1. INTRODUCTION

Injection of fluid into the subsurface can induce seismicity at various scales (Ellsworth, 2013). Current studies on induced seismicity mostly focus on the pre-seismic quasi-static poroelastic triggering but not the co-seismic dynamic fault rupture (Murphy et al., 2013; Chang & Segall, 2016a; Chang & Segall, 2016b; Fan et al., 2016; Deng et al., 2016; Jin & Zoback, 2016; Chang & Segall, 2017; Zbinden et al., 2017). However, only by completing the second step can one then provide proxies, e.g., peak ground velocity, for quantifying the associated risk. In principle, one can utilize an existing co-seismic rupture model to perform this task, by inserting the fault fluid pressure scaled by the static fault frictional coefficient as an equivalent initial shear traction on the fault (Richards‐Dinger & Dieterich, 2012; Dieterich et al., 2015). However, in the case of a poroelastic medium, this approach is no longer valid as it inherently cannot account for the fault poroelastic stress, which includes both a normal component and a shear component generated from the pre-seismic quasi-static and fluid-solid coupled triggering process. Because of the nonlinear nature of a spontaneous rupture problem, the accuracy in the initial stress state on the fault is crucial for a proper co-seismic

rupture model. Although a recent model has been proposed that considers the fluid-solid coupling prior to rupture (Jha & Juanes, 2014), it assumes quasi-dynamic as opposed to fully dynamic rupture, therefore cannot be utilized to investigate the seismic radiation pattern and the wavefield associated with an induced earthquake. A fully dynamic rupture model considering the fault poroelastic stress does exit, but with simplified model rupture physics, e.g., no dynamically evolving fault frictional strength, no rupture nucleation and propagation (Jin et al., 2017). We are therefore motivated in this study to develop a first reported computational model specifically of fluid-induced co-seismic fully dynamic spontaneous rupture considering the pre-seismic poroelastic stress on the fault, as is elaborated on below.

2. GOVERNING LAWS

Injection induced seismicity encompasses both a pre-seismic quasi-static poroelastic triggering process and a co-seismic fully dynamic spontaneous fault rupture process. The duration of the former process is typically orders of magnitude longer than that of the latter. The former process is usually ‘drained’ with the presence of a relative fluid-to-solid velocity; the gradient of the fluid pressure within the pores and the faults interact with the

ARMA 18–604

Modeling Induced Seismicity: Co-Seismic Fully Dynamic Spontaneous Rupture Considering Fault Poroelastic Stress

Jin, L. Stanford University, Stanford, CA, USA

Zoback, M.D. Stanford University, Stanford, CA, USA

Copyright 2018 ARMA, American Rock Mechanics Association

This paper was prepared for presentation at the 52nd US Rock Mechanics / Geomechanics Symposium held in Seattle, Washington, USA, 17–20 June 2018. This paper was selected for presentation at the symposium by an ARMA Technical Program Committee based on a technical and critical review of the paper by a minimum of two technical reviewers. The material, as presented, does not necessarily reflect any position of ARMA, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of ARMA is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 200 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of where and by whom the paper was presented.

ABSTRACT: Pre-seismic poroelastic stress generated by quasi-static fluid perturbation is a centerpiece of information specific to injection-induced seismicity. To this day, there exists no fault rupture model that considers the fault poroelastic stress in the co-seismic dynamic rupture process of an induced earthquake. To this aim, we develop a computational model of fluid-induced, fully dynamic and spontaneous co-seismic rupture under slip-weakening and an exact fault surface contact constraint. We show how the pre-seismic fault poroelastic stress enters the model formulation and discretization and how is it reflected on the computational procedures including linearization and preconditioner design. A numerical example is provided to demonstrate the impact of the fault poroelastic stress on the co-seismic fault rupture behavior and the associated dynamic changes of an induced earthquake. It is shown that including the fault poroelastic stress can lead to delayed nucleation of the rupture, promote the so-called supershear rupture and result in an overall stronger wavefield and a more profound stress loss near the fault. Additionally, when the pre-seismic transverse simple shear behavior of the fault is considered, the fault poroelastic stress with a steep gradient across the fault may also lead to asymmetric earthquake rupture propagation.

