Simulation of Coupled Single-Phase Flow and … · SIMULATION OF COUPLED SINGLE-PHASE FLOW AND...
Transcript of Simulation of Coupled Single-Phase Flow and … · SIMULATION OF COUPLED SINGLE-PHASE FLOW AND...
SIMULATION OF COUPLED SINGLE-PHASE
FLOW AND GEOMECHANICS IN
FRACTURED POROUS MEDIA
A REPORT
SUBMITTED TO THE DEPARTMENT OF PETROLEUM
ENGINEERING
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
By
Karine Levonyan
August 2011
I certify that I have read this report and that in my
opinion it is fully adequate, in scope and in quality, as
partial fulfillment of the degree of Master of Science in
Energy Resources Engineering.
Hamdi A. Tchelepi(Principal advisor)
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Abstract
Accurate predictions of the complex interactions between fluid-flow and mechanical
deformation in fractured geologic formations is of interest in a wide range of reservoir
engineering applications including subsurface CO2 sequestration, heavy-oil recovery,
and wellbore stability. In this report, we describe a fully implicit method for coupled
geomechanics and fluid flow in naturally fractured porous rocks. Specifically, we study
single-phase fluid flow and poroelastic mechanical deformation for two-dimensional
domains. A Discrete Fracture Model (DFM) with a static unstructured computational
grid is employed, in which the fractures are represented explicitly as low-dimensional
objects embedded in the matrix. The fracture segments lie at the interface between
matrix elements (control volumes). The combination of unstructured grids and a
DFM-based description allows for accurate representation of the complex geometry
and the wide range of length-scales often observed for naturally fractured formations.
We assume that the deformations are small, so that the computational grid remains
as a function of time. We also assume that all the fractures are already present
and represented explicitly, and that no failure (i.e., fracture propagation) will take
place. A low-order finite-volume method is used to discretize the mass conservation
equations of the matrix and the fractures. A finite-element method is used for the
mechanics problem, in which a double-node numbering scheme is used for the fracture
segments. The double nodes are enriched with additional degrees of freedom that are
chosen to enforce the equilibrium conditions at the two fracture surfaces represented
by the segment. The appropriate equilibrium conditions at discrete fracture segments
and how to enforce them in the numerical model are discussed in detail. Fully
implicit coupling of the flow and mechanics problems is performed by updating the
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primary unknowns, namely, displacement (vector) and pressure, until convergence
is achieved. We validate the numerical model using several simple test cases of
two-dimensional domains with one, or several intersecting, fractures for a variety
of boundary conditions and matrix-fracture properties.
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Acknowledgements
My time as a master student at Stanford has exceeded my wildest dreams. It has
been exciting and rewarding. I have had benefit of a company of a large number of
excellent people. I would like to take this opportunity to express my gratitude towards
several of them who have had the greatest impact on my academic endeavours.
First and foremost, I would like to genuinely acknowledge my advisor, Prof. Hamdi
Tchelepi. He has guided and inspired me throughout the last few years.
I am indebted to ERE faculty members, my friends, colleagues and officemates. I
would like to thank Denis Voskov for his support and mentorship; Jihoon Kim for our
discussions and for sharing the Matlab code he developed, which became the core of
current algorithm; and Timur Garipov for his constructive comments and invaluable
friendship.
I owe my warmest gratitude to Prof. Vladimir Entov. He was the first to introduce
me to the exciting world of geomechanical modeling in petroleum engineering. A
very special recognition goes to Prof. Entov’s wife, Liya Kaplinskaya, for her endless
support since my very first steps in the U.S. This work would not have been possible
without her encouragement.
I would also like to acknowledge SUPRI-B for financial support of this research.
Last but not least I wish to thank my family for their unconditional love and
support.
v
Contents
Abstract iii
Acknowledgements v
Table of Contents vi
List of Tables viii
List of Figures ix
1 Introduction 1
1.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Treatment of Fractures. Equilibrium Analysis . . . . . . . . . . . . . 3
1.2.1 Incorporating Fractures into the Numerical Reservoir Model . 4
1.2.2 Boundary Conditions on the Fracture . . . . . . . . . . . . . . 5
1.3 Fluid Flow in Fractures . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Mathematical Formulation 11
2.1 Mechanical Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Flow Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Solution Algorithm 16
3.1 Discretization of the Coupled Problem . . . . . . . . . . . . . . . . . 16
3.2 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
vi
4 Numerical Examples 22
4.1 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 Influence of crack pressure on the mechanics . . . . . . . . . . . . . . 28
4.3 Example with two wells . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Conclusions 35
Nomenclature 37
Bibliography 39
vii
List of Tables
4.1 Material parameters for the poroelasticity example model (Lamb et al.,
2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
viii
List of Figures
1.2.1 Idealization of a fracture in a porous medium, w is the ‘effective’ fracture
aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 2D medium with a single fracture. The ‘matrix’ is discretized using triangles,
and the fracture is represented as a one-dimensional (1D) object. . . . . . 5
1.2.3 The force equilibrium condition. The fracture element is in equilibrium
under external compression. Internal forces are positive at ’-’ surface and
negative at ’+’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.4 Geometrical representation of two adjacent control volumes in DFM (fig.
