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3D non-conforming mesh model for flow in
fractured porous media using Lagrange multipliers
Philipp Sch¨adle1, Patrick Zulian2, Daniel Vogler1,
Sthavishtha Bhopalam R.1, Maria G. C. Nestola2, Anozie Ebigbo1,
Rolf Krause2, and Martin O. Saar1
1Geothermal Energy and Geofluids Group, Department of Earth
Sciences, ETH Z¨urich, 8092 Z¨urich, Switzerland
2Institute of Computational Sci-
ence, USI Lugano, 6904 Lugano, Switzerland
Arxiv version from December 23, 2018 submission
Abstract
This work presents a modeling approach for single-phase flow in 3D
fractured porous media with non-conforming meshes. To this end, a
Lagrange multiplier method is combined with a parallel L2-projection
variational transfer approach. This Lagrange multiplier method en-
ables the use of non-conforming meshes and depicts the variable cou-
pling between fracture and matrix domain. The L2-projection vari-
ational transfer allows general, accurate, and parallel projection of
variables between non-conforming meshes (i.e. between fracture and
matrix domain).
Comparisons of simulations with 2D benchmarks show good agree-
ment, and the method is further validated on 3D fracture networks by
comparing it to results from conforming mesh simulations which were
used as a reference. Application to realistic fracture networks with hun-
dreds of fractures is demonstrated. Mesh size and mesh convergence
are investigated for benchmark cases and 3D fracture network appli-
cations. Results demonstrate that the Lagrange multiplier method, in
combination with the L2-projection method, is capable of modeling
single-phase flow through realistic 3D fracture networks.
Keywords: Embedded discrete fracture model, Flow in 3D fractured porous
media, Finite element method, Non-conforming grids
1
arXiv:1901.01901v1 [physics.geo-ph] 7 Jan 2019
1 Introduction
Fractured rock formations in the subsurface are of crucial importance in
a variety of reservoir applications, such as geothermal energy extraction,
CO2sequestration, nuclear waste storage, and unconventional oil and gas
recovery [51, 37, 11, 12, 48, 2]. As fluid flow velocities in fractures are often
magnitudes higher than in the rock matrix, individual fractures as well as
fracture networks commonly govern the overall fluid transport characteris-
tics of the entire fracture-dominated porous medium. Here, the geometric
fracture configuration and the hydraulic properties of individual fractures,
such as fracture permeability fields, largely determine, where preferential
fluid flow may occur, as fractures with particularly high or low permeability
can act as flow conduits or ”bottlenecks”, respectively [16, 56, 17]. Fur-
thermore, Ahkami et al. [1] describe and visualize experimentally that the
permeability of the porous-medium matrix influences fluid flow in the frac-
tures of a fractured porous medium.
An in-depth understanding of processes in fractured rock masses thus re-
quires knowledge about the hydraulic parameters of each fracture in a frac-
ture network. As these parameters are notoriously difficult to obtain in the
subsurface, reliance on stochastic investigations are often required, where
hundreds or more system realizations typically have to be performed to
assess uncertainties [7, 15, 28, 41]. To facilitate solving large numbers of nu-
merical simulations of fluid flow through fracture-network systems in three
dimensions (3D), highly efficient and accurate numerical methods and mesh
generation approaches are required.
Generally, in numerical models, two method classes are used when represent-
ing fractured porous media, i.e. fractures embedded in a porous-medium ma-
trix. Fractures are either represented by a continuum approach [55, 6, 33, 32]
or as discrete domains in a numerical mesh [42, 4]. In the continuum ap-
proach, the fractures and the porous-medium matrix share the same geomet-
ric mesh with separate continua. The respective flow properties are obtained
by upscaling and information needs to be transfered between the continua.
In contrast, the classic discrete-domain approach explicitly meshes both frac-
tures and porous media, with the two meshes conforming at the boundaries
of the domains (i.e. conforming numerical mesh). Since fracture configura-
tions in fracture networks can be arbitrarily complex, mesh generation for
discrete fracture networks (DFN) [30, 43, 44, 22] or discrete fracture models
(DFM) [14, 29, 9] with the background matrix can be very difficult and time
consuming. Due to the large length-to-width ratio of fractures and the need
2
to overcome very small elements in the fracture domain, fractures may be
represented by lower-dimensional elements, e.g. [31, 10, 39, 26]. Still, gen-
erating smooth matrix meshes on, and around, fracture intersections and
fracture tips can lead to very small elements and significant increases in the
number of degrees of freedom. In contrast, areas in the (porous) medium
that are void of fractures might contain elements with large edge lengths,
leading to large differences in element size, compared to elements close to
fractures. The strong influence of fractures on fluid mass and energy trans-
fer processes in a wide range of applications, as mentioned above, and the
associated difficulties in model generation, have rendered related method
improvements an area of active research.
The aforementioned shortcomings of classic discrete-domain approaches have
led to an increased focus on the development of numerical methods that al-
low the use of independent meshes for the fracture domain and the matrix
domain. Such non-conforming mesh approaches might be based on mor-
tar methods [21, 13], where the mesh for the discrete fractures and the
matrix are required to align geometrically, but consist of independent dis-
cretizations. Larger flexibility is offered by methods that handle fracture
and matrix meshes separately (i.e. no aligned geometries). Such methods
exist for finite volume schemes, e.g. (p)EDFM [25, 50, 38] and for XFEM-
based approaches [19, and references therein] for finite elements. A review
of existing mathematical and conceptual models for flow in fractured porous
media is given by Berre et al. [8]. Recently, K¨oppel et al. [35] proposed a
Lagrange multiplier method for a non-conforming finite element formula-
tion. This method enables the use of independent meshes for fractures and
matrices by applying variational transfer between the two mesh domains.
The variational transfer allows projection of variables between the fractures
and the matrix domain. With the geometric mapping between the fractures
and the matrix domain established, the Lagrange multiplier accounts for the
variable coupling at the domain boundaries. In contrast to XFEM methods,
this approach does not enrich the finite element space locally, so that the
pressure across the fractures is assumed to be continuous and the fractures
have a higher permeability than the matrix. K¨oppel et al. [35] developed
a general Lagrange multiplier method for 2D or 3D model domains. They
further show the uniqueness of the solution for the primal formulation of
the continuous problem. Due to the large technical complexities in 3D, they
focused on the implementation and verification in 2D.
