Extended-FEM Analysis of Injection-Induced Slip on a Fault with Rate-and-State Friction: Insights into Parameters that Control Induced Seismicity

Abstract

The extended finite element method (X-FEM) is utilized to simulate the behavior of a heterogeneous fault characterized by rate-state frictional rheology, embedded within a poroelastic medium. The displacement and pore-pressure fields that are discontinuous across the fault are computed using X-FEM, by enriching the standard finite element approximation with additional degrees of freedom for elements intersected by the fault. We investigate a $M_w$ 4.1 injection-induced earthquake in western Canada; this model incorporates depth-varying rate-slip behavior wherein a high-pressure zone due to hydraulic fracturing stimulation intersects the fault within a stable layer, producing aseismic slip that progressively loads an unstable fault region, thereby triggering dynamic rupture. Parametric studies using our numerical approach provide insights into the influence of rate-state parameters on fault activation, as well as hydraulic properties of a damage zone that surrounds the fault. Results confirm that aseismic slip near the injection zone propagates outwards to seismogenic unstable regions of the fault. The coseismic slip profile, seismic moment, and slip latency are determined by the difference $a-b$ for rate-state parameters of the unstable fault regions. Hydraulic diffusivity in the damage zone controls the rate of pore-pressure diffusion along the fault, which affects timing of the initial seismic event and aftershock productivity.

1 Introduction

Subsurface fluid injection can induce diverse fault-slip behavior. The documented range of fault-slip response to fluid injection includes fault creep (Guglielmi et al. 2015; Eyre et al. 2022), hybrid-frequency waveforms (Yu et al. 2021), earthquake swarms (Wicks et al. 2011; Eyre et al. 2020) and runaway rupture (Ellsworth et al. 2019; Atkinson et al. 2020; Lyakhovsky and Shalev 2021). The fault-slip response is influenced by many factors including the frictional rheology of the fault core (Guglielmi et al. 2015), characteristics of natural fracture systems (Igonin et al. 2021) including the fault damage zone (Eyre et al. 2019b), constitutive properties of the host rock (Segall and Lu 2015) and the characteristics of fluid injection (Eaton 2018), thermo-elastic stressing (Candela et al. 2018). Numerical simulation is essential to decipher the relationship between fault or medium parameters and fault-slip response.

Various numerical approaches have been applied to this problem. In some previous studies using the classical finite element method (FEM) approach (Poulet et al. 2017; Hageman and de Borst 2021; Jha and Juanes 2014; Rutqvist et al. 2015; Cueto-Felgueroso et al. 2017, 2018; Pampillón et al. 2018; Andrés et al. 2019; Chang and Segall 2016), it was shown that different constitutive laws for a fault lead to a variety of stick–slip patterns, ranging from stable sliding or a sequence of many small slip events leading to a single, larger seismic event after significant aseismic slip has occurred. In these studies, the fault is modeled using zero-thickness interface elements defined between faces of bulk elements, or it is represented as damaged bulk elements with a different constitutive law. Other studies have investigated the effect of permeability of the damage zone around a fault on its seismic response (Cappa 2011; Cappa et al. 2018; Yang et al. 2021; Rohmer et al. 2015; Rohmer 2014). Garagash and Germanovich (2012) used a semi-analytical approach to investigate nucleation and arrest of fault rupture based on a simplified slip-weakening frictional rheology. A recent study using a 3D physics-based, quasi-dynamic earthquake simulation code based on rate-state friction (Kroll and Cochran 2021) showed that higher initial shear stresses and smoother stress fields promote a fault-slip response in the form of runaway rupture. As an alternative approach for modeling the discontinuities, we used the extended finite element method (X-FEM) to investigate the rate-and-state frictional behavior of a fault during fluid injection. Since its introduction in the late 1990s by Belytschko and Black (1999) and Moës et al. (1999) to model crack growth, X-FEM has become one of the most widely used numerical methods for simulating domains with stationary or evolving discontinuities (e.g., fracture, fault, inclusion, and material interface). In the X-FEM approach, rather than predefining interface elements along a fault, the presence of the fault (or fracture) is captured by adding extra degrees of freedom to individual elements intersected by the discontinuity and the conventional FEM approximation is enriched with an appropriate function. The enriched FEM was presented in the modeling of crack growth in saturated and unsaturated porous media in pioneering works of de Borst et al. (2006) and Réthoré et al. (2007, 2008). Then, Mohammadnejad and Khoei (2013) and Gordeliy and Peirce (2013) developed X-FEM frameworks for coupled hydraulic fracture propagation problem. Dahi-Taleghani and Olson (2011) and Khoei et al. (2018) investigated the interaction of a propagating hydraulic fracture with existing natural fractures. Recently, Hosseini and Khoei (2021) and Hosseini and Khoei (2020) utilized X-FEM for modeling hydraulic fracturing which incorporates effects of proppant transport and interfacial boundary conditions on fracture faces. A comprehensive review on the applications of X-FEM can be found in Khoei (2014).

In comparison to the FEM solution to contact problems, less attention has been paid to the contact friction problem based on the X-FEM since most advances in the X-FEM solution of continuum problems have focused on cracks with opening/closing modes. In the context of frictional contact modeling, Dolbow et al. (2001) was the first to implement the X-FEM framework in a contact problem for modeling the frictional contact on the crack faces. Khoei and Nikbakht (2007) developed an X-FEM framework for frictional contact based on the penalty method for the frictional behavior between two bodies. Liu and Borja (2008, 2009, 2010, 2013) have developed the stabilized X-FEM framework for slow-rate and dynamic frictional faulting in solid mechanics. Recently, Schwartzkopff et al. (2021a, b) utilized X-FEM approach to investigate seismic behavior of a fault during fluid injection based on the framework of Khoei (2014). An important numerical issue for rate-and-state friction is that updating the friction coefficient and shear stress on the fault requires an appropriate iterative scheme, otherwise the numerical scheme will not converge. Such a scheme has been presented by Liu and Borja (2009) for X-FEM modeling of a rate-and-state fault in solid media. We have extended their approach to the case of porous media saturated with a fluid phase to investigate the unstable fault slip during fluid pressurization, considering dynamic ruptures in a quasi-dynamic sense, by introducing a radiation damping into fault cohesion. The study about the advantages of X-FEM over other methods requires a comparative study; Dias-da Costa et al. (2010) have conducted such a study on the different modeling approaches (enriched nodal, element techniques, and interface elements) for discontinuous fracture and noted that all approaches present advantages and disadvantages, such that their usefulness depends on the problem at hand.

Independent of the technique used to apply the contact condition (penalty or Lagrange multipliers approach), it is recognized that some combinations of discrete interpolation spaces for solid displacement and contact traction exhibit numerical instability in the normal contact traction (Hirmand et al. 2015). Numerous techniques can be considered to reduce these instabilities using additional basis functions known as bubble functions (Mourad et al. 2007), using different Lagrange multipliers space on the interface (Moës et al. 2006; Kim et al. 2007), or using a tuning stabilization parameter through the polynomial pressure projection method (Liu and Borja 2010). These techniques try to satisfy the Ladyzhenskaya–Babuska–Brezzi conditions for an over-constrained contact problem. The oscillatory instabilities are more apparent in the Lagrange multipliers method since contact constraints are imposed exactly (Liu and Borja 2010). In the penalty method, where the contact constraints are imposed approximately by assuming a spring with a high stiffness, instabilities can be controlled by reducing the values of the penalty parameter, but at the expense of accuracy in the magnitude of interpenetration of contact faces. However, if oscillations are observed even after penalty regularizations, or when a stiff spring constraint is imposed, one efficient approach that can be adopted, such as that proposed by Moës et al. (2006), will be construction of the correct discretization space for interfacial domain which is independent of that of the bulk domain.

The main objective of this paper is to apply X-FEM for parametric analysis of fault activation by fluid pressurization. We apply this approach to a well-documented case of an earthquake induced by pressure increase during hydraulic fracturing (HF) in western Canada (Eyre et al. 2019a), for which heterogeneous frictional rheology of the fault was inferred to play a dominant role. Eyre et al. (2019a) investigated the potential causes of a $M_w$ 4.1 event in this region using a 1D integral equation, showing that observations could be fit by aseismic fault creep in direct response to fluid injection that triggered dynamic rupture as a secondary effect. However, no accurate frictional parameters are available for the rocks in question and they did not consider the effects of different rate-state frictional parameters, nor did they consider characteristics of the inferred damage zone around the fault. In this paper, we apply a quasi-dynamic hydro-mechanical model using X-FEM to explain the mechanisms of fault activation and to explore factors that contributed to rupture during injection operations.

