Received: 12 April 2019 Revised: 20 August 2019 Accepted: 21 August 2019 DOI: 10.1002/nme.6208

RESEARCH ARTICLE

Coupled poromechanics-damage mechanics modeling of fracturing during injection in brittle rocks

Ahmed Bubshait Birendra Jha

Mork Family Department of Chemical Engineering and Materials Science, Summary University Southern of California, Los Subsurface injection of in a stress-sensitive naturally fractured rock faces Angeles, California the problem of near-wellbore fracture evolution and associated changes in rock Correspondence properties. Numerical modeling of the changes in permeability and poroelas- Birendra Jha, Mork Family Department of tic properties of the near-wellbore region is challenging due to the coupling Chemical Engineering and Materials Science, University of Southern between fracture dynamics and poromechanics across multiple length scales California, Los Angeles, CA 90089. of fractures and the host rock. We present a numerical framework to model Email: [email protected] anisotropic and dynamic evolution in rock properties based on a coupled formu- Funding information lation of flow, rock mechanics, and fracturing, where fracturing-induced Petroleum Research Fund, Grant/Award Number: PRF 58769-DNI9 damage is used to update the rock properties. A generalized fixed-stress method, which accounts for damage-induced anisotropy in flow and deformation pro- cesses, is developed to sequentially solve the equations of flow, mechanics, and fracture evolution. We demonstrate the usefulness of our framework in quantify- ing the effects of injection rate variability, initial fracture distribution, and in-situ stress state on the evolution in permeability, elastic stiffness, and the Biot param- eters. Our framework does not require the computational mesh to conform to existing or future fractures, allows simultaneous growth of multiple randomly distributed fractures, and can be implemented relatively easily in existing cou- pled flow-geomechanics simulators to extend them to model fracturing at the reservoir scale.

KEYWORDS anisotropy, damage mechanics, fluid injection, fracture modeling, geomechanics

1 INTRODUCTION

Fractures control strength and brittleness of rocks under hydromechanical loads.1,2 Fractures also act as conduits of fluid flow and heat transport in aquifers, geothermal rocks, and oil reservoirs.3 In many engineering applications, it is essential to understand the role of growing fractures on material strength or fluid flow. As a result, various constitutive models with different levels of complexity have been proposed over the years to model the flow and mechanical response of fractured rocks.4-14 However, many models struggle to capture the evolving complexity of propagating fractures of different orien- tations and length scales due to the role of local stresses, which is time dependent and difficult to capture via constitutive modeling. Concentration of locally tensile stresses, especially the deviatoric part, around grain and defect boundaries in rocks6,11,15,16 causes new cracks to originate from defect boundaries, which grow along the directions governed by the principal stress directions and the locations of other defects.17-21 This is further supported by lower fracture toughness values at the grain boundaries compared to the values inside the grains.22 The result of such high stress concentration

Int J Numer Methods Eng. 2019;1–21. wileyonlinelibrary.com/journal/nme © 2019 John Wiley & Sons, Ltd. 1 2 BUBSHAIT AND JHA at the small scale is that the rock fails at a far-field stress value that is much lower in magnitude than predicted by the Griffith theory and the molecular cohesive strength.23-26 In addition, as the crack propagates, local stresses near the crack tip change in magnitude and direction27 suggesting that one needs to solve poro-elastodynamics coupled with fracture propagation in order to capture the effect of such propagation on deformation and flow processes in brittle rocks. How- ever, the problems of dynamic fracture propagation and fluid flow through fractures can be highly nonlinear due to the geometry and scale interaction issues.2,25,28 To make the computational modeling of propagating fractures tractable, fractures are often treated as lower-dimensional objects within a higher-dimensional matrix, eg, one-dimensional fractures in a two-dimensional (2D) domain.29-31 Even with these simplifications, it is challenging to model the dynamics of flow and fracture propagation in naturally fractured rocks due to the complex geometry of the fracture network and the scale discrepancy between the fractures and the host matrix. A common approach in finite element analyses involves representing a fracture as a dis- placement discontinuity discretized with cohesive zone elements. The concept of a cohesive zone in fracture mechanics is proposed earlier.32,33 The cohesive zone acts as a fracture process zone where a resisting force or traction acts on each crack surface as a function of the displacement jump across the surfaces. Cohesive zone models have also been used to evaluate crack formation and growth in concrete.34 In cohesive zone models, the crack path is often defined a priori such that the locations of the cohesive elements are known. The discrete fracture model approach is similar in terms of its requirement to know the fracture locations and path a priori.35 These approaches lead to mesh-dependent fracture path solutions because the fractures can propagate only along element edges. Simulation of flow and deformation processes in fractured reservoirs using a mesh that explicitly represents multiple fractures and conform to individual fracture geometry is computationally prohibitive. There is another hurdle in simulating naturally fractured reservoirs. During production or injection periods, poromechanical properties might change around the well due to changing pressure and stress conditions. These changes impact near-wellbore permeability, which is an important hydrologic parameter that is used to evaluate the injectivity/productivity of the well and to determine the locations of future wells in the field. We present a framework to model flow and deformation in naturally fractured reservoirs based on a coupled formula- tion of fluid flow, rock mechanics, and fracturing-induced damage processes. We extend the fixed-stress method, proposed earlier for isotropic deformation,36-39 to model the anisotropic deformation and flow behaviors observed due to micro- cracking at the reservoir scale40,41 and the core scale.42 Role of anisotropy in altering flow patterns43 and mechanical strength of naturally fractured rocks44,45 has been emphasized recently. We use the Irwin-Griffith theory46,47 combined with the continuum damage mechanics (CDM) approach48-52 to model fracture propagation and associated changes in mechanical and hydraulic properties of the medium. The CDM approach allows one to capture the macroscopic effects of microscopic cracking, such as material anisotropy, inelasticity, pressure sensitivity, volumetric dilatation, and strength deterioration, into continuum scale models through constitutive equations based on a damage variable defined using the crack length. Recently, the damage mechanics approach has been used to simulate fracturing and permeability enhancement in brittle permeable rocks subjected to injection.53 The approach consists of solving the coupled poroelastic problem simultaneously using COMSOL Multiphysics followed by updating the tensor, permeability, and the Biot parameters as functions of the damage variable. Sequential iterations are conducted over the poroelastic solver and the damage-based updater with a convergence criterion on the global norm of the change in the elasticity tensor. This approach is expected to show poor convergence properties and smaller timestep requirements during hydromechanical loading of stress-sensitive rocks, compared to establishing convergence directly on the poroelastic solution of pressure and stress or displacement. In another implementation,54 a damage-based updater for permeability is used to model per- meability enhancement during injection for enhanced geothermal applications. However, in that implementation, the flow problem ignores the mechanical coupling arising from the change in volumetric stress, and therefore, it is not clear how the damage-induced permeability enhancement interacts with the volumetric deformation-induced and permeability changes,55 especially when the two types of coupling act against each other. Recent progress in using CDM approach motivates us to develop a sequentially coupled flow-mechanics-damage model to investigate flow-induced changes in hydromechanical properties of a reservoir containing networks of cracks. The capa- bility of sequentially coupled flow-mechanics simulators to simulate well operations in fields spanning tens of kilometers in space and multiple decades in time has been demonstrated recently.38,56-58 In our approach described in the following, we consider the changes in crack length and aperture due to flow-induced deformation and rock failure as damage to the hydromechanical properties—permeability, elastic stiffness, and the Biot parameters—which evolve in space and time as the damage variable evolves with the deviatoric stress. We present results from numerical studies using our simula- tor that validate our framework and illustrate the effects of well rate variability, fracture directionality, and the boundary stress ratio on rock properties. BUBSHAIT AND JHA 3

