 Applied Mathematics, 2011, 2, 541-550 doi:10.4236/am.2011.25071 Published Online May 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Pressure/Saturation System for Immiscible Two-Phase Flow: Uniqueness Revisited Koffi B. Fadimba University of South Carolina Aiken, Aiken, USA E-mail: SCKoffiF@usca.edu Received December 31, 2010; revised March 18, 2011; accepted March 21, 2011 Abstract 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. Keywords: Porous Medium, Uniqueness of a Solution, Degenerate Equation, Immiscible Two-Phase Flow, Regularization, Phase Mobility. 1. Introduction Consider the coupled nonlinear problem (1), with , which arises from modeling incompre- ssible 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. 00Sx1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 fun ction, 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 . nR=1n2n=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    10=idiv =in0,=0on[0, ]d=0forall [0,]=()0in (0,)=0on[0, ],0 =inuaSp TuQ TuTpxt TSfSu kSSQSTtSkS TSxS x  n0, (1) K. B. FADIMBA542 0120C  2,gbgaCHbHaba  (10) for all 01ab . Proof. We use a calculus argument. If , then the only value that =1ax can assume is 1, and (10) is obvious. For 00CThus, 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 , with some modification. For the proof, it suf- fices to bound the quantity    ,fxfx fakxx aKxKa  independently of and ax. Thanks to the symmetry implied by (3), we prove this for 10ax1 only, without lost of generality; the rest of the prove can be obtained by the change of vari- able xx , for 21ax1x, and by using the fact that for 2kx c2 . Using (7) and (3), we obtain  1111=d.1xxaadKxKa ksscsscxa (14) Therefore, since ,0xakx, andKis increa- sing,          11111111fxfxfafxfx fakxxaKx KaKx Kafxfx facxafx fafx xacxaxa   (15) By the Mean-Value Theorem, there exist such that <, 2axd. Hence >2cd (16) and >.2xd (17) Going back to (1 5 ) , w e get       11221122121||||12,Lfxfcfxfx fakxx aKxKacdfx fccddfx fcxccxcCf  (18) where we have used (16), (17), and the fact that 0=0f. Therefore the lemma is proved. 3. Uniqueness of a Solution and Convergence of the Regularized Problem 3.1. Uniqueness We give an existence and uniqueness result for the case when and satisfy (9), i.e. ak asas acCksscKs Kc (19) for all , and for all 0cxc. We also give a conver- gence result for a perturbation of Problem 1 to a nonde- generate case in the next subsection. Under condition (19) and the analogue for the fract- ional flow function f, its is easy to see, through the proof of Theorem 6.1 of , that the following holds. Theorem 3.1 Suppose the data , af, and are Lip- schitz continuous in their argument ks. Then Problem 1 has a solution ,pS , with *210, ,,0,1 ..0,SLTHandtSxtae T . (20) Furthermore, if the pairs ,fk and satisfy (9), respectively, and if we assume that ,ak. ,,aSpL L , then the solution is unique. Copyright © 2011 SciRes. AM K. B. FADIMBA 544 3.2. Convergence of the Regularized Problem To get around the difficulties from the degeneracies of the problem, we perturb the diffusion coefficient, , to kk in such that a way that kkstrongly as 0. Define  0=sKs kd. (21) Then under the condition (19), the family of solutions ,pS converges to the unique solution ,pS of (1). More precisely. Theorem 3.2 Under the conditions of Theorem 3.1, let be the s olutio n to (1). For ,pS>0 small, say 10< <2, let ,pSk be the solution of (1) when is replaced by k, with k as described above. Then 2222 0, ,0, ,()()( )(),LTLLTLaSp pCaSaS  (22) and 21*0, ,()00,1,() ()LTHTLSSKSKSS SCK K d (23) where 2=, with K and K 0defined by (7) and (21), respectively, and for some >. 