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Productivity Prediction Method for Fractured Wells inExtra- ow Permeability Oil Reservoirs

ZEN Ya-q n HUAN We NIU a -yunOil Gas Tec ology Research InstituteChangqing Oil eld Company, i' n Chin

E-mail: zyq2_cq@petrochina.com.cn

Abstract-In ow pe fo m n e el ion hip (IPR) i hefound ion of op im l de igning fo ope ion p me e of oilwell . p edi ion me hod i p opo ed fo ev lu ing he in owpe fo m n e el ion hip of f u ed well in ex lowpe me ili y defo med medi oil e e voi . Spe i lly, nellip i ow m hem i l model i on u ed in he i u ion

h he well in ow pe fo m n e of f u ed well e dividedin o wo p in whi h he eep ge ow f om oil l ye goe o he

f u e nd eep ge ow f om f u e he d o o om hole.ddi ion lly, he h nge of he pe me ili y nd uid vi o i y

wi h p e u e e ken in o oun . The fe i ili y of he oveme hod i hown y n ppli ion ex mple.

K r s-p odu ivi y p edi ion, ex low pe me ili yoil e e voi , f u ed well, ellip i ow m hem i l model

I NTRODUCT ONFract ring is used commonly n extra-low permeability

reservoir with low production, while he prediction me od ofin ow perfo nce relationship for act red wells is t eimportant basis for actur ng techniques parametersoptimization and production par meters optimization of oil

well.The reasonable me od to predict the IPR of the extra-low

permeability reservo should not only consider the seepage of non-line r i the deformed media oil layer, but also the ow in the acture. Up to now, production prediction for act r g have been st died by Raymond I. R [1] , Poe B. Jr.[2], et al.But none of em has considered e low velocity non-da cy percolation the characteristic of the ex a-low permeability reservoir. Song Fuquan[3], W ng Yuying nd others havestudied and analysis e production of the extra-low permeability reservoir by di erent me ods, but their studiesare assumed that the reservoir was homogeneous medium and

actured well did not enter nto calc lations.

A prediction me od of in ow performance relationship foract red wells n extra-low permeability oil reservo s is given based on an overall consideration of the seepage ow of nonl ear in the deformed media oil layer nd e seepage ow the acture.

This k is supp te by N ti n l N tu e Science F un ti nun e G nt 6087 021

978-1-61284-491-6 11 $26.00 ©2011 IEEE 382

ZHAO Zh -cheng YAN L - ngCollege of Petroleum EngineeringYangtzeu University, Jingzhou , ChinaE-mail: zhaozhicheng@263.sina.com

II. ER AT ON OF FUNDAMENTAL EQUAT ONS FORPRODUCT TY PR D CT ON

It is assumed that formation uid ow om oil layer intoactures homogeneously rst and then ow into e bot om of

the well during e in ow process of the actured well. The pressure distribution n reservo outside of e act re c n be treated as a family of conjugate elliptic isobaric, whose focusare the endpoints of t e acture. Let a nd b be e half-lengand half-wid of acture, respectively, r be the dist ce om

acture, e pressure at R the radius of supply boundary be Pr ,and e average pressure in acture be Pf

Flow om reser oir into actureshen the pressure gradient higher than e reshold

pressure gradient in extra-low permeability reservoir, e uid beg n to ow, e ow equation c n be written as ):

v=�d p (1_ )=�( d p G (1d r dP d rd rhere G is t reshold pressure gradient.

Considering e reservoir pe eability Kand uid viscositychange with pressure, hey can be expressed as follows,

respectively ):

Then je- (p, p)

(2

(3

Considering steady seepage, the seepage veloci c n beexpressed as:

V= -- : --h7[ b

here Q is well production, h is reservoir ic ess.

