Numerical Simulation of Fracking in Shale Rocks: Current State and Future Approaches
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Abstract
Extracting gas from shale rocks is one of the current engineering challenges but offers the prospect of cheap gas. Part of the development of an effective engineering solution for shale gas extraction in the future will be the availability of reliable and efficient methods of modelling the development of a fracture system, and the use of these models to guide operators in locating, drilling and pressurising wells. Numerous research papers have been dedicated to this problem, but the information is still incomplete, since a number of simplifications have been adopted such as the assumption of shale as an isotropic material. Recent works on shale characterisation have proved this assumption to be wrong. The anisotropy of shale depends significantly on the scale at which the problem is tackled (nano, micro or macroscale), suggesting that a multiscale model would be appropriate. Moreover, propagation of hydraulic fractures in such a complex medium can be difficult to model with current numerical discretisation methods. The crack propagation may not be unique, and crack branching can occur during the fracture extension. A number of natural fractures could exist in a shale deposit, so we are dealing with several cracks propagating at once over a considerable range of length scales. For all these reasons, the modelling of the fracking problem deserves considerable attention. The objective of this work is to present an overview of the hydraulic fracture of shale, introducing the most recent investigations concerning the anisotropy of shale rocks, then presenting some of the possible numerical methods that could be used to model the real fracking problem.
1 Introduction
Conventional shale reservoirs are formed when gas and/or oil have migrated from the shale source rock to more permeable sandstone and limestone formations. However, not all the gas/oil migrates from the source rock, some remaining trapped in the petroleum source rock. Such a reservoir has been named “unconventional” since it has to be fractured in order to extract the gas from inside. Hydraulic fracture, or “fracking”, has emerged as a alternative method of extracting gas and oil. Experience in the United States shows it has the potential to be economically attractive. Many concerns exist about this type of extracting operation, especially how far the fracture network will extend in shale reservoirs.

No two shale formations are alike. Shale formations vary spatially and vertically within a trend, even along the wellbore;

Shale “fabric” differences, combined with insitu stresses and geologic changes are often sufficient to require stimulation changes within a single well to obtain best recovery;

Understanding and predicting shale well performance requires identification of a critical data set that must be collected to enable optimization of the completion and stimulation design;

There are no optimum, onesizefitsall completion or stimulation designs for shale wells.
These points encapsulate well the uncertainties involved. Many models have been proposed over the years but they are either too simplified or they tend to focus on one key aspect of fracking (e.g. crack propagation schemes, influence of natural fractures, material heterogeneities, permeabilities). The scarcity of insitu data makes the study of fracking even more complicated.
The most usual concerns in fracking are addressed by Soeder et al. [252], where integrated assessment models are used to quantify the engineering risk to the environment from shale gas well development. Davies et al. [55] have investigated the integrity of the gas and oil wells, analysing the number of known failures of well integrity. The modelling of reservoirs is also a difficult task due to the lack of experimental data and oversimplification of the complex fracking problem [177].
Glorioso and Rattia [97] provide an approach more focused on the petrophysical evaluation of shale gas reservoirs. Some techniques are analysed, such as log responses in the presence of kerogen, log interpretation techniques and estimation methods for different volumes of gas insitu, among others. It is shown that volumetric analysis is imprecise for inplace estimation of shale gas; however, it is one of the few techniques available in the early stages of evaluation and development. The measurement of an accurate density of specimens is an important parameter in reducing the uncertainty inherent in petrophysical interpretations.
This paper provides an overview of the current state of fracking research. A stateoftheart review of fracking is performed, and several points are analysed such as the models employed so far, as well as the underlying numerical methods. Special attention is given to problems involving brittle materials and the dynamic crack propagation that must be taken into account in the fracking model. The hydraulic fracture modelling problem has been tackled in several different ways, and the shale rock has mostly been assumed to be isotropic. This simplification can have serious consequences during the modelling of the fracking process, since shale rocks can present high degrees of anisotropy.
This paper is organised as follows: a description of the shale rock including the most common simplifications is presented in Sect. 2, followed by the description of the fracking operation in Sect. 3. Section 4 presents a review of the analytical formulations for crack propagation and crack branching. Different types of models such as cohesive methods and multiscale approaches are tackled in Sects. 5 and 6. Numerical aspects are discussed in Sect. 7, including the boundary element method, the extended finite element method, the meshless method, the phasefield method, the configurational force method and the discrete element method. A recently proposed discretisation method is discussed in Sect. 8. The paper ends with conclusions and a discussion of possible future research directions in Sect. 9.
2 Description of the Shale Rock
Shale, or mudstone, is the most common sedimentary rock. It can be viewed as a heterogeneous, multimineralic natural composite consisting of sedimented clay mineral aggregates, organic matter and variable quantities of minerals such as quartz, calcite and feldspar. By definition, the majority of particles are less than 63 microns in diameter, i.e. they comprise silt and claygrade material. In the context of shale gas and oil, organic matter (kerogen) is of particular importance as it is responsible for the generation and, in part, the subsequent storage of oil and gas.
Mud is derived from continental weathering and is deposited as a chemically unstable mineral mixture with 70–80 % porosity at the sedimentwater interface. During burial to say 200 °C and 100 MPa vertical stress, it is transformed through a series of physical and chemical processes into shale. Porosity is lost as a result of both mechanical and chemical compaction to values of round 5 % [31, 32, 287]. At temperatures above 70 °C, clay mineral transformations, dominated by the conversion of smectite to illite (e.g. [121, 254]), lead to a fundamental reorganisation of the clay fabric, converting it from a relatively isotropic fabric to one in which the clay minerals are preferentially aligned normal to the principal (generally vertical) stress [56, 57, 120]. Although quantitative mechanical data are scarce for mineralogically wellcharacterised samples, it is likely that the clay mineral transformations strengthen shales [206, 264]. In muds which contain appreciable quantities of biogenic silica (opalA) and calcite, the conversion of opalA to quartz [134, 281], and dissolutionreprecipitation reactions involving calcite [259], will also strengthen the shale. Indeed, it is generally considered that finegrained sediments which are rich in quartz and calcite are more attractive unconventional oil and gas targets compared to clayrich media, as a result of their differing mechanical properties (e.g. [204]).
Shales with more than ca. 2 % organic matter act as sources and reservoirs for hydrocarbons. Between 100 and 200 °C kerogen is converted to hydrogenrich liquid and gaseous petroleum, leaving behind a carbonrich residue (e.g. [126, 144, 207]). The kerogen structure changes from more aliphatic to more aromatic, and its density increases [194]. Changes in the mechanical properties of kerogen with increasing maturity are not well documented. However, they may be quite variable, depending on the nature of the organic matter. For example, Eliyahu et al. [68] performed PeakForce QNM® tests with an atomic force microscope to make nanoscale measurements of the Young’s modulus of organic matter in a single shale thin section. Results ranged from 025 GPa with a modal value of 15 GPa.
Shales are heterogeneous on multiple scales ranging from submillimetre to tens of metres (e.g. [10, 204]). Hydrodynamic processes associated with deposition often result in a characteristic, ca. millimetrescale lamination [35, 157, 241], which can be disturbed close to the sedimentwater interface by bioturbation [63]. On a larger, metrescale, parasequences form within mudrich sediments, driven by orbitallyforced changes in climate, sealevel and sediment supply [35, 156, 157, 204]. Parasequence boundaries are typically defined by rapid changes in the mineralogy and grain size of mudstones, with more subtle variations within the parasequence. Stacked parasequences add further complexity to the shale succession and result in a potentially complex mechanical stratigraphy which depends on the initial mineralogy of the chosen unit and the way that burial diagenesis has altered physical properties on a local scale.
During the shale formation process bedding planes are formed, which may present sharp or gradational boundaries. This is the most regular type of deposition that occurs in shales. Deposition may not be uniform during the whole process, presenting discontinuities at some points or other type of deposition patterns. This makes the mechanical characterisation of shale a complex issue. Moreover, not all shale rocks are the same, so a prediction made for an specific shale rock probably is not valid elsewhere.
The works of Ulm and coworkers about nanoindentation in shale rocks [34, 198, 199, 200, 268, 269, 270] have been important developments in our ability to characterise the mechanical properties of shale rocks. From [268], it is seen that shales behave mechanically as a nanogranular material, whose behaviour is governed by contact forces from particletoparticle contact points, rather than by the material elasticity in the crystalline structure of the clay minerals. This assumption is valid for scales around 100 nm.
 1.
Shale building block (level I  nanoscale): composed of a solid phase and a saturated pore space, which form the porous clay composite. A homogeneous building block, which consists in the smallest representative unit of the shale material, is assumed at this scale. The material properties are composed of two constants for the isotropic clay solid phase, the porosity and the pore aspect ratio of the building block.
 2.
Porous laminate (level II  microscale): the anisotropy increases due to the particular spatial distribution of shale building blocks (considering different types of shale rocks). The morphology is uniform allowing the definition a Representative Volume Element (RVE).
 3.
Porous matrixinclusion composite (level III  macroscale): shale is composed of a textured porous matrix and (mainly) quartz inclusions of approximately spherical shape that are randomly distributed throughout the anisotropic porous matrix. The material properties are separated into six indentation moduli plus the porosity.
Nanoindentation results provide strong evidence that the nanomechanical elementary building block of shales is transversely isotropic in stiffness, and isotropic and frictionless in strength [34]. The contact forces between the spherelike particles activate the intrinsically anisotropic elastic properties within the clay particles and the cohesive bonds between the clay particles.
The determination of the mechanical microstructure and invariant material properties are of great importance for the development of predictive microporomechanical models of the stiffness and strength properties of shale.
3 The Hydraulic Fracturing Process and Its Modelling
 1.
The mechanical deformation induced by the fluid pressure on the fracture surfaces;
 2.
The flow of fluid within the fracture;
 3.
The fracture propagation
The shale measures in question are usually found at a distance of 1 to 3 km from the surface. A major concern relating to fracking is that the fracture network may extend vertically, allowing hydrocarbons and/or proppant fluid to penetrate into other rock formations, eventually reaching water reservoirs and aquifers that are found typically approximately 300 m below the surface.
Fracking can occur naturally, such as in magmadriven dykes for example. In the 1940s, when fracking started commercially in US, the hydraulic fracture was applied through a vertical drilling. In that case, the pressurised liquid was applied perpendicular to the bedding planes. It was known that the shale was a layered material due to its formation process, but technology of that time was very limited.
The horizontal drilling was not new to the industry, but it was fundamental for the success of shale gas developments. From 1981 to 1996, only 300 vertical wells were drilled in the Barnett shale of the Fort Worth basin, north central Texas. In 2002, horizontal drilling has been implemented, and by 2005 over 2000 horizontal wells had been drilled [40]. The Barnett shale formation found in Texas produces over 6 % of all gas in continental United States [273]. The application of this new drilling technique has turned the United States from a nation of waning gas production to a growing one [221].
To optimise the fracking process of shale, it is important to detect accurately the location of natural fractures. The anisotropic behaviour of the shale generates preferential paths through the shale fabric [136, 279]. Moreover, the alignment of the natural fractures can also induce anisotropic patterns of the fluid flow [86, 87].
3.1 Modelling of the Shale Fracture
Much of the work done so far in attempting to model shale fracture is very simple, taking into account only the influence of the crack and not the fluid. Only recently have a few researchers [3, 4, 62, 160, 161, 188, 205, 296] successfully developed more sophisticated methods including the fluidcrack interaction.
In the early stages following initial pressurisation, the volume of injected fluid is sufficiently small such that the the porous formation do not absorb the incoming fluid. As injection process continues, the fluid is accommodated locally in the pore space and consequently predicts leakoff. Once the system reaches steadystate, it again becomes independent of porosity system. This analytical formulation have issues when predicting the behaviour during the transient state [161].
Even though these models can represent complex processes occurring during fracking, they are still far from being accurate, mainly because shale is considered to be isotropic, which has been seen not to be true [268], and since the material presents nanoporosity, it is difficult to accurately model the mechanical properties of shale.

