574
Effects of fractures on reservoir deformation and
flow modeling
Mohammad Ali Bagheri and Antonin Settari
Abstract: Fracture opening and closure caused by changes in the effective stress on fracture planes is well known.
This phenomenon happens in aquifers and hydrocarbon reservoirs where production or injection events change the applied effective stress. Modeling this phenomenon requires a rigorous coupled geomechanics and flow simulator as well
as detailed information regarding fracture properties, spacing, and orientation. A method has been developed to account
for the effects of fractures on deformation in coupled reservoir and geomechanics simulation. The equivalent continuum
approach was used where the constitutive matrix of individual blocks was calculated by means of an analytical formula
that considers both rock and fracture deformation properties. The rock material is assumed to be isotropic and the normal stiffness of fractures varies with the applied effective stress according to a law that is typical for joints. The shear
deformation of the fractures can be incorporated easily in the future. The general pseudo-continuum model was verified
by comparing it with an approach using explicit modeling of the fractures. The fractured rock geomechanics model is
then coupled to a single porosity, multiphase flow model. The deformations produce changes in the permeability tensor
in both magnitude and orientation, which in turn influences compaction behavior.
Key words: fracture deformation, equivalent moduli, coupling, reservoir compaction, reservoir simulation.
Résumé : L’ouverture et la fermeture des fractures dues au changement de la contrainte effective sur le plan de fracture sont bien connues. Ce phénomène se produit spécialement dans les aquifères et les réservoirs d’hydrocarbure où la
production ou l’injection change les contraintes effectives appliquées. La modélisation de ce phénomène requiert un simulateur couplé de géomécanique et d’écoulement, de même qu’une information détaillée sur les propriétés des fractures, leur espacement et leur orientation. On a développé une méthode pour prendre en compte les effets des fractures
sur la déformation dans la simulation couplée de géomécanique et du réservoir. L’approche du continuum équivalent a
été utilisée là où la matrice constitutive des blocs individuels a été calculée au moyen d’une formule analytique qui
considère les propriétés de déformation de la roche de même que des fractures. On suppose que le matériau rocheux
est isotrope et que la rigidité normale des fractures varient avec la contrainte effective appliquée selon un loi typique
pour les joints. La déformation en cisaillement des fractures pourra être incorporée facilement dans le futur. Le modèle
général du pseudo-continuum a été vérifié en le comparant avec une approche qui utilise la modélisation explicite des
fractures. Le modèle géomécanique de la roche fracturée est ensuite couplé à un modèle d’écoulement multiphase à porosité simple. La déformation produit des changements dans le tenseur de perméabilité tant en amplitude qu’en orientation, qui à leur tour influencent le comportement du compactage.
Mots clés : déformation par fractures, modules équivalents, couplage, compactage de réservoir, simulation de réservoir.
[Traduit par la Rédaction]
Bagheri and Settari
Introduction
As fluids are being produced reservoirs experience a pressure drop, which causes fractures to close. The reverse phenomenon is observed in injection wells used to pressurize
the reservoirs. This closure and opening can be called “fracture breathing” and has been demonstrated by field measureReceived 18 February 2005. Accepted 15 December 2005.
Published on the NRC Research Press Web site at
http://cgj.nrc.ca on 12 May 2006.
M. Bagheri1 and A. Settari.2 Department of Chemical and
Petroleum Engineering, University of Calgary, 2500
University Dr., Calgary, AB T2N 1N4, Canada.
1
Present address: Sproule International Ltd., North Tower,
Suite 900, 140 4 Ave. SW, Calgary, AB T2P 3N3, Canada.
2
Corresponding author (e-mail: asettari@ucalgary.ca).
Can. Geotech. J. 43: 574–586 (2006)
586
ments. For instance, large variations in resistivity logs after
changes in drilling mud density seen in a Gulf Coast well
can be attributed to this phenomenon (Timko 1966).
Fracture breathing can contribute to compaction of the
reservoir and in turn to subsidence problems. From the point
of view of reservoir engineering (fluid flow in porous media), this phenomenon is also of importance because of its
effects on fracture porosity (fracture volume occupied by
fluid) and fracture network permeability. Usually most of the
reservoir fluid is stored in the rock matrix (as opposed to the
fractures) so variation in fracture porosity is less important
for pressure changes in the reservoir. In contrast, the
changes in fracture permeability can have major influences
on reservoir performance.
Over the past decades, experimental, theoretical, and numerical studies have been carried out to account for the fracture deformation caused by change in applied stresses. These
doi:10.1139/T06-024
© 2006 NRC Canada
Bagheri and Settari
575
Fig. 1. Effect of fracture on normal deformation (modified from
Bandis et al. 1983).
studies cover different aspects of fracture deformation and
provide some insight into its significance for fluid flow and
overall deformation of fractured rocks.
The stress–strain behavior of fractures is a nonlinear relationship depending crucially on the mechanical properties of
the fractures. This type of nonlinear behavior has been shown
in the experimental work of several investigators and has
also been modeled mathematically. Goodman (1976) proposed a hyperbolic function relating fracture closure to the
applied normal stress. Youshinaka and Yamabe (1986) modeled the normal closure of fracture by an exponential function, and Barton and Choubey (1977) modeled the shear
strength of fractures under shear stress. Bandis et al. (1983)
have experimentally studied and mathematically formulated
fracture deformation in detail. Their results have been utilized in this study.