However, flow through fractured rock formations is governed by 3D effects,
3
as strong heterogeneities affect flow properties in the fracture and porous-
medium domains [52, 16, 54, 53]. To apply the Lagrange multiplier method
to 3D fracture networks, the variational transfer between fractures and ma-
trix for non-conforming methods needs to be implemented both accurately
and with parallel processing capabilities. To this end, Krause and Zulian
[36] developed a general, accurate and parallel variational transfer approach,
which requires no prior knowledge about the relationship between the two
meshes. More specifically, an L2-projection variational transfer operator has
been shown [27] to provide better approximations than interpolations. Pre-
vious applications of this L2-projection include fluid–structure interaction
(FSI) problems and mechanical contact of rough fractures [40, 46, 47, 45].
Typically, these applications are solved in equi-dimensional domains. To
address the representation of fractures by lower-dimensional manifolds (i.e.
surface elements), the L2-projection algorithm has been extended to surface–
volume interactions. This enables transfer of information between surface el-
ements and volume elements for fractures and matrices, respectively. Build-
ing on the work of Krause and Zulian [36] and K¨oppel et al. [35], the aim
of this work is to demonstrate their methods’ applicability to steady-state,
single-phase fluid flow in 3D fractured porous media.
This paper presents an application of the Lagrange multiplier method in
combination with the L2-projection variational transfer operator in 3D. Sec-
tion 2 provides a brief overview of the method, by discussing the mathemat-
ical formulation, the discretization, the surface–volume interaction, and the
implementation. Section 3 first compares 2D and 3D results to state-of-
the-art benchmark results [20]. Next, the method is validated for 3D frac-
ture networks by comparing it to results from conforming mesh simulations,
which are used as a reference. For all cases described above, different mesh
sizes and mesh convergences are discussed. Finally, we apply the method
to realistic fracture networks with hundreds of fractures, demonstrating the
capability of the Lagrange multiplier method to model single-phase flow
through realistic 3D fracture networks. The presented findings are then
summarized in Section 4.
2 Method
In order to accommodate fluid flow through 3D fractured porous rock vol-
umes, the Lagrange multiplier formulation, proposed by K¨oppel et al. [35],
is applied and solved in 3D. The Lagrange multiplier formulation considers
4
fractures as lower-dimensional manifolds (i.e. surface elements). In 3D, this
results in surface domains for the fractures and one volume domain for the
rock matrix. Accurate transfer of fluid pressure between surface and volume
domains is accomplished by using the L2-projection variational transfer op-
erator [36] to discretize the Lagrange multipliers. The following subsections
discuss the mathematical formulation, followed by a description of the dis-
cretization, the L2-projection method for surface–volume interaction, and
the implementation.
2.1 Mathematical formulation
Following K¨oppel et al. [35], we formulate a continuous Lagrange multiplier
fracture problem.
The matrix domain is designated with Ω ⊂Rn,n= 2 or 3, and the fracture
domain with γ⊂Ω of dimension n−1. A normal vector, nγ, is defined
with respect to the fracture surface (Fig. 1). Steady-state fluid flow in the
porous-medium matrix, Ω, is governed by
∇ · (−K∇p)−λ=fin Ω ,
p= 0 on Γ = ∂Ω,(1)
where Kis the permeability tensor, pis the fluid pressure, and fis the
sink/source term.
Flow in the fracture, γ, is described by
∇γ·(−Kγ∇γpγ) + λ=fγin γ ,
pγ= 0 on Γ = ∂γ . (2)
Above, ∂Ω and ∂γ is the interface boundary between Ω and γ. Fluid ex-
change between Ω and γis given by λ=λ(x), x ∈γ.
The spaces VΩ, Vγ,V, and Λ are defined by:
VΩ=H1
0(Ω), Vγ=H1
0(γ),
V=VΩ×Vγ,Λ = H
1
2
0,0(γ),(3)
with the test functions q∈VΩ,qγ∈Vγ, and µ∈Λ. The variational
formulation is found by multiplying Eqs. (1) and (2) by the test functions,
integrating over Ω and γ, and using integration by parts on both equations.
From that, the variational formulation is given as follows:
Find (p, pγ)∈Vand λ∈Λ, such that
5
ZΩ
K∇p· ∇q+Zγ
Kγ∇γpγ· ∇γqγ−
Zγ
λ(q−qγ) = ZΩ
fq +Zγ
fγqγ,∀(q, qγ)∈V
(4)
and Zγ
(p−pγ)µ= 0 ,∀µ∈Λ.(5)
Here, Eq. (5) indicates the coupling conditions between the domains Ω and γ.
The Lagrange multiplier represents the fluid pressure gradient λ=K∇p·nγ
and ensures the continuity of the fluid pressure and the exchange of the
forces between the fracture domain, γ, and the matrix, Ω, in the direction
normal to γ.
K¨oppel et al. [35] show that there exists a unique solution to the variational
formulation of this problem. Further details on the mathematical proof can
be found in their work.
Ω
γ
nγ
Figure 1: 2D matrix domain Ω with an embedded 1D fracture domain γ
and normal vector nγon γ.
2.2 Discretization
Based on the variational formulation in Eq. (4), the discrete counterpart
is formulated in a finite element framework. In order to solve the discrete
formulation, three distinct meshes are defined to approximate the matrix Ω,
the fracture γ, and the Lagrange multiplier λin the fracture.
The meshes for the finite element discretization of Eq. (4) are defined to be
Min Ω, Mγin γ, and Mλin λ. The respective mesh widths hM,hM,γ ,
and hM,λ are defined by:
6
hM:= max
1≤M≤M hM, where hM= diam M,
hM,γ := max
1≤m≤Mγ
hm, where hm= diam m,
hM,λ := max
1≤n≤Mλ
hn, where hn= diam n.
To enhance readability, the respective mesh widths are reduced to h,hγ,
and hλfor the remainder of this paper.
The shape of elements in each mesh might be of any kind. Hence, the
approximation spaces Vh,Ω,Vh,γ, and Λhof continuous, piecewise-polynomial
functions on Ω, γ, and λare defined by:
Vh,Ω={q∈H1
0(Ω) : ∀M∈ M,
q|M∈(P2(M) if Mis: a triangle, pyramid, tetrahedron, or prism
Q2,2(M) if Mis: a quadrilateral or hexahedron ),
Vh,γ ={qγ∈H1
0(γ) : ∀m∈ Mγ,
qγ|m∈(P1(m) if mis: a triangle or line segment
Q1,1(m) if mis: a quadrilateral ),
Λh=Vh,γ.
