1. INTRODUCTION

Fluid injection into naturally fractured geological media

can induce seismicity over a wide range of scales. An

understanding of the physical processes of induced

seismic and micro-seismic events helps to better assess

potential seismic hazards associated with, e.g., CO2

sequestration and wastewater injection, as well as to

assist stimulation of hydrocarbon and geothermal

reservoirs with ultra-low permeability. The triggering

mechanism of fluid injection induced shear re-activation

on pre-existing fractures is fundamentally a coupled

hydro-mechanical process. The presence of a discrete

fracture network (DFN) imposes significant challenges

on numerical modeling of this coupled process, due to

not only the geometric complexity, but also two sets of

material properties and constitutive laws for both the

fluid and the solid, as well as discontinuous changes in

modeling targets.

In the fluid problem, a prevalent approach to capture the

contribution of fractures to flow is a so-called dual-

porosity double-permeability (DPDP) model (Barenblat

et al., 1960; Warren and Root, 1963). Two sets of

governing equations are formulated for the fracture

domain and matrix domain, respectively, and interact in

response to pressure gradient through mass exchange.

However, a prerequisite for using such a model is to

regularize the fractured medium into a sugar-cube

representation for calculation of certain up-scaled

properties, namely, shape factors (Lim and Aziz, 1995).

The distribution the DFN is not explicitly represented

unless in simple cases with repetitive fractures (e.g.,

Gilman and Kazemi, 1983). For preservation of DFN

distribution, an alternative, which is reminiscent to

DPDP model, is to split the fractures and matrix into two

separate computational domains and reduce the mass

exchange term into a source term (Norbeck et al., 2015).

However, this approach relies on a different set of

averaged properties. Another alternative, namely discrete

fracture models, has been proposed in which fracture

flow is well captured, but the matrix flow is typically

neglected (e.g., Erhel et al., 2009; Hyman et al. 2015),

empirically based (Unsal et al., 2010) or averaged

(Sandve, 2014). It is important to recognize that, when

coupled to poroelastic stressing for the study of injection

induced seismicity, it is desirable to conserve the

distribution of at least the large-scale heterogeneities,

e.g., faults and fracture, as these often can cause

statistically significant variations (Berkowitz, 2002;

Vujevic´ et al., 2014; Hirthe and Graf, 2015; Hardebol et

al., 2015). The pressure distribution with fidelity to pre-

exiting fractures fundamentally controls the induced

stress field and thus has important consequences on the

stress paths on fractures.

ARMA 16-175

Including a Stochastic Discrete Fracture Network into

One-Way Coupled Poromechanical Modeling of Injection-Induced

Shear Re-Activation

Jin, L. and Zoback, M.D.

Geophysics Department, Stanford University, Stanford, CA, U.S.A.

Copyright 2016 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 50th US Rock Mechanics / Geomechanics Symposium held in Houston, Texas, USA, 26-29 June 2016. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: We present a finite element method for the one-way coupled poromechanical modeling of injection-induced shear

re-activation in a porous medium embedded with a highly conductive pre-existing discrete fracture network (DFN). The fluid

problem is formulated over an integrated matrix-fracture domain by permitting two sets of flow constitutive laws and by admitting

discontinuities in normal fluid fluxes across fractures to account for matrix-fracture mass exchanges. Based on a transversal

uniformity assumption, a novel hybrid-dimensional approach is proposed where factures need not be explicitly meshed along

normal directions, but are modeled as linear line elements tangentially conforming to the edges of linear triangular elements

representing the porous matrix. Fracture nodes holds no additional degrees of freedom. A dimensional transformation matrix is

introduced during finite element interpolation, leading to three additional equi-dimensional modification terms to the mass and

stiffness matrices to account for contribution of fractures to flow. The gradient of the modeled fluid pressure is then passed as an

equivalent body force vector to the solid problem to solve for induced poroelastic stresses by assuming a single solid constitutive

law for the medium. Finally, Coulomb stresses on fractures are calculated for determining onset of shear re-activation.

