Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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199
CHAPTER
5
FILTRATION
OF
IMMISCIBLE
LIQUIDS
The present chapter investigates
in
solvability
of
boundary value
problems
for
the Musket-Leverette equations which describe the
process
of
filtration
of
two liquids
in
porous medium.
By
special
choice
of
the sought functions basic equations are reduced
to
a
quasi-linear composite system, which includes one uniformly ellip-
tic and one degenerative parabolic equations. Correctness of the
boundary value problems
for
the system has been studied in a
num-
ber of works
(Antontzev S.N., Monakhov
V.N.
Results presented here include the proof of a generalized solution
existence, the study
of
their differential properties as well as
the uniqueness problems.
1.
REDUCTION
OF
EQUATIONS,
FORMULATION
OF
PROBLEM
1.
Peculiarities
of
equations, the choice
of
the sought functions
The Musket-Leverette model
of
two-phase filtration of immiscible
(
pi=
const) in porous medium are characterized by the following
system
of
equations (equations
(1.50,)
-
(1.521,
ch.
1)
in respect
to
the filtration phase velocity
vi
presaure
pi
and satura-
tion
16
-
221.
Preliminary
analysis
of
a
linear model was
first
done by
i
onovalov
A.N.
“771.
s,,
s
(
s,+
s2
=
1
):
+
-#
2
+
m
Q-
pi
si+
divp
vi
=
0;
-vi
=
ici(
v
pi
+
pi
g
>,
i
=I,z,
at
0
>a
s1-
s1
p,-
p,=
p,
(x,
9)
(s
E
1-
s;-
s;
(1.1)
Here,
Ki
=
K;,koi(s)
is
a symmetric tensor
of
phase penetration.
$(x)
is
a filtration tensor
for
homogeneous
liquids,
1-
k
o1
.=
rikoi(s)
is
phase penetration for a homogeneous
isotropic
ground.
In papers
[
77,
78
]
the authors have presented problem
designs
for
filtration in potentials
Qi=
pi+
pi@
for which the system
(1.1)
takes the form
=
g
v
h
)
,
=
^a
div
(K,v
a,)
,
(l.l*)
200
Chapter
5
where
a
A
=
-
-
,
s=
pi1{
x,
Q2
-Q,+
(
p,
-
p,)h
}
.
The system
(l.l*)
being non-linear and degenerative on the set, where
spect to derivafives
Saturation
s(
x,
t
1
and a flow function
(h(xJ
.t>
of
the
as
and can not be solved
in
re-
d@;
OS1
PCl
{X'
Q2-
a,+
(
P,-
P,)h
at
,
i
=
1'2.
summarized stream have bee9
su
ht fun tiom for
a two-dimensional caae(%R ,n=!Y(Konovalov
A.IV.y77,
787).
In
other papers
L16
-
191
such functions have been
-
s(x,t)
and com-
ponents
(u,,
u2)
of
the summarized velocity vectors
3
=
"V,t
%z
a
This choice of the sought functions results in the system of two
(three) equations: degenerative parabolic
-
for
s(x,t)
and uni-
formly
elliptic
-
for
contain higher derivatives
of
other unknowns.
In the case
of
arbitrary dimension we add to the
unknown function
s(x,t>
an average ('lreducedff) pressure
p(x,t>
chosen to obtain the system
of
equations for
s(x,t)
and
p(x,t)
which ossesses the same properties as the system for
s(x,t>
and
(h?x,t
)
when
Add
up equations
of
continuity (initial equatfons
1.1
)
devided
into
pi=
const which brings to the correlation
ested as the
so
$(x,t)
of the
first
order elliptic set for
-3
(u,,u2)
=
v).
Equations for each
of
the sought f'unctions don't
n
I
1
(x
ERn)
n
=
2.
(1.2)
+++
6iv
3
=
o
v
=
v,+
v2,
-3
in which
v
is
a mixture filtration velocity vector
(
13
1
is
a
total specific mixture consumption).
Let's introduce a new function
-
a 9educed11 preseure
(1.3)
+
where
k
=
ko,+ ko2
functions, we
will
express with the help of
the
Darcy laws (the
second equations
(1.1))
a vector
3
by function gradients
p,
g
Vh=
c;
.
