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Numerical Manifold Method Modeling of Coupled Processes in Fractured Geological
Media at Multiple Scales
Mengsu Hu, Jonny Rutqvist
PII: S1674-7755(20)30052-4
DOI: https://doi.org/10.1016/j.jrmge.2020.03.002
Reference: JRMGE 660
To appear in: Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 26 January 2020
Revised Date: 5 March 2020
Accepted Date: 9 March 2020
Please cite this article as: Hu M, Rutqvist J, Numerical Manifold Method Modeling of Coupled Processes
in Fractured Geological Media at Multiple Scales, Journal of Rock Mechanics and Geotechnical
Engineering, https://doi.org/10.1016/j.jrmge.2020.03.002.
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Full Length Article
Numerical Manifold Method Modeling of Coupled Processes in Fractured Geological Media at
Multiple Scales
Mengsu Hu*, Jonny Rutqvist
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
Received 26 January 2020; received in revised form 5 March 2020; accepted 9 March 2020
Abstract: The greatest challenges of rigorously modeling coupled hydro-mechanical (HM) processes in fractured geological media at different scales
are associated with computational geometry. These challenges include dynamic shearing and opening of intersecting fractures at discrete fracture scales
as a result of coupled processes, and contact alteration along rough fracture surfaces that triggers structural and physical changes of fractures at micro-
asperity scale. In this paper, these challenges are tackled by developing a comprehensive modeling approach for coupled processes in fractured
geological media based on numerical manifold method (NMM) at multiple scales. Based on their distinct geometric features, fractures are categorized
into three different scales: dominant fracture, discrete fracture, and discontinuum asperity scales. Here the scale is relative, that of the fracture relative
to that of the research interest or domain. Different geometric representations of fractures at different scales are used, and different governing equations
and constitutive relationships are applied. For dominant fractures, a finite thickness zone model is developed to treat a fracture as a porous nonlinear
domain. Nonlinear fracture mechanical behavior is accurately modeled with an implicit approach based on strain energy. For discrete fractures, a zero-
dimensional model was developed for analyzing fluid flow and mechanics in fractures that are geometrically treated as boundaries of the rock matrix.
With the zero-dimensional model, these fractures can be modeled with arbitrary orientations and intersections. They can be fluid conduits or seals, and
can be open, bonded or sliding. For the discontinuum asperity scale, the geometry of rough fracture surfaces is explicitly represented and contacts
involving dynamic alteration of contacts among asperities are rigorously calculated. Using this approach, fracture alteration caused by deformation, re-
arrangement and sliding of rough surfaces can be captured. Our comprehensive model is able to handle the computational challenges with accurate
representation of intersections and shearing of fractures at the discrete fracture scale and rigorously treats contacts along rough fracture surfaces at the
discontinuum asperity scale. With future development of three-dimensional (3D) geometric representation of discrete fracture networks in porous rock
and contacts among multi-body systems, this model is promising as a basis of 3D fully coupled analysis of fractures at multiple scales, for advancing
understanding and optimizing energy recovery and storage in fractured geological media.
Keywords: dominant fractures; discrete fractures; discontinuum asperity scale; coupled processes; numerical manifold method (NMM)
1. Introduction
Fractures play key roles in subsurface energy recovery and storage,
including hydrocarbon and geothermal energy production, and nuclear
waste disposal. Fractures, with sizes ranging from microns to kilometers,
may act as conduits or seals for fluid flow in these various subsurface
energy activities (Rutqvist and Stephansson, 2003). In subsurface energy
recovery (e.g. hydrocarbon recovery and geothermal exploitation),
effectively creating a fracture network or making use of natural fractures
is the key for efficient production. In contrast, in subsurface energy
storage systems (such as carbon sequestration and nuclear waste
disposal), fractures or faults may act as unfavorable flow conduits that
may compromise the seal integrity of the storage facility. In geological
systems involving fluid-filled fractures and porous media, complex
coupled hydro-mechanical (HM) and thermo-hydro-mechanical (THM)
processes can occur, including alteration of natural fractures and creation
of new fractures that may intersect with natural fractures. Because
fractures play key roles in energy recovery and storage and because they
can be created and altered as a result of coupled processes, understanding
coupled processes in fractured geological media is essential.
At reservoir scale, fractures are often not isolated entities, but form
networks of interacting fractures. These fractures are usually very thin
(e.g. microns to millimeters) relative to their length (meters). These
E-mail address: mengsuhu@lbl.gov
fractures may dynamically alter in terms of their dimensions and physical
properties, and new fractures may be created as a result of coupled
processes. Since 1980s, a number of numerical models have been
developed for modeling coupled HM processes in fractured rock,
including equivalent continuum, dual-continuum, and discontinuous
models. As the arbitrarily oriented fractures may influence the
deformation and fluid distribution in a complex way that cannot be
simplified as a continuum, numerical modeling considering discrete
fractures with full coupling capability is of great importance. Noorishad et
al. (1982; 1992) developed a finite element model for fully coupled HM
and THM processes in deformable fractured rock masses. After that,
increasing demands for engineering solutions have inspired development
of many computer codes capable of modeling HM and THM behaviors of
fractured rock at various levels of sophistication, including those applied
for the analysis of nuclear waste disposal and geothermal energy recovery
(Rutqvist et al., 2001). Most of the aforementioned models were
developed based on finite element method (FEM). With the development
of discontinuous methods, fractures could be explicitly represented as a
displacement discontinuity, as they are modeled as interfaces between
contacting individual blocks. This includes models based on the distinct
element method (DEM), including the commercially available universal
distinct element code (UDEC) (Itasca, 2011) and 3DEC (Itasca, 2013), and
models based on discontinuous deformation analysis (DDA) that treat
coupled fluid flow and deformation in discrete fractures, but with rock
blocks being assumed impermeable (Kim et al., 1999; Jing et al., 2001).
Later, models based on enriched FEM were developed (e.g. Silvestre et al.,
2015) that included simplified jump terms to capture the mechanical
displacement discontinuity and hydraulic pressure continuity associated
with fractures. In this method, however, the mechanical effects on hydraulic
properties were not considered.
Models where fractures are explicitly simulated can be categorized
depending on the geometric representation of the fractures for fluid flow
and mechanics. For fluid flow in fractures, there are three types of
models: n-dimensional (i.e. for two-dimensional (2D) models, fractures
are represented by 2D elements), n−1 dimensional (i.e. for 2D models,
fractures are represented by one-dimensional (1D) elements), and zero-
dimensional models recently developed by Hu et al. (2016, 2017b). As
dimensions are associated primarily with degrees of freedom (DOFs) to
be solved in the numerical modeling, the zero-dimensional model for
discrete fractures means that no additional DOFs are required for the
fractures because they are treated as discontinuous boundaries of rock
matrix. For mechanics of fractures, there are two types of models: n-
dimensional solid element models and discontinuous models which treat
fractures as discontinuities between contacting rock blocks. The major
disadvantage of n-dimensional fluid flow models is the inaccuracy of
representation of fracture apertures and the changes they are subjected to,
in spite of the high computational efforts associated with the number of
elements required to approximate the real geometry of intersecting
fractures. The disadvantage of n−1 dimensional fluid flow models is that
they neglect fracture thickness and this results in additional required
DOFs that make them incompatible with mechanical models for fractures
(e.g. difficult to model initiation, shearing or re-opening of fractures).