The modeled pressure is subsequently considered as an

input for the geomechanical analysis of fluid-induced

seismicity, assuming other inputs, including initial in-

situ stresses and pore pressure, fracture orientations and

frictional strengths, are known a priori. The new state of

effective stress tensor is typically obtained through a

decoupled process by directly subtracting the new pore

pressure from the Cauchy total normal stresses, while

shear stresses remain unaffected. Such an approach,

which largely remains as the basis of many analyses of

induced seismicity (e.g., Terakawa, 2012), neglects the

poroelastic modification to the full stress tensor arising

from pore pressure gradient acting as a body force when

the pore pressure is spatially non-uniform. As has been

shown analytically, poroelastic coupling can have

substantial impacts on induced seismicity even for the

simplest possible cases (Segall et al., 1994; Altmann, et

al., 2014; Segall and Lu, 2015).

In this study, we numerically model one-way coupled

fluid flow and induced poroelastic stressing in a porous

medium embedded with a stochastic discrete fracture

network. The objective is to provide an efficient finite

element modeling strategy for arriving at accurate inputs

for the analysis of injection-induced seismicity, by

explicitly resolving the fractures in the flow modeling to

obtain a pressure field faithful to the DFN, and by

deriving a new effective stress state with considering

poroelastic effect. In the fluid flow problem, fractures

are assumed to be highly conductive and pressure is thus

continuous at fracture-matrix interfaces. An asymptotic

approach is used where fracture flow is assumed to be

transversally uniform, and fractures are modeled as

lower-dimensional linear elements conforming to the

edges of higher-dimensional linear elements of the

porous matrix. Fractures thus hold no additional degrees

of freedom. This allows a formulation of the

conservation law by treating fractures and matrix as one

integrated physical domain, and the mass exchange is

expressed via discontinuities in fluid flux (pressure

gradient) across fractures. This differs from a domain

separation approach as is employed in the DPDP model.

Fluid flow is ruled by Darcy’s law in porous matrix and

obeys Poiseuille equation for laminar flow in fractures.

An efficient dimensional transformation procedure is

proposed during finite element discretization, allowing

modification to elemental mass and stiffness matrices

while preserving a standard global matrices assembly

procedure. In the geomechanical problem, we are not

concerned with mechanical discontinuities, owing to our

goal towards the stress state prior to occurrence of slip,

and the fractured medium is considered as linearly

elastic, and a single constitutive relationship is assumed

for the whole medium. The new effective stress state is

solved by passing the pore pressure gradient as an

equivalent body vector into the conservation of linear

momentum equation, as opposed to a decoupled

approach. The new effective stress tensor is then used

for calculating Coulomb stress on each fracture, and the

mechanical process of slip triggering is demonstrated.

2. GOVERNING LAWS

Consider a 2D physical domain with a boundary .

consists of a fracture domain f and a porous matrix

domainm . Here f m , f m . Consider

also that the fracture domain is composed of a set of

fractures such that 1fn

f i i iU f b , where fn is the

number of fractures, if and ib represent the tangential

and transversal extension of the ith fracture if . In the

gridding domain, since i ib f , we let fractures degrade

into 1D entities and be absorbed as if to the boundary

such that m , iif f . In this manner,

ib will be implicitly represented.

2.1. The Fluid Problem Assume an incompressible fluid and linear type pressure

dependent porosities, and consider fractures as a primary

type of pores, the transient case of conservation of mass

in saturated fractured porous medium reads:

0 0m m f fp

C C q st

(1)

where is the constant fluid density; 0m and 0f are

the initial matrix porosity and fracture porosity, mC and

fC are the matrix compressibility and the fracture

compressibility, respectively, q is the fluid flux vector,

and s is the source/sink term.

The fluid flux is ruled by two sets of linear flow

constitutive laws. In the porous matrix domain, q is

given by Darcy’s law:

1

mmq q p p x m mκ k (2)

where is the fluid viscosity, mκ and mk are the

conductivity and permeability tensors, both of which can

be fully anisotropic.

In the fracture domain, q is assumed to be transversally

uniform. This entails only a tangential flow, which in

this study, is approximated using Poiseuille equation for

laminar flow:

1

f f ffq q p k p x

(3)

where is the tangential gradient operator, and f ,

fk are the fracture tangential conductivity and

tangential permeability. fk is related to the fracture

aperture b via cubic law (Witherspoon et al., 1980):

21

12fk b (4)

In addition, we stipulate that the external fluid source s

is provided only within the matrix domain m ,

0 fs x .

