a fully coupled hydro logical (1)

8/3/2019 A Fully Coupled Hydro Logical (1)

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A fully coupled method to model fracture permeability change in naturally

fractured reservoirs

Qingfeng Tao n, Ahmad Ghassemi, Christine A. Ehlig-Economides

Harold Vance Department of Petroleum Engineering, Texas A&M University, College Station, TX, USA

a r t i c l e i n f o

Article history:

Received 26 December 2009

Received in revised form2 August 2010

Accepted 27 November 2010Available online 18 December 2010

Keywords:

Naturally fractured reservoirs

Fracture permeability change

Effective stress

Joint deformation

Displacement discontinuity method

a b s t r a c t

We present a newapproachto model thefracturepermeability change in naturallyfractured reservoirsby

combining a finite difference method (FDM) for fluid diffusivity equation in a fracture network, a fully

coupled displacement discontinuitymethod (DDM)for theglobal relation of fracturedeformation andtheBartonBandis joint deformation model for the local relation of fracture deformation. The method is

applied to naturally fractured reservoirs under isotropic in-situ stress conditions and anisotropic in-situ

stress conditions, respectively. The application of the new approach shows that the fracture permeability

decreases with the pressure depletion under isotropic in-situ stress condition. Under highly anisotropic

stress, it is possible that shear dilation can increasefracture permeability evenas pore pressure decreases

with production.

Published by Elsevier Ltd.

1. Introduction

In many reservoirs, fractures are the main flow channels, and

the matrix provides the main storage capacity. Somereservoirs, e.g.tight gas reservoirs, are not possible to produce without the

existence of natural fractures (microfractures). Therefore, the

fracture permeability is critical to hydrocarbon production.

The dependence of formation permeability on pressure for a

single porosity system has been investigated [112]. The pressure

dependence of matrix permeability occurs as the porosity and

connectivity of pores decrease with the increase in effective

stress. But the permeability change in tight gas reservoirs mainly

results from the closure of microcracks with the increase of

effective stress [13].

Generally fractures are more deformable than the matrix in a

naturally fractured reservoir, and the permeability of fractures, not

the matrix, dominates the flow behavior. Furthermore, fractures

are more sensitive to pressure and stress change than the matrix,and the fracture deformation mechanism is much more compli-

cated than matrix deformation. The effect of stress on the aperture

and permeability of a single fracture has been well investigated in

laboratories [1417]. Experimental data show a nonlinear relation

between normal stress and fracture closure. Bandis et al. [16]

presented a hyperbolic formula to represent the normal stress

fracture closure relation. For shear deformation the experimental

data show an approximately linear relation between shear stress

and shear displacement before yielding, and then show a compli-

cated relation after yielding. Shear deformation can also induce

fracture opening as the opposed asperities of a fracture slide over

each other and cause an increase in aperture.In naturally fractured reservoirs, there are coupled interactions

between porous matrixand fluid, as well as between fractures. Biot

[18,19] developed a theory of poroelasticity for porous media

saturated with incompressible fluid to account for the coupled

diffusiondeformation mechanism. Rice and Cleary [20] extended

the theory for porous media saturated with compressible fluid.

Biots theory of poroelasticity is a continuum theory for a porous

medium consisting of an elastic matrix containing interconnected

fluid-saturated pores. The fluid diffusion in porous media induces

porous matrix deformation and stress redistribution, and porous

matrix deformation also induces fluid flow and fluid pressure

redistribution. If there is a discontinuous surface (fracture) in the

continuum porous medium, the deformation of the fracture (open-

ingor closing) will inducethe deformationof theporous matrixandalso pore pressure change and fluid flow.

In addition to the interactions of fluid, porous matrix and

fracture, there are interactions between fractures including

mechanical deformation and fluid flow. One fracture deformation

will cause stress change in the field and induce deformation of

other fractures [2123]. The fluid injection or production from one

fracture can also induce fluid pressure change in other fractures, as

well as mechanical deformation. Curran and Carvalho [21], Cheng

and Predeleanu [22] and Carvalho [23] developed a poroelastic

DDM for fluid-saturated porous media with many discontinuous

surfaces (fractures) in it. The poroelastic DDM can be applied to

model the coupled interactions of fractures, porous matrix and

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijrmms

International Journal ofRock Mechanics & Mining Sciences

1365-1609/$ - see front matter Published by Elsevier Ltd.

doi:10.1016/j.ijrmms.2010.11.012

n Corresponding author.

E-mail address: [email protected] (Q. Tao).