FIGURE 1 Left: a horizontal injector placed in a naturally fractured reservoir containing microcracks and larger fractures. Right:a two-dimensional conceptual model of a horizontal cross-section of the fractured reservoir showing the injector intersecting multiple pre-existing microcracks. The model domain containing the randomly distributed, noninteracting microcracks is considered to be undamaged before the well becomes active [Colour figure can be viewed at wileyonlinelibrary.com]

2 PHYSICAL MODEL

The model domain is assumed to be a 2D Lx × Ly horizontal section of a reservoir containing the wellbore as shown in Figure 1. The section is a fluid-saturated homogeneous rock embedded with randomly placed idealized line microcracks. Locations and geometric properties of such microcracks can be estimated from electrical resistivity or acoustic logs along boreholes,59 pressure transient tests,60 and seismic surveys,61 which are measurement techniques at different length scales and resolutions. The domain is subjected to compressive loads representing the two horizontal stresses on its top and right boundaries, 𝜎H and 𝜎h, respectively. The stress ratio is defined as SR = 𝜎h∕𝜎H. The bottom and left boundaries are fixed in the normal directions. No-flow conditions are prescribed on all boundaries. The model is in hydrostatic and mechanical equilibrium, and the state of the rock before fluid injection is regarded as the initial or undamaged state.

3 MATHEMATICAL MODEL

Consider the model problem shown in Figure 2. We classify the cracks into crack families defined using the surface normal vector of the cracks. Subscript k is the crack family index, and N is the number of crack families. A representative elementary volume (REV) containing a single elliptical crack with unit normal nk and subjected to total normal stresses 𝜎xx and 𝜎yy is shown. A local coordinate system is defined with nk and the unit tangent vector along the crack long axis. The

FIGURE 2 Stress field in a representative elementary volume (REV) around

a crack of half-length ak induced by injection in a horizontal well of half-length

Lw in a domain under tectonic compressions 𝜎h and 𝜎H. 𝜓 is the fracture

direction measured from the x-axis. 𝜎t is the tension at the crack tip due to the normal projections of the effective stress and the deviatoric stress tensors. p is pore pressure induced by injection. The distance between the well and the

crack is characterized with lengths r, r1,andr2 and with angles 𝜃, 𝜃1,and𝜃2. The REV size is exaggerated with respect to the well size [Colour figure can be viewed at wileyonlinelibrary.com] 4 BUBSHAIT AND JHA model boundary conditions are as follows. Zero normal displacements are prescribed on the left and bottom boundaries. Constant normal compressions are prescribed on the right and top boundaries equal to the minimum horizontal principal stress 𝜎h and the maximum horizontal principal stress 𝜎H. Zero fluid flux is prescribed on all the boundaries.

3.1 Poroelastic problem in an isotropic domain For the poroelastic host matrix, we use a classical continuum representation, where the fluid and skeleton are viewed as overlapping continua. The governing equations of fluid flow and mechanical deformation are obtained from the mass and linear momentum balance statements of a quasi-static domain undergoing infinitesimal deformations.25,62 These equations form the basis of Biot's self-consistent theory of poroelasticity,63 which describes the relation between the changes in total stress, the pore pressure, and the fluid mass content in space and time. Fluid mass conservation during isothermal single-phase flow through a poroelastic medium can be expressed as62

dm +Δ·w = 𝜌𝑓 q , (1) dt w 𝜌 𝜌 𝜇 where m is the fluid mass (per unit bulk volume), f is the fluid density, w = fv is the fluid mass flux, and v =−(k∕ )∇p is the Darcy velocity relative to the deforming skeleton. qw is the volumetric source term of the fluid phase at the reservoir pressure condition. 𝜇 is the dynamic fluid , k is the absolute permeability, and p is the pore pressure. t denotes time. Gravity is neglected. The quasi-static mechanical equilibrium of the medium in absence of body forces such as gravity is governed by the linear momentum balance equation

∇·𝝈 = 𝟎, (2) where 𝝈 is the Cauchy total stress tensor. The effective stress describes the relationship between the total stress and the pore fluid pressure, 𝝈 = 𝝈′ −𝛼p1, where 𝝈′ is the effective stress tensor, 0 ≤ 𝛼 ≤ 1 is the Biot coefficient (a scalar quantity for an isotropic material), and 1 is the rank-2 identity tensor. We assume that normal stresses are positive in tension, and pressure is positive in compression. The stress tensor can be decomposed into an isotropic part and a deviatoric part such that, in 2D, 𝝈 =(tr(𝝈)∕2)1 + S,whereS is the deviatoric stress tensor. The isotropic part is used to define the volumetric stress 𝜎v = tr(𝝈)∕2 =(𝜎xx + 𝜎yy)∕2. For time-dependent linear problems, where we are interested in solving for changes with respect to initial conditions, it is convenient to decompose quantities into their initial value, denoted with subscript 64 0, and the change from the initial value. For example, the stress tensor can be decomposed as 𝝈 = 𝝈0 + 𝜹𝝈, where 𝝈0 is the initial total stress and 𝜹𝝈 is the change in the total stress. The equilibrium equation becomes

′ ∇·𝜹𝝈 =−∇·𝝈𝟎 + 𝛼∇𝛿p. (3)

The constitutive equations of isotropic poroelasticity in terms of increments in stress and pressure are65

𝜹𝝈 = 𝜹𝝈′ − 𝛼𝛿p𝟏 (4)

𝛿m 1 = 𝛼𝜖v + 𝛿p, (5) 𝜌𝑓,0 M ′ where M is the Biot modulus, 𝛿𝝈 = Cdr ∶ 𝝐 is the increment in the effective stress tensor, Cdr = 𝜆𝛿ij𝛿kl +G(𝛿ik𝛿jl +𝛿jk𝛿il) is the rank-4 drained isotropic elasticity tensor corresponding to the undamaged state, 𝜆 and G (shear modulus) are the Lame parameters, and 𝛿ij is the Kronecker delta. Young's modulus E and Poisson's ratio 𝜈 canbeexpressedintermsof𝜆 and G. The linearized strain tensor is defined as 𝝐 ∶= (∇u +∇Tu)∕2 in terms of the displacement vector u. In Voigt notation, the T T stress and strain vectors under plane strain conditions can be written as 𝝈 =[𝜎xx,𝜎yy,𝜎xy] and 𝝐 =[𝜖xx,𝜖yy, 2𝜖xy] ,which T renders Cdr into a 3×3 rank-2 tensor. The 2D identity vector in Voigt notation becomes 1 =[1, 1, 0] . The volumetric strain is 𝜖v = tr(𝝐). Biot's theory of poroelasticity has two coupling coefficients: the Biot modulus M and the Biot coefficient 𝛼, which are scalars for an isotropic medium. They are related to rock and fluid properties as65

