Implementation of an Empirical Joint Constitutive Model into Finite-Discrete Element Analysis of the Geomechanical Behaviour of Fractured Rocks
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An empirical joint constitutive model (JCM) that captures the rough wall interaction behaviour of individual fractures associated with roughness characteristics observed in laboratory experiments is combined with the solid mechanical model of the finite-discrete element method (FEMDEM). The combined JCM-FEMDEM formulation gives realistic fracture behaviour with respect to shear strength, normal closure, and shear dilatancy and includes the recognition of fracture length influence as seen in experiments. The validity of the numerical model is demonstrated by a comparison with the experimentally established empirical solutions. A 2D plane strain geomechanical simulation is conducted using an outcrop-based naturally fractured rock model with far-field stresses loaded in two consecutive phases, i.e. take-up of isotropic stresses and imposition of two deviatoric stress conditions. The modelled behaviour of natural fractures in response to various stress conditions illustrates a range of realistic behaviour including closure, opening, shearing, dilatancy, and new crack propagation. With the increase in stress ratio, significant deformation enhancement occurs in the vicinity of fracture tips, intersections, and bends, where large apertures can be generated. The JCM-FEMDEM model is also compared with conventional approaches that neglect the scale dependency of joint properties or the roughness-induced additional frictional resistance. The results of this paper have important implications for understanding the geomechanical behaviour of fractured rocks in various engineering activities.
KeywordsFinite-discrete element method Joint constitutive model Fractures Roughness In situ stress
List of symbols
Cohesive bonding forces
External nodal forces
Internal nodal forces
- fn, ft
Normal and tangential contact forces
Unit base vectors
Lumped nodal mass matrix
Vector of coordinate difference
Vector of nodal displacements
Cauchy stress tensor
Mobilised friction coefficient
Mobilised tangential dilation angle
- GI, GII
Mode I, mode II energy release rates
Joint compressive strength
Joint compressive strength at the laboratory scale
Joint compressive strength at the field scale
Joint roughness coefficient
Joint roughness coefficient at the laboratory scale
Joint roughness coefficient at the field scale
Mobilised joint roughness coefficient
Initial joint normal stiffness
Current joint normal stiffness
- knt, ktn
Current joint normal–shear stiffness
Current joint shear stiffness
Laboratory joint length
Field joint length
Current shear displacement
Peak shear displacement
Accumulative joint normal displacement
Joint maximum allowable closure
Joint normal displacement induced by normal stress
Joint normal dilational displacement corresponding to the peak shear displacement
Joint normal displacement induced by shear dilation
Mesoscopic (grid scale) opening displacement
- σ′x, σ′y
Effective far-field principal stresses
Uniaxial compressive strength
Normal stress on fracture walls
Current shear stress
Peak shear strength
Internal friction angle
Residual friction angle
Fractures are ubiquitous in crustal rocks in the form of faults, joints, and veins over different length scales (Lei and Wang 2016). Conceptually, there are two physical domains involved: fractures with relatively low stiffness and high porosity, and rock matrix with the opposite characteristics. The hydromechanical properties of fractured rocks are often governed by both the behaviour of individual fractures and the interactions among the fracture population (Zimmerman and Main 2004). The dominant role of fractures in crustal processes, such as in localising deformation and fluid flow (Sibson 1994), has important implications for various engineering applications including hydrocarbon production, geothermal energy, groundwater management, nuclear repository safety, and ground engineering.
To describe the behaviour of individual fractures associated with intrinsic surface roughness, many studies have been reported in the past few decades. Goodman (1976) proposed a hyperbolic relation to characterise the nonlinear closure of fractures under normal compression and studied the effect of mismatch between opposite rough joint walls. Barton and Choubey (1977) introduced an empirical system based on three main index parameters, i.e. joint roughness coefficient (JRC), joint wall compressive strength (JCS), and residual friction angle, to predict the shear strength of natural fractures. These parameters can be measured in the laboratory from tilt tests or shear box experiments. Bandis et al. (1983) summarised a series of empirical equations to interpret the deformation characteristics of rock joints in normal loading and direct shear experiments. Size effect on shear strength and deformation characteristics of individual fractures were further investigated based on the laboratory experiment results conducted on natural fracture replicas that were cast at different sizes (Bandis 1980; Bandis et al. 1981; Barton 1981). Fractures having the same roughness morphology but different sizes may exhibit distinctly different mechanical responses (Barton and Bandis 1980; Barton 2013). The empirical joint constitutive model was recently improved to capture the stress dependency of peak shear displacement (Asadollahi and Tonon 2010). In addition to the laboratory experiments, numerical studies have also been performed to model the behaviour of single fractures with the roughness profile represented explicitly. Karami and Stead (2008) used a finite-discrete element code to simulate the shearing process of rough joint specimens. Joint shear behaviour, e.g. peak shear displacement, shear strength, dilational displacement, and asperity degradation, was found to be dominated by the roughness geometry and the applied normal stress. Bahaaddini et al. (2014) studied the scale effects on joint shear strength and peak shear displacement using a particle-based discrete element model. Their numerical results reveal that with the increase in joint length, the peak shear strength and shear stiffness decrease, while the peak shear displacement increases, similar to the phenomena observed in the laboratory experiments. Tatone and Grasselli (2012, 2015a) investigated the scale-dependent shear strength and the asperity damage mechanism of rough fractures in direct shear tests, using a numerical modelling approach that is based on a finite-discrete element model comprehensively calibrated for tensile and shear fracturing and employs explicit representation of fracture surfaces. The numerical solutions of shear strength and dilation were further compared with the experimental results from micro-CT imaging (Tatone and Grasselli 2015a).