from (Karimi-Fard et al., 2004)) . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Domain Ω intersected by a crack with boundary Γf . . . . . . . . . . . . 12
3.1.1 Typical element and shape functions for the displacement and pressure un-
knowns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.1.1 Displacements is the vertical (z) direction; comparison with the example
in (Lamb et al., 2010) [a] Lamb et al. (2010), FFM-XFEM; min dz =
0.00mm, max dz=4.782mm [b]Current model: double-node FEM - FV; min
dz = 0.00mm, max dz=4.874 mm . . . . . . . . . . . . . . . . . . . . . 24
4.1.2 Pressure Field, comparison with [a] Lamb et al. (2010), FFM-XFEM; [b]
Current model: Double-node FEM - FV . . . . . . . . . . . . . . . . . . 25
4.1.3 Displacements in the vertical (z) direction for four interconnected fractures;
comparison with the example in (Lamb et al., 2010) [a] Lamb et al. (2010),
FFM-XFEM; [b] Current model: double-node FEM - FV . . . . . . . . . 26
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4.1.4 The pressure field in a domain with four intersecting fractures for differ-
ent fracture permeabilities; the domain is compressed at the top [a] high-
permeability cracks kf = 104 km, [b] low-permeability cracks kf = 10−4 km. 27
4.2.5 Domain with a fracture, fracture deformation is magnified [a] initial (un-
loaded) configuration [b] deformed configuration for no effect of fluid pres-
sure in the fracture on mechanics [c] deformed configuration accounting for
the influence of fluid pressure inside the fracture on mechanics . . . . . . 29
4.2.6 Horizontal displacement field for deformed configuration (to scale) [a] pres-
sure in the crack does not affect the deformation [b] accounting for the
pressure in the crack holding its wall and thus transmitting stress . . . . . 30
4.2.7 Same as Fig. 4.2.6 for the vertical displacement field in the deformed config-
uration (to scale) [a] pressure in the crack does not affect the deformation [b]
accounting for the pressure in the crack holding its wall and thus transmitting
stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3.8 The pressure field for an injector-producer problem [a] high-permeability
fracture [b] low-permeability fracture . . . . . . . . . . . . . . . . . . . . 33
4.3.9 The horizontal-direction displacement field for an injector-producer problem
[a] high-permeable fracture [b] low-permeable fracture . . . . . . . . . . . 34
x
Section 1
Introduction
1.1 Review
Fractures in geologic porous formations usually occur at multiple scales with very
complex geometry. The contact surfaces of a fracture are often quite complex, and
they are usually described using an effective aperture. Thus, fractures are usually
treated as low-dimensional objects within the matrix. Even with such a simplified
fracture representation, it is quite challenging to model the dynamics of flow in
naturally fractured formations due to the complex geometry of the fracture network
and the scale discrepancy between the fractures (low-dimensional objects) and the
matrix formation they intersect.
For flow simulation, the so-called dual porosity model (Barenblatt et al., 1960;
Warren and Root, 1963), in which the matrix blocks provide fluid storage (large
pore volume) and the fracture network provides the conductivity (small pore-volume,
large permeability), is widely used. In addition to flow through the fractures, the
dual-permeability approach accounts for flow between matrix blocks. Recently, a
dual-permeability approach, in which for each computational element there is a
superposition of matrix and fracture elements carrying properties of both media,
was proposed (Lamb et al., 2010). In dual-porosity and dual-permeability models,
a ‘transfer function’ is used to characterize the interaction between the matrix and
the fracture. However, it is quite challenging to design a single transfer function that
1
SECTION 1. INTRODUCTION 2
captures the wide range of scales, properties, and geometry of naturally fractured
formations. Extensions to multiple continua have been developed (Wu et al., 2004).
For a given discrete representation of the naturally fractured geologic formation,
Lee et al., (2001) proposed a hierarchical approach that depends on the fracture
length relative to the scale of the computational gridblock. In their approach, small-
scale fractures are are treated in an average sense using homogenization methods, and
large-scale fractures are accounted for as source terms. Here, we employ a Discrete
Fracture Modeling (DFM) method (Karimi-Fard et al., 2004; Juanes et al., 2002), in
which a single static unstructured grid represents all the fractures as low-dimensional
objects (with complex network geometry) embedded within the matrix formation.
The unstructured grid captures the complex geometry of the discrete fracture network
by conforming grid elements along the fractures. So, in three dimensions (3D), matrix
elements are 3D and fractures are represented as the (2D) interfaces between matrix
elements. A Finite-Volume Method (FVM) is used to discretize the flow equations of
both the matrix and the fractures. FVM is locally mass conservation and allows for
modeling nonlinear multiphase flow with strong gravity or capillarity.
Several strategies have been used to model mechanical deformation of fractured
media (Mohammadi, 2008). These includes finite-element methods, the ‘smeared
crack’ model, discrete inter-element cracks, discrete cracked element, singular ele-
ments, enriched elements, boundary-element schemes, and various meshless methods.
The most widely used approach for incorporating fractures in mechanical modeling is
the Extended Finite Element Method (XFEM) (Stazi et al., 2003; Moes et al., 1999;
Borst et al., 2004; Borja, 2000), which allows for maintaining a static mesh. In this
method, the finite elements are locally enriched, and new shape functions are used
for the additional degrees of freedom. The Level Set Method (LSM) is usually used
together with XFEM in order to deal with complex fracture orientations with respect
to the computational grid. For an interesting example of XFEM for coupled flow and
mechanics in a fractured medium, see (Lamb et al., 2010).
The concept of equivalent media was proposed recently for modeling coupled flow
and mechanics in naturally fractured reservoirs (Bagheri and Settari, 2008), where a
third type of material, in addition to the matrix and fractures, is introduced. The
SECTION 1. INTRODUCTION 3
equivalent material is supposed to deform in a manner similar to the fractured-rock
system based on Huang’s principle of energy conservation (Huang et al., 1995). In
their framework, the contribution of the fractures to the total continuum is expressed
as an equivalent constitutive matrix, which is a function of the orientation of fracture
sets, average distance between fractures, aperture and stiffness. Similar methods have
been proposed elsewhere (Elsworth and Bai, 1992).
Here, we model coupled mechanical deformation and single-phase flow in naturally
fractured porous media using a single unstructured DFM grid. The computational
grid is constructed around the fracture, such that a fracture (long dimension) is dis-
cretized as an interface segment between two adjacent matrix elements in the mesh. A
double-node strategy, with additional degrees of freedom, is used for fracture segments
in order to represent the mechanical deformation problem. This strategy of combining
DFM with double-node numbering allows one to keep the same unstructured mesh
throughout the simulation, as long as the assumptions are honored, namely, that all
the fractures (scale and geometry) are already present in the computational model and
that no failure (fracture propagation) takes place during the time period of interest.
The details of numerical implementation and corresponding compatibility conditions
on fracture surfaces are discussed below in the text.