2.2. The Solid Problem We consider a one-way coupled poroelastic process.

Assuming a quasi-static state, the conservation of linear

momentum may be written as:

' 0bp f in σ 1 (5)

where p is the injected excess fluid pressure (pressure

above the initial pore pressure) obtained from the fluid

flow modeling, 'σ is the induced changes in effective

stress tensor, 1 is the identity tensor, ' pσ 1 is the

induced changes in Cauchy total stress tensor, and bf is

the body force vector.

The above equation can be further rearranged into:

' 0b pf f in σ (6)

where pf p is an equivalent body force vector. This

term enables a one-way coupled poroelastic process

through which the fluid pressure-induced effective stress

tensor is solved, which can be superimposed onto an

arbitrary initial effective stress state 0'σ . The result is

fundamentally different from 0' ' p σ σ 1 , a prediction

made by a typical approach that assumes the decoupling

between stresses and pore pressure.

Since we are concerned with the stress state before

occurrence of slip, is assumed to be linearly elastic,

and 'σ is related to the changes in displacement vector

u via the classic constitutive law:

' : : s u σ (7)

where s is the symmetric gradient operator, u is the

induced changes in displacement vector, and is the

elastic stiffness tensor, which reads the following under

a plane strain assumption:

2 0

2 0

0 0

(8)

where is the Lame’s constant, and is the shear

modulus.It is important to note that f should be

allowed to follow a separate constitutive law, especially

along fracture tangential directions, since mechanically

weak fractures can perhaps accommodate more shear

deformation even prior to slip. However, this process is

not considered separately in this study.

2.3. Boundary Condition We will study a 2D square reservoir with a circular

injection well at the center. Of our particular interests are

the changes in fluid pressure and induced stresses, which

can be superimposed onto an arbitrary initial state, since

both fluid and solid are ruled by linear constitutive laws.

To model the changes only, here a constant pressure and

zero displacement are prescribed at the well boundary,

whereas the reservoir boundary is subjected to zero flux

and zero traction condition.

, 0w w wp p p on (9)

0w wu u on (10)

0 \ ( )h wq n q on f (11)

0 \ ( )h wt n on f σ (12)

where wp is the excess injection pressure at the well

boundary, hq is the normal fluid flux, ht is the traction,

w is the well boundary, w .

2.4. Frictional Failure Having the new state of effective stresses, the normal

stress and shear stress resolved on a fracture are given

by:

0' ' :n f fn n σ σ (13)

1/2

22

0' ' f nn σ σ (14)

where: fn is the fracture normal vector.

The Mohr-Coulomb shear failure criterion is then used

for determining onset of induced seismicity. Assume that

fractures are cohensionless, the Coulomb Failure

Function (CFF) of a fracture then reads:

f nCFF (15)

where f is the frictional coefficient.

3. WEAK FORMULATION

3.1. The Fluid Problem The key of this study lies in the weak formulation of

Eq.(1), which can be completed upon two assumptions.

First, to facilitate a mixed-dimensional approach, we

make the following transversal uniformity assumption

across the thickness of the fracture:

( ) ( )f f

f d b f d

(16)

where b is the fracture aperture, and indicates fracture

tangential direction, and f is a function related to the

weighting residual and fluid flux. Computationally, this

assumption allows an implicit representation of the

fracture thickness that need not be meshed. See also

Karimi-Fard and Firoozabadi (2003) for a similar

assumption.

Second, we also assume that fractures are highly

conductive, thus fluid pressure is continuous, but fluid

flux/velocity needs not be continuous between fractures

and matrix (Martin et al., 2005).

Weak formulation over domain is decomposed into

two sets of formulations over fracture domain f and

matrix domain m , respectively. In the meantime,

recognize the transversal uniformity assumption in f ,

and admit a fluid flux discontinuity across a fracture by

applying extended divergence theorem (e.g., Martin et

al., 2005; Pouya, 2015), and finally substitute in the two

constitutive laws, and absorb the boundary condition, the

following integral form of Eq. (1) can be arrived:

0 0

\

i im i

im i

i

w m

m m i f ff

i

i ffi

fi fif

i

hf

w C pd b w C pd

w pd b w pd

w q n d

wq d wsd

mκ

(17)

where w is the weighting residual, fin is the norm

vector of ith fracture if , and is defined as:

fi fi fin n n (18)

The discontinuity in fluid flux reads:

fi fifiq q q (19)

Asymptotically, if degrades into 1D and is absorbed as

if to the boundary of the porous matrix, and the flux

discontinuity can further be written as:

m m fi

fi m mq p p κ κ (20)

Eq.(20) can also be viewed as the mass exchange

between if and the surrounding matrix, see also

Noetinger (2015).