how to appropriately adjust current (linear elastic) simulators to enable modelling of the propagation of hydraulic fractures in weakly consolidated and unconsolidated “soft” sandstones;

laboratory and field observations demonstrate that mode III fracture growth does occur, and this needs to be further researched.
Some works have analysed the crack propagation path in shales, including refracturing sealed wells. For example, Gale and coworkers [87] found that propagation of the hydraulic fracture over a natural fracture will cause delamination of the cement wall and the shale. The fluid enters the fracture and causes further opening of the fracture in a direction normal to the propagating hydraulic fracture while the pressure inside the fracture increases. After the fracture propagation at the natural fracture reaches a sealed fracture tip, the hydraulic fracture resumes growth parallel to the direction of maximum shear stress.
In an analytical work, Vallejo [271] has investigated the hydraulic fracture on earth dam soils, where shear stresses were seen to promote crack propagation on traction free cracks. Other analytical study about refracturing was carried out by [295], where the dynamic fracture propagation is characterised in lowpermeability reservoirs. The results are comparable to an experimental test with the same material parameters.
In summary, research works in hydraulic fracture formulation have considered a large number of variables and processes which occur during the actual operation: leakoff, shale permeabilities, crack opening and fluid interaction over a crack surface. However, the current analytical theories for hydraulic fracture do not include crack propagation conditions, especially dynamic crack propagation, neither crack branching, since material instabilities at the crack tip during crack propagation may cause the propagation path not to be unique. These concerns are summarised in the next section.
4 Crack Propagation and Crack Branching
Consider a homogeneous isotropic body under a known applied loading. The resulting elastic stress distribution over the body due to the applied force is generally smooth. However, introducing a discontinuity such as a crack imposes a singular behaviour to the stress distribution. It can be shown that the stress increases as it is measured closer to the crack tip, varying with \(1/\sqrt{r}\), where r is the distance from the crack tip. Irwin [125] proposed that the asymptotic stress field at the crack tip is governed by parameters depending on the geometry of the crack and the applied load. These parameters are known as Stress Intensity Factors (SIFs) and have been widely used as criteria for crack stability and propagation. The three SIFs, \(K_I, K_{II}, K_{III}\), each correspond to one of three modes of crack behaviour: mode I (opening), mode II (sliding) and mode III (tearing). In this paper we will confine ourselves mostly to mode I.
It can be postulated that crack growth will begin if the value of the SIFs increase to a certain value. If the SIF is higher than a critical fracture toughness parameter \(K_c\), which depends on the material properties, then the crack will propagate through the body. The situation becomes more complicated when the load is applied rapidly so that dynamic effects become important. This does not imply that the value of the dynamic fracture toughness will be independent of the rate of loading or that dynamic effects do not influence the fracture resistance in other ways [82].
The theoretical limiting speed of a tensile crack must be the Rayleigh wave speed. This was anticipated by Stroh [257] on the basis of a very intuitive argument [82].
Gao et al. [89] studied crack propagation in an anisotropic material, and presented expressions for the dynamic stresses and displacements around the crack tip. These predict that larger crack propagation velocities induce higher stress and displacement fields at the crack tip. The limiting speed in crack propagation is analysed in [88], where a local wave speed resulting from the elastic response near the crack tip also changes with the crack propagation velocity. A molecular dynamic model is used in this work, so crack propagation is modelled as bond breakage between the particles. The crack velocity is expressed using the Stroh formalism.
 1.Maximum tangential stress: This criteria was defined by Erdogan and Sih [69] and is based in two hypothesis:The crack propagates when the SIF is higher than a critical SIF \(K_c\), which depends on the materials properties. From [69], the crack propagation angle \(\theta _p\) can be obtained from the following relation
 a.
The crack extension starts at its tip in radial direction;
 b.
The crack extension starts in the plane perpendicular to the direction of greatest tension.
where \(K_I\) and \(K_{II}\) are the mode I and mode II SIF, respectively, and \(\theta _p\) is taken with respect to the horizontal axis. This crack propagation criteria was extended to anisotropic materials in [239].$$K_I \sin {\theta _p} + K_{II}(3\cos {\theta _p}1)=0$$(14)  a.
 2.Strain energy release rate: In this criteria, the crack propagates when energy release rate G reaches some critical value \(G_c\), taking the direction where G is maximum [123]. The energy release rate is defined aswhere W represents the strain energy and a is the halfcrack length. Equation (15) can be expressed in terms of mixed mode SIFs for an isotropic material as$$G = \frac{\partial W}{\partial a}$$(15)where \(\mu\) is the shear modulus.$$G = \frac{1\upsilon ^2}{E}\left( K_I^2 + K_{II}^2 \right) + \frac{1}{2\mu }K_{III}^2$$(16)
 3.Minimum strain energy: crack propagation occurs at the minimum value of the strain density S defined as [169, 247]where \(a_{ij}\) come from the material properties. The direction of propagation goes toward the region where S assumes a minimum value \(S_{min}\). The crack extension \(r_0\) is proportional to the minimum strain energy, such that the ratio \(\frac{S_{min}}{r_0}\) is constant along the crack front [169].$$S = a_{11}K_I^2 + 2a_{12}K_I K_{II}+a_{22}K_{II}^2+a_{33}K_{III}^2$$(17)
One can observe that all these criteria are related to the SIFs. These criteria are well consolidated in the fracture mechanics literature over the years. However they fail in one aspect, since they do not consider the possibility of crack branching, i.e., at some point of the crack propagation process, the crack may bifurcate in two or more new cracks. This issue is especially important when modelling highly heterogeneous materials such as the shale rock.
Yoffe [289] attempted to explain the branching of cracks from an analysis of the problem of a crack of constant length that translates with a constant velocity in an unbounded medium. From this solution she found that the maximum stress acted normal to lines that make an angle of \(60^\circ\) with the direction of crack propagation when the crack velocity exceeded 60 % of the shear wave speed. This fact might cause the crack to branch whenever the crack velocity exceeds that value. However, Yoffe did not consider that the maximum stress would be perpendicular to the crack path, so this assumption is not valid for brittle materials. Moreover, the \(60^\circ\) angles are quite large in comparison with the branching angles observed from experiments [223].
 1.
crack branching is the result of many interacting microcracks or microbranches;
 2.
only a few of the microbranches grow larger while the rest are arrested;
 3.
the branches evolve from the microcracks which are initially parallel to the main crack, but deviate smoothly from the original crack orientation;
 4.
the microbranches do not span the thickness of the plate, some occurring on the faces of the plate while others are entirely embedded in the interior of the plate.
Sih [247] made the hypothesis that the instability that occurs in crack bifurcation is associated with the fact that a high speed crack tends to change its direction of propagation when it encounters an obstacle in the material. The excess energy in the vicinity of such a change in direction is sufficient to initiate a new crack. This event occurs so quickly that the crack appears to have been split in two, or bifurcated.
