Volume 124, Issue 4 p. 3724-3743
Research Article
Free Access

Effect of Dilatancy on the Transition From Aseismic to Seismic Slip Due to Fluid Injection in a Fault

F. Ciardo

F. Ciardo

Geo-Energy Laboratory, Gaznat chair on Geo-Energy, École Polytechnique Fédérale de Lausanne, EPFL-ENAC-IIC-GEL, Lausanne, Switzerland

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B. Lecampion

Corresponding Author

B. Lecampion

Geo-Energy Laboratory, Gaznat chair on Geo-Energy, École Polytechnique Fédérale de Lausanne, EPFL-ENAC-IIC-GEL, Lausanne, Switzerland

Correspondence to: B. Lecampion,

[email protected]

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First published: 11 March 2019
Citations: 26

Abstract

Aseismic crack growth upon activation of fault slip due to fluid injection may or may not lead to the nucleation of a dynamic rupture depending on in situ conditions, frictional properties of the fault, and the value of overpressure. In particular, a fault is coined as unstable if its residual frictional strength τr is lower than the in situ background shear stress τo. We study here how fault dilatancy associated with slip affect shear crack propagation due to fluid injection. We use a planar bidimensional model with frictional weakening and assume that fluid flow only takes place along the fault (impermeable rock/immature fault). Dilatancy induces an undrained pore-pressure drop locally strengthening the fault. We introduce an undrained residual fault shear strength urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0001 (function of dilatancy) and show theoretically that under the assumption of small-scale yielding, an otherwise unstable fault (τr < τo) is stabilized when urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0002 is larger than τo. We numerically solve the complete coupled hydromechanical problem and confirm this theoretical estimate. It is important to note that the undrained residual strength is fully activated only if residual friction is reached. Dilatancy stabilizes an otherwise unstable fault if the nucleation of an unabated dynamic rupture—without dilatancy—is affected by residual friction, which is the case for sufficiently large injection pressure. We also discuss the effect of fault permeability increase due to slip. Our numerical results show that permeability increases lead to faster aseismic growth but do not impact the stabilizing effect of dilatancy with respect to dynamic rupture.

Key Points

  • Dilatancy above a critical value cancels the nucleation of dynamic rupture for injection pressure sufficient to reach residual friction
  • Dilatancy delays the onset of a dynamic rupture (if occurring) and slows down aseismic crack growth
  • Fault permeability increase with slip speed-up aseismic crack growth but does not affect the critical stabilizing value of dilatancy

1 Introduction

Seismic and aseismic ruptures associated with anthropogenic fluid injection at depth have been observed in variety of settings (Bao & Eaton, 2016; Cornet et al., 1997; Ellsworth, 2013; Hamilton & Meehan, 1971; Healy et al., 1968; Scotti & Cornet, 1994; Shapiro et al., 2006; Skoumal et al., 2015) to cite a few. Industrial applications involved range from waste water disposal to the stimulation of enhanced geothermal systems and hydraulic fracturing.

Injection of fluid into the subsurface alters the local stress state. Preexisting fractures/faults or intact rock mass can fail due to the local reduction of effective stresses associated with pore pressure increase. Shear fractures can thus be activated and propagate along favorably oriented planes of weaknesses/faults. In some cases, the aseismic slip may lead to the nucleation of a dynamic rupture (seismic event). A necessary ingredient for such a transition from aseismic to seismic slip is the reduction of fault frictional strength with slip, that is, when the frictional resistance decreases faster than the elastic unloading associated with slip (Cornet, 2015).

The transition from the activation of aseismic slip to the nucleation of a seismic event due to fluid injection has been discussed theoretically (Garagash & Germanovich, 2012) and observed in situ (Cornet et al., 1997; Cornet, 2016; Guglielmini et al., 2015; Scotti & Cornet, 1994; Wei et al., 2015). We investigate here the effect of fault/fracture dilatancy associated with slip on the transition from aseismic crack propagation to seismic slip in the context of fluid injection. The physical mechanism of dilatancy associated with sliding over fault's asperities leads to a pore pressure drop under undrained conditions and thus to a fault strengthening denoted as dilatant hardening (Rudnicki & Chen, 1988; Segall & Rice, 1995; Segall et al., 2010). Strong dilatant behavior has been observed during aseismic crack propagation in scaled laboratory experiments by D. A. Lockner and Byerlee (1994) and Samuelson et al. (2009) and inferred during field experiment of the stimulation of geothermal reservoir (Batchelor & Stubs, 1985) suggesting that dilatancy possibly plays an important role on shear fracture propagation in some cases.

Although the concept of dilatant hardening associated with undrained conditions has been studied on saturated rock masses (Rice, 1975) as well as on frictional weakening faults loaded by tectonic strain (Rudnicki, 1979; Segall & Rice, 1995; Segall et al., 2010; Shibazaki, 2005), the quantification of its effect on the transition from aseismic to seismic slip propagation in association with fluid injection remains elusive.

The interplay between fluid injection, in situ conditions, frictional properties, and evolution of the hydraulic properties of fault presents a highly coupled problem leading to a wide range of possible behavior even under “simple” homogeneous in situ conditions (Garagash & Germanovich, 2012; Viesca & Rice, 2012; Zhang et al., 2005). In this paper, we extend previous work to account for fault dilatancy and quantify its impact on the propagation of a shear crack induced by a constant pressure injection. For simplicity, we reduce to a 2-D configuration and model the fault/joint as a planar thin strip where both shear slip and fluid flow are localized. We adopt a simple linear weakening slip-dependent friction law (Ida, 1972) combined with a nonassociated flow rule to account for dilatancy, assume isothermal conditions, and neglect poroealastic stress changes in the surrounding rock. We pay particular attention to the verification of our numerical solver and discuss the different type of crack propagation (aseismic/seismic) as a function of in situ and injection conditions. We also put in perspective our results at the light of theoretical arguments under the small-scale yielding approximation (Palmer & Rice, 1973; Rice, 1968).

2 Problem Formulation

We consider an infinite planar fault in an infinite homogeneous isotropic elastic medium (see Figure 1) under plane-strain conditions. We also assume that the host rock has a much lower permeability than the fault. As a result, the fluid flow only occurs along the fault plane from a source injection located at the middle of the fault in the 2-D space—more precisely a line source in the out-of-plane direction to satisfy plane-strain conditions. Furthermore, we assume a uniform initial in situ stress and pore pressure field prior to the start of the injection. Such a homogeneous model is obviously only valid for small fault slippage length compared to the background in situ gradient, but it allows to isolate and understand the different type of responses in a clearer way. Although different type of injection conditions, either away or directly on the fault, can be investigated, we restrict here to the case of a constant pressure injection from a point source directly in the fault.

Details are in the caption following the image
Model of frictional weakening dilatant fault and loading conditions. Zoom represents schematically the dilatant process that occur during shear fracture propagation.

2.1 Equilibrium, Activation, and Dilatancy of Slip-Weakening Fault

We consider the occurrence of a mode II shear crack of length 2a on the fault plane due to a constant pressure fluid injection. Initially, we assume the fault to be in static equilibrium with the uniform in situ stress state. Upon activation of slip due to the increase of fluid pressure along the fault, the bidimensional quasi-static elastic equilibrium can be written as the following integral equations relating fault tractions and displacement discontinuities in the local normal and tangential frame along the fault plane (using the convention of summation on repeated indices):
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0003(1)
where ti = σijnj is the current traction vector acting along the fault, urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0004 its value under the initial in situ stress, and dj denotes the vector of displacement discontinuities on the fault:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0005(2)

The quasi-static fundamental elastic displacement-discontinuity tensor Kij is known in closed form for a bidimensional infinite medium (see, e.g., Crouch & Starfield, 1983; Hills et al., 1996; Gebbia, 1891). It is worthwhile to recall that for a planar crack, the shear and normal boundary integration uncouples as Ksn = Kns = 0. As a result, shear slip does not induce any changes in the normal stress along the planar fault. However, if shear slip induces plastic dilatancy, the corresponding normal displacement discontinuity associated with dilatancy modify the normal stress along the fault. We note that the use of a quasi-static approach will obviously overshoot any finite dynamic rupture. Although a quasi-dynamic approximation (Rice, 1993) would provide more realistic results without the expense of a complete dynamic simulation, we restrict ourselves mostly to the nucleation of a dynamic rupture for which a quasi-static approximation is granted.

