Modeling assumptions and parameters

FC Frédéric Cappa MS Marco Maria Scuderi CC Cristiano Collettini YG Yves Guglielmi JA Jean-Philippe Avouac

This protocol is extracted from research article:

Stabilization of fault slip by fluid injection in the laboratory and in situ

**
Sci Adv**,
Mar 13, 2019;
DOI:
10.1126/sciadv.aau4065

Stabilization of fault slip by fluid injection in the laboratory and in situ

Procedure

We used the 3DEC code (*46*), a distinct element method (*47*), to simulate the interaction between fluid flow and fault slip evolution, including hydromechanical coupling and rate- and state-dependent friction. The model incorporates the full coupling between fluid pressure diffusion and effective stress- and strain-dependent permeability. The method has been previously used to understand the hydromechanical behavior of fractured rocks during fluid injection (*48*) and to study earthquake rupture and off-fault fracture response (*49*).

Our 3D model considers a fluid injection into a fault (dip angle of 70°) in a homogeneous elastic and impervious medium (Fig. 1C). The fault geometry reproduces the conditions of the in situ experiment (*6*). The remote normal (σ_{n}) and shear stress (τ) resolved on the fault plane are constant. During injection, the fluid pressure is increased into the fault in a point source using the loading path applied during the in situ experiment (blue curve in Fig. 1B). The initial value of stress (σ_{no} = 4.25 MPa, τ_{o} = 1.65 MPa) and fluid pressure (*p*_{o} = 0) into the fault represents the conditions of the in situ experiment (*6*). At time *t* = 0, water is injected into the fault so that the effective normal stress within the fault is defined by σ_{n}′ = σ_{n} − *p*.

This model is based on the cubic law to describe the coupling between the fluid pressure and fault normal displacement (*50*)$$\mathit{Q}(\mathit{t})=\frac{{({\mathit{b}}_{\text{ho}}+{\mathit{u}}_{\mathrm{n}})}^{3}\cdot \mathit{w}\cdot \mathrm{\Delta}\mathit{p}}{12{\mathrm{\mu}}_{\mathrm{f}}\cdot \mathit{L}}$$(5)where *Q*(*t*) is the flow rate (in m^{3}/s), *w* denotes the fault width (in m), *b*_{ho} is the initial hydraulic aperture of the fault (in m), *u*_{n} is the fault normal displacement (in m), μ_{f} is the fluid dynamic viscosity (in Pa·s), *L* is the contact length of the fault surface (in m), and Δ*p* is the fluid pressure change (in Pa). The fluid pressure perturbation follows a diffusivity equation$$\mathrm{\Delta}\stackrel{.}{\mathit{p}}=\frac{\mathit{T}}{\mathit{S}}{\nabla}^{2}\mathrm{p}$$(6)where *T* is the transmissivity$$\mathit{T}=\frac{{({\mathit{b}}_{\mathrm{h}}+{\mathit{u}}_{\mathrm{n}})}^{3}}{12{\mathrm{\mu}}_{\mathrm{f}}}$$(7)and *S* is the storativity$$\mathit{S}=(\frac{{\mathit{b}}_{\mathrm{h}}}{{\mathit{K}}_{\mathrm{f}}}+\frac{1}{\mathit{K}+4/3\mathit{G}})$$(8)where *b*_{h} is the hydraulic aperture (*b*_{ho} + *u*_{n}) (in m), *K*_{f} is the fluid bulk modulus (in Pa), and *K* and *G* are the rock bulk and shear moduli (in Pa), respectively.

The magnitude of the pressure perturbation and the hydraulic properties affect the dynamics of the transient. We use an initial hydraulic aperture (*b*_{ho} = 9 × 10^{−6} m) consistent with an initial fault permeability of 7 × 10^{−12} m^{2} estimated in situ (*4*), *K*_{f} = 2 GPa, *K* = 20 GPa, *G* = 7.5 GPa, *g* = 9.81 m/s^{2}, ρ_{f} = 1000 kg/m^{3}, and μ_{f} = 1 × 10^{−3} Pa·s.

For the frictional behavior of the fault, the model is based on the rate-and-state friction law (Eq. 2) derived from laboratory experiments (*1*, *20*, *36*). The state evolution law is described by the aging law (*20*)$$\frac{\mathit{d}\mathrm{\theta}}{\mathit{d}\mathit{t}}=1-\frac{\mathit{v}\mathrm{\theta}}{{\mathit{d}}_{\mathrm{c}}}$$(9)

Consistent with our laboratory measurements at low slip velocity (*v* < 10 μm/s), here, we assume a rate-weakening fault with μ_{o} = 0.6, *a* = 0.001, *b* = 0.003, and *d*_{c} = 10 μm. The frictional resistance of the fault is then given by$$\mathrm{\tau}=\mathrm{\mu}(\mathit{v},\mathrm{\theta})({\mathrm{\sigma}}_{\mathrm{n}}-\mathit{p})$$(10)

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