 3.4.4, one obtains the existence of a subsequence δ & 0 (still denoted by δ) and of u ∈ W 1,2 (Q), T ∗ ∈
∗
∗
∗
W 1,r (0, TM ; W 1,r (Ω)) and p ∈ L2 (0, TM ; W 1,r (Ω)), such that
uδ −→ u strongly in L2 (Q),
(3.101)
∂t uδ * ∂t u weakly in L (Q),
(3.102)
∇uδ * ∇u weakly in L (Q),
(3.103)
2
2
T̃δ * T
weakly in W
∗
T̃δ −→ T
∗
∇∂t T̃δ * ∇∂t T
(Q),
(3.104)
strongly in L (Q),
(3.105)
q
∗
dr ∗
d−r ∗ ,
r∗
(3.106)
(Q),
(3.107)
weakly in L (Q),
pδ * p weakly in W
where q = +∞, if d = 1, q =
1,r ∗
1,r ∗
if d = 2 or d = 3 (see [48]).
In the remaining, we prove that T ∗ = T (u) a.e., and that (u, p) is a solution pair
to Problem P. But before doing so, we also prove that the limit u above is essentially
bounded by 0 and 1.
Theorem 3.4.1. The limit u ∈ W 1,2 (Q) satisfies 0 ≤ u ≤ 1 a.e. in Q.
Proof. Given t ∈ (0, TM ], let Ω−
δ, (t) be the support of [uδ (t, ·) + ]− (the negative cut
of uδ (t, ·) + ). As follows from Lemma 3.4.1 , a C > 0 exists such that, for all δ > 0,
one has
Z
Z Z uδ Z v
1
(z)dzdvdx
Ẽwδ (uδ )dx =
Ω
Ω CD CD kwδ
Z
(CD + δ)2−α
(CD + δ)1−α
+
+
CD dx
(α − 1)
Ω (1 − α)(2 − α)
≤C,
Z
Z Z
uδ
Z
v
Enδ (uδ )dx =
Ω
Ω
1
CD
τδ
(z)dzdvdx +
knδ
Z
Ω
δ 2−e
dx ≤ C.
(1 − e)(2 − e)
50CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
Since the constant arguments in the last two integrals are positive, this gives
Z Z uδ Z v
Z Z yZ v
1
τδ
(z)dzdvdx +
dzdvdx ≤ C.
k
k
Ω CD CD wδ
CD nδ
Ω 1
Then we get
Z Z
uδ
Z
v
C≥
Ω
0
−α
CD
1
kwδ
(z)dzdvdx
(CD + δ)1−α − δ 1−α
δ 2−α
uδ +
dx
(α − 1)
(1 − α)(2 − α)
Ω 2
Z
δ −α 2 δ 1−α − (CD + δ)1−α
δ 2−α
≥
+
+
dx.
2
(α − 1)
(1 − α)(2 − α)
Ω−
δ, (t)
Z
=
δ
u2δ +
Let now δ & 0, this immediately implies that
meas(Ω−
0, (t)) = 0,
−
with Ω−
0, (t) having the same definition as Ωδ, (t), but now for the function u. Since
uδ → u in C((0, TM ); L2 (Ω)) (by (3.102), (3.103) and the compact embedding see [2],
Theorem 4.12), thus uδ → u a.e. in Ω, forZall Zt. This
holds for every > 0, hence u ≥ 0.
yZ v
τδ
Similarly, if uδ > 1 + , use the bounds on
dzdv dx, we obtain u ≤ 1.
k
Ω 1
CD nδ
Remark 3.4.2. For two phase flow model, 0 ≤ u ≤ 1 means that the saturation remains
in the physical range. Note that this only holds due to the degeneracy encountered for
u = 0 or u = 1.
Finally, we obtain the existence of a solution for Problem P.
Theorem 3.4.2. Let assumptions (A1), (A2) and (A5) be satisfied, then there exists a
solution pair (u, p) for Problem P.
Proof. We start by identifying T ∗ as T (u). To do so, define [v]1− = min{1, v} and let
T (f ) = T ∗ a.e. We proceed as in [81] and consider first the inverse function of T −1 .
According to the definition of T and T̃δ , one has

1
1
 T ∗ − ω−1
,
if T ∗ < ω−1
,
1
ω−1
(3.108)
f = T −1 (T ∗ ) =
1
1
∗
 1−
, if T ≥ ω−1 .
(ω−1)T ∗
Clearly,
[uδ ]1−

1
 T̃δ − ω−1
+ δ,
1
ω−1
= Tδ−1 (T̃δ ) =
1
 1−
+ δ,
(ω−1)T̃
δ
if T̃δ <
1
ω−1 ,
if T̃δ ≥
1
ω−1 .
(3.109)
3.4. EXISTENCE OF WEAK SOLUTIONS FOR PROBLEM P
51
Now we prove that [uδ ]1− → f strongly in L2 (Q), and hence a.e. in Q. Since T −1 (·) is
Lipschitz continuous, by (3.109) one has
Z
Z
|f − [uδ ]1− |dxdt =
|T −1 (T ∗ ) − T −1 (T̃δ ) + T −1 (T̃δ ) − Tδ−1 (T̃δ )|dxdt
Q
Q
Z
Z
≤C
|T ∗ − T̃δ |dxdt +
|T −1 (T̃δ ) − Tδ−1 (T̃δ )|dxdt.
Q
Q
Clearly, the first integral above approaches 0 as δ & 0. For the second we note that, by
(3.108) and (3.109), one has Tδ−1 (T̃δ ) − T −1 (T̃δ ) = δ, for any argument T̃δ . With this,
the second integral also approaches 0 as δ & 0. This means that
[uδ ]1− → f
a.e. in Q.
However uδ → u a.e. in Q (by the strong convergence in L2 (Q)). This immediately gives
[uδ ]1− → [u]1− a.e. In the view of Theorem 3.4.1, we also have [u]1− = u a.e. Therefore,
u = f a.e., and consequently, we have
T (u) = T ∗ .
Having identified T ∗ by T (u), (3.106) gives ∇∂t Tδ (uδ ) * ∇∂t T (u).
Then, according to knδ (·) Lipschitz continuous, one has
knδ (uδ ) → knδ (u)
a.e. in Q.
Further, since knδ (u) converges pointwise to kn (u), and
|knδ (u)| ≤ C,
uniformly with w.r.t. δ. Then by Dominated Convergence Theorem, we obtain
knδ (u) → kn (u) a.e. in Q.
Therefore, we have
knδ (uδ ) → kn (u) strongly in
L2 (Q).
p
p
Similarly, we also have knδ (uδ ) converges to kn (u) strongly in L2 (Q). In the same
fashion, since kw , k and θ are Lipschitz continuous, on also obtains
kwδ (uδ ) → kw (u),
kδ (uδ ) → k(u),
θδ (uδ ) → θ(u),
52CHAPTER 3. EXISTENCE OF WEAK SOLUTIONS TO THE TWO-PHASE MODEL
strongly in L2 (Q).
By the (3.85), a g ∈ L2 (Q) exists, such that
weakly in L2 (Q),
p
kwδ (uδ )∇(pδ + ∂t Tδ (uδ )) * g
for δ & 0.
p
p
To identify g, we consider φ ∈ C0∞ (Ω) arbitrarily, and note that φ kwδ (uδ ) → φ kw (u)
strongly in Lq (Q), for any q ∈ [1, ∞). This follows as above, using the uniform boundedness of kwδ , kw and the pointwise convergence of kwδ (uδ ) to kw (u). Further, by (3.106)
∗
and (3.107), one gets ∇(pδ + ∂t Tδ (uδ )) → ∇(p + ∂t T (u)) weakly in Lr (Q). Taking q
such that 1/q + 1/r∗ = 1 gives by weak-strong convergence argument
Z
TM
Z
(kwδ ∇(pδ + ∂t Tδ (uδ )), ∇φ)dt →
0
TM
(kw ∇(p + ∂t T (u)), ∇φ)dt.
(3.110)
0
p
This is sufficient to identify g = kw (u)∇(p + ∂t T (u)).
We have now all the ingredients to pass to the limit (δ & 0) in the integrals appearing
in Problem Pδ . Using the convergence results above, it is straightforward to show that
(u, p) solves Problem P.
3.5
Conclusions
In this chapter, we have proved the existence of a weak solution to degenerate elliptic
parabolic system modeling two-phase flow in porous media, and including dynamic effects
in the capillary pressure. The major difficulty is the degeneracy of the non-linear third
order derivative term. We get the estimate for the third order derivative term by applying the structures of relative permeabilities, capillary pressure and the dynamic damping
factor. By compactness arguments, we show the existence of a solution for the original
problem.
Chapter 4
Finite volume scheme
4.1
Introduction
In this chapter, we define and analyze a finite volume method for a two phase flow model
in a porous medium:
∂t u − ∇ · (ko (u)∇p̄) = 0,
(4.1)
∂t (1 − u) − ∇ · (kw (u)∇p) = 0,
(4.2)
p̄ − p = pc (u) + τ ∂t u,
(4.3)
which are defined in Q := Ω × (0, T ], where Ω is a bounded subset of R2 , T is a given
maximal time. The unknowns u, p̄ and p are the non-wetting phase saturation, the nonwetting phase and wetting phase pressures. The equations (4.1), (4.2) are obtained by
combing the mass balance and the Darcy’s law (see [12, 61, 82]). The permeabilities
ko (·), kw (·) for non-wetting phase and wetting phase are given monotone functions. The
gravity is neglected in the model. In order to close the above system, we prescribe the
initial and boundary conditions
u(0, ·) = u0 ,
p̄ = p = 0,
in Ω,
at ∂Ω for
(4.4)
t > 0,
(4.5)
where u0 is a given function, which will be specified later.
Equation (4.3) expresses the phase pressure difference p̄ − p, as a function of u and
∂t u. In classical models (see [12, 61, 69]), one assumes
p̄ − p = pc (u),
This chapter is a collaborative work with S.F. Nemadjieu, I.S. Pop and has been submitted to SIAM
Journal of Numerical Analysis.
53
54
CHAPTER 4. FINITE VOLUME SCHEME
where pc , the capillary pressure is a monotone function of saturation u. This however,
holds only if measurements are obtained under equilibrium condition. Experiments (see
[17, 38]) have invalidated this assumption, whenever flow is more rapid. One possible
extension is (4.3) as proposed in [60], where τ is a positive damping factor (τ > 0).
Standard models, obtained for τ = 0 have been intensively investigated in the reservoir
simulation. In this sense, existence and uniqueness of the weak solutions are proved
in [69] , but assuming that initial and boundary conditions bounded away from 0. This
has been extended to the case of arbitrarily chosen saturation for initial and boundary
conditions. The existence can be found in [6, 35] and the uniqueness of weak solution
was proved in [35]. For numerical schemes, we refer to ( [7, 21, 23, 39, 47, 49, 83, 88,
89]), where finite element method, mixed finite element method, discontinuous Galerkin
method are analyzed, or linear iterative schemes are investigated. In particular, for finite
volume schemes, we refer to [29, 51, 76]. Whenever τ > 0, (4.1) - (4.3) becomes a
so-called non-equilibrium model. In this case, the existence and uniqueness of a weak
solution is obtained in [28, 30, 53, 77], but in a simplified context when the total flow
is assumed to be known. This allows reducing one equation in (4.1) - (4.3). In this
case, but in the heterogeneous case, if no entry pressure presents, numerical schemes are
discussed in [63]. Also, variational inequality approaches have been considered in [62]
for situations including an entry pressure. Further, we refer to [41] which gives the
coupling conditions analysis. In [85], they consider numerical algorithms for unsaturated
flow in highly heterogeneous media for this model. For the full model, the existence and
uniqueness of the weak solutions are proved in [32, 68], but assuming that the equations
are non-degenerate (i.e. all non-linearities are bounded away from 0 or +∞). The authors
in [66] present discontinuous Galerkin scheme for this case. The authors have given the
numerical investigations in [57] in heterogeneous case. In the degenerate case, we refer
to [31], which gives the existence of weak solutions for the model in a equivalent form.
In this chapter, we show that the approximate solution of the system (4.1) - (4.3)
obtained by a multi-point flux approximation finite volume scheme converges to its weak
solution. The rest of the chapter is organized as follows. In Section 4.2, we give the
assumptions on the data and the define the weak solution. We introduce the finite volume
scheme in Section 4.3, and show the existence of the numerical solution. In Section 4.4,
we prove the convergence of the scheme by compactness arguments. In the last section,
we present some numerical results that confirm the theoretically obtained convergence.
4.2
The weak solution
To investigate the system (4.1) - (4.5), we make the following assumptions
• (A1) Ω is an open, bounded and connected polygonal domain in R2 with Lipschitz
continuous boundary ∂Ω. Ω̄ denotes the closure of Ω.
• (A2) The functions ko and kw : R → R are C 1 , and there exists δ > 0 such that
4.2. THE WEAK SOLUTION
55
δ ≤ ko (u), kw (u) ≤ 1 for all u ∈ R. We assume ko to be an increasing function
with ko (u) = δ for u ≤ 0 and ko (u) = 1 for u ≥ 1. Also kw will be considered to
be a decreasing function with kw (u) = 1 for u ≤ 0 and kw (u) = δ for u ≥ 1.
• (A3) pc : R → R is an increasing function of u, pc ∈ C 1 , pc (0) = 0 and there exist
0
mp , Mp > 0 such that mp ≤ pc (u) ≤ Mp < ∞.
• (A4) τ > 0 is a positive constant.
• (A5) The initial condition u0 is in C 1 (Ω) ∩ W01,2 (Ω).
Remark 4.2.1. It is not necessary to take pc (0) = 0. We just expect to obtain a consistent
boundary condition for u|∂Ω = 0. If pc (0) 6= 0, one can impose p̄|∂Ω = pc (0) or define a
’new non-wetting phase pressure’ p̄ := p − pc (0) + pc (u) + ∂t u to make sure that u = 0
at the boundary (see [52]). Furthermore, the proofs here can be extended easily to other
types of boundary conditions like non-homogeneous Dirichlet or Neumann.
Remark 4.2.2. The choice of u0 ∈ C(Ω̄) is for the ease of presentation, since the
proposed discretization of the gradients involves continuous functions. While these are
available pointwise approximations due to the spaces where these are sought, taking
u0 ∈ W 1,2 (Ω)\C 0 (Ω̄) would not be sufficient to define its discrete gradient, This is
however, needed in the proof, but not for the scheme itself. If u0 is not continuous, then
one may take into convolution with a grid-size dependent mollifier.
Furthermore, we define Pc as
Z
Pc (u) =
u
pc (s)ds.
(4.6)
0
Clearly, by (A3), Pc is convex and for all u ∈ R
Pc (u) ≥ 0,
with Pc (0) = 0.
(4.7)
Also one has
pc (a)(a − b) ≤ Pc (a) − Pc (b) for all a, b ∈ R.
(4.8)
In the following, we define the solution for the system (4.1) - (4.5):
Definition 4.2.1. (u, p̄, p) is a weak solution of the model (4.1) - (4.5) if u ∈ W 1,2 (0, T ;
L2 (Ω)), p̄, p ∈ L2 (0, T ; W01,2 (Ω)), and for any φ, ψ ∈ L2 (0, T ; W01,2 (Ω)), λ ∈ L2 (0, T ;
L2 (Ω)) there hold
(∂t u, φ) + (ko ∇p̄, ∇φ) = 0,
(4.9)
−(∂t u, ψ) + (kw ∇p, ∇ψ) = 0,
(4.10)
(p̄ − p, λ) = (pc (u), λ) + τ (∂t u, λ).
(4.11)
As mentioned before, existence and uniqueness results can be found in [31, 32, 68].
Note that by (A3) we obtain u ∈ W 1,2 (0, T ; W01,2 (Ω)) (see [52]).
56
CHAPTER 4. FINITE VOLUME SCHEME
4.3
The finite volume scheme
In this section, we introduce a finite volume scheme for the system (4.1) - (4.5), then
give the a priori estimates.
4.3.1
Meshes and notations
To introduce the finite volume scheme to the system, we consider an admissible mesh
(see [50] pp. 38).
Definition 4.3.1. Let Ω be an open bounded polygonal subset of R2 . An admissible
finite volume mesh of Ω, denoted by T is a family of triangular disjoint subsets of Ω such
that two triangles may either be disjoint, or share a node, or a full edge. The set of all
edges including the boundary ones is denoted by E. The geometric centers of the triangles
form the set P. In other words:
• The closure of the union of all the triangles is Ω̄;
• For any K ∈ T , there exists a subset EK of E such that ∂K = K̄ \ K = ∪σ∈EK σ̄.
Furthermore, E = ∪K∈T EK .
• For any (K, L) ∈ T 2 with K 6= L, either the 1-dimension Lebesgue measure of
P
K̄ ∩ L̄ is 0 or K̄ ∩ L̄ =
σ̄ for some σ ∈ E. There exists a subset EK of E such
that ∂K = K̄ \ K = ∪σ∈EK σ̄.
• The family P = {xK }K∈T is such that xK ∈ K (for all K ∈ T ) and it is the
geometric center of the volume K.
Further, we assume:
p
• (A6) mp , Mp satisfy 4 mp Mp ≥ mp + Mp . The angles θ of any triangle K ∈ T
√
√
2 mp Mp
2 mp Mp
satisfy arccos( mp +Mp ≤ θ ≤ π − arccos( mp +Mp ).
Remark 4.3.1. If pc (·) is a linear function with respect to u, the assumption (A6) can
be relaxed to 0 < θ < π, which is practically fulfilled by any triangular mesh.
Throughout this chapter, the following notations are used: the mesh size is defined by
size(T ) = sup{diam(K), K ∈ T }. The sets of interior and boundary edges are denoted
by Eint , resp. Eext : Eint = {σ ∈ E; σ 6⊂ ∂Ω}, and Eext = {σ ∈ E; σ ⊂ ∂Ω}. For a triangle,
we denote by m(K) its measure. We introduce some notations for the triangle K ∈ T
(see Figure 4.1). Pi , Pj , Pk denote the vertices of the triangle K, xK is the geometric
center of K, Pi,j , Pj,k , Pk,i are the midpoints of the segments Pi Pj , Pj Pk , Pk Pi . Pi/2,j
is the point on Pi Pj which satisfies m(Pi Pi/2,j )/m(Pi Pj ) = 1/3, similar to Pi/2,k . We
let Pv stand for the set of all vertices Pi , PM for all edges midpoint, PT for all points
Pi/2,j introduced above. We use Kr (r = i, j, k) to denote the quadrilateral determined
1
by Pr , xK and the midpoints Pr,· , P·,r of the edges. Let σK
denote the segment Pi Pi,j ,
i
4.3. THE FINITE VOLUME SCHEME
57
−−−−−→
−−−−−−→
2
1
2
σK
denote the segment Pi Pk,i . Then we denote eσK
= xK Pi/2,j , eσK
= xK Pi/2,k as
i
i
i
1
2
the vectors. Let nσK
and nσK
be the normal vectors to Pi Pj and Pk Pi outward to Ki .
i
i
1 , µσ 2
1
Finally, we define two vectors µσK
as following (see [72, 80]). Observe that µσK
Ki
i
i
1 , respectively µσ 2
2
and nσK
and
n
are
parallel.
σK
K
i
i
i

