Generalised Rayleigh-Taylor condition

# A generalised Rayleigh-Taylor condition for the Muskat problem

Joachim Escher Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany. Anca-Voichita Matioc Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany.  and  Bogdan–Vasile Matioc Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany.
###### Abstract.

In this paper we consider the evolution of two fluid phases in a porous medium. The fluids are separated from each other and also the wetting phase from air by interfaces which evolve in time. We reduce the problem to an abstract evolution equation. A generalised Rayleigh-Taylor condition characterizes the parabolicity regime of the problem and allows us to establish a general well-posedness result and to study stability properties of flat steady-states. When considering surface tension effects at the interface between the fluids and if the more dense fluid lies above, we find bifurcating finger-shaped equilibria which are all unstable.

###### Key words and phrases:
Muskat problem; Rayleigh-Taylor condition; stability; bifurcation theory; finger-shaped equilibria
###### 2010 Mathematics Subject Classification:
35B35; 35B36; 35K55; 35R37

## 1. Introduction

The Muskat problem is a widely used model for the intrusion of water into oil sand. A linear analysis was performed in [21, 22, 24] where a relation, the so-called Rayleigh-Taylor condition, was found to determine two regimes for the problem: a stable regime, when a flat interface is stable under small deviations, and an unstable one, when fingering occurs.

Nonetheless, existence and uniqueness of classical solutions has been firstly proven in [25] by using Newton’s iteration method. In the last decade the problem has received more interest and was studied by means of complex analysis [23], energy estimates [2, 4, 5, 6], power series expansions [16], or abstract parabolic theory [13]. These different approaches cover a wide spectrum of questions related to the Muskat problem: local well-posedness, global existence of solution, singular solutions, stability properties of equilibria.

It is worth noticing that all these papers mentioned above consider the situation when there is only one moving boundary, namely the one separating the fluids. Either one prescribes boundary conditions at two boundaries which are kept fixed during the flow or so-called far-field boundary condition are imposed. This setting corresponds to an abstract equation with only one unknown - the interface between the fluids. In the present paper we consider the more involved situation when there are two moving boundaries, one separating the two fluids and one separating the wetting phase from air (assumed to be at uniform pressure equal to zero). The fluids are located in a porous medium (or a vertical Hele-Shaw cell) and are assume to fill together with the dry phase (air) the entire void medium. Moreover, we incorporate gravity and viscosity effects into the modeling as well as surface tension forces at both interfaces. The invertibility of a bounded operator permits us to re-write the problem as an abstract non-autonomous evolution equation

 ∂tZ=Φ(t,Z),Z(0)=Z0,

where the variable parametrises both unknown interfaces. The temporal variable is induced into the problem by the boundary condition for the pressure on the bottom of the cell. For this problem we find a generalised Rayleigh-Taylor condition in terms only of the boundary data , the viscosities , and densities of the fluids of the following form

 bμ++gρ+μ−>0andμ+−μ−μ++μ−(b−gρ+)+g(ρ+−ρ−)<0, (1.1)

which determines the parabolic character of the problem in the absence of surface tension effects. When including surface tension forces at both interfaces we may drop condition (1.1). We steadily use in this paper the subscript for the fluid on the bottom of the cell and for that above. After showing that the Fréchet derivative generates a strongly continuous and analytic semigroup, parabolic theory provides local well-posedness of the problem and the principle of linearised stability may be applied to study the stability properties of the unique flat equilibrium which is determined for a fixed amount of fluid (this quantity is preserved by the flow) and a certain constant boundary data.

When considering surface tension effects at the interface between the fluids and the more dense fluid lies above we re-discover the global bifurcation branches obtained in [14, 13] which consist only of finger-shaped equilibria of the Muskat problem. The exchange of stability theorem due to Crandall and Rabinowitz [8] applies to this particular problem and we show that all small equilibria are unstable.

The outline of the paper is as follows: we describe in Section 2 the mathematical model and present the main results. Section 3 is dedicated to the proof of the well-posedness result Theorem 2.1, and in the subsequent section we analyse the stability properties of the unique flat equilibrium as stated in Theorem 2.5. In Section 5 we prove our third main result, Theorem 2.7. The calculations leading to the representation of as a Fourier multiplication operator are done in the Appendix.

