## Publications

240 results found

Lutz MP, Zimmerman RW, 2021, The effect of pore shape on the Poisson ratio of porous materials, *Mathematics and Mechanics of Solids*, Vol: 26, Pages: 1191-1203, ISSN: 1081-2865

A brief review is given of the effect of porosity on the Poisson ratio of a porous material. In contrast to elastic moduli such as K, G, or E, which always decrease with the addition of pores into a matrix, the Poisson ratio ν may increase, decrease, or remain the same, depending on the shape of the pores, and on the Poisson ratio of the matrix phase, νo. In general, for a given pore shape, there is a unique critical Poisson ratio, νc, such that the addition of pores into the matrix will cause the Poisson ratio to increase if νo<νc, decrease if νo>νc, and remain unchanged if νo=νc. The critical Poisson ratio for spherical pores is 0.2, for prolate spheroidal pores is close to 0.2, and tends toward zero for thin cracks. For two-dimensional materials, νc=1/3 for circular pores, 0.306 for squares, 0.227 for equilateral triangles, and again approaches 0 for thin cracks. The presence of a “trapped” fluid in the pore space tends to cause νc to increase, and for the range of parameters that may occur in rocks or concrete, this increase is more pronounced for thin crack-like pores than for equi-dimensional pores. Measurements of the Poisson ratio therefore may allow insight into pore geometry and pore fluid. If the matrix phase is strongly auxetic, small amounts of porosity will generally not cause the Poisson ratio to become positive.

Burtonshaw J, Thomas R, Paluszny A, et al., 2021, Effect of Viscosity and Injection Rate on the Tensile and Shear Deformation Experienced By Interacting Hydraulic and Natural Fractures, 55th U.S. Rock Mechanics/Geomechanics Symposium - American Rock Mechanics Association (ARMA)

Alsuwaidi ES, Xi G, Zimmerman RW, 2021, Mechanical characterization of Laffan and Nahr Umr anisotropic shales, *Journal of Petroleum Science and Engineering*, Vol: 200, Pages: 1-16, ISSN: 0920-4105

Geomechanics-related wellbore instability has become a major source of non-productive time in highly deviated wells drilled in oil and gas fields in offshore Abu Dhabi. These wells are drilled through two highly anisotropic shale formations, namely the Laffan shale formation and the Nahr Umr shale formation. Most of the models used in the oil and gas industry do not account for the strength and elastic anisotropy of shale. Therefore, a laboratory study was conducted to examine the strength and elastic anisotropy of shales using uniaxial compression tests and triaxial compression tests.Jaeger's Plane of Weakness (JPW) model was used to understand the anisotropic failure behavior of highly laminated shales of the Laffan and Nahr Umr formations. This model assumes that for an anisotropic rock, there exists a plane of weakness that has strength properties (cohesion and angle of internal friction) different than the strength properties of the intact rock. By minimizing the Root Mean Square Error (RMSE), the experimental strength values of the samples, as measured at different orientations, were fitted to the JPW model.Elastic moduli were also measured on these shales, as a function of orientation angle. The results showed that the moduli vary with angle according to the expected tensor transformation law. Therefore, the transverse isotropy assumption is a reasonable model to be used when dealing with these laminated sedimentary rocks.

Bhullar AS, Stewart GE, Zimmerman RW, 2021, Perturbation solution for one-dimensional flow to a constant-pressure boundary in a stress-sensitive reservoir, *Transport in Porous Media*, Vol: 137, Pages: 471-487, ISSN: 0169-3913

Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow.

