Localization of solutions of the equations of filtration in poroelastic medium Текст научной статьи по специальности «Математика»

УДК 517.9

Localization of Solutions of the Equations of Filtration in Poroelastic Medium

Margarita A. Tokareva*

Faculty of Mathematics and Information Technologies

Altai State University Lenina, 61, Barnaul, 656049

Russia

Received 22.06.2015, received in revised form 10.08.2015, accepted 20.09.2015 A system of equations of 1D non-stationary fluid motion in poroelastic medium is considered. Localization of solutions of the equations has been established by the integral energy estimates method.

Keywords: filtration, Darcy's law, poroelasticity, localization, metastable localization. DOI: 10.17516/1997-1397-2015-8-4-467-477

1. Problem Statement. The Main Results

A quasi-linear system of equations of composite type is considered [1-3]:

^+d((1 - «<->=0 I<■■>■'>=»•

ф(vf- vs) = -к(ф)^ 'dpx - pf g) '

dvs a <A.\(dPe , dPe\

— = -Мф){Ж + vs

dx

dptot

Ptotg,

dx

Ptot = «Pf + (1 - «>Ps; Pe = Ptot - Pf; Ptot = (1 - «>Ps + «Pf .

This quasi-linear system of equations describes 1D non-stationary isothermal motion of fluid in poroelastic medium. The laws of conservation of mass for each phase, Darcy's law for fluid phase, the rheological Maxwell law and the equation of conservation of momentum for the system describe this process. Here ps,pf ,vs,vf are, respectively, real density and velocity of solid and fluid phases, « is the porosity, Pf ,Ps are, respectively, pressures of the fluid and solid phases; Pe is the effective pressure, Ptot is the total pressure, ptot is the density of the two-phase medium, g is the density of the mass forces, k(«> is the coefficient of filtration, ¡3t(«> is the coefficient of bulk compressibility (specified function). The problem is written in the Eulerian coordinates x, t. The real density of the fluid and solid particles pf, ps are assumed constant. The unknown quantities are vs, vf, Pf, Ps.

Local (with respect to time) solvability of the initial-boundary value problem for the system of equations under consideration has been established in [4], a self-similar solution has been

* [email protected] (c Siberian Federal University. All rights reserved

found in [5]. Numerical results for this system of equations are given in [1,2]. In these studies we

use Euler variables, additional assumptions about smallness of the speed of solid phase, and the

k

following relations between the functional parameters of the problem: k($) = — $n, ¡3t($) = Ps$b,

where n, b are positive environment parameters. In this paper a complete system of equations of filtration in a deformable medium is considered. This system of equations can be reduced to a degenerate parabolic equation using transition to Lagrange variables with respect to the speed of the solid phase. To this equation we apply the well-known technique for proving finiteness of the propagation speed of disturbances.

Rewrite the original system in Lagrange variables, following [6]. Suppose that x = x(r,x,t) is a solution of the Cauchy problem

dx

— = Vs(x,T), x \T=t= x. We set x = x(0; x,t) and take x and t for the new variables. Then [6] 1 — $(x,t) =

dx

(1 — 4>°{x)) J(x,t), where J(x,t) = —(x,t) is the Jacobian of the transformation, $0(x) = $\t=0.

dx

The system of equations in the new variables has the form

d(l — $) + (1 — dVs =0

dt +1 — $0 dx '

d$ , (1 — $) d x (1 — $) d$ at +T—W ex ($vf ) = VsT—W dx,

(1 — $) dp f

A—)dpf 99\

Since

$ (v s — v f) = —k(4>), 1 — dx

j1 — $) dVs r,(1 s9pe

T—$p dx =

(1 — $) dptot = .

