Homogenization of the degenerate
two-phase flow equations
Patrick Henning1, Mario Ohlberger1, and Ben Schweizer2
October 30, 2012
Abstract: We analyze two-phase flow in highly heterogeneous media.
Problems related to the degeneracy of the permeability coefficient
functions are treated with a new concept of weighted solutions. Instead
of the pressure variables we formulate the problem with the weighted
pressure function ψ, which is obtained as the product of permeability
and pressure. We perform the homogenization limit and obtain effective
equations in the form of a two-scale limit system. The nonlinear effective
system is of the classical form in the non-degenerate case. In the
degenerate case, the two-scale system uses again a weighted pressure
variable. Our approach allows to work without the global pressure
function. Even though internal interfaces are included, our approach
provides the homogenization limit without any smallness assumptions
on permeabilities or capillary pressures.
MSC: 76S05, 35B27
key-words: homogenization, two-phase flow, degenerate permeability
1
Introduction
Flow problems in porous media are usually modelled with time dependent partial
differential equations. Unsaturated media are often described with the Richards
equation, if two immiscible fluids must be modelled, then the two-phase flow system
is commonly used.
Two-phase flow in porous media
We consider two immiscible fluids, one is denoted with a subscript w, the other with
a subscript n. In the following, we use an index j that stands for either w or n, the
1
Westfälische Wilhelms-Universität Münster, Center for Nonlinear Sciences, Einsteinstr. 62,
D-48149 Münster, Germany.
2
Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87, D-44227 Dortmund, Germany.
2
Homogenization of the two-phase flow system
letters are chosen in view of applications where one fluid is wetting, the other is nonwetting. The saturations of the two fluids are denoted by sw and sn , the pressures of
the two fluids by pw and pn , the negative fluxes by qw and qn . The description of the
porous material consists in the choice of three coefficient functions: two permeability
functions kjtot and one capillary pressure function pc . We write the two-phase flow
system as
∂t sj = ∇ · qj
qj = kjtot (sj )∇pj
sw + sn = 1,
in Ω, for j = w, n
in Ω, for j = w, n
pw − pn ∈ pc (sw )
in Ω,
(1.1)
(1.2)
(1.3)
where Ω ⊂ Rn is the spatial domain and where the dependence on the time variable
t ∈ [0, T ) is not explicitly stated; below we use ΩT := Ω × (0, T ) to denote the spacetime cylinder. We wrote the system in such a way that it contains the six unknowns
sj , pj , qj (of which the fluxes are vector valued) and six equations (1.1)–(1.3) (of
which (1.2) is vector valued). The equations are for incompressible fluids, gravity is
neglected, porosity and densities are normalized.
Degenerate coefficient functions
Two degeneracies must be treated in the physical application, one regards the permeabilities kjtot , the other the capillary pressure function pc . Since in absence of
microscopic channels for fluid j, fluid j cannot be transported, we necessarily have
kjtot (0) = 0. Furthermore, the function pc in (1.3) should be regarded as a multivalued function: in a region without wetting fluid (sw = 0), the non-wetting pressure pn can be increased arbitrarily, hence pc (0) must be an interval of the form
(−∞, p̃c (0)) with a real number p̃c (0). The analogous reasoning applies for sw = 1,
where pc (1) = [p̃c (1), ∞).
Nevertheless, many applications permit simplifications. Often one may assume
a multiplicative decomposition of the total permeability as kjtot (x, s) = Kj (x)kj (s),
into an absolute permeability Kj that depends only on the position x, and a relative
permeability kj that depends only on the saturation s. We will make a similar, but
slightly less restrictive assumption below, assuming that the relative permeability
depends on macroscopic position and on the saturation, but that it is, microscopically,
piecewise constant.
Regarding the capillary pressure we note that, even in the multi-valued setting,
we can define the inverse map, the saturation function Θ = p−1
: R → [0, 1]. The
c
second relation of (1.3) may then be written in the form sw = Θ(pw − pn ). In this
formulation, we have to deal with flat parts in the monotone function Θ. A crucial
consequence is that, in regions with sw = 0, a wetting pressure pw is physically not
well-defined. Mathematically, we read relation (1.2) in regions with sw = 0 as the
statement that also the negative flux qw must vanish.
Since the pressure pw is not a well defined quantity in regions with sw = 0,
we choose a new set of variables to describe the system, and replace pw by the
weighted pressure ψw := kw pw . This product is globally well defined by setting ψw =
0 whenever kw = 0. Solutions of the corresponding equations are called weighted
solutions. A similar approach has been used in [32]. We emphasize that we do not
P. Henning, M. Ohlberger, B. Schweizer
3
change the number of variables and that we always keep the saturation sw as one
of our primary variables. This is in contrast to some numerical approaches, see e.g.
[9, 28] and references therein.
Heterogeneous media
In order to model a heterogeneous medium with a small length-scale ε > 0, we use the
standard approach of homogenization theory. We denote the unit cell by Y = [0, 1)n ;
we will always impose periodicity conditions on the cell boundaries, i.e. we regard Y
as a flat torus. In this context, one demands that the oscillatory coefficient functions
are given as
x x tot,ε
ε
tot
(1.4)
kj (x, s) := kj x, , s , pc (x, s) := pc x, , s ,
ε
ε
with kjtot , pc : Ω × Y × [0, 1] → R, possibly multi-valued. For oscillatory coefficient
functions, we denote solutions as sεj , pεj , qjε ; our interest is to study the behavior of
solutions in the limit ε → 0.
Related literature
Due to its importance in practical applications, there is a vast literature on flow
problems in porous media. If the medium is completely saturated by one fluid, the
flow is described by Darcy’s law and incompressibility, and no deep mathematical
theory must be used. When two immiscible phases are present in the medium, but
one of the fluids need not be modelled (a constant pressure can be assumed), then
one uses Richards equation. In this equation, existence questions are interesting
due to the degeneracy of the permeability, see e.g. [3, 32, 34]. When both phases
must be modelled, one typically uses the two-phase flow system that we investigate
here. Again, the degenerate permeabilites make existence results intricate, compare
[18, 19, 25]. Usually, existence results use the global pressure as an additional variable.
The global pressure is defined in such a way that it solves an elliptic equation and
estimates on the global pressure can be exploited in the further analysis. Nevertheless,
there are situations in which the global pressure is of very limited use; this is the
case e.g. in problems with different capillary pressure functions [14, 15, 16, 17], in
problems with hysteresis [6, 24] and in problems with outflow boundary conditions
[4, 26, 30, 31].
Homogenization. The homogenization problem for two-phase flow equations receives considerable attention. This is due to the fact that the limit problem has
qualitatively new features and can, in particular, include the double porosity model.
This model was investigated in [12] with formal calculations and with mathematical
rigour in [11] and [35]. A stochastic setting has been studied in [10] in the nondegenerate case and the problem with hysteresis has been studied in [6], again in the
nondegenerate case.
A consequence of a degenerate ellipticity coefficient is that no uniform bounds for
the pressure gradients in L2 -spaces are available. This problem is often treated with
4
Homogenization of the two-phase flow system
the help of the global pressure function, a tool of limited use in a setting with internal
interfaces. As a consequence, the above quoted homogenization results are restricted
to cases in which some smallness assumption is satisfied. In [8], this is an explicit
smallness assumption concerning the jump of the global pressure, see their inequality
(4.20). In [11], the smallness is imposed by using a permeability of order ε2 in one
of two media. We want to emphasize that this assumption is physically justified
in many situations. In [35], more general permeabilities in the second medium are
studied. The author uses the scaling ε̟ and treats all cases with ̟ > 0. In short,
we may say that our works extends [35] to the case ̟ = 0.
Finally, we mention [5] for a homogenization study in the compressible case. In
that contribution, no y-dependence of the coefficients λw (s) or ρg (p) is permitted.
2
Problem description and solution concepts
2.1
Boundary conditions and structural assumptions
Let Ω ⊂ Rn be bounded with Lipschitz boundary. Below, we will consider periodic
disjoint subdomains such that each subdomain is connected — which is only possible
for n ≥ 3. Therefore, restricting ourselfs additionally to physically relevant dimensions, our homogenization results treat only the case n = 3. For j = w, n, let Σj ⊂ ∂Ω
be a Dirichlet boundary with positive n − 1-dimensional measure, and let Γj ⊂ ∂Ω be
a Neumann boundary with Σj ∩Γj = ∅ and Σ̄j ∪ Γ̄j = ∂Ω. As boundary conditions we
impose a no-flux condition for fluid j on Γj , and the Dirichlet condition pj − Uj = 0
on Σj for two given functions Uj ∈ H 1 (Ω). We furthermore assume that initial values
Sj ∈ L∞ (Ω, [0, 1]) are given with the properties Sw + Sn ≡ 1 and ess supΩ Sw < 1.