We adopt the convention of normal stresses positive in compression. The normal and shear components of the traction vector on the fault plane ti = (tn, ts) (in the local s − n reference system on Figure 1) will be noted as σ = −tn = −(niσijnj) and τ = siσijnj for the normal and shear component, respectively. We will also write the normal displacement discontinuities as dn = w (positive for opening) and the shear displacement discontinuities (slip) as ds = δ (positive in a clockwise rotation).

2.1.1 Activation and Plasticity

We assume that the fault obeys a Mohr-Coulomb yield criterion without cohesion, accounting for a slip weakening of friction. The yield criterion is
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0006(3)
where urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0007 is the friction coefficient (ϕ the corresponding friction angle), which is supposed to weaken linearly with slip δ, from a peak value fp to a residual value fr for slip larger than δr (see Figure 2, bottom left). σ = σ − p(x,t) > 0 is the local effective normal stress acting on the fault plane. We will write the initial in situ conditions (prior to fluid injection) as urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0008 and τo for the ambient effective normal stress and shear stress, respectively.
Details are in the caption following the image
The Mohr-Coulomb plot (top) illustrates the evolution of the yielding surface urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0009 with weakening of friction coefficient as well as the evolution of plastic displacement discontinuity vector urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0010 with slip after shear fracture activation due to fluid overpressure ΔP. The linear evolutions of friction coefficient urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0011 (left) and dilatancy angle urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0012 (right) with slip δ are displayed in the bottom plots.
The fault is activated when the injection overpressure is sufficient to reach the Mohr-Coulomb criterion at peak initial friction (see Figure 2, top), and shear slip starts to occur on the fault. We model the fault as rigid plastic and account for a possible dilatant behavior. Using a nonassociative flow rule, the rate of displacement discontinuities (denoted with a dot) derive from a plastic potential function of the corresponding effective tractions when the yield criterion is satisfied (i.e., ϝ = 0):
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0013(4)
The (scalar) plastic multiplier λ is either greater than zero as long as the local stress state satisfies the Mohr-Coulomb yield criterion 3 or equals to zero for non-yielded stress state (for which ϝ(τ,σ) < 0 and the fault is not activated). Plastic slip takes place along the yielding surface (see plastic strain vector in Figure 2, top). This can be summarized by the following set of conditions (Lubliner, 2005; Maier et al., 1993):
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0014(5)
In order for plastic flow to occur, the tractions on the fault must persist on the yield surface ϝ = 0, while upon unloading, plastic flow stops as soon as ϝ < 0. This requirement is often denoted as a consistency condition and written as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0015(6)

It allows to obtain the plastic multiplier λ (see, e.g., Lubliner, 2005; Simo & Hughes, 1997, for more details).

We use a nonassociated Mohr-Coulomb criterion with a dilatancy angle ψ decreasing with accumulated slip δ. We write the plastic potential as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0016(7)
As a result, the rate of slip and opening displacement discontinuity are related to each other as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0017(8)
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0018(9)
We assume that the dilatancy coefficient (tangent of the dilatancy angle ψ) decreases linearly with slip from an initial peak value urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0019 to zero: The fault is assumed to reach a critical state (where the dilatancy angle is zero) over the same slipping distance δr at which it reaches residual friction (see Figure 2, bottom right). Such a choice is consistent with experimental observations that a critical state (where no change of volume occur) is reached after sufficient plastic deformation for most rocks and granular material. Integration of 8 and 9 provides the following quadratic evolution of fracture width due to the dilatancy induced by slip:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0020(10)
where Δw denotes here the maximal/final dilatant opening at residual friction:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0021(11)

A similar law for the dilatancy evolution with frictional slip has been proposed by Rudnicki and Chen (1988) to model uplift in sliding over asperities of a long homogeneous slab subjected to shear and normal mechanical loadings. Initially, the fault is supposed to have a constant uniform “hydraulic aperture” ωo which can be considered as the fault gouge thickness in such a model. The ratio between the maximal dilatant increment of fracture width Δw and this initial aperture ωo defines a dilatant strain urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0022 which can be related to the maximum change of fault porosity due to slip. Such a quantity will directly appear in the hydromechanical coupling.

Dilatancy is known to be function of effective normal stress—with lower dilatancy angle typically measured under larger confinement (Barton et al., 1985; Matsuki et al., 2010). Measured value of the peak dilatancy angle (at small slip) ranges from ∼40° at 5 MPa of confinement to ∼6° at 30 MPa for Inada granite (Matsuki et al., 2010), leading to values of ϵd in a range of 10−3–10−2. Laboratory experiments on quartz fault gouge (Samuelson et al., 2009) provide value of ϵd in the range 10−4–10−3 at effective confinement up to 20 MPa, values which appears of similar order than the one measured at larger confinement (Marone et al., 1990).

In what follows, for sake of simplicity, we do not explicitly account for the complete details of the dependence of dilatancy with normal effective stresses. The peak dilatancy angle can be implicitly taken to be a function of the level of in situ confinement prior to injection. Moreover, we acknowledge that a relatively large range of possible value of the dilatant strain ϵd may exist from 10−4 to 10−2.

2.1.2 Slip Weakening and Nucleation Length Scale

Following Garagash and Germanovich (2012) and Uenishi and Rice (2003), we introduce a characteristic nucleation patch length scale aw
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0023(12)
to scale the crack length. This characteristic nucleation length scale is obtained by normalizing the slip δ and shear stress τ in the elasticity equation  1 by urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0024 and τp = fp urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0025, respectively. δw denotes the amount of slip at which the friction coefficient goes to zero if an unlimited linear slip-weakening friction law is considered (see Figure 2, bottom left). Typically, δw is of the order the fault's asperities and thus ranges between 0.1 to 10 mm. τp = fp urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0026 defines the peak frictional strength at ambient conditions, its difference from the ambient shear stress τo quantify the fault stress criticality. Such a peak fault shear strength can vary widely with depth and fault orientation as well as hydrogeological conditions (normally pressurized versus overpressured formations) and thus can range between a fraction to hundred of megapascals. We thus deduce that the range of characteristic patch length scale aw (e.g., for a crystalline rock with E∼60 GPa) can approximately ranges between tens of centimeters to tens of meters depending on geological conditions.

2.2 Fluid Flow

Under the assumption of much smaller rock permeability compared to the longitudinal fault permeability, the fluid flow is confined within the fault zone. This case corresponds to an immature fault with little accumulated slip for which the extent of the damage zones around the fault core remains limited. For active and mature fault, the permeability structure around the fault cannot be neglected. Much larger permeabilities have indeed been measured in the damage zone (that can have decameters thickness) of active mature fault compared to the fault gouge unit (Lockner et al., 1999). Here we restrict to the case of an immature/young fault for which the flow is confined in the gouge. Such a hypothesis could also be valid for inactive mature fault that would have exhibited a plugging of their damage zone permeability (e.g., via long term thermo-hydro-chemical effects).

