SPE-179138-MS

Mesh-Free Numerical Simulation of Pressure-Driven Fractures in BrittleRocks

T. Douillet-Grellier, R. Pramanik, K. Pan, A. Albaiz, and B. D. Jones, MIT; H. Pourpak, Total E&P; J. R. Williams,MIT

Copyright 2016, Society of Petroleum Engineers

This paper was prepared for presentation at the SPE Hydraulic Fracturing Technology Conference held in The Woodlands, Texas, USA, 9–11 February 2016.

This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contentsof the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflectany position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the writtenconsent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations maynot be copied. The abstract must contain conspicuous acknowledgment of SPE copyright.

Abstract

A better understanding of failure in heterogeneous rock materials can benefit a wide range of areas, fromearthquake engineering to petroleum engineering. Study of such failure is of particular interest in the fieldof hydraulic fracturing. The prediction of this breakage phenomenon is a big challenge for the scientificcommunity. Traditional continuum modeling techniques have the advantage of using classical nonlinearmaterial models, however they often fail to accurately capture the complexity of the fractured geometryand path of multiple intersecting fractures. In particular, mesh dependence of the fracture path, 3Drepresentation of natural fractures and their intersections, closing of an opened fracture, or shear infractures, are difficult to accurately capture using these techniques.

The use of the smoothed particle hydrodynamics (SPH) method for simulation of fracture in solids isrelatively recent, where mesh free methods like SPH have the potential to overcome the previouslymentioned limitations of mesh based methods. Simulation of the initiation and propagation of pressure-driven fractures in brittle rocks is presented in this study. By exploiting techniques commonly used intraditional continuum methods, we have implemented an elasto-plastic SPH model, which is based on theDrucker-Prager yield criterion, and the Grady-Kipp damage model. Results show that SPH is able tocorrectly predict the evolution of fracture in brittle rocks.

The SPH method has been applied to the solution of crack propagation in a variety of test cases,including a pressurized borehole, 2D line crack, and 3D penny shaped crack. The influence of initialin-situ stresses was also accounted for. Comparison of SPH results for these cases to analytical solutionsshows that SPH may be applied to accurately simulate the evolution of fluid-driven fractures in brittlerocks. Such model is a vital tool in correctly predicting fracture propagation in highly heterogeneousformations, for instance, shale formations.

IntroductionHydraulic fracturing is a process typically applied to both conventional and non-conventional reservoirs.In recent years, the hydraulic fracturing of unconventional reservoirs became a topic of interest for the oiland gas industry as these reservoirs, which are characterized by a very low permeability and sometimesexhibit a pre-existing fracture network, could contain a large amount of hydrocarbon reserves. An example

for such reservoirs are shale gas formations where hydraulic fracturing is applied to increase the overallformation permeability, and thus production. Hydraulic fracturing is a very complex process that can bemore simply defined as the initiation and the propagation of cracks due to the injection of fluid at highpressure into the formation. It involves several interacting phenomena: (1) fluid-driven fracture propa-gation, (2) mechanical deformation of the rock, and (3) fluid advection along the fracture and diffusioninto the formation.

In the 2000’s, exploration and drilling operations in shale gas and shale oil reservoirs have stronglyincreased all over the world, especially in the USA. It is very clear that the economic viability of theseactivities is related to the efficiency of the fracturing process. It can be observed from field measurementsthat the complexity of the fracture network induced by hydraulic fracturing can be very tortuous due tolocal heterogeneities, in-situ stresses and pre-existing joints in the formation. Consequently, numericalmodeling of this phenomenon is very challenging (Economides and Nolte, 2000). Moreover, due to theinaccurate modeling capabilities in simulating complex fractures, stimulation of unconventional reservoirsrelies on estimations from microseismic observations and a trial-and-error approach. This is a majorlimitation, the oil industry should be able to use an engineering approach to optimize the fractureparameters and production according to reservoir properties.