Though discrete fracture network (DFN) models have been widely
developed, accurate and efficient modeling of interaction between
fractures and rock matrix is lacking, especially when geomechanics plays
an important role. Mechanically, n-dimensional solid element models
(Rutqvist et al., 2009) are excellent for representing fractures with certain
apertures, but when fracture apertures are far less than their lengths, the
use of n-dimensional solid elements becomes too computationally
expensive. Discontinuous mechanical models, on the other hand, are
promising for representing thin fractures where fracture surfaces could be
open, bonded or slip/shearing, but very few numerical models rigorously
consider all these mechanisms. Because of these limitations, most models
are not suitable for analyzing coupled processes at discrete fracture scales
without rigorous consideration of intersections and shearing of the
discrete fractures, particularly those that may be evolving dynamically as
a result of coupled processes.
For single fractures, several models have been developed recently for
analyzing micro-scale coupled processes in fractures as a part of the
international DECOVALEX project (Bond et al., 2016). The models were
categorized into 2D simplified models, statistical models, and
homogenized (in which a rough fracture surface is treated as a single
entity) models. For example, McDermott et al. (2015) presented a model
that combined numerical (for flow) and analytical (for chemo-
mechanical) methods to simulate small-scale coupled chemo-hydro-
mechanical processes, but the model is limited to parallel edge to edge
surface contacts. Pan et al. (2016) developed an elasto-plastic cellular
automaton model for a single fracture using grids with different apertures
to represent contacts and voids. In their model, flow and transport are
calculated using conventional numerical schemes. However, the
mechanics and contact alteration of the fracture are not considered in their
analysis. In these models, geometric features (such as rough boundaries
along fractures, pores and grains) are either not represented explicitly, or
they are approximated by spheres or rectangular grids. Thus, contacts
along rough surfaces cannot be accurately captured. Because of these
limitations, numerical modeling of coupled processes has to the author’s
knowledge never been attempted at the microscopic scale where
discontinuities are important.
In this paper, numerical approaches are presented using numerical
manifold method (NMM) that is able to overcome the limitations as
discussed above, in order to simulate coupled processes at multiple scales.
In Section 2, fractures are categorized into three different scales based on
their geometric features: dominant fracture, discrete fracture, and
discontinuum asperity scales. The analyses of the fundamentally different
physical constitutive behaviors and coupled processes of fractures at each
of these three scales are presented. In Section 3, NMM is introduced that
allows simulating fractures as continuities as well as discontinuities. In
Section 4, three different models for these three scales of fractures are
presented. These are (1) a finite thickness zone model for dominant
porous fractures with nonlinear mechanical behavior; (2) a zero-
dimensional fracture model for fluid flow and mechanics in fractures
networks, where fractures are boundaries of the rock matrix, and can be
open, bonded, or sliding while being fluid conduits or seals; and (3)
explicit model for micro-scale fractures with rigorous treatment of
contacts between fracture asperities and their dynamic alteration. In
Section 5, differences among these different models are summarized, and
the challenges associated with numerical modeling of coupled processes
in fracture at multiple scales are concluded.
2. Fracture across scales: Geometric and physical features
2.1. Three scales of fractures based on geometric features
Depending on multiple scales, fractures appear to have different
geometric and physical features. Fig. 1 shows discrete fractures at
reservoir (100 m-10 km), outcrop (1 m–100 m), and core (1 mm–1 cm,
Ajo-Franklin et al., 2018) scales. Herein, it can be seen that at the
reservoir and outcrop scales, fractures appear in groups, arbitrarily
oriented and intersecting with each other. At the core scale, a dominant
fracture appears, possibly filled with mineral fillings and connected with
smaller fractures at the surrounding rock. If looking at the microscale (1
μm–1 cm) or nanoscale, a single fracture appears to be quite rough and its
tensile failure could be controlled by failure of carbon nanotubes (Taloni
et al., 2018). Therefore, the three types of fractures can be geometrically
summarized: dominant fractures with a certain width, discrete thin
fractures, and discontinuum fractures with asperities forming rough
surfaces.
Fig. 1. Fractures at reservoir, outcrop, and core (Ajo-Franklin et al., 2018) scales.
Note that the scale is relative, meaning that the scales of fractures are
relative to the scale of interest of the problem or domain. For example, at
1 m-1 km scale, fractures/faults could also appear with similar geometric
features to that at core scale, and those different sized fractures could be
considered as dominant features of their respective scale of interest.
Another example is that fracture networks could exist at micro-scale, such
as micro-fractures in core samples, micro-crack populations associated
with subcritical crack growth. Regardless of different formation
mechanisms (Voigtländer et al., 2018), micro-fractures in rock cores
appear to have similar geometric features to meter-scale discrete fractures
at reservoir. Based on this ‘relative scale’ concept, it is also possible to
label micro-crack populations as discrete fracture scale, which makes it
reasonable to use similar types of numerical models to simulate discrete
fracture networks at reservoir.
Corresponding to geometric features, physical features of fractures at
different scales can also differ. For a dominant fracture, it can be porous,
connected with small cracks, or filled with minerals; across its width,
compression of the fracture can be more difficult for a given increment of
stress, resulting potentially in a fracture with a residual aperture. For
discrete fractures, they may serve as fluid conduits or seals depending on
their connectivity and flux exchange with the rock matrix. They may be
mechanically open, bonded, or shearing due to fluid pressure or stress. If
looking at a fracture at micro-scale, on the two sides of the rough
surfaces, a significant difference of the open fracture channel and rock
matrix can be identified. Fluid flow mostly occurs within the open
channel, while the mechanical deformation of the rock matrix impacts the
geometry of the open channel. Furthermore, the deformation of the
fracture surfaces can be significant, due to the lack of constraints even
though the surface-to-surface contacts exist.
2.2. General governing equations for coupled processes in porous media
Regardless of the scales considered, in a fluid-saturated porous
system (e.g. rock matrix or a dominant fracture), coupled HM processes
satisfy conservation of momentum and mass, described by Biot’s general
theory of three-dimensional (3D) consolidation (Biot, 1941):
where is the total stress tensor, is the body force vector, is the fluid
velocity vector, is the Biot-Willis coefficient (usually ranges between 0
and 1),
is
the volumetric strain of the porous media, is the Biot’s
modulus, is the fluid pressure, and t is time. The Biot-Willis coefficient
as a factor multiplied by fluid pressure in Eq. (1) signifies a modification
and generalization of Terzaghi’s effective stress law to
where σ′ is the effective stress tensor, m
T
= [1, 1, 1, 0, 0, 0] for three
dimensions or m
T
= [1, 1, 0] for two dimensions.
For mechanical analysis of linearly elastic porous media with small-
deformation, we have
!
"#
$
"%
&
&'
&
&(
&
&( &
&'
)
*
+
where is the elastic constitutive tensor, is the strain tensor, " is the
strain-displacement matrix, and # is the displacement vector.
For fluid flow in porous media, it is assumed that the fluid flow
satisfies Darcy’s law:
,-.
where , is the tensor of permeability coefficient, and - is the hydraulic
head or the piezometric head (i.e. the sum of the pressure head and the
elevation head).
Fig. 2. Hydro-mechanical direct (I) and indirect (II) couplings (Modified after
Rutqvist and Stephansson, 2003).
In the case of fractures, in contrast, different governing equations
and constitutive relationships are applied for fluid flow and mechanics.