The concept of equivalent continuum is invoked in this
work to treat fractured rock as a single media having the deformation properties of both intact rock and fractures. The
resulting elastic equation is no longer linear because of the
effect of nonlinear fracture deformation. Different approaches
have been reported in the literature to define the equivalent
properties of the continuum. From these, the method of
Huang et al. (1995), which appears to be the most general
and applicable, has been used to construct the elastic moduli
of the fractured rock in our study. The main attraction of this
method is that there are no constraints regarding the orientations of the fracture sets, which is important considering the
general diversity of fractures in reservoirs. Knowing the deformation of the equivalent continuum allows precise calculation of the stress field, which in turn results in the
calculation of deformation of fractures individually in space
and time. This paper focuses on the rock deformation and its
effect on the volumetric (porosity) behavior of reservoirs.
Fracture deformations also cause variation of fracture tensor
permeability. The permeability aspect of the coupling has
been studied with results published elsewhere (Bagheri and
Settari 2005).
where σn is the normal stress, ∆ν is the change in fracture
aperture, and “a” and “b” are constants that have physical
meaning. For a value of σn → ∞, the closure reaches its
maximum
Single joint deformation
[2]
Here we follow the joint mechanics models developed by
Bandis and co-workers (Bandis et al. 1983). In their laboratory investigations, interlocked block samples with natural
unfilled joints were subjected to a series of loading–unloading
cycles. During each test, the total deformation was recorded
as a function of normal stress. In all cases, a nonlinear behavior was observed. Figure 1 shows a typical behavior of a
single joint experiencing two cycles of a series of loading–
unloading cycles in comparison with the behavior of an intact rock. The difference in total deformation of these two
samples demonstrates the large influence of joints on normal
deformation of rocks (the ratio of total deformation to the
rock compression is around 3 at the end of compression).
The stiffness of the jointed rock and that of the intact rock at
high normal stresses becomes the same (i.e., the two curves
are parallel), which indicates a complete closure of the joint.
On decompression, a hysteretic behavior accompanied by a
permanent set is observed. Large hysteresis and large permanent set are always seen in the first load cycle, while diminishing on subsequent cycles. In situ fractures likely react to
Joint normal stiffness, which is the ratio of change of normal stress to normal closure, is defined as
stress changes in a fashion similar to the third or fourth cycles of lab experiments. In other words, the hysteresis effect
is not dominant in highly stressed in situ rocks. For our purposes, we neglected the hysteresis effect and used a single
formula for the deformation of a joint in loading and unloading, which is assumed to best represent the deformation path
of the joints.
Normal closure – normal stress curve for both artificial
and natural joints has a hyperbolic shape, given by the following equation:
[1]
[3]
σn =
∆ν
a − b∆ν
∆ν max ≅
kn =
a
b
dσn
d ∆ν
The physical meaning of “a” is found from the derivative
of eq. [1].
[4]
kn =
∂σ n
=
∂ ∆ν
1
b ⎞
⎛
a ⎜1 − ∆ν⎟
⎝
a ⎠
2
At zero normal stress, normal stiffness is called initial
normal stiffness and is shown to be
[5]
kni =
∂σ n
1
=
∂∆ν σ n = 0 a
Substituting eqs. [2] and [5] into eq. [4] will result in
© 2006 NRC Canada
576
[6]
Can. Geotech. J. Vol. 43, 2006
kn =
kni
⎛
∆ν ⎞
⎜1 −
⎟
∆
ν
⎝
max ⎠
2
Shear deformation of a rock sample containing a single
fracture is also well described in the literature. In this case,
the shear stiffness (which has a similar definition to the normal stiffness) is involved. However, including shear deformation of fractures in a full field study is somewhat difficult.
First of all, many joint model parameters are involved,
which requires extensive lab work. Secondly, in laboratory
shear tests, the walls of the joints can move freely relative to
each other and large shear deformations can be reached,
which is not the case in hydrocarbon reservoirs. In this
study, to verify the coded equivalent moduli of the jointed
rock mass against analytical solutions, finite shear stiffness
can be assigned to the joints. However, for reservoir application of this method, a very high stiffness has been chosen to
minimize the effect of shear deformation.
Equivalent moduli of a jointed rock mass
The concept of an equivalent medium of a jointed rock
mass (composed of two materials) is used to introduce a
third material that behaves the same way as the jointed rock
mass, under normal and shear stresses. The normal and shear
moduli of this third material are functions of the joints and
intact rock moduli. Normal and shear stiffness of joints
along with their orientations and spacings are key parameters for constructing the equivalent material’s constitutive
equation.
In this section the constitutive relation of single fracture
will be presented, and a general method for calculating the
equivalent properties of the jointed rock mass, developed by
Huang et al. (1995), is critically reviewed.
Constitutive equation of a single joint
The general constitutive equation for a joint has the following form:
[7]
∆␦ I = DIJ ∆␶ J
where ␦ I is the displacement vector (1 by 3), ␶J is the traction vector (1 by 3), and DIJ is the constitutive matrix (3 by
3).
The traction vector bears the direction and magnitude of
the stress acting on the joint surface, and can be expressed
by Cauchy’s formula
[8]
∆␶ j = ∆␴ijn i
where “n” is the vector normal to the joint surface.
The local coordinate system of every individual joint is
composed of three vectors in the direction of the normal,
strike, and dip angles. The components of the displacement
and traction vectors in joint local coordinate systems are in
these directions.
Joint structure restricts having more than one normal deformation and two shear deformations (dimensions of the
displacement vector). Generally, each of these deformations
can be produced by all three components of traction. For instance, a normal opening or closure can be caused by shear
stresses as well as normal stresses. Such normal deformation
of the fracture (increase of aperture) caused by shear stress
is known as “dilation” and results in an increase in fracture
volume. It is also possible that a normal stress acts like a
shearing factor on the joint surface and creates shear deformation. The rate of this deformation, as well as the rate of
dilatancy, is connected to the joint surface properties.
