Comput Geosci
DOI 10.1007/s10596-012-9335-x
ORIGINAL PAPER
Meshless numerical modeling of brittle–viscous
deformation: first results on boudinage and hydrofracturing
using a coupling of discrete element method (DEM)
and smoothed particle hydrodynamics (SPH)
Andrea Komoróczi · Steffen Abe · Janos L. Urai
Received: 10 April 2012 / Accepted: 6 December 2012
© Springer Science+Business Media Dordrecht 2013
Abstract We have developed a new approach for the
numerical modeling of deformation processes combining brittle fracture and viscous flow. The new approach
is based on the combination of two meshless particlebased methods: the discrete element method (DEM)
for the brittle part of the model and smooth particle
hydrodynamics (SPH) for the viscous part. Both methods are well established in their respective application
domains. The two methods are coupled at the particle
scale, with two different coupling mechanisms explored: one is where DEM particles act as virtual
SPH particles and one where SPH particles are treated
like DEM particles when interacting with other DEM
particles. The suitability of the combined approach is
demonstrated by applying it to two geological processes, boudinage, and hydrofracturing, which involve
the coupled deformation of a brittle solid and a viscous
fluid. Initial results for those applications show that
the new approach has strong potential for the numerical modeling of coupled brittle–viscous deformation
processes.
Keywords Numerical modeling ·
Coupled brittle–ductile deformation · DEM · SPH ·
Boudinage · Hydrofracturing
A. Komoróczic · S. Abe (B) · J. L. Urai
Structural Geology - Tectonics - Geomechanics,
RWTH Aachen University, Lochnerstr. 4-20,
52056 Aachen, Germany
e-mail: [email protected]
A. Komoróczic
e-mail: [email protected]
J. L. Urai
e-mail: [email protected]
1 Introduction
The coupled deformation of brittle and ductile materials plays an important role in numerous geological
processes. For example, the brittle deformation of carbonate or anhydrite layers embedded in a salt body
during tectonics is important in salt tectonics [25, 40, 41,
77, 82]. Another example is boudinage: a wide range
of boudins is observed in the nature [27]; these are
formed during deformation if there is a component
of lengthening parallel to a brittle layer in a ductile
matrix [88]. Another process of both basic and applied
importance is hydraulic fracturing, when viscous fluid is
pumped into a brittle reservoir rock in order to generate fractures [10, 14, 16, 20, 30]. Detailed understanding
on these large deformation and extension fracturing is
lacking, and these processes present a challenge to both
analog and numerical modeling.
Analog models are often used to investigate the parameters which control the geometry of these structures
[22, 71, 76, 95, 96], although many properties, such as
pressure inside the model, are unknown during the experiment [62], and complete scaling is rarely achieved.
Existing numerical models of brittle–ductile deformation have also limitations because of the complex
geometries and the difficulties regarding the fluid–solid
interactions. For example, in salt tectonics, mostly the
finite element or finite difference method is applied
[23, 24, 31, 34, 37, 41, 68, 69, 80, 94], which is a continuum method and has limited capabilities to model
discontinuous media. One possibility to improve the
modeling of brittle faulting within the context of the
finite element method (FEM) method is the use of split
nodes as demonstrated in [52]. However, this approach
makes it necessary to define the possible fault locations
Comput Geosci
a priori. There are particle-based models to simulate
hydraulic fracturing; however, in these models, only the
solid part is represented by particles, and the fluid part
is described by a fluid flow algorithm [12, 15, 81]. The
introduction of combined brittle–ductile behavior into
a particle-based method was shown to be possible [9],
but the exact modeling of a linear viscous material has
not been achieved yet using this approach.
In this paper, we present a new method, in which
both the ductile and the brittle material are modeled
using meshless, particle-based, Lagrangian methods.
The viscous fluid is represented by a smoothed particle hydrodynamics (SPH) model and the brittle solid
material is represented by a discrete element method
(DEM) model. Both approaches are well tested in their
respective domain and due to the similarities between
them, it is possible to couple the two approaches to
develop a hybrid method.
DEM was developed by Cundall and Strack [18] to
investigate the mechanical behavior of granular materials and extended by Mora, Place and others [57,
67, 92] with lattice solid model. This approach has
been proven very successful in the simulation of brittle
and elastic–brittle material behavior. Applications in
the geosciences include the study of fault mechanics
in a wide range of settings [3, 5], the mechanics and
evolution of granular fault gouge [2, 6, 7, 47, 48, 58, 59],
fracturing of brittle rocks [78], and the evolution of rock
joints [79]. A recent study of the boudinage process
[9] extends the discrete element model to include a
ductile material. However, the rheology of this ductile
material is better approximated by Bingham plasticity
rather than a linear viscous behavior.
In DEM simulations, particles interact only with
their nearest neighbors and with the walls. Different
materials are described by different particle interactions, such as frictional or brittle–elastic interactions.
The forces and moments acting on the particles are
calculated from these interactions. The motion of the
particles are calculated by updating the velocities of
each particle from the sum of the forces exerted on that
particle using Newton’s law.
The SPH technique was originally developed to
simulate non-axisymmetric phenomena in astrophysics
[26, 45]. Since then, the applications have covered a
wide variety of fields. For example, it has been used
to simulate multiphase flow [56], free surface flow [54],
flow in fractures and porous media [29], landslides [50]
underwater explosions [44] and many other problems
like blood flow [85] and traffic flow [72]. Takeda and
his group presented a model of viscous flow to simulate
two-dimensional Poiseuille flow and three-dimensional
Hagen–Poiseuille flow [84].