1
1

µσK
· eσK
= 1,


i
i


µ 1 · e 2 = 0,
σK
σK
i
(4.12)
i

1
µσ2 · eσK
= 0,


i

 Ki

2
2
µσK
· eσK
= 1.
i
i
Figure 4.1 A triangular finite volume and the associated nodes, edges and vectors.
4.3.2
The scheme
To define the scheme, some notations are needed.
Definition 4.3.2. Let Ω be an open bounded polygonal subset of R2 , T be an admissible
T
mesh in Section 4.3.1. h = N
denotes the time step for any N ∈ N and tn denotes the
time at t = nh for n ∈ {0, ..., N }. Let X(T , h) be the set of functions that are piecewise
constant in both time and space, i.e. v from Ω × (0, N h) to R such that there exists a
n
n
family of real values {vK
, K ∈ T , n ∈ {0, ..., N }}, with v(x, t) = vK
for a.e. x ∈ K,
K ∈ T and for a.e. t ∈ (nh, (n + 1)h], n ∈ {0, ..., N − 1}.
Further, for considering discrete gradients, additional values at edges σ will be needed.
To give the full discretization for the system (4.1) - (4.5), we use {unK , K ∈ T , n ∈
{0, ..., N }} to denote the discrete approximation of u, the value unK is the approximation
of u(xK , nh), and the same for pnK , p̄nK . For given K ∈ T and with Pr being one
of its nodes, r ∈ {i, j, k} a counterclockwise ordering, unσ1 , p̄nσ1 , pnσ1 denote the
Kr
Kr
Kr
58
CHAPTER 4. FINITE VOLUME SCHEME
approximations of u(xPr/2,r+1 , tn ), p̄(xPr/2,r+1 , tn ), p(xPr/2,r+1 , tn ) and it’s similar for
unσ2 , p̄nσ2 , pnσ2 .
Kr
Kr
Kr
Observe that, due to (4.12), given a vector v ∈ R2 one has
(4.13)
2 ) µσ 2 .
1 ) µσ 1
+ (v · eσK
v = (v · eσK
K
K
r
r
r
r
This inspires the definition of discrete gradient: for K ∈ T and r ∈ {i, j, k}, let the values
1 , vσ 2
be given, the discrete gradient in the quadrilateral Kr is
vK , vσ K
K
r
r
(4.14)
1
1
2
2 .
∇Kr vK := (vσK
− vK ) · µσK
+ (vσK
− vK ) · µσK
r
r
r
r
Then for any n = 0, 1, ..., N − 1, we define the scheme as follows
n+1
uK
− unK
(4.15)
h
X
n+1
n+1
n+1
n+1
1
1
2
1
=ko (un+1
m(σK
)
(p̄
−
p̄
)
µ
+
(p̄
−
p̄
)
µ
· nσK
1
2
σ
σ
K )
K
K
r
σ
σ
K
K
m(K)
r
Kr
r=i,j,k
r
Kr
r
n+1
n+1
n+1
2
1
2
2
+ m(σK
) (p̄n+1
−
p̄
)
µ
+
(p̄
−
p̄
)
µ
· nσK
1
2
σ
σ
K
K
r
σ
σ
K
K
r
Kr
r
Kr
!
,
r
un+1
− unK
K
(4.16)
h
X
1
1
2
1
=kw (un+1
m(σK
) (pn+1
− pn+1
+ (pn+1
− pn+1
· nσK
K )
K ) µσK
K ) µσK
r
σ1
σ2
−m(K)
r
Kr
r=i,j,k
r
Kr
r
2
1
2
2
+ m(σK
) (pn+1
− pn+1
+ (pn+1
− pn+1
· nσK
K ) µσK
K ) µσK
r
σ1
σ2
r
Kr
r
Kr
p̄n+1
− pn+1
= pc (un+1
K
K
K )+τ
un+1
− unK
K
,
h
!
,
r
(4.17)
for all K ∈ T . Similarly, at each edge σ ∈ Eint , we impose
p̄n+1
I
σKdr
−
pn+1
I
σKdr
=
pc (un+1
)
I
σKdr
n
un+1
Id − u Id
+τ
σKr
σKr
h
(Id = 1, 2),
and the flux continuity of each phase
n+1
1
2
ko (un+1
− p̄n+1
+ (p̄n+1
− p̄n+1
) µσK
· nK|L
1
2
K ) (p̄σK
K ) µσK
K
σ
r
r
Kr
r
n+1
n+1
1
2
+ko (un+1
+ (p̄n+1
− p̄n+1
· nL|K = 0,
L ) (p̄σ 1 − p̄L ) µσL
L ) µσL
σ2
Lr
r
Lr
r
(4.18)
(4.19)
4.3. THE FINITE VOLUME SCHEME
59
n+1
n+1
n+1
n+1
2
1
+
(p
−
p
)
µ
kw (un+1
)
(p
−
p
)
µ
2
1
σKr · nK|L
σKr
K
K
K
σKr
σKr
n+1
n+1
n+1
n+1
2
1
· nL|K = 0.
+
(p
−
p
)
µ
+kw (un+1
)
(p
−
p
)
µ
2
1
σ
σ
L
L
L
σ
σ
L
L
r
Lr
(4.20)
r
Lr
Here L is the neighboring element of K sharing the edge σ, and nL|K is the unit normal
1 , p̄σ 2 , pσ 1 , pσ 2
are set to
vector from L into K. Whenever σ ∈ Eext , the values p̄σK
K
K
K
r
r
r
r
Id
0. One takes un+1
Id = 0 (Id = 1, 2) for any K ∈ T and r = i, j, k such that σKr ∈ Eext .
σKr
Also, flux continuity holds each half edge σ and for the discrete gradient is used.
Initially, we take u0K = u0 (xK ) for any K ∈ T . This makes sense since u0 ∈ C(Ω̄). If
u0 ∈
/ C(Ω̄), then one takes as explanation in Remark 4.2.2, u0T = ηT ∗ u0 , where η is the
standard mollifier ( [48]). Clearly, since u0 ∈ W01,2 (Ω), one has ku0T − u0 kW 1,2 (Ω) → 0
as size(T ) → 0.
4.3.3
A priori estimates and existence of the fully discrete solution
In this section, we discuss the fully discrete solution to (4.15) - (4.20). We first provide
some elementary results that will be used later.
Lemma 4.3.1. Let m ≥ 1 and aj , bj ∈ Rm be an m-dimensional real vectors, j ∈
{0, ..., N }. We have the following identities:
N
N
N
N
X
X
X
X
bj i,
haj , bj i − ha0 ,
bn i =
haj − aj−1 ,
(4.21)
j=1
j=1
n=j
j=1
N
N
X
X
1
haj − aj−1 , aj i = (|aN |2 − |a0 |2 +
|aj − aj−1 |2 ),
2
j=1
j=1
(4.22)
N X
N
N
N
X
1 X j2 1X j2
h
aj , an i = |
a | +
|a | ,
2 j=1
2 j=1
j=1 j=n
(4.23)
where < ·, · > is the inner product in Rm .
Lemma 4.3.2. Discrete Gronwall inequality: If {yn }, {fn } and {gn } are nonnegative
sequences and
yn ≤ fn +
X
gk yk ,
f or all n ≥ 0,
0≤k<n
then
yn ≤ fn +
X
0≤k<n
fk gk exp(
X
gj ),
f or all n ≥ 0.
k<j<n
The existence of a solution to the discrete system (4.15) - (4.20) can be obtained by
a Leray - Schauder argument, as done in [76]. One has
60
CHAPTER 4. FINITE VOLUME SCHEME
Lemma 4.3.3. Let n ∈ {0, 1, ..., N − 1}, and assume un given. Then there exists a solun+1
n+1 n+1 n+1 n+1 n+1 n+1 n+1
tion (un+1
K , uσ 1 , uσ 2 , p̄K , p̄σ 1 , p̄σ 2 , pK , pσ 1 , pσ 2 )K∈T to the discrete system
Kr
Kr
(4.15) - (4.20).
Kr
Kr
Kr
Kr
Without entering into details, the proof requires a priori estimates, which are obtained
below.
Lemma 4.3.4. A C > 0 not depending on h or size(T ) exists such that, for any N ∗ ∈
{0, ..., N − 1} we have the following:
∗
N
X
X
X
h
ko (un+1
m(Kr )∇Kr p̄n+1
· ∇Kr p̄n+1
K )
K
K
n=0 K∈T
(4.24)
r=i,j,k
∗
N
X
X
X
+
h
kw (un+1
)
m(Kr )∇Kr pn+1
· ∇Kr pn+1
K
K
K
n=0 K∈T
r=i,j,k
N∗
X
un+1 − un 2 X
X
∗
+1
K
+τ
h
m(K) K
+
m(K)Pc (uN
) ≤ C.
K
h
n=0
K∈T
K∈T
Proof. We start by proving the following:
1
µσK
=
1
1
m(σK
) · nσK
r
r
r
m(Kr )
,
2
µσK
=
2
2
) · nσK
m(σK
r
r
m(Kr )
r
(4.25)
.
To see this, we refer to Figure 4.1 and take without losing of generality r = i. Note that
m(Pi xK Pk,i ) = m(Pi xK Pi,j ) = 16 m(Pi Pj Pk ) since xK is the geometric center and Pk,i ,
Pi,j are midpoints. This gives m(Ki ) = 31 m(Pi Pj Pk ). With θi being the angle spanned
Id
by eσId and µσId , the matching height of xK to σK
is |eσId |cosθi = |µ 1I | , due to
i
Ki
Ki
Ki
(4.12). Therefore, one has
m(Ki ) = 2m(Pi xK Pi,j ) =
1
Id
m(σK
)
i
|µσId |
σ d
Ki
(Id = 1, 2),
Ki
so
|µσId | =
Ki
Id
m(σK
)
i
.
m(Ki )
This immediately implies (4.25) since µσId and nσId are parallel and have the same sense.
Ki
Ki
n+1
1
2
Then multiplying (4.15) by p̄n+1
and m(σK
)p̄n+1
respectK , (4.19) by m(σKr )p̄σ 1
σ2
r
Kr
Kr
ively, adding the three equalities and summing the resulting over K ∈ T , we find that
−
X
K∈T
m(K)(un+1
− unK )p̄n+1
=h
K
K
X
K∈T
ko (un+1
K )
X
m(Kr )∇Kr p̄n+1
· ∇Kr p̄n+1
K
K .
r=i,j,k
(4.26)
4.3. THE FINITE VOLUME SCHEME
61
Similarly, we also obtain
X
m(K)(un+1
− unK )pn+1
=h
K
K
K∈T
X
kw (un+1
K )
K∈T
X
m(Kr )∇Kr pn+1
· ∇Kr pn+1
K
K .
r=i,j,k
(4.27)
Adding (4.26) and (4.27) gives
h
X
ko (un+1
K )
K∈T
+h
kw (un+1
K )
K∈T
+
m(Kr )∇Kr p̄n+1
· ∇Kr p̄n+1
K
K
(4.28)
r=i,j,k
X
X
X
X
m(Kr )∇Kr pn+1
· ∇Kr pn+1
K
K
r=i,j,k
m(K)(un+1
− unK )(p̄n+1
− pn+1
K
K
K ) = 0.
K∈T
n
Further, multiplying (4.17) by m(K)(un+1
K − uK ) and summing the resulting over K ∈ T
leads to
X
n+1
m(K)(p̄n+1
− pn+1
− unK )
K
K )(uK
(4.29)
K∈T
=
X
n+1
m(K)pc (un+1
− unK ) + τ
K )(uK
K∈T
X
K∈T
m(K)
un+1
− unK n+1
K
(uK − unK ).
h
Using this into (4.28) gives
X
h2
K∈T
X
+h2
+τ
m(Kr )∇Kr p̄n+1
· ∇Kr p̄n+1
K
K
(4.30)
r=i,j,k
kw (un+1
K )
K∈T
X
X
ko (un+1
K )
X
m(Kr )∇Kr pn+1
· ∇Kr pn+1
K
K
r=i,j,k
m(K)(un+1
− unK )2 + h
K
K∈T
X
n+1
m(K)pc (un+1
− unK ) = 0.
K )(uK
K∈T
Recalling (4.8), one gets
h2
X
K∈T
+h2
X
K∈T
+τ
X
K∈T
X
ko (un+1
K )
m(Kr )∇Kr p̄n+1
· ∇Kr p̄n+1
K
K
(4.31)
r=i,j,k
X
kw (un+1
K )
m(Kr )∇Kr pn+1
· ∇Kr pn+1
K
K
r=i,j,k
m(K)(un+1
K
− unK )2 + h
X
K∈T
m(K)Pc (un+1
K )≤h
X
K∈T
m(K)Pc (unK ).
62
CHAPTER 4. FINITE VOLUME SCHEME
Summing the above equation from 0 to N ∗ for any N ∗ ∈ {0, ..., N − 1} gives
∗
N
X
X
X
h
kw (un+1
m(Kr )∇Kr pn+1
· ∇Kr pn+1
K )
K
K
n=0 K∈T
(4.32)
r=i,j,k
∗
N
X
X
X
+
h
ko (un+1
m(Kr )∇Kr p̄n+1
· ∇Kr p̄n+1
K )
K
K
n=0 K∈T
r=i,j,k
∗
+τ
N
un+1 − un 2 X
X
X
X
∗
+1
K
+
m(K)Pc (uN
)≤
m(K)Pc (u0K ).
h
m(K) K
K
h
n=0
K∈T
K∈T
K∈T
This proof is then concluded by using the continuity of pc (·) and (A5), yielding
X
m(K)Pc (u0K ) ≤ C.
K∈T
To obtain estimates in terms of the discrete gradients of the saturation, we first prove
the result below:
Lemma 4.3.5. √
Given α, β ∈ [mp , Mp ]√
and two vectors a, b such that the angle in between
2 mp Mp
2 mp Mp
is γ ∈ [arccos( mp +Mp ), π − arccos( mp +Mp )], one has
α|a|2 + β|b|2 + (α + β)|a| |b|cosγ ≥ 0.
(4.33)
|a|
. Then, the proof reduces to
Proof. The case b = 0 is trivial. If b 6= 0, let x = |b|
showing that
αx2 + (α + β)cosγ x + β ≥ 0,
√
2 mp Mp
for all x ∈ R. Since |cosγ| ≤ mp +Mp , one has
∆ = (α + β)2 (cosγ)2 − 4αβ
4mp Mp
≤ (α + β)2
− 4αβ
(mp + Mp )2
m M
α 2 β
p p
= 4α2
(1
+
) −
.
(mp + Mp )2
β
α
Observing that
proof.
mp
Mp
≤
α
β
≤
Mp
mp ,
one immediately sees that ∆ ≤ 0, which concludes the
Now we can provide the a-priori estimates
4.3. THE FINITE VOLUME SCHEME
63
Lemma 4.3.6. If h < τ , for any N ∗ ∈ {0, ..., N − 1} it holds
X X
m(Kr )|∇Kr uN
K
∗
(4.34)
+1 2
| ≤ C,
K∈T r=i,j,k
where C is independent of h, size(T ), or N ∗ .
Proof. Subtracting (4.17) from (4.18) gives
n+1
n+1
h(p̄n+1
) − h(pn+1
)
Id − p̄K
Id − p K
σKr
σKr
=h(pc (un+1
)
I
σKdr
−
pc (un+1
K ))
n+1
n
+ τ (un+1
) − τ (un+1
Id − uK
Id − uK )
σKr
σKr
(Id = 1, 2).
(4.35)
Multiplying (4.35) by m(Kr )µσId ∇Kr un+1
K , adding the resulting for Id = 1 and 2 and
Kr
summing over r ∈ {i, j, k}, K ∈ T and n ∈ {0, ..., N ∗ } for any fixed N ∗ < N gives
∗
N
X
h
n=0
=
N∗
X
X X
n+1
m(Kr )(∇Kr p̄n+1
− ∇Kr pn+1
K
K ) · ∇Kr uK
K∈T r=i,j,k
h
n=0
X X
n+1
m(Kr )∇Kr pc (un+1
K ) · ∇Kr uK
K∈T r=i,j,k
∗
+
N
X
X X
τ
m(Kr )(∇Kr un+1
− ∇Kr unK ) · ∇Kr un+1
K
K .
n=0
K∈T r=i,j,k
Applying on the left the Cauchy-Schwarz inequality and using Lemma 4.3.1 for the last
term on the right gives
∗
N
∗
τ X X X
τ X X
+1 2
m(Kr )|∇Kr uN
| +
m(Kr )|∇Kr un+1
− ∇Kr unK |2
K
K
2
2 n=0
K∈T r=i,j,k
+
N∗
X
n=0
h
K∈T r=i,j,k
X X
n+1
m(Kr )∇Kr pc (un+1
K ) · ∇Kr uK
K∈T r=i,j,k
∗
N
τ X X
1X X X
2
≤
m(Kr )|∇Kr u0K |2 +
h
m(Kr )|∇Kr p̄n+1
− ∇Kr pn+1
K
K |
2
2 n=0
K∈T r=i,j,k
+
K∈T r=i,j,k
N∗
X
X X
1
2
h
m(Kr )|∇Kr un+1
K | .
2 n=0
K∈T r=i,j,k
By Lemma 4.3.4, the second term on the right is bounded uniformly in h, size(T ) and
64
CHAPTER 4. FINITE VOLUME SCHEME
N ∗ . This gives:
∗
N
τ X X
τ X X X
N ∗ +1 2
m(Kr )|∇Kr uK
| +
m(Kr )|∇Kr un+1
− ∇Kr unK |2
K
2
2 n=0
K∈T r=i,j,k
+
N∗
X
n=0
≤C +
h
K∈T r=i,j,k
X X
n+1
m(Kr )∇Kr pc (un+1
K ) · ∇Kr uK
K∈T r=i,j,k
N∗
X
X X
1
2
h
m(Kr )|∇Kr un+1
K | .
2 n=0
K∈T r=i,j,k
Furthermore, the third term on the left is positive. To see this, observe that for any
n ∈ {0, ..., N ∗ }, K ∈ T , r ∈ {i, j, k} there exist ξ1 , ξ2 ∈ R such that
∗
N
X
h
n=0
X X
n+1
m(Kr )∇Kr pc (un+1
K ) · ∇Kr uK
K∈T r=i,j,k
∗
=
N
X
h
n=0
X X
n+1
n+1
n+1
1
2
m(Kr ) pc (un+1
)
−
p
(u
)
µ
+
p
(u
)
−
p
(u
)
µ
1
2
c
c
c
σ
σ
K
K
σ
σ
K
K
r
Kr
K∈T r=i,j,k
·
r
Kr
1
2
un+1
− un+1
µσK
+ un+1
− un+1
µσK
K
K
σ1
σ2
r
Kr
r
Kr
∗
=
N
X
n=0
h
X X
2
2
0
0
n+1
n+1
n+1
2 |
1 |
u
−
u
|µ
−
u
|µ
+
p
(ξ
)
m(Kr ) pc (ξ1 ) un+1
2
1
2
σ
σ
c
K
K
σ
σ
K
K
r
Kr
K∈T r=i,j,k
r
Kr
!
n+1
1 | · u 2
2 | · cos(π − θ)
+ pc (ξ1 ) + pc (ξ2 ) un+1
− un+1
|µσK
− un+1
|µσK
.
K
K
σ1
σ
0
0
r
Kr
r
Kr
Note that to avoid an excess of notions, we omitted any additional indices for ξ1 , ξ2 ,
which actually depend on the particular n, K or r. Observing that
√ γ the angle between
2
mp Mp
1
2 , satisfies γ = π − θ and by (A6), |cosγ| ≤
µσK
and µσK
mp +Mp , using (A3) and
r
r
Lemma 4.3.5, one immediately gets that the above is positive. This gives
∗
N −1
1 X X X
τ −h X X
N ∗ +1 2
2
m(Kr )|∇Kr uK
| ≤C+
h
m(Kr )|∇Kr un+1
K | ,
2
2 n=0
K∈T r=i,j,k
K∈T r=i,j,k
and the conclusion is a direct consequence of Lemma 4.3.2.
4.4. CONVERGENCE OF THE SCHEME
4.4
4.4.1
65
Convergence of the scheme
Compactness results
To prove the convergence we recall Definition 3.2 and use the time-space discrete values
to construct a sequence of triples defined in Ω × (0, T ]:
for all
n
vT ,h (x, t) = vK
and t ∈ (nh, (n + 1)h],
x∈K
n = 0, ..., N − 1. (4.36)
Further, we will use the discrete version of the seminorm in the space L2 (0, T ; W 1,2 (Ω)).
Definition 4.4.1. (Discrete seminorms) For v ∈ X(T , h) enriched with values {(vσn1 ,
Kr
vσn+1
)|K ∈ T , r = i, j, k}, define
2
Kr
|v(·, t)|1,T =
X X
n 2
2
n
n 2
2
1 | + |v 2
2 |
m(Kr ) |vσn1 − vK
| |µσK
−
v
|
|µ
σ
K
σ
K
Kr
K r=i,j,k
for all t ∈ (nh, (n + 1)h],
Kr
r
!1/2
n = 0, ..., N − 1, and
N
X
X X
n 2
n 2
2
n
2
1 | + |v 2
2 |
−
v
|
|µ
| |µσK
h
m(Kr ) |vσn1 − vK
σ
K
σ
K
|v|1,T ,h =
n=0
,
r
Kr
K r=i,j,k
r
Kr
!1/2
.
r
Note that | · |1,T and | · |1,T ,h are the discrete counterparts of the gradient norms for
functions in W 1,2 (Ω), respectively L2 (0, T ; W 1,2 (Ω)). Clearly, these are seminorms in the
corresponding spaces. Following the Lemma 4.3.4 and Lemma 4.3.6, we have
Lemma 4.4.1. Under assumption (A3), if (uT ,h , p̄T ,h , pT ,h ) ∈ (X(T , h)))3 solves the
system (4.15) - (4.20), one has
|pT ,h |21,T ,h + |p̄T ,h |21,T ,h + |uT ,h |21,T ,h ≤ C,
and |uT ,h |21,T ≤ C
for all t ∈ (0, T ],
where C does not depend on size(T ) or h.
The following is a discrete counterpart of the Poincaré inequality. Before stating it,
1 , vσ 2 , vσ 1 , vσ 2 , vσ 1 , vσ 2 ) be given 7-tuples for any K ∈ T satisfying
let (vK , vσK
K
K
K
K
K
i
i
j
j
k
k
1
2
1
2
vσK
= vσ K
= 0 whenever σK
, σK
∈ Eext . Let XT0 be the space of piecewise function
r
r
r
r
v : Ω → R, v|K = vK , endowed with the discrete gradient ∇K v by using the additional
values, and k · kL2 (Ω) we mean the L2 -norm of the piecewise constant v. Then we have
Lemma 4.4.2. (Discrete Poincaré inequality) A constant C > 0 depending on Ω,
but not on size(T ) exists such that
kv(·)k2L2 (Ω) ≤ C|v|21,T .
66
CHAPTER 4. FINITE VOLUME SCHEME
Proof. We essentially apply the technique in [50]. For σ ∈ E, define χσ from R2 × R2 to
{0, 1} as
(
1, [x, y] ∩ σ 6= ∅,
χσ (x, y) :=
(4.37)
0, [x, y] ∩ σ = ∅.
Let e be a given vector and x ∈ Ω. Let Dx be the semi-line having the origin x and
continuing in the direction of e. Let y(x) be such that y(x) ∈ Dx ∩∂Ω and [x, y(x)] ⊂ Ω̄,
where [x, y(x)] = {βx + (1 − β)y(x), β ∈ [0, 1]} (i.e. y(x) is the first point where Dx
meets ∂Ω). For y(x) such that χσ (x, y(x)) = 1 and if σ ∈ Eint , let K|L be the triangles
adjacent to σ, where K is the one closest to x. Let vK , vL be the corresponding values.
Also, let vσ be the value from {vσId , I = 1, 2, r = i, j, k} corresponding to the past of
Kr
σ in K intersected by Dx . For σ ∈ Eext , one takes vL = vσ = 0. Also if the intersection
point happens to be vertex in P or edge midpoint in PM , then one takes arbitrary vσ ,
vK , vL . The choice is not important as finally, we integrate for x ∈ Ω.
Along [x, y(x)], we define
Dσ v :=


|vσ − vK | + |vσ − vL |,

if σ ∈ Eint and χσ (x, y) = 1,
|vσ − vK |, if σ ∈ Eext and χσ (x, y) = 1,


0, if χ (x, y) = 0.
(4.38)
σ
Let now K ∈ T arbitrary. For a.e. x ∈ K, one has
|vK | ≤
X
χσ (x, y(x))Dσ v.
(4.39)
σ∈E
Using the Cauchy - Schwarz inequality, this gives
|vK |2 ≤
X
χσ (x, y(x))
σ∈E
(Dσ v)2 X
χσ (x, x + η)dσ cσ ,
dσ cσ
(4.40)
σ∈E
for a.e. x ∈ R2 , where cσ = |nσ · e|, nσ denotes a unit normal vector to σ, and
dσ :=


