Applied Mathematics, 2011, 2, 541-550
doi:10.4236/am.2011.25071 Published Online May 2011 (http://www.SciRP.org/journal/am)
Pressure/Saturation System for Immiscible Two-Phase
Flow: Uniqueness Revisited
Koffi B. Fadimba
University of South Carolina Aiken, Aiken, USA
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.

0
0Sx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 .
n
R=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



 
  

 
1
0
=i
div =in0,
=0on[0, ]
d=0forall [0,]
=()0in (0,)
=0on[0, ]
,0 =in
uaSp T
uQ T
uT
pxt T
SfSu kSSQST
tS
kS T
SxS x




 






 








n0,

(1)

K. B. FADIMBA 542
0
1
2
<
conditions generalize the ones considered in [3] and in [4]
for the uniqueness of the problem and for the conver-
gence of a family of perturbed problems. This work con-
stitutes, 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).
 
kk01 (2)


1
21
32
,if0
,if<
1,if 1
c
c
c














k

(3)
where 12
1
0<< <<1
2
 , and 0< 2

, for some
1

and 2

.
The function f satisfies the following.



20,1fC

in the variable
s
(4)
and
 
0= 1=0.ff
 (5)
Conditions (2) thro ugh (5 ) imp ly the fo llowing (see [5,
6]).
   
2,
f
bfa CKbKaba  (6)
for all 01ab
where
 
0
=
s
Ksk d


 (7)
Condition (6) h as been used (as a sufficient condition,
among other conditions) for the proof of the wellposed-
ness for the saturation equ ation , the con vergence for a re-
gularization of that equation, and the convergence of
numerical approximations of the same equation [7-9].
The rest of the paper is articulated as follows. In Sect-
ion 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 p ressure satu-
ration 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 rela-
tive “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 2
L
v for

2
L

v,

22
L
L
v for


22
0, ;LTL

v,
the norm of the function as an function in
the variable t on

,vxt

2
L


0,T with values in


2
L

. If


12
=,,vvv,
n
v is a vector, we denote by p
L
v the
norm


N
p
L
v.
2. Sufficient Condition for Uniqueness for
the Pressure/Saturation System
Lemma 2.1 Let and be two functions
defined on the interval 0g0h


0,1 . Assume that
g
is conti-
nuously differentiable on


0,1 , and that is continu-
ous on h


0,1 . Define
 
0
=Hs h
sd


 (8)
and suppose that
 

  

,
2
C
g
xgxgahxxa Hx Ha

(9)
for all


,1
x
a and for all


0,1 ,a
for some constant , then
>0C
      
2,

g
bgaCHbHaba  (10)
for all 01ab .


Proof.
We use a calculus argument. If , then the only
value that =1a
x
can assume is 1, and (10) is obvious. For
0<a1

, define the function
 

 


2
=.
a
GxgxgaCHx Ha xa 

(11)
for
x
a.
Then,


=0
a
Ga , and

 
=2
'
a
Gxgxgx ga

 
(12)
Clearly, if (9) holds, then for all

0
'
a
Gx
x
a, i. e.,
a is decreasing on the interval G


,1a. Since


a
Ga=0,
we must have


0
a
Gx

for all . This is true
for any 1ax
0a1

. Hence the lemma is proved.
In [9], for one space variable and the unilateral case
(


0=0k), and in [6] for several variables and the
bilateral case (




0= 1=0kk ), it is proved that if
f
and satisfy conditions (2) through (5), then k
f
and
satisfy (6), for k




20,1fC. 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 con-
ditions (2) through (5) to hold, nor do we require that
f
be in


20,1C. However, if those conditions are satis-

K. B. FADIMBA543
fied, we have the following lemma.
Lemma 2.2 Under conditions (2) through (5), and un-
der the assumption that the function
f
is twice conti-
nuously differentiable on


0,1 , we have
    

 
,
f
xfxfaCkxxa Kx Ka

(13)
for all


,1
x
aand for all


0,1 ,a
for some constant . >0C
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 suf-
fices to bound the quantity
 

  
,
fxfx fa
kxx aKxKa

 
independently of and a
x
.
Thanks to the symmetry implied by (3), we prove this
for 1
0ax


1
only, without lost of generality; the
rest of the prove can be obtained by the change of vari-
able
x
x , for 21ax


1
x, and by using the fact that for

2
kx c2

  . Using (7) and (3),
we obtain
 

1
11 1
=d
.
1
xx
aa
d
K
xKa ksscss
cxa








(14)
Therefore, since


,0xakx, and
K
is increa-
sing,
  
  
 


 
  