The overall behaviour of fractured rocks involving multiple interconnected fractures has also been widely investigated based on various numerical methods. Sanderson and Zhang (1999) used the discrete element method (DEM) to analyse the deformation of naturally fractured rocks, and they found very large apertures are generated when the fracture system is under a critical stress state. Min et al. (2004) applied the DEM model to analyse the influence of far-field stresses on the distribution of fracture apertures with the effects of nonlinear normal deformation and shear dilation considered. Harthong et al. (2012) combined the particle-based DEM method with discrete fracture networks (DFNs) to study the strength characteristic of pre-fractured rocks under triaxial stress loading. Latham et al. (2013) employed the finite-discrete element method (FEMDEM) to simulate the geomechanical response of a natural fracture system with the consideration of bent fractures that accommodate high dilation, and crack propagation that can connect pre-existing fractures and increase network connectivity. Lei et al. (2014) examined the stress effect on the validity of synthetic networks for representing a two-dimensional (2D) naturally fractured rock in terms of geomechanical and hydraulic properties. The geomechanically induced fracture apertures captured by the 2D FEMDEM model were further scaled up for larger scale modelling of fluid flow in a self-referencing multi-scale system of fracture networks (Lei et al. 2015a). Some preliminary work on modelling the geomechanical behaviour of three-dimensional (3D) fracture networks has also been conducted using the 3D FEMDEM solver (Lei et al. 2015b, 2016).
In summary, previous research has focused on two separate scales at which fracture properties affect rock mass strength and deformation: (1) the level of the individual fracture where surface roughness is represented in detail and (2) the level of fracture network with emphasis on the overall properties. For more realistic modelling of fractured rocks, it is of importance to integrate the detailed characteristics of individual fractures into the simulation of complex fracture systems (Jing and Stephansson 2007). In the past few decades, a few attempts have been made to bridge these two scales in numerical modelling (Saeb and Amadei 1992; Cai and Horii 1992; Jing et al. 1994). Some pioneer work has also been done by Mahabadi and Grasselli (2010) to implement an empirical shear strength criterion into the FEMDEM model. However, to better characterise the nonlinear deformation of rock fractures, some important aspects, such as the mobilisation of friction coefficient during shearing processes and the scale dependency of fracture roughness properties, may need to be more adequately considered.
The objective of this study is to develop a methodology to incorporate realistic joint constitutive characteristics in the numerical simulation of fractured rock masses involving pre-existing and propagating fractures. A novel feature of the proposed method is its capability of simulating the important size effect of fracture wall properties as observed in laboratory experiments through a systematic characterisation of fracture network topologies. The paper is organised as follows. In Sect. 2, approaches for mechanical modelling of multi-body systems are briefly discussed covering the issues of solid deformation, interaction, and fracture propagation. The constitutive models for rock fractures are reviewed with respect to normal closure, shear deformation, and dilatancy. A scheme to couple the empirical joint constitutive model (JCM) with the FEMDEM computation is then formulated. Section 3 presents a verification of the proposed numerical method. In Sect. 4, a natural fracture pattern involving intersections, bends, and roughness-induced initial apertures is presented. Numerical experiments are designed to illustrate various in situ stress conditions for which effects of far-field stresses on fracture system properties are investigated. A comparison with conventional joint modelling approaches is also presented. Finally, a brief discussion is given, and conclusions are drawn.
2 Numerical Methods
2.1 Finite-Discrete Element Method (FEMDEM)
2.1.1 Governing Equation
2.1.2 Contact Force
2.1.3 Crack Propagation
Crack propagation induced by stress concentration is modelled by a smeared crack model (Munjiza et al. 1999) embedded in the FEMDEM formulation with both mode I and mode II brittle failure captured (Latham et al. 2013). Fracture initiation and propagation is characterised as occurring in three stages: (1) the continuum stage simulated by the finite element method through the solid constitutive law, (2) the transition stage described by the strain softening using the smeared crack model, and (3) the discontinuum stage in which elements along the new crack are physically separated with their interaction further modelled by the discrete element method (Munjiza 2004).