The report is organized as follows. First, the method used here to account for
embedded fractures in porous formations is described. Next, the equations governing
mechanical deformation and fluid-flow are presented. Then, the variational form of
the equations and the corresponding discrete forms for unstructured triangular grids
are detailed. That is followed by a detailed description of the fully coupled solution
scheme. The proposed method is tested using simple, yet challenging, numerical
examples with various boundary configurations. Our results are compared with the
Lamb’s recent work (Lamb et al., 2010). Finally, the results are summarized, and
several ideas of future development are outlined.
SECTION 1. INTRODUCTION 4
1.2 Treatment of Fractures. Equilibrium Analysis
A fracture is idealized as two parallel surfaces such that only normal displacement
is permissible, and across which discontinuity in the displacement (vector) may take
place (Fig. 1.2.1). Based on the permeability and displacement properties of the
model, one can describe various fracture types, such as faults, compaction bands,
joints, veins, and dikes. For more detailed geomechanical classification of fractures
see (Pollard and Aydin, 1988).
w
Figure 1.2.1: Idealization of a fracture in a porous medium, w is the ‘effective’ fracture
aperture
The equilibrium conditions for open fractures are governed by the mechanical
stresses and the pressure of the fluid occupying the space within the fracture. This
means that when the fractured porous medium is in equilibrium, the fracture has an
effective aperture. Moreover, for a fracture in equilibrium, load is transmitted not only
by the fluid, but also by solid-solid contact points between the two internal surfaces
of the fracture. In the model, the stresses and strains on the fracture due to the fluid
pressure and the contact points must be consistent with the state of equilibrium of
the overall rock-fluid system. Fracture properties, such as fluid conductivity, and the
normal, kn, and tangential, τn, stiffness coefficients must be captured by the model.
1.2.1 Incorporating Fractures into the Numerical Reservoir
Model
In our numerical model, a fracture is represented as the interface between two adjacent
mesh elements. A double-node technique is used such that each node along the
SECTION 1. INTRODUCTION 5
fracture is split into two nodes, each with its own degrees of freedom. This double-
node approach allows us to deal with the boundary conditions on the surfaces of the
‘open’ fracture (see Fig. 1.2.2).
Figure 1.2.2: 2D medium with a single fracture. The ‘matrix’ is discretized using triangles,
and the fracture is represented as a one-dimensional (1D) object.
To the best of our knowledge, this approach was first introduced by Goodman
(1976) for discrete fracture modeling using a finite-element method (FEM) with a
direct stiffness approach. Pak and Chan (2008) extended the idea of double-node
numbering to hydraulic fracture modeling, where during the fracture propagation
process the nodes between adjacent elements were split as needed.
We make the following important assumptions in this work:
- All the fractures are already present in the system, and no failure can occur due
to tension or compression;
- A fracture introduces a discontinuity, meaning that the displacement field across
the fracture is discontinuous, while its gradient, the stress field – is continuous
across the crack;
SECTION 1. INTRODUCTION 6
- The tangential component of the stress on the fracture wall is neglected.
1.2.2 Boundary Conditions on the Fracture
Consider a domain with a fracture inside it (see Fig. 2.1.1). For each element edge
along the fracture, we introduce a double-node with degrees of freedom associated
with each side of the fracture. Therefore, we need additional relations to account for
the additional degrees of freedom along fracture segments. These additional relations
come from enforcing equilibrium conditions on each interface.
When there is no fluid in the system and no contact forces between the fracture
sides, the fracture is represented as a single interface, and equilibrium requires equal
stresses to act through the matrix on the both sides of the ‘zero width’ fracture. How-
ever, in the presence of a fluid, or contact points between the two surfaces enclosing
the fracture, we need to specify two conditions, one for each surface. Mechanical
equilibrium requires that the traction be continuous across the fracture (Borja and
Regueiro, 2001), that is
[|σ|]n = 0, (1.2.1)
where [|σ|] is the stress jump defined as
[|σ|] = σ+n − σ−
n . (1.2.2)
Here σ+n and σ−
n are the normal components of total stress taken on opposite fracture
surfaces, and n+ and n− are the outward normal vectors at each face. For a fracture
with parallel walls (far from the tips) this condition simply becomes
(σ+n − σ−
n )n = 0. (1.2.3)
Depending on the physical problem, there are two ways to proceed. The first way
is to assume that fracture surfaces are traction free. That is
σ′+n n+ = −σ
′−n n− = 0, (1.2.4)
SECTION 1. INTRODUCTION 7
where σ′n is the effective stress on the corresponding boundary. Eq. 1.2.4 implies that
there are no contact forces between the fracture faces. Only the fluid pressure holds
the fracture surfaces apart. If there is no fluid in the system, the total stress is equal
to zero on both sides of the fracture. So, the fracture is an internal boundary with
respect to the matrix, whether a fluid is present or not. The ‘open’ nature of the
fracture leads to displacement discontinuity, while the stress is continuous as required
by equilibrium. In terms of total stress, Eq. 1.2.4 leads to
σ+n n
+ = −pFn+,
and
σ−n n
− = −pFn−,
where pF is the average pressure inside the fracture. This approach is most suitable
when the reservoir is under tension conditions, or in the case of hydraulically induced
fractures.
F+2
F+1
F-2
F-1
solid
F+1 = - F
+2
solid
Force balance
Figure 1.2.3: The force equilibrium condition. The fracture element is in equilibrium under
external compression. Internal forces are positive at ’-’ surface and negative at ’+’
The second approach takes into account the fact that the space between fracture
SECTION 1. INTRODUCTION 8
walls is not completely open, and there are contact points between the walls (e.g.,
via gouged rock material). However, at the macro-scale description of interest here,
the specific mechanical properties are treated as if the fracture simply encloses an
area bounded by the two parallel plates. So, a fracture segment has to be modeled
as an element with its own properties. However, since the fracture opening is
extremely small in comparison with the typical size of matrix elements, the fracture
is represented as a lower-dimensional object. In essence, the fracture is an internal
boundary. Fracture displacements are induced by changes in the effective stress field
acting along the fracture (Goodman, 1976). Basically, one can linearly relate the
normal displacements along the fracture to the changes in the normal effective stresses
as follows:
∆Un =∆σ′
n
kn, (1.2.5)
where kn is the stiffness of the fracture. The normal displacements, ∆Un, are
then related to the effective, or hydraulic, fracture aperture, wn, using experimental
relations like
wn = wni + f ·∆Un,
where wni is the initial aperture and the factor f accounts for the geometry and
roughness of the fracture (e.g., (Detournay, 1980)). Note, that in this case we are
speaking about a consistency condition between the different sides (+/-) of each
fracture boundary Γf1 and Γf2, rather than a boundary condition between the two
surfaces.
