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[coupled˙formulation˙paper – November 5, 2015]

Coupled Formulation and Algorithms for the Simulation of

Non-Planar Three-Dimensional Hydraulic Fractures Using the

Generalized Finite Element Method

P. Gupta and C.A. Duarte1

Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign

Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL, USA

SUMMARY

This paper presents a coupled hydro-mechanical formulation for the simulation of non-planar three-dimensional

hydraulic fractures. Deformation in the rock is modeled using linear elasticity and the lubrication theory is adopted

for the ﬂuid ﬂow in the fracture. The governing equations of the ﬂuid ﬂow and elasticity and the subsequent

discretization are fully coupled. A Generalized/Extended Finite Element Method (G/XFEM) is adopted for the

discretization of the coupled system of equations. A Newton-Raphson method is used to solve the resulting system

of nonlinear equations. A discretization strategy for the ﬂuid ﬂow problem on non-planar 3-D surfaces and a

computationally efﬁcient strategy for handling time integration combined with mesh adaptivity are also presented.

Several three-dimensional numerical veriﬁcation examples are solved. The examples illustrate the generality and

accuracy of the proposed coupled formulation and discretization strategies.

Keywords: Hydraulic fracturing, GFEM, XFEM, Multi-physics, Coupled formulation, Hydro-

mechanical coupling, Lubrication equation, Solid-ﬂuid coupling

1. Introduction

Hydraulic fracturing can be broadly deﬁned as the process by which a fracture initiates and propagates

due to hydraulic loading applied by a ﬂuid inside the fracture [1]. Hydraulic fracturing is widely used

in the oil and gas industry to increase the effective permeability of a reservoir. A successful fracturing

treatment may increase the production tens of times, making the technique economically attractive.

Yet there are concerns about the environmental impact of toxic ﬂuids used in reservoir treatment. The

potential of groundwater contamination from the hydraulic fracturing treatments has been one of the

major roadblocks for its rapid development. One of the main reasons for this concern is lack of a

thorough understanding of induced hydraulic fracturing propagation.

Recent examples of hydraulic fracturing diagnostic data suggest complex, tortuous, and non-planar

fracture geometry [1]. With the advent of real-time monitoring techniques during hydraulic fracturing,

1Correspondence to: C.A. Duarte, Department of Civil and Environmental Eng., University of Illinois at Urbana-Champaign,

Newmark Laboratory, 205 North Mathews Avenue, Urbana, IL 61801, USA. Tel.: +1-217-244-2830; Fax: +1-217-333-3821.

E-mail: caduarte@illinois.edu.

⋆Submitted to International Journal for Numerical and Analytical Methods in Geomechanics (November 5, 2015)

2 of 47 P. GUPTA AND C.A. DUARTE

there is a growing need for fully three-dimensional models that can be used to update treatment designs

in real time as information is fed back into the models. These updates should ideally be enabled

through the use of three-dimensional modeling. This will lead to better recovery rates with reduced

risk of environmental concerns. These requirements necessitate new computational techniques for the

simulation of three-dimensional non-planar hydraulic fracturing.

Numerical solution of even the most basic hydraulic fracturing model is challenging because it

involves the coupling of at least three processes:

1. The mechanical (rock) deformation induced by ﬂuid pressure on the fracture surface;

2. The ﬂow of fracturing ﬂuid within the fracture;

3. The fracture propagation dependent on the current stress state of the rock.

Rock deformation is usually modeled using the theory of linear elasticity [1, 47]. The ﬂuid ﬂow inside

the fracture is modeled using the lubrication theory [1, 47], which is a simpliﬁed model to represent

the ﬂow of an incompressible ﬂuid in a channel.

Mathematical modeling of ﬂuid-driven fractures aims to predict the evolution of treatment pressure,

induced fracture length, and the width and geometry of the fracture. Several numerical solutions for

this problem have been proposed. Classical papers on this subject by Khristianovic and Zheltov [31],

Geertsma and de Klerk [25], Weertman [51], and Spence and Turcotte [46] have used simpliﬁed

assumptions about either the fracture opening or the pressure ﬁeld. Such assumptions are necessary

because of the difﬁculties in modeling the complex fracture geometry growing under different stress

and well conditions.

An exhaustive summary of the numerical methods used to simulate hydraulic fracturing is given

by Adachi et al. [1] and Gupta and Duarte [27]. Many numerical techniques for three-dimensional

hydraulic fracturing have been developed by Clifton and Abou-Sayed [11], Sousa et al. [45], and others.

However, these techniques are either restricted to planar crack growth or evolution of the fracture to a

predetermined shape.

Rungamornrat et al. [42] present a numerical technique for the simulation of non-planar evolution of

hydraulic fractures. They developed a fully-coupled hydro-mechanical formulation using a Symmetric

Galerkin Boundary Element Method (SGBEM). Zielonka et al. [53] have also recently presented a

fully-coupled formulation to simulate 3-D hydraulic fracturing in porous media using cohesive zone

modeling and the XFEM. XFEM formulations for the simulation of hydraulic fracture propagation

in ﬂuid-saturated porous media are also presented in the works of R´

ethor´

e et al. [41] and Irzal et

al. [29]. They adopt a two-scale approach to couple the effects of the ﬂuid ﬂow in the fracture with

the deformation in porous media. Watanabe et al. [49] have introduced lower-dimensional interface

elements for hydro-mechanical coupling in fractured porous media.

A variety of hydraulic fracturing simulation software including, but not limited to, MFrac [4], MPwri

[4], MShale [4], TRIFRAC [10], GOHFER [5], Flac3D [30], FRANC3D [50], and HYFRANC3D

[12] have also been developed. However, all these software make certain assumptions, such as planar

fracture propagation, analytical solutions for stresses around the crack front, etc., which limit their

applicability for a true non-planar three-dimensional hydraulic fracturing simulation.

To our knowledge, a fully-coupled formulation for the simulation of non-planar 3-D hydraulic

fractures using the G/XFEM is not available in the literature. Gupta and Duarte [27] have presented

a methodology to model three-dimensional hydraulic fracture propagation using the G/XFEM. A

constant pressure distribution on the fracture surface and no ﬂuid lag were assumed in that work. These

assumptions are acceptable if the viscosity of the fracturing ﬂuid is neglegible and when fracturing deep

reservoirs with high conﬁning stress. Leak-off across the fracture faces can also be neglected for low

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 3 of 47

permeability reservoirs. Under these conditions, the hydraulic fracture propagates in storage-toughness

fracturing regime [9] with near constant pressure distribution on the fracture surface.

In this paper, we develop a fully-coupled system of equations for modeling non-planar three-

dimensional hydraulic fracturing. The proposed formulation extends the work of Gupta and Duarte [27]

by modeling ﬂuid ﬂow, for arbitrary ﬂuid viscosity and injection rate, inside the fracture. We restrict

the discussion in this paper to modeling of coupled linear elasticity and ﬂuid ﬂow equations for static

hydraulic fractures. An analysis of the coupled system of discretized equations is also presented and

demonstrates that the solution of the proposed coupled formulation is unique. A solution algorithm to

solve the fully-coupled nonlinear system of equations is presented. An efﬁcient time marching scheme

for the solution is also proposed. A novel discretization strategy for the 2-D ﬂuid ﬂow equation on

non-planar 3-D fracture surfaces is also presented. A subsequent paper with fracturing of the elastic

media coupled with solid deformation and ﬂuid ﬂow will be presented in the future.

The outline of the paper is as follows. Sections 3 and 4 present the governing equations and

the discretization of the proposed coupled formulation. Section 5 details the methodology for the

automatic generation of the ﬂuid ﬁnite element mesh and numerical modeling of the ﬂuid ﬂow on a

three-dimensional non-planar fracture. Then, the quintessence of the solution strategy for the coupled

formulation using a GFEM is described in Section 6. Section 7 demonstrates the generality and

accuracy of the proposed methodology through several 3-D hydraulic fracturing problems. Finally,

we close with a few concluding remarks in Section 8.

2. Generalized FEM for Three-Dimensional Fractures

The Generalized Finite Element Method (GFEM) is a Galerkin method based on the concept of a

partition of unity. Partition of unity methods have origins in the works of Babuˇ

ska et al. [2, 3, 34]

and Duarte and Oden [18, 19, 36]. The partition of unity for the GFEM is provided by Lagrangian

ﬁnite element shape functions. The same method is also known as the eXtended FEM (XFEM) [6, 35].

Recent reviews of the Generalized/eXtended FEM can be found in [7, 23].