Indeed, hydraulic fracturing simulation can provide useful information such as fracture dimensions,proppant distribution, fracture surface areas, etc. This information can be used to improve production andcannot be obtained only by microseismic measurements (Cipolla et al., 2011b). Nevertheless, thesimulation of the initiation and propagation of a hydraulic fracture network in a formation presents lotsof difficulties (Weng, 2015). The amount of research on this topic has increased over the past yearsshowing good progress, but certain aspects are not fully understood yet or are limited by computationalpower.

The first study on hydraulic fracture initiation and propagation was published by Haimson andFairhurst (1967) to determine in-situ stresses in earth’s crust. However there is prior evidence of hydraulicfracturing to increase hydrocarbon production (Clark, 1949). Despite the use of hydraulic fracturing as aprincipal stimulation technique for years (Howard and Fast, 1970), oil and gas companies are still dealingwith numerous problems like premature screen-outs or low proppant concentrations (Cleary et al., 1993;Hossain et al., 2000).

Computational simulations are heavily used to investigate the process of rock degradation and failureinvolved in hydraulic fracturing. Several numerical techniques have been used to address this question,such as (1) the lattice-based model (Blair and Cook 1998); (2) fast Lagrangian analysis of continua(FLAC) (Fang and Harrison 2002); (3) particle flow code (PFC) analysis (Potyondy and Cundall 2004);and (4) digital image approaches based on FLAC (Chen et al. 2004) and the FEM (Ortiz et al. 1987) forlocalized failure. Of course, the classical Finite-Element Method (FEM) (Williams, 1987, Mustoe et al.,1987, Tang et al., 2000) and Extended Finite Element Method (XFEM) (Taleghani, 2009) have beenwidely applied to analyze fracture propagation in rock materials.

These methods can successfully predict the failure process in one aspect or another; however, they alsohave limitations. For example, it is not an easy task to evaluate the parameters related with fracture(fracture toughness, crack density) that are needed to correctly calibrate the model. Besides, grid-basedmethods do not typically work well for simulation of large deformations, fractures, and fragmentation,especially if discontinuities occur in the failure process. The most widely used strategy to handle a movingdiscontinuity is to remesh the domain in each step such that the discontinuity is accurately represented inthe simulation mesh and exists along the boundaries of mesh elements. Of course, this introducesnumerical difficulties (loss of accuracy, complex implementation, computational time). Another commonmethod used to solve these problems is the Discrete Element Method (DEM) (Hocking et al., 1987, Nasehiand Mortazavi, 2013). However, this method is restricted to only small scales and requires calibration toestimate the parameters for solid mechanics (Wang and Tonon 2010).

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Meshless methods have gained popularity in the past decades and now offer the advantage of being ableto simulate large deformations, fracture propagation, and fragmentation that may occur during thehydraulic fracturing process (Ouchi et al. 2015 and others). Among the mesh-free methods, SmoothedParticle Hydrodynamics (SPH) has shown promising results in simulating the large deformation behaviorof solids and fluids. SPH was first developed for astrophysical dynamics (Gingold and Monaghan 1977;Benz et al. 1989). It is relatively easy to formulate SPH schemes from a system of equations describinga continuum, and thus the method has been applied successfully to a vast range of problems. Theseinclude, but are not limited to, elastic flow (Monaghan 1992), fluid flow (Monaghan 1994; Morris et al.1997), heat transfer problems (Cleary and Monaghan 1999), multiphase flow (Monaghan and Kocharyan1995; Tartakovsky et al., 2015), geophysical flow (Cleary and Prakash 2004), fluid-structure interaction(Pan et al. 2015) and post-failure of cohesive and non-cohesive soils (Bui et al. 2008). Benz and Asphaug(1995) first successfully extended SPH to the simulation of the fracture process in brittle solids,subsequently, Randles and Libersky (1996) and then Gray et al., (2001) made significant improvementsin extending SPH to elastic dynamics. Recently, Cleary and Das (2008, 2010), and Deb and Pramanik(2013) demonstrated the abilities of SPH in modeling the brittle fracturing process of rocks under impactand compressive loading. SPH is a gridless Lagrangian particle method that replaces the continuumequations of motion with particle equations. Because of the Lagrangian nature of particles, cracks initiateimmediately after the yielding of particles. Cracks that occur in between the yielded particles interact witheach other to form a dominant crack without any special treatment.