(6)
The definition of the approximation space Λhof the Lagrange multiplier
implies that hλ=hγ. To prevent a poorly conditioned system matrix, it is
necessary to satisfy hλ≤min(h, hγ), which – in combination with the above
– results in h≤hγ. Generally, by using qγ|m∈P1(m) and λ|n∈P1(n), the
Lagrange multiplier λis applied on the fracture mesh. This offers higher
flexibility regarding the meshing.
2.3 Volume–surface information transfer
The meshes associated with the spaces Vh,Ω(matrix) and Vh,γ (fracture net-
work) are generally non-matching (i.e. Fig. 2), which means that the surfaces
of the volume elements of the matrix do not necessarily coincide with those
of the surface elements of the fracture network. This leads to mutually
non-conforming discretizations which require the use of information transfer
techniques such as interpolation or L2-projections for handling the coupling
terms in Eq. (4). Here the L2-projection approach is adopted, as it has been
shown to have better approximation properties than interpolation [27].
Ultimately, intersections between the matrix mesh and the fracture mesh
7
Figure 2: Examples of matrix volume mesh (left) and inclined fracture sur-
face mesh (center). The two meshes are combined in a single model (right).
have to be found in order to perform quadrature on the coupling term in
Eq. (5) up to numerical precision. Then, integration is done on these in-
tersections. For any pair of a volume element M∈ M and a shell element
m∈ Mγ, the following procedure is completed:
•Computation of the intersection I=M∩mby using a variant of the
Sutherland–Hodgman clipping algorithm [49] as shown in Fig. 3. Here,
Mis interpreted as a set of half-spaces which are sequentially used to
clip m.
•If I6=∅,Iis meshed into the simplicial complex TI={S}, where Sis
a simplex.
•A suitable quadrature rule is mapped to each simplex Swhich is then
mapped to the reference configurations of Mand m.
The task of detecting pair-wise element intersections is accelerated by em-
ploying octree data structures. Additional acceleration can be gained by
M
I
m
M
I
m
Figure 3: Example of intersections Ibetween elements of meshed domains
Mand m. Left: 3D. Right: 2D.
8
applying techniques such as spatial-hashing [18]. The applied algorithms are
fully automated and no prior knowledge about the relation of the meshes is
required. Further details regarding the information transfer procedure can
be found in Krause and Zulian [36].
2.4 Implementation
The routines described in this paper are implemented within the open-source
software library Utopia [58]. Utopia uses libMesh [34] for the finite ele-
ment discretization, MOONoLith [57] for the intersection detection, and
PETSc [5] with MUMPS [3] for the linear algebra calculations. In the nu-
merical experiments illustrated in Section 3, the size of the algebraic linear
system of equations (4) reaches at most one million degrees of freedom,
which is solved with the MUMPS direct solver.
3 Numerical results & Discussion
This section discusses various numerical experiments performed to investi-
gate the Lagrange multiplier method in 2D and 3D. First, the implementa-
tion is verified in 2D by comparing the results to a benchmark case presented
by Flemisch et al. [20]. In a second experiment, the 2D case is extruded in the
third dimension. This enables testing the accuracy of the Lagrange multi-
plier method, combined with the L2-projection variational transfer operator
(LM–L2) in 3D by comparing it to the 2D benchmark results. Next, a het-
erogeneous 3D fracture network is built and the LM–L2method is validated
by comparing it with results from a conforming mesh model. Finally, the
LM–L2method is applied to a realistic fracture network with 150 randomly
distributed fractures. These final numerical experiments are conducted to
investigate different geometrical complexities as well as mesh convergence.
Throughout the numerical experiments, various element shapes are used,
following Eqs. (6). The implementation of the LM–L2method causes no
restrictions regarding the element shape, so that the matrix domain, Ω,
and the fracture domain, γ, can therefore be of arbitrary shape. However,
for simplicity, the matrix mesh, in all presented numerical experiments, is
composed of hexahedral (for 3D) and quadrilateral (for 2D) elements. The
respective element spaces for all experiments is second order in Ω and first
order in γand λ. Nevertheless, the implementation allows consideration of
zero-, first-, and second-order elements in all domains. To test convergence
and accuracy, various mesh sizes for Ω and γ/λare employed throughout
this study.
9
To facilitate comparison with existing studies [20, 35], all physical parame-
ters are normalized throughout this study. All experiments are conducted on
a unit domain with an edge length of 1. Further, the fracture permeability
is Kγ= 104Iγand the fracture aperture is a= 10−4, which is incorporated
in the applied fracture permeability and the Neumann boundary condition.
The permeability in the matrix domain is set to K=I. These choices en-
sure that, for the cases studied here, the matrix and fracture network both
contribute to a similar extent to the overall fluid flow, i.e. the ratio between
a) the average aspect ratio of the fractures and b) the permeability ratio
between the matrix and the fractures approximately equal to one [17].
3.1 Benchmark – 2D
First, the implementation of the LM–L2method is tested by comparing
the obtained results to 2D benchmark results presented by Flemisch et al.
[20]. Benchmark 1, which was first introduced by Geiger et al. [24], is cho-
sen as a representative example for this study. Emphasis is placed on the
validation of the presented method with reference results from Geiger et al.
[24], Flemisch et al. [20], while results of alternative approaches can be found
in the listed references. While K¨oppel et al. [35] used P0elements for their
benchmark comparison, this study employs P1elements for the Lagrange
multiplier mesh and hγ=hλ. The fracture elements consist of line segments
and the matrix of quadrilateral elements. Further, a mesh-size convergence
study is conducted for the fracture and the matrix mesh.
The fracture network at hand contains six fractures in perpendicular orienta-
tion as shown in Fig. 4. A non-homogeneous Neumann boundary condition
on the left domain boundary serves as a fluid source and a non-homogeneous
Dirichlet boundary condition on the right domain boundary serves as a fluid
sink, which yields fluid flow across the domain from the left to the right
boundary. Homogeneous Neumann boundary conditions at the top and
bottom boundaries enforce no-flow conditions across those domain bound-
aries.