Eq. 1.2.5 is all we need to fully specify the problem for a porous medium with a
fracture. To illustrate how this equation is incorporated into the mathematical and
numerical models, we write a force balance for a fracture element in weak form:
F1(force from upper matrix)− F2(force from lower matrix) =
Ff (average force acting from inside the fracture)
SECTION 1. INTRODUCTION 9
or, in a more formal notation,∫Γ+f1
wi[σn(u)·n+]dΓ−∫Γ+f2
wi[σn(u)·n−]dΓ = −∫Γ−f1
wi(kn∆UFn −pF )dΓ+
∫Γ−f2
wi(kn∆UFn −pF )dΓ.
(1.2.6)
After Eq. 1.2.6 is added to the global system, the coupled problem can be solved as
discussed later in 3.1.
1.3 Fluid Flow in Fractures
The discrete fracture model (DFM) (Karimi-Fard et al., 2004) is employed for ap-
proximating fluid flow using a low-order, finite-volume approach. With DFM, the
complex geometry of fracture network in the domain is represented using unstructured
grids. Then, the governing equations are written for explicitly defined fractures,
as well as, the matrix. DFM is quite different from the dual-porosity and dual-
permeability approaches, which employ a so-called transfer function to represent the
communication between the matrix and the fracture network. DFM provides more
accurate representation of fractures with various properties and variable aperture.
Moreover, DFM allows for more accurate representation of multiphase fluid flow
accounting for capillary and gravity effects. In this section, we briefly review the
main concepts of the DFM approach for flow modeling. For the details, please see
(Karimi-Fard et al., 2004; Lee et al., 2001; Kim and Deo, 2000).
DFM employs a Two-Point Flux Approximation (TPFA). Fractures in DFM are
aligned with the mesh boundaries. For 2D problems, it basically means that the
domain is discretized using 2D unstructured grids, and the fracture is a 1D object.
The unknown pressure in the DFM model is associated with the matrix blocks
and fractures. Proper treatment of fracture intersections is based on the star-delta
transformation used for electrical circuits. This formulation correctly defines the fluid
flow direction and eliminates boundary elements with small pore volumes.
More specifically, the geometric transmissibility between two adjacent control
SECTION 1. INTRODUCTION 10
volumes is calculated as the harmonic average of αi:
T12 =α1α2
α1 + α2
,
where
αi =AikiDi
fi · ni, i = 1, 2.
Here Ai is the area of interface between two control volumes, ki is the permeability
of control volume i, Di is the distance between the centroid of the interface and the
centroid of control volume i, ni is the unit normal to the interface, and fi is the
unit-vector along the direction of the line joining the control volume centroid to the
centroid of the interface (Fig. 1.3.4).
Figure 1.3.4: Geometrical representation of two adjacent control volumes in DFM (fig.
from (Karimi-Fard et al., 2004))
Section 2
Mathematical Formulation
In this section we develop the governing equations for coupled flow and geomechanics.
These equations are the momentum balance for the mechanical deformation problem
and mass conservation for the fluid flow problem. The respective boundary conditions
are formulated in detail.
2.1 Mechanical Problem
We consider a boundary-value problem for a domain with an explicitly defined
fracture. Consider a domain Ω in Rn with outer boundary Γ = ∂Ω and inner fracture
boundary Γf (Fig. 2.1.1). Essential boundary conditions are imposed on Γgi , i = 1, 2
as prescribed displacements and natural boundary conditions on Γhi, i = 1, 2 as
tractions, so that Γi = Γgi ∪ Γhi, i = 1, 2 and Γgi ∩ Γhi
= ⊘, i = 1, 2.
The total stress describes the relationship between the effective normal stress, σ′n,
normal stress, σn, and the fluid pressure, Pf :
σn = σ′n − bPf ,
where b is Biot’s coefficient. The strong form of the mechanical problem is stated in
11
SECTION 2. MATHEMATICAL FORMULATION 12
x
z
n
Figure 2.1.1: Domain Ω intersected by a crack with boundary Γf
terms of the total stress as follows:
σij,j + fi = 0
ui = ui on Γgi , i = 1, 2
σijnj = hi on Γhi, i = 1, 2
σijnj = −pδijni on Γfi , i = 1, 2
(2.1.1)
where in 2D
σij,j =∂σij
∂xj
=∂σi1
∂x1
+∂σi2
∂x2
For linear elasticity, the constitutive relation for effective stress σ′ij is given by Hooke’s
law
σ′
ij = Cijklϵkl(u),
where C is a fourth-order compliance tensor, and the total stress
σij = σ′
ij − pδij.
The deformation vector is the symmetric gradient of the displacement vector u:
ϵij(u) =1
2(∂ui
∂xj
+∂uj
∂xi
),
SECTION 2. MATHEMATICAL FORMULATION 13
where the infinitesimal deformation assumption is used and second-order terms are
neglected.
In the Voight notations for plane-strain in 2D, the stress and strain tensors are
reduced to a vector σ = (σxx, σyy, σxy) and ϵ = (ϵxx, ϵyy, 2ϵxy), respectively. The
elastic compliance tensor takes the following matrix form:
C =E
(1 + ν)(1− 2ν)
1−νν
1 1
1 1−νν
1
1 1 1−2ν2ν
(2.1.2)
where E is Young’s modulus, and ν is Poisson’s ratio.