In this paper, SPH has been used to study fracture initiation and propagation in the in-situ stress field.A softening elastoplastic constitutive model has been applied in the framework of SPH to simulate theyielding response of individual particles. The Drucker-Prager yield criterion is incorporated in theconstitutive model for plastic deformation and a non-associated flow rule is used to maintain inelasticdilatancy (Bui et al. 2008). For detection of tensile failure of a SPH particle, the Grady-Kipp damagemodel is used (Grady et al., 1980; Melosh et al., 1992). These models are coupled together so that shearand tensile failures can be treated during the same simulation. We first give a brief overview of thegoverning equations and constitutive model. We then introduce SPH equations for solids. Finally wedemonstrate the validity of our model, by comparing to analytical solutions.

Problem FormulationConsider a two-dimensional hydraulic fracture,�fs, driven by fluid pressure propagating in a homogeneousisotropic impermeable medium, �s, under the plane strain assumption (Figure 1). The boundary of thedomain consists of �t on which prescribed traction, , is imposed, �u on which velocity, , is applied,and fracture surface, �fs, which is subjected to fluid pressure, pf. In this problem, it is assumed that thefracture space is completely filled by the injected fluid. The objective of the problem is to determine stressconcentration inside the solid domain, deformation, fracture opening and its propagation.

Figure 1—Schematic of a two dimensional domain containing a hydraulic fracture

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Governing Equations:The stress field inside the domain, �, is related to the in-situ stress condition and the fluid pressure, pf.The governing equations for modelling rock media consist of conservation of mass and momentum in aLagrangian system as follows

(1)

(2)

(3)

(4)

(5)

Generally, density in a homogeneous rock specimen does not differ much from the original densityduring elastic deformation, therefore the density can remain constant without using the mass conservationequation. But when stain localization appears due to yielding, density changes significantly as a result oflocalized dilatancy. To account for the change in density, the continuity equation has been used in thegoverning equations.

Constitutive ModelUnder the assumption of small strain, the rate of strain tensor can be expressed as a kinematic relationshipwith velocity field such that

(6)

For an elasto-plastic material, the rate of strain tensor can be decomposed into an elastic componentand plastic component as

(7)

The material behavior in �s \�fs is governed by a linear elastic law, and the incremental form of theconstitutive model is given by

(8)

where D is the fourth order standard isotropic elastic tensor; G and K are, respectively the shear andbulk moduli; and are the deviatoric and volumetric parts of the elastic rate of strain tensor.

Non-linearities in rock behavior arise from two micro-mechanical changes, damage accumulation dueto the development of microstructure, and plastic flow. Plastic flow results in permanent deformation andis the consequence of a dislocation process along the development of the slip line, which is controlled bythe presence of local shear stress. Since both micro-cracking and plastic flow are present in the non-linearresponse of rock material, a suitable constitutive model should address equally the two physically distinctmodes of irreversible changes and should satisfy the basic postulates of mechanics.

For the plastic flow rule the Drucker-Prager plasticity model, F(�, q), with an isotropic softening lawis used with a flow potential, Q(�, q). Then the plastic strain increment, , is

(9)

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together with the loading-unloading conditions for each instance t �[t0, T]

(10)

where q is the component of internal variables associated with the phenomenon of softening, is theplastic consistency parameter.

In order to account for tensile failure, a generalized Grady and Kipp (1980) damage model for higherdimensions is incorporated to determine the damage variable, D�[0,1], when the material is in tension(Melosh et al. 1992). The damage growth satisfies the condition (D � 0), and is governed by the loadingsurface, in which is the effective tensile strain and denotes the threshold of damage.

Uniqueness of the function with regards to the stress state is ensured by choosing FD as a function ofstrain. To respect the loading condition, evaluation of damage is defined as

(11)

where f(�) is a positive function of strain.