mean stress and the volumetric strain of the hosting solid rock in a fully coupled manner. The latter process is typically considered as ‘undrained’ over the timescale relevant to a seismic event. The former is related to the latter in the following few ways. First, the former spatial-temporally alters the effective normal stress, shear stress and the resulting Coulomb stress on the fault, and therefore controls the time and location of nucleation of the rupture. Second and central to an induced seismicity problem, the accumulated fault poroelastic stress prior to nucleation of the rupture enters the co-seismic rupture process via the fault boundary condition (i.e., the consistency condition in plasticity theory). The fault stress profile when the rupture initiates has a fundamental control over the rupture style, rupture velocity and the associated seismic radiation pattern and wavefield, as has been amply demonstrated by many studies (e.g., Andrews, 1976; Dunham, 2007; Ampuero & Ben-Zion, 2008; Ben-David et al., 2010; Gabriel et al., 2012). Lastly, the poroelastic process also modifies the density of the fluid-solid mixture and must be taken into account in solving the fully dynamic rupture equation. Our Jin & Zoback (2017) paper describes in detail a nonlinear and fully coupled poromechanical model of an arbitrarily faulted and fluid-saturated quaso-static poroelastic system; the discretization and computational procedures are also given there. This computational model will be used for generating necessary inputs including the fault poroelastic stress for the study presented here. We hereinafter focus on the co-seismic fully dynamic spontaneous rupture problem only and we shall assume a plane-strain (in-plane fault slip) condition.

Denote the model domain as Ω, the location within Ω as �and the time as t, the conservation law we solve is the Cauchy equation of motion described by including the reference state at the end of the pre-seismic quasi-static process (i.e., when the rupture nucleates). It reads:

( , )

'( , *) ( , *) '( , ) ( , )

( , *) ( , ) , *

σ

σ σ

undrained

p s s

divergence free

mix s

s x t

x t p x t x t p x t

x t u x t x t t

1

(1)

where t* indicates the time of nucleation of the rupture, �(�, �∗) and ��′(�, �∗) are injection-induced (i.e.,

poroelastic) changes in the fluid pressure and the effective stress tensor at t*, ��′(�, �), ��(�, �), ��(�, �) and ��(�, �)are slip-induced changes in the effective stress tensor, the pore pressure, the Cauchy total stress tensor and the displacement vector, � is the Biot-Willis coefficient and ����(�, �∗) is the density of the fluid-solid mixture (fluid-saturated solid) evaluated at t*. Note that an initial quasi-static balance has been reached when the rupture nucleates and therefore the first part is divergence free as is indicated.

We have introduced that the co-seismic fluid-solid system is undrained. This motivates the use of an undrained linearly elastic constitutive law for the fluid-saturated solid (Rice & Cleary, 1976; Cleary, 1977). It resembles that of a pure elastic solid except that (1) it is the Cauchy total stress tensor ��(�, �) as opposed to the effective

stress tensor ��′(�, �) that is related to ��(�, �), and (2) an undrained counterpart for the Poisson’s ratio is used. It reads:

( , ) ( , )(1 )(1 2 )

( , ) ( , )2(1 )

σ us s

u u

T

s s

u

Ex t u x t

Eu x t u x t x

1

(2)

where �� is the undrained Poisson’s ratio of the fluid-saturated solid and E is the Young’s modulus.

The mixture density is calculated according to the following based on the volumetric fractions of the fluid and the solid at t*:

0

( , *) 1 ( , *) ( , *)

( , *) ( , *) 1 ( , *)

mix p s

p

x t x t u x t

x t u x t C p x t

(3)

where Φ(�, �∗) is the so-called partial porosity of the

faulted porous medium, �� and �� are the initial density

and the compressibility of the fluid, �� is the density of the solid. In equation (3), −�∇ ∙ ��(�, �) quantifies the

poroelastic modification to the porosity following a compressive strain positive notation. Details can be found in Jin & Zoback (2017).

During the co-seismic rupture, in addition to traction continuity condition, equation (1) is also subjected to the following contact constraint which we shall impose exactly using a Lagrangian multiplier method. It reads:

( , ) ( , ) ( , ) 0 , *

s s f s fu x t u x t n u x t n x f t t

(4)

where �� is the unit fault normal vector defined on the

negative side of the fault, and �� indicates the fault.

Lastly, the co-seismic fault shear traction and normal traction obey the consistency condition:

12 2 2

( , ) ( , ) :

( , ) : 0 , *

σ σ

σ

f f f

f f f

x t n x t n n

x t n n x f t t

(5)

Here,

0 ' '( , *) ( , ) σ σ σ σp sx t x t (6)

where ��′ is the initial in-situ effective stress tensor before the poroelastic modification.

and �� is the frictional coefficient of the fault. It is worth

noting that �� must be dynamically evolving as opposed

to be constant; this is the necessary condition for driving and resisting the spontaneous rupture along the fault. Here we opt for the widely used slip-weakening law:

( , )

( ) , ( , )( , ) ,

, ( , )

s s d c

cf

d c

s x ts x t d

ds x t x f

s x t d

(7)

where ��, �� are the static and dynamic frictional

coefficients, ���, �� ≔ �����, �� − ��

���, ��, � ∈ �� is

the fault slip vector with a magnitude ����, ���, and �� is the slip-weakening distance. Equation (7) renders the rupture problem nonlinear.