## 2. The mathematical model and the main results

Let us start this section by presenting the mathematical model of the setting described in the introduction. Given and the small Hölder space stands for the closure of the smooth functions in We let denote the unit circle and functions on are identified with -periodic functions on For later purposes we define as the subspace of consisting only of even functions, is the subspace of consisting only of functions with integral mean zero, and Furthermore, we define the set of admissible functions to be

 U:={f∈C2(S):|f|<1/2},

Each pair determines two open and simply connected subsets of the porous medium, seen as , as follows:

 Ω(f):={(x,y):−1

Let and be given such that, at each time the fluid is located at and the fluid at (see Figure 1).

The two fluids are assumed to be of Newtonian type and incompressible, and both interfaces are supposed to move along with the fluids. The problem is governed by the following system of partial differential equations:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩Δu+=0inΩ(f,h),Δu−=0inΩ(f),∂th+k√1+h′2μ+∂νu+=0onΓ(h),u+=gρ+(1+h)−γdκΓ(h)onΓ(h),u−=bonΓ−1,u+−u−=g(ρ+−ρ−)f+γwκΓ(f)onΓ(f),∂tf+k√1+f′2μ±∂νu±=0onΓ(f),f(0)=f0,h(0)=h0 (2.1)

for , where determines the initial domains occupied by the fluids. We used the variable for parametrising the interface between the two fluids and separates the fluid from air. The unit normal at [resp. ] is chosen such that, if is the tangent, the orthonormal basis has positive orientation. We also write and for the curvature of and , respectively. Moreover [resp. ] is the surface tension coefficient of the interface separating the fluids from air [resp. the fluids].

The potentials incorporate both pressure and gravity force , with the gravity constant. The velocity fields , which satisfy Darcy’s law

 →v±=−kμ±∇u±,

are presupposed to be equal on the boundary separating the fluid phases. Hereby, stands for the permeability of the porous medium. On the fixed boundary we prescribed the value of the velocity potential . For a precise deduction of (2.1) we refer to [11, 13, 25].

Let be fixed for the following. A pair is called classical Hölder solution of (2.1) if

 (f,h)∈C([0,T],V)∩C1([0,T],(h1+α(S))2), u+(t)∈\it buc2+α(Ω(f(t),h(t))) and u−(t)∈\it buc2+α(Ω(f(t))) for t∈[0,T],

and if satisfies the equations of (2.1) pointwise. We defined to be the subset of given by

 V1 :={f∈h2+2sign(γw)+α(S):f∈U}, V2 :={h∈h2+2sign(γd)+α(S):h∈U},

where and for The space is defined as closure of the smooth functions with bounded and uniformly continuous derivatives in . The space is defined similarly. Moreover, since the potentials are determined, when knowing , as solutions of elliptic problems (see Section 3) we also refer to to be the solution of (2.1). The first main result of this paper states:

###### Theorem 2.1.

Let be given.

There exist an open neighbourhood of the zero function in such that for all and problem (2.1) possesses a unique classical Hölder solution defined on a maximal time interval and which satisfies The mapping

 (t,f0,h0)↦F(t;(f0,h0))

has the same regularity as has.

###### Remark 2.2.

The conclusion of Theorem 2.1 remains valid if or with the following modifications: if we have to replace by and require that where is a small neighbourhood of the zero function in and satisfies

 cμ++gρ+μ−>0, (2.2) μ+−μ−μ++μ−(c−gρ+)+g(ρ+−ρ−)<0. (2.3)

When [resp. ] we replace by [resp. ] and request that the constant satisfies only equation (2.2) [resp. eq. (2.3)].

Relation (2.2) is a generalisation of the positive pressure condition imposed in [11, 10, 12] to ensure well-posedness and stability of the one-phase Hele-Shaw problem without surface tension. Indeed, if the fluids have the same densities and viscosity, (2.2) re-writes which is, up to a scaling, the same condition as in [11, 10, 12]. Moreover, it turns out that the Muskat problem without surface tension effects studied in [6, 13, 26] is similar to our problem if Indeed, we have:

###### Lemma 2.3.

The volume of fluid is preserved by the solutions of (2.1).

###### Proof.

The proof is similar to that of [10, Lemma 3.1]. ∎

In order to establish similarity between our problem when and that in [13, 26], we determine a special solution of (2.1) in the case when the volume of fluid is equal to , i.e.