This is an interpolation method, and the integral interpolant of any function is calculated using a
special smoothing function, called kernel. The SPH
formulation is derived by spatial discretization, the
Navier–Stokes equations, leading to a set of ordinary
differential equations with respect to time, which then
can be solved via time integration. However, SPH is not
suitable for modeling brittle failure because the state
of the media is described by field variables and not by
discrete elements.
Since DEM is not suitable to model viscous behavior
and SPH is not suitable for modeling brittle failure,
these two methods need to be combined in order to
model complex brittle–ductile deformation processes.
It is possible to couple these methods since they have
several similarities: both methods are meshless. The
material is represented by a set of particles or field variables, which are carrying mass, momentum, and other
physical properties; they interact with each other in
their vicinity and move due to the forces resulting from
these interactions. However, there are also differences
between the two methods; for example, the way in
which these interactions are calculated and the positions are updated. Therefore, some modifications are
necessary to fit the methods together. For this reason,
we did not use any existing codes but developed our
own code which couples the SPH and DEM techniques.
This paper demonstrates the initial results of an ongoing study to simulate coupled deformation of brittle
and ductile materials using combined DEM and SPH
techniques. This new numerical method is a truly meshfree Lagrangian method. The coupling of DEM and
SPH modeling techniques introduce a new approach
that is different from other methods found in the literature for coupled deformation of brittle and ductile
materials. The theory of this method is briefly described
below; a more detailed description of the methods can
be found separately in either the literature for DEM
[8, 18, 57, 67, 92] or SPH [26, 43, 45, 53]. The first part of
the paper includes the results of the validation test for
the new coupled method, while the second part presents the first results of two applications in geoscience: a
boudinage simulation, which is a geological application
of this method, and a hydraulic fracturing simulation,
which is an industrial application of this method.
2 Methods
2.1 Smoothed particle hydrodynamics
SPH is a mesh-free particle-based method; it does not
need a grid to calculate spatial derivatives. SPH is a
Comput Geosci
Lagrangian method: the Lagrangian description is a material description, the properties of every point of the
moving material is given at every time step. Therefore,
the fluid is represented by an ensemble of field variables (e.g., density, pressure, velocity, strain rate). Each
point is carrying mass, momentum, and hydrodynamic
properties such as density, pressure, and strain rate.
We did some modification so that we were able
to couple the SPH technique with DEM. In our new
method, the field variables are acting as particles. They
can interact with each other within a certain radius,
which is defined by the kernel function. To calculate
these interactions, we used SPH formalization.
One problem which has to be considered when using
the SPH approach for the simulation of viscous fluids
is the existence of a tensile instability [83] leading to
the unphysical clumping of particles in cases where
the pressure is negative. This issue is not specifically
addressed in this work because the method proposed
here is mainly targeted at geological applications where
the mean stress, or pressure, is positive in most cases.
However, there are some geological problems where
tensile stresses are possible in the viscous materials, for
example, in salt tectonics applications. In those cases, it
would be possible to extend the method by adding the
appropriate stabilizing terms to the SPH momentum
equation (Eq. 5) as described in [55] and [51] and
to enable the application of the method to problems
where negative pressures may arise in the SPH part of
the model.
the particles, which have position r j , mass m j, and
density ρ j. A j denotes the value of any quantity A at
position r j .
2.1.2 Kernel function
The kernel function, also called the smoothing function,
plays a very important role in the SPH approximations, as it determines the accuracy of the function
representation and efficiency of the computation. The
kernel function W(R, h) depends on two parameters:
the smoothing length h, which defines the radius of
the particle, and the non-dimensional distance between
the particles, which is given by R = r/h, where r is
the actual distance between the particles (Fig. 1). The
kernel function must satisfy certain conditions such
as normalization, positivity, compact support, and the
Dirac-delta function condition as h approaches zero. It
is supposed to decrease monotonically with increasing
distance from a given particle [53]. Throughout our
simulations, we used the two-dimensional form of the
Quintic kernel as described by [60].
⎧
(3 − R)5 − 6(2 − R)5 + 15(1 − R)5 if 0 R < 1
⎪
⎪
⎪
⎨(3 − R)5 − 6(2 − R)5
if 1 R < 2
W=F
5
if 2 R < 3
⎪(3 − R)
⎪
⎪
⎩
0
if R 3
(3)
where in the 2-D case the factor F is
2.1.1 SPH formalization
F=
The integral representation of any function A at a given
position r is defined by
A(r )W(r − r , h)dr
A(r) =
As (r) =
j
Aj
m j 2 W(r − r j, h)
ρj
(4)
(1)
where the integration is calculated over the entire space
and W is the smoothing kernel function.
By discretizing the computational volume into a
finite number of integration points, which can be
thought of as particles, the integral representation of a
function is approximated by a summation interpolant
7
.
478π h2
W
rij
r
(2)
where j is the summation index, which denotes the
particle label, and the summation is calculated all over
Fig. 1 Schematic drawing of an SPH kernel in 2-D, showing the
value of the kernel function W over the radial distance r. The
value rij is the distance between two particles for which the kernel
can be applied
Comput Geosci
2.1.3 Equation of motion
The velocity is determined through discretization of the
Navier–Stokes conservation of momentum equation.
The SPH representation of equation of momentum can
be given as follows:
N
p j ∂ Wij
Dvix
pi
=−
mj
+ 2
Dt
∂ xi
ρi2
ρj
j=1
+
N
j=1
mj
xy xy
μ jε j
μi εi
+
2
ρi
ρ 2j
∂ Wij
∂ yi
(5)
where vix is the x component of the velocity, pi is the
xy
pressure, and εi is the xy component of the strain rate
tensor of particle i. The y component of the velocity can
be calculated in a similar way.
