Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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Chapter
1
The capillary pressure
p,
is
defined by the curvature
of
saturation
s1
of
the wetting liquid, characteristics of the
porous medium and liquids and
is
expressed by Laplace's formula
r1,2
,
pc(x,
s>
=
Pc(x>j(s>, Fc(x>
=
u
cos
O(m/lKol
1/2,
(1.52*)
where
u
is
the coefficient
of
interfacialt tension, j(s)
is
the
Leverette's function, and the magnitude
lKol
determinant
{K,.}
if
KO
is
the symmetrical tensor
of
filtration
of
the anisotropic porous medium.
Generally, the relative phase penetrabilities
Eoi
and the Leve-
rettels function
j(s>
are defined from the experiments on
impregnation,
when the wetting phase, under the effect of capil-
lary forces, substitutes the non-wetting liquid that occupies the
whole
of
the porous material,
as a result
of
which the probable
effect of hysteresis
is
eliminated.
Set
of
equations
(1.50)
-
(1.521, respective to the characteris-
tics
v,,
p,,
p,
and
s
=
(sl
-
si)/(l
-
si
-
sz)
of
the
immi-
scible liquids moving in a porous medium,
in
the isothermal case
(the temperature in the flow
is
constant),
is
completed by for-
mulation
of
the equations
of
state
of
the liquids
Pi
=
P,(P,),
i
=
1,
2.
denotes the matrix
-3
(1.53)
From here on,
if
not stipulated otherwise, both liquids are
assumed to be incompressible, i.e.,
It
is
natural to call the obtained mathematical model
of
filtra-
tion of multiphase liquids (equations (1.50)
-
(1.53)) the model
of Masket-Leverette in honour
of
M.Masket who was the
first
to
propose generalization (1.51
of
Darcy law,
and ILLeverette who
was the
first
to use Laplace law (1.52)
(see [138]).
The functional parameters
m,
koi,
I<,
and
j
of the Masket-Leve-
rettels model are assumed to be prescribed by the functions
of
the respective variables x and
s
and all the numerical para-
meters
pi,
pi,
s:
and others)
-
to
be fixed.
In
this case the
phase penetrabilities
k,,(
s)
=
-
k,,(
s)
posses the properties
pi
=
const.
-
I-
pi
k,,(s>
>
0,
s
6
(0,
11,
k,l(o)
=
kon(l>
=
0.
Let us formulate the initial-boundary problem for equations
(1.50)
-
(1.53). Let the filtration
of
an inhomogeneous liquid
occur
in
the finite region
the boundary
and
surface
I"
Q
of the variable
x
=
(xl,
x2,
x,),
r
=
aQ
of
which consist
of
an impenetrable surfacero
corresponding to injection and operating wells
Models
of
Heterogeneous Media
19
and the specified boundaries
I"
c
less liquid (for example
ground water on
its
foot).
are
of
the form
with a homogeneous motion-
with air on the seam roof or with under-
The
conditions of the absence
of
flow on r'for both phases
(1.54)
00
+-i
(vi
n)
=
0,
(x,
t)
E
S
=
I?
x
(0,
T),
-9
where
n
is
the outward normal to
r,
S'
=
r'
x
(0,
T),
i
=
1,
2.
On the sections
r2
c
I?'
less liquid, the pressure in the wetting phase
is
prescibed, and
it
coincides with the hydrostatic pressure in a motionless liquid,
and the saturatipn value (for example,
on
the roof
the foot
s
=
1).
contiguous with the homogeneous motion-
and on
s
=
0
P,
=
Pl,(X, t>,
s
=
So(&
t),
(x,
t)
E
s2
(1.55)
Sometimes conditions
of
type (1.55) are prescribed
3180
on the
sections
r'
corresponding to the wells. Generally, the rate
of
flow of the mixture
(discharges of the wells) on
I?'
is
assumed
to be known
Besides,
if
the normal component of the gradient of capillary
pressure
p,
on
r1
is
neglected in comparison with the gradients
of phase pressures, and the gravitational forces are not taken
into account, then
(v;,
(1.511,
((kO2v1
-
k,,v,)
With the aid of (1.56), the obtained relation can be transformed
into
a
more convenient form:
-3
-9
n)
=
(vp,
.
n)
or, according to
3
-9
n)
=
0,
(x,
t)
E
S'.
demonstrating that the injection and extraction of the mixture are
conducted proportionally to the fluidity of the phases.
Besides the boundary conditions,
it
is
necessary to prescribe the
initial distribution of saturation, too
s(x,
0)
=
so(x,
O),
x
E
a.