Salimzadeh S, Zimmerman RW, Khalili N, 2020, Gravity hydraulic fracturing: a method to create self-driven fractures, *Geophysical Research Letters*, Vol: 47, ISSN: 0094-8276

In this study, we investigate the possibility of using a high‐density fluid to induce downward fracture growth in a hydraulic fracturing process. We propose a mathematical model to calculate the minimum amount of a dense fluid required to trigger downward fracture propagation under gravity forces, and we verify the calculated minimum volume of the fluid through numerical simulations. Results show that when the injected fluid exceeds the minimum amount, a steady downward growth of the hydraulic fracture is obtained. The fracture propagation consists of two distinct responses: The first response can occur under either toughness‐dominated, viscosity‐dominated, or an intermediate hydraulic fracturing regime, depending on fluid rheology, rock properties, and injection scenario. The second response occurs mainly under the toughness‐dominated regime, meaning the predominant energy dissipation mechanism is to overcome the fracture toughness and break the rock. In the latter, the speed of the downward fracture growth depends on the viscosity and fluid weight.

Setiawan NB, Zimmerman RW, 2020, A unified methodology for computing the stresses around an arbitrarily-shaped hole in isotropic or anisotropic materials, *International Journal of Solids and Structures*, Vol: 199, Pages: 131-143, ISSN: 0020-7683

A unified semi-analytical solution based on graphical conformal mapping and complex variable methods is proposed to calculate the in-plane stress around an arbitrarily-shaped hole in isotropic or anisotropic materials. The method requires only the outline coordinates of the hole, the elastic moduli of the material, and the magnitude and direction of the far-field stresses. Comparison with many published results for a wide range of shapes, such as triangles, squares, ovaloids, and ellipses, has been carried out to validate the method. The method has also been applied to a highly irregular geometry that has been observed in a breakout of a subsurface borehole. The solution is essentially closed-form, in the sense that it can be explicitly expressed in terms of the mapping coefficients, and parameters that depend only on the elastic moduli of the materials. With such a degree of flexibility, the method will be useful to study the effect of hole geometry on the stress distribution around holes in isotropic or anisotropic materials.

Paluszny A, Thomas RN, Saceanu MC,
et al., 2020, Hydro-mechanical interaction effects and channelling in three-dimensional fracture networks undergoing growth and nucleation, *JOURNAL OF ROCK MECHANICS AND GEOTECHNICAL ENGINEERING*, Vol: 12, Pages: 707-719, ISSN: 1674-7755

The flow properties of geomechanically generated discrete fracture networks are examined in the context of channelling. Fracture networks are generated by growing fractures in tension, modelling the low permeability rock as a linear elastic material. Fractures are modelled as discrete surfaces which grow quasi-statically within a three-dimensional (3D) volume. Fractures may have their locations specified as a simulation input, or be generated as a function of damage, quantified using the local variation in equivalent strain. The properties of the grown networks are shown to be a product of in situ stress, relative orientation of initial flaws, and competitive process of fracture interaction and growth. Fractures grow preferentially in the direction perpendicular to the direction of maximum tension and may deviate from this path due to mechanical fracture interaction. Flow is significantly channelled through a subset of the fractures in the full domain, consistent with observations of other real and simulated fractures. As the fracture networks grow, small changes in the geometry of the fractures lead to large changes in the locations and scale of primary flow channels. The flow variability and formation of channels are examined for two growing networks, one with a fixed amount of fractures, and another with nucleating fractures. The interaction between fractures is shown to modify the local stress field, and in turn the aperture of the fractures. Pathways for single-phase flow are the results of hydro-mechanical effects in fracture networks during growth. These are the results of changes to the topology of the network as well as the result of mechanical self-organisation which occurs during interaction leading to growth and intersection.

Paluszny Rodriguez A, Graham CC, Daniels K,
et al., 2020, Caprock integrity and public perception studies of carbon storage in depleted hydrocarbon reservoirs, *International Journal of Greenhouse Gas Control*, Vol: 98, ISSN: 1750-5836