International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268
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fluid in porous media with fractures. This methodhas been applied

to simulate the hydraulic fracturing in continuum porous media

[24]. But the poroelastic DDM has not been applied to model

the interactions of fracture, porous matrix and fluid in fractured

porous media. Asgian [25] applied an elastic DDM to model the

deformable fractured reservoirs and assumed the matrix is

impermeable, thus, ignored the fluid flow between matrix and

fracture, which is usually the most important contribution to the

hydrocarbon production.In this study, we present a new approach to model the fracture

permeability change by combining a finite difference method for

solving the coupled fluid diffusivity equation in a fracture network,

a displacement discontinuity method (DDM) basedon Biots theory

of poroelasticity and the nonlinear BartonBandis joint deforma-

tion model. If there is no cooler fluid injection into the reservoir to

maintain reservoir pressure and enhance the hydrocarbon produc-

tion, the reservoir temperature will be constant. As a result, the

thermal effect is not considered in this method.

2. Fully coupled displacement discontinuity method

Based on Biots theory of poroelasticity, Carvalho [23] gave the

fundamental solutions of induced stress and pore pressure for afinite thin fracture segment with a fluid injection/production

source in an infinite two-dimensional homogeneous and isotropic

porous medium saturated with a compressible single-phase fluid.

The induced stress and pore pressure by a single long fracture or

many fractures with fluid injection/production can be obtained by

discretizing the fracture or fractures into Nfracture segments and

summing the influences of all N fracture segments.

2.1. Constitutive equations of a porous medium saturated with a

compressible single-phase fluid

The relation of stress to strain and pore pressure for a linear

isotropic poroelastic medium is given by Biots theory of poro-

elasticity [18]

sij 2G eij diju

12u ekkh i

dijap 1

where s and e are stress and strain, respectively, G and u are shearmodulus and Poissons ratio, ekk is the volumetric strain, i and j are

either x, y, or z and dij is the Kronecker delta. Combining a static

force balance equation with Eq. (1) and ignoring the body force,

yields the Navier equation of poroelasticity

Gui,kk G

12nuk,kiap,i 0 2

The total volumetric deformation (ekk) consists of pore space

change (zp) and the solid grain deformation (zs). The solid grain

deformation is dueto fluidpressure andeffective stress loading: (i) the

effect of fluid pressure (the compression stress or strain is negative)

Bs1 p

Ks1f 3

(ii) the effect of effective stress loading

Bs2 sukk3Ks

4

where Ks is the bulk modulus of the solid grains and f is the

porosity. The average effective stress (skk0/3) has the followingrelation with the volumetric strain and pore pressure [23]

sukk3

suxx suxx suxx3

Kmekk KmKs

p 5

where Km is the bulk modulus of the whole system including fluid

andsolidgrains.Combining Eqs. (3)and (4), andsubstituting Eq.(5)

yield the solid grain deformation

Bs KmKs

ekk p

Ks

KmKs

1f !

6

Thepore space change is obtained by subtracting thesolid grain

deformation from the total volumetric strain and using the defini-

tion of Biots coefficient (a 1Km=Ks)

Bpaekk

af

p

Ks 7

2.2. Fluid diffusion equation in the porous media

Thefluidmass balance equation gives that thenet fluid flowrate

is equal to the sum of the increase of fluid mass in the pore space

and injected/produced fluid

rfqm,i @rfVf

@trfqs 8

where rfis the fluid density, qm is the fluid flow rate in matrix, Vf ispore volume, qs is the production or injection rate and tis the time.

The fluid is compressible and the fluid density is pressure depen-

dent

@rf@p

cfrf 9

where cfis thefluid compressibility. In a unit volumeporous media,

the pore space is f, and the pore space change is zp, and Eq. (8) is

rewritten as

rfqm,i f@rf@t

rf@Bp@t

rfqs 10

Darcys flow is assumed

qm kmp,i 11

where k is the matrix permeability and m is the fluid viscosity.Substituting Eqs. (7), (9) and (11) into Eq. (10), neglecting the term

with @p=@xi2

[26] and assuming small change in the pore volumeyield

p,ii

m

kM

@p

@t am

k

@ekk@t

mqsk

12

where 1=Mfcf af=Ks.

