SELF-SIMILAR SOLUTIONS OF PROBLEMS OF TWO-PHASE FLOW THROUGH POROUS MEDIA WITH ALLOWANCE FOR THE COMPRESSIBILITY OF ONE OF THE PHASES UDC 532.546

P. G. Bedrikovetskii, R. D. Kanevskaya, and M. V. Lur'e Self-similar problems of mutual displacement are solved with allowance for the compressibility of one of the phases. It is found that when an incompressible liquid is displaced by a gas the extent of displacement increases. The mutual displacement of gas-liquid mixtures results in the formation of not one but several saturation jumps, i.e., the structure of the displacement zone is much more complex.

Oil reservoir flooding processes are described by the solutions of problems of mutual displacement of immiscible liquids. The processes of displacement of oil by a gas and gas injection into water-bearing formations can be similarly described. Solutions of these problems are known for the case in which both phases are assumed to be incompressible [1--5]. However, when oil and water are displaced by a gas, the compressibility of the gas phase plays an important role. i. Formulation of the Problem The process of plane-parallel two-phase immiscible liquid-gas flow through a porous medium is described by the equations

o (s,p,')

m - Ot'

~

0 (p,'w,') Ox"

= O;

~=1,2,

s,=s,

s2=l-s,

.

.

p~ =const,

.

.

p2 = P (p2)

kl,(s,) @,' ~, Oz"

w,'=

p/--p/=p/

( ! . 1)

(1.2)

(s ) = e ' J ( s)

(1.3)

Here, x' is the volume of the reservoir reckoned from the injection point, t' is time, s i is the saturation o f t h e p o r e s p a c e by t h e i - t h p h a s e , m i s t h e p o r o s i t y , k is the permeability, fi is the relative phase permeability, ~i is the viscosity of the i-th phase, w~, p~, and p~ are the velocity, density and pressure of the i-th phase, respectively, p~ is the capillary pressure difference, ~' is its characteristic value, and J(s) is the dimensionless saturation function; the subscript i = 1 relates to the liquid, and i = 2 to the gas. In the large-scale approximation for large x' and t' we can neglect the capillary pressure difference and assume that Pl = P2 = P'. Problems of the mutual displacement of gas-liquid mixtures correspond to the following initial and boundary conditions: s(O, t ' ) = s ~ p'(O, t ' ) = p ~

s(x', O)=s0,

p ' ( x ' , O)=p0'

(1.4)

L e t F = (1 + f 2 ~ l / f l ~ 2 ) "z be t h e B u c k l e y - - L e v e r e t t f u n c t i o n ( F i g . 1 ) , and u ' = w i + w~ t h e t o t a l f i l t e r velocity; t h e n wi = F u ' , w~ = (1 -- F ) u ' . We i n t r o d u c e t h e d i m e n sionless variables rex' kmpo" t' p/ ~h~u" X ~-t------pi~-,, l g ~ - .Q ' ~i~ ~ ' po kmpo'' ~ti

p

p~

e

Po

Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 71-80, January-February, 1990. Original article submitted January 16, 1989.

0015-4628/90/2501-0059512.50 9 1990 Plenum Publishing Corporation

59

Fig. 1 Here, ~ is the characteristic value of the volume, comparable with the reservoir volume. The system of equations (i.i), (1.2) and conditions (1.4) take the form:

Os OaF --+ = Ot Ox

O,

Op(t-s) + apu(l-F) Ot

0

1.5

Ox

u = - (i~+~/:) ~

1.6

gX

s(O, t)=s ~ p(O, t ) = p ~

1.7

s(x, O)=so, p(x, O)=i

2. Conditions at the Discontinuities The system of equations (1.5), (1.6) admits discontinuous conditions at the discontinuity have the form [8]:

[s]D=[uF];

[p(l-s)]D=[pu(l-F)];

solutions.

