Three-phase relative permeability model

Overview

For the simulation of three-phase flow in porous media, it is common to use a specific treatment (i.e., different from the typical two-phase procedure) to evaluate the oil relative permeability. Specifically, the three-phase oil relative permeability is obtained by interpolation of oil-water and oil-gas experimental data measured independently in two-phase displacements.

Let \(k_{rw,wo}\) and \(k_{ro,wo}\) be the water-oil two-phase relative permeabilities for the water phase and the oil phase, respectively. Let \(k_{rg,go}\) and \(k_{ro,go}\) be the oil-gas two-phase relative permeabilities for the gas phase and the oil phase, respectively. In the current implementation, the two-phase relative permeability data is computed analytically using the Brooks-Corey relative permeability model.

The water and gas three-phase relative permeabilities are simply given by two-phase data and only depend on \(S_w\) and \(S_g\), respectively. That is,

\[k_{rw,wog}(S_w) = k_{rw,wo}(S_w),\]

\[k_{rg,wog}(S_g) = k_{rg,go}(S_g).\]

The oil three-phase relative permeability is obtained using a variant of the saturation-weighted interpolation procedure initially proposed by Baker. Specifically, we compute:

\[k_{ro,wog}(S_w,S_g) = \frac{ (S_w - S_{w,\textit{min}}) k_{ro,wo}(S_w) + S_g k_{rg,go}(S_g) }{ (S_w - S_{w,\textit{min}}) + S_g }.\]

This procedure provides a simple but effective formula avoiding the problems associated with the other interpolation methods (negative values).

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 <ElementRegion> node.

The following attributes are supported:

Name	Type	Default	Description
gasOilRelPermExponent	real64_array	{1}	Rel perm power law exponent for the pair (gas phase, oil phase) at residual water saturation The expected format is “{ gasExp, oilExp }”, in that order
gasOilRelPermMaxValue	real64_array	{0}	Maximum rel perm value for the pair (gas phase, oil phase) at residual water saturation The expected format is “{ gasMax, oilMax }”, in that order
name	string	required	A name is required for any non-unique nodes
phaseMinVolumeFraction	real64_array	{0}	Minimum volume fraction value for each phase
phaseNames	string_array	required	List of fluid phases
waterOilRelPermExponent	real64_array	{1}	Rel perm power law exponent for the pair (water phase, oil phase) at residual gas saturation The expected format is “{ waterExp, oilExp }”, in that order
waterOilRelPermMaxValue	real64_array	{0}	Maximum rel perm value for the pair (water phase, oil phase) at residual gas saturation The expected format is “{ waterMax, oilMax }”, in that order

Below are some comments on the model parameters.

• phaseNames - The number of phases should be 3. 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.

• waterOilRelPermExponent - The list of exponents \(\lambda_{\ell,wo}\) for the two-phase water-oil relative permeability data, with the water exponent first and the oil exponent next. These exponents are then used to compute \(k_{r \ell,wo}\) in the Brooks-Corey relative permeability model.

• waterOilRelPermMaxValue - The list of maximum values \(k_{\textit{r} \ell,wo,\textit{max}}\) for the two-phase water-oil relative permeability data, with the water max value first and the oil max value next. These exponents are then used to compute \(k_{r \ell,wo}\) in the Brooks-Corey relative permeability model.

• gasOilRelPermExponent - The list of exponents \(\lambda_{\ell,go}\) for the two-phase gas-oil relative permeability data, with the gas exponent first and the oil exponent next. These exponents are then used to compute \(k_{r \ell,go}\) in the Brooks-Corey relative permeability model.

• gasOilRelPermMaxValue - The list of maximum values \(k_{\textit{r} \ell,go,\textit{max}}\) for the two-phase gas-oil relative permeability data, with the gas max value first and the oil max value next. These exponents are then used to compute \(k_{r \ell,go}\) in the Brooks-Corey relative permeability model.

Example

<Constitutive>
   ...
 <BrooksCoreyBakerRelativePermeability name="relperm"
                                       phaseNames="{oil, gas, water}"
                                       phaseMinVolumeFraction="{0.05, 0.05, 0.05}"
                                       waterOilRelPermExponent="{2.5, 1.5}"
                                       waterOilRelPermMaxValue="{0.8, 0.9}"
                                       gasOilRelPermExponent="{3, 3}"
                                       gasOilRelPermMaxValue="{0.4, 0.9}"/>
   ...
</Constitutive>