Capture and subsurface storage of CO2 is widely viewed as being a necessary component of any strategy to minimise and control the continued increase in average global temperatures. Existing oil and gas reservoirs can be re-used for carbon storage, providing a substantial fraction of the vast amounts of subsurface storage space that will be required for the implementation of carbon storage at an industrial scale. Carbon capture and storage (CCS) in depleted reservoirs aims to ensure subsurface containment, both to satisfy safety considerations, and to provide confidence that the containment will continue over the necessary timescales. Other technical issues that need to be addressed include the risk of unintended subsurface events, such as induced seismicity. Minimisation of these risks is key to building confidence in CCS technology, both in relation to financing/liability, and the development and maintenance of public acceptance. These factors may be of particular importance with regard to CCS projects involving depleted hydrocarbon reservoirs, where the mechanical effects of production activities must also be considered. Given the importance of caprock behaviour in this context, several previously published geomechanical caprock studies of depleted hydrocarbon reservoirs are identified and reviewed, comprising experimental and numerical studies of fourteen CCS pilot sites in depleted hydrocarbon reservoirs, in seven countries (Algeria, Australia, Finland, France, Germany, Netherlands, Norway, UK). Particular emphasis is placed on the amount and types of data collected, the mathematical methods and codes used to conduct geomechanical analysis, and the relationship between geomechanical aspects and public perception. Sound geomechanical assessment, acting to help minimise operational and financial/liability risks, and the careful recognition of the impact of public perception are two key factors that can contribute to the development of a successful CCS project in a

Thomas RN, Paluszny A, Zimmerman RW, 2020, Permeability of three‐dimensional numerically grown geomechanical discrete fracture networks with evolving geometry and mechanical apertures, *Journal of Geophysical Research: Solid Earth*, Vol: 125, ISSN: 2169-9313

Fracture networks significantly alter the mechanical and hydraulic properties of subsurface rocks. The mechanics of fracture propagation and interaction control network development. However, mechanical processes are not routinely incorporated into discrete fracture network (DFN) models. A finite element, linear elastic fracture mechanics‐based method is applied to the generation of three‐dimensional geomechanical discrete fracture networks (GDFNs). These networks grow quasi‐statically from a set of initial flaws in response to a remote uniaxial tensile stress. Fracture growth is handled using a stress intensity factor‐based approach, where extension is determined by the local variations in the three stress intensity factor modes along fracture tips. Mechanical interaction between fractures modifies growth patterns, resulting in nonuniform and nonplanar growth in dense networks. When fractures are close, stress concentration results in the reactivation of fractures that were initially inactive. Therefore, GDFNs provide realistic representations of subsurface networks that honor the physical process of concurrent fracture growth. Hydraulic properties of the grown networks are quantified by computing their equivalent permeability tensors at each growth step. Compared to two sets of stochastic DFNs, GDFNs with uniform fracture apertures are strongly anisotropic and have relatively higher permeabilities at high fracture intensities. In GDFN models, where fracture apertures are based on mechanical principles, fluid flow becomes strongly channeled along distinct flow paths. Fracture orientations and interactions significantly modify apertures, and in turn, the hydraulic properties of the network. GDFNs provide a new way of understanding subsurface networks, where fracture mechanics is the primary influence on their geometric and hydraulic properties.

Thomas R, Paluszny A, Zimmerman RW, 2020, Growth of three-dimensional fractures, arrays, and networks in brittle rocks under tension and compression, *Computers and Geotechnics*, Vol: 121, ISSN: 0266-352X

Concurrent growth of multiple fractures in brittle rock is a complex process due to mechanical interaction effects. Fractures can amplify or shield stress on other fracture tips, and stress field perturbations change continuously during fracture growth. A three-dimensional, finite-element based, quasi-static growth algorithm is validated for mixed mode fracture growth in linear elastic media, and is used to investigate concurrent fracture growth in arrays and networks. Growth is governed by fracture tip stress intensity factors, which quantify the energy contributing to fracture extension, and are validated against analytical solutions for fractures under compression and tension, demonstrating that growth is accurate even in coarsely meshed domains. Isolated fracture geometries are compared to wing cracks grown in experiments on brittle media. A novel formulation of a Paris-type extension criterion is introduced to handle concurrent fracture growth. Fracture and volume-based growth rate exponents are shown to modify fracture interaction patterns. A geomechanical discrete fracture network is generated and examined during its growth, whose properties are the direct result of the imposed anisotropic stress field and mutual fracture interaction. Two-dimensional cut-plane views of the network demonstrate how fractures would appear in outcrops, and show the variability in fracture traces arising during interaction and growth.

