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Coupled Formulation and Algorithms for the Simulation of Non-Planar Three-Dimensional Hydraulic Fractures Using the Generalized Finite Element Method

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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.
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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
iR3,ˆ
u
u
unRNu
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
uH1(), such that
δ
u
u
uH1()
Z
s
δ
u
u
u:C
C
Csu
u
ud=Z
¯
t
t
t·
δ
u
u
udΓ+ZΓ+
c
¯
t
t
t+
c·J
δ
u
u
uKdΓc(3)
where and
are 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
, operator sis defined as
s=1
2+T,(4)
J
δ
u
u
uK=
δ
u
u
u+
δ
u
u
uis the virtual displacement jump across the crack surface Γc, and
¯
t
t
t+
c=pn
n
n+=pn
n
n
with n
n
nand 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
nand 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
¯
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=QIQL¯
x
x
xΓc(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
xΓc(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=QIQL(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=ZΓcQIQL
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
pQIQL
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
uis 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) = Z
s
δ
u
u
u:C
C
Csu
u
u d(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
¯
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
[QIQL]
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+1u
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+1tn=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,
Z
s
δ
u
u
u:C
C
C
α
su
u
un+1+ (1
α
)su
u
undZΓc
J
δ
u
u
uK·n
n
n
α
pn+1+ (1
α
)pndΓc=Z
δ
u
u
u·¯
t
t
td Γ
(26)
and
ZΓc
δ
pJu
u
un+1KJu
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,
Z
s
δ
u
u
u:C
C
C
α
su
u
un+1dZΓc
J
δ
u
u
uK·n
n
n
α
pn+1dΓc
=Z
s
δ
u
u
u:C
C
C[(1
α
)su
u
un]d+ZΓc
J
δ
u
u
uK·n
n
n[(1
α
)pn]dΓc+Z
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
ndΓcZΓc
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
Z
s
δ
u
u
u:C
C
Csu
u
un+1dZΓc
J
δ
u
u
uK·n
n
npn+1dΓc=Z
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
c2tK
K
Kn+1
pˆ
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
uK
K
Kn+1
c2T
K
K
Kn+1
c2tK
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
xΓc
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
uK
K
Kn+1
cT
K
K
Kn+1
ctK
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+1RNuis nonzero for any nonzero vector ˆ
p
p
pn+1RNp.
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
u1t
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
u1K
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
pK
K
Kn+1
cK
K
Kn+1
u1t
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
pK
K
Kn+1
cK
K
Kn+1
u1t
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
u1is 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+1TtK
K
Kn+1
p+K
K
Kn+1
cK
K
Kn+1
u1K
K
Kn+1
cTˆ
p
p
pn+1
=ˆ
p
p
pn+1TtK
K
Kn+1
pˆ
p
p
pn+1+ˆ
p
p
pn+1TK
K
Kn+1
cK
K
Kn+1
u1K
K
Kn+1
cTˆ
p
p
pn+1
The term ˆ
p
p
pn+1TtK
K
Kn+1
pˆ
p
p
pn+10 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
u1is 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
u1K
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
uK
K
Kn+1
cT
K
K
Kn+1
ctK
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
untQ
Q
Qn+1
pt¯
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
untQ
Q
Qn+1
pt¯
q
q
qn+1
p"K
K
Kn+1
uK
K
Kn+1
cT
K
K
Kn+1
ctK
K
Kn+1,i1
p#ˆ
u
u
un+1,i1
ˆ
p
p
pn+1,i1
(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,i1
uand K
K
Kn+1,i
c=K
K
Kn+1,i1
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
uK
K
Kn+1
cT
K
K
Kn+1
ct
K
K
Kn+1,i1
p
ˆ
u
u
utK
K
Kn+1,i1
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,i1
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
uK
K
Kn+1
cT
K
K
Kn+1
ctK
K
Kn+1,i1
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,i1+ˆ
u
u
un+1,i(48)
ˆ
p
p
pn+1,i=ˆ
p
p
pn+1,i1+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)
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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 106in 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=0n= 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=0i= 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,i1+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.
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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 . 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
a
¯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 is 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)
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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, =3333 s1and
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
=15to
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
=15to
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)
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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) = 104tm
This problem is simulated for 10 seconds using 10 time steps for four values for the fluid viscosity:
µ
=102cP, 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.
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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]
26 of 47 P. GUPTA AND C.A. DUARTE
Figure 13. Input GFEM discretization of the cuboid domain for planar penny-shaped fracture problem.
(a) (b)
Figure 14. (a) Discretization of geometrical crack surface and (b) the automatically generated fluid mesh.
[coupled˙formulation˙paper – November 5, 2015]
COUPLED FORMULATION AND ALGORITHMS FOR NON-PLANAR HYDRAULIC FRACTURES 27 of 47
injection rate, Q=5×105m3/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)KJu
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