The resulting pressure distribution for h= 1/129, and hγ=hλ= 1/128 is
depicted in Fig. 5, and shows good agreement with the reference benchmarks
[24, 20]. As expected, the central, horizontal fracture serves as a channel of
high permeability, facilitating faster fluid flow than the surrounding porous-
medium matrix. Matrix regions further removed from the high-permeability
fractures (e.g., top and bottom left corner) therefore result in the steepest
fluid pressure gradients.
Mesh-size dependency is subsequently investigated in a convergence study.
10
Figure 4: Numerical mesh used for the 2D benchmark case [20]. The
fracture network is shown in white. The enlarged region on the right shows
the fracture intersections. The displayed mesh has a discretization of h=
1/65. Points A, A0, B and B0mark the observation lines AA0and BB0along
which fluid pressure profiles are plotted.
Mesh widths are chosen such that non-conforming meshes for the fractures
and the matrix are ensured. The initial realization has the largest mesh
width of h= 1/33 and hγ=hλ= 1/32. With each refinement, the mesh
resolution is then increased by a factor of two, resulting in h= 1/65 and
hγ=hλ= 1/64, h= 1/129 and hγ=hλ= 1/128, and h= 1/257 and
hγ=hλ= 1/256.
Figs. 6a and 6b show the results for all realizations and the reference results
along the lines AA0and BB0, yielding good convergence and accuracy for all
realizations. However, for low resolutions of h(γ/λ), small deviations can be
observed along BB0between Arc length = 0.6 and Arc length = 1.0. This
can be attributed to a lack in resolution of the fracture intersections by the
non-conforming mesh configuration. This error increases with decreasing
matrix mesh resolution. K¨oppel et al. [35] suggest that results might be
more accurate for h≥hγ,λ. However, the considered second-order function
space in Ω reduces this effect.
Fig. 6c shows the RMSMin the matrix for h= 1/33, h= 1/65, and
h= 1/129, relative to h= 1/257, which is decreasing approximately lin-
early with increasing mesh resolution. The error between different results is
calculated by the root mean square (RMS) over all elements. To facilitate
comparison of non-conforming meshes, results from all resolutions are inter-
polated on a mesh with the finest resolution. The squared error on a single
11
Figure 5: 2D benchmark case, consisting of a pressure gradient from left to
right and six embedded, lower-dimensional fractures with higher permeabil-
ity [20]. The discretization is h= 1/129 and hγ=hλ= 1/128.
element in the mesh M(γ)and the RMSM,(γ)error are calculated following:
err2
n= (pn,ref −pn)2,
RMSM,(γ)=v
u
u
t
1
N
N
X
n=1
err2
n,(7)
where nis a node in Mor Mγ, and N= Σn.
3.2 Benchmark – 3D
The accuracy of the LM–L2method in 3D is tested by extruding the setup in-
troduced in Section 3.1 in the third dimension, which results in a 3D porous-
medium matrix domain and surface domains for the fractures (Fig. 4). This
extrusion enables comparison of 3D results and the 2D benchmarks from
Flemisch et al. [20]. The extrusion causes the observation lines AA0and
BB0(see Fig. 4) to be located anywhere in the direction of the extrusion.
For the present case, their position is chosen at half of the extrusion length.
Non-homogeneous Neumann boundary conditions are applied at the frac-
ture and the matrix on the left boundary. A non-homogeneous Dirichlet
boundary condition is applied on the right boundary. Homogeneous Neu-
mann boundary conditions are imposed on the remaining boundaries.
Various realizations with different mesh sizes are performed to study conver-
gence. Fig. 7 shows the pressure profile along the lines (a) AA0and (b) BB0
12
Figure 6: Numerical convergence with mesh refinement for the 2D bench-
mark case [20]. Shown are: (a) the fluid pressure profile along the line AA0
(see Fig. 4); (b) the pressure profile along the line BB0(see Fig. 4); and
c) the RMSMerror for h= 1/33, h= 1/65, and h= 1/129, relative to
h= 1/257 and hγ=hλ= 1/256 (see Eq. 7).
for the reference results in 2D and the 3D results for all realizations. The
initial mesh width of h= 1/33 and hγ=hλ= 1/32 is consecutively refined
by a factor of two, which results in h= 1/65 and hγ=hλ= 1/64, and
h= 1/129 and hγ=hλ= 1/128. The element shapes in this experiment are
hexahedrons in Ω and quadrilaterals in γ/λ. A more detailed convergence
study in 3D is presented for the more complex case in Section 3.3. Here, we
restrict ourselves to a comparison of the pressure results along the observa-
tion lines AA0and BB0with the reference results (Fig. 7).
While the pressure profile is captured for the mesh widths h= 1/65 and
hγ=hλ= 1/64, and h= 1/33 and hγ=hλ= 1/32, they display a slight
deviation between Arc length = 0.6 and Arc length = 1.0. As accuracy
improves with decreasing mesh size, the realization with h= 1/129 and
hγ=hλ= 1/128 only shows minor deviations from the reference results.
The more pronounced deviations for these coarser mesh realizations can be
explained with a lack of resolution around the fracture intersections in the
area of Arc length = 0.6 to Arc length = 1.0. This suggests that the matrix
mesh width around the fractures and in particular at the fracture intersec-
tions should be smaller. However, this effect in 3D is similar to the one in
2D systems (see Section 3.1).
13
Figure 7: Pressure profiles along the lines of: (a) AA0; and (b) BB0as shown
in Fig. 4. The two reference lines are located in the center of the extrusion
in the third dimension. Numerical convergence is shown by reference results
and three mesh refinements.
3.3 Heterogeneous Fracture Network – 3D
Here we generate an artificial heterogeneous fracture network to test the
LM–L2method with more complex geometries. This enables the investiga-
tion of accuracy and convergence for fractures with varying orientation, size,
and location. Fractures are often assumed to be disc-shaped [17]. Hence,
our numerical experiment also approximates fractures as circular surfaces.
Fracture tips are generally difficult to represent by non-conforming mesh
methods, as they characterize the boundary of a fracture domain. On the
tips, the Lagrange multiplier can act in multiple spatial directions, while
its primary direction in the center of the fracture is normal to the fracture
plane (i.e. flow normal to the fracture plane). This becomes particularly
important in 3D, if the fracture edge is not a line (e.g. corners or circular
fractures), as this results in a discontinuity of the direction of the Lagrange
multiplier. Throughout this experiment, only the matrix-mesh width, h,
is varied and the results are compared to results from a conforming mesh
simulation. This enables investigation of the influence of the matrix mesh
width on the accuracy of the results.