Often it is assumed that the deformation characteristic of
the joint in the dip direction is the same as in the strike direction. Using this assumption, and neglecting the effect of
dilatancy behavior and shear strain due to normal stress,
eq. [7] can be expanded in extremely simple form as
[9]
⎧∆δ N ⎫ ⎡ k1n
⎪
⎪ ⎢
⎨ ∆δ S ⎬ = ⎢ 0
⎪ ∆δ T ⎪ ⎢ 0
⎩
⎭ ⎢⎣
0
1
ks
0
0 ⎤ ⎧∆τ N ⎫
⎥⎪
⎪
0 ⎥ ⎨ ∆τ S ⎬
1 ⎥ ⎪ ∆τ ⎪
T⎭
ks ⎥
⎦⎩
This equation will be used to find the constitutive relation
of jointed rocks next.
Constitutive equation of jointed rock masses
A constitutive equation to calculate the overall deformation of rocks containing joints is desired because it helps to
simplify the overall expanded form of the equation and also
reduces the number of equations without neglecting the effect of joints.
Huang et al. (1995) proposed a stress–strain model for a
fractured rock mass containing several fracture sets. In their
method, the strains and stresses have been considered as tensors and so the final equation is more general compared to
some other methods. The following paragraphs briefly describe this method.
Changes in total strain due to stress changes consist of
two components: one from the intact rock and the second
from the fractures
[10]
∆␧ij = ∆␧ Iij + ∆␧ Jij
Equation [10] can be rewritten for both media to show the
relation of each component in response to the total stress
change.
[11]
∆␧ Iij = C Iijkl ∆␴kl
[12]
∆␧ Jij = C Jijkl ∆␴kl
Note that lower case subscripts indicate the global coordinate system. Each joint set has its own local coordinate system, which is indicated by upper case subscripts.
Based on the principle of energy conservation, strain energy stored in fractures due to stress is equal to the total
work done by the traction at each joint that caused joint
movement. Thus
M
[13]
␴ij∆␧ Jij =
1
␶ jf ∆␦ jf A f
∑
V f =1
where “M” is the total number of fracture sets, “ f ” indicates
individual fracture sets and “Af ” is the total area of the f th
fracture set, which is equal to the volume of the representative block divided by the average spacing between the frac© 2006 NRC Canada
Bagheri and Settari
577
tures in the set. Substituting Cauchy’s formula into eq. [13],
the strain of fractured rock due to movement of the fractures
is given by
M
[14]
∆␧Jij = ∑ n if ∆␦ jf
f =1
Fig. 2. Dimensions of the block used for linear verification of
the model: (a) intact rock, (b) fractured rock.
1
Sf
where “S” is the spacing between the fractures. The traction
and displacement vectors can be transformed between the
global coordinate system and the fracture set local coordinate system. For instance
[15]
∆␦ jf = L jJf ∆␦ Jf
[16]
∆ ␶ If = L Iif ∆ ␶ if
where “L” contains the direction cosines between the local
and global coordinate systems. Substituting eqs. [7], [9],
[15], and [16] into eq. [14], the general constitutive equation
for joints is obtained
M
[17]
f
f
C Jijkl = ∑ n if L jJf D JL
L Ll
n kf
f =1
1
Sf
The expanded form of this equation for a rock containing
three sets of fractures is presented in Appendix A, and differences found between the present work and that of Huang
et al. (1995) are discussed.
Model verification
In this section, the aim is to verify the 3D finite element
model developed using this approach in linear and nonlinear
conditions. Here, the term “linear” means that the normal
and shear stiffness values of fractures are assumed constant,
in contrast to the “nonlinear” condition where the normal
stiffness of fractures changes according to eq. [4].
Linear verification
Elastic deformation
Consider a quarter of symmetry of a cubic block of intact
rock as shown in Fig. 2a. The dimensions of the simulated
element are 0.5 m in the “x” and “y” directions and 1 m in
the “z” direction. Boundary condition of fixed surfaces is applied on three faces of the cube, which have a point in common including the bottom of the block. A typical Young’s
modulus of 100 GPa is assigned to the intact rock material.
A σn of 1 MPa creates a deformation along the z direction
equal to 0.01 mm. Now consider a horizontal fracture with a
constant (corresponding to the given stress level) normal
stiffness of kn intersecting the intact rock according to
Fig. 2b. Such a fracture increases rock mass deformation in
the z direction according to
[18]
⎛ 1
1⎞
δz = σn⎜
+ ⎟H
⎝ Skn E ⎠
where “S” is the spacing between the identical fractures in
the sample and “H” is the height of the sample. For instance,
if normal stiffness of the fracture is 20 GPa/m, the total
computed deformation with one fracture will be 0.06 mm.
It is apparent from eq. [18] that the higher the number of
fractures perpendicular to the z direction, the higher will be
the deformation. However, for this special case where only
one normal stress is applied, any configuration of vertical
fractures has no effect on the deformation in the vertical direction. This was verified for different types of vertical fractures. Further modeling verifications were also done by
adding 180° to the strike of the fractures and by reproducing
the results of the test problems with a multiblock numerical
model.
