Laboratory Simulation of Injection-Induced Shear Slip on Saw-Cut Sandstone Fractures under Different Boundary Conditions

Abstract

It is widely acknowledged that induced earthquakes may be linked to the fluid mass injection into the subsurface in a manner of reactivating preexisting natural faults. To better understand the potential effect of tectonic boundary conditions on induced fault slip behavior, we performed injection-induced shear slip experiments on critically stressed saw-cut sandstone samples under the conditions of constant piston displacement (CPD test) and of constant shear stress (CSS test) at confining pressure of 10 MPa and 15 MPa, respectively. In response to the low fluid pressurization rate of 0.1 MPa/min, the artificial fractures (faults) show a slow slip velocity (< 0.1 μm/s) accompanied by a linear drop of shear stress in CPD tests, whereas a faster slip velocity (< 5 μm/s) and a larger slip distance are observed in the subsequent CSS tests. Our experimental results indicate that the laboratory fractures slide stably in response to fluid overpressure during the two tests, supported by the experimentally measured rate-and-state frictional parameters and linear stability analysis. As the confining pressure applied increases from 10 to 15 MPa, the fractures show a considerable decrease in hydraulic diffusivity. Also, the fluid pressure heterogeneity is found to be affected by slip velocity in CSS tests at both confining pressures. We interpret this as a result of the local fluid depressurization that potentially occurs far away from injection end due to the rapid shear-enhanced dilation. In addition, the complete energy budget components associated with fluid-induced fracture slip in the context of CPD and CSS tests, respectively, have been quantitatively evaluated. Our experimental observations highlight that injection-induced fracture slip behavior is closely related not only to the applied effective normal stress, but also to tectonic boundary conditions.

Appendices

Appendix A: Fluid Pressure Distribution Across Intact and Fractured Samples

Under an assumption that the hydraulic diffusivity (D) is constant, a partial differential equation can be used to describe the temporal and spatial variations of fluid pressure in porous rocks (Rice and Cleary 1976):

$$\frac{{\partial P_{f} }}{\partial t} = D\nabla^{2} P_{f} ,$$

(A1)

where P_f is the fluid pressure and t is time. For 1D fluid pressure diffusion process through a cylindrical sample, the diffusion Eq. (A1) can be rewritten as

$$\frac{{\partial P_{f} (x,t)}}{\partial t} = D\frac{{\partial^{2} P_{f} (x,t)}}{{\partial x^{2} }},$$

(A2)

where x is the distance along the sample from the upstream boundary (i.e., the injection end in our study).

For simplicity, the initial fluid pressure (P₀) distribution in the sample is set as zero:

$$P_{f} (x,0) = 0,{\text{ for }}0 \le x \le L.$$

(A3)

By applying a constant pressurization rate (R) at the injection end (see Sect. 2.2), the temporal evolution of fluid pressure at the injection end can be described by

$$P_{f} (0,t) = Rt$$

(A4)

Considering an undrained boundary condition at the downstream boundary (i.e., monitoring end in our study), we have

$$\partial P_{f} /\partial x = 0,{\text{ at}}\;x = L.$$

(A5)

Under the above boundary conditions (i.e., Eqs. A3, A4 and A5), the analytical solution to Eq (A2) can be mathematically derived as follows (Polyanin 2002):

$$P_{f} (x,t) = R\frac{2}{L}\sum\limits_{n = 0}^{\infty } {\sin } (\lambda_{n} x)\frac{{(D\lambda_{n}^{2} t - 1) + \exp ( - D\lambda_{n}^{2} t)}}{{D\lambda_{n}^{3} }},{\text{ for }}0 \le x \le L,$$

(A6)

where $\lambda_{n} = \pi (2n + 1)/2L$.

Based on the measured hydraulic diffusivity of intact sandstone sample (D ≈ 8.92 × 10^–7 m²/s) and the constant pressurization rate of 0.1 MPa/min applied at the injection end (see Sect. 2.2), the normalized fluid pressure distribution (i.e., P_f(x,t)/P_f(0,t)) along the sample at given times can be calculated accordingly using Eq. A6, as shown in Fig. A1a. Clearly, fluid overpressure front requires about 900 s to reach the other sample end, and it requires around 4 × 10⁵ s to make the pore pressure distribute homogeneously within the whole rock matrix. Compared to the fluid injection duration in the CPD and CSS tests, it is reasonable to assume that the injected fluid volume is primarily constrained within the fracture zone.