(I .5a)
Investigation of the correctness of the nonlinear boundary-value
problems of filtration of two-phase liquids was undertaken in
works by
S.N.
Antontsev and V.N.Monakhov [I6 to 191.
At
the first
stage the case of plane-parallel motion was considered. For two-
dimensional flows, as the new unknown functions,
it
is
convenient
to take the saturation
s(x,
t)
and the stream function
(I,
for
the velocity of mixture
3
singular, at
s
=
0,
1
parabolic equation for
s(x,
t)
and the
uniformly elliptic equation of the
second order for
+(x,
t)
(A,N. Konovalov [77
).
Then the initial
both
model
is
reduced to the equivalent system cons
1
sting
of the
20
Chapter
7
equations having quadratic terms of the form
sXt*
+xi
.
With the
data of the problem, which ensure the non-singularity
of
the equa-
tion for
s(x,
t)
it was proved that the classical solution
existe,.including a number
of
problems with free (unknown) boun-
daries
18,
18,
191.
For the singular problems the Solvability was
established in the class of generalized solutions: firstly in the
simplest case of one-dimensional motion by
G.V.
Alekseyev and
N.V.
Husnutdinova
[6,
1361,
then by
S.N.
Antontsev
[7,
131
for plane-
parallel flows.
At
the next stage it was proved
[20
to
22, 1561
that in the ge-
neral three-dimensional case asw well, the Masket-Leverette's
equations are reduced
to
the elliptic-parabolic system, if the
saturation
s(x,
t)
and some mean pressure
p(x,
t)
are taken as
the unknown functions. Owing to this circumstance, the existence
of
the generalised solutions to the boundary-value problems was
proved and some significant qualitative properties of these solu-
tions were established: the maximum principle for the saturation
s(x,
t)
ensuring the satisfiability of the inequalities
0
I
s
I
1
the smoothness properties depending upon the Smoothness of the
data of the problem, the uniqueness of the solutions in the non-
singular case, hen
0
<
S
5
S(X,
t)
5
1
-
S
etc.
S.N.
Antontsev
and
A.A.
Papin
723
to
251
investigated the differential properties
of the generalized solutions nd the quest on8
of
uniqueness for
the singular problems. Works
89
to
12, 1361
show that such pro-
perties as the finite velocity of propagation of perturbations of
the saturation values
s(x,
t)
and the finite time of the stabili-
zation of Solutions with increasing the time, are characteristic
for
the solut on f the singular problems.
S.N.
Kruzhkov and
S.M.
Sukoryanskii
t85
pput forward the algorithms of the approximate
solution to the two-dimensional regular boundary-value problems
and proved their convergence. The substantiation
of
the conver-
gence
of
the approxima
e
meth ds for three-dimensional problems
is presented in works
i24,
851.
With this the presentation of hydrodynamic models is completed.
All the above formulations of the initial-boundary problems are
characterized by the common feature that the sets
of
equations
are non-linear and are
of
a compound type. When studying them in
the following chapters use is made
of the general methods of solu-
tion of the evolutionary boundary-value problems presented, for
example, in monographs by 0.A.Ladyzhenskaya
[go]
and J.-L.Lions.
While proving the existence theorems, major efforts are concentra-
ted
on obtaining a priori estimates, on the basis of which, with
the aid of familiar theorems from analysis
(Banach's principle for
the contractive mappings or Schauder's principle for completely
continuous operators
[
981)
or by the method
of
Bubnov-Galerkin
[
90
1
the solvability of the problems is proved.
2.
AUXILIARY INFORTL4TION FROM ANALYSIS
AND
DIFFERENTIAL EQUATIONS
1.
Functional spaces
Let
51
and denote the bounded regions
in
the Euclidean spaces
R",
x
and
t
denote the coordinates of the points from
51
or
Q.
x
=
(x,
,
x2,
x3)
are the Cartesian coordinates in
R3,
t
is the
time. A number of functional spaces on
Q
and
Q
is used.lirstly,
Lp(51),
15
p
5
is the set of realvalued functions, summed
Models
of
Heterogeneous Media
21
over with the index
p.
The norm in
L
(Q)
is
defined by the
formula
P
(2.1)
In
the
p
=
m
the space
L
M(Q)
consists of the essentially boun-
ded functions with the norm
(2.2)
At
p
=
2
the space L,(Q)
is
Hilbert space
with
the scalar
product
(f,
g)2,sz
=
If
th% case of
p
=
2
simply as
Jlf(l.