The chosen model setup for seven fractures is shown in Fig. 8. All fractures
are located within the matrix domain and are characterized by a random
diameter, orientation and location. At the boundaries BC0and BC1, the
applied Dirichlet condition is set to 1 and 0, respectively. The homogeneous
Neumann conditions imposed on the remaining boundaries ensures no-flow.
Convergence is studied with three realizations with a matrix-mesh width of
14
Figure 8: Model setup for the heterogeneous 3D case with seven embedded
fractures, shown in green. A slice through the porous-medium matrix mesh,
with h= 1/129, is depicted in red. Dirichlet boundary conditions, BC0
and BC1, are located at the left and right sides of the matrix mesh. Three
observation surfaces are located at (A) x= 0.5, (B) y= 0.38, and (C)
z= 0.5.
h= 1/33, h= 1/65, and h= 1/129, respectively. The fracture-mesh width
for all realizations is constant at hγ=hλ= 1/129. The results of these
realizations are compared to reference results which are obtained with a
conforming mesh using finite element methods in MOOSE [23] with a mesh
width of h=hγ= 1/129. For the conforming mesh, the fracture elements
are triangles and the matrix elements tetrahedrons. For the LM–L2method,
the element shapes in γare triangles and hexahedrons in Ω.
The observed RMS errors in the matrix and fracture show linear conver-
gence rates with an improved rate for h= 1/129 (see Fig. 9). Due to the
similar mesh width of this realization to the reference model, this error solely
compares the effect of non-conforming meshes. Furthermore, the observed
convergence rates are similar for matrix and fractures. Hence, the chang-
ing mesh width in the matrix mesh affects the accuracy in the fractures
and the matrix similarly. This further suggests that sufficiently high matrix
resolution around the fractures is crucial. Generally, the RMS error in the
fractures is higher because the entire domain is affected by the non-matching
meshes. In contrast, large matrix areas are further away from the fractures
and therefore not affected by the non-matching meshes.
Fig. 10 shows the local error at cross sections A, B, and C, which are de-
15
Figure 9: RMS error between the reference results and the results of the
LM–L2method. The errors are calculated separately for the matrix mesh,
M, and the fracture mesh, Mγ. Shown is the error for different 3
√#cells of
the matrix mesh M.
fined in Fig. 8. Overall, the error in planes A and B is small, although an
Figure 10: Error (qerr2
Min Eq. 7) between the reference results and the
LM–L2results for the realization with h= 1/129, hγ= 1/129, hλ=hγ.
The planes A, B, and C are defined in Fig. 8. The fractures, intersected by
the respective plane, are illustrated in black.
increased error is present around the fractures and in particular the fracture
tips. The biggest errors can be found on plane C, specifically at the fracture
tips of the upper most large fracture. This fracture is cut through its center
and has the largest diameter, thereby connecting high-pressure areas on the
left with low-pressure areas on the right. These relatively large errors can
also be observed for other fractures oriented parallel to the gradient, as they
form a shortcut for pressure in the system. This leads to the largest pressure
16
gradients at the fracture tips which are located close to the inflow and out-
flow of the domain, or which happen to be the farthest apart in the direction
of the gradient. This is also where the biggest errors in the fracture domain
are found. The difficulties encountered during the generation of the con-
forming mesh for the reference solution serve as further motivation for the
development of non-conforming methods. Particularly, very small elements
are necessary to represent the matrix between the fracture intersections with
low intersection angle. Specialized meshing tools are required to efficiently
mesh high-quality, conforming mixed-dimensional meshes [14, 29, 9].
3.4 Random Fracture Network – 3D
Fracture networks commonly contain hundreds or even thousands of frac-
tures within a rock volume of 100 m side length. They are geometrically
characterized by several distributions regarding the fracture size, orienta-
tion, density or location. It is commonly assumed that the fracture size
is distributed following a truncated power law [12]. Fracture orientation,
density, and location are site-specific and usually derived from geological
information.
These highly heterogeneous geometries are computationally demanding, when
generating meshes for numerical simulations. Using the LM–L2method, the
geometry only needs to be represented by the fracture mesh, as the porous-
medium matrix is represented by a regular grid. This numerical experiment
shows and investigates the application of the LM–L2method on a realistic
fracture network with 150 fractures.
In this experiment, the fracture radius distribution follows a power law,
with truncations at 0.1 and 0.4. The 150 fractures are circular, randomly
oriented, and distributed in a cubical model domain with a side length of
1. For simplicity, it is further ensured that no fracture intersects the model
domain boundary. Fig. 11 shows the fractures embedded in the porous-
medium matrix, of which only a single layer is displayed in red, illustrating,
how the fracture mesh (green) cuts through the porous-medium matrix mesh
(red). The mesh width for this numerical experiment is set to h= 1/33 in
the porous-medium matrix and hγ=hλ= 1/200 in the fractures. In ac-
cordance with the previous numerical experiment, the element shapes are
triangles in γand hexahedrons in Ω. Fluid flow is modeled from left to right
by non-homogeneous Dirichlet boundary conditions with values of 1 and 0,
respectively. The imposed boundary conditions on the remaining bound-
aries are homogeneous Neumann boundary conditions.
The resultant dimensionless fluid pressure distribution in the fractures and
17
the porous-medium matrix is depicted in Fig. 12. Additionally, three obser-
vation planes A, B, and C are defined to observe the fluid pressure in each
plane (Fig. 13). Although the fluid pressure field is clearly influenced by the
fracture network, Planes B and C show that the porous-medium matrix still
contributes considerably to the overall fluid flow through the entire frac-
tured porous medium. These results demonstrate the presented method’s
capability of modeling fluid flow through complex fractured porous media.
Figure 11: Mesh for the random fracture network. Shown in red is one layer
of the matrix mesh. The 150 fractures are depicted in green. The displayed
mesh discretization is h= 1/33 and hγ=hλ= 1/200.
Figure 12: Dimensionless fluid pressure field in the fracture network and the
porous-medium matrix. Cross sections through the matrix are given at (A)
x= 0.26, (B) y= 0.5, and (C) z= 0.5.
18
Figure 13: Pressure contours in the porous-medium matrix within Planes
A, B, and C, shown in Fig. 12. Fractures intersecting a plane are shown in
white.
