PorousFlowVanGenuchten Namespace Reference

van Genuchten effective saturation, capillary pressure and relative permeability functions. More...

Functions
Real	effectiveSaturation (Real p, Real alpha, Real m)
	Effective saturation as a function of porepressure. More...
Real	dEffectiveSaturation (Real p, Real alpha, Real m)
	Derivative of effective saturation wrt porepressure. More...
Real	d2EffectiveSaturation (Real p, Real alpha, Real m)
	Second derivative of effective saturation wrt porepressure. More...
Real	capillaryPressure (Real seff, Real alpha, Real m, Real pc_max)
	Capillary pressure as a function of effective saturation. More...
Real	dCapillaryPressure (Real seff, Real alpha, Real m, Real pc_max)
	Derivative of capillary pressure wrt effective saturation. More...
Real	d2CapillaryPressure (Real seff, Real alpha, Real m, Real pc_max)
	Second derivative of capillary pressure wrt effective saturation. More...
Real	relativePermeability (Real seff, Real m)
	Relative permeability as a function of effective saturation. More...
Real	dRelativePermeability (Real seff, Real m)
	Derivative of relative permeability with respect to effective saturation. More...
Real	d2RelativePermeability (Real seff, Real m)
	Second derivative of relative permeability with respect to effective saturation. More...
Real	relativePermeabilityNW (Real seff, Real m)
	Relative permeability for a non-wetting phase as a function of effective saturation. More...
Real	dRelativePermeabilityNW (Real seff, Real m)
	Derivative of relative permeability for a non-wetting phase with respect to effective saturation. More...
Real	d2RelativePermeabilityNW (Real seff, Real m)
	Second derivative of relative permeability for a non-wetting phase with respect to effective saturation. More...

Detailed Description

van Genuchten effective saturation, capillary pressure and relative permeability functions.

Note: effective saturation is provided as a function of porepressure, not capillary pressure. Note: capillary pressure and relative permeability are functions of effective saturation. The derivatives are therefore given wrt effective saturation. These derivatives must be multiplied by the derivative of effective saturation wrt the true saturation in objects using these relations.

Based on van Genuchten, M. Th., A closed for equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc., 44, 892-898 (1980).

Function Documentation

capillaryPressure()

Real PorousFlowVanGenuchten::capillaryPressure	(	Real	seff,
		Real	alpha,
		Real	m,
		Real	pc_max
	)

Capillary pressure as a function of effective saturation.

Parameters

seff	effective saturation
alpha	van Genuchten parameter
m	van Genuchten exponent
pc_max	maximum capillary pressure (Pa)

Returns

capillary pressure (Pa)

Definition at line 63 of file PorousFlowVanGenuchten.C.

{
  if (seff >= 1.0)
    return 0.0;
  else if (seff <= 0.0)
    return pc_max;
  else
  {
    Real a = std::pow(seff, -1.0 / m) - 1.0;
    return std::min(std::pow(a, 1.0 - m) / alpha, pc_max);
  }
}

Referenced by PorousFlowCapillaryPressureVG::capillaryPressureCurve().

d2CapillaryPressure()

Real PorousFlowVanGenuchten::d2CapillaryPressure	(	Real	seff,
		Real	alpha,
		Real	m,
		Real	pc_max
	)

Second derivative of capillary pressure wrt effective saturation.

Parameters

seff	effective saturation
alpha	van Genuchten parameter
m	van Genuchten exponent
pc_max	maximum capillary pressure (Pa)

Returns

second derivative of capillary pressure wrt effective saturation

Definition at line 93 of file PorousFlowVanGenuchten.C.

{
  if (seff <= 0.0 || seff >= 1.0)
    return 0.0;
  else
  {
    Real a = std::pow(seff, -1.0 / m) - 1.0;
    // Return 0 if pc > pc_max
    if (std::pow(a, 1.0 - m) / alpha > pc_max)
      return 0.0;
    else
    {
      Real d2pc = -std::pow(a, -1.0 - m) * std::pow(seff, -2.0 - 2.0 / m) +
                  ((1.0 + m) / m) * std::pow(a, -m) * std::pow(seff, -1.0 / m - 2.0);
      d2pc *= (1.0 - m) / m / alpha;
      return d2pc;
    }
  }
}

Referenced by PorousFlowCapillaryPressureVG::d2CapillaryPressureCurve().

d2EffectiveSaturation()

Real PorousFlowVanGenuchten::d2EffectiveSaturation	(	Real	p,
		Real	alpha,
		Real	m
	)

Second derivative of effective saturation wrt porepressure.

Parameters

p	porepressure
alpha	van Genuchten parameter
m	van Genuchten exponent

Returns

second derivative of effective saturation wrt porepressure

Definition at line 46 of file PorousFlowVanGenuchten.C.