3.2. The Solid Problem We are concerned with the stress state prior to fracture

slip, and do not consider fracture opening, thus traction

is continuous across a fracture:

0fit n n σ σ (21)

This assumption, combined with the assumption of

single constitutive law throughout domain (see 2.2),

allows weak formulation of Eq.(6) to lead to the classic

integral form shown below, except with an additional

coupling term resulted from the excess pore pressure

gradient:

\

: :

w

s s

b p hf

u d

f d f d t d

(22)

where is the weighting residual.

4. A HYBRID-DIMENSIONAL FINITE ELEMENT METHOD

4.1. Conforming Discretization and Dimensional Transformation

Spatially discretize the matrix domainm into a set of

linear elements in a fashion such that each lower-

dimensional fractures if conforms to the edges of a

subset of these elements. Denote the matrix node set as

mX and fracture node set as fX , then f mX X . In other

words, fracture nodes hold no additional degrees of

freedom. Elements containing fracture nodes are referred

to as ‘Hybrid elements’ in this study and they constitute H

m . H

m mf .

Consider the excess fluid pressure p and the changes in

displacement vector as primary variables. For a hybrid

element, let ˆHm and ˆH

f represent the vector containing

matrix nodal pressure and fracture nodal pressure,

respectively. Here ˆ ˆH Hf m . Define a ‘dimensional

transformation matrix’ Q such that:

ˆ ˆH Hm f Q (23)

Here Q is matrix composed of 0 and 1, and 1T QQ .

For example, for a 2D triangular element, Q takes the

following different forms, depending on the local matrix

element node number in relation to the fracture line

elements fc (see Table 1).

Table 1: Examples of dimensional transformation matrices

associated with linear triangular elements

For the fluid problem, let mN and fN represent the

shape functions associated with matrix linear 2D

elements and conforming linear fracture line elements,

respectively. One can verify the validity of the following

relationship along fracture tangential directions:

|Tf m fN N Q (24)

4.2. Finite Element Interpolation In general, without specifying if an element is a hybrid

element, we denote the excess fluid pressure on matrix

nodes, fracture nodes and well nodes as ˆm ,

ˆf and

ˆw ,

note ˆ ˆf m . Also denote the induced changes in nodal

displacement vector as d , and two arbitrary vectors as ĉ

and . We propose the following means of interpolation

of excess fluid pressure and its time derivative:

ˆ

ˆ

ˆ ˆ

ˆ ˆ

m m m

m m m

f f f m

f f f m

p N x

p N x

p N N x f

p N N x f

Q

Q

(25)

The equivalent body force vector thus reads:

ˆp m mf N (26)

Fluid flux discontinuity is interpolated as:

ˆ ˆm m

Hi

fi m m m m mq N N x κ κ (27)

where Him indicate the area composed of hybrid

elements associated with ith fracture if .

The rest of the variables are interpolated in standard

ways:

ˆ ,m m

s

m m

s

m

w N c

u d x

N

B

B

(28)

where Bm is the standard displacement-strain

transformation matrix, and mN is the shape function for

the solid problem.

4.3. System of Equations Substitute Eq.(9)~Eq.(12) and Eq.(25)~Eq.(28) into

Eq.(17) and Eq.(22), apply Galerkin approximation,

meanwhile, honor Eq.(24), one arrives at the following

system of equations in matrix form:

ˆ ˆ F M K (29)

d YG (30)

where:

0

0

m

i i i ii

T

m m m m

T T

i f f f ff

i

N C N d

b N C N d

M

Q Q (31)

m

i i ii

i fi m mi

T

m m

TT

i f f ff

i

T T T

f m mf

i

N N d

b N N d

N n N N d

mK κ

Q Q

Q κ κ

(32)

w

T

m w wF N N p d

mκ (33)

T d

G B B (34)

T

m pY f d

Ν (35)

Compared Eq.(31) and Eq.(32) with the formulations for

porous flow only, here three modification terms are

introduced. In M, the second term indicates the

contribution of fractures as a primary type of pores; In

K, the second term arises from the tangential flow along

fractures. We note that the meaning of this term concurs

with a global matrix superposition approach (Baca and

Arnett, 1984; Kim and Deo, 2000), as well as a so-called

multiscale finite element method (Zhang et al., 2013).

