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:

(136)#\[\begin{equation} \rho_s\frac{\partial^2 \vec{u}}{\partial t^2} = \vec{f}(t) + \nabla \cdot \boldsymbol{\sigma}(\vec{u},p). \end{equation}\]

Enforcing mass balance of the fluid gives

(137)#\[\begin{gather} \frac{\partial \zeta(\vec{u},p)}{\partial t} + \nabla \cdot \vec{q}(p) = \gamma(\vec{x},t) \text{ in } \Omega, \\ % \vec{q} \cdot \vec{n} = q_0(\vec{x},t) \text{ on }\Gamma_q, \\ % p = p_0(\vec{x},t) \text{ on }\Gamma_p, \end{gather}\]

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),

(138)#\[\begin{equation} \vec{q}(p) = -\frac{\boldsymbol{k}}{\mu_{f}}(\nabla p - \vec{f}_f), \end{equation}\]

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

(139)#\[\begin{equation} \boldsymbol{\sigma}(\vec{u},p) = \boldsymbol{C} : \boldsymbol{\epsilon} - \alpha p \boldsymbol{I}, \end{equation}\]

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:

(140)#\[\begin{equation} \boldsymbol{\sigma}(\vec{u},p) = \boldsymbol{C}:\boldsymbol{\epsilon} - \alpha p \boldsymbol{I} = \lambda \boldsymbol{I} \epsilon_{v} + 2 \mu \boldsymbol{\epsilon} - \alpha \boldsymbol{I} p \end{equation}\]

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,

(141)#\[\begin{gather} \zeta(\vec{u},p) = \alpha \mathop{\mathrm{Tr}}({\boldsymbol{\epsilon}}) + \frac{p}{M}, \\ % \frac{1}{M} = \frac{\alpha-\phi}{K_s} + \frac{\phi}{K_f}, \end{gather}\]

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

(142)#\[\begin{equation} \zeta(\vec{u},p) = \alpha (\nabla \cdot \vec{u}) + \frac{p}{M}. \end{equation}\]

Table 6 Mathematical notation for poroelasticity with infinitesimal strain.#
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
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