In this work, we adopt the GFEM for the 3-D non-planar fractures proposed by Gupta and Duarte

[27]. Details on the construction of GFEM shape functions for 3-D elasticity equations, selection of

enrichment functions for hydraulic fractures, and adaptive mesh reﬁnement along the crack front can

be found in [27]. Here, we summarize the notation adopted for a GFEM approximation of a vector

ﬁeld. This approximation is used in later sections.

At any given time tn, a GFEM approximation u

u

un

h(x

x

x,tn)of the solution of a 3-D elasticity problem

can be written as

u

u

un

h(x

x

x,tn) = ∑

α

∈Ih

N

α

(x

x

x)ˆ

u

u

un,

α

(tn)

|{z }

PoU

+∑

α

∈Ih

enr

N

α

(x

x

x)

n

α

enr

∑

i=1

L

α

iˆ

u

u

un,

α

i(tn)

|{z }

Enrichment

=

φ

φ

φ

n

u(x

x

x)ˆ

u

u

un(tn)(1)

where

φ

φ

φ

n

u=

N

N

N1

|{z}

PoU

N

N

N11 ··· N

N

N1n1

enr

|{z }

Enrichment

···

N

N

NnG

|{z}

PoU

N

N

NnG1··· N

N

NnGnnG

enr

|{z }

Enrichment

(2)

[coupled˙formulation˙paper – November 5, 2015]

4 of 47 P. GUPTA AND C.A. DUARTE

for

N

N

N

α

=

N

α

0 0

0N

α

0

0 0 N

α

and N

N

N

α

i=

φα

i0 0

0

φα

i0

0 0

φα

i

PoU stands for Partition of Unity. Vectors

ˆ

u

u

un,

α

,ˆ

u

u

un,

α

i∈R3,ˆ

u

u

un∈RNu

contain degrees of freedom, with Nubeing the number of degrees of freedom of the 3-D elasticity

problem. N

α

and L

α

iare an FEM shape function and a GFEM enrichment function, respectively.

Index

α

,

α

∈Ih={1,...,nG}, is the index of a node in a ﬁnite element mesh with nGnodes, while

i,i=1,...,n

α

enr, denotes the index of the enrichment function L

α

iat node

α

. Parameter n

α

enr is the

number of enrichments at node

α

.

φα

i(x

x

x) = N

α

(x

x

x)L

α

i(x

x

x) (no summation on

α

)

is a GFEM shape function at a node

α

∈Ih

enr. The set Ih

enr ⊂Ihhas the indices of the nodes with

enrichment functions.

3. Governing Equations for Coupled Hydro-Mechanical 3-D Hydraulic Fractures

In this work, we aim to ﬁnd the solution of the 3-D elasticity equations coupled with the ﬂuid ﬂow

equations in the fracture. We do not consider the ﬂuid ﬂow in the reservoir. The statement of the

principle of virtual work for the 3-D elasticity problem is given by the following:

Find u

u

u∈H1(Ω), such that ∀

δ

u

u

u∈H1(Ω)

ZΩ

∇s

δ

u

u

u:C

C

C∇su

u

udΩ=Z

∂

Ω

¯

t

t

t·

δ

u

u

udΓ+ZΓ+

c

¯

t

t

t+

c·J

δ

u

u

uKdΓc(3)

where Ωand

∂

Ωare the analysis domain and its boundary, respectively, C

C

Cis the elasticity tensor for

the deformable elastic body, ¯

t

t

tis the in-situ stress prescribed on

∂

Ω, operator ∇sis deﬁned as

∇s=1

2∇+∇T,(4)

J

δ

u

u

uK=

δ

u

u

u+−

δ

u

u

u−is the virtual displacement jump across the crack surface Γc, and

¯

t

t

t+

c=−pn

n

n+=pn

n

n−

with n

n

n−and n

n

n+being the unity normal vectors to Γ−

cand Γ+

c, respectively, and p the pressure of the

ﬂuid in the fracture. Furthermore, n

n

n−=−n

n

n+and ¯

t

t

t−

c=−¯

t

t

t+

c. A cross section of the analysis domain

and crack surface Γcis illustrated in Figure 1.

Hereafter, n

n

n−and Γ+

care written as n

n

nand Γc, respectively, for simplicity of the notation. A unique

solution to the Neumann problem deﬁned by Equation (3) is found by preventing rigid body motions

of the analysis domain through point constraints.

In hydraulic fracturing, the opening of a fracture is in general much smaller than other dimensions of

the fracture. Therefore, the ﬂuid velocity in the normal direction of the fracture is negligible compared

to the velocity in other directions. Consequently, the ﬂuid ﬂow problem can be reduced to a two-

dimensional problem. The ﬂuid ﬂow equation on a non-planar surface is derived below using local

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 5 of 47

n

n

n

−

Ω

¯

t

t

t

z

¯

t

t

t

¯

t

t

t

¯

t

t

t

¯

t

t

t

x

Γ−

c

Γ+

c

n

n

n

+

c

Figure 1. Cross section of fractured domain. In the ﬁgure, ¯

t

t

tcrepresents ¯

t

t

t−

cwhen applied to Γ−

c, or ¯

t

t

t+

cwhen applied

to Γ+

c.

coordinate systems associated with ﬁnite elements of the mesh adopted for the solution of the ﬂuid

equation, as illustrated in Figure 3.

The ﬂuid is assumed to be Newtonian and incompressible. The assumption of incompressibility is

acceptable for liquids (e.g. water, gas-free oil) under typical sub-surface conditions [20]. Gravitational

and inertial effects are neglected. The strong form of the ﬂuid ﬂow equation in the fracture is given by

the mass conservation law [1, 13, 15, 28, 32],

∇¯

x

x

x·q

q

q+

∂

w

∂

t=QI−QL∀¯

x

x

x∈Γc(5)

where QIis the ﬂuid injection rate per unit area and QLis the leak-off rate per unit area. Vector q

q

qis the

ﬂuid ﬂux within the fracture, which is related to the pressure, p, through Poiseuille’s cubic law, given

by

q

q

q=w3

12

µ

∇¯

x

x

xp∀¯

x

x

x∈Γc(6)

where wis the fracture opening,

µ

is the dynamic viscosity of the Newtonian ﬂuid, and ∇¯

x

x

xis given by

∇¯

x

x

x=

∂

∂

¯x1

¯

e

e

e1+

∂

∂

¯x2

¯

e

e

e2(7)

Figure 2 shows the schematic of the ﬂuid ﬂow inside a fracture Γcwith fracture opening wand ﬂuid

ﬂux q

q

q. Figure 3 shows examples of local coordinate systems on Γcand the boundary

∂

Γcof the fracture

surface. Each coordinate system with base vectors {¯

e

e

e1,¯

e

e

e2}and position vector ¯

x

x

x= ( ¯x1,¯x2)is associated

[coupled˙formulation˙paper – November 5, 2015]

6 of 47 P. GUPTA AND C.A. DUARTE

with a ﬁnite element of the mesh used for the solution of the ﬂuid equation. Further details on the

deﬁnition of this coordinate system and a strategy for the automatic generation of the ﬂuid mesh are

presented in Sections 5.2 and 5.1, respectively.

Figure 2. Schematic of a fracture surface with ﬂuid ﬂux, q

q

q, fracture opening, w, and the boundary of the fracture

surface,

∂

Γc.

Figure 3. Crack surface, Γc, showing local coordinate systems associated with ﬁnite elements of the mesh adopted

for the solution of the ﬂuid equation. The boundary of the crack surface,

∂

Γc, is also shown in the ﬁgure.

The ﬂuid ﬂow at the boundary

∂

Γcof the fracture surface is given by

q

q

q·n

n

nc=w3

12

µ

∇¯

x

x

xp·n

n

nc=w3

12

µ

∂

p

∂

n

n

nc

=¯q(s)∀s∈

∂

Γc(8)

where n

n

ncis the normal to the boundary of fracture surface, as shown in Figure 3. Flux ¯q(s)is zero

along the crack front. From Equations (5) and (6), we obtain

∇¯

x

x

xw3

12

µ

∇¯

x

x

xp+

∂

w

∂

t=QI−QL(9)

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 7 of 47

The weak form of the ﬂuid ﬂow equation can be obtained by ﬁrst multiplying Equation (9) by a test

function

δ

pand integrating the product over the problem domain Γc,

ZΓc∇¯

x

x

xw3

12

µ

∇¯

x

x

xp

δ

p dΓc=ZΓcQI−QL−

∂

w

∂

t

δ

p dΓc(10)

Integrating the left-hand side by parts using Green’s theorem,

ZΓc∇¯

x

x

xw3

12

µ

∇¯

x

x

xp

δ

p dΓc=Z

∂

Γc

w3

12

µ

∇¯

x

x

xp·n

n

nc

δ

p d s −ZΓc

∇¯

x

x

x

δ

p·w3

12

µ

∇¯

x

x

xp dΓc(11)

Substituting Equations (11) and (8) in Equation (10), the weak form of the equation governing the ﬂuid

ﬂow in the fracture is given by

ZΓc

w3

12

µ

∇¯

x

x

x

δ

p·∇¯

x

x

xpdΓc=ZΓc

δ

pQI−QL−

∂

w

∂

tdΓc+Z

∂

Γc

¯q(s)

δ

pds (12)

The governing equations for the solid and ﬂuid problem are coupled through the following relations:

¯

t

t

t+

c=pn

n

n(13)

w=Ju

u

uK·n

n

n(14)

where Ju

u

uK=u

u

u+−u

u

u−is the displacement jump across the crack surface. Variable wis dependent on

the solution u

u

uof the elasticity problem which, in turn, is dependent on the ﬂuid pressure, p, applied on

the fracture surface.