The mass balance equation width-averaged across the fault hydraulic aperture wh of the gouge layer thus reduces to
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0027(13)
where ρ is the fluid density and V is the averaged fluid velocity. The fault hydraulic aperture wh = ωo + w(δ) is the sum of its initial value ωo and the additional dilatant aperture function of slip (see equation 10).
By combining fluid compressibility (taken as liquid water) and pore compressibility of the fault gouge in an unique parameter β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0028, the width-averaged fluid mass conservation 13 along the fault (x axis) reduces to
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0029(14)
where q = whV is the unidimensional local fluid flux given by Darcy's law:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0030(15)
with μ the fluid viscosity urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0031 and kf(δ) the fault intrinsic permeability urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0032. The hydraulic diffusivity of the fault α urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0033 is defined as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0034(16)

In particular, the location of the fluid/pressure front evolves as urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0035 for such type of diffusion problem (Carslaw & Jaeger, 1959).

In conjunction with the increase of the fault aperture with dilatant slip 10, the longitudinal fault permeability may also evolve with shear slip. A number of different models have been proposed in the literature for the evolution of permeability with slip, from using the cubic law for the fault transmissivity (product of permeability urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0036 and hydraulic aperture wh) to an exponential dependence of permeability with normal stress or Cozeny-Karman type relations. Here we first make the assumption of a constant fault permeability urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0037 before relaxing such an approximation in section 6 in order to properly gauge its effect.

It is important to note that even in the absence of permeability evolution, the changes of hydraulic aperture induced by dilatancy still impact the fluid flow in a nontrivial and nonlinear way. This is notably due to the sink term urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0038 associated with slip-induced dilatancy. Fluid flow cannot be uncoupled from mechanical equilibrium and fault slip, contrary to the case of zero dilatancy (Garagash & Germanovich, 2012), where for a constant pressure injection ΔP, the pore pressure on the fault plane is simply given by urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0039. No simple analytical solution does exist for this complete nonlinear hydromechanical coupling.

The effect of slip-induced dilatancy leads to a pore-pressure drop under undrained conditions (denoted here as Δpu). At large slip rate, the change of hydraulic width from its initial value up to its maximum value ωo + Δw 11 will be sudden. In such an undrained limit, the fluid has no time to flow, and the associated pore pressure change can be directly obtained from mass conservation 14:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0040(17)

This undrained pore-pressure drop will be localized at the crack tips, where frictional slip weakening occurs. From the previously discussed range of the dilatant strain ϵd ∈ [10−4 − 10−2], for a compressibility coefficient β between the one of liquid water and usual pore compressibility (β ∈ [5 − 100] 10−10Pa−1), we obtain a range of values [0.01 − 20] MPa for such an undrained pore-pressure drop. The previous estimate corresponds to the maximum possible amount of undrained pore-pressure drop (sudden slip from zero to δr). A restrengthening of the fault is thus expected as the effective normal stress increase locally as a result of this undrained pore-pressure drop. Similar dilatant hardening is typically observed in fluid-saturated porous medium subject to undrained loading (Rice, 1975; Rudnicki, 1979). It is important to underline that such restrengthening effect is less pronounced for “mature" faults, for which pore fluid diffusion normal to fault plane (across the permeable units of damaged zone) may prevail against fluid diffusion along the fault gouge unit.

2.3 Initial and Injection Conditions

Initially, the (peak) fault strength urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0041 at ambient condition is everywhere larger than the in situ shear traction on the fault τo. In other words, the fault is initially stable (i.e., ϝ(τo, urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0042) < 0) and locked before the start of fluid injection. We consider here the case of a constant overpressure ΔP at the injection point:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0043(18)
We assume that the choice of the injection overpressure ΔP is such that the minimum principal effective stress urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0044 always remain compressive (positive) such that no hydraulic fracture type failure occurs: that is, urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0045. A constraint often enforced in practice for large-scale injection but also sometimes during hydraulic stimulation of geothermal reservoirs. We investigate here the activation of a shear crack that would occur if the overpressure ΔP at the injection point is sufficient to lower the effective normal stress and reach Mohr-Coulomb failure. Such a minimum overpressure ΔP for activation is directly related to the fault criticality:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0046(19)

The ratio τo/τp represents the stress criticality of the fault at ambient condition (quantifying how far the fault is from failure). For a critically stressed fault (τo/τp∼1), slip is activated for small overpressure. On the other hand, a fault whose initial uniform stress state is much lower than its peak frictional strength (τo/τp ≪ 1) requires a larger overpressure to activate a shear crack and is sometimes referred to as a marginally pressurized fault.

3 Activation and Transition Between Aseismic and Seismic Slip

3.1 Case of a Nondilatant Fault

We first briefly recall the results obtained for the case of a nondilatant fault by Garagash and Germanovich (2012) using the same linear frictional weakening model. This summary is required in order to properly put in perspective the effect of a dilatant fault behavior.

After activation of aseismic slip, there exist two ultimate fault stability behaviors depending on the relative value of the residual strength (defined at ambient condition) τr = fr urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0048 compared to the in situ background shear stress τo. Notably, if the residual frictional strength τr exceeds the ambient shear stress τo, the fault is ultimately stable. On the other hand, for a residual frictional strength τr lower than τo, the fault is unstable. Figure 3 summarizes the different behaviors, as a function of the dimensionless fluid overpressure ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0049, stress criticality τo/τp, and relative value of τr with respect to the initial shear stress τo. Region 1 on Figure 3 corresponds to the trivial case of an injection without activation of any slip.

Details are in the caption following the image
Phase diagram of Garagash and Germanovich (2012) that describes the different regimes of propagation for a nondilatant fault, as a function of the dimensionless fluid overpressure ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0047, and stress criticality τo/τp. Region 1 corresponds to the trivial case of an injection without fault reactivation. Regions 4a–4c correspond to the unstable fault case for which an unabated dynamic rupture occurs as the residual shear strength (defined at ambient conditions) is lower than the in situ shear stress (τr < τo). Regions 2 and 3 corresponds to the case of ultimately stable faults (τr > τo) for which most of the crack growth is aseismic although transient seismic slip may occur (region 2).

For an ultimately stable fault (for which the residual strength τr is larger than the ambient shear stress τo), it can be shown that for an overpressure sufficient to activate slip, at large time/large crack length, the shear crack grows quasi-statically (aseismically) as long as the fluid injection continues (regions 2 and 3 on Figure 3). However, because of the weakening of its frictional properties, an ultimately stable fault may host an episode of seismic slip followed by an arrest (region 2 on Figure 3). Such a “seismic event” depends on both the stress criticality and the amount of overpressure. For a moderate overpressure (sufficient to activate slip), the shear crack first lags behind the fluid diffusion front and, due to the interplay between fluid pressurization and frictional weakening, a dynamic event nucleates and grows until it catches up the fluid pressure diffusion front ahead of which the overpressure is minimal. The subsequent propagation is then a-seismic and tracks the fluid front as long as injection continues. In other words, depending on the value of fluid overpressure applied in the middle of the fault, the local accumulation of slip during the (aseismic) crack propagation varies. If the fluid overpressure induces a large local slip accumulation during the aseismic propagation (such that it exceeds the residual slip δr), the fault never exhibits a dynamic event (strictly aseismic propagation, region 3); otherwise, a nucleation of a dynamic rupture episode occurs (region 2 in Figure 3) .

The situation is different for unstable fault (τr < τo)—regions 4a–4c on Figure 3. It can be proved that an unabated dynamic rupture will always occur when τr < τo. The nucleation length (and time of nucleation) depends again on stress criticality, the value of overpressure, and, in some cases (region 4c), on the value of the residual friction fr. For criticality stress fault (region 4a, τo/τp∼1), the nucleation patch size ac is independent of the overpressure ac = 0.579 aw (Garagash & Germanovich, 2012). In these cases, even a small overpressure is sufficient to nucleate a dynamic rupture and the fluid front lies well within the crack when the unabated instability occurs. For unstable but marginally pressurized fault (moderate stress criticality), subjected to a moderate value of overpressure, a transient seismic event may occur and then arrest when the crack front catches up with the fluid front. However, here (region 4b in Figure 3) a renucleation always occurs (affected by residual friction) leading then to an unabated dynamic rupture. For larger of overpressure, a single nucleation of an unabated dynamic rupture occur (region 4c).