The Hugoniot

[p]=0

Here, D = dx/dt is the velocity of the discontinuity, and [A] symbolizes a jump in the value of A. Hence it follows that [u] = [p] = 0, i.e., the functions u and p are continuous; only s-jumps are allowed: D = u[F]/[s]. In order to obtain a unique generalized solution of the problem (1.7) for system (1.5), (1.6), we require that the discontinuity satisfy the admissibility conditions [9]. A discontinuity will be admissible if it is the limit of the continuous solutions of the complete system (1.1)-(1.3) as the capillary pressure difference e tends to zero. We will seek the solution of the complete system of equations in the neighborhood of the discontinuity of the shortened system in the form of a traveling wave s = s(~), u = u(w), p = p(m), ~ = (x -- Dt)E -l. This solution will satisfy the following matching conditions on the boundaries:

u(•

•

p2(•

~, s(•177

H e r e , u • p• and s • a r e t h e v a l u e s o f t h e c o r r e s p o n d i n g f u n c t i o n s nuity. We o b t a i n t h e s y s t e m o f o r d i n a r y d i f f e r e n t i a l equations

D ds

aF+f,

dJ ds'l

(t-F)Tss ~ - J = 0,

- D

dp(i-s)

dp~= (]~q-~tf,_) do)

[ --e

+

d

[ pu(t-F)-pf,

(2.1) at the disconti-

dJ ds

]

= 0 (2.2)

dJ ds ] u--[l ds do)

The condition of admissibility of the discontinuity is the existence of a continuous solution of the problem (2.1), (2.2). Since from the condition at the discontinuity it follows that u + = u- = u,, p+ = p- = p,, in the solution of the problem (2.1), (2.2) u(~) = u,, p2(~) = p,. Therefore the complete system of equations (2.1), (2.2) reduces to a single equation for the saturation and the problem of the structure of the discontinuity has the same form as when the fluids are incompressible [I0]. It can be shown

60

that it has a continuous solution when and only when for any value of s on the interval (s-, s+) Oleinik's inequalities [Ii]

F ( s ) - F ( s +) <~-D- ...--_,[F] <~. F ( s ) - F ( s - ) s--s + u, [s] s--sare satisfied. The function F(s) has a single point of inflection (Fig. i); tions obtained can be simplified: F~ (s +) ~<

D u,

[F]

--

[s]

therefore the condi-

dF

~
F~ = - -

ds

(2.3)

3. Self-Similar Solutions The problem (1.5)--(1.7) admits the self-similar solutions

s=s(~), p=p(~), u=U(~)/~2t, ~=x/~2t When these expressions are substituted in Eqs. (1.5), (i.6), the problem reduces to a boundary-value problem for a system of ordinary differential equations:

ds d U f . ~ d-~= d~ '

~ do ( l - s ) d~

u=-

= dpU ( l - F ) d~

(/, +~!2)-~

s(O)=s ~ p(O)=p~

(B.i)

(3.2)

s(~)=So, p ( ~ ) = l

(3.3)

The condition at the discontinuity takes the form D = U[F]/[s], where D is the value of the self-similar variable at which the saturation s has a discontinuity. In the planes (s, FU), (~, s) the self-similar solution of the problem corresponds to the path connecting the points s(0) and s(~), which may consist of the following elements: continuous intervals describing the solution of ordinary differential equations (3.1), (3.2), henceforth denoted by the letter S; saturation jumps, at which the Hugoniot condition and the admissibility condition (2.3) are satisfied, denoted by the letter J; and rest intervals, denoted by the letter P. Geometrically, the Hugoniot condition means that the slope of the segment of a straight line connecting points behind and ahead of the discontinuity in the plane (s, FU) is equal to D. From Eq. (3.1), written in the form $ = dFU/ds, it follows that the slope of the tangent to the continuous part of the path in the plane (s, FU) is equal to the self-similar variable $. In what follows each solution of the problem is represented by a structural formula consisting of a sequence of the elements S, J, and P. We write the system of equations (3.1), (3.2) in the following form:

ds

t dp U ( I - F ) / ( t - s ) - ~

dU.... -TV(, i dp d~ dp_

-F t-s

UF(t-s)