2 Mathematical Model

2.1 Governing Equations for Poroelasticity

The fundamental governing equations for linear isotropic poroelasticity include conservation of momentum and fluid mass. According to Biot’s theory, the momentum equilibrium equation for a porous medium can be expressed in the following form (Biot 1941; Coussy 2004; Cheng 2016)

$$\begin{aligned} \nabla \cdot \varvec{\sigma }+\rho {\textbf{g}}=0, \end{aligned}$$

(1)

where $\varvec{\sigma }$ is the total stress tensor, ${\textbf{g}}$ is the gravitational acceleration vector, $\rho$ is the bulk density of the porous medium. The bulk density can be defined by averaging densities of the solid and fluid phases as, $\rho =\phi \rho _f+(1-\phi )\rho _s$, where $\phi$ is the porosity, and $\rho _s$ and $\rho _f$ are the densities of solid and fluid phases, respectively. Considering negative sign convention for compression stress, the total stress tensor can be decomposed into two parts through

$$\begin{aligned} \varvec{\sigma }={\textbf{D}}:\varvec{\epsilon }-{\alpha }p{\textbf{I}}, \end{aligned}$$

(2)

where the first part of the right-hand side of the equation is called effective stress ($\varvec{\sigma '=D:\epsilon }$), in which, ${\textbf{D}}$ is the elasticity tensor characterized by two independent material constants, i.e., shear modulus (G) and Poisson ratio ($\nu$); $\varvec{\epsilon }$ is the strain tensor which is related to solid displacement through $\varvec{\epsilon =({\nabla }u+{\nabla }u^T)}/2$, where ${\textbf{u}}$ is displacement vector of solid phase. For the second term, p is the pressure of fluid phase, ${\textbf{I}}$ is the delta Kronecker tensor; and $\alpha$ is the Biot coefficient defined as $\alpha =1-K/K_s$, where K and $K_s$ are the bulk moduli of the porous skeleton and solid phase, respectively. In the limiting case of an incompressible solid phase or when $K\ll {K_s}$, the Biot coefficient is equal to unity.

The fluid mass conservation equation can be written as

$$\begin{aligned} {\dot{p}}/M+\alpha \nabla \cdot \dot{{\textbf{u}}}+\nabla \cdot {\textbf{w}}=Q, \end{aligned}$$

(3)

where ${\textbf{w}}$ is fluid velocity vector relative to solid phase, known as Darcy velocity; Q is the volumetric rate of injected fluid per unit volume of the porous medium; and M is the Biot modulus defined as the pore pressure increase due to a unit increase in volume of fluid within unit volume of the porous medium at constant strain conditions and given by $M=[(\alpha -\phi )/K_s+\phi /K_f]^{-1}$, where $K_f$ is the bulk modulus of fluid. The ${\dot{p}}$ and $\dot{{\textbf{u}}}$ terms stand for the derivative of corresponding variables with respect to time. According to Darcy’s law, the velocity vector ${\textbf{w}}$ can be written as

$$\begin{aligned} {\textbf{w}}=-(k/\mu _f)({\nabla }p-\rho _f{\textbf{g}}), \end{aligned}$$

(4)

where k is the intrinsic permeability, and $\mu _f$ is the viscosity of fluid phase. For the special case of an irrotational displacement field, it can be proved that the mass conservation equation becomes uncoupled so that the pressure field can be determined independently. It is required that changes in far-field stresses and pressure, away from the fault region, are zero. Another situation where the decoupling is possible is when the fluid is highly compressible with respect to the porous rock matrix, $K_f\ll {\phi {K}}$ (Cheng 2016). In an uncoupled model, Eq. (3), without body force, can be written as

$$\begin{aligned} S{\dot{p}}-\nabla \cdot \left( \frac{k}{\mu _f}{\nabla }p\right) =Q, \end{aligned}$$

(5)

in which, S is the storage coefficient given by $S=1/M+\alpha ^2(1-2\nu )/2G(1-\nu )$. Note that Eqs. (1) and (3) are two-way coupled equations such that a change in volumetric strain produces a change in pore pressure and the gradient of pore pressure induces deformation in the solid phase. However, Eqs. (1) and (5) define an uncoupled poroelastic behavior in which the effect of solid deformation on the pressure field is eliminated. Substituting constitutive laws (2) and (4) into governing Eqs. (1) and (3), a coupled system of momentum equilibrium and mass conservation equations can be derived in terms of primary variables, ${\textbf{u}}$ and p, given by

$$\begin{aligned}{} & {} \nabla \cdot ({\textbf{D}}:\varvec{\epsilon }-{\alpha }p{\textbf{I}})=-\rho {\textbf{g}}, \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \frac{{\dot{p}}}{M}-\nabla \cdot \left( \frac{k}{\mu _f}{\nabla }p\right) +\alpha \nabla \cdot \dot{{\textbf{u}}}=Q-\nabla \cdot \left( \frac{k}{\mu _f}\rho _f{\textbf{g}}\right) . \end{aligned}$$

(7)

Seven independent material constants, $G,\nu ,\alpha ,M,k/\mu _f,\rho ,$ and $\rho _f$ are required to fully characterize the above equations. Boundary conditions for the system of Eqs. (6) and (7) can be expressed as follows: for solid phase, the displacement field can be prescribed on a portion of the boundary, or the traction vector, defined as $\mathbf {T_S=\sigma .n_S}$, can be known on the boundary, where $\mathbf {n_S}$ is unit vector normal to the external boundary; for fluid phase, the pressure field can be prescribed on the boundary, or the normal Darcy velocity, defined as $\mathbf {w_S}=-(k/\mu _f)({\nabla }p-\rho _f{\textbf{g}}).\mathbf {n_S}$, can be known. On the fault interface, the displacement and pressure fields are discontinuous while the traction vector (due to Newton’s third law) and normal Darcy velocity (due to principle of mass conservation) are continuous. Given the boundary conditions on the external boundary and the hydro-mechanical constitutive law for the fault, the governing Eqs. (6) and (7) can be solved to determine the distribution of solid displacement and fluid pressure fields.

2.2 Hydro-mechanical Behavior of a Fault

Shear failure of a fault surface during fluid injection can occur through the following mechanisms. The fault itself has low permeability such that it hinders the fluid flow across the fault faces; consequently, pore pressure can increase on the fault surface. Therefore, during fluid injection, a reduction in effective normal stress on the fault occurs, which leads to a reduction in shear strength of the fault. The reduction in normal effective stress can lead to slippage on the fault that can either be a low-velocity aseismic slip or an abrupt coseismic slip (able to generate seismic waves that depend on rate-state parameters as discussed below). Figure 1a illustrates the stresses, displacements, fluid velocity and pore pressure acting on an idealized fault ($\Sigma$), where the displacement and pressure fields on the $\Sigma ^+$ side of the fault differ from those of $\Sigma ^-$ side. The resultant effective stress on the fault is defined as $\mathbf {T'}_\Sigma =T'_n{\textbf{n}}_\Sigma +T_t{\textbf{t}}_\Sigma$, which can be decomposed into normal ($T'_n$) and shear ($T_t$) stresses. The same traction vector with opposite direction is developed on the other face. The vector ${\textbf{n}}_\Sigma$ is the unit vector normal to the interface of the discontinuity, and vector ${\textbf{t}}_\Sigma$ is the unit vector tangential to the fault. It is noted that total traction vector (${\textbf{T}}_\Sigma$) is equal to the summation of effective traction and pore pressure as ${\textbf{T}}_\Sigma =\mathbf {T'}_\Sigma -p_\Sigma {\textbf{n}}_\Sigma$, in which based on suggestion of Jha and Juanes (2014), the fault pressure $p_\Sigma$ is the maximum either sides of the fault, $p_\Sigma =\textrm{max}(p^+,p^-)$. The relative displacement of fault faces is defined as $\mathbf {[u]=u^+-u^-}$, where the jump of displacement field over the fault is denoted by the bracket operator. Therefore, the normal gap between fault faces is equal to $g_N=[{\textbf{u}}].{\textbf{n}}_\Sigma$, and tangential movement between faces can be written as $g_T=[{\textbf{u}}].{\textbf{t}}_\Sigma$.

Fig. 1

Schematic representation of the a mechanical behavior of a fault surface ($\Sigma$) and positive sign convention for effective traction vector (${\textbf{T}}'_\Sigma$) and its normal ($T'_n$) and tangential ($T_t$) components; displacement jump ($[{\textbf{u}}]$) across the fault, and normal ($g_N=[{\textbf{u}}].{\textbf{n}}_\Sigma$) and tangential ($g_T=[{\textbf{u}}].{\textbf{t}}_\Sigma$) gaps between fault surfaces, b hydraulic behavior of the fault zone, including fault core and damage zone. Fluid velocity across the fault is equal to $w_\Sigma =-k_\Sigma [p]$, where $k_\Sigma$ is the hydraulic transmissibility of the fault core and [p] is the pressure jump across the fault

To prevent penetration of faces of the fault into each other, Kuhn–Tucker conditions are fulfilled (Wriggers and Laursen 2006). These conditions state that when two faces of the fault are in contact ($g_N=0$), some normal compression force is developed on the fault surface ($T'_n\le 0$), and when there is a positive gap between fault faces, $g_N>0$, the normal force will vanish on the fault ($T'_n=0$). Thus, the mathematical description of normal contact conditions can be written as

$$\begin{aligned} g_N\ge 0,\,T'_n\le 0,\,g_NT'_n=0. \end{aligned}$$

(8)

In this study, the penalty method is chosen for applying the contact constraint for the normal traction and tangential traction in the stick state, see “Appendix”. Based on the suggestion of Cueto-Felgueroso et al. (2018), we use a penalty parameter of the order $\varepsilon \sim (E/h)$, where E is the Young modulus and h is the mesh size along the fault, to provide an optimum penalty stiffness, resulting in a stabilized and accurate result.