Combining Equations (2 �(4 ,Eq. can be rewrit en as:

(4

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T k g e boundary conditions: R, P=Pr; 0, P=Pf,integral ofEq.5 can be obta ed:

here

1Ao =" n( a+b)1B o = -[(a +b) (a+b)l n(a+b) ]41 1 2Co= [ (a+b) (a+b)(a+b)8

+(a+b) 2 n(a+b) ]D o= _ [ (a+b)3i(a+b) (a+b 16

+ (a+b)2(a+b) (a+b)3n(a+b) ]A = - In(a+b+ R)

1B =4 [(a+b+ R) (a+b)ln(a+b+ R) ]1 1C1= [-(a+b+ R)2(a+b)(a+b+ R)8

+(a+b)2 n(a+b + R) ]D = [ (a+b+ R)3 (a+b)(a+b+ R) 26

(6

+ (a+b)2(a+b +R)(a+b)3 n(a+b +R) ]B The Flow om actures into the bottom of the well

The pressure at any point x in acture is Po,and e owfollow e Darcy theorem, then:

K dpoV =pd (7

The relationship of permeability of acture K with pressurec n be expressed as = K fe-a k (P r -P o) , e rela t io ns hip be t wee n uid viscosity and pressure is s milar to Eq.3.

Then! = Kf e-a(P r-po) , where Kf is t he initial fac ture o permeability, a = a -a

Supposing the end of crack closed, the production of ecrack initiat g terminal (i.e well bottom is Q, production

383

distribution along the crack is Q � ,therefore Eq.7ca be2 a

rewritten as:

(8

T king the boundary conditions:

oLo=

w' integral

of Eq.8 can be obta ed e acture pressure distributionfor ul r:

o x = r+ n P i Q _ x2+ x +e -a(P r-P w)]Kf h

(9

Then average pressure of the acture zone c n be obta ed t ough the follow g integral equation:

f o x d =

f d= _1_ f1n[a P i QI _ x 2 + x + e-a(P r-Pw)]d r Kf b h a

(10

Pre d ict i owperformance for oil wellsThe relationship between Pwand Pfunder some Q value is

calculated by solving Eq. wi numerical solution; encombining Eq.6, e relationship between Pwand Q c n beobtained, namely in ow perfo ance relationship.

III. ASE TUDY

A eld of ndamental data of a act r g well are:h= 1m,K= 0.5X10 3pm2, G=O.O MP, R =150m , a=80m, =O.O m, Kf=45pm 2,

K= 0.0 MPa ,P i= 1.07mP ·s r= 16.7 M,k 0.0 77MP , 0. 1 9 MP a •The in ow perfo nce relationship curve for the well is

obtained by us g this paper me od, shown Figure 1. It isfound om gure 1 at, a er e bottom pressure is lower n6 MPa, production almost stopped crease even reduced e

pressure still her. So, e proper bottom-hole owing

pressure is 6 MPa.IV. ONCLUS ON

A prediction me od of in ow performance relationship foractured wells in extra-low permeability-defomed media oil

reservoirs is given wi elliptic ow mathematical model as ewell in ow performance of act red well be divided nto two par s at is seepage ow om oil layer head to acture ndseepage ow om act re head to bottom-hole d thechanges of the pe eability and uid viscosity with pressure

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were taken into account. The application example showed that this new model can be used to calculate the in ow perfo anceof the act red well for an extra-low pe eability reservoand provide a basis for the proper bot om-hole ow g pressuredete ination.

14

-12

10

o L � � _ �

o 0.5 1.5 2.5 3.5Q(m3/d)

FigureI . n ow performance relationship curvefor the well

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EFERENCES[I) Raymond, . R. and Binder G. G, "Productivity of Wells in Ver ica

Fractured D aged Formations," Paper SPE 1454 (1967 .[ 2) Poe, B. D. Jr, "Production Performance Evaluation of Hydraulical

Fractured Wells," Paper SPE 59758, (2000 .[ 3) Song Fu quan, productivity nalysis for low permeable reservoi

media deformation , Special oil&Gas Reservoirs, vol. 9, 2002, pp. 33-36.

[4) W ng Yu ying, W ng Xiao dong and Wang Yi fei, Analysireasonable pressure di erence d productivity of oil wells in deformmedia reservoirs , Jourmal of Daqing Petroleum nstitute, vol. 29, 2pp.51-55.