Implementation of the regularized van Genuchten's capillary pressure / relative permeability <-> saturation relation. More...

#include <dumux/material/fluidmatrixinteractions/2p/regularizedvangenuchten.hh>

Inheritance diagram for Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >:

Description

template<class ScalarT, class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>
class Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >

Implementation of the regularized van Genuchten's capillary pressure / relative permeability <-> saturation relation.

This class bundles the "raw" curves as static members and doesn't concern itself converting absolute to effective saturations and vice versa.

In order to avoid very steep gradients the marginal values are "regularized". This means that in stead of following the curve of the material law in these regions, some linear approximation is used. Doing this is not worse than following the material law. E.g. for very low wetting phase values the material laws predict infinite values for \(\mathrm{p_c}\) which is completely unphysical. In case of very high wetting phase saturations the difference between regularized and "pure" material law is not big.

Regularizing has the additional benefit of being numerically friendly: Newton's method does not like infinite gradients.

The implementation is accomplished as follows:

check whether we are in the range of regularization
- yes: use the regularization
- no: forward to the standard material law.

An example of the regularization of the capillary pressure curve is shown below:

See also: VanGenuchten

Public Types
using	Params = ParamsT

using	Scalar = typename Params::Scalar

Static Public Member Functions
static Scalar	pc (const Params &params, Scalar swe)
	A regularized van Genuchten capillary pressure-saturation curve. More...

static Scalar	sw (const Params &params, Scalar pc)
	A regularized van Genuchten saturation-capillary pressure curve. More...

static Scalar	endPointPc (const Params &params)
	The capillary pressure at Swe = 1.0 also called end point capillary pressure. More...

static Scalar	dpc_dswe (const Params &params, Scalar swe)
	A regularized version of the partial derivative of the \(\mathrm{p_c(\overline{S}_w)}\) w.r.t. effective saturation according to van Genuchten. More...

static Scalar	dswe_dpc (const Params &params, Scalar pc)
	A regularized version of the partial derivative of the \(\mathrm{\overline{S}_w(p_c)}\) w.r.t. cap.pressure according to van Genuchten. More...

static Scalar	krw (const Params &params, Scalar swe)
	Regularized version of the relative permeability for the wetting phase of the medium implied by the van Genuchten parameterization. More...

static Scalar	dkrw_dswe (const Params &params, Scalar swe)
	A regularized version of the derivative of the relative permeability for the wetting phase in regard to the wetting saturation of the medium implied by the van Genuchten parameterization. More...

static Scalar	krn (const Params &params, Scalar swe)
	Regularized version of the relative permeability for the non-wetting phase of the medium implied by the van Genuchten parameterization. More...

static Scalar	dkrn_dswe (const Params &params, Scalar swe)
	A regularized version of the derivative of the relative permeability for the non-wetting phase in regard to the wetting saturation of the medium as implied by the van Genuchten parameterization. More...

Member Typedef Documentation

◆ Params

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

using Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::Params = ParamsT

◆ Scalar

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

using Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::Scalar = typename Params::Scalar

Member Function Documentation

◆ dkrn_dswe()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::dkrn_dswe	(	const Params &	params,
		Scalar	swe
	)

inlinestatic

A regularized version of the derivative of the relative permeability for the non-wetting phase in regard to the wetting saturation of the medium as implied by the van Genuchten parameterization.

Parameters

swe	The mobile saturation of the wetting phase.
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Note: Instead of undefined behaviour if pc is not in the valid range, we return a valid number, by clamping the input.

◆ dkrw_dswe()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::dkrw_dswe	(	const Params &	params,
		Scalar	swe
	)

inlinestatic

A regularized version of the derivative of the relative permeability for the wetting phase in regard to the wetting saturation of the medium implied by the van Genuchten parameterization.

Parameters

swe	The mobile saturation of the wetting phase.
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Note: Instead of undefined behaviour if pc is not in the valid range, we return a valid number, by clamping the input.

◆ dpc_dswe()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::dpc_dswe	(	const Params &	params,
		Scalar	swe
	)

inlinestatic

A regularized version of the partial derivative of the \(\mathrm{p_c(\overline{S}_w)}\) w.r.t. effective saturation according to van Genuchten.

regularized part:

low saturation: use the slope of the regularization point (i.e. no kink).
high saturation: connect the high regularization point with \(\mathrm{\overline{S}_w =1}\) by a straight line and use that slope (yes, there is a kink :-( ).

For not-regularized part:

This is equivalent to \(\mathrm{ \frac{\partial p_C}{\partial \overline{S}_w} = -\frac{1}{\alpha} (\overline{S}_w^{-1/m} - 1)^{1/n - } \overline{S}_w^{-1/m} / \overline{S}_w / m }\)

Parameters

swe	Effective saturation of the wetting phase \(\mathrm{\overline{S}_w}\)
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Note: Instead of undefined behaviour if swe is not in the valid range, we return a valid number, by clamping the input.

