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opm-simulators/opm/material/binarycoefficients/brine_co2.hh

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// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
// vi: set et ts=4 sw=4 sts=4:
/*****************************************************************************
* Copyright (C) 2012 by Andreas Lauser *
* *
* This program is free software: you can redistribute it and/or modify *
* it under the terms of the GNU General Public License as published by *
* the Free Software Foundation, either version 2 of the License, or *
* (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
* GNU General Public License for more details. *
* *
* You should have received a copy of the GNU General Public License *
* along with this program. If not, see <http://www.gnu.org/licenses/>. *
*****************************************************************************/
/*!
* \file
*
* \copydoc Opm::BinaryCoeff::Brine_CO2
*/
#ifndef OPM_BINARY_COEFF_BRINE_CO2_HH
#define OPM_BINARY_COEFF_BRINE_CO2_HH
#include <opm/material/components/brine.hh>
#include <opm/material/components/h2o.hh>
#include <opm/material/components/co2.hh>
#include <opm/material/idealgas.hh>
namespace Opm {
namespace BinaryCoeff {
/*!
* \ingroup Binarycoefficients
* \brief Binary coefficients for brine and CO2.
*/
template<class Scalar, class CO2Tables, bool verbose = true>
class Brine_CO2 {
typedef Opm::H2O<Scalar> H2O;
typedef Opm::CO2<Scalar, CO2Tables> CO2;
typedef Opm::IdealGas<Scalar> IdealGas;
static const int lPhaseIdx = 0; // index of the liquid phase
static const int gPhaseIdx = 1; // index of the gas phase
public:
/*!
* \brief Binary diffusion coefficent [m^2/s] of water in the CO2 phase.
*
* According to "Diffusion of Water in Liquid and Supercritical Carbon Dioxide: An NMR Study",Bin Xu et al., 2002
* \param temperature the temperature [K]
* \param pressure the phase pressure [Pa]
*/
static Scalar gasDiffCoeff(Scalar temperature, Scalar pressure)
{
//Diffusion coefficient of water in the CO2 phase
Scalar const PI=3.141593;
Scalar const k = 1.3806504e-23; // Boltzmann constant
Scalar const c = 4; // slip parameter, can vary between 4 (slip condition) and 6 (stick condition)
Scalar const R_h = 1.72e-10; // hydrodynamic radius of the solute
Scalar mu = CO2::gasViscosity(temperature, pressure); // CO2 viscosity
Scalar D = k / (c * PI * R_h) * (temperature / mu);
return D;
}
/*!
* \brief Binary diffusion coefficent [m^2/s] of CO2 in the brine phase.
*
* \param temperature the temperature [K]
* \param pressure the phase pressure [Pa]
*/
static Scalar liquidDiffCoeff(Scalar temperature, Scalar pressure)
{
//Diffusion coefficient of CO2 in the brine phase
return 2e-9;
}
/*!
* \brief Returns the _mol_ (!) fraction of CO2 in the liquid
* phase and the mol_ (!) fraction of H2O in the gas phase
* for a given temperature, pressure, CO2 density and brine
* salinity.
*
* Implemented according to "Spycher and Pruess 2005"
* applying the activity coefficient expression of "Duan and Sun 2003"
* and the correlations for pure water given in "Spycher, Pruess and Ennis-King 2003"
*
* \param temperature the temperature [K]
* \param pg the gas phase pressure [Pa]
* \param salinity the salinity [kg NaCl / kg solution]
* \param knownPhaseIdx indicates which phases are present
* \param xlCO2 mole fraction of CO2 in brine [mol/mol]
* \param ygH2O mole fraction of water in the gas phase [mol/mol]
*/
static void calculateMoleFractions(const Scalar temperature,
const Scalar pg,
const Scalar salinity,
const int knownPhaseIdx,
Scalar &xlCO2,
Scalar &ygH2O)
{
Scalar A = computeA_(temperature, pg);
/* salinity: conversion from mass fraction to mol fraction */
const Scalar x_NaCl = salinityToMolFrac_(salinity);
// if both phases are present the mole fractions in each phase can be calculate
// with the mutual solubility function
if (knownPhaseIdx < 0) {
Scalar molalityNaCl = molFracToMolality_(x_NaCl); // molality of NaCl //CHANGED
Scalar m0_CO2 = molalityCO2inPureWater_(temperature, pg); // molality of CO2 in pure water
Scalar gammaStar = activityCoefficient_(temperature, pg, molalityNaCl);// activity coefficient of CO2 in brine
Scalar m_CO2 = m0_CO2 / gammaStar; // molality of CO2 in brine
xlCO2 = m_CO2 / (molalityNaCl + 55.508 + m_CO2); // mole fraction of CO2 in brine
ygH2O = A * (1 - xlCO2 - x_NaCl); // mole fraction of water in the gas phase
}
// if only liquid phase is present the mole fraction of CO2 in brine is given and
// and the virtual equilibrium mole fraction of water in the non-existing gas phase can be estimated
// with the mutual solubility function
if (knownPhaseIdx == lPhaseIdx) {
ygH2O = A * (1 - xlCO2 - x_NaCl);
}
// if only gas phase is present the mole fraction of water in the gas phase is given and
// and the virtual equilibrium mole fraction of CO2 in the non-existing liquid phase can be estimated
// with the mutual solubility function
if (knownPhaseIdx == gPhaseIdx) {
//y_H2o = fluidstate.