3.2 Effect of Dilatancy

3.2.1 Undrained Fault Response

At high slip rate, the undrained response associated with dilatancy causes a pore-pressure drop 17. Scaling the fluid pressure by effective normal in situ stress, we express the undrained response via the following dimensionless undrained pressure change
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0050(20)

The dimensionless ratio urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0051 quantifies the effect of dilatancy in terms of pore pressure drop under undrained conditions with respect to the initial confining stress. For a realistic range of effective in situ normal stress of [1–200] MPa, whose extremes may represent the case of normally pressurized and overpressurized fault located approximately between 0.1 and 5 km below the Earth's surface, and for the previously reported range for undrained pore-pressure drop ϵd/β, the dimensionless dilatancy parameter urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0052 ranges between 0.01 (deeper conditions/low dilatancy) and 20 (shallow conditions/large dilatancy).

Dilatancy is mobilized in the frictional weakening zone. Moreover, its impact on pore pressure is modulated by the slip rate ∂δ/∂t (see equation 14). In proximity of a dynamic event when the slip rate increases rapidly, the undrained pore pressure drop leads to a local strengthening at the crack tip (dilatant hardening). In the case where the slip rate and crack velocity is larger than the fluid flux, the undrained dilatant pore-pressure drops will be at its maximum 17 and will persist inside the crack away from the crack tip. We can thus quantify the associated strengthening by adding its effect to the fault residual strength—defining an undrained residual shear strength urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0053 as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0054(21)

From the ranges of value previously discussed, we see that the undrained shear strength can be from 1.01 to 2 times larger than the drained residual strength.

On the other hand, the shear-induced dilation impact (via the nonassociated flow rule 9) the distribution of normal stress along the fault through the effect of corresponding opening displacement discontinuity in the elasticity equation 1. Inside the crack, the opening of the fault leads to an increase of compressive normal stress, whereas ahead the crack tips, it induces tensile stresses therefore reducing the fault frictional strength. There is thus an interplay between the nonlocal stress-induced perturbation due to fault opening and dilatant hardening. The tensile stresses ahead of the tip have however a lower magnitude than the undrained pore-pressure drop. For instance, if we suppose that the weakening region is small compared to the whole crack size (small-scale yielding conditions), the mechanical opening is uniform and equal to Δw along the whole crack. We can thus estimate the tensile normal stress ahead of the crack front using the solution for an edge dislocation (e.g., Hills et al., 1996) of intensity Δw. Scaling the distance urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0055 from the dislocation by the nucleation length scale aw, we have
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0056(22)

The corresponding stress intensity for such a singular field is thus EΔw/(4πaw). For a maximum increment of dilatant width Δw of the order of few millimeters and corresponding estimate of the nucleation patch size aw gives an order of magnitude of about ∼1 MPa or less for such stress perturbation. Taking the ratio of such stress intensity with the estimate of the undrained pore-pressure drop 17, after rearranging, one obtain Eβωo/(4πaw) which will be always smaller than one as EβO(10) and ωo/awO(10−2). We therefore conclude that the mechanical effect of dilatancy induced tensile stresses ahead of the crack tip is lower than the undrained pore pressure change Δpu. The dilatant hardening effect dominates. This is confirmed by our fully coupled numerical simulations (see section 5).

3.2.2 Small-Scale Yielding and Stability Condition

Following the work of Garagash and Germanovich (2012), we extend their ultimate stability condition to account for the effect of dilatant hardening. This stability condition can be obtained under the assumption of small-scale yielding which holds when the shear crack of half-length a is sufficiently larger than the characteristic length scale aw such that all the frictional weakening occurs in a small zone near the crack tip. Such a localization of the frictional weakening in a small zone near the crack tip can be observed on our numerical results (see section 5.1). Under such assumption, the fracture energy Gc (Palmer & Rice, 1973; Rice, 1968) for the linear frictional weakening model can be estimated as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0057(23)
under the assumption that the effective normal stress σ(a) is constant within the weakening region. The condition for quasi-static crack growth of such a shear crack reduces to the classical linear elastic fracture mechanics criteria. The driving force for propagation G (the energy release rate) must equal Gc for quasi-static growth to occur. A criteria which for such a shear crack reduce to
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0058(24)
where KII is the stress intensity factor for the given loading and crack size.
As all the cracks—besides the small weakening zone at the tips—are at residual frictional strength, the stress intensity factor can be obtained by superposition of the effect of the loading of the crack by (i) the residual frictional strength at ambient condition urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0059 minus the far-field in situ shear stress τo (which are both uniform along the crack) and (ii) the effect of the overpressure due to the fluid injection on the decrease of shear strength frΔp(x,t). The stress intensity factor for such a configuration is thus given as (Rice, 1968; Tada et al., 2000)
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0060(25)
On the contrary to the nondilatant case, the exact solution for the overpressure Δp(x,t) evolution along the fault is not known analytically. It is the complete solution of the coupled hydromechanical problem in the dilatant case. However, in order to obtain an ultimate stability condition for large crack length, it can be approximated as follow. If the shear crack a is much larger than both the slipping patch length scale aw (which is required for the small-scale yielding approximation to be valid) and the diffusion length scale urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0061, the overpressure can be approximated as the sum of a point source term of intensity ΔP for the effect of the injection (as urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0062) and the response of the undrained pore-pressure drop. Moreover, if the crack velocity is much larger than the fluid velocity—which would be true in all cases if the crack accelerates—the undrained pore-pressure drop can be assumed to remain constant and equal to Δpu (equation 17) over the entire crack. Under those conditions, the stress intensity factor reduces to
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0063(26)
where the undrained shear strength previously introduced appear. This expression is strictly similar to the one derived in Garagash and Germanovich (2012) pending the replacement of the residual shear strength τr by its undrained counterpart urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0064 21. As previously anticipated, the effect of dilatancy is akin to an increase of the residual shear strength.
The reasoning of Garagash and Germanovich (2012) for the ultimate stability can thus be directly transposed to the dilatant case. In the limit of infinitely large crack a, one directly see that the stress intensity factor tends to either + if urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0065 and − if urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0066. In other words, if the initial shear stress τo is larger than the undrained residual strength, the fault is ultimately unstable as the stress intensity factor diverges for large crack length: The nucleation of a dynamic rupture will thus always appear. On the other hand, the fault is ultimately stable when urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0067. We therefore see that as the undrained residual shear strength urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0068 is larger than τr, sufficient dilatancy may stabilize a fault that otherwise would be unstable. The minimal/critical amount of dilatancy urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0069 required for such a stabilization to occur is simply given as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0070(27)

It is interesting to note that it directly depends on the residual stress criticality τo/τr and the in situ effective normal stress.

It is important to note that—obviously—in the ultimately stable case ( urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0071), the stress intensity factor does not tend to − in reality as the propagation can only occur for G = Gc. Upon continuous fluid injection, a stable quasi-static growth will occur and will be modulated by the fluid diffusion: That is, the crack will decelerate for large crack length at constant injection. It is actually possible to devise an approximated solution for such a quasi-static growth by hypothesizing that the crack length evolves as a factor of the fluid front: urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0072. An approximation of the pore-pressure evolution accounting for the undrained pore-pressure drop at the tip can be obtained and used in equations 24 and 25 to obtain an estimate of γ. Such an approximated solution is detailed and compared to our numerical results in Supporting Information S1. Such a refined (but still largely approximated) solution for the pore-pressure evolution gives the exact same limit for the stability condition at large crack length as well as critical dilatancy than the simpler profile postulated previously.

