Comput GeosciDOI 10.1007/s10596-012-9335-x

ORIGINAL PAPER

Meshless numerical modeling of brittle–viscousdeformation: first results on boudinage and hydrofracturingusing 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 thenumerical modeling of deformation processes combin-ing brittle fracture and viscous flow. The new approachis based on the combination of two meshless particle-based methods: the discrete element method (DEM)for the brittle part of the model and smooth particlehydrodynamics (SPH) for the viscous part. Both meth-ods are well established in their respective applicationdomains. The two methods are coupled at the particlescale, with two different coupling mechanisms ex-plored: one is where DEM particles act as virtualSPH particles and one where SPH particles are treatedlike DEM particles when interacting with other DEMparticles. The suitability of the combined approach isdemonstrated by applying it to two geological pro-cesses, boudinage, and hydrofracturing, which involvethe coupled deformation of a brittle solid and a viscousfluid. Initial results for those applications show thatthe new approach has strong potential for the numer-ical modeling of coupled brittle–viscous deformationprocesses.

Keywords Numerical modeling ·Coupled brittle–ductile deformation · DEM · SPH ·Boudinage · Hydrofracturing

A. Komoróczic · S. Abe (B) · J. L. UraiStructural Geology - Tectonics - Geomechanics,RWTH Aachen University, Lochnerstr. 4-20,52056 Aachen, Germanye-mail: s.abe@ged.rwth-aachen.de

A. Komoróczice-mail: a.komoroczi@ged.rwth-aachen.de

J. L. Uraie-mail: j.urai@ged.rwth-aachen.de

1 Introduction

The coupled deformation of brittle and ductile mate-rials plays an important role in numerous geologicalprocesses. For example, the brittle deformation of car-bonate or anhydrite layers embedded in a salt bodyduring tectonics is important in salt tectonics [25, 40, 41,77, 82]. Another example is boudinage: a wide rangeof boudins is observed in the nature [27]; these areformed during deformation if there is a componentof lengthening parallel to a brittle layer in a ductilematrix [88]. Another process of both basic and appliedimportance is hydraulic fracturing, when viscous fluid ispumped into a brittle reservoir rock in order to gener-ate fractures [10, 14, 16, 20, 30]. Detailed understandingon these large deformation and extension fracturing islacking, and these processes present a challenge to bothanalog and numerical modeling.

Analog models are often used to investigate the pa-rameters which control the geometry of these structures[22, 71, 76, 95, 96], although many properties, such aspressure inside the model, are unknown during the ex-periment [62], and complete scaling is rarely achieved.

Existing numerical models of brittle–ductile defor-mation have also limitations because of the complexgeometries and the difficulties regarding the fluid–solidinteractions. For example, in salt tectonics, mostly thefinite element or finite difference method is applied[23, 24, 31, 34, 37, 41, 68, 69, 80, 94], which is a con-tinuum method and has limited capabilities to modeldiscontinuous media. One possibility to improve themodeling of brittle faulting within the context of thefinite element method (FEM) method is the use of splitnodes as demonstrated in [52]. However, this approachmakes it necessary to define the possible fault locations

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a priori. There are particle-based models to simulatehydraulic fracturing; however, in these models, only thesolid part is represented by particles, and the fluid partis described by a fluid flow algorithm [12, 15, 81]. Theintroduction of combined brittle–ductile behavior intoa particle-based method was shown to be possible [9],but the exact modeling of a linear viscous material hasnot been achieved yet using this approach.

In this paper, we present a new method, in whichboth the ductile and the brittle material are modeledusing meshless, particle-based, Lagrangian methods.The viscous fluid is represented by a smoothed parti-cle hydrodynamics (SPH) model and the brittle solidmaterial is represented by a discrete element method(DEM) model. Both approaches are well tested in theirrespective domain and due to the similarities betweenthem, it is possible to couple the two approaches todevelop a hybrid method.

DEM was developed by Cundall and Strack [18] toinvestigate the mechanical behavior of granular ma-terials and extended by Mora, Place and others [57,67, 92] with lattice solid model. This approach hasbeen proven very successful in the simulation of brittleand elastic–brittle material behavior. Applications inthe geosciences include the study of fault mechanicsin a wide range of settings [3, 5], the mechanics andevolution of granular fault gouge [2, 6, 7, 47, 48, 58, 59],fracturing of brittle rocks [78], and the evolution of rockjoints [79]. A recent study of the boudinage process[9] extends the discrete element model to include aductile material. However, the rheology of this ductilematerial is better approximated by Bingham plasticityrather than a linear viscous behavior.

In DEM simulations, particles interact only withtheir nearest neighbors and with the walls. Differentmaterials are described by different particle interac-tions, such as frictional or brittle–elastic interactions.The forces and moments acting on the particles arecalculated from these interactions. The motion of theparticles are calculated by updating the velocities ofeach particle from the sum of the forces exerted on thatparticle using Newton’s law.

The SPH technique was originally developed tosimulate non-axisymmetric phenomena in astrophysics[26, 45]. Since then, the applications have covered awide variety of fields. For example, it has been usedto simulate multiphase flow [56], free surface flow [54],flow in fractures and porous media [29], landslides [50]underwater explosions [44] and many other problemslike blood flow [85] and traffic flow [72]. Takeda andhis group presented a model of viscous flow to simulatetwo-dimensional Poiseuille flow and three-dimensionalHagen–Poiseuille flow [84].