These will be presented in detail in Section 4. In general, two types of
couplings between fluid flow and mechanics exist: direct and indirect
couplings (see Fig. 2, as a concept presented by Rutqvist and Stephansson
(2003)). Direct couplings refer to as solid deformation perturbing
conservation of mass that impacts pore fluid pressure (i.e. pore-volume
coupling), while fluid pressure impacts the effective stress. Direct
couplings are included in Biot’s general theory (Eqs. (1)–(3)). When
volumetric strain is very small, however, direct coupling will be reduced
to one way. Indirect coupling means that hydraulic or mechanical
properties change with deformation and/or fluid pressure, where the
fracture permeability is very sensitive to deformation, and this requires
accurate calculation of this indirect coupling.
3. Fundamentals of NMM
NMM (Shi, 1991) is based on the concept of “manifold” in topology.
In NMM, independent meshes for interpolation and integration are
defined separately. Based on this approach, an initially one-time
generated, non-conforming mesh (not necessarily conforming to the
physical boundaries) can be used and flexible local approximations can be
constructed and averaged to establish global approximations for both
continuous and discontinuous analyses.
In NMM, independent mathematical and physical covers are defined.
A mathematical cover is a set of connected patches covering the entire
material domain. For example, a quadrilateral/circular/rectangular patch
can be used as a mathematical cover (see A, B, C in Fig. 3, respectively).
Features such as density and shape of these mathematical patches define
the precision of the interpolation. The physical patches are mathematical
patches divided by boundaries and discontinuities, determining the
integration fields. The union of all the physical patches forms a physical
cover. For example, physical patch C is the entire model domain, while
physical patch B is divided from mathematical patch B by boundaries.
Physical patch A (divided from mathematical patch A by boundaries) is
further divided into physical patches A
1
and A
2
by the inner discontinuity.
The overlapping areas by multiple physical patches are defined as
elements. As a result, the model domain Ω is discretized into five
elements: A
1
BC (the overlap of physical patches A
1
, B and C), A
1
C, A
2
C,
BC, and C. From Fig. 3, it can be seen that the shape of the mathematical
patches can be arbitrary; the relative location of the mathematical patches
to the model domain can also be arbitrary (only if satisfying /0
12324), and the number of physical patches on each element can be
arbitrary.
Fig. 3. NMM mathematical and physical meshes.
On each physical patch, a local function is assigned, such as constant
one, linear one, or anyone that is able to capture the solution behavior on
the patch. The weighted average of the local patch functions forms the
global approximation. For example, if using linear local functions, a
global second-order approximation could be constructed (Fig. 4a, Wang
et al., 2016); if using a local function with a jump of the first derivative, a
material interface crossing patches and elements could be simulated (Fig.
4b, Hu et al., 2015b). Or most commonly, if using discontinuous local
functions, fractures can be simulated (Fig. 4c, Hu et al., 2016, 2017b).
With this dual-mesh concept, NMM is capable of simulating both
continuum and discontinuum problems with accurate geometric
representation and flexible numerical approximation. With the advantages
of dual-mesh concept, NMM also has been successfully applied to
analyzing moving interface problems, such as free surface flow (Wang et
al., 2014, 2016; Zheng et al., 2015; Yang et al., 2019).
Fig. 4. Flexible choice of local approximation functions: (a) linear function, (b) a
jump junction for a weak discontinuity, (c) a discontinuous function for a fracture.
NMM with the concept of global approximation can be related to
other numerical methods, as shown in Fig. 5. If using a bilinear weight
function on rectangles with a constant local function, NMM can be
simplified to the FEM. If using a piecewise linear weight function in each
direction with a constant local function, NMM is then simplified to the
finite volume method (FVM, Hu and Rutqvist, 2020). If using a constant
weight function with a constant local function, NMM is simplified to the
DEM. If using a constant weight function with a linear patch function
(resulting in a notable difference from DEM with constant patch
function), NMM is simplified to DDA. Nevertheless, a comparison of
various numerical methods on all aspects is not attempted to be made
herein, including (1) interpolation/approximation, (2) construction of
global equilibrium (transforming differential to integral equations), (3)
approaches of integration, and (4) solving of linear or nonlinear global
equations. Only the first aspect is compared, i.e.
interpolation/approximation, as it defines fundamentals of a numerical
method. NMM provides a flexible and general approach to include
continuous and discontinuous methods in a unified form.
In this study, constant patch functions and linear weight functions
composed of shape functions of triangular mathematical meshes are used
to approximate temperature, fluid pressure and displacements, which are
generally expressed as follows:
56
*
7
89
:
where 5, 6 and 7
89
are the field variables (such as hydraulic head or
fluid pressure and displacements), weight function, and physical patch
functions, respectively.
Fig. 5. Relating NMM to other numerical methods.
4. Modeling coupled processes in fractures at multiple scales:
Numerical models and applications
4.1. A finite-thickness nonlinear poroelastic model for dominant
fractures
In fractured rock masses, it is common to see individual dominant
fractures or faults with asperities and mineral fillings that cannot be easily
simplified to parallel planes. The main flow feature may be a complex
geologic feature, consisting of multiple branching fractures intermingled
with mineral-filled sections and damaged host rocks adjacent to fracture
surfaces (Fig. 6a). Another related key property is the nonlinear
relationship between stress and fracture aperture. Moreover, the flow
feature is also associated with a mechanical weakness that may allow for
inelastic shear slip along its plane.
In a study focusing on the larger scale (larger than a single
fracture/fault), a porous finite thickness zone was developed to deal with
this type of system (Hu et al., 2017a). This finite thickness zone is porous
and has strong nonlinear properties reflecting fracture flow and fracture
opening and/or shear behavior, in consideration of fracture filling effects.
The thickness of this equivalent porous media flow feature in the model
may far exceed the real fracture width including open fracture parts and
filling. It can include part of the host rocks on each side of the flow
feature, still retaining the key features of potential fracture flow and
nonlinear deformation behavior. The model for such a feature is depicted
in Fig. 6b. It is a porous medium of thickness I
d
that includes a dominant
fracture flow path as well as other materials such as fracture filling and
part of the host rocks.
Fig. 6. Schematic of the simplified porous fractured rock model.
With the concept of the porous finite thickness zone that has
nonlinear flow and mechanical features, this section describes its
mathematical formulation, numerical implementation, and shows an
application with such a model.
4.1.1. Mathematical statement
In order to account for the nonlinear feature of the porous fracture, a
reformulation of Bandis’ (1983) equation (Rutqvist et al., 1998, 2000) is
used to describe the nonlinear relationship of the effective fracture normal
stress ;
<
with the mechanical aperture =
>
(as shown in Fig. 7):
;
<
?
@
A
;
<B
C
where ;
<B
is related to a Bandis’ parameter, which is user-defined; and D
is a constant defined as
D=
>E
;
<E
;
<B
Fig. 7. Nonlinear relationship between F
<
and =
>
(Modified from Rutqvist et al.,
2000).
The following relationship describes the behavior of fracture shear
displacement under shear stress:
;
G
HI
J
KLMHI
J
where N and O are the constants; ;
G
is the shear stress; and HP
G
is the
shear displacement. Examining Eq. (11), it is concluded that when O,
the linear relation between shear stress and shear displacement is retained.
For fluid flow in fractures, the hydraulic conductivity Q
R
of a fracture
is related to a hydraulic fracture aperture =
S
, which can be defined
according to Witherspoon et al. (1980):
Q
R
@
T
U
V
W
X
YZ
W
where [
R
and
\
R
are the fluid density and dynamic viscosity, respectively;
] is the gravitational acceleration; and =
S
is the hydraulic aperture
assumed to be
=
S
=
S^
_=
>
where =
S^
is the residual hydraulic aperture when the fracture is
mechanically closed, and _ is a factor that compensates for the deviation
of flow in a natural rough fracture from the ideal parallel smooth fracture
surfaces.