To formulate the weak form of the problem, we define a collection of trial functions
S:S = ui, ui ∈ H1(Ω), ui = u on Γgi , i = 1, 2
and a collection V of weighted functions, vanishing on the essential boundaries:
V = wi, wi ∈ H1(Ω), wi = 0 on Γgi , i = 1, 2
where H1(Ω) defines a Sobolev space of functions over Ω with square-integrable
derivatives. The general weighted-residual form is then recovered from the strong
form after integrating Eq. 2.1.1 by parts, and applying the divergence theorem:
0 =
∫Ω
wi · σij,jdΩ = −∫Ω
wi,j · σijdΩ +
∫∂Ω
wi · (σijnj)dΓ (2.1.3)
After accounting for the boundary conditions, the variational equation takes the form:∫Ω
wi · σijdΩ =
∫Γh
wi · hjdΓ +
∫Γf1+f2
wi · (σF′ijnj − pF δijnj)dΓ, (2.1.4)
where σF and pF are the effective stress and pressure within the fracture, respectively.
Or, more compactly,
a(w, u) = (w, u)|Γh+ (w, u)|Γf1+f2
, (2.1.5)
SECTION 2. MATHEMATICAL FORMULATION 14
where
a(w, u) =
∫Ω
wi · σij(u)dΩ
(w, u)|Γh=
∫Γh
wi · σij(u)njdΓ
(w, u)|Γf1+f2=
∫Γf1+f2
wi · (σ′Fij (u)− pF δij)njdΓ
(2.1.6)
2.2 Flow Problem
A porous medium consists of the solid skeleton and the fluid. The mass balance
for a single-phase fluid under small transformation theory and isothermal conditions
(neglecting the skeleton particle velocity) can be described as:
∂m
∂t+Div q = ρff, (2.2.7)
where m is the skeleton mass content per unit area, q is fluid mass flux per unit area,
f is a volumetric source term and ρf refers to the fluid density.
The fluid velocity v = q/ρf is given by Darcy’s law
v = − 1
Bf
k
µ(Gradp− ρfg),
where Bf = ρf0/ρf is a volume-factor accounting for fluid compressibility, k is the
absolute permeability, and µ is fluid viscosity.
The variation of mass content, δm, is caused by volume deformation, ϵv = tr(ϵ),
and pressure variation, δp, so the poroelasticity expression takes the following
form (Coussy, 2004; Kim et al., 2011):
1
ρf0(m−m0) = bϵv +
1
M(p− p0),
where subscript 0 means the reference state, M is Biot’s modulus and b is Biot’s
SECTION 2. MATHEMATICAL FORMULATION 15
coefficient given by the compatibility relations (Coussy, 2004) :
1
M= ϕ0cf +
1− ϕ0
Ks
, (2.2.8)
b = 1− Kdr
Ks
.
Ks is the bulk modulus of the solid grains, Kdr is the drained bulk modulus and fluid
compressibility
cf =1
ρ
∂ρ
∂p.
Substitution of Eq. 2.2.8 into Eq. 2.2.7 gives the continuity equation in terms of
pressure and the volumetric strain tensor:
1
M
∂p
∂t+ b
∂ϵv∂t
+Div v = f. (2.2.9)
For simplicity, it is assumed that changes in the volumetric strain do not affect
the flow in the fracture elements. In addition, the storage capacity of fractures is
neglected. Then, the mass balance Eq. 2.2.9 for fractures reduces to the following
equation:
Div v = f. (2.2.10)
Section 3
Solution Algorithm
In this section the solution procedure for coupled mechanics and flow a fractured
porous medium is described. We start with the weak formulation of the governing
equations, which allows for incorporation of the boundary conditions. A fully implicit
method is used to discretize the coupled equations, and the solution is achieved using
the Newton-Raphson method.
3.1 Discretization of the Coupled Problem
In this section, the discretization strategy of the coupled flow and poromechanical
problem is described. The coupled system can be written as follows:div(σ) + ρtg = 0,
∂∂t(ρtϕ) + div(ρfw) = 0.
(3.1.1)
The main requirements for the approximations are (1) local mass conservation (2)
continuous displacement field within elements that are not adjacent to a fracture, and
(3) a stable convergent numerical scheme using a single unstructured computational
grid. We use different discretization strategies for flow and mechanics because of
different regularity conditions (Jha and Juanes, 2007). Thus, we use the Finite-
Element Method (FEM) for mechanics and a low-order, Finite Volume Method
16
SECTION 3. SOLUTION ALGORITHM 17
(FVM) for flow. The equations are approximated using standard triangular isopara-
metric elements. To develop finite-element and finite-volume discrete equations that
take into account the contact condition on the fracture faces, we introduce a finite
dimensional approximation of trial, Sh ⊆ S, and variational, Vh ⊆ V , spaces for
displacements. Likewise, Ph ⊆ P and Qh ⊆ Q are finite-dimensional spaces that
approximate P and Q for trial and variational spaces of pressure respectively (Kim
et al., 2011; Jha and Juanes, 2007):
Sh = uhi (., t), u
hi (., t) ∈ H1(Ω), uh
i (., t) = uh on Γgi , i = 1, 2
Vh = whi , w
hi ∈ H1(Ω), wh
i = 0 on Γgi , i = 1, 2
for displacements, and
Ph = phi (·, t), phi (·, t) ∈ H1(Ω), phi (·, t) = ph on Γgi , i = 1, 2
Qh = ηhi , ηhi ∈ H1(Ω), ηhi = 0 on Γgi , i = 1, 2
for pressure unknowns. Note, that while the trial function spaces Sh and Ph depend
on time, Vh and Qh are time independent.
d2
N1
p
d3
d1
NeNp
Figure 3.1.1: Typical element and shape functions for the displacement and pressure
unknowns.
In order to develop the matrix form of the system, first, the domain Ω is partitioned
SECTION 3. SOLUTION ALGORITHM 18
into nel elements:
Ω = ∪nele=1Ω
e.
Then, the displacement and pressure unknowns are approximated using shape func-
tions on the element level. Since ph has to be only square integrable, it may be
discontinuous across element boundaries. However, to guarantee convergence it can-
not be arbitrary (Hughes, 2000). Since linear triangular elements are used, pressure is
chosen to be constant within the elements (Fig. 3.1.1). The following approximation
is written for the fluid pressure function ph ∈ Q as
ph =
nelem∑j=1
ϕjPj (3.1.2)
where nelem is the set of pressure node numbers, ϕj is the pressure shape function
associated with pressure node number j, Pj is the value of fluid pressure at the center
of element j, and the continuous part of the displacement function uh ∈ Sh as
uh =
nnodes∑a=1
NaUa (3.1.3)
where nnodes is the set of displacements node numbers, Na is the displacement shape
function associated with the node a and Ua is the displacement in the node a.
