4. V. V. Kadet and V. I. Selyakov, "Fluid percolation in a medium containing an elliptical hydraulic fracture," Izv. Vyssh. Ucheb. Zaved. Neft Gaz, No. 5, 54 (1988). 5. J. L. Elbel, "Consideration for optimum fracture geometry design," SPE Prod. Eng.,

~, 323 (1988). 6. V. P. Pilatovskii, Fundamentals of the Hydromechanics of Thin Strata [in Russian], Nedra, Moscow (1966). 7. I. M. Abdurakhmanov, "Effect of a single fracture on the flow through a porous medium," Prikl. Mat. Mekh., 33, 871 (1969). 8. I. M. Abdurakhmanov and M. G~ Alishaev, "Two-dimensional steady flow in a reservoir divided by a straight fracture," Izv. Akad. Nauk SSSR, Mekh. Zhidk; Gaza, No. 4, 173 ( 1 9 7 3 ) . 9. M. A. Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable [in Russian], Fizmatgiz, Moscow (1958). i0. S. M. Belotserkovskii and I. K. Lifanov, Numerical Methods in Singular Integral Equations and Their Applications in Aerodynamics, Theory of Elasticity and Electrodynamics [in Russian], Nauka, Moscow (1985).

NONISOTHERMAL VISCOPLASTIC FLUID FLOW THROUGH POROUS MEDIA S. V. Panako

UDC 532.546

The Alishaev model [i] is extended to the case of nonisothermal flow. Neglecting conductive heat transfer, it is shown that for the model in question in the plane of the complex potential not only are the problems linear but the decoupling of the thermal and hydrodynamic problems is also allowed. The latter is reduced to a mixed problem for an analytic function. This makes it possible to use the wellknown methods and results of the theory of limiting equilibrium pillars for isothermal flow [2--5]. It is also established that the solutions of the unsteady problems tend asymptotically to the solutions of the corresponding steady-state problems and can be obtained from the latter by simpler conversion. The effectiveness of the approach proposed is illustrated with reference to the problem of a source-sink system [i--4]. i. In the exploitation of viscous and paraffin-base oil deposits nonisothermal effects are expressed as a lowering of the viscosity and a decrease in the limiting gradient for the oil with increase in temperature [6]. In [2] the dependence of the limiting gradient on the coordinates of points on the oil--water interface was given (and interpreted as a nonisothermal effect) in such a form that with the aid of an auxiliary analytic function, recoverable from the boundary values, a number of problems could be considered for the model [i]. In particular, for a source--sink system [i--4] it was found that there is an increase in the flow region as compared with the case in which the limiting gradient is constant. The results of the calculations for a five-point scheme also indicate an increase in oil output [6]. The equations of motion of an incompressible fluid in a homogeneous porous medium with a nonlinear flow law [3] can reasonably be extended to the nonisothermal case as follows: V H = - ~ (w, T) w, div w=0, r (1.1) w Here, H is the reduced head, w is the filter v e l o c i t y , G(T) i s t h e l i m i t i n g g r a d i e n t f o r t h e o i l , and T i s t h e a b s o l u t e t e m p e r a t u r e . To ( 1 . 1 ) we add t h e t r a n s p o r t e q u a t i o n , w h i c h , i n a c c o r d a n c e w i t h [ 7 ] , we w r i t e in t h e f o r m :

Tomsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117-122, July-August, 1990. Original article submitted June 2, 1989.

0015-4628/90/2504-0593512.50 9 1991 Plenum Publishing Corporation

593

?(wVT) = div(xVT)+q(T),

pcp C+mpcv'

?

As a r e s u l t we o b t a i n a s y s t e m o f e q u a t i o n s flow through porous media. In (1.2) q(T) is the heat of the skeleton material, Cp i s t h e s p e c i f i c diffusivity.