_ ~] U(i-s)

L+~h U

(3.4)

(3.s) (3.6)

Since in the process of displacement U d 0, from (3.6) it follows that p decreases monotonically from the value p~ at ~ = 0 to unity when $ = ~. When dp/dp > 0 the sign of the right side of Eq. (3.5) will coincide with the sign of the expression in parentheses. Therefore U(~) will increase when 0 ~ $/U < (i -- F)/(I -- s) and decrease when

Uu

> (i

-

F)/(i

-

s).

61

[

!

d

i

C

)p

b

//

a

0[

J/ "

/

1

o

Fig. 2

Fig. 3

We will consider Eq. (3.4) as an autonomous system in the plane (~, s), assuming that the functions U($) and p(~) are known in advance from the solution of the problem and vary as indicated above. The isoclines along which ds/d$ = 0 and ds/dg = ~ are given by the equations ~/U=(I-F)/(i-Q; ~/U=F~ (3.7)

I f t h e f u n c t i o n ~/U i s monotonic, then t h e i s o c l i n e s

( 3 . 7 ) w i l l have t h e c h a r a c t e r -

istic

form represented by the broken curves in Fig. 2. They divide the phase plane into four domains, in each of which the sign of ds/d~ is constant. The isoclines (3.7) intersect along two intervals {~=0, se[0, s.]} and {~=0, !~[r, ill and, moreover, at the point (~f, sf). The points of intersection of the curves (3.7) are singular points of the autonomous system: the point (~f, sf) is a saddle point, the point (0, s*) a degenerate focus, and the point (0, s*) a degenerate node. When s~[0, S.] we have ds/dg = 0, and the integral curves are straight lines. When s~[s', i] Eq. (3.4) can be integrated using (3.6) and boundary conditions (3.3):

~=i-(i-so)polp

(3.8)

Let us consider the asymptotic behavior of the solution as we tend to infinity along the arbitrary trajectory ~. In all cases, apart from s 0 = I, the pressure p tends to P0 = i. Then from Eq. (3.6) it follows that U + 0. Starting from a certain $ the inequality ds/d~ < 0 is satisfied; therefore the function s(~), being both monotonic and bounded, will have a limit equal to s o as ~ + ~. Therefore from (3.5) and (3.6) as u ~ = we have

P-Po~Erfc(v~),

U~2W(f,(so)-F~tf~(So))exp(-w~2), 7n

v=(

i--so

lim--I --/dP\'h

2(fi(so)+~tf2(so) ) p~pop dp/

(3.9)

The reservoir-average saturation and gas phase density <0(i -- s)> are important characteristics. Expressions for determining them can be obtained as a result of integrating Eqs. (1.5), which express the liquid phase volume balance and the gas phase mass balance, over the region of the plane (x, t) bounded by the contour (0, 0) § (0, t) (L, t) ~ (0, 0), where (L, t) ~ (0, 0) is a segment of the curve x = ~ , ~ = L/~ [13]: L

0 L


!o

oo(t-F

VO-p (l-e (s))V]

4. Solutions of Problems of the Displacement of a Liquid by a Gas and a Gas-Liquid Mixture If only incompressible phase is present in the reservoir before displacement begins, then s o = i. Let the displacing phase saturation s ~ satisfy the condition 0 ~ s ~ ~ s,. The region in which ds/d$ > 0 is covered by the integral curves departing from the confluent