To
explain this choice
of
the sought
and
s
:
vs
+
+
2
+
ap,
1COZ
c
Ki
(
vpi
+
pi
6
)
=
ldC0(
v
p,
+
-
-
v
=
-
1
as
k
2
-3
1
ap,
ko2
+-
ko2
Vp,
)
+
C
Ki
pi
c1;
=
kKo
V(p,
-
1
-
-d5)
+
k
1
s
as
ic
Filtration
of
Immiscible Liquids
20
1
Thus, with the help
of
substitution
(1.3)
a
vector
5
is
represen-
ted by
VP
and
s
and
it
is
independent
of
vs
:
+
++
(1.4)
V
=
-@Vp
+
f
)
Ev(
S,
p),
1Z
=
kKo
,
1
+
s
as
+
where
f
=
IC
1
v
ap,
Ir,,
d
<+
E,
V
p,+
I<
(
p,
-
p,
)
g
and
V
is
used over the variable
x
explicitly occuring
(for
In the same way and in view
of
(1.3)
we have
we obtain
-+
-D
(1.5)
-
+
-
v
=
KO
avs
+
Xlvp
+
fo
=
-
v,(s,
p
)
.
1
+-#
Using
(1.4)
we find
lCl
vp
=
-
K~K-’(~+
f)
and notice that in
accordance with the definition
I{
1=
ko
,K~
and
i.;
=
klio.Therefore
K1K-’
=
kO,K-l
E
b(s)
and the representation (1.5) can take the
f
om
-+
--t
++
+
-+
-vl=KoaVs-bv+~,P=f
0
-bf.
(1.6)
Substituting (1.5)
in
the continuity equation for the
first
phase
we are led
to
the equation system
in
respect to
{s,
p
}:
(1.7)
+
+
m
as
=
div
(KO
a
V
s
+
IClvp
+
fo
1
B
-
div vl(s,
p
>,
at
+
+
--
div
(
KVp
+
f
)
P
-
div
v
(s,
p)
=
0,
(1.8)
And substituting
(1.6)
we have
an
equivalent system in respect
to
{
s,
v
9%):
(1.9)
++
m
=
div
(l:oaVs
-
b
v
+
P
);
m
=
(1-
s1
0
-
sz
).
-+
-9
-D
at
div
(KVp
+
f
)
=
0,
-
v
=
Kvp
+
f
.
(1.10)
Note that
a
filtration tensor ~z,(x)is assumed
to
be symmetric
202
Chapter
5
and positive,
i.e.
-1
sv
Ig,12.
u>o
(1.11
1
(15
1
=
C
Si
)
but
a
capillary pressure and relative phase pene-
i
trations possess the properties
(1.12)
<
0
,
k
=
k,,
+
ko2
>
0
,
a*c
-
as
Therefore, owing to properties
koi(s)
there
will
be
a(x,s)
>
0
at
s
E(0,I)
and
a(x,O)
=
a(x,
1
)
=
0
.
Thus (1.71,
(1.8)
are quasi-linear system, which comists
of
a
uniformly elliptic equation for
p(x, t)
and a singular parabolic
equation for
s
=
0,l
if
s(
x,
t
).
2.
Initial boundary value problem
We
consider
a
filtration flow in the given boundary domain
Q
with
a piecewise smooth boundary
r
=an
According to
$1
Ch.
1
and in
conformity with different types
of
boundary conditions, the boun-
dary
r
can be splitted into some united components
ri.
Let
q
=
QX
[
o,'p]
,
si
=ri
[
0,
TJ,nlS an external normal to
Rewrite the boundary values (1.54)
-
(1.58) from
9
1 ch.
1
conformably to the functions
sip.
Non-penetration conditions
on
r'for both phases are equivalent to the following:
-+*
-+-+
v
n
=
vln
=
o
,
(x,t>
E
so
=
ro
x
Lo,
TJ.