From Equations (3), (12), (13), and (14), the coupled hydro-mechanical equations are given by

A(u

u

u,

δ

u

u

u) + B(p,J

δ

u

u

uK) = Lu(

δ

u

u

u)(15)

C

∂

Ju

u

uK

∂

t,

δ

p+D(p,

δ

p) = Lp(

δ

p)(16)

where the integral operators are deﬁned by

From (3)

A(u

u

u,

δ

u

u

u) = ZΩ

∇s

δ

u

u

u:C

C

C∇su

u

u dΩ(17)

From (3) and (13)

B(p,J

δ

u

u

uK) = −ZΓc

pJ

δ

u

u

uK·n

n

n dΓc(18)

From (12) and (14)

C

∂

Ju

u

uK

∂

t,

δ

p=ZΓc

δ

p

∂

Ju

u

uK

∂

t·n

n

n dΓc(19)

From (12)

D(p,

δ

p) = ZΓc

w3

12

µ

∇¯

x

x

x

δ

p·∇¯

x

x

xp dΓc(20)

and the linear forms Luand Lpare given by

From (3)

Lu(

δ

u

u

u) = Z

∂

Ω

¯

t

t

t·

δ

u

u

u dΓ(21)

[coupled˙formulation˙paper – November 5, 2015]

8 of 47 P. GUPTA AND C.A. DUARTE

From (12)

Lp(

δ

p) = ZΓc

[QI−QL]

δ

p dΓ+Z

∂

Γc

¯q(s)

δ

p ds (22)

Hereafter, QLis assumed to be zero. The formulation above is related to the one obtained in [42] for a

3-D Boundary Element Method. This formulation is also related to the 2-D one presented in Watanabe

et al. [49].

The coupled equations (15) and (16) are nonlinear because of the nonlinear term in Equation (20).

The discretization of these equations is discussed in Section 4.

4. Discretization in Space and Time

In this section, the discretization of the coupled equations (15) and (16) is discussed. The equations

are discretized ﬁrst in time and then in space. With this strategy, the resulting equations can properly

handle time-dependent shape functions [37]. The GFEM shape functions are time dependent in the

case of propagating fractures since the location of enrichment functions evolve over time. In addition,

changes in discretization due to mesh adaptivity can also be handled like a time-dependent basis.

While the numerical examples in this paper consider only static cracks, the proposed formulation and

discretization can be used for the simulation of propagating fractures, and thus the need to properly

account for the time dependency of shape functions.

4.1. Time Discretization

We adopt the generalized trapezoidal rule, or

α

method, for our time-marching scheme,

∂

u

u

u

∂

t=u

u

un+1−u

u

un

∆t(23)

u

u

un+

α

= (1−

α

)u

u

un+

α

u

u

un+1(24)

pn+

α

= (1−

α

)pn+

α

pn+1(25)

where tn+1−tn=∆t. Assuming that the in-situ stress, ¯

t

t

t, is not time dependent and plugging Equations

(23)–(25) into Equations (15) and (16) gives the temporally discretized equations,

ZΩ

∇s

δ

u

u

u:C

C

C

α

∇su

u

un+1+ (1−

α

)∇su

u

undΩ−ZΓc

J

δ

u

u

uK·n

n

n

α

pn+1+ (1−

α

)pndΓc=Z

∂

Ω

δ

u

u

u·¯

t

t

td Γ

(26)

and

ZΓc

δ

pJu

u

un+1K−Ju

u

unK

∆t·n

n

ndΓc+ZΓc

w3

12

µ

∇¯

x

x

x

δ

p·

α

∇¯

x

x

xpn+1+ (1−

α

)∇¯

x

x

xpndΓ

=ZΓc

δ

p

α

Qn+1

I+ (1−

α

)Qn

IdΓc+Z

∂

Γc

α

¯qn+1(s) + (1−

α

)¯qn(s)

δ

pds

(27)

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 9 of 47

Rearranging Equations (26) and (27) such that the unknown terms u

u

un+1and pn+1are moved to the

left-hand side and all known terms are moved to the right-hand side,

ZΩ

∇s

δ

u

u

u:C

C

C

α

∇su

u

un+1dΩ−ZΓc

J

δ

u

u

uK·n

n

n

α

pn+1dΓc

=−ZΩ

∇s

δ

u

u

u:C

C

C[(1−

α

)∇su

u

un]dΩ+ZΓc

J

δ

u

u

uK·n

n

n[(1−

α

)pn]dΓc+Z

∂

Ω

δ

u

u

u·¯

t

t

td Γ

(28)

and

ZΓc

δ

pJu

u

un+1K

∆t·n

n

ndΓc+ZΓc

w3

12

µ

∇¯

x

x

x

δ

p·

α

∇¯

x

x

xpn+1dΓc

=ZΓc

δ

pJu

u

unK

∆t·n

n

ndΓc−ZΓc

w3

12

µ

∇¯

x

x

x

δ

p·[(1−

α

)∇¯

x

x

xpn]dΓc

+ZΓc

δ

p

α

Qn+1

I+ (1−

α

)Qn

IdΓc+Z

∂

Γc

δ

p

α

¯qn+1(s) + (1−

α

)¯qn(s)ds

(29)

where ¯q(s)is the normal ﬂux, which is given as boundary condition.

Equations (28) and (29) are discretized in time. Assuming

α

=1 for an unconditionally stable

Backward Euler scheme, we get

ZΩ

∇s

δ

u

u

u:C

C

C∇su

u

un+1dΩ−ZΓc

J

δ

u

u

uK·n

n

npn+1dΓc=Z

∂

Ω

δ

u

u

u·¯

t

t

td Γ(30)

and

ZΓc

δ

pJu

u

un+1K

∆t·n

n

n dΓc+ZΓc

w3

12

µ

∇¯

x

x

x

δ

p·∇¯

x

x

xpn+1dΓc

=ZΓc

δ

pJu

u

unK

∆t·n

n

ndΓc+ZΓc

δ

pQn+1

IdΓc+Z

∂

Γc

δ

p¯qn+1(s)ds

(31)

4.2. Spatial Discretization of Governing Equations

For spatial discretization of coupled equations (15) and (16), we use generalized ﬁnite element shape

functions for the displacement ﬁeld and ﬁnite element shape functions for the ﬂuid pressure. Following

the notation introduced in Section 2, at a given time, tn, we deﬁne u

u

un(x

x

x,tn) =

φ

φ

φ

n

u(x

x

x)ˆ

u

u

un(tn)where

ˆ

u

u

un(tn)is the vector of degrees of freedom and

φ

φ

φ

n

u(x

x

x)are generalized ﬁnite element shape functions at

tnfor the solid problem. We also deﬁne pn(¯

x

x

x,tn) =

φ

φ

φ

n

p(¯

x

x

x)ˆ

p

p

pn(tn)where ˆ

p

p

pn(tn)is the vector of degrees

of freedom and

φ

φ

φ

n

p(¯

x

x

x)is the vector of ﬁnite element shape functions for the ﬂuid problem.

It is important to choose the test function properly because of the time-dependency nature of the

spatial discretization [37]. In Equations (28) and (29),

δ

u

u

u=

δ

u

u

un+1x

x

x,tn+1=

φ

φ

φ

n+1

u(x

x

x)

δ

ˆ

u

u

un+1tn+1

and

δ

p=

δ

pn+1¯

x

x

x,tn+1=

φ

φ

φ

n+1

p(¯

x

x

x)

δ

ˆ

p

p

pn+1tn+1.

