A Numerical Method for Fractured Reservoir Poromechanics Using a Mixed-Continuum Embedded Fracture Model

Transcription

1 GRC Transactions, Vol. 40, 2016 A Numerical Method for Fractured Reservoir Poromechanics Using a Mixed-Continuum Embedded Fracture Model Jack H. Norbeck and Roland N. Horne Department of Energy Resources Engineering, Stanford University, Stanford, California, USA jnorbeck@stanford.edu Keywords Poroelasticity, embedded fracture model, fracture mechanics, reservoir simulation ABSTRACT In this paper, we introduce a numerical modeling framework that is useful for understanding behavior in fractured reservoir systems. We extended a model called CFRAC that coupled fluid flow, fracture deformation, and fracture propagation to incorporate the effects of poroelastic deformation of the rock surrounding the fractures. The model was based on an embedded fracture modeling framework, which allowed fracture propagation to be handled efficiently. Fluid flow was solved with a finite volume approximation, fracture deformation was solved with a boundary element approximation, and poroelastic deformation was solved with a finite element approximation. This novel approach leveraged the advantages of each numerical method while making limiting assumptions where appropriate in order to achieve an optimal balance between a proper descriptions of physical complexity and computational performance. We verified the accuracy of CFRAC by comparing solutions with another fractured geomechanics reservoir simulator, called AD-GPRS, for a synthetic scenario of fluid circulation in a fractured reservoir. 1. Introduction Fracture deformation is often thought to give rise to nonlinear flow regimes because the transmissivity and storativity of fractures can change significantly even for small changes in fracture aperture. Throughout the operational life of a reservoir, different physical processes can dominate the mechanical behavior of fractures. For example, during hydraulic fracture stimulation in low permeability formations pressurization within the fractures may dominate if leakoff into the surrounding rock is negligible. For long-term water circulation in a geothermal reservoir the pressure field may be at or close to steady-state, but poroelastic stress due to fluid leakoff or thermoelastic stress due to reservoir cooling may cause the state of stress throughout the reservoir to evolve in a transient manner. In addition, the timescales over which each process occurs can vary over orders of magnitudes. Hydraulic stimulation may occur over a period of several days, while long-term operation can last for decades. These issues can make developing a comprehensive numerical model of a field site difficult. In this work, we developed a numerical modeling framework for general purpose fractured reservoir simulation. The paper is organized as follows. In Section 2, we present an overview of the physical processes considered in the model as well as the numerical formulation of the model. In Section 3, we compare the results of our model with another fractured reservoir geomechanics model in order to verify its accuracy. In Section 4, we discuss several practical applications for 1 this type of model and lay out some of the model s limitations. In Section 5, we present our concluding remarks. 2. Poroelastic Fractured Reservoir Model In this section, we introduce a numerical modeling framework that is able to couple fluid flow and mechanical deformation in fractures and the surrounding rock. This type of model has application for understanding behavior in frac- 911

2 tured reservoir systems such as hydrothermal reservoirs, enhanced geothermal systems, or unconventional hydrocarbon reservoirs. The main advantage of our model is that it is able to consider the nonlinear evolution of reservoir permeability as fractures deform, fail in shear, or fail in tension and propagate as hydraulic fractures. The model is an extension of previous versions of CFRAC developed by McClure (2012) and Norbeck et al. (2015). A novel technique for coupling poroelastic deformation that results from mass exchange between the fractures and matrix rock with fracture deformation is described. Other models have been developed previously that are able to couple fluid flow and mechanical deformation in fractures and the surrounding rock. For example, Safari and Ghassemi (2016) introduced a model that used a boundary element method for the matrix flow, poroelastic deformation, and fracture deformation problem. A finite element method was used to solve for flow in the fractures. An advantage of the boundary element approach is that the rock surrounding the fractures does not require discretization and the semianalytical nature of the numerical method is ameanable for obtaining highly accurate solutions. A tradeoff is that general heterogeneity cannot be considered. The elastic properties