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Computers and Geotechnics 85 (2017) 151–176

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Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Review

The use of discrete fracture networks for modelling coupledgeomechanical and hydrological behaviour of fractured rocks

http://dx.doi.org/10.1016/j.compgeo.2016.12.0240266-352X/� 2016 Published by Elsevier Ltd.

⇑ Corresponding author.E-mail addresses: [email protected] (Q. Lei), [email protected] (J.-P. Latham), [email protected] (C.-F. Tsang).

Qinghua Lei a,⇑, John-Paul Latham a, Chin-Fu Tsang b,c

aDepartment of Earth Science and Engineering, Imperial College London, London, UKbDepartment of Earth Sciences, Uppsala University, Uppsala, SwedencEarth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 June 2016Received in revised form 3 November 2016Accepted 24 December 2016

Keywords:Fractured rockDFNGeomechanical modellingFluid flowHM coupling

We present a discussion of the state-of-the-art on the use of discrete fracture networks (DFNs) for mod-elling geometrical characteristics, geomechanical evolution and hydromechanical (HM) behaviour of nat-ural fracture networks in rock. The DFN models considered include those based on geological mapping,stochastic generation and geomechanical simulation. Different types of continuum, discontinuum andhybrid geomechanical models that integrate DFN information are summarised. Numerical studies aimingat investigating geomechanical effects on fluid flow in DFNs are reviewed. The paper finally provides rec-ommendations for advancing the modelling of coupled HM processes in fractured rocks through morephysically-based DFN generation and geomechanical simulation.

� 2016 Published by Elsevier Ltd.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522. Representation of natural fracture networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

2.1. Geologically-mapped fracture networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1522.2. Stochastically-generated fracture networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1532.3. Geomechanically-grown fracture networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1562.4. Summary with a view towards HM modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3. Modelling of the geomechanical behaviour of fracture networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.1. Continuum/extended-continuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.2. Block-type discontinuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
3.2.1. Distinct element method (DEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.2.2. Discontinuous deformation analysis (DDA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3.3. Particle-based discontinuum models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.4. Hybrid finite-discrete element models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

3.4.1. ELFEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.4.2. Y-code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.5. Summary with a view towards HM modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4. Modelling of the hydrological behaviour of fracture networks under geomechanical effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.1. Fluid pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1654.2. Permeability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1664.3. Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.4. Comments on the use of DFNs for HM modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

5. Outstanding issues and concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

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152 Q. Lei et al. / Computers and Geotechnics 85 (2017) 151–176

1. Introduction

Fractures such as joints, faults, veins and bedding planes areubiquitous in crustal rocks. These naturally occurring discontinu-ities often comprise complex networks and dominate the geome-chanical and hydrological behaviour of subsurface rocks [1].Understanding of the nontrivial effect of fractures is a challengingissue which is relevant to many engineering applications such asunderground construction, enhanced geothermal systems, uncon-ventional shale gas production (fracking), groundwater manage-ment and radioactive waste disposal [2–4]. The importance ofthe presence of natural fractures, which can result in heteroge-neous stress fields [5] and channelized fluid flow pathways [6] inhighly disordered geological formations, has promoted the devel-opment of robust fracture network models for numerical simula-tion of fractured rocks [7].

The first difficulty in modelling fractured rocks is the geometri-cal representation of complex three-dimensional (3D) discontinu-ity systems. Natural fractures form under certain mechanicallyself-organised dynamics, where breakage and fragmentation canoccur at all scales [8]. They are subject to in-situ stress fields atdepth and can form intricate topologies, such as cross-cutting,abutting, branching, termination, bends, spacing and clustering[9,10]. However, direct observation of the detailed 3D structureof fracture networks deep in the crust is impossible. Field dataare usually collected from lower dimensional limited exposures,e.g. one-dimensional (1D) borehole logging and two-dimensional(2D) outcrop mapping [11]. Seismological surveys may be able tolocate 3D large-scale structures but the current technology canhardly detect widely-spreading medium and small fractures dueto the resolution limit. The description of natural fracture geome-tries, therefore, has to largely rely on extrapolations, from 1D/2Dto 3D and from small samples to the whole study domain. Hence,the question of how to create realistic fracture networks remainsan unresolved issue. Often simplifications have to be made byignoring some details of less importance for the particular ques-tions at hand.

The second fundamental issue in modelling fractured rocks is tosolve the fracture and solid mechanics problems of the discontinu-ous geological media under complex boundary conditions, i.e. thefractured rock mass mechanical response to boundary stresses ordisplacements. When the spacing of natural fractures is compara-ble to the scale of interest of the problem to be modelled, the con-ventional continuum approach may not be adequate to capturesome important mechanical behaviour of fractured rocks [12], suchas fracturing of intact rocks [13], interaction of multiple fractures[5], and opening, shearing and dilatancy of rough fracture walls[14]. Thus, many computational schemes based on extended con-tinuum, discontinuum, or hybrid continuum-discontinuum meth-ods have been developed to solve a numerical system withfracture geometries explicitly represented.

Another important question is about the hydrological behaviourof fractured rocks under geomechanical conditions. Fractured rocksmay deform in response to geological and/or man-made perturba-tions (e.g. tectonic events, underground excavations), resulting inchanges of bulk permeability and fluid migration [2]. The intricacyof such coupled hydromechanical (HM) processes is increased ifthe presence of natural fractures associated with topological com-plexities is to be considered [1,15]. The quest for a means of quan-tifying the influence of in-situ stresses on the permeability offractured reservoirs has been driven largely by the motivation frompetroleum engineering [16]. The understanding of contaminantmigration through tectonically strained fractured formations isalso crucial for the groundwater community [17] and nuclearwaste management [3,4].

The objective of this paper is to review the current state-of-the-art of fracture network models and to provide some discussionsand recommendations for HM modelling of fractured rocks. In thiscontext, the issues arising in the aforementioned three key subjectareas (i.e. geometry, geomechanics and hydromechanics) will beexamined progressively. The paper will present a summary of var-ious approaches used in developing fracture network models thatrepresent natural fracture geometries often with different degreesof simplification, and different numerical frameworks that inte-grate discrete fracture representations for modelling geomechani-cal and HM behaviour of fractured rocks. One important focus ofthis work is to discuss the features or characteristics of a fracturenetwork model needed to properly simulate coupled HM processesin fractured rocks. The rest of the paper is organised as follows.Section 2 reviews the methods of representing natural fracturegeometries by geological mapping, stochastic generation orgeomechanical simulation. Section 3 provides a brief overview ofcontinuum and discontinuum models that integrate explicit frac-ture geometries for geomechanical modelling of fractured rocks.A short summary is presented in each of Sections 2 and 3 with aview towards HM modelling. Section 4 summarises the numericalstudies of geomechanical effects on fluid flow in fractured rocksand further provides some suggestions on choosing appropriateDFNs and geomechanical models for HM modelling. Finally, out-standing issues are discussed and some concluding remarks aremade. It has to be mentioned that this review is focused on HMbehaviour of fractured rocks and is not exhaustive. Only limitedreferences are cited for brevity. For more extensive descriptionsof fracture network models, the reader is referred to the reviewsby Dershowitz and Einstein [18] and Liu et al. [19], and the text-books by Adler and Thovert [20] and Adler et al. [21]. Extensivereviews of different numerical methods developed in the field ofrock mechanics can be found in Jing and Hudson [22], Jing [12],Yuan and Harrison [23], Bobet et al. [24], and Lisjak and Grasselli[25]. More in-depth discussions about stress effects on fluid flowin fractured rocks can be found in Zhang and Sanderson [26], andRutqvist and Stephansson [2].

2. Representation of natural fracture networks

A ‘‘discrete fracture network” (DFN) refers to a computationalmodel that explicitly represents the geometrical properties of eachindividual fracture (e.g. orientation, size, position, shape and aper-ture), and the topological relationships between individual frac-tures and fracture sets. Unlike the conventional definition ofDFNs that corresponds to stochastic fracture networks, the termDFN here represents a much broader concept of any explicit frac-ture network model. A DFN can be generated from geological map-ping, stochastic realisation or geomechanical simulation torepresent different types of rock fractures including joints, faults,veins and bedding planes.

2.1. Geologically-mapped fracture networks

Fracture patterns can be mapped from the exposure of rock out-crops or man-made excavations (e.g. borehole, quarry, tunnel androadcut). These geologically-mapped fracture networks werewidely used to understand the process of fracture formation[5,27], interpret the history of tectonic stresses [28–30], and derivethe statistics and scaling of fracture populations [8,31,32]. How-ever, digitised outcrop analogues (Fig. 1) can also be used to buildDFNs for numerical simulations. For example, a series of discretefracture patterns were mapped from limestone outcrops at thesouth margin of the Bristol Channel Basin, UK [33]. The tracedDFNs were used to study the connectivity [34], multiphase flow

Fig. 1. Geologically-mapped DFN patterns based on (a) a limestone outcrop at the south margin of the Bristol Channel Basin, UK [33], (b) sandstone exposures in theDounreay area, Scotland [43], and (c) fault zone structures in the Valley of Fire State Park of southern Nevada, USA [53].

Q. Lei et al. / Computers and Geotechnics 85 (2017) 151–176 153

[35–37], solute transport [38] and HM behaviour [15,39–41] ofnatural fracture systems. Similar outcrop-based DFNs have alsobeen constructed by many other researchers for modelling naturalfracture systems [42–48]. It is noteworthy that the terrestrialremote sensing techniques (e.g. LIDAR and photogrammetry) haverecently been introduced to better capture large-scale trace fea-tures [49–52]. Fracture apertures may be determined from adetailed field mapping [53,54] and further calibrated by comparingflow simulation results with in-situ measurements [55]. However,apertures were more commonly assumed to be constant or to fol-low an a priori statistical distribution (sometimes correlated withtrace lengths).