We discretize each of the individual terms from Equations (30) and (31) in Appendix I. Using the

deﬁnitions introduced in that appendix, Equations (30) and (31) can be written as

[coupled˙formulation˙paper – November 5, 2015]

10 of 47 P. GUPTA AND C.A. DUARTE

K

K

Kn+1

uˆ

u

u

un+1+K

K

Kn+1

c1ˆ

p

p

pn+1=t

t

tn+1

u(32)

and

1

∆tK

K

Kn+1

c2ˆ

u

u

un+1+K

K

Kn+1

pˆ

p

p

pn+1=1

∆tK

K

Kn+1,n

c2ˆ

u

u

un+Q

Q

Qn+1

p+¯

q

q

qn+1

p(33)

Multiplying Equation (33) by ∆tgives

K

K

Kn+1

c2ˆ

u

u

un+1+∆tK

K

Kn+1

pˆ

p

p

pn+1=hK

K

Kn+1,n

c2iˆ

u

u

un+∆tQ

Q

Qn+1

p+∆t¯

q

q

qn+1

p(34)

In matrix form, the coupled system can be written as

K

K

Kn+1

uK

K

Kn+1

c1

K

K

Kn+1

c2∆tK

K

Kn+1

p ˆ

u

u

un+1

ˆ

p

p

pn+1="t

t

tn+1

u

K

K

Kn+1,n

c2ˆ

u

u

un+∆tQ

Q

Qn+1

p+∆t¯

q

q

qn+1

p#(35)

Equation (35) can also be written as

"K

K

Kn+1

u−K

K

Kn+1

c2T

K

K

Kn+1

c2∆tK

K

Kn+1

p#ˆ

u

u

un+1

ˆ

p

p

pn+1="t

t

tn+1

u

K

K

Kn+1,n

c2ˆ

u

u

un+∆tQ

Q

Qn+1

p+∆t¯

q

q

qn+1

p#(36)

It is noted that the system of equations (36) is non-symmetric. From here on, we use the notation K

K

Kcto

represent K

K

Kc2.

5. Fluid Formulation in 3-D

In this section, we discuss the solution of the ﬂuid ﬂow equation (12) on a non-planar fracture surface.

Section 5.1 presents a methodology for the automatic generation of the ﬂuid ﬁnite element mesh.

Section 5.2 discusses some implementation aspects related to the solution of the 2-D ﬂuid ﬂow equation

on a non-planar surface in 3-D.

5.1. Generation of the Fluid Finite Element Mesh

In this paper, explicit representations of crack surfaces are adopted. They are composed of ﬂat triangles

(facets) and were ﬁrst used in the GFEM presented in Duarte et al. [16, 17] and Pereira et al.

[38, 39]. Their use in the context of hydraulic fracture problems is discussed in Section 5.1 of [27].

Recent improvements of explicit crack surface representation and update with application to 3-D crack

coalescence are presented in [24].

Explicit crack surface representations, such as those shown in Figures 14(a) and 21(a), are referred

to as geometrical crack surfaces. The facets of a geometrical crack surface are not suitable for the

deﬁnition of a ﬁnite element mesh used for the solution of the ﬂuid ﬂow equation. A ﬂuid ﬁnite element

mesh must satisfy the following requirements:

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 11 of 47

1. It can be generated automatically from a given geometrical crack surface;

2. It can approximate well the geometrical crack surface;

3. It should be reﬁned near the crack front to capture strong pressure gradients in that region;

4. It must be a valid ﬁnite element mesh and thus cannot have any hanging nodes;

5. It should be suitable for the integration of the coupling term of the stiffness matrix given by

K

K

Kn+1

c=ZΓc

φ

φ

φ

n+1

pTn

n

nTJ

φ

φ

φ

n+1

uKdΓc(37)

The integrand involves shape functions from both the ﬂuid and solid meshes. No ﬂuid ﬁnite

element should cross solid element faces or edges; otherwise, Gaussian quadratures deﬁned over

ﬂuid elements may lead to severe integration errors.

Geometrical crack surfaces can be reﬁned to meet the third requirement, but they in general violate the

last requirement and thus cannot be used as a ﬂuid ﬁnite element mesh.

The concept of a computational crack surface is introduced in Section 5.2 of Gupta and Duarte

[27] in order to address the numerical integration issue listed above. This surface is deﬁned from faces

of integration sub-elements that ﬁt the geometrical crack surface. However, the computational crack

surface deﬁned in [27] may have hanging nodes and thus does not form a valid ﬁnite element mesh.

This is the case since the algorithm proposed in [27] is local: the sub-elements, and therefore the facets

of the computational crack surface, are created on an element-by-element basis without enforcing any

sort of continuity between neighboring computational elements. Below, we present a new algorithm for

the generation of computational crack surfaces that can be used to deﬁne ﬂuid ﬁnite element meshes

meeting the above requirements.

5.1.1. Generation of Computational Crack Surfaces The proposed algorithm for the generation

of computational crack surfaces follows the same main steps as in the approach presented in [27]:

Step 1: Compute intersections between 3-D solid elements and the geometrical crack surface;

Step 2: Generate integration sub-elements from a Delaunay tetrahedralization of intersection points at

each 3-D element;

Step 3: Use the faces of integration sub-elements to deﬁne facets of the computational crack surface.

In the proposed algorithm, intersections with the crack surface in Step 1 are computed as follows:

1. Compute the signed distance from nodes of the 3-D solid mesh to the geometric crack surface.

The signed distance of a point x

x

xis given by

S(x

x

x) = min

¯

x

x

x∈Γc

kx

x

x−¯

x

x

xksign(n

n

n+·(x

x

x−¯

x

x

x))

where sign is the sign function and n

n

n+is a vector normal to the positive side of the geometrical

crack surface.

2. For all elements that are not intersected by the crack front, compute the intersection of their

edges with the crack surface using the signed distances of the end nodes of the edges. An edge

intersects the crack surface if it has nodes with positive and negative signed distances to the

crack surface. The intersection is taken as the point whose distance to the surface is zero. In this

approach, all the set of intersection points for any element are on a plane. This step is related to

the crack surface approximation proposed by Prabel et al. [40] and Fries and Baydoun [22]. This

step of the algorithm is illustrated in Figure 4(b) for the case of a 2-D problem.

[coupled˙formulation˙paper – November 5, 2015]

12 of 47 P. GUPTA AND C.A. DUARTE

(a) (b) (c)

Figure 4. The red line represents the geometric crack surface. Blue circles represent intersections computed using

the signed distance function. The yellow triangle represents the intersection of the crack front (tip in this case)

and the element. The green line represents the approximation of the explicit geometric crack surface using the

proposed approach. It deﬁnes the computational crack surface. The reader is referred to the web version of this

article for interpretation of colors in this ﬁgure.

3. Compute the intersections between element faces and crack front edges and between crack front

vertices and element interior. These intersections are not computed using signed distances as in

[22, 40]. The use of actual intersections preserves the exact geometry of the crack front. This

results in an accurate representation of the crack front geometry regardless of volume (solid)

element mesh, as shown in Figure 21. This step of the algorithm is illustrated in Figure 4(c) for

the case of a 2-D problem.

It is noted that in the above algorithm the intersections between elements and edges of the crack

surface not on the crack front are ignored.

Computation of Signed Distance: The signed distance to the geometric crack surface is computed

using functions from the Computational Geometry Algorithms Library (CGAL) [48]. These functions

provide robust and efﬁcient computation of signed distances with arbitrary precision. No equations are

solved like in the level-set methodology. Intersections which are very close to a 3-D solid element

node are snapped to the node. This improves the aspect ratio of the ﬂuid ﬁnite elements created

from computational crack surface facets, as described below. For the purpose of this paper, an edge

intersection is snapped to a node if its distance to the node is less than 2% of the edge length. This

guarantees that the worst possible aspect ratio of a ﬂuid element is 1:50. Further improvement to the

aspect ratio can be achieved by snapping to the crack surface ﬁnite element nodes that are close to it.

It should also be noted that the ﬂuid pressure variation away from the crack front and ﬂuid injection

points is typically quite smooth. Therefore, it can be approximated well even in the presence of ﬂuid

elements with a high aspect ratio, in particular if high order polynomial shape functions are adopted.

The most signiﬁcant difference between the computational crack surface identiﬁed by the above

strategy and the one proposed in [27] is that the geometry of computational crack surface away from

the crack front is dependent on the volume mesh density. The computational crack surface is planar

inside elements that are not intersected by the crack front. Figure 4 illustrates this in a 2-D setting.

This approximation is caused by the use of signed distances to compute intersections away from the

crack front. The payoff of this approximation is that it leads to a computational crack surface without

hanging nodes. Its facets can thus be used to deﬁne a ﬂuid ﬁnite element mesh, as discussed below.