62

node (0, s,). Let us consider a trajectory passing through the singular point (sf, UjFs(s.))j . We denote the corresponding continuous interval of the solution by ~f We now consider the jump from the point s- of the region ds/d~ > 0 to the point s i. From the admissibility condition (2.3) it follows that s- e sf. From the Hugoniot condition we have s- = sf. In accordance with (2.3) jumps from trajectories on which 0 ~ s ~ ~ s, to the trajectory Sf are permissible only when ~ = 0. Thus, the solution of the problem consists of a stationary jump from the point (0, s ~ ) to the point (0, s,), a continuous interval Sf, on which the saturation increases from s, at g = 0 to sf at ~f = UFs(sf), a jump to the point s = i, ~ = ~f and a rest interval s = i when ~(~j, ~). The corresponding path is shown in the plane (~, FU) in Fig. i. The structural formula describing this solution has the form:

(4.1)

s~

In Fig. 3 this path is represented in the plane (~, s) by the heavy line a. The path corresponding to the solution of the same problem when both phases are assumed to be incompressible differs only with respect to the continuous interval: instead of the curve Sf it contains a segment of the curve ~ = UF s (broken line in Fig. 3). Thus, disregarding the compressibility of the displacing phase leads to the underestimation of the saturation values, i.e., the extent of displacement. A comparison of the results of the calculations showed that the neglected volume of injected gas increases with increase in the injection pressure: when p~ = 2 it is 22%, and when pO/pf = 5 39%. The compressibility of the gas phase leads to an increase in the rate of advance of the displacement front, the ratio Uf/U increasing with the injection pressure: when p~ = 2 we have Uf/U ~ = 1.2, and when p~ = 5 Uf /U ~ = 1.38. However, the frontal saturation value remains the same as in the case of two incompressible phases. Now let the displacing phase saturation satisfy the condition s, < s ~ < s I. The value of s I is determined for ~ = 0 from the integral curve passing through the singular point (gf, sf) (Fig. 2). From the point (0, s ~ it is possible to depart along a single integral curve, on which ds/d~ < 0. Assuming that s- varies along this curve, we construct the dependence s + = s(~, s-), which expresses the Hugoniot condition at the discontinuity. This dependence is represented by the heavy line S~ in Fig. 4. For any we have s + # i, and accordingly the solution contains at least two jumps. In accordance with the admissibility condition (2.3), the intermediate integral curves must pass from the region ~ ~ UF s into the region ~ ~ UF s as $ increases. Moreover, the last of them must intersect the Hugoniot adiabatic curve ~ = U(I - F)/(I -- s). Therefore the jump can be made only to the trajectory Sf. We will show that the curves s + = s+(~, s-) and Sf intersect at a single point. Let s- = s-($) and s+ = s+($, s-) be the limiting values of the saturation at the points of discontinuity, related by the Hugoniot condition, and let ds/d$ be determined along the integral curve intersected by the curve s + = s+(~, s-). We introduce the new independent variable p. From the relation $ = U[F]/[s], valid at the discontinuity, we calculate ds+/dp

ds + dp

I d~ s+-s U dp F ~ + - ( F + - F - ) / ( s + - s -)

t dU F+-F d s - F z _ ( y + - y - ) / ( s + _ s -) U dp F . + - ( F + - F - ) / ( s + - s -) T d p F F _ ( F + ~ F _ ) / ( s + _ s _ )

We assume that s- varies along the trajectory of Eq. (3.4). Then the values of ds-/ dp and dU/dp are determined from the system (3.4)--(3.6). Taking into account the fact that $ satisfies the relation at the discontinuity, we obtain

ds + ~ dp

~

t d~ s+--s U dp F~+ - ( F + - F -) / (s+-s -)

ds dp

4---

We now go over to the independent variable $. From the conditions of admissibility of the discontinuity (2.3) it follows that F + -- (F + -- F-)/(s + -- s-) <_ 0. Since U > 0, the sign of the difference ds+/d$ -- ds/d~ will be opposite to that of the expression s+ -- s-, i.e.,

63

2

d

6

7

19 7z

I I

s,

Fig.