From [223], one can see that the velocity with which the crack propagates is determined by the SIF at initiation. Cracks propagating at low speeds may undergo a change in the crack velocity if stress waves are present. Cracks propagating at high speeds do not change crack velocity, but may exhibit crack branching.
Crack branching formulations can be found in [78, 131, 247, 289], to cite just a few works. In all cases, only the isotropic material case is considered. For anisotropic crack branching, numerical methods have to be employed.
5 Cohesive Methods
The fracture process is usually considered only at the crack tip. In such cases, the fracture process zone is considered to be small compared to the size of the crack [17, 66, 67].
In Linear Elastic Fracture Mechanics (LEFM), the stress becomes infinite at the crack tip. Since no material can withstand such high stress, there will be a plastification/fracture zone around the crack tip.
 1.
The fracture process zone is lumped into the crack line and is characterised as a stressdisplacement law with softening;
 2.
Inelastic deformations in the process zone are smeared over a band of a given width, imagined to exist in front of the main crack.
Most of the work done in cohesive cracks makes use of the former approach, otherwise known as the Dugdale–Barrenblatt model, fictitious crack model or stress bridging model [178].
Dugdale [66] and Barenblatt [17] models are the basis of many cohesive models. The Dugdale cohesive crack model is very simplistic and is best used for ductile materials. A uniform traction equal to the yield stress is used to describe the softening in the fracture process zone.
Most of the cohesive models are developed for isotropic materials (see [67] for example). However, there are some models for heterogeneous materials [11, 216, 244] and composite [148, 181, 261, 272] materials. Nevertheless, the material models are quite simple, usually considering different types of isotropic materials instead of a full anisotropic model. To the authors’ best knowledge, there is no anisotropic cohesive crack model to this date.
Cohesive models have been also applied in multiscale problems, where cracks are significantly smaller than the RVE. In [210], a microelastic cohesive model is developed for quasibrittle materials. The stability of crack growth is analysed, and it is concluded that macroscopic strength is not necessarily correlated to crack propagation, and may be caused by unstable growth of cohesive zones ahead of nonpropagating cracks. The initial cohesive zone has a significant influence on the macrostrength of quasibrittle materials.
A number of different approaches for cohesive models have been proposed over the years. Enriched formulations for delamination problems were analysed by Samimi [236, 237, 238]. A stochastic approach for delamination in composite materials was proposed in [181], where the imperfections of the material were considered in the cohesive model. The cohesive crack has been extensively studied as can be seen in [59, 75, 102, 152, 209, 297] to cite just a few works. Crack propagation in cohesive models was recently discussed in [152, 298] for example.
A rudimentary model for hydraulic fracture for isotropic materials using the finite element software ABAQUS was considered in [288]. Papanastasiou [203] has evaluated the fracture toughness in hydraulic fracturing, modelling the rockfluid coupling through a finite element model with cohesive behaviour. The MohrCoulomb criterion is used to take plasticity into account in the rock deformation. The plastic behaviour that develops around the crack tip provides an effective shielding, resulting in an increase in the effective fracture toughness.
6 Multiscale
Another important part of multiscale modelling is the coupling of stresses and strains from the local and global models. Numerical homogenisation is a popular technique and is an alternative to the traditional analytical homogenisation. It is especially used for monophasic heterogeneous materials, where the balance and constitutive equations are considered at the RVE level. The first work in numerical homogenisation is due to Ghosh et al. [96].
Zeng et al. [292] proposed a multsicale cohesive model for geomaterials. At the macroscopic scale, a sample of polycrystalline material is considered as a continuum made of many material points. The estimation of the material properties at the microscale is performed by statistical homogenisation, since the RVE represents a number of different constituents or phases, as mineral grains and voids, and is therefore composed of randomly distributed constituents.
This formulation is referred to as the local QC method, a simplification of the continuum system when interface and surface energies may be neglected. The general nonlocal QC potential energy may lead to some nonphysical effects in the transition region. The derivatives of the energy functional to obtain forces on atoms and finite element nodes may lead to socalled ghost forces in the transition region between the macro and microscale, and it has several issues that remain to be resolved, such as the computation of approximations in the macroscale far from microscale defects [245] and the correct balance of energy which needs to be ensured between macro and microscales [174].
Since the connections between atoms are modelled through bonds, this multiscale cohesive formulation is able to capture the crack branching behaviour during crack propagation.
In [299], the RVE properties of a hydrogeologic reservoir are averaged through statistical parameters. The main reason is that the heterogeneity of the reservoir can be more easily modelled through the mean and standard deviation of the rock properties. The site scale represents the entire solution domain used for modelling global flow and transport. The layer scale represents geologic layering in the vertical direction. Within a layer, relatively uniform properties are present in both vertical and lateral direction, in comparison with the larger variations between different layers that may vary significantly in thickness. The local scale represents the variation of properties within a hydrogeologic layer.
In [164], a multiscale model for the shale porous network is proposed. Permeability is assumed as an intrinsic porous medium property independent of fluid properties (such as viscosity) or thermodynamic conditions. The porous medium was modelled as networks of pores connected by throats. This simplification neglects the physics of the real porous network. Permeability further depends on the relative size of the void spaces as well as the fraction of pores belonging to each length scale. Unlike absolute permeability in conventional reservoirs, gas permeability depends on absolute pressure values in individual pores (and not only the gradient). Specifically, smaller pressures result in (somewhat counterintuitively) an increase in permeability.
A number of multiscale models for brittle materials can be mentioned: [2, 83, 130, 209, 274] just to cite some of the most recent works.
7 Discrete Numerical Methods
In this section we will present a brief description of different elementbased numerical methods that can be used in the modelling of fracking problems. The boundary element method (BEM) has been used in brittle anisotropic problems including crack propagation. The extended finite element method (XFEM) has been developed recently and is also a good choice for fracture mechanics problems, and can be easily applied in cohesive models. Meshless methods are becoming popular in fracture mechanics problems. The discrete element method (DEM) is particularly used in problems with rock materials. The phasefield method and the configurational force method are also reviewed in this section.
7.1 Boundary Element Method (BEM)

