Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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288
Chapter
5
IIV
Plls,rJ
5
CllV
Pollq,Q
9
c
=
const.
(9.70)
Theorem 9.6. Let
(8,
PI
be a generalized solution of the problem
(9.631, (9.64) and the conditions (9.65)
-
(9.69) to be satisfied.
Then there exists a finite time
to(Iul)
<
m
such
that
s(x,
t)
0,
x
E
E-,
t
2
to.
(9.71)
Proof. For the solution being considered
s(x,
t)
an equality
is
established
by
a
usual procedure. On account of (9.651, (9.661,
(9.70) the right side (9.72)
is
estimated
in
the following way.
(9.73)
1
2n=
A,
+
hZ
+
1
Here
I1
8
M
l12q(q-2),Q
5
C(llSll,,Q)
A
5
c(lIs112,Q
)
(2--cL)/;
(9.74)
A
=
min((Al
+
A2
+
IY2,
(q
-
A/q)
L
(2
-
o!)/2
=
(I
+
A,
-
~,)/2.
Owing to the inequality (9.41)
and
(9.67)
we
have
IIs112,Q
5
clls-o!'2.
vs11;p
a)
5
;11fi
-
vsIl;$-
(9.75)
Introducing a designation
Y(t)
=
Ils(t)ll:,a
and summing (9.72)
-
(9.75) we finfilly come to inequality
(9.76)
Integration
of
the last from
0
to
in evaluation
to
with
(9.67) in view results
Filtration
of
Immiscible Liquids
Consequently, for
289
a
to
2
c-’(
-
c
c1‘
+
Y+(O))
2l
we have
y(t>
=
//SI/;,~
=
0.
The theorem has been proved.
Remark
4.
There
is
a similar statement
for
values
s
=
I
if
res-
pective conditions
(9.62)
-
(9.68)
are satisfied for
s
=
1
-
s
(9
=
0).
This statement can be interpreted in the following way.
Let the region
12
be initially occupied by
a
mixture
(for
example
water
-
oil where
s(x,
t)
is
water saturation and
0
5
s(x,O)
5
I),
fort
>
0
has a boundary with the region, occupied
by
water
(a
E
1).
Then water forces oil out of
Q
at a finite time.
Remark
5.
Properties of
a
problem solution of two-phase filtration
in regions unbounded by space variables have been investigated by
the method
similar
to
5,
6
in work
[
lo].
10.
APPROXIMATION
SOLUTION
OF
TWO-PFASE
FILTRATION
PROBLEMS
1.
Introduction
This paragraph suggests
two
approximation methods
of
two-phase
filtration problem solution (regular and singular) with the help
of
properties
of
these problem solutions deternined earlier, their
convergence
is
proved[24,
25,
85
!.
A
linear differential problem,
which can be approximated be the known difference schemes
is
sol-
ved
in
each of theae methods, this allows
to
establish the estimate
of
convergence velocity of difference solutions to the solution of
initial non-linear equations. One should note that numerical rea-
lization
of
these methods in two-phase filtration problems have
been made
in
[77,
781.
2.
Regular problem
Consider the following initial-boundary problem for
(9,
p)formula-
ted in
81,
2:
as
-+
rn
-
=
div(KoaD
s
+
dl
Vp
+
fo),
(x,
t)
E
Q,
(10.1)
div(Kvp
+
Y)
=
0,
(x,
t)
E
Q,
(10.2)
at
*
S(X,
t)
=
s0(x,
t),
p
=
p0(x,
t),
(x,
t)
E
S
=rx
[O,
T],
(10.3)
(10.4)
The following a proximation methods are suggested
for
the solution
of the problem
f10.1)
-
(10.4).
Method
I.
s(x,
0)
=
qx,
O),
x
E
n.
asi+i
LISi+l
-
-
+
div(E(x,
si>
v
&+I)
+
at
i+i
+
B(x,
si)
V
si+’
+
D(x,
sirJpi+’
*vs
=o,
(10.5)
290
Chapter
5
Si+l(X,
t)
=
so(x,
t),
(x,
t)
E
s,
(10.6)
(10.7)
(10.8)
(10.9)
,i+i
(x,
0)
=
soh
01,
x
E
52,
(x,
t>
=
P0(X,
t),
(x,
t)
E
s.