( !. 2 )

X=h?pc~

of nonisothermal non-Newtonian fluid heat release rate, c is the specific heat of the fluid, and X is the thermal

The boundary conditions for the system (i.i), (1.2) are formulated in the same way as in problems of nonlinear flow [3]: only the condition T = T(s) or T n = Tn(s), where s is the arc length of the boundary and n is the normal to it, is added. In the plane of the complex potential W = --H + i~, where ~ is the stream function, the flow region is a half-strip 0 < $ < Q, - ~ < --H < 0, in some cases multivalent [2--5], irrespective of the arrangement of the wells. It is therefore natural to choose H and as the independent variables. Then in the plane of the complex potential the system (I.I), (1.2) takes the form:

0

OT

wE OT ] r o,

OT -=~ (1.3)

o O0

w'--

o,

=

Ow

0 1

O0

OH wb-r162 r 1 6 2 -

(w, T)

where 0 is the angle of inclination of the velocity to the x axis. In accordance with the known solution of the system (1.3) the return to the flow plane is accomplished by means of the conversion formula

dz=eiO [ _ dH , .d~ ] T.~T]

(1.4)

The analysis of the system (1.3) for an arbitrary dependence %(w, T) presents considerable difficulties in view of its nonlinearity. Accordingly, we will make the following simplifying assumptions~ We neglect the heat conduction, i.e., set X = 0. This assumption is justified if the thickness of the reservoir and the filter velocity are large enough [6] and convective heat transfer is decisive. Moreover, as the flow law we will take the generalization of the model [i]:

IVHI

w=o,

_~k G

(r),

H=-VH,

i

k

IVH[>--G(T)

(1.5)

From (1.5) it follows that the boundary of the pillar will no longer be a line of constant absolute value of the velocity. On the basis of these assumptions the system (1.3) can be considerably simplified and takes the form:

O0 0r

Ow OH'

Om 0,

O0 OH

20T ?w - - - . . , - - . ) OH § q(T'=O

(1.6)

The heat transfer equation in (1.6) shows that the isotherms coincide with the streamlines only in the absence of heat sources. On the boundary of the pillar, in addition to the limiting equilibrium condition w = w(T), the no-fiow condition ~ = 0 must be satisfied. Bearing in mind the last of Eqs. (1.6), we find that along this boundary

w=w(T(H,O)),

20T

?w ~

(H,O)+q(T(H,O))=O

(1.7)

For given w(T) and q(T) the temperature and filter velocity distributions on the pillar boundary can be found from the solution of the first-order ordinary differential equation (1.7).

594

In particular, if

q(T)=a(r-To), where a,

~0, and T o a r e known c o n s t a n t s ,

w=~0[t+2a(T-T0) ] -'~

('t .8)

"then we h a v e

w:=XoZ(t-mexp+bH),

G~(T)=Go2(i-mexp-bH) (1.9)

GZ(Tl)

m=~

r b= - ~Co~ '

a ~(ro) '

kGo ~.o = - -

Here, T I is the given temperature on the supply contour. Similarly, taking as G(T) the dependence recommended in [6]

G(T)=Go e x p - ~ ( T - T o )

(1.10)

which satisfactorily approximates the experimental data, and assigning the heat release rate in the following form: q (T) = ~ - [1 - exp-213 (T-To) ]

( 1.11 )

we obtain

exp-2~(T-To)=l-m

exp-bH,

w==h0~(i-m e x p - b H )

(1.12)

From relations (1.9) and (1.12) it follows that for the model in question the hydrodynamic and thermal problems can be decoupled. We will first find the solution of the hydrodynamic problem -- an ordinary mixed problem for the function F(~) analytic in the half-strip 0 < ~ < Q, - ~ < --H < 0:

f(~)=ln(w/lo)-iO,

~=~+i~=-H+i~,

ReF=t/21n(i-mexpb~),

-Ho<~<0,

ImF=0,

-~<~<-Ho,

~=0, ImF=-0o,

-=<~<0,

~=0

(1.13)