 ∫Sf0−h0dx=0. (2.4)

If initially and depends only on time, then

 f′(t)=−kgρ−μ−f(t)+gρ+−bgρ−f(t)+μ++μ−μ−,f(0)=f0, (2.5)

and, by Lemma 3.1, as long as the solution exists. If and we obtain from (2.3) that if , then , thus is positive if is close to zero, meaning that the more viscous fluid drives upwards the less viscous one in the medium. This condition has been found also in [13, 26] to guarantee well-posedness of the Muskat problem studied therein. Moreover, if the Atwood number

 Aμ:=(μ+−μ−)/(μ++μ−)

is zero, then (2.3) tells us that the more dense fluid must lay beneath in order to guarantee well-posedness of (2.1) when result similar to that in [6, 13].

Corresponding to the result in [13], where an optimal value for the normal velocity at which water may replace oil in the absence of surface tension effects was found, we obtain herein an optimal value for the pressure on the bottom of the medium:

###### Remark 2.4.

If the fluid below is water and that above oil, and we neglect the surface force at the interface between them, we find from (2.3) an optimal value

 pmax:=g(ρ++ρ−)−g(ρ+−ρ−)A−1μ (2.6)

for the pressure on the bottom of the porous medium below which water may drive upwards oil in a stable regime (no fingering occurs).

###### Proof.

Relation (2.6) is obtained form (2.3) in view of on The optimal value for the potential is and if the boundary value is close to this value we find that the solutions of (2.5) fulfill thus water drives oil upwards. This last assertion follows from

 gρ+−bmax=g(ρ+−ρ−)A−1μ<0

since it is well-known [3] that and (oil is less dense and more viscous than water). ∎

We infer from (2.5) that if and , then for all Concerning the stability properties of the stationary solution , which is the unique flat stationary solution of (2.1) for and which satisfies (2.4), we state:

###### Theorem 2.5.

Let . Then:

• If then the flat equilibrium of (2.1) is exponentially stable. More precisely, there exists positive constants and such that if and satisfies (2.4), then the solution of (2.1) exists globally and

 ∥(f(t),h(t))∥h2+2sign(γw)+α(S)×h2+2sign(γd)+α(S)+∥(∂tf(t),∂th(t))∥(h1+α(S))2
• If then is unstable.

###### Remark 2.6.

When we study the stability of equilibria in Theorem 2.5 and Theorem 2.7 below we fixed a volume of fluid equal to meaning that the initial data of (2.1) are presupposed to satisfy (2.4). This setting is imposed by Lemma 2.3, since the volume of fluid is preserved by the solutions of (2.1).

Theorem 2.5 is related to the exponential stability result established in [13, Theorem 5.3] for the Muskat problem with only one free boundary and is stronger than that in [16], where only stability is shown. Notice that if then the flat solution is always stable, since is exactly the condition (2.3) which guarantees well-posedness of (2.1). Concerning the unstable case, numerical experiments [17] show that the interface between the fluids becomes very ramified, and dendrite like structures occur as time evolves if

If and the volume of fluid is equal to , there exist also other stationary solutions of (2.1). They appear only in the unstable regime or sufficiently close to it, that is when and the more dense fluid lies above in the cell. We show that for certain small there exist finger-shaped stationary solutions of (2.1), and therefore we shall refer also to to be solution of (2.1). Given we define

 ¯¯¯γl:=g(ρ+−ρ−)l2.
###### Theorem 2.7.

Let and . If is a stationary solution of (2.1) satisfying (2.4), then and is a solution of the Laplace-Young equation

 γwκ(f)+g(ρ+−ρ−)f=0. (2.7)

The solution of (2.7) are, up to a translation, even and all even solutions of (2.7) can be represented as a disjoint union

 ∪∞l=1{(γl(ε),fl(ε)):ε∈R}⋃∪∞l=1(¯¯¯γl+1,¯¯¯γl)×{0}⋃(¯¯¯γ1,∞)×{0},

with continuous functions

 (γl,fl):R→(0,∞)×{f∈h4+α0,e(S):∥f∥C(S)<1},1≤l∈N,

which, near , are real analytic and satisfy:

 γl(ε) =¯¯¯γl+3g(ρ+−ρ−)8ε2+O(ε4),fl(ε) =εcos(lx)+O(ε2).

While is even and

 lim|ε|↗∞γl(ε)≤2π2g(ρ+−ρ−)B2(3/4,1/2)l2,

either or

Additionally, the equilibrium of problem (2.1) is unstable if is small. When we have to assume too.

Here stands for Euler’s beta function. Notice that the stationary solutions of (2.1), which satisfy (2.4) (see in Figure 2), are the same with the stationary solutions of the Muskat problem studied in [13, 6], where just one moving boundary is considered ( is chosen a priori to be zero). For a precise description of the global bifurcation branches we refer to [14]. It is shown there that the situation may occur only for small integers .