Drucker-Prager model:The Drucker-Prager yield criterion with zero tensile cutoff is used in this study for modelling shearbehavior and strain localization. The surface of the Drucker-Prager cone is an approximation of theMohr-Coulomb criterion and a modification of the von Mises yield criterion (Figure 2). The yield criterionand the plastic potential functions are respectively expressed as

(12)

where and p(�) are, the second invariant and mean of the stress tensor respectively. c is the

cohesion of the material, the parameters � and � at the outer edges are given by

(13)

where � is the friction angle and can be found by replacing � with the dilatancy angle, �, in theexpression for �.

Using the above failure surface, the plastic strain takes the form

Figure 2—The � plane section of Drucker-Pager yield surface

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(14)

In the present study, associative isotropic strain softening is included by letting the cohesion, c, whichcontributes to the yield function, be a linear function of the accumulated plastic strain

(15)

Considering the non-associative law, the accumulated plastic strain is then defined by

(16)

where the scalar parameter k is associated with isotropic hardening.The consistency condition for the Drucker-Prager surface is generally nonlinear, however at time,

t��t, it reduces to a linear function of �� (Deb and Pramanik, 2013)

(17)

where s and p are trial deviatoric and hydrostatic stresses respectively.

Damage model:Tensile failure occurs principally due to stress induced micro-cracks. The growth of these micro-cracksrenders portions of the rock volume unable to carry loads, which is then reflected in the decrease of thestiffness of the material. To update the stress field, a rational approach similar to that of Melosh et al.(1992) is adopted here to treat only the tensile stress components, the compressive part is kept unaltered.

Total stress tensor, �, is decomposed into tensile,��, and compressive, ��, parts as

(18)

where the fourth-order projection tensors, P� and P�, are expressed as Faria et al. (1998)

(19)

where I is the fourth-order identity tensor; H(�i) denotes the Heavyside function computed for the ith

eigenvalue, �i, of �; the second order symmetric tensor, Pij, is defined as

(20)

where ni is the ith normalized eigenvector corresponding to �i.Now, the activation of microstructural flaws (defects) should be described in the damage model. It

could be based on a Griffith criterion or a dynamic fracture criterion due to rapid loading. However, earlierstudies on the brittle fracture of rock material have shown that Weibull statistics provide a satisfactorydescription of the inherent flaws leading to tensile failure (Jaeger et al., 2009; Grady and Kipp, 1980). Thisconcept is pursued and the number of flaws activated for a given tensile strain can be described by the twoparameter Weibull distribution,

(21)

where n is the number of flaws which can activate at or below a tensile strain level �; k and m are thematerial properties that characterize fracture activation.

Assuming a constant crack growth velocity, cg, damage, D, can be evaluated by the following integralequation (Grady and Kipp, 1980)

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(22)

where the current damage is governed by a micro-structural law,

(23)

which determines the growth of cracks activated at a past time, �.The constitutive description governing the damage growth is given by an approximate differential form

of the damage model as

(24)

where

(25)

The crack growth speed, cg, can also be regarded as an additional tensile fracture property whichgoverns the rate of damage growth during dynamic failure. A common value is cg 0.4cs where cs isthe speed of sound in the material (Melosh et al. 1992).

The state of damage in the materials is then characterized by means of a damage threshold with thefollowing form

(26)

where is the effective tensile strain; is the threshold value; V is the

volume of the SPH particle. Thus, if the effective tensile strain associated with maximum principal stressis less than the critical threshold value, no damage occurs in the rock particles. On the other hand, if theeffective tensile strain exceeds the threshold value, damage has occurred in the rock particle and isevaluated from equation (24). After evaluating damage, the tensile part in equation (18) is scaled downby the factor (1�D) and added with compressive part to have the modified stress tensor for tensiledamage.

SPH Formulation of the governing equationsIn SPH, the state of particles is represented by a set of points with fixed mass, which possess materialproperties and interact with the all neighboring particles by a weighting function typically referred to asthe smoothing kernel, W (Gingold & Monaghan, 1977). This function is required to be continuous,differentiable, and must satisfy the normalization, delta function, and compactness properties. Eachparticle has a support domain, �a, @a � �s specified by a smoothing length, ha. The value of a functionat a typical particle is obtained by interpolating values of that function at all particles in its supportdomain, weighted by the smoothing kernel. Gradients that appear in the flow equation are obtained viaanalytic differentiation of the smoothing kernel.