1
1
|µσK | + |µσL | ,
 1 , if σ ∈
|µσK |
if σ ∈ Eint ,
Eext .
(4.41)
As in [50], we show that for a.e. x ∈ Ω
X
χσ (x, y(x))dσ cσ ≤ diam(Ω).
(4.42)
σ∈E
Given now e and x and assuming that Dx does not go through any vertex. Assuming
σL ∈ L̄ ∩ ∂Ω, L ∈ T , let xσx ∈ Ω̄ be the perpendicular foot of XL xσx to σL . Since the
4.4. CONVERGENCE OF THE SCHEME
67
control volumes are convex, there exists xC ∈ K such that
X
χσ (x, y(x))dσ cσ ≤ |(xC − xσx ) · e|.
(4.43)
σ∈E
Further, using xC , xσx ∈ Ω̄ gives (4.42). Integrating (4.40) over Ω and using to (4.42)
this gives
XZ
|vK |2 dx
K
K∈T
(|vσ − vK | + |vσ − vL |)2
dσ cσ
σ∈Eint Ω
Z
X
|vσ − vK |2 +
χσ (x, y(x))dx
.
dσ cσ
Ω
≤diam(Ω)
X Z
χσ (x, y(x))dx
σ∈Eext
Since
R
Ω
χσ (x, y(x))dx ≤ diam(Ω)m(σ)cσ , one has
XZ
K∈T
|vK |2 dx
K
≤2(diam(Ω))2
X
m(σ)(|µσK ||vσ − vK |2 + |µσL ||vσ − vL |2 )
σ∈Eint
+
X
m(σ)(|µσK ||vσ − vK |2 .
σ∈Eext
Recalling (4.25), Definition 4.4.1 and Lemma 4.4.1, we have
kvk2L2 (Ω) ≤ C|v|21,T .
With this lemma, one has uniformly with respective to T and h
kuT ,h k2L2 (0,T ;L2 (Ω)) + kp̄T ,h k2L2 (0,T ;L2 (Ω)) + kpT ,h k2L2 (0,T ;L2 (Ω)) ≤ C.
Now we show the following lemma about space translations.
Lemma 4.4.3. Given the trianglarization T and v ∈ XT0 , let ṽ be the extension of v by
0 to the entire R. Then for any η ∈ R2 , one has
kṽ(· + η) − ṽ(·)k2L2 (R2 ) ≤ 2|v|21,T |η|(|η| + Csize(T )),
with C > 0 only depending on Ω and not on v, η or T .
(4.44)
68
CHAPTER 4. FINITE VOLUME SCHEME
Proof. For σ ∈ E, using χσ as defined in (4.37), and for η ∈ R2 , one has
|ṽ(x + η) − ṽ(x)| ≤
X
χσ (x, x + η)Dσ v, for a.e. x ∈ Ω,
σ∈E
where K, L are the volumes adjacent to σ. Following again [50], but defining dσ as in
(4.41), one obtains
|ṽ(x + η, t) − ṽ(x, t)|2
X
(|vσ − vK | + |vσ − vL |)2
≤
χσ (x, x + η)
dσ cσ
σ∈Eint
X
|vσ − vK |2 X
+
χσ (x, x + η)
·
χσ (x, x + η)dσ cσ ,
dσ cσ
σ∈Eext
(4.45)
σ∈E
η
for a. e. x ∈ R2 . Here cσ = |nσ · |η|
|, and nσ denotes a unit normal vector to σ. First,
by [50] there exists C > 0, only depending on Ω such that
X
χσ (x, x + η)dσ cσ ≤ |η| + Csize(T ), for a.e. x ∈ R2 .
(4.46)
σ∈E
Further, observe that for all σ ∈ E,
Z
χσ (x, x + η)dx ≤ m(σ)cσ |η|.
R2
Therefore, integrating (4.45) over R2 and using (4.25) one gets
kṽ(· + η, ·) − ṽ(·, ·)k2L2 (R2 )
Z X
(|vσ − vK | + |vσ − vL |)2 X
≤
χσ (x, x + η)
χσ (x, x + η)dσ cσ dx
dσ cσ
R2 σ∈E
σ∈E
int
Z X
|vσ − vK |2 X
+
χσ (x, x + η)
χσ (x, x + η)dσ cσ dx
dσ cσ
R2 σ∈E
σ∈E
ext
X m(σ)
X m(σ)
≤
(|vσ − vK | + |vσ − vL |)2 +
|vσ − vK |2 |η|(|η| + Csize(T ))
dσ
dσ
σ∈Eint
σ∈Eext
X
≤2
m(σ)(|µσK ||vσ − vK |2 + |µσL ||vσ − vL |2 )
σ∈Eint
+
X
m(σ)|µσK ||vσ − vK |2 |η|(|η| + Csize(T ))
σ∈Eext
=2|v|21,T |η|(|η| + Csize(T )),
which completes the proof.
The result in Lemma 4.4.3 extends straightforwardly to the case where v is time
4.4. CONVERGENCE OF THE SCHEME
69
dependent as well, namely if v is piecewise constant in the space-time volumes as in the
case of X(T , h) elements. Clearly, when estimating the L2 (0, T ; L2 (R2 )) norm, in this
case the norm |v|21,T ,h will appear on the right. We continue with the estimates for the
time translations:
Lemma 4.4.4. Let {un+1
K , n = 0, ..., N − 1} be the u components of the solution of
(4.15) - (4.20) and uT ,h the extension to Ω × (0, T ] defined in (4.36). A C > 0 exists
such that for any ξ ∈ (0, T )
kuT ,h (·, · + ξ) − uT ,h (·, ·)k2L2 (Ω×(0,T −ξ)) ≤ C.
Proof. Letting
B(t) =
Z 2
uT ,h (x, t + ξ) − uT ,h (x, t) dx,
Ω
for t ∈ (0, T − ξ), one has obviously
Z
T −ξ
Z
2
uT ,h (x, t + ξ) − uT ,h (x, t) dxdt =
Ω×(0,T −ξ)
B(t)dt.
0
With n0 (t), n1 (t) ∈ {0, ..., N − 1} such that n0 (t)h ≤ t ≤ (n0 (t) + 1)h and n1 (t)h ≤
t + ξ ≤ (n1 (t) + 1)h, B rewrites
B(t) =
X
2
X
n (t)
n (t)
m(K) uK1 − uK0
=
m(K)
K∈T
K∈T
n1 (t)−1
X
un+1
− unK
K
2
.
(4.47)
n=n0 (t)
Using the Cauchy-Schwarz Inequality gives
n1 (t)−1
B(t) ≤ N
X
X
m(K)
K∈T
un+1
− unK
K
2
.
n=n0 (t)
Defining χ(n; t, t + ξ) as
χ(n; t, t + ξ) =
(
1,
if nk ∈ (t, t + ξ],
0,
if nk ∈
/ (t, t + ξ],
(4.48) becomes
B(t) ≤ N
N
−1
X
n=0
χn (t, t + ξ)
X
K∈T
2
m(K) un+1
− unK .
K
(4.48)
70
CHAPTER 4. FINITE VOLUME SCHEME
Since 0 ≤
R T −ξ
0
χn (t, t + ξ)dt ≤ ξ, we have
Z
T −ξ
B(t)dt ≤ N ξ
0
N
−1
X
X
2
m(K) un+1
− unK .
K
n=0 K∈T
Following (4.31) and according to (A5), one has
τ
N
−1
X
X
2
X
m(K) un+1
− unK ≤ h
m(K)Pc (u0K ),
K
n=0 K∈T
K∈T
which gives
Z
T −ξ
B(t)dt ≤ CT ξ,
0
and the proof is concluded.
4.4.2
Convergence results
In this section, we show the convergence of the finite volume scheme. Following the a
priori estimates obtained above, one has
Theorem 4.4.1. There exists a sequence (Tm , hm ) such that size(Tm ) → 0, hm → 0
as m → ∞, and the triple (uTm ,hm , p̄Tm ,h , pTm ,hm ) converges weakly in L2 (Q) to the
solution (u, p̄, p) in the sense of Definition 4.2.1. Moreover, uTm ,hm converges strongly
to u in L2 (0, T ; L2 (Ω)).
Proof. Lemma 4.4.1 gives that (uT ,h , p̄T ,h , pT ,h ) is bounded uniformly in L2 (0, T ; L2 (Ω)).
This gives immediately the existence of a sequence (Tm , hm ) and of a triple (uTm ,hm , p̄Tm ,h ,
pTm ,hm ) such that it converges weakly to a triplet (u, p̄, p) in L2 (Q). Then, by Lemma
4.4.3, Lemma 4.4.4 and Theorem 3.11 in [50], we obtain u ∈ W 1,2 (0, T ; W01,2 (Ω)), p̄, p ∈
L2 (0, T ; W01,2 (Ω)). Furthermore, Lemma 4.4.3 and Lemma 4.4.4 and the KolmogorovM. Riesz-Frécht Theorem (Theorem 4.26 in [20]) also give the strong convergence:
uTm ,hm → u as m → ∞. In the following, we show (u, p̄, p) is the weak solution of
Problem P. To do so, we let φ, ψ ∈ C 2 (Ω̄ × [0, T ]) such that φ = ψ = 0 on ∂Ω × [0, T ],
φ(·, T ) = ψ(·, T ) = 0. For λ, we make the assumption as λ ∈ C 1 (Ω̄ × [0, T ]),
λ(·, T ) = 0, which means that pointwise values make sense. Multiplying (4.17) by
hm λTm ,hm (xK , (n + 1)hm )m(K), summing the resulting for n ∈ {0, ..., N − 1} and
4.4. CONVERGENCE OF THE SCHEME
71
K ∈ T gives
N
−1
X
n=0
=
m(K)(p̄n+1
− pn+1
K
K )λ(xK , (n + 1)hm )
(4.49)
K∈Tm
N
−1
X
X
hm
n=0
+τ
X
hm
m(K)pc (un+1
K )λ(xK , (n + 1)hm )
K∈Tm
N
−1
X
X
m(K)(un+1
− unK )λ(xK , (n + 1)hm ).
K
n=0 K∈T
We denote the last term of (4.49) by T1 and rewrite it as
T1 =τ
N
−1
X
X
m(K)(un+1
− unK )λ(xK , (n + 1)hm )
K
n=0 K∈Tm
=τ
N
−1
X
X
m(K)unK (λ(xK , nhm ) − λ(xK , (n + 1)hm ))
n=1 K∈Tm
+τ
X
0
m(K) uN
K λ(xK , T ) − uK λ(xK , hm ) .
K∈Tm
First We have λ(xK , T ) = 0. Then according to the property of the initial condition, one
has
Z
X
m(K)u0K λ(xK , hm ) −→
u0 (x)λ(x, 0)dx as m −→ ∞.
Ω
K∈Tm
Further, since λ ∈ C 1 (Ω̄ × [0, T ]), λ(·, T ) = 0, one has
N
−1
X
X
m(K)un+1
K (λ(xK , (n − 1)hm ) − λ(xK , nhm )) →
n=0 K∈Tm
Z
0
T
Z
u(x, t)∂t λ as m → ∞.
Ω
Similarly, since p̄Tm ,hm − pTm ,hm converges weakly to p̄ − p, one has
N
−1
X
X
(p̄n+1
K
−
pn+1
K )λ(xK , nh)
n=0 K∈Tm
Z
T
Z
−→
(p̄ − p)λdtdx
0
as
m −→ ∞.
Ω
From the above, one gets that (u, p, p̄) satisfies (4.11).
Furthermore, given ϕ ∈ (C0∞ (Ω × [0, T )))2 , for any K ∈ T and n ∈ {0, ..., N } set
ϕnK = ϕ(xK , tn ),
divK ϕnK =
2
X X
1
Id
m(σK|L
) ϕnK · nK|L .
m(K)
(4.50)
L∈NK Id =1
Letting χK×[tn ,tn+1 ) be characteristic function of K × [tn , tn+1 ), we use (4.50) to define
72
CHAPTER 4. FINITE VOLUME SCHEME
ϕT ,h : Ω × (0, T ) as
ϕT ,h =
N
−1
X
X
ϕn+1
K χK×(tn ,tn+1 ) .
(4.51)
divK ϕn+1
χK×(tn ,tn+1 ) .
K
(4.52)
n=0 K∈T
Further, its discrete divergence is:
divT ϕT ,h =
N
−1
X
X
n=0 K∈T
With the definitions (4.50), (4.51) and (4.52), we have ϕTm ,hm → ϕ and divTm ϕTm →
divϕ uniformly as m → ∞. By the compactness results and Lemma 4.3.6 there exists a
RT
RT
ζ such that 0 ∇T uTm divϕ → 0 ζdivϕ as m → ∞.
Now we identify the discrete gradient limit ζ with ∇u:
Z
=
T
Z
uTm ,hm divT ϕT ,h
0
Ω
N
−1
X
hm
n=0
=
N
−1
X
hm
N
−1
X
X
N
−1
X
hm
=−
2
X X
hm
L∈NK Id =1
X
(un+1
− un+1
L
K )
X
2
X
Id
m(σK|L
) ϕn+1
· nK|L
K
Id =1
2
X
Id
n+1
m(σK|L
) ϕn+1
· nK|L (un+1
− un+1
− un+1
Id ) + (u Id
K
L
K )
σK|L
K|L∈Eint Id =1
hm
n=0
X
2
m(σ Id )
X
L|K
d
|µIL|K
|
K|L∈Eint Id =1
+
N
−1
X
Id
m(σK|L
) ϕn+1
· nK|L
K
K|L∈Eint
n=0
N
−1
X
un+1
K
K∈T
n=0
=−
n+1
m(K)un+1
K (divK ϕK )
K∈T
n=0
=−
X
σK|L
Id
(un+1
− un+1
Id
L )|µL|K | nL|K
σL|K
Id
m(σK|L
)
d
|µIK|L
|
Id
n+1
n+1
(un+1
−
u
)|µ
|
n
Id
K|L · ϕK
K
K|L
σK|L
2
m(L)
X
m(K) n+1
Id
n+1
n+1
d
(un+1
− vL
) µIL|K
+
(u Id − un+1
)
µ
Id
K
K|L · ϕK
σL|K
σK|L
3
3
n=0
K|L∈Eint Id =1
Z TZ
Z TZ
=−
∇Tm uTm ,hm · ϕK → −
ξ · ϕ.
=−
0
hm
Ω
X
0
Ω
Therefore ∇u = ξ in the sense of distributions, and in particular, u ∈ L2 (0, T ; W 1,2 (Ω)).
Similarly, we can also obtain ∇Tm p̄Tm ,hm → ∇p̄, ∇Tm pTm ,hm → ∇p weakly in L2 (0, T ; L2 (Ω))
as m → ∞.
4.4. CONVERGENCE OF THE SCHEME
73
RT R
Now we concentrate on A = 0 Ω ko (u)∇p̄ · ∇φdxdt. To do so, we define the
discretization and approximation of φ denoted by Φ and φT ,h :

n+1


ΦK = φ(xK , (n + 1)h),
Φn+1
 σ

φ
T ,h
= φ(xσ , (n + 1)h),
=
Φn+1
K ,
x ∈ K,
K ∈T,
σ ∈ E,
n ∈ {0, ...N − 1},
(4.53)
n ∈ {0, ...N − 1},
t ∈ (nh, (n + 1)h) for all n = {1, ..., N − 1}.
Then multiplying (4.15) by hm φn+1
:= hm φ(xK , (n + 1)h) and (4.19) by hm φn+1
:=
Id
K
σKr
hm φ(xσId , (n+1)hm ) and summing over K ∈ Tm and n ∈ {0, N −1}, with r = Pi , Pj , Pk
Kr
and Id = 1, 2, one has
ATm ,hm
N
−1
X
X Z
X
=
n=0 K∈Tm r=i,j,k
tn+1
tn
Z
Kr
n+1
n+1
1
2
ko (un+1
+ (p̄n+1
− p̄n+1
·
K ) (p̄σ 1 − p̄K ) · µσK
K ) · µσK
σ2
r
Kr
r
Kr
n+1
n+1
n+1
1
2
(φn+1
−
φ
)
·
µ
+
(φ
−
φ
)
·
µ
dxdt.
1
2
σ
σ
K
K
σ
σ
K
K
r
Kr
Z
T
Kr
r
Z
ko (uTm ,hm )∇Tm p̄Tm ,hm ∇Tm φTm ,hm dxdt.
=
0
Ω
Then we have
T2 =ATm ,hm − A
Z TZ
Z
=
ko (uTm ,hm )∇Tm p̄Tm ,hm · ∇Tm φTm ,hm −
0
Z
Ω
T
|0
T
Z
ko (uTm ,hm )∇Tm p̄Tm ,hm ∇Tm φTm ,h −
Ω
{z
Ω
Z
Ω
ko (u)∇T p̄Tm ,hm · ∇Tm φTm ,hm
}
Ω
ko (u)∇p̄ · ∇φ .
}
0
T21
Z
+
|0
T
Z
Z
ko (u)∇Tm p̄Tm ,hm · ∇Tm φTm ,hm −
Ω
{z
T
Z
ko (u)∇p̄ · ∇φ
0
Z
=
T
Z
0
T22
By the assumption (A2), the compactness of ∇T p̄T ,h , the regularity of φ and uTm ,hm → u
as m → ∞, we easily obtain
T21 → 0
as m → ∞.
(4.54)
Furthermore, for T22 since we have ∇Tm p̄Tm ,hm → ∇p̄, it is also easily obtained that
T22 → 0
as m → ∞,
(4.55)
which together with (4.54) implies ATm ,hm converges weakly to A as m → ∞. In the
74
CHAPTER 4. FINITE VOLUME SCHEME
same way, one gets the convergence for
proof.
4.5
RT R
0
Ω
kw (u)∇p · ∇φdxdt, which concludes the
Numerical results
We start with a simple test problem defined in Ω = (0, 1) × (0, 1) and for t > 0. Further,
we take ko (u) = kw (u) = 1 and pc (u) = u, leading to
∂t u − ∆p̄ = 0,
(4.56)
∂t (1 − u) − ∆p = 0,
(4.57)
p̄ − p = u + τ ∂t u.
(4.58)
To close the system, we prescribe the boundary conditions:
p̄ = p = 0 at ∂Ω,
the initial condition
u(x, y, 0) = sin(2πx) · sin(3πy).
In this case, an explicit solution can be found:
u(x, y, t) = exp(
p̄(x, y, t) = exp(
−13π 2 t
) · sin(2πx) · sin(3πy),
2 + 13τ π 2
−13π 2 t
1
13τ π 2
)
·
sin(2πx)
·
sin(3πy)
·
(
−
),
2 + 13τ π 2
2 2(2 + 13τ π 2 )
p(x, y, t) = exp(
and
−13π 2 t
1
13τ π 2
)
·
sin(2πx)
·
sin(3πy)
·
(−
+
).
2 + 13τ π 2
2 2(2 + 13τ π 2 )
This is used to test the convergence of the method. Recall that the convergence is proved
based on compactness arguments, without having rigorous error estimates. Nevertheless,
for this specific example, since an explicit solution is known, we can estimate the order of
the scheme as follows.
After constructing a mesh and taking uniform time step, we refine it uniformly three
times by halving the mesh size and time step. First we consider a uniform mesh as shown
in Figure 4.2 (left). For each of the discretization parameters, we compute the L2 and
W 1,2 errors at t = 1/16:
ETu ,h =
Z
(u(x, t)−uT ,h (x, t))2 dx
1/2
Ω
The results in Table 4.1 refer to
ETu ,h
,
, ETp ,h =
ETp ,h ,
Z
Ω
1/2
(∇p̄(x, t)−∇T p̄T ,h (x, t))2 dx
.
(4.59)
which are representative. All other errors
4.5. NUMERICAL RESULTS
75
have similar behavior. We estimate the order by computing
α = log2 (
ETu ,h
ETu /2,h/2
),
β = log2 (
ETp ,h
ETp /2,h/2
(4.60)
).
Based on this the scheme is first order convergent in both L2 and W 1,2 norm. Observe
that the order in the approximation of the gradient is the same as the L2 -order, this being
a consequence of the multipoint flux approximation.
One of the advantage of the proposed scheme is that, theoretically, it is robust with
respect to the meshing. Since pc is linear, (A6) brings no restriction for the meshing. To
evaluate the behavior of the scheme for non-uniform meshes and anisotropic cases, we
use as starting point the non-uniform mesh in Figure 4.2 (right). Note that this mesh is
built without any connection with the solution, such as rapid changes in the magnitude of
the gradient. The results presented in Table 4.2 show practically no change in the order
of the scheme.
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
Figure 4.2 The uniform mesh (left) and nonuniform mesh (right).
No. of cells
43 × 8
44 × 8
45 × 8
46 × 8
ETu ,h
8.6546× 10−4
4.3880× 10−4
2.2097× 10−4
1.1088× 10−4
α
–
0.9799
0.9897
0.9949
ETp ,h
5.7211× 10−3
2.8433× 10−3
1.4191× 10−3
7.0912× 10−4
β
–
1.0087
1.0026
1.0009
Table 4.1 Convergence results for uniform mesh, τ = 1.
0.8
1
76
CHAPTER 4. FINITE VOLUME SCHEME
No. of cells
43 × 8
44 × 8
45 × 8
46 × 8
ETu ,h
8.7661× 10−4
4.4138× 10−4
2.2160× 10−4
1.1104× 10−4
α
–
0.9899
0.9941
0.9969
ETp ,h
7.8924× 10−3
3.8945× 10−3
1.9390× 10−3
9.6800× 10−4
β
–
1.0190
1.0061
1.0022
Table 4.2 Convergence results for nonuniform mesh, τ = 1.
The first test involved a scalar permeability. However, the multipoint flux approximation considered here applies to anisotropic cases too. To see this, we consider the
following problem:
∂t u − ∇ · (K∇p̄) = 0,
(4.61)
∂t (1 − u) − ∇ · (K∇p) = 0,
(4.62)
p̄ − p = u + τ ∂t u.
(4.63)
With k1 = 1, k2 = 5, K is defined as
K=
k1
0
0
k2
!
(4.64)
.
The boundary conditions remain unchanged:
p̄ = p = 0 at ∂Ω,
as the initial condition
u(x, y, 0) = sin(2πx) · sin(3πy).
Again, an explicit solution can be found:
u(x, y, t) = exp(
p̄(x, y, t) = exp(
−49π 2 t
) · sin(2πx) · sin(3πy),
2 + 49τ π 2
−49π 2 t
1
49τ π 2
) · sin(2πx) · sin(3πy) · ( −
),
2
2 + 49τ π
2 2(2 + 49τ π 2 )
p(x, y, t) = exp(
and
49τ π 2
−49π 2 t
1
).
) · sin(2πx) · sin(3πy) · (− +
2
2 + 49τ π
2 2(2 + 49τ π 2 )
For the numerical tests we carried out the same steps as before: two meshes (uniform
and non-uniform) are refined successively three times. We compute the same errors, and
observe that even in the anisotropic case, the scheme still remains first order convergence
for both meshes and practically does not loose accuracy. The results are given in Tables
4.3 and 4.4.
4.5. NUMERICAL RESULTS
No. of cells
43 × 8
44 × 8
45 × 8
46 × 8
ETu ,h
8.7790× 10−4
4.4681× 10−4
2.2544× 10−4
1.1324× 10−4
77
α
–
0.9777
0.9869
0.9934
ETp ,h
1.8338× 10−3
9.0883× 10−4
4.5329× 10−4
2.2647× 10−4
β
–
1.0127
1.0035
1.0011
Table 4.3 Convergence results for uniform mesh, τ = 1 and in the anisotropic case.
No. of cells
43 × 8
44 × 8
45 × 8
46 × 8
ETu ,h
8.8311× 10−4
4.4791× 10−4
2.2569× 10−4
1.1330× 10−4
α
–
0.9794
0.9889
0.9942
ETp ,h
3.1188× 10−3
1.4995× 10−3
7.3982× 10−4
3.6810× 10−4
β
–
1.0565
1.0192
1.0071
Table 4.4 Convergence results for nonuniform mesh, τ = 1 and in the anisotropic case.
For the final test, we mention that one of the known features of the model (4.1)
- (4.3) is that their solution does not satisfy a maximum principle. Instead, effects like
saturation overshoot can be observed both experimentally [17,38] and analytically [40,99].
To investigate this aspect, we present some numerical experiments carried out with the
relative permeability functions
kw (u) = u1.5 ,
ko = (1 − u)1.5 .
These are commonly encountered in modeling two-phase flows in porous media, when u
denotes the water saturation. For the equilibrium capillary pressure, we still take a linear
function
pc (u) = 1 − u.
We consider the domain Ω = (−5, 10) × (0, 10). The initial condition is shown in Figure
4.3,
u0 = (ur − ul )/(1 + exp(−4x) + ul ), for all (x, y) ∈ Ω.
Here ul , ur ∈ [0, 1] are two constant values, ul = 0.9, ur = 0.1. Observe that u0 does
not depend on y.
78
CHAPTER 4. FINITE VOLUME SCHEME
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
10
5
0
5
0
−5
10
Figure 4.3 The initial saturation.
At the lateral boundaries, we assume that the normal flux is 0,
−ko (s)p̄y = −kw (s)py = 0,
along {(−5, 10) × {0}} and {(−5, 10) × {10}}.
At the inflow and outflow boundary, we assume a given, constant total flux in the x
direction:
−ko (s)p̄x − kw (s)px = 1,
along {{−5} × (0, 10)} and {{10} × (0, 10)}.
Further, we assume that the saturation is given there,
u(−5, y) = ul ,
for any y ∈ (0, 10),
u(10, y) = ur
and compute the pressures accordingly.
The numerical approximation of the saturation is displayed in Figure 4.4 for two times.
Observe that the saturation exceeds its maximal initial and boundary values, which is the
so-called overshoot effect. The results are in good agreement with the profiles in the
one-spatial dimensional case obtained e.g. in [40].
1
0.9
1
0.8
0.9
0.7
0.8
0.6
0.7
0.5
0.6
0.4
0.5
0.3
0.4
0.2
0.3
0.1
0.2
0
0.1
10
10
5
5
0
−5
0
5
10
0
−5
0
5
Figure 4.4 The saturation at t = 1 (left) and t = 3 (right) with τ = 1.
10
Chapter 5
Two phase flow in
heterogeneous porous media
5.1
Introduction
Two-phase (wetting/non-wetting) flows are widely encountered in various real-life processes. A few prominent examples are water driven oil recovery, or geological sequestration
of CO2 . These processes involve large spatial scales, and (rock) heterogeneities appear
naturally. In this chapter we consider a simplified situation, where the medium consists of
two different homogeneous blocks with different permeabilities (coarse and fine), which
are separated by an interface. This makes the transition from one material to another
not smooth, and appropriate conditions have to be imposed at the interface for coupling
the models written in each of the two blocks. In particular, when the underlying models
are involving entry pressures to describe the dependency of the capillary pressure on the
phase saturations, the non-wetting phase may remain trapped in the coarse block at the
interface.
This situation has been analyzed in [42, 46], but for the case when the phase pressure
difference depends on the, say, wetting phase phase saturation and the medium itself.
These are standard models, for which the dependency between various quantities are determined under equilibrium conditions. Therefore, such models are also called equilibrium
models. In the paper mentioned above, regularization arguments (i.e. approximating the
interface by a thin porous layer ensuring a smooth transition between the two homogeneous blocks) are employed to derive appropriate coupling conditions between the models
in the two sub-domains. The resulting conditions are the flux continuity and an extended
pressure condition. We refer to [14, 25, 27] for the mathematical analysis of such models,
where the existence of weak solutions has been analyzed. Further, the case of many layers
This chapter is a collaborative work with C.J. van Duijn, I.S. Pop and it has been published in
Transport in Porous Media, 110(2015): 1-26.
79
80
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
is studied in [43,45], where homogenization techniques are applied for deriving an effective
model. Such kind of models are also studied in [3, 19, 54, 55, 88, 90], where appropriate
numerical schemes are studied.
Various experiments [17,38] have invalidated the equilibrium assumptions, and motivated considering non-equilibrium approaches. Here we focus on models involving dynamic
effects in the phase pressure difference, as proposed in [60]. In this case, we follow the
ideas in [42,46] and derive the coupling conditions at the interface separating the two homogeneous blocks. When compared to the equilibrium case, a striking difference appears.
In the former the non-wetting phase can only flow into the fine block if its saturation at
the coarse block side of the interface exceeds an entry value, in the latter situation this
flow can appear for lower saturation values. This is due to the dynamic effects in the
phase pressure difference and can reduce the amount of non-wetting phase that remains
trapped at the interface.
The models including dynamic effects in the phase pressure difference lead to socalled pseudo-parabolic problems. For such models, but posed in homogeneous domains,
existence and uniqueness of weak solutions are obtained in [30, 52, 77]. The case of
vanishing capillary effects and the connection to hyperbolic conservation laws is studied
in [40, 44]. For dynamic capillarity models in the heterogeneous case, but in the absence
of an entry pressure, numerical schemes are discussed in [61]. This situation is similar
to the case analyzed in [36], where the interface is replaced by a discontinuity in the
initial conditions. Also, variational inequality approaches have been considered in [62] for
situations including an entry pressure. However, the conditions are simply postulated and
no derivation is presented.
In Section 5.2 we present the mathematical model. For simplicity we consider the case
when only the absolute permeability is different in the two blocks, all other parameters
being the same. In Section 5.3, we derive the coupling conditions at the interface. These
are the flux continuity and an extended pressure continuity. In Section 5.4, we discuss different numerical approaches, and present numerical experiments that confirm the analysis
in Section 5.3. We give the conclusion in the last section.
5.2
Mathematical model
We consider the flow of two immiscible and incompressible phases in a one-dimensional
heterogeneous porous medium. Letting sw denote the saturation of the wetting fluid, and
sn the saturation of the non-wetting fluid, one has 0 ≤ sw , sn ≤ 1. The porous medium
is assumed to be saturated by the two fluids,
sw + sn = 1.
(5.1)
5.2. MATHEMATICAL MODEL
81
Mass balance holds for each phase (see [61, 82]),
φ
∂sα
∂qα
+
= 0,
∂t
∂x
α = w, n,
(5.2)
where φ is the porosity assumed constant, and qα denotes the volumetric velocity of the
phase α. These velocities satisfy the Darcy’s law
qα = −k̄(x)
krα (sα ) ∂pα
,
µα
∂x
α = w, n,
(5.3)
where k̄(x) is the absolute permeability of the porous medium, pα the pressure, µα
the viscosity, and krα the relative permeability of the α phase. The functions krα are
assumed known. Gravity effects are disregarded, as they have no influence on the interface
conditions derived here. Substituting (5.3) into (5.2) gives
∂
krα ∂pα
∂sα
−
k̄(x)
= 0,
α = w, n.
(5.4)
φ
∂t
∂x
µα ∂x
In standard models, the phase pressure difference depends on the saturation, which is
determined experimentally. An example in this sense is the Leverett relationship
s
pn − pw = pc (x, sw ) = σ
φ
J(sw ),
k̄(x)
(5.5)
where σ is the interfacial tension, and J a decreasing function.
The relationship in (5.5) is determined by measurements carried out under equilibrium
condition. In other words, before measuring the pressure and saturation in a representative
elementary volume, fluids have reached equilibrium and are at rest. However, processes
of interest may not satisfy this condition, and dynamic effects have to be included. Alternatively to (5.5), in [60] the following model is proposed
pn − pw = pc (x, sw ) − τ̄
∂sw
.
∂t
(5.6)
The damping factor τ̄ is assumed to be known and constant. Summing the two equations
in (5.4) and using (5.1), one gets
∂ q̄
= 0,
(5.7)
∂x
k̄(x)krn ∂pn
rw ∂pw
where q̄ = − k̄(x)k
µw
∂x −
µn
∂x denotes the total flow. In the one-dimension case,
this means that q̄ is constant in space. Here we assume it is constant in time as well, and
is positive. This allows reducing the two-phase flow model to a scalar equation in terms
of, say s = sw . After rescaling the space x with L, the time t with T , and using the
82
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
reference value K for the absolute permeability and σ
q
φ
K
for the pressure, one obtains
∂s ∂F
+
= 0,
∂t
∂x
(5.8)
where the F denotes the dimensionless flux of the wetting phase
∂
F = qfw (s) + Nc k(x)λ̄(s)
∂x
q̄T
> 0, and k(x) =
Here q = φL
wetting phase,
k̄(x)
K .
fw =
∂s
J(s)
p
− Nc τ
∂t
k(x)
!
(5.9)
.
Further, fw is the fractional flow function of the
krw (s)
,
krw (s) + krn (s)/M
√
with the mobility ratio M = µn /µw , the capillary number Nc = σµnφK
qL , the dimensionless
2
q̄
and λ̄(s) = krn (s)fw (s). For simplicity, we assume here
damping factor τ = τ̄ µn σφ
Nc = 1, as different values of Nc would not have any influence on the conditions derived
below.
Throughout this work, we make the following assumptions
(A1) krw , krn : [0, 1] → [0, 1] are continuous differentiable functions satisfying
a) krw is strictly increasing such that krw (0) = 0 and krw (1) = 1;
b) krn is strictly decreasing such that krn (0) = 1 and krn (1) = 0;
0
(A2) J is (0, 1] → R is continuous differentiable, decreasing which satisfies J < 0 on
(0, 1], J(1) ≥ 0 and lim J(s) = +∞.
s&0
We consider a simple heterogeneous situation, where two adjacent homogeneous blocks
are separated by an interface located at x = 0. For the ease of presentation, we assume
that all parameters and functional dependencies except the absolute permeability are the
same. For the latter, we have
(
k − , if x < 0 (the coarse medium),
k(x) =
k + , if x > 0 (the fine medium).
Here k − > k + > 0 are given.
Throughout this chapter, the non-wetting phase may be oil, or CO2 or any other
phase with non-wetting phase properties. For simplicity, below, we use as oil non-wetting
phase. Also, by pressure we actually mean the phase pressure difference.
5.3. CONDITIONS AT THE INTERFACE
5.3
83
Conditions at the interface
The model (5.8)-(5.9) is a parabolic equation, where the factor √1k appears under a second
order spatial derivative. Since k has a jump discontinuity at the interface, the model is
only valid in each of the two blocks, and coupling conditions at x = 0 have to be derived.
Commonly, the pressure continuity is taken as a second condition. However, this is shown
to be inappropriate for entry-pressure models in the absence of dynamic effects (τ = 0).
This statement is made rigorous in [42] by regularizing k. Specifically, the interface is
replaced by a thin layer in which k decays continuously from k − to k + . Next to k, we
will use the quantity
p
h(x) = k(x).
Clearly,
(
h(x) =
√
h− = k − , if x < 0,
√
h+ = k + , if x > 0,
and h− > h+ > 0.
For given > 0, we approximate the interface x = 0 by the interval [−, ] (the thin
layer) and h by a smooth function h , such that h is monotonic in the small interval
[−, ]. Specifically, the discontinuous function h is now approximated by the smooth
function h , such that