(1.13)
Boundary value conditions (1.55)
-
(1.57) from
gl
oh. 1 are con-
sequently reduced to
p
=
po(x,
t),
s
=
so(x,t),
(x,t)
E
s2=
x
[o,
T],
(1.14)
(1.15)
-+
+
-++
-(KVp
+
f)
nrv
n
=
R(x,t),
(x,t)
cS1
=
1''
x [
0,
p]
,
+
-+
-+-+
(1.16)
-
(IC,
ags
+
L,vp
+
f,)
nevln
=
bli(x,t), (x,t)
E
S
1
Since (1.151, (1.16) for
R(x,t)
20
are equivalent to
(1.131,one
must include
ro
and
I?'
and assume
r
to consist
of
several com-
ponents,
in
some parts of which
R
=
0
.
Therefore,
r=
r"
U
r2
.
The set
of
equations (1.71,
(1.8)
is
li-
kely to satisfy the Cauchy-Kovalevskay conditions (the second equa-
tion does not contain
ap
/at)so
it
is
quite sufficient to state
an initial condition
for
the saturation
s(x,
t)
:
We regard components
Iilor
P2
as
being absent
on
1'
i.e.
it
is
1
s(x,
0)
=
so
(x,
01,
x
E
a
.
(1.17)
Filtration
of
Immiscible Liquids
203
2
1
possible that
r
E
r1
or
r
i
r
In this case when
r
P
r
the
law
of
conservation
of
mixture mass in the domain leads
to
the
following necessary condition:
I
p(x,
t)
dx
=
1
H(x,
t)dx=O,
t
E
[0,
T]
.
51
r
(1.18)
Finally, we give formulas expressing coefficients
of
equations
(1.7)
-
(1.10)
and boundary conditions
(1.3)
--
(1.6)
:
,k=k
o1
+
ko2
,
b
= -=
a=--
apc
-
1'0
$0
2
as
k
k
ko2
ls
as
k
-t
f
=
K
jV
-
-
d
5
,
Ki
=
koi
KO,
i
=
1,2
,
0
(1.19)
3.
Outline
of
conditione which provide summarized filtration
If
coefficients
K
=
~~(x)
k(s)and
f(x,s>
dependent
of
s
the equation system
(1.71,
(1.8)
disintegrates and
admits the sequential determining
of
the velocity field
%
and
the phase saturation
si(x,
t).Formulas
(1.19)
make it possible to
formulate these conditions
in
terms
of
functional parameters
of
the Musket
-
Laverette model in the following way.
1)
k
=
kol(s)
+
ko2(s)= const
this assumption
to
a sufficient
degree of accuracy is realized for miscible liquids, for which
kol=
As
,
ko2(s)
=
A(1-
s),
A=
const
.
liquids essential deviation from a constant is observed
in
the
vicinity of limiting values
s=
0,1
of
the reduced saturation.
velocity
be
independent
of
saturation
equations
(1.8)
are in-
In the case
of
immiscible
det
Xo(x)=
const
here we have
p,
=
P,(s)
i.e.
=
0.
a
xi
2)
-
iii(
X)
3)
Gravity is not taken into account
(for instance, in plan fil-
tration) or liquids have the same densities
As seen from (1.19) asewnptions
2),
3)
provide the conditions
to
be satisfied
p,=p2
.
-+
af
=
0
.
as
2.
DETERMINATION
OF
GENERALIZED SOLUTIONS. PROBLEM REGULARIZATION,
MAXIMUM
AND COMPACTNESS PRINCIPLES
1.
Generalized solutions
We regard all the given functions
over which the coefficients
of
equations (1.71,
(1.8)
are expres-
sed as being determined for
all
(XIS)
E
;*=
x
[O,
I]
and
koi
(:.)
,
m(s>,
pc(x,s
)
,Ko(x)
204
Chapter
5
Functions
P,(x,
t)
and
so(x,t)
each occurring
in
boundary
-
condi-
tion8
-
(1.14)
on
s2
are assumed
to
be
known for
(x,t)
E
q
=
=
Q
X
[o,P]
and together with
H(x,t), (x,t)
ES1=
I"
x
[O,
,
from
(1.15)
and
(1.16)
they have properties
where
(1
?dn-l>/
n, n
>2;
q
>
1
for
n=2,
n-
is space dimen-
sions over the variable
x
=
(x,,x2,
...,
5
),
15n5
3;
(iv)
05
6,
5
so
(x,t
)
5
I
-
6,
5
1
.