Accurate mathematical representations of the underlying physics are employed, resulting in the ability of the BEM to provide highly accurate solutions;

The problem is defined only at the boundaries, which gives a reduction of dimensionality in the mesh (linear for 2D problems and surface for 3D problems), therefore resulting in a reduced set of linear equations to be solved;

In spite of the boundaryonly meshing, results at any internal point in the domain can be calculated once the boundary problem has been solved;

There is a great advantage in certain classes of problem that can be characterised by either (1) infinite (or semiinfinite) domains, or (2) discontinuous solution spaces. These advantages have resulted in the BEM gaining popularity for acoustic scattering, fracture mechanics, reentry corners and other stress intensity problems, where domain discretisation methods have poorer convergence.

The system of equations is both nonsymmetric and fully populated, which may lead to longer computing times (compared to FEM for example), especially in 3D problems. In this case, techniques such as the fast multipole method [228] have been introduced to accelerate the solution in largescale problems;

A Fundamental Solution (FS) or Green’s function, describing the behaviour of a point load in an infinite medium of the material properties is required as part of the kernel of the method. This can make the use of BEM infeasible in problems where a FS is not available;

calculation of the FS must be computationally efficient, which makes explicit FS formulations very desirable in this sense. Dynamic problems usually have implicit formulations, see [60, 227, 277] for instance, where the FS is expressed in a integral form by means of the Radon transform;

The BEM formulation requires the evaluation of weakly singular, strongly singular and sometimes hypersingular integrals which must be carefully treated. This can be done through a variety of methods, including singularity substraction, e.g. [100], or analytical regularisation, e.g. [91];