We consider oscillatory coefficient functions as in (1.4), where kjtot (x, y, s) =
Kj (x, y)kj (x, y, s) and pc (x, y, s) = pc (y, s) are prescribed by functions
Kj ∈ L∞ (Ω × Y, R),
kj : Ω × Y × [0, 1] → [0, ∞) is given as kj (x, y, s) = kjm (x, s) for y ∈ Ym
(2.1)
(2.2)
for M functions kjm : Ω × [0, 1] → R, m = 1, . . . , M.
pc : Y × [0, 1] → R is given as pc (y, s) = pm
(2.3)
c (s) for y ∈ Ym
m
for M functions pc : [0, 1] → R, m = 1, . . . , M.
S
Here, we decomposed the unit cell in subdomains as Y = M
m=1 Ȳm , such that the
open subsets Ym are pairwise disjoint. The number M ∈ N denotes the number of
different materials that are distributed in a fine mixture. We emphasize the important
restriction in the above setting: the capillary pressure coefficient does not vary in an
arbitrary fashion in y, but a y-independent coefficient function pm
c is used in the
subdomain corresponding to Ym . To describe the heterogeneous material, we use
x x x
, kjε (x, s) := kj x, , s , pεc (x, s) := pc
,s .
(2.4)
Kjε (x) := Kj x,
ε
ε
ε
2.2
Assumptions on the coefficient functions
Our assumptions on the coefficients are very general and our results cover most
situations of physical interest. As the most restrictive assumption we regard the non-
5
P. Henning, M. Ohlberger, B. Schweizer
degeneracy of the non-wetting fluid in (2.5). The lower bound for kn is physically not
justified, but it is obviously a valid assumption in flow regimes, where a vanishing
saturation sn = 0 does not occur.
Permeabilities. We assume that, for numbers 0 < c0 < C0 < ∞, 0 < ρ0 < ∞, and
j = w, n, there holds, independent of x and y,
c 0 ≤ Kj ≤ C 0 ,
kj ≤ C0 ,
k n ≥ c0 ,
kw (0) = 0,
kw strictly monotone with
p
0 < kw (s), c0 sρ0 −1 ≤ ∂s kw (s) ≤ C0 kw (s) ∀s ∈ (0, 1].
(2.5)
(2.6)
(2.7)
The strict positivity assumption kn ≥ c0 on the non-wetting permeability allows
us to concentrate here on the degeneracy in s = 0, which simplifies notation. The
growth-conditions of (2.7) are satisfied, e.g., for kw (s) = sρ with ρ ≥ 2.
We will derive the two-scale limit system first in the non-degenerate case. In this
case we assume that also the second relative permeability is bounded from below,
k w ≥ c0 .
(2.8)
Capillary pressure. We assume that, for 0 < c1 , there holds, for every index m
1
p̃m
c ∈ C ([0, 1), R),
c1 ≤ ∂s p̃m
c on [0, 1) ,
(
{p̃m
c (s)}
pm
(s)
=
c
(−∞, p̃m
c (0)]
for s > 0
for s = 0,
p̃m
c (s) → ∞ for s → 1.
(2.9)
(2.10)
We remark that the second assumption in (2.10) introduces a slightly asymmetric behavior, but our assumption is in accordance with experimental data. It simplifies the
notation in the following since it helps to concentrate the analysis to the degeneracy
at the point s = sw = 0.
We furthermore introduce a mild regularity assumption on the product kw (s) ·
pc (s), which we regard here not as a function in s, but as a function in kw (s), i.e.
Ψ(κ) = κ · p̃c (kw−1 (κ)). We assume that, for every m ≤ M and y ∈ Ym , after an
extension to boundary points, the maps
Ψ(x, y, .) : [0, kw (x, y, 1)) → R, κ 7→ κ · p̃c (kw−1 (κ))
[
∂
[(x, y) × [0, kw (x, y, 1))] → R are continuous.
Ψ :
Ψ,
∂κ
(2.11)
(x,y)∈Ω̄×Ȳm
Here, continuity is demanded in all the three variables x, y, and s, but we recall that
the subdomain index m is fixed. Under our other assumptions, (2.11) is satisfied
e.g. for polynomials kw . Indeed, in this case kw (s)(∂s kw (s))−1 can be extended
continuously to s = 0 and (2.11) follows from ∂κ Ψ(κ) = p̃c (s) + κ∂s p̃c (s)(∂s kw (s))−1
for s ∈ (0, 1) and κ = kw (s).
6
Homogenization of the two-phase flow system
Y1
Y2
Y
Figure 1: Left: The periodicity cell Y with two subdomains Ym , m = 1 and m = 2.
In the two subdomains, different coefficient functions describe the material response.
Right: Sketch of two strictly monotone and multi-valued pc -curves, s 7→ p1c (s) for
subdomain m = 1 and s 7→ p2c (s) for subdomain m = 2. The entry pressures are
ordered such that max p2c (0) > max p1c (0), the multi-valued functions satisfy pm
c (0) =
2 −1
1
(−∞, max pm
(0)].
The
transformation
map
Φ̃
=
(p
)
◦
p
is
continuous
on
[0, 1].
2,1
c
c
c
Interface condition maps. In order to encode interface conditions, we additionally introduce transformation functions as follows. By relabeling if necessary, we may
assume in the following without loss of generality that the entry pressures max pm
c (0)
m
m′
′
′
are ordered, max(pc (0)) ≤ max(pc (0)) for every pair 1 ≤ m, m ≤ M with m < m,
the argument of the maximum is an interval of the form (−∞, ρ] (cf. Figure 1). For
all pairs m′ < m we introduce the transformation maps
′
−1
◦ pm
Φ̃m,m′ (x, .) := (pm
c (x, .))
c (x, .) : [0, 1] → [0, 1],
m′
Φm,m′ (x, .) := kwm (x, .) ◦ Φ̃m,m′ (x, .) ◦ (kw (x, .))−1 .
(2.12)
(2.13)
The maps are defined such that the following holds. If x is a point on ∂Ωεm ∩ ∂Ωεm′
and both pressure functions pw and pn are continuous, then the two traces sm
w (x) and
′
m′
m
m
sw (x) of the wetting saturation satisfy sw (x) = Φ̃m,m′ (x, sw (x)). The other transformation map Φm,m′ maps the permeability value of one side to the corresponding
permeability value of the other side. We assume that, for all m′ < m,
[
∂
[(x, y) × [0, kw (x, y, 1)]] → R are continuous. (2.14)
Φm,m′ :
Φm,m′ ,
∂κ
(x,y)∈Ω̄×Ȳm
2.3
Solutions in heterogeneous media
Our aim is to study solutions of the two-phase flow system (1.1)–(1.3) for the oscillatory coefficients of (2.1)–(2.3). For a given length scale ε > 0 we hence study the
following system of equations.
∂t sεj = ∇ · qjε
qjε = Kjε kjε (sεj )∇pεj
sεw + sεn = 1,
pεw − pεn ∈ pεc (sεw )
in Ω for j = w, n
(2.15)
in Ω for j = w, n
(2.16)
in Ω,
(2.17)
where we omitted once more the dependence on t ∈ [0, T ). The initial and boundary
conditions are, using the normal vector ν on ∂Ω,
ν · qjε = 0 on Γj ,
pj = Uj on Σj ,
sj (., t = 0) = Sj ,
for j = w, n,
(2.18)
7
P. Henning, M. Ohlberger, B. Schweizer
where the no-flux condition is imposed as the natural boundary condition through the
weak formulation without boundary integral, the other two conditions are imposed
in the sense of traces. We start our analysis by giving a precise description of our
solution concept in the non-degenerate case.
Definition 2.1 (Strong solutions). We say that the six functions (sεj , pεj , qjε ) are a
strong solution to system (2.15)–(2.17) if the following holds. The functions are of
the classes
sεj ∈ L∞ (ΩT ; [0, 1]),
qjε ∈ L2 (ΩT ; Rn ),
pεj ∈ L2 (0, T ; H 1(Ω)),
(2.19)
relation (2.15) holds in the distributional sense and both relations (2.16) and (2.17)
hold pointwise almost everywhere.
The above choice of function spaces already encodes interface conditions at internal boundaries. Relation (2.15) contains, in a weak form, a continuity condition for
the fluxes, the H 1 (Ω) condition for pressures encodes, in a weak form, a continuity
of the pressures across interfaces. We call a solution in the sense of Definition 2.1 a
strong solution, since the regularity properties allow to impose the pointwise relation
(2.16) in the original variables (in contrast to weighted solutions and the relation
(2.24) below).
Proposition 2.1 (Existence of strong solutions in the non-degenerate case). Let the
assumptions (2.5)–(2.10) on the coefficients be satisfied, in particular, we assume with
(2.8) the non-degeneracy of the permeability of the wetting fluid. For data as described
above, there exists a strong solution to the two-phase flow system (2.15)–(2.18). With
a constant C that depends only on initial and boundary data, there holds an estimate
X Z
j∈{w,n}
0
T
Z
Ω
kjε (sεj )|∇pεj |2 ≤ C.
(2.20)
Proof. We derive the a priori estimate (2.20) in Section 4.1, see inequality (4.2).
The inequality implies, in the non-degenerate case, an estimate for both pressure
functions, pεj ∈ L2 (0, T ; H 1(Ω)).