With the above concepts and equations, the aperture of the dominant
fracture flow path is used to calculate the hydraulic conductivity as given
in Eq. (12). The deformation behavior of the finite thickness zone is
contributed to two parts: the nonlinear behavior of the fracture described
in Eq. (9), and linear elastic deformation of the solid fracture fillings and
adjacent host rocks. The HM coupling includes direct pore-volume
coupling, as well as indirect coupling with changes of mechanical and
hydraulic properties induced by flow and deformation, respectively.
4.1.2. Numerical approach for the nonlinear constitutive behavior
As shown in Fig. 6, dominant fractures are represented as a finite
thickness zone with a given width and constitutive behavior differing
from rock matrix. Geometrically, the fracture zone has the same
dimension as the rock matrix (i.e. n-dimensional representation).
Therefore, the geometric discretization for dominant fractures is the same
as that for the rock matrix. Here an approach is described to model the
nonlinear physical constitutive behavior of this zone.
As the deformation of the finite thickness zones consists of linear
deformation of the mineral fillings and adjacent host rocks and nonlinear
deformation of the fracture, the normal strain is expressed as the
summation of the two:
<EE
`a;
<EE
@
A
bb
c@
A
bbde
f
g
bbde
!
where the superscript hh represents the ii-th time step, and ` represents the
compliance of fillings and adjacent host rock within the fracture zone.
Combined with Eq. (9), Eq. (14) becomes
<EE
`a;
<EE
i
jk
lbbdjkm
l
c
i
jk
lbbdedjkm
l
f
g
bbde
$
In order to accurately account for the nonlinear behavior of the finite
thickness fracture zone, an implicit approach was developed by Hu et al.
(2017a). In this approach, instead of using an approximation with linear
segments to the nonlinear relationship, the nonlinear mechanical behavior
is fully incorporated using strain energy for the material under
deformation. Therefore, those nonlinear relationships are directly
introduced to strain energy as described in the following subsections for
normal and shear deformations.
The normal constitutive model expressed in Eq. (15) can be rewritten
as
;
<EE
Yn
op`q;
<EEc
;
<B
rstup`q;
<EEc
;
<B
rst
Y
!`
?
f
gbbde
v+
where
s
<EE
?
pw
k
lbbde
cw
km
l
tf
g
bbde
.
The strain energy in the porous medium representing a fracture zone
x
yR<
is expressed as
x
yR<
z;
<EE
{
<EE
{/
k
bb
B
:
Combined with Eq. (16), Eq. (18) becomes
x
yR<
Yn
zp
Y
|
YEE
}
<EE
~
Y
|
EE
<EE
t{•{€C
where
|
EE
•
bbde
f
g
bbde
`q;
<EEc
;
<B
ru‚`q;
<EEc
;
<B
r
•
bbde
f
g
bbde
ƒ
Y
!`
?
f
gbbde
(20)
|
YEE
c„nqw
k
lbbde
cw
km
l
rL
…bbde
†g
bbde
‡
ˆ„nqw
k
lbbde
cw
km
l
rL
…bbde
†g
bbde
‡
U
c‰n
i
†g
bbde
(21)
Š
EEc
?
w
k
lbbde
cw
km
l
(22)
Transforming coordinates from global x-y to local s-n coordinate
system yields
x
yR<
Yn
zp
Y
|
YEE
*
‹‹
*
|
EE
‹
*
tŒ{s{•
where C
T
= (sin
2
Ž, cos
2
Ž, −sinŽcosŽ), Œ•
•
' •
(
‘
' ‘
(
•, and
<EE
‹
*
.
The shear constitutive model expressed by Eq. (11) can be further
expressed as
;
G
’
Jbb
K @
A
bbde
“LM’
Jbb
!
where
”
GEE
HP
GEE
=
>
EEc
• $
Similar to the approach for fracture normal mechanical behavior, the
associated strain energy using Eq. (11) can be written as
x
yRG
–—
M
”
GEE
K @
A
bbde
“M
U
˜™šN =
>
EEc
“O”
GEE
›œ{•+
With Taylor expansion and coordinate system transformation, we
have
x
yRG
Y@
A
bbde
K
zHP
GEE
ž
*
‹
*
‹
ž
HP
GEE
Œ{s{•.
where
‹
*
= (−sinŽcosŽ, sinŽcosŽ, cos
2
Ž, −sin
2
Ž) (28)
ž
Ÿ
'
(
(
'
*
(29)
Fluid flow in the fracture zone is governed by Eq. (2), where the
volumetric strain perturbs conservation of mass (direct coupling). In
addition, the hydraulic conductivity of the fracture zone is expressed by
Eqs. (12) and (13), where hydraulic aperture should be updated
dynamically as a result of the deformation of the fracture zone (indirect
coupling).
As the fracture zones are modeled in this approach as porous media
with nonlinear properties that differ from the surrounding rock, the
boundaries of the fracture zones are regarded as material interfaces. The
displacement continuity across these material interfaces is realized by the
penalty method (Shi, 1991) combined with the continuity of hydraulic
head, and the normal flux is fulfilled by the Lagrange multiplier method
developed by Hu et al. (2015a).
With the energy-work based theorem (Wang et al., 2014), work and
energy associated with fluid flow and mechanical deformation are
implicitly constructed and updated at each time step. Minimization of the
total energy at each time step leads to the global equilibrium equations.
4.1.3. Example: Coupled processes at a single dominant fracture
In order to demonstrate the formulation in consideration of both
direct and indirect coupled HM processes in rock with fractures, a
rectangular rock domain is simulated, which contains a fracture zone
subjected to instantaneous vertical load and a constant pressure fluid
injection. The simulated domain is 1 m wide and 2 m high, with a 1 m ×
0.1 m horizontal fracture zone at the vertical center (Fig. 8). The
mechanical fracture aperture for the assumed dominant fracture is 1 ×
10
−4
m (0.1 mm) with an equivalent hydraulic aperture of 5 × 10
−5
m (50
µm). For the rock matrix, the Young’s modulus is 4 GPa, the Poisson’s
ratio is 0.2, and the permeability coefficient is 5 × 10
−9
m/s. For the
fracture zone, Bandis’ parameter ;
<B
is 5 MPa, the shear constants N is
10
−11
Pa
−1
and O is 0, and the factor f is 0.5. Initially, the total vertical
stress is −8 MPa (compressive stress) and fluid pressure is 0.
First, an instantaneous 10 MPa vertical downward load is applied on
the top of the domain. A mechanical analysis without fluid injection was
conducted. This results in an instantaneous closure of the fracture as a result
of its nonlinear normal closure behavior with changing normal stiffness. The
simulation indicates a mechanical fracture aperture of 6 × 10
−5
m (60 µm) at
the final steady state, which is accurate according to Eq. (9). Then a
simulation was conducted considering only indirect coupling, i.e. the fluid-
solid interaction terms for direct coupling is deactivated. In this case, the
coupling occurs only in one way, i.e. mechanical deformation affects
permeability but there is no influence of fluid pressure on the mechanical
field. The changes of mechanical and hydraulic properties of the fracture
under load and injection with constant pressure of 8 MPa at the left end of
the fracture zone and the pressure at the right end of the fracture zone are
fixed at zero. Finally, the full package is run, considering both direct and
indirect couplings with results shown in Fig. 8.
Fig. 8. Distribution of fluid pressure (MPa) for (a) flow analysis without considering
coupled effects, 30 d after injection; (b) only considering indirect coupling; and (c)
considering both direct and indirect couplings. The dimensions of the simulated
domain are in meter.
