Download - A Fully Coupled Hydro Logical (1)
-
8/3/2019 A Fully Coupled Hydro Logical (1)
1/10
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
http://-/?-http://www.elsevier.com/locate/ijrmmshttp://dx.doi.org/10.1016/j.ijrmms.2010.11.012mailto:[email protected]://dx.doi.org/10.1016/j.ijrmms.2010.11.012http://dx.doi.org/10.1016/j.ijrmms.2010.11.012mailto:[email protected]://dx.doi.org/10.1016/j.ijrmms.2010.11.012http://www.elsevier.com/locate/ijrmmshttp://-/?- -
8/3/2019 A Fully Coupled Hydro Logical (1)
2/10
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
-
8/3/2019 A Fully Coupled Hydro Logical (1)
3/10
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
-
8/3/2019 A Fully Coupled Hydro Logical (1)
4/10
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.
Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268262
-
8/3/2019 A Fully Coupled Hydro Logical (1)
5/10
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].
Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268 263
-
8/3/2019 A Fully Coupled Hydro Logical (1)
6/10
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.
Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268264
-
8/3/2019 A Fully Coupled Hydro Logical (1)
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.
Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268 265
-
8/3/2019 A Fully Coupled Hydro Logical (1)
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.
Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268266
-
8/3/2019 A Fully Coupled Hydro Logical (1)
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.
Induced pore pressure
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.
Induced pore pressure
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
Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268 267
-
8/3/2019 A Fully Coupled Hydro Logical (1)
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
[1] Gray DH, Fatt I, Bergamini G. The effect of stress on permeability of sandstonecores. SPE J 1963;3:95100.
[2] Vairogs J, Hearn CL, Dareing DW, Rhoades VW. Effect of rock stress on gasproduction from low-permeability reservoirs. J Pet Tech 1971;23:11617.
[3] Thomas RD, Ward DC. Effect of overburden pressure and water saturation ongas permeability of tight sandstone cores. J Pet Tech 1972;24:1204.
[4] Raghavan R, Scorer JDT, Miller FG. An investigation by numerical methods ofthe effect of pressure-dependent rock and fluid properties on well flow tests.SPE J 1972;12:26775.
[5] Vairogs J, Rhoades VW. Pressure transient test in formations having stress-sensitive permeability. J Pet Tech 1973;25:96570.
[6] Samaniego VF, Brigham WE, Miller FG. Performance-prediction procedure fortransient flow of fluidsthroughpressure-sensitive formations. In: Proceedingsof the 51st annual technical conference and exhibition, New Orleans, 36 Oct1976, paper SPE 6051.
[7] Samaniego-V F, Brigham WE, Miller FG. An investigation of transient flow ofreservoir fluidsconsidering pressure-dependentrock andfluid properties. SPEJ1977;17:14050.
[8] Jones FO, Owens WW. A laboratory study of low-permeability gas sands. J Pet
Tech 1980;32:163140.[9] Samaniego F, Cinco-Ley H. On the determination of the pressure-dependent
characteristics of a reservoir through transient pressure testing. In: Proceed-
ings of the annual technical conference and exhibition, San Antonio, 811 Oct
1989, paper SPE 19774.[10] Buchsteiner H, Warpinski NR, Economides MJ. Stress-induced permeability
reduction in fissured reservoirs. In: Proceedings of the annual technical
conference and exhibition, Houston, 36 Oct 1993, paper SPE 26513.[11] Chin, LY, Raghavan R, Thomas LK. Fully coupled analysis of well responses in
stress-sensitive reservoirs. In: Proceedings of the annual technical conferenceand exhibition, New Orleans, 2730 Sept 1998, paper SPE 48967.[12] Davies JP, Davies DK. Stress-dependent permeability: characterization and
modeling. In: Proceedings of 1999 SPE annual technical conference and
exhibition, Houston, Texas, 36 October 1999, paper SPE 56813.[13] Ostensen RW. The effect of stress-dependent permeability on gas production
and well testing. SPE Form Eval 1986;1:22735.[14] Iwai K. Fundamental studies of fluid flow through a single fracture, PhD
dissertation, Univ Calif, Berkeley, 1976.[15] GoodmanRE. Methodsof Geological Engineeringin DiscontinuousRocks. New
York: West Pub; 1976.[16] BandisSC, LumsdenAC, BartonNR. Fundamentalsof rockjointdeformation.Int
J Rock Mech Min Sci Geomech Abstr 1983;20:24968.[17] Barton NR, Bandis SC, Bakhtar K. Strength, deformation and conductivity
coupling of rock joints. Int J Rock Mech Min Sci Geomech Abstr 1985;22:
12140.[18] Biot MA. General theory of three-dimensional consolidation. J Appl Phys
1941;12:15564.[19] BiotMA. General solutions of theequations of elasticity andconsolidation fora
porous material. J Appl Mech 1956;78:916.[20] Rice JR, Cleary MP. Some basic stress diffusion solutions for fluid-saturated
elastic porous media with compressible constituents. Rev Geophys 1976;14:
22741.[21] Curran,JH, CarvalhoJL. A displacementdiscontinuitymodelfor fluid-saturated
porous media. In: Proceedings of the sixth international congress on rock
mechanics, Montreal, 1987, p. 738.[22] ChengAHD, Predeleanu M. Transient boundaryelementformulation for linear
poroelasticity. Appl Math Modell 1987;11:28590.[23] Carvalho JL. Poroelastic effects and influence of material interfaces on
hydraulic fracturing. PhD dissertation, Univ Toronto, Toronto, 1990.[24] Vandamme LM,RoegiersJC. Poroelasticity in hydraulicfracturingsimulators. J
Pet Tech 1990;42:1199203.[25] Asgian M. A numerical-model of fluid-flow in deformable naturally fractured
rock masses. Int J Rock Mech Min Sci 1989;26:31728.[26] Lee J, Rollins J, Spivey J. Pressure Transient Testing. Richardson, Texas: Soc
Petrol Eng; 2003.[27] WitherspoonPA,Wang JSY,IwaiK, Gale JE. Validityof cubiclawfor fluid flowin
a deformable rock fracture. Water Resour Res 1980;16:101624.[28] Tao Q. Numerical modeling of fracture permeability change in naturally
fractured reservoirs using a fully coupled displacementdiscontinuity method.
PhD Dissertation, Texas A&M Univ, College Station, Texas, 2010.
Q. Tao et al. / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 259268268