ThreePhaseRelativePermeability.rst

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 /coreComponents/constitutive/docs/BrooksCoreyRelativePermeability.

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 <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:

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_ℓ, 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 λ_ℓ, 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_rℓ, wo in the /coreComponents/constitutive/docs/BrooksCoreyRelativePermeability.
• waterOilRelPermMaxValue - The list of maximum values k_{rℓ, wo, 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_rℓ, wo in the /coreComponents/constitutive/docs/BrooksCoreyRelativePermeability.
• gasOilRelPermExponent - The list of exponents λ_ℓ, 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_rℓ, go in the /coreComponents/constitutive/docs/BrooksCoreyRelativePermeability.
• gasOilRelPermMaxValue - The list of maximum values k_{rℓ, go, 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_rℓ, go in the /coreComponents/constitutive/docs/BrooksCoreyRelativePermeability.

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>