Van Genuchten capillary pressure model

Overview

In GEOS, the oil-phase pressure is assumed to be the primary pressure. The following paragraphs explain how the Van Genuchten capillary pressure model is used to compute the water-phase and gas-phase pressures as:

$p_w = p_o - P_{c,w}(S_w),$

and

$p_g = p_o + P_{c,g}(S_g),$

The Van Genuchten model computes the water-phase capillary pressure as a function of the water-phase volume fraction as:

$P_c(S_w) = \alpha_w ( S_{w,scaled}^{-1/m_w} - 1 )^{ (1-m_w)/2 },$

where the scaled water-phase volume fraction is computed as:

$S_{\textit{w,scaled}} = \frac{S_w - S_{\textit{w,min}} }{1 - S_{\textit{w,min}} - S_{\textit{o,min}} - S_{\textit{g,min} }}.$

The gas-phase capillary pressure is computed analogously.

Parameters

The capillary pressure constitutive model is listed in the <Constitutive> block of the input XML file. The capillary pressure 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:

XML Element: VanGenuchtenCapillaryPressure

Name	Type	Default	Description
capPressureEpsilon	real64	1e-06	Saturation at which the extremum capillary pressure is attained; used to avoid infinite capillary pressure values for saturations close to 0 and 1
name	groupName	required	A name is required for any non-unique nodes
phaseCapPressureExponentInv	real64_array	{0.5}	Inverse of capillary power law exponent for each phase
phaseCapPressureMultiplier	real64_array	{1}	Entry pressure value for each phase
phaseMinVolumeFraction	real64_array	{0}	Minimum volume fraction value for each phase
phaseNames	groupNameRef_array	required	List of fluid phases

Below are some comments on the model parameters:

phaseNames - The number of phases can be either 2 or 3. The phase names entered for this attribute should match the phase names specified in the relative permeability block, either in Brooks-Corey relative permeability model or in Three-phase relative permeability model. The capillary model assumes that oil is always present. 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. The values entered for this attribute have to match those of the same attribute in the relative permeability block.
phaseCapPressureExponentInv - The list of exponents $m_{\ell}$ for each phase is specified in the same order as in phaseNames. The parameter corresponding to the oil phase is not used.
phaseCapPressureMultiplier - The list of multipliers $\alpha_{\ell}$ for each phase is specified in the same order as in phaseNames. The parameter corresponding to the oil phase is not used.
capPressureEpsilon - The parameter $\epsilon$ . This parameter is used for both the water-phase and gas-phase capillary pressure. To avoid extremely large, or infinite, capillary pressure values, we set $P_{c,w}(S_w) := P_{c,w}(\epsilon)$ whenever $S_w < \epsilon$ . The gas-phase capillary pressure is treated analogously.

Example

<Constitutive>
   ...
   <VanGenuchtenCapillaryPressure name="capPressure"
                                  phaseNames="{ water, oil }"
                                  phaseMinVolumeFraction="{ 0.1, 0.015 }"
                                  phaseCapPressureExponentInv="{ 0.55, 0 }"
                                  phaseCapPressureMultiplier="{ 1e6, 0 }"
                                  capPressureEpsilon="1e-7"/>
  ...
</Constitutive>