Advantages of such an outcrop-based DFN model includepreservation of natural fracture features (e.g. curvature and seg-mentation) and unbiased characterisation of complex topologies(e.g. intersection, truncation, arrest, spacing, clustering and hierar-chy). However, it is often constrained to 2D analysis and may notbe applicable for general 3D problems involving obliquely dippingfractures. The LIDAR survey combined with assumptions of frac-ture shape/persistence [56,57] may provide an estimation of 3Dstructural variations in near surface, but it is not applicable to bur-ied geological complexities. Extrapolation from borehole imagingcan provide an estimation of 3D fracture distributions but confi-dence can only be guaranteed for the areas close to boreholes[58]. Seismic data can be used to build 3D maps of large-scale geo-logical structures, for which, however, the limited resolution oftenobscures detailed features such as the segmentation of faults andimpedes the detection of small cracks widely spreading in subsur-face rocks [59].

2.2. Stochastically-generated fracture networks

Due to the difficulty of performing a complete measurement of3D natural fracture systems, stochastic approaches using statisticsfrom limited sampling have been developed and widely used [18].The stochastic DFN method emerged in the 1980s with the aims tostudy the percolation of finite-sized fracture populations [60–62]and fluid flow in complex fracture networks [63–67].

The general stochastic DFN approach assumes fractures to bestraight lines (in 2D) or planar discs/polygons (in 3D), and treatsthe other geometrical properties (e.g. position, frequency, size, ori-entation, aperture) as independent random variables obeying cer-tain probability distributions [68] derived from fieldmeasurements such as scanline [69] or window sampling [70] ofoutcrop traces as well as borehole imaging [71]. The orientationdata can be processed using a rosette or stereogram so that frac-tures can be grouped into different sets with their orientationscharacterised by e.g. a uniform, normal or Fisher distribution[11]. Fracture sizes may exhibit a negative exponential, lognormal,gamma or power law distribution [72,73]. Fracture frequency canbe described in terms of fracture density or fracture intensityand denoted using the Pij system, where i gives the dimension ofthe sample and j the dimension of the measurement [74]. Fracturedensity measures the number of fractures per unit volume (P30),area (P20) or length (P10), while fracture intensity represents thetotal fracture persistence per unit volume (P32), area (P21) or length(P10). Fracture spacing is related to fracture frequency (P10) [75],which is both a density and intensity measure. Fracture spacingmay follow a negative exponential, lognormal or normal distribu-tion depending on the degree of fracture saturation in the network[76]. Fracture apertures usually obey a lognormal [77] or powerlaw distribution [78,79], and may be related to fracture sizes bya power law [73] with a linear [5] or sublinear [80] scaling relation-ship. The measurement data (e.g. those based on outcrop sam-pling) may be biased under the truncation and censoring effectsand requires to be amended to determine the underlying statisticaldistributions [81–83]. The statistics of 3D fracture systems (e.g.size, spacing, density distributions) may be extrapolated from1D/2D sampling data based on stereological analysis [84,85]. Inthe stochastic simulation, fractures can be assumed randomlylocated (represented by their barycentres), while the geometricalattributes can be sampled from the corresponding probability den-sity functions [18]. Such a random fracture network modellingapproach, termed the conventional Poisson DFN model (or Baechermodel) (Fig. 2), has been implemented within the commercial soft-ware FracMan [86] and also adopted by many research codes tostudy the connectivity, fragmentation, deformability, permeability

Fig. 2. The conventional Poisson DFN models: (a) a 2D random fracture pattern conditioned by field data from the Sellafield site, Cumbria, UK [95] and (b) a conceptual model(no scale specified) of 3D random fracture network with three orthogonal sets of disc-shaped fractures [64].


and transport properties of fractured rocks in the past three dec-ades [87–107] (only a few among many others).

However, the conventional Poisson DFN model tends to havelarge uncertainties due to its assumption of a homogeneous spa-tial distribution, simplification of fracture shape using linear/pla-nar geometries, and negligence of the correlations betweendifferent geometrical properties as well as disregard of the diversetopological relations (e.g. ‘‘T” type intersections). Severalresearchers have examined the conventional Poisson DFN modelby comparing it with an original natural fracture network withrespect to geometrical, geomechanical and hydrological proper-ties, and significant discrepancies were observed [36,39,108–110]. Several improvements on the Poisson DFN model have beendeveloped and include considerations of: (i) the inhomogeneity offracture spatial distribution based on a geostatistically-deriveddensity field [67,111] or a spatial clustering process [112–114],(ii) the correlation between fracture attributes (i.e. length, orien-tation and position) based on an elastic energy criterion[115,116], (iii) the unbroken areas inside individual fractureplanes based on a Poisson line tessellation and zone marking pro-cess [18,117], (iv) the topological complexity based on a charac-terisation of fracture intersection types [118] and thetermination of fractures at intersections with pre-existing frac-tures [86], and (v) the mechanical interaction between neighbour-ing fractures based on a stress shadow zone model [76,119,120].It has to be mentioned that an alternative to the Poisson DFNmodel may use the spacing distribution to locate fractures inthe stochastic generation [121], which is however consideredmore suitable for highly persistent fracture systems. For moredetailed summary of the Poisson model and other improved mod-els, the reader is referred to Dershowitz and Einstein [18], Staubet al. [122] and Fadakar-Alghalandis [114].

A more systematic characterisation of the hierarchy, clusteringand scaling of natural fracture systems may involve the methods offractal geometry and power law models [73]. Extensive field obser-vations suggest that fracturing occurs at all scales in the crust andcreates a hierarchical structure that can exhibit long-range correla-tions from macroscale frameworks to microscale fabrics[8,124,125]. The spatial organisation of natural fracture networkscan be characterised by the fractal dimension D, which quantifiesthe manner whereby fractals cluster and spread in the Euclideanspace and can be measured using the box-counting method[31,126,127] or the two-point correlation function [128,129]. Thedensity and length distribution of a fracture population can be thendescribed by a statistical model [32,130] given as: n(l,L) = aLDl�a,

for l 2 ½lmin; lmax�, where n(l,L)dl gives the number of fractures withsizes belonging to the interval [l, l + dl] (dl� l) in an elementaryvolume of characteristic size L, a is the power law length exponent,a is the density term, and lmin and lmax are the smallest and largestfracture sizes. The exponent a, which defines the respective pro-portion of large and small fractures, can be derived from the cumu-lative distribution or density distribution of fracture lengths[72,82]. In theory, D is restricted to the range of [1,2] for 2D and[2,3] for 3D, and a is restricted to [1,1] for 2D and [2,1] for 3D.However, extensive measurements based on 2D trace maps revealthat generally D varies between 1.5 and 2 and a falls between 1.3and 3.5 [73] in natural fracture systems. The D and a measuredfrom 1D/2D samples can be extrapolated to derive 3D parametersbased on stereological relationships [123]. The density term a isrelated to the total number of fractures in the system and variesas a function of fracture orientations [130]. The extent of the powerlaw relation is bounded by an upper limit lmax that is probablyrelated to the thickness of the crust, and a lower limit lmin that isconstrained by a physical length scale (e.g. grain size) or the reso-lution of measurement [73]. For numerical simulations, the modelsize L usually meets lmin � L� lmax [131]. A fractal spatial distribu-tion of fracture barycentres can be modelled through a multiplica-tive cascade process governed by a prescribed D value, whilefracture lengths can be sampled from a power law distributionhaving an exponent a [131]. Fracture orientations can be assignedisotropically or based on a statistical distribution. Fractal fracturenetworks can then be generated through a synthesis of the differ-ent geometrical attributes modelled by independent random vari-ables (Fig. 3). A D value of 2 (in 2D) and 3 (in 3D) represents ahomogeneous spatial distribution, i.e. ‘‘space filling”. As Ddecreases, the fracture pattern becomes more clustered associatedwith more empty areas. A small a value corresponds to a systemdominated by large fractures, while a?1 relates to a pattern withall fractures having an equal size (i.e. lmin). The D and a values aswell as their relationship may control the connectivity, permeabil-ity and strength of fractured rocks [131–134]. More interestingly,when a = D + 1, the fracture network is self-similar and the connec-tivity properties are scale invariant [131]. A self-similar fracturepattern statistically exhibits a hierarchical characteristic wherebya large fracture inhibits the propagation of smaller ones in its vicin-ity, but not the converse [32,130]. Implementation of such a hier-archical rule together with subcritical fracture growth laws leadsto a new DFNmodel that can simulate the sequential stages of frac-ture network formation associated with nucleation, propagationand arrest processes (Fig. 4a) [135].

Fig. 4. Some new stochastic DFN models: (a) trace map views of a 3D sequential DFN model that simulates the nucleation, growth and arrest processes of natural fractures(the ratio of the domain size L to the characteristic nuclei length lN is 100) [135] and (b) a 2D self-referencing DFN model that extrapolates a larger scale pattern recursivelyfrom a smaller one through random walks with the preservation of natural fracture curvatures [40].

Fig. 3. 2D and 3D fractal fracture networks generated with different values of the fractal dimension D and the power law length exponent awhile the ratio of the domain sizeto the minimum fracture length L/lmin = 1024 [123].


The assumption of a linear/planar fracture shape in the Poissonmodel and fractal DFN model may be simplistic. Many field obser-vations have shown that natural fracture geometries can be curvedand irregular [9]. To simulate the non-planarity of natural frac-tures, Lei et al. [40] developed a novel DFN model that accommo-dates discrete-time random walks in recursive self-referencinglattices to extrapolate fracture geometries into larger domainsbased on a small sample (Fig. 4b). This model was developedmainly for generating cross-cutting fracture sets. For more com-plex networks involving abutting relations, the random walk tech-

nique proposed by Horgan and Young [136] for simulatingpolygonal crack patterns in soil may be employed. Furthermore,the invasion percolation method that has been used to modelchannel networks [137] may provide a way to simulate somehighly branched and tortuous fracture systems.