Equations (5)~(7) are the key steps where the fault poroelastic stress enters the co-seismic system and alters its behavior. Finally, because we are solving for slip-induced changes relative to the reference state at t*, the initial conditions can be trivially set up by simply letting these changes and their speed of change as 0. A Lsymer absorbing boundary condition (Lysmer & Kuhlemeyer, 1969) is prescribed at the domain outer boundary for absorbing nonphysical reflections. Details are omitted.

3. DISCRETIZATION AND COMPUTATION

Equation (1) is discretized in space using a low-order split-node finite element method operated on a mesh conforming to the fault; the contact constraint equation (4), which is imposed exactly using the Lagrangian multiplier method, is also discretized; together they lead to a saddle-point block structure of the semi-discrete form of the collectively-written equations. The discrete Lagrangian space is constructed in a manner such that the inf-sup stability constraint (a sufficient and necessary condition for the stability in the co-seismic fault normal traction) can be satisfied. Additionally, a Rayleigh damping matrix is artificially introduced to the semi-discrete form for suppressing potential spurious oscillations of high frequencies. Subsequently, the semi-discrete form is further discretized in time using an implicit, unconditionally stable and second-order accurate finite difference method from the Newmark family. The use of the implicit time stepping scheme renders the fully discrete form nonlinear. We then perform the Newton-Raphson iterative linearization scheme, leading to the following critical step at time step (n+1) and Newton iteration step (k) where the major computational cost arises:

( 1, 1) ( 1, )( 1, ) ( 1, )

11 12

( 1, )21 22 ˆ ˆ

n k n kn k n k

d d d

n kl l l

a a R

a a R

J J

J J (8)

Here,

( 1, )

( 1, ) 211

20

( 1, )

( 1, )

( 1, )

1 1

2 4

1 ˆ2 ,4

/ 2

,

/ 2

n k

n k A R

Ts f s n l d

f

n k

c

n k

c

n k

t t

t t f N l d e

d d

d d

Q

J M C C K

N 1

0

(9)

( 1, ) 212

( 1, )2

( 1, )

2

( 1, )

1

4

11 2 ,

4

/ 2

1,

4

/ 2

n k

n kTs f s lf

n k

c

Ts d lf

n k

c

t

t d N d

d d

t N d

d d

J G

N 1

N 1

(10)

( 1, ) 221 21

1

4 n k TtJ J G (11)

( 1, )22 22

n kJ J 0 (12)

( 1, )

( 1, )

2 2( 1, )

2

( 1)( 1, ) ( 1, )

( 1)

1 1 1

2 4 4ˆ1

4

ˆ,

ˆ0̂

ˆ

n k

d

n k

l

A Rn k

d

T l

nn k n k A Rd

l

n

T

R

R

t t ta

at

vF d l

v

d

l

M C C K G

G 0

C C 0

0 0

K G

G 0

(13)

In equations (8)-(13), M and K are the mass and elastic stiffness matrices of the undrained fluid-solid system, Gis the contact matrix resulted from an exact contact constraint, CA is the boundary absorbing matrix, CR is the Rayleigh damping matrix, 1 is the unit identity, F is the external nodal force vector that carries information on the pre-seismic fault poroelastic stress, � and �� are

interpolation functions associated the mesh, � and �� are the discrete displacement and Lagrangian multiplier (fault

normal traction), ��, ��� are �̇, ��̇, ��, ��� are �̈, ��̈, the cap ‘~’ indicates intermediate predictions used in the

Newmark scheme, �� is the time step, �� is a simplifying

notation related to equation (7), ��� is the initial in-situ fault normal traction, � is the poroelastic modification to ���, �� is the unit vector specifying the slip direction on the negative side of the fault and is calculated as the direction of the maximum fault shear stress, and �� is a vector containing sub-vectors specifying the direction of motion of nodes of a fault-conforming element.