Under the stationary conditions (iii) will correspondingly take
the
form
(iii
*I
(
II
v
so
112,
Q;
llp0,
v
Y,
112,
Q;
IIR
Il4,ri
)I
b-
The latter may not be specified because, for
u
(x,
t)
E
u(x)
Ut
E
0,
u
Il2&
=
9
IIU
112,
Q
9
IIU
112,
m,y
=
IIU
112,
Q
We note that having defined
a,
zo,
3
and
3
in
formulae
(1.19)
and conditions (i) we have
3
3 3
(2.1
1
f
F
-
15
IUo(Ll),
(x,s)
E
Q*
f0
Iln
-.-.-.
-
a
kOIkO2
koqko2
k02
'
kOlk02
We will further refer to the problem
(1.71, (1.81, (1.14)
-
(1.17)
as problem
I.
Definition
2.1.
We call functions
s(x,t), p(x,
t)
bounded,dimen-
sional
in
Q
as a generalized solution
of
the problem
I,
if
a)
0
5
s(x,t)
5
1
almost everywhere in
.c,
b)
vp
E
LE,
m(~),
avs
~L,(y)where
for
o
5
s(x,t)
5
I
case
a
20)
the function
a
vs
is defined by
(in this
Filtration
of
Immiscible
Liquids
205
c) boundary conditions
(1.14)
hold on
S2
P
=
Po
(XYt),
u
=
u[
so(x,t)l
U0(X,t),
(x,t>
E
s2
9
d)
for
arbitrarily assumed functions such that
cp
(x,-t)
EW;(Q
;
+(XI
E
tl;(
a>,
cp(x,
t)
=
(1,
(x) f0,
(x,t)
E
sz
,
for
which
all
t
E
[O,
T]
the equations hold
where
3,=
31(s,
p)
ponding formulas
$1,
t+=
Q
x
(O,t),
St
=ri
x
(O,t),
i=
ly2;
R
is the function known from
(1.151, (1.16)
and
and
$
=
"V(s,p)
are' defined by the corres-
(ins,
cpl
It
=
(us
(XYt),
cp>
Q-
(
uso
(XYO),
cp)
Q
*
0
In the stationary case
d)
must
be
read as:
d*) for the arbitrary given functions such that
cp(x),
+(XI
E
w;(
Q>
and
cp(x)
=
+(XI
=
0,
x
E
r2
,
the following equations hold
+
x,
(Vl,V(P)
Q
=
(bR
y
CP)
y
(2.2*)
+
r'
,gz
(v,
v
$1
Q
=
(H,
9)
(2.3*)
Integral identities (2.21, (2.3)
are
obtained by integrating over
domains
%
and
Q
of equations
(1.71,
(1.8)
multiplied by
v(x,t)
and
$
(x)
here Gauss-Ostrogradskii formulas are applied and bo-
undary and initial conditions
(1.15)
-
(1.17)
are taken into
account.
Note
that with the help of representation
(1.6)
for
vl(s,
p)
integral identity (2.2) can be given another equivalent form.
Suppose that for the sought solution
s(x,
t)the inequalities
hold, and thus
+
0
<
do
5
S(X,
t)
51
-
6,
<
1,
(x,t)
E
J.
a(x,s)
2
6
>
0
and according to the condition b)
VS
E
L2(
a>.
+
In this case we represent
formula
(1.6)
and in the relation obtained we transform with the
help of
(1.6)
the addend
vl(s,
p)
in (2.2) in respect to the
-9
-D
(b
v
9
V
91%
=
-
(bsV
v
S,
9)
+
(bR,
cp)
,,
b
E
C'[O,I].
Introducting this expreseion in (2.2) with
$;,(s,p)
in
the form
of
(1.6)
finally we find:
%
"t
206
Chapter
5
and in the stationary case respectively
9
+
$1
-
(KO
avs
+
E',
V9>,
-
(bs
VV
S,
cp)
,=
0
.
(2.4*)
2.