Fig. 4

.

s~

s8

5

In accordance with (4.2), when s+ < s- we have ds+/d$ > ds/d~. In the case in question ds/dg > O; therefore all the integral curves intersected by the curve s+ = s+($, s-) have a single common point with the latter. We denote the value of s+ at the point of intersection of the curves s+ = s+(~, s-) and Sf by s ~ , and the corresponding value of s ~ by s ~ . The solution of the problem is represented by heavy line b in Fig. 3. It is described by the structural formula

s~176176

P

(4.3)

Taking the compressibility o f one o f t h e phases i n t o a c c o u n t l e d t o t h e a p p e a r a n c e o f a second s a t u r a t i o n jump and t h e f o r m a t i o n ahead o f t h e d i s p l a c e m e n t f r o n t o f a zone w i t h an e l e v a t e d gas c o n t e n t l e a d i n g t h e l i q u i d . Now l e t s 1 < s ~ < s * . From t h e p o i n t ( 0 , s ~ ) t h e r e d e p a r t s o n l y one i n t e g r a l trajectory. It intersects the Hugoniot curve constructed for the value s+ = i at a certain point ($, s~). The solution consists of a continuous interval, a jump and an interval of constancy s ~ I. It is represented in Fig. 3 by the heavy line c. It corresponds to the following structural formula:

s~

(4.4)

If in the injected mixture the mobility of the gas is equal to zero, i.e., s* ~ s ~ i, then the solution will consist of a jump at $ = 0 and a zone of constant saturation s ~ i. This corresponds to the path represented by the heavy line d in Fig. 3. 5. Breakdown of Arbitrary Discontinuity When the relation between the saturations s o of the gas-liquid mixture originally present in the reservoir and s ~ of the mixture injected into the reservoir is arbitrary, we get the problem of the breakdown of an arbitrary discontinuity for the initial system of equations. With each problem of the breakdown of a discontinuity we associate a point in the plane (s ~ , so). The solutions will be of the same type if they are described by the same structural formulas. The solutions are classified by type in Fig. 5. The existence and uniqueness of a solution to the problem of the breakdown of an arbitrary discontinuity correspond to the domains in question covering a square (s~ i]X[0, i] and not intersecting. In Figs. 4 and 5 we have used the following notation: Sp is the limiting value of the saturation as ~ + = along the integral curve passing through the point (UFs(Sp), sp); Sp is the abscissa of the point of inflection of the curve F(s); 3 is the point of in t tersection of the trajectory Sf and the loop ~ = UFs, s 3 > Sp, s~ is the root of the equation (F(s~) -- F(s3))/(s 4 -- s3) = F3(s3) , in which case ~ = ~3; s S and s 6 are the limiting values of the saturations as $ + = along the integral curves passing through the points 4 and ($f, sf), respectively; s 7 is the root of the equation Fs(s v) = F(sv)/s 7, in which case $7 = UFs(Sv); s 8 is the saturation value for $ = 0 on the integral curve passing through the point 7, s 8 < s*, $s = 0. We will show that when the admissibility conditions (2.3) are satisfied the boundaryvalue problem (3.3) for the system (3.4)--(3.6) has a unique self-similar solution for

64

any values of s ~ and s o . Let s 2 ~ s o < I. The integral curves along which s + s o a s 2 as $ + ~ can be reached by a jump either from the values s* ~ s ~ 1 when ~ = 0 or from trajectories on which s I s(0) < s" or from the trajectory Sf. Jumps from integral curves departing from the points (0, s~ s, < s ~ < s I and from trajectories on which 0 ~ s(~) < s 6 to the trajectories in question are impossible by virtue of the Hugoniot relation and conditions (2.3), respectively. It is therefore possible to distinguish four different types of solutions depending on the value of s ~ (domains 1--4 in Fig. 5). The initial part of the path corresponding to solutions of types 1 and 2 is described by structural formulas (4.1) and (4.3), respectively, but the last jump is made from the trajectory Sf when s- > sf. The value of s- is uniquely determined. In fact, let svary along Sf from the value sf to s 3. The relation ~ = U[F]/[s], satisfied at the jump, determines the dependence s + = s+($, s-). It is nonunique and in Fig. 4 is represented by the two nonintersecting heavy lines s~(~, s-) ~ s- and s~($, s-) ~ s-. The values of s+ = s~($, s-) decrease from s~ = 1 when $ = ~f to s~ = s 3 when ~s~)$ From (4.2) it follows that ds~/d~ < ds/d$ < 0. Therefore the curve s + = s~(~, has a unique common point with each of the integral curves s = s(~) it intersects. We denote the value of s~ at the point of intersection with the integral curve along which s + s o as ~ + ~ by s~ and the corresponding value of s- by s~. Then the structural formulas for solutions of types 1 and 2, respectively, take the following form:

s~

(5.1)

sO_S_s~176

(5.2)

Let us consider the solutions of type 3. We assume that s- varies along the integral curve departing from the point (0, s~ We construct the dependence s + = s+($, s-) corresponding to the condition at the jump and consider the part on which s + e s-. It is represented in Fig. 4 by the heavy line S~. We denote the value of s+ at the point of intersection of the curve s + = s+($, s-) and the integral trajectory along which s + s o as ~ + = by s~ and the corresponding value of s- by s~. Then the path corresponding to solutions of the third type is given by the structural formula

s~

(5.3)

The solutions of the fourth type consist of a jump from the value s ~ to the value s~ when ~ = 0 and a continuous interval on which the saturation varies from s~ when $ = 0 to s0 as $ ~ ~. Now let 0 ~ s o < s 2. We will consider the various jumps to the integral curves along which s + s o < s 2 as g + ~. For trajectories departing from the points (0, s~ s* < s ~ < i when ~ > 0 only jumps from points on the loop ~ = UF s are admissible. Jumps to the curves in question from trajectories departing from the points (0, s~ s s < s ~ < s* are impossible by virtue of the Hugoniot conditions. Not all the curves in question can be reached by a jump from trajectories departing from the points (0, s~ s I ~ s ~ ~ s 8. For each of the integral curves departing from the points (0, s ~ ) when s i ~ s ~ ~ s 8 we will determine the value of the saturation s-(s ~ at the point of intersection with the loop g = UF s and the coordinate of the point of intersection $(s ~ = UFs(s-(s~ From the Hugoniot conditions we find the value s+(g(s~ s-s(~ = s+(s ~ < a-(s~ We will determine the limiting value of the saturation s~(s ~ as $ ~ ~ along the integral curve passing through the point ($(s~ s+(s~ In Fig. 5 the curve s o = s~(s ~ connects the points (as, 0) and (s~, as). From the trajectory departing from the point (0, s~ where s I < s ~ < as, it is possible to jump to each of the integral curves along which s ~ s o ~ s~(s ~ as ~ + ~. The integral curves along which s + s o > s~(s ~ as ~ + cannot be reached by a jump from this trajectory. From trajectories departing from the points (0, s~ where 0 < s ~ < sl, it is possible to jump only to those curves on which 0 ~ s(~) ~ s 6 and when 0 ~ s(0) ~ s, the jumps take place at $ = 0. Thus it is possible to distinguish eight more types of solutions of the problem of the breakdown of an arbitrary discontinuity (domains 5--12 in Fig. 5). In order to describe the solutions of types 5 - 8 we construct the dependence s + = s+($, s-), assuming that s- varies along the part of the loop $ = UF s on which Sp < s ~ s 7

65

(Fig. 4). The values of s+ = s+($, s-) increase from 0 at ~ = ~v to Sp when ~ = UFs(sp). Along the integral curves intersected by the curve s + = s+($, s-) the saturation values do not increase. Therefore the curve s + = s+($, s-) has a single common point with each integral curve. We denote by s~ the value of s+ = s+($, s-) at the point of intersection of this curve and the integral curve along which s + s0 as ~ ~ ~, and by s~ the corresponding value of s-. Since trajectories along which s + s o as $ ~ ~ can be reached only by a jump from the trajectories departing from the points (0, s~ where s* < s ~ < i, the initial part of the paths corresponding 6o the solutions of types 5--8, respectively, will be the same as in the solutions of types 1--4. Therefore the structural formulas describing the solutions of types 5--8 will have the following form:

s~

j - - s - - 7 ~ s + - S - s o - - - J ~ so+--S-so

sO_S_sO-_j~s O*-Sf-s--J~s+-S--so---J~So+--S-so s~

s~

Here, the point (U[F]/[s], s +) lies on the trajectory passing through the point (UF s (s~), s~); the point (U[F]/[s], s-) lies on the integral curve departing from the point (0, s,) for solutions of types 5 and 6 and from the point (0, s ~ ) for solutions of type 7. The saturations s + and s- and the coordinate of the jump are found using the constructions described above. Let us consider the solutions of types 9 and i0. These solutions contain one more jump. In accordance with (2.3), as ~ increases the intermediate integral trajectories must go over from the domain ~ ~ UF s into the domain ~ < UF s. Accordingly, the solutions must contain an interval Sf. For each value of s- on the trajectory Sf there are two values s~ > s- and s~ < s- to which jumps are possible (Fig. 4). From the integral curves intersected by the curve S~ it is possible to jump only to trajectories along which s 5 s(=) < s 2. Therefore the solutions of types 9 and i0 include a jump from the trajectory Sf to values s~ < s- on the curve S~. Along the curve S~ the saturation decreases from sf when $ = ~f to s 4 when ~ = $4. From (4.2) we have ds/d~ < ds~/d$ < 0. Therefore the curve S~ lies above the trajectory along which s ~ s 6 as ~ ~ ~. The curve S~ shares a single common point with each of the integral curves it intersects. The structural formula for solutions of type 9 has the form (5.1), and that for solutions of type i0 the form

(5.2). The solutions of type Ii consist of a jump at ~ = 0 and a continuous which s + s o as ~ ~ =. If s o ~ s,, then s ~ s o when $ > 0.

interval on

In order to construct the solutions of type 12 we assume that s- varies along the trajectory departing from the point (0, s~ We construct the dependence s+ = s+($, s-) corresponding to the Hugoniot conditions. When s, < s ~ < s I this dependence is singlevalued, s+ ~ s-, and its graph has the characteristic form S 3 in Fig. 4. When s~ ~ s ~ ~ s 8 for each value of s- there exist two values of s+. Along the curve s+ = s+($, s-) > sthe saturation decreases from 1 when ~ = U(I -- F)/(I -- s) to s- when $ = UFs(s+). From the trajectories intersected by this curve it is possible to jump only to the integral curves along which s + s' > s o as g + ~. The graph of the dependence s + = s+($, s-) < shas the characteristic form S 4 (Fig. 4). It intersects all the trajectories on which s(=) < s0(s~ Therefore the solutions of type 12 include a jump to the value s+ < s-. From (4.2) it follows that the curves S 3 and S~ share a single common point with each of the integral curves they intersect. W e denote the corresponding value of s+ by s~ and the corresponding value of s- by s~. Then the solutions are described by the structural formula (5.3). The problem of the breakdown of an arbitrary discontinuity for the case of incompressible phases was investigated in [12]. When the compressibility of one of the phases is taken into account, of the displacement zone becomes much more complex: in the solutions of there are two saturation jumps and in the solutions of types (5) and (6) The undisturbed zone ahead of the displacement front typical of the case phases is observed only when s o = 1 and 0 < s o < s,. Otherwise ahead of front the saturation decreases. The authors are grateful to V. I. Maron for useful discussions.