The distribution of fluid pressure in the following three cases is
compared: (1) without considering coupling, (2) only considering indirect
coupling, and (3) considering both direct and indirect couplings in Fig. 8.
The difference of fluid pressure distribution in Fig. 8a and b is not
obvious, indicating that a steady state is reached for the case of indirect
coupling only after 30-d injection. However, in Fig. 8c, a steady state has
not been reached and fluid continues to dissipate from the left to right.
This difference can be explained by the fact that in Fig. 8b (with only
indirect coupling), a steady state is reached when mechanical deformation
no longer occurs, whereas in the case for Fig. 8c, the final steady state
will be reached only after the interaction between the mechanical and
fluid flow fields is balanced. The slower convergence to steady state
shown in Fig. 8c is caused primarily by pore-volume direct coupling in
the relatively low permeability rock matrix. As observed in the
simulation, fluid flow reaches steady state much earlier in the fracture
zone (see Fig. 8c). Overall, the effect of pressure on solid deformation is
not obvious, which is also reflected by the rapid convergence of fracture
aperture to its final values for both cases (2) and (3). Further, the aperture
change is compared with time at the injection point under these two
conditions. The aperture at the final stage reduces to 6 × 10
−5
m (60 µm)
when only considering indirect coupling. This value is the same as the
one in the case of pure mechanical analysis, proving its verification.
However, when considering both direct and indirect couplings, the
aperture remains steady at 6.5 × 10
−5
m (65 µm) under the effect of fluid
pressure on the solid skeleton. Based on this simple example, it is known
that pore-volume direct coupling may play a significant role for dominant
fractures, therefore n-dimensional models which consider the direct
coupling and nonlinear behavior of the fracture zone are necessary for
analyzing coupled processes for this type of fractures.
4.2. Zero-dimensional model for discrete fractures
At discrete fracture scales where fractures may be oriented and
intersecting arbitrarily, n-dimensional and n−1 dimensional models have
limitations. A zero-dimensional model was developed to simulate fluid
flow, geomechanics, and their couplings. In this section, the general
mathematical formulation, geometric representation, and numerical
implementation are summarized and an example is given to demonstrate
its efficiency and capability.
4.2.1. General mathematical formulation
The basic idea of a zero-dimensional fracture model is to use the
surrounding elements of a fracture to represent the fracture, avoiding the
need for additional dimensions or DOFs for the fractures themselves.
Since the fractures are treated as discontinuous boundaries of the rock
matrix, no additional DOFs are required to represent them. For fluid flow
and mechanics in porous media, Eqs. (1)–(7) are still applicable. As for
fractures, flux occurs both along fracture and normal to fracture
directions. Fig. 9 shows how this is accomplished numerically for the two
flow directions (where φ represents variables, i.e. hydraulic head for flow
and temperature for heat transfer).
Fig. 9. Fluid flow (a) along fracture and (b) normal to fracture directions in zero-
dimensional model.
Fig. 9a shows fluid flow along fractures. Because the fractures in this
case are very thin relative to their length (e.g. microns to millimeters
aperture for meter-sized fractures), it can be assumed that hydraulic head
within the fracture is uniform across its thickness:
5•5
R
•5¡•
where 5 and 5
denote the field variables on different elements that are
divided by the fracture, 5
R
• is field variable within a fracture, and s
represents local coordinate along the fracture. Thereby, fluid along a very
thin fracture is represented by flow along its two surfaces:
¢
GR
R&£
&G
R&£
&•
where
f
is the permeability coefficient for Darcy flow, and ¢
GR
is the flux.
Here parallel plate flow in fractures is assumed as given in Eq. (12).
Fig. 9b shows the fracture-matrix interaction, i.e. fluid or heat
exchange normal to a fracture. A fracture that is bounded by its two
surfaces from the surrounding rock is explicitly represented by these two
surfaces, which belongs to different elements. As the fractures are
assumed to be thin and unfilled, the distribution of the field variable on
each surface of a fracture and within a fracture is uniform in the direction
normal to the fracture surfaces. Therefore, these two fracture surfaces are
considered to consist of two Dirichlet boundaries:
—5
B
•5•¤5
R
•5
•
5
B
•5
•¤5
R
•5•
The Eqs. (30)–(32) include all the possibilities of fluid flow in a
fracture, which may act as a fluid conduit or seal. In contrast, the
mechanical state of a fracture is more complicated. A fracture may have
several segments and each of the segments could be open, bonded, or
sliding, as shown in Fig. 10. The contact state is impacted by fluid flow
and mechanical deformation, and may be dynamically changing. In view
of the mechanical states, these three states have different boundary
constraints.
Fig. 10. Mechanical states of each segment of a fracture in networks: open, bonded
and sliding.
When the segment of a fracture is open, it is considered that it has a
linear constitutive behavior:
a
R
¥
R
¦#
R
§
where
R
denotes a tensor of effective stress in both normal and tangential
directions of a segment of a fracture, ¥
R
is the stiffness tensor of the
segment, and ¦#
R
§ is the jump of displacements in both normal and
tangential directions of the fracture segment.
When a segment of a fracture is bonded, the distances between the
two sides of the segment should be zero:
¨
R
©!
where ¨
R
is the vector of distance, including the normal distance ª
R«<
between the two surfaces of the fracture segment, and relative distance
ª
R«G
along the two surfaces. ¨
R
is calculated by accounting for original
distances, and the relative displacements of the two surfaces while they
are moving and deforming.
When a segment of a fracture is sliding, the Coulomb’s law of
friction is satisfied in the tangential direction, while the normal distance
between the two surfaces of the fracture segment should be zero:
¬
R
¬
<
-®™5¯°™}ª
R«G
~¤ª
R«<
$
4.2.2. Implementation of zero-dimensional model in NMM
The key issues for modeling coupled processes in discrete fractures
are how to handle (1) the geometrical representation of intersecting
fractures, (2) fracture flow and fracture-matrix flow interaction, (3)
deformation and dynamic contacts involving slip and opening of the
fractures, and (4) couplings between flow and mechanics.
(1) Geometric representation
In order to explicitly simulate fluid flow and mechanics of discrete
fractures as well as their interactions (flux exchange with rock matrix),
both fractures and rock matrix need to be geometrically represented. In
NMM, a triangular mathematical mesh is used to overlap the entire
simulation domain. Once the mathematical mesh is generated, the same
tree-cutting algorithm could be used (Shi, 1991; Hu et al., 2017b) to
generate the physical covers and elements, divided by fractures and
boundaries, considering locations of triangle edges. Fig. 11 shows
different meshes for fractured porous rock domains with 20, 60 and 150
fractures. The triangular mathematical meshes are independent from the
fracture geometry. Indeed, fractures and boundaries divide mathematical
meshes into arbitrarily shaped physical covers and elements. By using
two coincided lines to represent a fracture, all fractures are accounted for
in the calculation, including isolated fractures that do not intersect with
other fractures where their interaction with the rock matrix cannot be
ignored. It is also interesting that with the tree-cutting algorithm, cases of
fluid flow in porous media with sealed fractures are readily modeled.
Corresponding to the mechanical states of fractures described by Eqs.
(33)–(35), each fracture is discretized into several line segments and these
segments may have different contact states. Thus, it is possible to
consider complicated behaviors such as shear dilation or uneven opening
of a fracture. The density of fracture discretization is consistent with the
global meshing density (see Fig. 11), which can be selected flexibly.
Fig. 11. NMM mesh generated for fractured porous rock with (a) 20, (b) 60, and (c) 150 fractures.
An important issue for calculating discrete fractures is how to simulate
intersections of fractures. Fig. 12 demonstrates a geometric representation of
two fractures that intersect with each other as well as with one triangular
mathematical mesh. As can be seen, the two intersecting fractures divide the
triangle into four different parts (A, B, C, D). Then fluid flow and contact
states (satisfying constraints described by Eqs. (33)–(35)) will be applied on
the four pairs of parallel interfaces (interfaces between A and B, C and D, B
and C, and A and D) to account for the opening, bonded and sliding states of
the surfaces of each fracture.
Fractures in open and bonded states or alteration between these two
states are easier to simulate because this does not require changes of
contact pairs. Zero-dimensional fracture model assumes that at the initial
stage, a fracture is approximated by two surfaces parallel with each other
at the beginning, but these two surfaces can be non-parallel after
deformation and motion, or opening of the fracture caused by fluid
pressure. This capability is included in the algorithm.
Fig. 12. Geometric representation of an element with one intersection by two
fractures.
A significant challenge for modeling coupled processes in fracture
network is to simulate shearing along fractures, as this leads to dynamic
changes of contacts between different elements. As shown in Fig. 13, when
the four blocks A, B, C and D are in contact (when the fractures are
completely bonded), the contact pairs are A and B, B and C, C and D, and A
and D. But when sliding (slip) occurs at one of the fractures, the contact
pairs become A and B, B and C, B and D, C and D, and D and A. By using
a rigorous contact algorithm that updates contact pairs at each time step,
sliding along fractures can be rigorously and explicitly represented.
Fig. 13. Geometric representation of open, bonded and sliding contact states for
elements divided by intersecting fractures.
The concept of a zero-dimensional fracture model was proposed by Hu
et al. (2016) for the first time and it is a practical method to simulate discrete
fracture networks. The zero-dimensional fracture model is distinct from
zero-thickness models, as the zero-dimensional model considers the
thickness of a fracture and its change under deformation (i.e. the fractures
have a width). With no need to introduce additional DOFs for fractures,
such an approach is very flexible to model existing and initiated fractures as
well as shearing and re-opening of fractures.
(2) Zero-dimensional model for coupled fluid flow and mechanics in
discrete fractures
The explicit geometric representation of discrete fractures with no
need to introduce additional DOFs lays the basis for the zero-dimensional
model for fluid flow and mechanics of discrete fractures embedded in a
porous rock matrix. By using a zero-dimensional fluid model, both along-
fracture flow and fracture-matrix interaction can be handled. In this
model, effect of a residual hydraulic aperture on fluid conduction is
considered even when the fractures are mechanically closed. When the
mechanical aperture of a fracture is changing due to continuous opening
or the change of contact state, its hydraulic aperture and hydraulic
conductivity are updated. Moreover, as a result of deformation, fluid flow
and their coupling, the contact state may change dynamically, and the
corresponding constitutive behavior should be described differently. Such
a stringent scheme that guarantees the accuracy of the results when
considering complex fractured rock and HM coupling is important. The
scheme also accounts for indirect couplings. Thus, coupled HM responses
are able to be modeled in a complex intersecting fracture network.
An energy-work based formulation for flow is used to establish the
global equilibrium equations by minimizing the total potential energy
associated with each component of work. The work associated with
along-fracture flow ±
GR
is represented as
±
GR
Y
²=
S
–³
GR´
•
³
GR´
l
•
{•+
where ³
GR
represents flux in the direction along a fracture; - and -
are
hydraulic heads on the two surfaces of a fracture.
Combining with Eqs. (7) and (36) yields
±
GR
Y
²=
S
–pQ
R
´
•
Y
Q
R
´
l
G
Y
t{•.
The work associated with fracture-matrix interaction flow ±
<R
is
±
<R
²–Q
R«<
p
&´
l
&‘
-
-
&´
&‘
--
t{•:
where ds denotes integration on each discretized segment of a fracture;
and Q
R«<
is the permeability coefficient in the direction normal to the
fracture.
For calculation of the contact between two fracture surfaces, there
are different terms of potential energy associated with different contact
states described in Eqs. (33)–(35).
When a fracture segment is open, the linear fracture constitutive
behavior (Eq. (33)) is represented by its corresponding strain energy:
x
R«µ8y<
–
R
{
R
{/
Y
–¦#
R
§
*
¥
R
¦#
R
§{•C
When a segment of a fracture is bonded, the continuity of relative
distance described as Eq. (36) is enforced by using the penalty method:
x
R«9¶µGy·
Y
¨
R*
¸
R
¨
R
!
where ¸
R
represents the stiffness of penalty springs in normal and
tangential directions.
When the two surfaces of a fracture segment are sliding, the
continuity of displacement in the normal direction should be satisfied and
this is realized with normal component of Eq. (40). In the tangential
direction, potential energy associated with sliding state for the two
surfaces corresponding to Eq. (35) is
¹
R«G¶º·º<»
]
R«<
ª
R«<
¯°™}ª
R«G
~-®™5¨
R«G
!
where the vector signs of ¹
R«G¶º·º<»
and ¨
R«G
represent the two sides of one
fracture segment that may have different shearing displacements. By
accounting for this together with geometric representation as shown in Fig.
13, sliding (slip) along fracture surfaces can be accurately simulated.
4.2.3. Example: Coupled processes in discrete fractured porous rock
The zero-dimensional fracture model is applied to a rather soft 100
m × 100 m porous domain containing 25 arbitrary fractures. The domain
is subjected to a 10 kPa traction on the top surface with a hydraulic head
of 0 m. The other three boundaries are impermeable. The initial hydraulic
head is assumed to be h = 100 m over the entire domain. For the rock
matrix, the Young’s modulus is 4 MPa, the Poisson’s ratio is 0, and
permeability coefficient is 2.5 × 10
−8
m/s. For the discrete fractures, the
initial mechanical aperture is 0, the shear and normal stiffnesses are both
1 × 10
6
Pa/m, the factor f is 0.5, and the residual aperture b
hr
is 10 µm.
In this example, only the major responses to the load are presented as
a demonstration of the zero-dimensional fracture model. The simulation
ran for 1000 d with the results shown in Fig. 14. The traction load from
the top has caused a significant vertical compaction of the fracture porous
media (Fig. 14b). Complex hydromechanical responses in the fractured
porous media result in heterogeneous vertical compaction, with
subsidence of the top surface varying from about 0.3 m to 0.8 m. This
vertical compaction does not occur instantaneously, but gradually as
water needs to be squeezed from the fully saturated porous fractured
media and out through the top surface. The heterogeneously and
dynamically changing hydraulic properties along with strong
hydromechanical pore-volume coupling are reflected in the heterogeneous
pressure field (see Fig. 14c). Finally, Fig. 14a indicates the locations of
fracture opening marked with red circles. Thus, this application
demonstrates a case of load and drainage of a saturated fractured media
that may lead to long-term large and heterogeneous ground surface
subsidence.
(a) (b) (c)
Fig. 14. Simulated (a) horizontal and (b) vertical displacements (m), and (c) pore pressure (Pa) at t = 1000 d.