Regularization
We extend each function
f(x,
s>
in
the equality (2.4) beyond the
0
5s
5
1
interval by the formula
f(x,
01,
s
5
0'
f(x,l),
s
21,
f,
(x,s)
=
f(x,s),
0
<
s<
1,
hO
1
and besides, replace
and
v
and
b
by their average
vh
over
x
and
b
over
s.
Then
(2.4),
(2.4*) are reduced
a(x,s)bx+
a(x,s>=
B
*(x,s)+
E,
E
>
0,
+
E
+
+
(2.5*)
$1
=
-(KO
av
s
+
p,
v
cp)
-
(c
Vh
v
s,
cp)
0
Qt
ho
'
where
C
=
(bs
)
*,
bo
=
b
De inition
2.2.
Let's call bounded dimensional functions
s(x,t>,
P&,t)
as a generalized solution
of
an auxiliary problem
I
if
they have the properties (b
-
d)
of
the definition 2.1
in
which
(2.3)
has been replaced by
(2.5).
At
first,
the solvability of the regularized problem
will
be estab-
lished and then, the theorem
of
existence
of
a
regularized initial
problem
I
solution by limiting transition over
E
and
h
in iden-
tity (2.5?
will
be proved
3.
The maximum principle
Notice that for the solution of the equation (1.7) the maximum
principle in a ''pure formtf
min
{inin
s(x,o),
rnin
s(x,t))ss
Smax{max
s(x,o),
max
s(x,t
)
}
a
S
B
S
+
is
true only
if
dif,
i?
=
O.Besides, boundary value conditions
occurring in real problems don't make
us
conclude that
For
instance, impermeability of the boundary
I'
for both phases
brings to the boundary condition
0
5s
51.
ah
-
a
p"*co
v
s%
=
g(
p,
-
p,
)
,
as
Filtration
of
Immiscible Liquids
207
ah
from which extremum on
r
That's why it's important to find out coefficients and boundary
conditions which rovide for the unknown solution the Inequalities
bi
>
d)*
The following statement is valid for a
generalized solution of an auxiliary problem.
Theorem
2.1
(the maximum principle). Let thea assumptions
(i-iv)
be solved, s(x,t)
,
p(x,t)
-
is a generalized solution
of
prob-
lem
I,.
Then almost everywhere
in
Q
can not be reached only if
~(p,
-~,>~y
=O.
6,
5
s
5
1-
6,
0
<s(x,t)
5
1,
(x,t>
E
u.
(2.6)
If
the additional conditions are solved
-t
--f+
div
F
=
0,
(x,t)
E
%;
F
n
=
0,
(x,~)
E
1;'
,
(2.7)
X
then almost everywhere in
Q
0 5
6
=
min
s
(x,t)
<s(x,t)
<ma
so(x,t)
=
I-
6,
5
I
.
(2.6*)
0
0
Proof. Consider a non-stationary case. Introduce a cut-off function
s(xst)
=
lmx
{
s(x,t>
-1,
0
}.
Due to the cut-off properties
vs
(2.5*)
it follows that
s,s
for almostalltthe following is valid
-
-
-
L2(Wt)9
S
I
rz
=o,
s
1
t=O=
0,
and from the integral identity
st
EL,[o,T;f!;l
(Q)]
and consequently, for
-
With
cp
E
in
(2.5)
we come
to
the equality
since due
to
the definition
;f;
only over the domain
52;
=
{(x,t)
E
%,
s
>I
},
VGcJa,
F(x,s)
=
0.
(a
=
a+
E
2
E
)
we obtain:
all integrals in
(2.8)
are taken
in which
s
=s-I,
Estimating
in
(2.8)
the second addend below
-?
and the last one by the Couchy inequality above,
t
~(t)
=
11
+S
[I:,
5
E-'
F
j
y(
t)dt,
jr(3)
=
0.
0
Hence, it follows that y(t)=
0.
In complete analogy with the use
of
s
=ma
,O
}
inequality
s
2
0
can be proved. To prove equa-
lity
(2.6*\-'
,
having substituted
cp
ZS
=
U~X
{
S-
l+
b,,
0
}
we
transform
(2.5)
to the
form