Apphed Mathematics and Mechamcs (Enghsh Edmon, Vol 12, No 3, Mar 1991)

THE T R A N S I E N T

Pubhshed by SUT, Shanghm, China

TWO-DIMENSIONAL

DOUBLE

FLOW THROUGH

POROUS MEDIA*

Lm Cl-qun ( 3 0 ] ~ ) (Research lnstttute of Porous Flow of Mechamcs, Stmca Academza Langfang)

(Received May

1990)

Abstract This paper presents the analytwal soluttons m Laplace domam for two-dtmen~tonal nonsteadv flo)i of shghtly compressible hqutd m porous medta with double porostty by using the methods of mtegral transforms and vartables separation The effects of the ratto of storatlvltws co mterporostty flow parameter ~ on the pressure behaviors for a vertwally fractured ~ ell )ttth mfintte conducttwty are mvesttgated by usmg the method of numerwal mverston The new log-log dtagnosts graph of the pressures ts gtven and analysed

Key words

I.

double porous me&a, two-&menslonal porous flow, transient pressure, nurnerlcal inversion

Introduction

Hydrofracturing technology through producing well can create a symmetrical vertical fracture with height h and length 2 x s m naturally fissured carbonate reservoir to improve oil and gas recovery The flows toward the fractured well with mfimte conductwity vertical plane are elhptlc, and their flow nets are composed ofconfocal ellipses for equal pressure and confocal hyperbolas for stream hnes which are orthogonal to each other Therefore it Is s~gmficantly to study the two-&mens~onal transient flow of shghtly compressible liquid through double porous media under the assumption of unsteady-state for lnterporosity flow between the matrix blocks and fracture systems, th~s paper presents the new solutions for the twodimensional transient flow m double porous me&a

II.

M a t h e m a t i c a l Model o f T w o - D i m e n s i o n a l T r a n s i e n t F l o w

The vertically fractured well is averaged m the center of a closed rectangular reservlor with double porosity behaviour When ~t works with constant flow rate the translent two-d~mensmnal flow wdi occur under the assumption of unsteady state for mterporoslty flow between the matrix and fracture systems, the governing equatmns m d~menslonless form arett 21

Dedicated to the Tenth Anm~ersary and One Hundred Numbers of AMM (HI) 265

266

Llu Cl-qun

OZPfo §

OZPto Ou~

Opro . Op,.o [ --O.)'5~--." - - A ~ ato Ozo I~o= 0

~

O2p.,__o= 1-co

Op,.o

(2.1) (2 2)

The uniform flux conditions for the constant rate of vertically fractured well are given by

~176 ago lyv=0-- [

~v 2x,.1

(o<~xv<~Xto)

0

(xto
(2 3)

The averaged pressure for fractured plane

p~,o(to) ---- xs---~j 1 [~/o ~

Pso(xo .0 .to)dxo

(2.4)

The Initial and outer boundary conditions for closed rectangular reservoir

(2 5)

p~o(xo,yo,o)=l,.,o(zo,o)=o aPtv( +_1 ,yo ,to) Oxo =0

(2.6)

op, o( xo, + v . o , t o ) =0

(2.7)

0yo

The Interface conditmn for naturally fissured reservoir

( z. 8)

p,~o( o,to)=p,~( xo,u~,to) The symmetrical condmon of layered matrix systems

_ OP.,o(1,to) azv -----0

(2.9)

where dlmensmnless coordinates x ~o=---~,

_

uo-

U xso----'%, ,,---7'

~/eo -

- ~ -Ue7 '

dimensionless pressure drop Pjo = '

e~rksht(P,PD #Q~

(j----/,re,W),

dimensionless time k ! . lt

t o = t~(r162 + r the ratio of storativltles

(•.fc! co =

4'sct + 4,,.c,.

The Transient Two-D~mensmnal Flow through Double Porous Medm

267

the lnterporos~ty-flow parameter for unsteady state km

x',

=-~-, "h.--g--ff., Eqs (2 2)-21 9) const~sUtute the mathemahcal model for two-dlmensmnal transient flow of shghtly compressible hqmd through double porous medm, which can be solved by using the methods of Laplace transform and variable separaUon IlL

Solutions in Laplace Transform Domain

In regard to mmal condmon (2 5), the Laplace transforms of Eqs (2 1)-(2 9) are

a~.,. i

Oz~s~ 4- 8ZPs~

(3.1) (3.2)

dz~

a~i. I

_-,

igbf~ I u,----o

= 2:Qo

I s

(o~
(3.3)

0 1 rSi~ ~w.(s)=-~-~$~J, ~,.(x.,O,s)dx.