Now consider a block intersected by three orthogonal
fracture sets. These fracture sets (presented as single lines)
are arranged according to Fig. 3. The objective is to find the
overall strain components of the rock mass under a uniaxial
normal stress of 1 MPa on the top of the specimen. Amadei
and Goodman (1981) proposed an analytical solution to this
problem using transformation of the stress–strain relationship. The strain components of the resulting deformation are
⎡ sin 2 2α ⎛ 1
1
1
1 ⎞
[19a] ε x = ⎢
+
−
−
⎜
⎟
⎢⎣ 4 ⎝ kn1S1 kn2 S2 ks1S1 ks2 S2 ⎠
−
v⎤
σn
E ⎥⎦
⎛ v⎞
[19b] ε y = ⎜ − ⎟ σ n
⎝ E⎠
⎡ 1 cos 4 α sin 4 α
[19c] ε z = ⎢ +
+
kn1S1
kn 2 S2
⎣E
+
1 ⎞⎤
sin 2 2α ⎛ 1
+
⎜
⎟ ⎥ σn
4
⎝ ks1S1 ks2 S2 ⎠ ⎥⎦
⎡
⎛ cos 2 α sin 2 α ⎞
[19d] γ xz = ⎢sin 2α ⎜
−
⎟
kn 2 S2 ⎠
⎝ kn1S1
⎢⎣
−
sin 4α ⎛ 1
1 ⎞⎤
+
⎜
⎟ ⎥ σn
4 ⎝ ks1S1 ks2 S2 ⎠ ⎥⎦
To calculate the strain components with the numerical
stress model, different values of joint properties were assigned to the fracture sets, which are shown in Table 1,
along with the other relevant data. Boundary conditions of
fixed surfaces were applied on the right (y = Ly) and bottom
(z = 0) faces to restrict them from moving in the “y” and “z”
directions, respectively. In addition, the front edge (x = Lx
© 2006 NRC Canada
578
Can. Geotech. J. Vol. 43, 2006
Fig. 3. Three orthogonal joint sets intersecting an intact rock
(test model of Amadei and Goodman 1981).
Table 2. Comparison of numerical and analytical results in terms
of strain components.
Uniaxial test
εx
εy
εz
γ xz
Triaxial test
Numerical
0.006 859
Analytical
0.006 859 1
–0.000 490
0.059 965
0.051 140
–0.000 490 0
0.059 965 4
0.051 139 8
Numerical
Analytical
0.036 205
0.000 131
0.066 335
0.092 728
0.036 205 4
0.000 131 1
0.066 334 6
0.092 728 0
⎡⎛ 1 − 2 v ⎞ cos 4 α sin 4 α
[20c] ε z = ⎢⎜
+
⎟+
kn1S1
kn2 S2
⎣⎝ E ⎠
+
Table 1. Rock sample properties (modified
from Amadei and Goodman 1981).
Lx = Ly = Lz (m)
E (kPa)
ν
α (°)
kn1 (kPa/m)
kn2 (kPa/m)
kn3 (kPa/m)
ks1 (kPa/m)
ks2 (kPa/m)
ks3 (kPa/m)
S1 (m)
S2 (m)
S3 (m)
1.0
1.0×106
0.49
36
1.0×107
2.0×108
3.0×109
1.5×107
1.5×108
1.5×109
0.001
0.002
0.003
and z = 0) of the bottom face of the block was also restricted
from moving in the “x” direction.
The resulting strain components from the numerical model
and those from the analytical solution are identical, as shown
in Table 2. Again, further tests were run using the same configurations in different coordinate systems obtained by cyclic substitution of the axes. The previous results were also
reproduced with a large number of computational elements.
In a triaxial compression test where three identical stresses
of 1 MPa are applied in the “x,” “y,” and “z” directions, the
strain components found by Amadei and Goodman (1981)
are
⎡⎛ 1 − 2 v ⎞ sin 4 α cos 4 α
[20a] ε x = ⎢⎜
+
⎟+
kn1S1
kn2 S2
⎣⎝ E ⎠
+
1 ⎞⎤
sin 2 2α ⎛ 1
+
⎜
⎟ ⎥ σn
4
⎝ kn1S1 kn 2 S2 ⎠ ⎥⎦
⎡⎛ 1 − 2 v ⎞
1 ⎤
[20b] ε y = ⎢⎜
⎟+
⎥ σn
⎝
⎠
E
k
n 3S3 ⎦
⎣
1 ⎞⎤
sin 2 2α ⎛ 1
+
⎜
⎟ ⎥ σn
4
⎝ kn1S1 kn 2 S2 ⎠ ⎥⎦
⎡⎛ 1
⎤
1 ⎞
[20d] γ xz = ⎢⎜
−
⎟ sin 2α ⎥ σ n
⎢⎣⎝ kn1S1 kn2 S2 ⎠
⎥⎦
In this case the computed strain components are again
identical to those obtained from analytical relations
(eq. [20]) as shown in Table 2. Again, further tests were run
using the cyclic substitution of axes and a large number of
elements.
Poroelastic deformation
Another method of linear verification is the poroelasticity
approach, which governs the elastic behavior of porous media saturated with a fluid. The general approach is the theory
of poroelasticity as first presented by Biot (1941). When the
porous medium is not an isotropic material, the classic isotropic poroelasticity theory is no longer valid and the more
general anisotropic poroelasticity theory must be invoked.
Consider an isotropic saturated porous rock having the
same dimensions as the intact rock of Fig. 2a, intersected by
three mutually perpendicular fracture sets whose normal
vectors are parallel to the coordinate axes. This configuration is shown in Fig. 4.