The stochastic DFN method, in essence, treats problems in aprobabilistic framework and regards the real physical system asone possibility among simulated realisations sharing the samestatistics. Hence, a sufficient number of realisations based on aMonte Carlo process are necessary to predict a bounded range. In


practice, a balance exists between the benefits of collectingdetailed information to create more representative DFNs and theincreased cost of field measurements. The uncertainty can bereduced by constraining the random process with deterministicdata, e.g. forcing the 3D DFN generator to reproduce a 2D trace pat-tern such as one exposed on tunnel walls [138]. Calibration andvalidation of stochastic DFN models are important for solving realproblems and can be conducted based on the in-situ data fromfield mapping and/or hydraulic tests [138–144]. The random nat-ure of the stochastic DFN method may be regarded as an advanta-geous aspect, because uncertainty is unavoidable when analysingcomplex geological systems, for which single-valued predictionsfrom deterministic methods may be more risky [7]. However, itis still very important to continue improving the realism and accu-racy of stochastic DFN models, because the predicted results (e.g.permeability, strength, fragmentation, etc.) featured by upperand lower bounds based on unrealistic DFNs may be systematicallybiased from the truth. Understanding of the variability and uncer-tainty in data collection and DFN generation are thus critical forengineering applications [145]. Developments are needed towards(i) a more thorough characterisation of the underlying statistics,such as multifractals for which a single scaling exponent is not suf-ficient [110], and (ii) a more precise and efficient generator to cre-ate DFNs respecting more details of real fracture systems, such asthe diversity of individual fracture shapes and morphology [9],the topological complexity in fracture populations [114,146]and the correlation between geometrical properties [129,147,148].The important difference between 2D and 3D fracture networkswith respect to connectivity and permeability [64,90,105] rendersanother advantage of the stochastic method, which has an intrinsiccapability of generating 3D networks.

2.3. Geomechanically-grown fracture networks

Extensive studies have been conducted to interpret the geolog-ical history and the formation mechanism behind field observa-tions of natural fracture systems (e.g. patterns, statistics andminerals) [8,9,28–30,149]. The increased knowledge of fracturemechanics [5] promoted the development of geomechanically-based DFN models that incorporate the physics of fracture growthand simulate fracture network evolution as a geometrical responseto stress and deformation. By applying a geologically-inferredpalaeo-stress/strain condition, natural fracture patterns may bereproduced by such a DFN simulator that progressively solvesthe perturbation of stress fields and captures the nucleation, prop-agation and coalescence of discrete fractures. Different numericalmethods have been proposed and the one based on linear elasticfracture mechanics (LEFM) is frequently adopted.

In the LEFM model, fracture patterns can be simulated by fourmain steps in an iterative fashion [150,151]: (1) generation of ini-tial flaws to mimic the process that natural fractures initiate frommicrocracks, (2) calculation of the perturbed stress field in the rockcaused by the presence and evolution of fractures under animposed boundary condition, (3) derivation of the stress intensityfactor (KI) at the tip of each fracture, and (4) propagation offractures which satisfy a growth criterion, e.g. a subcritical lawKO 6 KI 6 KIC, where KO is the stress corrosion limit and KIC is thematerial toughness [152]. The stress field and stress intensity fac-tor can be calculated based on analytical solutions [150] or (mostcommonly) numerical methods such as the boundary elementmethod (BEM) [153] and finite element method (FEM) [151,154].The propagation length in each growth iteration can be derivedaccording to a power law relation with the energy release rate G(related to KI) associated with a velocity exponent j (or subcriticalindex n) [152], while the propagation angle may be computed if

the curvature and coalescence effects are important, especiallywhen the tectonic stress field is quite isotropic [29].

The development of fracture networks is a sophisticatedfeedback-loop process, in which the complexity of growth dynam-ics is directly related to the complexity of the developing struc-tures. Specifically, the propagation of a fracture is influenced bythe mechanical interaction with others, and the propagated frac-ture geometries can conversely generate stress perturbations intothe system. The mechanical interaction of fractures was foundstrongly dependent on the velocity exponent j: an increased jtends to promote a localised fracture pattern [150,153,155–157].The fracture pattern evolution is also affected by the density[150,157] and orientation [158] of the initial flaws, and the 3Dlayer confinement effect [155,156]. Such a fracture mechanicsmodel has been developed to mimic the evolution of a 2D singleset of straight fractures (Fig. 5a) [150,153,155], 2D orthogonal setsof straight fractures [157], 2D curved fracture patterns (Fig. 5b)[29,151,156,158,159], and 3D curved fracture geometries (Fig. 5c)[160]. The generated DFN pattern can be further used to evaluatethe connectivity [91,157], permeability [161] and solute transportproperties [162] of natural fracture systems.

Apart from the LEFM approach, fracture patterns can also besimulated using other numerical methods. Cowie et al. [163,164]developed a lattice-based rupture model to simulate the anti-plane shear deformation of a tectonic plate and the spatiotemporalevolution of a multifractal fault system. Tang et al. [165] used adamage mechanics FEM model to simulate the evolution of paral-lel, laddering or polygonal fracture patterns formed under differentboundary conditions, i.e. uniaxial, anisotropic and isotropic tec-tonic stretch, respectively. Spence and Finch [166] employed thediscrete element method (DEM) to simulate the fracture patterndevelopment in a sedimentary sequence embedded with stratifiednodular chert rhythmites. Asahina et al. [167] coupled the finitevolume multiphase flow simulator (i.e. TOUGH2) and a lattice-based elasticity and fracture model (i.e. Rigid-Body-Spring Net-work) to simulate the desiccation cracking in a mining waste mate-rial under a hydromechanically coupled process. The finite-discreteelement method (FEMDEM) has also been used to simulate the for-mation and evolution of geological fracture patterns [168,169].

The geomechanically-grown DFN model, as a process-orientedapproach, has the advantage of linking the geometry and topologyof fracture networks with the conditions and physics of their for-mation. Another merit is the automatic correlation between thegeometrical attributes of individual fractures (e.g. length, orienta-tion, aperture and shear displacement) linked by the governingphysics. To solve practical problems, such a DFN generator canbe constrained by the measurement of rock properties (e.g. thesubcritical index measured from core samples) and the informa-tion of geological conditions (e.g. stress, strain, pore pressure anddiagenesis) to achieve rational predictions [170]. However, diffi-culty and uncertainty still exist in creating fracture patterns con-sistent with the real systems for which coupled tectonic,hydrological, thermal and chemical processes may be involved.

2.4. Summary with a view towards HM modelling

The three types of DFN models have each its own strength andalso suffer from some limitations, as listed in Table 1. Thegeologically-mapped DFNs can preserve many realistic featuresof fractures but it is hard to apply this method to characterise deeprocks and 3D structures. The stochastic DFN approach has the mer-its of simplicity and efficiency as well as applicability for 3D prob-lems. However, the strong geological hypotheses of fracturegeometries and topologies used in its construction may ignoresome important underlying mechanical and tectonic constraintsand thus possibly lead to errors or large uncertainties. The geome-

Table 1Comparison of different approaches for DFN geometrical representation.

Numericalmodels

Key inputs Strengths Limitations

Geological DFNs Analogue mapping, borehole imaging, aerialphotographs,LIDAR scan or seismic survey

(i) Preservation of geologicalrealisms.

(ii) Deterministic description of a frac-ture system.

(i) Limited feasibility for application to deeprocks.

(ii) Difficulty in building 3D structures.(iii) Constraints from measurement scale and

resolution.

Stochastic DFNs Statistical data of fracture lengths, orientations,locations,shapes and their correlations

(i) Simplicity and convenience.(ii) Efficient generation.(iii) Applicability for both 2D and 3D.(iv) Applicability for various scales.

(i) Oversimplification of fracture geometriesand topologies.

(ii) Uncertainties in statistical parameters.(iii) Requirement of multiple realisations.(iv) Negligence of physical processes.

GeomechanicalDFNs

Palaeostress conditions, rock and fracturemechanicalproperties

(i) Linking geometry with physicalmechanisms.

(ii) Correlation between different frac-ture attributes.

(i) Uncertainties in input properties and tec-tonic conditions.

(ii) Large computational time.(iii) Negligence of hydrological, thermal and

chemical processes.

Fig. 5. Geomechanically-grown DFN patterns based on linear elastic fracture mechanics: (a) evolution of a 2D fracture set [150], (b) development of a 2D polygonal fracturepattern [159] and (c) growth of 3D layer-restricted fractures [160].


chanical DFN models, which may capture somemechanical charac-teristics of natural fractures, are sensitive to the assumed/mea-sured rock properties and inferred palaeostress fields. If they areto be improved to become the preferred useable model, couplingwith hydrological, thermal and chemical mechanisms may beneeded to reproduce actual geological systems. To generate betterfracture networks, a future research direction that is attractingmuch effort is the development of hybrid DFN models that assim-ilate the advantages of different approaches. Some of the currentDFN models have already exhibited some of such features. Forexample, Kattenhorn and Pollard [59] used mechanical simulation

to correct the 3D fault structures interpreted from seismic surveydata. In the sequential stochastic DFN model proposed by Davyet al. [135], fractures develop following the subcritical growthlaw and their interactions are governed by an arrest mechanism.In the self-referencing DFN model by Lei et al. [40], ageologically-mapped fracture pattern is populated to largerdomains based on a random walk algorithm that preserves keygeometric and geomechanical attributes.