The first part of the above equation describes the
isotropic pressure, while the second part describes the
viscous stress. For Newtonian fluids, the viscous shear
stress is proportional to the shear strain rate denoted
by ε̇ through the dynamic viscosity μ. The particle
approximation of the shear strain rate can be given
xy
ε̇i =
N
j=1
m j y ∂ Wij m j x ∂ Wij
v
+
m j vij
ρ j ij ∂ xi
ρj
∂ yi
j=1
state to determine fluid pressure. Several equations can
be used. In our simulations, we used the equation of
Monaghan [54], where the relationship between pressure and density is assumed to follow the expression
p= B
ρ
ρ0
γ
−1
(8)
where γ = 7 , B = 1.013 · 105 Pa, and ρ0 is the reference
density of the fluid.
2.1.6 Time integration
Any appropriate integration method can be used since
the equations of motion are ordinary differential equations. In our code, we used a predictor–corrector
method for the velocity, displacement, and density calculations. Assuming that f(y, t) is the time derivative
of y, the predictor–corrector scheme first predicts the
evolution of the quantity y at half time steps (n+1/2)
N
mj
⎛
⎞
N
2 ⎝ m j
−
vij∇ j∂ Wij⎠ δij
3 j=1 ρ j
yn+1/2 = yn + f (yn , tn )
(6)
where vij = vi − vi and ∇ j∂ Wij is the total derivative of
the kernel.
t
.
2
(9)
Then these values are corrected at the end of the
time step.
yn+1 = yn + f (yn+1/2 , tn+1/2 )t.
(10)
2.1.4 Density calculation
There are several approaches to calculate the particle
approximation of density [42]. In our simulations, we
use the continuity density approach, which approximates the density according to the continuity equation
using the concepts of SPH approximations [53].
Dρi
=
m jvij∇i Wij
Dt
j=1
N
(7)
The advantage of the continuity density approximation is that it conserves the mass exactly since the integration of the density over the entire problem domain
is exactly the total mass of all the particles.
2.1.7 Boundary conditions
On the boundaries of the model, virtual boundary
particles are used. They have the same properties as
the ordinary (real) SPH particles; the only difference
is that the virtual particles are not moving. The support domain of the kernel function is truncated by the
boundary however these virtual particles fill the kernel.
The virtual particles are also used to exert a repulsive
boundary force to prevent the interior particles from
penetrating the boundary.
2.2 Discrete element method
2.1.5 Equation of state
The fluid in the SPH formalism is treated as weakly
compressible. This facilitates the use of an equation of
In the coupled model presented in this work, we are
using simple 2-D implementation of the DEM which
is based on the initial “lattice solid model” approach
Comput Geosci
force due to a dashpot-type interaction between particles touching is given by
described in [19, 57]. For more realistic models of brittle
deformation, more complicated implementations of the
particle interactions would be preferable. In particular,
bonded particle interactions which support shear and
bending stiffness such as those proposed by [70] and
[92] would result in a more realistic fracture behavior
of the models. However, to demonstrate the feasibility
and use of the coupled DEM–SPH model, the simple
implementation used here is sufficient and can easily
be extended in the future.
where A is the damping coefficient, dij is the distance
between the particles, vij is the velocity difference
between the particles, and rij is the average radius of
the particles.
2.2.1 Particle interactions
2.2.2 Particle movement
The DEM particles in the coupled model are interacting with each other in two different ways. Particles
which are connected by a “bonded interaction” are subject to a repulsive–attractive interaction force, whereas
particles which are touching, but not connected by a
bond, are interacting by a repulsive–elastic force. Besides, all touching particles are interacting by a “dashpot” damping force. The brittle–elastic bonds between
the particles do break if a defined breaking strain is
exceeded. Similar to the DEM implementations presented in [19] and [57], the force due to the bonded
interactions is calculated as
The calculation of the particle movement is performed
according to Newton’s second law, i.e.,
Fijbonded
=
kij(rij − (r0 )ij)eij
0
rij ≤ (rbreak )ij
rij > (rbreak )ij ,
(11)
where kij is the bond stiffness, rij is the distance between
the particles i, and j, (rbreak )ij the breaking distance
for the bond and e is a unit vector in the direction of
the interaction. The equilibrium distance between the
particles is the sum of the particle radii, i.e., r0 = ri + r j.
The individual bond stiffness kij for each bond can be
calculated from a global stiffness parameter k and the
radii of the particles connected by the bond as kij =
k(ri + r j)/2 where ri and r j are the radius of particles
i and j, respectively. The breaking distance for a bond
between two particles is calculated from a breaking
strain εb and the equilibrium distance between the
particles as (rbreak )ij = (1 + εb )(r0 )ij = (1 + εb )(ri + r j).
The force due to a non-bonded elastic interaction between two particles is calculated very similarly as
Fijelastic
kij(rij − (r0 )ij)eij
=
0
damping
Fij
mi
=
−Adij
vij
rij
0
rij ≤ ri + r j
rij > ri + r j ,
∂ 2x
= Fi
∂t2
(13)
(14)
where mi is the mass of particle i, x is the position of
the particle, and Fi is the sum of all forces acting on the
particle due to interactions with other particles. In the
implementation used here for the demonstration of the
coupled DEM–SPH model, a simple first-order scheme
for the time integration of the particle movement is
used, i.e.,
ai =
Fi
mi
(15)
vit+1 = vit + ai t
(16)
xit+1 = xit + vit+1 t
(17)
where ai is the acceleration of the particle i, vit and
vit+1 are the particle velocities at time steps t and t + 1,
respectively, and t is the size of the time step. xit
and xit+1 are the particle positions at times t and t + 1.