The existence result is classical, see [18, 25]. It can be derived, e.g., with a
discretization scheme exploiting an a priori estimate of the form (2.20) for the approximate solutions.
Definition 2.2 (Weighted solutions). We say that the six functions (sεj , qjε , pεj ) provide
a weighted solution to system (2.15)–(2.17) if the following holds. The capillary
pressure inclusion holds as an equality, pεw − pεn = p̃εc (sεw ) and sεw + sεn = 1 hold almost
everywhere, on the M open disjoint subdomains Ωεm the functions are of the class
sεj ∈ L∞ (ΩT ; [0, 1]),
qjε ∈ L2 (ΩT ; Rn ),
pεn ∈ L2 (0, T ; H 1(Ω)), pεw ∈ L2 (ΩT ; R),
kwε (sεw )|Ωεm , ψwε := kwε (sεw )pεw |Ωεm ∈ L2 (0, T ; H 1(Ωεm )),
(2.21)
(2.22)
(2.23)
8
Homogenization of the two-phase flow system
the conservation law (2.15) holds in the distributional sense, the negative flux relation
(2.16) for j = n holds almost everywhere. Instead of (2.16) for j = w we demand
that
qwε = Kwε (∇[kwε (sεw )pεw ] − ∇[kwε (sεw )]pεw )
(2.24)
holds in each open subdomain Ωεm . In order to encode continuity of the pressure pw
across interfaces, we additionally demand
tracem (kwε (sεw )) = Φm,m′ (x, tracem′ (kwε (sεw )))
(2.25)
on ∂Ωεm′ ∩ ∂Ωεm with m′ < m. Here tracem denotes the trace from the interior of Ωεm .
We note that the regularity assumption (2.23) on kwε (sεw ) allows to evaluate the
traces in (2.25). Furthermore we observe that in (2.24) the right hand side vanishes
almost everywhere in the set {(x, t) : kwε (sεw (x, t)) = 0}. This follows from Stampacchia’s lemma since both functions kwε pεw and kwε of which gradients are evaluated,
vanish on this set. In this sense, in regions with a vanishing wetting saturation, we
impose a vanishing flux, but we do not select a value for the wetting pressure.
In the subsequent existence result and its proof we regard the small scale size
ε > 0 as fixed and omit the index ε. On the other hand, the solutions carry an index
δ which indicates a regularization parameter of a non-degenerate approximation.
Proposition 2.2 (Existence of weighted solutions in the degenerate case). Let assumptions (2.5)–(2.7) and (2.9)–(2.14), but not necessarily (2.8). For a sequence
δ ց 0 let kwδ → kw (uniformly on [0, 1]) be approximations of the permeability in
C 0 ([0, 1], R) such that (2.5)–(2.10) hold for kwδ with kwδ (0) = δ. Let sδj , qjδ , and pδj
be solutions to the corresponding non-degenerate problem. We assume that, for a
constant C independent of δ, there holds kpδj kL∞ (ΩT ) ≤ C for j = w and j = n.
Under these conditions, there exists a subsequence δ → 0 and limit functions sj ,
qj , and pj which are a weighted solution to the degenerate two-phase flow system
(2.15)–(2.18) with coefficient kw .
Since our interest here regards the homogenization procedure, we do not investigate conditions that guarantee the boundedness of the pressures. From a physical
point of view, for bounded pressure values along the domain boundaries, one will always expect a boundedness of the pressures in the interior. Regarding mathematical
proofs of the boundedness assumption we mention that in [26] uniform bounds are
derived with the help of maximum principles.
Proof. Solutions to the non-degenerate two-phase flow system (2.15)–(2.18) satisfy
the pressure bounds of (2.20) as noted in Proposition 2.1. Additionally, under the
assumption of uniform L∞ -bounds on the pressure function, with a constant C depending only on initial and boundary data and this uniform bound, there holds an
estimate
Z TZ
Z TZ
δ
δ
2
|∇ψwδ |2 ≤ C.
(2.26)
|∇[kw (sw )]| +
0
Ωm
0
Ωm
for every m ≤ M. This estimate is shown in Section 4.2, see inequality (4.3). The
estimate allows to choose a subsequence and limit functions such that, in the limit
δ → 0,
qjδ ⇀ qj in L2 (ΩT ),
ψwδ ⇀ ψw and kjδ ⇀ kj
in L2 (0, T ; H 1(Ωm )) .
(2.27)
9
P. Henning, M. Ohlberger, B. Schweizer
Similarly, we find weak L2 (ΩT ) limit functions pn , sw , and sn , satisfying sw + sn = 1.
We claim that some additional strong convergences can be concluded, namely
kwδ → kw , and sδj → sj in L2 (ΩT ) .
(2.28)
The strong convergence of sδw can be concluded with the growth conditions (2.7) of
the functions kwδ . Indeed, let us consider the new family of functions f δ = (sδw )ρ with
ρ ≥ ρ0 of (2.7). This sequence satisfies that
|∇f δ | = ρ |(sδw )ρ−1 ∇sδw | ≤ C|∂s kwδ (sδw )| |∇sδw | = C|∇[kwδ (sδw )]| ∈ L2 (ΩT )
is bounded. Additionally, and as a consequence, for ρ ≥ ρ0 + 1, the sequence
∂t f δ = ρ (sδw )ρ−1 ∂t sδw = ρ (sδw )ρ−1 ∇ · qwδ = ∇ · [ρ (sδw )ρ−1 qwδ ] − ρ ∇[(sδw )ρ−1 ] · qwδ
is bounded in L2 (0, T ; H −1(Ω)) + L1 (ΩT ). The interpolation compactness lemma of
Lions-Aubin provides the compactness of L2 (0, T ; Y ) ∩ W 1,1 (0, T ; Z) → L2 (0, T ; X)
for Y ⊂ X compact and X ⊂ Z continuous. This yields the compactness of the
sequence f δ in L2 (ΩT ). The strong convergence f δ → f implies the pointwise convergence almost everywhere and thus the strong convergences of (2.28).
We can now use the weak limit pn and the strong limit sw to define pw := pn +
p̃c (sw ). The weak limit ψw satisfies ψw = kw pw by the strong convergence of kwδ . We
exploit here that we can define the pressure value pw arbitrarily on the set where
kw = ψw = 0 holds. On the remaining set we have the strong convergence sδj → sj
and hence pδw ∈ pδn + pc (sδw ) ⇀ pn + p̃c (sw ) = pw . We note that the limit equations
are in the appropriate function spaces as demanded in (2.21)–(2.23).
The distributional relation (2.15) holds in the limit δ → 0 since derivatives commute with distributional limits. Furthermore, (2.16) for j = n holds by the strong
convergence of knδ . The interface condition (2.25) is satisfied for the approximate
solutions (δ > 0) and continues to hold in the limit δ → 0 due to the strong convergence of the traces. The latter is obtained from the above argument on the strong
convergence of kwδ (sδw ).
It remains to verify (2.24). We start from the strong relation (2.16), which we
write with the product rule of H 1 -functions as
qwδ = Kw (∇ψwδ − pδw ∇kwδ ) .
The difficulty in this expression is that we have only the weak convergences for both
pδw and ∇kwδ . We can argue as follows. We multiply the relation with kwδ , such that
weak limits exist for both sides,
kw qw ↼ kwδ qwδ = Kw (kwδ ∇ψwδ − ψwδ ∇kwδ ) ⇀ Kw (kw ∇ψw − ψw ∇kw )
in L2 (ΩT ). In the last convergence, we used that ψwδ ∇kwδ = ∇[ψwδ kwδ ] − [∇ψwδ ] kwδ
converges (by the strong convergence kwδ → kw ) in the sense of distributions to its
formal limit. On the set of points (x, t) with kw (x, t) > 0, we divide by kw (x, t) and
obtain the pointwise relation (2.24). For the remaining set with kw (x, t) = 0 we have
already observed that the right hand side of (2.24) vanishes almost everywhere by
δ
δ
δ
the
p lemma
p of Stampacchia. The flux
p function is the weak limit of qw = kw ∇pw =
δ
δ
2
kwδ · kwδ ∇pw where the factor kwδ ∇pw is bounded in L (ΩT ). This provides
qw = 0 on the set with kw (x, t) = 0 and hence (2.24) holds also on this set.
10
Homogenization of the two-phase flow system
3
Main homogenization results
We study a sequence ε = εi → 0 and a corresponding family of solutions (sεj , pεj , qjε )
to system (2.15)–(2.17). In the case of a non-degenerate relative permeability kw we
study a family of strong solutions and obtain for their two-scale limits the two-scale
limiting problem (ND) described in Definition 3.1. In the case of a degenerate relative
permeability kw we study weighted solutions and obtain the limiting system (D)
described in Definition 3.2. The limiting problems (ND) and (D) coincide formally,
but the formulation of problem (D) is taylored to the fact that the pressures need
not have gradients of class L2 .
3.1
The limit problem in the non-degenerate case
We start with a formulation of the homogenized problem in the non-degenerate case.