The planar approximation away from the crack front also results in a simple algorithm to identify the

faces of integration elements that are on the crack surface. Algorithm 1 from Gupta and Duarte [27] is

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 13 of 47

used for this purpose.

Creation of Computational Crack Surface and Fluid Finite Element Mesh After the intersections

between 3-D solid elements and the geometrical crack surface are computed as described above, Steps

2 and 3 of the previous algorithm are used to generate integration sub-elements and to identify facets

deﬁning the computational crack surface, respectively. Details on these steps can be found in [27, 38].

There are facets on the positive and negative sides of the crack surface. They deﬁne exactly the same

computational crack surface and either one of them can be used to deﬁne a ﬁnite element mesh for

the solution of the ﬂuid ﬂow equation. This mesh meets all the requirements listed in Section 5.1, in

particular it can be used to efﬁciently integrate K

K

Kn+1

cgiven in Equation (37).

The geometry of the computational crack surface, and therefore the geometry of the ﬂuid ﬁnite

element mesh, depends on the volume mesh density. They do not exactly match the geometrical crack

surface in the case of non-planar surfaces. The volume mesh is adaptively reﬁned close to the crack

front in order to provide accurate solutions [27, 38]. As a result, the ﬂuid ﬁnite element mesh is very

close to the geometrical crack surface near the crack front. Furthermore, like the volume mesh, the

ﬂuid mesh is strongly graded close to the crack front. This is illustrated in Figure 14(b). Thus, the

ﬂuid mesh is suitable for capturing strong pressure gradients near the crack front. Away from the crack

front the volume mesh is typically coarse and therefore the geometry of the ﬂuid mesh can be a poor

approximation of the geometrical crack surface. This can be controlled by locally reﬁning the volume

mesh around the crack surface.

Figure 21 shows a cylindrical geometrical crack surface and various ﬂuid ﬁnite element meshes

automatically generated using the above algorithm. It is noted that the crack front is represented

accurately regardless of the 3-D solid element size. However, the approximation of the crack surface

away from the crack front is dependent on the solid element size. The ﬁgure shows the improvement

of the geometry of the ﬂuid mesh as a result of the reﬁnement of the volume mesh around the crack

surface.

5.2. Formulation of Fluid Flow Equations on a Surface in 3-D Space

The gradient operator ∇¯

x

x

xgiven in Equation (7) is deﬁned on a Cartesian coordinate system associated

with a ﬁnite element of the ﬂuid mesh introduced in the previous section. Figure 5(b) illustrates this

system for a triangular element. It is noted that the elements in the ﬂuid mesh must be ﬂat; otherwise,

the gradient operator adopted in Equation (5) would have to account for the curvature of the ﬁnite

element, which leads to a more complex implementation. The global stiffness matrix, K

K

Kn+1

p, associated

with the ﬂuid ﬂow equation is given in Equation (70). This matrix is assembled from ﬂuid element

stiffness matrices as usual. It is noted that since each ﬂuid element has its own coordinate system, as

illustrated in Figure 3, the gradient of the shape functions in Equation (70) is computed using these

element systems on an element-by-element basis. However, there is no need to transform the ﬂuid

element stiffness matrix or load vector to, e.g., nodal coordinate systems, as typically done with shell

ﬁnite elements, since we are dealing with a scalar quantity–the ﬂuid pressure. This greatly facilitates the

implementation and computation of K

K

Kn+1

p. A veriﬁcation example involving ﬂuid ﬂow on a non-planar

surface modeled using the proposed discretization technique is presented in Section 7.1.

[coupled˙formulation˙paper – November 5, 2015]

14 of 47 P. GUPTA AND C.A. DUARTE

(a) (b)

Figure 5. (a) Master coordinate system, and (b) element coordinate system, for a ﬂuid ﬁnite element on a non-

planar surface.

6. Solution Algorithm for the Coupled Nonlinear Equations

The system of equations given by (36) is non-symmetric and nonlinear. Hence, an iterative strategy is

required for its solution. In this section, we present a strategy to solve this coupled system.

Staggered schemes for the solution of the coupled multi-physics problem of hydraulic fracturing are

discussed in, e.g., [1, 47]. In this class of solution algorithms, an additional constraint is required to

solve the ﬂuid ﬂow problem as it is a pure Neumann problem–the solution for pressure is unique only up

to an arbitrary constant. To remove this indeterminacy, a constraint based on the global conservation of

mass is added to the problem. This additional condition is also called the solvability condition [1, 47].

Alternatively, a unique solution for pressure can be obtained by prescribing its value at a point on the

fracture boundary

∂

Γc. However, this prescribed value is not always physically meaningful. Adachi et

al. [1] also reports that additional stabilization techniques, such as Picard iterations, are required for

staggered solution schemes. Nonetheless, convergence is still not guaranteed. They also report that in

some situations, Picard iterations or similar schemes typically converge well initially but degenerate to

spurious oscillations later.

In this work, the coupled system of equations (36), or its symmetrized form (42), is solved

monolitically. Section 6.1 presents an analysis of the coupled system (36) showing that its solution is

unique. Mesh adaptivity is adopted in this work in order to control discretization errors. Therefore,

the GFEM discretization can potentially change between time steps. Section 6.2 discusses some

implementation aspects related to time integration with mesh adaptivity–a non-trivial problem. Section

6.3 presents the Newton-Raphson algorithm to solve the nonlinear system of equations for static

hydraulic fractures. Section 6.4 presents the overall solution algorithm to solve the coupled hydraulic

fracture equations (15) and (16) using the GFEM.

6.1. Uniqueness of the Solution of the Coupled System

In this section, we present an analysis of the coupled system of equations given by (36) and show that

its solution is unique.

In Equation (36), the coupled matrix is given by

K

K

Kn+1

coupled ="K

K

Kn+1

u−K

K

Kn+1

cT

K

K

Kn+1

c∆tK

K

Kn+1

p#(38)

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 15 of 47

Lemma 6.1. Vector K

K

Kn+1

cTˆ

p

p

pn+1∈RNuis nonzero for any nonzero vector ˆ

p

p

pn+1∈RNp.

Proof Vector K

K

Kn+1

cTˆ

p

p

pn+1represents the load on crack faces corresponding to a ﬂuid pressure. This

load vector is nonzero for any nonzero pressure distribution on the crack faces

Theorem 6.2. The coupled system of equations given by (36) has a unique solution if rigid body

motions of the reservoir are prevented.

Proof The ﬁrst set of equations in (36) gives

ˆ

u

u

un+1=K

K

Kn+1

u−1t

t

tn+1

u+K

K

Kn+1

cTˆ

p

p

pn+1(39)

Substituting the above equation in the second set of equations in (36) leads to

∆tK

K

Kn+1

p+K

K

Kn+1

cK

K

Kn+1

u−1K

K

Kn+1

cTˆ

p

p

pn+1=K

K

Kn+1,n

cˆ

u

u

un+∆tQ

Q

Qn+1

p+∆t¯

q

q

qn+1

p−K

K

Kn+1

cK

K

Kn+1

u−1t

t

tn+1

u

(40)

which can be rewritten as

S

S

SK

K

Kn+1

uˆ

p

p

pn+1=K

K

Kn+1,n

cˆ

u

u

un+∆tQ

Q

Qn+1

p+∆t¯

q

q

qn+1

p−K

K

Kn+1

cK

K

Kn+1

u−1t

t

tn+1

u(41)

where S

S

SK

K

Kn+1

uis the Schur complement [8, 52] of K

K

Kn+1

uin K

K

Kn+1

coupled.

Sub-matrix K

K

Kn+1

uis symmetric and positive deﬁnite if rigid body motions of the reservoir are

prevented. Thus, Equation (39) gives a unique solution for ˆ

u

u

un+1since K

K

Kn+1

u−1is positive deﬁnite.

Similarly, Equation (41) gives a unique solution for ˆ

p

p

pn+1, if the Schur complement, S

S

SK

K

Kn+1

u, is positive

deﬁnite. To prove that, pre and post multiply the Schur complement by an arbitrary nonzero vector,

ˆ

p

p

pn+1,

ˆ

p

p

pn+1TS

S

SK

K

Kn+1

uˆ

p

p

pn+1=ˆ

p

p

pn+1T∆tK

K

Kn+1

p+K

K

Kn+1

cK

K

Kn+1

u−1K

K

Kn+1

cTˆ

p

p

pn+1

=ˆ

p

p

pn+1T∆tK

K

Kn+1

pˆ

p

p

pn+1+ˆ

p

p

pn+1TK

K

Kn+1

cK

K

Kn+1

u−1K

K

Kn+1

cTˆ

p

p

pn+1

The term ˆ

p

p

pn+1T∆tK

K

Kn+1

pˆ

p

p

pn+1≥0 for any nonzero ∆tand nonzero vector ˆ

p

p

pn+1, since K

K

Kn+1

p

is positive semi-deﬁnite. Since K

K

Kn+1

u−1is positive deﬁnite and, from Lemma 6.1, K

K

Kn+1

cTˆ

p

p

pn+1is

nonzero, the second term, ˆ

p

p

pn+1TK

K

Kn+1

cK

K

Kn+1

u−1K

K

Kn+1

cTˆ

p

p

pn+1>0, for any nonzero vector ˆ

p

p

pn+1.