of intact rock do not span wide ranges and don t behave in extremely nonlinear fashion, which is why boundary element methods are popular for fracture deformation problems. However, permeability of rock can vary over many orders of magnitude even within the same formation. For example, Kurtoglu et al. (2014) calculated the permeability experimentally for four core samples from the Middle Bakken formation and found that permeability ranged from 0.01 microdarcy to 100 microdarcy, depending on minerology and the density of fine-scale fractures. Therefore, the assumption of homogeneous and constant hydraulic properties is a severe limitation for models that solve the flow problem with a boundary element method. In addition, because of the transient nature of the fluid flow problem, a complex convolution is required in the boundary element method (Kikani, 1989). The time convolution can be intensive computationally and is difficult to incorporate into a general method that utilizes adaptive timestepping. The method introduced by Safari and Ghassemi (2016) required the timestep to remain constant throughout the duration of the simulation, which is a limiting assumption for application to general purpose reservoir simulation. In an alternative approach, Garipov et al. (2016) introduced a model that was based on the finite element method. In that work, deformation of the fractures and surrounding rock were coupled rigorously using a contact formulation. Attractive aspects of this approach are that flow and geomechanics can be solved in a fullyimplicit manner, and general nonlinearity and heterogeneity of hydraulic and mechanical properties can be considered. Some disadvantages are that fracture deformation is highly sensitive to the level of discretization, stresses near the crack tips are difficult to resolve, and fracture propagation cannot be handled easily. In our model, we exploit different numerical techniques to solve each part of the problem efficiently and robustly. The fluid flow problem is solved using the embedded fracture approach, which is an extension of a traditional finite volume discretization strategy. Because the embedded fracture approach does not require conforming grids for the fractures and matrix rock, fracture propagation is a trivial issue in terms of numerical discretization. The fracture deformation problem is solved using the displacement discontinuity (DD) method, which is a type of boundary element method tailored to crack-like problems. The DD method is useful for solving problems with many mechanicallyinteracting fractures and is capable of obtaining accurate near-tip stress solutions which are necessary for assessing fracture propagation criteria. Poroelastic deformation of the rock surrounding the fractures is solved using a finite element discretization strategy. The stresses induced throughout the matrix rock due to changes in fluid 2 pressure are then resolved onto the fracture surfaces using the finite element shape functions and incorporated as additional boundary conditions that are satisfied in the fracture deformation calculations. The different model components are coupled using an iterative sequential-implicit strategy. In the following subsections, we review the set of physical mechanisms that we consider in the model and describe the numerical techniques used to solve the underlying differential equations. In our notation, lowercase superscripts refer to different domains (i.e., fracture volume or matrix volume) and capital superscripts refer to different loading or deformation mechanisms (i.e., remote loading, poroelastic stress, or fracture-mechanics-induced stress). 2.1 Fluid Flow We consider a fractured subsurface reservoir system that is fully saturated with a single-phase slightly compressible fluid. We follow the embedded fracture modeling approach introduced originally by Lee et al. (2001) and Li and Lee (2008), and later extended by others including Karvounis and Jenny (2016) and Norbeck et al. (2015). Fractures that are expected to influence fluid flow at a particular scale of interest are represented as discrete features. The rock surrounding the fractures is treated effectively as a continuum. This modeling approach is similar conceptually to a dual-porosity description, except for that flow in the fractures is modeled explicitly. This is necessary in order to consider nonlinear evolution of fracture transmissivity and storativity that arise due to mechanical deformation, as well as to preserve fracture geometrical complexity which can influence fracture deformation significantly. In the embedded fracture approach, it is useful to consider the fractured volume and the intact matrix rock volume as separate, but coupled, computational domains. Mass conservation can be described, in the matrix domain, as: 912