In our model, tensile rupture mode of the fault is not considered, thus it is assumed that $T'_n<0$. To derive a constitutive law for slip behavior along the fault, a slip criterion is introduced. According to Coulomb’s frictional law, the shear strength ($T_f$) of fault can be given by

$$\begin{aligned} T_f=\mu |T'_n |+c, \end{aligned}$$

(9)

in which, $\mu$ is the coefficient of friction of the fault, and c is the cohesive strength of the fault. If the magnitude of shear stress on the fault is less than its shear strength, $|T_t |<T_f$, then no relative movement parallel to the fault faces occurs (the fault is in stick state). If $|T_t |=T_f$, the fault slips and there is a relative movement between faces of the fault (slip state). Moreover, based on suggestion of Rice (1993), a velocity-dependent cohesion, $c=(G/2c_s){\dot{g}}_T$ is introduced into the fault’s shear stress, where $c_s$ is the shear wave speed. Introduction of this damping term prevents unbounded slip rate during coseismic events. For a rate-and-state friction model (Dieterich 1979; Ruina 1983), the coefficient of friction along the fault is function of slip rate ($V\equiv |\dot{g_T}|$) and state variable ($\theta$) given by

$$\begin{aligned} \mu =\mu _0+a\ln {\left( \frac{V}{V_0}\right) }+b\ln {\left( \frac{\theta }{\theta _0}\right) },\,\,\textrm{where}\,\,{\dot{\theta }}=1-\frac{V\theta }{D_c}, \end{aligned}$$

(10)

where $\mu _0$ is the steady-state friction coefficient at the reference slip rate $V_0$; a and b are rate-state parameters, $D_c$ is the characteristic slip length; and the reference state variable is defined as $\theta _0={D_c}/{V_0}$. The parameter a controls the change in the coefficient of friction due to a change in slip rate; parameter b captures the evolution of the contact population driven by slip. $V_0$ represents the threshold velocity for transition from static to dynamic friction. A lower bound value of V should be applied so that $V/V_0\ge {1}$. The evolution of the state variable is determined based on an aging law, where the initial condition for the state variable is $\theta =\theta _0$, which is consistent with the initial steady-state static condition (${\dot{\theta }}=0$).

A simple model for fault zone structure involves a fault core that, due to shear strain localization, is composed of gouge material that has relatively small permeability (Fig. 1b). This fault core is surrounded by a damage zone, which has higher permeability in comparison to intact rock (Faulkner et al. 2010). Since it is assumed that there is no tensile gap between fault surfaces, the principle of mass conservation implies that the normal component of fluid velocity ($w^\pm =-(k/\mu _f)({\nabla }p^\pm -\rho _f{\textbf{g}}).{\textbf{n}}_\Sigma$) to the fault core must be continuous, which means that mass flux on either side of the fault must be equal to the mass flux across the fault core. The fluid velocity across the fault can be determined as (de Borst 2017)

$$\begin{aligned} w_\Sigma =-k_\Sigma [p], \end{aligned}$$

(11)

in which, $k_\Sigma$ is the hydraulic transmissibility [m/Pa.s] of the fault core, and $[p]=p^+-p^-$ is the jump in pressure field across the fault. If the fault core has a low transmissibility, $k_\Sigma \rightarrow 0$, a strong pressure discontinuity can develop across the fault as observed in some areas of induced seismicity (Fox and Watson 2019), while for high transmissibility, $k_\Sigma \rightarrow \infty$, pressure will be continuous across the fault (Fig. 1b). Thus, this model is applicable to a range of fault core transmissibility, and the magnitude of $k_\Sigma$ parameter determines the strength of the pressure field discontinuity.

2.3 The Extended Finite Element Discretization

As discussed above, the X-FEM method introduces extra degrees of freedom at each node of the finite element mesh using an enrichment function. To capture the jump in displacement and pressure fields resulting from a fault, the Heaviside function $H({\textbf{x}})$ is typically used as an appropriate enrichment function (Khoei 2014). This function has the value of $+1$ on side $\Sigma ^+$ of the fault, and the value of $-1$ on the side $\Sigma ^-$ of the fault (see Fig. 2). Considering ${\mathcal {N}}$ is the set of all nodal points in the discretized domain, and $\tilde{{\mathcal {N}}}$ is the set of enriched nodes intersected by the fault (Fig. 2), then, the approximation of displacement and pressure fields can be written as

$$\begin{aligned}{} & {} {\textbf{u}}({\textbf{x}},t){\simeq }{\sum _{{\mathcal {N}}}}N_{uI}({\textbf{x}}){\bar{u}}_I(t)+{\sum _{\tilde{{\mathcal {N}}}}}N_{uI}({\textbf{x}})(H({\textbf{x}})-H({\textbf{x}}_I)){\tilde{u}}_I(t), \end{aligned}$$

(12)

$$\begin{aligned}{} & {} p({\textbf{x}},t){\simeq }{\sum _{{\mathcal {N}}}}N_{pI}({\textbf{x}}){\bar{p}}_I(t)+{\sum _{\tilde{{\mathcal {N}}}}}N_{pI}({\textbf{x}})(H({\textbf{x}})-H({\textbf{x}}_I)){\tilde{p}}_I(t), \end{aligned}$$

(13)

in which, ${\bar{u}}_I(t)$ and ${\bar{p}}_I(t)$ are the nodal unknowns, with ${\tilde{u}}_I(t)$ and ${\tilde{p}}_I(t)$ the enriched nodal degrees of freedom, for displacement and pressure fields, respectively. $N_{uI}({\textbf{x}})$ and $N_{pI}({\textbf{x}})$ denote the standard FEM shape functions for displacement and pressure fields, respectively; and ${\textbf{x}}_I$ is the coordinate of node I. Substituting the X-FEM approximations into the weak form of the governing equations, the discretized form of the governing equations can be obtained. The resulting system is nonlinear and a Newton–Raphson iterative algorithm is implemented at each time step, and the status of stresses on the fault surface is updated at each iteration. The implementation of X-FEM method on governing equations and solution algorithm is discussed comprehensively in the “Appendix”.

Fig. 2

Set of standard and enriched degrees of freedom in extended finite element method (left), the Heaviside function H(x) which imposes discontinuity into primary unknowns (right)

Numerical considerations: To improve accuracy and numerical stability, the finite element mesh is refined near the fault. The characteristic length for an element along the fault (h) should be less than the critical nucleation length given by $h<{GD_c/|T'_n |b}$, which determines transition from static to dynamic rupture in elastic media (Dieterich 1992). Our sensitivity analyses show that when the mesh size is smaller than the critical nucleation length, the numerical convergence is achieved. To optimize model run time while capturing the response of the fault during high slip rates (small time steps), an adaptive time step is used given by $\Delta {t}\ll {D_c/V}$. An X-FEM MATLAB code (Hosseini et al. 2023) is developed for numerical simulations of this study which is available in the Data Availability section. Main parts of the code (X-FEM discretization, hydro-mechanical solver, contact algorithm) are successfully verified for benchmark problems in our previous works (Hosseini and Khoei 2020; Hosseini et al. 2020).

3 Numerical Simulation: Injection-Induced Slip of a Heterogeneous Strike-Slip Fault

3.1 Model Definition

In this section, we simulate the slip behavior of a fault activated by fluid injection during hydraulic fracturing. We revisit a previously investigated case in western Canada, where the underlying model and injection parameters are well documented in Eyre et al. (2019a). As depicted in Fig. 3a, a vertical strike-slip fault with a length of 1017 m in the Y direction (spanning the depth of 2793–3810 m) cuts through several geological layers. The hydraulic fracturing (HF) stimulation was conducted within the Duvernay Formation. The Ireton, Duvernay, and Beaverhill Lake layers are composed of shale rocks (assigned average clay content of $\sim 33\%$). These units are overlain by carbonate rocks of the Wabamun Formation. The average vertical and horizontal in-situ stress gradients are equal to $\sigma _V=23.75\,\mathrm {kPa/m}$ and $\sigma _H=25.14\,\mathrm {kPa/m}$, respectively (Soltanzadeh et al. 2015). A shear stress gradient of $\tau =7.9\,\mathrm {kPa/m}$ is also applied on the fault surface due to its deviation from the principal stress directions. By assuming a hydrostatic pressure gradient of $p_{stat}=9.81\, \mathrm {kPa/m}$, the value of the in-situ stress ratio is equal to $\tau /(\sigma _H-p_{stat})=0.52$ such that the Duvernay rock is in critically stressed state, and near failure (based on static Coulomb failure criterion).

