Infinitesimal Strain and Prescribed Fault Slip#

We apply the prescribed fault slip formulation that we used for elasticity to poroelasticity. We add a boundary condition to the poroelasticity equation prescribing the jump in the displacement field across the fault,

\[\begin{gathered} -\vec{u}^+ + \vec{u}^- + \vec{d}(\vec{x},t) = \vec{0} \text{ on }\Gamma_f, \end{gathered}\]

where \(\vec{u}^+\) is the displacement vector on the “positive” side of the fault, \(\vec{u}^-\) is the displacement vector on the “negative” side of the fault, \(\vec{d}\) is the slip vector on the fault, and \(\vec{n}\) is the fault normal which points from the negative side of the fault to the positive side of the fault. We enforce the jump in displacements across the fault using a Lagrange multiplier corresponding to equal and opposite tractions on the two sides of the fault.

Warning

In this formulation, the fault acts as a barrier to fluid flow. That is, there is no coupling in fluid pressure across the fault.

Table 10 Mathematical notation for poroelasticity with infinitesimal strain and prescribed fault slip.#

Category

Symbol

Description

Unknowns

\(\vec{u}\)

Displacement field

\(\vec{v}\)

Velocity field

\(p\)

Pressure field (corresponds to pore fluid pressure)

\(\epsilon_{v}\)

Volumetric (trace) strain

\(\vec{\lambda}\)

Lagrange multiplier for fault

Derived quantities

\(\boldsymbol{\sigma}\)

Cauchy stress tensor

\(\boldsymbol{\epsilon}\)

Cauchy strain tensor

\(\zeta\)

Variation of fluid content (variation of fluid vol. per unit vol. of PM), \(\alpha \epsilon_{v} + \frac{p}{M}\)

\(\rho_{b}\)

Bulk density, \(\left(1 - \phi\right) \rho_{s} + \phi \rho_{f}\)

\(\vec{q}\)

Darcy flux, \(-\frac{\boldsymbol{k}}{\mu_{f}} \cdot \left(\nabla p - \vec{f}_{f}\right)\)

Common constitutive parameters

\(\rho_{s}\)

Solid (matrix) density

\(\rho_{f}\)

Fluid density

\(\mu_{f}\)

Fluid viscosity

\(\phi\)

Porosity

\(\mu\)

Shear modulus

\(K_{d}\)

Drained bulk modulus

\(\alpha\)

Biot coefficient, \(1 - \frac{K_{d}}{K_{s}}\)

\(M\)

Biot modulus

\(\boldsymbol{k}\)

Permeability

Source terms

\(\vec{f}\)

Body force per unit volume, for example: \(\rho_{b} \vec{g}\)

\(\vec{f}_{f}\)

Fluid body force, for example: \(\rho_{f} \vec{g}\)

\(\gamma\)

Source density; rate of injected fluid per unit volume of the porous solid