Higher-order schemes such as the one implemented for
the movement of the SPH particles in Section 2.1.6 or
the Velocity-Verlet Scheme [13] employed by Donze
et al. [19] and Mora [57] could be used but tests have
shown that this is not necessary here.
2.2.3 Boundary conditions
rij ≤ ri + r j
rij > ri + r j .
(12)
Damping is also implemented in the code in order
to prevent the buildup of kinetic energy. The damping
The model boundaries consist of infinite rigid planar
walls. They are specified by a point and a normal vector.
The wall can move in two ways: they can be either
displacement-controlled or force-controlled. The actual
position of a displacement control wall is calculated
Comput Geosci
based on a given applied velocity. While the position
of a force-controlled wall is calculated based on the
resultant force acting on the wall, which is the sum
of the forces caused by particles and a user-defined
confining force.
The particles interact with the wall via elastic interaction. The elastic force acting on particle i touching the
wall is given by
Fiwall = kw di nw
(18)
where kw is the elastic stiffness of the wall, di is the
distance of particle i and the wall, and nw is the normal
vector of the wall. The same force is acting on the wall
as well.
2.3 Coupling
Fig. 2 Simplified flow diagram for one time step of the coupled
SPH–DEM model. Some details such as calculations of forces
due to particle–boundary interactions are left out for clarity
interaction between the SPH and the DEM particles.
The advantages of the second approach is that it makes
it possible to use arbitrary particle sizes for the DEM
DEM
particles
SPH
particles
Force controlled DEM wall
Displacement controlled DEM wall
Force controlled DEM wall
In order to couple the viscous (SPH) and the brittle–
elastic (DEM) behavior, we implemented two different
kinds of coupling. This coupling is the key part of our
models.
The first, “SPH type” coupling is implemented by
using DEM particles as virtual SPH particles, which
interact with SPH particles according to the SPH discretization of the Navier–Stokes equation and with
DEM particles according to the DEM nearest-neighbor
interactions. The forces due to both types of interactions are combined, and the hybrid particles are moved
according to Newton’s law. On the SPH side of the
model, this approach is similar to the “virtual particle”
approach used to implement solid boundary conditions
in SPH models.
The second, “DEM type” coupling is implemented
by using SPH particles as DEM particles, which interact
with nearest-neighbor DEM particles based on DEM
force calculation, and with SPH particles according
to the SPH discretization of the Navier–Stokes equation. Between DEM and SPH particles, elastic and
dashpot interactions are identical to those described in
Section 2.2.1 (Eqs. 11–13) for interaction within the
DEM model.
Both type of coupling have specific advantages and
disadvantages. The fist type of coupling, i.e., using
the DEM particles in contact with the SPH particles
as virtual SPH particles, has the advantages that the
SPH approximation of pressure field is maintained
across the DEM–SPH boundary. In the second coupling approach, however, the boundary between SPH
and DEM particles is better described as a surface of
the SPH model, whose position is determined by the
force balance between the SPH particles. The disadvantage of this is that it makes the introduction of an additional parameter necessary which describes the elastic
Displacement controlled DEM wall
Fig. 3 Initial geometry setup for the pure shear simulations.
SPH particles are shown in blue, DEM particles are shown in
yellow (particles connected to the top and bottom wall), and
green (particles connected to the left and right wall)
Comput Geosci
Fig. 4 Displacement field observed during a pure shear simulation using a differential stress σdiff = 120 Pa and a viscosity of the
SPH particles of μ = 3,000 Pas. Particles are colored by horizontal
displacement. Total displacement is shown by arrows attached to
each particle. The overlap between the boundary particles which
can be seen near the corners of the model is there because the
DEM particles assigned to different boundaries do not interact
part of the model, while maintaining a single particle
size for the SPH particles, which is easier to implement
and has numerical advantages.
2.4 Implementation
While the SPH equations (Eqs. 5–8) are usually formulated in terms of a kernel function, they can also
be expressed as sums of particle–particle interactions
more similar to the approach generally taken in the
implementation of DEM models. For this purpose, the
summation is removed from Eq. 6 to calculate the
xy
individual contributions εij to the strain rate tensor at
a given particle i due to its interaction with another
0.16
0.14
0.12
0.1
Strain
Fig. 5 The evolution of strain
over time for a set of pure
shear simulations with the
same viscosity (μ =
1,000 Pas) but different
differential stresses ranging
from σdiff = 60 Pa to σdiff =
140 Pa
particle j. All those contributions are added incremenxy
tally to the total strain rate tensor εi for particle i.
Similarly, Eq. 7 is adapted to calculate the individual
D
contributions to the density change Dtij due to each
interacting particle pair.
After the contributions to the strain rate tensor and
the density change have been calculated for all interacting particle pairs, the acceleration due to the viscous
forces acting on each SPH particle can be calculated
according to Eq. 5. These accelerations are then used
to update the positions of the SPH particles using the
predictor–corrector scheme described in Section 2.1.6.
This approach has been implemented so that a list of
all interacting particle pairs is first generated by finding
all pairs of particles which are within a defined range
of the SPH interactions. The interaction calculations
described above are then performed by iterating over
the list of interacting particle pairs.
In contrast, the calculation of particle accelerations
and the position updates operate on particles rather
than interacting particle pairs and are therefore performed by iterating over a list of SPH particles. Similarly, in the DEM part of the model, the forces
(Eqs. 11–13) acting on the particles are first calculated
by iterating over the list of all DEM interactions and
then the positions of the DEM are updated according
to Eq. 14. The whole process, which is similar in structure to the program flow implemented in some DEM
software (see, for example [1]) is visualized in Fig. 2.