1 𝛼 − 𝜙 Kdr = 𝜙c𝑓 + ,𝛼= 1 − , (6) M Ks Ks BUBSHAIT AND JHA 5

𝜌 𝜌 where cf =(1∕ f)d f∕dp is the fluid compressibility, Ks is the bulk modulus of the solid grain, Kdr is the drained bulk modulus of the , and 𝜙 is porosity that evolves with pressure and stress. In 2D, K = 1 𝟏TC 𝟏,whichfor dr 4 dr 𝜆 𝛿𝜎 𝛿𝜎′ 𝛼𝛿 𝛿𝜎′ 𝜖 66 plane strain yields Kdr = + G. Therefore, v = v − p with v = Kdr v.WecanrewriteM as

Ks M = ( ) ( ). (7) Kdr 1 − − 𝜙 1 − c𝑓 Ks Ks

Substituting Equation (5) into Equation (1) and assuming a slightly compressible fluid, the fluid mass balance equation in a homogeneous isotropic rock, in the pressure-stress formulation, becomes38

( ) 2 𝜕 𝜕𝜎 1 𝛼 p 𝛼 v k 2 + + − ∇ p = qw. (8) M Kdr 𝜕t Kdr 𝜕t 𝜇

We can use the linear elastic fracture mechanics theory to approximately estimate the change in poroelastic stresses due to the presence of an injection well. Under the undrained deformation assumption (𝛿m = 0), valid for early times in low permeability rocks, the injection-induced poroelastic stresses 𝛿𝜎xx, 𝛿𝜎yy,and𝛿𝜎xy at a point far from the well can be given as67 [ ( )] r 𝜃1 + 𝜃2 𝛿𝜎xx + 𝛿𝜎𝑦𝑦 = 2𝛿pw 1 − √ cos 𝜃 − = 2𝛿𝜎v, r1r2 2 2 rLw 3 𝛿𝜎𝑦𝑦 − 𝛿𝜎 = 2𝛿p sin 𝜃 sin (𝜃 + 𝜃 ), (9) xx w 3∕2 1 2 (r1r2) 2 2 rLw 3 𝛿𝜎 𝑦 = 𝛿p sin 𝜃 cos (𝜃 + 𝜃 ), x w 3∕2 1 2 (r1r2) 2 𝛿 where pw is the injection-induced pressure, above the minimum principal stress, in the reservoir rock surrounding the wellbore.

3.2 Crack propagation We use the classical Irwin-Griffith theory25 to derive the crack propagation criterion in terms of the stress intensity factor. We use the CDM theory to update the petrophysical properties—permeability and stiffness—of our continuum model as anisotropic functions of the damage tensor, which itself is defined using a crack length-based damage variable. The following subsections describe these concepts in sequence and in detail.

3.2.1 Stress intensity factor 2,25 According to the linear elastic fracture mechanics theory, the mode-I (tensile failure mode) stress intensity factor, KI, around the tip of a crack of half-length ak and unit normal nk in an infinite body subjected to the total stress 𝝈 and pore pressure p is √ √ E K (𝝈, p, a , n )=𝜎 𝜋a = G , (10) I k k t k c 1 − 𝜈2 where 𝜎t is the tensile stress in the crack-tip region, and Gc is the mechanical energy release rate (energy per unit crack surface area). In case of a finite size body containing multiple cracks, such as the physical model in Figure 1, the above relation is modified accordingly, and it is valid only under the assumption of noninteracting cracks. Here, we make that assumption because the models that account for crack interaction are mathematically more complex68 and may not scale up for field-scale simulation. Therefore, to use Equation (10), we need an accurate estimate of 𝜎t because it controls the initiation and growth of new cracks. At late times, it also controls coalescence and localization of cracks into macrofrac- tures, which lead to failure, strength reduction, seismicity, and permeability enhancement at macroscopic scales. The lack of knowledge of 𝜎t around cracks and fractures is one of the major hurdles in understanding and predicting brittle 6 BUBSHAIT AND JHA processes in rocks. Based on some experimental observations, it has been proposed69,70 that the local stress near the crack tip can be expressed in terms of the effective normal traction and the normal component of the deviatoric stress on the crack surface as follows: 𝜎 𝜎′ 𝑓 , t = N + (ak)SN (11) 𝜎′ 𝜎 𝜎 𝝈 where N = N + p is the effective normal traction, N = nk · · nk is the total normal traction, SN = nk · S · nk is the deviatoric tension on the crack surface, and f(ak) is a scalar function that accounts for the contribution of the deviatoric tension in inducing crack growth. For nk =[−sin 𝜓,cos 𝜓] shown in Figure 2,

SN =[(𝜎𝑦𝑦 − 𝜎xx)∕2] cos(2𝜓)−𝜎x𝑦 sin(2𝜓), 𝜎′ 𝜎′, N = SN + v (12) 𝜎 𝑓 𝜎′. t =(1 + )SN + v

71 Hence, 𝜎t and, consequently, KI can be decomposed into an isotropic part and a deviatoric part similar to the decom- position of 𝝈 above. At early times, the injection-induced tractions can be evaluated analytically using Equation (9). For example, 2 𝛿pwrL 3𝜃 + 3𝜃 − 4𝜓 𝛿S = w sin 𝜃 sin 1 2 , (13) N 3∕2 (r1r2) 2 which tells us that SN can be different in the northeast (NE) and northwest (NW) quadrants of Figure 2 due to different signs of 𝜃 in the two quadrants. Within a quadrant, it will be different for cracks with different values of direction 𝜓. We use the above relation, which is derived under the linear elastic fracture mechanics theory, to validate and interpret results from our nonlinear simulations that are based on the solution algorithm presented in Figure 3. For a crack not oriented along the principal stress directions, eg, 𝜓 ≠ 0◦, 90◦, or for any crack under the injection-induced state of stress that is nonprincipal (Equation (9)), the crack is under a mixed mode loading, eg, both mode-I and mode-II loading. Under these conditions, it is known that the crack grows following a curved path with the time-dependent crack direction 𝜓(t) 2,27 given by the maximum of Gc(𝜓,ak) corresponding to 𝜕Gc∕𝜕𝜓 = 0, which changes with ak. Here, we assume only mode-I failure to avoid the difficulty associated with estimating KII, and we do not model the crack path to maintain the computational benefits of our CDM approach for field-scale simulation. When KI starts to increase and reaches the equilibrium condition of mode-I crack propagation, KI = KIC,whereKIC is the mode-I fracture toughness (a material property), the crack starts to grow under mode-I.4,25,71 Propagation of a crack, and associated evolution of the damage, in a REV is governed by the propagation condition, which requires evaluation of 𝜎t in the REV. 𝜎t must be positive (tensile) in order for the fracture length to propagate; otherwise, it will stabilize. This > 𝜎′ 𝑓 implies SN − N ∕ for crack propagation. Once the crack length reaches a certain critical length b, crack coalescence begins and the material is considered to have failed due to self-sustaining mechanical instability. The function f(ak) in Equation (11) plays a role in crack coalescence because it can model local concentration of stresses around the crack as the 50 crack length increases. We also note that f should decrease with ak to model the drop in local tensile stress during crack 70,71 propagation. f(ak) has also been compared with the hardening-softening parameter in plastic deformation models. Here, we choose the following model72,73 for f: { 𝜔b∕ak, if (ak < b), 𝑓(ak)= (14) 𝜔, if (ak ≥ b),

69 where 𝜔 is a material constant. A nonmonotonic form of f(ak) has also been proposed to capture both stress relaxation at early times and stress amplification for ak ≥ b due to interaction between the growing crack and its neighbors.