  
11
1
11
1
1
1
f
xfxfafxfx fa
kxxaKx KaKx Ka
fxfx fa
cxa
fx fa
fx xa
cxa
xa






  
 








(15)
By the Mean-Value Theorem, there exist
such that <, <acdx
 
=
fx fa
f
c
xa


and

11
=1
xa d
xa

Because the function
s
s

 is increasing, its mean
value occurs on the interval


,ax after the midpoint
2
ax

, so >2
ax
d

. Hence
>2
c
d (16)
and
>.
2
x
d (17)
Going back to (1 5 ) , w e get
 


  
 
 
 

1
122
11
22
1
2
1
||||
1
2
,
L
f
xfcfxfx fa
kxx aKxKacd
fx fc
cdd
fx fc
xc
cxc
Cf








 




 
(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
x
c. 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 [3], that the following holds.
Theorem 3.1 Suppose the data , a
f
, and are Lip-
schitz continuous in their argument k
s
. Then Problem 1
has a solution


,pS , with




 
*
21
0, ,,
0,1 ..0,
SLTHand
tSxtae T




 
.

(20)
Furthermore, if the pairs

,
f
k and satisfy
(9), respectively, and if we assume that

,ak

.













,,aSpL L


 , then the solution is unique.

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 k
k

in such that a way that kk


strongly as 0

. Define
 
0
=
s
Ks 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
1
0< <2  , let

,pS



k
be the solution of (1) when
is replaced by
k

, with k

as described above. Then




22
22 0, ,
0, ,
()()( )(),
LTL
LTL
aSp pCaSaS



 
(22)
and








2
1*
0, ,()
0
0,1
,
() ()
LTH
T
L
SS
KSKSS S
CK K

 


d







 (23)
where 2 =



, with
K
and
K

0
defined 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,
a
f
, 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
k
k



 
12
12
d
=
dc
ksks p
ks ks kss,
(25)
where is the capillary pressure. Assuming
c
pdd
c
ps
is bounded and bound ed away from 0, we will def ine, for
this analysis,



 
12
12
=
ksks
ks ks ks, (26)
dropping, in this manner, the factor dpds. The fract-
ional flow func tio n is defined by
 
 
1
12
=ks fs ks ks
 (27)
Figure 1. Fractional Flow.
Figure 2. Mobilities.

K. B. FADIMBA545
Figure 3. Conductivity of the Medium.
and , the total mobility, is given by (24).
a
For numerical modeling of immiscible two-ph ase flow
through porous media, it has been used the following
mobilities (see [13], 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


2
1=e ,
ss
ks s


 (30)
for the wetting phase, and
 

2
2=1 e ,
ss
ks 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
=1e
ss
as ss




 , (32)
while the conductivity of the medium (up to the same
multiplicative constant
K
) is given by
 


2
1e
=,
1
ss
ss
ks ss







 (33)
and the fractional flow function is given by
 
=
1
s
fs .
s
s



 (34)
It is clearly seen that , defined by (26), satisfies (2)
and (3), and that k
f
and satisfy (5) for 1< a2


.
One also checks that if =2

, then


 

 
22
0,10,1 . LL
af

 C

 
(35)
Therefore, for defined by (33), k
f
defined by
(34), and defined by (32), we have that the couple a


,ak and


,
f
k satisfy (6), by Lemma 2.1 and Lem-
ma 2.2. Hence the following.
Corolla ry 4.1 Under the conditions (26) through (34),
if =2

, Problem 1 has a unique solution . Fur-
thermore, the family of regularized solutions

,pS



pS,


,
as defined by Theorem 3.2, converges to the unique
solution


,ps of (1).
Conditions (32) through (35) and Corollary 4.1 show
that condition (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 re-
gularization of the same type of problems ([3,4,6,7,14].
4.2. 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 =0s=1s
values of

,1< <2

, namely for the values =32

and =43

, 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


Gs and


F
s on the interval


0,1, for a given
1< <2

by,


 


  


*
1
:
=
Gs
asasaccksscKsKc

(36)
and, in the same way,


      
*
2
:
=,
Fs
f
sfsfccksscKsKc

(37)
for a parameter 0c1

. Here and are fixed
positive constants that are independent of , but could
depend on
*
1
c*
2
c
c

.