Thus, the solution of the coupled system of equations (36) is unique if rigid body motions of the

reservoir are prevented

The coupled formulation results in a non-symmetric matrix. However, the matrix can be

symmetrized by multiplying Equation (34) by -1. The coupled system of equations is then given by

"K

K

Kn+1

u−K

K

Kn+1

cT

−K

K

Kn+1

c−∆tK

K

Kn+1

p#ˆ

u

u

un+1

ˆ

p

p

pn+1=t

t

tn+1

u

−K

K

Kn+1,n

cˆ

u

u

un−∆tQ

Q

Qn+1

p−∆t¯

q

q

qn+1

p(42)

[coupled˙formulation˙paper – November 5, 2015]

16 of 47 P. GUPTA AND C.A. DUARTE

6.2. Time Integration with Mesh Adaptivity

Consider the term

−K

K

Kn+1,n

cˆ

u

u

un=−ZΓc

φ

φ

φ

n+1

pTn

n

nTJ

φ

φ

φ

n

uKdΓcˆ

u

u

un(43)

from the right-hand side of the system of equations (42). This quantity requires shape functions from

time steps tnand tn+1. Mesh adaptivity is adopted in this work in order to control discretization errors.

Thus, the GFEM volume and ﬂuid ﬁnite element meshes may change between time steps. This can

be true even for the case of static hydraulic fractures if different volume mesh reﬁnement is used at

different time steps. Therefore, the GFEM shape functions

φ

φ

φ

n

uare not necessarily the same as

φ

φ

φ

n+1

u,

which are the shape functions used at time step tn+1. One option to address this is to keep the GFEM

mesh from time step tnavailable at time step tn+1, but this will lead to large memory requirements.

Another alternative is to volume map the solution u

u

unonto the GFEM space at time tn+1. However,

volume mapping at every time step is computationally expensive.

In this work, we propose a novel strategy to compute the right-hand side of (43). We start by re-

writing it as

−K

K

Kn+1,n

cˆ

u

u

un=−ZΓc

φ

φ

φ

n+1

pTn

n

nTJu

u

unKdΓc(44)

The proposed strategy is based on the following observations and properties of Equation (44):

1. The jump in displacement, Ju

u

unK, at the previous time step is required only on the crack surface

Γcat time tn+1;

2. The geometrical crack surface used at time tn+1is available at the end of time step tn. This is

true for explicit crack surface representations, even for the case of propagating fractures.

As discussed in Section 5.1, the geometrical crack surface is composed of ﬂat triangles and its

vertices. After the computation of the solution at time tn, the jump in displacement, Ju

u

unK, can be

computed and saved at the vertices of the geometrical crack surface. This jump vector can then be

used at time tn+1to compute Equation (44). This mapping on a surface is much less computationally

demanding than a volume mapping of the solution from the previous time step. The memory

requirements of this strategy are also quite low.

Similarly, the fracture opening wdeﬁned in Equation (14) can also be saved on the geometrical crack

surface vertices and used as an initial guess for the solution algorithm. Details on this are presented in

Section 6.4. This strategy illustrates one more beneﬁt of having an explicit representation of a crack

surface.

6.3. Newton-Raphson Algorithm

In this work, we employ the Newton-Raphson method in such a way that the solid and ﬂuid ﬂow

problems are solved simultaneously for a prescribed time step, ∆t. There is no restriction of the

magnitude of time step ∆t, as far as stability of the algorithm is concerned. This iterative strategy

is also employed by Rungamornrat et al. [42] with a symmetric Galerkin BEM method to discretize

the solid problem and a Galerkin FEM to discretize the ﬂuid ﬂow problem.

The residual vector, R

R

Rn+1,i, at iteration iat time tn+1=tn+∆tof the Newton-Raphson scheme is

deﬁned as

R

R

Rn+1,i=R

R

Rn+1,i

u

R

R

Rn+1,i

p=t

t

tn+1

u

−K

K

Kn+1

cˆ

u

u

un−∆tQ

Q

Qn+1

p−∆t¯

q

q

qn+1

p−"K

K

Kn+1

u−K

K

Kn+1

cT

−K

K

Kn+1

c−∆tK

K

Kn+1,i−1

p#ˆ

u

u

un+1,i−1

ˆ

p

p

pn+1,i−1

(45)

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 17 of 47

where ˆ

u

u

unis the solution vector for displacements computed at time step n. It is noted that K

K

Kn+1,i

u=

K

K

Kn+1,i−1

uand K

K

Kn+1,i

c=K

K

Kn+1,i−1

c. The Newton-Raphson iteration index is thus dropped from these

matrices.

The Jacobian, K

K

Kn+1,i, of the residual vector R

R

Rn+1,ifor a full Newton-Raphson algorithm is given by

K

K

Kn+1,i=

K

K

Kn+1

u−K

K

Kn+1

cT

−K

K

Kn+1

c−∆t

∂

K

K

Kn+1,i−1

p

∂

ˆ

u

u

u−∆tK

K

Kn+1,i−1

p

(46)

The coupling terms of the Jacobian matrix for a full Newton-Raphson algorithm are non-symmetric.

To restore symmetry, the contribution of the term −∆t

∂

K

K

Kn+1,i−1

p

∂

ˆ

u

u

uA similar procedure is adopted by

Rethore et al. [41] to restore the symmetry of the coupled system of equations governing the ﬂuid ﬂow

in a fractured porous media. is not considered. This results in a modiﬁed Newton-Raphson algorithm

with the following system of linear equations:

"K

K

Kn+1

u−K

K

Kn+1

cT

−K

K

Kn+1

c−∆tK

K

Kn+1,i−1

p#∆ˆ

u

u

un+1,i

∆ˆ

p

p

pn+1,i=R

R

Rn+1,i

u

R

R

Rn+1,i

p(47)

It is noted that the modiﬁed Newton-Raphson scheme may result in the loss of quadratic

convergence. The above system of equations is symmetric and solved using the direct solver PARDISO

[44]. The updated solution is obtained by

ˆ

u

u

un+1,i=ˆ

u

u

un+1,i−1+∆ˆ

u

u

un+1,i(48)

ˆ

p

p

pn+1,i=ˆ

p

p

pn+1,i−1+∆ˆ

p

p

pn+1,i(49)

6.4. Solution Algorithm

In this section, we present the solution algorithm for the proposed formulation. This algorithm utilizes

the ﬂuid formulation and implementation presented in Section 5 along with the strategy to handle time

integration proposed in Section 6.2.

One important issue to note is that the ﬂuid stiffness matrix, K

K

Kn+1,i

p, is dependent on the fracture

opening, w, from the Reynold’s lubrication equation as given by Equation (70). However, the fracture

opening wis identically zero at time t=0. This will result in a zero ﬂuid stiffness matrix. To overcome

this issue, an initial step is solved without the coupled formulation but with an assumed constant

pressure on the crack surface. The constant pressure is chosen such that the opening w>0 along

the entire crack surface. The formulation from Gupta and Duarte [27] is used to solve this single-

physics problem. This results in a ﬁnite fracture opening which is saved on the crack surface vertices

and is used as the initial guess to start the modiﬁed Newton-Raphson iterative solution algorithm. For

all numerical examples in this paper, a constant pressure p=1.0 MPa is applied at this initial solution

step. An alternative strategy is to assume a constant opening won the fracture surface at the start of the

Newton-Raphson iteration. Lecampion and Desroches [33] report that an initial opening of w=1.0

µ

m

can be attributed to the radial defects present around the wellbore. This initial opening is used as an

initial guess for the Newton-Raphson algorithm.