◆ dswe_dpc()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::dswe_dpc	(	const Params &	params,
		Scalar	pc
	)

inlinestatic

A regularized version of the partial derivative of the \(\mathrm{\overline{S}_w(p_c)}\) w.r.t. cap.pressure according to van Genuchten.

regularized part:

low saturation: use the slope of the regularization point (i.e. no kink).

high saturation: connect the high regularization point with \(\mathrm{\overline{S}_w =1}\) by a straight line and use that slope (yes, there is a kink :-( ).

For not-regularized part:

Parameters

pc	Capillary pressure \(\mathrm{p_C}\) in \(\mathrm{[Pa]}\)
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Note: Instead of undefined behaviour if pc is not in the valid range, we return a valid number, by clamping the input.

◆ endPointPc()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::endPointPc ( const Params & params )

inlinestatic

The capillary pressure at Swe = 1.0 also called end point capillary pressure.

Parameters

params A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

◆ krn()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::krn	(	const Params &	params,
		Scalar	swe
	)

inlinestatic

Regularized version of the relative permeability for the non-wetting phase of the medium implied by the van Genuchten parameterization.

regularized part:

below \(\mathrm{\overline{S}_w =0}\): set relative permeability to zero
above \(\mathrm{\overline{S}_w =1}\): set relative permeability to one
for \(\mathrm{0 \leq \overline{S}_w \leq 0.05}\): use a spline as interpolation

Parameters

swe	The mobile saturation of the wetting phase.
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Note: Instead of undefined behaviour if pc is not in the valid range, we return a valid number, by clamping the input.; See e.g. Dury, Fischer, Schulin (1999) for application of Mualem model to non-wetting rel. perm.

◆ krw()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::krw	(	const Params &	params,
		Scalar	swe
	)

inlinestatic

Regularized version of the relative permeability for the wetting phase of the medium implied by the van Genuchten parameterization.

regularized part:

below \(\mathrm{\overline{S}_w =0}\): set relative permeability to zero
above \(\mathrm{\overline{S}_w =1}\): set relative permeability to one
between \(\mathrm{0.95 \leq \overline{S}_w \leq 1}\): use a spline as interpolation

For not-regularized part:

Parameters

swe	The mobile saturation of the wetting phase.
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Note: Instead of undefined behaviour if pc is not in the valid range, we return a valid number, by clamping the input.

◆ pc()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::pc	(	const Params &	params,
		Scalar	swe
	)

inlinestatic

A regularized van Genuchten capillary pressure-saturation curve.

regularized part:

low saturation: extend the \(\mathrm{p_c(S_w)}\) curve with the slope at the regularization point (i.e. no kink).
high saturation: connect the high regularization point with \(\mathrm{\overline{S}_w =1}\) by a straight line (yes, there is a kink :-( ).

For not-regularized part:

Van Genuchten's empirical capillary pressure <-> saturation function is given by \(\mathrm{ p_C = (\overline{S}_w^{-1/m} - 1)^{1/n}/\alpha }\)

Parameters

swe	Effective saturation of the wetting phase \(\mathrm{\overline{S}_w}\)
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Note: Instead of undefined behaviour if swe is not in the valid range, we return a valid number, by clamping the input. Note that for pc(swe = 0.0) = inf, have a look at RegularizedVanGenuchten if this is a problem.

◆ sw()

template<class ScalarT , class ParamsT = RegularizedVanGenuchtenParams<ScalarT>>

static Scalar Dumux::RegularizedVanGenuchten< ScalarT, ParamsT >::sw	(	const Params &	params,
		Scalar	pc
	)

inlinestatic

A regularized van Genuchten saturation-capillary pressure curve.

regularized part:

low saturation: extend the \(\mathrm{p_c(S_w)}\) curve with the slope at the regularization point (i.e. no kink).
high saturation: connect the high regularization point with \(\mathrm{\overline{S}_w =1}\) by a straight line (yes, there is a kink :-( ).

The according quantities are obtained by exploiting theorem of intersecting lines.

For not-regularized part:

This is the inverse of the capillary pressure-saturation curve: \(\mathrm{ \overline{S}_w = {p_C}^{-1} = ((\alpha p_C)^n + 1)^{-m} }\)

Parameters

pc	Capillary pressure \(\mathrm{p_C}\) in \(\mathrm{[Pa]}\)
params	A container object that is populated with the appropriate coefficients for the respective law. Therefore, in the (problem specific) spatialParameters first, the material law is chosen, and then the params container is constructed accordingly. Afterwards the values are set there, too.

Returns: The effective saturation of the wetting phase \(\mathrm{\overline{S}_w}\)

Note: Instead of undefined behaviour if pc is not in the valid range, we return a valid number, i.e. sw(pc < 0.0) = 0.0, by clamping the input to the physical bounds.

The documentation for this class was generated from the following file:

regularizedvangenuchten.hh

Description

Public Types

Static Public Member Functions

Member Typedef Documentation

◆ Params

◆ Scalar

Member Function Documentation

◆ dkrn_dswe()

◆ dkrw_dswe()

◆ dpc_dswe()

◆ dswe_dpc()

◆ endPointPc()

◆ krn()

◆ krw()

◆ pc()

◆ sw()