While the interactions between particles in the SPH
models are not restricted to nearest neighbors in the
strict sense, the interaction range is still restricted to
a finite distance by the smoothing length of the kernel. This has two advantages. One is that the same
0.08
0.06
0.04
0.02
0
0
200
400
600
800
1000
Timesteps ( x 1000)
Differential stress:
σ = 60 Pa
σ = 80 Pa
σ = 100 Pa
σ = 120 Pa
σ = 140 Pa
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300
y = 295.88x
R² = 0.9925
250
Differential stress [Pa]
Fig. 6 Differential stress vs
steady-state strain rate shown
for all simulations with the
viscosity between μ = 500 Pas
and μ = 3,000 Pas and
differential stresses ranging
from σdiff = 60 Pa to σdiff =
140 Pa. Models with the same
viscosity are plotted using the
same symbol and color. The
straight lines are linear fits for
each group of models with
the same viscosity
200
y = 190.62x
R² = 0.9889
y = 98.516x
R² = 0.9909
150
y = 50.202x
R² = 0.9799
100
50
0
0
0.5
1
1.5
2
2.5
3
Strain rate [1/s]
Viscosity:
acceleration schemes for speeding up the determination
of all interacting particle pairs can be used as in DEM.
The other advantage, which will become important for
larger scale implementations of a coupled DEM–SPH
model is that, due to the finite interaction range, the
coupled model is also amenable to parallelisation using
a space partitioning approach which has been used for
DEM models before [1, 8].
3 Validation test—pure shear
We performed pure shear tests in order to investigate
the viscous behavior of the SPH material and also to
test the coupling we implemented in the model. Pure
shear is an coaxial flow involving shortening in one direction and extension in a perpendicular direction such
that the volume remains constant [21]. For Newtonian
fluid, the applied differential stress is directly proportional to the shear strain rate through the dynamic
viscosity
σdiff = με̇.
(19)
The aim of this test is to verify that this linear relationship is satisfied in our model.
Fig. 7 Boudinaged
amphibolite in marble
(Naxos, Greece). For details,
see [75]
η=3000 Pas
η=2000 Pas
η=1000 Pas
η=500 Pas
Here, we used a rectangular block of SPH particles.
The SPH particles are all of the same size and initially
packed in a regular triangular lattice. On the boundaries, there are double layers of DEM particles which
interact with SPH particles via SPH interaction, and
with the walls via elastic DEM interactions (Fig. 3).
In order to enable the boundary conditions needed for
this specific model, in particular to avoid problems in
the corners, the DEM particles assigned to different
boundaries do not interact with each other. Between
the SPH and DEM particles, the SPH type of coupling,
i.e., DEM particles acting as virtual SPH particles, was
applied. During the simulation, the horizontal walls
move towards each other at constant velocity, except
for an initial phase of the movement where the velocity
is ramped up linearly.
We used force-controlled side walls in order to keep
the differential stress constant, i.e., the forces acting
on the displacement-controlled horizontal walls are
recorded and used to calculate the forces which need to
be applied to the side walls. In order to ensure that the
correct force is acting on the side walls, they are moved
according to the stiffness of the DEM particle–wall
interactions and the chosen force. The strain (εxx + ε yy )
is calculated based on the positions of the walls, which
Comput Geosci
Displacement controlled wall
Less competent - SPH
Force controlled wall
Force controlled wall
Fig. 8 Schematic drawing of
the initial geometry setup for
the boudinage simulation.
SPH particles are shown in
blue, DEM particles are
shown in gray
Competent -DEM
Less competent-SPH
Displacement controlled wall
are also recorded during the simulation. The strain
rate is calculated accordingly. Twenty-six tests have
been run with various differential stress and viscosity
values. The differential stress values varied from 40 to
Fig. 9 Evolution of
boudinage in a model using a
viscosity of μ = 1 Pas. ε yy is
the amount of vertical
shortening. The SPH particles
are shown in gray and the
DEM particles are colored
according to their horizonal
displacement (blue is −x/left,
red is +x/right)
300 Pa, while the viscosity values varied from 500 to
3,000 Pas.
A typical example of a displacement field recorded
during one of the pure shear simulations can be seen
Comput Geosci
in Fig. 4. The evolution of the strain recorded during
the simulations (Fig. 5) shows that after an initial compaction, the increase of strain is linear with time. Consequently, the strain rate is constant during the later
steady-state part of the simulation. We also observe
that the length of the compression phase is dependent
on the stress conditions applied to the model. Because
the same displacement rate was applied to the top
and bottom boundaries in all simulations, the models with higher differential stress have a lower mean
stress and therefore show more compaction. Using the
steady-state strain rates, we found a linear relationship between the calculated strain rate and the applied
differential stress for each group of simulations using
the same viscosity in the SPH part of the model. The
slope of the fitted line, which is the calculated viscosity,
is proportional to the applied viscosity and independent
of mean stress (Fig. 6) as expected.
4 Geological application—boudinage
One important example of a coupled brittle–ductile
deformation in a geological context is the process
of boudinage. Boudinage occurs in mechanically layered rocks, where the layers are disrupted by layerparallel extension. A competent layer embedded between weaker layers is separated by tensile or shear
fracturing into rectangular fragments (torn, domino,
gash, or shearband boudins), or by necking or tapering into drawn boudins or pinch-and-swell structures
[27, 71]. One example of the drawn tapering boudins
observed in Naxos, Greece [75] is shown in Fig 7). In
this case, a boudinaged amphibolite layer is in a marble
matrix.