3 ( ) +!m wm +! Ψ fm = t ρλk m p m ( ρϕ m ), (1) and, in the fractured domain, as: ( ) +!m wf +! Ψ mf = t ρλk f p f ( ρe), (2) In Eqs. 1 and 2, ρ is fluid density, λ is inverse fluid viscosity, k is permeability, p is pressure, ϕ is porosity, e is fracture hydraulic aperture, E is fracture void aperture, is a normalized mass source term related to wells, and is a normalized mass source term related to fracture-matrix mass transfer. For rock and fluid properties typical of geothermal reservoirs, Eq. 1 is usually only slightly nonlinear due to the relatively low compressibility of the pore fluid and matrix rock, but Eq. 2 can be highly nonlinear because the hydraulic and void aperture of fractures and faults can be affected significantly by changes in fluid pressure, mechanical opening, and shear dilation. The flux term in Eq. 2 is controlled by the transmissivity, T = ek f = e 3 / 12, which was assumed to behave according to the cubic law. The void aperture affects the storage term in Eq. 2. Equations 1 and 2 were discretized using a finite volume approximation. 2.2 Poroelastic Deformation For a porous rock volume subjected to a change in pressure from a reference state, Δ p m, Hooke s law for isotropic material properties is: σ P = 2Gε P + Λ trace ( ε P ) + αδp m I. (3) Here, G is shear modulus Λ is Lamé coefficient, α is Biot coefficient is the strain tensor, is the stress tensor, and I is the identity matrix. It is important to recognize that is the change in total stress that is generated subject to all boundary conditions. Assuming infinitesimal strains and using Eq. 3 as the constitutive relationships between stress and strain, conservation of momentum can be expressed as: G 2 u P + ( G + Λ) ( u P ) = α ( Δp m ), (4) where u P is material displacement vector caused by poroelastic deformation. Previous versions of CFRAC incorporated a treatment for poroelasticity that used a finite difference approximation based on elastic potential theory for the mechanics calculations (Norbeck and Horne, 2015, 2016). In this work, we implented the finite element formulation described by Smith et al. (2014). For the remainder of this section only, we use standard finite element notation for vectors and matrices rather than conventional spatial notation. For example, u P for a two-dimensional four-node quadrilateral element now represents the collection of displacements at the finite element nodes: u P = u x1 u y1 u x2 u y2 u x3 u y3 u x4 u y4 P. (5) Upon discretization with the finite element method, Eq. 4 is reduced to a system of equations involving the displacement vectors at the finite element nodes: ku P = f. (6) The finite element stiffness matrix, k, is calculated as: k = B T DB dxdy. (7) 913

4 The strain-displacement matrix involves shape function derivatives. For a four-node quadrilateral element: N 1 N 0 2 N 0 3 N x x x x N B = 0 1 N 0 2 N 0 3 N 0 4. (8) y y y y N 1 N 1 N 2 N 2 N 3 N 3 N 4 N 4 y x y x y x y x In Eq. 8, N are the nodal shape functions. The stress strain matrix for each element involves only elastic properties. In two-dimensional plane strain: D = Y ( 1 ν ) ( 1+ν )( 1 2ν ) 1 ν 1 ν ν 1 ν ν 2 1 ν ( ). (9) where Y is Young s modulus and ν is Poisson s ratio. For an unconstrained material, Eq. 3 can be inverted to calculate the strains that would occur due to a change in fluid pressure: ε * = ε xx ε yy 2ε xy * α 3K Δpm = α 3K Δpm 0, (10) where K is bulk modulus. The equivalent nodal forces that would cause those strains to occur are: f = B T Dε * dxdy. (11) A significant issue to deal with here is that a finite volume grid is used for the flow problem where the fluid pressures are calculated as cell-centered values and represent the average pressure over each control volume. If we use, for example, a conventional four node quadrilateral finite element for the poroelastic problem, the fluid pressure is assumed to be continuous at the finite element nodes at the corners of each element. One approach could be to use staggered grids so that the finite element nodes coincide with the finite volume cell centers. In this work, we chose instead to use the same grids for the finite element and finite volume discretizations. Bilinear interpolation is performed to map the fluid pressure distribution to approximate values at the finite element nodes. Then, Eq. 11 is evaluated for each finite element using numerical integration. Equation 6 is assembled into a global system of equations and solved for the nodal displacements u P. As a postprocessing step, the stresses can be calculated anywhere within a finite element by interpolating with the shape functions. At the element level, the total strains can be calculated as: ε P = Bu P. (12) The B matrix relates displacements at the finite element nodes to strain within the element, and is therefore a function of location within the element. In our application to fractured reservoir problems, we evaluate B at each discrete fracture element. In this way, we are able to approximate nonuniform distributions of induced stress along fracture surfaces even if there are many fracture elements contained in a single grid block, which is commonly the case in practical applications of our embedded fracture model. Hooke s law is used to calculate the poroelastic stress change: σ P = D( ε P ε * ). (13) The embedded fracture discretization concept is illustrated in Fig

