Pressure / Saturation System for Immiscible Two-Phase Flow : Uniqueness Revisited

We give a sufficient condition for uniqueness for the pressure/saturation system. We establish this condition through analytic arguments, and then construct “mobilities” (or mobility-like functions) that satisfy the new condition (when the parameter  is 2). For the constructed “mobilities”, we do graphical experiments that show, empirically, that this condition could be satisfied for other values of 1 < < 2  . These empirical experiments indicate that the usual smoothness condition on the fractional flow function (and on the total mobility), for uniqueness and convergence, might not be necessary. This condition is also sufficient for the convergence of a family of perturbed problems to the original pressure/saturation problem.


Introduction
Consider the coupled nonlinear problem (1), with , which arises from modeling incompressible two-phase immiscible (water/oil, for example) flow through a porous medium (see [1,2], for instance).The problem considered, here, is in one of its simplified problem.

 
The conductivity of the medium is denoted by k while u is the total Darcy's velocity for the two-phase flow, f is the fractional flow function, S the saturation of the invading fluid (or wetting phase), P is the global pressure, and  the porosity of the medium.For the present analysis and for simplicity, we let 1 The set  is a sufficiently smooth bounded domain of , , 2 or 3, although this analysis focuses more on the case .
n R = 1 n 2 n = Obviously, Problem 1 cannot, in general, be solved analytically: One needs to proceed through numerical approximations.Before attempting any solution method, one needs to investigate whether the problem has a solution and, if it does, whether the solution is unique.The main purpose of this paper is to revisit the uniqueness question of Problem 1, exhibit sufficient conditions for which the problem has a unique solution, and construct examples for which these conditions are satisfied.Those conditions generalize the ones considered in [3] and in [4] for the uniqueness of the problem and for the convergence of a family of perturbed problems.This work constitutes, in some way, a complement to [3].In addition, and on the applied side, the mobility-like functions that we construct can be used in testing codes for two-phase flow through porous media.
The following conditions are usually imposed on the data (see [3], for instance). where , and 0 < 2   , for some 1  and 2  .
The function f satisfies the following.
Condition (6) has been used (as a sufficient condition, among other conditions) for the proof of the wellposedness for the saturation equation, the convergence for a regularization of that equation, and the convergence of numerical approximations of the same equation [7][8][9].
The rest of the paper is articulated as follows.In Section 2, we establish a new sufficient condition for (6) to hold, therefore for uniqueness of a solution of Problem 1.We also show that conditions (2) through (5) imply this new condition.In Section 3, we revisit the pressure saturation problem, to show, indeed, that, under this new condition (defined in Section 2), there is uniqueness for Problem 1.In Section 4, we construct examples of relative "mobilities" (mobility-like functions) and show that we have uniqueness under the special case = 2


, with  defined as in (3).We also explore experimentally, through graphs, the uniqueness problem for the pressure/ saturation problem, for these examples, for other values of  , 1 < < 2  , though condition (4) is not satisfied for the corresponding total mobility , or fractional flow function a f .
In this work we use standard notations.In particular, we use the norm of the function as an function in the variable t on , n v is a vector, we denote by

Sufficient Condition for Uniqueness for
the Pressure/Saturation System Lemma 2.1 Let and be two functions defined on the interval and suppose that and for all for all 0 1 a b .  

Proof.
We use a calculus argument.If , then the only value that = 1 a x can assume is 1, and (10) is obvious.For 0 < a 1  , define the function   (11)   for x a  . Then, , and Clearly, if (9) holds, then for all . This is true for any Hence the lemma is proved.
In [9], for one space variable and the unilateral case ), and in [6] for several variables and the bilateral case (     In the next lemma, we show that if (2) through (5) hold, then the couple   , f k satisfy (9), and therefore (6).

Note:
The above lemma is more general than what is known so far, since we do not require any of the conditions (2) through (5) to hold, nor do we require that f be in . However, if those conditions are satis-fied, we have the following lemma.Lemma 2.2 Under conditions ( 2) through (5), and under the assumption that the function f is twice conti- and for all   0,1 , a  for some constant .> 0 C Thus, the combination of Lemma 2.1 and Lemma 2.2 gives an alternative way of proving that (6) holds, which in turns leads to uniqueness for Problem 1.
Proof.We follow the lines of the proof of Proposition 3.2 of [6], with some modification.For the proof, it suffices to bound the quantity and  a x .Thanks to the symmetry implied by (3), we prove this for 1 0 a x     1 only, without lost of generality; the rest of the prove can be obtained by the change of variable , and by using the fact that for 7) and (3), we obtain Therefore, since By the Mean-Value Theorem, there exist such that where we have used ( 16), (17), and the fact that . Therefore the lemma is proved.

Uniqueness
We give an existence and uniqueness result for the case when and satisfy (9), i.e. a k for all , and for all 0 c  x c  .We also give a convergence result for a perturbation of Problem 1 to a nondegenerate case in the next subsection.Under condition (19) and the analogue for the fractional flow function f , its is easy to see, through the proof of Theorem 6.1 of [3], that the following holds.

S L T H and t S x t a e T
Furthermore, if the pairs  , f k and satisfy (9), respectively, and if we assume that  , a k , then the solution is unique.

Convergence of the Regularized Problem
To get around the difficulties from the degeneracies of the problem, we perturb the diffusion coefficient, , to k k  in such that a way that k k   strongly as 0 Then under the condition (19), the family of solutions converges to the unique solution   , p S of (1).More precisely.
Theorem 3.2 Under the conditions of Theorem 3.1, let be the solution to (1).For k be the solution of (1) when is replaced by k  , with k  as described above.Then and where 2 =     , with K and K  0 defined by ( 7) and (21), respectively, and for some >  .