For Sandstone #2, the fluid pressure at the monitoring end (P_f(L,t)) is found to be lower than the injection pressure (P_f(0,t)) throughout the whole pressurization period in CPD test, indicating the low hydraulic diffusivity of the fracture at P_c = 15 MPa. For simplicity, we assume that the hydraulic diffusivity of the fracture in the CPD test of Sandstone #2 remains unchanged during progressive fluid injection although the shear-induced dilation indeed occurs (see Sect. 4.1) potentially causing an increase in fracture permeability over time. By comparing the experimentally measured normalized fluid pressure increase at monitoring end of sample ($\Delta P_{f} (L,t)/\Delta P_{f} (0,t)$) since fluid injection with the modelled ones at given two fixed hydraulic diffusivity values (Fig. A1b), the hydraulic diffusivity of Sandstone #2 (P_c = 15 MP) in the CPD test may be bounded in the range between 7 × 10^–6 and 1 × 10^–5 m²/s. The significant discrepancy between measured and modelled fluid pressure values at the beginning of fluid injection (see Fig. A1b) might be the consequence of the spatial heterogeneity of diffusivity, which was not considered in Eq. (A2). The constrained upper and lower bounds of hydraulic diffusivity of Sandstone #2 at P_c = 15 MPa is about one order of magnitude higher than that of the intact sample.

Fig. A1

a Snapshots of fluid pressure distribution along the intact sample with a length of L = 0.1 m at given times (t = 100 s, 500 s, 900 s, 10000 s and 400000 s, respectively). b Inverted upper and lower bounds of hydraulic diffusivity of Sandstone #2 at P_c = 15 MPa in the CPD test based on the fluid pressure evolution measured at injection and monitoring ends (i.e., x = 0 and 0.1 m, respectively). In the x-axis, the initial time (t = 1550 s) corresponds to the beginning of fluid injection in the CPD test (see stage 3 in Fig. 3b).

Appendix B: Normal Displacement of a Fracture Caused by Stress and Fluid Pressure Changes

We consider an experimental configuration in which a cylinder sample that is split into two equal halves and subjected to initial confining pressure of $\sigma_{3}$, initial axial stress of $\sigma_{1}$ and initial fluid pressure of P_p (see Sect. 2.1). Now we attempt to quantify the radial deformation resulting from the changes of axial stress, of confining pressure ($\Delta \sigma_{3}$) and of fluid pressure (ΔP_f). Note that dilation is negative during the following derivation.

Since the permeability of rock matrix is very low (k ≈ 10^–19 m²), it is reasonable to assume that the fluid overpressure is primarily constrained within the fracture (fault) zone. In this case, the radial strain change ($\Delta \varepsilon_{3}$) may be estimated by

$$\Delta \varepsilon_{3} = \frac{1}{E}\left[ {\Delta \sigma_{3} - \nu (\Delta \sigma_{1} + \Delta \sigma_{3} )} \right] - \frac{{\Delta P_{f} }}{{K_{n} }},$$

(B1)

where E is Young’s modulus, $\nu$ is Poisson’s ratio, and K_n is fracture (fault) normal stiffness.

Moreover, considering that the stress level applied to the sample is low, the purely elastic response of bulk rock sample in our experiments is expected to occur. For simplicity, $\sigma_{1}$ is estimated by the axial force divided by the cross-section area of a cylinder. Since $\Delta \sigma_{3} = 0$ holds during the CPD and CSS tests, Eq. (B1) may be further simplified as

$$\Delta \varepsilon_{3} = - \frac{\nu }{E}\Delta \sigma_{1} - \frac{1}{{K_{n} }}\Delta P_{f} ,$$

(B2)

In this sense, the component of fracture normal displacement due to stress and fluid pressure changes may be finally estimated by $\Delta \varepsilon_{3} \times d$ where d is the diameter of a cylinder. The value of $\frac{\nu }{E}$ (≈ 2.62 × 10^–5 MPa⁻¹) in Eq. (B2) can be available from the Poisson’s ration and Young’s modulus measured in the uniaxial compression tests (see Table 1). To experimentally constrain the unknown parameter K_n in Eq. (B2), we focused on the initial stage of CSS test in Sandstone #1 (i.e., stage 6 in Fig. 2a) considering the fact that the laboratory fault remained locked and the axial load was kept constant (Δσ₁ = 0) during this period. In the meantime, the uniform fluid pressure was simultaneously distributed over the fracture zone. Such observations suggest that the component of fracture normal displacement due to fluid pressure changes is solely responsible for the measured fracture normal displacement in this stage. Using Eq. (B2), the parameter of 1/K_n was therefore estimated to be ~ 8.7 × 10^–5 MPa⁻¹.

On the basis of the determined parameters above, the evolving component of fracture normal deformation caused by stress and fluid pressure changes throughout experiments can be finally evaluated using Eq. (B2).

Appendix C: Calculation of Energy Budget

For the rock sample in our experimental setup, the energy balance associated with injection-induced shear slip since fluid injection can be formulated as (Goodfellow et al. 2015)

$$W + E_{I} + U_{\text{ini}} = U_{\text{fin}} + E_{f} + E_{d} + I,$$

(C1)