2.3. Fundamental solutionsfor a single fracture segment in an infinite

two-dimensional porous medium

For a plane strain condition, there is a constant discontinuity in

the media and also constant flow rate (injection or production)

along a thinfracturewitha lengthof 2a from t0 (Fig.1). Theinitialconditions aredefined in Eq.(13) andthe inner andouterboundary

conditions are defined in Eqs. (14) and (15). Since only the induced

solutions for changes in stress, displacement and pore pressure are

needed, the initial values of stress, displacement and pore pressure

are set as zero. The initial conditions are given by

at t 0, forall x,y : p 0, ux uy 0, sxx syy sxy 013

The boundary conditions at the inner boundary are given by

at y 0, for xj jra : uxx,0uxx,0 Ds, uyx,0uyx,0 Dn, qs 2aq0 14

where q01 m3/s. The outer boundary condition is

as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 y2q -1 : ux uy 0, sxx syy sxy p 0 15

Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268260


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Using the initial and boundary conditions, Eqs. (2) and (12) can

be solved for separate inner boundary conditionsconstant

volume flow rate injection/production (ux(x, 0)ux(x, 0+)0,uy(x, 0

)uy(x, 0+)0, qs 2aq0) at the inner boundary andconstant displacement discontinuity (DD)(ux(x, 0

)ux(x, 0+)Ds,uy(x, 0

)uy(x, 0+)Dn, qs0) at the inner boundary [23]. Theinduced displacement, pore pressure and stress at any point (x, y)

andtime tby the constant volume injection/production rate and by

the displacement discontinuities including normal and shear

displacement discontinuities through the fracture segment are

given in Appendix A and Appendix B, respectively [23]. The final

fundamental solutions for poroelastic DDM are obtained by

combining the solutions of the constant volume rate fluid injec-

tion/production and the constant displacement discontinuities in

the fracture segment.

Induced pore pressure

px,y,t pdnx,y,t Dn pdsx,y,t Ds pqx,y,t qint 16

Induced displacement

uxx,y,t udnx x,y,tDn udsx x,y,tDs uqxx,y,tqintuyx,y,t udny x,y,tDn udsy x,y,tDs uqyx,y,tqint 17

Induced stress

sxxx,y,t sdnxx x,y,tDn sdsxxx,y,tDs sqxxx,y,tqintsyyx,y,t sdnyy x,y,tDn sdsyyx,y,tDs sqyyx,y,tqint

sxyx,y,t

sdn

xy x,y,t

D

n sds

xyx,y,t

D

s sq

xyx,y,t

q

int 18

where Dn and Ds are the normal and shear displacement disconti-

nuity sources, and qint is the fluid source term in a fracture

(interface flow rate between fracture and matrix), and the super-

scripts dn, ds and q denote normal displacement discontinuity

source, shear displacement discontinuity source and fluid source,

respectively. The induced pore pressure, pq, displacement in x

direction, uxq and in y direction, uy

q, stress components, sxxq , syy

q and

sxyq by the constant rate fluid injection/production from a fracture

segment are listed in Appendix A. The induced pore pressure, pdn

and pds, displacement in x direction, uxdn and ux

ds, and in y direction,

uydn and uy

ds, stress components, sxxdn, syy

dn, sdnxy , sxxds, syy

ds and sxyds by the

constant normal and shear discontinuous displacement of a

fracture segment are listed in Appendix B.

3. Joint deformation model

In this study, we only consider those fractures with two rough

surfaces to contact each other and the fluid flow can flow through

the void spaces between them as illustrated in Fig. 2. The fracture

can deform normally and laterally to the fracture surface as the

stress acting on it changes.

Bandis et al. [16] presented a hyperbolic model for the normal

deformation of fracture based on a large body of experimental data

sun Dn

aabDn19

wheres0n is theeffective normalstress and Dn is thenormal closureof fracture; aa and b are constants and related with the experi-

mentally determined parameters initial normal stiffness (Kni) and

the maximum possible closure (Dnmax) as aa1/Kni and aa/bDnmax. Eq. (19) can be rewritten by substituting Kni and Dnmaxfor aa and b

sun KniDn1Dn=Dnmax

20

The normal stiffness (Kn) is therefore derived as follows as a

function of Dn or s0n

Kn @sun@Dn

Kni1Dn=Dnmax 2 21

or

Kn Kni1sun= KniDnmax sun 2 22

The change of shear stress has a linear relationship with the

change of shear displacement before yielding

Dss KsDDs 23where Ks is the shear stiffness. The fracture slips when the shear

stress exceeds the shear strength (sc) of the fracture. The tworough surfaces slide each other and cause an increase in apertures,

which is known as shear dilation [17]. We used an approximate

linear relation for the aperture increase (DDn-dilation) due to shear

movement

DDndilation DDstanfd 24where fd is the dilation angle.

4. Fluid flow in the fracture network

The aperture of real fracture varies in space (Fig. 2) and the fluid

flow inside is also very complicated due to the rough surfaces. But

Darcys law is still valid and the rough fracture can be represented

by a fracture with the average fracture aperture, which has been

verified by Witherspoon et al. [27]. The fluid balance equation in

x

y

Dn+

Ds+

qs

Fig. 1. A thin line fracture in an infinite two-dimensional elastic porous medium,

and the line fracture starts from (a,0) and ends at (a,0).