-f
L2pi+'
E
div(K(x,
si)
vpi+'
+
f(x,
si))
=
0,
pi+i
-
-
-3
--f
Here
mK
=
Koa;
mB
=
KVm
+
f;
-
bf;
mD
=
kb'.
Method
11.
Let's divide the time interval
[
0,
T]
into
N
parts
(z
=
T/N)
and for each time layert
E
[i-c,
(i
+
l)-c.],i
=
0,
N
-
1,
we'll solve the following initial-boundary proble respe
t
to
the functions S~+~(X,
t),
P~+~(X),
(si(x,
iz)
=
g"c"(x),
sto)
z
=
so(x,
0)):
%+I)
+
L,S~+~
E
m
asi+<
-
+
div(E(x,
s(~))
V
at
+
B(x,
s'~')
D
si+l
+
D(x,
s
(i)
V
si+,,
=
0,
(10.10)
(10.11)
(10.12)
L,+P~+~E
div(K(x,
+
f(x,
s(~))
=
0
(10.13)
(10.14)
Si+JX,
t)
=
so(x,
t),
x
E
a,
si+l(x,
iz)
=
s(i),
x
E
52,
-+
Pi+i
(XI
=
P,(x,
id,
x
E
r.
A
linear elliptic problem
(10.8), (10.9)
is
first
solve in both
methods
((10.13), (10.14)
respectively)
in
respect
to
pi+'(x,
t),
(pi+,(x))
p(x,
t)
determined at
i
+
1
step are introduced then in coeffi-
cients of a parabolic equation, and
to
define
sifl(x,
t)(si+l)
the respective non-linear problem
(10.5)
-
(10.9), (10.10)
-
(10.12)
is
solved. The found functions (si+,) are used to
find
pi+
on.
Introduce new sought functions, having
for
a
given saturation value.
A
'*reduced" pressure
(or
pi+2
respectively) at the next time layer and
so
P$,
t>
5
P,(X),
s,Cx,
t>
2i
Si(X,
t),
x
E
n
,
-
t
E
[(i
-
I)
z,
iz],
i
=
I,
N.
The last are defined over all
Q
and satisfy, in a general sense,
the following problem:
Filtration
of
Immiscible Liquids
291
L
s
z
-
m
asz
+
div(x
(x,
t)Vsz)
+
B,(x,
t)
Vst
+
at
5t
(10.15
(10.16)
(10.17)
(
10.18
1
(10.19)
-T
Here
Kt
=
Kt
(x,
s(~),+B~
=
B(x,
s(i),
DT
=
D(x,
~(~3,
Further for uniformity
a
couple of functions
w:+’=(si:’
,
pi;’
)
is
understood under
(si+’
,pi+’
)in
the case of method
I
and
(sz
,pT
respectively in the case of method
11,
and
function si(x,t)or
t(i)(x>
for
t
E
(i
z,
(i+
I>
z).
As
usual
we
assume that
+
K
=
~(x,
s(i3
,
ft
=
f(x,
s(’3,
t
E
[it
,(i+l)t
1
.
7
sk
is
a
if’
5
(m,
k,
(KO
5,
5)
/(
5,
5))
5
M
(10.20)
A
generalized solution ui+’
is
determined for approximation
methods
I
and
11,
respectively, for problem (10.15)
-
(10.15)
commonly (see
Q
2
t
si+i
z
E
L~[o,T;
w:~
RII,
pi;’
E
L
[o,T;
w,‘<
RII.
Xxistence theorem, maximum principle, Hb’lder continxity and
further smoothness are established for this solution in
a
comulete
analogy to
gy5,8
and convergence to initial non-linear problem
solution
is
proved. There assumptions can be formulated in the
following way
[24,
251.
Theorem
1.
[24, lemma
1
and theorem
11.
Let in addition to assump-
tions (10.20) and (i
-
iv)
2
the following conditions be
satis-
f ied
:
+
(
/If
11
q,Q
i
11
vP,
119,
Qi
11
P
Ila,,Q
IMY
’
*
(10.21
1
2
/rl
+
n/
q,
=
I-
w
<
I,
r1
E
[z/(I-
n),m
3.