where W is the external work of axial force F ($W = \int\nolimits_{{x_{1} }}^{{x_{2} }} {Fdx}$, [x₁ x₂] is the initial and final load point displacement) on the sample; E_I (= $\int\nolimits_{{t_{1} }}^{{t_{2} }} P Qdt$, Q is the injection rate, P is the injection pressure and [t₁ t₂] is the time interval) is the hydraulic energy (or fluid injection energy); U_ini and U_fin (= $\left[ {\frac{1 - 2\nu }{{6E}}(\sigma_{1} + \sigma_{3} )^{2} + \frac{1 + \nu }{{3E}}(\sigma_{1}^{2} + \sigma_{3}^{2} - \sigma_{1} \sigma_{3} )} \right]V$, $\nu$ is the Poisson’s ratio and V is the volume of bulk sample) are the initial and final strain energy, respectively, and $\Delta U = U_{\text{ini}} - U_{\text{fin}}$ is the change of elastic strain energy; E_f (= $\int\nolimits_{{x_{1} }}^{{x_{2} }} {\tau_{r} Adx}$, $\tau_{r}$ is the residual shear stress during each test and A is the fracture surface area) is the frictional energy; E_d (= $\int\nolimits_{{h_{1} }}^{{h_{2} }} {(P_{c} - P_{f} } )Adh$, P_c is the confining pressure, P_f is the mean fluid pressure used in this calculation and h is the opening height of the fracture) is the corresponding deformation energy due to fracture dilation and I accounts for other additional energy loss terms, such as the radiated ultrasonic waves due to the failure of micro-asperities, fluid viscous dissipation, etc.

Finally, the calculated results are presented in Table C1.

Table C1 Energy budget of two fluid injection experiments at different confining pressures

Appendix D: Velocity-Step** Tests

To experimentally constrain the evolution of frictional constitutive parameters with variation of normal stress, we performed a series of velocity-step** tests on water-saturated split sandstone blocks (a central saw-cut fracture along the axial direction of a cylinder sandstone sample, see Sect. 2.1 for more details) using the tri-axial direct-shear apparatus (see Fig. 1). Three different confining pressures of 4, 8 and 12 MPa, respectively, were employed to simulate the stress conditions applied in CPD and CSS tests.

Before conducting the velocity-step** tests, the fractured sandstone samples were sufficiently saturated with deionized water (imposing 0.1 MPa water pressure for saturation) under a given confining pressure of 4, 8 and 12 MPa, respectively. To avoid the potential occurrence of local fluid (de)pressurization (e.g., due to normal compaction or dilation) during the shearing process that may impact the measurement of frictional constitutive parameters, the imposed 0.1 MPa fluid pressure was removed before shearing. The fracture was initially sheared at a constant shear velocity of 1 μm/s until the steady state friction regime was reached. Subsequently, the shear velocity was stepped from 1 to 5 μm/s, anticipating the evaluation of velocity dependence of fracture friction. It should be pointed out that the step** slip velocity of 5 μm/s selected here corresponds to the upper bound of slip velocity observed in the injection-induced slip tests.

The rate-and-state friction (RSF) law (Dieterich 1979; Ruina 1983) allows us to assess the velocity dependence of friction from the velocity-step** tests. To be more precise, we first pre-processed the measured friction coefficient by detrending the minor hardening behavior with slip displacement (Skarbek and Savage 2019). Following the general framework of RSF law (Dieterich 1979), the evolving friction coefficuent μ at the instantaneous shear velocity v can be quantified by

$$\mu = \mu_{0} + a\ln \left( {\frac{v}{{v_{0} }}} \right) + b\ln \left( {\frac{{\theta v_{0} }}{{D_{c} }}} \right),$$

(D1)

where a and b are the dimensionless constitutive parameters; D_c is the critical slip distance required to reach a new steady state; μ₀ denotes the steady-state friction coefficient at the reference shear velocity v₀, and $\theta$ is a state variable. We employed the slip law (Dieterich 1979; Ruina 1983) for the evolution of state variable θ with time t:

$$\frac{d\theta }{{dt}} = - \frac{v\theta }{{D_{c} }}\ln \left( {\frac{v\theta }{{D_{c} }}} \right).$$

(D2)

For a steady-state friction, the state variable θ does not change with time t, and thus $d\theta /dt = 0$. In this case, the frictional stability parameter (a–b) can be given from Eq. (D1) at steady state by:

$$a - b = \frac{{\mu - \mu_{0} }}{{\ln (v/v_{0} )}}.$$

(D3)

A positive value of (a–b) indicates that friction coefficient increases with increasing shear velocity (i.e., velocity-strengthening behavior), and thus results in inherently stable sliding. Conversely, a negative value of (a–b) represents velocity-weakening behavior, potentially facilitating unstable sliding.

Using Eqs. (D1, D2 and D3), we ultimately constrained the constitutive parameters a, b and D_c from the velocity-step** tests conducted at three different confining pressures, as shown in Fig. D1. Apparently, the fractured sandstone samples undergo a transition from velocity-strengthening behavior ((a–b) ≈ 0.00053) at P_c = 4 MPa, weakly velocity-strengthening behavior ((a–b) ≈ 0.00005) at P_c = 8 MPa to weakly velocity-weakening behavior ((a–b) ≈ − 0.00058) at P_c = 12 MPa.

Fig. D1

Estimates of frictional parameters (a, b and D_c) from velocity-step** tests (1 μm/s to 5 μm/s) conducted at effective normal stress of 4 MPa (a), 8 MPa (b) and 12 MPa (c), respectively. The frictional parameter values fitted by least squares method with their standard deviations are given in the corresponding box