Examples of Uniqueness
In this Section, we describe the physical meanings of the parameters in Problem 1 and give an example that satisfies conditions (2) through (3).These are purely mathematical examples that might not correspond exactly to models derived through physical experiments.Nevertheless, the shapes of the graphs of the mobilities, the fractional flow function, and the conductivity, as functions of the saturation , resemble the ones obtained through experiments.See Figures 1-3, for S = 3 2  .For more details on the physical meanings of these parameters, see [1,2,10-12], for instance.We retain the simplicity of the examples below for the mathematical analysis in this paper.For these examples, the diffusion coefficient (also called the total mobility) of the pressure equation of (1), as well as the fractional flow function, a f , satisfy (5).Physically where 1 is the mobility of the wetting phase, and the 2 the mobility of the nonwetting phase.The conductivity of the porous medium is defined by where is the capillary pressure.Assuming c p d d c p s is bounded and bounded away from 0, we will define, for this analysis, dropping, in this manner, the factor dp ds .The fractional flow function is defined by   and , the total mobility, is given by (24).a For numerical modeling of immiscible two-phase flow through porous media, it has been used the following mobilities (see [13], for example).
for the wetting, and for the nonwetting phase, up to multiplicative constants (or bounded functions).For a mathematical analysis purpose, and in order to get an example of uniqueness of a solution of Problem 1, we multiply both (28) and (29) by a bounded function of on the interval s   0,1 .

A case of Uniqueness
We define our new mobilities (up to the same multiplicative constant) by the following.For 1 < 2 for the wetting phase, and for the non wetting phase.Then, the total mobility (up to a multiplicative constant K , the absolute permeability, which we take here to be 1) is given by while the conductivity of the medium (up to the same multiplicative constant K ) is given by and the fractional flow function is given by It is clearly seen that , defined by (26), satisfies ( 2) and (3), and that k f and satisfy (5) for 1 < a 2   .
One also checks that if = 2  , then  9) is not empty, neither is condition (6), which is often used in the proof of the well-posedness of problem 1 or the like and for the convergence of the regularization of the same type of problems ([3,4,6,7,14].

Graphical Experiments for Uniqueness
One can check, through computations, that and a f , as defined by ( 32) and (34), respectively, are not twice continuously differentiable, for 1 < < 2  . They fail to be twice differentiable at and .For some , namely for the values = 3 2  and = 4 3  , we show graphically, experimenting with several values of , that condition (19) seems to hold for and c a f defined by (32) and (34).So this is an indication that Corollary 4.1 could hold for these values of  (and, maybe, for 1 < < 2  ).We emphasize that this does not constitute a rigorous mathematical proof that Corollary 4.1 holds for these values of  , but it does point to the conjecture that this could be the case.
For our graphical illustrations, we define the functions (36) and, in the same way, for a parameter 0 c 1   .Here and are fixed positive constants that are independent of , but could depend on


In this subsection, we use other arguments to show that the hypotheses of Lemma 2.1 seem to hold for the functions and a f , respectively, for 1 < < 2  .We consider the following functions.
and    On the other hand, by the Mean-Value Theorem, we have and where i  , 1 4 i   , are between x and and where we have obtain from an ) that y , d (43 used (7).We (42) hope, in a future work, to be able to prove this claim or give a counterexample that disproves it.If this claim happens to be true, that would functions and We give examples of a f que that are not very smooth but for which the s uni ness for the problem (1).

Conclusions
In this paper, we have revisited the problem of uniquer the pressure/saturation system.A new sufficient condition for uniqueness has been established and we have showed that the old conditions for un ditions (3), ( 5


. For the general case 1 < < 2  , we have illustrated graphically (without a rigorous pro there of) that co uld be uniqueness for these cases.A sequel of this paper should concern itself with a rigorous proof (or disproof) of this claim.It should also concern itself with the general cas , especially the case of convection dominated flow.

Figure 3 .
Figure 3. Conductivity of the Medium.

4 . 1 2 ,
defined by (32), we have that the couple a   , a k and   , f k satisfy (6), by Lemma 2.1 and Lemma 2.2.Hence the following.Corollary Under the conditions (26) through (34), if = Problem 1 has a unique solution .Furthermore, the family of regularized solutions  , defined by Theorem 3.2, converges to the unique solution   , p s of (1).Conditions (32) through (35) and Corollary 4.1 show that condition (

For a given 1
< < 2  , for (19) to hold for the pairs and  , a k  ,  f k respectively, we need that   0 in the Figures 4 and 5, we show that this seems to be the case, at least for the chosen values of  .Here we do this just for two values  ( = for each such  , four values of and only for the function .However, one can check our claim, by plotting c these graphs,   0 G s  for s c  for the chosen values of c.

Figure 9
Figure 9. Case: Clearly, condition(9) holds for f and a , if the functions F and , defined respectively by (38) and (39) are bounded above independently of G x and on the region enclosed by the triangle with vertices (0,0), (1,0), and (1,1) i.e. the region y common denominator of both functions is positive in the interior of the region .See Figure 12 below.Functions F and G are very complex by their definition, especially for non integer values of  .They involve the integral-defined function K .They e difficult to handle algebraically.For the present work, we sketch the surfaces representing the two functions, above the region R , for some valu of ar es  , using Maple Soware, in order to analyze their boundedness.This is illustrated through the Figures 13 through 18.We notice ft the smoothness of the surfaces corresponding to the case = 2 Figure 12.Region R.

,
G x y are bounded on .Th Conjecture.Corollary 4.1 also holds for R is proves the lemma.