4 Conclusion
This study builds on previous work to combine a Lagrange multiplier method
with an L2-projection variational transfer operator to numerically model
single-phase fluid flow problems in 3D fractured porous media. The perfor-
mance of the method is assessed by comparison with benchmark cases and
numerical experiments in 2D and 3D. We also demonstrate the method’s
suitability for large-scale, realistic fracture-network realizations in 3D.
Our comparison with benchmark simulations shows good agreement with
reference results from the literature. In general, the accuracy of non-conforming
methods at fracture intersections is known to depend on the width of the
porous-medium matrix mesh. These findings are confirmed in this study
by all of our numerical experiments, as we also observe small deviations in
fluid pressures at fracture intersections for larger mesh widths. However,
the RMS error in the porous-medium matrix mesh decreases linearly with
increasing mesh size, as desired.
To study the presented method in 3D, the 2D benchmark case is extruded
in the third direction, thereby enabling a comparison of 3D simulations with
2D benchmark results. Although the 3D case provides additional complex-
ity for the transfer operator, results agree well with the benchmark results.
Analogous to the 2D results, the 3D results also show the largest deviations
from the reference results at fracture intersections.
A 3D fracture network, containing seven fractures, is presented, demonstrat-
ing the applicability of circular fractures with random orientations, sizes,
and fracture tips inside the model domain. These simulations are compared
to reference results obtained with conforming mesh simulations. A con-
vergence study refined the porous-medium matrix mesh while leaving the
19
fracture mesh unaltered. The observed convergence rates show a linear be-
havior in the porous-medium matrix and the fracture domains, with slightly
improved convergence for the highest mesh resolution. The largest errors
occur at the location of the steepest fluid pressure gradients, for example, at
the tips of fractures that are aligned parallel to the fluid pressure gradient.
More specifically, the errors occur at the fracture tips farthest apart from
one another and in the direction of the fluid pressure gradient.
Finally, a numerical experiment of a realistic discrete fracture network (DFM),
with 150 fractures, demonstrates the capability to model single-phase flow
through highly complex fractured porous media. The obtained fluid pres-
sure contours in the 3D domain show the expected behavior for a fractured
porous medium, with high-permeability fractures, largely influencing the
fluid flow through the numerical model domain. Despite a coarse porous-
medium matrix mesh, the presented method achieves detailed representa-
tions of fractures embedded in a porous-medium matrix domain.
Generally, the Lagrange Multiplier–L2-projection method shows good agree-
ment with benchmark cases and reference results, while yielding good con-
vergence. In all cases, our results suggest that the matrix mesh of the
porous-medium should be locally refined when accuracy is to be improved,
particularly around fracture tips and at fracture intersections.
Further research will focus on the development of mesh adaptivity algo-
rithms that allow pressure-gradient- and fracture-location-dependent porous-
medium matrix-mesh refinements. Additional extensions of the numerical
approach presented here will target transient fluid flow computation for
which the parallel-processing algorithm, employed in this study, can be op-
timized.
Acknowledgment
M.O.S., P.S., A.E., D.V., and S.B.R. thank the Werner Siemens Founda-
tion for their endowment of the Geothermal Energy and Geofluids group at
the Institute of Geophysics, ETH Zurich. P.Z., M.G.C.N, and R.K. thank
the SCCER-SoE program. We gratefully acknowledge the discussion with
Markus K¨oppel regarding his implementation of the Lagrange multiplier
method.
20
Computer Code Availability
All methods and routines, used for this study, are implemented with the
open-source software library Utopia [58]. Utopia’s lead developer is co-
author Patrick Zulian at USI Lugano, Switzerland. Co-developers are Alena
Kopaniˇc´akov´a, Maria Chiara Giuseppina Nestola, Andreas Fink, Nur Fadel,
Victor Magri, Teseo Schneider, and Eric Botter.
The contact address and e-mail of Patrick Zulian are as follows:
Institute of Computational Science
Universit`a della Svizzera italiana (USI - University of Lugano)
Via Giuseppe Buffi 13
CH-6904 Lugano
patrick.zulian@usi.ch
Utopia was first available in 2016, the programming language is C++ and
it can be accessed through a git repository or a docker container on:
https://bitbucket.org/zulianp/utopia (18.6 MB),
https://hub.docker.com/r/utopiadev/utopia.
The software dependencies are as follows:
PETSc (https://www.mcs.anl.gov/petsc/),
must be compiled with MUMPS enabled
libMesh for the FE module (https://github.com/libMesh)
There are no hardware requirements given by Utopia. Potential hardware
or software requirements of the underlying libraries libMesh and PETSc are
not stated here.
ORCID
P. Sch¨adle - https://orcid.org/0000-0002-5485-2028
P. Zulian - https://orcid.org/0000-0002-5822-3288
D. Vogler - https://orcid.org/0000-0002-0974-9240
S. Bhopalam R. - https://orcid.org/0000-0001-7221-695X
M.G.C. Nestola - https://orcid.org/0000-0002-5700-0306
A. Ebigbo https://orcid.org/0000-0003-3972-3786
R.H. Krause - https://orcid.org/0000-0001-5408-5271
M.O. Saar - https://orcid.org/0000-0002-4869-6452
21
References
[1] M. Ahkami, T. Roesgen, M. O. Saar, and X.-Z. Kong. High-resolution
temporo-ensemble piv to resolve pore-scale flow in 3d-printed fractured
porous media. Transport in Porous Media, pages 1–17, 2018.
[2] F. Amann, V. Gischig, K. Evans, J. Doetsch, R. Jalali, B. Valley,
H. Krietsch, N. Dutler, L. Villiger, B. Brixel, et al. The seismo-
hydromechanical behavior during deep geothermal reservoir stimula-
tions: open questions tackled in a decameter-scale in situ stimulation
experiment. Solid Earth, 9(1):115–137, 2018.
[3] P. R. Amestoy, I. S. Duff, J. Koster, and J.-Y. L’Excellent. A fully
asynchronous multifrontal solver using distributed dynamic scheduling.
SIAM Journal on Matrix Analysis and Applications, 23(1):15–41, 2001.
[4] R. G. Baca, R. C. Arnett, and D. W. Langford. Modelling fluid flow in
fracturedporous rock masses by finiteelement techniques. International
Journal for Numerical Methods in Fluids, 4(4):337–348, 4 1984.