To conclude, before moving to the complete numerical solution of the problem, a word of caution is required with respect to the stability condition urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0073. Such a stability condition holds on the premise that most of the crack is at residual friction pending a small weakening zone (small-scale yielding approximation). It is valid for sufficiently large crack length compared to aw and peak slip larger than δr. Only under this assumption, the maximum undrained pore pressure 20 can be achieved. If a dynamic rupture nucleates for slip smaller than the residual δr, the small-scale yielding is invalid: The undrained pore-pressure response will not be fully activated and thus not sufficient to quench the nucleation of a dynamic rupture. However, upon reaching larger crack length (and thus residual friction), the complete undrained pore pressure will ultimately kicks in such that a dynamic rupture should arrest if the ultimate undrained small-scale yielding stability condition ( urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0074) is satisfied.

4 Numerical Scheme Description

Although approximation of the complete problem has allowed to highlight the stabilizing effect of dilatancy on the nucleation of a dynamic rupture associated with fluid injection, a full numerical solution is needed to investigate the complete parametric space and test the concept of a critical dilatancy.

The complete problem described in section 2 is fully coupled due to the dilatant fault behavior as well as nonlinear due to the evolution of the fault hydraulic width (even if the fault permeability remains constant). It also involves the tracking of the moving crack tips. The shear crack evolves in space and time along the fault, paced by pore pressure evolution. Equation 1, which links tractions ti on the fault plane with displacement discontinuities dj, evolves in time due to the moving crack domain Γ. The developed numerical scheme solves this coupled problem by determining simultaneously the plastic multipliers λ in the “active” zone of the domain (i.e., where ϝ(τ, urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0075) = 0) through equations (1-89) and the increment of pore pressure Δp along the whole fault (through equation 14 and Darcy's law 15). We then recompute the increment of tractions (due to both increment of slip and the associated increment of hydraulic width) along the rest of the domain via the nonlocal elasticity equation 1. We have chosen a backward-Euler (implicit) time integration scheme for stability and robustness. A choice that stems from the restrictive Courant-Friederichs-Lewy (CFL) condition on the time step for diffusion problem (e.g., Quarteroni et al., 2000)—which even deteriorates for strong nonlinear variation of permeability similar to the hydraulic fracturing case (Lecampion et al., 2018).

We discretize the elasticity equations using the displacement discontinuity method (Bonnet, 1995; Crouch & Starfield, 1983) with piecewise linear element (Crawford & Curran, 1982). Because of the singular nature of the elastic kernel, the integral equation is collocated at points inside the displacement discontinuity segments. Knowing the effect in terms of traction of a single piecewise linear displacement discontinuity, the problem reduces to the one of determining the distribution of displacement discontinuities that generates tractions along the fault such that equilibrium with initial tractions and the failure criterion is satisfied (Crouch & Starfield, 1983). The fluid flow equation combining fluid mass conservation and Darcy's law is discretized via a vertex-centered finite volume scheme. The fluid pore pressure is assumed to vary continuously and linearly between element vertex.

In all the simulations reported here, the fault is discretized by N straight equal-sized elements (of size h)—with a total mesh extent of 20 × aw. We therefore have 2Na unknown shear displacement discontinuities (more precisely the plastic multiplier) for the Na active elements (and N + 1 unknowns for pore pressure for all the element in the grid). After discretization, we obtain a nonlinear system of size 2Na + N + 1, whose unknowns are composed of the plastic multipliers λ (which are linked to increment of slip Δδ through equation 8) in the Na active-yielded elements and increment of fluid pressure Δp at every nodes of the grid (N + 1 unknowns). The size of such a nonlinear system evolves with the shear crack propagation as more elements yield mechanically. The nonlinearities of such a system are related to shear-induced dilatancy and frictional weakening. For a given set of active elements, we use a fixed point iterative scheme to solve for this nonlinear system—ending iterations when subsequent estimates of both the increment of slip and pore pressure are within a relative tolerance of 10−6.

The yielding/active set of element is then rechecked using the Mohr-Coulomb criteria. It is worth noting that an element is at failure when the Mohr-Coulomb criteria is reached for both collocation points in the piecewise linear displacement discontinuity element.

Over one time step, such a fully implicit algorithm thus solves the coupled problem by means of two nested loops. The outer loop identifies the shear crack position by enforcing implicitly the friction weakening Mohr-Coulomb criterion 3 along the whole fault. The inner loop solves the aforementioned coupled nonlinear hydromechanical system of equations for a trial set of active/yielded elements.

For numerical efficiency, the time step is adjusted based on the current crack velocity vn, which is calculated via finite difference:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0076(28)
where h is the element size and ζ is a user-defined constant parameter. This allows to better capture the response of the system during high slip rate and increase time-step size during slow a-seismic growth. However, a constraint is required to avoid a too small time step that would necessarily occur when the shear crack is approaching a dynamic instability, for which the slip rate and crack velocity diverge. Notably, in our simulation, if the variation is such that Δtn + 1 < 0.8Δtn, we set Δtn + 1 = 0.8Δtn. Similarly, time step should remain reasonable in order to avoid sampling the pore pressure evolution too coarsely. In our simulations, if Δtn + 1 > 3Δtn, then the time step change is constrained to Δtn + 1 = 3Δtn, and the initial time step is taken as a small fraction of the characteristic diffusion time scale.

Thanks to the semianalytical results of Garagash and Germanovich (2012) for the nondilatant case, we have performed a thorough benchmarking of this numerical solver. Some of these verifications are described in Supporting Information S1 together with a mesh convergence study. Notably, our mesh convergence study have shown that the mesh size h must be such that at least 15 elements cover the characteristic length scale aw to obtain accurate results (i.e., h ≤ aw/15). All the simulations reported herein have been performed with h = aw/25.

4.1 Characteristic Scales for Dimensionless Governing Problem

By introducing properly chosen characteristic scales to normalize the governing equations, relevant physical processes can be systematically investigated. As already stated in section 2.1.2, we follow the scaling of Uenishi and Rice (2003) and Garagash and Germanovich (2012) in order to normalize the elasticity equation 1 and friction weakening Mohr-Coulomb criterion 3. We thus scale the slip δ and the tractions ti by the slip-weakening scale δw and the peak fault strength urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0077, respectively. By doing so, one can identify the nucleation patch length scale aw (see equation 12), which is used to scale all the spatial variables: half crack length a and longitudinal spatial coordinate x. We scale the time t by the characteristic fluid diffusion time scale urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0078. The characteristic scale for the fluid overpressure is taken as the in situ effective normal stress urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0079. Upon scaling the governing equations with the previous characteristic scales, the normalized solution is given by δ/δw, τ/τp, urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0080, urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0081, and 2a/aw and is function of only the following four dimensionless parameters (besides space and time):
  1. normalized injection overpressure urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0082;
  2. dimensionless frictional weakening ratio fr/fp;
  3. fault stress criticality τo/τp at in situ conditions (prior injection); and
  4. dimensionless dilatancy coefficient urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0083.

All the numerical results of the following sections have been obtained and will be presented in dimensionless form. For all simulations, we fix the dimensionless frictional weakening ratio to fr/fp = 0.6 and explore the effect of the remaining dimensionless parameters: urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0084 and urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0085.