To honor the nonlinearity of the system introduced by fractures, the problem is
formulated using a Jacobian-residual form. For each degree of freedom, the following
discretization of the mechanics equation is formulated, accounting for body forces,
non-zero traction on the external boundaries, and the fracture boundary (compati-
bility) condition on the internal boundaries:
Rua =
∫Ωa
BTa σadΩ−
∫Ωa
NaρtgdΩ−∫Γh
NatdΓ−∫Γf
NapFk ndΓ (3.1.4)
∀a = 1, .., nnodes.
The residual for the pressure unknowns contains pressure derivatives, which are
SECTION 3. SOLUTION ALGORITHM 19
treated implicitly:
Rpi =
∫Ωi
1
M(pn+1
i −pni )dΩ−∫Ωi
b(ϵn+1v − ϵnv )dΩ−∆t
nfaces∑j=1
V n+1h,ij −∆t
∫Ωi
fn+1dΩ (3.1.5)
∀i = 1, .., nelem, and for fracture elements
Rpfi = −∆t
nfaces∑j=1
V n+1h,ij −∆t
∫Ωi
fn+1dΩ, (3.1.6)
∀fi = 1, .., nfelem, where Vn+1h,ij is the sum of the fluxes into element i over all adjacent
elements j. Then, in Galerkin form, the problem is stated as follows: find (uh, ph) ∈Sh × Ph such that Eqs. 3.1.4, 3.1.5 and 3.1.6 are satisfied.
3.2 Solution Strategy
The matrix form the coupled discretized system can be expressed as follows:
K −LT F
L Q+ T∆t
0. . . 0 0
. . . 0 T∆t
︸ ︷︷ ︸
J
δu1
...
δun
δp1...
δpm
δpf1...
δpfm
n+1,k+1
= −
Ru
...
Rp
...
Rpf
n+1,k
, (3.2.7)
SECTION 3. SOLUTION ALGORITHM 20
where
Keab =
∫Ωe
BTa CBbdΩ (3.2.8)
F eaj =
∫Γe
Na(bpFk n)dΓ (3.2.9)
Leib =
∫Ωe
ϕimT b(GradNb)
TdΩ (3.2.10)
Qeij =
∫Ωe
ϕim−1ϕjdΩ (3.2.11)
(3.2.12)
J is the Jacobian matrix; K is the element stiffness matrix; matrix F reflects the
influence of fluid pressure in the fracture on the boundary of a matrix element
that is adjacent to this fracture; L is the coupling poromechanics matrix; Tij is
the transmissibility matrix between blocks i and j; superscripts n and n + 1 are the
previous and current time levels respectively, and k is the current newton iteration,
B = ∇xN .
The matrices written on the element level are then assembled into a global
matrix, which is then solved iteratively. All integrals are performed analytically,
since the shape functions of linear triangular elements can be integrated exactly.
After fully implicit discretization on a triangular unstructured grid, the fully-coupled
system 3.2.7 is solved using the Newton method until convergence with respect to the
primary variables – displacement u(x, y, t) and pressure p(x, y, t) is achieved. Then,
the secondary variables, such as strain and stress field, are calculated by a FEM
discretization, namely ϵ = Bu and σ = Dϵ− bpI.
Remark Note that we are generating an unstructured triangular grid, which enjoys
the property that the largest angle of each triangle is at most 30 degrees. This
property is advantageous from two points of view. First, for the mechanical problem
it guarantees that the Jacobian resulting from the iso-parametric mapping is positive
SECTION 3. SOLUTION ALGORITHM 21
definite (it allows us to invert the Jacobian during the FEM formulation). Moreover,
the TPFA scheme used to reduce the approximation error requires element centroids
to lie on almost the same line, which is also satisfied by this grid.
Section 4
Numerical Examples
4.1 Model Validation
To validate the proposed mathematical and numerical models, we compare our results
using Lamb’s recent example 2D problem of a fractured porous medium (see (Lamb
et al., 2010) ).
A 2D rectangular domain is fully saturated with a single-phase slightly-compressible
fluid and includes a crack with a 45 degree tilt located in the middle of the domain.
Initially the model is in equilibrium. The compressive force is applied to the top of
the domain, and the fluid is allowed to drain freely from the top. The bottom is fixed
and rollers are attached to the sides of the domain, so that only vertical displacement
is allowed there. No-flow conditions are prescribed on the other boundaries. The
contact forces within the fracture are neglected, so the fracture boundaries are free
of tension. The properties of the model are listed in Table 4.1.
22
SECTION 4. NUMERICAL EXAMPLES 23
Table 4.1: Material parameters for the poroelasticity example model (Lamb et al.,2010)
E, Young Modulus 40 MPaν, Poisson’s Ration 0.3km, Matrix Permeability 0.05 Dkf , Fracture Permeability 0.05 · 104 Dcf , Fluid Compressibility 10−9 Pa−1
ρr, Rock Density 2400 kg/m3
ρf , Fluid Density 1000 kg/m3
µf , Fluid Viscosity 0.001 Pa · spini, Initial Fluid Pressure 2.125 · 106 Pab, Biot’s Coefficient 1
SECTION 4. NUMERICAL EXAMPLES 24
Fig. 4.1.1[a-b] shows the displacement field in the vertical direction when the
sample undergoes compression due to the application of a force on the top surface.
As it can be seen from the figures, the results are in an excellent agreement. Fig. 4.1.2
shows the pressure field obtained from the two models. The presence of a highly
permeable fracture leads the pressure field to deviate from the linear gradient field.
Figure 4.1.1: Displacements is the vertical (z) direction; comparison with the example
in (Lamb et al., 2010) [a] Lamb et al. (2010), FFM-XFEM; min dz = 0.00mm, max
dz=4.782mm [b]Current model: double-node FEM - FV; min dz = 0.00mm, max dz=4.874
mm
The pressure field in the domain, which has four mutually parallel fractures is
highly dependent upon the fracture permeability as shown in Fig. 4.1.4.