The transformation of the set of governing equation Eq. (1) – (2) into the particle approximation yieldsthe following set of SPH equations

(27)

(28)

SPE-179138-MS 7

where is the artificial stress term in which is the initial particle spacing.

Gray et al. (2001) suggested that n4, �0.3 produces best result for elastic solids.

Numerical results and discussionTwo numerical examples have been simulated to show the efficiency of the current procedure bycomparing SPH results to analytical solutions. The first example deals with an internally pressurizedcircular hole in a rock medium under in-situ stress conditions, the second considers an internallypressurized penny shaped crack. The goal of these two simulations is to show the promising results of SPHfor the modeling of failure in brittle rocks and its application to hydraulic fracturing.

Material properties, model parameters and modeling strategiesThe material used in these simulations is representative of reservoir sandstone, the properties of which aredetailed in Table 1 and are as reported by Nasehi and Mortazavi (2013). In order to determine theseproperties, Nasehi and Mortazavi averaged the properties of a high-strength sandstone reservoir.

The Poisson’s ratio, v, Young’s modulus, E, and the speed of sound, cs, are all computed using thestandard elasticity relations. For the hole in a plate problem and the 2D pressurized crack, the Weibull’sparameters are k5.53 1029 and m7.5 whereas for the 3D penny shaped crack, the values are k1.17981029 and m7.5.

Pressurized circular holeThe injection of fracturing fluid into a borehole is one of the main processes during hydraulic fracturing.The fracture initiation and propagation are strongly related to the stress state around the borehole and theorientation of the in-situ stress field in the reservoir. In order to use a simplified model to represent thisphenomenon, the ‘hole in a plate’ problem has been considered. The analysis is focused on a 2D horizontalplane strain section perpendicular to the well axis. The radius of the hole is 0.1 m while the dimensionsof the plate are 5 m 5 m.

For these models the vertical effective stress, �yy, and the horizontal effective stress, �xx, are variedbetween the following values: 5 MPa, 10 MPa and 20 MPa. The reservoir is therefore constrained undera confining stress state depending on the horizontal compressive stress and the vertical compressive stress.Moreover, a constant radial pressure P is applied in the borehole (Figure 3). This model consists ofapproximately 90,000 particles.

Table 1—Material properties used in this study

Property Value

Bulk modulus (K) 13.9 GPa

Shear modulus (G) 9.1 GPa

Density (0) 2500 kg/m3

Cohesion (c0) 24.8 MPa

Friction angle (�) 32.5°

Tensile Strength (�t) 8.3 MPa

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Validation of the establishment of the in situ stressesBefore studying the crack initiation and propagation, it is necessary to validate the establishment of acorrect equilibrium stress state following the application of various combinations of borehole pressure andconfining stresses. In order to achieve this, only the pure elastic deformation of the material is considered,which means that the Drucker-Prager and the Grady-Kipp models are both disabled. Results are comparedagainst the analytical solution for this case.

For a cylindrical hole in an infinite isotropic elastic medium under plain strain conditions, the solutionfor stress distribution is given by Howard and Fast (1970) and is presented in the following equations.

(29)

(30)

where a is the radius of the hole, �H max(�xx,�yy) and �h min(�xx, �yy). The concentration of stressfield along the X and Y line (Figure 3) is considered for comparison where

Comparison of the SPH method against the analytical solution shows good agreement. In cases whereborehole pressure is relatively low (P 50 MPa) the average observed error is less than 2%. As boreholepressure is increased to 200 MPa, the average error increases to approximately 8%. Despite this increasein average error, the overall trend of the SPH results fits well with that of the analytical solution, as shownin Figure 4. The largest errors occur near the borehole, which is explained by the fact that thediscretization of the borehole does not exactly represent a circle, which leads to unrealistic stressconcentrations in some areas. It is expected that these errors would decrease with increasing modelresolution.