{
  if (p >= 0.0)
    return 0.0;
  else
  {
    Real n = 1.0 / (1.0 - m);
    Real inner = 1.0 + std::pow(-alpha * p, n);
    Real dinner_dp = -n * alpha * std::pow(-alpha * p, n - 1.0);
    Real d2inner_dp2 = n * (n - 1.0) * alpha * alpha * std::pow(-alpha * p, n - 2.0);
    Real d2seff_dp2 = m * (m + 1.0) * std::pow(inner, -m - 2.0) * std::pow(dinner_dp, 2.0) -
                      m * std::pow(inner, -m - 1.0) * d2inner_dp2;
    return d2seff_dp2;
  }
}

Referenced by PorousFlowCapillaryPressureVG::d2EffectiveSaturation().

d2RelativePermeability()

Real PorousFlowVanGenuchten::d2RelativePermeability	(	Real	seff,
		Real	m
	)

Second derivative of relative permeability with respect to effective saturation.

Parameters

seff	effective saturation
m	van Genuchten exponent

Returns

second derivative of relative permeability wrt effective saturation

Definition at line 143 of file PorousFlowVanGenuchten.C.

{
  // Guard against division by zero
  if (seff <= 0.0 || seff >= 1.0)
    return 0.0;
 
  const Real a = 1.0 - std::pow(seff, 1.0 / m);
  const Real da = -1.0 / m * std::pow(seff, 1.0 / m - 1.0);
  const Real d2a = -(1.0 / m) * (1.0 / m - 1.0) * std::pow(seff, 1.0 / m - 2.0);
  const Real b = 1.0 - std::pow(a, m);
  const Real db = -m * std::pow(a, m - 1.0) * da;
  const Real d2b = -m * (m - 1.0) * std::pow(a, m - 2.0) * da * da - m * std::pow(a, m - 1.0) * d2a;
 
  return -0.25 * std::pow(seff, -1.5) * Utility::pow<2>(b) + 2.0 * std::pow(seff, -0.5) * b * db +
         2.0 * std::sqrt(seff) * db * db + 2.0 * std::sqrt(seff) * b * d2b;
}

d2RelativePermeabilityNW()

Real PorousFlowVanGenuchten::d2RelativePermeabilityNW	(	Real	seff,
		Real	m
	)

Second derivative of relative permeability for a non-wetting phase with respect to effective saturation.

Parameters

seff	effective saturation
m	van Genuchten exponent

Returns

second derivative of relative permeability wrt effective saturation

Definition at line 190 of file PorousFlowVanGenuchten.C.

{
  // Guard against division by zero
  if (seff <= 0.0 || seff >= 1.0)
    return 0.0;
 
  const Real a = std::pow(1.0 - seff, 1.0 / m);
  const Real da = -1.0 / m * a / (1.0 - seff);
  const Real d2a = 1.0 / m * (1.0 / m - 1) * std::pow(1.0 - seff, 1.0 / m - 2.0);
  const Real b = std::pow(1.0 - a, 2.0 * m);
  const Real db = -2.0 * m * b / (1.0 - a) * da;
  const Real d2b =
      -2.0 * m * (db / (1.0 - a) * da + b * Utility::pow<2>(da / (1.0 - a)) + b / (1.0 - a) * d2a);
 
  return -0.25 * std::pow(seff, -1.5) * b + std::pow(seff, -0.5) * db + std::sqrt(seff) * d2b;
}

dCapillaryPressure()

Real PorousFlowVanGenuchten::dCapillaryPressure	(	Real	seff,
		Real	alpha,
		Real	m,
		Real	pc_max
	)

Derivative of capillary pressure wrt effective saturation.

Parameters

seff	effective saturation
alpha	van Genuchten parameter
m	van Genuchten exponent
pc_max	maximum capillary pressure (Pa)

Returns

derivative of capillary pressure wrt effective saturation

Definition at line 77 of file PorousFlowVanGenuchten.C.

{
  if (seff <= 0.0 || seff >= 1.0)
    return 0.0;
  else
  {
    Real a = std::pow(seff, -1.0 / m) - 1.0;
    // Return 0 if pc > pc_max
    if (std::pow(a, 1.0 - m) / alpha > pc_max)
      return 0.0;
    else
      return (m - 1.0) * std::pow(a, -m) * std::pow(seff, -1.0 - 1.0 / m) / m / alpha;
  }
}

Referenced by PorousFlowCapillaryPressureVG::dCapillaryPressureCurve().

dEffectiveSaturation()

Real PorousFlowVanGenuchten::dEffectiveSaturation	(	Real	p,
		Real	alpha,
		Real	m
	)

Derivative of effective saturation wrt porepressure.

Parameters

p	porepressure
alpha	van Genuchten parameter
m	van Genuchten exponent

Returns

derivative of effective saturation wrt porepressure

Definition at line 31 of file PorousFlowVanGenuchten.C.