5 2.3 Fracture Deformation In this work, we routinely took advantage of the fact that superposition of displacements and stresses holds for a linear elastic material. For example, the state of stress at a particular location in the material reflects the superposition of the remote tectonic stress, σ R, mechanically-induced stress caused by fracture deformation, σ M, and poroelastic effects: σ = σ R +σ M +σ P. (14) For a fracture surface with outward facing unit normal vector n, the stress tensor can be used to calculate the traction vectors acting on the fracture surface: t = σ n. (15) The traction vector can then be decomposed into its normal and shear components (Pollard and Fletcher, 2005): t = t n + t s = ( t n)n + n ( t n). (16) It is common to refer to the magnitudes of t n and t s as the normal and shear stresses acting on the fracture. In two dimensions: σ n = t n n, (17) σ s = t s s, (18) where s is the unit vector in the direction tangential to the fracture plane. In three dimensions, t s can be further decomposed into two orthogonal in-plane components (strike and dip, for example). The normal and shear stresses used as boundary Figure 1. Illustration of the embedded fracture discretization concept. (top) A structured Cartesian mesh is used for the fluid flow and poroelastic calculations. The red diamonds represent the cell centers for the finite volume discretization, the blue circles represent the finite element nodes, and the black line represents the fracture. (bottom) The fracture is discretized using the displacement discontinuity method. Poroelastic stresses are resolved onto the fractures using the finite element shape functions, which allows for the description of nonuniform loading conditions when multiple fracture elements exist at the sub-grid scale. conditions for the fracture deformation problem. In two dimensions, the two primary variables of interest are the opening mode and sliding mode displacement discontinuities, δ n and δ s, respectively. This system requires two equations that describe mechanical equilibrium. For opening mode deformation, mechanical equilibrium ensures that the internal fluid pressure acting on the fracture walls is balanced by the external loading due to remote loading, poroelastic stress, or the mechanical deformation of the fracture or its neighbors: R n = σ n 0. (19) Here, σ n = σ R n +σ M n +σ P n p f is the overall effective normal stress. For the shear deformation problem, we use a Mohr-Coulomb-type failure criterion to describe mechanical equilibrium: R s = σ s fσ n 0, (20) where σ s = σ s R +σ s M +σ s P is the total shear stress acting on the fracture and f is the coefficient of friction. At the initial condition, fractures are assumed to be at equilibrium with the remote tectonic loading. We use a displacement discontinuity (DD) method to relate fracture deformation to the changes in the state of stress along a fracture. In the DD method, fracture deformation is related linearly to the stress (traction) boundary conditions: σ P M n +σ n = σ P M s +σ s A nn A sn A ns A ss δ n δ n. (21) Here, vector and matrix notation represent the discretized system of DD fracture elements. The A submatrices are the DD interaction coefficients. Equations 19 and 20 are solved in a nonlinear framework using Newton s method to obtain and. 915

6 3. Model Verification and Discretization Refinement Study We compared the results of the present model with the fractured reservoir geomechanics model AD-GPRS introduced by Garipov et al. (2016). We considered isothermal fluid circulation between two wells connected by a vertical fracture that was 1000 m long and 500 m in vertical extent. Fluid was injected at the left end of the fracture a constant bottomhole pressure of 10 MPa above the initial reservoir pressure. Fluid was produced at the right end of the fracture at a constant bottomhole pressure equal to the initial reservoir pressure. The matrix rock surrounding the fracture had a permeability of k m = m 2 (10 md), so fluid was able to leakoff from the fracture during circulation. Increased fluid pressure in the matrix rock induced compressive poroelastic stresses and caused a backstress to develop on the fracture surface. For simplicity, the fracture aperture was assumed to be constant so that the changes in stress did not affect fluid flow. We assumed Table 1. Model parameters. Parameter Value Unit Parameter Value Unit R σ xx 100 MPa k m m 2 R σ yy 100 MPa m ϕ R σ xy 0 MPa c ϕ MPa -1 p 0 30 MPa ρ kg m -3 p inj. 40 MPa c ρ MPa -1 p prod. 30 MPa µ MPa s α µ 500 m ν H 1000 m G 15 GPa L f m Figure 2. Change in matrix fluid pressure cause by leakoff from the fracture at 1 day, 10 days, and 100 days after the start of injection. The white line is the fracture and the black dots are the wells. two-dimensional plane strain conditions. For the comparison with AD-GPRS, we solved the poroelastic deformation problem using 8-node quadrilateral elements. The model geometry, well controls, fluid properties, and rock properties are listed in Table 1. In Fig. 2 we show our model s solution for the change in matrix fluid pressure at several different times. The contours represent change in fluid pressure, the white line is the fracture, and the black dots represent the locations of the wells. Although the domain extended from ±4000 m in both directions to minimize boundary effects, we show a smaller portion of the domain to Figure 3. Matrix fluid pressure for the AD-GPRS model after 100 days. The colorbar scale is in units of MPa 10. Note the spatial axis range is larger than in Fig. 2. Figure 4. Induced stress σ P xx 1 day, 10 days, and 100 days after the start of injection. The white line is the fracture and the black dots are the wells. 916

7 highlight detail. Because fluid leakoff was driven by the pressure drop between the fracture and matrix rock, the leakoff rate was higher at early times and near the injection well. Similar to the case of single-well injection into an infinite conductivity fracture, leakoff tended to be one-dimensional away from the fracture at early times and transitioned towards radial flow at late times (Horne, 1995). The AD-GPRS solution at 100 days is shown in Fig. 3 for comparison. Note the different axis range. We observed a good qualitative and quantitative match between the two models. The poroelastic stress σ P xx is shown in Fig. 4. This is the normal stress component acting in the direction parallel to the fracture. The largest induced compressive stresses occurred at the fracture surface, and the perturbations were confined largely to the zone of perturbed pressure. Nonlocal effects caused tensile stresses to be generated further out in the reservoir, but their magnitudes were small. For comparison, the AD-GPRS solution for σ P xx is shown in Fig. 5. Note the opposite sign convention for stresses between the two models. The magnitudes and distributions of the stress changes were nearly identical for the two models. P Figure 5. Induced stress σ xx for the AD-GPRS model after 100 days. The colorbar scale is in units of MPa. Note the spatial axis range is larger than in Fig. 4. P Figure 6. Induced stress σ yy at 1 day, 10 days, and 100 days after the start of injection. The white line is the fracture and the black dots are the wells. The poroelastic stress σ P yy is shown in Fig. 6. This is the normal stress component acting in the direction normal to the fracture. This stress component would have a significant impact on the fluid flow if a nonlinear joint stiffness relationship were used to relate fracture transmissivity and effective stress. The increased compressive backstress would tend to close the fracture. The magnitude of σ P yy was largest near the injection well, and decreased along the length of the fracture. Near the production well, nonlocal effects actually caused small tensile stresses that would act to pull the fracture open. For comparison, the AD-GPRS solution for σ P yy is shown in Fig. 7. Note the opposite sign convention for stresses between the two models. A good agreement between the models was observed. In the embedded fracture modeling approach, the fracture and matrix discretizations do not conform. In this study, we used a conventional discrete fracture discretization strategy for the fracture and a structured Cartesian grid for the matrix. In our fractured reservoir simulations, it is common for many fracture elements to exist within a common matrix grid block. One of the main advantages of combining the embedded fracture approach for fluid flow with a finite element approach for poroelastic deformation is that nonuniform stress Figure 7. Induced stress σ P yy for the AD-GPRS model after 100 days. The colorbar scale is in units of MPa. Note the spatial axis range is larger than in Fig. 6. distributions can be resolved onto the fracture elements at the sub-matrix-grid scale using the finite element shape functions. In Fig. 8, we illustrate this concept by showing the resolved poroelastic normal stress, σ n P along the fracture. In this numerical example, the 1 km long fracture was discretized into 116 elements. The fracture intersected roughly 13 matrix grid blocks. The type of finite element used will determine the order of the interpolation that is possible at the sub-grid scale. For example, in Fig. 8(a) we show the stress interpolation using four-node quadrilateral element which resulted in 917

