PorousFlowBroadbridgeWhite Namespace Reference

Broadbridge-White version of relative permeability, and effective saturation as a function of capillary pressure. More...

Functions
Real	LambertW (Real z)
	Provides the Lambert W function, which satisfies W(z) * exp(W(z)) = z. More...
Real	effectiveSaturation (Real pc, Real c, Real sn, Real ss, Real las)
	Effective saturation as a function of capillary pressure If pc>=0 this will yield 1, otherwise it will yield <1. More...
Real	dEffectiveSaturation (Real pc, Real c, Real sn, Real ss, Real las)
	Derivative of effective saturation wrt capillary pressure. More...
Real	d2EffectiveSaturation (Real pc, Real c, Real sn, Real ss, Real las)
	Second derivative of effective saturation wrt capillary pressure. More...
Real	relativePermeability (Real s, Real c, Real sn, Real ss, Real kn, Real ks)
	Relative permeability as a function of saturation. More...
Real	dRelativePermeability (Real s, Real c, Real sn, Real ss, Real kn, Real ks)
	Derivative of relative permeability with respect to saturation. More...
Real	d2RelativePermeability (Real s, Real c, Real sn, Real ss, Real kn, Real ks)
	Second derivative of relative permeability with respect to saturation. More...

Detailed Description

Broadbridge-White version of relative permeability, and effective saturation as a function of capillary pressure.

Valid for Kn small. P Broadbridge, I White `‘Constant rate rainfall infiltration: A versatile nonlinear model, 1 Analytical solution’'. Water Resources Research 24 (1988) 145–154.

Function Documentation

d2EffectiveSaturation()

Real PorousFlowBroadbridgeWhite::d2EffectiveSaturation	(	Real	pc,
		Real	c,
		Real	sn,
		Real	ss,
		Real	las
	)

Second derivative of effective saturation wrt capillary pressure.

Parameters

pc	capillary pressure
c	BW's C parameter
sn	BW's S_n parameter
ss	BW's S_s parameter
las	BW's lambda_s parameter

Returns

second derivative of effective saturation wrt capillary pressure

Definition at line 114 of file PorousFlowBroadbridgeWhite.C.

{
  if (pressure >= 0.0)
    return 0.0;
  const Real x = (c - 1.0) * std::exp(c - 1.0 - c * pressure / las);
  const Real lamw = LambertW(x);
  return -(ss - sn) * Utility::pow<3>(c) / Utility::pow<2>(las) * lamw * (1.0 - 2.0 * lamw) /
         Utility::pow<5>(1.0 + lamw);
}

Referenced by PorousFlowCapillaryPressureBW::d2EffectiveSaturation().

d2RelativePermeability()

Real PorousFlowBroadbridgeWhite::d2RelativePermeability	(	Real	s,
		Real	c,
		Real	sn,
		Real	ss,
		Real	kn,
		Real	ks
	)

Second derivative of relative permeability with respect to saturation.

Parameters

s	saturation
c	BW's C parameter
sn	BW's S_n parameter
ss	BW's S_s parameter
kn	BW's K_n parameter
ks	BW's K_s parameter

Returns

second derivative of relative permeability wrt saturation

Definition at line 155 of file PorousFlowBroadbridgeWhite.C.

{
  if (s <= sn)
    return 0.0;
 
  if (s >= ss)
    return 0.0;
 
  const Real coef = (ks - kn) * (c - 1.0);
  const Real th = (s - sn) / (ss - sn);
  const Real krelpp = coef * (2.0 / (c - th) + 4.0 * th / Utility::pow<2>(c - th) +
                              2.0 * Utility::pow<2>(th) / Utility::pow<3>(c - th));
  return krelpp / Utility::pow<2>(ss - sn);
}

dEffectiveSaturation()

Real PorousFlowBroadbridgeWhite::dEffectiveSaturation	(	Real	pc,
		Real	c,
		Real	sn,
		Real	ss,
		Real	las
	)

Derivative of effective saturation wrt capillary pressure.

Parameters

pc	capillary pressure
c	BW's C parameter
sn	BW's S_n parameter
ss	BW's S_s parameter
las	BW's lambda_s parameter

Returns

derivative of effective saturation wrt capillary pressure

Definition at line 104 of file PorousFlowBroadbridgeWhite.C.

{
  if (pressure >= 0.0)
    return 0.0;
  const Real x = (c - 1.0) * std::exp(c - 1.0 - c * pressure / las);
  const Real lamw = LambertW(x);
  return (ss - sn) * Utility::pow<2>(c) / las * lamw / Utility::pow<3>(1.0 + lamw);
}

Referenced by PorousFlowCapillaryPressureBW::dEffectiveSaturation().

dRelativePermeability()

Real PorousFlowBroadbridgeWhite::dRelativePermeability	(	Real	s,
		Real	c,
		Real	sn,
		Real	ss,
		Real	kn,
		Real	ks
	)

Derivative of relative permeability with respect to saturation.

Parameters

s	saturation
c	BW's C parameter
sn	BW's S_n parameter
ss	BW's S_s parameter
kn	BW's K_n parameter
ks	BW's K_s parameter

Returns

derivative of relative permeability wrt saturation

Definition at line 140 of file PorousFlowBroadbridgeWhite.C.

