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).

Another option can be triggered using threePhaseInterpolator to set interpolation model to be STONEII described by:

$k_ro = k_rocw ((k_row/k_rocw + k_rw)(k_rog/k_rocw + k_rg) - k_rw - k_rg)$

…

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:

XML Element: BrooksCoreyBakerRelativePermeability

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	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
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>