[5] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith. Efficient man-
agement of parallelism in object oriented numerical software libraries.
In E. Arge, A. M. Bruaset, and H. P. Langtangen, editors, Modern Soft-
ware Tools in Scientific Computing, pages 163–202. Birkh¨auser Press,
1997.
[6] G. Barenblatt, I. P. Zheltov, and I. Kochina. Basic concepts in the the-
ory of seepage of homogeneous liquids in fissured rocks [strata]. Journal
of applied mathematics and mechanics, 24(5):1286–1303, 1960.
[7] B. Berkowitz. Characterizing flow and transport in fractured geological
media: A review. Advances in Water Resources, 25(8):861–884, 2002.
[8] I. Berre, F. Doster, and E. Keilegavlen. Flow in fractured porous media:
A review of conceptual models and discretization approaches. arXiv
preprint arXiv:1805.05701, 2018.
[9] D. Blessent, R. Therrien, and K. MacQuarrie. Coupling geological and
numerical models to simulate groundwater flow and contaminant trans-
port in fractured media. Computers & Geosciences, 35(9):1897–1906,
2009.
22
[10] I. Bogdanov, V. Mourzenko, J.-F. Thovert, and P. Adler. Two-phase
flow through fractured porous media. Physical Review E, 68(2):026703,
2003.
[11] C. E. Bond, R. Wightman, and P. S. Ringrose. The influence of fracture
anisotropy on co2 flow. Geophysical Research Letters, 40(7):1284–1289,
2003.
[12] E. Bonnet, O. Bour, N. E. Odling, P. Davy, I. Main, P. Cowie, and
B. Berkowitz. Scaling of fracture systems in geological media. Reviews
of Geophysics, 39(3):347–383, 8 2001.
[13] W. M. Boon, J. M. Nordbotten, and I. Yotov. Robust discretization of
flow in fractured porous media. SIAM Journal on Numerical Analysis,
56(4):2203–2233, 2018.
[14] M. Cacace and G. Bl¨ocher. Meshita software for three dimensional
volumetric meshing of complex faulted reservoirs. Environmental Earth
Sciences, 74(6):5191–5209, 2015.
[15] M. C. Cacas, E. Ledoux, G. Marsily, B. Tillie, A. Barbreau, E. Durand,
B. Feuga, and P. Peaudecerf. Modeling fracture flow with a stochastic
discrete fracture network: calibration and validation: 1. the flow model.
Water Resources Research, 26(3):479–489, 3 1990.
[16] J.-R. Dreuzy, Y. M´eheust, and G. Pichot. Influence of fracture scale
heterogeneity on the flow properties of three-dimensional discrete frac-
ture networks (dfn). Journal of Geophysical Research: Solid Earth, 117
(B11), 2012.
[17] A. Ebigbo, P. S. Lang, A. Paluszny, and R. W. Zimmerman. Inclusion-
based effective medium models for the permeability of a 3d fractured
rock mass. Transport in Porous Media, 113(1):137–158, 2016.
[18] C. Ericson. Real-Time Collision Detection (The Morgan Kaufmann
Series in Interactive 3D Technology). Morgan Kaufmann Publishers
Inc., San Francisco, CA, USA, 2004.
[19] B. Flemisch, A. Fumagalli, and A. Scotti. A Review of the XFEM-Based
Approximation of Flow in Fractured Porous Media. In Advances in
Discretization Methods, pages 47–76. Springer International Publishing,
2016.
23
[20] B. Flemisch, I. Berre, W. Boon, A. Fumagalli, N. Schwenck, A. Scotti,
I. Stefansson, and A. Tatomir. Benchmarks for single-phase flow in
fractured porous media. Advances in Water Resources, 111:239 – 258,
2018.
[21] N. Frih, V. Martin, J. E. Roberts, and A. Saˆada. Modeling fractures as
interfaces with nonmatching grids. Computational Geosciences, 16(4):
1043–1060, sep 2012.
[22] A. Fumagalli, E. Keilegavlen, and S. Scial`o. Conforming, non-
conforming and non-matching discretization couplings in discrete frac-
ture network simulations. Journal of Computational Physics, 376:694–
712, 2019.
[23] D. Gaston, C. Newman, G. Hansen, and D. Lebrun-Grandi´e. MOOSE:
A parallel computational framework for coupled systems of nonlinear
equations. Nuclear Engineering and Design, 239(10):1768–1778, 2009.
[24] S. Geiger, M. Dentz, and I. Neuweiler. A novel multi-rate dual-porosity
model for improved simulation of fractured and multiporosity reser-
voirs. SPE Journal, 18(04):670–684, 2013.
[25] H. Hajibeygi, D. C. Karvounis, and P. Jenny. A hierarchical fracture
model for the iterative multiscale finite volume method. Journal of
Computational Physics, 230(24):8729–8743, 2011.
[26] R. Helmig et al. Multiphase flow and transport processes in the subsur-
face: a contribution to the modeling of hydrosystems. Springer-Verlag,
1997.
[27] C. Hesch and P. Betsch. A comparison of computational methods for
large deformation contact problems of flexible bodies. ZAMM - Journal
of Applied Mathematics and Mechanics / Zeitschrift f¨ur Angewandte
Mathematik und Mechanik, 86(10):818–827, 2006.
[28] A. Hob´e, D. Vogler, M. P. Seybold, A. Ebigbo, R. R. Settgast, and
M. O. Saar. Estimating fluid flow rates through fracture networks using
combinatorial optimization. Advances in Water Resources, 122:85 – 97,
2018.
[29] R. Holm, R. Kaufmann, B.-O. Heimsund, E. Øian, and M. S. Espedal.
Meshing of domains with complex internal geometries. Numerical Lin-
ear Algebra with Applications, 13(9):717–731, 2006.
24
[30] J. D. Hyman, S. Karra, N. Makedonska, C. W. Gable, S. L. Painter, and
H. S. Viswanathan. dfnworks: A discrete fracture network framework
for modeling subsurface flow and transport. Computers & Geosciences,
84:10–19, 2015.
[31] M. Karimi-Fard, L. J. Durlofsky, K. Aziz, et al. An efficient discrete
fracture model applicable for general purpose reservoir simulators. In
SPE Reservoir Simulation Symposium. Society of Petroleum Engineers,
2003.