Nonlinear problems (e.g. material nonlinearities) are difficult to model;
In this latter equation, the free term has been set to unity due to the additional singularity arising from the coincidence of the two crack surfaces. The inconvenience of this approach is that the BEM formulation will now involve integrals including both strong singularities which require special treatment. Numerous hypersingular approaches have been developed, in particular to anisotropic materials under static [90, 91, 150, 282] and timeharmonic [6, 93, 94, 226, 232, 283, 294] loadings. The use of a hypersingular formulation does not limit at all the crack shape, being valid for curved and branched cracks, for example. However, it is commonplace to make use of discontinuous boundary elements to ensure that all collocation points lie on the smooth surface within the body of an element; this is required to satisfy the Hölder continuity requirement of the hypersingular BIE.
 1.
Quarterpoint: Developed by Henshell and Shaw [119] and Barsoum [19] for finite elements, it consists in moving the midside node of a quadratic boundary element from the centre to 1/4 of the element length from the crack tip. It was shown that the mapping between the element in real space and in the space of the intrinsic coordinates automatically captures the asymptotic displacement behaviour of \(1/\sqrt{r}\) present in the vicinity of the crack tip (refer to [231] for further explanations).
 2.Jintegral: Proposed by Rice [224], a path independent integral (assuming a noncurved crack) is used to evaluate the energy release rate due to the presence of the crack,where \(n_1\) is the component of the outward unit normal vector in the \(x_1\) direction, \(u_i\) are the displacement and \(t_i\) are the tractions. The term \(W = \frac{1}{2}\sigma _{ij}\varepsilon _{ij}\) is the strain energy density.$$J = \int _{\Gamma _j} \left( W n_1  t_i \frac{\partial u_i}{\partial x_1} \right) d\Gamma$$(33)
 3.Interaction integral: the Jintegral can be decomposed into 3 parts [110, 176]where \(J^{(1)}\) is the Jintegral of the socalled principal state, which represents the energy release rate of the material; \(J^{(2)}\) is the Jintegral of the auxiliary state, which depends on the displacements around the crack tip; \(M^{(1,2)}\) is the interaction integral containing terms of the principal and auxiliary state, and is defined as$$J = J^{(1)} + J^{(2)} + M^{(1,2)}$$(34)where A is the area inside the contour \(\Gamma _j\) surrounding the crack tip, and \(W^{(1,2)}\) is given as$$M^{(1,2)}=\int _{A}(\sigma _{ij}^{(1)}u_{i,1}^{(2)}+\sigma _{ij}^{(2)}u_{i,1}^{(1)}W^{(1,2)}\delta _{1j})q_{,j}\ dA$$(35)Let us remark that the indices (1) and (2) correspond to the principal and auxiliary states, respectively.$$W^{(1,2)}=\frac{1}{2}(\sigma _{ij}^{(1)}\varepsilon _{ij}^{(2)}+\sigma _{ij}^{(2)}\varepsilon _{ij}^{(1)})$$(36)
The quarterpoint approach allows a direct extrapolation of the SIF by using the crack opening displacement. The Jintegral is more cumbersome numerically since the displacements and tractions at the closed path integral are part of the BEM domain and have to be evaluated first; however it is more accurate than the direct extrapolation.
Chen [47] has analysed mixed mode SIFs of anisotropic cracks in rocks with a definition of the Jintegral for anisotropic materials and the relative displacements at the crack tip. In Ke et al. [133], the authors have suggested a methodology to obtain the fracture toughness of anisotropic rocks through experimental measurements of the elastic parameters and further comparison with a BEM code. In another work, Ke et al. [132] have proposed a crack propagation model for transversely isotropic rocks. Let us remark that all the previously mentioned works have used the Lekhnitskii formalism [145] in order to model the anisotropy of the material. The Lekhnitskii formalism is a polynomial analogy form of the matricial Stroh formalism.
Crack propagation problems have also been studied under the BEM framework. Portela et al. [214] used the maximum stress criterion as crack growth criteria in a dual BEM. Quasistatic 3D crack growth is analysed in [169].
Cohesive models have also been developed with the BEM: Oliveira and Leonel [195, 196] have proposed a cohesive crack growth model, where the zone ahead of the crack tip is modelled as a fictitious crack model. This formulation gives rise to a volume integral, which must be regularised. The cohesive stresses are dependent on the crack tip opening displacement.
Yang and RaviChandar [286] have proposed a cohesive model where the singledomain dual integral equations are used as an artifice to avoid the mathematical degeneration of the formulation imposed by the crack. In this case, the domain is divided in two subdomains, where the crack is in the fictional domain division. Moreover, the cohesive zone is modelled as an elastic spring connecting both crack faces. Normal and tangential crack tip opening displacements are considered, and the crack growth is obtained from successive iterations of the nonlinear system of equations, where the stiffness of the cohesive zone is taken into account.
Saleh and Aliabadi [233, 234, 235] and Aliabadi et al. [7] have studied the crack propagation problem in concrete using a fictitious crack tip zone. The cohesive zone is modelled with additional boundary elements at the fictitious crack tip that satisfy a softening cohesive law. A major drawback of this methodology is that the crack growth path has to be known a priori.
7.1.1 Fast Multipole Method (FMM)
 1.
Discretise the boundary \(\Gamma\);
 2.
Determine a tree structure of the elements. For example, in a 2D domain, define a square containing the entire boundary and call this square the cell of level 0. Then, divide the square into 4 equal cells and call them level 1. Repeat until each cell contains a predetermined number of elements (in Fig. 7, each cell has one element). Cells with no children cell are called leafs. For 3D cases, the same principle applies using cubic cells instead of square cells;
 3.
Compute the moments on all cells for all levels \(l \ge 2\) and trace the tree structure (shown in Fig. 8). The moment is the term from Eq. (39) that is independent from the collocation point. The moment of parent cells is calculated from the summation of the moments of its 4 children cells;
 4.
Compute the local expansion coefficients on all cells starting from level 2 and tracing the tree structure downward to all leaves. The local expansion of the cell C is the sum of the contributions from the cell in the interaction list of the cell and the far cells. The interaction list is composed by all the cells from the level l that do not share any common vertices with other cells at the same level, but their parent cells do share at least one common vertex at level \(l1\). Cells are said to be far cells of C if their parent cells are not adjacent to the parent cell of C;
 5.
Compute the integrals from element in leaf cell C and its adjacent cells as in standard BEM. The cells in the interaction list and the far cells are calculated using the local expansion;
 6.
Obtain the solution of \({\mathbf {Ax}}={\mathbf {b}}\). The iterative solver updates the unknown solution of \({\mathbf {x}}\) and goes to step 3 to evaluate the next matrix vector product \({\mathbf {Ax}}\) until the solution converges within a given tolerance.
The FMM has been used in 3D fracture mechanics problems as can be seen in [192, 290], and some recent works on GPU can be found in [101, 108, 278]. The FMM is largely detailed in [149].
7.1.2 Adaptive Cross Approximation (ACA)
The Adaptive Cross Approximation (ACA) approach uses a different technique in order to reduce the complexity of the BEM with respect to the storage and operations. ACA uses the concept of hierarchical matrices introduced by Hackbusch [107], where a geometrically motivated partitioning into subblocks takes place, and each subblock is classified as either admissible or inadmissible according to the separation of the node clusters within them.
In hierarchical matrices, the near and far fields have to be separated. The index sets I for row and J for columns so that elements far away will have indices with a large offset.