SECTION 4. NUMERICAL EXAMPLES 25
Figure 4.1.2: Pressure Field, comparison with [a] Lamb et al. (2010), FFM-XFEM; [b]
Current model: Double-node FEM - FV
SECTION 4. NUMERICAL EXAMPLES 26
Figure 4.1.3: Displacements in the vertical (z) direction for four interconnected fractures;
comparison with the example in (Lamb et al., 2010) [a] Lamb et al. (2010), FFM-XFEM;
[b] Current model: double-node FEM - FV
SECTION 4. NUMERICAL EXAMPLES 27
0 2 4 6 8 100
2
4
6
8
10
12
14
16
0.05
0.1
0.15
0.2
0.25
0.3
0 2 4 6 8 100
2
4
6
8
10
12
14
16
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
[a] [b]
Figure 4.1.4: The pressure field in a domain with four intersecting fractures for different
fracture permeabilities; the domain is compressed at the top [a] high-permeability cracks
kf = 104 km, [b] low-permeability cracks kf = 10−4 km.
SECTION 4. NUMERICAL EXAMPLES 28
4.2 Influence of crack pressure on the mechanics
One of the main differences in our formulation compared with that in (Lamb et al.,
2010) is that aligning the fracture with the mesh allows us to define compatibility
conditions on each fracture surface, so that the total-stress for an element edge is equal
to the pressure in the adjacent fracture, while in (Lamb et al., 2010) the contact forces
on the fracture surfaces are neglected, and the fracture surfaces are free of tension,
and thus they are not transmitting the stresses.
To illustrate the point in this example problem, the domain, which is initially in
equilibrium (Fig. 4.2.5[a]) is stretched, and the deformed configurations are presented
in Fig. 4.2.5[b,c]. When the fluid pressure in the fracture does not affect the mechani-
cal deformation, as in Fig. 4.2.5[b], one can observe symmetrical displacement of both
fracture surfaces, which are free of tension. While in Fig. 4.2.5[b], the fluid pressure
inside the fracture leads to less overall displacement. (Note, that the fracture width
is magnified for illustration purposes). The corresponding displacement field in the
vertical and horizontal directions are shown on Fig. 4.2.6[a,b] and Fig. 4.2.7[a,b].
SECTION 4. NUMERICAL EXAMPLES 29
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20
22
x
z
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20
22
x
z
0 2 4 6 8 100
2
4
6
8
10
12
14
16
18
20
22
x
z
[a] [b] [c]
Figure 4.2.5: Domain with a fracture, fracture deformation is magnified [a] initial (un-
loaded) configuration [b] deformed configuration for no effect of fluid pressure in the fracture
on mechanics [c] deformed configuration accounting for the influence of fluid pressure inside
the fracture on mechanics
SECTION 4. NUMERICAL EXAMPLES 30
1 2 3 4 5 6 7 8 9 10
x
−0.2
−0.15
−0.1
−0.05
0
0.05
[a]
1 2 3 4 5 6 7 8 9 10
x
−0.2
−0.15
−0.1
−0.05
0
0.05
[b]
Figure 4.2.6: Horizontal displacement field for deformed configuration (to scale) [a] pressure
in the crack does not affect the deformation [b] accounting for the pressure in the crack
holding its wall and thus transmitting stress
SECTION 4. NUMERICAL EXAMPLES 31
1 2 3 4 5 6 7 8 9 10
x
−0.2
−0.1
0
0.1
0.2
0.3
0.4
[a]
1 2 3 4 5 6 7 8 9 10
x
−0.2
−0.1
0
0.1
0.2
0.3
0.4
[b]
Figure 4.2.7: Same as Fig. 4.2.6 for the vertical displacement field in the deformed con-
figuration (to scale) [a] pressure in the crack does not affect the deformation [b] accounting
for the pressure in the crack holding its wall and thus transmitting stress
SECTION 4. NUMERICAL EXAMPLES 32
4.3 Example with two wells
We consider an example where two wells – a producer and an injector, are located
symmetrically with respect to the fracture. Fluid is injected from the left (injector)
and produced from the right (producer), both with a given constant rate. Dependent
on the fracture permeability relative to the permeability of the matrix, the fracture
can be either a barrier to flow (for a low permeability case), or as a highly conductive
path connecting the two wells. In both cases, proper modeling is required to detect
fluid breakthrough time. While the pressure fields for these two cases are quite
intuitive (Fig. 4.3.8), the displacement field in the horizontal direction Fig. 4.3.9
shows an interesting behavior (there are minor differences in the displacement in the
vertical direction). In this situation, the injected fluid acts as a force that induces
deformation in the horizontal direction. It should be noted that this kind of behavior
is seen only when the fluid inside the fracture transmits stress to the mechanical
problem, which enhances the deformation to the right of the fracture.
SECTION 4. NUMERICAL EXAMPLES 33
100 200 300 400 500 600 700
100
200
300
400
500
600
700
100 200 300 400 500 600 700
100
200
300
400
500
600
700
[a] [b]
Figure 4.3.8: The pressure field for an injector-producer problem [a] high-permeability
fracture [b] low-permeability fracture
SECTION 4. NUMERICAL EXAMPLES 34
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
x
z
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
x
z
[a] [b]
Figure 4.3.9: The horizontal-direction displacement field for an injector-producer problem
[a] high-permeable fracture [b] low-permeable fracture
Section 5
Conclusions
We studied the problem of coupled poroelastic mechanics and single-phase fluid flow in
naturally fractured porous formations. A discrete fracture model (DFM) with a single
unstructured grid is used. The fractures are represented in the computational grid
as low-order objects and do not require remeshing. The fluid-flow problem is solved
using the DFM approach, which employs a Finite-Volume Method on 2D unstructured
grids, while the Galerkin Finite-Element Method with a double-node approach is used
for the mechanical problem. The double-not scheme allows for using the appropriate
consistency conditions on the fracture surfaces. A fully-implicit approach is used to
solve the coupled flow and mechanics problem. The coupled problem is linearized
and solved by the Newton method.