This is an interpolation method, and the inte-gral interpolant of any function is calculated using aspecial smoothing function, called kernel. The SPHformulation is derived by spatial discretization, theNavier–Stokes equations, leading to a set of ordinarydifferential equations with respect to time, which thencan be solved via time integration. However, SPH is notsuitable for modeling brittle failure because the stateof the media is described by field variables and not bydiscrete elements.

Since DEM is not suitable to model viscous behaviorand SPH is not suitable for modeling brittle failure,these two methods need to be combined in order tomodel complex brittle–ductile deformation processes.It is possible to couple these methods since they haveseveral similarities: both methods are meshless. Thematerial is represented by a set of particles or field vari-ables, which are carrying mass, momentum, and otherphysical properties; they interact with each other intheir vicinity and move due to the forces resulting fromthese interactions. However, there are also differencesbetween the two methods; for example, the way inwhich these interactions are calculated and the posi-tions are updated. Therefore, some modifications arenecessary to fit the methods together. For this reason,we did not use any existing codes but developed ourown code which couples the SPH and DEM techniques.

This paper demonstrates the initial results of an on-going study to simulate coupled deformation of brittleand ductile materials using combined DEM and SPHtechniques. This new numerical method is a truly mesh-free Lagrangian method. The coupling of DEM andSPH modeling techniques introduce a new approachthat is different from other methods found in the lit-erature for coupled deformation of brittle and ductilematerials. The theory of this method is briefly describedbelow; a more detailed description of the methods canbe found separately in either the literature for DEM[8, 18, 57, 67, 92] or SPH [26, 43, 45, 53]. The first part ofthe paper includes the results of the validation test forthe new coupled method, while the second part pre-sents the first results of two applications in geoscience: aboudinage simulation, which is a geological applicationof 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 notneed a grid to calculate spatial derivatives. SPH is a

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Lagrangian method: the Lagrangian description is a ma-terial description, the properties of every point of themoving material is given at every time step. Therefore,the fluid is represented by an ensemble of field vari-ables (e.g., density, pressure, velocity, strain rate). Eachpoint is carrying mass, momentum, and hydrodynamicproperties such as density, pressure, and strain rate.

We did some modification so that we were ableto couple the SPH technique with DEM. In our newmethod, the field variables are acting as particles. Theycan interact with each other within a certain radius,which is defined by the kernel function. To calculatethese interactions, we used SPH formalization.

One problem which has to be considered when usingthe SPH approach for the simulation of viscous fluidsis the existence of a tensile instability [83] leading tothe unphysical clumping of particles in cases wherethe pressure is negative. This issue is not specificallyaddressed in this work because the method proposedhere is mainly targeted at geological applications wherethe mean stress, or pressure, is positive in most cases.However, there are some geological problems wheretensile stresses are possible in the viscous materials, forexample, in salt tectonics applications. In those cases, itwould be possible to extend the method by adding theappropriate stabilizing terms to the SPH momentumequation (Eq. 5) as described in [55] and [51] andto enable the application of the method to problemswhere negative pressures may arise in the SPH part ofthe model.

2.1.1 SPH formalization

The integral representation of any function A at a givenposition r is defined by

A(r) =∫

A(r′)W(r − r′, h)dr′ (1)

where the integration is calculated over the entire spaceand W is the smoothing kernel function.

By discretizing the computational volume into afinite number of integration points, which can bethought of as particles, the integral representation of afunction is approximated by a summation interpolant

As(r) =∑

j

m jA j

ρ2j

W(r − r j, h) (2)

where j is the summation index, which denotes theparticle label, and the summation is calculated all over

the particles, which have position r jr jr j, mass m j, anddensity ρ j. A j denotes the value of any quantity A atposition r jr jr j.

2.1.2 Kernel function

The kernel function, also called the smoothing function,plays a very important role in the SPH approxima-tions, as it determines the accuracy of the functionrepresentation and efficiency of the computation. Thekernel function W(R, h) depends on two parameters:the smoothing length h, which defines the radius ofthe particle, and the non-dimensional distance betweenthe particles, which is given by R = r/h, where r isthe actual distance between the particles (Fig. 1). Thekernel function must satisfy certain conditions suchas normalization, positivity, compact support, and theDirac-delta function condition as h approaches zero. Itis supposed to decrease monotonically with increasingdistance from a given particle [53]. Throughout oursimulations, we used the two-dimensional form of theQuintic kernel as described by [60].

W = F

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(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

(3 − R)5 if 2 � R < 3

0 if R � 3

(3)

where in the 2-D case the factor F is

F = 7

478πh2. (4)

W

rij r

Fig. 1 Schematic drawing of an SPH kernel in 2-D, showing thevalue of the kernel function W over the radial distance r. Thevalue rij is the distance between two particles for which the kernelcan be applied

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2.1.3 Equation of motion

The velocity is determined through discretization of theNavier–Stokes conservation of momentum equation.The SPH representation of equation of momentum canbe given as follows:

Dvxi

Dt= −

N∑j=1

m j

(pi

ρ2i

+ pj

ρ2j

)∂Wij

∂xi

+N∑

j=1

m j

(μiε

xyi

ρ2i

+ μ jεxyj

ρ2j

)∂Wij

∂yi(5)

where vxi is the x component of the velocity, pi is the

pressure, and εxyi is the xy component of the strain rate

tensor of particle i. The y component of the velocity canbe calculated in a similar way.

