A generalised RayleighTaylor condition for the Muskat problem
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 RayleighTaylor condition characterizes the parabolicity regime of the problem and allows us to establish a general wellposedness result and to study stability properties of flat steadystates. When considering surface tension effects at the interface between the fluids and if the more dense fluid lies above, we find bifurcating fingershaped equilibria which are all unstable.
Key words and phrases:
Muskat problem; RayleighTaylor condition; stability; bifurcation theory; fingershaped equilibria2010 Mathematics Subject Classification:
35B35; 35B36; 35K55; 35R371. 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 socalled RayleighTaylor 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 wellposedness, 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 socalled farfield 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 HeleShaw 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 rewrite the problem as an abstract nonautonomous evolution equation
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 RayleighTaylor condition in terms only of the boundary data , the viscosities , and densities of the fluids of the following form
(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 wellposedness 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 rediscover the global bifurcation branches obtained in [14, 13] which consist only of fingershaped 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 wellposedness 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
Each pair determines two open and simply connected subsets of the porous medium, seen as , as follows:
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:
(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
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
and for 
and if satisfies the equations of (2.1) pointwise. We defined to be the subset of given by
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
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
(2.2)  
(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 wellposedness and stability of the onephase HeleShaw problem without surface tension. Indeed, if the fluids have the same densities and viscosity, (2.2) rewrites 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.
(2.4) 
If initially and depends only on time, then
(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 wellposedness of the Muskat problem studied therein. Moreover, if the Atwood number
is zero, then (2.3) tells us that the more dense fluid must lay beneath in order to guarantee wellposedness 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
(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
since it is wellknown [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.
Remark 2.6.
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 wellposedness 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 fingershaped stationary solutions of (2.1), and therefore we shall refer also to to be solution of (2.1). Given we define
Theorem 2.7.
Let and . If is a stationary solution of (2.1) satisfying (2.4), then and is a solution of the LaplaceYoung equation
(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
with continuous functions
which, near , are real analytic and satisfy:
While is even and
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 rewrite 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
respectively
One can easily check that and are diffeomorphisms for all These diffeomorphisms induce pullback and pushforward 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
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 :
which, seen as bounded operators into , depend analytically on and as well. Lastly, we define a boundary operator on Given we let
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:
(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
(3.2) 
Further on, we define by writing for the unique solution of the problem
(3.3) 
It is convenient to write where
respectively with
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
which can be writen equivalently
(3.4)  
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
for Respectively, if then may be expanded as follows
for Combining these two relations and taking the normal derivative yields that
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
(3.5) 
with a nonlinear and nonlocal operator defined by the relation
(3.6)  
Furthermore, from and (3.5) we obtain that is solution of the equation
(3.7) 
where the operator is given by
(3.8)  
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
(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
(3.10)  
(3.11)  
(3.12) 
(3.13) 
Proof.
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.
for some By using the interpolation properties of the small Hölder spaces
(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].