Brooks-Corey relative permeability model

Overview

The following paragraphs explain how the Brooks-Corey model is used to compute the phase relative permeabilities as a function of volume fraction (i.e., saturation) with the expression:

$k_{r\ell} = k_{\textit{r}\ell,\textit{max}} S_{\ell,\textit{scaled}}^{\lambda_{\ell}},$

where the scaled volume fraction of phase $\ell$ is computed as:

$S_{\ell,\textit{scaled}} = \frac{S_{\ell} - S_{\ell,\textit{min}} }{1 - \sum^{n_p}_{m=1} S_{\textit{m,min}} }.$

The minimum phase volume fractions $S_{\ell,\textit{min}}$ are model parameters specified by the user.

Parameters

The relative permeability constitutive model is listed in the <Constitutive> block of the input XML file. The relative permeability model must be assigned a unique name via name attribute. This name is used to assign the model to regions of the physical domain via a materialList attribute of the <ElementRegions> node.

The following attributes are supported:

Name	Type	Default	Description
name	groupName	required	A name is required for any non-unique nodes
phaseMinVolumeFraction	real64_array	{0}	Minimum volume fraction value for each phase
phaseNames	groupNameRef_array	required	List of fluid phases
phaseRelPermExponent	real64_array	{1}	Minimum relative permeability power law exponent for each phase
phaseRelPermMaxValue	real64_array	{0}	Maximum relative permeability value for each phase

Below are some comments on the model parameters.

phaseNames - The number of phases can be either two or three. Note that for three-phase flow, this model does not apply a special treatment to the intermediate phase relative permeability (no Stone or Baker interpolation). Supported phase names are:

Value	Phase
oil	Oil phase
gas	Gas phase
water	Water phase

phaseMinVolFraction - The list of minimum volume fractions $S_{\ell,min}$ for each phase is specified in the same order as in phaseNames. Below this volume fraction, the phase is assumed to be immobile.
phaseRelPermExponent - The list of exponents $\lambda_{\ell}$ for each phase is specified in the same order as in phaseNames.
phaseMaxValue - The list of maximum values $k_{\textit{r} \ell,\textit{max}}$ for each phase is specified in the same order as in phaseNames.

Examples

For a two-phase water-gas system (for instance in the CO2-brine fluid model), a typical relative permeability input looks like:

<Constitutive>
   ...
   <BrooksCoreyRelativePermeability
     name="relPerm"
     phaseNames="{ water, gas }"
     phaseMinVolumeFraction="{ 0.02, 0.015 }"
     phaseRelPermExponent="{ 2, 2.5 }"
     phaseRelPermMaxValue="{ 0.8, 1.0 }"/>
   ...
</Constitutive>

For a three-phase oil-water-gas system (for instance in the Black-Oil fluid model), a typical relative permeability input looks like:

<Constitutive>
   ...
   <BrooksCoreyRelativePermeability
     name="relPerm"
     phaseNames="{ water, oil, gas }"
     phaseMinVolumeFraction="{ 0.02, 0.1, 0.015 }"
     phaseRelPermExponent="{ 2, 2, 2.5 }"
     phaseRelPermMaxValue="{ 0.8, 1.0, 1.0 }"/>
   ...
</Constitutive>