Willis-Richards Permeability Model

Overview

In the Willis-Richards permeability model, the stress-aperture relationship is derived based on Barton–Bandis constitutive model. In this model, fracture hydraulic aperture is assumed to be a function of effective normal stress acting on the fracture surface and shear displacement along the fracture surface (Willis-Richards et al., 1996).

$a = \frac{ a_{m} + U_{s} \text{tan} \left( \phi_{dil} \right) }{ 1 + 9 \frac{ \sigma_{n} }{ \sigma_{ref} }}$

Based on the assumption of parallel plates, the correlation between fracture hydraulic aperture and its corresponding permeability is defined as:

$k = \frac{a^2}{12}$

where: $a$ is the fracture hydraulic aperture; $a_{m}$ is the fracture aperture at zero contact stress; $U_{s}$ is the relative shear displacement; $\phi_{dil}$ is the shear dilation angle; $\sigma_{n}$ is the effective normal stress acting on the fracture surface; $\sigma_{ref}$ is the effective normal stress that causes a 90% reduction in the fracture hydraulic aperture.

Parameters

The Willis-Richards permeability model is called in the <Constitutive> block of the input XML file. This permeability model must be assigned a unique name via the name attribute. This name is used to attach the model to regions of the physical domain via a permeabilityModelName attribute in the <CompressibleSolidWillisRichardsPermeability> block.

The following attributes are supported:

Name	Type	Default	Description
dilationCoefficient	real64	required	Dilation coefficient (tan of dilation angle).
maxFracAperture	real64	required	Maximum fracture aperture at zero contact stress.
name	groupName	required	A name is required for any non-unique nodes
refClosureStress	real64	required	Effective normal stress causes 90% reduction in aperture.

Example

<Constitutive>
   ...
   <WillisRichardsPermeability
     name="fracturePerm"
     maxFracAperture="0.005"
     dilationCoefficient="0.01"
     refClosureStress="1.0e7"/>
   ...
</Constitutive>