Equation (8) must be solved iteratively using a Krylov sub-space iterative solver, due to the sparse nature of the Jacobina matrix J. As can be seen, our algorithm also leads to an asymmetric structure and indefiniteness of the J. Accordingly, we opt for the GMRES solver suitable for this class of linear problems. To accelerate the convergence, we design a physics-based non-stationary preconditioner at each step (n+1, k). It reads:

( 1, )

( 1, )1

( 1, ) ( 1, )21 21 12

ˆˆ

ˆ

n k

n kt n k n k

A 0P

J J A J(14)

where the first block submatrix is reads:

( 1, )

2

( 1, )( 1, )

( 1, )

ˆ 1 0.5 ( ) 0.5 ( )

0.5 0.25 ( )

, / 2

, / 2

n k A

n kn kc

n k

c

t a lump t diag

t b t diag

diag d d

d d

A M C

K

Q

0

(15)

Here, Q is indicated in equation (9) and it contains information about the pre-seismic fault poroelastic stress, and a and b are the Rayleigh damping coefficients.

Upon solving equation (8), one can then obtain the particle velocity and the nodal displacement by carrying the remaining steps of the Newmark scheme. The co-seismic dynamic stress tensor can be calculated from Standard finite element relationships. Details on the above discretization, linearization and computational procedures are not elaborated on here.

4. NUMERICAL EXPERIMENT SET-UP

For this numerical exercise, we construct a 200m×200m

model domain. A fault 50√2m long and dipping at -45° is generated symmetrically with respect to the domain center. An injection point immediately off the fault at [-25m, 20m] is introduced. The injection pressure is 5MPa, ��′ is [6.0 -2.7; -2.7 11.2] MPa, ��, �� are 1000 kg/m3 and 3000 kg/m3, ��, �� are 0.6 and 0.4, �� is 0.3 and E is 40 GPa, and dc is 0.1 mm (notice this is a very low value that can be sufficiently resolved using our computational model compared to most earthquake rupture models). The above parameters are required for the co-seismic problem, the parameters used for the pre-seismic quasi-static

poroelastic problem can be found in Jin & Zoback (2017), and the rest parameters are all calculated.

For comparison, two cases are constructed, including (1) a reference case without considering the fault poroelastic stress and the rupture is due to solely a tectonic loading prescribed as an uniform increase (0.0465 MPa) in the fault shear traction, and (2) a case in which the fault rupture is due to the pre-seismic fault poroelastic stress. In nucleating the rupture, the same overstress (0.1MPa) is prescribed. The nucleation patch size is calculated from the standard formula based on the fault peak and residual strengths and the given mechanical parameters of the undrained fluid-solid system under plane strain. The overstress is also smoothed within the first 4ms of the dynamic rupture.

5. RESULTS

5.1. Wavefield Figures 1-3 show the co-seismic wavefield, i.e., particle velocity at 7 ms, 15 ms and 29 ms since nucleation of the rupture, for the two cases without and with considering the fault poroelastic stress, respectively. Specifically, Figure 1 shows the magnitude and Figures 2-3 show the fault-tangential and -normal components. We observe that the seismic radiation pattern and the fault rupture velocity is radically different between the two cases. In the former case, the nucleation reverberation decays while the rupture front picks up and propagates along the fault following a pattern typically observed for a tectonic event. In the latter case, however, more abrupt nucleation and rupture propagation are observed; the rupture front travels at a much faster speed along the fault, the wave radiates into a wider range of area with sharply defined wavefront, and the magnitude is much stronger. This modeled seismic radiation pattern agrees well with some laboratory observations (e.g., Xia et al., 2004).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1. Magnitude of the particle velocity showing the co-seismic wavefield. (a)-(c) The case without considering the fault poroelastic stress, (d)-(f) the case with considering the fault poroelastic stress. Three time-slices are shown in each case and the time since nucleation of the rupture is shown at the top of each plot.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2. Same as Figure 1, but for the fault-tangential component of the particle velocity. The positive fault-tangential direction is [1, -1]T.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3. Same as Figure 1, but for the fault-normal component of the particle velocity. The positive fault-normal direction is [1, 1]T.

5.2. Rupture velocity Figure 4 is the classic space-time plot of the slip velocity on the fault, the magnitude of which is indicated by the color, for the two cases with and without considering the fault poroelastic stress. The slope of the delineated front of the colored region is the slowness (inverse of the speed) at which the rupture tip propagates along the fault, i.e., the inverse of the rupture velocity. As a reference, the blue line indicates the S-wave velocity Vs (slowness Cs) and the green line indicates the P-wave velocity Vp (slowness Cp). We observe that in the former case, the rupture velocity is below Vs (≈ 0.9 Vs), and this is the so-called sub-Rayleigh shear rupture. In the latter case, however, the rupture velocity is above Vs (≈ Vp), and this is the so-called supershear rupture, which is less commonly observed. Our model therefore implies the possibility of fluid-induced earthquakes as supershear rupture events. We also point out that in this example, specifically in Figure 4a, a transition from the sub-Rayleigh regime to the supershear regime is observed, as also has been

documented by some other numerical studies (e.g., Gabriel et al., 2012). Additionally, notice also that the nucleation is delayed in the latter case, as is reflected by the larger interception of delineated rupture front on the vertical axis.

