Real
PorousFlowFluidStateFlash::vaporMassFraction(std::vector<Real> & zi, std::vector<Real> & Ki) const
{
// Check that the sizes of the mass fractions and equilibrium constant vectors are correct
if (Ki.size() != zi.size() + 1)
mooseError("The number of mass fractions or equilibrium components passed to rachfordRice is "
"not correct");
Real v;
// If there are only two components, an analytical solution is possible
if (Ki.size() == 2)
v = vaporMassFraction(zi[0], Ki[0], Ki[1]);
else
{
// More than two components - solve the Rachford-Rice equation using
// Newton-Raphson method
// Initial guess for vapor mass fraction
Real v0 = 0.5;
unsigned int iter = 0;
while (std::abs(rachfordRice(v0, zi, Ki)) > _nr_tol)
{
v0 = v0 - rachfordRice(v0, zi, Ki) / rachfordRiceDeriv(v0, zi, Ki);
iter++;
if (iter > _nr_max_its)
break;
}
v = v0;
}
return v;
}
void
PorousFlowWaterNCG::saturationTwoPhase(Real pressure,
Real temperature,
Real Z,
std::vector<FluidStateProperties> & fsp) const
{
FluidStateProperties & liquid = fsp[_aqueous_phase_number];
FluidStateProperties & gas = fsp[_gas_phase_number];
// Mass fractions
const Real Xncg = liquid.mass_fraction[_gas_fluid_component];
const Real dXncg_dp = liquid.dmass_fraction_dp[_gas_fluid_component];
const Real dXncg_dT = liquid.dmass_fraction_dT[_gas_fluid_component];
const Real Yncg = gas.mass_fraction[_gas_fluid_component];
const Real dYncg_dp = gas.dmass_fraction_dp[_gas_fluid_component];
const Real dYncg_dT = gas.dmass_fraction_dT[_gas_fluid_component];
// Calculate the vapor mass fraction
const Real K0 = Yncg / Xncg;
const Real K1 = (1.0 - Yncg) / (1.0 - Xncg);
const Real vapor_mass_fraction = vaporMassFraction(Z, K0, K1);
// Liquid density is a function of gas saturation through the capillary pressure
// curve, so approximate liquid pressure as gas pressure
const Real liquid_pressure = pressure;
Real liquid_density, liquid_ddensity_dp, liquid_ddensity_dT;
_water_fp.rho_from_p_T(
liquid_pressure, temperature, liquid_density, liquid_ddensity_dp, liquid_ddensity_dT);
// The gas saturation in the two phase case
gas.saturation = vapor_mass_fraction * liquid_density /
(gas.density + vapor_mass_fraction * (liquid_density - gas.density));
const Real dv_dZ = (K1 - K0) / ((K0 - 1.0) * (K1 - 1.0));
const Real denominator = (gas.density + vapor_mass_fraction * (liquid_density - gas.density)) *
(gas.density + vapor_mass_fraction * (liquid_density - gas.density));
const Real ds_dZ = gas.density * liquid_density * dv_dZ / denominator;
const Real dK0_dp = (Xncg * dYncg_dp - Yncg * dXncg_dp) / Xncg / Xncg;
const Real dK0_dT = (Xncg * dYncg_dT - Yncg * dXncg_dT) / Xncg / Xncg;
const Real dK1_dp =
((1.0 - Yncg) * dXncg_dp - (1.0 - Xncg) * dYncg_dp) / (1.0 - Xncg) / (1.0 - Xncg);
const Real dK1_dT =
((1.0 - Yncg) * dXncg_dT - (1.0 - Xncg) * dYncg_dT) / (1.0 - Xncg) / (1.0 - Xncg);
const Real dv_dp =
Z * dK1_dp / (K1 - 1.0) / (K1 - 1.0) + (1.0 - Z) * dK0_dp / (K0 - 1.0) / (K0 - 1.0);
Real ds_dp = gas.density * liquid_density * dv_dp +
vapor_mass_fraction * (1.0 - vapor_mass_fraction) *
(gas.density * liquid_ddensity_dp - gas.ddensity_dp * liquid_density);
ds_dp /= denominator;
const Real dv_dT =
Z * dK1_dT / (K1 - 1.0) / (K1 - 1.0) + (1.0 - Z) * dK0_dT / (K0 - 1.0) / (K0 - 1.0);
Real ds_dT = gas.density * liquid_density * dv_dT +
vapor_mass_fraction * (1.0 - vapor_mass_fraction) *
(gas.density * liquid_ddensity_dT - gas.ddensity_dT * liquid_density);
ds_dT /= denominator;
gas.dsaturation_dp = ds_dp;
gas.dsaturation_dT = ds_dT;
gas.dsaturation_dZ = ds_dZ;
}