2.2 Joint Constitutive Model (JCM)
2.2.1 Joint Normal Deformation
2.2.2 Joint Shear Deformation
Dimensionless model for shear stress–shear displacement modelling (Barton et al. 1985)
−ϕ r/[JRCn log(JCSn/σ n)]
2.2.3 Joint Shear Dilatancy
2.2.4 Coupled Joint Normal and Shear Behaviour
2.3 Combined JCM-FEMDEM Formulation
2.3.1 Joint Element
2.3.2 Characterisation of Fracture Systems Based on a Binary-Tree Search
Due to the scale dependency of fracture parameters such as JRC, JCS, and peak shear displacement (as discussed in Sect. 2.2), it is important to precisely characterise the distribution of effective fracture lengths (i.e. size of a block edge between fracture intersections) in the numerical modelling of a disordered, interconnected fracture system. One critical numerical difficulty related to effective fracture lengths is to distinguish the sophisticated topological relations of what is very often a complex system containing numerous joint elements, in which some pre-existing fracture joint elements may connect with each other to form a continuous fracture wall (i.e. block edge) and would act together as an equivalent individual fracture with two facing walls.
2.3.3 Coupling Between JCM and FEMDEM
3 Numerical Verification
The empirical constitutive laws are implemented in the FEMDEM framework at the joint element scale, but the consistency between the simulated macroscopic fracture behaviour and the empirical formulations requires a detailed verification. The consistency between the empirical formulations and the laboratory experiments has been well demonstrated in the literature (Bandis et al. 1981, 1983; Barton et al. 1985; Olsson and Barton 2001). Hence, the validity of the numerical model will be examined by comparing numerical results with the empirical solutions, i.e. Eq. (13) for the shear stress, the integral of Eq. (18) for the dilational displacement, and Eq. (5b) for the normal closure.
To sum up, the consistency of the numerical results with the empirical solutions demonstrates the performance of the combined JCM-FEMDEM formulation for capturing realistic shear strength and normal closure behaviour of single fractures, although it is recognised that it would be ideal to further test the model over a parameter space with different JRC, JCS, normal stresses, etc. In the following section, the numerical model will be applied to simulate the geomechanical behaviour of a complex fracture network under in situ stresses.
4 Application to Geomechanical Modelling of a Natural Fracture Network
4.1 The Natural Fracture Network
Material properties of the fractured limestone
Young’s modulus E
Poisson’s ratio υ
Internal friction angle ϕ int
Tensile strength f t
Mode I energy release rate G I
Mode II energy release rate G II
Residual friction angle ϕ r
Laboratory sample length L 0
Initial aperture a 0
In the geological setting, fractures often exhibit displacements perpendicular and/or parallel to the discontinuity surface, i.e. aperture and shear displacement. In this study, fractures are represented with no initial phase of shearing before the phases of far-field stress application. However, an initial aperture is considered significant and is assigned a priori to all fractures equally to enable the introduction of a potentially realistic joint aperture that is in turn controlled by the roughness characteristic. This aperture is assigned a value based on the empirical relation (Bandis et al. 1983) given by Eq. (10). Fractures are modelled as uncemented interfaces, i.e. they have no cohesion or tensile strength.
4.2 Model Set-Up and Simulation Results
4.3 Comparison with Conventional Joint Modelling Approaches
Stress-dependent heterogeneity of fracture opening and shear displacement in a naturally fractured rock has been captured by the 2D JCM-FEMDEM model that incorporates both the network-scale mesoscopic effect (e.g. orientations, spacing, junctions, dilational bends, and jogs) and the roughness-scale microscopic effect (e.g. roughness-controlled aperture closure and dilatancy). Integration of the realism of joint constitutive characteristics is considered to give more realistic results compared to conventional approaches that neglect the scale dependency of joint properties and/or the roughness-induced additional frictional resistance as well as its shearing-dependent degradation. The results of the model in Phase II-B (Fig. 9c, f) with a critical far-field stress ratio, i.e. σ′ x /σ′ y = 3, are of particular interest. The system finds equilibrium by activating sliding with local extremes of shear displacement on the favourably orientated joint set 2 as highlighted in Fig. 9c (see further discussion of joint orientation effects in Lei et al. 2014). Locally, the sliding on the two sets has created large apertures in some active fractures as well as their intersections (Figs. 9f, 10), which shows consistency with the field observation from boreholes that critically stressed faults with favourable orientations appear to have larger apertures and higher hydraulic conductivity (Zoback 2007).