Setiawan NB, Zimmerman RW, 2020, Semi-analytical method for tracking the evolution of borehole breakouts

In response to the high-stress concentration caused by excavation, rocks at the wellbore wall will collapse if the shear failure limit is exceeded. When this occurs, the shape of the wellbore enlarges from circular to roughly elliptical (breakout). The standard stress calculation method assumes a wellbore with a circular cross-section, and hence can no longer be used. Yet, knowledge of the near-wellbore stress state after the wellbore has been damage may still be required for various purposes, such as stability assessment of the open hole. In this paper, a semi-analytical solution is presented for the calculation of the stresses around a wellbore having an essentially arbitrary cross-sectional shape, in an isotropic formation. The zone of initial shear failure around a circular wellbore is calculated for a given far-field stress, using the standard solution for a circular wellbore. The rock in the failed region is then assumed to be removed, and the stress state around this new wellbore contour is then calculated using the method of conformal mapping and Kolosov-Muskhelishvili complex stress potentials. This process is iterated until a shape is obtained for which the breakout will not progress any further, at which point a stable stress state, and stable borehole shape, is reached. The method is validated against several sets of experimental data from the literature. This semi-analytical solution allows a stability assessment around the wellbore to be carried out after the breakout develops. Similarly, in an inverse manner, the method can be used to predict the in situ stress state around a damaged wellbore, if the cross-sectional hole shape can be reconstructed by methods such as electrical logging tools.

Saceanu MC, Paluszny A, Zimmerman RW, et al., 2020, Numerical modelling of spalling around a nuclear waste storage deposition borehole using a fracture mechanics approach

Fracture growth leading to mechanical spalling around a deposition borehole for disused nuclear fuel waste is modelled numerically. Simulations are conducted using a finite-element-based discrete fracture growth simulator, which computes deformation in the system based on the mechanical properties of the rock. Fractures are grown by computing stress intensity factors at each fracture tip, and the mesh is re-generated to accommodate the changing fracture geometries at every growth step. Several numerical models are created to explore the effect of boundary conditions on the initiation and development of spalling fractures at Forsmark, where the Swedish repository for nuclear waste is planned to be constructed. It is shown that the reported uncertainty in the principal stress magnitudes and orientations will affect the predicted fracture nucleation and growth patterns, and implicitly the final repository design. The potential effect of spalling on the structural integrity of the deposition borehole is illustrated for each stress scenario.

Almulhim OA, Paluszny A, Thomas RN, et al., 2019, Fully-coupled three-dimensional finite element simulations of the interaction between a hydraulic fracture and a pre-existing natural fracture

Copyright 2019 ARMA, American Rock Mechanics Association. The effects of approach angle and matrix poroelasticity on three-dimensional fracture interactions are investigated by examining changes in stress intensity factors around hydraulic fracture tips. Additionally, the effect of the compressive stress induced by the hydraulic fracture (the “stress shadow effect”) on the natural fracture is explored. Fracture interaction is captured using three-dimensional interaction maps based on two interaction measures that quantify the magnitude and type of interaction. The results show that the stress conditions at the hydraulic fracture tip are more favorable for growth when interacting fractures have shallow approach angles, and the interaction reduces as the approach angle increases. For “high” permeability reservoirs, fluid leak-off causes the rock matrix to dilate, and generates a tensile stress that amplifies the local stress field. The increase in stress in the region ahead of the hydraulic fracture (the stress amplification zone) can cause natural fractures to open in tension before the hydraulic fracture intersects the natural fracture. The increase in matrix permeability causes earlier activation of natural fractures due to the additional stress induced by the dilated rock matrix.