3.2.2 A model for crack length The propagation condition, which also represents a surface of constant state of damage in the mean stress-deviatoric stress space,4 becomes √ ( ) 𝜋 𝜎′ 𝑓 , ak N + (ak)SN − KIC = 0 (15) BUBSHAIT AND JHA 7

FIGURE 3 The coupled poromechanics-damage( mechanics ) −1 𝜶T −1𝜶 sequential solution algorithm. Here, C𝟏 =(Ae∕Δt) M + Cdr and 𝜶T −1 C𝟐 =(Ae∕Δt)( Cdr ). See text for definitions of other quantities. Time level information (n or n + 1) is shown only for pressure, displacement, stress, and the fracture length. Damage mechanics method (DMM) is used to update the poromechanical properties. FEM, finite element method; FVM, finite volume method

which can be used to update the half-length ak. For early time growth when the cracks are below the critical length and do not interact, ie, ak < b,weobtain

[ √ ] R a =−𝜔bR + I R ± R2 − 4𝜔bR , (16) k N 2 I I N

√ 𝜎′ K2 𝜎′ 𝜋 𝜎′ N < < 1 IC where RN = SN ∕ N and RI =(1∕ )KIC∕ N . For real positive values of ak, − 𝜔 SN 𝜋𝜔 𝜎′ . Using Equation (12), the b 4 b N −𝜎′ lower bound condition can also be expressed as v < S . In the subsurface, the cracks are generally under compression, 1+𝜔b N 𝜎′ < ie, N 0 except during high pressure fluid injection (eg, hydraulic fracturing). A lithostatic state of stress (isotropic principal state of stress)√ such as the the one used below to initialize our base case yields SR = 1andSN = 0, which implies 𝜋𝜎′ 2 RN = 0andak =[KIC∕( N )] as per Equation (16). This can also be obtained from Equation (10), Equation (11), and the fracture equilibrium condition KI = KIC.Onceak is large enough for the cracks to start interacting among themselves through their local stress fields, unstable propagation and coalescence of cracks ensues culminating into a macroscopic fracture and structural failure of the material. We draw an analogy between the above model for computing the crack length ak during mode-I failure and a commonly used model for computing the fault slip s during mode-II (shear) failure based on the Mohr-Coulomb condition. The Mohr-Coulomb failure condition is 𝜏 ≥ 𝜏f,where𝜏 = |𝝈nk − 𝜎Nnk| is the shear traction magnitude on the fault/fracture ′ 𝜏𝑓 𝜏 𝜇𝑓 𝜎 𝜏 𝜇 plane, = c − N is the friction stress, c is cohesion, and f is the coefficient of friction. After substitution, the shear failure equilibrium condition becomes

𝜏 𝜇 𝜎′ 𝜏 . + 𝑓 N − c = 0 (17) 8 BUBSHAIT AND JHA

𝜏 𝜎′ 𝜇 Equation (17) resembles Equation√ (15) after the following substitutions in the former equation: with N , f with f(ak), 𝜎′ 𝜏 𝜋 71 4 N with SN,and c with KIC∕( ak). Therefore, as suggested earlier for plastic failure and frictional failure, the linear Mohr-Coulomb failure surface in Equation (17) is analogous to the linear mode-I failure surface in Equation (15). Such analogies have been performed earlier; see, eg, the work of Hoek and Martin,74 which states that the mode-I failure surface for a closed Griffith crack in the Mohr space of shear and effective normal tractions becomes a straight line given by the angle of friction and the shear strength. For open Griffith cracks, the mode-I surface becomes parabolic given by the tensile strength and the uniaxial compressive strength. Dragon and Mroz71 recognized the deviatoric stress S as a 𝜏 𝜇 type of effective stress in crack development. As cohesion √c or friction f changes, the Mohr-Coulomb failure surface moves in the stress space. Similarly, as ak changes in KIC∕( 𝜋ak) or in f(ak), which are the respective counterparts of 𝜏 𝜇 𝜇 c and f, the mode-I failure surface moves. In fault modeling, f is usually modeled as a slip-dependent function, eg, 38 𝜇 𝜇 𝜇 𝜇 𝜇 𝜇 the slip-weakening model, where f(s)= s −[( s − d)s∕sc], s is the static friction coefficient, d is the dynamic 𝜇 friction coefficient, and sc is the critical slip distance. This model of f(s) resembles the model of f(ak) in Equation (14), which further supports the analogy between these mode-I and mode-II failure models. A typical implementation of the Mohr-Coulomb failure model38,75 solves a contact problem to calculate the excess slip 𝛿s required to accommodate the excess shear, 𝛿𝜏 = 𝜏 − 𝜏f. Since the excess slip must be associated with the displacement discontinuity across the crack surface, the contact problem is coupled to the elasticity problem in Equation (2). Our mode-I model proposed above for 𝜎′ calculating ak (Equation (16)) assumes that N and SN do not change during crack propagation. This approximation is computationally much less expensive than solving the contact problem and is valid for microcracks that do not grow longer than the critical length and remain within the REV.

3.3 Damage in mechanical properties We use the CDM theory48 to define the damage tensor in terms of the geometry and properties of propagating fractures. We assume an isotropic distribution of the microcracks in the initial undeformed configuration. In 2D, the 2 × 2 damage tensor is defined as50,70,76 ∑N D ∶= mkdk(nk ⊗ nk), (18) k=1 where mk = Nk∕Ω denotes the number of cracks in family k per unit volume Ω of the REV, and dk is the dimensionless 2 2 2 scalar damage variable. For the 2D case considered here, dk =(ak − a0)∕a0,wherea0 is the average initial length of the uniformly distributed fractures in the domain. Considering the initial state of the material containing a certain spatial density and dimensions (eg, the half-lengths) of the cracks as the undamaged state, we can initialize dk = 0 at the begin- ≤ ≤ 2 2 2 ning of the simulation. Therefore, dk is nonnegative and bounded: 0 dk (b −a0)∕a0, ∀k. The damage variable evolves with the evolving state of stress through the crack length. Due to the growth of fractures and associated anisotropic dam- age, it is required to introduce the anisotropic elasticity tensor, which can be defined as a function of the damage tensor as follows53:

Cdr,i𝑗kl(D)=𝜆𝛿i𝑗 𝛿lk + G(𝛿ik𝛿𝑗l + 𝛿𝑗k𝛿il)+A(𝛿i𝑗 Dkl + Di𝑗 𝛿kl)+ (19) B(𝛿ikD𝑗l + 𝛿ilD𝑗k + Dik𝛿𝑗l + Dil𝛿𝑗k), ∀i,𝑗,k, l = 1, 2, where subscript k is not the fracture family index. Expanding the terms for plane strain yields

[ ] 𝜆 + 2G + 2AD11 + 4BD11 𝜆 + A(D22 + D11) AD12 + 2BD12 Cdr(D)= 𝜆 + A(D22 + D11) 𝜆 + 2G + 2AD22 + 4BD22 AD12 + 2BD12 . (20) AD12 + 2BD12 AD12 + 2BD12 G + B(D22 + D11)

A and B are the damage parameters, quantifying the damage-induced modification of the strain energy, which can be 50,51,71 measured via lab experiments. Their values should be chosen to ensure positive definiteness of Cdr.Thedamage tensor component D11 indicates crack opening-induced damage along the x-direction on crack planes with unit normals in the x-direction. D12 quantifies sliding-induced damage along the y-direction on cracks with unit normals along x.This introduces anisotropy in the mechanical and flow properties of the damaged rock. Following the idea of microisotropy at the grain scale, we can define the symmetric 2D Biot coefficient tensor as 𝛼ij = 𝛿ij −(Cdr,ij11 + Cdr,ij22)∕(2Ks) such that its BUBSHAIT AND JHA 9 components, as functions of the damage tensor, become66