Here and in the following, we denote the average of an integrable function u : A → Rl
on a measurable set A ⊂ RL by
Z
Z
Z
1
u,
|A| :=
1 the L-dimensional Lebesgue measure.
− u :=
|A| A
A
A
We write ϕ ∈ Cc∞ (A, RL ) if ϕ : A → RL is infinitely often continuously differentiable
in the interior of A, if all derivatives can be extended to continuous functions on A,
and if ϕ has a compact support {ϕ 6= 0} ⊂ A, the closure taken in Rl . This means,
in particular, that ϕ must vanish in boundary points x 6∈ A, but need not vanish
in boundary points x ∈ A. The space of test-functions ϕ ∈ Cc∞ (A, RL ) is used to
define distributions. In the special case of the flat torus Y , we use periodic smooth
∞
functions ϕ ∈ Cper
(Y, RL ) to define distributions.
Definition 3.1 (Strong limit problem in the non-degenerate case). We denote the
following two-scale problem as problem (ND). We seek for eight functions sj,0 (x, y),
pj,0(x), pj,1(x, y), qj,0 (x, y) in the function spaces
pj,0 ∈ L2 (0, T ; H 1(Ω)),
sj,0 ∈ L∞ (Ω × Y × (0, T ); [0, 1]),
1
pj,1 ∈ L2 ((0, T ) × Ω; Hper
(Y )),
qj,0 ∈ L2 (Ω × Y × (0, T ); Rn).
The equations are
Z
Z
∂t − sj,0 (x, y) dy = ∇x · − qj,0(x, y) dy,
Y
(3.1)
(3.2)
(3.3)
Y
qj,0(x, y) = Kj (x, y)kj (x, y, sj,0(x, y)) (∇xpj,0 (x) + ∇y pj,1(x, y)),
∇y · qj,0(x, y) = 0,
sw,0(x, y) + sn,0 (x, y) = 1,
pw,0(x) − pn,0 (x) ∈ pc (x, y, sw,0(x, y)).
(3.4)
(3.5)
(3.6)
(3.7)
We demand (3.3) and (3.5) in the distributional sense on Ω×(0, T ) and Ω×Y ×(0, T ),
respectively. The other relations are demanded as pointwise (almost everywhere)
equalities or inclusions, we emphasize that the regularities (3.1)–(3.2) permit all necessary point evaluations, in particular in (3.4).
11
P. Henning, M. Ohlberger, B. Schweizer
Concerning the solvability of the limit problem (ND), we remark that the eight
unknowns can, formally, be related to the eight equations. Relation (3.7) allows
to reconstruct sw,0 (x, y) from the pressures pj,0(x). In turn, relation (3.6) provides
sn,0 (x, y). The negative fluxes qj,0(x, y) are given by (3.4) and we regard (3.5) as an
elliptic equations for pj,1 (x, y)). With this reasoning, we have seen that (3.4)–(3.7)
can determine all other quantities from the two macroscopic pressures pj,0 (x). If this
is the case, the two macroscopic equations (3.3) have the structure
∂t Sj (pw,0(x), pn,0 (x)) = ∇x · Fj (pw,0 (x), pn,0(x), ∇x pw,0 (x), ∇x pn,0(x))
for some functions Sj and Fj . In the one-dimensional case, the above arguments can
be made precise, see the effective one-dimensional system (5.7) in Section 5.
Theorem 1 (Non-degenerate homogenization). Let the domain Ω, the time-interval
[0, T ], the boundary conditions Uj and the initial conditions Sj be as described above.
Let the coefficients satisfy assumptions (2.5)–(2.10), hence, in particular, the nondegeneracy condition (2.8). Let pεj , sεj , qjε be a family of strong solutions to the
two-phase flow system (2.15)–(2.17) and let pj,0, pj,1, sj,0, and qj,0 be two-scale limit
functions. Then the limit functions satisfy the two-scale limit system (ND) as described in Definition 3.1.
We note that the definition of the two-scale limit functions is made precise in (4.9)–
(4.12). The limit equations must be complemented with the boundary conditions that
are consequences of (2.18), formally
Z
Z
ν · − qj,0(x, y) dy = 0 on Γj , pj,0 = Uj on Σj , − sj,0 (x, y, 0) dy = Sj (x) ,
(3.8)
Y
Y
for j = w, n. While the Dirichlet condition for pj,0 can be imposed in the sense of
traces, the other conditions are once more formulated together with (3.3) in the weak
form
Z
Z
−
∂t ϕ(x, t) − sj,0 (x, y) dy dx dt
Ω×(0,T )
Y
Z
Z
Z
(3.9)
+
∇x ϕ(x, t) · − qj,0 (x, y) dy dx dt =
ϕ(x, 0)Sj (x) dx
Ω×(0,T )
Y
Ω
for every infinitely often continuously differentiable test-function ϕ, i.e. ϕ ∈ Cc∞ ((Ω∪
Γj ) × [0, T ), R). For all functions, periodicity in y ∈ Y is assumed. The theorem is
shown in Subsection 4.3.
3.2
The limit problem in the degenerate case
Definition 3.2 (Weighted limit problem in the degenerate case). We denote the following two-scale problem as problem (D). The unknowns are seven functions sj,0 (x, y),
qj,0 (x, y), pj,0(x), pn,1 (x, y) as in the non-degenerate case, with function spaces as in
(3.1)–(3.2). Two further unknowns are ψw,1(x, y) and kw,1(x, y) in the space
ψw,1 |Ym , kw,1|Ym ∈ L2 ((0, T ) × Ω; H 1 (Ym )).
12
Homogenization of the two-phase flow system
We use kj,0(x, y) := kj (x, y, sj,0(x, y)) and ψw,0 (x, y) := kw,0 (x, y)pw,0(x) as abbreviations. The system replacing (3.3)–(3.7) is, for j = w, n,
Z
Z
∂t − sj,0(x, y) dy = ∇x · − qj,0 (x, y) dy,
(3.10)
Y
Y
qn,0 (x, y) = Kn (x, y)kn,0(x, y) (∇xpn,0 (x) + ∇y pn,1 (x, y)),
qw,0 (x, y) = Kw (x, y)(∇xψw,0 (x) + ∇y ψw,1 (x, y)
− pw,0 (x)[∇x kw,0(x, y) + ∇y kw,1(x, y)])
∇y · qj,0 (x, y) = 0,
sw,0(x, y) + sn,0 (x, y) = 1,
pw,0 (x) − pn,0(x) = p̃c (x, y, sw,0(x, y)).
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
Since ψw,1 may jump across material interfaces, equation (3.12) is understood as a
distributional equation in each subdomain Ym . To encode continuity of the wetting
pressure in orders 0 and 1 we demand
on ∂Ym ∩ ∂Ym′
tracem (kw,0) = Φm,m′ (tracem′ (kw,0))
∂
Φm,m′ (tracem′ (kw,0)) · tracem′ (kw,1) on ∂Ym ∩ ∂Ym′
tracem (kw,1) =
∂κ
(3.16)
(3.17)
for every pair 1 ≤ m′ < m ≤ M, traces are with respect to the variable y in Ym .
While ψw,1 (x, y) is only a replacement of pw,1 (x, y), the additional scalar unknown
kw,1(x, y) must be determined from the additional equation
ψw,1 (x, y) =
∂
Ψ(kw,0(x, y)) · kw,1(x, y)
∂κ
+ pn,0(x)kw,1 (x, y) + kw,0(x, y)pn,1(x, y),
(3.18)
where we used the function Ψ of assumption (2.11).
We recall that the function Ψ = id · (p̃c ◦ kw−1 ) of assumption (2.11) maps a
permeability value kw to the corresponding capillary pressure contribution of ψw .
Relation (3.18) should be regarded as the first order variant of the pressure condition
(3.15). The functions ψw,0 and ψw,1 will appear as the two-scale limits
2
ψwε := kwε pεw ⇀ ψw,0 ,
2
∇ψwε ⇀ ∇x ψw,0 + ∇y ψw,1 ,
ψw,0 : Ω × Y → R,
(3.19)
ψw,1 : Ω × Y → R.
(3.20)
Theorem 2 (Homogenization in the degenerate case). Let the domain Ω, the timeinterval [0, T ], the boundary conditions Uj and the initial conditions Sj be as described
above with Ωm
ε connected for all m. Let the coefficients satisfy assumptions (2.5)–
(2.7) and (2.9)–(2.14), but not necessarily (2.8). Let sεj , qjε , pεj be a family of weighted
solutions of the two-phase flow system (2.15)–(2.17). We assume that the pressures
pεj are uniformly bounded in L∞ .
Let sj,0, qj,0 , and pj,0 be two-scale limit functions in L2 (Ω × Y × (0, T )). Then
these limit functions satisfy, with an appropriate choice of three additional unknowns
pn,1 (x, y), ψw,1 (x, y), and kw,1(x, y), the two-scale limit system (D) as described in
Definition 3.2.
P. Henning, M. Ohlberger, B. Schweizer
13
Theorem 2 on the degenerate homogenization result is shown in Subsection 4.4.
