.. _VanGenuchtenCapillaryPressure:

######################################
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:

.. math::
    p_w = p_o - P_{c,w}(S_w),

and

.. math::
    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:

.. math::

  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:

.. math::

   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:

.. include:: /docs/sphinx/datastructure/VanGenuchtenCapillaryPressure.rst

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 :doc:`/coreComponents/constitutive/docs/BrooksCoreyRelativePermeability` or in :doc:`/coreComponents/constitutive/docs/ThreePhaseRelativePermeability`. 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 :math:`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 :math:`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 :math:`\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 :math:`\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 :math:`P_{c,w}(S_w) := P_{c,w}(\epsilon)` whenever :math:`S_w < \epsilon`. The gas-phase capillary pressure is treated analogously.

Example
======================

.. code-block:: xml

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