Fig. 2. A fluid filled fracture subject to normal and shear stress.

Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268 261


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the fracture includes theflow from theconnected fractures andthe

interface flow from the connected matrices

rfqf,i @ rfwfDL

@tDLrfqintrfqs 25

where qf is the flow rate in the fracture per unit formation

thickness, qint is the interface flow rate per fracture length per

unit formationthickness,DL is thelength of fracture segment andqs

is theproductionrate per unit formationthickness. Theflow rate inthe fracture can be obtained by using Darcys law

qf kf wfm

rp 26

where kf is the fracture permeability and calculated from the

fracture aperture using the Cubic law and wf is the fracture

aperture. Combining Eqs. (9) and (26), and ignoring the second

order of the small term p,i [26], the net fracture flow rate is

rfqf,i rfkfwf

mp

,ii 27

The fluid mass change in the fracture includes two parts, one is

due to fracture volume change and another one is due to fluid

density change. The fracture volume change is mainly from thefracture aperture change

@Vf@t

DL @wf@t

28

Eq. (28) can be rewritten by substituting Dn for wf

@Vf@t

DL @Dn@t

29

The fluid mass change due to fluid density change is

@m

@t wfDL

@rf@t

30

Substituting Eq. (9) into Eq. (30) yields

@m

@t cfrfwfDL

@p

@t 31

Combing Eqs. (25)(31) yields the fluid diffusion equation in

fracture network

kfwfm

p,ii wfDLcf

@p

@tDL @Dn

@tDLqintqs 32

5. Numerical implementation

The fractured reservoir is treated as a fracture network in a

porous medium saturated with a compressible single-phase fluid.

As in dual-porosity models, the fracture network provides the main

flow channels and the porous media provides the main storage

media. On production, the fluid flows from matrixto fractures, then

in fractures to the well. The fracture network is discretized intosmall fracture segments and the change of aperture and perme-

ability of every fracturesegment withproduction aredeterminedby

combining the BartonBandis joint deformation model, the DDM,

and a FDM solving the diffusivity fluidequation in fracturenetwork.

5.1. Local relation between stress and displacement to fracture

deformation

Forany fracture in thefracturenetwork (Fig.3), the deformation

must comply with the fracture deformation model. The relation

between effective normal stress change Dsn0 and normal displace-

ment DDn of the ith fracture segment is

Dsnui

Ki

nDDni

33

The normal stiffness Kn is a coefficient that is dependent on the

fracture. The effective stress (tension is treated as positive) is

defined as

snu sn ap 34where a 1Km=Ks. For a fracture, when the bulk modulus ofsystem Km is much less than the solid bulk modulus Ks, the Biot

coefficient becomes unity, and the effective stress is given by

snu sn p 35

Substituting Eq. (35) for effective stress in Eq. (33) yields (for

each fracture segment)

Dsi

n Dpi KnDDi

n 36The relation of shear stress change Dss and shear displacement

DDs is

Dsi

s Ki

sDDi

s 37The shear stiffness is a constant before yielding and reduces to

zero after yielding. The normal deformationDDn-dilationdueto shear

dilation is

DD

i

ndilation DDi

s tanfd 38Eq. (36) must be rewritten when the normal deformation

induced by shear dilation is considered

Dsi

n Dpi Ki

n DDi

n DDi

stanfd

39

5.2. Global relation between stress and displacement to fracture

deformation

In the fracture network with m fracture segments, there are

interactions among fractures. The stress change of the ith fracture

segment is influenced by the deformation of all the fracture

segments in the system. The dependence of change of normaland shear stresses ofith fracture segment on the normal and shear

deformation,and theinterface flowrateof allfracturesegmentsare

derived from Eqs. (16) and (18) [28]

Dsi

n Xm

j 1A

ij

DDj

n Xm

j 1Bij

DDj

s Xm

j 1Cij

qj

int

Dsi

s Xm

j 1Eij

DDj

n Xm

j 1Fij

DDj

s Xm

j 1Kij

qj

int

Dpi

Xmj 1

Lij

DDj

n Xm

j 1Hij

DDj

s Xm

j 1Nij

qj

int 40

whereAij

, Bij

, Cij

, Eij

, Fij

, Kij

, Lij

, Hij

, and Nij

are the influence coefficients ofjth

fracture element on the ith fracture element.

n

s

Fig. 3. Local relation of fracture deformation.