The first part of the above equation describes theisotropic pressure, while the second part describes theviscous stress. For Newtonian fluids, the viscous shearstress is proportional to the shear strain rate denotedby ε̇ through the dynamic viscosity μ. The particleapproximation of the shear strain rate can be given

ε̇xyi =

N∑j=1

m jm j

ρ jvy

ij

∂Wij

∂xi+

N∑j=1

m jm j

ρ jvx

ij∂Wij

∂yi

−2

3

⎛⎝ N∑

j=1

m j

ρ jvij∇ j∂Wij

⎞⎠ δij (6)

where vij = vi − vi and ∇ j∂Wij is the total derivative ofthe kernel.

2.1.4 Density calculation

There are several approaches to calculate the particleapproximation of density [42]. In our simulations, weuse the continuity density approach, which approxi-mates the density according to the continuity equationusing the concepts of SPH approximations [53].

Dρi

Dt=

N∑j=1

m jvij∇iWij (7)

The advantage of the continuity density approxima-tion is that it conserves the mass exactly since the inte-gration of the density over the entire problem domainis exactly the total mass of all the particles.

2.1.5 Equation of state

The fluid in the SPH formalism is treated as weaklycompressible. This facilitates the use of an equation of

state to determine fluid pressure. Several equations canbe used. In our simulations, we used the equation ofMonaghan [54], where the relationship between pres-sure 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 referencedensity of the fluid.

2.1.6 Time integration

Any appropriate integration method can be used sincethe equations of motion are ordinary differential equa-tions. In our code, we used a predictor–correctormethod for the velocity, displacement, and density cal-culations. Assuming that f(y, t) is the time derivativeof y, the predictor–corrector scheme first predicts theevolution of the quantity y at half time steps (n+1/2)

yn+1/2 = yn + f (yn, tn)�t2

. (9)

Then these values are corrected at the end of thetime step.

yn+1 = yn + f (yn+1/2, tn+1/2)�t. (10)

2.1.7 Boundary conditions

On the boundaries of the model, virtual boundaryparticles are used. They have the same properties asthe ordinary (real) SPH particles; the only differenceis that the virtual particles are not moving. The sup-port domain of the kernel function is truncated by theboundary however these virtual particles fill the kernel.The virtual particles are also used to exert a repulsiveboundary force to prevent the interior particles frompenetrating the boundary.

2.2 Discrete element method

In the coupled model presented in this work, we areusing simple 2-D implementation of the DEM whichis based on the initial “lattice solid model” approach

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described in [19, 57]. For more realistic models of brittledeformation, more complicated implementations of theparticle interactions would be preferable. In particular,bonded particle interactions which support shear andbending stiffness such as those proposed by [70] and[92] would result in a more realistic fracture behaviorof the models. However, to demonstrate the feasibilityand use of the coupled DEM–SPH model, the simpleimplementation used here is sufficient and can easilybe extended in the future.

2.2.1 Particle interactions

The DEM particles in the coupled model are inter-acting with each other in two different ways. Particleswhich are connected by a “bonded interaction” are sub-ject to a repulsive–attractive interaction force, whereasparticles which are touching, but not connected by abond, are interacting by a repulsive–elastic force. Be-sides, all touching particles are interacting by a “dash-pot” damping force. The brittle–elastic bonds betweenthe particles do break if a defined breaking strain isexceeded. Similar to the DEM implementations pre-sented in [19] and [57], the force due to the bondedinteractions is calculated as

Fbondedij =

{kij(rij − (r0)ij)eij rij ≤ (rbreak)ij

0 rij > (rbreak)ij ,(11)

where kij is the bond stiffness, rij is the distance betweenthe particles i, and j, (rbreak)ij the breaking distancefor the bond and e is a unit vector in the direction ofthe interaction. The equilibrium distance between theparticles is the sum of the particle radii, i.e., r0 = ri + r j.The individual bond stiffness kij for each bond can becalculated from a global stiffness parameter k and theradii of the particles connected by the bond as kij =k(ri + r j)/2 where ri and r j are the radius of particlesi and j, respectively. The breaking distance for a bondbetween two particles is calculated from a breakingstrain εb and the equilibrium distance between theparticles as (rbreak)ij = (1 + εb )(r0)ij = (1 + εb )(ri + r j).The force due to a non-bonded elastic interaction be-tween two particles is calculated very similarly as

Felasticij =

{kij(rij − (r0)ij)eij rij ≤ ri + r j

0 rij > ri + r j .(12)

Damping is also implemented in the code in orderto prevent the buildup of kinetic energy. The damping

force due to a dashpot-type interaction between parti-cles touching is given by

Fdampingij =

{−Adij

�vij

rijrij ≤ ri + r j

0 rij > ri + r j ,(13)

where A is the damping coefficient, dij is the distancebetween the particles, �vij is the velocity differencebetween the particles, and rij is the average radius ofthe particles.

2.2.2 Particle movement

The calculation of the particle movement is performedaccording to Newton’s second law, i.e.,

mi∂2x∂t2

= Fi (14)

where mi is the mass of particle i, x is the position ofthe particle, and Fi is the sum of all forces acting on theparticle due to interactions with other particles. In theimplementation used here for the demonstration of thecoupled DEM–SPH model, a simple first-order schemefor the time integration of the particle movement isused, i.e.,

ai = Fi

mi(15)

vt+1i = vt

i + ai�t (16)

xt+1i = xt

i + vt+1i �t (17)

where ai is the acceleration of the particle i, vti and

vt+1i are the particle velocities at time steps t and t + 1,

respectively, and �t is the size of the time step. xti

and xt+1i are the particle positions at times t and t + 1.