The created DFN geometries can serve as important inputs formodelling the geomechanical and/or hydrological behaviour offractured rocks. To reduce the numerical difficulties induced by


DFN complexities whilst capturing the key characteristics, simpli-fications may be applied. In conventional hydrological studies,inactive fractures (i.e. dead-end or isolated fractures) and verysmall fractures are often removed from original DFNs based onan assumption of their negligible contribution to the overall con-nectivity and permeability [67,87,95,99]. Such treatments may beapplicable for low permeability rocks with well-connected fracturetopologies, inside which the matrix flow can be neglected. How-ever, when the fractures are poorly connected and/or the rockmatrix is highly permeable, the effects of fracture-matrix fluidtransfer and flow in the matrix can significantly affect (or evengovern) the hydrological processes in fractured rocks [171] andthus the features of inactive fractures may need to be preserved.Attentions may also be drawn when setting a truncation thresholdfor deleting small-sized fractures which were commonly believedto have little contribution to the flow. However, several studieshave suggested that when a > D + 1, the connectivity of fracturenetworks is ruled by fractures much smaller than the system size[131,172] and thus the removal of very small fractures may leadto important biases. Apart from these potential hydrologicaleffects, even more importantly, the presence of inactive fracturesmay also influence the mechanical behaviour of fractured rocks(e.g. elastic or brittle deformation). New fractures may propagatefrom the dead-end tips of pre-existing fractures under stress con-centrations and link with other fractures to form critical pathwaysfor fluid flow. More discussions about choosing appropriate DFNsfor geomechanical and HM modelling will be presented in the fol-lowing sections.

3. Modelling of the geomechanical behaviour of fracturenetworks

The numerical methods for geomechanical modelling of frac-tured rocks can be categorised as continuum and discontinuumapproaches with the classification based on their treatment of dis-placement compatibility [12]. The preference for a continuum ordiscontinuum modelling scheme depends on the scale of the prob-lem and the complexity of the fracture system [22]. The continuumapproach has the advantage of its greater efficiency to handlelarge-scale problems, whereas the discontinuum method can morestraightforwardly incorporate complex fracture networks andmodel multi-fracturing and fragmentation processes. In this sec-tion, commonly used models for simulating geomechanical beha-viour of fractured rocks will be reviewed: continuum/extended-continuum, block-type discontinuum, particle-based discontin-uum and hybrid finite-discrete element approaches. It is worthmentioning that the classification here is not intended to be abso-lute, since the boundary between the continuum and discontin-uum methods has become very vague. Some advancedcontinuum techniques have included contact algorithms and frac-ture mechanics to consider discontinuities, while many discontin-uum models are able to deal with continuous deformations.

3.1. Continuum/extended-continuum models

The conventional continuum approach treats a rock domain as acontinuous body that can be solved by the finite element method(FEM) or finite difference method (FDM). It may be applicable toa fractured rock with only a few or a large number of fractures[12]. If the system consists of only a few discontinuities associatedwith only a small amount of displacement/rotation, the discretefractures can be modelled by special ‘‘interface elements” (or ‘‘jointelements”) that are forced to have fixed connectivity with the solidelements [173]. However, such a treatment renders difficult tohandle the dynamics and large displacement problems of natural

fracture systems. When the density of DFN fractures is very high,the modelling domain may be divided into a finite number of gridblocks assigned with equivalent properties derived fromhomogenisation techniques (Fig. 6a). The equivalent properties,such as bulk modulus and strength parameters, are usually calcu-lated using empirical formulations that consider the degradationeffect caused by pre-existing fractures [174,175] or analytical solu-tions based on the crack tensor theory [176,177]. The crack tensortheory calculates the volume averaged parameters of all fractureswith respect to their geometrical properties (e.g. length, orienta-tion and aperture) and was extended to consider the couplingbetween stress and fluid flow [46,178]. Such a crack tensor methodhas been integrated into the FEM [179,180] and FDM [101,181] sol-vers to model the geomechanical and HM behaviour of fracturedrocks. The simulation results may be sensitive to the grid block dis-cretisation especially when the block size is significantly smallerthan the representative elementary volume (REV) [101]. Thehomogenisation-based continuum model may not adequately con-sider the connectivity effect of very long fractures that penetratenumerous grid blocks and the results may be even worse if aper-tures are very heterogeneous and positively correlated with frac-ture sizes. Thus, it may not be applicable to a fractal fracturesystem with high variability in its density distribution (i.e. a smallfractal dimension D) and/or a large proportion of long fractureshaving a size comparable to the problem domain (i.e. a smallpower law length exponent a). The two conventional continuumschemes (i.e. the one employing interface elements and the oneusing homogenised properties) may be combined to explicitlymodel large discontinuities (e.g. dominant faults) using interfaceelements and then to characterise each isolated block as contin-uum bodies with their bulk properties parameterised accordingto the distribution of small fractures. Furthermore, the crack tensormethod cannot consider the interaction between fractures andblocks as well as the resulting localised deformation and damagein the rock.

To more explicitly capture the effects of discrete fractures, anextended-continuum model has been developed by assuming frac-tures to have certain numerical width (for connectivity preserva-tion) and representing them as arrays of grid elements withsoftening and weakening properties based on a very fine finite dif-ference mesh (Fig. 6b) [41,182]. It treats the fractured rock as acomposite elasto-plastic solid system, in which the failure of intactrocks or stress-displacement behaviour of fractures can be mod-elled by a Mohr-Coulomb criterion with tension cut-off. Similar‘‘weak material” representation of fractures has also been imple-mented in the rock failure process analysis (RFPA) code (a damagemechanics FEM model) [183] and the cellular automation model[184]. Such a composite continuummodel with explicit DFN repre-sentations may be more suitable for simulating weakly cementedfractures (i.e. mineral filled veins), whereas the physical rationaleis not intuitive if being applied to unfilled discontinuities havingclean wall surfaces. Furthermore, another important extended-continuum model is the extended finite element method (XFEM),which can represent fracture propagations without re-meshingand has been recently developed for simulating fluid-driven frac-turing [185,186].

3.2. Block-type discontinuum models

The block-type discontinuum models include the distinct ele-ment method (DEM) with an explicit solution scheme and the dis-continuous deformation analysis (DDA) method with an implicitsolution form. In this discontinuummodelling framework, the frac-tured rock is represented as an assemblage of blocks (i.e. discreteelements) bounded by a number of intersecting discontinuities.The geometry of the interlocking block structures can be identified

Fig. 6. (a) A continuummodelling scheme: a fracture network is divided into a finite number of grid blocks with equivalent properties determined analytically or numerically[101] and (b) an extended continuum modelling scheme: the domain is discretised by a regular finite difference grid and fractures are represented by softening andweakening the grid elements intersected by fracture traces [41].


first by e.g. employing the techniques of combinatorial topology[187]. In the subsequent mechanical computations, these blockscan be treated as rigid bodies or deformable subdomains (furtherdiscretised by finite difference/volume grids) with their interac-tions continually tracked by spatial detections during their defor-mation and motion processes.

3.2.1. Distinct element method (DEM)The DEM method was originated by Cundall [188,189] and

gradually evolved to the commercial codes UDEC and 3DEC forsolving 2D and 3D problems, respectively [190,191]. Its basic com-putational procedure can be summarised as four steps [192]: (1)the contacts between blocks are identified/updated through aspace detection, (2) the contact forces between discrete bodiesare computed based on their relative positions, (3) the accelerationinduced by force imbalance for each discrete element is calculatedusing Newton’s second law, and (4) the velocity and displacementare further derived by time integration with new positions deter-mined. An explicit time marching scheme is applied to solve theproblem iteratively until the block interaction process to be simu-lated has been completed. The mechanical interaction betweenblocks is captured by a compliant contact model that accommo-dates virtual ‘‘interpenetrations” governed by assumed finite stiff-nesses to derive normal and tangential contact forces. Theempirical joint constitutive laws derived from laboratory experi-ments [14,193] can be implemented into the interaction calcula-tion in an incremental form to simulate joint normal and

shearing behaviour [194–197]. A viscous damping parametermay be introduced to reduce dynamic effects for modellingquasi-static conditions [198].

The DEM approach is able to capture the stress-strain character-istics of intact rocks, the opening/shearing of pre-existing fracturesand the interaction between multiple blocks and fractures. Com-bined with DFN models, it has been widely applied to study themechanical behaviour of fractured rocks. Zhang and Sanderson[44] studied the critical deformation behaviour of fractured rocksand observed an abrupt increase in the deformability associatedwith a power law scaling when the fracture density exceeds themechanical percolation threshold (slightly higher than the geo-metrical threshold) (Fig. 7). Min and Jing [199] examined the scaledependency of the equivalent elastic properties of a fractured rockbased on multiple DFN realisations conditioned by the same frac-ture statistics (Fig. 8). In their study, a technique to derive thefourth-order elastic compliance tensor has also been developedfor equivalent continuum representations. Min and Jing [200] fur-ther found that the equivalent mechanical properties (i.e. elasticmodulus and Poisson’s ratio) of fractured rocks may also be stressdependent. With the increase of stress magnitudes, the equivalentelastic modulus significantly increases, while the Poisson’s ratiogenerally decreases but can be well above 0.5 (i.e. the upper limitfor isotropic materials). Noorian-Bidgoli et al. [201] extended thisDEM-DFN model to a more systematic framework to derive thestrength and deformability of fractured rocks under different load-ing conditions, which is further applied to study the anisotropy

Fig. 7. Deformation of fractured rocks with (a) a relatively low fracture density of 5.25 m/m2 and (b) a high fracture density of 7.77 m/m2 under a uniaxial compressionloading. (c) The deformability Bs of the fractured rock exhibits a power law scaling behaviour when the fracture density d exceeds the mechanical percolation threshold of6.5 m/m2, which is higher than the geometrical threshold of 4.0–5.5 m/m2 of the study networks [44].

Fig. 8. Variation of the elastic modulus of fractured rocks with the increase of the model size. The ratio of shear stiffness to normal stiffness of fractures is assumed to be 0.2.Results are computed using the block-type DEM simulator (UDEC) based on multiple DFN realisations [199].