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The DDM solutions are both space and time dependent, and the

fundamental solutions are based on constant displacement dis-

continuities and constant interface or source flow rates. However,

for practical applications, the displacement discontinuities and

interface flow rates are timedependent. The timemarchingscheme

shown in Fig. 4 is used to allow source strengths (the displacement

discontinuities and interface flow rate) to change with time.

Starting each boundary integration from an initial homogeneous

status avoids the needfor volumetricintegration [23]. Therefore,allthe previous increments of source strengths must be included

while numerically integrating the effect of source strengths at each

time step. The induced stress and pore pressure on the ith fracture

segment by the increments of source strengths are

Dsi

nt Xm

j 1A

ij

ttxDDjx

n Xm

j 1Bij

ttxDDjx

s Xm

j 1Cij

ttxDqjx

int

Xx1

h 0

Xmj 1

Aij

tthDDjh

n Xx1

h 0

Xmj 1

Bij

tthDDjh

s Xx1

h 0

Xmj 1

Cij

tthDqjh

int

Dsi

st Xm

j 1Eij

ttxDDjx

n Xm

j 1Fij

ttxDDjx

s Xm

j 1Kij

ttxDqjx

int

Xx1

h 0 Xm

j 1Eij

t

th

DD

jh

n

Xx1

h 0 Xm

j 1Fij

t

th

DD

jh

s

Xx1

h 0 Xm

j 1Kij

t

th

Dq

jh

int

Dpi t

Xmj 1

Lij

ttxDDjx

n Xm

j 1Hij

ttxDDjx

s Xm

j 1Nij

ttxDqjx

int

Xx1

h 0

Xmj 1

Lij

tthDDjh

n Xx1

h 0

Xmj 1

Hij

tthDDjh

s Xx1

h 0

Xmj 1

Nij

tthDqjh

int

41where DD

jx

n, DDjx

s and Dqjx

int are the source strength increments for

the jth fracture segment at the current time step, x; DDjh

n, DDjh

s and

Dqjh

int are the previous source strength increments of for the jth

fracture segment at time step h, which indexed from 1 to x1.

Aij

t

th

, B

ij

t

th

, C

ij

t

th

, E

ij

t

th

, F

ij

t

th

, K

ij

t

th

, L

ij

t

th

, H

ij

t

th

and N

ij

tth arethe influence coefficients ofjth fracture element onthe ith fracture element at time step h.

5.3. Solution method

Discretizing the Eqs. (37) and (39) for the local relation of

fracture deformation in time, and combining the local and global

relations yields

pi t

Xmj 1

Aij

ttxDDjx

n Ki

nDDix

n Xm

j 1Bij

ttxDDjx

s Ki

ntanfdDDix

s

X

m

j

1

Cij

ttxqjx

int Xx1

h

0 X

m

j

1

Aij

tthDDjh

nXx1

h

0 X

m

j

1

Bij

tthDDjh

s

Xx1

h 0

Xmj 1

Cij

tthqjh

intKi

n

Xx1h 0

DDih

n tanfdXx1

h 0DD

ih

s

pi 0

Xmj 1

Eij

ttxDDjx

n Xm

j 1Fij

ttxDDjx

sKi

sDDix

s Xm

j 1Kij

ttxqjx

int

Xx1

h 0 Xm

j 1

Eij

tthDDjh

n Xx1

h 0 Xm

j 1

Fij

tthDDjh

s Xx1

h 0 Xm

j 1

Kij

tthqjh

int

Ki

s

Xx1h 0

DDih

s

pi t Xm

j 1Lij

ttxDDjx

n Xm

j 1Hij

ttxDDjx

s Xm

j 1Nij

ttxqjx

int

pi 0Xx1

h 0

Xmj 1

Lij

tthDDjh

nXx1

h 0

Xmj 1

Hij

tthDDjh

s

Xx1

h 0

Xmj 1

Nij

tthqjh

int 42

The diffusivity equation, Eq. (32), is discretized in space and

time for a given fracture network using an implicit finite differencemethod. For the ith fracture segment at the time step, x

Xmj 1

Cpij

pj tDLDD

ix

n DLDqix

int wfDLcfpi txDL

Xx1h 0

Dqih

intXx

h 0qih

s

43where Cp

ij

is the fluid coefficient matrix [28]. The production rate

from ith fracture segment qih

s is also discretized in time in Eq. (43).

All left terms in Eqs. (42) and (43) are unknown and all right terms

areknown. When the production rate andinitial reservoirpressure

are given, the normal and shear fracture displacement, interface

flow rate, and fluid pressure can be obtained by solving the linear

equation Eqs. (42) and (43).