Setiawan NB, Zimmerman RW, 2019, The implications of using anisotropic elasticity and fully-triaxial failure criteria for borehole stability analysis in shales

Setiawan and Zimmerman (IJRMMS, 2018) recently developed a computational toolkit for analyzing the stability of boreholes drilled at an arbitrary angle through an anisotropic formation. The borehole stresses can be computed using either the Hiramatsu-Oka solution based on isotropic elasticity, or the Lekhnitskii-Amadei anisotropic elasticity solution. Shear failure along the borehole wall can be modeled using two variants of the Jaeger plane of weakness model, using either the Mohr-Coulomb or the Mogi-Coulomb failure criterion for failure along planes other than a bedding plane. In this paper, the implications of using these various models are investigated. All four variants of the above-mentioned models have been evaluated over a wide range of the relevant parameter space, including the elastic anisotropy ratio, the well deviation angle, the in situ stress anisotropy, etc. The breakout pressure will be influenced by the elastic anisotropy when the ratio Ev/Eh exceeds about 2.3, even in a nearly-vertical wellbore. This effect is further amplified as the degree of elastic anisotropy increases, or the wellbore becomes more deviated. Due to its incorporation of the strengthening effect of the intermediate principal stress, the Mogi-Coulomb criterion predicts lower values of the minimum required mud weight than does the Mohr-Coulomb criterion.

Almulhim OA, Paluszny A, Thomas RN, et al., 2019, Fully-coupled three-dimensional finite element simulations of the interaction between a hydraulic fracture and a pre-existing natural fracture

The effects of approach angle and matrix poroelasticity on three-dimensional fracture interactions are investigated by examining changes in stress intensity factors around hydraulic fracture tips. Additionally, the effect of the compressive stress induced by the hydraulic fracture (the “stress shadow effect”) on the natural fracture is explored. Fracture interaction is captured using three-dimensional interaction maps based on two interaction measures that quantify the magnitude and type of interaction. The results show that the stress conditions at the hydraulic fracture tip are more favorable for growth when interacting fractures have shallow approach angles, and the interaction reduces as the approach angle increases. For “high” permeability reservoirs, fluid leak-off causes the rock matrix to dilate, and generates a tensile stress that amplifies the local stress field. The increase in stress in the region ahead of the hydraulic fracture (the stress amplification zone) can cause natural fractures to open in tension before the hydraulic fracture intersects the natural fracture. The increase in matrix permeability causes earlier activation of natural fractures due to the additional stress induced by the dilated rock matrix.

Setiawan NB, Zimmerman RW, 2019, The implications of using anisotropic elasticity and fully-triaxial failure criteria for borehole stability analysis in shales

Copyright 2019 ARMA, American Rock Mechanics Association. Setiawan and Zimmerman (IJRMMS, 2018) recently developed a computational toolkit for analyzing the stability of boreholes drilled at an arbitrary angle through an anisotropic formation. The borehole stresses can be computed using either the Hiramatsu-Oka solution based on isotropic elasticity, or the Lekhnitskii-Amadei anisotropic elasticity solution. Shear failure along the borehole wall can be modeled using two variants of the Jaeger plane of weakness model, using either the Mohr-Coulomb or the Mogi-Coulomb failure criterion for failure along planes other than a bedding plane. In this paper, the implications of using these various models are investigated. All four variants of the above-mentioned models have been evaluated over a wide range of the relevant parameter space, including the elastic anisotropy ratio, the well deviation angle, the in situ stress anisotropy, etc. The breakout pressure will be influenced by the elastic anisotropy when the ratio Ev/Eh exceeds about 2.3, even in a nearly-vertical wellbore. This effect is further amplified as the degree of elastic anisotropy increases, or the wellbore becomes more deviated. Due to its incorporation of the strengthening effect of the intermediate principal stress, the Mogi-Coulomb criterion predicts lower values of the minimum required mud weight than does the Mohr-Coulomb criterion.