## 3. The evolution equation

In order to solve problem (2.1) we re-write it as an abstract evolution equation on the unit circle. To do that we first transform system (2.1) into a system of equations on fixed domains by using the unknown functions Let and Given we define the mappings by

 ϕf(x,y):=(x,y+(1+y)f(x)),(x,y)∈Ω−,

respectively

 ϕf,h(x,y):=(x,y(1+h(x))+(1−y)f(x)),(x,y)∈Ω+.

One can easily check that and are diffeomorphisms for all These diffeomorphisms induce pull-back and push-forward operators (see e.g. [11]) which we use to transform the differential operators involved in system (2.1) into operators on the domains and their boundaries, respectively. Each pair induces linear elliptic operators

 A(f):\it{buc}2+α(Ω−)→% \it{buc}α(Ω−), A(f)v:=Δ(v−∘ϕ−1f)∘ϕf, A(f,h):\it{buc}2+α(Ω+)→%bucα(Ω+), A(f,h)v:=Δ(v+∘ϕ−1f,h)∘ϕf,h,

which depend, as bounded operators, analytically on and . Denote by the trace operator with respect to . We associate problem (2.1) the following trace operators on :

 B(f)v− :=kμ−1−tr0(⟨∇(v−∘ϕ−1f)|(−f′,1)⟩∘ϕf),v−∈% \it{buc}2+α(Ω−), B(f,h)v+ :=kμ−1+tr0(⟨∇(v+∘ϕ−1f,h)|(−f′,1)⟩∘ϕf,h),v+∈\it{buc}2+α(Ω+),

which, seen as bounded operators into , depend analytically on and as well. Lastly, we define a boundary operator on Given we let

 B1(f,h)v+:=−kμ−1+tr1(⟨∇(v+∘ϕ−1f,h)|(−h′,1)⟩∘ϕf,h),v+∈\it{buc}2+α(Ω+),

whereby is the trace operator with respect to .

With this notation one can easily verify that if is a solution of (2.1), then solves the following system of equations:

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩A(f,h)v+=0inΩ+,A(f)v−=0inΩ−,∂th=B1(f,h)v+inΓ1,v+=gρ+(1+h)−γdκ(h)onΓ1,v−=bonΓ−1,v+−v−=g(ρ+−ρ−)f+γwκ(f)onΓ0,∂tf+B(f)v−=0onΓ0,∂tf+B(f,h)v+=0onΓ0,f(0)=f0,h(0)=h0 (3.1)

for all , where the transformed curvature operator is defined by The notion of solution of (3.1) is defined analogously to that of (2.1). Notice that the parametrisation is left invariant by the transformation above. In fact, one can see, cf. [11, Lemma 1.2] that each solution of (3.1) corresponds to a unique solution of (2.1).

We introduce now solution operators corresponding to the system (3.1). Given and , we let denote the solution of the linear, elliptic mixed boundary value problem

 ⎧⎪⎨⎪⎩A(f)v−=0inΩ−,B(f)v−=qonΓ0,v−=ponΓ−1. (3.2)

Further on, we define by writing for the unique solution of the problem

 ⎧⎪⎨⎪⎩A(f,h)v+=0inΩ+,v+=ponΓ1,v+=ronΓ0. (3.3)

It is convenient to write where

 T1(f)q:=(A(f),B(f),tr)−1(0,q,0),T2(f)p:=(A(f),B(f),tr)−1(0,0,p),

respectively with

 S1(f,h)p: =(A(f,h),tr,tr)−1(0,p,0), S2(f,h)r: =(A(f,h),tr,tr)−1(0,0,r).

The operators and are bounded linear operators and they depend, in the norm topology, analytically on and too.

The key point of our analysis is the following observation. If is a classical solution of (3.1) for to the initial data , then it must hold:

• and

Let us now show that from we can determine the derivative as a function of , , and only. Indeed, we plug into and into to obtain the equation

 ∂tf+B(f,h)S1(f,h)[gρ+(1+h)−γdκ(h)] space=+B(f,h)S2(f,h)[tr0T(f,−∂tf,b)+g(ρ+−ρ−)f+γwκ(f)]=0,

which can be writen equivalently

 (idh1+α(S)−B(f,h)S2(f,h)tr0T1(f))∂tf+B(f,h)S2(f,h)tr0T2(f)b (3.4) space=+B(f,h)S(f,h,gρ+(1+h)−γdκ(h),g(ρ+−ρ−)f+γwκ(f)).