{
  if (p >= 0.0)
    return 0.0;
  else
  {
    Real n = 1.0 / (1.0 - m);
    Real inner = 1.0 + std::pow(-alpha * p, n);
    Real dinner_dp = -n * alpha * std::pow(-alpha * p, n - 1.0);
    Real dseff_dp = -m * std::pow(inner, -m - 1) * dinner_dp;
    return dseff_dp;
  }
}

Referenced by PorousFlowCapillaryPressureVG::dEffectiveSaturation().

dRelativePermeability()

Real PorousFlowVanGenuchten::dRelativePermeability	(	Real	seff,
		Real	m
	)

Derivative of relative permeability with respect to effective saturation.

Parameters

seff	effective saturation
m	van Genuchten exponent

Returns

derivative of relative permeability wrt effective saturation

Definition at line 128 of file PorousFlowVanGenuchten.C.

{
  // Guard against division by zero
  if (seff <= 0.0 || seff >= 1.0)
    return 0.0;
 
  const Real a = 1.0 - std::pow(seff, 1.0 / m);
  const Real da = -1.0 / m * std::pow(seff, 1.0 / m - 1.0);
  const Real b = 1.0 - std::pow(a, m);
  const Real db = -m * std::pow(a, m - 1.0) * da;
 
  return 0.5 * std::pow(seff, -0.5) * Utility::pow<2>(b) + 2.0 * std::sqrt(seff) * b * db;
}

Referenced by PorousFlowRelativePermeabilityVG::dRelativePermeability().

dRelativePermeabilityNW()

Real PorousFlowVanGenuchten::dRelativePermeabilityNW	(	Real	seff,
		Real	m
	)

Derivative of relative permeability for a non-wetting phase with respect to effective saturation.

Parameters

seff	effective saturation
m	van Genuchten exponent

Returns

derivative of relative permeability wrt effective saturation

Definition at line 175 of file PorousFlowVanGenuchten.C.

{
  // Guard against division by zero
  if (seff <= 0.0 || seff >= 1.0)
    return 0.0;
 
  const Real a = std::pow(1.0 - seff, 1.0 / m);
  const Real da = -1.0 / m * a / (1.0 - seff);
  const Real b = std::pow(1.0 - a, 2.0 * m);
  const Real db = -2.0 * m * b / (1.0 - a) * da;
 
  return 0.5 * std::pow(seff, -0.5) * b + std::sqrt(seff) * db;
}

Referenced by PorousFlowRelativePermeabilityVG::dRelativePermeability().

effectiveSaturation()

Real PorousFlowVanGenuchten::effectiveSaturation	(	Real	p,
		Real	alpha,
		Real	m
	)

Effective saturation as a function of porepressure.

Note: seff = 1 for p >= 0

Parameters

p	porepressure
alpha	van Genuchten parameter
m	van Genuchten exponent

Returns

effective saturation

Definition at line 16 of file PorousFlowVanGenuchten.C.

{
  Real n, seff;
 
  if (p >= 0.0)
    return 1.0;
  else
  {
    n = 1.0 / (1.0 - m);
    seff = 1.0 + std::pow(-alpha * p, n);
    return std::pow(seff, -m);
  }
}

Referenced by PorousFlowCapillaryPressureVG::effectiveSaturation().

relativePermeability()

Real PorousFlowVanGenuchten::relativePermeability	(	Real	seff,
		Real	m
	)

Relative permeability as a function of effective saturation.

Parameters

seff	effective saturation
m	van Genuchten exponent

Returns

relative permeability

Definition at line 114 of file PorousFlowVanGenuchten.C.

{
  if (seff <= 0.0)
    return 0.0;
  else if (seff >= 1.0)
    return 1.0;
 
  const Real a = 1.0 - std::pow(seff, 1.0 / m);
  const Real b = 1.0 - std::pow(a, m);
 
  return std::sqrt(seff) * Utility::pow<2>(b);
}

Referenced by PorousFlowRelativePermeabilityVG::relativePermeability().

relativePermeabilityNW()

Real PorousFlowVanGenuchten::relativePermeabilityNW	(	Real	seff,
		Real	m
	)

Relative permeability for a non-wetting phase as a function of effective saturation.

Parameters

seff	effective saturation
m	van Genuchten exponent

Returns

relative permeability

Definition at line 161 of file PorousFlowVanGenuchten.C.

{
  if (seff <= 0.0)
    return 0.0;
  else if (seff >= 1.0)
    return 1.0;
 
  const Real a = std::pow(1.0 - seff, 1.0 / m);
  const Real b = std::pow(1.0 - a, 2.0 * m);
 
  return std::sqrt(seff) * b;
}

Referenced by PorousFlowRelativePermeabilityVG::relativePermeability().