The equivalent continuum resulting from the presence of
fractures intersecting an isotropic rock is an orthotropic
medium. In this case, we apply the general form of poroelasticity equations
[21]
␧ = C␴ – (C′Bp/3)
where ␧ is the strain tensor, ␴ is the stress tensor, p is the
pore pressure, C is the compliance matrix for orthotropic
material (fractures plus isotropic rock), C ′ is a function of
the sum of all terms in the compliance matrix, porosity, fluid
compressibility etc. (Cheng 1997), and B is the general representation of Skempton’s pore pressure parameter and has
different values in different directions as follows:
[22a] Bx =
3 (C11 + C12 + C13) − C s
C′
[22b] By =
3 (C 21 + C 22 + C 23) − C s
C′
© 2006 NRC Canada
Bagheri and Settari
579
Fig. 4. Orthotropic porous rock for verification of the model using poroelasticity theory.
Table 3. Results of poroelasticity verification using either
Cheng’s approach (1997) or the current study.
∆σx
∆σy
∆σz
[22c] Bz =
[23]
−v
E
1
+ S 1k
E
2 n2
−v
E
0
εx
εy
εz
0.0
–0.000 893 0
–0.000 797 9
Note that “Cs” is the grain (solid) modulus. These equations are based on micromechanical analysis (e.g., Cheng
1997). Following the discussion presented in the section entitled “Constitutive equation of jointed rock masses”, the
orthotropic compliance matrix for the specimen to be substituted in eq. [21] is derived as
3 (C 31 + C 32 + C 33) − C s
C′
⎡1 + 1
⎢ E −vS1 kn1
⎢
E
⎢ −v
E
C=⎢
0
⎢
⎢
0
⎢
⎢
0
⎣
636 kPa
0.0
0.0
−v
E
−v
E
1
+ S 1k
E
3 n3
0
0
0
0
0
0
0
2(1 + v )
+ S 1k + S 1k
E
1 s1
2 s2
0
0
0
0
2(1 + v )
+ S 1k + S 1k
E
2 s2
3 s3
0
0
0
0
Young’s modulus of 1 GPa, Poisson’s ratio of 0.2, and a
very high grain (solid) modulus of 3.0 × 1013 kPa are assigned to the porous rock. Each fracture set has a normal
stiffness of 2 × 108 kPa/m. Fracture spacings of 0.05 m,
0.03 m, and 0.07 m are assigned to the fracture sets whose
normals are parallel to either the “x,” “y,” or “z” axis, respectively. Moreover, a uniform initial compressive stress of
6000 kPa is imposed on the system. The specimen experiences an increase in pore pressure while its front (x = Lx),
back (x = 0), right (y = Ly) and bottom (z = 0) faces are restricted from moving in the “x,” “x,” “y,” and “z” directions,
respectively. The analysis is isothermal.
A single step increase in pore pressure from its initial
value of 1 kPa to 1000 kPa creates changes in stresses and
strains, as shown in Table 3. The model and the analytical
method of Cheng (1997) are in perfect agreement.
Nonlinear verification
In the previous section, joints had constant parameters and
there was no limitation on the amount of deformation a joint
could undergo. As a result of constant joint parameters, the
overall compliance matrix was also constant, and a single
step load was sufficient to find the overall deformation.
While convenient for testing, this can result in unrealistic behavior.
In this section, a nonlinear joint model (discussed in the
section entitled “Single joint deformation”) is assigned to
the joints and as a result, normal stiffness is no longer a constant. High shear stiffness is assumed to eliminate its effect
⎤
⎥
0
⎥
⎥
0
⎥
0
⎥
⎥
0
⎥
2(1 + v )
1
1 ⎥
+
+
E
S 1 ks1
S 3 ks3 ⎦
0
on deformation. A linear variation of normal stress with time
is applied on the top of a rock sample containing a single
horizontal fracture. The Young’s modulus of the intact rock
and the size of the block were picked from a range of data
given in Bandis et al. (1983) for their experiments. The initial stress was zero.
Figure 5 shows the effect of fracture on the overall deformation of the rock mass and qualitatively verifies results of
the model. Normal deformation of intact rock was assumed
to be linear in the model, while the experiment has a small
deviation from a straight line. When the system is at low
stress, a small increase in normal load will create a relatively
high total deformation, which demonstrates high deformability of fracture at low stress. At high stresses the slope of
the rock mass deformation curve is almost equal to that of
intact rock, indicative of reaching the maximum closure of
the fracture. This can be seen from the fracture deformation,
which is also plotted in Fig. 5. The vertical trend of this
curve indicates asymptotically decreasing closure across the
fracture (mathematically an infinite normal load is required
for a complete closure).
Accuracy of the nonlinear solution
The dynamic problem must be solved using load steps.
Depending on the initial effective stress level, choosing
proper load increments may be very important for the accuracy of the computations, owing to the nonlinearity of the
stiffness. In our implementation, the fracture normal stiffness is updated one load step behind the load increment. In
© 2006 NRC Canada
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Can. Geotech. J. Vol. 43, 2006
Fig. 5. Verification of the model using Bandis experiment (1983).
Fig. 6. Effect of load steps on the overall deformation of rock mass (Bandis experiment 1983).
other words, equivalent properties at each load increment are
computed from the rock and fracture stiffnesses at the end of
the preceding increment. In a loading cycle, a higher deformation than actual will result, owing to increasing stiffness
of fractures with time. The reverse behavior is expected in
the case of load decrement where smaller expansion would
be computed. To ensure an accurate deformation solution, a
sensitivity analysis has been performed. Figures 6 and 7
show the effect of load step size on the accuracy of the deformation of jointed rock mass and of the joint itself. Larger
load steps obviously result in larger errors, and in some
cases a negative fracture aperture was calculated (e.g., for a
load increment of 1 MPa the final fracture closure is about
0.1 mm, which is larger than the maximum possible closure
of 0.08 mm). A load step of 10 kPa is sufficiently accurate
as seen in comparison with the 5 kPa step. Obviously, the
optimum step depends on the joint properties used and the
range of the load. It would be desirable to develop a
dimensionless criterion for automatic selection of the load
steps.