The
class
d
(a),
1
ie
natural,
1
5
p
6
m
,
consists of the
functions aving generalized derivatives in the sense of
S.L.
So-
bolev
L121j
up to the order of
l
including, which belong to
g&.For the sake
of
brevity, the norm in
P
L
(Q)
is
often represented as
llfll
y,Q,
1
5
p
5
m
,
while in
P
LJQ)
t
where
C
denotes the summation over all the derivatives of the
order
f:(
=
al
+
.
..
+
a,.If
p
=
2
then
W;(Q)
is
the Hilbert
space
with
the scalar product
(f,
g)(')
=
C
(D
kk
f,
D
g),
.
(k(=O
(k)
NOW,
C(Q)
is
a set of continuous in
5
(the bar denotes
the closure of the set) functions
with
the norm
(2.4)
The space
e(Q>,
0
C
a
5
I
to Helder
with
the exponent
a
of the functions, the norm
in
is
the set of continuous according
f(Q)
is
introduced as the
sum
JIflJca(n)
=
IIfll~(sz)
+
H"(f),
(2.5)
where
f(f>
is
called the Helder's constant (at
a
=
1
it
is
called the Lipshits constant) and,
by
definition,
is
P(f>
=
SUP
{lf(x,)
-
f(x,)l.lx,
-
xpl-ol).
(2.6)
X,
,X2EQ
The norm in
"(a)
,
for the sake of brevity,
is
denoted as
in particular, at
a
=
0
the space
Co(Q)
coincides with
22
Chapter
1
Besides,
Ck*(Q),
k
functions having derivatives up
to
the order
of
k
including,
continuous according to Helder with the index
ci.
is
natural,
0
5
ci.
5
I
is
the space of
Use
is
also made of the spaces of the functions having different
differential properties with respect
to the spatial variables and
and the time.
If
x
=
(x,,
...,
xn) are the spatial coordinates,
x
E
Q
c
R"
and
t
is
the time,
t
E
(0,
T),
then
q
=
Q
Then
Lp
(%I,
1
5
p,
q
I
rn
is
the space of functions
norm
99
The space
<l'l(.;)
,
consists of the ele-
ments
L
(%I
having generalized derivatives
of
the
form
Dz
D:f
from
LI)(&)
relation
2r
+
1
SI
5 21.
The norm
in
yVp
(%)
is
introduced
by the equality
1
is
natural,
p
L
1
P
with any
r
and
s
=
(s,
,
. .
.
,
s
)
satisfying the
21:1
Ckm'u'p(i.l)
is
the set of functions that with respect to the
variables of
x
have derivatives up to the order
k
with respect
to the
t
up to the order
m
(
k
and
m
are nonnegative integers),
these derivatives being continuous according to Helder over x
with the exponentci.,
0
5
c1
5
1
and over
t
with the exponent
p,
O5!3Il,
(2.10)
+
tc(D:f)
+
H!(Dzf)
+
<(DTf>
+
<(D;f).
Here
Dif
are the derivatives of the order
11
I
=
1,
+
.
. .
+
ln
with respect to
X.
Dtf
is
the derivative with respect to
t
of
the order
j
the
symbols
<
and
€I!
denote the constants
of
Helder's continuity with respect to
x
and to
t
i.e.,
Models
of
Heterogeneous Media
23
t,
,tzE(O,T)
Lastly,
Hz(D:f)
sible derivatives
of
the order
I
k(
is
the
sum
Helder's constants over
t
.
It
should be noted that all the enumerated spaces are complete,
i.e., Banach's spaces.
In
a number of cases the functions that depend on the variables
x
E
Q
and
t
E
(0,
T) are treated
as
the functions
of
the argu-
ment
t
with the values
in
the Banach space defined
on
Q.
For
example,
Lq(O,
T;
W1
(Q))
on
(0,
TI
and operating in
W1
($2)
with the norm
is
the
sum
of Helder's constants over all pos-
with respect to
x9
((0:f)
is
a set of functions
f(t)
prescribed
P
P
(2.11)
It
is
evident that the space
Lq(O,
T;
Lp(Q))
and
Lp,q(G2),
Q
=
Q
x
(0,
TI
coincide.
Similarly,
der with the index
p,
0
5
p
5
l
functions on
[0,
?!]
with values
in
ca(Q),
0
5
CL
5
'1.