8 a linear distribution. A stress discontinuity occurred at the transition across finite element boundaries because in the finite element method the displacements are continuous at element boundaries but the stresses are not. In contrast, in Fig. 8(b) we show the stress interpolation using 8-node quadrilateral elements which resulted in a quadratic interpolation. 4. Discussion (a) Figure 8. Poroelastic normal stress resolved on the fracture plane for simulations with (a) 4-node and (b) 8-node quadrilateral finite elements. We are currently participating as part of a collaborative effort to compare different geothermal reservoir simulation codes led by the Department of Energy Geothermal Technologies Office (White et al., 2016). The collective group decided to use the Fenton Hill Enhanced Geothermal System (EGS) test site as the basis for a challenging numerical exercise. The group has been tasked with using multiple interdisciplinary data sets (including rate/pressure data, microseismic data, and diagnostic fracture injection test data) to develop conceptual models of the reservoir structure and test various hypotheses aimed at understanding the interaction between fluid flow, heat transfer, and mechanical deformation in the reservoir. Ultimately, the goal of that project is to inform future EGS design and operational strategies by gaining an improved understanding of the physics that govern EGS reservoir behavior. This version of CFRAC is particularly well-suited to address this type of problem. Several different physical processes were believed to influence behavior at Fenton Hill, including mechanical opening and shear failure of preexisting fractures, hydraulic fracture propagation, and thermal stress effects. In addition, the operational strategies adopted at the site were rather complex. For example, many multi-day hydraulic stimulation treatments with varying injection rate were carried out on both of the main wells over the course of several years. The extremely large volumes of water injected during the hydraulic stimulation treatments was believed to have leaked off into the surrounding rock while the wells were shut-in between injection experiments. Multiple water circulation tests were attempted that ranged from days, to weeks, to months depending on the experiment. The circulation tests were generally not performed under ideal constant operational conditions, but rather sought to identify reservoir responses to changes in production well backpressure, for example. We are currently developing a unified reservoir model of the Fenton Hill site that integrates the stimulation phase, circulation phase, and a hypothetical long-term production scenario. The initial stages of this effort were presented by Norbeck et al. (2016). In the future, this model could be improved in several useful ways. In this study, we were interested in understanding how poroelastic deformation of the matrix rock influenced fracture deformation. We accounted for this by resolving the poroelastic stresses back onto the fractures and including those stresses as additional boundary conditions in the boundary element calculations. However, the full equations of poroelasticity include the effect of the matrix rock deformation on the matrix porosity. In this work, we assumed that matrix porosity remained constant so that the poroelastic coupling was effectively one-way. Sequential methods for solving poroelasticity have been shown to be unstable when matrix porosity is not constant. Some techniques have been developed previously to overcome this limitation, and could be adopted within our framework (Ganis et al., 2014; Garipov et al., 2016; Kim et al., 2011; White et al., 2016). Furthermore, the porosity update should ideally include the effects of fracture deformation-induced volumetric strain in the matrix. This could be calculated in each coupling iteration as a post-processing step following the boundary element solution by calculating the volumetric strain at each matrix control volume center. In Section 3, we demonstrated that our model yielded accurate (b) 918

9 solutions when the induced poroelastic stresses resolved on the fracture surface were compressive. Further experiments must be performed in order to verify the method s accuracy for cases in which the induced stresses cause the overall effective normal stress acting on the fracture to become tensile, thereby causing fracture opening to occur. 