The full evolution of the boudins is not completely
understood yet. Existing models focus on either the
brittle–elastic processes of the fracture initiation phase
µ = 0.5 Pas
µ = 1 Pas
µ = 2 Pas
µ = 5 Pas
µ = 10 Pas
Fig. 10 Comparison of the boudinage structures which have
developed in simulations with different viscosities of the SPH
material. The model is rotated by 90◦ with respect to Figs. 8
and 9. The viscosity is increasing from left μ = 0.5 Pas to right
μ = 10 Pas. SPH particles are shown in gray, whereas the DEM
particles in the competent layer are colored according to their
layer-parallel displacement (red, +x/up; blue, −x/down). The
snapshots of the different models are all taken at a the same
deformation state with strains εxx = 0.65 and ε yy = 0.43
Comput Geosci
apart, but some of the parts are still connected to each
other with boudin necks. During further extension, the
necks become thinner and separate boudins form.
As the viscosity of the incompetent layers increases
while the material properties of the competent layer
remain the same, the shape of developing boudins
changes from long, well-separated boudins to shorter
boudins connected by necks (Fig. 10). At even higher
viscosity of the incompetent layers, i.e., decreasing mechanical contrast between the layers, pinch-and-swell
structures develop. The number of boudin blocks also
increases with increasing viscosity of the incompetent
layers as would be expected from theoretical considerations (cf. [71]).
5 Industrial application—hydrofracturing
Hydraulic fracturing is the process of initiation and
propagation of a fracture due to hydraulic pressure in
a cavity. During hydraulic fracturing, viscous fluid is
pumped into a wellbore at high-pressure. This highpressure fluid breaks the rock in tension and forces
Moving SPH
boundary particles
Fixed SPH
boundary particles
Fixed DEM wall
Force controlled DEM wall
Fixed DEM wall
Force controlled DEM wall
[35, 79] or ductile processes of the post-fracture phase
[46, 49, 65, 87].
Although Abe et al. [9] simulated full boudinage
which described the process from the brittle failure
through the separation of boudins to post-fracture
movements (e.g., rotation) of the fragments, the quasiviscous matrix material used in this model does not
allow full quantitative analysis of the influence of the
matrix rheology.
There are several approaches to model fracturing
in layered rocks. On the one hand, saturation models
assume no slip interface coupling between the elastic
matrix and the brittle–elastic competent layer. On the
other hand, models with interface slip provide better
approximation, although these models are only valid if
the ratio of tensile stress in the competent layer (T) to
interface shear strength (τ ) is high (T/τ > 3.0 ) [35, 79].
Models incorporating viscous coupling between the
viscous matrix and the brittle–elastic competent layer
show qualitatively correct behavior compared to observations in nature [9]. Therefore, we used this approach
in our boudinage model.
In our boudinage simulation, the initial setup consists of three layers (Fig. 8). The ductile top and bottom
layers are made up of SPH particles, while the competent central layer is made up of DEM particles. Because
the SPH particles all have the same radius, they are
initially placed in a regular arrangement. The DEM
particles are packed in a random arrangement because
of their heterogeneous radius distribution. Within the
competent layer, the DEM particles are connected by
brittle–elastic bonds which can break if a failure criterion is exceeded. The SPH and DEM particles interact
with each other via the DEM type of coupling, i.e., in
case of an interaction between SPH and DEM particles, the SPH particles act as DEM particles. On the
boundaries of the model, there are four infinite walls,
which interact with both types of particles via elastic
DEM interactions. During the boudinage simulation,
we applied vertical displacement, which resulted in
horizontal extension. The upper and the lower walls are
moved at a constant velocity, while left and right walls
were force-controlled to apply a constant confining
stress. Five simulations have been run with different
viscosity values, namely 0.5, 1, 2, 5, and 10 Pas. While
these viscosity values are much lower than what would
be expected in a geological context, they are within
the range of those used in analog modeling (see also
Section 6).
The evolution of boudinage can be seen in Fig. 9.
Due to extension, cracks develop in the competent layer.
The incompetent layer then flows into the cracks as
they widen. The fragments of the competent layer move
Fixed DEM wall
Fig. 11 Initial geometry setup for the simulation of hydraulic
fracturing. DEM particles are shown gray, real SPH particles
in blue, moving boundary SPH particles in green, and fixed
boundary SPH particles in yellow
Comput Geosci
it to open. The fractures open in the direction of the
minimal principal stress and propagate perpendicular
to the minimal principal stress [36, 63].
The viscosity of pumped fluid plays an important
role in the hydraulic fracturing processes. The viscosity
is one parameter, which controls the widths of the
generated fractures: fluid with higher viscosity initiates
wider cracks, whereas fluid with lower viscosity initiates
narrow cracks [66]. It can also influence the geometry of
the fractures: fluid with high viscosity tends to generate
planar cracks with only a few branches, whereas lowviscosity fluid, such as water, tends to generate wavelike
cracks with many secondary branches [33, 81].
Hydraulic fracturing was first developed by Clark in
1949 [17], and, since then, it has become a widely used
and efficient technique in the petroleum industry to
increase the near-well permeability and to enhance oil
and gas extraction from a reservoir [10, 30, 33, 36]. It is
also used in the geothermal energy industry to increase
the heat exchange surfaces and the permeability of the
hot dry rock [74].
Hydrofracturing simulations have traditionally applied either boundary element method [11, 14] or finite
element method [64, 91, 97] approached. The problem
with these models is that they do not always show
Fig. 12 Evolution of a crack
due to fluid injection in a
simulation of hydraulic
fracturing. The brittle solid
(DEM) particles are shown in
gray, the fluid (SPH) particles
in blue. To visualize the SPH
particles in a compact form,
we use the smoothing length
h as radius for the particle
glyphs. The viscosity used for
the SPH fluid is μ = 0.5 Pas.