~=Q

Here, 8 0 is determined by the arrangement of the wells (e 0 = ~ for a two-well system, 8 o = v/4 for a five-point scheme, etc.). The parametric equation of the pillar boundary is found by means of the conversion formula dH

dz

w (H, 0) exp i0 (H, 0)

The temperature distribution along the pillar boundary is already known from (1.9) and (1.12). The temperature field within the flow region is found by integrating the last of Eqs. (1.6) with the initial condition T(0, ~) = TI, after substituting the solution of problem (1.13) for w(H, $). Thus, for the model (1.5) introduced, as for the model [I], it is possible to use the apparatus of the theory of analytic functions and the theory of jets [8, 9]. The assumption that the flow is steady narrows the region of application of the results obtained above. They can be used in those cases in which preliminary heating of the reservoir has taken place and the heat carried by the sources is transmitted to the displaced fluid changing its properties. 2. Leaving aside the well-studied case of one-dimensional flows [I0, Ii] and neglecting, as in the previous section, conductive heat transfer, instead of the system (i.i), (1.2) we have the following:

V H = - H ( w , T ) w, w If

in (2.1)

we go o v e r t o t h e v a r i a b l e s

form: w2 _O_0

a~

=

~w

O0

OH

OH

H--,

=

w - -0- - ,i

-Or - + ~wVT=q(T) at

divw=0,

O~ H

t,

(2.1)

H, and $, we a r r i v e

aT

OF

-- --?wHa~

=q(r)'

at a system of the

O=O(w,T)

(2.2)

Correspondingly, for the model (1.5) we have

595

O0

Ow

w0r

O0

0~'

Ow

W0H

OT

0r

OT

0--7-~ w ~ = q(r)

(2.3)

From ( 2 . 3 ) i t i s i m m e d i a t e l y c l e a r , t h a t , as i n t h e s t e a d y c a s e , t h e t h e r m a l and h y d r o dynamic p r o b l e m s can be d e c o u p l e d . S e t t i n g $ = 0 and w = ( k / u ) G ( T ) on t h e p i l l a r bounda r y , i n a c c o r d a n c e w i t h t h e l a s t o f Eqs. ( 2 . 3 ) we a r r i v e a t t h e n o n l i n e a r d i f f e r e n t i a l equation

Ot

tain

~

G(T)

~ - ~ = q(T)

(2.4)

S o l v i n g i t f o r g i v e n G ( T ) , q ( T ) and t h e i n i t i a l c o n d i t i o n T ( 0 , H, ~) = T 1, we o b t h e t e m p e r a t u r e and f i l t e r velocity distributions on t h e p i l l a r boundary.

I f G(T) and q ( T ) a r e t a k e n f r o m ( 1 . 8 ) , the velocity have the form:

the solution

T=To+ t m e x p - b H + e x p - a t 2~ t + m exp - at Similarly,

f o r G(T) and q ( T ) d e t e r m i n e d

t - m e x p - bH

(2.4)

and t h e v a l u e

of

w 2 ~o~ t - m e x p - b H t+m exp-at f r o m Eqs.

T=T0 + "-2 8 t + m exp - at

o f Eq.

w2=~0 z

(1.10)

(2.5)

and ( 1 . 1 1 ) ,

we f i n d

l - m exp - bH t + m exp - a t

(2 6)

From ( 2 . 5 ) and ( 2 . 6 ) i t f o l l o w s t h a t f o r a p p a r e n t l y d i f f e r e n t , but asymptotically equivalent models the velocity distribution along the pillar b o u n d a r y o b e y s t h e same law. The h y d r o d y n a m i c p r o b l e m i n t h e u n s t e a d y c a s e d i f f e r s from that in the corresponding steady-state c a s e o n l y w i t h r e s p e c t t o t h e s u b s i d i a r y t e r m , s i n c e by v i r t u e o f ( 2 . 5 ) , ( 2 . 5 ) on t h e p i l l a r boundary the condition w i . l-mexp-bH Re F = In ~ 0 = - - m 2 t + m exp - at is satisfied~ In accordance with (1.14), in the parametric equation of the pillar boundary an additional cofactor appears. As a result we arrive at the following equation for determining the pillar boundary, written in dimensionless form: ~o

dg=Q~

state

(t+me-~gdh'

Qo

T h u s , f o r t h e G(T) and q ( T ) i n d i c a t e d case, that is to the problem (1.13).