1 𝛼11 = 1 − (2(𝜆 + G)+2(A + 2B)D11 + A(D11 + D22)) 2Ks 1 𝛼22 = 1 − (2(𝜆 + G)+2(A + 2B)D22 + A(D22 + D11)) (21) 2Ks 1 𝛼12 =− (2(A + 2B)D12) . 2Ks

Anisotropy in the damage and elasticity tensors imply that we cannot separate the volumetric deformation from the deviatoric deformation response. However, one can define a generalized drained bulk modulus,66 K∗ (D)= ( ) dr 1 C , + C , + C , + C , = 𝜆+G+(A+B)(D +D ),whereC , C ,andC = C are the 4 dr 1111 dr 2222 dr 1122 dr 2211 11 22 dr,1111 dr,2222 dr,1122 dr,2211 components of plane strain elasticity tensor from Equation (20). The scalar Biot modulus can be updated with the effects ∗ of damage by substituting Kdr in Equation (7)

K M(D)=[ s ] . (22) 𝜆 ( ) +G+(A+B)(D11+D22) 1 − − 𝜙 1 − c𝑓 Ks Ks

We can rewrite Equations (3)-(5) to account for the anisotropy in poromechanical properties as follows:

′ ∇·𝜹𝝈 =−∇·𝝈𝟎 +∇·(𝜶𝛿p)=−∇·𝝈𝟎 + 𝜶 ·∇𝛿p + 𝛿p∇·𝜶, (23)

𝜹𝝈′ = 𝜹𝝈 + 𝜶𝛿p, (24)

𝛿m 1 = 𝜶 ∶ 𝝐 + 𝛿p. (25) 𝜌𝑓,0 M This has three implications compared to the isotropic Equations (3)-(5). First, appearance of ∇·𝜶 asasourcetermin Equation (23) suggests that spatial variation in the Biot coefficient, which emerges during spatially heterogeneous micro- crack growth around a well, can contribute to deformation and enhance the heterogeneity in the effectives stress. The simulation results discussed below validate this hypothesis of enhanced heterogeneity in the stress field. Second, if 𝛼12 ≠ 0, then, as per Equation (24), a change in pressure can modify shear stresses, not just normal stresses. Third, appearance of 2𝛼12𝜖12 in the product of the Biot coefficient and the strain tensors in Equation (25) suggests a contribution to the change in fluid mass due to shear strain, which is a commonly missed mechanism because our understanding of the poroelastic response is often biased by the assumption of isotropy.

3.4 Damage in flow properties Consider the REV Ω defined earlier in terms of the porous matrix volume and the volume occupied by the microcracks, Ωc. The Darcy flow velocity v is assumed to be given by the sum of a matrix part and a crack part, v =−k ·∇p∕𝜇, where k = k0I + kc is the total permeability tensor, k0 is the initial (undamaged state) isotropic permeability, kc is the microcracking-induced permeability tensor (2 × 2 in 2D), and ∇p is the global pressure gradient driving the mean flow through the domain. Here, it is assumed that the local pressure gradient driving the flow through a crack is the same as the global pressure gradient. This assumption is valid for randomly oriented cracks with relatively high permeabilities. k0 does not change with damage and remains constant. We assume that kc(D) determines permeability changes caused by changes in the length, orientation, average aperture, and number of microcracks. To determine kc(D), we express the crack part of v as the average velocity through all cracks within Ω70

k 1 v =− 0 ∇p + v dΩ , (26) 𝜇 Ω∫ c c Ωc where vc is the velocity in an individual crack, and integration over Ω is equivalent to integration over Ωc because cracks are only present in Ωc part of Ω. We assume the flow inside the crack to be laminar and parallel to the crack plane direction. 10 BUBSHAIT AND JHA

For a given crack of orientation nk, vc can be expressed as

e2 v =−𝛾 k (I − n ⊗ n )∇p, (27) c 12𝜇 k k where we used the assumption that the local pressure gradient is equal to the global pressure gradient, and there is no flow in the direction transverse to the crack plane. The positive scalar 0 <𝛾 ≤ 1 is introduced to account for the fact that not all parts of a crack may contribute to flow. The classic cubic law for flow rate through a crack created by two perfectly parallel surfaces, ie, in plane Poiseuille flow, is recovered with 𝛾 = 1. We assume a linear relation between the average aperture of microcracks and the local tensile stress such that ek(𝜎t, ak)=(𝜋ak𝜎t)∕E0, where E0 is Young's modulus of the undamaged material. Evaluation of the integral term in Equation (26) requires estimation of the crack volume Ωc, which depends on the number of cracks of all orientations within the REV that are contributing to flow. Let 0 ≤ Rk ≤ 1 be the crack connectivity coefficient that represents the fraction of cracks contributing to flow. Above, we defined Nk as the number of cracks oriented along nk. Therefore, RkNk is the number of cracks oriented along nk and contributing to , 𝜋 2 flow. The infinitesimal crack volume for cracks oriented along nk can be calculated as dΩc k = RkNk · ek · ak in 3D and dΩc,k = RkNk · ek · 2ak in 2D. This has to be averaged over all orientations 0 ≤ 𝜓 ≤ 2𝜋 (all crack families) to obtain the average crack velocity. Therefore, we can rewrite the velocity in Equation (26) as

2𝜋 k0 2Nk 1 𝜓. v =−𝜇 ∇p + 𝜋 ∫ Rkekakvc d (28) Ω 2 0

The crack permeability tensor kc is regarded as a function of the crack orientation nk, average aperture ek,andNk.For 70 flow through a single crack of uniform aperture ek and unit normal nk,weobtain

𝛾 Nk 1 3 ⊗ 𝜓. kc = Rkakek(I − nk nk) d (29) 6 Ω 2𝜋 ∫𝜓

Since the crack family index k varies with the crack direction 𝜓, the integration over 𝜓 can be approximated with a summation over k 𝛾 ∑N k (D)= m R a e3(I − n ⊗ n ), (30) c 6 k k k k k k k=1 where we used mk = Nk∕Ω defined earlier. This model provides the components kxx, kyy,andkxy of the k tensor defined for the REV. Following the work of Zhang et al,77 we express the tensor in terms of minimum and maximum principal permeabilities, kmin and kmax,where

kxx + k𝑦𝑦 |kx𝑦∕ sin(2𝜃k)| −2kx𝑦 kmax = + , tan 2𝜃k = . (31) 2 2 kxx − k𝑦𝑦

We will analyze kmax in the Results section in the following.