This has potentially large implications for the computing
efficiency of the coupled solutions of fluid flow and
© 2006 NRC Canada
Bagheri and Settari
581
Fig. 7. Effect of load steps on single fracture deformation (Bandis experiment 1983).
geomechanics in fractured rock masses. If the stress path includes the nonlinear region of joint deformation, an accurate
solution will require either a small load (i.e., time) increments or iterative updating of the stiffness matrix during the
load increment. This situation is similar to other nonlinear
problems of computational rock mechanics.
Fig. 8. Hypothetical reservoir coupled with the stress model.
Application in reservoir engineering
The aforementioned stress model has been coupled with a
single porosity flow model to study the flow behavior of hydrocarbon reservoirs. The resulting simulator solves both
flow equations and stress–strain equations iteratively. The
coupling term in this simulator is the porosity term. The details of the coupling method for the simulator used in this
work are found in Settari and Mourits (1994).
As the effective stress acting on the fracture plane
changes, fracture will deform according to eq. [1]. This results in a variation in fracture normal stiffness over time,
which in turn makes the compliance matrix of the fractured
rock mass a nonlinear function of stress. In the coupled flow
and geomechanics simulator, this matrix is recalculated in
the stress module of the system iteratively within a time
step.
The ability of the model to handle fracture deformations
will produce more accurate elemental volumetric strain as
well as a more realistic stress field. The former leads to a
more realistic determination of the porosity variation and total deformation of the reservoir and overburden rocks. The
latter provides the necessary input to calculate variation in
fracture aperture and permeability by some typical law. The
main interest in modeling the variation of the aperture of
fractures in time is to allow for the calculation of the fracture full permeability tensor, which is likely to have changing orientation and will evolve with time. Inclusion of
fracture deformation in a full field coupled simulator also al-
lows the contribution of fractures to compaction and in turn
to reservoir subsidence to be considered.
The presence of deforming fractures alters the calculation
of volumetric strain and in turn the porosity in the flow
model. The volumetric strain of each fractured element in
the stress model is updated every time step based on new
values of normal stiffness. Volumetric strain εv corresponding to a given confining stress state σkl is
[24]
εv = εx + εy + εz
3
⎡ ⎛ 1 − 2v ⎞
⎤
= ⎢3 ⎜
⎟ σ avg + ∑ C ijkl δ ij δ kl σ kl ⎥
m =1
⎢⎣ ⎝ E ⎠
⎥⎦
where m = i, j, k, l.
For a hydrostatic triaxial confining stress, the equivalent
bulk modulus of fractured rock is
[25]
3
⎡ ⎛ 1 − 2v ⎞
⎤
k * = ⎢3 ⎜
+
⎟ ∑ C ijklδ ij δ kl ⎥
⎢⎣ ⎝ E ⎠ m =1
⎥⎦
−1
© 2006 NRC Canada
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Can. Geotech. J. Vol. 43, 2006
Fig. 9. Effect of fractures on vertical deformation of deformable matrix containing incompressible oil.
Fig. 10. Effect of fractures on vertical deformation of nondeformable matrix containing incompressible oil.
In this work, we restrict ourselves to coupled problems
where the flow equations are solved as a single porosity media. In this case, the equivalent bulk modulus, k*, is sufficient to calculate the porosity changes and compressibility
of the equivalent media in the flow model. This is in contrast
to the dual porosity flow equations where two sets of equations are solved, for fluid in the fractures and in the matrix.
In such a system, one needs to calculate the bulk moduli and
compressibilities of the fractures and the matrix separately
from the stress solution. An extension of this work to the
dual porosity flow systems is described in Bagheri and
Settari (2005). The development of the pseudo-continuum
presented here in terms of total stresses is also valid in terms
of effective stresses, provided that the fluid pressure difference between the matrix and the fractures is small (see
Bagheri (2006) for details). Treatment of nonlinearity in the
equivalent properties of jointed rocks in the coupled simulator was implemented differently from what was described
for a single joint in the section entitled “Accuracy of the
nonlinear solution.” The normal stiffness used in the equiva© 2006 NRC Canada
Bagheri and Settari
583
Fig. 11. Effect of fractures on reservoir pressure and production of incompressible oil.
Fig. 12. Effect of fractures on vertical deformation of deformable matrix containing compressible oil.
lent moduli of fractured rock mass is the average of the old
and the new (current) normal stiffness. This parameter is updated at every iteration loop during a time step.
Averaging the normal stiffness of fractures results in
better accuracy and will alleviate the problem of controlling
the load (time) steps. However, monitoring of stress (or pressure) changes over the time step is necessary to ensure accuracy. It is also noted that fractures in reservoirs usually do
not start deforming from zero effective stress (except in
highly overpressured reservoirs). In most cases fractures
start deforming from the less sensitive part of the fracture
deformation path.