Fig. 3

a Geological layers, in-situ stresses, and variation of frictional parameters with depth for the strike-slip fault located in the Duvernay formation, b geometry and boundary conditions for the pseudo-3D X-FEM model

Hydraulic fracturing was conducted in multiple stages along a horizontal well in the Z direction. In our modeling, a 2D plane strain section of the fault in the XY plane is modeled (for one HF stage) in which an out-of-plane displacement field, independent of the Z coordinate, is included (pseudo-3D model depicted in Fig. 3b). An uncoupled set of poroelastic equations is solved in order to render the same assumption made in numerical analysis of Eyre et al. (2019a). The finite element domain is a rectangle with dimensions of 1 km $\times$ 2.017 km, in which a 500 m offset to the fault line is considered to minimize boundary effects on the fault domain. For simplicity, it is assumed that the high-pressure zone of HF intersects the fault, and pore pressure rises to 50 MPa + hydrostatic over the entire height of the Duvernay layer immediately. Thus, a higher permeability is assigned to the HF zone compared to the surrounding low-permeable shale rock, and the constant HF pressure is applied to the left boundary of the HF zone. The impermeable fault core ($k_\Sigma =0$) is surrounded by a 10 m wide damage zone (on each side) that transmits high-pressure HF fluid along the fault. The permeability of the damage zone is set to a value which corresponds to a hydraulic diffusivity of $k/\mu _fS=0.05 \, \mathrm {m^2/s}$.

Poroelastic properties of the rock matrix and pore fluid are summarized in Table 1, which are based on the typical values of shale rock and water. The rate-and-state frictional parameters vary along the fault according to the rock type (Table 2). At present, no measured rate-state frictional parameters are available for the rocks in question. Therefore, the range of rate-state frictional parameters were based on laboratory studies carried out on shales, Kohli and Zoback (2013), Ikari et al. (2011) Chen et al. (2015) and Carpenter et al. (2014) for carbonates. For shale rocks (such as the Duvernay), the rate-state parameters are dependent on the clay content of the rock. Shales with clay and organic content above $\sim 30\%$ (by weight) show velocity-strengthening behavior ($a-b>0$), and these values were assigned to rock depths > 3365 m (Duvernay and below). For carbonate-rich rocks, a negative value for parameter $a-b$ is observed that highlights a velocity-weakening behavior and therefore gives rise to unstable behavior. In this case study, the large seismic event occurred within the Wabamun carbonate unit at a depth of $\sim$ 3055 m, therefore $a-b<0$ were assigned to rocks at a depth < 3055 m. To ensure a smooth transition in the rate-state parameters from velocity-strengthening to velocity-weakening, the $a-b$ parameter linearly decreased from the shale end-member at a depth of 3365 m to the carbonate end-member at 3055 m (see Fig. 3a), such that neutral velocity dependence of friction ($a-b=0$) occurs at the contact between shales and carbonates. This is a first-order model and the exact transition of the frictional parameters is not known. Although the thickness of the transition zone can influence the behavior of the fault near the $a-b=0$ line, the general response of the problem remains unchanged. A static friction coefficient ($\mu _0$) 0.5 was assigned to the Duvernay layer, and 0.6 for rocks above 3055 m and rocks below 3470 m, with a linear taper used in between these end-members. A constant value for $D_c$ and $V_0$ was assigned to all formations.

Table 1 Poroelastic properties of rock and fluid

Table 2 Frictional parameters of the fault for base case

The mesh size and penalty parameter along the fault are determined so that numerical convergence is achieved. We used a mesh size of $h=0.5$ m along the fault, which is much smaller than the average critical nucleation length for the fault (5 m), implying the smooth transition from stick to slip state. The penalty parameter used for applying contact conditions in normal and tangential tractions along the fault is chosen to be $\varepsilon =5E/h$ [Pa/m]. Our sensitivity analysis on the penalty parameter shows that this magnitude leads to stabilized results with enough accuracy, as discussed in the next section.

3.2 Long-Term Geological Creep Arising from Hydrocarbon Generation

The Duvernay formation in the region where the large seismic event occurred is currently in a state of overpressure due to hydrocarbon generation which has occurred over the last 50 Myr, and is assumed to have led to changes in shear stresses and slip along the fault due to creep resulting from this overpressure. To account for this, the poroelastic model is initialized by imposing the far-field stresses depicted in Fig. 3b with hydrostatic pore pressure. The Duvernay formation is then subjected to an increasing pressure from hydrostatic pressure up to the estimated overpressure that now exists. Following Eyre et al. (2019a), the pore pressure increased linearly from a hydrostatic pressure of $\sim$ 33 MPa (at the middle of Duvernay layer) up to the current-day pressure of 60 MPa (Soltanzadeh et al. 2015; Eaton and Schultz 2018), as shown by blue line in Fig. 4a. Transition ramps were used between the overpressure in the Duvernay layer and hydrostatic pressures at depths of 3055 and 3470 m. These transition zones are important in view of stability for the solution of the numerical problem. Imposing the overpressure causes long-term creep, with the slip rate during the development of overpressure very low compared to $V_0$, such that the static coefficient of friction is maintained. As shown in Fig. 4a, the overpressure leads to an decrease in effective stress along the fault (since the total stress remains constant and equal to far-field stress), which in turn leads to failure of the Duvernay layer and shear stress relaxation in the creeping shale rocks. However, a stress concentration is developed inside the carbonate rock at the location of the creep front, which corresponds to the transition from slip to stick state on the fault. Since the Wabamun and Beaverhill Formations have higher shear strength due to their higher friction coefficient, the slip profile tapers to zero in these formations, as depicted in Fig. 4b, and remains in the stick state during long period of overpressure.

Fig. 4

Distribution of a effective stress, shear stress, and pore pressure, b slip along the fault after 50 Myr geological overpressure

Using an optimum penalty parameter to achieve stable and accurate results for contact stresses is crucial. A sensitivity analysis for this parameter is shown in Fig. 5 which reveals that a large penalty parameter (500E/h) can generate spurious oscillations in the normal stress profile, especially near the tips of the fault. On the other hand, a low penalty parameter (0.05E/h) can lead to an inaccurate normal stress, particularly near the sharp edges of the stress profile, smoothing out the results.

Fig. 5

Effect of different penalty parameters on the accuracy and stability of the normal traction along the fault

3.3 Effect of Frictional Parameters on the Fault Behavior

Numerical simulations were conducted by varying the rate-state frictional parameters to investigate their impact on fault behavior during HF treatment, which is characterized by the spatio-temporal distributions of shear stress, slip, and friction coefficient along the fault. For this purpose, the evolution of relevant variables is investigated at two points along the fault: point A, located at the center of the Duvernay layer (stable velocity-strengthening zone), which is in direct contact with instantaneous HF pressure rise; and point B, located at a depth of 3055 in the unstable velocity-weakening zone ahead of the geological creep front, which is initially in a stick state. The occurrence time of a seismic event in the unstable zone is characterized by an abrupt slip jump at point B. The moment magnitude of a seismic event is determined by the relation $M_w=(2/3)(\log {M_0}-9.1)$, in which $M_0$ is the seismic moment obtained by $M_0=G\int {g_T\textrm{d}\Sigma }$ over a width of 1 km.

3.3.1 General Response of Different Regions on the Fault (Base Case)

First, the response of the fault for the base frictional parameters (Table 2) is investigated, as shown in Fig. 6. The fault surface is initially at a steady-state sliding velocity $V_0$, where the coefficient of friction is $\mu _0$. After imposing HF pressure, the sliding rate is increased at point A, which was initially in a slip state; consequently, the coefficient of friction instantaneously increases, as shown with blue line in Fig. 6a, which is called direct effect and is controlled by the parameter a. As the fault surface moves, the slip rate decreases to its initial value and the direct-effect decreases over time. Meanwhile, the state variable evolves to a new steady-state value, resulting in a net increase in friction coefficient for point A ($a-b>0$). Specifically, the friction coefficient increases from an initial value of 0.5 to a higher value of $\sim 0.54$ due to velocity-strengthening behavior (see Fig. 6a). As the pore pressure rises abruptly from a value of 60 MPa (initial overpressure) to $\sim 83$ MPa (as a result of hydraulic fracturing) at point A, the effective stress drops by an amount of $~83-60=23\,$ MPa, leading to a net decrease in shear stress over the Duvernay layer by an amount of $\sim (0.54+0.5)23/2=12$ MPa, which is depicted in Fig. 6b. It is noted that the main reason for shear stress relaxation over the Duvernay layer is the increase in pore pressure. At other parts of the stable zone, the effective stress remains constant because fluid does not have enough time to propagate along the damage zone; however, the friction coefficient increases which leads to a slight increase in shear stress in other parts of the stable zone (see Fig. 6b, c).