=

Q~ 2~kl ,

z Z = - -l,

zH

h = - -Q

a b o v e , we c a n c o n f i n e

ourselves

(2.7)

to the steady-

3. I n o r d e r t o s o l v e t h e b o u n d a r y - v a l u e p r o b l e m ( 1 . 1 3 ) we map t h e h a l f - s t r i p the upper half-plane and u s e t h e K e l d y s h - - S e d o v f o r m u l a s [ 8 ] . Bearing we find

i n mind t h e b o u n d e d n e s s o f t h e v e l o c i t y F(;)=_i0o+20Oln

a

Y~+a+~;+i.+

Ya--i

at the points -i

nt

r~Ito

~ = --1,

](x)d_._~ -. R(x) (x--~)

R(;) ~

RC~)=~'-(g+a) (~+1), I(~)=iAm[t-(-x+Y~-t)"] aW

~ = --a,

onto

(3.1)

czl

From (3.1) and the conversion formula (1.14) we find the parametric equation of the pillar boundary, the coordinates determining the position of the characteristic points, and the expression relating the dynamic parameter Q0 and the parameters m, a, and n:

596

2

3

1 Z Fig. 1

=~u

z/L

expI(;)d;=Qo~ f,(;)d~, I ( ; ) = / ~ (;~; ) O( R('O ](~)d'~ ('~-;) -~

i

~

--i

(3.2)

!

ffl.

We will construct the asymptotic solution, taking as the small parameter the quantity m, which characterizes the deviation of the limiting gradient from the isothermal value, and assuming, to be specific, that n = i. Taking into account the relations

in [ l - m ( - ~ + Y ~ - i )

] ~ m ( - ~ + Y ~ - i ) =I(~) --|

m exp - I(~) ~ + ~ r ,

m r (;) = t + ~-77~ (~) J --G

1,(~)--'~ [--z~+Vl-'~ l a ~ + A --

l (~) d~

R('O

(~) "k"R(~)aretgA (--t)] -A(~)

(3.3)

A(~)= w ~ 1 ' '

and confining ourselves to the case @ o = z, we arrive at the following pillar boundary equation for a source--sink system [1--4]:

:

(a-t) Y~z--1

(3.4)

We used Eqs. (3.2)--(3.4) to c a l c u l a t e the p i l l a r boundary for the values of the parameter m = 0.05, 0.1 and a = 3. The shape of the boundary is shown in Fig. 1, where the broken curve represents the boundary in the isothermal case (m = 0). C l e a r l y , when fluid at a temperature higher than the reservoir temperature is injected, displacement leads to a decrease in the residual oil pillar. If G(T I) > G(T0), then the temperature on the supply contour will be lower than the reservoir temperature. Then m < 0 and, in accordance with (3.4), the area of the pillar will increase as compared with the isothermal value. LITERATURE CITED i. M. G. Alishaev, G. G. Vakhitov, M. M. Gekhtman, and I. F. Glumov, "Some aspects of the reservoir flow of Devonian oil at low temperatures," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3~ 166 (1966). 2. L. M. Kotlyar and E. V. Skvortsov, Two-Dimensional Steady-State Problems of InitialGradient Flow Through Porous Media [in Russian], Izd. Kazan. Univ., Kazan' (1978). 3. M. G. Bernadiner and V. M. Entov, Hydrodynamic Theory of Anomalous-Fluid Flow through Porous Media [in Russian], Nauka, Moscow (1975). 4. M. G. Alishaev, M. G. Bernadiner, and V. M. Entov, "Effect of limiting gradient on water-displaced oil losses,'! in: Problems of Nonlinear Flow Through Porous Media and Oil Recovery in Connection with the Exploitation of Oil and Gas Deposits [in Russian], Moscow (1972), p. 15.