Then there
exists
a
solution
(10.19) and for
it
the estimate% are valid
of
the
problem (10.15)
-
292
Chapter
5
where
ct
6
(O,l),
q’
=
2+
E
and a constant
C
is
independent of
it
z
.
11
KO
,
vmt
f(x,s),
f&,s),
Ms),
a(x,s>
llcl(Q*)
5
s
(10.25)
If
in additionj
.+
n,
=
{(x,s),
x
E
Q
,
o<
s<l
},
q
>
2 (n+2>,
1
20,
r
E
H~+~
,
and sequence conditions to the order
[l
/2]
are satisfied, then
i+i
and
begi.ming with
1
L
2
(10.27)
(10.28)
i+i
i+i
I1
Pz
,
ST
Ilc(Q)l+
.,(l-l+
.)/2
5
c
with
a
constant
C
independent
of
-C
8
i
Theorem
2.
[
24, theorem
21.
Let the conditions
of
lema
2
with
1=
2,
be
satisfied. Then
for
=
(aL:’
t
P
the initial problem (10.1
)
-
(10.4) where
i
+
a,
z
--f
0
functions
ui+’
=
1;
If
1
)converge to
a
classical solution
w
=
(s,
p)
of
(‘I
II
+
II
G+I-
8
II~~(G~
5
E
(10.29)
Ill1
P:+I
-
P
II
2,
Q
m
where
E
=
C(
(8’)
’I2
in method
I
and
E=
c
z
in method
I1
res-
pectively
c
=
C(
pt
M)
t
B
E
(0,
1)
.
One may point out
121
(in conditions (10.251,
(10.26))
such that
uh
converge
to
the solution
of
the initial non-linear problem
(10.1)
-
(10.4) and
-
llw
-
w‘;
llG(Qh)
<
-
c
EP’
+
ti(<)
(10.30)
with some
>
0.
Filtration
of
Immiscible Liquids
293
The proof results directly from evaluations (10.291, (10.38) and
the inequality
of
a tria.ng1e
Remark. Some difference schemes are considered in works
[24,
771
for problems (10.10)
-
(10.12).
3.
Degenerating problem
Now let in a two-phase filtration the saturation
s(x,
t)
be able
to achieve residual values
s
=
0,
I
and, consequently, parabollic
equation be able to degenerate. Consider the system (10.1),(10.2)
with the following boundary and initial conditions:
--t
+*
(Koav
8,
n)
=
0,
(Kg
p
+
f,
n)
=
H,
(x,
t)
E
S,
(10.31
s(x,
0)
=
So(&
0).
We deal with
only
those solutions of the problem (10.11, (10.21,
(10.31
for
which
1
Remember that the conditions providing inequalities (10.32) to
hold have been mentioned in
4
99
.
Tfiethods
I
and
I1
suggested in
2
can be of use for the solution of
the formulated singular two-phase filtration problem and for
solu-
tions of approximated problem
(
10.15
-
(1 0.19) the analogue of
the theorem
2
is
valid
[25,
theorems
3
-
51.
11.
UNSOLVED
PROBLENS
I.
Of
great interest are problems of liquid filtration possessing
non-newtonean properties. The Darsy laws for immiscible mixture
components in the case
of
non-linear
law
of
medium resistence
may
be generalized in analogy with Navier-Stokes model by introducing
interaction forces of these components or in analogy with homoge-
neous liquid at the expense of dependence of phase penetrations
on
component pressure gradients.
In
respect to this two forms of non-
-linear Darsy laws are suggested for consideration:
+-+
.--t
--t
+--t
--t
vi
=
ui
+
(-I>’F,,
F,
=
Q(S,
W)W;
vi
=
m(w>ui,
i
=
I,
2,
to which corresponds the system of equations relative to
s(x,
t)
and
p(x,
t)
similar to the linear case:
as
--t+
-+
m
-
=
div iPn[Koa
vs
-
bv
+
F
+
(I
-
n)Fo],
at
(11.1)
--t
div
Qn(Kv
p
+
f)
=
0,
n
=
0,
I.