The effect of production on deformation of fractures will
now be studied using test examples. The effect of several parameters, including intact rock stiffness and fluid compressibility, was studied. Figure 8 shows a hypothetical single
phase oil reservoir, which contains dead oil. The grid is 4 by
4 by 1 with ∆x = ∆y = ∆z = 10 m. The initial in-situ porosity
is 35%, and the reservoir permeabilities are 1000 D in all directions to cause uniform pressure changes throughout the
© 2006 NRC Canada
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Can. Geotech. J. Vol. 43, 2006
Fig. 13. Effect of fractures on reservoir pressure and production of compressible oil.
entire reservoir. Oil has a constant formation volume factor
of 1.0 (zero oil compressibility), viscosity of 1.0 cP (1 P =
0.1 Pa·s), and a density of 800 kg/m3. The initial fluid pressure is 14 450.8 kPa at a depth of 805 m.
The rock properties were based on data from Bandis et al.
(1983). The elastic modulus is 51 GPa, and the Poisson’s ratio is 0.2. The initial horizontal stresses are 16 000 kPa and
the initial vertical stress is 18 000 kPa over the entire reservoir depth. A set of equal roughness horizontal fractures is
assumed to intersect all grid blocks of the model. These
fractures have a normal stiffness of 19.35 GPa/m at zero
stress, a shear stiffness of 5 × 1099 kPa/m (a static parameter
as discussed earlier), and an aperture of 0.149 655 mm. It is
assumed that the fracture can close completely if enough
stress is applied (zero roughness fracture). Several configurations of fracture intensity and rock properties have been
considered. Corresponding initial apertures of the fractures
and initial normal stiffnesses were calculated using the relations from Bandis et al. (1983). It is worth noting that these
correlations may produce unrealistic data for low strength
materials (especially for initial aperture and maximum joint
closure).
A vertical well was located in the cell (1,1,1), producing
1 m3 of oil at surface conditions until the reservoir pressure
was depleted to 500 kPa. Depending on the case, the model
was run for 50 or 500 days. Figure 9 depicts the deformation
of the structure for various fracture intensities. In these figures (Figs. 9–13), a conventional reservoir with no fractures
is indicated as “Conventional Res.”; a fractured reservoir
with 1 m fracture spacing is labeled “Low Intensity”; the
case with fracture spacing of 10 cm is labeled “High Intensity”; and the case with 1 cm spacing is denoted as “Very
High Intensity.” As fracture intensity increases, the overall
deformation also increases. The amount of contribution of
the fractures shows the importance of considering fractures
in bulk deformation. At the end of the reservoir life, the cal-
culated fracture contribution to the deformation of the high
intensity fractured reservoir lies exactly on the top of a
curve, which is obtained by subtraction of the overall deformation of the conventional reservoir from that of the high
intensity case. These two curves are not equal at the beginning of production simply because of the fracture contribution to reservoir compaction, which causes a difference in
pressure decline of these two reservoirs. The gap between
these two curves is due to the rate of matrix deformation.
For a nonfractured reservoir, the rate of pressure change is
higher than that in a fractured reservoir. As a result, the matrix will deform faster in the conventional reservoir. However, because of the contribution of fracture deformation, the
overall rate of the deformation in the high intensity case is
higher. This makes the difference in the overall deformations of these two cases lower than the calculated fracture
contribution in deformation. This difference disappears
when very high Young’s moduli of rock matrix are used.
Figure 10 shows such a case and demonstrates that for
nondeformable rocks, all of the deformation in the model is
caused by fractures.
The effect of fractures on compaction can also be observed from their impact on fluid flow. Higher intensity of
fractures increases the compaction of the reservoir, which in
turn results in a slower pressure decline and maintains longer production. This fact is demonstrated in Fig. 11 for an
incompressible oil reservoir.
In a more realistic reservoir where fluid is compressible,
the effect of fractures on solid deformation remains the
same; but their impact on fluid flow parameters tends to diminish. Figure 12 shows the impact of fracture intensity on
the same reservoir containing oil with compressibility of
10–5 kPa–1. Slower pressure decline in a compressible system
masks the influence of fractures on pressure maintenance.
As a result, the gap observed in an incompressible system is
not seen here. Comparison of this figure and Fig. 9 reveals
© 2006 NRC Canada
Bagheri and Settari
that the overall deformations in both systems are exactly the
same. When a fluid system is compressible, fractures do not
play a significant role in pressure decline until their density
becomes large. Figure 13 shows the effect of fracture compaction on fluid flow. In this figure, the solid line curve,
which has the lowest slope, represents pressure drop in a
fractured reservoir with spacing equal to 1 cm (very high
intensity fractured reservoir, which is too dense for most actual reservoirs). Fracture spacing must be very small to
result in an appreciable influence on reservoir pressure. This
fact emphasizes that the main effect of fractures on fluid
flow coupling is not the effect of fracture deformation on
compressibility. The main coupling to fluid flow is through
the effect of fracture deformation on reservoir permeability
as shown in Bagheri and Settari (2005). However, fracture
deformation, as shown by the examples herein, plays an important role in the overall deformation of the system, regardless of system compressibility.
Conclusions
A reliable methodology for the determination of equivalent moduli of jointed rock mass was presented and implemented for modeling fracture contribution to deformation
and flow solution in a coupled geomechanical and reservoir
simulator. The methodology is flexible enough to accommodate any number of fracture sets of any fracture orientation
and properties. The interesting feature is that the deformation characteristics of fractures are included without treating
them explicitly. In addition to modeling the role of fractures
in overall deformation, fracture aperture can be calculated as
a parameter needed for the calculation of the dynamic permeability tensor on an element basis.
Deformation of fractures plays an important role in the
overall deformation of fractured rocks, but has only minor
impact on the flow behavior of fractured reservoirs, if its effect on permeability is not considered. Neglecting the contribution of fractures on overall deformation could lead to
unrealistic predictions of deformation, especially in highly
fractured rocks. The effect of fracture deformation on reservoir permeability also plays a significant role in fluid flow
of fractures reservoir. This coupling will have appreciable
influence especially in a low compressibility system.