K. B. FADIMBA 546
For a given 1< <2

, for (19) to hold for the pairs
and

,ak
 
,

f
k respectively, we need that


0Gs

and for all

Fs 0
s
c and for all, fixed,0<1c

.
Through the graphs in the Figures 4 and 5, we show that
this seems to be the case, at least for the chosen va lues of

. Here we do this just for two values

(=32

and =43

), and for each such

, four values of
and only for the function . Ho we ver, one can check
our claim, by plotting
c

Gs



F
s and for other values
of

Gs


and different choices for each chosen value of c

.
Here we have used the value 1 for both values
*=10c
=32

and=43

. One can see, thro ugh Figures 4-7,
for =32

, and Figures 8-11, for =43

, that, for
these graphs,


0Gsfor
s
cfor the chosen values of c.
Figure 4. Case. =32

and =14c.
Figure 5. Case. =32

and =12c.
Figure 6. Case. =32

and =34c.
Figure 7. Case: =32

and =910c.
Figure 8. Case: =43

and =14c.

K. B. FADIMBA547
Figure 9. Case: =43

and =12c.
Figure 10. Case: =43

and =34c.
Figure 11. Case: =43

and =910c.
4.3. Other Graphical Experiments for
Uniqueness for <<12

In this subsection, we use other arguments to show that
the hypotheses of Lemma 2.1 seem to hold for the fun-
ctions and
a
f
, respectively, for 1< <2

. We con-
sider the following functions .





  
,:= ,
fxfx fy
Fxy kx x yKxKy

  (38)
and





 
,:= . (39)
axax ay
Gxy kxx yKxKy

 
Clearly, conditio n (9) hold s for
f
and a, if the fun-
ctions
F
and , defined resp ectively by (38) an d (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




2,:=>xy,Rx 0,1y
R
.
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
ar
es

, using Maple So-
ware, in order to analyze their boundedness. This is illu-
strated through the Figures 13 through 18.
We notice
ft
the smoothness of the surfaces correspond-
ing to the case =2

. This suggests that the two funct-
ions are definitended in this case. For =2 ly 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


,
x
y over the region R.
Proof of Lemma From (32) and (34), i 4.2.t is easily
se

en that
 


1
1
2
1
=1
xx
fx
xx









(40)
and


 


2
2
2
=
112123
xx
xx xxxx e








 .
(41)
s
ax

K. B. FADIMBA 548
Figure 12. Region R.
Figure 13. Surface

=,zFx
y
, over region R, for =32

.
Figure 14. Surface


=,
z
Gxy
x
Figure 15. Surface


=,
z
Fxy, over region R, for=43

.


=,
z
Gxy
over region R, for =32  .
Figure 16. Surface, over region R, for=43

.


=,zFx
y
=2

. Figure 17. Surface, over region R, for

K. B. FADIMBA549
Figure 18. Surface


=,zGx
y
, over region R, for =2

.
On the other hand, by the Mean-Value Theorem, we
have
  




 
1
2
,= =
fxf xyfxf
Fxykxx ykx ykxk
1
2




 


(42)
and
 
 




 
3
4
,= =
axax yaxa
Gxy kxx ykx ykxk
3
4




 


(43)
where i

, 14i , are between
x
and and wh ere
we haveobtain from an) that
y,
d (43 used (7). We (42)
  

1
2
2
fxf
yk ,Fx xk



 (4
an
4)
d
 


3
4
,2
axa
Gxy kxk



 (45
Combining (33),(40 ), (41), (44), and (45), we obta
)
in
 
11
22
,= 1Fxy Oxx

 




(4)
and
6
 
11
22
=1 ,Ox 



(47) ,Gxy x



as , . Hence, if yx<yx=2

,

,
F
xy and

,Gxy are bounded on. Th
Conjecture. Corollary 4.1 also holds f or
Ris proves the lemma.
3<2
2
 .
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 thes uniness for the problem (1).
5. Conclusions
In this paper, we have revisited the problem of unique-
r 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), and (35)) imply the new condition. An
mple of familndingn a param
re i
ness fo
iqueness (con-
exa ay of data (depe oeter
) has

been constructedthat gives uniqueness for
=2

. For the general case 1< <2

, we have illustra-
ted graphically (without a rigorous prothere of) that
co
e
[3] K stence an
ysis, Vol. 5, 2001, pp. 503-521.
“Regularization and Numerical Methods
Porous Medium Equations,” PhD Thesis,
University of South Carolina, Columbia, 1993.
uld be uniqueness for these cases. A sequel of this
paper should concern itself with a rigorous proof (or dis-
proof) of this claim. It should also concern itself with the
general cas, especially the case of convection domina-
ted flow.
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K. B. FADIMBA 550
s for Flows through Po-
ater Flow and
umerical
a, “Computational Methods
mylie, “A near Optimal Order Approximation to a
eak Solu-
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[10] J. Bear and A. Verruijt, “Modeling Groundw
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[13] D. L. S
Class of Two-Sided Nonlinear Parabolic Partial differen-
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[14] Z. Chen, “Degenerate Two-Phase Incompressible Flow-I:
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