−

for x < −,

h ,
h (x) =
ĥ( x ),


h+ ,
for − < x < ,
for
x > .
The function ĥ is smooth and monotonic on [−, ] (see Figure 5.1).
Coarse block
Fine block
Figure 5.1 The function h .
84
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
With the given the solution corresponding to the regularized problem is denoted by
s , the corresponding flux by F . In the expression for the flux, we replace h by h . By
taking y = x we rescale [−, ] to [−1, 1]. We define v (y, t) = v ( x , t) = s (x, t), and
investigate its behavior when & 0. First, for x ∈ [−, ] (and thus y ∈ [−1, 1]), from
(5.8) one gets
∂
1 ∂
v (y, t) +
F (y, t) = 0.
(5.10)
∂t
∂y
Assuming
∂v
∂t
bounded uniformly with respect to , passing to 0 gives lim ∂F
∂y (y, t) = 0,
&0
implying
lim F (−1, t) = lim F (1, t).
&0
&0
In fact, this is nothing but the flux continuity at the interface, which is as expected.
For the second condition we consider again y ∈ [−1, 1]. From (5.9), one has
F − qfw (v ) = h2 λ̄(v )
1 ∂ J(v )
∂v −τ
.
∂y
h
∂t
As before, let now & 0, assume that v (y, t) → v(y, t) and that during the limit process,
the flux F is bounded uniformly in . Then, this gives
λ̄(v)
∂v ∂ J(v)
−τ
= 0,
∂y h(y)
∂t
in Ω,
where Ω denotes the strip Ω = {(y, t) : −1 < y < 1, t > 0} (see Figure 5.2).
Coarse
block
Fine
block
Figure 5.2 The strip Ω.
Along the boundary of Ω, we define
s− (t) := s(0− , t) = v(−1, t),
s+ (t) := s(0+ , t) = v(1, t),
for t > 0.
(5.11)
5.3. CONDITIONS AT THE INTERFACE
85
The goal is to understand how s− (t) and s+ (t) are related, as well as p− (t) =
∂s− (t)
∂t ,
J(s+ (t))
h+
∂s+ (t)
∂t .
J(s− (t))
h−
−
τ
p (t) =
−τ
These are nothing but the left and right saturation
and pressure at the interface. Note that (5.11) is an ordinary differential equation in
t, where y can be seen as a parameter. To define an initial condition, we let s0 (y) be
a smooth function s0 : [−1, 1] → R satisfying s0 (−1) = s− (0) and s0 (+1) = s+ (0).
Clearly, the choice of s0 is not unique. Below we investigate the relation between s− (t)
and s+ (t), and its dependence of the regularization of h. Since λ̄(v) > 0, for 0 < v < 1,
and λ̄(0) = λ̄(1) = 0, from (5.11), one has
+
v = 0,
or if 0 < v < 1,
or v = 1,
∂ J(v)
∂v −τ
= 0.
∂y h(y)
∂t
(5.12)
In other words
•
∂v
J(v)
−τ
h(y)
∂t
is constant in y, whenever 0 < v < 1;
• v(·, t) is continuous with respect to y, for y[−1, 1];
• v(y, ·) is C 1 with respect to t, for t ≥ 0.
For the sake of understanding, we consider some particular cases.
5.3.1
Constant saturation at the coarse side of the interface
We let s− ∈ (0, 1) and assume s− (t) = s− for all t. Let s0 : [−1, 1] −→ (0, 1) be
the given initial value, not necessarily compatible to s− : s0 (−1) 6= s− in general. We
construct a solution for which the set Ωc = {(y, t) ∈ Ω, 0 < v(y, t) < 1} is connected.
From (5.12), if (y, t) ∈ Ω, v solves the autonomous initial value problem:

1 h(y)
 ∂v
=
J(v) − − J(s− ) ,
for t > 0,
τ
∂t
h(y)
h
Py
(5.13)

v(y, 0) = s0 (y).
By (A2) and the assumption on h, Py has a unique solution locally. Let s∗ be defined
by
J(s∗ )
J(1)
= + .
(5.14)
−
h
h
We consider the cases s− > s∗ and s− ≤ s∗ , separately.
• Case 1: s− > s∗
86
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
Note that
J(s− ) J(s∗ ) J(1)
<
=
.
h−
h−
h+
Since h is decreasing function in y, there exists a unique y ∗ ∈ (−1, 1) such that
J(1)
J(s− )
J(1)
=
< + .
h−
h(y ∗ )
h
(5.15)
We study now the long time behavior of v(y, t). We distinguish the following sub-cases.
a) For y ∈ (−1, y ∗ ], define s∞ (y) as the unique solution of
J(s∞ (y))
J(s− )
=
.
(5.16)
h(y)
h−
Clearly, s∞ (y) is the equilibrium point for Py and satisfies s∞ > s− (see Figure 5.3).
Further, standard phase plane arguments show that, regardless of s0 (y) ∈ (0, 1),
lim v(y, t) = s∞ (y). If s0 (y) > s∞ (y), the solution v(y, t) decreases continuously from
t→∞
v(y, 0) = s0 (y) to v(y, ∞) = s∞ (y). If s0 (y) < s∞ (y), the solution v(y, t) increases
continuously from v(y, 0) = s0 (y) to v(y, ∞) = s∞ (y). Note that s∞ (−1) = s− , and
that s∞ (y) is strictly increasing in y up to s∞ (y ∗ ) = 1. For y > y ∗ , we have s∞ (y) = 1.
b) For y > y ∗ , one has
τ
J(v) J(s− )
∂v
J(1) J(s− )
=
−
−
≥
> 0.
∂t
h(y)
h−
h(y)
h−
(5.17)
This implies the solution v(y, t) increases in t and reaches v = 1 in finite positive time
(see Figure 5.4)
τ (1 − s0 (y))
0 < t∗ (y) < J(1) J(s− ) .
h(y) − h−
Note: if s0 (y) is non-decreasing, then t∗ (y) is decreasing to t∗ (1) > 0.
5.3. CONDITIONS AT THE INTERFACE
Figure 5.3 The functions
J(v)
h(y)
87
for various y ∈ [−1, 1] for s− > s∗ .
Figure 5.4 The solution for s− > s∗ .
The long time behavior of v is summarized in
Proposition 5.3.1. Assume s0 (y) ∈ (0, 1) for all y ∈ [−1, 1] and let s− > s∗ , where s∗
is defined in (5.14). With y ∗ in (5.15), one has:
a) if y ∈ [−1, y ∗ ) then lim v(y, t) = s∞ (y), where s∞ (y) ∈ (0, 1] is given by (5.16).
t→∞
Further, s∞ (·) is strictly increasing from s− = s∞ (−1) to 1 = s∞ (y ∗ );
b) if y ∈ (y ∗ , 1], then there exists t∗ (y) > 0 such that v(y, ·) is increasing in t for all
t < t∗ < ∞, and v(y, t) = 1 for all t ≥ t∗ (y). Moreover, if s0 (·) is non-decreasing,
then t∗ (·) is decreasing to t∗ (1) > 0.
Corollary 5.3.1. In particular, at y = 1, we have for s+ (t) = v(1, t). s+ (t) increases
J(s− )
+
to 1 for t ∈ (0, t∗ (1)), where 0 < t∗ (1) < τ (1 − s0 (y))/( J(1)
h+ − h− ), s (t) = 1 for
∗
t ≥ t (1), as presented in Figure 5.5.
88
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
Figure 5.5 Behavior of s+ (t) for t > 0: s+ is increasing to 1 for t < t∗ (1), and s+ (t) = 1 for
t ≥ t∗ (1).
This allows constructing an extended pressure condition, similar to [42]. Consider the
pressure
∂v
J(v(y, t))
−τ ,
(5.18)
p(y, t) :=
h(y)
∂t
observe that
p(−1, t) = p− (t) =
J(s− )
,
h−
and p(+1, t) = p+ (t) =
J(s+ )
∂s+
−τ
.
+
h
∂t
With the entry pressure
p+
e :=
J(1)
,
h+
(5.19)
since s− > s∗ , one has
J(s− )
< p+
e .
h−
By Corollary 5.3.1, we obtain the condition:
(
p− (t) = p+ (t),
for
0 < t < t∗ (1),
s+ (t) = 1,
for
t ≥ t∗ (1).
(5.20)
In other words, the pressure remains continuous for t < t∗ (1) and oil keeps flowing into the
fine material although p− is below the entry pressure. This effect is due to incorporating
dynamic effects in the phase pressure difference, and would not be possible in their absence
(τ = 0).
• Case 2: s− < s∗
5.3. CONDITIONS AT THE INTERFACE
89
For any y ∈ [−1, 1], with s∞ (y) defined by (5.16), one has
J(s∞ (y))
J(s∗ )
J(1)
J(s− )
>
= + .
=
−
−
h(y)
h
h
h
(5.21)
Therefore, there exists an s̃ ∈ (s− , 1), such that
J(s̃)
J(s∞ (y))
J(s− )
= + ,
=
h(y)
h−
h
which, in view of the monotonicity of h and J, gives s∞ < s̃ < 1. Consequently, this
case is similar to the sub-case y < y ∗ before.
Proposition 5.3.2. Assume s0 (y) ∈ (0, 1) for all y ∈ [−1, 1] and let s− < s∗ (see
(5.14)). Then,
lim v(y, t) = s∞ (y) for all y ∈ (−1, 1),
t→∞
∞
with s (y) given by (5.16). In particular, for y = 1, one can define s∞
+ = s (1) as the
unique solution of
J(s∞
J(s− )
+)
=
.
h+
h−
∞
Corollary 5.3.2. At y = 1, we have for s+ (t) = v(1, t):
lim s+ (t) = s∞
+ ∈ (0, 1).
t→∞
In other words, the pressure remains continuous at the interface for all t > 0:
p+ (t) =
J(s+ (t))
J(s− )
∂s+
=
−τ
= p− .
+
h
∂t
h−
(5.22)
Note that unlike in (5.20), the pressure remains continuous for all t > 0.
All results refer to the case J(1) > 0, hence to the entry pressure model. If instead,
J(1) = 0 (no entry pressure), s∗ = 1 and the analysis before leads to s− < 1, and
p− (t) = p+ (t) for all t > 0 (see [61]).
Another special case is when τ & 0. Then, the time t∗ (y) in Proposition 5.3.1 and
Corollary 5.3.1 approaches to 0, and if s− > s∗ the pressure becomes discontinuous
instantaneously. This is, in fact, exactly the behavior in [42] for equilibrium models.
5.3.2
Non-constant saturation at the coarse side of the interface
The results before are obtained for a constant saturation at the coarse side of the interface.
±
±
Here we generalize these results. We let p± (t) = J(sh±(t)) − τ ∂s∂t(t) be the phase pressure
difference at the two sides of the interface and derive an extended pressure condition
similar to (5.20) and (5.22). With the entry pressure p+
e defined in (5.19), we assume
−
p (t) given, and distinguish the following cases.
90
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
• Case 1: p− (t) > p+
e for all t > 0
Again, for y ∈ (−1, 1), v(y, t) solves

∂v
 J(v(y, t))
−τ
= p− (t),

h(y)
∂t
Py

v(y, 0) = s0 (y).
for t > 0,
(5.23)
Assume s0 (y) ∈ (0, 1), then the equation holds in the neighborhood of t = 0. We show
that in this case, v(y, t) < 1 for any t, and hence p− = p(y, t), for any y ∈ (−1, 1).
Assume v(y, t∗ ) = 1 for some t∗ < ∞. For t < t∗ one has
τ
1 ∂v
=
J(v(y, t)) − h(y)p− (t) .
∂t
h(y)
By the monotonicity of h,
h(y)p− (t) > h+ p− (t) > h+ p+
e = J(1),
implying that, at t = t∗ , one has
τ
∂v ∗
J(1)
(t ) =
− p− (t) < 0.
∂t
h(y)
This shows that v(y, ·) cannot grow to 1 for t % t∗ . Therefore no finite t∗ exists, such
that v(y, t∗ ) = 1, implying v(y, t) ∈ (0, 1) for all t. We have proved
Proposition 5.3.3. Assume s0 (y) ∈ (0, 1) for all y ∈ [−1, 1] and let p− (t) > p+
e for all
t > 0. Then for all y ∈ [−1, 1] and t > 0, one has v(y, t) ∈ (0, 1).
Corollary 5.3.3. At y = 1, we get s+ (t) = v(1, t) ∈ (0, 1) for all t > 0, and therefore
the pressure remains continuous at both sides of the interface for all t > 0
p− (t) = p+ (t).
p+
e
(5.24)
As in Subsection 5.3.1, if the model involves no entry pressure (J(1) = 0), one has
= 0. Then p− (t) ≥ p+
e = 0, and the pressure is continuous for all t > 0.
• Case 2: 0 < p− (t) < p+
e for all t > 0
We assume first that an exists such that 0 < p− (t) < p+
e − for all t > 0. Further,
we assume that the initial condition s0 : (−1, 1) → R is non-decreasing and smooth, and
satisfies s0 (y) ∈ (0, 1) for any y. We have
Proposition 5.3.4. Assume there exists > 0 such that 0 < p− (t) < p+
e − for all t > 0,
5.3. CONDITIONS AT THE INTERFACE
91
and that s0 is non-decreasing and smooth. Then there exists a t∗ > 0 such that
(
p− (t) = p+ (t),
for 0 < t < t∗ ,
s+ (t) = 1,
for t ≥ t∗ .
Proof. Clearly, if v(y, t) < 1, then v(y, ·) solves

∂v
 J(v(y, t))
−τ
= p− (t),

h(y)
∂t
Py

v(y, 0) = s0 (y).
(5.25)
for t > 0,
(5.26)
First we prove the monotonicity of v(·, t). Specifically, for all t such that v(·, t) < 1
uniformly in y, one also has v(·, t) is non-decreasing. To see this, we differentiate (5.26)
with respect to y to obtain
0
0
J (v) ∂v
h
∂2v
−
= − 2 J(v).
τ
∂t∂y
h(y) ∂y
h
For a fixed y, this has the general form:
τ u̇ = f u + b,
0
0
0
(v)
∂v
with u = ∂y
, f = Jh(y)
< 0, b = − hh2 J(v). Clearly, u(0) = s0 ≥ 0. Assuming that a
t̄ > 0 exists such that u(t̄) = 0 and u(t) > 0 for any t ∈ (0, t̄], one gets:
u̇(t̄) = f (t̄)u(t̄) + b(t̄) = b(t̄) > 0.
On the other hand, one has
u̇(t̄) = lim
∆t&0
u(t̄) − u(t̄ − ∆t)
≤ 0,
∆t
which contradicts the above.
Now we proceed by proving the conclusion of the proposition. Since p− (t) < p+
e − ,
by the continuity of h, there exists δ > 0 such that
h(y) −
p (t) < p+
e − /2,
h+
for 1 − δ < y < 1.
Hence, for y ∈ (1 − δ, 1) and t > 0, we have
+
+
J(v) − h(y)p− (t) > J(v) − h+ p+
e + h /2 = J(v) − J(1) + h /2.
This implies
τ h(y)
∂v
> J(v) − J(1) + h+ /2 > h+ /2.
∂t
92
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
Clearly, a finite t = t(y) > 0 exists such that v(y, t(y)) = 1 and v(y, t) for t < t(y). By
the monotonicity of v and s0 , taking t∗ = t(1), one has v(y, t) < 1 for any y ∈ [−1, 1]
and t < t∗ . Therefore, v(y, t) solves (Py ) for all t < t∗ and all y, hence p(y, t) remains
continuous, in particular, p− (t) = p+ (t).
Observe that, for fixed y ∈ (−1, 1), v(y, t) solving Py and w(y, t) solving

∂w
τ h(y)
= J(w) − h(y)(p+
e − ),
∂t
Py
w(y, 0) = s (y),
0
t > 0,
0
are ordered. Specifically, as long as both v and w remain less 1, one has w(y, t) ≤ v(y, t).
To see this, we let u = v − w, and define u− = min{0, u}. Then, subtracting the
0
equations in Py and Py , and multiplying the result by u− gives
h(y) ∂
(u− )2 = (J(v) − J(w))u− − h(y)(p− (t) − (p+
e − ))u− ≤ 0.
2 ∂t
Here we have used the monotonicity of J. Since at t = 0 one has u− (y, 0) = 0, integrating
in time gives u− (y, t) = 0, implying the ordering. Hence w is a lower bound for v, and
therefore analyzing w makes sense. Let y ∗ ∈ (−1, 1) be defined by
h(y ∗ ) =
J(1)
J(1)
= h+
.
J(1)
− /2h+
−
/2
p+
e
Then for y > y ∗ , a δ > 0 exists such that h(y) = h(y ∗ ) − δ < h(y ∗ ), as long as
w(y, t) < 1, one has
τ
1
∂w
J(w)
J(1)
1 δ
(y, t) =
−(p+
−(p+ −)+
−
J(1) =
J(1) > 0.
e −) ≥
∗
∂t
h(y)
h(y )
h(y) h(y ∗ )
h(y)h(y ∗ )
This shows that w reaches the cut-off value w = 1 at finite time t = t(y). If the function
s0 (y) is constant, it is straightforward to show that t(y) is strictly decreasing to t(1) > 0
(see Figure 5.6). At y = y ∗ , w = 1 satisfies the equation. Therefore, w ≡ 1 is an
0
equilibrium solution, implying that w(y ∗ , t) solving Py satisfies w(y ∗ , t) < 1 for all t,
and thus
lim∗ t(y) = ∞.
y&y
Remark 5.3.1. In fact, the argument above is an alternative proof for Proposition 5.3.4.
Since s+ (t) = v(1, t) > w(1, t), and w(1, t(1)) = 1 for t(1) < ∞, (5.25) follows immediately.
5.3. CONDITIONS AT THE INTERFACE
93
Figure 5.6 The function t(y).
Based on the analysis before, we discuss particular examples where oil trapping may
∂v
occur. Defining p(y, t) := J(v)
h(y) − τ ∂t , in the transition region (the blown up interface)
Ω = {(y, t) : −1 < y < 1, t > 0}, we have