294
Chapter
5
-+
+
-+
Here
ui
=
-
Ki( pi
+
pig),
w
=
1
wl
and
w
coincides with one of
-3 -3
ui
with
v
or with
KO
vpc which conforms to the assumption on
-3
realization of common non-linear Darsy law for a component
to the supposition of proportionality of interaction forces to the
total mixture consumption, or to the assumption on the primary in-
fluence
of
capillary forces on the component interaction process.
The
form
of coefficients a(s,
x),
b(s),
Y
and
3
,i”
equationa
(10.1)
doesn‘t change in comparison with the case
Fo
=
0,
2.
It
would be very important to investigate filtration models,
accounting the compressible properties
of
separate components of a
mixture. In this case densities of each component are the sought
functions and for system closure are used the equations of
state
3.
Besides, changes
of
filtration properties of porous ground are
always observable
in
actual processes. These changes are caused by
hydrodynamic reactions
of
a liquid. One can assume that the change
of a porous layer
is
ropertional to an average pressure of a
li-
quid in pores:
m
=
mo?~)
+
P(p
-
p,) where
mo
is
porosity under
a certain standard pressure
po,
p
is
an average pressure in a
liquid,
is
a coefficient
of
ground misciblity. In these anmp-
tions the equations
of
a model form
a
parabolic system relative
to and
the solvability of a problem in the class
of
generalized solutions
has been proved. The problems of uniqueness and smoothness have
not been studied yet.
4.
Multicomponent mixtures being considered, a great number of
problems arise. Let
vi, pi,
pi,
si,
i
=
1,
...,
n
+
1
correspond-
ingly, velocities of filtration, pressure, density and rela ive
sa-
turations
of
components,
3
=
(sl,
sc,
...,
sn),
sn+l
=
1
-
C
si.
For each
of
the constituents hold the Darsy laws and continuity
equations, similar
to
a
two-component case, and the following Lap-
lace laws related to the singled out phase pn+l
-
pi
=
(dlKol
)l’Lx
Hi(s),
i
=
I,
...,
n.
Irere,
some mixture constituents
can possess the compressibility. BIoreover, different liquids may
interact in the complicated way. (The case
of
immiscible liquids
has been considered everywhere before, but a diffusion exchange,
which plays an important role for some substances, must be taken
into account in a model.)
5.
We
also
note, that temperature changes must not be neglected in
some problems of practical importance. In this case energy equa-
tions should be added to filtration equations
ui
E
1.
pi
=
pi(p.),
i
=
1,
2.
1
,
their investigation has been started in the work
[31]:
+
14
i=
1
(11.2)
aQ
-t
cp
-
=
div
(M
00)
-
pu
-00.
at
Here
0
is
temperature of a mixture, c-+and
H
are heat capacity
and heat conductivity of a mixture,
U
is
an average velocity.
6.
In the work
[
79
]
an approximated model of filtration of immiscib-
le liquids, based on the assumptions similar to the conditions the
Filtration
of
Immiscible
Liquids
295
boundary layer theory has been suggested. The equation for saturs-
tion
E
in this case would have the form
(a
-
)
+
b'v
-
,
(11.3)
as
as
a
as
as
at
ax
ay ay
aY
10.
-
-
btu
-
=
-
where
x
and y are coordinates along and across the layer res-
pectively,
u
and
v
are components of mixture velocity. The main
difference of the given model
is
connected with the fact that the
equation obtained
is
an evolutionary one over a variable
x.
The question of the boundary problem design for the given model
is
of great interest.
7.
Finally, we note a range of problems with free boundaries. These
problems arise when, for example, filtration in layers not fully
filled with liquid are considered. The system of quasi-conformal
mappings can be successfully used in two-dimensional problems
.
Three-dimensional, moreover non-stationary problems with free
P
oun-
daries and for two-phase flows are to be investigated. Some of
their modal designs for two-dimensional non-stationary flows have
been considered in
(18,
191.
This Page Intentionally Left Blank
297
LITERATUW
1. Agmon
S.,
Douglis
A.,
Nirenberg
L.