[32] H. Kazemi, L. Merrill Jr, K. Porterfield, P. Zeman, et al. Numerical
simulation of water-oil flow in naturally fractured reservoirs. Society of
Petroleum Engineers Journal, 16(06):317–326, 1976.
[33] H. Kazemi et al. Pressure transient analysis of naturally fractured reser-
voirs with uniform fracture distribution. Society of petroleum engineers
Journal, 9(04):451–462, 1969.
[34] B. S. Kirk, J. W. Peterson, R. H. Stogner, and G. F. Carey. libMesh:
A C++ Library for Parallel Adaptive Mesh Refinement/Coarsening
Simulations. Engineering with Computers, 22(3–4):237–254, 2006.
[35] M. K¨oppel, V. Martin, J. Jaffr´e, and J. E. Roberts. A lagrange mul-
tiplier method for a discrete fracture model for flow in porous media.
Computational Geosciences, 2018.
[36] R. Krause and P. Zulian. A parallel approach to the variational trans-
fer of discrete fields between arbitrarily distributed unstructured finite
element meshes. SIAM Journal on Scientific Computing, 38(3):C307–
C333, 2016.
[37] M. W. McClure and R. N. Horne. Correlations between formation
properties and induced seismicity during high pressure injection into
granitic rock. Engineering Geology, 175:74 – 80, 2014.
[38] A. Moinfar, A. Varavei, K. Sepehrnoori, and R. T. Johns. Development
of an Efficient Embedded Discrete Fracture Model for 3D Compositional
Reservoir Simulation in Fractured Reservoirs. SPE Journal, 19(02):
289–303, apr 2014.
[39] J. Monteagudo and A. Firoozabadi. Control-volume method for nu-
merical simulation of two-phase immiscible flow in two-and three-
dimensional discrete-fractured media. Water resources research, 40(7),
2004.
25
[40] M. Nestola, B. Becsek, H. Zolfaghari, P. Zulian, D. Obrist, and
R. Krause. An immersed boundary method based on the variational
l2-projection approach. Proceedings DD24 2016, 2017.
[41] S. P. Neuman. Trends, prospects and challenges in quantifying flow
and transport through fractured rocks. Hydrogeology Journal, 13(1):
124–147, 2005.
[42] J. Noorishad and M. Mehran. An upstream finite element method
for solution of transient transport equation in fractured porous media.
Water Resources Research, 18(3):588–596, 6 1982.
[43] G. Pichot, J. Erhel, and J. de Dreuzy. A mixed hybrid mortar method
for solving flow in discrete fracture networks. Applicable Analysis, 89
(10):1629–1643, 2010.
[44] G. Pichot, J. Erhel, and J. de Dreuzy. A generalized mixed hybrid
mortar method for solving flow in stochastic discrete fracture networks.
SIAM Journal on Scientific Computing, 34(1):B86–B105, 2012.
[45] C. Planta, D. Vogler, X. Chen, M. Nestola, M. O. Saar, and R. Krause.
Simulation of hydro-mechanically coupled processes in rough rock frac-
tures using an immersed boundary method and variational transfer op-
erators. ArXiv e-prints, Dec 2018.
[46] C. Planta, D. Vogler, M. Nestola, P. Zulian, and R. Krause. Variational
parallel information transfer between unstructured grids in geophysics-
applications and solutions methods. PROCEEDINGS, 43rd Workshop
on Geothermal Reservoir Engineering, Stanford, CA, pages 1–13, 2018.
[47] C. Planta, D. Vogler, P. Zulian, M. O. Saar, and R. Krause. Solution
of contact problems between rough body surfaces with non matching
meshes using a parallel mortar method. ArXiv e-prints, Nov 2018.
[48] A. Rasmuson and I. Neretnieks. Radionuclide transport in fast channels
in crystalline rock. Water Resources Research, 22(8):1247–1256, 1986.
[49] I. E. Sutherland and G. W. Hodgman. Reentrant polygon clipping.
Commun. ACM, 17(1):32–42, Jan. 1974.
[50] M. T¸ ene, S. B. Bosma, M. S. Al Kobaisi, and H. Hajibeygi. Projection-
based embedded discrete fracture model (pedfm). Advances in Water
Resources, 105:205–216, 2017.
26
[51] J. W. Tester, B. Anderson, A. Batchelor, D. Blackwell, R. DiPippo,
E. Drake, J. Garnish, B. Livesay, M. C. Moore, K. Nichols, et al. The
future of geothermal energy: Impact of enhanced geothermal systems
(egs) on the united states in the 21st century. Massachusetts Institute
of Technology, 209, 2006.
[52] C.-F. Tsang and I. Neretnieks. Flow channeling in heterogeneous frac-
tured rocks. Reviews of Geophysics, 36(2):275–298, 1998.
[53] D. Vogler, S. Ostvar, R. Paustian, and B. D. Wood. A hierarchy of
models for simulating experimental results from a 3d heterogeneous
porous medium. Advances in Water Resources, 114:149 – 163, 2018.
[54] D. Vogler, R. R. Settgast, C. Annavarapu, C. Madonna, P. Bayer, and
F. Amann. Experiments and simulations of fully hydro-mechanically
coupled response of rough fractures exposed to high pressure fluid in-
jection. Journal of Geophysical Research: Solid Earth, 123:1186–1200,
2018.
[55] J. Warren and P. Root. The behavior of naturally fractured reservoirs.
Society of Petroleum Engineers Journal, 3(3):245–255, 1963.
[56] R. Zimmerman, S. Kumar, and G. Bodvarsson. Lubrication theory
analysis of the permeability of rough-walled fractures. International
Journal of Rock Mechanics and Mining Sciences & Geomechanics Ab-
stracts, 28(4):325 – 331, 1991.
[57] P. Zulian. ParMOONoLith: parallel intersection detec-
tion and automatic load-balancing library. Git repository.
https://bitbucket.org/zulianp/par moonolith, 2016. URL
https://bitbucket.org/zulianp/par_moonolith.
[58] P. Zulian, A. Kopaniˇc´akov´a, M. C. G. Nestola, A. Fink, N. Fadel,
V. Magri, T. Schneider, and E. Botter. Utopia: A C++ embed-
ded domain specific language for scientific computing. Git reposi-
tory. https://bitbucket.org/zulianp/utopia, 2016. URL https://
bitbucket.org/zulianp/utopia.
27