[202] and randomness [203] of the strength/deformability ofstochastic DFNs. Recently, Le Goc et al. [204] integrated 3D DFNsinto the 3DEC simulator and investigated the effects of fracturedensity, sizes and orientations on the magnitude and scaling ofthe equivalent elastic modulus of fractured rocks. Havaej et al.[205] conducted 3D DEM-DFN modelling to analyse the effects ofgeological structures on the quarry slope failure mechanism. Inaddition, a considerable number of similar DEM-DFN models havebeen developed where the main motive is to study the effect ofstresses on fluid flow. Such models were applied to explicitly cap-ture the fracture opening, closing, shearing and dilational charac-teristics in complex fracture networks under in-situ stresses,after which the fluid flow implications were investigated [43–45,96,98,206–209] (more details are presented in Section 4). Further-more, some models that incorporate the pore fluid pressure effectshow that the pore pressure level can also significantly influence

the mechanical and hydrological properties of fractured rocks[47,210].

The classic DEM formulation cannot simulate the propagationof new fractures in intact rocks driven by stress concentrations,although plastic yielding can capture some aspects of the rockmass failure process [211]. Such a shortcoming was recentlyaddressed by introducing a Voronoi or Trigon discretisation inmatrix blocks that allows fracturing along the internal ‘‘grain”boundaries governed by tensile and shear failure criteria [212–216]. The DFN representation can also be integrated into such aDEM model using a dual-scale tessellation, in which the primarygrid represents natural fractures and the secondary discretisationmimics microstructures [217]. The uncertainty of the Voronoi/Tri-gon DEMmodel due to mesh dependency indicates the necessity ofrealistic representation of natural grain shapes and textures[218].

Fig. 9. (a) A fractured rock with a geologically-mapped DFN pattern and (b) its dynamic collapsing process modelled by the DDA method [227].


3.2.2. Discontinuous deformation analysis (DDA)The DDA method was proposed by Shi and Goodman [219,220]

to compute the deformation and motion of a multi-block system.The discretisation of a DDA system is quite similar to the one forthe DEM, i.e. the medium is dissected into blocks by intersectingdiscontinuities. However, a fundamental difference between thetwo methods lies in their computational frameworks. The DEMtreats the kinematics of each block separately based on an explicittime-marching scheme, while the DDA calculates the displacementfield based on a minimisation of the total potential energy of thewhole blocky system and an implicit solution to the establishedsystem of equations through a matrix inversion. Thus, the DDAmethod has an important advantage of fast convergence withunconditional numerical stability compared to the DEM methodthat requires a time step smaller than a critical threshold [12].Important extensions of the original DDAmethod include the finiteelement discretisation of rock matrix [221], the sub-block tech-nique (similar to the Voronoi DEM) for simulating fracturing pro-cesses [222], the formulation for modelling coupled soliddeformation and fluid flow [223,224], and the development of 3Dmodels [225]. The mechanical behaviour of fractured rocks has alsobeen investigated based on DDA-DFN simulations, with emphasison studying the stability of underground excavations and slopeengineering (Fig. 9) [226–228]. Recently, Tang et al. [229] com-bined the RFPA (a damage mechanics FEM model) with the DDAmethod to capture crack propagations and block kinematics of arock slope system.

3.3. Particle-based discontinuum models

The particle-based discontinuum model was originally intro-duced by Cundall and Strack [230] to simulate granular materialssuch as soils/sands and gradually evolved to a commercial code,i.e. particle flow code (PFC) [231]. Similar to the block-type DEMmethod, PFC calculates the inertial forces, velocities and displace-ments of interacting particles by solving Newton’s second lawthrough an explicit time-marching scheme. The discrete particlesare assumed to be rigid with a circular (in 2D) or spherical (in3D) shape, and can have variable sizes which are usually much lar-ger than the physical grain scale. To extend PFC to model rockmaterials, Potyondy and Cundall [232] developed a bonded-particle model (BPM), in which intact rocks are represented asassemblages of cemented rigid particles and the macroscopic frac-turing is simulated as a result of the breakage and coalescence ofthe inter-particle cohesive bonds. The interaction between parti-cles can be characterised by two types of bond models in PFC, i.e.the contact bond model and the parallel bond model. A contactbond serves as a linear elastic spring with normal and shear

strengths, and transmits forces via the contact point betweentwo particles. A parallel bond with certain normal and shear stiff-nesses joins two particles to resist against separation under ten-sion, shear and rotation. The parallel bond model is moresuitable for simulating rock materials as it can capture thecement-like behaviour with resistance to bending moments. Toovercome the original deficiency of PFC in reproducing a realisticrock strength ratio (i.e. the ratio of uniaxial compressive strengthto tensile strength) and macroscopic friction angle, a hierarchicalbonding structure can be built based on a cluster logic [232] or aclump logic [233]. The cluster/clump approach mimics the inter-locking effect of irregular grains by defining a higher value ofintra-cluster bond strength (i.e. the strength between particles inthe same cluster) than that of the strength between clusterboundaries.

The BPM representation using particles with idealised circular/-spherical shapes can introduce unphysical asperities on disconti-nuity surfaces, resulting in an additional resistance to frictionalsliding. To suppress such an artificial roughness effect, a smooth-joint contact model (SJM) was proposed to simulate fracture wallbehaviour based on the actual geometry and morphology of dis-continuities and independent of the arrangement of local particlesin contact [234]. The smooth contact is assigned to all particle pairslying on the fracture interface but belonging to opposite matrixblocks, so that they can overlap and pass through each other(Fig. 10a). The contact forces are calculated based on the relativedisplacements and the smooth-joint stiffness in the normal andtangential directions of the local surfaces. The SJM was found tobe able to capture the shear strength and dilational behaviour ofnatural fractures associated with significant scale effects[235,236]. By simulating intact rock matrix using the BPM and dis-crete fractures using the SJM, a synthetic rock mass (SRM) mod-elling approach (Fig. 10b) has been developed to characterise themechanical properties of fractured rocks including peak strength,damage, fragmentation, brittleness, anisotropy and scale effects[234]. Compared to the Hoek-Brown empirical approach for pre-sumed isotropic rock masses, the SRM method that integratesexplicit DFN representations has an advantage to derive theorientation-specific strength of naturally fractured rocks and con-sider the influence of fracture length distribution and connectivity[237]. The SRM model has been applied to reproduce the failurebehaviour of veined core samples under uniaxial compression tests[238], to estimate the mechanical REV of a jointed rock mass nearan underground facility [239], to simulate fracture propagation injointed rock pillars [240,241], and to evaluate the stability ofwedges around a vertical excavation in a hard rock [242].

An open source particle-based modelling platform named YADE[243,244] has recently been developed as an alternative to the

Fig. 10. Integration of (a) a smooth-joint contact model in PFC to achieve (b) synthetic rock mass (SRM) modelling of fractured rocks with stochastic DFN geometries [234].

Fig. 11. Integration of (a) a fractal DFN into (b) the YADE bonded-particle model (BPM) for mechanical modelling of fractured rocks [134].


commercial code PFC. YADE represents intact rocks using glueddiscs/spheres and models the fracturing process based on the rup-ture of inter-particle bonds with the contact bond algorithms fol-lowing a similar logic to PFC. To reproduce the high ratio of thecompressive strength to the tensile strength and the non-linearfailure envelope of brittle rocks, the concept of ‘‘interaction range”was introduced by Scholtès and Donzé [245]. They mimic themicrostructural complexity by assembling constitutive particlesin neighbouring zones (not only the particles in direct contact). Ajoint contact logic equivalent to the SJM has also been imple-

mented in YADE to avoid the particle interlocking effect betweensliding fracture surfaces [246]. By integrating 3D fractal DFNs intothe YADE BPM model associated with the smooth joint contacttreatment (Fig. 11), Harthong et al. [134] studied the influence offracture network properties (i.e. fractal dimension D, power lawlength exponent a and fracture intensity P32) on the mechanicalbehaviour of fractured rocks. The strength and elastic modulus ofrock masses decrease if P32 increases (i.e. more fractures) or adecreases (i.e. higher proportion of larger fractures), while the spa-tial heterogeneity and scaling of the mechanical properties are


affected by D. Such a combined BPM-DFN model has also beenapplied to analyse the stability of fractured rock slopes, which iscontrolled by the strengths of both pre-existing fractures andintact rocks [247,248].

3.4. Hybrid finite-discrete element models

The hybrid finite-discrete element method (FEMDEM or FDEM)combines the finite element analysis of stress/deformation evolutionwith the discrete element solutions of transient dynamics, contactdetection and interaction. In such a discontinuummodelling scheme,the internal stress field of each discrete matrix block is calculated bythe FEM solver, while the translation, rotation and interaction ofmultiple rock blocks are traced by the DEM algorithms. Pre-existing fractures in rocks are treated as the internal boundaries ofrock volumes. The FEMDEM approach also provides a natural solu-tion route to modelling the transitional behaviour of brittle/quasi-brittle materials from continuum to discontinuum (i.e. fracturingprocesses) by integrating fracture mechanics principles into the for-mulation. This section will review the two most commonly usedFEMDEM models, i.e. the commercial software ELFEN [249] and anopen source platform Y-code [250], which have been broadly usedto simulate the mechanical processes in geological media containingpre-existing discontinuities. Although the two FEMDEM modelsoriginated from the same concept [251], a fundamental differenceexists in their discontinuum implementations: ELFEN is based on apartially-discontinuous discretisation with joint elements insertedonly in pre-existing or propagated fractures, while Y-code is basedon a fully-discontinuous mesh where joint elements are embeddedbetween the interfaces of all finite elements (i.e. along fracturesand inside intact domains). Some other differences also lie in thecomputation of intact material deformation and determination oflocal failure. For more details, the reader is referred to the work byLisjak and Grasselli [25].