However, here the effect of fracture thickness is included

through rigorous weak formulation. In addition, we also

include the third term in K, which introduces asymmetry,

to accounts for the discontinuity in pressure normal

gradient, which fundamentally arises from the high

conductivity of the fractures such that they act as the

preferred flow channels as well as fluid sources to the

Local node number Q

1 0 0

0 1 0

1 0 0

0 0 1

0 1 0

0 0 1

surrounding matrix. This term is especially important for

studying poroelastic stressing, as it predicts an

equivalent body force vector acting towards the fractures

that could potentially increase the normal stresses on

fractures. Elementwise, these three terms appear for

hybrid elements and vanish for others, thus allowing the

development an independent subroutine, and the

standard global matrices assembly process can be

employed.

4.4. Time Discretization and One-Way Sequential Coupling

The transient flow problem Eq.(29) is solved using a

fourth-order explicit Runge-Kutta method. A one-way

sequential coupling scheme is adopted. At selected time

steps, the excess pressure distribution is used to calculate

the gradient vector and then passed to Eq.(30) to solve

for nodal displacement changes, before calculating the

induced strain and stresses.

5. NUMERICAL EXAMPLE

5.1. Model Set-up, Conforming Meshing and Nominal Parameters

A 2D porous reservoir m embedded with a

stochastic discrete fracture network f composed of 100

1D zero-thickness fractures is generated, see Fig.1. The

reservoir is 200m by 200m, with a cylindrical hole w

of radius 5m at the center. Centers of fractures are

distributed following a non-uniform distribution and are

more concentrated in the inner reservoir; Fracture

lengths and orientations are generated following uniform

distributions between [20m, 50m] and between [0°,

360°], respectively. The fracture network is partially

interconnected.

Several meshing tools are available to generate

conforming meshes as are required by this study (e.g.,

Erhel et al., 2009; Hyman et al. 2015). Here an open-

source code DistMesh (Persson and Strang, 2004) is

modified to incorporate the above stochastic DFN and

discretize the reservoir into a set of linear triangular

elements, and fractures are resolved as conforming linear

line elements. The final representation of the DFN might

be slightly different, but a satisfactory preservation is

maintained, see Fig.2. The highlighted elements are

identified hybrid elements (two of the three nodes are

also fracture nodes) colored by their corresponding

fracture indices.

The permeability tensor of a matrix element mk is

assumed to be simply anisotropic: [ 0; 0 ]m mx myk kk . A

list of the nominal parameters used in this study is given

in Table 2.

Fig.1. A 2D reservoir embedded with a stochastic DFN

Fig.2. Conforming meshing and hybrid elements

Table 2 Model parameters

Variables Values

0f 1

0m 0.24~0.26, random

fC 10-7 Pa-1

mC 10-9 Pa-1

fk 1000D (b=0.11mm)

mxk 0.9~1.1mD, random

myk 0.9~1.1mD, random

10-3 Pa·s

wp 5 MPa

16 GPa

16 GPa

f 0.6

https://scholar.google.com/citations?user=STt3tpcAAAAJ&hl=en&oi=sra

5.2. Excess Fluid Pressure The simulated injection time is 93 minutes in total. Fig.3

gives the distribution of excess fluid pressure obtained

by solving Eq.(29). Specifically, (a) shows the spatial-

temporal distribution of , with a well delineated

pressure front constrained in between two solid lines that

represent the analytical pressure fronts in a

homogeneous porous medium when the hydraulic

diffusivity is equal that of matrix and of fractures,

respectively. The heterogeneous distribution shows the

contribution of the pre-existing DFN to fluid flow. (b),

(c) and (d) show the spatial distributions of at time is

equal to 6 minutes, 40 minutes and 93 minutes since

onset of the injection. As can been seen, our hybrid

dimensional approach well captures the flow along the

DFN, which acts a preferred fluid channel and a fluid

source to the surrounding matrix due to its high

conductivity. The canyon circle indicates the analytical

pressure front of a pure matrix flow.

Fig.3. Hybrid-dimensional finite element solution for the

excess fluid pressure . (a) Spatial-temporal distribution with

analytical constraints; (b)~(d): Spatial distribution at 6 min,

40min and 93min since onset of injection.