6. Applications

6.1. Fracture permeability change under isotropic in-situ stress

conditions

In this section the fracture permeability change during produc-

tion and its effect on transient wellbore pressure are investigated

for a well with constant production rate (2 m3/day) from a unit

reservoir thickness (1 m) in formation with a fracture network

consisting of two sets of orthogonal vertical fractures surrounded

by an effectively infinite porous medium as in Fig. 5. Only two-

dimensional flow and deformation are considered, and change in

the vertical direction is ignored. The in-situ stress field before

production is assumed to be isotropic with compression set to

21 MPa. To better illustrate the geomechanic effects during pro-

duction, the reservoir pressure is set very close to the in-situ stress

at 20 MPa. The two joint parameters, initial normal stiffness and

maximum closure, characterizing the normal deformation of

fracture are 0.5 GPa/m and 0.8 mm, respectively. The nonlinear

relationship between effective normal stress under compression

and fracture closure is shown in Fig. 6. The fracture aperture at the

initial condition (zero effective normal stress) is assumed as

0.8 mm. Thefractureapertureunder theinitial in-situ stress before

production is assumed as 0.229 mm for all fractures. Other para-

meters are listed in Table 1. Because the fracture permeability

dominates the reservoir permeability, changes in matrix perme-

ability are neglected and assumed as constant during production.

(

xj,yj,

)

(xj, yj, 0)

(xj, yj, )

0 1 t

(xj, yj, 1)

Fig. 4. Time marching scheme, w represents Dn, Ds or qint [23].



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Fig. 7 shows the reservoir pressure distribution after 360 days

on production. In this case, thelowest pressure is 14.2 MPa, andthe

highest pressure is 18.6 MPa. Fig. 8 shows that the fracture

permeability declines with production, and changes from 4428

to 280 D at the well, and from 4428 to 1920 D at the boundary. The

slope ofthe trend in Fig. 6 gives the normal fracture stiffness, which

changes with the effective stress, from a small value at small

effective stress to a rapidly increasing value at high effective stress.

As such, the fracture is more deformable when the reservoir

pressure is close to the in-situ stress than when there is a large

contrast between them. To study the influence of a higher stiffness,

considerthe sameinitialfracture aperture of 0.229 mm and fracture

permeability of 4428 D before production, but set the initial in-situ

stress to a value that increases the effective stress while all other

properties remain same. Fig. 9 shows the fracture permeabilitychange at the well for different effective in-situ stress conditions.

The influence of production on the fracture permeability change

strongly depends on the initial effective stress condition, and

decreases rapidly with increase in the effective in-situ stress. The

fracture permeabilityonly reduces 3.5%of the initial permeabilityof

4428 D for the case with an effective in-situ stress of 10 MPa.

However fracturepermeability loss for the case with an effective in-

situ stress of 1 MPa is 93.7% of the initial permeability.

The weakness of fracture is critical to the influence of production

on fracture permeability change. If all other conditions are the same,

the weaker the fracture, the more fracture permeability reduction.

The initial normal stiffness and maximum closure characterize the

normal fracture deformation. For the above reservoir and production

conditions, the fracture permeability change is simulated for threedifferentfractures withdifferent kniand Dnmax (weakest: kni0.8 mm,Dnmax0.5 GPa/m; medium: kni0.357 mm, Dnmax5 GPa/m; stron-gest: kni0.247 mm, Dnmax50 GPa/m). For the three kinds offractures, all fracture apertures at the normal effective stress of

1 MPa are 0.229 mm. As a result, they have the same initial fracture

permeability of 4428 D. The fracture aperture reduces with the

increase of the effective normal stress. The weakest fracture has

the maximum reduction and the strongest one has the minimum

reduction (Fig. 10). Fig. 11 shows the comparison of the fracture

permeability change with time for the three different kinds of

fracture. For the weakest fracture, the fracture permeability of the

fracture at the well decreases by 93.7% of the initial permeability,

while the fracture permeability for the strongest fracture only

decreases by 21% of the initial permeability.

6.2. Fracture permeability change under high anisotropic in-situ

stress conditions

We consider anisotropic in-situ stress conditions in this case.

The shear deformation of a fracture is approximately linear before

yielding and is treated as linear here, as is characteristic of a

constant shear stiffness value. The shear stiffness is abruptly

reduced to zero after yielding. The yielding stress can be calculated

using the following formula:

tpeak snu tanfi 44

where fi is the internal friction angle. For reservoirs already at

the critical stress conditions the fractures are already yielded.

Well

Fig. 5. Well located atthe center ofa fracturedfield, whichis surrounded by matrix

rock of effectively infinite extent.

0

2

4

6

8

10

0

Fracture closure (mm)

Effectivenormalstress(MPa)

0.2 0.4 0.6 0.8

Fig. 6. Nonlinear fracture normal deformation.