J(v)
∂
∂v

λ̄(v)
= 0,
−
τ

∂y
h(y)
∂t

p(t) = p− (t), t > 0,



v(y, 0) = s0 (y), −1 < y < 1,
which implies that either v = 0 or v = 1, or
h(y)τ
∂v
= J(v) − h(y)p− (t).
∂t
(5.27)
To understand the trapping, we now take s0 (y) = 1, for −1 < y < 1, and give the
pressure at the coarse side of the interface for t > 0. We assume the following behavior
(see Figure 5.7):
Figure 5.7 The function p− (t).
94
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
−
a) There exists T1 > 0 such that 0 < p− (t) < p−
e for t ∈ (0, T1 ), where pe =
J(1)
h− .
In this case we have
−
J(1) − h(y)p− (t) = h− p−
e − h(y)p (t) > 0.
Let y ∈ [−1, 1] be fixed. Assume that a finite t̃ > 0 and δ > 0 exist such that v(y, t) = 1
for all t < t̃ and v(y, t̃) < 1 for t ∈ (t̃, t̃ + δ), then one has ∂v
∂t (y, t̃) ≤ 0. Whereas by
assumption for t ∈ (t̃, t̃ + δ), one has
∂v
1 J(1)
(y, t̃) =
− p− (t̃) > 0.
∂t
τ h(y)
This gives a contradiction.
−
+
b) There exists a T2 > T1 such that p−
e < p (t) < pe , for t ∈ (T1 , T2 ).
−
+
Since p−
e < p (t) < pe , one has
J(1)
J(1)
< p− (t) < + .
−
h
h
For T1 < t < T2 , we define ỹ(t) by
h(ỹ(t)) :=
J(1)
.
p− (t)
−
p (t)
1
1
−
+
Note that the definition makes sense, as p−
e < p (t) < pe implies h− < J(1) < h+ , and
+
−
−
h(·) is a monotone, continuous interpolation between h and h . Since p is increasing
in [T1 , T2 ], this implies that ỹ(·) is increasing and ỹ(T1 ) = −1, ỹ(T2 ) = 1. Then, for
y > ỹ(t), we have
J(1) − h(y)p− (t) > J(1) − h(ỹ(t))p− (t) = 0.
(5.28)
Similarly, for y < ỹ(t), we have
(5.29)
J(1) − h(y)p− (t) < 0.
Furthermore, since v(y, T1 ) = 1 for all y ∈ (−1, 1), from (5.28), (5.29), one get for all
t ∈ (T1 , T2 )
v(y, t) = 1,
for y > ỹ(t),
and v(y, t) < 1,
for y < ỹ(t).
Thus y = ỹ(t) defines a free boundary in the transition region, separating regions where
v = 1 from regions where v < 1.
Remark 5.3.2. In this context, for t < T2 one has s+ (t) = v(1, t) = 1, and oil remains
trapped in the coarse medium at the interface.
5.3. CONDITIONS AT THE INTERFACE
95
c) There exists T3 > T2 such that p− > p+
e for t ∈ (T2 , T3 ).
Based on the above analysis, we have v(y, T2 ) < 1 for all y ≤ 1, since ỹ(T2 ) = 1. Further,
one also has
+ +
J(1) − h(y)p− (t) < J(1) − h(y)p+
e < J(1) − h pe = 0.
As before, one cannot obtain v(y, t̃) = 1 for some t̃ > T2 , since at t̃, it holds
∂v
1 J(1)
−
(y, t̃) =
− p (t̃) < 0.
∂t
τ h(y)
Therefore, v(y, t) < 1
for − 1 < y < 1 and T2 < t < T3 . Consequently, we have
p− (t) = p+ (t) for T2 < t < T3 .
Remark 5.3.3. Next to the pressure continuity, this shows that oil starts flowing into the
fine region for t > T2 .
d) For all t > T3 , p− < p+
e .
+
We assume p− (·) decreasing and lim p− (t) = p∞ ∈ (p−
e , pe ). Given y ∈ [−1, 1], we
t→∞
compare the solution v(y, t) and w(y, t) of
∂v
= J(v) − h(y)p− (t),
∂t
∂w
= J(w) − h(y)p∞ ,
h(y)τ
∂t
h(y)τ
for all t > T3 , with v(y, T3 ) = w(y, T3 ) < 1.
Since J is decreasing and p− (t) > p∞ , one gets v(y, t) ≤ w(y, t) for −1 < y < 1, t >
T3 . Further, there exists y ∗ ∈ (−1, 1) such that
h(y ∗ ) =
J(1)
∈ (h+ , h− ).
p∞
This gives for y > y ∗
J(1) − h(y)p∞ > 0.
As before, there exists t(y) < ∞, such that w(y, t) = 1 for t > t(y).
Similarly, for y < y ∗ , one has
J(1) − h(y)p∞ < 0,
implying w(y, t) < 1 for all t > T3 . In this case,
lim w(y, t) = s∞ (y),
t→∞
96
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
where s∞ (y) is defined by
J(s∞ (y)) = h(y)p∞ .
h(y)
∞
Observe that, since h(y) > h(y ∗ ), J(s∞ (y)) = h(y
(y) ∈ (0, 1). For
∗ ) J(1) > J(1), so s
y = −1 one has
h−
h−
J(s∞ (−1)) =
J(1) < + J(1) = J(s∗ ),
∗
h(y )
h
giving s∞ (−1) > s∗ . Further, if p∞ = p+
e =
J(1)
h+ ,
then s∞ (−1) = s∗ .
+
This analysis shows that, if p− (t) behaves as in Figure 5.7, and p∞ ∈ (p−
e , pe ), a
+
∗
−
+
T < ∞ exists such that s (t) = 1, for t > T , and p (t) = p (t) for T3 < t < T ∗ .
This means up to T ∗ , oil flows into the fine material, while trapping occurs for t > T ∗ .
This behavior is sketched in Figure 5.8.
∗
Remark 5.3.4. Compared to the equilibrium case (τ = 0), a striking difference appears.
If τ = 0 for a pressure p− (t) behaving as in Figure 5.7, no oil flows into the fine layer for
any t > T3 . If τ > 0, oil continues flowing for t > T3 and up to a time T ∗ < ∞, the
+
delay time. This delay appears, as discussed, if lim p− (t) = p∞ ∈ (p−
e , pe ).
t→∞
The following result extends the statement in the remark to a more general situation.
R∞ +
−
Proposition 5.3.5. Let p− (t) ≤ p+
e be such that 0 (pe − p (t))dt > τ and let s0 (y) ∈
(0, 1). Further, let s+ be the solution of

+
h+ τ ds = J(s+ ) − h+ p− (t),
dt
 +
s (0) = s0 .
for t > 0,
Then there exists T ∗ < ∞ such that s0 < s+ (t) < 1 for all 0 < t < T ∗ and s+ (T ∗ ) = 1.
Proof. Since p− (t) ≤ p+
e =
J(1)
h+ ,
and J(s+ ) is strictly decreasing, we have
J(s+ ) − h+ p− (t) > 0,
for s+ ∈ (0, 1).
Therefore s+ (t) is strictly increasing whenever s+ < 1. Furthermore, we have
h+ τ
ds+
−
> h+ (p+
e − p (t)),
dt
giving for all t such that s+ (t) < 1 and
1
s (t) > s0 +
τ
+
Z
t
−
(p+
e − p (ζ))dζ.
0
Rt
−
For convenience, define f (t) = τ1 0 (p+
e − p (ζ))dζ. Clearly, f (0) = 0, ∂t f ≥ 0 and
∗
f (∞) > 1. Hence, there exists t < ∞ for which f (t∗ ) = 1 − s0 . Consequently, there
5.3. CONDITIONS AT THE INTERFACE
97
exists T ∗ < t∗ for which s+ (T ∗ ) = 1, and since s+ is increasing, one has s0 < s+ (t) < 1
for all t < T ∗ .
Remark 5.3.5. A lower bound for s+ is w = w(t), the solution of

h+ τ dw = J(w) − J(1),
dt
w(0) = s .
for t > 0,
0
Since J : (0, 1] → R+ is locally Lipschitz, then for all t > 0, w(·) is strictly increasing,
w(t) < 1, and lim w(t) = 1. By a comparison argument, s(t) > w(t) for all t > 0.
t→∞
5.3.3
Comparison of extended pressure conditions with static case
In this section, we show the difference in the pressure conditions appearing in the equilibrium and non-equilibrium models between static case and dynamic case. As proved
in [42], if s− ≥ s∗ , then one has
s+ = 1,
and no oil flows into the fine medium. Further, if s− < s∗ , then s+ < 1, but the pressure
continuous:
[p] = p− (t) − p+ (t) = 0.
Then oil flows into the fine medium.
Conversely, given p− (t) =
assuming that
J(s− (t))
h−
the pressure at the interface on the coarse side and
p− (t) ≤ p+
e =
it implies
J(s− ) ≤
J(1)
,
h+
h−
J(1) = J(s∗ ),
h+
and therefore s− ≥ s∗ . In this case, one also has s+ = 1.
−
∗
Similarly, p− (t) > p+
e implies s < s , and the pressure is continuous:
p− (t) = p+ (t).
Referring to Figure 5.7, if p− (t) ≥ p+
e for t > T2 , the matching conditions in static
and dynamic case are the same. Specifically, s+ = 1 for t < T2 , and p− (t) = p+ (t)
for t ∈ (T2 , T3 ). Assuming now p− (t) is decreasing for t > T3 , with p− (T3 ) = p+
e and
98
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
+
lim p− (t) = p∞ ∈ (p−
e , pe ), we have (also see Figure 5.8):
t→∞
s+ (t) = 1, for t > T3 in the equilibrium case τ = 0,
s+ (t) < 1, for t ∈ (T3 , T ∗ ), and s+ (t) = 1 for t ≥ T ∗ in the non-equilibrium case.
In other words, a delay (T ∗ − T3 ) appears in the non-equilibrium case before trapping
occurs.
In fact, this delay can be infinite. An extreme situation, when s(t) < 1 for all t > 0,
can be constructed. To see this, we assume J : (0, 1] → R locally Lipschitz, and study
the behavior of s+ solving

+
h+ τ ds = J(s) − h+ p− (t), for t > 0,
dt

s(0) = s0 ∈ (0, 1),
with appropriately chosen p− satisfying p− (t) < p+
e for all t > 0.
First, note that an L > 0 exists such that for all s ∈ [s0 , 1]
0 ≤ J(s) − J(1) ≤ L(1 − s).
Hence, an upper bound to s+ is the solution u of

τ du = L (1 − u) + (p+ − p− (t)),
e
dt
h+
u(0) = s .
for t > 0,
0
Let now v = 1 − u. Then
dv
L
−
+ + v = −(p+
e − p (t)), t > 0.
dt
h
τ
This gives
L
v(t) = (1 − s0 )e− τ h+ t −
1
τ
t
Z
L
−
e τ h+ (z−t) (p+
e − p (z))dz,
0
and the upper bound for s reads
+
L
− τh
+t
u(t) = 1 − (1 − s0 )e
1
+
τ
Z
t
L
−
e τ h+ (z−t) (p+
e − p (z))dz.
0
Thus, if p− (t) < p+
e is such that
Z
∞
L
−
e τ h+ z (p+
e − p (z))dz ≤ (1 − s0 )τ,
0
we obtain s+ (t) < 1 for all t > 0, and consequently, the pressure remains continuous for
all t > 0, whereas oil flows into the fine layer.
5.4. NUMERICAL SCHEMES AND EXAMPLES
99
Figure 5.8 Saturation inside the thin layer appximating the interface.
5.4
Numerical schemes and examples
In this section, we provide some numerical examples to illustrate how the dynamic effects
influence the flow of the oil across the interface between two homogeneous blocks. For
simplicity, in (5.8)-(5.9), we take φ = 1. This gives
∂s ∂F
+
= 0,
∂t
∂x
F = qfw (s) + h2 λ̄(s)
∂ J(s)
∂s −τ
,
∂x h(x)
∂t
(5.30)
(5.31)
for t > 0, and x ∈ (−l, l). The boundary and initial conditions are given below.
5.4.1
Linear numerical scheme
For the discretization of (5.30) and (5.31) we decompose the interval (−l, l) into 2N + 1
cells : −l = x−N −1/2 < x−N +1/2 < ... < x−1/2 < x1/2 < ... < xN −1/2 < xN +1/2 = l,
where the grid is uniform with ∆x = 2N2l+1 , we let xj = j∆x for j ∈ {−N − 1/2, ..., N +
1/2}. The discontinuity of h(x) is at x = 0. With ∆t > 0 a given time step, the fully
discrete scheme is:
n
sni−1/2 − sn−1
F n − Fi−1
i−1/2
=− i
,
∆t
∆x
100
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
n
where sn−1
i−1/2 is the approximation of s(x, t) at x = xi−1/2 and at t = t = n∆t. Since
n
n
q > 0, if i 6= 0 the upwind flux Fi at x = i∆x and t = t is defined as
n−1
sn−1
i−1/2 + si+1/2
0 sn−1 + sn−1
sni+1/2 − sni−1/2
i−1/2
i+1/2
)· J (
)(
)
2
2
∆x
n−1
n
sni−1/2 − sn−1
h+/− τ si+1/2 − si+1/2
i−1/2
−
(
−
) .
∆x
∆t
∆t
+/−
Fin = qfw (sn−1
λ̄(
i−1/2 )+h
Here h+/− means h− if i < 0, or h+ if i > 0.
At i = 0, we introduce two saturation unknowns s−,n , s+,n and define the F −,n , and
F +,n as
!
"
s−,n − sn−1/2
0
n−1
−,n
−
−,n−1
−,n−1
F
= qfw (s−1/2 ) + h λ̄(s
) · J (s
)
∆x/2
!#
n
s−1/2 − sn−1
h− τ s−,n − s−,n−1
−1/2
−
−
,
∆x/2
∆t
∆t
"
0
F +,n = qfw (s+,n−1 ) + h+ λ̄(s+,n−1 ) · J (s+,n−1 )
h+ τ
−
∆x/2
sn+1/2 − sn−1
+1/2
sn+1/2 − s+,n
∆x/2
s+,n − s+,n−1
−
∆t
∆t
!
!#
.
At the interface we also define the left and right discretized pressures
p+,n =
J(s+,n )
s+,n − s+,n−1
−τ
+
h
∆t
and p−,n =
J(s−,n )
s−,n − s−,n−1
−τ
.
−
h
∆t
By using the extended pressure condition discussed before one has
(p−,n − p+,n )(1 − s+,n ) = 0.
(5.32)
Defining
g(s−,n , s+,n ) =
J(s−,n ) J(s+,n )
s−,n − s+,n
−
−τ
,
−
+
h
h
∆t
and
C n−1 := C(s−,n−1 , s+,n−1 ) =
τ +,n−1
(s
− s−,n−1 ),
∆t
(5.32) implies either s+,n = 1, or the pressure continuity
g(s−,n , s+,n ) − C(s−,n−1 , s+,n−1 ) = 0.
(5.33)
5.4. NUMERICAL SCHEMES AND EXAMPLES
101
Obviously, ∂1 g > 0 and ∂2 g < 0. Further, given s− ∈ (0, 1), one has
lim g(s− , s+ ) = −∞,
s+ &0
and
g(s− , 1) =
J(s− ) J(1)
τ
J(1) J(1)
− + +
(1 − s− ) ≤ − − + ,
−
h
h
∆t
h
h
since g(s− , ·) is strictly increasing and continuous. For any C n−1 ∈ (−∞, g(s− , 1)], there
exists a unique s+ = s+ (s− ) such that
g(s− , s+ (s− )) = C n−1 .
Also, note that g(s− , 1) is decreasing in s− ,
lim g(s− , 1) = +∞,
s− &0
g(1, 1) =
J(1) J(1)
− + ,
h−
h
and therefore g(·, 1) : (0, 1] → [g(1, 1), +∞) is one to one. Observe that if cn−1 >
g(s− , 1), discretized pressure becomes discontinuous at the interface x = 0. By (5.32),
one obtains s+n = 1. In this way, we have actually constructed the curves in the
(0, 1] × (0, 1] square:
n
o
(s− , s+ )|s− ∈ (0, D(C n−1 )], g(s− , s+ ) = C n−1
n
o
∪ (s− , 1)|s− ∈ (D(C n−1 ), 1] ,
n
o
if C n−1 ≤ g(1, 1), then Γ(C n−1 ) =
(s− , s+ )|s− ∈ (0, 1], g(s− , s+ ) = C n−1 ,
(5.34)
where D(·) is the inverse of g(·, 1).
Below we give a property of the discretized extended pressure condition:
if C n−1 > g(1, 1), then Γ(C n−1 ) =
Proposition 5.4.1. If
J(1)
J(1)
h− − h+ .
τ
+,n−1
∆t (s
− s−,n−1 ) >
J(1)
h−
−
J(1)
h+ ,
then
τ
+,n
∆t (s
− s−,n ) >
Proof. If p−,n 6= p+,n , one has s+,n = 1, and obviously,
τ
1 − s−,n
J(1) J(1)
≥0> − − + .
∆t
h
h
If p−,n = p+,n , then we have
J(s−,n ) J(s+,n ) τ (s+,n − s−,n )
J(1) J(1)
−
+
= C n−1 ≥ − − + .
h−
h+
∆t
h
h
In this case, if s+,n ≥ s−,n , one has
τ (s+,n − s−,n )
J(1) J(1)
≥0> − − + ,
∆t
h
h
(5.35)
102
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
otherwise, s+,n < s−,n implies
J(s−,n ) J(s+,n )
−
< 0.
h−
h+
Together with (5.35), we yield
τ (s+,n − s−,n )
J(1) J(1)
> − − + ,
∆t
h
h
which concludes the proof.
5.4.2
Fully implicit scheme
Here a nonlinear, implicit scheme is discussed as an alternative to the linear one. Next
to improved stability properties, for this scheme we can prove that s±,n , the saturation
at the interface, remain between 0 and 1, a property that is not guaranteed for the linear
scheme. To construct the scheme we first define the decreasing function β : R → R by
Z
s
0
λ̄(z)J (z)dz,
β(s) =
0
and rewrite the flux in (5.31) as
F = qfw (s) + h
∂β(s)
∂ ∂s − τ h2 λ̄(s)
.
∂x
∂x ∂t
As before, for i 6= 0 we write
sni−1/2 − sn−1
i−1/2
∆t
=−
n
Fin − Fi−1
,
∆x
(5.36)
but now the upwind flux Fin becomes
h± n
β(si+1/2 ) − β(sni−1/2 )
∆x
τ (h± )2
−
λ̄ (sn−1
+ sn−1
)/2 (sni+1/2 − sn−1
) − (sni−1/2 − sn−1
) .
i+1/2
i−1/2
i+1/2
i−1/2
∆x∆t
Fin = qfw (sni−1/2 )+
As before, by h± we mean h− if i < 0, or h+ if i > 0. Further, if i = 0, the flux is
defined on each side of the interface as
2h− −,n
β(s ) − β(sn−1/2 )
2
∆x
− 2
2τ (h )
−
λ̄(s−,n−1 ) (s−,n − s−,n−1 ) − (sn−1/2 − sn−1
)
−1/2 ,
∆x∆t
F −,n = qfw (sn− 1 ) +
5.4. NUMERICAL SCHEMES AND EXAMPLES
103
and
2h+ n
β(s+1/2 ) − β(s+,n )
∆x
2τ (h+ )2
+,n
+,n−1
)
−
(s
λ̄(s+,n−1 ) (sn+1/2 − sn−1
−
s
)
.
−
+1/2
∆x∆t
F +,n = qfw (s+,n ) +
For having a conservative scheme, the two expressions should be equal. Combined with
the pressure condition (5.32), and viewing sn±1/2 as well as the saturation values sn−1
i+1/2
and s±,n−1 as known, this provides a nonlinear system with s±,n as unknowns. Below we
show that this system has a unique solution pair in the square [0, 1]2 .
The condition F −,n = F +,n can be written as
(5.37)
R(s−,n , s+,n ) = B(sn−1/2 , sn+1/2 ),
where
2
h+ β(s+,n ) + h− β(s−,n )
R(s−,n , s+,n ) = qfw (s+,n ) − ∆x
2τ
+ ∆x∆t
(h+ )2 λ̄(s+,n−1 )s+,n + (h− )2 λ̄(s−,n−1 )s−,n ,
(5.38)
2
h− β(sn−1/2 ) + h+ β(sn+1/2 )
B = qfw (sn−1/2 ) − ∆x
2τ
+ ∆x∆t
(h− )2 λ̄(s−,n−1 )(s−,n−1 + (sn−1/2 − sn−1
−1/2 ))
+(h+ )2 λ̄(s+,n−1 )(s+,n−1 + (sn+1/2 − sn−1
+1/2 )) .
Using (5.36), B becomes
2
h− β(sn−1/2 ) + h+ β(sn+1/2 )
B = qfw (sn−1/2 ) − ∆x
∆t
2τ
n
(h− )2 λ̄(s−,n−1 ) s−,n−1 − ∆x
(F −,n − F−1
)
+ ∆x∆t
+ 2
+,n−1
+(h ) λ̄(s
+,n−1
) s
−
∆t
n
∆x (F1
−F
+,n
(5.39)
) .
Obviously, R is increasing in both arguments and one has
0 = R(0, 0)
≤ R(s− , s+ )
≤ R(1, 1) = q −
2
2τ
(h+ + h− )β(1) +
[(h+ )2 λ̄(s+,n−1 ) + (h− )2 λ̄(s−,n−1 )].
∆x
∆x∆t
Note that, if both s−,n−1 , s+,n−1 take the values 0 or 1, the last terms in (5.39) vanish,
giving
B =qfw (sn−1/2 ) −
2 −
h β(sn−1/2 ) + h+ β(sn+1/2 ) .
∆x
Since β is decreasing, in this case one has 0 ≤ B ≤ R(1, 1). Further, if s−,n−1 is not
104
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
equal to 0 or to 1, with ∆t small enough one gets
0 ≤ s−,n−1 −
∆t −,n
n
(F
− F−1
) ≤ 1,
∆x
and analogously for s+,n−1 . From (5.39), this shows again that 0 ≤ B ≤ R(1, 1). This
gives the following:
Lemma 5.4.1. For a sufficiently small time step ∆t, with C n−1 = C(s−,n−1 , s+,n−1 )
defined in (5.33) and for R and B in (5.37) and (5.39), the system
−,n
(g(s
+,n
,s
R(s−,n , s+,n ) = B,
)−C
)(1 − s+,n ) = 0,
n−1
(5.40)
has a unique solution pair (s−,n , s+,n ) ∈ [0, 1]2 .
Proof. The set Γ(C n−1 ) introduced in (5.34) contains pairs satisfying the pressure con+,n
dition (5.32). In this set, we seek a pair (s−,n
0 , s0 ) such that
+,n
−,n +,n
R(s−,n
0 , s0 ) = B(s0 , s0 ).
If such a pair exists, it solves the system (5.40). Since g is increasing in the first argument
and decreasing in the second one, long as both s−,n and s+,n are below 1, the curve
Γ(C n−1 ) is a graph of a general non-decreasing function (see Figure 5.9). Similarly, for
B ∈ [0, R(1, 1)], the set R(·, ·) = B is the graph of a decreasing function in the s−,n s+,n plane. Also, observe that both curves are continuous. Therefore, these two curves have
at most one intersection point inside the square [0, 1]2 , implying that (5.40) has a unique
solution pair.
Note that if the solution pair provided above lies inside [0, 1)2 , then one has pressure
continuity across the interface, p−,n = p+,n . Otherwise, the solution pair lies on the
boundary of the square [0, 1]2 . Moreover, assuming that initially one has s−,0 ≤ s+,0 ,
implying that at the interface separating the two blocks more oil is present at the coarse
material side then at the fine material side, repeating the proof of Proposition 5.4.1 one
obtains that s−,n ≤ s+,n for all n. This means that pressure discontinuity at the interface
can only occur if no oil is present at the fine material side, s+,n = 1, which is precisely
the discrete pressure condition in (5.32).
5.4. NUMERICAL SCHEMES AND EXAMPLES
105
Figure 5.9 The curves g(·, ·) = C and R(·, ·) = B.
Remark 5.4.1. The construction above assumes that sn−1/2 , sn1/2 are known. In fact,
these are part of the solution computed implicitly, at time step tn . This means that,
actually, sn−1/2 , sn1/2 and consequently B depend on s−,n , s+,n . However, decoupling the
calculation of the solution pair s±,n from the effective time stepping suggests an iterative
procedure for the implicit scheme: using, say, the values s±,n−1 as starting point, compute
sni+1/2 by solving (5.36) away from the interface, and use sn−1/2 , sn1/2 to update s±,n .
5.4.3
Numerical results
In this section, we give some numerical results obtained with the semi-implicit method
in Subsection 5.4.1, which is much easier to implement. Here we used the following
functions and parameters:
krw (s) = s2 , krn (s) = (1 − s)2 , J(s) = s−1 , Nc = 1, M = 1, h− = 1, h+ = 0.5.
1
Oil saturation
0.8
0.6
entry
o
s
0.4
0.2
0
−1
so− s
+
o
−0.5
0
x
0.5
Figure 5.10 Initial oil saturation s0o .
1
106
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
The tests are done in the interval (−1, 1). Further, we present the results in terms
of the oil/nonwettig phase saturation so = 1 − s, as this is the phase for which trapping
may occur. The initial oil saturation is hat shaped (see Figure 5.10)
so (x, 0) =
s0o
:=