5 Dilatant Hardening Effect on a Fault Characterized by Constant Permeability

5.1 Case of Unstable Fault Without Dilatancy τo > τr

We first investigate numerically the effect of dilatancy on otherwise unstable fault, that is, for which the in situ shear stress is larger than the residual shear strength and the nucleation of a runaway dynamic rupture is always expected in the absence of dilatancy. We display the time evolution of the different variables (crack length and maximum slip) using the square root of dimensionless time urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0086 as the x axis. Such a choice stems from the fact than the injection is diffusion controlled and urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0087 is directly the ratio between the diffusion front over the nucleation length scale.

Figure 4 (top left and top right) displays the time evolution of half-crack length and peak slip for different values of the dimensionless dilatancy coefficient ϵd/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0088) for the case of a rather critically stressed fault τo/τp = 0.75 and a moderate injection overpressure ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0089 = 0.5. The theoretical estimate of the critical dilatancy sufficient 27 to stabilize the fault is ϵd,c/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0090) = 0.25 in that particular case. We clearly observe that an increase of dilatancy delays the occurrence of a dynamic rupture (highlighted by a red dot on these plots) for values below the critical dilatancy. However, for values of dilatancy equal or larger than the critical one, no nucleation is observed: The propagation is always aseismic. This can be better observed on the time evolution of crack velocity (Figure 4, bottom), where we see how dilatancy larger than the critical value kills the acceleration preceding the nucleation of a dynamic rupture.

Details are in the caption following the image
Time evolution of normalized half crack length a/aw (top left) and normalized peak slip δ/δw at the middle of the fault (top right), that is, at x = 0, for an otherwise unstable fault (τo/τp = 0.75), subjected to moderate overpressure ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0091 = 0.5. The friction weakening ratio considered here is fr/fp = 0.6. The corresponding time evolution of normalized crack velocity vaw/α is showed in the plot at the bottom. We vary the dimensionless dilatancy parameter urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0092 below and above the critical stabilizing value—which is ϵd,c/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0093) = 0.25 in this particular case. Red dots point the onset of unabated dynamic ruptures (color online).

Figure 5 displays the profile along the fault of the fluid overpressure, friction coefficient, shear slip, effective normal stress, and shear fault strength at different times for two distinct values of dilatancy, below and above the critical value. For insufficient dilatancy (left panel on Figure 5), although an undrained pore-pressure drop can be seen in the weakening zone close to the crack tip, it is not strong enough to stabilize the fault, and the last profiles reported in these plots are right before the nucleation of an unabated dynamic rupture. For this particular case without any dilatancy, the nucleation occurs early and is not influenced by residual friction. We see that a dilatancy lower than the critical value delays the occurrence of nucleation which is now occurring when a significant part of the crack is at residual friction. For a value of dilatancy larger than the critical one (right panel on Figure 5), the crack growth is always quasi-static. The undrained pore-pressure drops is now well developed, and its minimum reaches the critical value Δpu/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0096 = −ϵd,c/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0097) = −0.25 locally at the tip. The local fault restrengthening can be observed on the corresponding effective normal stress profiles as well as on the corresponding shear strength versus slip results of our simulation (Figure 5). Under undrained conditions, near the crack tips dilatancy leads to a slip hardening phase before the onset of weakening, a response often observed in healed fault rocks. Brantut and Viesca (2015) used a nonmonotonic, piecewise linear slip-dependent strength constitutive law (accounting for a strengthening phase followed by a weakening phase) to investigate earthquake nucleation in healed rocks. They solved semianalytically an uncoupled problem for which stress perturbation is obtained through a mechanical loading whose time and space dependency is known analytically. They notably show that the strengthening phase that occur before the slip-weakening phase considerably increases the critical nucleation size. This is in line with our numerical results for increasing values of dilatancy parameter (see Figure 5e and the crack length at nucleation time for increasing values of dilatancy in Figure 4). We can also observe on these profiles that the weakening zone at the crack tip is small such that the stability condition derived previously under the assumption of small-scale yielding is valid.

Details are in the caption following the image
Spatial profile of (a) dimensionless pore pressure, (b) friction coefficient, (c) slip, and (d) effective normal stress for a critically stressed dilatant fault (τo/τp = 0.75), subjected to a moderate overpressure Δp/σo = 0.5, at different (normalized) time snapshots. Subfigures (e) show the evolution of normalized shear strength with slip, at the same time snapshots. Results for two dimensionless dilatancy parameters are reported: (left) ultimately unstable fault for which ϵd/β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0094 is lower than the critical stabilizing value for that particular set of parameter (ϵd,c/β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0095 = 0.25) and (right) ultimately stable for a dimensionless dilatancy above the stabilizing value. Red curves (color online) denote the numerical results at nucleation time for the unstable case (left).

These simulations confirm the fact that dilatancy can stabilize an otherwise unstable fault if it is above the critical theoretical dilatancy previously derived in section 3.2.2. It is worth noting that this would have been difficult to demonstrate solely numerically even with very long simulations.

5.1.1 Effect of the Injection Overpressure ΔP

For the same value of stress criticality τo/τp = 0.75, placing ourselves at critical dilatancy (ϵd,c/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0098) = 0.25 in that case, we test the influence of the amount of overpressure. Figure 6 displays the time evolution of crack length and peak slip for different amount of injection overpressure ΔP. As expected, the larger the injection overpressure, the faster the crack grows and the propagation remains quasi-static (aseismic). However, an interesting situation occurs for lower value of overpressures (here ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0099 = 0.4 and lower) where the nucleation of an unrestricted dynamic rupture is observed. This somehow invalidates the existence of an universal value of stabilizing dilatancy independent of the overpressure. However, we can clearly see that, for these low overpressure cases, the peak slip at the instant of nucleation is lower than the residual slip. In other words, the whole crack is weakening and has not yet reached residual friction. As a result, the undrained pore pressure is not fully developed and not sufficient to stabilize the fault via dilatant hardening. In these cases, the small-scale yielding assumption (small weakening zones at the crack tip) is invalid, and the stability condition previously derived for large crack length compared to the characteristic nucleation patch size does not hold. It is worth noting that the nucleation of an unrestricted dynamic rupture is a consequence of the assumption of quasi-static elastic equilibrium. The shear crack velocity at nucleation time diverges instantaneously. Such an unbounded slip rate at nucleation will disappears if inertial terms are accounted for (full elastodynamic or quasi-dynamic formulation): Energy dissipation via radiation of elastic waves always ensures a finite crack velocity. In Figure S13, we show that using a quasi-dynamic formulation (with a rather large damping for illustrative purpose), the slip rate remains bounded, and the crack propagation eventually slows down at later time compared to the quasi-static formulation where a divergence of the slip rate occurs at nucleation.

Details are in the caption following the image
Effect of dimensionless overpressure ΔP/σo on a critically stressed dilatant fault (τo/τp = 0.75), in terms of time evolution of dimensionless half crack length a/aw and dimensionless peak slip δ/δ|x = 0. The dimensionless dilatancy parameter equals the critical value for such configuration: ϵd,c/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0100) = 0.25.

In summary, if residual friction is reached during a-seismic crack propagation, the dilatant hardening effect stabilizes the fault for urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0101, and the shear crack always propagates quasi-statically. This always occurs for sufficiently large values of overpressure ΔP, which promotes larger initial aseismic slip rate thus maximizing the effect of dilatant hardening (i.e., sink term associated with dilatancy in the fluid mass conservation 14 is proportional to slip rate— urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0102). On the contrary, a lower overpressure significantly slows down the initial aseismic crack growth, and the beneficial effect of dilatancy cannot develop sufficiently to avoid the nucleation of a dynamic rupture even when urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0106. If inertia effects are included during crack acceleration (fully dynamic or quasi-dynamic elastic equilibrium), the slip rate will remain bounded, and the full effect of dilatant hardening would eventually kick in for sufficient crack length (larger than ac)—therefore leading to an arrest of the dynamic rupture due to sufficient dilatancy ( urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0107). Full elastodynamic simulations would be needed to confirm that the dynamic rupture would indeed arrest upon full activation of dilatant hardening under those conditions of low injection overpressure and large dilatancy.