By means of a distance based hierarchical subdivision of I and J cluster trees \(T_I\) and \(T_J\) are created. In each step of this procedure, a new level of son clusters is inserted into the cluster trees. A son cluster is not further subdivided and is said to be a leaf if its size reaches a prescribed minimal size \(b_{min}\). Usually one of two different approaches is considered. First, a subdivision based on bounding boxes splits the domain into axisparallel boxes which contain the son clusters. Alternatively, a subdivision based on principal component analysis splits the domain into wellbalanced son clusters leading to a minimal cluster tree depth.
A block b is said to be admissible if it satisfies this admissibility criterion. Otherwise, the admissibility is recursively verified for each son cluster, until the block becomes admissible or reaches the minimum size.
 1.
Define \(k=0\) where \({\mathbf {S}}_0={\mathbf {0}}\) and \({\mathbf {R}}_0={\mathbf {A}}\) and the first scalar pivot to be found is \(\gamma _1 = (R_0)^{1}_{ij}\), and i, j are the row and column indices of the actual approximation step;
 2.For each step \(\upsilon\), obtain$$v_{\upsilon +1}= \gamma _{\upsilon +1} (R_\upsilon )_i$$(44)$$u_{\upsilon +1}= (R_\upsilon )_j$$(45)$${\mathbf {R}}_{\upsilon +1}= {\mathbf {R}}_\upsilon  {\mathbf {u}}_{\upsilon +1} {\mathbf {v}}_{\upsilon +1}^t$$(46)where the operators \(()_i\) and \(()_j\) indicate the ith row and the jth column vectors, respectively;$${\mathbf {S}}_{\upsilon +1}= {\mathbf {S}}_\upsilon + {\mathbf {u}}_{\upsilon +1} {\mathbf {v}}_{\upsilon +1}^t$$(47)
 3.
The next pivot \(\gamma _{\upsilon +1}\) is chosen to be the largest entry in modulus of the row \((R_\upsilon )_i\) or the column \((R_\upsilon )_j\)
 4.The approximation stops when the following criterion holds$$u_{\upsilon +1}_F\ v_{\upsilon +1}_F < \varepsilon\ {\mathbf {S}}_{\upsilon +1}_F$$(48)
The main advantage comparing to the FMM method is that ACA can be implemented directly into an existing BEM code. Moreover, due to its inherently parallel data structure, parallel programming can be readily implemented, increasing the computational efficiency. However, the original matrix \({\mathbf {A}}\) will not be entirely recovered.
Note that it is not necessary to build the whole matrix beforehand. The respective matrix entries can be computed on demand [20]. Working on the matrix entries has the advantage that the rank of the approximation can be chosen adaptively while kernel approximation requires an a priori choice.
A few recent works on ACA implementation can be found in [81, 99]. Use of the method for problems in 3D elasticity can be found in [28, 158] and the application of ACA in crack problems was shown for the first time in [137].
7.2 Enriched Formulations
7.2.1 eXtended Finite Element Method (XFEM)
The motivation that lay behind the development of XFEM was to eliminate some of the deficiencies of standard FEM for crack modelling, most importantly the requirement for highly refined meshing around the crack tips and the mandatory remeshing for crack growth problems. The partition of unity [15] is a general approach that allows the enrichment of finite element approximation spaces so that the FEM has better convergence properties. In XFEM, the partition of unity method allows element enrichment such that degrees of freedom (dofs) are added to represent discontinuous behaviour. In this framework, the mesh is independent from the discontinuities, so that cracks may now pass through elements rather than being constrained to propagate along elment edges. This gives the FEM much more flexibiility to model crack growth without remeshing.
Two types of enrichment function are applied in the XFEM: the Heaviside enrichment function, responsible for characterising the displacement discontinuity across the crack surfaces, and a set of crack tip enrichment functions (CTEFs), responsible for capturing the displacements asymptotically around the crack tip. This latter presents complex behaviour, varying for different constitutive laws (see [12, 79, 193], for some different CTEF). In this sense, it is similar to the FS, necessary in BEM formulations.
Since the CTEFs describe the displacements at the crack tip zone through the addition of several dofs, the stress concentration around the crack tip can be found more accurately with a significantly coarser mesh compared to the mesh used with standard FEM in a similar problem.
The presence of blending elements, which do not contain the crack but contain enriched nodes, is also important and has to be considered. These elements were analysed by Chessa et al. [48], and some studies have improved the convergence of blending elements (see [84], for instance). The XFEM convergence rate can also be increased through the use of geometrical enrichment [142], where a number of elements around the crack tip receive the CTEF instead of a single element (this latter named topological enrichment).
XFEM has been widely used with cohesive models in the last few years. Some authors [45, 51, 175] have used a typical XFEM formulation to model the cohesive crack, i.e., a Heaviside enrichment function is used to model the jump between the crack surfaces and a crack tip enrichment function is used to model the asymptotic behaviour at the crack tip.
The crack growth is modelled considering some rules, for example, if the level of stress at the crack tip is above the material tensile strength [178, 262].
In [139], a 2D cohesive model for an isotropic material was presented, where both fluid and porous material interact. The pressure inside a crack is also modelled. The Heaviside enrichment function is employed, as well as a pressure enrichment function, which allows the continuity of steep gradients without enforcing this condition. The crack propagation criteria depends on the stress state at the crack tip. The fluid behaviour can retard crack initiation and propagation. A local change of the flow can be seen immediately after crack propagation. The deformation around the crack causes fluid to flow mostly from the crack itself since the crack permeability is much higher than the medium permeability. This flow from the crack to the crack tip causes closing of the crack. However, a delamination test has shown that if the stiffness and permeability are high, the fluid does not influence crack growth.
More methods for crack propagation in XFEM can be found in [151, 167, 182, 183, 225] for brittle fracture and [168, 179, 291] for cohesive cracks.
7.2.2 Enriched BEM
Let us emphasise that the anisotropic enrichment functions can also be used for isotropic materials, since this is a degenerated case from anisotropic materials. For more details please refer to reference [110].
7.3 Meshless Method
7.3.1 Meshless Methods for Fracture
Ever since their initial development in the 1990s meshless methods have been applied to crack modelling [22, 24, 197], to dynamic fracture [26] and crack propagation [25]. The key advantage of meshless methods over standard FE methods for fracture is removal of the need to remesh during crack propagation. Another positive feature of meshless methods is that smooth stress results can be obtained for high stress gradients around crack tips [37] thus requiring less effort in postprocessing compared with the XFEM. As with all numerical methods applied to fracture we have to find ways of dealing with the stress singularities at the crack tips and the discontinuities introduced by the crack surfaces. The former can be dealt in meshless methods by enriching the approximation space just as is done in XFEM and other enriched methods, e.g. [27], based on the the partition of unity (PU) concept [16, 166] where the jump discontinuity is included in the displacement approximation exactly as already laid out for XFEM above in Eq. (49). “Extrinsic” techniques like this have more recently been developed into meshless “cracked particle” methods in a number of references [37, 219, 220, 303]. Extrinsic enrichment like this can however lead to an illconditioned global stiffness matrix [21] as is the case with many other PU methods, due to the additional unknowns at nodes which do not correspond to the physical degrees of freedom [44]. The cracked particle methods are examples of smeared approaches to modelling cracks, i.e. the exact crack face/surface geometry is approximated, but this clashes with the requirement for an accurate description of the crack geometry since it governs the accuracy of field solution, and hence the crack growth magnitude and direction. Extrinsic approaches which attempt to improve on this have used piecewise triangular facets [37, 64] which however suffer from discontinuous crack paths and requires user input to “repair” the mesh of facets.