The solution algorithm is detailed in Algorithm 1. We deﬁne U

U

Un+1,ias

U

U

Un+1,i=ˆ

u

u

un+1,i

ˆ

p

p

pn+1,i(50)

[coupled˙formulation˙paper – November 5, 2015]

18 of 47 P. GUPTA AND C.A. DUARTE

and ∆U

U

Un+1,ias

∆U

U

Un+1,i=∆ˆ

u

u

un+1,i

∆ˆ

p

p

pn+1,i(51)

The convergence tolerance,

ε

, is taken as 10−6in all the examples presented in Section 7. The

time increment ∆tis chosen by the user. Although the formulation allows us to choose an arbitrary

time increment, a larger time increment results in a higher temporal discretization error. A larger time

increment also requires more Newton-Raphson iterations for the algorithm to converge, as the change

in solution from tnto tn+1is more pronounced. Thus, CPU time required for the assembly and solving

of the coupled system of equations increases for a larger time increment ∆t.

Algorithm 1 Solution algorithm to solve the nonlinear system of coupled equations

1: Step: INITI AL GU E SS O F F RACT URE O PEN IN G ,wn=0⊲n= Time step index

2: Solve problem with a constant pressure applied on the fracture faces.

3: Save fracture opening, wn=0, at each vertex of the geometrical crack surface, as discussed in

Section 6.2.

4: end Step:

5: t0=0

6: for n=0,n<number of time steps do ⊲Loop over time steps

7: Update time tn+1=tn+∆t

8: Step: MOD IFI ED NEW TO N-RAP H SO N A LGO RIT HM FO R COU PLE D SOL UT I ON

9: i=0⊲i= Newton iteration counter

10: U

U

Un+1,i=0=U

U

Un

11: Compute the residual vector R

R

Rn+1,i=0

12: Compute the L2norm of the residual: kR

R

Rn+1,i=0kL2

13: Save the norm of the residual: kR

R

Rinit kL2=kR

R

Rn+1,i=0kL2

14: while kR

R

Rn+1,ikL2

kR

R

Rinit kL2

>

ε

do

15: Compute modiﬁed tangent stiffness matrix, as shown in Equation (46).

16: Solve for ∆U

U

Un+1,iusing (47).

17: Update the solution as U

U

Un+1,i=U

U

Un+1,i−1+∆U

U

Un+1,i

18: Save the fracture opening on the geometrical crack surface (Section 6.2).

19: Update the residual and its L2norm: kR

R

Rn+1,ikL2

20: i=i+1

21: end while

22: Update the fracture opening and the solution vector (used for time integration at next time

step) on each of the geometrical crack surface vertices.

23: end Step:

24: n=n+1

25: end for

7. Numerical Examples

In this section, we present several examples to show the ﬂexibility and accuracy of the proposed

formulation. These examples are compared with analytical solutions when available.

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 19 of 47

7.1. Pressure Distribution in a Journal Bearing

In this section, a manufactured solution based on the analytical solution for the pressure distribution

of the lubricant in a journal bearing is used to verify the proposed ﬂuid ﬁnite element formulation and

implementation for 3-D surfaces.

A journal bearing is a hollow cylinder enclosing a solid shaft that rotates about its axis at an angular

speed Ω. The journal bearing radius, a, is larger than that of the shaft by an amount ¯wthat is very

small compared to the radius of the bearing. With an external force applied on the bearing, the shaft

can move laterally becoming eccentric with respect to the journal bearing. The clearance wbetween

the shaft and the bearing varies with angular position

θ

[21],

w(

θ

) = ¯w(1−

η

cos

θ

)(52)

where the eccentricity

η

is the ratio of the lateral displacement of the shaft to ¯w. This setup is shown

in Figure 6.

Figure 6. Schematic of the journal bearing problem. A shaft rotating in a journal bearing moves laterally to an

eccentric position, thus varying the height for the lubricant in the bearing.

The pressure distribution of the lubricant in a journal bearing is governed by the Reynolds equation

for lubrication in polar coordinates [21]. The pressure distribution in a bearing with a length much

larger than its radius can be assumed, far from the bearing ends, to depend on the angle

θ

only. The

pressure distribution p(

θ

)of the lubricant on these long bearings is given by [21],

p(

θ

) = p(0)−6

µ

Ωa

¯w2

η

sin

θ

(2−

η

cos

θ

)

(2+

η

2)(1−

η

cos

θ

)2(53)

where

µ

is the viscosity of the ﬂuid and Ωis the angular speed. A detailed procedure to obtain the

pressure variation is given in [21].

The pressure variation (53) can be used to compute the value of ﬂuid source QIusing Equation (9).

Assuming that the opening between the bearing and the shaft does not vary with time and that there is

no ﬂuid leak-off, Equation (9) reduces to

∇¯

x

x

xw3

12

µ

∇¯

x

x

xp=QI(54)

[coupled˙formulation˙paper – November 5, 2015]

20 of 47 P. GUPTA AND C.A. DUARTE

(a) (b)

Figure 7. (a) Mesh with 18 elements along the circumference of the cylinder and (b) mesh with 36 elements along

the circumference of the cylinder.

Rewriting (54) in cylindrical coordinates and since the pressure (53) is a function of

θ

only, we obtain

1

2

µ

1

a

d

d

θ

w31

a

d p (

θ

)

d

θ

=QI(

θ

)(55)

Plugging (53) into (55) and adopting a=1 m, ¯w=0.001 m,

η

=0.5, Ω=3333 s−1and

µ

=1 Pa-s

gives a ﬂuid source

QI(

θ

) = 9999

12000 sin

θ

(56)

If this ﬂuid source is prescribed, the exact solution of (54) is given by (53).

In the problem considered in this section, the ﬂuid source given by Equation (56) is applied on a

ﬂuid FEM mesh. A Dirichlet boundary condition with prescribed pressure p(0) = 2×1010 Pa was

applied at a single node at

θ

=0◦, where

θ

is deﬁned in Figure 6. The problem is solved using

the ﬁnite element meshes shown in Figure 7. The mesh shown in Figure 7(a) has 18 elements along

the circumference of the cylinder, while the mesh shown in Figure 7(b) has 36 elements along the

circumference. Two element types, 3-node linear and 6-node quadratic triangles, are used to verify the

proposed formulation. This results in a total of four discretizations. Tri3 18 and Tri3 36 denote the

linear triangle meshes with 18 and 36 elements along the circumference of the cylinder, respectively.

Similarly, Tri6 18 and Tri6 36 denote quadratic triangle meshes with 18 and 36 elements along the

circumference, respectively.

Figure 8 shows the analytical and computed values of the lubricant pressure for all four

discretizations considered. Figure 8(b) shows the zoomed-in results from

θ

=15◦to

θ

=75◦, where

the pressure variation is maximum. As can be seen from the plots, the computed solution matches very

well with the analytical solution, thus verifying the proposed ﬂuid ﬂow formulation for non-planar

surfaces.

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 21 of 47

060 120 180 240 300 360

θ (Degrees)

5e+09

1e+10

1.5e+10

2e+10

2.5e+10

3e+10

3.5e+10

Pressure

Analytical, James A. Fay

Tri3_18

Tri3_36

Tri6_18

Tri6_36

(a)

15 20 25 30 35 40 45 50 55 60 65 70 75

θ (Degrees)

5e+09

6e+09

7e+09

8e+09

9e+09

1e+10

1.1e+10

1.2e+10

1.3e+10

1.4e+10

1.5e+10

Pressure

Analytical, James A. Fay

Tri3_18

Tri3_36

Tri6_18

Tri6_36

(b)

Figure 8. (a) Variation of lubricant pressure along the surface of the cylinder for a long journal bearing. The black

line is the analytical solution computed using Equation (53). Results for four discretizations are also shown. Angle

θ

is measured as shown in Figure 6. (b) zoomed-in results from

θ

=15◦to

θ

=75◦. The reader is referred to the

web version of this article for interpretation of colors in this ﬁgure.

[coupled˙formulation˙paper – November 5, 2015]

22 of 47 P. GUPTA AND C.A. DUARTE

7.2. 3-D Fully-Cut Cuboid Domain with Constant Fluid Flow

In this section, we present an example where the fracture opening and ﬂuid pressure can be computed

exactly as a function of time. The example is used to verify the proposed discretization and solution

algorithm for the coupled equations (15) and (16).

The problem setup is as follows: a cuboid domain is fully cut by a planar fracture, as shown in Figure

9; the top and bottom (y=0 and y=2L) faces of the domain are ﬁxed in the ydirection; point Dirichlet

boundary conditions are applied to prevent rigid body motions; a constant ﬂuid source QIis applied in

the fracture. This setup allows the computation of the exact solution of the coupled problem governed

by (15) and (16).

As a constant ﬂuid source is applied on the entire fracture, the fracture opening wis constant over

the fracture faces. Assuming a linear elastic material for the domain, the fracture opening is given by

w(t) = 2d(t) = p(t)

E2L(57)

where pis the pressure on the fracture surface, Lis the height of the top/bottom cubic domain, and E

is the Young’s modulus of the solid material. It is noted that the solution is independent of the viscosity

of the injected ﬂuid.