3.4.1. ELFENThe ELFEN code models the degradation of an initial continuous

domain into discrete bodies by inserting cracks into a finite ele-

Fig. 12. Integration of DFN geometries into the FEMDEM model of ELFEN for modellinstrength of intact rocks, P21 denotes the fracture intensity, i.e. total length of fractures p

ment mesh. A nodal fracture scheme was introduced by construct-ing a non-local failure map for the whole system [252]. Thefeasibility of local failure is determined based on the evolution ofnodal damage indicators. The fracturing direction (if failure occurs)is calculated based on the weighted average of the maximum fail-ure strain directions of all surrounding elements. A new discretefracture is then inserted along the failure plane with the local meshtopology updated through either the ‘‘intra-element” or ‘‘inter-element” insertion algorithm with adaptive mesh refinementapplied if necessary [253]. ELFEN provides various material consti-tutive models including the elastic, elasto-plastic and visco-plasticlaws, and many brittle/quasi-brittle failure models including therotating crack model, the Rankine material model, and the com-pressive fracture model (i.e. Mohr-Coulomb failure criterion cou-pled with a tensile crack model) [253,254]. The capability ofELFEN for modelling 3D fracturing processes has also been demon-strated through the simulation of laboratory strength tests [255].The explicit DFN fracture geometries generated from e.g. FracMancan be imported into the ELFEN platform by embedding fractureentities into rock solids and representing each fracture as opposedfree surfaces [256,257]. To mesh such complex systems, specialgeometrical treatments may be involved to avoid ill-posed ele-ments caused by subparallel fractures intersecting at a very smallacute angle or a fracture tip terminating at the vicinity of anotherfracture [249]. Both pre-existing and newly propagated fracturesare assigned with contact properties, e.g. fracture stiffness and fric-tion coefficient, to simulate solid interactions through discontinu-ity surfaces [257]. The degradation of natural fractures duringshearing can also be modelled by introducing roughness profiles[258].

The combined FEMDEM-DFN model has been applied to tacklethe geomechanical problems for various engineering applications[259]. The presence of natural fractures may dominate the strengthof slender pillars but have a reduced influence for wider pillars(Fig. 12a) [256]. The orientation and length distribution of DFNfractures also affect the failure mode of the pillar structures, whichcan exhibit splitting with lateral kinematic releases or shearing ofcritically inclined pre-existing fractures linked by new cracks

g strength of (a) a prefractured pillar [256] (note: rci is the uniaxial compressiveer unit area) and (b) an open pit slope [265].


through intact rock bridges [260]. This synthetic numerical modelhas also been used to investigate the progressive failure of rockslopes (Fig. 12b) [261,262], which in reality is usually triggeredby both the reactivation of natural fractures and the propagationof new cracks [263]. The FEMDEM model is well suited to mimicthe staged failure processes of rock slopes including initiation,transportation/comminution and deposition, which involve yield-ing and fracturing of intact materials, shearing of fracture surfacesand translational/rotational instabilities [264]. The rock mass fab-rics and rock bridge properties can have important influences onthe stability of large-scale open pit slopes [261]. The caving-induced rock mass deformations and associated surface subsidencemay be controlled by the orientation of joint sets and the location/inclination of major faults [265]. The FEMDEM-DFN model has alsobeen applied to capture the progressive failure of the roof strata ofa coal mine tunnel [266]. All these engineering applications high-lighted the advantage of the FEMDEM-DFN technique with thecapability of simulating the reactivation/interaction of pre-existing fractures and initiation/propagation of new cracks.

3.4.2. Y-codeDuring the 1990s, many algorithmic solutions for 2D and 3D

FEMDEM simulation were developed by Munjiza et al. [251,267]and Munjiza and Andrews [268,269]. Extensive developmentsand applications of the FEMDEM model have been conducted afterthe release of the open source Y-code [250], and different versionshave emerged including the code developed collaboratively byQueen Mary University and Los Alamos National Laboratory[270–272], the Y-Geo program by Toronto University[25,273,274], and the VGeST (recently renamed ‘‘Solidity”) plat-form by Imperial College London [15,39,275–281]. The Y-codeFEMDEM model accommodates the finite strain elasticity coupledwith a smeared crack model and is able to capture the complexbehaviour of discontinuous rock masses involving deformation,rotation, interaction, fracturing and fragmentation.

In the Y-code, the fractured rock is represented by a discontin-uous discretisation of the model domain using three-noded trian-gular (in 2D) or four-noded tetrahedral (in 3D) finite elementsand four-noded (in 2D) or six-noded (in 3D) joint elements embed-ded at the interface between finite elements. An important differ-ence with the ELFEN code is that the joint elements are inserted forall edges (in 2D) or surfaces (in 3D) of finite elements. The defor-mation of the bulk material is captured by the linear-elasticconstant-strain finite elements with the impenetrability enforcedby a penalty function and the continuity constrained by a constitu-tive relation [267], while the interaction of matrix bodies throughdiscontinuity interfaces is simulated by the penetration calculation[269]. The joint elements are created and embedded between tri-angular/tetrahedral element pairs before the numerical simulation,and no further remeshing process is performed during later com-putations. Pre-existing fractures can be represented by a series ofjoint elements which are initially overlapped (but opposite sidesare separately defined) free surfaces [39]. The brittle failure ofintact materials is governed by both fracture energy parameters(for mode I and mode II failure) and strength properties (e.g. tensilestrength, internal friction angle and cohesion) [25]. A numericalcalibration can be conducted to achieve consistency between theinput material strength parameters and simulated macroscopicresponse [282]. Code development for modelling 3D crack propa-gation has also been achieved by different research groups[272,283–286].

The advantage of the FEMDEM model for simulating the degra-dation of continuum into discrete pieces promoted the applicationto tackle various engineering problems, such as rock blasting [287],fracture development around excavations in isotropic/anisotropicintact rocks [288–291] and mountain slope failure [292]. It has also

been used to model the mechanical behaviour of fractured rocksembedded with pre-existing fractures. Lisjak et al. [293] integratedDFN crack arrays into the FEMDEMmodel to imitate the anisotropyof an argillaceous rock. The FEMDEM technique has been applied tomodel geologically-mapped DFNs under various stress conditions[15,39,40,280,294]. It can capture realistic geomechanical phe-nomena such as deformation and rotation of matrix blocks, open-ing, shearing, and dilation of pre-existing fractures as well as thepropagation of new cracks [281]. Due to the presence of naturalfractures, strongly heterogeneous stresses are distributed in therock mass (Fig. 13a) [39]. The in-situ stress can introduce newcracks into the natural fracture system which potentially increasesthe network connectivity (Fig. 13b) [15,280]. Highly variable aper-tures can also be generated (Fig. 13c), which is related to the com-plexity of DFN geometries and the condition of far-field stresses[39,278,280]. The stress-dependent mechanical response of frac-tured rocks (e.g. deformation and rotation of matrix blocks, open-ing, shearing and dilation of pre-existing fractures and new crackpropagation) can significantly affect the fluid flow in fracture net-works, which will be discussed in more details in Section 4.

3.5. Summary with a view towards HM modelling

As can be seen from above, geomechanical modelling of frac-tured rocks can be achieved by continuum or discontinuumapproaches, which have important differences in the conceptuali-sation of geological media and the treatment of displacement com-patibility [12]. Table 2 presents a comparison of the continuum anddiscontinuum models. The continuum modelling scheme mainlyreflects the material deformation of a geological system from amore overarching view and attempts to bypass the geometricalcomplexity by using specific constitutive laws and equivalentmaterial properties derived from homogenisation techniques.However, it cannot adequately consider the effects of stress varia-tions, fracture interactions, block displacements and rotations.More importantly, the applicability of a homogenisation processis based on the assumption of an REV, which may not exist for nat-ural fracture systems [73]. On the other hand, the discontinuumscheme treats the system as an assemblage of interacting individ-ual components and permits the integration of complex constitu-tive laws for rock materials and fracture interfaces. Adiscontinuum model can be established at a specific scale of inves-tigation without presuming the existence of an REV. However,some of the input parameters (e.g. bonding strength, joint stiff-ness) may need to be determined by indirect numerical calibra-tions rather than from physical measurements. Furthermore, thecomputational time for solving discontinuous problems can beconsiderably larger than that for continuum models. To takeadvantage of the two modelling technologies for tackling practicalissues, a discontinuum model may be used to derive the REV size(if it exists) as the onset to treat a geological system as acontinuum.

In the conventional block-type DEM and DDA modelling, therock mass is dissected into blocks through topological identifica-tions, during which the dead-ends and isolated fractures are usu-ally removed for simplicity [44,199,206,221,227]. Suchtreatments may be acceptable for modelling extremely hard rocksor highly fragmented rocks, for which the effect of new crack prop-agations may be negligible. However, for most rock types, fracturesare likely to propagate from the tips of pre-existing fractures dri-ven by concentrated local stresses. This fracture propagation andcoalescence process can create new pathways for fluid migrationand thus affect the hydrological properties (e.g. permeability) offractured rocks, as has been revealed in numerical simulations[15,39]. Furthermore, the curved natural fractures, which are oftenassumed to be straight line (in 2D) or planar disc (in 3D) in the

Fig. 13. FEMDEM-DFN modelling results. (a) Heterogeneous distribution of local maximum principal stresses in fractured rocks represented by a geologically-mapped DFN(the upper array of figures) and a statistically-equivalent random DFN (the lower array of figures) under in-situ stresses applied at different angles to the model [39]. (b)Variation of fracture apertures in a geologically-mapped DFN network that accommodates further new crack propagations in response to a biaxial stress condition [15]. (c)Highly inhomogeneous aperture distribution within a single fracture of a 3D persistent DFN system under a polyaxial stress condition [278].


Poisson DFN model, may accommodate localised dilations underhigh differential stresses with localised increases in aperture andthis may have important HM effects [15,47,280]. Hence, towardsa better HM modelling of fractured rocks, the capability of pre-cisely modelling the geomechanical response is prerequisite, forwhich the important characteristics of fracture dead-ends and cur-vatures need to be appropriately treated.