5. Concluding Remarks We extended the CFRAC modeling framework to include the effects of poroelasticity on fracture deformation. The novel coupling strategy is based on the embedded fracture approach for fluid flow, the displacement discontinuity method for fracture mechanics, and the finite element method for poroelastic deformation. Some of the advantages gained (or retained) by invoking this coupling strategy include: Efficient and accurate fracture deformation calculations for problems with dense fracture networks using the DD method No remeshing for fracture propagation because fracture-matrix mass transfer is handled by simple source terms in the embedded fracture method Nonuniform sub-grid-scale distributions of poroelastic stress can be resolved onto fracture elements using finite element shape function interpolation (when many fracture elements cut through a single matrix grid block) Heterogeneous matrix permeability Adaptive timestep control The accuracy of the present model was verified against AD-GPRS, another fractured reservoir geomechanics model. We expect that this model will be useful for understanding behavior in fractured reservoir systems such as hydrothermal reservoirs, EGS reservoirs, or unconventional hydrocarbon reservoirs. Acknowledgements The financial support of the industrial affiliates of the Stanford Center for Induced and Triggered Seismicity is gratefully acknowledged. Part of this work was supported by the Department of Energy Geothermal Technologies Office geothermal code comparison study under grant number DE-EE For the poroelastic component of the model, we modified the open source finite element software package associated with the book, Programming the Finite Element Method, by I.M. Smith, D.V. Griffiths, and L. Margetts which can be obtained at The authors thank Timur Garipov his help with the comparison study between CFRAC and AD-GPRS. The CFRAC model can be licensed for research purposes by contacting J.H. Norbeck. References Ganis, B., M. Mear, A. Sakhaee-Pour, M. Wheeler, and T. Wick (2014), Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method, Computational Geosciences, 18, , doi: /s Garipov, T., M. Karimi-Fard, and H. Tchelepi (2016), Discrete fracture model for coupled flow and geomechanics, Computational Geosciences, 20 (1), , doi: / s z. Horne, R. (1995), Modern Well Test Analysis, 2nd ed., Petroway Inc., Palo Alto. Karvounis, D., and P. Jenny (2016), Adaptive hierarchical fracture model for enhanced geothermal systems, Multiscale Modeling and Simulation, 14 (1), Kikani, J. (1989), Application of boundary element method to streamline generation and pressure transient testing, Ph.D. thesis, Stanford University. Kim, J., H. Tchelepi, and R. Juanes (2011), Stability, accuracy, and efficiency of sequential methods for coupled flow and geomechanics, SPE Journal, 16 (2), , doi: / pa. Kurtoglu, B., H. Kazemi, and R. Rosen (2014), A rock and fluid study of Middle Bakken Formation: Key to enhanced oil recovery, in Proc., SPE/ CSUR Unconventional Resources Conference, Calgary, Canada. Lee, S., M. Lough, and C. Jensen (2001), Hierarchical modeling of flow in naturally fractured formations with multiple length scales, Water Resources Research, 37 (3), Li, L., and S. Lee (2008), Efficient field-scale simulation of black oil in a naturally fractured reservoir through discrete fracture networks and homogenized media, SPE Reservoir Evaluation and Engineering, August, McClure, M. (2012), Modeling and characterization of hydraulic stimulation and induced seismicity in geothermal and shale gas reservoirs, Ph.D. thesis, Stanford University. Norbeck, J., and R. Horne (2015), Injection-triggered seismicity: An investigation of porothermoelastic effects using a rate-and-state earthquake model, in Proc., Fourtieth Workshop on Geothermal Reservoir Engineering, Stanford, California, USA. 919

10 Norbeck, J., and R. Horne (2016), Physical mechanisms related to microseismic-depletiondelineation field tests with application to reservoir surveillance, SPE Journal, Published online January 2016, doi: / pa. Norbeck, J., M. McClure, J. Lo, and R. Horne (2015), An embedded fracture modeling framework for simulation of hydraulic fracturing and shear stimulation, Computational Geosciences, 20 (1), 1 18, doi: /s Norbeck, J., M. McClure, and R. Horne (2016), Revisiting stimulation mechanism at Fenton Hill and an investigation of the influence of fault heterogeneity on the Gutenberg-Richter b-value for rate-and-state earthquake simulations, in Proc., 41st Workshop on Geothermal Reservoir Engineering, Stanford, California, USA. Pollard, D., and R. Fletcher (2005), Fundamentals of Structural Geology, Cambridge University Press. Safari, R., and A. Ghassemi (2016), Three-dimensional poroelastic modeling of injection induced permeability enhancement and microseismicity, International Journal of Rock Mechanics and Mining Sciences, 84, 47 58, doi: /j.ijrmms Smith, I., D. Griffiths, and L. Margetts (2014), Programming the Finite Element Method, 5th ed., John Wiley and Sons Ltd. White, S., S. Kelkar, and D. Brown (2016), Bringing Fenton Hill into the digital age: Data conversion in support of the Geothermal Technologies Office comparison study challenge problems, in Proc., 41st Workshop on Geothermal Reservoir Engineering, Stanford, California, USA. 920