4 NUMERICAL FORMULATION

We use a sequential solution scheme where we solve the fluid mass balance and the mechanical equilibrium equations iteratively using the fixed-stress method36 followed by the solution of the crack propagation problem. We extend the isotropic fixed-stress method to its anisotropic equivalent form to capture the cracking-induced anisotropic damage. We begin by extending the mass balance equation, Equation (8), to account for anisotropic 𝜶, Cdr,andk ( ) ( ) 𝜕 1 [ ] 𝜕 ( ) k(D) + 𝜶T(D)C−1(D)𝜶(D) p + 𝜶T(D)C−1(D)𝝈 +∇· − ·∇p = q . (32) 𝜕t M(D) dr 𝜕t dr 𝜇 w

We solve this equation using a finite volume method with a piecewise constant discretization of the pressure field, two-point flux approximation of the Darcy flux term (second-order accurate) and the Backward Euler (first-order accurate) BUBSHAIT AND JHA 11 time integration method.38 The system of equations for the element-centered pressure vector P can be written as follows: ( ) ( ) n+1,𝑗 Ae −1 𝜶T −1𝜶 n+1,𝑗+1 M + Cdr + T P = Δt (33) ( ) (( ) ( ) ) A n A n+1,𝑗 n e M−1 + 𝜶TC−1𝜶 Pn − e 𝜶TC−1𝝈 − 𝜶TC−1𝝈 + A qn+1. Δt dr Δt dr dr e w

Here, superscript j is the iteration counter of the sequential loop at the new timestep n + 1andT is the transmissibility matrix composed of harmonic means of fluid mobilities across element faces and, hence, depends on the permeability tensor k(D). M is the matrix of element-centered Biot moduli in the mesh. Vector P also includes unknown element pressures pw in the elements subjected to injection qw.Areaofith element is Ae,i =ΔxiΔyi. We discretize the rate of change of stress term in the mass balance equation, which is the solid-to-fluid coupling term, using the stress tensor at theprevioussequentialiteration,𝝈n+1,j. Hence, it appears on the right-hand side of Equation (33). We nondimensionalize time in our simulations with the hydraulic diffusion time scale

2 Δx 𝜇St t tflow = , t = , (34) c d flow k0 tc

𝛼2 where 𝛥x is the characteristic mesh size, k is the initial permeability value, and t is the dimensionless time. S = 1 + 0 d t M 𝜆+2G is the storativity coefficient defined in terms of initial or undamaged values. We use the standard finite element method with bilinear quadrilateral elements to solve the mechanics problem for the nodal displacement vector U. The discretized system of equations is

n+1,𝑗 n+1,𝑗+1 n+1,𝑗 n+1,𝑗+1 K U = Fu + Q P , (35)

T T where K = ∫ B CdrB dAe is the stiffness matrix, Q = ∫ Np𝜶 B dAe is the coupling matrix, and Fu is the load vector composed of traction integrals on Neumann boundaries. B is the strain-displacement matrix. Np is the vector of piecewise constant shape functions corresponding to the finite volume approximation of the pressure equation. The crack propagation process influences the mechanics problem only through the change in the rock properties, which is different compared to the approach based on discretized fractures, eg, cohesive zone, extended finite element, or inter- face element method. In the discretized fracture approach, a contact problem is solved on each crack surface at each timestep to obtain the traction and slip vectors on the surface. This increases the accuracy of the solution, however, at a significant cost because the tractions are treated as unknowns similar to the nodal displacements. This approach may not be suitable for modeling simultaneous propagation of hundreds of fractures in a field-scale simulator. It is also not required because discretizing individual fractures implies that the initial position, length, and direction of the fractures are known precisely, which is rarely the case. Our numerical simulator, which is implemented in MATLAB, solves the discretized system of equations to determine P and U. The element-averaged stresses are obtained by post-processing the displacement solution. The fracture length ak is updated by solving the damage-based propagation condition, Equation (15). k, Cdr, 𝜶,andM are updated to their new values using Equations (30), (20), (21), and (22), respectively. The flow, mechanics, and fracture propagation problems are solved again at the same timestep with the new values of the properties and the state variables P and U.Thissequential solution procedure is repeated until convergence at which point the system advances to the next timestep. Figure 3 shows our solution algorithm.

5 RESULTS AND DISCUSSION

We present results from multiple simulation cases to verify our computational framework and illustrate the effect of frac- turing on deformation and flow. In each case, we analyze the spatial distribution and temporal evolution of pressure, stress, and permeability in order to understand the impact of flow-mechanics coupling on the damage in poromechanical properties. We evaluate the sensitivity of induced damage to the following model parameters: well flow rate (qw), frac- ture orientation (𝜓), number of fracture families (N), and the boundary stress ratio SR. We consider that each element contains one randomly oriented fracture. We assume the values listed in Table 1 for important physical properties in the 12 BUBSHAIT AND JHA

TABLE 1 Rock-fluid properties. Subscript 0 indicates t = 0, ie, initial values Mathematical symbol Numerical value

Lx 100 m Ly 100 m 𝜙 0 0.1 k0 1md 𝜌 3 f 1000 kg/m 𝜇 1cp

p0 2000 psi 𝜈0 0.21 E0 68 GPa 𝛼0 0.8 a0 0.1 m 𝜔 0.6 √ KIC 1.1 MPa m b 1m

Overburden compression, 𝜎ob 12 MPa Horizontal compressions, 𝜎h, 𝜎H 7.385 MPa Damage parameter A 20 MPa m3 Damage parameter B −80 MPa m3 simulations. These values are representative of a tight formation. This implies a principal state of stress initially such that, at t = 0, 𝜎xx = 𝜎yy, 𝜎xy = 0, and SN = 0 (Equation (12)).

5.1 Case I: injection well at the center (base case) We distribute the fractures in the domain with orientation 𝜓 distributed randomly as shown in Figure 4. This distribution corresponds to all fractures oriented in the general northwest-southeast (NW-SE) direction of the domain and belonging to the same fracture family k = 1. We inject water at 50 bbl/day for 88 days uniformly along the 100 ft length of the horizontal injector. Figure 5 shows the diffusion of pressure over 3 months. At the beginning (t = 1 day), the pressure diffuses uniformly around the injector because of an isotropic distribution of the poromechanical properties. At t = 0, the stress field is homogeneous isotropic with 𝜎xx = 𝜎yy. As pressure increases (Figure 5), the near-wellbore region expands away from the well and applies compression on the far region. Figure 6 shows the distribution of stress field at two timesteps. The effect of 𝜃 in Equation (13) leads to anisotropy in the stress and permeability fields between northeast-southwest (NE-SW) and NW-SE corridors. This leads to preferential flow along the NW-SE corridor and anisotropy in the pressure contours, which orient along the NW-SE corridor (Figure 5). Stress anisotropy (𝜎xx ≠ 𝜎yy) and rotation of the principal stress directions from the x-andy-axes (𝜎xy ≠ 0) causes the magnitude of SN to increase from zero (Equation (13)). The actual increase depends on both the quadrant position 𝜃 and the fracture direction 𝜓. For example, for a REV location in 𝜃 ◦ 𝜃 ◦ 𝜃 ◦ 𝜓 ◦ 𝛿 2 3∕2 the NE corridor with =−45 , 1 =−60 , 2 =−30 ,and = 135 , SN = pwrLw∕(2(r1r2) ), which is positive. For a location in the NW corner, with similar values of the angles, SN becomes negative. The fractures start to propagate in regions with high SN. The heterogeneity in 𝜓 introduces heterogeneity in 𝜎xx, 𝜎yy, 𝜎xy, SN, ak, k,andCdr. The permeability growth field is shown in Figure 7. Heterogeneity and anisotropy of the permeability and elasticity tensor fields enter the

FIGURE 4 Case I: distribution of the initial fracture direction 𝜓(t = 0) in the domain. All the fractures are oriented along the general northwest-southeast direction. All the fractures belong to one single family, ie, N = 1, and each mesh element contains one fracture, ie, mk = 1∕Ae [Colour figure can be viewed at wileyonlinelibrary.com] BUBSHAIT AND JHA 13