For a given set of parameters, the exact minimum value of overpressure required to fully stabilize the fault cannot be estimated analytically but can be estimated numerically via a series of simulations varying the injection overpressure. Figure 7 displays such an estimation for different stress criticality τo/τp below or equal to the undrained residual strength—that is, the domain where dilatant hardening can stabilize an otherwise unstable fault. More precisely, Figure 7 displays both the maximum overpressure for which a nucleation of finite dynamic event occur and the minimum overpressure for which the propagation is solely aseismic (the fault is stabilized). A linear increase of the required overpressure as the stress criticality decreases can be clearly observed. This can again be understood as a larger driving force is required to reach residual friction for lower stress criticality.

Details are in the caption following the image
Numerical estimation of the minimum amount of overpressure required to activate the full benefit of dilatant hardening and stabilize an otherwise unstable fault (τr < τo) for different stress criticality between the ultimately stable limit (τr/τp = fr/fp = 0.6 in that case) and the undrained dilatant residual strength ( urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0103 in that case). The black filled circle denotes the maximum value of overpressure below which a finite dynamic event always nucleate (for overpressure above the slip activation limit), while the empty circle corresponds to the minimum normalized overpressure required to stabilize such a fault (aseismic slip only for larger overpressure). The minimum overpressure required for slip activation ( urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0104) is also displayed as empty square/continuous line. Stress criticality τo/τp larger than urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0105 always result in the nucleation of unrestricted dynamic rupture for any value of overpressure larger than the activation limit.

5.2 Case of an Ultimately Stable Fault Even Without Dilatancy (τo < τr)

We now turn to the case of ultimately stable fault (τo < τr), where only a transient seismic episode occurs for moderate overpressure (region 2 of Figure 3), while crack growth is strictly aseismic for large overpressure (region 3 of Figure 3).

For a configuration representative of region 2 in Figure 3 (τo/τp = 0.55,ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0111 = 0.5), a transient seismic episode occurs for a low accumulated slip: The residual friction is not yet reached anywhere in the crack. Such a seismic event is directly linked with the crack “catching” up the fluid front in association with frictional weakening. Figure 8 displays the crack evolution and peak slip for such configuration for different values of dilatancy. Increasing dilatancy slows down the initial quasi-static crack growth and thus delays the nucleation of this finite seismic slip episode. Interestingly, because with larger dilatancy, the quasi-static crack lags even more behind the fluid diffusion front prior to nucleation, the dynamic run-out increases for larger dilatancy. After this finite seismic slip episode, upon continuous injection, the shear crack propagates quasi-statically on par with the evolution of the diffusion front urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0112. Larger dilatancy slows down the quasi-static crack growth. The corresponding profiles of overpressure, friction coefficient, slip, and effective normal stress along the fault at different time snapshots are reported in Figure S6. The finite seismic episode can be clearly seen where we observe that prior to nucleation, the weakening zone occupies the whole crack. Because the fault is ultimately stable, beside the seismic episode, the fault propagates quasi-statically: Due to the low slip rate, dilatancy does not significantly alter the pore-pressure profile although the effect can be observed on the effective normal stress profiles (see Figure S6).

Details are in the caption following the image
Time evolution of normalized half crack length a/aw and normalized peak slip δ/δw at the middle of the fault, that is, at x = 0, for an ultimately stable fault (τo/τp = 0.55 and τo < τr, for fr/fp = 0.6), subjected to moderate overpressure ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0108 = 0.5. We span several dilatancy cases by varying the dimensionless dilatancy parameter ϵd/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0109). Red dots denote the onset of dynamic event, which is always characterized by a nucleation followed by an arrest (red arrow). The run-out distance increases with increase values of dimensionless dilatancy parameter ϵd/β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0110.

Finally, for large overpressure (region 3 of Figure 3), the crack growth is always quasi-static (aseismic). Results for such aseismic growth are reported in Supporting Information S1. For similar stress and fault strength conditions, an increase of the fault dilatancy slows down the crack velocity as expected.

6 Effect of Shear-Induced Permeability Changes

The results presented so far are based on the assumption of a constant fault permeability—although in our numerical results, the fault transmissivity whkf is changing in conjunction with the dilatant behavior. Experimental (e.g., Lee & Cho, 2002; Li et al., 2008; Makurat et al., 1985) and field evidences (Evans, Genter, & Sausse, 2005; Evans, Moriya, et al., 2005) have shown that deep fractures under fluid-induced slip exhibit an increase of fault permeability (Cornet, 2015; Evans, Genter, & Sausse, 2005; Evans, Moriya, et al., 2005; McClure & Horne, 2014). It is important to note that, although possibly significant, the increase of permeability with slip remains small compared to the drastic increase observed when the fracture opens (i.e., when the effective normal stress becomes tensile) such as in hydraulic fracturing. Like previously, we restrict here to the case of compressive effective normal stress, where permeability changes with slip are strictly associated with shear-induced dilatancy.

Several empirical models have been proposed and used in literature for permeability evolution. Some of them account for porosity changes, while some others include explicit dependency on effect stress changes (see, e.g., Rutqvist & Stephansson, 2003, for a review). For example, Rice (1992) used an effective stress-dependent permeability law, in which the permeability is a nonlinearly decreasing function of the local (compressive) effective normal stress:
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0113(29)
where k is the maximum fault permeability urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0114 and σ is a normalizing stress level urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0115 which ranges between 3 to 40 MPa (Rice, 1992)—see also Seront et al. (1998). Another common choice is to use the cubic law for the fault transmissivity (kfwh), relating the fault permeability directly to the changes of aperture—that is, a parallel plate idealization of the fluid flow in the fault (e.g., Bawden et al., 1980; McClure & Horne, 2014; Ucar et al., 2018):
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0116(30)

Under this assumption, the maximum constant fault permeability that is exerted when the slip δ is larger than the critical value δr is directly function of dilatant strain ϵd as urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0117. Such a maximum increase of longitudinal permeability with respect to its initial value urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0118 is actually rather small since the dilatant strain ϵd ranges between 10−4 and 10−2. This is clearly in contradiction with experimental and field evidences which mention much larger permeability increase (Evans, Genter, & Sausse, 2005; Evans, Moriya, et al., 2005; Makurat et al., 1985).

In order to investigate cases in which fault dilatancy induces significant increases of fault permeability with inelastic deformations (for instance due to change of fault porosity, for which Δkf ∝ Δφ7 − 8 for dense rocks—see Bernabé et al., 2003), we generalize the fault permeability evolution law as
urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0119(31)
where a and b are two constant parameters. Note that when a = 1 and b = 2, the fault transmissivity obeys the cubic law. By varying these two parameters, one can obtain tenfold permeability increase associated with shear slip at maximum dilatancy compared to the initial value urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0120. We use this permeability law 31 to gauge the impact of permeability change with slip on the stabilization by dilatancy of an otherwise unstable fault. In particular, our aim is to see if an increase of permeability affect the stabilizing effect of the undrained pore pressure drop associated with dilatancy.

We focus on the case of an otherwise unstable fault τo/τp = 0.75 (fr/fp = 0.6) and a moderate overpressure case ΔP = 0.5 with a dilatancy equal to the critical stabilizing value ϵd,c/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0121) = 0.25 for these conditions. Figure 9 displays the crack length and peak slip evolution for the case of a constant permeability as well as for different values of a and b for the permeability evolution law 31. We span a = 1, b = 2 (cubic law), and a = 2, b = 3, 5, 8, and 10 which entails, respectively, a 1.5 (cubic law), 3.3, 7.6, 25.6, and 57.6-fold increases of permeability at maximum dilatancy.