0,
−1 < x < −0.34,
−0.34 ≤ x ≤ −0.12,
0.9,


0,
−0.12 < x < 1.
At x = ±1 we take homogeneous boundary conditions, ∂x s(±1, t) = 0. This mimics the
situations when an oil blob in the coarse layer is displaced by water. After a certain time,
the oil reaches the interface. Note that initially no oil is present in the fine medium.
1
2
0.15
p+
p−
1.8
so−
0.4
soentry
0.05
Fo−
Fo+
0
0.2
0
−1
Pressure
0.1
0.6
Oil flux
Oil saturation
0.8
1.6
1.4
1.2
so+
−0.5
0
x
0.5
1
−1
−0.5
0
x
0.5
1
−1
−0.5
0
x
0.5
1
Figure 5.11 τ = 0, t = 0.7: oil saturation (left), oil flux (middle) and pressure difference (right).
Before discussing the results we recall that the saturation s∗ defined in (5.14) is the
limit saturation allowing for pressure continuity in the equilibrium models (τ = 0). This
also defines an entry saturation for the oil, sentry = 1 − s∗ . For the equilibrium model, oil
flows into the fine material only if so > sentry at the coarse material side of the interface,
and it remains trapped if so ≤ sentry . Figure 5.11, displaying the results obtained for
τ = 0 at t = 0.7, confirm this statement. At the coarse side of the interface, one has
entry
s−
(picture on the left) and the oil flux is 0 there (picture in the middle). This
o < s
means that no oil enters into the fine medium. Further, the pressure is discontinuous at
the interface (picture on the right).
1
0.3
0.4
3
Pressure
soentry
0.2
0.15
0
−1
0.05
s−o
−0.5
s+o
0
x
0.5
1
0
−1
2.5
2
0.1
0.2
p− p+
3.5
0.25
0.6
Oil flux
Oil saturation
0.8
Fo−
−0.5
1.5
Fo+
0
x
0.5
1
−1
−0.5
0
x
0.5
1
Figure 5.12 τ = 1, t = 0.7: oil saturation (left), oil flux (middle) and pressure difference (right).
The case τ = 1, presented in Figure 5.12, shows a different situation. In the left
entry
picture, although s−
, one still has s+
o <s
o > 0, meaning that oil has already entered
5.4. NUMERICAL SCHEMES AND EXAMPLES
107
in the fine medium. This is also confirmed by the middle picture, displaying a non-zero
the oil flux at the interface. Finally, the picture on the right shows that the pressure is
continuous.
1
0.01
p −p +
2
0.005
so−
soentry
0.4
0
1.6
1.4
−0.005
0.2
0
−1
1.8
Fo− F +
o
Pressure
0.6
Oil flux
Oil saturation
0.8
1.2
so+
−0.5
0
x
0.5
−0.01
−1
1
−0.5
0
x
0.5
1
−1
−0.5
0
x
0.5
1
Figure 5.13 τ = 0, t = 4: oil saturation (left), oil flux (middle) and pressure difference (right).
1
2.6
0.05
2.4
0.4
p− p+
2.2
Fo− F +
o
Pressure
soentry
0.03
0.02
2
1.8
1.6
0.01
s+o
0.2
0
−1
0.06
0.04
s−o
0.6
Oil flux
Oil saturation
0.8
−0.5
0
x
1.4
0
0.5
1
−0.01
−1
1.2
−0.5
0
x
0.5
1
−1
−0.5
0
x
0.5
1
Figure 5.14 τ = 1, t = 4: oil saturation (left), oil flux (middle) and pressure difference (right).
1
1
0.8
0.8
0.6
so−
soentry
0.4
0.2
0
−1
s−o
0.6
soentry
0.4
s+o
0.2
so+
−0.5
0
x
0.5
1
Oil saturation
1
0.8
Oil saturation
Oil saturation
Figure 5.13, presents the results for τ = 0 and at t = 4. Then oil has flown into the
fine medium (picture on the left). The flux and the pressure are both continuous (middle
and right pictures). At the same time, but with τ = 1, we observe that more oil has flown
into the fine media (left picture of Figure 5.14). As expected, the flux and pressure are
continuous as well.
0
−1
−0.5
0
x
0.6
soentry
0.4
0.2
0.5
1
0
−1
s−o
−0.5
s+o
0
x
0.5
1
Figure 5.15 t = 4, oil saturation: τ = 0 (left), τ = 1 (middle) and τ = 10 (right).
The results above suggest that the amount of oil flowing into the fine material increases
with τ . To understand this behaviour we compare the oil saturation obtained for τ = 0,
τ = 1 and τ = 10, all at the same time t = 4. The profiles in Figure 5.15 show that, for
τ = 0, little oil has flown into the fine media, and this amount is higher for τ = 1. In
108
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
both cases, s−
o , the oil saturation at the coarse side of the interface, already exceeds the
entry
entry saturation sentry . However, for τ = 10, s−
, but oil has still flown into the
o <s
fine material. On expects that the oil flow into the fine material will take longer for the
largest value of τ , in agreement with Corollary 5.3.1.
Finally, we observe that in the equilibrium case τ = 0 one can determine the maximal
amount of oil that can be trapped at the interface, see [42]. Having this in mind, we
choose again a hat-shaped initial saturation


−1 < x < −0.34,

0,
so (x, 0) = s0o :=
−0.34 ≤ x ≤ −0.12,
0.4,


0,
−0.12 < x < 1,
1
1
0.8
0.8
0.8
0.6
so−
soentry
0.4
0.2
Oil saturation
1
Oil saturation
Oil saturation
where the total amount of oil equals the maximal amount that can be trapped for equilibrium models. With this initial data, we compute the numerical solutions for three values
of τ , namely 0, 10 and 30. Figure 5.16 shows the results at t = 400, when practically all
solutions have reached a steady state and no oil flow is encountered anymore.
0.6
soentry
s−o
0.4
0.2
so+
0
−1
−0.5
0
x
0.5
1
0
−1
0.6
soentry
s−o
0.4
0.2
s+o
−0.5
0
x
0.5
1
0
−1
s+o
−0.5
0
x
0.5
1
Figure 5.16 t = 400, oil saturation: τ = 0 (left), τ = 10 (middle) and τ = 30 (right).
The left picture shows the result for τ = 0. In this case, the entire amount of oil is
trapped in the coarse medium, and the oil saturation s−
o is matches the entry saturation
sentry . No oil has flown at all into the fine material. As following from the middle picture,
entry
for τ = 10, one has s−
, and the oil remaining trapped in the coarse material is
o < s
less than in the equilibrium case. The situation becomes more obvious for the solution
corresponding to τ = 30. The oil saturation s−
o has decayed further, and the trapped oil
is less than in the previous cases. This is again in agreement with the analysis in Section
5.3.2.
5.5
Conclusions
We have considered a non-equilibrium model for two-phase flow in heterogeneous porous
media, where dynamic effects are included in the phase pressure difference. A simple
situation is considered, where the medium consists of two adjacent homogeneous blocks.
We obtain the conditions coupling the models in each of the two sub-domains. The first
5.5. CONCLUSIONS
109
condition is, as expected, flux continuity, whereas the second is an extended pressure
condition extending the results in [46] for the standard two-phase flow model.
In the equilibrium case, if an entry pressure model is considered, oil can flow into the
fine material only if its saturation exceeds an entry point. In the non-equilibrium case
instead, the non-wetting phase may flow even if the oil saturation at the coarse side of the
interface is below the entry point, and amount of oil remaining trapped at the interface
is less than in the case of equilibrium models.
Finally, two different numerical schemes are discussed, and different numerical experiments are presented to sustain the theoretical findings.
110
CHAPTER 5. TWO PHASE FLOW IN HETEROGENEOUS POROUS MEDIA
Chapter 6
Uniqueness of a solution for the
hysteresis model
6.1
Introduction
In this chapter, we consider the full non-equilibrium mathematical model for two-phase
flow in a porous medium. Two immiscible fluid phases (for example, water – the wetting
phase, and oil – the non-wetting one) are flowing through a porous medium occupying a
bounded, connected domain Ω ⊂ Rd (d ≥ 1). Ω̄ and ∂Ω denote the closure, respectively
boundary of Ω. We let t ∈ (0, T ] be the time variable, where T > 0 is a given maximal
time. The phase pressures are denoted by pw , pn . The non-wetting phase saturation is s.
We assume the porous medium is saturated by the two phases, so no other flowing phase
is present. This means, the wetting phase saturation is 1 − s. Then from the Darcy’s law
and mass conservation for each fluid, one obtains (see [12, 61])
−
−∂t s − ∇ · (kw (s)∇pw ) − ∇ · (kw (s)→
g1 ) = 0,
(6.1)
−
∂t s − ∇ · (kn (s)∇pn ) − ∇ · (kn (s)→
g2 ) = 0.
(6.2)
−
−
−
Here →
g1 , →
g2 ∈ Rd are the gravity vectors in direction −→
ed = (0, ..., 0, −1) ∈ Rd . kw (s), kn (s)
are the relative permeabilities, two nonlinear functions depending on s. The system is
closed by the relation between the phase pressures and saturation. Standardly, one assumes that
pn − pw = pc (s),
pc being a given increasing function in s (see [69]). These are so-called equilibrium models.
However, experiments [17, 38, 64] have invalidated such models. In particular, one should
distinguish between infiltration, when the wetting phase is displacing the non-wetting
This chapter is a collaborative work with I.S. Pop and it has been published in Computers & Mathematics with Applications, 69 (2015): 688-695.
111
112 CHAPTER 6. UNIQUENESS OF A SOLUTION FOR THE HYSTERESIS MODEL
phase, and the opposite process, drainage. The switch between the two situations above
is achieved in so-called hysteresis models. Further, experiments for which classical, equilibrium models would predict monotone infiltration saturation profiles, actually revealed
non-monotone profiles (saturation overshoot) (see [40]). Such profiles are instead allowed
in dynamic capillarity models. Both effects mentioned above are included in:
pn − pw ∈ pc (s) + γ(x)sign(∂t s) + τ ∂t s.
(6.3)
The second term on the right hand side models play-type hysteresis (see [13, 103]), while
the last one accounts for dynamic capillarity (see [60]). In (6.3), γ ≥ 0, τ > 0 are given,
sign is the multi-valued graph:


if ξ > 0,

1
sign(ξ) =
if ξ = 0,
[−1, 1]


−1
(6.4)
if ξ < 0.
Following [68], we also consider the Lipschitz continuous function Ψ : R × Ω → R
Ψ(ξ, x) =

ξ−γ(x)