Estimates near boundary for
solutions of elliptic partial differential equations satisfying
general boundary conditions (Russian).- Moscow, 1962.- 203 pp.
fluid flow in
a
weakly curvilinear channel (Russian)
//
Dinamika
Sploshn.Sredy: Sb. nauch.
tr.
/
Akad. Nauk
SSSR,
Sibirsk. Otd.,
Institut Gidrodinamiki.- 1969.- Vyp.68.- 5.115-121.
tionary hydrodynamical problems (Russian)
//
Dinamika Sploshn.
Sredy: 5b. nauch.
z./
Akad. Uauk
SSSR,
Sibirsk. Otd., Institut
Gidrodinamiki.- 1972.- Vyp.10.- S.5-27.
vortical flow of ideal fluid (Russian)// Dinamika Sploshn. 3redy:
Sb. nauch.
tr./
Alcad. PIauk 3SSR, Sibirsk. Otd., Institut Gidro-
dinamiki.- 1972.- Vyp.15.- 3.7-17.
5.
Alekseev
G.V.
On
the solvability of inhomogeneous boundary value
problem
lor
two-dimensional nonstationary equations of ideal
fluid dynamics (Russian)// Dinamika Sploshn. Sredy: Sb. nauch.
tr./
Akad. PIaulc SSSR, Sibirsk. Otd., Institut Gidr0dinamik.l.-
2, Alekseev
G.V.
On the existence and uniqueness of conductive
3.
Alekseev
G.V.
On
the vanishing viscosity
in
two-dimensional
sta-
4.
Alelcseev
G.V.
On
the uniqueness and smoothness of the plane
1976.- Vyp.24.- S.15-35.
6.
Alelcseev
G.V.,
Khusnutdinova
N.V.
On the solvability of the
first boundary value problem
for
the one-dimensional equation
of two-phase filtration (Russian)// Dokl.. Akad. ?!auk
S3SR.-
7.
lntoncev
S.11,
On
the solvnbility of boundary value problems for
degenerating equations of two-phase filtration (Russian)//
Di-
namika Sploshn. Sredy: Sb. nauch.
tr./
Akad. Nauk SSSR, Sibirslc.
Otd., Institut Gidrodinamiki.- 1972.- Vyp.10.- 5.28-53.
8.
Antoncev S.N. Stationary problems of two-phase filtration with
unknown boundaries
(Russian)//
Dinamj.ka Sploshn. 3redy;
Sb.
nauch.
tr./
Akad. Nauk
XYR,
Sibirsk. Otd., Institut Gidrodina-
mi1ci.-
1978.- Vyp.3G.- S.3-10.
9.
hntoncev
S.N.
On finite rate of disturbance progagation for
plane problems in two-phase filtration (Russian)// Dinamika
Sploohn. Sredy: 3b. nauch.
tr./
Akad. Nauk
SSSR,
Sibirsk. Otd.,
Institut Gidrodinamiki.- 1979.- Vyp.39.- 5.23-29.
(Russian)// Numerical methodJ for solutions of filtration prob-
lems in multiphase incompressible fluid/ Proceedings of the
IVth All-lJnion Seminar.- Novosibirsk, 1980.-
P.
13-22.
11. Antoncev
S.N.
Finite rate of disturbance propagation in
multi-
dimensional problems of two-phase filtration (Russian)// Boun-
dary
value problems of mathematical physics and related questions
in
the theory
of
functions/ Zap, Nauch.
Sem.
Leningrad. Otd.
Alat.
Inst. (LOMI) Akad. Nauk SSSR.-
7980.-
T.96.-
S.3-12.
12. Antoncev
S.rI.
On
localization of solutions of nonlinear degene-
ratinE elliptic and parabolic equations (Russian)//
Dokl.
Akad.
13. Antoncev S.N., Kazhilchov
A.V.
blathematical problems in dynamics
1972.- V.202.-
N
2.- S.310-312.
10. Antoncev S.N. Two-phase filtration problems in unbounded domains
Nauk 3SSR.- 1981.- T.260.-
N
6.-
5,1289-1293.
of inhomogeneous fluids: Lecture Notes.(Russian)/ Novosibirsk
State University.- 1973.- 121 pp.