Table 1

Rock and fracture parameters in the modeling.

Area (m2) 1000 1000Shear modulus G (GPa) 5.9

Possoins ratio u 0.16

Undrained Possoins ratio uu 0.31

Matrix permeability (md) 0.8Matrix porosity f 0.2

Biots coefficient a 0.83Fluid viscosity m (cp) 1Fluid compressibility (/MPa) 6.8 104

Fig. 7. Pore pressure distribution after 360 days production.



7/10

Therefore, the fractures are week and the shear stress disturbance

can result in large shear deformation. The shear deformation will

induce some normal deformation by dilation.

In Fig. 12 a fractured reservoir with high anisotropic in-situ

stress (s130 MPa, s323 MPa)has are two sets of fractures withanangleof601. Theshearstiffness before yielding is 100 GPa/m,the

internal friction angle is 301, the dilation angle is 51, and all other

parameters are the same as those in the isotropic case listed in

Table 1. All fractures are already yielded before production and the

production with a constant rate of 2 m3/day induces not only the

normal deformation but also large shear deformation.

Fig. 13 shows the direction and magnitude of the shear displace-

ment after 360 days production. If the shear dilation induces moreopenness of the fracture than the closure induced by the increase of

the effective normal stress, the fracture permeability will increase

with production instead of reduction. Fig. 14 shows the fracture

permeability distribution after 360 days production. There is still

reduction of fracture permeability for those fractures in dark

blue. But the fracture permeability for other fractures increases

compared with the initial fracture permeability of 4428 D. The

fracture permeability and shear displacement are compared and

show consistent change (Fig. 14). Fig. 15 shows that the fracture

permeability increases with production both for the fracture inter-

sected by the well and for a fracture at the boundary with the

maximum enhancement at early time. But the permeability of the

fracture intersected by the welldecreaseslater as the effective stress

keeps increasing.And thepermeabilityof thefracture at thetop rightcorner increases until very latetime and changes the trend whenthe

effect of compression exceeds the effect of dilation. Therefore, under

highly anisotropic stress conditions production may enhance the

fracture permeability.

7. Conclusions

Production in naturally fractured reservoirs will cause reservoir

pressure change, thereby changing the stress. The stress change

will change the fracture aperture and permeability, thereby

influencing the production. In this study, we developed a new

approach to model the fracture permeability change in naturally

fractured reservoirs by combining a finite difference method

0

1000

2000

3000

4000

5000

0

Time (hr)

Fractureaperture(mm)

fracture at boundary

fracture intersected with Well

1000 2000 3000 4000 5000 6000 7000 8000 9000

Fig. 8. Fracture permeability declines with time.

0

1000

2000

3000

4000

5000

0

Time (hr)

Fracturepermea

bility(darcy)

Effective in situ stress = 1 MPa

2 MPa

5 MPa

10 MPa

1000 2000 3000 4000 5000 6000 7000 8000 9000

Fig. 9. Effect of initial effective in-situ stress on the fracture permeability change.

Fig. 10. Relation of fracture aperture with the effective normal stress for differentfractures.



8/10

solving the coupled fluid diffusivity equation in fracture network,

the nonlinear BartonBandis joint deformation model, and the

poroelastic displacement discontinuity method (DDM).

Fig. 11. The influence of weakness of fracture on the fracture permeability change with the fluid production.

Fig. 12. Well located at the center of a fractured field under anisotropic stress field

and the fractured network are surrounded by matrix rock.

Fig. 13. Shear displacement distribution after 360 days production for the case

fractures are already yielded before production. The arrow represents the shear

direction.

Fig. 14. Distribution of fracture permeability and shear displacement (shown with

arrows) after 360 days production for the case fractures are already yielded before

production.

2000

4000

6000

8000

10000

12000

14000

0

Time (hr)

Fractureperm

eability(darcy)

Fracture at the top right corner

Fracture intersected by well

2000 4000 6000 8000 10000

Fig. 15. Fracturepermeability changeswith production forthe case thefracture are

already yielded before production.