Note that we do not show here an existence result for the homogenized system.
To obtain a complete the set of equations, one has to complement the system
with boundary and initial conditions as in (3.8). Unfortunately, due to the weak
regularity properties, it is not clear how a Dirichlet condition for pw,0 on Σw can be
imposed. In the case Γw = Γn and Σw = Σn , the problem can easily be solved by
calculating from the two prescribed pressures Uj on Σj the resulting saturation and
then the Dirichlet conditions for ψj,0 on Σj . In the opposite case, if on some part
of the boundary pw satisfies a Dirichlet condition and pn a Neumann condition, the
values of ψw,0 cannot be determined. We remark that a related problem occurs in
the global pressure formulation when in boundary portions the two phases satisfy
different boundary conditions.
In the above theorem, we assumed the connectedness of all subdomains Ωm
ε . This
is done for ease of notation, it is possible to extend the result to the case when only
one subdomain is connected, but in this case one has to assume that the connected
domain is the best conducting medium. Conditions on the capillary pressures must
insure in such a case that the control of the pressure in the connected subdomain
provides a control for all pressure functions. As mentioned before, our assumptions
make the theorem applicable in the case of multiple subdomains only in three space
dimensions.
3.3
Formal equivalence of the solution concepts
We want to include in this subsection two observations on the formal equivalence of
different formulations of the limit system. Our first observation regards the comparison of the degenerate limit system (D) and the non-degenerate limit system (ND).
Lemma 3.1. We consider a solution to the degenerate homogenized problem (D) and
assume the additional regularity property that pw,1 := pn,1 + ∂κ (p̃c ◦ kw−1 )(sw,0 )kw,1 is
in the function space as given in (3.2). Then, the solution to the degenerate problem
(D) provides also a solution to problem (ND).
Proof. We observe that the set of non-degenerate equations (3.3)–(3.7) coincides exactly with the set of degenerate equations (3.10)–(3.15), with the only exeption that
(3.4) for j = w is replaced by (3.12). We therefore only have to verify (3.4) for
qw,0 (x, y).
With the function pw,1 := pn,1 + ∂κ (p̃c ◦ kw−1 )(sw,0)kw,1 we calculate
(3.18)
∇y ψw,1 = ∂κ Ψ(kw,0) · ∇y kw,1 + pn,0 ∇y kw,1 + kw,0 ∇y pn,1
= p̃c (sw,0) + kw,0 ∂κ (pc ◦ kw−1 )(sw,0 ) · ∇y kw,1 + pn,0 ∇y kw,1 + kw,0 ∇y pn,1
(3.15)
= pw,0∇y kw,1 + kw,0 ∂κ (p̃c ◦ kw−1 )(sw,0)∇y kw,1 + ∇y pn,1
= pw,0 ∇y kw,1 + kw,0 ∇y pw,1 .
We can investigate the expression for qw,0 in (3.12). We use the product rule and
14
Homogenization of the two-phase flow system
insert the above calculation to obtain
∇x ψw,0 + ∇y ψw,1 − pw,0 [∇x kw,0 + ∇y kw,1 ]
= ∇x kw,0 pw,0 + kw,0∇x pw,0 + kw,0 ∇y pw,1 + ∇y kw,1 pw,0 − pw,0[∇x kw,0 + ∇y kw,1]
= kw,0 [∇x pw,0 + ∇y pw,1 ] ,
which confirms (3.4). We emphasize that the calculation is done for each subdomain Ym separately, in the subdomains vanishes the gradient ∇y kj,0(x, y) identically
because of our structural assumptions in (2.2)–(2.3).
In the non-degenerate case, the equation (3.18) could be written in the following
more intuitive and more symmetric form, using four new auxiliary variables pw,1(x, y),
kn,1 (x, y) and sj,1(x, y). We emphasize that, due to the degeneracy of the problem, the
meaning of these auxiliary variables cannot necessarily be made precise and system
(3.21)–(3.24) has only formal character.
kj,1(x, y) = ∂s kj (sj,0 (x, y))sj,1(x, y) for j = w, n,
ψw,1(x, y) = pw,0(x)kw,1 (x, y) + kw,0(x, y)pw,1(x, y)
sw,1(x, y) + sn,1(x, y) = 0,
pw,1 (x, y) − pn,1(x, y) = ∂s p̃c (x, y, sw,0(x, y))sw,1(x, y).
(3.21)
(3.22)
(3.23)
(3.24)
Our next observation regards the formal equivalence of the degenerate limit system
(D) with the symmetric formulation of (3.21)–(3.24).
Lemma 3.2. Let the four functions kj , ∂s kj , ∂s pc : [0, 1] → R be continuous and
positive and let sw,0 be bounded from below by a positive number. Then equation
(3.18) is equivalent to the five equations (3.21)–(3.24), which include the four auxiliary
variables sj,1, kn,1 , and pw,1.
Proof. Let kj,1, pj,1 and sj,1 be a solution of system (3.21)–(3.24). We evaluate with
k = kw (s) the derivative of Ψ as ∂κ Ψ(k) = p̃c (s) + kw ∂s p̃c (s)(∂s kw (s))−1 and calculate
for the right hand side of (3.18)
∂κ Ψ(kw,0) · kw,1 + pn,0 kw,1 + kw,0pn,1
= p̃c (sw,0) + kw,0 ∂s p̃c (sw,0)(∂s kw (sw,0))−1 · kw,1 + pn,0 kw,1 + kw,0 pn,1
(3.21)
= pw,0kw,1 + kw,0 ∂s p̃c (sw,0 )sw,1 + kw,0 pn,1
(3.24)
= pw,0kw,1 + kw,0(pw,1 − pn,1 ) + kw,0pn,1
(3.22)
= pw,0 kw,1 + kw,0 pw,1 = ψw,1 .
We have thus obtained (3.18). We emphasize that we have used also the macrorelation (3.15).
For the other implication we start with a solution of (D), in particular, kw,1 satisfies
(3.18). We define sw,1 with (3.21) for j = w, then sn,1 with (3.23), kn,1 with (3.21)
for j = n, and finally pw,1 with (3.24). It remains to check for these quantities that
relation (3.22) is satisfied. This follows with the same calculation as above.
15
P. Henning, M. Ohlberger, B. Schweizer
4
4.1
Proofs
Energy estimates for the pressure
Energy estimates are obtained by using pεj − Uj as a test-function in (2.15), where
Uj is the function that provides the boundary data. We add the equations for j = w
and j = n. This provides
X Z
X Z
ε
ε
pj ∂t sj +
∇pεj · Kjε kjε (sεj )∇pεj
j∈{w,n}
Ω
Ω
j∈{w,n}
X Z
=
j∈{w,n}
Ω
∇pεj
·
Kjε kjε (sεj )∇Uj
+
X Z
j∈{w,n}
Ω
(4.1)
∂t sεj Uj
.
On the right hand side we use the Cauchy-Schwarz inequality, exploiting Uj ∈ H 1 (Ω)
and the boundedness Kj , kj ≤ C0 to find
Z
1/2
Z
ε ε ε
ε 2
∇pε · K ε k ε (sε )∇Uj ≤ C
Kj kj (sj )|∇pj |
.
j
j j j
Ω
Ω
With the Young inequality, this term can be absorbed into the second sum of the left
hand side. For the further treatment of the first sum on the left hand side of (4.1),
we use the algebraic relations of (2.17). The saturation relation sεn + sεw = 1 provides
∂t sεn = −∂t sεw . Exploiting additionally the capillary pressure relation, we can write
the first sum of the left hand side as a total time derivative:
X
d ε ε
[P (s )] ,
pεj ∂t sεj = (pεw − pεn )∂t sεw = pεc (sεw )∂t sεw =
dt c w
j∈{w,n}
where we used the family of primitive functions Pcε with ∂s Pcε (x, s) = pεc (x, s) and the
normalization Pcε (s = 0) = 0. Equation (4.1) therefore leads to
X Z TZ
ε ε
[Pc (sw )] (T ) +
Kjε kjε (sεj )|∇pεj |2 ≤ C + [Pcε (Sw )] .
j∈{w,n}
0
Ω
The functions pεc are monotone in s and bounded from below. This implies that
the (convex) primitives Pcε are uniformly bounded from below. By our assumption
ess supΩ Sw < 1 on the initial values Sw , the right hand side is bounded. Since the
functions Kjε were assumed to be strictly positive, we obtain the a priori estimate
X Z TZ
kjε (sεj )|∇pεj |2 ≤ C ,
(4.2)
j∈{w,n}
0
Ω
with C depending on initial and boundary values, but not on ε.
4.2
A priori estimates for kw and ψw .
The fundamental observation for the definition of a weighted solution is the following
H 1 -bound for the two quantities kwε := kwε (sεw ) and ψwε := kwε pεw ,
Z TZ
Z TZ
ε 2
|∇kw | +
|∇ψwε |2 ≤ C.