5.3. Induced Poroelastic Stressing Taking fluid flow simulation result, fluid-induced

changes in displacements are then solved from Eq.(30)

at all selected time steps. At each selected time step,

poroelastic stresses (newly induced effective stresses)

are calculated according to Eq.(7). Fig.4 gives an

example showing different stress components at 93

minutes under a zero traction boundary condition. It can

be seen that the changes are most prominent around the

DFN, but also noticeable outside the excess pressure

front even though this area is not in direct contact with

the fluid. In addition, the result also shows anisotropic

changes in normal stresses and an additional shear stress

field. All these changes can only be predicted by

including poroelastic coupling.

Pressure front of

pure fracture flow

Pressure front of

pure matrix flow

(a)

(b)

(c)

(d)

Fig.4. Newly induced stresses due to poroelastic effect at

93min since onset of injection. (a) Effective normal stress

along x; (b) Effective normal stress along y; (c) Shear stress.

5.4. Induced Shear Failure We assume an initial compressive effective stress tensor

0' σ [6MPa 0; 0 2.5MPa] such that all fractures are

initially below the shear failure line. The newly induced

changes from 5.3 are then superimposed onto 0'σ to

calculate the Coulomb stresses on all fractures according

to Eq.(13)~Eq.(15), and the locations of their stress state

in relation to the shear failure line are determined. Fig.5

gives the result at three selected time steps. The result

shows complex modification to the stress state on

fractures due to poroelastic effect. Each circle to the left

of the shear failure line represents a fracture that is

induced to slip as a result of fluid flow and poroelastic

stressing. Locations of these shear re-activated fractures

are shown in Fig.6.

Fig.5: Changes in effective normal stress and shear stress

resolved on all fractures in relation to the shear failure line,

colored by their Coulomb failure functions. (a) 6min; (b)

40min; (c) 93min

(a)

(b)

(c)

(a)

(b)

(c)

Fig.6. Re-activated fractures (green) and stable fractures (blue)

upon injection and poroelastic stressing; Background color

indicates excess fluid pressure. (a) 6min; (b) 40min; (c) 93min

6. DISCUSSION AND CONCLUSIONS

In the fluid flow problem, although the existing

framework of finite element method can be used by

modeling fractures as equi-dimensional elements (e.g.,

Geiger, 2004), explicit representation of fracture

thickness requires extremely fine meshing within the

fracture domain. This is computationally impractical,

especially when there is a large number of fractures of

thicknesses usually orders of magnitude smaller than

typical mesh sizes. Our novel hybrid-dimensional

approach addresses this problem by modeling fractures

as lower dimensional elements tangentially conforming

to matrix elements, while implicitly accounting for the

fracture transversal dimensions. Our weak formulation

and dimensionally-compensated finite element

interpolation show that, compared to a pure porous

matrix flow, the changes associated with fractures

translate to a few additional terms in the flow mass and

stiffness matrices M and K. At an element level, a

subroutine can easily be developed to modify the

element-wise Me and Ke, and a standard global matrices

assembly procedure can be employed before solving the

system of equations. In addition, compared to the DPDP

model, this method offers two important advantages.

First, the problem can be formulated within one

integrated domain, without introducing two sets of

governing equations that interact via a mass exchange

term. Second, the distribution of the DFN can be

accurately preserved, without the need for characterizing

and regularizing the media and calculating averaged or

up-scaled hydraulic properties.

In the solid problem, we considered fluid-to-solid

coupling, which was shown to have considerable impact

on Coulomb stresses on fractures. However, solid-to-

fluid coupling, which can be expressed by the

volumetric strain acting as a source term in the fluid

equation, was not included in this study. In addition, a

single solid constitutive law was assumed for the

fractured medium. However, we note that the hybrid-

dimensional approach proposed for the flow problem

can also be employed for the solid problem to allow an

additional tangential constitutive law for the fractures.

The finite element modeling framework we provided

here is shown capable of modeling of a distribution of

fluid pressure and induced poroelastic stresses with high

fidelity to the pre-existing discrete fracture network, and

can be used for arriving at some of the most critical

inputs for the mechanical analysis of injection-induced

shear re-activation in fractured porous media.

ACKNOWLEDGEMENT

We thank the Stanford Rock Physics and Borehole

Geophysics Project for the financial support.

(a)

(b)

(c)

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