4.3. Explicit modeling of contacting rough fracture surfaces at
discontinuum asperity scales
In order to understand how fractures respond to coupled processes
and thereby derive reasonable mechanical and hydraulic constitutive laws,
it is necessary to model coupled processes with a detailed representation
of surface geometry. In this context, two challenges are settled. First,
fractures with asperities (for example at a microscale) have more
complicated geometric features that cannot be simplified easily; therefore,
these geometric features lead to discontinuities in each physical field
(flow, transport, mechanics). Second, pore-scale processes are described
by a different set of governing equations for flow, transport, and
mechanics. In this section, a discontinuum asperity model with explicit
representation of geometric features is developed for accurate modeling
of coupled processes in fractures. This approach can be used for micro-
scale analysis.
4.3.1. Mathematical statement
At micro-scale, Darcy’s law is insufficient to describe flow in the
open channels of fractures, and the Navier-Stokes equation in
combination with mass conservation equation are required:
&¼
&
½¾¿!
&
&
&
&
!
where ³ is the fluid velocity.
In addition to the force balance, small-strain assumption, and
continuum constitutive equations, translational and rotational
displacements need to be considered. These processes are described by
the following equations:
À
&
U
#
&
U
!!
##
Á^
#
^
– !$
In addition, the force is a result of not only the internal or external
load, but also the interactive forces between elements (i.e. the contact
force):
ÂÂ
9µ<ÁÃ9Á
Â
º<Á
Â
yÄÁ
!+
Correspondingly, continuum constitutive laws (Eqs. (4), (9) and
(11)) are not sufficient to describe the stress-strain/stress-displacement
relationships. Instead, contact forces are the functions of
displacements/location discontinuity:
¸ «Â
9µ<ÁÃ9Á
Ŧ#§!.
Depending on the states of contact, different constitutive
relationships may apply, such as Eqs. (33)–(35). The most challenging
additional equations involve calculation of contact forces between
multiple blocks. Different from discrete fracture analysis where fractures
are initially assumed as combinations of parallel surfaces’ segments,
rough fractures often contain multiple asperities with arbitrary shapes.
The challenge of computing contacts is to identify when and where
contacts occur among many blocks, which is made complicated by the
fact that these blocks are moving, deforming, and in some cases breaking
apart. In turn, motion, deformation and breakage of blocks are impacted
by contact forces, which constitute a serial process. Thus, inaccurate
contact calculation may lead to a completely different overall system
behavior.
In order to carry out contact calculation, simplifications are
generally made, by either assuming contact pairs are fixed, or using
simple shapes such as spheres or rectangles to approximate them. The
mechanisms involved as well as the errors caused by these geometric
approximations are shown in Fig. 15.
Fig. 15. Simplified contact calculation using (a) spheres and (b) predefined
rectangles for contacts.
The coupling between fluid flow and geomechanics within open channels
(pores, channels of fractures) is mostly in one way: mechanical
deformation leads to boundary changes of these fluid channels. The
coupling between fluid flow and mechanics in rock matrix and open
channels is carried out by ensuring the continuity of pore pressure at the
fracture surfaces between the channel Navier-Stokes flow and the porous
Darcy’s flow through flux terms. Therefore, fluid impact on mechanics in
the rock matrix is considered using the Biot’s equation.
4.3.2. Numerical implementation
In NMM (Shi, 1991), an algorithm was developed that rigorously
incorporates contact detection, contact enforcement and open-close
iteration. As shown in Fig. 16, among a number of moving and deforming
blocks, NMM first computes the possible contact blocks. This involves
certain estimation because the motion and deformation of blocks may
occur at any time. Setting a range of possibilities enables precise and
complete detection of all possible contact blocks. Between each two
potentially contacting blocks, there are many possibilities where exact
contact occurs. All of these possibilities are accounted in the code. They
are categorized into three possibilities: vertex-to-vertex, vertex-to-edge,
and edge-to-edge contacts. Edge-to-edge contact is a special case where
the surfaces of two contacting blocks are parallel (e.g. discrete fractures
where the surfaces are parallel). Vertex-to-vertex contacts can be
transformed to vertex-to-edge contact with the possible contact pairs in
Fig. 16. Criteria for identifying the contact pairs from all these possible
contact pairs are included in the code. After contact pairs are identified,
the contact pairs may be open, bonded or sliding, as described by Eqs.
(33)–(35). For bonded and sliding states, Eqs. (34) and (35) can be used to
calculate the contact forces. For the open state, however, it is assumed
that there is no interaction between contact pairs (i.e. complete open space
between boundaries). Therefore, there are no constraints applied.
Fig. 16. Schematic of contact calculation for micro-scale mechanical analysis.
At each time step, open-close iteration may be carried out several
times until the enforced contacts reach convergence. As a result of the
coupled processes, contact pairs may change; for the same contact pairs,
these three states may transfer dynamically. Thus, the iterations involving
detection, enforcement of contact constraints, and open-close iteration
need to be executed to ensure convergence.
4.3.3. Example: Calculation of rough fractures with explicit geometric
representation
By using the contact model with explicit geometric representation of
the fracture asperities, an example of contact alteration along rough
surfaces under the impact of load is given. In this example, a single
fracture partially contacting along their rough surfaces is laterally
confined with two plates and fixed on the bottom. It is assumed that there
is no flux exchange between the fracture channel and the rock matrix on
the top and bottom, and fluid dissipates from both sides. Therefore, only
one-way coupling exists: the deformation and contacts of fracture
asperities impact the geometry of the fluid channel. Such an assumption
allows us to decouple the problem. Fig. 17 shows a mechanical simulation
of the fracture alteration impacted by load. When load is applied (Fig.
17a), the upper rough surface deforms and moves toward the bottom
rough surface until they fully contact (Fig. 17b). This example involves
large deformation (the upper surface), dynamic change of contact pairs
(such as contact pairs alteration between the left plate and the upper
surface), and contact states transferring (such as the contact state
transferring from open to bonded between the upper and bottom surfaces).
Fig. 18a–c show results of vertical displacement, vertical stress, and
shear stress at the steady state. Stress concentration occurs at both
contacting asperities, which can lead to different responses depending on
the materials, such as plasticity, growth of new fractures, or coupled
thermal and chemical responses. Such an extreme case demonstrates the
capability of NMM for tackling problems of contacts between multiple
bodies and/or asperities. From this example, it can be seen that contact
alteration could lead to dramatic structural changes of fractures, which
further impacts upscale physical (mechanical and flow) features of the
fractures. On the other hand, localized stress concentration can lead to
further geometric and geophysical changes of the fracture. Therefore, this
modeling capability is essential for understanding fundamental behavior
of fractures at micro-scale.
(a) (b)
Fig. 17. Modeling of a rough fracture: (a) before load and (b) after load reaching steady state.
Fig. 18. Results of (a) vertical displacement, (b) vertical stress, and (c) shear stress at the steady state.
5. Discussion
It is quite challenging to rigorously simulate coupled HM processes
in fractured geological media because of computational geometry
associated with fractures at different scales. Because of limitations in the
current models, there is a gap insofar, as coupled processes at discrete
fracture scales could not be analyzed without consideration of
intersections and shearing of the discrete fractures that might be
dynamically evolving as a result of coupled processes. In addition,
numerical modeling of coupled processes has never been attempted at
micro-scale when the contact evolution of discontinuities is important. In
this study, these computational challenges are tackled by developing a
comprehensive numerical model with NMM for analyzing coupled
processes in fractures embedded in porous geological media at multiple
scales. Depending on the ‘relative scale’ of fractures (the scale of
fractures relative to the scale of interest of the problem or research), the
fractures are categorized into three different scales: dominant fracture,
discrete fracture, and discontinuum asperity scales. For these different
scales, different governing equations as well as fracture constitutive
behaviors are applied in terms of HM coupling. Correspondingly,
numerical implementation varies between different scales, including
implementation of those different governing equations, constitutive
relationships, and HM couplings. With all these components, the NMM
model presented herein is able to simulate fractures ranging from micro-
scale to reservoir scale (summarized in Table 1).