Between the spaces
C'(0,
T;
C"(Q))
and
ca"(Q>
valid are the relations
C
B
(0,
T;
Ca(Q))
are the continuous according
to
Hel-
(2.12)
(a>),
&o,
'2;
c%>>
c
C"@(,),
C%P(,)
c
Chqo,
ll;
c
(1-h)CL
where
A
is
arbitrary,
0
<
A
<
1.
In a number of cases use
is
made of the class
of
functions
01
W,(Q,
r0),
r0
c
as1
obtained by closing the set of finite in
the neighbourhood
Po
infinitely differentiable functions in the
norm
w
',(a).
If
roo= r
=
a8
then
$',
(a,
r,)
is
denoted
as
#
'@).
iii(~)
by parts
is
of
the
form
The space
\V:(Q,
ro)
may also be defined
as
the subset
for the elements
f
of which the formula for integration
1
-+
where
n
is
the unit vector of the outward normal to
P
,
g
E
W2(Q)
is
the arbitrary function. Since for the spaces
W1(Q),
1
I
1,p
>
1
in
the regions with a iece-smooth boundary the,notion of trace
on
aQ
is
the
set
of
functions from
W:(Q)
zero. The scalar product and the norm in
iz(Q,
Po)
P
is
defined (see
f21]),
one may say that
W$(Q,
r,)
the traces of which
on
ro
equal
can be
24
Chapter
I
defined by the formulas
In
conclusion let
us
note that the functions of
the
class
p
Z
1
Lp(Q),
have the property
of
continuity on the average:
A-
0
6f(A>
=
Ilf(x
+
A>
-
f(x>llp,a
-0
where
f(x
+
A>
0
at
x
+
A
#
52-
Besides, the existence
of
the generalized derivatives
is
related
to the modulus of continuity
of
the function
8f(A).
Namely,
if
8f(A)
I
c
A
then
f(x)
E
iVi
(a)
and for any strictly internal
subdomain
9'
c
G?
af
f(x
+
Ai>
-
f(x)
3
0,
P,Q'
II-
-
II
ax,
A
where
f'(X
+
A)
=
f(X,,
...,
Xi-l
I
Xi
+
A,
Xi+l
9
*
I
xn>
2.
Special inequalities and embedding theorems
Let
us
cite some inequalities we shall need later.
First,
it
is
the Cauchy inequality
--t
valid for any non-negative form of
a,
at arbitrary
5
E
R",
+
E
R".
Second,
it
is
the
Young's
unequality
I
1
11
P
4
P9
ab
5
-
cPaP
+
-
L-qbbY,
E
>
0,
-
+
-
=
1,
P
>
1,
(2.15)
valid for any
a
2
0,
b
L
0.
For the functions
f
and
g
given in the domain
Q
=
SZ
x
(0,
T)
with the anisotropic properties over the variables
x
E
Q
and
t
E
(0,
T)this inequality can have the form
Models
of
Heterogeneous
Media
25
where
q
2
1,
r
2
1,
l/h
+
l/A1
=
1,
l/p
+
l/pt
=
1,
h
2
1,
p
21.
Finally, let
us
write the Grownwall’s inequality, which
is
often
used when obtaining a priori estimates in non-stationary problems.
If
the function
y(t)
being non-negative in
(0,
TNbeys the
in-
e qua1
i
t
y
+
Y(t)
5
C
+
i
[A(z)
Y
(7)
+
B(z)]dt
,
where
C
=
const
>
0,
A(t)
and
B(t)
are the given non-negative
functions
of
class
L,(O,
T)
then the Grownwallls lemma states
that
Let us now formulate a number of inequalities from the theory of
embedding of functional spaces.
Lemma 2.1. Let
Q
-smooth boundary and
I’
be the
Q
crossing with any r-dimen-
sional smooth hypersurfacg,
r
I
a.
(In
particular,
if
r
=
n
F
=
Q
at
r
=
n
-
1
one Can take
15aQ
as
rr
Then for
any
function
f
EW~
(Q),
12
I
there exists a trace
f[,_
on
Ilr
in which case
be a bounded domain in
R”
with
a
piecewise-
then
is
natural, p
>
1,
at
n
2
pl
r>n-pl
(at
n
=
gl
the number
q
is
any,
q
<
m).
If
p1
>
n
then
f
is
a continuous, by Helder, function of class
=
1
-
I
-
[dpj,
ct
=
’1
+
[dp]
-
n/p
if
Ck*(Q)
n/p
is
not an integer,
where
k
=
ct
<
1
is
any number,
if
n/p
is
an integer, in which case
The
symbol
[dp]
while the constants
the
f.
The proof of the Lemma can be found in [121].