9/10

Fracture aperture and permeability decreases with pressure

reduction caused by the production in naturally fractured reser-

voirs under isotropic stress conditions. But the magnitude of the

change is dependent on the initial effective in-situ stress. For low

initial effective in-situ stress (the reservoirpressure is very close to

the magnitude of stress), the normal stiffness is small if the initial

normal stiffness is small, i.e., weak fractures. The small change of

reservoir pressure and effective stress can induce large fracture

closure andpermeability loss. Butfor hard rock (high initial normalstiffness) or high effective in-situ stress, the normal stiffness is

large. The change of fracture aperture and permeability is small

even for large reservoir pressure change. Therefore, whether the

reservoir is stress sensitive can be decided by laboratory tests on

the properties of fractures and field tests of the in-situ stress. For

stress sensitive fractured reservoirs, the method developed in this

study can be applied to evaluate the change of fracture perme-

ability during production and its influence on production. How-

ever, underhighly anisotropic in-situ stress condition, the fractures

can be at the critical stress condition, and a small change of the

shear stress can induce large shear displacement. The fracture

aperture and permeability can be enhanced due to shear dilation

while the reservoir pressure is decreasing.

Appendix A. Fundamental solutions for fluid source

Induced pore pressure pq, displacement uq and stress sq bycontinuous unit fluid source along a line fracture segment

r2 xxu2 y2 A:1where x0 varies from a to +a. Recall that

E1x Z1

x

eu

udu A:2

The induced pore pressure is given by

pq

m

4pkZ

a

aEi

x

2

dxu

A:3

where

x r2ffiffiffiffi

ctp A:4

The induced displacements are

uqx am12u

16pkG1u 2ctex2 r

2E1x22

2ct lnr2 E1x2 " #a

aA:5

uqy am12u

16pkG 1u 4ctarctanxxu

y

!aa

&

yZa

aE1x2dxu4cty

Zaa

ex2

r2dxu

)

A:6

The induced stresses are

sqxx am12u8pk1u xx

1

x2 e

x2

x2E1x2

" #aa

2Za

aE1x2dxu

( )

A:7

sqyy am12u8pk1u xxu

1

x2 e

x2

x2

E1x2 " #a

aA:8

sqxy am 12u 8pk 1

u

y 1x

2 e

x2

x2

E1x2 " #a

a

A:9

Appendix B. Fundamental solutions for displacement

discontinuities source

1. Induced pore pressure, displacement and stress by the

continuous unit normal displacement discontinuity along a line

fracture segment.


pdn

G

uu

u

2par212u1uu 2

x

xu

r2 1ex2

!a

a B:1Induced displacement

udnx 1

4p 1u 12uln9r9uuu1uu

lnr E1x

22

1ex2

2x2

" #(

y2

r21 uuu

1uu

1 1

x2

ex2

x2

" #)aa

B:2

udny 1

4p1u 21uarctanxxu

y

ln9r9

&

xxuyr2

1 uuu1uu 1

1

x2

ex2

x2 " #)

a

a

B:3

Induced stress

sdnxx G

2p1uxxu3xxuy2

r4 uuu

1uu

(

xxu3xxuy2

r4 3xxuy

2xxu3r4

1ex2

x2

2xxuy2ex

2

r4

" #)aa

B:4

sdnyy G

2p1uxxu3 3xxuy2

r4 uuu

1uu

(

xxu3 3xxuy2

r4 xxu

33xxuy2r4

1ex2

x2 2xxu

3ex2

r4

" #)aa

B:5sdnxy

G

2p1uxxu2yy3

r4 uuu

1uu

(

xxu2yy3

r4 3xxu

2yy3r4

1ex2 x2

2xxu2yex

2

r4

" #)aa

B:6

where vu is the undrained Poissons ratio.

2. Induced pore pressure, displacement and stress by the

continuous unit shear displacement discontinuity along a line

fracture segment.


pds

Guuu

2par2

12u1uu

2y

r2

1

ex

2

!a

a B:7

Induced displacement

udsx 1

4p1u

(21uarctan xxu

y

ln rj j

xxuyr2

1 uuu1uu

1 1

x2

ex2

x2

" #)aa

B:8

udsy 1

4p 1u 12u ln rj juuu1uu

lnr E1x

22

1ex2

2x2

" #(

y2

r21 uuu

1uu 1

1

x

2 e

x2

x

2 " #)a

a

B:9



10/10

Induced stress

sdsxx G

2p 1u

( 3xxu

2yy3r4

uuu1uu

3xxu2yy3

r4 3xxu

2yy3r4

1ex2 x

2 2y

3ex2

r4

" #)aa

B:10

sdsyy G

2p 1u

(xxu2yy3

r4 uuu

1uu

xxu2yy3

r43xxu

2yy3r4

1ex2 x

2 2xxu

2yex2

r4

" #)aa

B:11

sdsxy G

2p1uxxu3xxuy2

r4 uuu

1uu

(

xxu3xxuy2

r4 3xxuy

2xxu3r4

1ex2 x2

2xxuy2ex

2

r4

" #)aa

B:12References

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PhD Dissertation, Texas A&M Univ, College Station, Texas, 2010.


a fully coupled hydro logical (1)

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