Setiawan NB, Zimmerman RW, 2018, Wellbore breakout prediction in transversely isotropic rocks using true-triaxial failure criteria, *International Journal of Rock Mechanics and Mining Sciences*, Vol: 112, Pages: 313-322, ISSN: 0020-7624

This paper presents a unified approach through which the influence of the elastic and strength anisotropy on wellbore instability can be thoroughly examined. The stresses at the wellbore wall are first calculated using the Lekhnitskii-Amadei solution, which accounts for elastic anisotropy. Then, shear failure is treated by combining the Mogi-Coulomb criterion for intact rock, with the Jaeger plane of weakness concept. The developed model accounts for all three principal stresses in predicting the onset of shear failure.The results of the specific case investigated show that rock elastic anisotropy induce higher stress concentrations. The difference, compared with the stresses found using the isotropic elastic model, could reach as high as 25% for the highest degree of anisotropy that might be expected for rocks of practical interest. The strengthening effect of the intermediate stress, as reflected in the Mogi-Coulomb criterion, reduces the required mud weight density by approximately 1.0 pounds-per-gallon (ppg). Furthermore, it is demonstrated that the risk posed by bedding slippage, for a wellbore with an inclination between 15° and 50° from the vertical, is masked when an isotropic elastic stress model is used. In contrast, the fully anisotropic model shows that an extra mud weight of approximately 4.5 ppg would be required, in order to avoid bedding plane slippage for the case under investigation. Although these results apply for a particular choice of strength properties and elastic properties, they give an indication of the implications of fully accounting for anisotropy and the effect of the intermediate stress when doing borehole stability analysis.

Lang P, Paluszny Rodriguez A, Morteza N,
et al., 2018, Relationship between the orientation of maximum permeability and intermediate principal stress in fractured rocks, *Water Resources Research*, Vol: 54, Pages: 8734-8755, ISSN: 0043-1397

Flow and transport properties of fractured rock masses are a function of geometrical structures across many scales. These structures result from physical processes and states and are highly anisotropic in nature. Fracture surfaces often tend to be shifted with respect to each other, which is generally a result of stress‐induced displacements. This shift controls the fracture's transmissivity through the pore space that forms from the created mismatch between the surfaces. This transmissivity is anisotropic and greater in the direction perpendicular to the displacement. A contact mechanics‐based, first‐principle numerical approach is developed to investigate the effects that this shear‐induced transmissivity anisotropy has on the overall permeability of a fractured rock mass. Deformation of the rock and contact between fracture surfaces is computed in three dimensions at two scales. At the rock mass scale, fractures are treated as planar discontinuities along which displacements and tractions are resolved. Contact between the individual rough fracture surfaces is solved for each fracture at the small scale to find the stiffness and transmissivity that result from shear‐induced dilation and elastic compression. Results show that, given isotropic fracture networks, the direction of maximum permeability of a fractured rock mass tends to be aligned with the direction of the intermediate principal stress. This reflects the fact that fractures have the most pronounced slip in the plane of the maximum and minimum principal stresses, and for individual fractures transmissivity is most pronounced in the direction perpendicular to this slip.

Thomas RN, Paluszny A, Hambley D,
et al., 2018, Permeability of observed three dimensional fracture networks in spent fuel pins, *Journal of Nuclear Materials*, Vol: 510, Pages: 613-622, ISSN: 0022-3115

The three-dimensional fracture network within a spent fuel pin is characterised using sequential grinding, and its permeability is numerically estimated. Advanced Gas-cooled Reactors (AGRs) produce spent fuel pins consisting of an outer steel cladding enclosing ceramic uranium dioxide (UO2) pellets. During irradiation, fuel pellets may undergo fracturing due to thermal, densification and swelling effects. Fracture patterns are usually observed on the surface of the pellet or through a cross section or longitudinal plane along the pellet. In this work, the three-dimensional fracture pattern within the pellet is characterised using an optical microscope. The pellet is progressively ground and polished, providing sequential cross sections, which together yield a three-dimensional discrete fracture pattern. Multiple large fractures grow to connect the cladding to the internal region of the pellet. Multiple surface fractures are observed that do not penetrate into the matrix of the pellet. The porosity of the UO2 and apertures of the fractures are estimated by sampling microscopic images. Darcy flow is numerically solved using the finite element method, computing flow through the matrix and fractures simultaneously. The equivalent tensorial permeability of the system is estimated for various approximate fracture apertures. The fracture network raises the permeability of the pellet by an order of magnitude.

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.

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

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.

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, 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.

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.

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