FIGURE 5 Case I: snapshots of the pressure field over approximately 3 months with a timestep of Δt = 11 days between the snapshots. Pressure diffuses uniformly for the first three timesteps such that the pressure contours are elliptic in shape with the well lying at the center along the north-south long axis of the ellipse. Later, the contours start to orient along the northwest-southeast direction following the permeability enhancement in that direction due to injection-induced fracture growth [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 6 Case I: stresses (MPa) at two different timesteps showing the creation of anisotropy and heterogeneity in the stress field, which is homogeneous isotropic at t = 0. Anisotropy and shear, resulting from the effect of 휃 in Equation (9), lead to an increase of the deviatoric stress S and its

crack-normal projection SN , which affects crack propagation through Equation (16). The heterogeneity in the stress field results from the heterogeneous distribution of 휓,

which makes SN and, subsequently, k and

Cdr heterogeneous [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 7 Case I: the change in the

permeability field Δkmax at the same 8 timesteps as in Figure 5. The change in permeability increases in the beginning (changing color from blue to red along the northwest-southeast corridor) and decreases later (red to yellow) as the induced stresses relax due to pressure diffusion and fracture growth processes [Colour figure can be viewed at wileyonlinelibrary.com] pressure solution through the solid-to-fluid coupling term and the rock compressibility term as shown in Equation (32). The pressure field is relatively smooth because of the diffusive nature of the mass balance equation. We select two locations in the domain: Location (1) in the NW corner and Location (2) in the NE corner (Figure 8) to conduct a more detailed analysis. The volumetric compression −휎v increases monotonically at both locations due to near-wellbore expansion under injection. However, deviatoric tension SN behaves differently at the two locations due to 휃<0inNEand휃>0 in NW corners (Equation (9)). Figure 9 shows that, as the fracture grows at a location, stiffness in the x-direction decreases and the Biot coefficient increases. This suggests a loss of poroelastic strength and an increase in the drained compressibility of the medium, which agrees with lab observations and predictions from empirical models of strength of jointed rock mass, eg, the Hoek-Brown model.7 14 BUBSHAIT AND JHA

FIGURE 8 Locations of the two fractures selected for detailed investigation. The two fractures are located in northwest and northeast corners of the domain as shown by the circles. The fracture lengths are different in space because the plots are at the end of Case I simulation [Colour figure can be viewed at wileyonlinelibrary.com]

10 2 1.5 Loc (1) 1.45 1 Loc (2) FIGURE 9 Case I: (A) The deviatoric stress S is positive (tensile) Time 1.4 N 0 1.35 at Loc (2) and negative (compressive) at Loc (1) due to the difference Time -1 in signs of 휃, 휃1,and휃2 in Equation (13). At both locations, SN 1.3 evolves toward zero as the stresses relax with fracture propagation. -2 1.25 -8.44 -8.42 -8.4 -8.38 -8.36 -8.34 0 0.02 0.04 0.06 0.08 The volumetric stress 휎v at both locations are negative (compressive) 10 because of near-wellbore expansion. This is also evident from the 3 negative sign of 훿휎v in the first equation of Equation (9) for both 0.81 positive and negative signs of 휃. (B) Normalized stiffness in the 2.5 x-direction, Cdr,1111∕E0, decreases with time due to fracture growth, 2 Time (C) 훼11 increases with time suggesting reduction in the drained bulk 0.8 modulus, (D) normalized permeability kmax∕k0 vs normalized 1.5 stiffness Cdr,1111∕E0 curves show the two implications of fracturing: 0.79 1 increase in permeability and decrease in stiffness [Colour figure can 0 0.02 0.04 0.06 0.08 1.3 1.4 1.5 be viewed at wileyonlinelibrary.com]

5.2 Case II: effect of variable injection rate In many oilfields or wastewater disposal sites, the injection well flow rate varies with time due to various operational constraints, eg, availability of injection pumps, decrease in injectivity due to calcite scale build-up on the wellbore wall, and the need to control the breakthrough of injected fluid at producers. It is also a common practice to slowly ramp up the injection rate in a new injection well to avoid unintentional fracturing or activation of “thief” zones around the well. Effect of injection rate variability on poroelastic stresses induced on nearby faults is also important from induced seismicity point of view78 because many wastewater disposal wells, which are tied to induced seismicity in Oklahoma and other states, have time-dependent flow rates. Analysis of rate variability can provide insights on how to mitigate induced seismicity by operating the wells on special injection/production rate schedules.58 Here, we evaluate the effect of variability in the injection rate on fracture propagation and permeability evolution. To that effect, we compare results from constant rate and variable rate injection simulations. The final cumulative injected volume and the injection duration are the same between the two cases. Figure 10 shows the cumulative injection profile BUBSHAIT AND JHA 15

FIGURE 10 Cumulative injection profile in the constant and variable rate injection cases. The variable rate well starts off at an injection rate (given by the slope of the dotted curve above) lower than that of the constant rate well. Later it gradually increases and surpasses the constant rate. The total injected volume is the same in both cases. 1 bbl = 0.159 cubic meter [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 11 Case II: evolution of pressure over 3 months with a timestep of Δt = 11 days during variable rate injection. Compare with Figure 5 [Colour figure can be viewed at wileyonlinelibrary.com]

FIGURE 12 Case II: changes in the

permeability field, Δkmax, at different times with a constant timestep of Δt = 11 days. Compare with Figure 7 [Colour figure can be viewed at wileyonlinelibrary.com] of the two cases. Figure 11 illustrates that, similar to Case I, the Case II pressure map develops anisotropy with time as a result of anisotropic fracture growth. However, the two pressure fields are very different in terms of their pressure values. This difference results from the difference in the injected mass at any given time and the difference in fracture propagation with time. One observation is that the near-wellbore pressure is significantly higher in Case II. This is related to a slower increase in fracture length and permeability due to a lower injection rate in the beginning of the variable rate case; compare Figure 12 with Figure 7. Constant rate injection has resulted in higher permeability values. Due to a slower fracture growth in Case II, which is discussed in the following with Figure 13, water starts to accumulate around the wellbore, which leads to larger near-wellbore pressures. In Figure 13, we analyze the effect of rate variability on fracture propagation and damage at the two selected locations of Figure 8. By definition, fracture propagation is a direction-dependent process and depends on local stresses. At both 16 BUBSHAIT AND JHA

0.4 CR: Loc (1) 0.8 CR: Loc (2) 0.3 VR: Loc (1) 0.6 VR: Loc (2) 0.2 0.4

0.2 0.1 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08

105 4500 FIGURE 13 Fracture length (ak)and 4 permeability (k ) enhancement processes 4000 max 2 are dependent on the location and the well 3500 injection rate. CR and VR denote constant 0 3000 rate and variable rate in Cases 1 and 2, -2 respectively. Loc (1) and Loc (2) denote the 2500 -4 northwest and northeast corner locations 2000 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 selected in Figure 8 [Colour figure can be viewed at wileyonlinelibrary.com]

locations, the fracture length ak for Case I is larger than that for Case II, which is mimicked by the maximum permeability kmax plot. An interesting observation is that the evolution in the location pressure does not mimic the evolution in ak or kmax exactly. For example, ak for Case I Location (2) is lower than ak for Case II Location (1). A lower fracture length or permeability should result into a higher pressure. However, p for Case I Location (2) is lower than p for Case II Location (1). The reason is that the pressure equation is diffusive and has a global character. Since the permeability around wellbore is lower in Case II than in Case I, the pressure is higher in Case II at all locations. The SN plot in Figure 13 provides further insights into the coupling between fracture propagation and stress evolution processes. We observe a nonmonotonic trend in SN in Case I, which is lost in Case II over the chosen duration of the simulation because of its slower injection. The effect of location, ie, 휃 on SN is twofold. First, 휃<0 at Location (2) makes SN positive (tensile) as per Equation (13). Second, a smaller pressure p at Location (2) compared to Location (1) makes SN smaller in magnitude. This causes a slower and smaller growth in the fracture length ak at Location (2) (Figure 13A, top left).