Figure 3—Schematic of the circular opening problem

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Crack propagationNext, the results obtained for 2 sets of in-situ stress states and a borehole pressure of 200 MPa arepresented to show the influence of in-situ stresses on the fracture path. This pressure value does notrepresent the reality of the field practice, but was considered in order to simulate the highly dynamicfracture propagation in a small period of time. In this case, the Drucker-Prager (with non-associative flowrule and trilinear cohesion softening) and the Grady-Kipp models were enabled. Before applying thepressure in the borehole, the simulation is conducted for a certain period of time to allow the stress stateto reach equilibrium state, typically 5000 time steps.

Pressure is continuously applied in both the bore hole, and the newly formed fracture surfaces as thefractures propagate. It is considered that when a particle is fully damaged and/or the aperture is largeenough, the water can flow inside the fracture and the hydraulic pressure is applied on the fracturesurfaces.

As one can expect, the fracture is initiated around the wellbore after an initially plastic deformation(Figure 5a and Figure 6a). Because the applied pressure is high, fractures are initiated first in everydirection around the borehole. Then, the direction perpendicular to the minimum in situ stress value ispreferred and the fractures start to grow specifically in this direction (Figure 7a and Figure 7b).

Figure 4—Comparison of the stress distributions against analytical solutions

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Figure 5—2D Circular opening with �xx � 10 MPa; �yy � 20 MPa; P � 200 MPa

Figure 6—2D Circular opening with �xx � 20 MPa; �yy � 10 MPa; P � 200 MPa

SPE-179138-MS 11

It can be seen that the propagation of the crack is not axial symmetric, because the packing of particlesin these models is hexagonal and thus not necessarily symmetric. Moreover, as the borehole is alsodiscretized with hexagonal packing, its circular shape cannot be exactly represented. It is clear that thediscretization affects the propagation of damage and the appearance of bifurcations, but one needs toremember that damage does not necessarily mean crack opening. According to the aperture of cracks, ithas been observed that only one crack is actually opening (Figure 7c and 7d).

A high value has been used for the applied pressure in the borehole in order to accelerate the simulationand to obtain more visually instructive results. This does not represent the values that are used in the fieldfor hydraulic fracturing, where much lower net pressures are typically applied. This model is thereforeintended as a demonstration of model capabilities.

Penny shaped crackThe penny shaped crack problem is a well-known test case in the field of fracture mechanics. It allowsvalidation of fracture initiation, propagation, and also stress concentrations at the tip of the crack.

In the 2D case, the initial crack is 0.4 m long and is located at the center of a 5 m 5 m plate (Figure8). For 3D, the initial crack is a disk with a 0.4 m diameter located at the center of a 5 m 5 m 5 mcube. Material properties in both cases are as in Table 1.

Figure 7—Comparison of the two fracture profiles

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To model the initial crack, two layers of fully damaged particles are specified along the initial fracturesurface, the pressure, P, is initially applied only here (Figure 8 and Figure 9). The whole model iscomposed of approximately 85000 particles in 2D and assumes plane strain conditions.

2D ValidationThe analytical solution for the penny shaped crack in 2D (which corresponds to the so called Line Crackproblem) with constant pressure is well known and given as

(31)

(32)

where P is the applied pressure in the crack, and c, the length of the initial crack. The analytical solutionis obtained based on a linear elastic model and it has to be noted that the shape of the displacement curveis elliptical and the stress tends to infinity near the tip of the crack (stress singularity).

Figure 8—Schematic of a 2D line crack problem

Figure 9—SPH model of a 2D line crack problem

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Shown in Figure 11, the results are in good agreement with analytical solutions especially when itcomes to the stress concentration at the crack tip. Average error is less than 2% for stresses where fordisplacement it is less than 4%. Naturally, numerical models cannot allow infinite stresses at the crack tip,as in the analytical solutions, the general variation of stresses is however accurately captured, even as thecrack tip is approached.