(a)

(b)

Figure 4. Space-time plot of the magnitude of the slip velocity on the fault. The slope of the delineated front of the colored region is the inverse of the rupture propagation velocity along the fault. The two reference lines indicating P- and S-wave velocities are also shown. (a) The case without considering the fault poroelastic stress, and (b) the case with considering the fault poroelastic stress. To exclude the nucleation reverberation effect, only the part outside the nucleation patch on the fault is plotted.

5.3. Dynamic Coulomb Stress Figure 5 illustrates the slip-induced stress changes 50 ms since nucleation of the rupture, for the two cases without

and with considering the fault poroelastic stress. Specifically, the normal stress and shear stress are calculated in the fault-normal and -tangential coordinates, and the resulting Coulomb stress is also shown. As can be seen, the overall stress (proportionate to the symmetric gradient of the displacement) distribution is similar, despite the differences in the wavefield (time derivative of the displacement). However, by considering the fault poroelastic stress, the shear stress drop due to the rupture is more profound surrounding the fault, and the stress lobes near the tips of the fault are stronger in magnitude, i.e., stronger compressive stress, stronger extensional stress and stronger shear stress in their respective areas.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5. Snapshots of the spatial distribution of the slip-induced fault-normal stress, shear stress and the resulting Coulomb stress at the end of dynamic simulation (50ms since nucleation). (a)-(c) The case without considering the fault poroelastic stress, and (d)-(f) the case with considering the fault poroelastic stress.

6. DISCUSSION

In the above work, the poroelastic fault stress accumulated over the pre-seismic quasi-static process is modeled by assuming a single solid constitutive law, i.e., the fault follows the same rheology as the hosting rock prior to the rupture. As a result, the poroelastic stress within the domain vary smoothly without any sharp gradient. While this is a reasonable assumption, in our Jin & Zoback (2017) computational model, we have demonstrated that including a transverse simple shear behavior of the fault (continuous shear deformation of the exceedingly thin fault zone) before the occurrence of fault slip can lead to a sharp variation in the shear stress across the fault (termed as apparent discontinuity), if the injection is on one side of the fault. Here, we test how this apparent discontinuity may alter the co-seismic rupture behavior, as is illustrated in Figure 6. The magnitude and the fault-normal and -tangential components are plotted at 15 ms and 19 ms since nucleation of the rupture. Interestingly, the resulting wavefield becomes asymmetric. The fault surface on the hanging wall side (injection takes place in the footwall) experiences delayed nucleation and weaker dynamic signals. Asymmetric fault rupture has been documented by many studies (e.g., Bhat et al., 2010). Here, rather than being based on a bi-material model, our work provides an alternative possible mechanism of such events.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6. Example of an injection-induced seismic event with asymmetric co-seismic rupture. (a)-(c) the magnitude and the fault-tangential and -normal components of the particle velocity at 15 ms since nucleation, (d)-(f) same as (a)-(c) excepted the time is 19 ms since nucleation.

7. CONCLUSION

We have developed a computational model of fully dynamic spontaneous rupture specifically due to external fluid perturbation. Central to our work is that we first solve the pre-seismic, fully coupled and quasi-static poroelastic problem, and then carry the fault poroelastic stress into the co-seismic dynamic rupture modeling. This is not resolved in any current fault rupture model. To highlight the potential impact, we compared the results, including the co-seismic wavefield, fault rupture velocity and slip-induced stress with their counterparts from a reference cases that is designed to simulate a tectonic seismic event. Depending on the model configuration, we demonstrated that considering the pre-seismic fault poroelastic stress can lead to supershear rupture rather than sub-Rayleigh rupture; nucleation of the rupture can be delayed in time, and the rupture is accompanied with radically different seismic radiation pattern, stronger amplitude in the waves and stronger slip-induced stress changes. We have also provided a potential mechanism of asymmetric fault rupture, by considering the pre-seismic transverse simple shear deformation of the fault and carrying the resulting poroelastic stress with a sharp variation across the fault into the dynamic rupture modeling. Our computational model serves as a general tool for the modeling of fluid-induced seismic rupture and the findings presented here have important implications for distinguishing between induced and natural earthquakes.

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