Higher-order schemes such as the one implemented forthe movement of the SPH particles in Section 2.1.6 orthe Velocity-Verlet Scheme [13] employed by Donzeet al. [19] and Mora [57] could be used but tests haveshown that this is not necessary here.

2.2.3 Boundary conditions

The model boundaries consist of infinite rigid planarwalls. They are specified by a point and a normal vector.The wall can move in two ways: they can be eitherdisplacement-controlled or force-controlled. The actualposition of a displacement control wall is calculated

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based on a given applied velocity. While the positionof a force-controlled wall is calculated based on theresultant force acting on the wall, which is the sumof the forces caused by particles and a user-definedconfining force.

The particles interact with the wall via elastic inter-action. The elastic force acting on particle i touching thewall is given by

Fwalli = kwdinw (18)

where kw is the elastic stiffness of the wall, di is thedistance of particle i and the wall, and nw is the normalvector of the wall. The same force is acting on the wallas well.

2.3 Coupling

In order to couple the viscous (SPH) and the brittle–elastic (DEM) behavior, we implemented two differentkinds of coupling. This coupling is the key part of ourmodels.

The first, “SPH type” coupling is implemented byusing DEM particles as virtual SPH particles, whichinteract with SPH particles according to the SPH dis-cretization of the Navier–Stokes equation and withDEM particles according to the DEM nearest-neighborinteractions. The forces due to both types of interac-tions are combined, and the hybrid particles are movedaccording to Newton’s law. On the SPH side of themodel, this approach is similar to the “virtual particle”approach used to implement solid boundary conditionsin SPH models.

The second, “DEM type” coupling is implementedby using SPH particles as DEM particles, which interactwith nearest-neighbor DEM particles based on DEMforce calculation, and with SPH particles accordingto the SPH discretization of the Navier–Stokes equa-tion. Between DEM and SPH particles, elastic anddashpot interactions are identical to those described inSection 2.2.1 (Eqs. 11–13) for interaction within theDEM model.

Both type of coupling have specific advantages anddisadvantages. The fist type of coupling, i.e., usingthe DEM particles in contact with the SPH particlesas virtual SPH particles, has the advantages that theSPH approximation of pressure field is maintainedacross the DEM–SPH boundary. In the second cou-pling approach, however, the boundary between SPHand DEM particles is better described as a surface ofthe SPH model, whose position is determined by theforce balance between the SPH particles. The disadvan-tage of this is that it makes the introduction of an addi-tional parameter necessary which describes the elastic

Fig. 2 Simplified flow diagram for one time step of the coupledSPH–DEM model. Some details such as calculations of forcesdue 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 makesit possible to use arbitrary particle sizes for the DEM

For

ce c

ontr

olle

d D

EM

wal

l

Displacement controlled DEM wall

For

ce c

ontr

olle

d D

EM

wal

l

Displacement controlled DEM wall

DEMparticles

SPHparticles

Fig. 3 Initial geometry setup for the pure shear simulations.SPH particles are shown in blue, DEM particles are shown inyellow (particles connected to the top and bottom wall), andgreen (particles connected to the left and right wall)

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Fig. 4 Displacement field observed during a pure shear simula-tion using a differential stress σdiff = 120 Pa and a viscosity of theSPH particles of μ = 3,000 Pas. Particles are colored by horizontaldisplacement. Total displacement is shown by arrows attached toeach particle. The overlap between the boundary particles whichcan be seen near the corners of the model is there because theDEM particles assigned to different boundaries do not interact

part of the model, while maintaining a single particlesize for the SPH particles, which is easier to implementand has numerical advantages.

2.4 Implementation

While the SPH equations (Eqs. 5–8) are usually for-mulated in terms of a kernel function, they can alsobe expressed as sums of particle–particle interactionsmore similar to the approach generally taken in theimplementation of DEM models. For this purpose, thesummation is removed from Eq. 6 to calculate theindividual contributions ε

xyij to the strain rate tensor at

a given particle i due to its interaction with another

particle j. All those contributions are added incremen-tally to the total strain rate tensor ε

xyi for particle i.

Similarly, Eq. 7 is adapted to calculate the individualcontributions to the density change Dij

Dt due to eachinteracting particle pair.

After the contributions to the strain rate tensor andthe density change have been calculated for all inter-acting particle pairs, the acceleration due to the viscousforces acting on each SPH particle can be calculatedaccording to Eq. 5. These accelerations are then usedto update the positions of the SPH particles using thepredictor–corrector scheme described in Section 2.1.6.This approach has been implemented so that a list ofall interacting particle pairs is first generated by findingall pairs of particles which are within a defined rangeof the SPH interactions. The interaction calculationsdescribed above are then performed by iterating overthe list of interacting particle pairs.

In contrast, the calculation of particle accelerationsand the position updates operate on particles ratherthan interacting particle pairs and are therefore per-formed by iterating over a list of SPH particles. Sim-ilarly, in the DEM part of the model, the forces(Eqs. 11–13) acting on the particles are first calculatedby iterating over the list of all DEM interactions andthen the positions of the DEM are updated accordingto Eq. 14. The whole process, which is similar in struc-ture to the program flow implemented in some DEMsoftware (see, for example [1]) is visualized in Fig. 2.