Table 1. Overview of NMM modeling of coupled processes in fractures at multiple scales.
Dominant fractures Discrete fractures Discontinuum asperity scale
Governing equations
Mechanics:
Fluid flow:
Æ
´
Mechanics:
Fluid flow:
Æ
´
Mechanics:
À
&
U
#
&
U
Fluid flow:
Ç
ÇȾH
Ç
Ç
È
Ç
Ç
È
Fracture constitutive
behavior
Mechanics:
<EE
`a;
<EE
=
>
EE
=
>
EEc
É
·EEc
;
G
”
GEE
N=
>
EEc
ÊO”
GEE
Fracture flow:
Q
R
@
T
U
V
W
X
Y
Z
W
Mechanics:
Open:a
R
¥
R
¦#
R
§
Closed: ¨
R
©
Sliding:¬
R
¬
<
-®™5¯°™}ª
R«G
~¤ª
R«<
Fracture flow:
Q
R
=
SY
[
R
]
\
R
Mechanics:
##
Á^
#
^
Ë
ÂÂ
9µ<ÁÃ9Á
Â
º<Á
Â
yÄÁ
Â
9µ<ÁÃ9Á
Ŧ#§
Open: a
R
¥
R
¦#
R
§
Closed:¨
R
©
Sliding:
¬
R
¬
<
-®™
5
¯°™
}
ª
R
«
G
~
¤
ª
R
«
<
HM coupling
Two-way direct coupling: rock matrix
and fractures;
Two-way indirection coupling:
fractures
Two-way direct coupling: rock matrix;
Two-way indirect coupling: fractures
Two-way direct coupling: rock matrix;
One-way indirect coupling: fracture
channels
Geometric
representations of
fracture, intersection,
and shearing
Solid element
Zero-dimensional elements
Explicit geometric representation
For dominant fractures, Biot’s equation in combination with
nonlinear constitutive relationships is used. Such a system involves both
direct coupling in rock matrix and fractures, and indirect coupling in
fractures. As fractures are geometrically represented by solid elements,
intersections of fractures are intersections of solid elements, which is
straightforward to treat. Shearing of fractures can be implicitly realized
with different nonlinear laws (Eq. (11)). The most important requirement
of a numerical model for dominant fractures is to accurately represent
their nonlinear behavior. To accomplish this, an implicit approach that
accounts for their strain energy was developed and verified.
For discrete fractures, Biot’s equation can be used for describing
fluid mass conservation and balance of force in the rock matrix, and the
associated direct coupling. However, the fractures, which may be oriented
or intersected arbitrarily, are very thin and may be open, bonded or sliding
dynamically. These fractures have flux exchange with rock matrix. Fluid
flow in these fractures, which is described by a reformulated cubic law, is
highly sensitive to mechanical changes (open, bonded or sliding).
Similarly, the mechanical changes are highly sensitive to the fluid flow,
thus a two-way indirect coupling should be considered. In order to
account for such complex behavior, a zero-dimensional fracture model
was developed by considering fractures as boundaries of solid rock
matrix. Fluid flow in fractures and real-time flux exchange with the rock
matrix are implicitly considered. Permeability is updated each time as a
function of mechanical aperture, depending on the mechanical states. The
mechanical states of each fracture segments are rigorously considered in
three types: open, bonded and sliding with different constraints and
constitutive behaviors. Such a complex system is complicated further with
fracture intersections and shearing. By using a tree-cutting algorithm with
discontinuous surfaces approximating each fracture, each fracture
intersection is able to be calculated considering contacts between each
two sides around the intersection. As each contact pair (two surfaces of
each fracture segment) is updated at each time step, shearing along a
fracture segment can be explicitly calculated. As each fracture is
discretized into several line segments and these segments may have
different contact states, complex behaviors such as shear dilation or
uneven opening of a fracture can be calculated.
For fractures where the geometry of asperities cannot be simplified
(e.g. at a microscale), a discontinuum asperity model is developed that
explicitly represents the boundaries of asperities. For matrix of fractures,
Biot’s equation is used. But for the open channel bounded by the rough
surfaces of a fracture, Navier-Stokes equation in combination with
conservation of mass is recommended to calculate fluid flow. Direct
coupling occurs at the rock matrix. However, in the fluid channel, the
coupling is majorly in one way: mechanical deformation and contacts
impact the geometry of the fluid channel and thus the flow. In this
situation, it is more reasonable to treat the hydrological and mechanical
process as decoupled. Intersections of fractures are explicitly represented
as for a single fracture. As for shearing in fractures, it occurs mostly at the
asperity scale rather than at the scale of a single fracture. The challenge of
modeling this micro-scale behavior is to capture when and where contacts
occur. By using a rigorous contact detection algorithm, fracture alteration
and shearing due to severe contact alteration are able to be simulated.
6. Summary and perspectives
In summary, modeling at different scales requires a different set of
governing equations, constitutive behaviors, and geometric
representations. By defining relative scales, it is possible to use the same
type of model at different scales as long as the physical behaviors are
described well. Between those differences for different relative scales,
computational geometry plays an important role and provides the basis for
numerical modeling, determining discretization to be used, accuracy for
modeling of fracture intersections, and capability to treat complex
behaviors such as shearing, or multi-body contacts. To date, due to the
limitations of computational geometry for describing 3D fracture
networks in porous rock and the limitations of geometric representation of
contacts among multi-body systems, 3D computation of fracture
modeling at discrete fracture or micro-scale is still rare. Development of
3D computational geometry targeting at these two problems will be
essential and promising for 3D fully coupled analysis of fractures at
multiple scales for advancing fundamental understanding and optimized
control of energy recovery from fractured geological systems.
Declaration of Competing Interest
We confirm that there are no known conflicts of interest associated
with this publication and there has been no significant financial support
for this work that could have influenced its outcome.
Acknowledgements
This work was supported by Laboratory Directed Research and
Development (LDRD) funding from Berkeley Lab, and by the Spent Fuel
and Waste Disposition Campaign, Office of Nuclear Energy of the U.S.
Department of Energy, and the Office of Science, of the U.S. Department
of Energy under Contract No. DE-AC02-05CH11231 with Berkeley Lab.
This work was partially supported by Open Fund of the State Key
Laboratory of Geomechanics and Geotechnical Engineering, Institute of
Rock and Soil Mechanics, Chinese Academy of Sciences (Grant No.
Z017004). Editorial review by Dr. Carl Steefel at the Berkeley Lab is
greatly appreciated.
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Mengsu Hu is a Research Scientist at the Lawrence Berkeley National Laboratory (LBNL). She received a BEng in Civil
Engineering in 2010 and a PhD in Geotechnical Engineering in 2016. Her research interests covers numerical modeling of
coupled thermal-hydro-mechanical-chemical (THMC) processes in the energy geosciences, including: (1) development of
computational methods including the extended finite volume method, and the numerical manifold method; (2) numerical
modeling of microscale mechanical-chemical processes such as carbonate compaction, pressure solution, fracture alteration and
fracture healing; (3) multi-scale, long-term analysis of THM processes in energy-geosciences applications such as nuclear
waste disposal.
Declaration of interests
☒
The authors declare that they have no known competing financial interests or personal relationships
that could have appeared to influence the work reported in this paper.
☐The authors declare the following financial interests/personal relationships which may be considered
as potential competing interests:
