Publications
254 results found
Salimzadeh S, Paluszny Rodriguez A, Zimmerman RW, 2018, Effect of cold CO2 injection on fracture apertures and growth, International Journal of Greenhouse Gas Control, Vol: 74, Pages: 130-141, ISSN: 1750-5836
The injection of cold CO2 is modelled in three dimensions using a two-stage coupled thermoporoelastic model, with the aim of evaluating changes in apertures and potential growth of fractures. Non-isothermal flow is considered within the fractures and the rock matrix, and the two flow domains are coupled through a mass transfer term. The numerical model has been developed using standard finite elements, with spatial discretisation achieved using the Galerkin method, and temporal discretisation using finite differences. A full-scale field case geometric model, based on the Goldeneye depleted hydrocarbon reservoir in the North Sea, is developed and used for simulations. The in situ faults are modelled discretely as discontinuous surfaces in a three-dimensional matrix, including basement, reservoir, caprock and overburden layers. The faults are assumed initially to be low-permeable faults, with the same permeability as the caprock. However, the simulations show that their apertures (and as a result, their permeabilities) vary due to the thermoporoelastic effects caused by the injection of the relatively cold CO2. The change in the fracture apertures is mainly due to thermal effects; the reservoir layer undergoes contractions due to the cooling, significantly increasing fault aperture in the region of the fault within the reservoir, whereas the fault aperture is reduced in regions within the caprock. Propagation of fractures under thermoporoelastic loading is investigated. Results show that the distance to the injection well, as well as spatial orientation of fractures with respect to the injection well, affect aperture evolution and potential growth of fractures. A sensitivity analysis is performed on the parameters affecting the fracture growth: minimum normal stress acting on the fracture plane, dip angle of the fracture, and the contact friction coefficient. It is found that low friction, low normal contact stress, or high in situ shear stress on the fracture surface
Thomas RN, Paluszny A, Zimmerman RW, 2018, Effect of fracture growth velocity exponent on fluid flow through geomechanically-grown 3d fracture networks, 2nd International Discrete Fracture Network Engineering Conference
Copyright 2018 ARMA. Geomechanical discrete fracture networks (DFNs) are grown using a 3D finite element-based fracture mechanics simulator. The influence of the fracture growth rate exponent (β) on the resulting fracture geometry and hydraulic properties of networks is investigated. Previous work has found that β has a complex relationship with the final geometry of geomechanically-grown 2D DFNs. Realistic features evolve during the growth of DFNs as a result of the orientation of the principal stress axis and fracture interaction. High values of β cause interaction effects to be more pronounced, and irregular shaped fractures to be more common. Low values of β are found to produce networks with a balance between selective growth on preferentially oriented and interacting fractures, and significant increases in fracture surface area with computation time. The permeability of DFNs is significantly influenced by anisotropy, which develops in the axes perpendicular to the principal stress direction. For fracture networks with different β values, permeabilities along the principal axes are similar for the same total fracture void space.
Karmakar GP, Grattoni CA, Zimmerman RW, 2018, An oil-based gel system for reservoir rock permeability modification, Advances in Materials and Processing Technologies, Vol: 4, Pages: 669-679, ISSN: 2374-068X
Water production during gas and oil recovery is a major problem for the oil industry, as the average worldwide production is more than five barrels of water per barrel of oil. Among the many attempted remedies, water-based polymers and cross-linked gels are often injected into the reservoir to control excessive water production. Recently, oil-based gelant systems have been proposed which are oil-soluble. These systems react with the reservoir water to form a rigid water-based gel during the shut-in period, thereby drastically reducing the permeability of the reservoir to water. The aim of this paper is to improve the understanding of how the flow of oil and water are affected by one of the oil-based gelant systems, TMOS. The gelation behaviour and gel characteristics were studied under static and dynamic conditions. Two-dimensional transparent glass models were used to study the effect of gelant flow, and evaluate the effectiveness of the gel in modifying the oil and water permeability. The ability of the gel to modify the water flow at different flow velocities yields a velocity-dependent permeability. New insights are presented that may help reservoir and production engineers to select and design better gel treatments for a given reservoir.
Paluszny A, Thomas RN, Zimmerman RW, 2018, Finite element-based simulation of the growth of dense three-dimensional fracture networks, 52nd US Rock Mechanics / Geomechanics Symposium
Copyright © 2018 ARMA, American Rock Mechanics Association. The growth of fractures within a quasi-brittle rock is computed numerically with the aim of generating high-density geomechanically realistic three-dimensional discrete fracture patterns. Patterns are generated with a finite element-based discrete fracture propagation simulator, in which deformation and flow are numerically computed. These detailed multi-fracture growth simulations study the emergence of patterns as a function of the interaction of fractures and the mechanical effects of pattern evolution on the distribution of apertures in response to in situ stresses.
Zimmerman RW, 2018, The Imperial College lectures in petroleum engineering, ISBN: 9781786344991
This book presents, in a self-contained form, the equations of fluid flow in porous media, with a focus on topics and issues that are relevant to petroleum reservoir engineering. No prior knowledge of the field is assumed on the part of the reader, and particular care is given to careful mathematical and conceptual development of the governing equations, and solutions for important reservoir flow problems. Fluid Flow in Porous Media starts with a discussion of permeability and Darcy's law, then moves on to a careful derivation of the pressure diffusion equation. Solutions are developed and discussed for flow to a vertical well in an infinite reservoir, in reservoirs containing faults, in bounded reservoirs, and to hydraulically fractured wells. Special topics such as the dual-porosity model for fractured reservoirs, and fluid flow in gas reservoirs, are also covered. The book includes twenty problems, along with detailed solutions. As part of the Imperial College Lectures in Petroleum Engineering, and based on a lecture series on the same topic, this book provides the introductory information needed for students of the petroleum engineering and hydrology.
Paluszny A, Salimzadeh S, Zimmerman RW, 2018, Finite-Element Modeling of the Growth and Interaction of Hydraulic Fractures in Poroelastic Rock Formations, Hydraulic Fracture Modeling, Pages: 1-19, ISBN: 9780128129982
This chapter presents a finite element-based method for simulating the hydraulic fracturing process in porous rocks. The finite-element method is used to compute the mechanical deformation of the rock, and it accounts for the effects of poroelasticity, thermoelasticity, and fluid flow in both the fractures and the rock matrix, in a fully coupled manner. The fractures are represented in the mesh as fully three-dimensional objects, with evolving local apertures that are computed as part of the solution. Fracture growth is modeled using stress intensity factors, and the direction and rate of growth is evaluated individually at each fracture tip node. Specifically, the direction of fracture growth at each tip node is governed by the maximum circumferential stress criterion, and the extent of fracture growth is approximated using a Paris-type growth law. Contact between opposing fracture surfaces is handled using a gap-based augmented Lagrangian approach. Fracture growth is computed independently of the underlying mesh, and the fracture path is not constrained to follow the mesh. Instead, a new mesh is constructed after each time step, if the fracture has grown in that time step. This numerical framework is then applied to the growth of multiple hydraulic fractures in impermeable and permeable formations to investigate the effects of matrix permeability, matrix poroelasticity, and temperature contrast (between the rock and the injected fluid) on the growth and interaction of hydraulic fractures.
De Simone S, Jackson SJ, Zimmerman RW, et al., 2018, Analysis of the use of superposition for analytic models of CO2 injection into reservoirs with multiple injection sites
Large scale CCS is crucial to reduce the cost associated with minimizing climate change. Energy system models should thus include CCS at regional or global scale with a proper evaluation of pressure limitations and injectivity, which are currently ignored. To this aim, the use of simplified analytical solutions is highly useful because they provide fast evaluation of pressure and plume evolution without the computational costs of the numerical models. Application of these solutions to assess storage capacity has been extended to cases of multiple well injection. In these cases, the pressure build-up is evaluated as the superposition of the analytical solutions for pressure associated with each individual well. In this study we investigate the validity of the superposition procedure, given the non-linearity of the multiphase flow. We quantify the error associated with the application of superposition to estimate reservoir pressurisation in different scenarios of.multi-site CO2 injection in a large regional aquifer. We find that the error associated with the adoption of this procedure increases with time and with the number of wells in proportion to the area invaded by CO2 in the reservoir.
Zimmerman RW, Ambrose J, Setiawan NB, 2018, Failure of anisotropic rocks such as shales, and implications for borehole stability
In anisotropic rocks such as shale, the value of the maximum principal stress required to cause shear failure depends not only on the other two principal stresses, but also on the angle between the maximum principal stress and the normal to the bedding plane. According to Jaeger’s plane of weakness model, for near 0° or 90°, failure will occur at a stress determined by the failure criterion for the “intact rock”, and the failure plane will cut across the bedding planes. At intermediate angles, failure will occur along a bedding plane, at a stress determined by the strength parameters of the bedding plane. Data were analyzed from a set of triaxial (23) compression tests conducted on a suite of shale samples, at different confining stresses, and a range of angles , and it was found that the data could be fit reasonably well with the four-parameter plane of weakness model. Based on these results, a model has been developed for the stability of boreholes drilled in shales. The fully anisotropic Lekhnitskii-Amadei solution is used to compute the stresses around the borehole wall. The Mogi-Coulomb failure criterion is used for the strength of the “intact rock”, and the plane of weakness model is used for the strength of the bedding planes. The model can be used to predict the minimum mud weight required to avoid shear failure, for arbitrary borehole orientations and anisotropy ratios. The results show the importance of using a fully anisotropic elastic model for the stresses, and using a true-triaxial failure criterion, in borehole stability analysis.
Salimzadeh S, Paluszny Rodriguez A, Nick HM, et al., 2018, A three-dimensional coupled thermo-hydro-mechanical model for deformable fractured geothermal systems, Geothermics, Vol: 71, Pages: 212-224, ISSN: 0375-6505
A fully coupled thermal-hydraulic-mechanical (THM) finite element model is presented for fractured geothermal reservoirs. Fractures are modelled as surface discontinuities within a three-dimensional matrix. Non-isothermal flow through the rock matrix and fractures are defined and coupled to a mechanical deformation model. A robust contact model is utilised to resolve the contact tractions between opposing fracture surfaces under THM loadings. A numerical model has been developed using the standard Galerkin method. Quadratic tetrahedral and triangular elements are used for spatial discretisation. The model has been validated against several analytical solutions, and applied to study the effects of the deformable fractures on the injection of cold water in fractured geothermal systems.Results show that the creation of flow channelling due to the thermal volumetric contraction of the rock matrix is very likely. The fluid exchanges heat with the rock matrix, which results in cooling down of the matrix, and subsequent volumetric deformation. The cooling down of the rock matrix around a fracture reduces the contact stress on the fracture surfaces, and increases the fracture aperture. Stress redistribution reduces the aperture, as the area with lower contact stress on the fracture expands. Stress redistribution reduces the likelihood of fracture propagation under pure opening mode, while the expansion of the area with lower contact stress may increase the likelihood of shear fracturing.
Tsang C-F, Lippmann M, Dobson P, et al., 2018, Commemorating Dr. Gudmundur "Bo" Bodvarsson (1951-2006), a leader of the deep unsaturated flow and transport investigations, Water, Vol: 10, ISSN: 2073-4441
Salimzadeh S, Nick HM, Zimmerman RW, 2018, Thermoporoelastic effects during heat extraction from low permeability reservoirs, Energy, Vol: 142, Pages: 546-558, ISSN: 0360-5442
Thermoporoelastic effects during heat extraction from low permeability geothermal reservoirs are investigated numerically, based on the model of a horizontal penny-shaped fracture intersected by an injection well and a production well. A coupled formulation for thermo-hydraulic (TH) processes is presented that implicitly accounts for the mechanical deformation of the poroelastic matrix. The TH model is coupled to a separate mechanical contact model (M) that solves for the fracture contact stresses due to thermoporoelastic compression. Fractures are modelled as surface discontinuities within a three-dimensional matrix. A robust contact model is utilised to resolve the contact tractions between opposing fracture surfaces. Results show that due to the very low thermal diffusivity of the rock matrix, the thermally-induced pore pressure partially dissipates even in the very low-permeability rocks that are found in EGS projects. Therefore, using the undrained thermal expansion coefficient for the matrix may overestimate the volumetric strain of the rock in low-permeability enhanced geothermal systems, whereas using a drained thermal expansion coefficient for the matrix may underestimate the volumetric strain of the rock. An “effective” thermal expansion coefficient can be computed from the drained and undrained values to improve the prediction for the partially-drained matrix.
Salimzadeh S, Usui T, Paluszny A, et al., 2017, Finite element simulations of interactions between multiple hydraulic fractures in a poroelastic rock, International Journal of Rock Mechanics and Mining Sciences, Vol: 99, Pages: 9-20, ISSN: 0020-7624
A fully coupled three-dimensional finite-element model for hydraulic fracturing in permeable rocks is utilised to investigate the interaction between multiple simultaneous and sequential hydraulic fractures. Fractures are modelled as surface discontinuities within a three-dimensional matrix. This model simultaneously accounts for laminar flow within the fracture, Darcy flow within the rock matrix, poroelastic deformation of the rock, and the propagation of fractures using a linear elastic fracture mechanics framework. The leakoff of fracturing fluid into the surrounding rocks is defined as a function of the pressure gradient at the fracture surface, the fluid viscosity, and the matrix permeability. The coupled equations are solved numerically using the finite element method. Quadratic tetrahedral and triangle elements are used for spatial discretisation of volumes and surfaces, respectively. The model is validated against various analytical solutions for plane-strain and penny-shaped hydraulic fractures. Several cases of simultaneous fracturing of multiple hydraulic fractures are simulated in which the effects of the various parameters (the in situ stresses, the distance between fractures, the permeability of the matrix, the Biot poroelastic coefficient, and the number of the fractures in a group) are investigated. The results show that the stress induced by the opening of the fractures, and the stress induced by the fluid leakoff, each have the effect of locally altering the magnitudes and orientations of the principal stresses, hence altering the propagation direction of the fractures. Opening of a fracture induces excessive compression (also known as the “stress shadow”) that causes adjacent fractures to curve away from each other. This excessive compression competes against the differential in situ stresses, which tend to cause the fracture to grow in the plane normal to the minimum in situ stress. The stress shadow effect is reduced by increasing th
Thomas RN, Paluszny A, Zimmerman RW, 2017, Quantification of fracture interaction using stress intensity factor variation maps, Journal of Geophysical Research. Solid Earth, Vol: 122, Pages: 7698-7717, ISSN: 2169-9356
Accurate and flexible models of fracture interaction are sought after in the fields of mechanics and geology. Stress intensity factors (SIFs) quantify the energy concentrated at the fracture tips and are perturbed from their isolated values when two fractures are close to one another. Using a three-dimensional finite element fracture mechanics code to simulate static fractures in tension and compression, interaction effects are examined. SIF perturbations are characterized by introducing three interaction measures: the circumferential and maximum SIF perturbation provide the “magnitude” of the effect of interaction, and the amplification to shielding ratio quantifies the balance between increased and decreased SIFs along the tip. These measures are used to demonstrate the change in interaction with fracture separation and to find the separation at which interaction becomes negligible. Interaction maps are constructed by plotting the values of the interaction measures for a static fracture as a second fracture is moved around it. These maps are presented for several common fracture orientations in tension. They explore interaction by highlighting regions in which growth is more likely to occur and where fractures will grow into nonplanar geometries. Interaction maps can be applied to fracture networks with multiple discontinuities to analyze the effect of geometric variations on fracture interaction.
Zimmerman RW, 2017, Pore Volume and Porosity Changes under Uniaxial Strain Conditions, Transport in Porous Media, Vol: 119, Pages: 481-498, ISSN: 0169-3913
Expressions for the changes that occur in the pore volume and the porosity of a porous rock, due to changes in the pore pressure, overburden stress, and temperature, are derived within the context of the linearised theory of poroelasticity. The resulting expressions are compared to the commonly used equations proposed by Palmer and Mansoori, and it is shown that their expressions are consistent with the exact expressions if their factor f is identified with (1+ν)/3(1−ν)(1+ν)/3(1−ν) , where νν is the Poisson’s ratio of the porous rock. Finally, the first derivation is given, within the context of the fully coupled linearised theory of poroelasticity, that under uniaxial strain, the partial differential equation that governs the evolution of the pore pressure is a pure diffusion equation, with a total compressibility term that (exactly) equals the sum of the fluid compressibility and the uniaxial pore volume compressibility.
Usui T, Salimzadeh S, Paluszny A, et al., 2017, Effect of poroelasticity on hydraulic fracture interactions, Sixth Biot Conference on Poromechanics 2017, Publisher: American Society of Civil Engineers, Pages: 2008-2015
This study investigates, by performing finite element-based simulations, the influence of fluid leak-off and poroelasticity on growth of multiple hydraulic fractures that initiate from a single horizontal well. In this research, poroelastic deformation of the matrix is coupled with fluid flow in the fractures, and fluid flow in the rock matrix, in three dimensions. Effects of the fluid leakoff and poroelasticity on the propagation of the neighboring fractures are studied by varying the matrix permeability, and the Biot coefficient. Simulation results show that the stress induced by the opening of the fractures, and the stress induced by the fluid leak-off, each have the effect of locally altering the magnitudes and orientations of the principal stresses, hence altering the propagation direction of the fractures. The stress induced by the opening of the fractures tends to propagate both of the fractures away from each other in a curved trajectory, whereas the effects of fluid leak-off and poroelasticity (i.e., a higher Biot coefficient) tend to straighten the curved trajectory.
Paluszny A, Zimmerman RW, 2017, Modelling of primary fragmentation in block caving mines using a finite-element based fracture mechanics approach, Geomechanics and Geophysics for Geo-Energy and Geo-Resources, Vol: 3, Pages: 121-130, ISSN: 2363-8419
The growth of fractures around an undercut of a block cave is simulated. A finite element based approach is used, in which fractures are represented as non-planar 3D surfaces that grow in response to boundary stresses and interaction. A new mesh is recreated at each step to compute the displacement field. Stress intensity factors are computed around fracture tips using a technique that computes the interaction integral over a virtual disk. Fracture geometry is updated using Paris and Schöllmann propagation laws, and a geometric fracture pattern ensues from the simulation. The growth of fractures is examined in the lower 20 m of a mine at 812 m depth. The growth of 30, 60 and 90 fractures is examined. Realistic extraction schedules for over 100 draw points control the rate of mass extraction. The effect of rock bridges as overburden stress shields is investigated. Bridges are modelled by constraining the vertical displacement of the top boundary. This case is compared to a Neumann-type overburden stress boundary condition in which the overburden is felt throughout the top of the cave. In both cases, fractures grow to form a dome shape above and around the cave during extraction. For the case of a fixed top boundary, fracture growth is observed away from the cave, while in the direct overburden stress case, fractures tend to grow close to the cave. Over-arching fractures concentric to the undercut continue to grow as the cave progresses.
Salimzadeh S, Paluszny A, Zimmerman RW, 2017, Three-dimensional poroelastic effects during hydraulic fracturing in permeable rocks, International Journal of Solids and Structures, Vol: 108, Pages: 153-163, ISSN: 0020-7683
A fully coupled three-dimensional finite-element model for hydraulic fractures in permeable rocks is presented, and used to investigate the ranges of applicability of the classical analytical solutions that are known to be valid in limiting cases. This model simultaneously accounts for fluid flow within the fracture and rock matrix, poroelastic deformation, propagation of the fractures, and fluid leakage into the rock formation. The model is validated against available asymptotic analytical solutions for penny-shaped fractures, in the viscosity-dominated, toughness-dominated, storage-dominated, and leakoff-dominated regimes. However, for intermediate regimes, these analytical solutions cannot be used to predict the key hydraulic fracturing variables, i.e. injection pressure, fracture aperture, and length. For leakoff-dominated cases in permeable rocks, the asymptotic solutions fail to accurately predict the lower-bound for fracture radius and apertures, and the upper-bound for fracture pressure. This is due to the poroelastic effects in the dilated rock matrix, as well as due to the multi-dimensional flow within matrix, which in many simulation codes is idealised as being one-dimensional, normal to the fracture plane.
Defoort T, Paluszny A, Zimmerman RW, et al., 2017, Numerical study of the contact between a chamfered cylindrical PDC cutter and the rock, Pages: 3147-3153
Numerical rock cutting simulations are performed for different cutter configurations. Special attention is given to the contact region between the cutter and the rock. Frictional contact is solved using a Gap Based Augmented Lagrangian approach, in order to minimize the interpenetration error that is present in the less expensive penalty approach, while reducing finite element accumulation time compared to the standard Lagrangian method. Numerical experiments are performed with a single cylindrical cutter of radius 16 mm, a back rake of 15°, a depth of cut of 4 mm, and two different chamfers: a 0.3 mm chamfer at 15°, and a 0.5 mm chamfer at 45°. The number of Uzawa iterations required for the interpenetration error to have minimal impact on cutting forces is estimated, and the accuracy of the penalty approach is emphasized. Contact tractions are shown, suggesting areas where the mesh should be refined. A suitable mesh size is given for both chamfer geometries. The influence of the friction coefficient on cutting forces is evaluated, suggesting a cutter configuration insensitive to the value of the friction coefficient within a realistic range.
Bond D, Krevor SC, Muggeridge AH, et al., 2017, Imperial College Lectures In Petroleum Engineering: Topics In Reservoir Management, ISBN: 9781786342850
Paluszny A, Salimzadeh S, Zimmerman RW, 2017, Finite-Element Modeling of the Growth and Interaction of Hydraulic Fractures in Poroelastic Rock Formations, Hydraulic Fracture Modeling, Pages: 1-19, ISBN: 9780128129999
This chapter presents a finite element–based method for simulating the hydraulic fracturing process in porous rocks. The finite-element method is used to compute the mechanical deformation of the rock, and it accounts for the effects of poroelasticity, thermoelasticity, and fluid flow in both the fractures and the rock matrix, in a fully coupled manner. The fractures are represented in the mesh as fully three-dimensional objects, with evolving local apertures that are computed as part of the solution. Fracture growth is modeled using stress intensity factors, and the direction and rate of growth is evaluated individually at each fracture tip node. Specifically, the direction of fracture growth at each tip node is governed by the maximum circumferential stress criterion, and the extent of fracture growth is approximated using a Paris-type growth law. Contact between opposing fracture surfaces is handled using a gap-based augmented Lagrangian approach. Fracture growth is computed independently of the underlying mesh, and the fracture path is not constrained to follow the mesh. Instead, a new mesh is constructed after each time step, if the fracture has grown in that time step. This numerical framework is then applied to the growth of multiple hydraulic fractures in impermeable and permeable formations to investigate the effects of matrix permeability, matrix poroelasticity, and temperature contrast (between the rock and the injected fluid) on the growth and interaction of hydraulic fractures.
Paluszny A, Salimzadeh S, Tempone P, et al., 2017, Evaluating natural fracture growth in shale caprocks during cold CO<inf>2</inf> injection at the Heletz pilot site, Pages: 2988-2995
The potential growth of multiple fractures is investigated during the deformation of the caprock of a reservoir during the subsurface sequestration of CO2. Multiple scenarios of the interaction of pre-existing natural fractures are investigated, as a function of induced stress changes due to the temperature contrast between the injected CO2 and the reservoir. Simulations are performed at the field scale, on a simplified geologically-informed model of the Heletz pilot test site, Israel, in the framework of the EU-sponsored TRUST project. The potential reactivation and growth of pre-existing natural fractures is investigated. During cold injection, changes in the strain field around the well are tracked. The progression of the temperature plume is observed to cause a local release of stresses that in turn causes short propagation episodes, which mostly focus on downward growth. Although some upwards growth is modeled, growth is expected to prominently occur towards the reservoir.
Zimmerman RW, 2017, Introduction to Rock Properties, Imperial College Lectures In Petroleum Engineering, The - Volume 3: Topics In Reservoir Management, Pages: 1-46, ISBN: 9781786342850
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Lang PS, Paluszny A, Zimmerman RW, 2016, Evolution of fracture normal stiffness due to pressure dissolution and precipitation, International Journal of Rock Mechanics and Mining Sciences, Vol: 88, Pages: 12-22, ISSN: 1873-4545
The normal stiffness of a fracture is a key parameter that controls, for example, rock mass deformability, the change in hydraulic transmissivity due to stress changes, and the speed and attenuation of seismic waves that travel across the fracture. Non-linearity of normal stiffness as a function of stress is often attributed to plastic yield at discrete contacts. Similar surface-altering mechanisms occur due to pressure solution and precipitation over larger timescales. These processes partition the fracture surfaces into a flattened contact region, and a rough free surface that bounds the void space. Under low loads, contact occurs exclusively over the flattened part, leading to rapid, exponential stiffening. At higher loads, contact occurs over the rough surface fraction, leading to the conventional linear increase of stiffness with stress. It follows that a relationship exists between the history of in situ temperature and stress state of a rock fracture, and its subsequent deformation behavior.
Ebigbo AOD, Lang PS, Paluszny A, et al., 2016, Inclusion-based effective medium models for the permeability of a 3D fractured rock mass, Transport in Porous Media, Vol: 113, Pages: 137-158, ISSN: 1573-1634
Effective permeability is an essential parameter for describing fluid flow through fractured rock masses. This study investigates the ability of classical inclusion-based effective medium models (following the work of Sævik et al. in Transp Porous Media 100(1):115–142, 2013. doi:10.1007/s11242-013-0208-0) to predict this permeability, which depends on several geometric properties of the fractures/networks. This is achieved by comparison of various effective medium models, such as the symmetric and asymmetric self-consistent schemes, the differential scheme, and Maxwell’s method, with the results of explicit numerical simulations of mono- and poly-disperse isotropic fracture networks embedded in a permeable rock matrix. Comparisons are also made with the Hashin–Shtrikman bounds, Snow’s model, and Mourzenko’s heuristic model (Mourzenko et al. in Phys Rev E 84:036–307, 2011. doi:10.1103/PhysRevE.84.036307). This problem is characterised by two small parameters, the aspect ratio of the spheroidal fractures, αα , and the ratio between matrix and fracture permeability, κκ . Two different regimes can be identified, corresponding to α/κ<1α/κ<1 and α/κ>1α/κ>1 . The lower the value of α/κα/κ , the more significant is flow through the matrix. Due to differing flow patterns, the dependence of effective permeability on fracture density differs in the two regimes. When α/κ≫1α/κ≫1 , a distinct percolation threshold is observed, whereas for α/κ≪1α/κ≪1 , the matrix is sufficiently transmissive that such a transition is not observed. The self-consistent effective medium methods show good accuracy for both mono- and polydisperse isotropic fracture networks. Mourzenko’s equation is very accurate, particularly for monodisperse networks. Finally, it is shown that Snow’
Nejati M, Paluszny A, Zimmerman RW, 2016, A finite element framework for modeling internal frictional contact in three-dimensional fractured media using unstructured tetrahedral meshes, Computer Methods in Applied Mechanics and Engineering, Vol: 306, Pages: 123-150, ISSN: 0045-7825
This paper introduces a three-dimensional finite element (FE) formulation to accurately model the linear elastic deformation of fractured media under compressive loading. The presented method applies the classic Augmented Lagrangian(AL)-Uzawa method, to evaluate the growth of multiple interacting and intersecting discrete fractures. The volume and surfaces are discretized by unstructured quadratic triangle-tetrahedral meshes; quarter-point triangles and tetrahedra are placed around fracture tips. Frictional contact between crack faces for high contact precisions is modeled using isoparametric integration point-to-integration point contact discretization, and a gap-based augmentation procedure. Contact forces are updated by interpolating tractions over elements that are adjacent to fracture tips, and have boundaries that are excluded from the contact region. Stress intensity factors are computed numerically using the methods of displacement correlation and disk-shaped domain integral. A novel square-root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. Tractions and compressive stress intensity factors are validated against analytical solutions. Numerical examples of cubes containing one, two, twenty four and seventy interacting and intersecting fractures are presented.
Paluszny A, Tang XH, Nejati M, et al., 2016, A direct fragmentation method with Weibull function distribution of sizes based on finite- and discrete element simulations, International Journal of Solids and Structures, Vol: 80, Pages: 38-51, ISSN: 0020-7683
A direct method is proposed to rapidly fragment bodies during impulse-based discrete element method simulations of multiple body interactions. The approach makes use of patterns and size distributions obtained both from experiments, and from numerical models that rigorously compute fragmentation by growing fractures explicitly. Weibull parameters approximate the fragment size distributions as a function of body size and relative contact velocity. Structured domain decomposition is applied on the colliding bodies directly, resulting in a low-cost fragmentation calculation that depends on relative velocity, but which does not require the computation of fracture growth nor explicit element de-bonding, but instead classifies pre-existing mesh elements into the newly fragmented sub-domains. The method is applied to the fragmentation of a single spherical rock fragment, to an irregularly shaped rock fragment, and to the crushing of an array of spherical rock fragments.
Defoort T, Paluszny A, Zimmerman RW, 2016, Comparison of fracture mechanics and damage mechanics approaches to simulate three-point bending and double-notch shear experiments on rock samples, Pages: 1265-1271
Three-point-bending and double-notch shear experiments are modeled using a continuum damage mechanics approach, and an explicit fracture mechanics approach, for both homogeneous and heterogeneous rocks. The local damage approach uses the Mazars isotropic damage model. In the explicit fracture simulations, fractures are represented as explicit surfaces within a three-dimensional unstructured mesh comprised of isoparametric quadratic tetrahedra and quarter point tetrahedra. Heterogeneities in strength, stiffness and toughness are introduced in a random manner within ±50% of the average values. A characteristic length is used to define the size of element clusters having uniform properties. Both approaches are used to evaluate the influence of material heterogeneity on crack propagation and interaction. Both models reproduce well the experimental results for homogeneous rocks. While in the damage model the mesh is fixed and refined globally, the discrete approach only requires refinement around the fracture tips. In the three-point-bending and double-notch shear simulations, the damage mechanics approach is more realistic in that it leads to rougher crack surfaces. However, the fracture mechanics model predicts lower curvature of the fracture, which better corresponds to experimental observations. Both approaches predict that the presence of heterogeneities seems to diminish fracture interaction.
Salimzadeh S, Paluszny A, Zimmerman RW, 2016, Thermal effects during hydraulic fracturing in low-permeability brittle rocks, Pages: 45-50
A three-dimensional finite-element model for hydraulic fracturing has been developed that accounts for local thermal non-equilibrium between the injected fluid and host rock. The model also accounts for fluid flow and heat transfer within the fracture, heat conduction through the solid rock, deformation of the rock, and propagation of the fracture. Fluid flow through the fractures is modeled using the lubrication equation, and is fully coupled to the thermoelastic mechanical model through the pressure exerted by the fluid on the fracture walls, as well as by ensuring compatibility of fracture volumetric strains. Fractures are discretely modeled using triangular surfaces in an unstructured three-dimensional mesh. The growth of fractures is modeled using linear elastic fracture mechanics (LEFM), with the onset and direction of growth based on stress intensity factors that are computed for unstructured triangle-tetrahedral meshes. The model has been verified against analytical solutions available in the literature for penny-shaped (3D) fractures. A radial hydraulic fracture from a horizontal well is simulated to investigate the effects of the thermal non-equilibrium between the fracturing fluid and the host rock. For the case of very low matrix rock permeability, results show very little influence of thermal effects on the creation of hydraulic fractures.
Nejati M, Paluszny A, Zimmerman RW, 2015, A disk-shaped domain integral method for the computation of stress intensity factors using tetrahedral meshes, International Journal of Solids and Structures, Vol: 69-70, Pages: 230-251, ISSN: 0020-7683
A novel domain integral approach is introduced for the accurate computation of pointwise J-integral and stress intensity factors (SIFs) of 3D planar cracks using tetrahedral elements. This method is efficient and easy to implement, and does not require a structured mesh around the crack front. The method relies on the construction of virtual disk-shaped integral domains at points along the crack front, and the computation of domain integrals using a series of virtual triangular and line elements. The accuracy of the numerical results computed for through-the-thickness, penny-shaped, and elliptical crack configurations has been validated by using the available analytical formulations. The average error of computed SIFs remains below 1% for fine meshes, and 2–3% for coarse ones. The results of an extensive parametric study suggest that there exists an optimum mesh-dependent domain radius at which the computed SIFs are the most accurate. Furthermore, the results provide evidence that tetrahedral elements are efficient, reliable and robust instruments for accurate linear elastic fracture mechanics calculations.
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