a~..(+1,u.,s) axo

#pt~(:r

___y.~,

~., ~(0 , s ) = ~ t~

(3.4)

--o

(3.s)

~-=o ~

(3.6)

,u. ,s)

(3. z)

atD.o(z,s) =0

(3.8)

0go

where

~jv~? ffjD'eIp[~tD']dtlj

(jffi/,m,W)

Subjected to boundary condmons (3 7) and (3 8), the solution of ordinary dlfferenual Eq (3 2) is

,.o---,,~

eh(J--~s-)

(3.9)

Differentiating Eq (3 9), subsututmg ~t mto partml dlfferentml Eq (3 I) yields twod~mensmnal Po~sson equauon

(3.1o)

268

Lm Cl-qun

where

8

c h ( - ) , th( ) are separately hyperbohc cosine and tangent functmns Whens~ o o , f ( s ) ~ co~s-->O,f(s)--> LFrom this we may know the pressure behavior at early period hkcs it m naturally fracture system, and at late period hkcs it m homogeneous porous media In regard to boundary conditions (3 5) and (3 6) of closed rectangular reservoir, by usmg the method of variable separation the elementary solution of Potsson equatton Is

ch(q.(Ve~- Y~))"cos(n~rxo)

(3

11)

where q . = [ s . / ( s ) + n Z~rz'] ",}-

Expanding the dlscontmuous boundary condztlon (3 3) into Fourier series

al~,o(xo,0,s)_ o~yv

~ [ 1+ ~

2:s

2

u-I

n'R'~.f D

sm(nrcxtv).cos(mrxv)

]

(3.12)

From Eqs*(3 2) and (3 12) the Imaginary functmn for pressure ofverucaily fractured well wlth mfimte conduct~wty m double porous media can be obtained as

II,D(xo , y D , s ) = - ~ y -

[ ch(q0(y.~-yo)) qo.sh(qoyo~)

+ n~---L--2 sln(n~xs~) cos(n~Zo) -I n~X,f D I

..

~

chq,(y.b--yD)

q..sh(q.y.,,) ]

(3 13)

I-eth(q0yeo)

s-I

eth(qnye~) ] q~

nz~gX~ D

(3 14)

where cth( ) is a hyperbolic cotangent funcuon When s ) ) l , q , ~ - , ~ / -cos - , eth(q~yeD)~J , Eq (3 14) reduces to ~r

A

(3.15)

where

A =1 + 2 ~ n-1

smZ(n~xl~ n2~g'~ D

To analytically mvert Eq (3 15) the formula for pressure of fractured well at early period zs

(3.16)

The Transient Two-D~mensionai Flow through Double Porous Medm When s <<1 , q o ~ , , / - ~ ' ,

q.~n~r

,

269

Eq (3 14) reduces to

(3.J7) where

To invert Eq (3 17) the formula for pressure of fractured well at late period is

(a 18) IV

Numemcal

Inversion

of Laplace

Transforms

Based on the Laplace transform solution (3 14) for fractured well in closed rectangular reservoir with double porosity, the data of pressure behaviors are calculated by numerical Inversion methodt31 The log-log pre~sure-time graph is plotted in Fig l DII~

/

t01

,1.-----l0"~

IQItO~

--- ' ,

,

*

10-6 10-~ 10-" lf~~

-

102

f

.....

104" &

Fxg 1 lnfinle conductivity fracture m dual-porosity system, transient interporostty flow The pressure behavior of Fig 1 may be divided into three flow stages early hnear flow stage characterized by Eq (3 16), middle transition lnterporosity flow stage, late pseudo-radial flow stage depicted by Eq (3 18) The shape and form of pressure curve is determined by the fracture l e n g t h x and parameters of double porous media co and 2 c = compressibility, Pa-1 , h = formation thickness, m, k = permeability, m 2, p = pressure, Pa, PN = initial pressure, Pa, Q = f l o w rate, m~/s,

Nomenclature s = Laplace space variable, t = time, s, x , y , z = coordinates, u =v~scoslty. Pa s, = porosity,

Subscripts f = related to fissures, m = related to matrix,

D = d~mensmnless, W = welibole

270

Lm Cl-qun

References [ 1 ] Grmgarten, A C et al, Unsteady state pressure dlstrlbutaons created by a well with a single mfimte conductivity vertical fracture, Soc Pet Eng J , Aug (1971), 3 1 7 - 3 6 4 [ 2 ] Houxze, O P et al, Pressure transient response of an mfimte-conductlvJty vertical-porosity behavior, SPE Formatwn Evaluation Sept (1988), 510 - 518 [ 3 ] Stehfest, H, Numerical inversion of Laplace transforms, Commumcatlons of the A CM, Jan (1974), 47 - 49