1 — $0 dx = pg '

d$ d x x8vs

ox ox ox

it follows that the continuity equation for liquid phase can be reduced to the form

1 d$ 1 d \ 1 -dvs

(7—$yTt + ex \$(vf — vs)) + $dx =

Using the continuity equation for the solid phase, we find that

*(^ ) + d —^=0

Finally, passing from (x, t) to the mass Lagrangian variables (y, t) by the rule

,-x

(1 — $0(x))dx = dy, y(x) = (1 — $0(n)) dn € [0,1],

0

and formally replacing y by x, we obtain

i (t-«) + dx (*(v>- v->>=0 (1)

^+(1 - «>2 ds=»■ (2)

«(vs - Vf > = k(«> ((1 - «> dx - Pfg) , (3)

(r - *> dxs = « t • (4)

(1 - «> = -ptotg- (5)

Introduce a function G(«> defined by the equation dG(« = 1- . Therefore, from

d« 3t(«>(1 - «)

(2) and (4), we obtain

dPe _ dG(«>

dt dt '

and hence

Pe = -G(«) + Go + p0, Go = G(«°), «\t=0 = pe\t=0 = p0e ■ (6)

Therefore, from (1) and (3), we have

Kéi) = K™«1 - *)If - P'»)■

Taking into account the equality Pf = Ptot - Pe and equations (5), (6) we have

d (i-y=¿(H g) - ¿ ^ - *)» - *»>)).

Further on, we introduce a new function s = «/(1 — «) instead of « G [0,1), and assume [2] that

k

k(4>) = -r, ßt(4>) =

where k is the permeability, n is the dynamic viscosity of the fluid, ß^ is the coefficient of bulk compressibility of solid phase, b,n are positive environment parameters (in what follows it is assumed that 0 < n + b — 2, 0 < n — b < 2). Then the equation for s can be expressed as

ds d \ ds \ df (s) (7) dt = axAd(s) dx +-aT> (7)

it is assumed that there is a constant M > 0 such that we have the following estimates

kk

0 < s < M< ro, ~^sn-b(1 + M)b-n-2 < d(s> g > 0,

k

\f (s)| < -sng (Ps + (1 + 2M>pf ) . ¡i

The main result of this paper can be formulated as follows: let s(x,t> be a weak solution

of (7) in Kpo (x0> x (0, ro), Kpo (x0> = {(x,x0> : \x - x0\ < p0} such that s0(x> = s(x, 0) = 0

in Kpo(x0). Then there exist T > 0 and p(t) € (0,p0) such that s(x,t) = 0 for all t < T and x € Kp(x0). Under additional assumptions on the character of vanishing of s0(x) it is proved that s(x,t) = 0 in KP0 (x0). Questions of the existence of the corresponding solution are not considered here. The local energy method developed in the papers [7, 8] is used for the proof .

On Q and QT we consider several function spaces following the notation from [9]. Suppose that || • ||q_n is the norm on the Lebesgue space Lq(Q), q € [1, to]. For brevity, let || • ||q = || • ||q_n,

o

|| • || = || • ||2,n . We also use the space C of infinitely differentiable functions with compact support in Q, and the Sobolev spaces Wp(Q), where l is a natural number and p € [1, to], with i

norms ||f||wl(n) = E ^D^fHp,n.

x

m=0

Definition 1. By a weak solution of the equation (7) with initial condition s0(x) we mean a non-negative bounded measurable function s(x,t) (0 ^ s(x,t) ^ M) on Q x (0, œ), if VT > 0 and any open subset Q1 C R1 the following conditions are fulfilled

s G Lo(0,T,W- (Q)), l(sn-b+1) G L2[(0,T) x fiij, (8)

lim sdx = / sodx, (9)

1^0jQ jQ

and y^(x,t) GC0((0,T) x Q-t)

r0r\) Osdv df (s) ■

Jo In r(s)dXdX — "â^.

We introduce the notation

dxdt = f í Sd^dxdt + Í s(x, 0) ^(x, 0) dx. (10)

Jo Jn dt Jn

A(p,t) = [ s2(x, t)dx, B(p,t) = [ sn-b ( ^^ dx,

JkJ x o) JkJ x o) \dxJ

s (x,t)dx, B(p,t) = s

>Kp(x0) JKp(xo)

and without loss of generality we assume that x0 = 0.

Lemma. Suppose that (8), (9) are fulfilled. Then for s(p,t) we have the estimates

s° (p,t) < CA — (p,t)[B1 (p,t)+ p-s A1 (p,t)]e, i = 1,2, (11)

where

a = n - b + 1 > 0, e = s = 1. 2 2 ' 2 + r' er

If i = 1 then r € (1, 2), 0 <n - b< 2,

Ci = CM-2-m.ax(a,M—),

and if i = 2, n — b = 2,

4

n — b + 2

C is a positive constant independent of the radius p.

= 1, C2 = C max(a, 1),

Proof follows [10]. For all u G Wiq(Kp(0)> we have the estimate [11]

U(p)\ < C ■ (Kll,,*,«,) + p~s\U\r,KP(0))e hwlj^y (12)

e =---, S = 1, q > 1, 1 < r < œ.

q — r + qr er

Take in (12) u = s" and q = 2, then

(2 1 1\ e 1-9

< Lof-*(M x '+p"(L dx) j ( U"d*)'■(13)

Let us strengthen the right-hand side of (13). If 0 < n — b < 2, then

4

sr" = s2 sr"-2 < Mr"-2 s2, r € ( 1,

' 1 'n — b + 2

fi,—4—V

V n—b+2J

If n — b = 2, then take r = 4/(n — b + 2) = 1 in (13), and, given that sr" = s2 sr"-2 < s2, we deduce

' , 2 \i a I / sn-b I ds ) dx) +

>KP(0)

s" Pt) < CM — im ^d^

) e

If s2 dx) \JKp(0) J

-s \ s2 dx) I / s2 dx

'Kp(O) J J \jKp(0)

if 0 < n — b < 2, 1 < r < 2, and if n — b = 2, r = 1, then

(2 1 1\ 9 Ç--V)

-(L>dv + pxLsdiï) (L)sdx) ' •

that is, we come to (11). □

Theorem 1. Assume that the conditions (8)-(10) are fulfilled and additionally t G [0, T], T ^ T*, where

T* < min(^M2-b-nF-2 (min^L,-^ (1 + M)b-n-2 - 2)) j >

( i 1+26 1+26\ (2 - ty^Pcp 1-29/ \

- P (25 + 1)4kK2 W (p0,t))

i = 1, 2.

If s(x,t) is a weak solution of (7) and s0(x) = 0 in KPo (x0), 0 < p0 < dist(x0,dG), then s(x,t) = 0 almost everywhere in Kpi(t)(x0), 0 ^ t ^ T ^ T*. Moreover

i

)1+2i

,

where if 0 < n — b < 2, then

L = 4C2 ■ Q(r), r € (1, 2),

and if n — b = 2, then

L = 4C2 ■ Q(r), r =-= 1.

n — b + 2

In both cases

29

>M= sup<t £ B(po, s)ds, Q(r) = ^ilKl Po + T1M 2(s-l)ps0-1) ,

w(P0,t) = sup B(po, s)ds, Q(r)

Ít^tj0 2U — 1 V 2

Ki = Ci

2P0S + T1 po- M2(—)

, i = 1, 2, Fi = ^ (pa + (1 + 2M)pf

and constants C\ and C2 are determined in (11).

Theorem 2. Assume that in addition to the conditions of Theorem 1 we have

t , ) 2+r Ct C / \ 2-r

/ B(p,T )dT < Co, sz0(x)dx < K3[ p - po) , yp G (po,R). (14)

Jo JKp(x0) \ )

Then there exists T0 depending on the data of the problem such that s(x,t) = 0 for almost all x G KP0 (x0) and t G [0, To].

9

2. Finite Propagation Speed of Disturbances

1 pt+n

Proof of Theorem 1. Suppose in (10) p(x,t) = ^n(\x — x0\)£k(t) — Tl(s(x,T))dr, where

h G (0,T — t), t

Tl(s) = min(\s\, l)sign s,

1, r G \0,p — 1/n], <fin(r) = { n(p — r), r G [p — 1/n,p],

0, r G [p,po],

1, r G [0,t — 1/k], Ck(r)= { k(t — r), r G [t — 1/k,t],

0, r G [t,T*].

We have

'0 JKP0 (xo)

d(s) ^k(r) ¿(Vn h J^ Tl(s(x,^))d^ — <f(x,T) d ( 1 't+h

dx

dx dr =

í í t( T (s(x,^))d'^)dxdr + Í s(x, 0) p(x, 0) dx. (15)

J0 JKpn (xo) dT \ hJ t ) JKpn (xo)

to jKP0 (x0) dT V hj t J JKp0 (x0)

Taking into account the Lebesgue theorem with k ^ x> we get

lim

0 Jkp0 (xo) "h

Of 1 ft+n s^n h J Ti(s(x,^))d^ ) dxdT =

0

lim [ [ s<pn— [ T(s(x,^))d^dxdT+

k^J0 Jkpo (xo) dT h Jt

+ lim [ [ s^nCkUti(s(x,T + h)) — Ti(s(x,T))) dxdT

k^J0 Jkpo (xo) h\ J

CO

CO

,t , i ,t+h — lim k s*n — Ti(s(x,^))d^ dx dr+

Jt-i/UKp (xo) h Jt

+ II s^n T\Ti(s(x,r + h)) — T (s(x,r))\dx dr =

Jo JKP0 (xo) h\ )

I s^n T¡ T(s(x,^))d^ dx +1 I s^n T\Ti(s(x,r + h)) — Ti(s(x,r))\ dx dr

JKrjn (xo) h J t J0 JKrjn (xo) h\ J

'Kp0 (xo) hJ t Jo JKp0 (xo)

and, therefore, at h ^ 0 the identity (15) can be written as

t

0 Kpo(xo)

d(s)

ds d*n dx dx

Tl + (ds) di*n T — dM)

y dx n dx dx J

dx dr

¡ s^nTi dx + / so(x)pnTt(s(x, 0)) dx.

■'KPo (xo) KPo (xo)

Therefore, after passing to the limit as l ^ to, we obtain

0 J Kpo (xo)

'M,dsd¥nl(M ,(ds\ 2 df (s)\

sd(s)ax^x + lvd(s\dx) *n — xx)'fns

f S2*n dx +f sl(x)*n dx.

JKnn (xo) JKnn (xo)

dx dr

'KPo (xoo) jKPo (xo)

Finally, passing to the limit as n ^ to, we have

t

lim sd(s)

n^J 0 J Kpo (xo)

ds d*n dx dx

dx dr ■

ftf ds ft ds

lim n sd(s)— dx dr = — sd(s)— dr.

n^ Jo J dx Jo dx

p-l/n<\x-xo\<p

Therefore, we arrive at the equality(xo = 0)

rp ^ fP

/ s2dx + / /

'o Jo Jo

u wds N2 df

d(s)( dx) — sdx

r P rt ds

dx dr = so dx + sd(s)—(p,r) ds] dr.

Jo Jo dx

Let

a(p,t) = sup A(p,r).

o<r <t

It follows from (16) that

kk a(p,t) + — (1 + M)b-n-2w(p,t) < —A + I2,

where

s(p, T)

n-b+l

o

ds

dx (p,T)

dT,

I2 = f ÍP s(p,T)

oo

df (s)

dx

dxdr.

(16)

(17)

t

l

Applying the Holder inequality and (11), we obtain

Ct f)o

h = J '(p,T)n-b+1^(p, s)ds Í

< I °n-"(p-T>{ S <p>T>) *)'( l sn-"+2(p-T)

t 1 t — t 1-0 ai(p, t) ^ sn-b+2 (p, T)) 2 < C^^ (b 1 (p, T) + p- A1 (p, T)) 2 d^ ( ^ Ar (p, T)dt) 2 <

< Ci(( f B(p,T )d^j 2 + p-s( f A22 (p,T d) t— a — (p,t) <

I

00

r 1-0

< c1p-st

12 ^w1 (p,t) a1-0 (p,t)ps + T1 a1 (p,t^ .

But p < po, and moreover

1 1-e i i

2w2 a re =2w2 a2 < a + w, a re (p,t) < a(p,t) a5-1 (po,t) < a(p,t) po5-1 M2(5-1), 5> 1.

Consequently,

( 5 \8

ai(p,t) < Cp-5t^( (a + w) + T2apSo-1M20(5-1)\ < < Cp-51^ (a + w)^2po + T2pSo-1M2(5-1^ ".

Correspondingly,

10 id Ii < Kit^ p- [a(p,t)+ w(p,t)]9( dp) ,

' 3w\ 2 dp)

where

n , , , < 2„ -J9

i = 1, 2.

1 ] 9

2p0S + T2 p0S-1 M2(s-1) ,

Now we estimate I2. We have

df

dx

ds dx

Fi,

where

Fi = —g(p a + (1 + 2M) pf).

Therefore,

/ />t fP f c)e\ 2 \2/ />t fP \2

I2 < FA — sn-bdxdA sn+bdxdA <

\J0 J0 \dxJ J \J0 J0 J

1 n+b-2 ( ÍP o \ 2 n + b-2 111 1 n+b-2 1 , _

< F1w2 M^^ Í j j s2dxdTj < F1M^^ w2 a212 < F1Mt2 (a + w).

Consequently, (17) takes the form

k k mm(1,— (1 + M)b-n-2)(a(p,t) + w(p,t)) < —h + I2 < р/Зф !лрф

k

рЗф

ídw\

\дР)

1 b+n

< —гKiP(a(p,t)+ w(p,t))°[ — ) +-FMt2 (a + w), i = 1, 2.

Now choose t in such a way that

1F1M^12 < mn(1, (1 + M)b-n-2) - 1.

2 рЗф 2

Therefore,

1 , , k i—ex, sa f dw \ 2

2 + W) < Жф Кг^ P-S (a + W)\dp

f

Accordingly,

p0w1-0 < (a + w)1-0p0 < 2 — Kt— —

рЗф

i—eí dw \ 2

Пw -

and hence

20 2(1-0) ^ is*. 1-0 dw

p w ( ) < K* t ——, dP

(18)

к

where K* = 4(—Ki)2, i = 1,2. fфф

Integrating (18) by p from p\ to p0, we find that (1 < r < 2)

1

225 + 1

(f0+2S - p\+2°) < Kit1-0(w20-1(po,t) - w20-1(pi,t))

1

2в- 1'

Therefore, we have

„1+2S „1+2S , 25 + 1 TS* +1-0,20-1/ , 25 + 1 TS* +1-0„,,20 — 1/ i\ (19)

Pi - Po + 2в- 1 Kit w ( ) ^ 226—TKit w (P1-t)■ (19)

26- 1

Choosing t such that the equality

pI*" = p0+M - K* t- w29-q(P0,t),

holds, we obtain that w(p,t) = 0 for all p < pq, i.e. s(x,t) = 0 almost everywhere in Kp(0) for p < pq and

0 < t < min[ 4M2-b-nFi-^-ßr (1 + M)b~n~2

k

рЗф1

((a

1+2S _ 01+2&\ 26 - 1 w1-20 ( t) P >{25 + 1)K*w (P0, )

f)

2

2

3. Metastable Localization of Solutions

Proof of Theorem 2. Following the initial reasoning in Theorem 1 and formally replacing in (15) p by R, for all p € (0, R) we deduce the initial equality (16). According to the conditions of the theorem, s0(x) = 0 in the ball KP0(x0). Therefore, the first integral on the right hand side of the equality (16) (of s0 ) is in fact (for p € (p0 ,R)) taken over the interval (p0,p), and hence the estimate (14) is valid. Other terms of the right hand side of (16) are estimated in the same way as in Theorem 1. So, instead of (19) we have for all t <T

1 k 1-8 ( dw\ 2 ( \ 26-1

2(a + w) * —Ki(a + w)e p-S t— {dp) + K3[ p - poj .

1 . . k

2(a + W) * ^

The first term of the right hand side is estimated by using Young's inequality

1. . , , ,1 - 9 i _ (dw) ( k \ — / )

2 + W) * £W9(a + W) + ^12 p1-9 UJ Ki) + K\p - po)

Choosing el/e = -1 > 0, we obtain 49

i i i 1, , 1 - 9 i _ s (dw\ 2(1-o) ( k „ \i-° ( \ 20-i

4(a+w * 12 pi-0 \jp) KV + K\p -p0) ■

Using the inequality ap + wp ^ (a + w)p, 0 < p < 1, a,w ^ 0, we have

2 2(i-o)

w2(1-°) * (4(1 - 9))2(1-d)e-2 p-2 K^ t— ^ + (4K3)2(—>(p - p^"1 . The result is a special case of the inequality

w * CtKw'p + c(p - p^y , 0 < a < 1, p € [po,R], (20)

studied in [12]. As shown in the cited paper, (20) implies the equality w(p0,t) = 0 for all t € [0,to], where t0 is calculated from the relation

to = ((1 - a) 2—, R/CC-)2/a, C = (4(1 - 9))2(1-d)e-2 p-2S (J^K^ ,j = 1, 2. Therefore s(x,t) = 0 for almost all x € Kp0(x0) and t € [0, T0], T0 = mm(t0,T*).

References

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[2] J.A.D.Connolly, Y.Y.Podladchikov, Compaction-driven fluid flow in viscoelastic rock, Geodin. Acta, 11(1998), 55-84.

[3] J.Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.

[4] A.A.Papin, M.A.Tokareva, The Problem of Motion of a Compressible Fluid in a Deformable Rock, Izvestiya Altaiskogo Gosudarstvennogo Universiteta, 72(2011), no. 1/2, 36-43 (in Russian).

[5] A.A.Papin, M.A.Tokareva, Dynamics of melting deformable snow-ice cover, Vestnik Novosi-birskogo Gosudarstvennogo Universiteta. Seriya Matematika, Mekhanika, Informatika, 12(2012), no. 4, 107-113 in (Russian).

[6] S.N.Antontsev, A.V.Kazhikhov, V.N.Monakhov, Boundary-Value Problems of the Mechanics of Inhomogeneous Fluids, Nauka, Sibirskoe Otdelenie, Novosibirsk, 1983 (in Russian).

[7] S.N.Antontsev, J.I.Diaz, S.Shmarev, Energy Methods for Free Boundary Problems. Applications to Nonlinear PDEs and Fluid Mechanics, Progress in Nonlinear Differential Equations and Their Applications, Washington D.C., 2002, 331.

[8] A. Favini, G. Marinoschi, Degenerate Nonlinear Diffusion Equations, Springer, 2012, 143.

[9] O.A.Ladyzhenskaya, V.A.Solonnikov, N.N.Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Moscow, Nauka, 1967 (in Russian).

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[11] S.N.Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akademiya Nauk SSSR Sibirskoe Otdelenie Instituta Gidrodinamiki, Novosibirsk, 1986 (in Russian).

[12] S.N.Antontsev, Metastable localization of solutions of degenerate parabolic equations of general form. Dinamika Sploshnoy Sredy, Novosibirsk, 83(1987), 138 -144 (in Russian).

Локализация решений уравнений фильтрации в пороупругой среде

Маргарита А. Токарева

В работе рассматривается система уравнений одномерного нестационарного движения жидкости в пороупругой среде. Методом интегральных энергетических оценок устанавливается локализация решений уравнений.

Ключевые слова: фильтрация, закон Дарси, пороупругость, локализация, метастабильная локализация.