4. Examples of Uniqueness In this Section, we describe the physical meanings of the parameters in Problem 1 and give an example that satis- fies conditions (2) through (3). These are purely mathe- matical examples that might not correspond exactly to models derived through physical experiments. Neverthe- less, the shapes of the graphs of the mobilities, the fract- ional flow function , and the conductivity, as functions of the saturation , resemble the ones obtained through experiments. See Figures 1-3, for S=32. 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 pre- ssure equation of (1), as well as the fractional flow funct- ion, af, satisfy (5). Physically 12=asksks (24) where 1 is the mobility of the wetting phase, and the 2 the mobility of the nonwetting phase. The con- ductivity of the porous medium is defined by kk 1212d=dcksks pks ks kss, (25) where is the capillary pressure. Assuming cpddcps is bounded and bound ed away from 0, we will def ine, for this analysis,  1212=ksksks ks ks, (26) dropping, in this manner, the factor dpds. The fract- ional flow func tio n is defined by   112=ksfs ks ks (27) Figure 1. Fractional Flow. Figure 2. Mobilities. Copyright © 2011 SciRes. AM K. B. FADIMBA545 Figure 3. Conductivity of the Medium. and , the total mobility, is given by (24). aFor numerical modeling of immiscible two-ph ase flow through porous media, it has been used the following mobilities (see , for exam ple). 1=ks s (28) for the wetting, and  2=1ks s (29) 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 . 4.1. A case of Uniqueness We define our new mobilities (up to the same multipli- cative constant) by the following. For 1< 2, let 21=e ,ssks s (30) for the wetting phase, and  22=1 e ,ssks s (31) for the non wetting phase. Then, the total mobility (up to a multiplicative constant K, the absolute permeabi- lity, which we take here to be 1) is given by  2=1essas ss , (32) while the conductivity of the medium (up to the same multiplicative constant K) is given by  21e=,1ssssks ss (33) and the fractional flow function is given by  =1sfs .ss (34) It is clearly seen that , defined by (26), satisfies (2) and (3), and that kf and satisfy (5) for 1xy,Rx 0,1yR. Notice that the common denominator of both functions is positive in the interior of the region . See Figure 12 below. Functions F and Gare very complex by their defi- nition, especially for non integer values of . They in- volve the integral-defined function K. They e diffi- cult to handle algebraically. For the present work, we sketch the surfaces representing the two functions, above the region R, for some valu of ares, using Maple So- ware, in order to analyze their boundedness. This is illu- strated through the Figures 13 through 18. We notice ftthe smoothness of the surfaces correspond- ing to the case =2. This suggests that the two funct- ions are definitended in this case. For =2ly bou, we show directly that this is indeed the casethat Corollary 4.1 holds. W e prove this through the following lemma. Lemm, i.e. a 4.2 For =2, functions F and , defined by tiv G (38) and (39), resely, are bounded indepen- dently of pec,xy over the region R. Proof of Lemma From (32) and (34), i 4.2.t is easily seen that  1121=1xxfxxx (40) and  222=112123xxxxxxxx e .(41) s axCopyright © 2011 SciRes. AM K. B. FADIMBA 548 Figure 12. Region R. Figure 13. Surface=,zFxy, over region R, for =32. Figure 14. Surface=,zGxy x Figure 15. Surface=,zFxy, over region R, for=43. =,zGxyover region R, for =32. Figure 16. Surface, over region R, for=43. =,zFxy=2. Figure 17. Surface, over region R, for Copyright © 2011 SciRes. AM K. B. FADIMBA549 Figure 18. Surface =,zGxy, over region R, for =2. On the other hand, by the Mean-Value Theorem, we have    12,= =fxf xyfxfFxykxx ykx ykxk12  (42) and   34,= =axax yaxaGxy kxx ykx ykxk34  (43) where i, 14i , are between x and and wh ere we haveobtain from an) that y,d (43 used (7). We (42)  122fxfyk,Fx xk (4 an 4)d  34,2axaGxy kxk (45 Combining (33),(40 ), (41), (44), and (45), we obta)in  1122,= 1Fxy Oxx (4)and 6  1122=1 ,Ox  (47) ,Gxy x as , . Hence, if yx