The value t at the bottom
right corner of each frame
shows 1,000 time steps, i.e., t
= 100 shows the model at
time step 100.000
good agreement with the recorded seismicity data [73].
One reason for this is that the models assume tensile
fracture growth but usually shear-type seismic events
are recorded [32]. The other reason is that hydrofracture growth is not symmetrical around the borehole.
One option to fix this problem is the use of DEM. The
existing DEM models [12, 81] use particles only for
the solid rock; the fluid is described using fluid flow
algorithms. Here, we present a new model where both
the solid rock and the fluid are represented by particles.
In our hydraulic fracturing simulation, the viscous
fluid is represented by SPH particles, while the solid
rock is represented by DEM particles. The initial geometry (Fig. 11) consists of a block of DEM particles with a
notch on the top. This notch is filled with SPH particles.
The notch is continuing upwards into a column (“the
pump”), which is also filled with SPH particles in order
to drive fluid into the notch and the fracture. The DEM
type of coupling described in Section 2.3 is used here,
i.e., SPH and DEM particles interact through DEM
interactions.
The boundaries of the DEM block are walls, the top
and bottom walls are fixed, while the side walls are
force-controlled. The DEM particles interact with the
walls by elastic DEM interactions. The borders of the
1.
0%
t=0
2.
46%
t=100
3.
52%
t=112
4.
69%
t=148
5.
70%
t=151
6.
74%
t=160
7.
83%
t=179
8.
87%
t=187
9.
100%
t=215
Comput Geosci
Fig. 13 Snapshots of the
propagating crack in the
hydrofracture model at the
same time step but with
different fluid viscosities
used. Viscosity is
μ = 10−3 Pas (top left frame),
μ = 10−2 Pas (top right
frame), μ = 0.1 Pas (bottom
left frame), and μ = 1.0 Pas
(bottom right frame)
SPH column are virtual SPH particles. On the side, the
boundary particles are fixed, while the particles of the
top border can move only vertically. The different kinds
of boundary particle do not interact with each other.
During the simulation, we moved the top SPH border
µ=0.01 Pas
µ=0.1Pas
µ=1 Pas
with a constant velocity. Twelve simulations that have
been run with viscosity values varied between 10−3 Pas
(similar to water) and 5 Pas.
The evolution of the induced crack is shown on
Fig. 12. First, the original notch is expanding in vertical
4.5
4
3.5
Crack depth [mm]
Fig. 14 Progress of the
fracture propagation with
time in hydrofracture models
with different viscosities.
Initial crack tip position
defined in the geometry setup
is at y = 4.25 with the crack
propagating downwards
µ=0.001 Pas
Viscosity:
0.001 Pas
3
0.01 Pas
2.5
0.1 Pas
2
0.2 Pas
0.5 Pas
1.5
1 Pas
1
2 Pas
5 Pas
0.5
0
0
50
100
150
Timestep ( x 5000)
200
250
300
Comput Geosci
350
Breaking time step
Fig. 15 Time of the fracture
reaching the bottom of the
model depending on the
viscosity of the SPH fluid
300
250
200
150
100
0.001
0.01
0.1
1
10
Viscosity [Pas]
direction. Due to this expansion, the surrounding solid
blocks start to rotate, and, due to the resulting stresses,
the solid material is starting to fracture. When the developed crack is large enough, the fluid is flowing into
it and force the crack to expand. This results in further
propagation of the fracture. The position of the crack
tip can be determined by observing the breaking of the
bonds between the DEM particles. Analyzing that the
locations where the DEM bonds break is starting near
the tip of the initial notch and moving away from it, we
can plot the progress of the fracture process (Figs. 13
and 14).
The velocity of the crack propagation depends on
the viscosity of the fluid: the lower the viscosity is, the
faster the crack is propagating. This means that the rock
breaks faster if lower viscosity fluid is injected into it
(Fig. 15). The reason is that with lower viscosities, the
rock splits into two parts at once; while with higher viscosities, the fracture develops in more stages (Fig. 14).
It should be noted that the geometry of the developing
fractures is influenced by the relatively small model
size and the details of the boundary conditions. The
geometry of the developed fracture is also affected by
the viscosity (Fig. 13). The aspect ratio of the fracture
changes with viscosity (Fig. 16): the lower the viscosity,
the longer and wider cracks develop.
The combination of the DEM and SPH represents a
new approach to the problem of numerical modeling
of coupled brittle–ductile deformation processes. Both
methods are well established in their respective domains. An advantage of coupling DEM and SPH is that
the two methods are sufficiently similar in their numerical approach, so that a combined implementation can
be made seamless. In particular, because both methods
are Lagrangian and particle-based, there is no complex
interpolation needed to transfer data between the two
fluid (DEM) and solid (SPH) part of the model. This
contrasts to other studies coupling particle-based and
mesh-based methods [4, 20].
The results of the validation test (Section 3) demonstrate that the correct viscous behavior of the SPH
model is maintained when the deformation is driven
by moving DEM particles coupled to the boundaries of
the SPH volume. The model also validates the correct
transfer of stress and strain between the SPH and the
DEM part of the model. At the interface between
the top and bottom DEM “wall” (Fig. 3), which is
displacement-controlled, the deformation is transferred
from the moving DEM particles to the SPH particles.
At the SPH–DEM interface along the side walls, the
25
Aspect ratio of the crack
Fig. 16 Aspect ratios of the
cracks developed at time step
125.000 in hydrofracture
simulations with different
viscosities
6 Discussion
20
15
10
5
0
0.001
0.01
0.1
Viscosity [Pas]
1
10
Comput Geosci
movement of the SPH particles, driven by the viscous
deformation of the SPH volume, forces the movement
of the DEM wall particles in response to the force
balance at the interface.