(4.3)
0
Ωεm
0
Ωεm
16
Homogenization of the two-phase flow system
In order to obtain (4.3), we first express the saturation sεw in terms of the pressures
with the help of (2.17) as sεw = (pεc )−1 (pεw − pεn ). Due to the strict monotonicity
assumption (2.10), the inverse function (pεc )−1 is well-defined also for a multi-valued
function pεc . We obtain
2
1
1
ε
ε
ε 2
|∇sw | = ε ε (∇pw − ∇pn ) ≤ 2 |∇pεw − ∇pεn |2 .
∂s pc (sw )
c1
We recall that in regions with sεw = 0, the pressures are not well-defined; since the
saturation satisfies ∇sεw = 0 in these regions, we can nevertheless insert the estimate
below. We calculate with property (2.7)
|∇kwε |2 = |∂s kwε (sεw )|2 |∇sεw |2 ≤ C02 kwε (sεw )|∇sεw |2
≤ (C0 /c1 )2 kwε (sεw )|∇pεw − ∇pεn |2 ≤ 2(C0 /c1 )2 kwε (sεw )(|∇pεw |2 + |∇pεn |2 ).
The non-degeneracy of the non-wetting permeability in (2.5) and the boundedness of
the wetting permeability allow to conclude from estimate (4.2) the boundedness of
integrals over kwε (sεw )|∇pεn |2 . Therefore, estimate (4.2) implies the a priori bound
Z TZ
(4.4)
|∇kwε |2 ≤ C.
0
pεw
Ω
Based on this observation, we can now calculate for ψwε = kwε pεw in the case that
is uniformly bounded
Z TZ
Z TZ
ε 2
|∇ψw | ≤ C
(|∇kwε |2 + kwε |∇pεw |2 ) ≤ C
(4.5)
0
Ω
0
Ω
by (4.2) and (4.4).
4.3
Two-scale limits and proof in the non-degenerate case
We use two-scale convergence as described in [1]; the (weak) two-scale convergence
2
uε ⇀ u allows to extract εY -periodic behavior of the sequence of functions uε = uε (x)
and encodes it in the function u = u(x, y) with y ∈ Y . We furthermore use strong
two-scale convergence as was investigated e.g. in [27, 36].
Definition 4.1. We say that a sequence uε ∈ L2 (Ω) is strongly two-scale convergent
to u ∈ L2 (Ω × Y ), iff
2
uε ⇀ u
and
kuε kL2 (Ω) → kukL2 (Ω×Y ) .
(4.6)
We note that, just as in the case of two-scale convergence, the concept can be
transferred directly to space-time domains ΩT , in which the fine-scale expansion is
performed only in the spatial parameters x ∈ Ω and not in the time variable t ∈ [0, T ].
Below, we always treat the time variable as a parameter and omit dependences on t.
An equivalent definition of strong two-scale convergence can be given as follows.
2
A sequence uε ⇀ u is strongly two-scale convergent, if, for every (weakly) two-scale
2
convergent sequence v ε ⇀ v, there holds
Z
Z Z
ε
ε
lim u (x)v (x) dx =
u(x, y)v(x, y) dy dx.
(4.7)
ε→0
Ω
Ω
Y
P. Henning, M. Ohlberger, B. Schweizer
17
The full strength of strong two-scale convergence becomes clear with the following
2
result. Let uε ⇀ u be a strongly two-scale convergent sequence. Then there holds
x ε
lim u (x) − u x,
= 0.
(4.8)
ε→0
ε L2 (Ω)
We note that, in the setting of [36], a Caratheodory property of the limit function
u is assumed in the derivation of (4.8). On the other hand, no such assumptions are
necessary for the Lebesgue measure setting in [27]; nevertheless, the authors of the
latter work emphasize that also the appropriate choice of a space of test-functions is
important. Vice versa, property (4.8) also implies the strong two-scale convergence,
hence (4.8) can be used as a characterization of strong two-scale convergence.
In the proofs of our main theorems below we will use the following facts.
Lemma 4.1 (Piecewise H 1 -bounds,
S strong two-scale convergence, nonlinear funcε
tions). Let Ym ⊂ Y and Ωm = Ω ∩ k ε(k + Ym ) be subdomains such that every Ωεm
is a connected Lipschitz-subdomain of Ω.
(i) Let uε be a bounded sequence in H 1(Ωεm ), we identify uε with its trivial extension setting uε = 0 in Ω \ Ωεm such that we may also write uε ∈ L2 (Ω). Then, along
a subsequence and for a limit function u ∈ L2 (Ω, H 1 (Ym )), the sequence uε converges
strongly in two scales to u.
2
(ii) Let the sequence uε ⇀ u be strongly two-scale convergent and let q : R → R be
a Lipschitz continuous function. We consider v ε := q ◦ uε ∈ L2 (Ω) and v := q ◦ u ∈
2
L2 (Ω × Y ). Then there holds the strong two-scale convergence v ε ⇀ v.
(iii) Let the sequence uε : Ω → [a, b] have values in the interval [a, b]. Let q :
[a, b] → R be a continuous nonlinear function, continuously differentiable on (a, b)
with q ′ > 0 and q ′ bounded on every interval [a + δ, b] with δ > 0. Then the strong
2
two-scale convergence uε ⇀ u implies the strong two-scale convergence v ε → v for
v ε = q ◦ uε and v = q ◦ u.
√
In item (iii) we think, e.g., of a square-root function, q(ξ) = ξ, q : [0, 1] →
[0, 1]. The connectedness of the subdomains is important: A sequence of functions,
which is piecewise constant on disconnected subdomains, can be bounded in H 1 , but
nevertheless oscillatory and only weakly two-scale convergent.
Sketch of proof for Lemma 4.1. Property (i) is a fact that is used in various forms for
homogenization problems in domains with holes. Early contributions use extension
operators and require regular subdomains. For the statement above we refer to the
approach of [2].
Regarding (ii) it suffices to calculate with the Lipschitz constant Cq of the function
q and with property (4.8)
Z Z x 2
x 2
ε
ε
2
u (x) − u x,
≤ Cq
→ 0.
v (x) − v x,
ε
ε
Ω
Ω
This shows property (4.8) for the sequence v ε and the function v and thus the strong
two-scale convergence of v ε .
To check property (iii), we assume without loss of generality a = 0. We introduce,
for the arbitrarily small parameter δ > 0, the set of good points Gε,δ = {x ∈ Ω :
18
Homogenization of the two-phase flow system
uε (x) ≤ 2δ and u(x, x/ε) ≤ 2δ}; loosely speaking, both uε and the limit function
are small on this set. Secondly, we use the set of points Hε,δ = {x ∈ Ω : uε (x) ≥
δ and u(x, x/ε) ≥ δ}, where both functions uε and u are large. The two sets do not
necessarily cover Ω, since one function may be small while the other is large. We
1
introduce the corresponding “bad” sets Bε,δ
:= {x ∈ Ω : uε (x) ≥ 2δ and u(x, x/ε) ≤
2
δ} and Bε,δ
:= {x ∈ Ω : uε (x) ≤ δ and u(x, x/ε) ≥ 2δ}. Estimate (4.8) for strongly
two-scale convergent sequences implies for the bad sets the smallness in measure,
l
|Bε,δ
| → 0 for ε → 0, any δ > 0, l = 1 and l = 2.
For points x ∈ Gε,δ we exploit the fact that both v ε = q(uε (x)) and v(x, x/ε) =
q(u(x, x/ε)) are close to q(0). For points x ∈ Hε,δ we use the argument of (ii),
exploiting the Lipschitz continuity of q : [δ, b] → R with constant Cq,δ . With the
indicated decomposition of the domain we calculate
Z Z
x 2
ε
(|v ε (x) − q(0)|2 + |v(x, x/ε) − q(0)|2 )
v (x) − v x,
≤2
ε
Ω
Gε,δ
Z
x 2
ε
1
2
2
| + C|Bε,δ
|.
+ Cq,δ
+ C|Bε,δ
u (x) − u x,
ε
Hε,δ
The right hand side is arbitrarily small if we choose first δ > 0 small enough to obtain
smallness of the first integral, and then ε > 0 small in order to have smallness of the
second integral and of the measures of the bad sets.
Due to the characterization in (4.8), this implies the strong two-scale convergence
of the sequence v ε .
Proof of Theorem 1. We study here the non-degenerate case, i.e. we assume that the
positivity assumption kj ≥ c0 of (2.8) is satisfied for both fluids j = w and j = n.
In this case, the a priori estimate (4.2) provides uniform estimates for both pressure
functions pεj ∈ L2 (0, T ; H 1(Ω)). We can extract two-scale limits which read, omitting
the time dependence in all expressions,
2
pεj ⇀ pj,0 ,
2
∇pεj ⇀ ∇x pj,0 + ∇y pj,1 ,
2
sεj ⇀
2
qjε ⇀
sj,0 ,
qj,0 ,
pj,0 : Ω → R,
(4.9)
pj,1 : Ω × Y → R,
(4.10)
sj,0 : Ω × Y → R,
(4.11)
qj,0 : Ω × Y → Rn .
(4.12)
All two-scale convergences are with respect to the L2 -norm. In particular, we have
now defined the eight variables of (3.1)–(3.2). It remains to verify the relations
(3.3)–(3.7).
We observe that an additional regularity of the saturation is available. The strict
monotonicity of pc in the variable s allows to express, in each subdomain Ωεm , the
−1 ε
ε
saturation as sεw (x) = (pm
c ) (pw (x) − pn (x)). This implies a uniform estimate for
sεw ∈ L2 (0, T ; H 1(Ωεm )). In particular, the two scale limits sj,0 are independent of y in
each subdomain Ym . By the y-independence of kw , see (2.2), also the two-scale limit
kw,0 of the permeabilities are y independent in each subdomain Ym .
We have an additional information on the time derivative from the evolution equation (2.15): the family ∂t sεw ∈ L2 (0, T ; H −1(Ωεm )) is bounded. Combining a LionsAubin Lemma (as in the proof of Proposition 2.2) with the two-scale compactness
19
P. Henning, M. Ohlberger, B. Schweizer
statement of item (i) in Lemma 4.1, we can conclude that the two scale convergence
in (4.11) is indeed a strong two scale convergence. A proof of this fact can be given
along the lines of the above-mentioned results, we omit the technicalities here.
The weak limits of sεj and qjε are recovered as the averages of the two-scale limits,
Z
Z
ε
ε
sj ⇀ − sj,0 (x, y) dy,
qj ⇀ − qj,0 (x, y) dy,
Y
Y
hence (3.3) follows by taking the weak limit in (2.15).
In order to derive (3.4), we first observe that (2.16) implies
2
(Kjε kjε (sεj ))−1 qjε = ∇pεj ⇀ ∇x pj,0 + ∇y pj,1 .
(4.13)
On the left hand side, the factor qjε converges (weakly) in two-scales, see (4.12). In
the other factor, we exploit the strong two scale convergence of sεj , non-degeneracy
of kjε and item (ii) of Lemma 4.1, to conclude the strong two scale convergence of
2
(Kjε kjε (sεj ))−1 ⇀ (Kj (x, y)kj (x, y, sj,0(x, y)))−1.
To be more precise, we use, in this conclusion, item (ii) of Lemma 4.1 M times,
separately for each subdomain. With the help of property (4.7) of strong two-scale
convergence, we can take the (weak) two-scale limit of the product (Kjε kjε (sεj ))−1 qjε on
the left hand side of (4.13). We find that it converges in two scales to the function
(Kj (x, y)kj (x, y, sj,0(x, y)))−1 qj,0(x, y). This provides (3.4).
We next want to verify (3.5), that the distribution ∇y · qj,0 vanishes. By (2.15),
for an arbitrary function φ ∈ Cc∞ (Ω×Y ×(0, T ), R) and the corresponding oscillatory
function φε (x, t) = φ(x, x/ε, t) there holds
Z TZ Z
h∇y · qj,0 i (φ) =
qj,0 (x, y, t) ∇y φ(x, y, t)
0
Ω Y
Z TZ
Z TZ
ε
ε
= lim
qj ε∇φ = lim
sεj ε∂t φε = 0.
ε→0
0
Ω
ε→0
0
Ω
Relations (3.6) and (3.7) are immediate consequences of the algebraic conditions
in (2.17). For the limit consideration leading to the nonlinear term in (3.7) we exploit,
as in the derivation of (3.4), the strong two-scale convergence of sεw to sw,0.
This concludes the proof of Theorem 1.
4.4
Proof in the degenerate case
Proof of Theorem 2. In the degenerate case we consider, in addition to the primal
variables of saturations sεj and pressures pεj , the wetting permeability kwε = kwε (sεw )
and the weighted pressure ψwε = kwε pεw as dependent variables.
Due to the H 1 estimates on subdomains, obtained in (4.3), we find two-scale limits
2
kwε ⇀ kw,0,
2
∇kwε ⇀ ∇x kw,0 + ∇y kw,1,
kw,0 : Ω × Y → R,
(4.14)
kw,1 : Ω × Y → R,
(4.15)
20
Homogenization of the two-phase flow system
and analogous limits for sεw and ψwε = kwε pεw . We claim that the convergences sεw to
sw,0 and kwε to kw,0 are strong two-scale convergences. Without time dependence,
this could be concluded directly from item (i) of Lemma 4.1. The time dependence
makes an argument as in the proof of Proposition 2.2 necessary: with the exponent
ρ ≥ ρ0 + 1 (compare the growth condition of (2.7)), we consider the new family of
functions f ε = (sεw )ρ . The functions f ε are bounded in L2 (0, T ; H 1(Ωεm ) for each
subdomain. Furthermore, the time derivative is controlled in L1 (0, T ; Z) for the
dual space Z of a Sobolev space of regular functions. A Lions-Aubin lemma in the
two-scale convergence setting implies the strong two-scale convergence of f ε . Since
nonlinear functions can be applied to strongly two-scale convergent sequences (see
item (ii) and (iii) of Lemma 4.1), we can conclude the strong two-scale convergence
of sεw to sw,0 and of kwε to kw,0. As in the non-degenerate case, we find that sw,0 and
kw,0 are independent of y in each subdomain Ym .
Step 1. Consequences of the strong two-scale convergence. The algebraic relation
ε
kw = kwε (sεw ) allows not only to conclude the strong two-scale convergence of both
variables, but item (iii) of Lemma 4.1 also provides the characterization of the limit,
kw,0(x, y) := kw (x, y, sw,0(x, y)).
Regarding the pressure pεw (x) we had to make a choice in regions with vanishing
saturation. We made this choice by demanding pεw −pεn = p̃εc (sεw ) in Definition 2.2. Accordingly, we find the two-scale limit function pw,0 (x, y) = pn,0 (x) + p̃c (x, y, sw,0(x, y))
by the strong two-scale convergence of the saturation. In particular, (3.15) is satisfied.
The strong two-scale convergence of kwε now implies also relation ψw,0 (x, y) =
kw,0(x, y)pw,0(x, y) for the product term. The fact that pw,0 (x, y) is independent of
y in each subdomain Ym follows from the y-independence of kw,0 and ψw,0 in each
subdomain. Once the trace relation (3.16) is established, we conclude that p̃c (sw,0) is
without jumps at interfaces and (3.15) for pw,0(x, y) implies that the latter is indeed
independent of y ∈ Y .
Step 2. Further limit equations. We have to verify the remaining equations of
the system (3.10)–(3.18). Relations (3.10)–(3.14) with the exeption of (3.12) follow
exactly as in the non-degenerate case, exploiting again the strong two-scale convergence of sεw . Relation (3.12) must be derived from the flux relation in the weighted
solution setting, i.e. (2.24). This relation reads
(Kwε )−1 qwε = ∇ψwε − (∇kwε )pεw .
We multiply with kwε and write the result as
(Kwε )−1 kwε qwε = kwε ∇ψwε − (∇kwε )kwε pεw = 2kwε ∇ψwε − ∇(kwε ψwε ) .
The strong two-scale convergence kwε → kw,0(x, y) found in Step 1 allows to take the
two-scale limit in both product terms. Dividing the limit relation by kw,0(x, y) (in
the domain with sw,0(x, y) 6= 0), we find (3.12).
The jump conditions (3.16) and (3.17) follow from (2.25), which provides
tracem (kwε ) = Φm,m′ (x, tracem′ (kwε )) .
The piecewise L2 (0, T ; H 1)-bound for kwε and the trace theorem allow to transfer this
relation to the two-scale limit.
P. Henning, M. Ohlberger, B. Schweizer
21
Finally, the relation between ψw,1 and kw,1 of (3.18) follows from the definition of
Ψ(κ) = κ p̃c (kw−1 (κ)) in (2.11), which allows to write
ψwε = kwε pεw = Ψ(kwε ) + kwε pεn ,
and hence also
∇ψwε = ∂κ Ψ(kwε )∇kwε + ∇(kwε pεn ) .
In this expression, the last term converges weakly in two scales to some limit function;
the fact that the last term is a total derivative and that kwε converges strongly in two
scales allows to identify the limit with the formal limit. In the two-scale limit we find
∇x ψw,0 + ∇y ψw,1 = ∂κ Ψ(kw,0)(∇x kw,0 + ∇y kw,1)
+ ∇x (kw,0 pn,0 ) + kw,0 ∇y pn,1 + pn,0 ∇y kw,1 .
We consider a fixed subdomain Ym and subtract from the above the x-gradient of the
zero-order limit relation ψw,0 (x) = kw,0(x)pw,0 (x, y) = Ψ(kw,0(x)) + kw,0(x)pn,0 (x).
This provides that the y-gradients of both sides in (3.18) coincide. Upon changing
ψw,0 (x, y) by an additive constant, we conclude (3.18). Hence Theorem 2 is shown.
5
Effective model in a one-dimensional case
As an example for the non-degenerate limit problem of Definition 3.1, we study in this
section the one-dimensional situation. We investigate a setting that had been treated
before in [13, 20, 21, 33]. Setting the space dimension to n = 1, we use Y = [−1, 1)
as a periodicity cell, set Kj := 1, and choose, for x ∈ Ω = (−L, L), permeability and
capillary pressure as periodically oscillating functions by setting
(
(
p̃+
kj+ (x, s) for y ∈ [0, 1)
c (s) for y ∈ [0, 1)
.
,
p̃
(y,
s)
=
kj (x, y, s) =
c
−
p̃−
kj (x, s) for y ∈ [−1, 0)
c (s) for y ∈ [−1, 0)
Our aim is to investigate the non-degenerate limit problem of Definition 3.1 in this
case. To simplify further, we assume that pw,0(x) − pn,0 (x) = p̃c (x, y, sw,0(x, y))
holds as an equality in (3.7). This one-dimensional setting has been also studied in
[21], where an effective system has been suggested by formal asymptotics, the correct
effective system has been derived in [20], again by formal asymptotics, and it has been
verified analytically in [33]. Regarding existence results and, in particular, interface
conditions, we refer to [7, 13].
5.1
Effective system in one space dimension
Omitting once more the time dependence, the unknowns in (3.1)–(3.2) are pj,0(x),
sj,0(x), pj,1(x, y), qj,0 (x, y). Relation (3.5) implies that qj,0 (x, y) = qj,0(x) is independent of y. Furthermore, (3.7) yields that sw,0 (x, y) is independent of y in the interval
(−1, 0) and in the interval (0, 1). We can therefore abbreviate sw,0(x, y) = sw,± (x) =
s± (x) for ±y > 0, from (3.6) we know that the non-wetting saturation can be recovered as sn,0 (x, y) = sn,± (x) = 1 − s± (x) for ±y > 0. Our last structural observation
22
Homogenization of the two-phase flow system
is that (3.4) yields for the effective pressure gradient ∂x pj,0 (x) + ∂y pj,1 (x, y) = vj,± (x)
for ±y > 0. After these reductions, we have as unknowns
Z
pj,0 (x) and sj,0(x) := − sj,0 (x, y) dy as macroscopic variables, and
s± (x), vj,± (x) as auxiliary variables describing microscopic properties.
In order to find a one-dimensional two-phase flow system, we have to express the
six variables s± (x) and vj,± (x) by the macroscopic variables. From the definition of
sj,0(x) and relation (3.7) we obtain
s+ + s− = 2sw,0(x),
−
p̃+
c (s+ ) = p̃c (s− ).
(5.1)
(5.2)
Given sw,0(x), this system can be solved for s± = s± (x) due to the monotonicity
of p̃c . The facts that pj,1(x, y) is periodic and that the left hand side in (3.4) is
y-independent, imply
vj,+ + vj,−
+
kj (x, sj,+ (x)) vj,+
= 2∂x pj,0(x),
= kj− (x, sj,− (x)) vj,− .
(5.3)
(5.4)
Given ∂x pj,0(x), this system can be solved for vj,± = vj,± (x) by positivity of the
factors kj± . We have thus seen that the six equations (5.1)–(5.4) determine the six
auxiliary variables.
The macroscopic equations (3.3) can now be written in a compact form. We can
reconstruct the saturations from the pressures with the two maps Sw and Sn ,
Sj : Ω × R2 → R,
Sn := 1 − Sw ,
(5.5)
1 −
1
−1
−1
(x,
.))
(p
−
p
)
+
(p̃
(x,
.))
(p
−
p
).
Sw : (x, pw , pn ) 7→ (p̃+
w
n
w
n
2 c
2 c
With the saturation sw,0(x) = Sw (x, pw , pn ), we can define the flux functions as
Fj : Ω × R2 × R2 → R,
(x, pw , pn , δpw , δpn ) 7→ kj+ (x, sj,+ (x)) vj,+
(5.6)
∂t Sj (x, pw (x), pn (x)) = ∂x Fj (x, pw (x), pn (x), ∂x pw (x), ∂x pn (x))
(5.7)
where the quantities sw,± (x) = s± (x), sn,± (x) = 1 − sw,± (x) and vj,± (x) are calculated from (5.1)–(5.4) with sw,0 (x) = Sw (x, pw , pn ) and with ∂x pj,0 (x) = δpj . The
macroscopic effective two-phase flow equations (3.3) for (pw (x, t), pn (x, t)) now read
for j = w and j = n.
The above reduction can be compared directly with the procedure in [33]. Our
equations (5.1)–(5.2) coincide with equations (3.1) and (3.2) of [33], except that we
write here s± instead of u± for saturation values. Our equation (5.3) relates to the
geometric slope condition (3.6) of [33], our flux continuity (5.4) relates to the flux
continuity (3.5) of [33]. In the contribution at hand we have chosen not to exploit
the fact that the total flux is constant and given by the boundary condition. Instead,
we stick with two effective equations, using the two indices j = w and j = n, which
makes the effective system more symmetric and more similar to the original system.
As a consequence, we can avoid the additional unknowns u±,x (x) of [33] in the set of
microscopic variables.
23
P. Henning, M. Ohlberger, B. Schweizer
5.2
One-dimensional numerical experiment
In order to validate the homogenized system (5.1)–(5.7), we consider the one dimensional trapping test problem introduced in Section 4.1 of [21], and Section 4 of [20].
The test problem investigates the permeabilities and capillary pressures
kj (x, y, s) =
(
k + krj (s) for y ∈ [0, 1)
,
k − krj (s) for y ∈ [−1, 0)
p̃c (y, s) =
(
√1 J(s)
k+
√1 J(s)
k−
for y ∈ [0, 1)
for y ∈ [−1, 0)
with
k − = 0.5,
k + = 1.0,
krw (s) = (1 − s)2 ,
krn (s) = (s)2 ,
J(s) = √
1
,
1−s
where s now denotes the saturation of the non-wetting phase. We consider the problem in Ω = (−1, 1) with initial saturation s = 1, the total flux at the inflow boundary
(x = −1) equals one, and the saturation boundary datum at the inflow boundary is
s(−1) = 0 for all times.
For numerical purposes, we rewrite the two equations of (5.7) in a global flux
formulation. Subtracting and adding the equations for both phases, we obtain
2∂t Sn = ∂x (Fn − Fw ),
0 = ∂x (Fn + Fw ) .
(5.8)
The second equation of (5.8) yields the x-independence of the total flux. Since the
total flux at the inflow is prescribed as 1, the second equation of (5.8) provides
Fn + Fw = 1.
It remains to solve the first equation of (5.8), the conservation law for the macroscopic saturation, where the fluxes are determined by (5.6). This homogenized global
flux formulation coincides with the effective model that was derived formally in [20]. A
formal distinction is that the definition of the effective macroscopic capillary pressure,
given here through equation (5.5), is expressed in terms of the microscopic variables
in [20]. However, taking into account (5.1)–(5.2), it is clear that both definitions
coincide.
The numerical results shown in Figures 2 and 3 are obtained with an explicit
upwind finite volume approximation of the homogenized system (5.1)–(5.7), based on
the global flux formulation. Obviously, they are in agreement with the results for the
effective model from [20]. In particular, Figure 2 illustrates the oil-trapping effect of
the fine scale simulation with 80 periodic cells. The zoom in of Figure 3 shows a close
to perfect matching between the numerical solution of the homogenized equation on
a coarse grid with 640 grid cells with the cell-averaged fine scale simulation obtained
on a fine grid with 3200 cells. In both cases grid convergence is ensured with the
chosen mesh sizes.
We emphasize that we present here fine-scale simulations and calculations for the
effective system, not the results of numerical multi-scale methods. The work at hand
opens the way to investigate, in a direct comparison, numerical multi-scale methods
for two-phase flow in porous media as developed, e.g., in [22, 23, 29].
24
Homogenization of the two-phase flow system
1
Saturation s(x)
0.8
0.6
0.4
0.2
detailed solution
averaged solution
homogenized solution
0
-1
-0.5
0
0.5
1
x
Figure 2: One-dimensional trapping effect. The graph shows saturation distributions
at a fixed time instance. The solid line shows the result of a detailed simulation with
80 periodicity cells. The dashed line shows its averages over one period, the dotted
line shows the solution of the homogenized problem.
6
Conclusions
Using the concept of weighted solutions, we have derived a rigorous homogenization
result for degenerate two phase flow in porous media. Our result covers multiple space
dimensions and the situation in which no scaling law is assumed between the local
permeability values. Our contribution therefore closes a gap in the homogenization
analysis of two phase flow in porous media that was left open in [35], where one of the
local media coefficients was supposed to scale with ε̟ , ̟ > 0. In the one dimensional
case, the resulting homogenized system coincides with the effective model that was
formally derived in [20] and rigorously obtained in [33]. In this sense, our result can
be regarded as a generalization of [33] to the multi-dimensional case.
Acknowledgement
Support by DFG-grants SCHW 639/3-1 and OH 98/4-2 is gratefully acknowledged.
The authors thank the referees for the very relevant and helpful comments on a first
version of this article.
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25
P. Henning, M. Ohlberger, B. Schweizer
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