 τ

0


 ξ+γ(x)
τ
for
ξ > γ(x),
for
ξ ∈ [−γ(x), γ(x)],
for
ξ < −γ(x).
(6.5)
Clearly, for a. e. ξ ∈ R, one has
0 ≤ ∂ξ Ψ(ξ, x) ≤ 1/τ.
(6.6)
∂t s = Ψ(pn − pw − pc (s), x).
(6.7)
With this, (6.3) rewrites
The model (6.1), (6.2), (6.7) is complemented by the initial and boundary conditions:
(6.8)
s(0, ·) = s0 ,
pn = pw = 0
at ∂Ω,
for all
t ≥ 0.
(6.9)
Remark 6.1.1. Other boundary conditions are possible (see Remark 6.2.1), but for clarity,
we restrict the presentation to (6.9).
We make the following assumptions:
A1: The functions kw , kn : R → R are Lipschitz continuous. Further, δ, Mk > 0 exist,
such that δ ≤ kn (s), kw (s) ≤ Mk < +∞, for all s ∈ R.
A2: pc (·) ∈ C 1 (R) is increasing and Lipschitz continuous, there exist mp , Mp > 0, such
that mp ≤ kpc kLip ≤ Mp < +∞, for all s ∈ R.
6.2. UNIQUENESS OF THE WEAK SOLUTION
113
A3: Ω is a C 1,β domain with 0 < β ≤ 1.
A4: γ(x) ∈ C 0,1 (Ω̄).
A5: s0 ∈ C 0,β (Ω̄).
Remark 6.1.2. Commonly, the permeabilities and capillary pressure model encountered
in the literature ( [22, 24]) are
kn (s) = sp ,
kw (s) = (1 − s)q ,
and
1
pc (s) = (1 − s)− µ ,
with p, q > 1,
µ > 1, for s ∈ [0, 1].
Then (A1) and (A2) are not satisfied when s approaches 0 or 1. We consider here a
regularized approximation of these functions.
Below we use standard notation in the theory of partial differential equations. For any
h ∈ C 0,β : Ω → R, we define the norm
khkC(Ω̄) := sup|h(x)|,
x∈Ω
the β th - Hölder semi-norm of h
[h]C 0,β (Ω̄) :=
|h(x) − h(y)|
,
|x − y|β
x,y∈Ω,x6=y
sup
and the β th - Hölder norm of h
khkC 0,β (Ω̄) := khkC(Ω̄) + [h]C 0,β (Ω̄) .
The existence of a solution for such models is studied in [68, 71, 96], for both onephase and two-phase models. The analysis covers degenerate cases, i.e. when k(s) = 0,
or pc (s) = ∞ for some s. For γ = 0, existence for such models is obtained in [52, 77].
Uniqueness results are much fewer. In [52], this is obtained, but only for a scalar model
with linear higher order term. Also for a scalar nonlinear model, but in a form that does
not match such porous media flow models, uniqueness is obtained in [14]. Closest to the
present work is [30], where uniqueness is obtained for a scalar, non-degenerate model.
Here we prove the uniqueness of a (weak) solution to the model (6.1), (6.2) and
(6.7), describing two-phase porous media flow. The proof uses essential bounds for the
pressures, which are obtained in the first part.
6.2
Uniqueness of the weak solution
In this section, we provide a rigorous proof of the uniqueness of weak solutions to (6.1),
(6.2), (6.7). We use common notations for function spaces, namely, L2 , W 1,2 , W01,2 ,
114 CHAPTER 6. UNIQUENESS OF A SOLUTION FOR THE HYSTERESIS MODEL
and Bochner space L2 (0, T ; X). We follow [68] and consider weak solutions solving
Problem P: Given s0 satisfying (A5), find pn ∈ L2 (0, T ; W01,2 (Ω)), pw ∈ L2 (0, T ; W01,2 (Ω))
and s ∈ W 1,2 (0, T ; L2 (Ω)), such that s(·, 0) = s0 in Ω, and
−
(∂t s, φ) + (kn (s)∇pn , ∇φ) + (kn (s)→
g2 , ∇φ) = 0,
(6.10)
−
(−∂t s, ψ) + (kw (s)∇pw , ∇ψ) + (kw (s)→
g1 , ∇ψ) = 0,
(6.11)
(∂t s, ρ) = (Ψ(pn − pw − pc (s), x), ρ),
(6.12)
for any φ, ψ ∈ L2 (0, T ; W01,2 (Ω)) and ρ ∈ L2 (0, T ; L2 (Ω)).
In [68], the hysteresis is modeled by considering (6.3) valid a.e. This immediately
implies that (6.7) holds a.e. and further (6.12). The weak solution introduced by Problem
P is, in fact, equivalent to the weak solution concept in [68]. Here to prove the uniqueness
of the weak solution, some intermediate results are needed. We start with essential bounds
for the gradients of pn and pw .
Lemma 6.2.1. Let (pn , pw , s) be a weak solution to Problem Pe , then one has ∇pn , ∇pw ∈
L∞ ((0, T ] × Ω).
Proof. First we show that k∇pn kL2 (Ω) ∈ L∞ (0, T ) and k∇pw kL2 (Ω) ∈ L∞ (0, T ). Taking
φ = pn in (6.10), ψ = pw in (6.11) and adding the resulting equations give
p
kn (s)∇pn k2L2 (Ω) + k kw (s)∇pw k2L2 (Ω)
−
−
+ (kn (s)→
g2 , ∇pn ) + (kw (s)→
g1 , ∇pw ) = 0.
(∂t s, pn − pw ) + k
p
(6.13)
For the first term of (6.13), we note that (6.3) holds almost everywhere. Then since
sign(ξ)ξ ≥ 0 for any ξ ∈ R, one has
Z
Z
Z
Z
1
τ
|pc (s)|2 dx.
∂t s(pn − pw ) ≥
τ |∂t s|2 dx +
pc (s)∂t sdx ≥ k∂t sk2L2 (Ω) −
2
2τ
Ω
Ω
Ω
Ω
(6.14)
Further, since s ∈ L∞ (0, T ; L2 (Ω)) (see [68,71]), by using the Cauchy-Schwarz inequality,
(A1) and (A2), (6.13) gives
k∂t sk2L2 (Ω) + k∇pn k2L2 (Ω) + k∇pw k2L2 (Ω) ≤ C,
for a.e. t ∈ (0, T ].
(6.15)
Then for almost every time t, we have the following elliptic equations with respect to
pressure pn , pw :
−
−∇ · (kn (s)∇pn ) = −∂t s + ∇ · (kn (s)→
g2 ),
(6.16)
−
−∇ · (kw (s)∇pw ) = ∂t s + ∇ · (kw (s)→
g1 ).
(6.17)
−
−
Then, since for almost every time t, we have pn , pw ∈ W01,2 (Ω), kn (s)→
g2 , kw (s)→
g1 ∈
6.2. UNIQUENESS OF THE WEAK SOLUTION
115
L∞ (Ω), ∂t s ∈ L2 (Ω), and (A3), by using Theorem 14.1 in [70] gives for almost every t,
kpn kC 0,α (Ω̄) + kpw kC 0,α (Ω̄) ≤ C,
(6.18)
here α is independent on t.
Furthermore, we use pn , pw , s to define
P = pn − pw − pc (s).
(6.19)
Based on this, we also define χ:
( Ψ(P
1 ,x)−Ψ(P2 ,x)
P1 −P2
χ(P1 ,P2 ,x) =
if P1 6= P2 ,
if P1 = P2 .
0,
(6.20)
Obviously, one has 0 ≤ χ(P1 ,P2 ,x) ≤ τ1 .
Since pc ∈ C 1 , from (6.7), for almost every x, y ∈ Ω (x 6= y) and t > 0, a ζ depending
on x, y, t exists, such that
Ψ P (t, y), x − Ψ P (t, y), y
s(t, x) − s(t, y) Ψ P (x, t), x − Ψ P (t, y), x
=
+
∂t
|x − y|α
|x − y|α
|x − y|α
P (t, x) − P (t, y) Ψ(P (t, y), x) − Ψ(P (t, y), y)
=χ(P (x,t),P (y,t),x)
+
|x − y|α
|x − y|α
p (t, x) − p (t, y) p (t, x) − p (t, y)
n
n
w
w
−
=χ(P (x,t),P (y,t),x)
|x − y|α
|x − y|α
0
s(t, x) − s(t, y) − pc (ζ) ·
+ Γ(t, x, y),
(6.21)
|x − y|α
where
Γ(t, x, y) =
Ψ(P (t, y), x) − Ψ(P (t, y), y)
.
|x − y|α
(6.22)
By (A2) - (A4), and since pn , pw ∈ C 0,α (Ω̄) for almost every t, one gets
|Γ(t, x, y)|+
|pn (t, x) − pn (t, y)|
|pw (t, x) − pw (t, y)|
+ sup
≤ C, (6.23)
α
|x
−
y|
|x − y|α
x,y∈Ω,x6=y
x,y∈Ω,x6=y
sup
for some C > 0 not depending on t, x, y.
With w : (0, T ] × Ω2 → R as
w=
s(t, x) − s(t, y)
,
|x − y|α
(6.24)
one has
∂t w = f w + g,
(6.25)
116 CHAPTER 6. UNIQUENESS OF A SOLUTION FOR THE HYSTERESIS MODEL
where
0
f (t, x) = −χ(P (x,t),P (y,t),x) · pc (ζ),
and
g(t, x) = χ(P (x,t),P (y,t),x) (
pn (t, x) − pn (t, y) pw (t, x) − pw (t, y)
−
) + Γ(t, x, y).
|x − y|α
|x − y|α
Note that (6.20), (6.23) and (A2) give f, g ∈ L∞ ((0, T ] × Ω). Multiplying (6.25) by w
and integrating from 0 to t lead to
1 2
w (t) =
2
Z
t
f w2 (z)dz +
0
t
Z
gw(z)dz +
0
1
2
s0 (x) − s0 (y)
|x − y|α
2
.
(6.26)
Since f, g ∈ L∞ ((0, T ] × Ω) and s0 ∈ C 0,α (Ω̄) from (A5), we have
w2 (t) ≤ C(1 +
t
Z
w2 dz),
for every t.
(6.27)
0
Using Gronwall’s inequality yields w ≤ C, implying that
|s(t, x) − s(t, y)|
≤ C,
|x − y|α
for almost every x, y ∈ Ω, for every t.
(6.28)
Let Ωc be the subset of Ω, where (6.28) holds everywhere. Clearly, Ω\Ωc is zero measured.
For any x ∈ Ω\Ωc , we consider a sequence {xn }n∈N ∈ Ωc converging to x, and define
s(t, x) = lim s(t, xn ).
n→+∞
(6.29)
In the view of (6.28), s(t, x) does not depend on the choice of {xn }n∈N . With this choice,
s ∈ C 0,α (Ω̄) (see [48]).
Finally, by Theorem 8.33 and Corollary 8.35 in [59] (see also [35]), and recalling the
uniform time estimates in (6.18), we get for almost every t:
kpn kC 1,α (Ω̄) ≤ C(kpn kC(Ω̄) + kΨkC(Ω̄) + kkn (s)kC 0,α (Ω̄) ),
(6.30)
kpw kC 1,α (Ω̄) ≤ C(kpw kC(Ω̄) + kΨkC(Ω̄) + kkw (s)kC 0,α (Ω̄) ),
(6.31)
here C is independent on t, then, the above estimates imply ∇pn , ∇pw ∈ L∞ ((0, T ] ×
Ω).
Remark 6.2.1. Actually, the uniqueness proof only requires that pn , pw have L∞ (Ω ×
(0, T ]) gradients. Such results can be obtained with other boundary conditions than
homogeneous Dirichlet. We refer to Theorem 16.1 and Theorem 16.2 in [70] for similar
results, in the case of homogeneous Neumann boundary conditions.
Here we show the solution to Problem P is unique. To do so, we use the weak solution
6.2. UNIQUENESS OF THE WEAK SOLUTION
117
pair (Ga−b , G̃a−b ) to the elliptic system (see [28, 30, 52]):
−∇ · (kn (b)∇Ga−b ) + χ(a,b,x) (G̃a−b + Ga−b ) = χ(a,b,x) (a − b),
(6.32)
−∇ · (kw (b)∇G̃a−b ) + χ(a,b,x) (Ga−b + G̃a−b ) = χ(a,b,x) (a − b),
(6.33)
Ga−b = G̃a−b = 0,
at ∂Ω,
(6.34)
with a, b ∈ L2 (Ω).
The existence and uniqueness follow the Lax-Milgram Theorem. Using the properties
of χ, kw , kn , one immediately gets
kGa−b k2W 1,2 (Ω) ≤ Cka − bk2L2 (Ω) ,
and kG̃a−b k2W 1,2 (Ω) ≤ Cka − bk2L2 (Ω) .
(6.35)
These functions are introduced because the uniqueness proof below is based on contradiction. Specially, assuming the existence of two solutions, we estimate their difference. In
doing so, we need to deal with difference as in the derivatives of nonlinearities applied to
these solutions, or when these appear as multiplications. The functions introduced above
will be used in these estimates.
Theorem 6.2.1. Problem P has at most one solution.
Proof. Let (u, pun , puw ) and (v, pvn , pvw ) be the two solutions of Problem P, and since Ψ is
Lipschitz, for almost every (x, t) ∈ ΩT , then one has
(∂t (u − v), φ) + (kn (v)∇(pun − pvn ), ∇φ)
−
+((kn (u) − kn (v))∇pun , ∇φ) + ((kn (u) − kn (v))→
g2 , ∇φ) = 0,
(6.36)
− (∂t (u − v), ψ) + (kw (v)∇(puw − pvw ), ∇ψ)
−
+ ((kw (u) − kw (v))∇puw , ∇ψ) + ((kw (u) − kw (v))→
g1 , ∇ψ) = 0,
(6.37)
(∂t (u − v), ρ) = χ(u,v,x) (pun − pvn ) − (puw − pvw ) − (pc (u) − pc (v)) , ρ ,
(6.38)
and
for any φ, ψ ∈ L2 (0, T ; W01,2 (Ω)), ρ ∈ L2 (0, T ; L2 (Ω)).
According to the functions defined in (6.32) - (6.33), and the estimates in (6.35), we
know that Gu−v , G̃u−v ∈ W01,2 (Ω) and
(χ(u,v,x) G̃u−v , λ) + (χ(u,v,x) Gu−v , λ) + (kn (v)∇Gu−v , ∇λ) = (χ(u,v,x) (u − v), λ),
(6.39)
(χ(u,v,x) Gu−v , λ̃) + (χ(u,v,x) G̃u−v , λ̃) + (kw (v)∇G̃u−v , ∇λ̃) = (χ(u,v,x) (u − v), λ̃),
(6.40)
1,2
for any λ, λ̃ ∈ W0 (Ω).
118 CHAPTER 6. UNIQUENESS OF A SOLUTION FOR THE HYSTERESIS MODEL
Testing with φ = Gu−v in (6.36), and ψ = G̃u−v in (6.37), one has
(∂t (u − v), Gu−v ) + (kn (v)∇(pun − pvn ), ∇Gu−v )
−
+((kn (u) − kn (v))∇pun , ∇Gu−v ) + ((kn (u) − kn (v))→
g2 , ∇Gu−v ) = 0,
(6.41)
(∂t (u − v), G̃u−v ) − (kw (v)∇(puw − pvw ), ∇G̃u−v )
−
−((kw (u) − kw (v))∇puw , ∇G̃u−v ) − ((kw (u) − kw (v))→
g1 , ∇G̃u−v ) = 0.
(6.42)
Choosing λ = pun − pvn in (6.39) and λ̃ = puw − pvw in (6.40) gives
(kn (v)∇Gu−v , ∇(pun − pvn ))
=(χ(u,v,x) (u − v), pun − pvn ) − (χ(u,v,x) G̃u−v , pun − pvn ) − (χ(u,v,x) Gu−v , pun − pvn ),
(6.43)
(kw (v)∇G̃u−v , ∇(puw − pvw ))
=(χ(u,v,x) (u − v), puw − pvw ) − (χ(u,v,x) Gu−v , puw − pvw ) − (χ(u,v,x) G̃u−v , puw − pvw ).
(6.44)
Substitute (6.43) into (6.41) and (6.44) into (6.42) to replace the terms (kn (v)∇Gu−v ,
∇(pun − pvn )) and (kw (v)∇G̃u−v , ∇(puw − pvw )), we find that
(∂t (u − v), Gu−v ) − (χ(u,v,x) G̃u−v , pun − pvn )
−(χ(u,v,x) Gu−v , pun − pvn ) + (χ(u,v,x) (u − v), pun − pvn )
−
+((k (u) − k (v))∇pu , ∇G
) + ((k (u) − k (v))→
g , ∇G
n
n
n
u−v
n
n
2
u−v )
= 0,
(6.45)
= 0.
(6.46)
(∂t (u − v), G̃u−v ) + (χ(u,v,x) Gu−v , puw − pvw )
+(χ(u,v,x) G̃u−v , puw − pvw ) − (χ(u,v,x) (u − v), puw − pvw )
−
−((k (u) − k (v))∇pu , ∇G̃
) − ((k (u) − k (v))→
g , ∇G̃
w
w
w
u−v
w
w
1
u−v )
Taking ρ = u − v into (6.38) yields
− (χ(u,v,x) (pun − pvn ), u − v) + (χ(u,v,x) (puw − pvw ), u − v)
= − (∂t (u − v), u − v) − (χ(u,v,x) (pc (u) − pc (v)), u − v).
(6.47)
Adding (6.45), (6.46), and (6.47), we eliminate (χ(u,v,x) (pun − pvn ), u − v) − (χ(u,v,x) (puw −
6.2. UNIQUENESS OF THE WEAK SOLUTION
119
pvw ), u − v) and have the following result:
(∂t (u − v), Gu−v ) + (χ(u,v,x) Gu−v , puw − pvw ) − (χ(u,v,x) Gu−v , pun − pvn )
+(∂t (u − v), G̃u−v ) + (χ(u,v,x) G̃u−v , puw − pvw ) − (χ(u,v,x) G̃u−v , pun − pvn )
+(∂t (u − v), u − v) + (χ(u,v,x) (pc (u) − pc (v)), u − v) + ((kn (u) − kn (v))∇puw , ∇Gu−v )
−
−((k (u) − k (v))∇pu , ∇G̃
) + ((k (u) − k (v))→
g , ∇G
)
w
w
n
u−v
n
n
u−v
2
−
−((kw (u) − kw (v))→
g1 , ∇G̃u−v ) = 0.
(6.48)
Further, taking ρ = Gu−v and ρ = G̃u−v in (6.38), respectively, one has
(∂t (u − v), Gu−v ) + (χ(u,v,x) Gu−v , puw − pvw ) − (χ(u,v,x) Gu−v , pun − pvn )
(6.49)
= − (χ(u,v,x) Gu−v , pc (u) − pc (v)),
and
(∂t (u − v), G̃u−v ) + (χ(u,v,x) G̃u−v , puw − pvw ) − (χ(u,v,x) G̃u−v , pun − pvn )
(6.50)
= − (χ(u,v,x) G̃u−v , pc (u) − pc (v)).
Substituting (6.49) and (6.50) into (6.48) leads to
(∂t (u − v), u − v) + (χ(u,v,x) (u − v), pc (u) − pc (v))
−(χ(u,v,x) Gu−v , pc (u) − pc (v)) − (χ(u,v,x) G̃u−v , pc (u) − pc (v))
+((kn (u) − kn (v))∇pun , ∇Gu−v ) − ((kw (u) − kw (v))∇puw , ∇G̃u−v )
−
−
+((k (u) − k (v))→
g , ∇G
) − ((k (u) − k (v))→
g , ∇G̃
) = 0.
n
n
u−v
2
w
w
1
u−v
For any t ∈ (0, T ], integrating (6.51) from 0 to t gives us
Z
t
Z
t
(∂t (u − v), u − v) +
(χ(u,v,x) (u − v), pc (u) − pc (v))
Z t
t
−
(χ(u,v,x) Gu−v , pc (u) − pc (v)) −
(χ(u,v,x) G̃u−v , pc (u) − pc (v))
0
0
Z t
Z t
+
((kn (u) − kn (v))∇pun , ∇Gu−v ) −
((kw (u) − kw (v))∇puw , ∇G̃u−v )
0
0
Z t
Z t
−
−
+
((kn (u) − kn (v))→
g2 , ∇Gu−v ) −
((kw (u) − kw (v))→
g1 , ∇G̃u−v ) = 0.
0
0
Z
0
(6.51)
0
We proceed by estimating each term called T1 , T2 , T3 , T4 , T5 , T6 , T7 , T8 . For T1 , since
u(0) = v(0) = s0 , one has
Z tZ
∂t (u − v)(u − v)dxdz =
0
Ω
1
k(u − v)(t)k2L2 (Ω) .
2
(6.52)
120 CHAPTER 6. UNIQUENESS OF A SOLUTION FOR THE HYSTERESIS MODEL
For the terms T2 , T3 , T4 , according to (6.6), (A2) and (6.35), we obtain
Z
t
(χ(u,v,x) (u − v), pc (u) − pc (v)) ≥ 0,
(6.53)
0
t
Z
|
t
Z
(χ(u,v,x) Gu−v , pc (u) − pc (v))| ≤ C
0
(6.54)
k(u − v)(·, z)k2L2 (Ω) dz.
(6.55)
0
t
Z
k(u − v)(·, z)k2L2 (Ω) dz,
t
Z
(χ(u,v,x) G̃u−v , pc (u) − pc (v))| ≤ C
|
0
0
Furthermore, for T5 and T6 , since ∇pn , ∇pw ∈ L∞ ((0, T ] × Ω), then by using Cauchy Schwarz inequality, one has the following estimates:
t
Z
((kn (u) − kn (v))∇pun , ∇Gu−v )| ≤ C
|
0
k(u − v)(·, z)k2L2 (Ω) dz,
(6.56)
k(u − v)(·, z)k2L2 (Ω) dz.
(6.57)
0
t
Z
t
Z
t
Z
((kw (u) − kw (v))∇puw , ∇G̃u−v )| ≤ C
|
0
0
Similarly, one also has the estimates for T7 and T8 :
t
Z
|
−
((kn (u) − kn (v))→
g2 , ∇Gu−v )| ≤ C
0
Z
|
t
t
Z
k(u − v)(·, z)k2L2 (Ω) dz,
(6.58)
k(u − v)(·, z)k2L2 (Ω) dz.
(6.59)
0
−
((kw (u) − kw (v))→
g1 , ∇G̃u−v )| ≤ C
0
Z
t
0
Finally, summarizing the above leads to
k(u −
v)(·, t)k2L2 (Ω)
Z
≤C
t
k(u − v)(·, z)k2L2 (Ω) dz.
(6.60)
0
By Gronwall’s inequality, we have k(u − v)(·, t)k2L2 (Ω) = 0. Since t is arbitrary, this gives
u = v a.e. in Ω and for all t ∈ (0, T ].
To show that pun = pvn , puw = pvw , we use (6.36) and (6.37). Since u = v, one has
(kn (u)∇(pun − pvn ), ∇φ) = 0,
(6.61)
(kw (u)∇(puw − pvw ), ∇ψ) = 0,
(6.62)
for any φ, ψ ∈ W01,2 (Ω), for almost every t. The rest of the proof follows straightforwardly
by taking φ = pun − pvn , ψ = puw − pvw , and recalling that pun , pvn , puw , pvw have equal traces
on ∂Ω.
6.3. CONCLUSIONS
6.3
121
Conclusions
In this chapter, we have proved the uniqueness of weak solutions to a non-degenerate
system which models two-phase flow in porous media including hysteresis and dynamic
effects in the capillary pressure. In doing so, we use arguments based on Green’s function.
122 CHAPTER 6. UNIQUENESS OF A SOLUTION FOR THE HYSTERESIS MODEL
Chapter 7
Moisture transport in concrete
7.1
Mathematical model
In this chapter, the moisture transport (one phase flow) in concrete is described based on
the Darcy’s law and mass balance
k̄
v = − kr ∇pc
µ
(Darcy),
(7.1)
where v is the volumetric velocity (m/s), k̄ the intrinsic permeability of the concrete
(m2 ), kr the relative permeability (−), µ the viscosity (kg · m−1 · s−2 ).
∂
(φs) + ∇ · v = 0
∂t
(mass balance),
(7.2)
where φ is the porosity of the concrete (−), s the water saturation (−), and t the time
(s). Commonly, kr and pc are monotone functions of the saturation s, see [86].
By substituting (7.1) into (7.2), we have
∂
k̄
(φs) − ∇ · ( kr ∇pc ) = 0.
∂t
µ
(7.3)
Furthermore, the following is assumed in this chapter:
• The sample is homogeneous.
• There is no flow at the boundaries except the boundary at the right.
• The initial saturation is constant.
This chapter is a collaborative work with A. Taher, I. Pop, A.J.J van der Zanden, H.J.H Brouwers
and it has been published in Chemistry and Materials Research, 5(2013): 86-90.
123
124
CHAPTER 7. MOISTURE TRANSPORT IN CONCRETE
For the boundary at the right, a periodic repetition of wetting and drying cycles is assumed.
In view of these assumptions, the lateral flow is 0. This allows reducing the model to one
dimension. The sample occupies then the interval.
The following conditions are imposed
at t = 0,
s = si ,
k̄ ∂pc
kr
= 0,
µ ∂x
s = sb (t),
in (0, L),
x = 0,
x = L,
t > 0,
t > 0,
(7.4)
(7.5)
(7.6)
where si is the initial saturation, sb (t) the saturation at the right boundary. In the context
of a one-dimensional model, (7.3) becomes
!
∂ k̄ ∂pc
∂
(φs) −
kr
= 0.
(7.7)
∂t
∂x µ ∂x
To make the equation dimensionless, the following notations are introduced
x :=
x
t
pc
, t :=
, pc := ,
L
Tr
P
(7.8)
where L, Tr , P are characteristic values for the length, time, and capillary pressure.
Substitute (7.8) into (7.7) and set
then (7.7) becomes
Tr
φµ
,
=
2
L
k̄P
(7.9)
!
∂s
∂
∂s
=
D(s)
.
∂t
∂x
∂x
(7.10)
∂pc (s)
.
∂s
(7.11)
Here
D(s) = kr (s) ·
By using the Kirchhoff transformation
Z
β(s) =
s
D(z)dz,
(7.12)
0
(7.10) becomes
∂s
= ∂xx β(s),
∂t
while the initial and boundary conditions are
s(x, 0) = si ,
∂s
(0, t) = 0,
∂x
(7.13)
s(1, t) = sb (t),
(7.14)
7.2. APPROACH
125
where si is a constant number between 0 (in case of dried concrete) and 1 (in case of full
saturated concrete). sb (t) is a periodic function, which simulates wetting/drying cycles
sb (t) =
(
1
0
t ∈ (0, Tw ] + k · Tp ,
t ∈ (Tw , Td ] + k · Tp ,
(7.15)
where Td is the dimensionless drying time, Tw is the dimensionless wetting time, Tp =
Td + Tw , is the period of one cycle, and k is any natural numbers.
7.2
Approach
Two modeling approaches for the moisture transport in concrete are compared in this
chapter. The first model uses the same diffusion coefficient for both wetting and drying
phase. The second model uses two diffusion coefficients, one for the drying D− (s) and
one for the wetting D+ (s).
In the literature [74], these following coefficients are determined experimentally
D+ (s) = 10−10 · e6s ,
D− (s) = 10−10 · 0.025 +
0.975 1−s 6 ,
1 + ( 0.208
)
(7.16)
(7.17)
which are plotted in Figure 7.1.
7.2.1
Standard model: one diffusion coefficient
A standard model considers the same diffusion coefficient for both wetting and drying. In
view of (7.16) and (7.17), the following averages are considered
D+ (s) + D− (s)
,
2
p
Dg (s) = D+ (s) · D− (s).
Da (s) =
(7.18)
(7.19)
Here Da (s) is the arithmetic average of the diffusion coefficients, Dg (s) the geometric
average of the diffusion coefficients.
126
CHAPTER 7. MOISTURE TRANSPORT IN CONCRETE
Figure 7.1 Various diffusion coefficients as a function of saturation.
The coefficients introduced in (7.18) and (7.19) are then used for the Kirchhoff transformation in (7.10). For the resulting model, (7.13), an implicit numerical scheme [88] is
combined with a linear iterative procedure [90].
7.2.2
Hysteretic model
Here the hysteretic model in [13] is adapted
∂s
= ∂xx p.
∂t
(7.20)
In the above, p has to include the switch between two diffusion coefficients, in this case,
p reads
p ∈ pc (s) + γ(s)sign(∂t s),
(7.21)
where pc (s), γ(s) and sign(∂t s) are defined as
pc (s) =
β + (s) + β − (s)
,
2
(7.22)
γ(s) =
β + (s) − β − (s)
,
2
(7.23)
and
sign(∂t s) =
(
1
∂t s > 0,
−1
∂t s < 0.
(7.24)
β +/− (s) are the Kirchhoff transformations for the wetting/drying diffusion coefficients
β
+/−
Z
(s) =
0
s
D+/− (z)dz.
(7.25)
7.2. APPROACH
127
When solving the system, sign(∂t s) is replaced by the regularization signδ (∂t s) (see Figure
7.2)

δ∂t s + δ 2 − 1
if ∂t s < −δ,


∂
s
t
signδ (∂t s)
(7.26)
if − δ < ∂t s < δ,
δ


δ∂ s + δ 2 + 1
if ∂ s > δ,
t
t
where 0 < δ << 1 is the regularization parameter.
1
1.5
1
0.5
signδ
sign
0.5
0
0
−0.5
−0.5
−1
−1
−2
−1
0
z
1
−1.5
−2
2
(a)
−1
0
z
1
2
(b)
Figure 7.2 The graphs of (a.) sign(z) and (b.) signδ (z).
Solving the above equations directly using finite difference method leads to incorrect
results. Alternatively, the inverse φδ (r) of the regularized signδ (∂t s) is used as considered
in [97]