Figure 9. Schematic of the 3-D fully-cut cuboid domain with constant ﬂuid ﬂow.

The fracture opening can also be computed using the volume of injected ﬂuid over time. The volume

of ﬂuid in the fracture is given by

V(t) = QIL2t=L2w(t)

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 23 of 47

(a) (b)

Figure 10. (a) GFEM mesh for the cuboid and the geometrical fracture surface and (b) ﬂuid ﬁnite element mesh.

It is noted that the geometrical crack surface does not need to end at domain boundaries while the ﬂuid FEM mesh

does.

where QIis the volume of ﬂuid injected per second, per unity area of fracture surface. Thus, the fracture

opening as a function of time is given by

w(t) = QIt(58)

From Equations (57) and (58), the ﬂuid pressure as function of time is given by

p(t) = QIEt

2L(59)

The geometrical and material properties adopted are as follows: L=1 m, E=5×104MPa,

ν

=0.2,

and ﬂuid volume injected per second, per unit area of fracture surface, QI=0.0001 m/sec. Plugging

these parameters into the equations above gives

p(t) = 2.5tMPa w(t) = 10−4tm

This problem is simulated for 10 seconds using 10 time steps for four values for the ﬂuid viscosity:

µ

=10−2cP, 1.0 cP, 105cP, and 1010 cP.

Figure 10(a) shows the GFEM discretization of the cuboid domain. It is modeled using 12 tetrahedral

linear elements. The fracture surface fully cutting the cuboid is also shown in the ﬁgure. Figure 10(b)

shows the ﬂuid ﬁnite element mesh, which was automatically generated using the algorithms presented

in Section 5.1. The resulting coupled system of equations is solved using the solution strategy detailed

in Section 6. The solution is obtained with one Newton-Raphson iteration at each time step for this

example.

A comparison of the analytical solution and the computational solution is shown in Figure 11.

The computational solution exactly matches the analytical solution irrespective of the viscosity of the

injected ﬂuid, as expected. This problem serves as a veriﬁcation example of the coupled formulation

and its implementation.

[coupled˙formulation˙paper – November 5, 2015]

24 of 47 P. GUPTA AND C.A. DUARTE

0 1 2 3 4 5 6 78 9 10

Time (seconds)

0

5

10

15

20

25

30

Pressure (MPa)

Analytical Solution

Numerical Sol, µ = 10-2cP

Numerical Sol, µ = 1.0 cP

Numerical Sol, µ = 105 cP

Numerical Sol, µ = 1010 cP

Figure 11. Evolution of pressure with time on fracture faces for 3-D fully-cut cuboid domain.

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 25 of 47

7.3. Planar Penny-Shaped Fracture

In this section, we present an example of pressure evolution with time for a penny-shaped fracture in

a cubic domain. A cut-away schematic for the example is shown in Figure 12. A Newtonian ﬂuid with

viscosity

µ

is injected at the center of the fracture at a constant volume injection rate Q0. The wellbore

radius is neglected for this simulation. It is also assumed that there is no lag between the fracture front

and ﬂuid front, i.e. ﬂuid reaches the end of the fracture. By neglecting the lag, the solution does not

depend on the far ﬁeld stress,

σ

0[43].

Figure 12. 1/4th cut-away schematic of the 3-D cuboid domain with a planar penny-shaped fracture of radius R.

A constant volumetric injection rate, Q0, is applied at the center of the fracture.

The cubic domain used for GFEM simulation is shown in Figure 13. An initial discretization of

20 ×20 ×20(×6)tetrahedrals is used. The discretization of the geometrical crack surface is shown

in Figure 14(a). The dimension of the cube is taken as 2L=10m and the radius of the penny-

shaped fracture is R=0.5m. The GFEM mesh is then locally reﬁned using the strategy described

in Section 4.2 by Gupta and Duarte [27]. The maximum and minimum element size along the crack

front is hmax/R=0.0044 and hmin /R=0.0027, respectively. The polynomial order of the GFEM

shape functions is taken as two.The ﬂuid mesh, which was automatically generated using the strategy

presented in Section 5.1, is shown in Figure 14(b). This mesh is generated after the reﬁnement of the

volume GFEM mesh.

Material properties for this simulation are adopted from Zielonka et al. [53] with Young’s modulus

E=17 GPa and Poisson’s ratio

ν

=0.2. This problem is simulated for three values of viscosities:

µ

=25 cP, 50 cP, and 100 cP. The time step, ∆t, is taken as 1 second for all simulations. The ﬂuid

[coupled˙formulation˙paper – November 5, 2015]

COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 27 of 47

injection rate, Q=5×10−5m3/sec, is applied at the center of the fracture, as shown in Figure 12.

Savitski and Detournay [43] presented an asymptotic solution for a penny-shaped fracture

propagating at constant speed in an inﬁnite domain. The pressure variation in the fracture depends

on the viscosity of the injected ﬂuid and the volumetric rate of injection [9, 14, 26]. If the viscosity

of the ﬂuid is very low, it results in a near constant pressure along the entire fracture. However, if the

viscosity is high, the pressure variation along the fracture can be signiﬁcant [43, 53]. Here we solve a

static crack and compute the evolution of ﬂuid pressure with time. Nonetheless, some features of the

solution presented in [43] can also be observed in the GFEM solutions computed here.

Figure 16 shows the variation of pressure with time along the diameter of the penny-shaped fracture

for different values of ﬂuid viscosity. It can be observed that the gradient of the pressure reduces with

time in analogy to the transition of the solution of a propagating crack as it moves from a viscosity-

dominated regime to a toughness-dominated regime [43]. It should also be noted that the pressure at

the injection point for

µ

=100 cP is higher at t=2 seconds than at t=3 seconds. At early time steps,

strong pressure gradients near the injection point and the crack front are observed for all values of ﬂuid

viscosity. It is noted that the pressure ﬁeld for a propagating crack has a logarithmic singularity at the

injection point and along the crack front [43].

Figure 17 compares the effect of viscosity on pressure with time. The gradient in pressure is higher

for higher viscosity ﬂuids at all times. However, it can be observed that the effect of viscosity on

pressure variation diminishes with time. Thus, the pressure variation along the fracture surface at 6

seconds is similar for ﬂuids with different viscosities.

The number of Newton-Raphson iterations required for the convergence of the solution of system

(47) varies from seven, for the highly nonlinear pressure variation at t=2 seconds and

µ

=100 cP, to

four iterations, for the case of an almost constant pressure variation along the diameter of the fracture

at t=6 seconds and

µ

=25 cP. The convergence proﬁles for these two cases are shown in Figure 15.

1 2 3 4 5 6 78 9 10

Number of Newton-Raphson iterations

1e-13

1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

Relative norm of the residual

100 cP, 2 Seconds

25 cP, 6 Seconds

Figure 15. Convergence proﬁles for the cases

µ

=100 cP at t=2 seconds and

µ

=25 cP at t=6 seconds.

Figure 18 shows the fracture opening, Ju

u

uK=u

u

u+−u

u

u−, along the diameter of the fracture for different

values of ﬂuid viscosities. It can be observed that the gradient of fracture opening is more pronounced

for higher viscosity ﬂuids at early time steps. However, the fracture opening is similar at the later time

steps as the effect of ﬂuid viscosity on pressure variation reduces with time.

[coupled˙formulation˙paper – November 5, 2015]

28 of 47 P. GUPTA AND C.A. DUARTE

Figure 19 shows a comparison between the injected ﬂuid volume and the volume of the fracture

opening for different values of ﬂuid viscosity. The volume of the fracture opening is computed using

Vopen ≈

50

∑

i=1

π

h(i)

3RL(i)2+RL(i)RU(i) + RU(i)2(60)

Each term in the summation represents the volume of a frustum of a right circular cone with

height h(i), radius of the lower base RL(i), and radius of the upper base RU(i). The height h(i) =

Ju

u

u(x

x

xi)K−Ju

u

u(x

x

xi+1)K, where Ju

u

u(x

x

xi)Kis the fracture opening and x

x

xi,i=1,...,51, are points spaced by

∆R=R/50 =0.01 m in the radial direction of the penny-shaped crack with x

x

x1representing the injection

point. The radii of the lower and upper bases of frustum iare RL(i) = i×∆Rand RU(i) = (i−1)×∆R,

respectively. It can be observed that the volume of the fracture opening matches very well with the

volume of the injected ﬂuid. The error between the injected volume and fracture opening is less than

0.25% for all cases. The small difference can be attributed to approximations in (60).

[coupled˙formulation˙paper – November 5, 2015]