The boudinage simulation (Section 4) shows the
applicability of the coupled simulation approach to a
problem in structural geology. The test models presented here show a qualitatively correct behavior,
which, if the lower resolution and the simplified fracture mechanics is taken into account, is not too dissimilar from other DEM simulations of boudinage [9].
A more realistic fracture behavior of the brittle layer
could be obtained by implementing a more complex
bond model between the DEM, i.e., “parallel bonds” as
described by [70] or the rotational bond model by [92].
The advantage of a coupled SPH–DEM simulation
of boudinage compared to the pure DEM models is that
a linear viscous material law can be used for the ductile
matrix material in the SPH model, whereas in the pure
DEM model, viscosity could only be approximated [9].
In addition, it is possible to extend the SPH model to
non-linear viscous materials, for example, to include
power law viscosity which would be important for the
simulation of deformation processes involving rocks
deforming by dislocation creep [38, 39, 89].
The viscosity of the matrix material used in the
boudinage model is much lower than that is expected
in a realistic geologic setting (1016 vs. 10 Pas). This discrepancy is due to the fact that the large range of time
scales involved in a boudinage process with the correct
geologic material properties would make the model
numerically intractable. The time scale of ductile deformation would be in the range of thousands or even millions of years (≈ 1010 . . . 1014 s), whereas the time scale
of the fracture processes would be in the sub-second
to millisecond range, requiring microsecond time steps
to resolve the crack propagation. This problem can
be alleviated by applying techniques such as “density
scaling” [86] to move the time scales of these processes
closer together. The material parameters used here in
our numerical models are much closer to those typical
for analog models, i.e., paraffine (≈ 106 Pas [61]) or
silicone putty (≈ 105 Pas [28]). Another material, which
has been used in analog modeling is honey with a
viscosity of ≈ 10 − 20 Pas [39] or polydimethylsiloxane
[90], which has a viscosity of ≈ 10−2 to 0.15 Pas [93]
which is well within the range of viscosities used in
the numerical models making our model comparable to
these physical models.
The hydrofracturing models (Section 5) demonstrate
the use of the coupled model for applications where
the brittle deformation of the solid material is directly
driven by fluid pressure. The details of the fracture
propagation in this model depend on the material heterogeneity of the solid and the distribution of pressure applied on the solid material by the fluid. The
fracture propagation generates new pathways for the
fluid flow, and the details of how the fluid moves
into these pathways are resolved. This depends on the
viscous properties of the fluid and will determine the
redistribution of the fluid pressure and thus the further
propagation of the fractures. For a productive application of hydrofracture modeling, more details about
the material properties need to be included into the
model; a detailed calibration of the fracture behavior
of the DEM part of the model in particular would be
necessary. A further possible extension of these model
would be the inclusions of solid grains made up of DEM
particles into the SPH fluid, similar to the sand grains
used as proppant hydrofracturing shale gas reservoirs.
Both applications of the coupled DEM–SPH approach presented in this work have shown that it can
capture the combined deformation of elastic–brittle
and viscous materials, even for large local strains. An
apparent limitation which can be observed, for example, in Figs. 9 and 12, is that narrow cracks which appear
in the brittle solid (DEM) material are not immediately filled by the viscous (SPH) material, leading to
the appearance of voids even in models with strongly
compressive mean stress, such as the boudinage example. This is largely a model resolution issue. The SPH
particles cannot move into the voids as long as entrance
to these voids are smaller than the particle size, i.e.,
the minimum aperture of a crack into which the fluid
can flow is limited by the model resolution. While this
might be a limitation for specific applications requiring
fluid flow through cracks with very high aspect ratios, it
does not impact the applicability of the method to many
other problems.
7 Conclusions
We present a new Lagrangian method for the simulation of brittle–ductile deformation using meshless particle-based methods. This method couples
smoothed particle hydrodynamics and discrete element
method.
We implemented a variation of SPH, which allows
the coupling with DEM in order to describe geological
deformations.
In a plane strain model, the linear relationship between the applied differential stress and the strain rate
is validated to show that so we are able to describe real
viscous behavior qualitatively.
Comput Geosci
We have shown two examples of possible applications of this method, namely boudinage and hydraulic
fracturing. Boudinage simulations demonstrated that as
the viscosity of the incompetent layer increases, the
number of boudins increases, and, with even higher viscosity, pinch-and-swell structures develop. Hydrofracturing simulations have shown that the viscosity of
the injected fluid affects the evolution of the induced
crack: the lower the viscosity is, the faster the crack
propagates and the wider the crack is.
This approach can be adopted to model the deformation of carbonate layers embedded in a viscous salt
body during tectonics. In the future, we are planning to
parallelize the code and extend it to 3-D code as well.
Acknowledgements This study was carried out within the
framework of DGMK (German Society for Petroleum and Coal
Science and Technology) research project 718 “Mineral Vein
Dynamics Modelling”, which is funded by the companies
ExxonMobil Production Deutschland GmbH, GDF SUEZ E&P
Deutschland GmbH, RWE Dea AG, and Wintershall Holding GmbH, within the basic research program of the WEG
Wirtschaftsverband Erdöl- und Erdgasgewinnung e.V. We thank
the companies for their financial support and their permission to
publish these results. We thank RWTH Aachen University for
enabling the performance of the main computations at the High
Performance Computing Cluster at RWTH Aachen University.
We also thank an anonymous reviewer for the helpful comments.
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