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the structure types I, 2, 7--10 three such jumps. of incompressible the displacement

LITERATURE CITED I. M. Muskat and M. W. Meres, "Flow of heterogeneous fluids through porous media," Physics, ~, 346 (1936). 2. S. E. Buckley and M. C. Leverett, "Mechanism of fluid displacement in sands," Trans. AIME, 146, 107 (1942). 3. M. Muskat, Physical Principles of Oil Production, McGraw-Hill, New York (1949). 4. Development of Soviet Research on the Theory of Flow Through Porous Media (1917-1967) [in Russian], Nauka, Moscow (1969). 5. I. A. Charnyi, Subterranean Hydrogasdynamics [in Russian], Gostoptekhizdat, Moscow (1963). 6. G. A. Osipova, G. V. Rassokhin, and G. P. Tsybul'skii, "One-dimensional problem of the displacement of gas by water with allowance for capillary forces," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. i, 68 (1972). 7. M. V. Filinov, V. I. Maron, and I. M. Rokhlin, "Allowance for compressibility in multiphase flow through porous media," Tr. Mosk. inst. Neft. Gaz. Prom-sti., No. 79,

37 (1969). 8. L. I. Sedov, Continuum Mechanics, Vol. I [in Russian], Nauka, Moscow (1973). 9. A. A. Barman and A. G. Kulikovskii, "Discontinuous solutions in continuum mechanics," in: Some Problems of Continuum Mechanics [in Russian], Published by Moscow State University, Moscow (1978), p. 70. i0. V. M. Ryzhik, I. A. Charnyi, and Chen' Chzhunsyan, "Some exact solutions of the equations of nonsteady two-phase flow through porous media," Izv. Akad. Nauk SSSR, Mekh. Mashinostr., No. i, 121 (1961). ii. O. A. Oleinik, "A class of discontinuous solutions of a first-order quasilinear equation," Nauch. Dokl. Vyssh. Shk. Fiz.-Mat. Nauki, No. 3, 91 (1958). 12. M. V. Lur'e, V. M. Maksimov, and M. V. Filinov, "Investigation of various cases of mutual displacement of immiscible fluids in a porous medium," Inzh.-Fiz. Zh., 41,

656 (1981). 13. P. G. Bedrikovetskii, "Two-phase three-component flow in the displacement of oil by an active solution," Zh. Prikl. Mekh. Tekh. Fiz., No. 5, 121 (1983).

PROPAGATION OF NONLINEAR BODY WAVES IN A MAGNETIC TUBE A. L. Molotovshchikov

UDC 532.592:537.84

The propagation of slow symmetrical small-amplitude body waves in a cylindrical magnetic tube is investigated on the basis of the nonlinear equation obtained in [3, 4]. The breaking of periodic disturbances of a certain type in a finite time is numerically demonstrated. It is noted that the equation in question does not have solutions in the form of solitary waves. The observational data show that the magnetic fields in the solar atmosphere have a sharply expressed structure and are concentrated mainly in formations of the extended magnetic tube type. In [ii the propagation of waves in cylindrical magnetic tubes was investigated in the linear approximation, dispersion relations for such waves were obtained, and a classification of the main wave types corresponding to differences in their conditions of existence, the geometry of the disturbed flow and, moreover, the value of the characteristic propagation velocity of the disturbances was introduced. The possibility of comparing the properties of the individual types of wave motions in a magnetic tube with the available results of observations was examined in [2]. The waves described in [i] as slow symmetrical body waves are characterized by a subsonic sub-Alfv~nic propagation velocity cT, small deformations of the surface of the magnetic tube and a nonuniform axisymmetric distribution of the fluid parameters over the tube cross section. In [3, 4] a nonlinear equation describing the propagation of slow symmetrical small-amplitude body waves in the long-wave approximation was derived for the case of a cylindrical magnetic Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. i, pp. 81-84, January-February, 1990. Original article submitted December 26, 1988.

0015-4628/90/2501-0067512.50 9 1990 Plenum Publishing Corporation

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