While the interactions between particles in the SPHmodels are not restricted to nearest neighbors in thestrict sense, the interaction range is still restricted toa finite distance by the smoothing length of the ker-nel. This has two advantages. One is that the same

Fig. 5 The evolution of strainover time for a set of pureshear simulations with thesame viscosity (μ =1,000 Pas) but differentdifferential stresses rangingfrom σdiff = 60 Pa to σdiff =140 Pa

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 200 400 600 800 1000

Str

ain

Timesteps ( x 1000)

σ = 60 Pa σ = 80 Pa σ = 100 Pa σ = 120 Pa σ = 140 PaDifferential stress:

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Fig. 6 Differential stress vssteady-state strain rate shownfor all simulations with theviscosity between μ = 500 Pasand μ = 3,000 Pas anddifferential stresses rangingfrom σdiff = 60 Pa to σdiff =140 Pa. Models with the sameviscosity are plotted using thesame symbol and color. Thestraight lines are linear fits foreach group of models withthe same viscosity

y = 295.88x

R² = 0.9925

y = 190.62x

R² = 0.9889 y = 98.516x

R² = 0.9909

y = 50.202x

R² = 0.9799

0

50

100

150

200

250

300

0 0.5 1 1.5 2 2.5 3D

iffe

ren

tial s

tres

s[P

a]

Strain rate [1/s]

Viscosity: η=3000 Pas η=2000 Pas η=1000 Pas η=500 Pas

acceleration schemes for speeding up the determinationof all interacting particle pairs can be used as in DEM.The other advantage, which will become important forlarger scale implementations of a coupled DEM–SPHmodel is that, due to the finite interaction range, thecoupled model is also amenable to parallelisation usinga space partitioning approach which has been used forDEM models before [1, 8].

3 Validation test—pure shear

We performed pure shear tests in order to investigatethe viscous behavior of the SPH material and also totest the coupling we implemented in the model. Pureshear is an coaxial flow involving shortening in one di-rection and extension in a perpendicular direction suchthat the volume remains constant [21]. For Newtonianfluid, the applied differential stress is directly propor-tional to the shear strain rate through the dynamicviscosity

σdiff = με̇. (19)

The aim of this test is to verify that this linear rela-tionship is satisfied in our model.

Here, we used a rectangular block of SPH particles.The SPH particles are all of the same size and initiallypacked in a regular triangular lattice. On the bound-aries, there are double layers of DEM particles whichinteract with SPH particles via SPH interaction, andwith the walls via elastic DEM interactions (Fig. 3).In order to enable the boundary conditions needed forthis specific model, in particular to avoid problems inthe corners, the DEM particles assigned to differentboundaries do not interact with each other. Betweenthe SPH and DEM particles, the SPH type of coupling,i.e., DEM particles acting as virtual SPH particles, wasapplied. During the simulation, the horizontal wallsmove towards each other at constant velocity, exceptfor an initial phase of the movement where the velocityis ramped up linearly.

We used force-controlled side walls in order to keepthe differential stress constant, i.e., the forces actingon the displacement-controlled horizontal walls arerecorded and used to calculate the forces which need tobe applied to the side walls. In order to ensure that thecorrect force is acting on the side walls, they are movedaccording to the stiffness of the DEM particle–wallinteractions and the chosen force. The strain (εxx + εyy)is calculated based on the positions of the walls, which

Fig. 7 Boudinagedamphibolite in marble(Naxos, Greece). For details,see [75]

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Fig. 8 Schematic drawing ofthe initial geometry setup forthe boudinage simulation.SPH particles are shown inblue, DEM particles areshown in gray

Less competent - SPH

Less competent-SPH

Competent -DEM

Displacement controlled wall

Displacement controlled wall

For

ce c

ontr

olle

d w

all

For

ce c

ontr

olle

d w

all

are also recorded during the simulation. The strainrate is calculated accordingly. Twenty-six tests havebeen run with various differential stress and viscosityvalues. The differential stress values varied from 40 to

300 Pa, while the viscosity values varied from 500 to3,000 Pas.

A typical example of a displacement field recordedduring one of the pure shear simulations can be seen

Fig. 9 Evolution ofboudinage in a model using aviscosity of μ = 1 Pas. εyy isthe amount of verticalshortening. The SPH particlesare shown in gray and theDEM particles are coloredaccording to their horizonaldisplacement (blue is −x/left,red is +x/right)

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in Fig. 4. The evolution of the strain recorded duringthe simulations (Fig. 5) shows that after an initial com-paction, the increase of strain is linear with time. Con-sequently, the strain rate is constant during the latersteady-state part of the simulation. We also observethat the length of the compression phase is dependenton the stress conditions applied to the model. Becausethe same displacement rate was applied to the topand bottom boundaries in all simulations, the mod-els with higher differential stress have a lower meanstress and therefore show more compaction. Using thesteady-state strain rates, we found a linear relation-ship between the calculated strain rate and the applieddifferential stress for each group of simulations usingthe same viscosity in the SPH part of the model. Theslope of the fitted line, which is the calculated viscosity,is proportional to the applied viscosity and independentof mean stress (Fig. 6) as expected.

4 Geological application—boudinage

One important example of a coupled brittle–ductiledeformation in a geological context is the processof boudinage. Boudinage occurs in mechanically lay-ered rocks, where the layers are disrupted by layer-parallel extension. A competent layer embedded be-tween weaker layers is separated by tensile or shearfracturing into rectangular fragments (torn, domino,gash, or shearband boudins), or by necking or taper-ing into drawn boudins or pinch-and-swell structures[27, 71]. One example of the drawn tapering boudinsobserved in Naxos, Greece [75] is shown in Fig 7). Inthis case, a boudinaged amphibolite layer is in a marblematrix.