References
Amadei, B., and Goodman, R.E. 1981. A 3-D constitutive relation
for fractured rock masses. In Proceedings of the International
Symposium on the Mechanical Behavior of Structured Media,
Ottawa. pp. 249–268.
Bagheri, M. 2006. Modeling geomechanical effects on the flow
properties of fractured reservoirs. Ph.D thesis, University of Calgary, Calgary, Alta.
Bagheri, M., and Settari, A. 2005. Modeling of geomechanics in
naturally fractured reservoirs. In Proceedings of the Reservoir
Simulation Symposium, The Woodlands, Tex., 31 January –
2 February 2005. Society of Petroleum Engineers, Richardson,
Tex.
Bandis, S.C., Lumsden, A.C., and Barton, N.R. 1983. Fundamentals of rock joint deformation. International Journal of Rock Me-
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chanics and Mining Science and Geomechanics Abstracts, 20:
249–268.
Barton, N.R., and Choubey, V. 1977. The shear strength of rock
joints in theory and practice. Rock Mechanics, 10: 1–54.
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Goodman, R.E. 1976. Methods of geological engineering in discontinuous rocks. West Publication Company.
Huang, T.H., Chan, C.H., and Yang, Z.Y. 1995. Elastic moduli for
fractured rock mass. Rock Mechanics and Rock Engineering,
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Settari, A., and Mourits, F.M. 1994. Coupling of geomechanics and
reservoir simulation models. In Proceedings of Computer
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Morgantown, W. Va., 22–28 May 1994. Edited by H.J.
Siriwardane and M.M. Zaman. A.A. Balkema, Rotterdam, The
Netherlands. Vol 3, pp. 2151–2158.
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List of symbols
a, b constants
A total area of a fracture set in a block
B matrix of Skempton’s pore pressure parameters
Cs grain (solid) modulus
C compliance matrix
D constitutive matrix
E Young’s modulus
G shear modulus
H height of sample
k joint stiffness
k* bulk modulus
L dimensions of the sample
L transformation matrix
n joint normal vector
p pore pressure
S joint spacing
v Poisson’s ratio
V volume
α dip angle
δ Kronocker delta
␦ joint deformation vector
ε normal strain
␧ strain tensor
γ shear modulus
ν normal deformation
σ stress
␴ stress tensor
␶ traction vector
Superscripts
f individual fracture set
I intact rock
J joint
© 2006 NRC Canada
586
Can. Geotech. J. Vol. 43, 2006
M number of joint sets
Fig. A1. Test example used to expand eq. [A1].
Subscripts
avg mean (average)
i initial
i,j,k,l tensor indices
I,J,L tensor indices
j joint
max maximum
n normal
N,S,T joint coordinate system (normal, strike, dip)
s shear
t total (joint + rock)
v volumetric
x,y,z directions
Appendix A
Figure A1 shows an element of intact rock, which is intersected by three sets of fractures. The first two sets of fractures are orthogonal to the third set, and have an angle of θ =
2α between them. These two sets are assumed to have the
same normal and shear stiffness, kn, ks, and spacing, S,
which are different from that of the third joint set parameters, kn3, ks3, S3. This example is presented in Huang et al.
(1995), where eq. [A1] has been used to derive the equivalent moduli, neglecting the dilatancy effect.
M
[A1]
f
f
C Jijkl = ∑ n if L jJf D JL
L Ll
n kf
f =1
1
Sf
The resulting moduli according to Huang et al. (1995) are
as follows:
[A8]
1
1
2 sin(2α) cos 2 α
=
+
G xy
S3ks3
Sks
[A9]
2 sin 2 (2α)(ks + kn )
1
=
G yz
Skn ks
[A10]
1
1
2 sin(2α) sin 2 α
=
+
G zx
S3ks3
Sks
where G is the shear modulus of the rock mass.
It seems however that eqs. [A8]–[A10] are incorrect. This
can be demonstrated by setting α = 0 and neglecting the
third fracture set for clarity without losing the generality of
the discussion. In this case all shear terms are equal to zero,
while we have two horizontal sets of fractures intersecting
the “z” axis with an angle of 90°. The shear modulus in “zx”
and “zy” should be affected by the fracture shear deformation term, 1/Sks, where in this case it has to be multiplied by
a factor of 2 owing to the presence of two sets of fractures
with the same properties.
Equations [A8]–[A10] have been re-derived in this work
using the same method, as follows:
[A2]
1
1
=
Ex
kn3S3
[A3]
1
2 sin 2 α(kn cos 2 α + ks sin 2 α)
=
Ey
ks kn S
[A4]
1
2 cos 2 α(kn sin 2 α + ks cos 2 α)
=
Ez
ks kn S
[A11]
1
1
2 sin 2 α
=
+
G xy
S3ks3
Sks
[A5]
−
kn − ks
sin 2 (2α)
2 kn ks S
[A12]
2 [sin 2 (2α)ks + cos 2 (2α)kn ]
1
=
G yz
Skn ks
[A6]
−
[A13]
1
1
2 cos 2 α
=
+
G zx
S3ks3
Sks
[A7]
−
vyz
Ez
vxy
Ex
=−
=−
vzy
Ey
vyx
Ey
=−
=0
vzx
v
= − xz = 0
Ez
Ex
These new formulae give the expected shear terms for the
case of α = 0 and in other simple cases where eqs. [A8]–
[A10] are unable to do so.
© 2006 NRC Canada