4. Modelling of the hydrological behaviour of fracture networksunder geomechanical effects

The presence of fractures can generate stress perturbations inthe rock, such as rotation of stress fields, stress shadows arounddiscontinuities and stress concentration at fracture tips [5]. Theresulting heterogeneous stress distribution may lead to variablelocal normal/shear stresses loaded on different fractures havingdistinct sizes and orientations, and produce various fractureresponses such as opening, closing, sliding, dilatancy and propaga-tion. Since the conductivity of fractures is critically dependent onthe third power of fracture apertures [295], the geomechanical

conditions can considerably affect the hydrological properties offractured rocks including fluid pathways, bulk permeability andmass transport [207]. Numerical models that integrate explicitDFNs and non-linear rock/fracture behaviours provide powerful(and so far irreplaceable) tools to investigate the geomechanicaleffects on fluid flow in complex fracture networks [296].

4.1. Fluid pathways

Fracture networks usually serve as the major pathways for fluidtransport in subsurface rocks, especially if the matrix is almostimpermeable compared to the fractures [297]. The partitioning offluid flow within a fracture population relies on the spatial connec-tivity of fracture geometries and the transmissivity of individualfractures, both of which can be affected by the geomechanicalconditions.

Zhang et al. [45] used the UDEC DEM code to study the defor-mation of a fractured rock based on a geologically-mapped DFNpattern and found the closure of fractures under applied in-situstresses can re-organise the fluid pathways. They also found the

Table 2Comparison of different numerical models for DFN geomechanical modelling.

Numerical models Key inputs Strengths Limitations

Continuum models Equivalent material properties (i) Simplicity of geometries.(ii) Efficient calculation.(iii) Suitability for large-scale industrial

applications.

(i) No consideration of fracture inter-action, block displacement/inter-locking/rotation.

(ii) Complexity in deriving equivalentmaterial parameters and constitu-tive laws.

(iii) Valid only if an REV exists.

Block-type &particle-baseddiscrete models

Material properties for both fractures androcks, damping coefficient, bonding strengths

(i) Explicit integration of DFNs.(ii) Simple particle/grain bonding logic.(iii) Integrated constitutive laws for rocks/

fractures.(iv) Capturing the interaction of multiple

fractures.

(i) Limited data on joint stiffnessparameters.

(ii) Calibration of input particle bond-ing properties.

(iii) No fracture mechanics principle.(iv) Large computational time.

Hybrid FEMDEMmodels

Material properties for both fractures androcks, fracture energy release rate, dampingcoefficient

(i) Explicit integration of DFNs.(ii) Fracture propagation is based on both

the strength criterion and fracturemechanics principles.

(iii) Integrated constitutive laws for rocks/fractures.

(iv) Capturing the interaction of multiplefractures.

(i) Limited data on joint stiffnessparameters.

(ii) Calibration of fracture energyrelease rates.

(iii) Large computational time.


closed joints of one set can hinder fluids from passing through notonly themselves but open fractures of another set due to the neteffect. Zhang and Sanderson [43] applied the same technique todifferent types of outcrop DFN patterns with/without systematicfracture sets. The fluid flow in systematic fracture networks tendto be dominated by the primary joint set if it is relatively open,while for non-systematic networks containing fairly randomly ori-ented small fractures, the flow channels tend to align the directionof the maximum principal stress. Min et al. [206] incorporated thefracture shear dilation behaviour in the DEM modelling of astochastic DFN having a power law distribution of fracture lengthsand a uniform distribution of initial (i.e. zero stress) apertures.They observed that, under high differential stresses, a small portionof fractures which have critical/near-critical orientations, goodconnectivity and long lengths would dilate and form large flowchannels (Fig. 14a). The ‘‘critical orientations” here correspond toa range of fracture orientations that would allow discontinuitieswith no cohesive strength to slide under the given in-situ differen-tial stress condition. The localised features would be augmented ifthe initial apertures are broadly distributed (e.g. following a log-normal distribution) and correlated with fracture lengths [98].Latham et al. [15] employed the FEMDEM method combined witha smeared crack model to study the geomechanical response andfluid flow in an outcrop-based DFN. They found that bent naturalfractures under high differential stresses may exhibit evident dila-tional jogs and can be linked by newly propagated cracks to formmajor fluid pathways. Similar localised flow channels created bythe connection of pre-existing fractures have also been observedby Figueiredo et al. [41] using a FDM simulator. Sanderson andZhang [47,96] calculated the fluid flow in the third dimension ofsedimentary rocks using an analytical pipe formula based on thedeformed 2D fracture networks. They found vertical flow becomesextremely localised when the pore fluid pressure exceeds a criticallevel and very large aperture channels emerge (especially in somepreferentially-oriented and well-connected curved fractures)(Fig. 14b). The flow distribution also exhibits significant multifrac-tality when the loading condition approaches the critical state. Leiet al. [278] studied the fluid flow in a 3D persistent fracture net-work under various polyaxial stress conditions. They observed atransition from homogeneous flow under an isotropic stress fieldto highly localised patterns under a deviatoric stress condition

(Fig. 14c). Recently, Lei et al. [280] also studied the impacts ofstress on reorganising vertical fluid flow through a 3D sedimentarylayer.

4.2. Permeability

There are two different notions of rock mass permeability, i.e.equivalent permeability and effective permeability. The equivalentpermeability is defined as a constant tensor in Darcy’s law to rep-resent flow in a heterogeneous medium, while the effective perme-ability is an intrinsic material property based on the existence of anREV at a large homogenisation scale [298]. Permeability heremainly refers to the equivalent permeability of a fractured rockat a specific study scale.

The permeability tensor was observed to be highly dependenton both the geometrical attributes of fracture networks (e.g. den-sity, lengths and orientations) and the in-situ stress conditions(e.g. direction, magnitude and ratio of the principal stresses) [45].When the differential stress ratio is relatively low, the permeabilitydecreases with the increase of burial depth (or mean stress) of thefractured rock due to the closure of most fractures [43,206]. Thenon-linear relationship between normal stress and fracture closureresults in a phenomenon that the permeability is more sensitive atshallower depths (i.e. smaller mean stresses) and approaches aminimum value when most fractures are closed to their residualapertures under high mean stresses (Fig. 15a) [206]. The perme-ability anisotropy of a fracture network with non-systematic frac-tures is more dependent on the ratio and direction of the appliedprincipal stresses than that of a network with systematic fracturesets which is more controlled by the fracture set orientations[43]. With the increase of differential stresses, the permeabilityexhibits a decrease and then an abrupt increase separated by a crit-ical stress ratio that begins to cause continued shear dilationsalong some preferentially oriented fractures (Fig. 15b) [206]. Asimilar variation of permeability occurs when the pore fluid pres-sure is elevated [41]. Simultaneously, the permeability anisotropyis also enlarged by the increased stress ratio [206]. If initial aper-tures are correlated with fracture lengths, the permeability of frac-tured rocks is dominated by larger fractures with wider apertures.This model tends to exhibit a permeability value much higher thanthe model assigned with a constant initial aperture (Fig. 15c) [98].

Fig. 14. Fluid pathways in fracture networks under in-situ stresses. (a) With the increase of the boundary stress ratio, fluid flow becomes more concentrated in only part ofthe fractures in the network due to the shear dilation effect [206]. (b) The vertical fluid flow through a jointed layer exhibits a highly localised pattern when the fractured rockis deformed under a critical stress state [47]. (c) The vertical flow in a 3D persistent fracture network shifts from a homogeneous to localised pattern with the increase of thehorizontal stress ratio [278].


The permeability tensor may be destroyed but then reestablishedwith the increase of differential stresses [98]. In addition to theshear dilation of fractures, the increase of network connectivitycaused by brittle failure and crack propagation under geomechan-ical loading can also significantly raise the permeability of frac-tured rocks [15,41,151,157]. More interestingly, the emergence ofdilational jogs/bends associated with curved fractures in responseto high differential stresses may lead to even more increase of per-meability in the third dimension [47,96,280] and result in a highlyanisotropic permeability tensor [278]. The permeability of 3D frac-ture networks also exhibits a transition stage having steep growththat occurs when the stress ratio is approaching the critical state(Fig. 15d) [278]. It has to be mentioned that an increased fracturedensity may not always lead to an increased permeability undersome tectonic conditions (e.g. an extensional regime), becausefractures may be closed due to mechanical interactions when theyare too densely spaced [299].

4.3. Transport

Mass transport in fractured rocks is governed by various mech-anisms including advection, dispersion, matrix diffusion, interfacesorption and chemical reaction [17,297]. The heterogeneous fluidvelocity fields in geological media consisting of distributed frac-tures and porous rocks can result in complex transport phenomenain the system. Computational models employing solute compo-nents or tracked particles have been developed to simulate migra-tion processes in fractured rocks [6,300]. Recently, numericalstudies have also been conducted to investigate the effects of thestress/deformation on the transport properties of fracture net-works, as summarised below.

Zhao et al. [208] coupled the UDEC DEM code and a randomwalk particle tracking code PTFR and investigated the stress effectson the hydrodynamic dispersion of contaminant solutes in astochastic DFN system. They found that compressive stresses canclose fracture apertures and attenuate the dispersivity, but anincreased differential stress ratio could greatly intensify thespreading phenomenon if it exceeds certain threshold for trigger-ing shear dilations. Zhao et al. [209] extended this modelling tech-nique to incorporate the effects of matrix diffusion and sorption,and conducted a systematic study of the solute transport undervarious stress conditions. Their results showed that the stress cansignificantly affect the solute residence time in the fracture net-work (Fig. 16a). When the stress ratio is increased but not veryhigh (<3), the breakthrough curve shifts to the right side (i.e. theaverage residence time increases) compared to that of the initialzero-stress condition, due to the closure of fractures under rela-tively isotropic stresses. However, as the stress ratio exceeds 3,fluid velocity is raised drastically in some dominant channelsformed by dilated fractures due to shearing, and thus the residencetime decreases with the breakthrough curve shifting backward.They also observed that the breakthrough curve for interactingtracers (i.e. with matrix diffusion) exhibits longer tails than thatfor non-interacting tracers (i.e. without matrix diffusion) due tothe meandering of a small amount of particles passing through tor-tuous fluid pathways. Such long tail phenomena were more signif-icant when the pressure gradient is small, for which the matrixdiffusion tends to play a dominant role in solute transport(Fig. 16b). Rutqvist et al. [101] used an extended multiple interact-ing continua model combined with the crack tensor approach tosimulate the advection-dominated transport (under a high hydrau-lic gradient) and diffusion-retarded transport (under a low hydrau-lic gradient). In addition to the stress-dependent transport

Fig. 15. Variation of permeability of fractured rocks in response to the change of stress conditions. (a) Permeability change versus stress change with a fixed principal stressratio of 1.3 [206]. (b) Permeability change with the increase of stress ratio for a DFN with a constant initial aperture [206]. (c) Permeability change with the increase of stressratio for a DFN with a lognormally distributed and length correlated apertures under rotated stress fields [98]. (d) Increase of vertical permeability with the increase ofhorizontal stress ratio for a 3D persistent fracture network [278].