The full evolution of the boudins is not completelyunderstood yet. Existing models focus on either thebrittle–elastic processes of the fracture initiation phase

µ = 10 Pas

µ = 5 Pas

µ = 2 Pas

µ = 1 Pas

µ = 0.5 Pas

Fig. 10 Comparison of the boudinage structures which havedeveloped in simulations with different viscosities of the SPHmaterial. The model is rotated by 90◦ with respect to Figs. 8and 9. The viscosity is increasing from left μ = 0.5 Pas to rightμ = 10 Pas. SPH particles are shown in gray, whereas the DEM

particles in the competent layer are colored according to theirlayer-parallel displacement (red, +x/up; blue, −x/down). Thesnapshots of the different models are all taken at a the samedeformation state with strains εxx = 0.65 and εyy = 0.43

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[35, 79] or ductile processes of the post-fracture phase[46, 49, 65, 87].

Although Abe et al. [9] simulated full boudinagewhich described the process from the brittle failurethrough the separation of boudins to post-fracturemovements (e.g., rotation) of the fragments, the quasi-viscous matrix material used in this model does notallow full quantitative analysis of the influence of thematrix rheology.

There are several approaches to model fracturingin layered rocks. On the one hand, saturation modelsassume no slip interface coupling between the elasticmatrix and the brittle–elastic competent layer. On theother hand, models with interface slip provide betterapproximation, although these models are only valid ifthe ratio of tensile stress in the competent layer (T) tointerface shear strength (τ ) is high (T/τ > 3.0 ) [35, 79].

Models incorporating viscous coupling between theviscous matrix and the brittle–elastic competent layershow qualitatively correct behavior compared to obser-vations in nature [9]. Therefore, we used this approachin our boudinage model.

In our boudinage simulation, the initial setup con-sists of three layers (Fig. 8). The ductile top and bottomlayers are made up of SPH particles, while the compe-tent central layer is made up of DEM particles. Becausethe SPH particles all have the same radius, they areinitially placed in a regular arrangement. The DEMparticles are packed in a random arrangement becauseof their heterogeneous radius distribution. Within thecompetent layer, the DEM particles are connected bybrittle–elastic bonds which can break if a failure crite-rion is exceeded. The SPH and DEM particles interactwith each other via the DEM type of coupling, i.e., incase of an interaction between SPH and DEM parti-cles, the SPH particles act as DEM particles. On theboundaries of the model, there are four infinite walls,which interact with both types of particles via elasticDEM interactions. During the boudinage simulation,we applied vertical displacement, which resulted inhorizontal extension. The upper and the lower walls aremoved at a constant velocity, while left and right wallswere force-controlled to apply a constant confiningstress. Five simulations have been run with differentviscosity values, namely 0.5, 1, 2, 5, and 10 Pas. Whilethese viscosity values are much lower than what wouldbe expected in a geological context, they are withinthe range of those used in analog modeling (see alsoSection 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 asthey widen. The fragments of the competent layer move

apart, but some of the parts are still connected to eachother with boudin necks. During further extension, thenecks become thinner and separate boudins form.

As the viscosity of the incompetent layers increaseswhile the material properties of the competent layerremain the same, the shape of developing boudinschanges from long, well-separated boudins to shorterboudins connected by necks (Fig. 10). At even higherviscosity of the incompetent layers, i.e., decreasing me-chanical contrast between the layers, pinch-and-swellstructures develop. The number of boudin blocks alsoincreases with increasing viscosity of the incompetentlayers as would be expected from theoretical consider-ations (cf. [71]).

5 Industrial application—hydrofracturing

Hydraulic fracturing is the process of initiation andpropagation of a fracture due to hydraulic pressure ina cavity. During hydraulic fracturing, viscous fluid ispumped into a wellbore at high-pressure. This high-pressure fluid breaks the rock in tension and forces

Fixed DEM wall

For

ce c

ontr

olle

d D

EM

wal

l

Fixed DEM wall Fixed DEM wallF

orce

con

trol

led

DE

M w

all

Fixed SPH boundary particles

Moving SPH boundary particles

Fig. 11 Initial geometry setup for the simulation of hydraulicfracturing. DEM particles are shown gray, real SPH particlesin blue, moving boundary SPH particles in green, and fixedboundary SPH particles in yellow

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it to open. The fractures open in the direction of theminimal principal stress and propagate perpendicularto the minimal principal stress [36, 63].

The viscosity of pumped fluid plays an importantrole in the hydraulic fracturing processes. The viscosityis one parameter, which controls the widths of thegenerated fractures: fluid with higher viscosity initiateswider cracks, whereas fluid with lower viscosity initiatesnarrow cracks [66]. It can also influence the geometry ofthe fractures: fluid with high viscosity tends to generateplanar cracks with only a few branches, whereas low-viscosity fluid, such as water, tends to generate wavelikecracks with many secondary branches [33, 81].

Hydraulic fracturing was first developed by Clark in1949 [17], and, since then, it has become a widely usedand efficient technique in the petroleum industry toincrease the near-well permeability and to enhance oiland gas extraction from a reservoir [10, 30, 33, 36]. It isalso used in the geothermal energy industry to increasethe heat exchange surfaces and the permeability of thehot dry rock [74].

Hydrofracturing simulations have traditionally ap-plied either boundary element method [11, 14] or finiteelement method [64, 91, 97] approached. The problemwith these models is that they do not always show

good agreement with the recorded seismicity data [73].One reason for this is that the models assume tensilefracture growth but usually shear-type seismic eventsare recorded [32]. The other reason is that hydrofrac-ture growth is not symmetrical around the borehole.One option to fix this problem is the use of DEM. Theexisting DEM models [12, 81] use particles only forthe solid rock; the fluid is described using fluid flowalgorithms. Here, we present a new model where boththe solid rock and the fluid are represented by particles.