5.3 Case III: effect of fracture direction The fracture direction 휓 plays an important role in selecting the natural fractures that propagate during injection because it determines the magnitude of 휎t at the fracture tip. This was mentioned in Section 3.2.1 through 휓-dependent quan- 휎′ 휓 tities SN and N (Equation (11)) and also indicated through Figure 8. Therefore, we hypothesize that affects the evolution and distribution of poromechanical properties during injection. To test our hypothesis over a larger range of fracture directions, we conduct multiple simulations by varying 휓 at the two locations across the following range: 휓 = 90◦, 110◦, 135◦, 150◦, 180◦, while keeping 휓 fixed at all other locations. We analyze the simulation results in Figure 14 휎′ for Location (1) and Figure 15 for Location (2). We observe that the growth in permeability is a function of v and the deviatoric tension SN as expected from Equation (12). Figure 15 shows that, at Location (2), the magnitude of SN is smaller compared to the values at Location (1). This explains the smaller permeability observed at Location (2) for all 휓.

5.4 Case IV: multiple fracture families In this case, we consider a random distribution of the fracture direction 휓 overtheentirerangefrom0◦ to 180◦ as shown in Figure 16. This can be understood as a case with multiple fracture families with N ≥ 2. Equation (30) predicts that the change in permeability will be larger in this case because of the increase in the total number of fractures, which creates additional connectivity. The injection rate is the same as in Case I. In Figure 17, we see the evolution of the permeability field. Compared to Case I with the fracture family k = 1, we see that the permeability enhancement is azimuthally more uniform. This is expected from the effect of 휓 on SN. The result is that the stimulated reservoir volume (SRV) is larger and spatially more uniform than in Case I. This highlights the importance of knowing the fracture direction in the reservoir region away from the injector. BUBSHAIT AND JHA 17

10 o 8 =180 0 150o 6 135o 110o -2 4 90o 2 -4

01230123

1.1 휓 8 FIGURE 14 Effect of the fracture direction on evolution in the 1.05 deviatoric tension SN , the local stress ratio 휎xx∕휎yy, and the 6 maximum permeability k at Location (1) at early times. S is 1 max N 4 nonmonotonic with 휓 because of the sin 2휓 term in Equation (12). 0.95 Growth in kmax is proportional to the growth in SN magnitude. kmax 2 0.9 vs SN shows that, as the magnitude of deviatoric stress increases, the 0123 -4 -2 0 permeability increases for all 휓 [Colour figure can be viewed at 10 wileyonlinelibrary.com]

10 3 2 =180o o 2.5 150 1 135o 2 110o 0 90o 1.5 -1

1 -2 01230123

1.1 3

1.05 2.5

1 2 FIGURE 15 Effect of the fracture direction 휓 on evolution in the 0.95 1.5 deviatoric tension SN , the local stress ratio 휎xx∕휎yy, and the 0.9 1 maximum permeability kmax at Location (2) at early times. kmax is 0123-2 0 2 proportional to SN magnitude [Colour figure can be viewed at 10 wileyonlinelibrary.com]

FIGURE 16 Case IV: distribution of the fracture direction for the two fracture families present in the domain. The fracture orientations are randomly distributed [Colour figure can be viewed at wileyonlinelibrary.com]

5.5 Case V: boundary stress ratio Boundary stresses can significantly impact the onset, type, and magnitude of failure in rocks subjected to hydraulic stimu- lation because they determine the solution of the equilibrium equation, Equation (35). We focus on the effect of anisotropy in the boundary stresses on permeability growth and poroelastic damage. We conduct four simulations with the following 18 BUBSHAIT AND JHA

FIGURE 17 Case IV: snapshots of the change in permeability field for a fracture family with fracture directions distributed over the entire range of 0◦ to 180◦ [Colour figure can be viewed at wileyonlinelibrary.com]

105 105 7 7 Location (1) 15 Location (2) 15 6 6

FIGURE 18 Case V: evolution of the 10 10 5 5 normalized permeability kmax∕k0 and the deviatoric tension SN with the stress ratio 4 4 (SR) at the two locations. The plot shows 5 5 that, as SR increases, ie, the boundary stress 3 3 anisotropy decreases the permeability 0 0 increases at Location (1) and decreases at 2 2

Location (2). SN decreases with SR at both 1 1 locations [Colour figure can be viewed at 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 wileyonlinelibrary.com] values of the boundary stress ratio, SR = 1, 1/2, 1/3, and 1/4 obtained by increasing the boundary stress in the y-direction, 휎H. Figure 18 shows the effect of the stress anisotropy on permeability growth at the two selected locations in the domain. At Location (1) (휃>0), as SR decreases from 1, ie, the stress anisotropy increases, the permeability growth decreases and the deviatoric tension increases such that it become tensile (SN > 0) in rocks with anisotropic in-situ stress. At Location (2), as the stress anisotropy increases, both the permeability growth and the deviatoric tension increase. This is an interesting result that cannot be deduced from the permeability vs deviatoric tension behavior of the isotropic stress case (SR = 1) shown in Figures 14 and 15.

6 CONCLUSION

We presented a novel numerical framework to model fracturing-induced changes in rock properties—the elasticity tensor, the permeability tensor, the Biot coefficient tensor, and the Biot modulus—of a naturally fractured rock. The framework couples Biot's poroelasticity theory, Griffith's brittle failure theory, and the CDM theory to model propagation of fractures with randomly distributed angles. Our sequential coupling method extends the fixed-stress method to model dynami- cally evolving anisotropy in the properties. Results from multiple representative simulations are presented to validate the proposed framework and investigate the effects of injection rate variability, initial distribution of fracture direction, and in-situ stress ratio on evolution in the properties. Our analysis elucidates the role of deviatoric stress in evolution of the poromechanical properties. Information on the hydraulically SRV can be used to optimize the well injection schedule, eg, to ensure a more uniform sweep of the reservoir and to choose hydraulic fracturing job parameters to create a more connected fracture network. Results from the variable injection rate analysis can be used in induced seismicity studies to determine the role of injection rate variability on poroelastic stress changes induced on faults. Finally, our solution algorithm shows how a damage mechanics-based solver module can be inserted inside the sequential loop of an existing coupled flow-geomechanics simulator to extend its capability in modeling fracturing at the reservoir scale.

ACKNOWLEDGEMENT This research was funded by American Chemical Society Petroleum Research Fund Number 58769-DNI9. BUBSHAIT AND JHA 19

ORCID Birendra Jha https://orcid.org/0000-0003-3855-1441

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Howtocitethisarticle: Bubshait A, Jha B. Coupled poromechanics-damage mechanics modeling of fracturing during injection in brittle rocks. Int J Numer Methods Eng. 2019;1–21. https://doi.org/10.1002/nme.6208