Fig. 6

a Time history of friction coefficient, profiles of b shear stress, and c friction coefficient along the fault at time 30 min (base case). The dashed line stands for initial condition of the fault at the beginning of fluid injection

Point B is initially in a stick state and is located in the vicinity of a stress concentration ahead of the rupture front as depicted in Fig. 4. The stress concentration is generated due to high slip rates at the rupture fronts. As the rupture front propagates up the fault, the available shear stress at point B overcomes the static shear strength ($\sim 0.6\times 47=28$ MPa) after which the fault at this point starts to slip in an unstable manner, arrested finally by the upper edge of the fault. The pre-existing fault plane has a length of  1 km; thus, when the unstable rupture front reaches the upper tip of the fault, it can induce stress singularities in the surrounding intact rock, which can in turn lead to propagation of the fault into rock. However, the possibility of fault propagation is not considered in this study, which requires more data about rock behavior. As shown in Fig. 6a, c, the friction coefficient after 30 min is $\sim 0.54$ at point B, which is lower than its initial value ($=0.6$) due to velocity-weakening behavior ($a-b<0$). Since the effective stress remains constant ($\sim 47$ MPa), a $\sim (0.6-0.54)47=2.8$ MPa drop in shear stress occurs at this point after the seismic event (Fig. 6b). Thus, the main reason for the drop in shear stress in the unstable zone is not the increase in pore pressure but the propagation of the rupture front, consistent with the results of Eyre et al. (2019a) who used a different numerical approach.

3.3.2 Effect of $a-b$ on Seismic Magnitude

The a parameter in a rate-and-state frictional model relates to the initial abrupt increase in friction coefficient due to an increase in the magnitude of the slip rate. In contrast, the rate-state b parameter relates to the healing of the fault and restoration of coefficient of friction as a function of the state variable. However, the net change in $a-b$ determines the final post-seismic slip profile of the fault. Since experimental data for this parameter is limited, numerical simulations were conducted by varying $a-b$ over the experimental range of $(\pm )\,0.0025$ to 0.0075 to investigate the effect of this parameter on both stable and unstable zones. Figures 7a–c and 8a, b show the results for variations in $(-)\,a-b$ for the unstable zone, while the stable zone remains constant at the base value of $+0.005$. As the magnitude of $(-)\,a-b$ increases, a larger reduction in the friction coefficient of this zone after the seismic event occurs (i.e., point B), which corresponds to a larger reduction in shear stress over this zone. The timing of the seismic event that occurs for each case is the same ($\sim 3$ min after fluid injection), but the magnitude of the jump in slip, as well as the corresponding seismic magnitude, is larger for higher $(-)\,a-b$ values. The magnitude of each seismic event ($M_w$) is shown in Fig. 8. The seismic magnitude for $a-b=-0.0025$ is much lower ($M_w=3.6$) than the other cases because the seismic slip is low and the rupture front does not propagate over all of the unstable zone. Aseismic slip in the stable zone continues until the end of the fluid pumping. The results for changes in $(+)\,a-b$ value for the stable zone, while maintaining $a-b=-0.005$ in the unstable zone, are highlighted in Figs. 7d–f and 8c, d. It can be seen that the post-seismic slip profile, as well as seismic magnitude, are relatively insensitive to the magnitude of the $(+)\,a-b$ along the stable zone. However, the post-seismic friction coefficient for the stable zone becomes higher for high values of $a-b$, resulting in a fault with higher shear strength. Consequently, it takes a longer time for the aseismic slip front to propagate into the upper unstable zones, resulting in a delay in the occurrence of a seismic event. Simulations were also performed, wherein $a-b$ is varied for both stable and unstable zones, to consider the combined effect (Figs. 7g–i, 8e, f). It can be observed that a higher $(-)\,a-b$ in the unstable zone leads to a seismic event occurring with a higher intensity, while a higher $(+)\,a-b$ in the stable zone results in a delay in the onset of the seismic event.

Fig. 7

Distribution of shear stress (first column), slip (second column), and friction coefficient (third column) along the fault at time 30 min after applying HF pressure for difference values of parameter $a-b$. a–c $a-b$ in stable zone is equal to $+0.005$ while $a-b$ changes in unstable zone, d–f $a-b$ in unstable zone is equal to $-0.005$ while $a-b$ changes in stable zone, g–i magnitude of $a-b$ changes in both stable and unstable zones. The dashed line stands for initial condition (I.C.) of the fault at the beginning of fluid injection

Fig. 8

Time history for slip (first column) and friction coefficient (second column) at two points on the fault for different values of parameter $a-b$. a, b $a-b$ in stable zone is equal to $+0.005$ while $a-b$ changes in unstable zone, c, d $a-b$ in unstable zone is equal to $-0.005$ while $a-b$ changes in stable zone, e, f magnitude of $a-b$ changes in both stable and unstable zones. Point A is located at the center of Duvernay layer (stable zone) and point B is located at depth of 3055 m (unstable zone). Time axis is evaluated after applying HF pressure on the fault

Fig. 9

a Effect of $a-b$ parameter in the unstable zone on the moment magnitude of seismic event, b effect of $a-b$ parameter in the stable zone on the occurrence time of seismic event

Figure 9 depicts how the magnitude of the $a-b$ parameter in both the stable and unstable zones can affect the moment magnitude and occurrence time of seismic events. It is observed that moment magnitude ($M_w$) is directly proportional to $|a-b|^\beta$, where $|a-b|$ parameter is related to the unstable zone and power is $0<\beta <1$, which indicates that this increasing function has a decreasing positive slope. On the other hand, the occurrence time of a seismic event is directly proportional to $a-b$ of the stable zone as $(a-b)^\gamma$, where the power $\gamma >1$, indicating that the slope of this fitted function is increasing.

Simulations were also conducted where a is varied over the range of $0.01-0.02$ while other parameters remain unchanged. As can be seen in time histories of Fig. 10, the coseismic slip and final drop in friction coefficient at point B is the same for all cases. However, the transient increase in friction coefficient (at the beginning of the seismic event) is higher for large values of a. Thus, the absolute values of a and b have no effect on the magnitude of a seismic event; it is only the difference that determines the magnitude. However, the time at which a seismic event occurs is dependent on a, which is mainly due to the aseismic processes in the stable zone. The initial and post-seismic distributions of out-of-plane displacement and shear stress fields around the fault (in a 1 $\times$ 1 km area) are depicted in Fig. 11 for the base case. As can be seen, the displacement field is discontinuous near the fault (Figs. 11a, b), and shear stress concentrations correspond to the high slip rate on the fault (Figs. 11c, d), and high stress drops occur in the location of Duvernay layer due to high pressure rise.

Fig. 10

Time history for a slip, b friction coefficient at two points on the fault for different values of direct-effect parameter a, while other parameters are assigned to their reference values

Fig. 11

Contours of a, b anti-plane displacement field and c, d out-of-plane shear stress over a 1 $\times$ 1 km area in XY plane around the fault for base case. The left figures show the initial distribution and right figures show the contours at time 30 min

3.3.3 Effect of $D_c$ Parameter on Fault Healing

In the rate-and-state friction law, the critical slip length $D_c$ relates to the rate of fault healing process, or the time required for the fault to regain its strength after a seismic event. Thus, lower $D_c$ parameter leads to faster evolution in friction coefficient in the post-seismic transient response of the fault. Figure 12b shows the variation of friction coefficient at point B for different values of $D_c$ in the range of 10–100 μm. Change in friction coefficient is a sum of two terms: (1) the direct-effect term, which after a sudden increase at the beginning of seismic event, diminishes over time due to post-seismic decrease in slip rate; and (2) the aging term, which increases over post-seismic time due to evolution of state variable. Figure 12a shows variation of these two terms with time, solid lines for direct effect and dashed lines for aging effect. As can be seen, at the onset of the seismic event, the direct-effect is dominant leading to a sharp increase in friction coefficient. But during post-seismic time, the aging term controls the magnitude of the friction coefficient.

Fig. 12

Time history for a direct-effect and aging terms of the rate-and-state model, b friction coefficient at point B for different values of critical slip length $D_c$. Other parameters are assigned base values listed in Table 2

Fig. 13

Time history for slip and friction coefficient at point B for $a-b\,(\pm )=0.0025$ and $D_c=10\,\upmu {\text{m}}$, while other parameters are equal to base values

As previously discussed, in the case of $a-b=-0.0025$, the rupture front does not propagate the entire length of the unstable zone and is arrested at some point before the upper end of the fault. Rupture in the intact part of the fault could occur during the 2.5 h of HF treatment in this case, with a low magnitude for $D_c$ parameter. The time histories of friction coefficient and slip for this case are depicted in Fig. 13. This scenario indicates that for faults with low magnitudes of $a-b$ and $D_c$, some series of low-magnitude seismic events could be expected, instead of one high-magnitude event. For such repeated, arrested fault activation, the interval time between seismic events is controlled by the $D_c$ parameter while the magnitude of events is determined by the parameter difference, $a-b$.