The visibility criterion is simpler to implement, especially for 3D problems, but leads to spurious crack extension (thus impairing accuracy) while the diffraction method has no spurious crack extension problem but its implementation leads to high computational complexity especially in 3D or with multiple cracks.
7.4 PhaseField
The solution of Eq. (69) is found through the minimisation of \(E_\epsilon ({\mathbf {u}},\Gamma )\). To avoid the minimisation problem to be illposed, the small term \(k_\epsilon\) has been added to the formulation. For more details see [39].
The phasefield formulation has been modified through the years to be more general, consider more cases of interface interaction and different types of loading conditions to the problem. The work of Amor et al. [9] has considered the compression into the formulation, avoiding the interpenetration between crack surfaces. The proposed idea consisted in separating the elastic energy density according to the deviatoric and volumetric contributions.
A different phasefield formulation was proposed by [172, 173], defined as a “thermically consistent” formulation. The regularised phasefield variable d is defined as 0 for the unbroken state and 1 for the fully broken state.
A downside of the phasefield formulation is that it can result in unrealistic solutions. An example analysed by [8] consists of the case when the principal strains are negative, which is not considered in the model of [9] for instance. Nevertheless, a strongly nonlinear strain relation is used, which requires higher computational charges as compared to [9].
The advantage of this new form is that the irreversibility of the crack phasefield evolution is put into a more general form, allowing loading/unloading conditions, besides allowing a better numerical treatment of the phasefield.
Crack branching effects are studied with phasefield in [117] for a 2D fracture problem. The instabilities are seen to appear at the critical crack speed of \(0.48c_s\), where \(c_s\) is the shear wave speed. It is worth to note that this relation is valid for perfect brittle materials only. Moreover, it was observed that, as the crack speed increases, the curvature of the area around the crack tip increases, splitting into two cracks when a critical value for crack speed is attained. In [118], a 3D study of crack branching stability is performed by means of fractographic patterns. The authors conclude that the instability is either restricted to a portion of the crack front or a quasi2D branches.
A phasefield model is applied for damage evolution in composite materials in [29]. The evolution equation of the phasefield model was able to include difficult topological changes during damage evolution, such as void nucleation and crack branching and merging. Moreover, no meshing was required by the used phasefield model.
In [141], the formulation used in [39] is complemented by a GinzburgLandau type evolution equation, where an additional variable M is responsible for the crack propagation behaviour. If M is too small, the crack propagation may be delayed, while for sufficiently high values, the crack propagation is not affected by M. The FEM was coupled with the phasefield theory. This work was extended by [242] for dynamic brittle fracture.
Numerical aspects of the phasefield models used with finite differences, FEM and multipole expansion methods are discussed in [211].
More information about phasefield methods can be found in [38, 50, 217, 242, 253, 275].
7.5 Configurational Force Method
Numerical implementations of brittle fracture propagation are relatively rare in the computational mechanics literature. One of the most promising numerical techniques developed within a conventional finiteelement framework over the last decade is based on configurational forces. Within this setting, the most recent application of the configurational force methodology to the modelling of fracture is the work of Kaczmarczyk et al. [129], which focuses on large, hyperelastic, isotropic threedimensional problems.
Kaczmarczyk et al.’s paper [129] is largely based on the work of Miehe and coworkers [103, 170, 171]. Miehe and Gürses [170] presented a twodimensional large strain local variational formulation for brittle fracture with adaptive Rrefinement, the simplification of this framework to small strain problems was presented by Miehe et al. [171]. The approach was extended to threedimensions for the first time by Gürses and Miehe [103].
All of the works in this area are based on Eshelby [70, 73] and Rice’s [224] concept of material configurational forces acting on a crack tip singularity. A more general overview can be obtained from several sources [104, 105, 135, 162, 256]. Within this setting several local variational formulations have been proposed, for example see the works of [163, 258], and fracture initiation defects of the classical Griffithtype brittle fracture overcome by global variational formulations [54, 80]. Several researchers have numerically determined the material configurational forces at static fracture fronts [61, 116, 185, 255]. Before the works of Miehe and coworkers [103, 170, 171], there were several other attempts towards the implementation of fracture propagation in the configurational mechanics context, including: Mueller and Maugin [186] within the conventional finiteelement context, Larsson and Fagerström [74, 143] in XFEM and Heintz [115] within a discontinuous Galerkin (DG) setting. The framework has also recently been applied to materials with nonlinear behaviour, see for example the works of Runesson et al. [229] and Tillberg and Larsson [265] on elastoplasticity and Näser et al. [189, 190] on timedependent materials and the review by Özenç et al. [202]. In the following a configurational force approach to modelling fracture propagation is outlined based on the notation of Kaczmarczyk et al. [129].
As noted by Kaczmarczyk et al. [129], three possible solutions to Eq. (77): zero crack growth with \(\dot{\varvec{W}}=0\); force balance \((\gamma \varvec{A}_{\partial \Gamma } \varvec{G}_{\partial \Gamma })=0\); or that the crack front velocity is orthogonal to \((\gamma \varvec{A}_{\partial \Gamma } \varvec{G}_{\partial \Gamma })\). However, there is insufficient information in Eq. (77) to dictate the evolution of the crack front. Such an evolution law can be obtained by considering the second law of thermodynamics, supplemented by a material constitutive law and the principal of maximum energy dissipation.
In the work of Kaczmarczyk et al. [129], this fracture methodology was combined with a mesh quality control algorithm based on the work of Scherer et al. [240]. Within this, the nodal positions of the elements are modified based on a shapebased (volume to length) measure of element quality through the determination of a pseudo force vector. This pseudo force features in the discretised material nodal force equilibrium equation and is solved using a NewtonRaphson process. Note, that this modification to the discrete equilibrium equation only influences the stability of the solution and not the crack propagation criterion [129]. This mesh quality control procedure reduces the progressive degradation of the solution with fracture propagation.
Kaczmarczyk et al. [129] note that their approach could easily be extended to anisotropic materials. However, one limitation of the approach is that it is currently unable to capture nonsmooth crack kinking [171]. Also, crack branching and multiple crack coalescence has yet to have been demonstrated, or even formulated.
7.6 Discrete Element Method (DEM)