Concerning the opening of the fracture, an elliptical aperture shape is obtained except at the crack tips.Instead, the shape of the aperture directly at the tip is pointed. Though this is incorrect with respect to theanalytical solution, it is likely that the elliptical assumption is not valid directly at the crack tip, and servesonly to simplify the derivation of a solution. Further, the analytical solution should be treated withskepticism as it is based on LEFM (Linear Elastic Fracture Mechanics) and implies singularities thatwould not exist in reality.

Fracture propagation 2DIn order to study the propagation of the fracture, this model is again simulated with material non-linearity.Figure 12 shows the variation of vertical stress beyond the crack tip in the medium as the fracturepropagates.

Figure 10—SPH 2D line crack problem

Figure 11—Comparison with the analytical solution of the 2D elastic line crack problem P�15 MPa (top: vertical stress, bottom: verticaldisplacement)

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Though the analytical solutions in equations (31) and (32) allow for variation of fracture radius, it isnot appropriate to compare these results to predictions from these equations. This is because the analyticalsolution assumes a pseudo-static state, and does not consider the dynamics of fracture propagation.

3D CaseThe same problem is then studied in 3D. The entire model is composed of approximately 265,000 particlesand modeled using the same techniques as in 2D. The cube is 5 m 5 m 5 m and the crack is circularwith a radius of 0.2 m and located at the center. The Y axis is the vertical axis in this case.

In 3D, the same analytical solution could not be used because it assumes plane strain. Therefore Figure13 and Figure 14 show a curve fit of the vertical stress and the vertical displacement in 3D along X andZ. The form of this curve fit is equivalent to the expressions in equations (31) and (32), with additionalcoefficients included. The vertical stress distribution is in agreement with what one could expect: itincreases exponentially as it approaches the tip of the crack and it tends towards zero as it reaches thelimits of the domain. A similar conclusion can be made for the vertical displacement; it shows an ellipticalshape in both directions, which means that the global crack has the shape of a penny as expected.Nevertheless, as in 2D, it can be observed that the tips of the crack are not as smooth as an ellipsoid, butsharp instead. As in 2D, the axial symmetry is not exactly achieved, due to the fact that the hexagonalpacking is not perfectly axial symmetric. The distribution of vertical stress as the fracture propagates isdepicted in Figure 15 at different time steps. The stress concentration at the tip can be observed in Figure16 as the crack propagates gradually.

Figure 12—Vertical stress distribution as fracture propagate in the SPH 2D line crack problem (exaggerated displacements)

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Figure 13—Curve fit 3D elastic penny-shaped crack problem along X P�15 MPa (top: vertical stress, bottom: vertical displacement)

Figure 14—Curve fit 3D elastic penny-shaped crack problem along Z P�15 MPa(top: vertical stress, bottom: vertical displacement)

Figure 15—Vertical stress distribution during fracture propagation in the SPH 3D penny shape crack problem (exaggerateddisplacements)

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ConclusionsThree numerical problems (hole in a plate in 2D, line crack problem in 2D and penny shaped crack in 3D)have been investigated to demonstrate the ability of the proposed SPH framework to simulate hydraulicfracturing problems. An elasto-plastic damage model has been employed in this paper to model materialnon-linearity, fracture initiation, and propagation, within the SPH framework.

The results predicted from SPH have been compared to analytical solutions and found to be in goodagreement. It shows that the proposed SPH method is a promising numerical tool to predict crack initiationand propagation. The main advantage of this method is its meshless property, leading to adaptability, andallowing crack growth and bifurcation in any direction without special treatment. Moreover, SPH isalready confirmed to produce good results in studying the behavior of fluids and solids which makes ita good choice to study hydraulic fracturing, where strong interactions between fluids, solids and fractureoccur.

The next step would be to compare the results against experimental results and to couple fluids andsolids models within the SPH framework to be able to simulate the actual phenomena of the hydraulicfracturing process.

AcknowledgmentsThe authors would like to thank Total E&P for its financial and scientific support.

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Figure 16—Contour plot of the vertical stress distribution at the crack tip during fracture propagation in the SPH 3D penny shape crackproblem

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