It
should be noted
that the embedding of the spaces
iiil(Q)
into
Lq(r,)
at
n
2
pl
is
compact,
if
y
<
pr/(n
-
$1)
In case when pl
>
n then the
embedding
in
Ck
(n)
is
compact. More subtle dependences between
the spaces
Wi
(a)
and
L
(Q)
are reflected by the so-called
interpolation (multiplicative) inequalities. Here are two classes
of such relations.
Lemma 2.2. Let
Q
c
Rn
and
f
E
W1(Q)
n
Lq(Q),
1
<
p,
q
I
m
denotes here the integer part
of the number n/p
C
in
(2.19) and
(2.20)
are independent of
P
9
P
26
Chapter
1
k
Then
f
E
Wr(Q>
and
llfll'":I
cllflla
"f";;.
2
(2.21
1
w;
where
l/r
=
k/n
+
a(l/p
-
l/n)
+
(1
-
a)/q,
k/l
5
a
<-
1.
Exclusive
is
the case when the number
and
1
<
p
<
M.
Then
a
<
1.
It
should be noted that here the domain
Q
is
not necessarily
bounded, in particular,
(2
=
H*
.
Let
UB
especially single out the inequalities
of
type (2.21) for
the functions
of
class
Wi(Q),
m
5
1.
Lemma 2.3. Let u(x)
E
fJ:
(Q)
or
u(x)
E
W:
(Q)
but
1
-
k
-
n/p
is
a non-negative integer,
0
I
U(x)h
=
0.
a
In
this case
(2.22)
1
-a
IIUIIq,Q
5
CllV
ull:,a
llUIlr,Q
where
then
q
E[r,
mn/(n
-
m)] at
r
I
mn/(n
-
m) while at
r
>
mn/(n
-
-
m>
we can take
q
E
[mn/(n
-
m),
r].
If
II:
2
n
then
q
E.[r,
m]
is
any, in which case at
m
>
n
inequality (2.22)
is
also valid
for
q
=
m
.
The inequalities similar to (2.22) are also valid for the functions
of class
W
f;l(Q).
a
=
(l/r
-
l/Ci)(l/r
-
(n
-
m)/mn)"in which
case
if
m
<
n
To
deduce them, one should make a substitution
1
V(X)
=
U(X)
-
rvi,
IA
=
-
I
u(x)dx.
mes
Q Q
In
this case, obviously,
J
v(x)
dx=O
and
for
v(x)
relations
(2.22) hold. Using the Young's inequality and allowing
for
the fact
that
[Mimes
51
I
l[ulll
,Q
a
conclude
:
Another class of interpolation relations, called the Nierenberg-
-Gaslyardo inequalities [I83
]
relates the spaces
Lp
(a)
and
Models
of
Heterogeneous
Media
cka(
Let us assume
27
Lemma
2.4.
Let
5
v
<
oo.Then
A,
p
and
v
be
arbitrary numbers,
-
m
<
A
5
p
5
In particular, it yields the inequality
Here
are
three
more statements which can be useful when obtaining
estimates
in
Helder spaces.
Lemma
2.5.
Let the integrable in
Q
function
u(x)
for
all
k
1
2
ko
>
0
obey the inequalities
(2.26)
J
(u
-
k)dx
5
yk"
(mes
A,>
1+6
.
,
Ak
where
A
=
{X
E
QIu(x)
>k},
k
,
y,
01
and,
6
are the constants,
in
which"case
6
>
0,
0
5
ct
1Io+
6.Then Iu(x)1
5
Ll
where
hl
depends on
y,
01,
6,
k,
and
mes
Q.
Lemma
2.6.
Let
u(x)
E
a;(Q>,
rn
>
1,
((~(l~,~
5
!,I
for any sphere
1(
the inequalities
and
=
{x
E
52
I
I
x
-xoI<
p,
xo
E
$2)
c
51
valid are
P
where
A
=
Ak
n
X
1
<
m
5
n
<
q,
k
>
,<(x)
is an
arbitrary non-negative smooth function which equals zero on the
boundary of the sphere
I:
Then for any strictly internal sub-
domain
52'
c
Q
there exists a,
0
<
ct
<
1
such that the norm of
IU~~,~,
m,
lil,
IIuII~,~
and on the distance =
aa.
If
(2.27)
holds
for the spheres
K
crossing F then
IU~~,~
is also limited.
Lemma
2.7.
Let
U(X,
t)
k,P
P'
P'
is estimated by the constant which is dependent ony,
q,
a'
to
P
be a limited
(IIUllm,q
5
bl)
in