{
  if (s <= sn)
    return 0.0;
 
  if (s >= ss)
    return 0.0;
 
  const Real coef = (ks - kn) * (c - 1.0);
  const Real th = (s - sn) / (ss - sn);
  const Real krelp = coef * (2.0 * th / (c - th) + Utility::pow<2>(th) / Utility::pow<2>(c - th));
  return krelp / (ss - sn);
}

Referenced by PorousFlowRelativePermeabilityBW::dRelativePermeability().

effectiveSaturation()

Real PorousFlowBroadbridgeWhite::effectiveSaturation	(	Real	pc,
		Real	c,
		Real	sn,
		Real	ss,
		Real	las
	)

Effective saturation as a function of capillary pressure If pc>=0 this will yield 1, otherwise it will yield <1.

Parameters

pc	capillary pressure
c	BW's C parameter
sn	BW's S_n parameter
ss	BW's S_s parameter
las	BW's lambda_s parameter

Returns

effective saturation

Definition at line 94 of file PorousFlowBroadbridgeWhite.C.

{
  if (pressure >= 0.0)
    return 1.0;
  const Real x = (c - 1.0) * std::exp(c - 1.0 - c * pressure / las);
  const Real th = c / (1.0 + LambertW(x)); // use branch 0 for positive x
  return sn + (ss - sn) * th;
}

Referenced by PorousFlowCapillaryPressureBW::effectiveSaturation().

LambertW()

Real PorousFlowBroadbridgeWhite::LambertW

(

Real

z

)

Provides the Lambert W function, which satisfies W(z) * exp(W(z)) = z.

Parameters

z	z value, which must be positive

Returns

W

Definition at line 16 of file PorousFlowBroadbridgeWhite.C.

{
  /* Lambert W function.
     Was ~/C/LambertW.c written K M Briggs Keith dot Briggs at bt dot com 97 May 21.
     Revised KMB 97 Nov 20; 98 Feb 11, Nov 24, Dec 28; 99 Jan 13; 00 Feb 23; 01 Apr 09
 
     Computes Lambert W function, principal branch.
     See LambertW1.c for -1 branch.
 
     Returned value W(z) satisfies W(z)*exp(W(z))=z
     test data...
        W(1)= 0.5671432904097838730
        W(2)= 0.8526055020137254914
        W(20)=2.2050032780240599705
     To solve (a+b*R)*exp(-c*R)-d=0 for R, use
     R=-(b*W(-exp(-a*c/b)/b*d*c)+a*c)/b/c
 
     Test:
       gcc -DTESTW LambertW.c -o LambertW -lm && LambertW
     Library:
       gcc -O3 -c LambertW.c
 
     Modified trivially by Andy to use MOOSE things
  */
  mooseAssert(z > 0, "LambertW function in RichardsSeff1BWsmall called with negative argument");
 
  const Real eps = 4.0e-16; //, em1=0.3678794411714423215955237701614608;
  Real p, e, t, w;
 
  /* Uncomment this stuff is you ever need to call with a negative argument
  if (z < -em1)
    mooseError("LambertW: bad argument ", z, "\n");
 
  if (0.0 == z)
    return 0.0;
  if (z < -em1+1e-4)
  {
    // series near -em1 in sqrt(q)
    Real q=z+em1,r=std::sqrt(q),q2=q*q,q3=q2*q;
    return
     -1.0
     +2.331643981597124203363536062168*r
     -1.812187885639363490240191647568*q
     +1.936631114492359755363277457668*r*q
     -2.353551201881614516821543561516*q2
     +3.066858901050631912893148922704*r*q2
     -4.175335600258177138854984177460*q3
     +5.858023729874774148815053846119*r*q3
      -8.401032217523977370984161688514*q3*q;  // error approx 1e-16
  }
  */
  /* initial approx for iteration... */
  if (z < 1.0)
  {
    /* series near 0 */
    p = std::sqrt(2.0 * (2.7182818284590452353602874713526625 * z + 1.0));
    w = -1.0 + p * (1.0 + p * (-0.333333333333333333333 + p * 0.152777777777777777777777));
  }
  else
    w = std::log(z); /* asymptotic */
  if (z > 3.0)
    w -= std::log(w); /* useful? */
  for (unsigned int i = 0; i < 10; ++i)
  {
    /* Halley iteration */
    e = std::exp(w);
    t = w * e - z;
    p = w + 1.0;
    t /= e * p - 0.5 * (p + 1.0) * t / p;
    w -= t;
    if (std::abs(t) < eps * (1.0 + std::abs(w)))
      return w; /* rel-abs error */
  }
  /* should never get here */
  mooseError("LambertW: No convergence at z= ", z, "\n");
}

Referenced by d2EffectiveSaturation(), dEffectiveSaturation(), and effectiveSaturation().

relativePermeability()

Real PorousFlowBroadbridgeWhite::relativePermeability	(	Real	s,
		Real	c,
		Real	sn,
		Real	ss,
		Real	kn,
		Real	ks
	)

Relative permeability as a function of saturation.

Parameters

s	saturation
c	BW's C parameter
sn	BW's S_n parameter
ss	BW's S_s parameter
kn	BW's K_n parameter
ks	BW's K_s parameter

Returns

relative permeability

Definition at line 125 of file PorousFlowBroadbridgeWhite.C.

{
  if (s <= sn)
    return kn;
 
  if (s >= ss)
    return ks;
 
  const Real coef = (ks - kn) * (c - 1.0);
  const Real th = (s - sn) / (ss - sn);
  const Real krel = kn + coef * Utility::pow<2>(th) / (c - th);
  return krel;
}

Referenced by PorousFlowRelativePermeabilityBW::relativePermeability().