The proposed double-node approach is an alternative to the commonly used
XEFM on structured grids. However, the double-node approach used here does not
introduce additional shape functions and allows for using the same grid as for the flow
problem. The test cases with single and interconnected fractures in porous media are
validated by comparing with the examples of (Lamb et al., 2010). The numerical
results show an excellent agreement between the two models.
One of the directions of future work is to account for more complicated fracture
networks and large-scale leaky faults. In addition, the algorithm should be generalized
to handle complex physics, such as plasticity, nonlinear elasticity, and multiphase flow
and transport. Possible extensions may include large-deformation formulations, as
35
SECTION 5. CONCLUSIONS 36
opposed to the current infinitesimal transformation assumption. We also plan to study
different coupling strategies, such as sequential-implicit algorithms and their stability
properties. This will allow us to implement the proposed strategies in GPRS. In the
long term, efficient upscaling and multiscale formulations will have to be investigated.
37
SECTION 5. CONCLUSIONS 38
Nomenclature
σ total (Cauchy) stress tensor
σ′ effective stress tensor
ϵ strain tensor
u displacement vector
E Young’s modulus
ν Poisson’s ratio
ρt total density
p fluid pressure
k absolute permeability
t total value
km, matrix permeability
kf , fracture permeability
ρr, rock density
ρf , fluid density
µf , fluid viscosity
kn, fracture stiffness
w fracture aperture
n outward normal vector
Bf fluid volume factor
q fluid mass flux per unit area
N, ϕ shape functions for flow and mechanics problem, respec-
tively
V , Q collection of weighted functions
δij Kronecker’s delta
Bibliography
Bagheri, M.A. and Settari, A. [2008] Modeling of geomechanics in naturally fractured
reservoirs. PE Res. Eval. Eng., 108–118.
Barenblatt, G., Zheltov, I. and Kochina, I. [1960] Basic concepts in the theory of
seepage of homogeneous liquids in fissured rocks [strata]. Prikl. Mat. Mekh. (J.
Appl. Math. Mech.), 24, 1286–1303.
Borja, R.I. [2000] A finite element model for strain localization analysis of strongly
discontinuous fields based on standard galerkin approximation. Computer Methods
in Applied Mechanics and Engineering, 190, 1529–1549.
Borja, R.I. and Regueiro, R.A. [2001] Strain localization of frictional materials exhibit-
ing displacement jumps. Computer Methods in Applied Mechanics and Engineering,
190(20-21), 2555–2580.
Borst, R., Remmers, J.C., Needleman, A. and Abellan, M. [2004] Discrete vs smeared
crack models for concrete fracture: bridging the gap. Int. J. Numer. Anal. Meth.
Geomech.
Coussy, O. [2004] Poromechanics. Chichester, England: John Wiley and Sons.
Detournay, E. [1980] Hydraulic conductivity of closed rock fractures: an experimental
and analytical study. Proc. 13th Canadian Rock Mech. Symp. on Underground Rock
Engineering, 168–173.
Elsworth, D. and Bai, M. [1992] Coupled flow-deformation response to a dual porosity
media. Journal Geotech. Eng., 118(1), 107–124.
Goodman, R.E. [1976] Methods of geological engineering in discontinuous rocks. St.
Paul : West Pub. Co.
39
BIBLIOGRAPHY 40
Huang, T.H., Chang, C.S. and Yang, Z.Y. [1995] Elastic moduli for fractured rock
mass. Rock Mechanics and Rock Engineering, 28(3), 135–144.
Hughes, T.G.R. [2000] The finite element method: linear static and dynamic finite
element analysis. Dover Publications.
Jha, B. and Juanes, R. [2007] A locally conservative finite element framework for
the simulation of coupled flow and reservoir geomechanics. Acta Geotechnica, 2(3),
139–153.
Juanes, R., Samper, J. and Molinero, J. [2002] A general and efficient formulation
of fractures and boundary conditions in the finite element method. International
Journal for Numerical Methods in Engineering, 54(12), 1751–1774.
Karimi-Fard, M., Durlofsky, L.J. and Aziz, K. [2004] An efficient discrete fracture
model applicable for general purpose reservoir simulator. SPE Journal, 9(2), 227–
236.
Kim, J., Tchelepi, H. and Juanes, R. [2011] Stability, accuracy and efficiency of
sequential methods for coupled flow and geomechanics. SPE Journal, 16(2), 249–
262.
Kim, J.G. and Deo, M.D. [2000] Finite element, discrete-fracture model for multiphase
flow in porous media. AIChE J, 46(6), 1120–1130.
Lamb, A., Gorman, G., Gosselin, O. and Onaisi, A. [2010] Coupled deformation and
fluid flow in fractured porous media using dual permeability and explicitly defined
fracture geometry. 72nd EAGE Conference & Exhibition.
Lee, S.H., Lough, M.F. and Jensen, C.L. [2001] Hierarchical modeling of flow in
naturally fractured formations with multiple scale lengths. Water Resour. Res.,
37(443).
Moes, N., J.Dolbow and Belytschko, T. [1999] A finite element method for crack
growth without remeshing. Int. J. Numer. Meth. Engng., 46, 131–150.
Mohammadi, S. [2008] Extended finite element method for fracture analysis of struc-
tures. Wiley-Blackwell.
Pollard, D.D. and Aydin, A. [1988] Progress in understanding jointing over the past
one hundred years. Geological Society of America Bulletin.
BIBLIOGRAPHY 41
Stazi, F.L., Budyn, E., Chessa, J. and Belytschko, T. [2003] An extended finite
element method with higher-order elements for curved cracks. Computational Me-
chanics, 31, 38–48.
Warren, J. and Root, P. [1963] The behavior of naturally fractured reservoirs. Soc.
Petroleum Engr. J., 3, 245–255.
Wu, Y.S., Liu, H.H. and Bodvarsson, G.S. [2004] A triple-continuum approach for
modeling flow and transport processes in fractured rock . original research article.
Journal of Contaminant Hydrology, 73(1-4), 145–179.