3.4 Effect of Hydraulic Diffusivity of the Damage Zone

Hydraulic diffusivity for the damage zone (DZ), defined as $k/\mu _f{S}$, controls pore pressure diffusion along the fault. Intrinsic permeability (k) of shale rocks typically ranges from $10^{-16}$ to $10^{-18}\,\textrm{m}^2$, but permeability of a damage zone that is extensively fractured can increase to a range of $10^{-14}{-}10^{-16}\,\textrm{m}^2$, (Faulkner et al. 2010; Cappa and Rutqvist 2011). On the other hand, storage coefficient (S) depends on poroelastic properties of the damage zone, i.e., porosity, fluid and bulk compressibility. Thus, to investigate the effect of hydro-mechanical properties of the damage zone on the pore pressure diffusion, three limit cases for hydraulic diffusivity were investigated by varying the k and S parameters of DZ: (1) diffusivity $=0.05\,\mathrm {m^2/s}$, in which all parameters are equal to values listed in Table 1, (2) diffusivity $=0.5\,\mathrm {m^2/s}$, in which intrinsic permeability has a higher magnitude of $2.89\times {10^{-14}}\,\textrm{m}^2$, (3) diffusivity $=2.17\,\mathrm {m^2/s}$, in which permeability is equal to $2.89\times {10^{-14}}\,\textrm{m}^2$ and storage coefficient has a lower magnitude, where Biot’s modulus $M\rightarrow \infty$, which means the limit case of incompressible fluid or low-porosity rock (Shapiro 2015).

Figure 14 shows the time variations of slip and friction coefficient for different values of hydraulic diffusivity of the damage zone. The results reveal that the occurrence time of the seismic event decreases when the diffusivity of the damage zone increases (Fig. 14a). The reason for this phenomenon can be found by investigation of the pore pressure profile along the fault, as depicted in Fig. 15b. In the case of a high-diffusive damage zone, the pore pressure front is distributed over a longer distance within the stable zone; consequently, the effective stress decreases in a longer portion of this zone, which in turn decreases the shear strength of the fault. Figure 14a shows that post-seismic slip rate at the stable zone (point A) is higher for the high-diffusive case; as a result, a higher post-seismic slip profile over this zone happens, as depicted in Fig. 15c. Another point in the time histories of Fig. 14 is that for very high diffusivity, after the first main seismic event, some after-shocks with lower intensity happen, which is the result of a propagating pore pressure front. The local concentrations of stress and friction coefficient along the fault, which are depicted in Fig. 15a, d show the transition point between slip and stick states, where the slip rate has a local maximum.

Fig. 14

Time history for a slip and b friction coefficient at two points of the fault for different values of hydraulic diffusivity of the damage zone

Fig. 15

Distribution of a shear stress, b pore pressure, c slip and d friction coefficient along the fault at time 30 min after applying HF pressure for different values of of hydraulic diffusivity of the damage zone

4 Conclusion

This paper describes an extended finite element method (X-FEM) simulation of injection-induced slip on a heterogeneous rate-state fault embedded within a heterogeneous poroelastic medium. We used the X-FEM framework for a parametric study of an earthquake triggered by fluid injection during hydraulic fracturing treatment in western Canada. Parameter sensitivity analysis using a suite of layered models, with varying rate-state friction parameters and varying poroelastic diffusivity within a damage zone that surrounds the fault core, provides new insight into fault-slip response for this scenario event.

• In our model, the difference of rate-state parameters $a-b$, which is well established as a primary controlling factor for fault stability (Dieterich 1992), determines the slip profile of the fault. Increasingly negative values of $a-b$ within the velocity-weakening (unstable) zone of the fault leads to seismic events with higher moment, and the moment magnitude ($M_w$) can be estimated as a power function with a positive power less than unity. On the other hand, higher positive values of $a-b$ within the velocity-strengthening (stable) zone of the fault results in a delay in the onset of the seismic event, and the occurrence time of the seismic event can be forecasted using a power function having power greater than unity.

• For fault systems having low magnitudes of both $a-b$ and critical slip distance, $D_c$, runaway rupture is inhibited. The results in a tendency for ongoing injection to generate sequences of smaller earthquakes rather than a single large event.

• For the scenarios considered in this study, the main mechanism for shear stress drop over the fault zone is increased pore pressure, since the hydraulic fracture directly intersects the fault. However, similar to Eyre et al. (2019a), the mechanism for activation of unstable zones of the fault is the propagation of aseismic slip front away from the injection zone.

• Hydraulic diffusivity of the damage zone determines the rate of pore-pressure diffusion along the fault. For highly diffusive damage zones, pore pressure increases over a longer area of the fault, accelerating occurrence of the initial seismic event and promoting more complex sequences that are richer in after-shocks.

Our numerical results point to the high importance of increasing the number of experimental studies, to constrain frictional parameters of the fault as well as poroelastic properties of the host rock that control injection-induced earthquakes.

In this study, the X-FEM framework for induced seismicity during fluid injection is developed in its general form. Although this framework is used for the investigation of rupture propagation on a pre-existing vertical fault, it can be readily used for non-planar inclined faults without any modification. Moreover, it can be extended to study the intersected faults using Junction enrichment function (Khoei et al. 2018) at elements intersected by more than one fault. The possibility of fault propagation into surrounding rock can also be incorporated into this framework by including an appropriate fault propagation criterion and enriching newly cracked elements (Gordeliy and Peirce 2013).

Appendix: Numerical Method: Implementation of X-FEM on Governing Equations

1.1 Weak Form of Governing Equations

To discretize the governing equations with extended finite element method, the weak form of the Eqs. (6) and (7) should be derived. We consider a porous medium, $[{\textbf{x}},t]=\Omega \times [0,\infty )$, saturated with single fluid phase having an external boundary (${\mathcal {S}}$) and a fault ($\Sigma$) as an internal boundary. The momentum equation (6) is multiplied by a trial function for displacement field ($\delta {{\textbf{u}}}$) and then integrated over the domain $\Omega$ as follows (Khoei 2014)

$$\begin{aligned} \int _\Omega \nabla \delta {{\textbf{u}}}:({\textbf{D}}:\varvec{\epsilon }-\alpha {p}{\textbf{I}}) \, \textrm{d}\Omega + \int _\Sigma [\delta {{\textbf{u}}}].{\textbf{T}}_\Sigma \, \textrm{d}\Sigma = \int _\Omega \delta {{\textbf{u}}}.\rho {\textbf{g}} \, \textrm{d}\Omega + \int _{S} \delta {{\textbf{u}}}.{\textbf{T}}_S \, \textrm{d}{S}. \end{aligned}$$

(14)

In the above equation, some integrals are appeared on external and internal boundaries by applying the Divergence theorem. The right-hand side of Eq. (14) includes the body force on $\Omega$ and traction on external boundary, respectively. The traction vector over the fault surface, ${\textbf{T}}_\Sigma$, is appeared in the last integral on the left-hand side of this equation. The contact conditions (8) on the fault surface can be modeled by employing the penalty method, in which the relative displacement between fault surfaces is prevented by embedding very stiff springs on the contact surface. The normal contact traction is obtained from multiplication of the penalty coefficient ($\varepsilon$) and normal gap as, $T'_n=\varepsilon g_N$. However, the shear traction over the fault should be obtained incrementally based on the status of stresses (see Sect. 4).

To derive the weak form of mass conservation equation, the Eq. (7) should be multiplied by a trial function for pressure field ($\delta {p}$) and then integrated over the whole domain as

$$\begin{aligned} \begin{aligned}&\int _\Omega \delta {p} \frac{1}{M} {\dot{p}} \, \textrm{d}\Omega + \int _\Omega \nabla \delta {p} \frac{k}{\mu _f} \nabla {p} \, \textrm{d}\Omega + \int _\Omega \delta {p} \alpha \nabla \cdot \varvec{{\dot{u}}} \, \textrm{d}\Omega - \int _\Sigma [\delta {p}]\, w_\Sigma \, \textrm{d}\Sigma \\ {}&\quad = \int _\Omega \nabla \delta {p}\frac{k}{\mu _f}\rho _f {\textbf{g}} \, \textrm{d}\Omega - \int _{S} \delta {p}\, w_S \, \textrm{d}{S}. \end{aligned} \end{aligned}$$

(15)

The right-hand side of Eq. (15) includes integrals for the fluid body force and normal fluid velocity on the external boundary, respectively. The last integral on the left-hand side of above equation should be determined along the fault surface by substituting relation (11) for fluid velocity across the fault as $w_\Sigma =-k_\Sigma [p]$.