Details are in the caption following the image
Effect of permeability increase on a critically stressed (τo/τp = 0.75, fr/fp = 0.6) dilatant fault under moderate overpressure (ΔP/ urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0122 = 0.5): time evolution of normalized half crack length a/aw and corresponding peak slip δ|x = 0/δw. The dimensionless dilatancy parameter ϵd/(β urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0123) is set to the corresponding critical stabilizing value 27, equal here to 0.25. The effect of permeability evolution following the slip-dependent law 31 is investigated for five different values of the parameters (a, b) spanning small and large permeability increase from 1.5 to ∼60 times the initial fault permeability.

We observe that although the increase of permeability directly enhances the crack velocity, the propagation always remains aseismic. The permeability increase has a very significant effect on aseismic growth, and this effect increases with the value of b as expected. For example, for the strongest permeability variation with dilatancy (a = 2, b = 10 resulting in kf,max/kfo∼57.6), we observe a ∼550% increase in crack length at urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0124 compared to the constant permeability case (see Figure 9). A difference that will obviously continue to grow with time. For such an evolution of permeability with slip 31, the permeability profile is similar to the dilatancy strain: constant at is maximum value all along the crack except in the weakening zone near the tip—see Figure S10. The large permeability increases with slip, however, do not modify the stabilizing effect of undrained dilatant hardening. As the permeability accelerates quasi-static crack growth, the undrained pore-pressure response remains strong at the crack tip (see the pore-pressure profiles in Figure S10). Moreover, due to the quasi-static acceleration with increasing permeability, residual friction is reached earlier such that the undrained shear strength is fully mobilized—even possibly for smaller value of overpressure compared to the constant permeability case. Note that similar results are obtained with other type of permeability evolution (such as the one described by equation 29) as reported in Figures S11 and S12. In conclusions, in the case of an impermeable surrounding, the increase of permeability with slip along the fault does not affect the ultimate stability condition ( urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0125).

7 Conclusions

We have investigated the effect of dilatancy on the transition from seismic to aseismic slip due to sustained fluid injection regulated at a constant pressure in a fault. Although simple in its nature (planar bidimensional fault, uniform in situ stress and rock properties, and linear weakening friction), the model investigated properly couples, via nonassociated plasticity, the hydromechanical interplay between slip, dilatancy, frictional weakening, and fluid flow. We have developed a robust fully implicit numerical scheme—which properly reproduces existing semianalytical solutions for the case of nondilatant fault (Garagash & Germanovich, 2012). We notably would like to emphasize the necessity of numerical model verification for such type of nonlinear fracture propagation problem which—similarly to hydraulic fracturing problem—necessitates to resolve multiple scales (weakening zone and diffusion length scale here).

We have shown that dilatant hardening can stabilize an otherwise unstable fault (τo > τr), as long as the weakening of friction occurs in a small zone near the tip of the shear crack (small-scale yielding). This is captured by an ultimate stability condition defined with an undrained residual strength urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0126 function of the dilatant strain of the fault at critical state (when dilatancy saturates). We have demonstrated theoretically that under the assumption of small-scale yielding, dilatancy ultimately stabilizes the fault if urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0127 . In other words, for a given fault criticality, there exists a critical dilatancy above which the fault will remain stable and shear slip is solely aseismic. However, the hypothesis behind small-scale yielding (small frictional weakening zone near the crack tips) must be satisfied for such a ultimate stability condition to hold. This is the case if and only if the injection overpressure is sufficient to propagate quasi-statically the shear crack/slipping patch fast enough to reach residual friction and activate the beneficial effect of dilatancy prior to the crack reaching the nucleation length of the nondilatant case. For injection pressures below a limiting value, the crack propagates too slowly initially. The nucleation of a dynamic rupture occurs prior to reaching residual friction such that the maximum dilatancy is not activated prior to nucleation. For such small injection overpressure, dilatancy cannot prevent the nucleation of a dynamic rupture for an unstable fault (τo > τr) even for large dilatancy urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0128 (see Figure 7 for the evolution of the minimum overpressure). However, such a dynamic rupture for low overpressure and a priori sufficient dilatancy ( urn:x-wiley:jgrb:media:jgrb53362:jgrb53362-math-0129)—which occurs prior to reaching residual friction—will eventually arrest as the dilatant hardening effect kicks in for sufficient crack length. Although observed with quasi-dynamic damping (see Supporting Information S1), a fully elastodynamics simulation would be required to confirm such arrest.

For an ultimately stable fault (τo < τr), our numerical results indicate that dilatancy delays the occurrence of a finite episode of dynamic slip for moderate overpressure. Such a finite seismic event is associated with the abrupt catch up of the diffusion front by the crack front and the fact that the residual friction has not yet reached all along the crack. For large overpressure and stable fault, an increasing dilatant behavior simply slows down the quasi-static propagation (strictly aseismic slip).

Permeability increases with slip lead to faster aseismic crack growth for the different permeability evolution tested. However, it does not affect the critical dilatancy stabilizing an otherwise unstable fault. It appears evident that the details of the slip-permeability law greatly influence aseismic growth—a discussion on the most appropriate permeability model clearly require more investigation and necessarily better controlled hydromechanical laboratory experiments for sufficient slippage length.

The strengthening effect of dilatancy discussed here must be put perspective with some well-known dynamic weakening mechanisms that may occur as the crack accelerates: notably, thermal pressurization and flash heating of asperities. Although the effect of these weakening mechanisms have been already studied (Garagash & Germanovich, 2012; Garagash & Rudnicki, 2003; Rice, 2006; Segall & Rice, 2006) in the scope of earthquake nucleation via remote loading, a complete investigation of such competition would be required in the context of fluid injection. This is out of the scope of this paper. We note however that both of these dynamic weakening mechanisms requires dynamic slip rates (m/s and above), while dilatant hardening is activated quasi-statically. A probably more important point with respect to the stabilizing effect of dilatant hardening is related to the assumption of an impermeable host rock. Although possibly acceptable for young fault/fractures, this is highly doubtful for most mature fault structure. With a permeable surrounding (of say hydraulic diffusivity αr), the undrained pore-pressure drop associated with the fault dilatant behavior may be short-lived as fluid will be sucked in the fault and repressurize it. The importance of dilatant hardening will directly depend on the ratio between the changes due to dilatancy (which scales with slip rate) and the influx of fluid from the rock mass (which scales as αr/hw with hwωo the gouge thickness). A thorough investigation for the case of injection-induced slip is required to clarify that competition further, along the lines of Segall and Rice (1995) and Segall et al. (2010) in the context of the seismic cycle. In the sequel, we have also used a simple linear weakening friction law compared to a more elaborate rate-state model. It is nevertheless worthwhile to note that some work (Uenishi & Rice, 2003; Viesca, 2016a; 2016b) have demonstrated a correspondence between linear weakening friction and rate and state at the onset of nucleation. Investigations of the combined effect of rate and state and dilatancy in the case of fluid injection combined with proper scaling and stability analysis would surely produce a more refined understanding of the mechanisms of induced seismicity.

Finally, we conclude by recalling the decreases of dilatancy with confinement, such that the effect of dilatant hardening is likely to be more prominent mostly at shallow depths. Additional experimental data of fault dilatant behavior in conjunction with frictional properties would enable to further decipher its impact on fluid induced a-seismic and seismic slip with the help of the type of model presented here.

Acknowledgments

This work was funded by the Swiss Federal Office of Energy (Stim-Design project, SI/501354-01) and the Swiss National Science Foundation under grant 160577. The analytical formulae and numerical methods described in the main text and supporting information are sufficient to reproduce all the results presented in the paper.