The formation of large apertures along displacing and dilating fractures illustrated by the 2D model implies that localised flow might occur in the vertical direction and a higher permeability is expected in the third dimension of the strike-slip faulting system (Sibson 1994), as demonstrated in the work by Sanderson and Zhang (1999) using analytical solutions for vertical flow rate calculation based on the cubic law and the pipe formula. In the 3D geological setting of limestone–shale sequences, aperture variability and even impersistence may exist along the fracture walls normal to the layering, e.g. caused by inhomogeneous filling of calcite minerals.
Geological processes, such as episodes of delamination between layers and fracturing through shales, may make the flow in 3D even more complex, and furthermore, fluid flows are known to be channelized within the bedding planes and fractures, rather than flowing as if between parallel plates. Hence, further work is needed to integrate the empirical JCM model into a 3D modelling scheme. Such a scheme has been developed to model fluid flow through a 3D persistent fracture network (Lei et al. 2015b), where channelized flow and the significance of fracture intersections are highlighted. A 3D crack propagation module (Guo et al. 2014) has been also combined with the JCM-FEMDEM model to capture the brittle deformation response including local concentrations of critically high tensile or differential stresses, together with realistic fracture opening and shearing behaviour on both pre-existing and newly propagated fractures (Lei et al. 2016). Such capability opens the way to modelling 3D flows in geomechanically realistic multi-layer systems with both ‘strata bound’ and ‘non-strata bound’ fractures as well as plutonic rock masses.
One limitation of this research is the assumption that deformation of the solid skeleton was determined by the effective stress condition and the direct influence of local internal fluid pressure was not explicitly included. The immersed shell method (Viré et al. 2012, 2015) and the multiphase flow modelling (Su et al. 2015) that have been recently developed in the research group at Imperial College will be coupled with the 2D and 3D JCM-FEMDEM geomechanical models to capture the non-trivial two-way coupling process involving the transient response of rock solid and pore fluid pressure as well as the dynamic fluid–solid interaction. Some preliminary results have been presented in Obeysekara et al. (2016).
Compared to other discrete element modelling approaches, e.g. the particle-based synthetic rock mass approach (Mas Ivars et al. 2011) and the grain-based Voronoi tessellation method (Damjanaca et al. 2007; Ghazvinian et al. 2014), the FEMDEM model is able to capture the realistic fracturing behaviour of brittle rocks governed by fundamental fracture mechanics principles associated with strength and fracture energy parameters. A detailed review about the FEMDEM method and various other discrete modelling techniques can be found in the paper by Lisjak and Grasselli (2014). The addition of the JCM module to the FEMDEM framework further permits the simulation of the sophisticated shearing behaviour of pre-existing rough fractures based on experimentally derived constitutive laws. Unlike the work conducted with an explicit representation of the fracture roughness profile (Karami and Stead 2008; Bahaaddini et al. 2014; Tatone and Grasselli 2012, 2015a) that models the underlying process of asperity failure and roughness degradation, the proposed method integrates the well-established empirical joint constitutive laws directly as the criteria for implicit microscale modelling and can be advantageous in applications for large-scale engineering problems. However, these discrete modelling approaches based on an explicit time marching scheme may all suffer from potential dynamic effects in numerical experiments. Although a large damping coefficient can help significantly attenuate the dynamic oscillation and approximate a quasi-static condition (Mahabadi 2012; Tatone and Grasselli 2015b), further development in computational formulation and efficiency (e.g. implicit solution and parallel computing) is still required to more realistically model the physical conditions in laboratory experiments.
To conclude, an empirical joint constitutive model that captures the overall behaviour of sheared or compressed individual fractures as observed in laboratory experiments was implemented in the finite-discrete element analysis framework for 2D geomechanical modelling of fractured rocks. The combined JCM-FEMDEM model is able to achieve compatibility for both the fracture and matrix fields with respect to stress and displacement. The numerical model exhibits realistic shear strength and displacement characteristics with the recognition of fracture length influence, which was demonstrated by a comparison with the experimentally derived empirical solutions. 2D plane strain geomechanical modelling based on the combined JCM-FEMDEM formulation was conducted on an outcrop-based fracture network. The fracture system response to different stress phases led to a wealth of different local fracture-dominated deformational behaviour. The numerical experiments include the specific local developments of fractures apertures due to the fracture closing, opening, shearing, dilatancy, and propagation. With the increase in stress ratio, significant deformation enhancement occurs in the vicinity of fracture tips, intersections, and bends, where large apertures can be generated. The JCM-FEMDEM model is considered to give more realistic results compared to conventional approaches that neglect the scale dependency of joint properties and/or the asperity effect. The results of this paper have important implications for many rock engineering applications where in situ stress and pore fluid pressure is disturbed including underground construction, geothermal energy, nuclear repository safety, and petroleum recovery.
The authors would like to thank the sponsors of the itf-ISF project ‘Improved Simulation of Faulted and Fractured Reservoirs’ and to acknowledge the Janet Watson scholarship, awarded to the first author by the Department of Earth Science and Engineering, Imperial College London.
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