Finite displacements and rotations of the bodies is permitted, which includes complete detachment;

New contacts (or the absence thereof) are recognised automatically as the calculation progresses.
In practice, DEM is used in problems with a large number of elements, each element representing a body in contact. The formulation itself can be quite simplified compared to other discretisation methods, but it allows the simulation of complex behaviour, including material heterogeneities.
The DEM can be decomposed into several subclasses, which differ in some aspects such as the contact treatment, material models, number of interacting bodies, fracturing, and integration schemes [30].
In this framework, each element is a particular body which can be in contact with a number of surrounding elements. This implies that contact detection is one of the main problems that can arise, since missing a contact between elements can result in non representative behaviour of the model. Moreover, inspecting the elements for possible contact can require large amounts of computational processing time. The most common contact search algorithms are based on socalled body based search, where the vicinity of a given discrete element is searched for possible contact, and repeated after a number of iterations to check if the elements are still in contact. The Region Search algorithm [263] is an example of this kind of contact detection. Other contact detection algorithms use space search rather than a body search, and some examples are based on binary trees [30, 36, 208].
The next step is to obtain the contact forces. The calculation is usually performed with penalty based methods or Lagrange multiplier based methods. A review of contact algorithms evaluation can be found in [112].
The modelling of fracture using DEM has been mostly confined to element interfaces, where the breakage of the link between elements determines the appearance or propagation of the damage [30]. Particles can be bonded into clusters, where the bond stiffnesses are the equivalent to the continuum strain energy. Bond failure is assumed when the strength has exceed the maximum tension the bond can handle. Consistent breakage of the particle bonds define the fracture shape in the material. In [18, 187], a combination of the FEM with DEM has been used to model fracture starting from a continuum representation of the finite elements, and as the damage appears it is restated in the discrete element framework. A multifracture FEM/DEM scheme has been proposed by [212], where sliver elements arising from poor intraelement fracturing were avoided using local adaptive mesh refinement.
Discontinuous deformation analysis (DDA) is a variation of the DEM proposed originally by Shi and Goodman [246] to simulate the dynamics, kinematics, and elastic deformability of a system contacting rock blocks. While each block is treated separately in DEM, in DDA the total energy of the system is minimised in order to obtain a solution; a linear system of equations is obtained, resembling the finite element formulation. In fact, displacements and strains are taken as variables and the stiffness matrix of the model is assembled by differentiating several energy contributions including block strain energies, contacts between blocks, displacement constraints and external loads [146]. In the basic DDA implementation, each block is simply deformable as the strain and stress fields are constant over the entire block area, while the contacts are solved using regular contact algorithms that allow interpenetration between bodies [112]. To conclude, DDA is an implicit formulation while DEM uses an explicit procedure to solve the equilibrium equations. DDA has been used extensively in rock mechanics applications, as can be seen in [113, 114, 155, 267] for example.
The influence of the bond parameters defined at the microscale and how they affect the response on the macroscale are analysed in detail in [49] for rock model analysis. It is shown that using a clumpedparticle model, i.e. the particles rotate in a cluster instead of each particle being allowed to rotate, can reduce the limitations of the model, such as the overestimated ratio between tensile and compressive strengths, and the friction angles of the failure envelope.
A combined Lattice Boltzmann method (LBM) and DEM have been used to simulate fluidparticle interactions by [76]. The fluid field is solved by an extended 3D LBM with a turbulence model, while particle interactions are modelled using the DEM. Simulation results have matched experimental measurements.
There are available codes for the DEM, as the universal distinct element code (UDEC) [124], the ELFEN [218], the Yade [138] and YGeo [159]. More information on the discrete element framework can be found on [30, 146] and some applications in [33, 127, 128].
8 Peridynamics
We will now introduce a new numerical method called peridynamics, which appears to be very promising for fracking problems. The main difference between the peridynamic theory and classical continuum mechanics is that the former is formulated using integral equations as opposed to derivatives of the displacement components. This feature allows damage initiation and propagation at multiple sites, with arbitrary paths inside the material, without resorting to special crack growth criteria. In the peridynamic theory, internal forces are expressed through nonlocal interactions between pairs of material points within a continuous body, and damage is a part of the constitutive model. Interfaces between dissimilar materials have their own properties, and damage can propagate when and where it is energetically favourable for it to do so.
8.1 Definitions
Boundary conditions in peridynamics are not completely alike to the classical theory. Although the essential boundary condition is still present (displacements), there are no natural boundary conditions (tractions) in the peridynamics framework. Forces at the surface of a body must be applied as body forces \({\mathbf {b}}\) acting through the thickness of some layer under the surface. Usually, the thickness is taken to be the horizon \(\delta\). The displacement boundary conditions also have to be imposed as a volume rather than a surface. For more details see [248].
8.2 Constitutive Modelling
8.3 Anisotropic Materials in Peridynamics
The peridynamics formulation was initially presented for isotropic materials, in order to make some simplifying assumptions. It is expected then that the spring stiffness of the bonds does not vary over the direction of \(\varvec{\xi }\). It was demonstrated in detail in [248] that for isotropic materials, the Poisson’s ratio in the peridynamics formulations is constrained to take the constant value of 1/4. The constant Poisson’s ratio is a consequence of the Cauchy relation for a solid composed of a lattice of points that interact only through a central force potential [153].
Refinements of the peridynamics theory can allow the dependence of strain energy density on local volume change in addition to twoparticle interactions [154].
The fracture behaviour of the material is fully defined by using the mode I energy release rates. Hence, mode II energies are not independent from mode I, which is another consequence of the bondbased peridynamic theory.
An important issue has been highlighted in [95, 201], concerning the use of “unbreakable” bonds near to the regions where a traction boundary condition is applied. The possible reason for this would be crack initiation and propagation close to these regions, due to the high stresses that could be present. It is important to understand the physics of the analysed problem properly in order to use this type of assumption during a peridynamic simulation.
8.4 Statebased Formulation
The peridynamics formulation assumes that any pair of particles interacts only through a central potential which is independent of all the other particles surrounding it. This oversimplification has led to some restrictions of the material’s properties, such as the aforementioned fixed Poisson’s ratio of 1/4 for isotropic materials. Also, the pairwise force is responsible for modelling the constitutive behaviour of the material, which is originally dependent on the stress tensor. To overcome this limitation, Silling et al. [250] have extended the peridynamics formulation to include vector states. The vector states allow us to consider not only a particle, but a group of particles in the peridynamics framework. Moreover, the direction of the vector states would not be conditioned to be in the same direction of the bond, as in the bondbased theory. This property is fundamental to consider truly anisotropic materials.
The concept of a vector state is similar to a second order tensor in the classical theory, since both map vectors into vectors. Vector states may be neither linear nor continuous functions of \(\varvec{\xi }\). The characteristics of the vector states are listed in [250], and they imply the vector states mapping of \({\mathscr {H}}\) may not be smooth as in the usual peridynamic model, including the possibility of having a discontinuous surface.
The processing of mapping a stress tensor as a peridynamic force state is the inverse of the process of approximating the deformation state by a deformation gradient tensor. A peridynamic constitutive model that uses stress as an intermediate quantity results in general in bond forces which are not parallel to the deformed bonds. This type of modelling was called “nonordinary” by Silling [250].
8.5 Numerical Discretisation
Convergence in peridynamics is affected by two parameters: the grid spacing \(\Delta x\) and the horizon \(\delta\). Reducing the horizon size for a fixed grid spacing will lead to the peridynamics solution approximating the solution using classical theory. However, fixing the horizon size while increasing the grid spacing will lead to the exact nonlocal solution for that particular horizon size [122]. As for domain discretisation methods, it is important to balance the size of the horizon so the damage features in the analysed body are properly considered, and the grid spacing should be sufficiently small for the results to converge to the nonlocal solution. Usually, it ranges from 1/3 to 1/5 of the size of the horizon.
In recent works, the peridynamics formulation is used conjointly with other discretisation methods, such as meshless formulation [249] and finite element formulation [154].
In [201], peridynamics is used only to obtain the prediction of failure of the composite material, where an FEM code is employed to solve the global problem. This type of combined approach is often necessary since the peridynamic formulation can demand significant computational power, a common problem in molecular dynamics simulations as well.
9 Conclusions and Prospective Work
We have seen that the hydraulic fracture problem presents several characteristics which makes its study complicated: the shale is not a homogeneous material, it is not isotropic, the nanoporosity may retard crack propagation as the fluid penetrates the rock, and a large fracture network has to be considered in which cracks develop at multiple length scales, all of which can greatly increase the computational solving time. Moreover, most current analytical and numerical methods do not take into account crack branching, a key factor in order to obtain a correct estimation of the extended fracture network.
The current fracture models for brittle rocks and fracking have been useful as a first step in offering a more realistic fracking model. There are of course other limitations attached to each of the numerical models discussed earlier: for instance, in cohesive models, the cohesive zone model is not a parameter to be found, so the crack propagation path is already known a priori. Most works on XFEM and BEM models consider that the crack propagation path is unique; only recently have some works appeared considering crack branching [184, 244, 285].
Fracking models developed so far have not considered the full complexity of shale rocks. Ulm and coworkers [268, 269, 270] have established that shales are likely to be transversely isotropic materials, with the direction perpendicular to the bedding planes taken as the symmetry axis. This is mainly due to the deposition process. It was also stated that the shale anisotropy is due more to the interaction between the particles than the elastic behaviour of the shale components.
It was seen in [139] that the fluid penetrating the crack may retard crack propagation, so the material’s porosity has to be taken into account in the numerical model.
9.1 Future Works
The main challenges researchers are facing with respect to the development of a new numerical formulation for modelling hydraulic fracture are: (1) the multiscale characteristic of the fracking in shale rocks, and (2) the requirement for the numerical method to deal with a large number of cracks simultaneously propagating and possibly branching.
For crack propagation and crack branching, the peridynamics formulation has been shown to have excellent results. A few issues have been raised about the method, such as how to choose the grid spacing (interval between particles) and the horizon (area of influence of a given particle). Even though an orthotropic formulation for 2D materials was developed by [95], there are some limitations over these formulations, since a direct bond force formulation is used. To overcome this limitation, a statebased formulation for anisotropic materials should be developed.
A multiscale model must be able to consider how a crack entering the RVE interacts with the voids that are present. Moreover, there must be a coupling between the microscale (anisotropic) and the macroscale (transversely isotropic). The peridynamics formulation could be used to model the microscale, so the crack branching inside the RVE can be properly considered. Once the crack propagation path is obtained, another numerical method (XFEM/XBEM) can be employed to model the crack in the macroscale. Crack branching has already been considered in peridynamics in [106]. A comparison against experimental results of XFEM, cohesive models and peridynamics in dynamic fracture is done in [5], where it is observed that the peridynamics model is able to capture the physical behaviour seen in experiments.
A stochastic approach is likely to be the most useful way to model the extended fracture network, since the natural variability in geological conditions makes us unlikely to be able to obtain a deterministic model of the fracture system induced around any particular well. Moreover, the crack propagation obtained with the peridynamics formulation may change significantly if changes to the grid spacing or horizon size are made.
Notes
Acknowledgments
The first author acknowledges the Faculty of Science, Durham University, for his Postdoctoral Research Associate funding. Figures 4 and 5 have been reprinted with permission from Elsevier Limited, and Figs. 10, 11, 12, 14 and 15 have been reproduced with permission from John Wiley and Sons.
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