In our hydraulic fracturing simulation, the viscousfluid is represented by SPH particles, while the solidrock is represented by DEM particles. The initial geom-etry (Fig. 11) consists of a block of DEM particles with anotch on the top. This notch is filled with SPH particles.The notch is continuing upwards into a column (“thepump”), which is also filled with SPH particles in orderto drive fluid into the notch and the fracture. The DEMtype of coupling described in Section 2.3 is used here,i.e., SPH and DEM particles interact through DEMinteractions.

The boundaries of the DEM block are walls, the topand bottom walls are fixed, while the side walls areforce-controlled. The DEM particles interact with thewalls by elastic DEM interactions. The borders of the

Fig. 12 Evolution of a crackdue to fluid injection in asimulation of hydraulicfracturing. The brittle solid(DEM) particles are shown ingray, the fluid (SPH) particlesin blue. To visualize the SPHparticles in a compact form,we use the smoothing lengthh as radius for the particleglyphs. The viscosity used forthe SPH fluid is μ = 0.5 Pas.The value t at the bottomright corner of each frameshows 1,000 time steps, i.e., t= 100 shows the model attime step 100.000

1. 2. 3.

4. 5. 6.

7. 8. 9. t=215t=187

t=160t=151

t=179

t=148

t=0 t=100 t=1120% 46% 52%

69% 70% 74%

83% 87% 100%

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Fig. 13 Snapshots of thepropagating crack in thehydrofracture model at thesame time step but withdifferent fluid viscositiesused. Viscosity isμ = 10−3 Pas (top left frame),μ = 10−2 Pas (top rightframe), μ = 0.1 Pas (bottomleft frame), and μ = 1.0 Pas(bottom right frame)

µ=0.001 Pas µ=0.01 Pas

µ=0.1Pas µ=1 Pas

SPH column are virtual SPH particles. On the side, theboundary particles are fixed, while the particles of thetop border can move only vertically. The different kindsof boundary particle do not interact with each other.During the simulation, we moved the top SPH border

with a constant velocity. Twelve simulations that havebeen run with viscosity values varied between 10−3 Pas(similar to water) and 5 Pas.

The evolution of the induced crack is shown onFig. 12. First, the original notch is expanding in vertical

Fig. 14 Progress of thefracture propagation withtime in hydrofracture modelswith different viscosities.Initial crack tip positiondefined in the geometry setupis at y = 4.25 with the crackpropagating downwards

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150 200 250 300

Cra

ck d

epth

[m

m]

Timestep ( x 5000)

0.001 Pas

0.01 Pas

0.1 Pas

0.2 Pas

0.5 Pas

1 Pas

2 Pas

5 Pas

Viscosity:

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Fig. 15 Time of the fracturereaching the bottom of themodel depending on theviscosity of the SPH fluid

100

150

200

250

300

350

0.001 0.01 0.1 1 10

Bre

akin

g

tim

e st

ep

Viscosity [Pas]

direction. Due to this expansion, the surrounding solidblocks start to rotate, and, due to the resulting stresses,the solid material is starting to fracture. When the de-veloped crack is large enough, the fluid is flowing intoit and force the crack to expand. This results in furtherpropagation of the fracture. The position of the cracktip can be determined by observing the breaking of thebonds between the DEM particles. Analyzing that thelocations where the DEM bonds break is starting nearthe tip of the initial notch and moving away from it, wecan plot the progress of the fracture process (Figs. 13and 14).

The velocity of the crack propagation depends onthe viscosity of the fluid: the lower the viscosity is, thefaster the crack is propagating. This means that the rockbreaks faster if lower viscosity fluid is injected into it(Fig. 15). The reason is that with lower viscosities, therock splits into two parts at once; while with higher vis-cosities, the fracture develops in more stages (Fig. 14).It should be noted that the geometry of the developingfractures is influenced by the relatively small modelsize and the details of the boundary conditions. Thegeometry of the developed fracture is also affected bythe viscosity (Fig. 13). The aspect ratio of the fracturechanges with viscosity (Fig. 16): the lower the viscosity,the longer and wider cracks develop.

6 Discussion

The combination of the DEM and SPH represents anew approach to the problem of numerical modelingof coupled brittle–ductile deformation processes. Bothmethods are well established in their respective do-mains. An advantage of coupling DEM and SPH is thatthe two methods are sufficiently similar in their numer-ical approach, so that a combined implementation canbe made seamless. In particular, because both methodsare Lagrangian and particle-based, there is no complexinterpolation needed to transfer data between the twofluid (DEM) and solid (SPH) part of the model. Thiscontrasts to other studies coupling particle-based andmesh-based methods [4, 20].

The results of the validation test (Section 3) demon-strate that the correct viscous behavior of the SPHmodel is maintained when the deformation is drivenby moving DEM particles coupled to the boundaries ofthe SPH volume. The model also validates the correcttransfer of stress and strain between the SPH and theDEM part of the model. At the interface betweenthe top and bottom DEM “wall” (Fig. 3), which isdisplacement-controlled, the deformation is transferredfrom the moving DEM particles to the SPH particles.At the SPH–DEM interface along the side walls, the

Fig. 16 Aspect ratios of thecracks developed at time step125.000 in hydrofracturesimulations with differentviscosities

0

5

10

15

20

25

0.001 0.01 0.1 1 10

Asp

ect

rati

o o

f th

e cr

ack

Viscosity [Pas]

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movement of the SPH particles, driven by the viscousdeformation of the SPH volume, forces the movementof the DEM wall particles in response to the forcebalance at the interface.