1.2 Spatial Discretization Using X-FEM

In this section, the weak forms of momentum equation and mass conservation, Eqs. (14) and (15), are discretized in spatial domain employing the X-FEM approach, in which the conventional FEM approximation is enriched using Heaviside function. For convenience in deriving the discretized form of governing equations, the vector form of displacement and pressure approximations can be used as follows (Khoei 2014)

$$\begin{aligned} {\textbf{u}}({\textbf{x}},t)=\, & {} {\textbf{N}}_u({\textbf{x}})\, \varvec{{\bar{U}}}(t)+\varvec{{\tilde{N}}}_u({\textbf{x}})\, \varvec{{\tilde{U}}}(t), \end{aligned}$$

(16)

$$\begin{aligned} p({\textbf{x}},t)=\, & {} {\textbf{N}}_p({\textbf{x}})\, \bar{{\textbf{P}}}(t)+\varvec{{\tilde{N}}}_p({\textbf{x}})\, \tilde{{\textbf{P}}}(t), \end{aligned}$$

(17)

in which, ${\textbf{N}}_u$ and ${\textbf{N}}_p$ are the matrices of standard shape functions for displacement and pressure fields, respectively; $\varvec{{\tilde{N}}}_u$ and $\varvec{{\tilde{N}}}_p$ are the matrices of enriched shape functions for displacement and pressure fields, respectively; $\bar{{\textbf{U}}}$ and $\tilde{{\textbf{U}}}$ are the vectors of standard and enriched degrees of freedom for displacement field; $\varvec{{\bar{P}}}$ and $\varvec{{\tilde{P}}}$ are the vectors of standard and enriched degrees of freedom for pressure field. Moreover, the strain vector can be written in terms of nodal values of displacement field as $\varvec{\epsilon }({\textbf{x}},t)={\textbf{B}}_u\, \varvec{{\bar{U}}}+\varvec{{\tilde{B}}}_u\, \varvec{{\tilde{U}}}$, in which ${\textbf{B}}_u$ and $\varvec{{\tilde{B}}}_u$ are the matrices of spatial derivative of the standard and enriched shape functions for displacement, respectively. The gradient of pressure field can also be written as $\nabla {p}({\textbf{x}},t)={\textbf{B}}_p\, \varvec{{\bar{P}}}+\varvec{{\tilde{B}}}_p\, \varvec{{\tilde{P}}}$, in which ${\textbf{B}}_p$ and $\varvec{{\tilde{B}}}_p$ are the matrices of spatial derivative of the standard and enriched shape functions for pressure, respectively. The trial functions for the displacement and pressure fields are defined in the same enriched approximating space as the approximate displacement and pressure fields. The displacement jump across the fault can be written in terms of the enriched degrees of freedom for displacement as $[{\textbf{u}}({\textbf{x}},t)]=2{\textbf{N}}_u \varvec{{\tilde{U}}}$; the normal gap ($g_N$) and tangential relative movement ($g_T$) over the fault is determined as $g_N=[{\textbf{u}}].{\textbf{n}}_\Sigma <0$ and $g_T=[{\textbf{u}}].{\textbf{t}}_\Sigma$. Accordingly, the pressure jump across the fault can be written as $[p({\textbf{x}},t)]=2{\textbf{N}}_p \varvec{{\tilde{P}}}$. Substituting the vector form of the X-FEM approximations (16) and (17) into the integral Eqs. (14) and (15), the discretized form of the governing equations is obtained as follows

$$\begin{aligned}{} & {} {\textbf{K}}{\textbf{U}}-({\textbf{Q}}+{\textbf{Q}}_\Sigma ){\textbf{P}}+{\textbf{F}}_\Sigma ={\textbf{F}}_{ext}, \end{aligned}$$

(18)

$$\begin{aligned}{} & {} {\textbf{Q}}^{\textbf{T}}\varvec{{\dot{U}}}+{\textbf{S}}\varvec{{\dot{P}}}+{\textbf{H}}{\textbf{P}}={\textbf{q}}_{ext}. \end{aligned}$$

(19)

In above system of equations, ${\textbf{U}}$ and ${\textbf{P}}$ are the vectors for both standard and enriched degrees of freedom for primary unknowns defined as ${\textbf{U}}=[\varvec{{\bar{U}}},\varvec{{\tilde{U}}}]$ and ${\textbf{P}}=[\varvec{{\bar{P}}},\varvec{{\tilde{P}}}]$; ${\textbf{K}}$ is the stiffness matrix, ${\textbf{S}}$ is the storage matrix, ${\textbf{H}}$ is the permeability matrix, ${\textbf{Q}}$ is the coupling matrix, and ${\textbf{Q}}_\Sigma$ is the fault coupling matrix. ${\textbf{F}}_{\Sigma }$ is the fault force vector, ${\textbf{F}}_{ext}$ is the external force vector, and ${\textbf{q}}_{ext}$ is the external source vector. These matrices and vectors are defined as follows

$$\begin{aligned}{} & {} {\textbf{K}}=\int _\Omega \begin{bmatrix} {\textbf{B}}^\textrm{T}_u {\textbf{D}} {\textbf{B}}_u &{} {\textbf{B}}^\textrm{T}_u {\textbf{D}} \varvec{{\tilde{B}}}_u \\ \varvec{{\tilde{B}}}^\textrm{T}_u {\textbf{D}} {\textbf{B}}_u &{} \varvec{{\tilde{B}}}^\textrm{T}_u {\textbf{D}} \varvec{{\tilde{B}}}_u \end{bmatrix} \textrm{d}\Omega ,\\{} & {} {\textbf{Q}}=\int _\Omega \begin{bmatrix} {\textbf{B}}^\textrm{T}_u \alpha {\textbf{m}} {\textbf{N}}_p &{} {\textbf{B}}^\textrm{T}_u \alpha {\textbf{m}} \varvec{{\tilde{N}}}_p \\ \varvec{{\tilde{B}}}^\textrm{T}_u \alpha {\textbf{m}} {\textbf{N}}_p &{} \varvec{{\tilde{B}}}^\textrm{T}_u \alpha {\textbf{m}} \varvec{{\tilde{N}}}_p \end{bmatrix} \textrm{d}\Omega ,\\{} & {} {\textbf{Q}}_\Sigma =2\int _\Sigma \begin{bmatrix} {\textbf{0}} &{} {\textbf{0}} \\ {\textbf{N}}^\textrm{T}_u {\textbf{n}}_\Sigma {\textbf{N}}_p &{} {\textbf{N}}^\textrm{T}_u {\textbf{n}}_\Sigma \varvec{{\tilde{N}}}_p \end{bmatrix} \textrm{d}\Sigma ,\\{} & {} {\textbf{S}}=\int _\Omega \frac{1}{M} \begin{bmatrix} {\textbf{N}}^\textrm{T}_p {\textbf{N}}_p &{} {\textbf{N}}^\textrm{T}_p \varvec{{\tilde{N}}}_p \\ \varvec{{\tilde{N}}}^\textrm{T}_p {\textbf{N}}_p &{} \varvec{{\tilde{N}}}^\textrm{T}_p \varvec{{\tilde{N}}}_p \end{bmatrix} \textrm{d}\Omega ,\\{} & {} {\textbf{H}}=\int _\Omega \frac{k}{\mu _f} \begin{bmatrix} {\textbf{B}}^\textrm{T}_p {\textbf{B}}_p &{} {\textbf{B}}^\textrm{T}_p \varvec{{\tilde{B}}}_p \\ \varvec{{\tilde{B}}}^\textrm{T}_p {\textbf{B}}_p &{} \varvec{{\tilde{B}}}^\textrm{T}_p \varvec{{\tilde{B}}}_p \end{bmatrix} \textrm{d}\Omega + 4\int _\Sigma \begin{bmatrix} {\textbf{0}} &{} {\textbf{0}} \\ {\textbf{0}} &{} {\textbf{N}}^\textrm{T}_p k_\Sigma {\textbf{N}}_p \end{bmatrix} \textrm{d}\Sigma ,\\{} & {} {\textbf{F}}_\Sigma =2\int _\Sigma \begin{bmatrix} {\textbf{0}} \\ {\textbf{N}}^\textrm{T}_u {\textbf{T}}'_\Sigma \end{bmatrix} \textrm{d}\Sigma ,\\{} & {} {\textbf{F}}_{ext}=\int _\Omega \begin{bmatrix} {\textbf{N}}^\textrm{T}_u \rho {\textbf{g}} \\ \varvec{{\tilde{N}}}^\textrm{T}_u \rho {\textbf{g}} \end{bmatrix} \textrm{d}\Omega + \int _{S} \begin{bmatrix} {\textbf{N}}^\textrm{T}_u {\textbf{T}}_s \\ \varvec{{\tilde{N}}}^\textrm{T}_u {\textbf{T}}_s \end{bmatrix} \textrm{d}S,\\{} & {} {\textbf{q}}_{ext}=\int _\Omega \frac{k}{\mu _f} \begin{bmatrix} {\textbf{B}}^\textrm{T}_p \rho _f {\textbf{g}} \\ \varvec{{\tilde{B}}}^\textrm{T}_p \rho _f {\textbf{g}} \end{bmatrix} \textrm{d}\Omega - \int _{S} \begin{bmatrix} {\textbf{N}}^\textrm{T}_p w_S \\ \varvec{{\tilde{N}}}^\textrm{T}_p w_S \end{bmatrix} \textrm{d}S. \end{aligned}$$

Remark 1

An uncoupled system of equations can be derived by setting the coupling matrix to zero, ${\textbf{Q}}={\textbf{0}}$, in Eqs. (18) and (19). However, to restore the hydraulic diffusivity to the coupled one, the coefficient 1/M in the storage matrix (${\textbf{S}}$) should be replaced by the storage coefficient (S) defined in Sect. 2.

