Poroelasticity with Infinitesimal Strain and No Faults#
We base this formulation for poroelasticity on Zheng et al. and Detournay and Cheng (1993). We assume a slightly compressible fluid that completely saturates a porous solid, undergoing infinitesimal strain.
We begin with the conservation of linear momentum, including inertia, borrowed from linear elasticity:
Enforcing mass balance of the fluid gives
where \(\zeta\) is the variation in fluid content, \(\vec{q}\) is the rate of fluid volume crossing a unit area of the porous solid, \(\gamma\) is the rate of injected fluid per unit volume of the porous solid, \(q_0\) is the outward fluid velocity normal to the boundary \(\Gamma_q\), and \(p_0\) is the fluid pressure on boundary \(\Gamma_p\).
We require the fluid flow to follow Darcy’s law (Navier-Stokes equation neglecting inertial effects),
where \(\boldsymbol{k}\) is the intrinsic permeability, \(\mu_f\) is the viscosity of the fluid, \(p\) is the fluid pressure, and \(\vec{f}_f\) is the body force in the fluid. If gravity is included in a problem, then usually \(\vec{f}_f = \rho_f \vec{g}\), where \(\rho_f\) is the density of the fluid and \(\vec{g}\) is the gravitational acceleration vector.
Constitutive Behavior#
We assume linear elasticity for the solid phase, so the constitutive behavior can be expressed as
where \(\boldsymbol{\sigma}\) is the stress tensor, \(\boldsymbol{C}\) is the tensor of elasticity constants, \(\alpha\) is the Biot coefficient (effective stress coefficient), \(\boldsymbol{\epsilon}\) is the strain tensor, and \(\boldsymbol{I}\) is the identity tensor. For this case, we will assume that the material properties are isotropic, resulting in the following formulation for the stress tensor:
where \(\lambda\) and \(\mu\) are Lamé’s parameters, \(\lambda = K_{d} - \frac{2 \mu}{3}\), \(\mu\) is the shear modulus, and the volumetric strain is defined as \(\epsilon_{v} = \nabla \cdot \vec{u}\).
For the constitutive behavior of the fluid, we use the volumetric strain to couple the fluid-solid behavior,
where \(1/M\) is the specific storage coefficient at constant strain, \(K_s\) is the bulk modulus of the solid, and \(K_f\) is the bulk modulus of the fluid. We can write the trace of the strain tensor as the dot product of the gradient and displacement field, so we have
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 |
|
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, \(-\frac{\boldsymbol{k}}{\mu_{f}} \cdot \left(\nabla p - \vec{f}_{f}\right)\) |
|
\(M\) |
Biot Modulus, \(\frac{K_{f}}{\phi} + \frac{K_{s}}{\alpha - \phi}\) |
|
Common constitutive parameters |
\(\rho_{f}\) |
Fluid density |
\(\rho_{s}\) |
Solid (matrix) density |
|
\(\phi\) |
Porosity |
|
\(\boldsymbol{k}\) |
Permeability |
|
\(\mu_{f}\) |
Fluid viscosity |
|
\(K_{s}\) |
Solid grain bulk modulus |
|
\(K_{f}\) |
Fluid bulk modulus |
|
\(K_{d}\) |
Drained bulk modulus |
|
\(\alpha\) |
Biot coefficient, \(1 - \frac{K_{d}}{K_{s}}\) |
|
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 |