The linear operator which is evaluated at is invertible, so that we obtain, by applying its inverse to (3.4), an equation expressing the derivative in dependence of and . Indeed, we have:

###### Lemma 3.1.

The set contains an open neighbourhood of with the property that

is an isomorphism for all

###### Proof.

The proof is based on a continuity argument. Namely, all the operators defined in this section depend analytically on their variables and then so does too. Thus, it suffices to show that is an isomorphism. To do that, we represent as a Fourier multiplication operator. Given we let denote its Fourier series expansion. A Fourier series ansatz yields for the following expression

 T1(0)q(x,y)=μ−k(1+y)ˆq(0)+μ−k∑m∈Z∖{0}ememy−e−me−mym(em+e−m)ˆq(m)eimx

for Respectively, if then may be expanded as follows

 S2(0,0)r(x,y)=(1−y)ˆr(0)+∑m∈Z∖{0}e2me−my−emye2m−1ˆr(m)eimx

for Combining these two relations and taking the normal derivative yields that

 G(0,0)q=μ−+μ+μ+q,∀q∈h1+α(S),

thus is an isomorphism. ∎

In virtue of Lemma 3.1, if the pair maps into we may apply the inverse of to (3.4), and get

 ∂tf=Φ1(t,f,h), (3.5)

with a nonlinear and nonlocal operator defined by the relation

 Φ1(t,f,h):= −G−1(f,h){B(f,h)S2(f,h)tr0T2(f)b (3.6) +B(f,h)S(f,h,gρ+(1+h)−γdκ(h),g(ρ+−ρ−)f+γwκ(f)).

Furthermore, from and (3.5) we obtain that is solution of the equation

 ∂th=Φ2(t,f,h), (3.7)

where the operator is given by

 Φ2(t,f,h):= B1(f,h)S(f,h,gρ+(1+h)−γdκ(h),g(ρ+−ρ−)f+γwκ(f)) (3.8) +B1(f,h)S2(f,h)tr0T(f,−Φ1(t,f,h),b).

By Lemma 3.1 and relations (3.5)-(3.8) we found that all the solutions of (3.1) which are contained in solve the following abstract evolution equation

 ∂tZ=Φ(t,Z),Z(0)=Z0, (3.9)

where and we introduced the new variable . Concerning the operator we state:

###### Theorem 3.2.

The operator has the same regularity as , it is analytic in the variable , and if then and are Fourier multipliers with symbols and respectively, given by

 λf1(m):=[Aμ(c−gρ+)+g(ρ+−ρ−)−γwm2]k|m|(μ++μ−)tanh(|m|), (3.10) λf2(m):=[c(μ+−μ−)+2gρ+μ−μ++μ−−gρ−−γwm2]k|m|(μ++μ−)sinh(|m|), (3.11) λh1(m):=−[gρ+μ−+cμ+μ−+μ++γdm2]k|m|(μ−+μ+)sinh(|m|), (3.12)
 λh2(m):=−[cμ++gρ+μ−μ−+μ++γdm2]k|m|μ+tanh(|m|)−μ−μ+λh1(m)cosh(m). (3.13)
###### Proof.

The regularity assertion is obvious. That the first order partial derivatives of with respect to and are Fourier multipliers follows from (6.6)-(6.9), relations proven in the Appendix. ∎

We give now a proof of our first main result.

###### Proof of Theorem 2.1 .

We verify that the assumptions of [19, Theorem 8.4.1] are fulfilled by Theorem 2.1 is then a consequence of this result. For continuity reasons it suffices in fact to show only that the derivative generates a strongly continuous and analytic semigroup in i.e.

 −∂ZΦ(0)∈H(h2+2sign(γw)+β(S)×h2+2sign(γd)+β(S),(h1+β(S))2)

for some By using the interpolation properties of the small Hölder spaces

 (hσ0(S),hσ1(S))θ=h(1−θ)σ0+θσ1(S), (3.14)

if and we find then all assumptions of [19, Theorem 8.4.1] to be fulfilled. Here denotes the interpolation functor introduced by Da Prato and Grisvard [9].

Let us first notice that derivative maps continuously into for some . This property can be verified easily by using [11, Theorem 3.4], which is a multiplier theorem based on some generalized Marcinkiewicz conditions. Since by (3.14)

 h2+2sign(γd)+˜β(S)=(h1+β(S),h2+