Fig. 16. Breakthrough curves for interacting tracers (i.e. with matrix diffusion) in a 2D DFN network stressed by various ratios of horizontal to vertical stresses (i.e. K) under ahorizontal hydraulic pressure gradient of (a) 1 � 104 Pa/m, and (b) 10 Pa/m [209]. (Note: c0 is the initial concentration along upstream boundary, c is the concentrationobserved at the downstream boundary, K = 0 denotes a zero stress condition, and the two figures have different time scales).



behaviour, they also observed a delayed breakthrough in low pres-sure gradient scenarios due to the residence of solutes in the por-ous rock matrix. Wang et al. [301] further applied thismulticontinuum method to demonstrate the contribution of inac-tive fractures (i.e. isolated cracks or dead-ends of fractures) forstagnating solutes by providing additional surface areas for diffu-sive transfer into/out of matrix pores. Zhao et al. [102] comparedthe stress-flow models of five different research groups, whichshowed consistency in predicting the stress dependency of masstransport in a fractured rock. Recently, Kang et al. [302] studiedthe impact of in-situ stresses and aperture changes on the anoma-lous transport in a fractured rock based on an outcrop-based DFNmodel. Apart from the stress-induced aperture change that canaffect the transport behaviour, Nick et al. [162] found that thepropagation of fractures under tectonic loading can also vary thebreakthrough properties due to the increased fracture densityand network connectivity.

4.4. Comments on the use of DFNs for HM modelling

The geomechanical effects on fluid flow in fractured rocks asobserved in numerical simulations demonstrate the importanceof using appropriate fracture network representations and con-ducting systematic geomechanical computations for characterisingthe coupled HM behaviour of fractured geological formations. Sim-ulation results rely strongly on the suitability and accuracy of theconstructed DFNs and the geomechanical models of rocks andfractures.

Natural fracture networks have been observed to exhibit fractalcharacteristics and long-range correlations [8,31,32,73,124–127,130]. The fractal (or clustered) characteristics have been found tohave significant impacts on the connectivity, permeability and flowpattern of fracture networks [131–133]. The scattered distributionof fracture sizes (e.g. power law distribution) also considerablyaffects the hydrological [92–94], geomechanical [134] and HM[40] behaviour of fractured rocks. These important features mayneed to be prudently incorporated when creating DFNs. Thehydraulically inactive fractures (i.e. dead-end and isolated frac-tures) were found playing important roles in propagating new frac-tures that can link with pre-existing fractures[15,39,41,151,161,280], and in transferring fluids and solutesinto/out of matrix pores [101,171,301]. Thus, they may need tobe preserved in DFNs for HMmodelling of fractured rocks. The bentfractures that can accommodate large shear dilations and inducelocalised fluid flows [15,47] may also need to be adequately char-acterised in DFN representations. Such curvature characteristicsmay provide additional surface areas for diffusive transfer andare therefore important for solute transport modelling. Specialattention is needed when defining fracture apertures in DFNs forHM modelling. Fracture apertures have been suggested to haveintrinsic correlations with fracture sizes [5,80], which tend to exhi-bit a significant power law scaling over a broad spectrum of lengthscales [8,73]. This long-range correlation has been found to havecritical influences on the permeability and flow distribution of frac-ture networks [93,94,97,99] as well as their stress dependency[40,98]. In addition, the geomechanical simulation tool also needsto be carefully adopted or developed for HMmodelling. The follow-ing aspects may be of concern: (1) whether the elastic/elasto-plastic deformation of rock matrix can be modelled; (2) whetherthe non-linear deformation of natural fractures (opening/closing,shearing and dilation) can be captured; (3) whether the fracturepropagation can be simulated; and (4) whether the geomechanicalmodel can be coupled with fluid flow solvers.

It can be noted that this section is focused on the research ofmodelling geomechanical impacts on the hydrological behaviourof fractured rocks. The reverse effect of fluid flow on rock mass

deformations is also a very important issue which is relevant tomany engineering problems such as hydraulic fracturing, CO2sequestration and reservoir compaction. Recent developments inexploitation of shale gas resources have highlighted the demandfor a better understanding of the interaction between hydraulicfractures and natural fractures. Several studies have been done todevelop coupled HM models to simulate fluid-driven fracturepropagations in pre-existing DFN networks [185,186,303–308].

5. Outstanding issues and concluding remarks

Modelling the geomechanical evolution and the resultinghydrological characteristics of fractured rocks is a challengingissue. The previous HM simulation results have shown consistencywith field measurements: (i) only a small portion of fractures areconductive [6,144], (ii) permeability is less sensitive to depth indeep rocks [2], and (iii) critically stressed faults tend to have muchhigher hydraulic conductivity [16,309]. However, a number ofimportant issues of difficulty still exist in applying DFNs for HMmodelling of fractured rocks. The first task will be the developmentof realistic DFNs that can represent the important geometrical andgeostatistical characteristics of natural fracture systems (e.g. clus-tering, scaling, connectivity, intersection, termination, curvatureand correlation), and that can further be used to capture the keygeomechanical and hydrological characteristics of fractured rocks(e.g. strength, deformation, permeability and mass transport).The geometry of the created DFNs needs to be consistent withthe geomechanics such as the in-situ stress field so that thegeometrically- and geomechanically-dependent fluid flow can beproperly modelled. Another critical issue is to develop appropriateupscaling approaches to DFN models to evaluate the large-scalebehaviour, and this would require the preservation of geostatisticaland geomechanical characteristics. Some techniques have beenproposed to construct large-scale heterogeneous models basedon upscaled stress-dependent permeability tensors of local gridblocks [310–312] or using self-referencing random walks associ-ated with stress-dependent aperture information [40]. More effortsare also needed in the future for many other aspects, such as devel-oping more advanced ‘‘two-way” coupling schemes, modellinggeomechanical effects on multiphase flow and, importantly, exten-sion to complex 3D systems.

The development of calibration and validation procedures forDFN models based on field measurements is a crucial issue if thenumerical models are to be used for practical applications. TheDFNmodel can be examined by comparing the geometrical proper-ties of a simulated 2D trace network with those of the trace patternexposed on an outcrop or excavation wall [117,138,141]. Mine-byexperiments in underground research laboratories [313,314] pro-vide also valuable patterns of fracture traces that intersect tunnelwalls and measurement data of rock mass mechanical responses(e.g. breakout characteristics, spalling depth, tunnel radial defor-mation, etc.). Such data might be very useful for the DFN calibra-tion study. Hydraulic test data such as groundwater inflowrecorded at boreholes/shafts, pressure variations and derivativesobtained from well/interference tests, and particle concentrationmonitored during tracer tests can also be used to tune the hydro-logical performance of DFN models [138–140,142–144]. Further-more, the oil recovery case history information in reservoirengineering [315] offers useful references for the calibration ofDFN models used for multiphase flow simulation. Recently, aninverse model constrained by in-situ hydraulic data from sequen-tial pumping tests has been developed to derive the spatial distri-bution of flow channels on the bedding plane of a karstic formation[316]. This new technique can be employed to calculate best-fitaperture distributions for DFN models based on an optimisation


process using multiple objective functions. Other possible calibra-tion approaches may include those based on heat transfer experi-ments and electrical resistivity measurements of fractured rocks.

In summary, the present paper began by presenting an over-view of various discrete fracture network (DFN) models for simu-lating the geometry and topology of natural discontinuitysystems. Different continuum and discontinuum approaches thatintegrate DFN geometries to model the geomechanical behaviourof fractured rocks were then surveyed. Numerical results of thefracture-dependent mechanical response and stress-dependenthydrological characteristics of fractured rocks suggest that it isimportant to use appropriate DFN representations and conductgeomechanical computations to better characterise the bulk beha-viour (e.g. strength, deformation, permeability and mass transport)of highly disordered geological media embedded with naturallyoccurring discontinuities. Some of the outstanding issues that needto be addressed in the near future will be those related to genera-tion of more realistic DFNs, improvement of computational effi-ciency, study of coupled multiphysics processes, calibration andvalidation of numerical models as well as upscaling for large-scale practical applications.

Acknowledgements

We thank the industry consortium for supporting part of thelead author’s PhD research on the project of ‘‘Improved simulationof faulted and fractured reservoirs, itf-ISF Phase 3: gravity assistedrecovery processes”. The lead author also acknowledges the JanetWatson scholarship awarded by the Department of Earth Scienceand Engineering, Imperial College London. The second author isgrateful for the NERC RATE HYDROFRAME project of ‘‘Hydrome-chanical and biogeochemical processes in fractured rock massesin the vicinity of a geological disposal facility for radioactive waste”and the EU H2020-SURE project of ‘‘Novel productivity enhance-ment concept for a sustainable utilization of a geothermalresource”. The third author gratefully acknowledges the supportof the Swedish Geological Survey (SGU) through its grant, number1724, to Uppsala University.

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