597

5. V. M. Entov and M. G. Odishariya, "Some problems of determining the dimensions of limiting-equilibrium pillars in the displacement of viscoplastic oil by water," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3, 88 (1974). 6. M. G. Alishaev, M. D. Rozenberg, and E. V. Teslyuk, Nonisothermal Flow in the Exploitation of Oil Deposits [in Russian], Nedra, Moscow (1985). 7. G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Motion of Liquids and Gases in Porous Strata [in Russian], Nedra, Moscow (1984). 8. M. A. Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1973). 9. M. I. Gurevich, Theory of Jets in Ideal Fluids [in Russian], Nauka, Moscow (1979). i0. V. M. Entov, "Physicochemical hydrodynamics of processes in porous media (mathematical models of enhanced oil recovery techniques)," Preprint No. 161 [in Russian], Institute of Problems of Mechanics, USSR Academy of Sciences, Moscow (1980). ii. A. F. Zazovskii and K. M. Fedorov, "Displacement of oil by steam," Preprint No. 267 [in .Russian], Institute of Problems of Mechanics, USSR Academy of Sciences, Moscow

(1986).

STEADY-STATE WAVE FLOWS OF A WEAKLY CONDUCTING GAS IN TRANSVERSE ELECTROMAGNETIC FIELDS Yu. N. Gordeev and V. V. Murzenko

UDC 532.59:537.84

Gas flow in a MHD channel in transverse magnetic and electric fields is considered. The steady-state flows associated with the establishment of equilibrium between the hydraulic resistence and ponderomotive forces are investigated. The conditions of existence and the properties of such a steady-state flow regime are analyzed. The motion of a weakly conducting gas (Rem ~ i) in a MHD channel was investigated theoretically and experimentally in [1--3]. In [i] as a result of the numerical modeling of shock wave propagation in a weakly conducting gas the formation of a retardation shock was demonstrated. The formation of such a shock was analytically investigated in [2] by solving the self-similar problem. Experimental observations on the formation and development of a retardation shock wave in connection with the motion of a weakly conducting gas through a transverse magnetic field were reported in [3]. i. Formulation of the Problem We will consider the flow of a viscous conducting gas in constant transverse electric (E0) and magnetic (H0) fields (Ho=(0, H0, 0), E0=(0, 0, -E0); the channel walls are perpendicular to the y axis (Fig. i). When A 0 ~ 1 (A 0 = 4zqEoZ(H0c) -I, q is the conductivity of the gas, c is the velocity, and s the characteristic dimension of the flow region) the magnetic Reynolds number Re m ~ I; therefore the induced electromagnetic fields can be neglected. We will assume that the thermal conductivity of the gas is sufficiently high and the removal of heat through the walls sufficiently intense for Joule and viscous dissipation not to prevent the establishment of a uniform temperature distribution in the gas flow. If we average the flow characteristics over the channel cross section yz, then the equations of magnetohydrodynamics take the form [i]:

0-7p + p

a +

Ox

~-~

Ox ~"

(p~)=o,

P=c~p

+c-i<(jXH)e~>+<~=>,

(1.1)

j=~(Eo+c-~uXHo)

(1.2)

Here, P, p, and T are the pressure, density, and temperature of the gas averaged over the flow cross section; u is the gas velocity component along the x axis; cT is Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 123-129, July-August, 1990. Original article submitted May 25, 1989.

598

0015-4628/90/2504-0598512.50 9 1991 Plenum Publishing Corporation