The boudinage simulation (Section 4) shows theapplicability of the coupled simulation approach to aproblem in structural geology. The test models pre-sented here show a qualitatively correct behavior,which, if the lower resolution and the simplified frac-ture mechanics is taken into account, is not too dis-similar from other DEM simulations of boudinage [9].A more realistic fracture behavior of the brittle layercould be obtained by implementing a more complexbond model between the DEM, i.e., “parallel bonds” asdescribed by [70] or the rotational bond model by [92].

The advantage of a coupled SPH–DEM simulationof boudinage compared to the pure DEM models is thata linear viscous material law can be used for the ductilematrix material in the SPH model, whereas in the pureDEM model, viscosity could only be approximated [9].In addition, it is possible to extend the SPH model tonon-linear viscous materials, for example, to includepower law viscosity which would be important for thesimulation of deformation processes involving rocksdeforming by dislocation creep [38, 39, 89].

The viscosity of the matrix material used in theboudinage model is much lower than that is expectedin a realistic geologic setting (1016 vs. 10 Pas). This dis-crepancy is due to the fact that the large range of timescales involved in a boudinage process with the correctgeologic material properties would make the modelnumerically intractable. The time scale of ductile defor-mation would be in the range of thousands or even mil-lions of years (≈ 1010 . . . 1014s), whereas the time scaleof the fracture processes would be in the sub-secondto millisecond range, requiring microsecond time stepsto resolve the crack propagation. This problem canbe alleviated by applying techniques such as “densityscaling” [86] to move the time scales of these processescloser together. The material parameters used here inour numerical models are much closer to those typicalfor analog models, i.e., paraffine (≈ 106 Pas [61]) orsilicone putty (≈ 105 Pas [28]). Another material, whichhas been used in analog modeling is honey with aviscosity 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 inthe numerical models making our model comparable tothese physical models.

The hydrofracturing models (Section 5) demonstratethe use of the coupled model for applications wherethe brittle deformation of the solid material is directlydriven by fluid pressure. The details of the fracture

propagation in this model depend on the material het-erogeneity of the solid and the distribution of pres-sure applied on the solid material by the fluid. Thefracture propagation generates new pathways for thefluid flow, and the details of how the fluid movesinto these pathways are resolved. This depends on theviscous properties of the fluid and will determine theredistribution of the fluid pressure and thus the furtherpropagation of the fractures. For a productive appli-cation of hydrofracture modeling, more details aboutthe material properties need to be included into themodel; a detailed calibration of the fracture behaviorof the DEM part of the model in particular would benecessary. A further possible extension of these modelwould be the inclusions of solid grains made up of DEMparticles into the SPH fluid, similar to the sand grainsused as proppant hydrofracturing shale gas reservoirs.

Both applications of the coupled DEM–SPH ap-proach presented in this work have shown that it cancapture the combined deformation of elastic–brittleand viscous materials, even for large local strains. Anapparent limitation which can be observed, for exam-ple, in Figs. 9 and 12, is that narrow cracks which appearin the brittle solid (DEM) material are not immedi-ately filled by the viscous (SPH) material, leading tothe appearance of voids even in models with stronglycompressive mean stress, such as the boudinage exam-ple. This is largely a model resolution issue. The SPHparticles cannot move into the voids as long as entranceto these voids are smaller than the particle size, i.e.,the minimum aperture of a crack into which the fluidcan flow is limited by the model resolution. While thismight be a limitation for specific applications requiringfluid flow through cracks with very high aspect ratios, itdoes not impact the applicability of the method to manyother problems.

7 Conclusions

We present a new Lagrangian method for the sim-ulation of brittle–ductile deformation using mesh-less particle-based methods. This method couplessmoothed particle hydrodynamics and discrete elementmethod.

We implemented a variation of SPH, which allowsthe coupling with DEM in order to describe geologicaldeformations.

In a plane strain model, the linear relationship be-tween the applied differential stress and the strain rateis validated to show that so we are able to describe realviscous behavior qualitatively.

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We have shown two examples of possible applica-tions of this method, namely boudinage and hydraulicfracturing. Boudinage simulations demonstrated that asthe viscosity of the incompetent layer increases, thenumber of boudins increases, and, with even higher vis-cosity, pinch-and-swell structures develop. Hydrofrac-turing simulations have shown that the viscosity ofthe injected fluid affects the evolution of the inducedcrack: the lower the viscosity is, the faster the crackpropagates and the wider the crack is.

This approach can be adopted to model the defor-mation of carbonate layers embedded in a viscous saltbody during tectonics. In the future, we are planning toparallelize the code and extend it to 3-D code as well.

Acknowledgements This study was carried out within theframework of DGMK (German Society for Petroleum and CoalScience and Technology) research project 718 “Mineral VeinDynamics Modelling”, which is funded by the companiesExxonMobil Production Deutschland GmbH, GDF SUEZ E&PDeutschland GmbH, RWE Dea AG, and Wintershall Hold-ing GmbH, within the basic research program of the WEGWirtschaftsverband Erdöl- und Erdgasgewinnung e.V. We thankthe companies for their financial support and their permission topublish these results. We thank RWTH Aachen University forenabling the performance of the main computations at the HighPerformance Computing Cluster at RWTH Aachen University.We also thank an anonymous reviewer for the helpful comments.

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