r+1

if r < −δ,

−δ + δ
(7.27)
φδ (r) :=
δr
if − δ < r < δ,


δ + r−1
if r > δ.
δ
For the time discretization, let ∆t be the time step, and pm ≈ p(m · ∆t), sm ≈
s(m · ∆t). The implicit discretization of (7.20) and (7.21) are:
sm+1 − sm
= ∂xx pm+1 ,
∆t
(7.28)
pm+1 − p (sm+1 ) sm+1 − sm
c
= φδ
,
∆t
γ(sm )
(7.29)
where m = 0, 1, ..., s0 = si . Further, at x = 0, ∂x pm+1 = 0 and at x = L, sm+1 =
sb ((m + 1)∆t).
In the equations above, the right side of (7.29) is implicit for sm+1 , due to the
nonlinear function of pc (s). Therefore, pc (sm+1 ) is simplified for a linear function as
investigated in [92] by a Taylor expansion given by
0
pc (sm+1 ) ≈ pc (sm ) + pc (sm ) · (sm+1 − sm ).
(7.30)
128
CHAPTER 7. MOISTURE TRANSPORT IN CONCRETE
Then the discretized form becomes
sm+1 − sm
= ∂xx pm+1 ,
∆t
(7.31)
0
pm+1 − p (sm ) − p (sm ) · (sm+1 − sm ) sm+1 − sm
c
c
.
= φδ
∆t
γ(sm )
(7.32)
For discretizing the spatial derivatives, finite difference method is applied.
7.2.3
Numerical results
The model is implemented in Matlab. Here, only one cycle is computed, which is one
day for wetting and six days for drying. The initial condition (si ) in this case is assumed
to be 0.5 saturation. Further, the following is used in the numerical scheme: δ = 10−5 ,
∆x = 10−3 and ∆t = 10−6 . The comparison between the results of the standard model
with one diffusion coefficient and the results of the hysteretic model with two diffusion
coefficients are shown in Figure 7.3.
Figure 7.3 Comparison of the models.
The results show that there is a large difference between the three models. To validate
the models, experimental work is needed, which will be explained in the next section.
7.3
Experiment
Mortar specimens with a water cement ratio of 0.5 and cement type of CEM I 42.5 N are
used in this chapter to validate the models. Mortar specimens are prepared by casting
them in PVC tubes with a diameter of 100 mm. After one day, the mortar is demoulded
7.3. EXPERIMENT
129
and cured for 28 days. The side of the specimens is sealed with epoxy to ensure a one
dimensional flux. At one of the two open surfaces of the specimen, the condition is
changed to simulate cycles. Wetting is simulated by contacting the surface with water
during one day and drying by blowing dry air with a certain flow at the surface during
6 days. After one drying/wetting cycle (one week), the moisture content is determined
by weighing the specimen. This mass is compared with the calculated mass from the
moisture profiles of the models for validation. Figure 7.4 shows the experimental setup.
Figure 7.4 Sealed mortar specimens encountered with wetting cycle for one day and drying
cycle for six days.
130
CHAPTER 7. MOISTURE TRANSPORT IN CONCRETE
Summary
Mathematical and Numerical Analysis for Non-Equilibrium
Two Phase Flow Models in Porous Media
This thesis deals with models for the two phase flow in porous media under non-equilibrium
conditions. Porous media models are derived by the mass balance and Darcy’s law, and
completed by the relationship between phase saturation and pressures. For the standard
case, these constitutive equations are obtained under equilibrium assumptions: e.g. for
a given medium and at a given phase saturation, the capillary pressure is fixed and the
so-called capillary pressure is a monotone function of saturation.
This obtained equilibrium approach is invalidated by many experimental results. For
example, non-monotonic saturation profiles are observed for infiltration processes in a
homogeneous medium whereas standard models rule out such profiles. In such a case,
non-equilibrium models which include hysteretic effects are proposed. There are at least
two kinds of hysteresis: the play-type hysteresis and the one including dynamic effects.
Without play-type hysteresis and under the assumption of a known total flow, the original
model has been transformed into a scalar model — a pseudo parabolic equation. In such
a case, the existence of weak solutions for this model is a known result. However, the
uniqueness was still an open question. This is answered positively in Chapter 2 of this work.
Next, for the original two phase flow model including dynamic effects, we investigate an
elliptic-parabolic system. We do this even in the degenerate case, when nonlinearities
multiplying the higher order term in the model may vanish, depending on the unknowns
of the model. We first deal with the regularized model when all nonlinearities are assumed
to be bounded away from 0 by a factor δ. Finally, the existence of weak solutions in degenerate case is obtained when letting the regularization parameter go to zero.
In Chapter 4, we propose a multipoint flux approximation finite volume scheme to approximate the solution of the models. By giving energy estimates and compactness arguments,
the convergence of the numerical approximation to the weak solution has been proved, as
131
132
SUMMARY
the mesh size and time step tend to zero. Finally, we present some numerical results to
confirm the convergence proved rigorously. Moreover, we calculate the numerical errors
for saturation and gradient of pressures, which show that they all have a first order convergence, even in the anisotropic cases and for irregular meshes. At the end, we present
some numerical experiments to simulate the saturation overshoot, which has been observed both experimentally and analytically.
In Chapter 5, we investigate the two phase flow in heterogeneous media, where dynamic
effects are taken into account. The two phase flow model reduces to one equation in the
case of one spatial dimension. We consider a domain which combines two homogeneous
subdomains. After doing so, the behavior of the solution compared to the one for equilibrium models is discussed and illustrated numerically.
For the two-phase flow model including dynamic effects and play-type hysteresis, the existence of weak solutions for such non-equilibrium models under non-degenerate cases is
also a known result, but the uniqueness was again an open question. In Chapter 6, we
give the proof for the uniqueness. The difficulty is the strong coupling of the two phase
flow differential equations and the play-type hysteresis term. We prove the uniqueness of
a weak solution by means of an auxiliary dual system.
In the last chapter of this thesis, we consider a practical situation related to the durability
analysis of concrete. Here we investigate two different diffusion coefficients which should
be considered while drainage, respectively, imbibition. Finally, we propose a model including play-type hysteresis in diffusion coefficients and give some numerical calculations for
this model.
Curriculum Vitae
Xiulei Cao was born on April 02, 1985 in Jilin, China. After finishing his schooling from
Qianan 7th School, he started his Bachelor studies in Computational Mathematics in
Jilin University, Jilin, China. After his bachelor studies, he pursued his Master degree
of Computational Mathematics in Jilin University. During his Master studies, he was
awarded first prize for two times.
From October 2011, he started a PhD project at the Center for Analysis, Scientific computing and Applications (CASA) of the Department of Mathematics and Computer Science,
Eindhoven University of Technology, the Netherlands under the supervision of Prof. I. S.
Pop and Prof. C. J. van Duijn. The results of this PhD research are presented in this
thesis.
133
134
CURRICULUM VITAE
List of Publications
Journal articles:
1. A. Taher, X. Cao, I.S. Pop, A.J.J. van der Zanden and H.J.H. Brouwers. Moisture transport in concrete during wetting/drying cycles. Chemistry and Materials
Research. 5:86-90, 2013.
2. X. Cao and I.S. Pop. Uniqueness of weak solutions for a pseudo-parabolic equation
modeling two phase flow in porous media. Appl. Math. Lett., 46:25-30, 2015.
3. X. Cao and I.S. Pop. Two-phase porous media flows with dynamic capillary effects
and hysteresis: uniqueness of weak solutions. Comput. Math. Appl., 69(7):688695, 2015.
4. C.J. van Duijn, X. Cao and I.S. Pop. Two-phase flow in porous media: dynamic
capillarity and heterogeneous media. Transp. Porous Med., 110(3):1-26, 2015.
5. X. Cao and I.S. Pop. Degenerate two-phase porous media flow model with dynamic
capillarity. J. Differ. Equ., 260(3):2418-2456, 2016.
Preprints/Work in process:
1. X. Cao, S.F. Nemadjieu and I.S. Pop. A multipoint flux approximation finite volume
scheme for two phase porous media flow with dynamic capillarity. CASA Report
15-33, Eindhoven University of Technology, preprint, 2015.
2. X. Cao, M. Kimura and I.S. Pop. Finite element approximation of non-equilibrium
flow in porous media. In preparation.
135
136
LIST OF PUBLICATIONS
Acknowledgments
First of all, I would like to thank my supervisor Prof. Iuliu Sorin Pop who is also my
first promoter for his constant patience, advice and trust for the last four years. His encouragement has given me self-confidence. Without him, this thesis would not have been
possible. I would like to express my gratitude to him for inspiring me and guiding me
tirelessly for the entire period of my PhD. I would also like to thank my second promoter
Prof. Hans van Duijn for sharing his invaluable knowledge when we were working on a
paper.
I would like to take this opportunity to thank Prof. Harald van Brummelen, Prof. Rainer
Helmig, Prof. Barry Koren, Prof. Ben Schweizer and Prof. Barbara Wohlmuth for being
in the committee of my defense ceremony. I feel honored that you have agreed to referee
this thesis.
I am thankful to Prof. Majid Hassanizadeh and Prof. Rainer Helmig for many nice suggestions and for giving me scientific motivation. The NUPUS environment was inspiring.
I would also like to thank Prof. Mark Peletier, who has taught me how to improve my
presentation skills. I want to thank Prof. Masato Kimura for giving me the opportunity
to visit Kanazawa University and his supervision there. Thanks are also to Prof. Michal
Beneš who read the thesis and gave helpful comments during my stay in Japan. I would
like to thank Prof. Florin Adrian Radu and Prof. Kundan Kumar for many discussions
and useful suggestions and for their help in broadening my network. Thanks are to my
collaborators: Azee, Ton van der Zanden, Prof. Jos Brouwers, Prof. Hans van Duijn and
Simplice.
I want to thank all the CASA members, who provided a nice working atmosphere. In
particular, many thanks to the secretaries, Enna and Marèse, who helped in all kinds of
administrative work. I am thankful to my office mates Sangye, Ian, Sarah, René and my
former office mate Bogdan for keeping up the good and positive atmosphere in our office.
Thanks are to Upanshu, Thomas and Sangye whom I have talked a lot with and learned
a lot from. Koondanibha, it was a pleasure to travel with you to Spain for the conference
and I had a wonderful time. Thanks are to Sarah, Deepesh, Nitin, Nikhil and Pranav for
137
138
ACKNOWLEDGMENTS
the nice party time. Behnaz and Prof. Mikko Karttunen, thank you for the help when
looking for a postdoctoral position.
Thanks are to Liu Lei, Lu Shengnan, Yang Yusen and Wang Jiquan for being wonderful
roommates, for the fun time together. Hou Qingzhi, Li Guangliang, Ma Ming, Sangay,
Sangye and Thomas, thank you for those lovely dinners.
I would like to thank my Chinese friends in the Netherlands: Wang Wenhan, Sui Huapeng,
Zhang Shiqiang and Ran Shenghai. Fan Yabin, Zhuang Luwen, Yin Xiaoguang and Qin
Chaozhong, I thank all of you that we share a lot of happy time when we have attended
the NUPUS conferences. I would also like to thank all my friends in China: Gu Yuanye,
Liu Jiaji, Cao Xianhui, Guo Hexin, Li Yonggang, Niu Chunhu, Sun Duo, Wu Yongdi and
Xu Jian for enriching my personal life.
Last but not the least, I express my gratitude to my parents, my brother and cousins, who
have always helped and encouraged me a lot. I would like to thank my wife Zhao Xiukun
for her support during all these years.
Xiulei Cao.
Eindhoven, February 2016.
Bibliography
[1] I. Aavatsmark. An introduction to multipoint flux approximations for quadrilateral
grids. Comput. Geosci., 6(3):405-432, 2002.
[2] R. A. Adams. Sobolev spaces. Academic Press, New York, 1975.
[3] B. Andreianov and C. Cancès. A phase-by-phase upstream scheme that converges
to the vanishing capillarity solution for countercurrent two-phase flow in two-rocks
media. Comput. Geosci., 18(2):211-226, 2014.
[4] B. Andreianov, R. Eymard, M. Ghilani and N. Marhraoui. Finite volume approximation of degenerate two-phase flow model with unlimited air mobility. Numer. Meth.
Part Differ. Equ., 29(2):441-474, 2013.
[5] S. N. Antontsev. On the solvability of boundary value problems for degenerate twophase porous flow equations. Dinamika Splos̆noı̆ Sredy Vyp., 10:28-53, 1972.
[6] T. Arbogast. The existence of weak solutions to single porosity and simple dualporosity models of two-phase in-compressible flow. Nonlinear Anal., 19(11):10091031, 1992.
[7] T. Arbogast, M. F. Wheeler and N. Y. Zhang. A nonlinear mixed finite element
method for a degenerate parabolic equation arising in flow in porous media. SIAM
J. Numer. Anal., 33(4):1669-1687, 1996.
[8] J. Arfvidsson. A new algorithm to calculate the isothermal moisture penetration for
periodically varying relative humidity at the boundary. Nordic Journal of Building
Physics, 2, 1999.
[9] B. C. Aslan, W. W. Hager and S. Moskow. A generalized eigenproblem for the
Laplacian which arises in lighting. J. Math. Anal. Appl., 341(2):1028-1041, 2008.
[10] F. Bagagiolo and A. Visintin. Hysteresis in filtration through porous media. Z. Anal.
Anwendungen, 19(4):977-997, 2000.
[11] G. I. Barenblatt, I. P. Zheltov and I. N. Kochina. Basic concepts in the theory of
seepage of homogeneous liquids in fissured rocks (strata). J. Appl. Math. Mech.,
24(5):1286-1303, 1960.
139
140
BIBLIOGRAPHY
[12] J. Bear. Dynamics of Fluids in Porous Media. Elsevier, New York, 1972.
[13] A. Y. Beliaev and S. M. Hassanizadeh. A theoretical model of hysteresis and dynamic
effects in the capillary relation for two-phase flow in porous media. Transp. Porous
Media, 43(3):487-510, 2001.
[14] M. Bertsch, F. Smarrazzo and A. Tesei. Pseudoparabolic regularization of forwardbackward parabolic equations: a logarithmic nonlinearity. Anal. PDE, 6(7):17191754, 2013.
[15] M. Bertsch, R. Dal Passo and C. J. van Duijn. Analysis of oil trapping in porous
media. SIAM J. Math. Anal., 35(1):245-267, 2003.
[16] M. Böhm and R. E. Showalter. A nonlinear pseudoparabolic diffusion equation. SIAM
J. Math. Anal., 16(5):980-999, 1985.
[17] S. Bottero. Advances in theories of capillary in porous media. Ph.D. Thesis, Utrecht
University, 2009.
[18] A. Bourgeat and M. Panfilov. Effective two-phase flow through highly heterogeneous porous media: Capillary nonequilibrium effects. Comput. Geosci., 2(3):191-215,
1998.
[19] K. Brenner, C. Cancès and D. Hilhorst. Finite volume approximation for an immiscible two-phase flow in porous media with discontinuous capillary pressure. Comput.
Geosci., 17(3):573-597, 2013.
[20] H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations.
Springer Science+Business Media, LLC 2011.
[21] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer Ser.
Comput. Math. 15, Springer-Verlag, New York, 1991.
[22] R. H. Brooks and A. T. Corey. Hydraulic properties of porous media, in hydrology
Papers, Vol. 3. Colorado State University, Fort Collins, 1964.
[23] F. Brunner, F. A. Radu and P. Knabner. Analysis of an upwind-mixed hybrid finite
element method for transport problems. SIAM J. Numer. Anal., 52(1):83-102, 2014.
[24] N. Burdine. Relative permeability calculations from pore size distribution data. T.
AM. I. MIN. MET. ENG., 198:71-78, 1953.
[25] F. Buzzi, M. Lenzinger and B. Schweizer. Interface conditions for degenerate twophase flow equations in one space dimension. Analysis, 29(3):299-316, 2009.
[26] G. Camps-Roach, D. M. O’Carroll, T. A. Newson, T. Sakaki and T. H. Illangasekare.
Experimental investigation of dynamic effects in capillary pressure: Grain size dependency and upscaling. Water Resour. Res., 46:W08544, 2010.
141
BIBLIOGRAPHY
[27] C. Cancès and M. Pierre. An existence result for multidimensional immiscible
two-phase flows with discontinuous capillary pressure field. SIAM J. Math. Anal.,
44(2):966-992, 2012.
[28] C. Cancès, C. Choquet, Y. Fan and I. S. Pop. Existence of weak solutions to a
degenerate pseudo-parabolic equation modeling two-phase flow in porous media.
CASA Report 10-75, 2010.
[29] C. Cancès, I. S. Pop and M. Vohralík. An a posteriori error estimate for vertexcentered finite volume discretizations of immiscible incompressible two-phase flow.
Math. Comput., 83:153-188, 2014.
[30] X. Cao and I. S. Pop. Uniqueness of weak solutions for a pseudo-parabolic equation
modeling two phase flow in porous media. Appl. Math. Lett., 46:25-30, 2015.
[31] X. Cao and I. S. Pop. Degenerate two-phase porous media flow model with dynamic
capillarity. J. Differ. Equ., 260(3):2418-2456, 2016.
[32] X. Cao and I. S. Pop. Two-phase porous media flows with dynamic capillary effects
and hysteresis: uniqueness of weak solutions. Comput. Math. Appl., 69(7):688-695,
2015.
[33] G. Chavent and J. Jaffré. Mathematical Models and Finite Elements for Reservoir
Simulation. North-Holland, Amsterdam, 1978.
[34] J. Chen, J. W. Hopmans and M. E. Grismer. Parameter estimation of two-fluid capillary pressure-saturation and permeability functions. Adv. Water Resour., 22(5):479493, 1999.
[35] Z. Chen. Degenerate two-phase incompressible flow I. Existence, uniqueness and
regularity of a weak solution. J. Differ. Equ., 171(2):203-232, 2001.
[36] C. M. Cuesta and I. S. Pop. Numerical schemes for a pseudo-parabolic Burgers
equation: discontinuous data and long-time behaviour. J. Comput. Appl. Math.,
224(1):269-283, 2009.
[37] M. J. Cunningham. Moisture diffusion due to periodic moisture and temperature
boundary conditions - an approximate steady analytical solution with non-constant
diffusion coefficients. Build. Environ., 27(3):367-377, 1992.
[38] D. A. DiCarlo. Experimental measurements of saturation overshoot on infiltration.
Water Resour. Res., 40:W04215, 2004.
[39] J. Jr. Douglas, R. E. Ewing, and M. F. Wheeler. The approximation of the pressure
by a mixed method in the simulation of miscible displacement. RAIRO Anal. Numér.,
17(1):17-33, 1983.
142
BIBLIOGRAPHY
[40] C. J. van Duijn, Y. Fan, L. A. Peletier and I. S. Pop. Travelling wave solutions for
degenerate pseudo-parabolic equation modelling two-phase flow in porous media.
Nonlinear Anal., 14(3):1361-1383, 2013.
[41] C. J. van Duijn, X. Cao and I. S. Pop. Two-phase flow in porous media: dynamic
capillarity and heterogeneous media. Transp. Porous Med., 110(3):1-26, 2015.
[42] C. J. van Duijn, J. Molenaar and M. J. de Neef. The Effect of capillary forces on
immiscible two-phase flow in heterogeneous porous media. Transp. Porous Media,
21(1):71-93, 1995.
[43] C. J. van Duijn, A. Mikelić and I. S. Pop. Effective equations for two-phase flow with
trapping on the micro scale. SIAM J. Appl. Math., 62(5):1531-1568, 2002.
[44] C. J. van Duijn, L. A. Peletier and I. S. Pop. A new class of entropy solutions of the
Buckley-Leverett equation. SIAM J. Math. Anal., 39(2):507-536, 2007.
[45] C. J. van Duijn, H. Eichel, R. Helmig and I. S. Pop. Effective equations for twophase flow in porous media: the effect of trapping on the microscale. Transp. Porous
Media, 69(3):411-428, 2007.
[46] C. J. van Duijn and M. J. de Neef. Similarity solution for capillary redistribution of
two phases in a porous medium with a single discontinuity. Adv. Water. Resour.,
21(6):451-461, 1998.
[47] L. J. Durlofsky. Coarse scale models of two phase flow in heterogeneous reservoirs:
Volume averaged equations and their relationship to existing upscaling techniques.
Comput. Geosci., 2(2):73-92, 1998.
[48] L. C. Evans. Partial Differential Equations, graduate studies in mathematics. Amer.
Math. Soc., Providence, 1975.
[49] R. Ewing and M. Wheeler. Galerkin methods for miscible displacement problems in
porous media, SIAM J. Numer. Anal., 17(3):351-365, 1980.
[50] R. Eymard, T. Gallouët and R. Herbin. The finite volume method, in Handbook for
Numerical Analysis, Ph. Ciarlet and J. L. Lions, eds, North-Holland, Paris, 2000.
[51] R. Eymard, R. Herbin and A. Michel. Mathematical study of a petroleum-engineering
scheme. ESAIM Math. Model. Numer. Anal., 37(6):937-972, 2003.
[52] Y. Fan and I. S. Pop. Equivalent formulations and numerical schemes for a class of
pseudo-parabolic equations. J. Comput. Appl. Math., 246:86-93, 2013.
[53] Y. Fan and I. S. Pop. A class of pseudo-parabolic equations: existence, uniqueness
of weak solutions, and error estimates for the Euler-implicit discretization. Math.
Methods Appl. Sci., 34(18):2329-2339, 2011.
143
BIBLIOGRAPHY
[54] R. Fučík and J. Mikyška. Discontinuous Galerkin and mixed-hybrid finite element
approach to two-phase flow in heterogeneous porous media with different capillary
pressures. Procedia Computer Science, 4:908-917, 2011.
[55] R. Fučík and J. Mikyška. Mixed-hybrid finite element method for modelling two-phase
flow in porous media. Journal of Math-for-Industry, 3:9-19, 2011.
[56] R. Fučík and J. Mikyška. Numerical investigation of dynamic capillary pressure in
two-phase flow in porous medium. Math. Bohem., 136(4):395-403, 2011.
[57] R. Fučík, J. Mikyška, T. Sakaki, M. Beneš and T. H. Illangasekare. Significance of
dynamic effect in capillarity during drainage experiments in layered porous media.
Vadose Zone J., 9(3):697-708, 2010.
[58] M. T. van Genuchten. A closed-form equation for predicting the hydraulic conductivity of unsaturated soil. Soil Sci. Soc. Am. J., 44(5):892-898, 1980.
[59] D. Gilbarg and N. Trudinger. Elliptic Partial Differential Equations of Second Order.
Springer-Verlag, Berlin, 1977.
[60] S. M. Hassanizadeh and W. G. Gray. Thermodynamic basis of capillary pressure in
porous media. Water Resour. Res., 29(10):3389-3405, 1993.
[61] R. Helmig. Multiphase Flow and Transport Processes in the Subsurface: a Contribution on the Modeling of Hydrosystems. Springer, Berlin, 1997.
[62] R. Helmig, A. Weiss and B. I. Wohlmuth. Variational inequalities for modeling flow
in heterogeneous porous media with entry pressure. Comput. Geosci., 13(3):373-389,
2009.
[63] R. Helmig, A. Weiss and B. I. Wohlmuth. Dynamic capillary effects in heterogeneous
porous media. Comput. Geosci., 11(3):261-274, 2007.
[64] V. Joekar-Niasar and S. M. Hassanizadeh. Effect of fluids properties on nonequilibrium capillarity effects: Dynamic pore-network modeling. Int. J. Multiph. Flow,
37(2):198-214, 2011.
[65] J. Kačur. Method of Rothe in evolution equations. Teubner Texts Math. 80, Teubner
Verlagsgellschaft, Leipzig, 1985.
[66] S. Karpinski and I. S. Pop. Analysis of an interior penalty discontinuous Galerkin
scheme for two phase flow in porous media with dynamic capillarity effects. CASA
report 15-27, Eindhoven University of Technology, 2015.
[67] S. Khomrutai. Uniqueness and grow-up rate of solutions for pseudo-parabolic equations in Rn with a sublinear source. Appl. Math. Lett., 48:8-13, 2015.
144
BIBLIOGRAPHY
[68] J. Koch, A. Rätz and B. Schweizer. Two-phase flow equations with a dynamic capillary pressure. Eur. J. Appl. Math., 24(1):49-75, 2013.
[69] D. Kroener and S. Luckhaus. Flow of oil and water in a porous medium. J. Differ.
Equ., 55(2):276-288, 1984.
[70] O. A. Ladyzenskaja and N. N. Uralćeva. Linear and Quasilinear Elliptic Equations.
Translation of Mathematical Monographs, Vol. 46 Amer. Math. Soc., Providence,
1968.
[71] A. Lamacz, A. Rätz and B. Schweizer. A well posed hysteresis model for flows in
porous media and applications to fingering effects. Adv. Math. Sci. Appl., 21(1):3364, 2011.
[72] M. Lenz, S. F. Nemadjieu and M. Rumpf. A convergent finite volume scheme for
diffusion on evolving surfaces. SIAM J. Numer. Anal., 49(1):15-37, 2011.
[73] M. Lenzinger and B. Schweizer. Two-phase flow equations with outflow boundary conditions in the hydrophobic-hydrophilic case. Nonlinear Anal., 73(4):840-853,
2010.
[74] C. Li, K. Li and Z. Chen. Numerical analysis of moisture influential depth in concrete
during drying-wetting cycles. Tsinghua Sci. Technol., 13(5):696-701, 2008.
[75] S. J. H. Meijers, J. M. J. M. Bijen, R. de Borst and A. L. A. Fraaij. Computational
results of a model for chloride ingress in concrete including convection, drying-wetting
cycles and carbonation. Mater. Struct., 38(2):145-154, 2005.
[76] A. Michel. A finite volume scheme for two-phase immiscible flow in porous media.
SIAM J. Numer. Anal., 41(4):1301-1317, 2003.
[77] A. Mikelić. A global existence result for the equations describing unsaturated flow
in porous media with dynamic capillary pressure. J. Differ. Equ., 248(6):1561-1577,
2010.
[78] A. Mikelić and H. Bruining. Analysis of model equations for stress-enhanced diffusion
in coal layers. Part I: Existence of a weak solution. SIAM J. Math. Anal., 40(4):16711691, 2008.
[79] E. Milne. The diffusion of imprisoned radiation through a gas. J. Lond. Math. Soc.,
1(1):40-51, 1926.
[80] S. F. Nemadjieu. A stable and convergent O-method for general moving hypersurfaces. CASA Report. 14-21, 2014.
[81] T. L. van Noorden and I. S. Pop. A Stefan problem modelling crystal dissolution and
precipitation. IMA J. Appl. Math., 73(2):393-411, 2008.
145
BIBLIOGRAPHY
[82] J. M. Nordbotten and M. A. Celia. Geological storage of CO2 : modeling approaches
for large-scale simulation. John Wiley & Sons, New Jersey, 2012.
[83] M. Ohlberger. Convergence of a mixed finite elements-finite volume method for the
two phase flow in porous media. East-West J. Numer. Math., 5:183-210, 1997.
[84] L. Pel. Moisture transport in porous building materials. Ph.D. Thesis, Einhoven University of Technology, 1995.
[85] M. Peszynska and S. Y. Yi. Numerical methods for unsaturated flow with dynamic
capillary pressure in heterogeneous porous media. Int. Int. J. Numer. Anal. Model.,
5:126-149, 2008.
[86] F. G. Pinder and G. W. Gray. Essentials of Multiphase Flow and Transport in Porous
Media. John Wiley Sons, Inc., 2008.
[87] I. S. Pop, M. Sepúlveda, F. A. Radu and O. P. Vera Villagráne. Error estimates for
the finite volume discretization for the porous medium equation. J. Comput. Appl.
Math., 234(7):2135-2142, 2010.
[88] F. A. Radu, I. S. Pop and P. Knabner. Error estimates for a mixed finite element
discretization of some degenerate parabolic equations. Numer. Math., 109(2):285311, 2008.
[89] F. A. Radu, J. M. Nordbotten, I. S. Pop and K. Kumar. A robust linearization
scheme for finite volume based discretizations for simulation of two-phase flow in
porous media. J. Comput. Appl. Math., 289:134-141, 2015.
[90] F. A. Radu, I. S. Pop and P. Knabner. Newton type methods for the mixed finite
element discretization of some degenerate parabolic equations, in Numerical Mathematics and Advanced Applications. Springer-Verlag Heidelberg, 2006.
[91] F. A. Radu and W. Wang. Convergence analysis for a mixed finite element scheme
for flow in strictly unsaturated porous media. Nonlinear Anal., 15:266-275, 2014.
[92] A. Rätz and B. Schweizer. Hysteresis models and gravity fingering in porous media.
Z. Angew. Math. Mech., 94(7-8):645-654, 2012
[93] F. Rezanezhad, H.-J. Vogel and K. Roth. Experimental study of fingered flow through
initially dry sand. Hydrol. Earth Syst. Sci. Discuss., 3:2595-2620, 2006.
[94] T. Sakaki, D. M. O’Carroll and T. H. Illangasekare. Direct quantification of dynamic
effects in capillary pressure for drainage-wetting cycles. Vadose Zone J., 9(2):424437, 2009.
[95] N. Seam and G. Vallet. Existence results for nonlinear pseudoparabolic problems.
Nonlinear Anal., 12(5):2625-2639, 2011.
146
BIBLIOGRAPHY
[96] B. Schweizer. The Richards equation with hysteresis and degenerate capillary pressure. J. Differ. Equ., 252(10):5594-5612, 2012.
[97] B. Schweizer. Instability of gravity wetting fronts for Richards equations with hysteresis. Interface Free Bound., 14(1):37-64, 2012.
[98] A. Selander. Hydrophobic Impregnation of Concrete Structures: Effects on Concrete
Properties. Ph.D. Thesis, Royal Institute of Technology, 2010.
[99] K. Spayd and M. Shearer. The Buckley-Leverett equation with dynamic capillary
pressure. SIAM J. Appl. Math., 71(4):1088-1108, 2011.
[100] R. Temam. Navier-Stokes Equations: Theory And Numerical Analysis. NorthHolland Publishing Company-Amsterdam, New York, Oxford, 1977.
[101] J. M. Torrenti, L. Granger, M. Diruy and P. Genin. Modeling concrete shrinkage
under variable ambient conditions. ACI Mater. J., 96(1):35-39, 1999.
[102] R. S. Varga. Matrix Iterative Analysis. Springer-Verlag Berlin Heidelberg, 2000.
[103] A. Visintin. Differential Models of Hysteresis. Springer, Berlin, 1994.

Download Report

Mathematical and numerical analysis for non-equilibrium two

Paperzz.com

Your Paperzz