Simulation of Coupled Single-Phase Flow and … · SIMULATION OF COUPLED SINGLE-PHASE FLOW AND GEOMECHANICS IN FRACTURED POROUS MEDIA ... superposition of matrix and fracture elements

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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)

ii

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

iii

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.

iv

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

ix

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


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.


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


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;


- 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)


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


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)


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


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

),


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)


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


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


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


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)


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


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


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.


Figure 4.1.2: Pressure Field, comparison with [a] Lamb et al. (2010), FFM-XFEM; [b]

Current model: Double-node FEM - FV


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


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.


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].


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


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


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


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.


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


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

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Simulation of Coupled Single-Phase Flow and … · SIMULATION OF COUPLED SINGLE-PHASE FLOW AND GEOMECHANICS IN FRACTURED POROUS MEDIA ... superposition of matrix and fracture elements