Remark 2

The fault coupling matrix (${\textbf{Q}}_\Sigma$) is responsible for triggering the fault failure according to the pressure field. This matrix should be evaluated on the fault interface, but the enriched shape function $\varvec{{\tilde{N}}}_p$ has different values on each side of the fault. Thus, this shape function should be evaluated on the side of the fault that has higher pressure.

1.3 Solution Strategy

The fully coupled system of Eqs. (18) and (19) should be then discretized in the time domain; and due to the contact force vector (${\textbf{F}}_{\Sigma }$), the resulting system is nonlinear; Thus, the Newton–Raphson iterative algorithm should be implemented, and the status of stresses on the fault should be updated at each iteration. To advance the solution over the time domain, the link between the successive values of the unknown field variables at current time $t^{n+1}$ and the known field variables at previous time $t^n$ should be established by applying the backward implicit method as follows

$$\begin{aligned} \dot{{\textbf{X}}}^{n+1}=\frac{{\textbf{X}}^{n+1}-{\textbf{X}}^n}{\Delta {t}},\quad {\textbf{X}}\in \{{\textbf{U}},{\textbf{P}}\}, \end{aligned}$$

(20)

where $\Delta {t}=t^{n+1}-t^n$ is the time increment. Equation (20) can be used to eliminate time derivatives of displacement and pressure in Eq. (19) and thus the governing equations are expressed only in terms of primary unknowns at time $t^{n+1}$ in the following form

$$\begin{aligned} \varvec{\Psi }_u^{n+1}=\, & {} {\textbf{K}}{\textbf{U}}^{n+1}-(\mathbf {Q+Q_\Sigma }){\textbf{P}}^{n+1}+{\textbf{F}}^{n+1}_{\Sigma }-{\textbf{F}}_{ext}={\textbf{0}}, \end{aligned}$$

(21)

$$\begin{aligned} \varvec{\Psi }_p^{n+1}= & {} \mathbf {Q^T}(({\textbf{U}}^{n+1}-{\textbf{U}}^{n})/\Delta {t})+{\textbf{S}}(({\textbf{P}}^{n+1}-{\textbf{P}}^{n})/\Delta {t})+{\textbf{H}}{\textbf{P}}^{n+1}-{\textbf{q}}_{ext}={\textbf{0}}. \end{aligned}$$

(22)

The set of above equations is nonlinear due to presence of contact force vector ${\textbf{F}}_\Sigma$ in $\varvec{\Psi }_u$. Thus, the iterative Newton–Raphson scheme should be used to linearize this system of equations. The Jacobian matrix, which is defined as the partial derivatives of nonlinear Eqs. (21) and (22) with respect to the unknowns at time $t^{n+1}$, can be written as

$$\begin{aligned} {\textbf{J}}^{n+1}=\begin{bmatrix} \partial {\varvec{\Psi }_u}/\partial {{\textbf{U}}} &{} \partial {\varvec{\Psi }_u}/\partial {{\textbf{P}}} \\ \partial {\varvec{\Psi }_p}/\partial {{\textbf{U}}} &{} \partial {\varvec{\Psi }_p}/\partial {{\textbf{P}}} \end{bmatrix}^{n+1}= \begin{bmatrix} {\textbf{K}}+{\textbf{K}}_{\Sigma }^{n+1} &{} -(\mathbf {Q+Q_\Sigma }) \\ \mathbf {Q^T}/\Delta {t} &{} {\textbf{S}}/\Delta {t}+{\textbf{H}} \end{bmatrix}. \end{aligned}$$

(23)

Using above Jacobian matrix, the primary unknowns ${\textbf{U}}^{n+1}$ and ${\textbf{P}}^{n+1}$ in Eqs. (21) and (22) can be determined in an iterative scheme until the convergence is achieved (see Algorithm 1). We have used a relative tolerance of $TOL=10^{-5}$ as a convergence criterion for this iterative scheme. The Jacobian matrix and contact force vector are updated at each Newton–Raphson iteration based on whether the fault is in the stick or slip state. The term ${\textbf{K}}_{\Sigma }$ in the Jacobian matrix is equal to partial derivative of fault force vector with respect to the displacement and can be determined as follows

$$\begin{aligned} {\textbf{K}}_{\Sigma }=\frac{\partial {{\textbf{F}}_\Sigma }}{\partial {{\textbf{U}}}}= 4\int _\Sigma \begin{bmatrix} {\textbf{0}} &{} {\textbf{0}} \\ {\textbf{0}} &{} {\textbf{N}}^\textrm{T}_u {\textbf{D}}_\Sigma {\textbf{N}}_u \end{bmatrix} \textrm{d}\Sigma , \end{aligned}$$

(24)

where, the contact stiffness matrix, ${\textbf{D}}_\Sigma$, is defined as the derivative of effective traction vector with respect to the displacement jump on the fault as follows

$$\begin{aligned} {\textbf{D}}_\Sigma =\frac{\partial {\mathbf {T'}_\Sigma }}{\partial {[{\textbf{u}}]}}={\textbf{n}}_\Sigma \otimes \frac{\partial {T'_n}}{\partial {[{\textbf{u}}]}}+{\textbf{t}}_\Sigma \otimes \frac{\partial {T_t}}{\partial {[{\textbf{u}}]}}=\varepsilon {\textbf{n}}_\Sigma \otimes {\textbf{n}}_\Sigma +{\textbf{t}}_\Sigma \otimes \frac{\partial {T_t}}{\partial {[{\textbf{u}}]}}. \end{aligned}$$

(25)

The derivative of shear traction with respect to displacement jump, $\partial {T_t}/\partial {[{\textbf{u}}]}$, is evaluated based on the work of Liu and Borja (2009), see Algorithm 1.

1.4 Updating Shear Stress and Slip over the Fault

The theory of plasticity can be used for determination of shear stresses and slip over the fault. The approach utilized for this purpose is mainly based on the method developed previously by Liu and Borja (2009). At each time step, the changes in tangential relative movement ($\Delta {g_T}$) along the fault can be decomposed into elastic and plastic parts as $\Delta {g_T}=\Delta {{g_T}_{elastic}}+\Delta {{g_T}_{plastic}}$; and changes in shear stress on that point can be determined using relation $\Delta {T_t}=\varepsilon \Delta {{g_T}_{elastic}}$ where the operator $\Delta$ is defined as $\Delta {\Theta }=\Theta ^{n+1}-\Theta ^n$. Note that penalty coefficient for tangential movement is set equal to the normal penalty coefficient. Thus by defining slip ($\zeta$) as plastic part of relative movement, $\zeta \equiv {g_T}_{plastic}$, the following constitutive law for shear stress can be obtained

$$\begin{aligned} T_t^{n+1}=T_t^{n}+\varepsilon (\Delta {g_T}-\Delta \zeta ). \end{aligned}$$

(26)

For stick condition, we get $\Delta \zeta =0$, while in slip condition, $\Delta \zeta \ne 0$ and slip at current time step, $\zeta ^{n+1}$, should be found such that slip criterion ($|T^{n+1}_t |=T_f$) is satisfied. As friction coefficient is function of slip rate, an iterative scheme must be employed for this purpose (see Algorithm 2). A tolerance of $TOL=10^{-5}$ is used for convergence criterion of this inner Newton–Raphson scheme. Note that $g_T$ can only equal $\zeta$ in the limit as penalty coefficient approaches infinity ($\varepsilon \rightarrow \infty$). In order to proceed in time, slip rate ($V\equiv {\dot{\zeta }}$), and state variable evolution (Eq. 10) should be discretized in time as

$$\begin{aligned}{} & {} {\dot{\zeta }}^{n+1}=\frac{\zeta ^{n+1}-\zeta ^{n}}{\Delta {t}}, \end{aligned}$$

(27)

$$\begin{aligned}{} & {} \theta ^{n+1}=(\theta ^n+\Delta {t})/\left( 1+\frac{\Delta {t}{\dot{\zeta }}^{n+1}}{D_c}\right) . \end{aligned}$$

(28)

Thus, friction coefficient (Eq. 10) can be expressed in terms of slip $\zeta ^{n+1}$ alone as follows

$$\begin{aligned} \mu ^{n+1}(\zeta ^{n+1})=\mu _0+a\ln \left( \frac{{\dot{\zeta }}^{n+1}}{V_0}\right) +b\ln \left( \frac{\theta ^{n+1}}{\theta _0}\right) . \end{aligned}$$

(29)

Remarks for Algorithms: Function $\mu '(\zeta )$ used in algorithms is derivative of friction coefficient (29) with respect to slip,$\zeta$, defined as $\mu '=a/({\dot{\zeta }}\Delta {t})-\chi {b}/((1+\chi {{\dot{\zeta }}})\Delta {t})$ where $\chi =\Delta {t}/D_c.$