xlCO2 = 1 - x_NaCl - ygH2O / A;
}
}
/*!
* \brief Henry coefficent \f$\mathrm{[N/m^2]}\f$ for CO2 in brine.
*/
static Scalar henry(Scalar temperature)
{ return fugacityCoefficientCO2(temperature, /*pressure=*/1e5)*1e5; }
/*!
* \brief Returns the fugacity coefficient of the CO2 component in a water-CO2 mixture
*
* (given in Spycher, Pruess and Ennis-King (2003))
*
* \param T the temperature [K]
* \param pg the gas phase pressure [Pa]
*/
static Scalar fugacityCoefficientCO2(Scalar T, Scalar pg)
{
Valgrind::CheckDefined(T);
Valgrind::CheckDefined(pg);
Scalar V = 1 / (CO2::gasDensity(T, pg) / CO2::molarMass()) * 1.e6; // molar volume in cm^3/mol
Scalar pg_bar = pg / 1.e5; // gas phase pressure in bar
Scalar a_CO2 = (7.54e7 - 4.13e4 * T); // mixture parameter of Redlich-Kwong equation
static const Scalar b_CO2 = 27.8; // mixture parameter of Redlich-Kwong equation
static const Scalar R = IdealGas::R * 10.; // ideal gas constant with unit bar cm^3 /(K mol)
Scalar lnPhiCO2, phiCO2;
lnPhiCO2 = std::log(V / (V - b_CO2));
lnPhiCO2 += b_CO2 / (V - b_CO2);
lnPhiCO2 -= 2 * a_CO2 / (R * std::pow(T, 1.5) * b_CO2) * log((V + b_CO2) / V);
lnPhiCO2 +=
a_CO2 * b_CO2
/ (R
* std::pow(T, 1.5)
* b_CO2
* b_CO2)
* (std::log((V + b_CO2) / V)
- b_CO2 / (V + b_CO2));
lnPhiCO2 -= std::log(pg_bar * V / (R * T));
phiCO2 = exp(lnPhiCO2); // fugacity coefficient of CO2
return phiCO2;
}
/*!
* \brief Returns the fugacity coefficient of the H2O component in a water-CO2 mixture
*
* (given in Spycher, Pruess and Ennis-King (2003))
*
* \param T the temperature [K]
* \param pg the gas phase pressure [Pa]
*/
static Scalar fugacityCoefficientH2O(Scalar T, Scalar pg)
{
Scalar V = 1 / (CO2::gasDensity(T, pg) / CO2::molarMass()) * 1.e6; // molar volume in cm^3/mol
Scalar pg_bar = pg / 1.e5; // gas phase pressure in bar
Scalar a_CO2 = (7.54e7 - 4.13e4 * T);// mixture parameter of Redlich-Kwong equation
static const Scalar a_CO2_H2O = 7.89e7;// mixture parameter of Redlich-Kwong equation
static const Scalar b_CO2 = 27.8;// mixture parameter of Redlich-Kwong equation
static const Scalar b_H2O = 18.18;// mixture parameter of Redlich-Kwong equation
static const Scalar R = IdealGas::R * 10.; // ideal gas constant with unit bar cm^3 /(K mol)
Scalar lnPhiH2O, phiH2O;
lnPhiH2O = log(V / (V - b_CO2)) + b_H2O / (V - b_CO2) - 2 * a_CO2_H2O
/ (R * pow(T, 1.5) * b_CO2) * log((V + b_CO2) / V) + a_CO2
* b_H2O / (R * pow(T, 1.5) * b_CO2 * b_CO2) * (log((V + b_CO2)
/ V) - b_CO2 / (V + b_CO2)) - log(pg_bar * V / (R * T));
phiH2O = exp(lnPhiH2O); // fugacity coefficient of H2O
return phiH2O;
}
private:
/*!
* \brief Returns the molality of NaCl (mol NaCl / kg water) for a given mole fraction
*
* \param salinity the salinity [kg NaCl / kg solution]
*/
static Scalar salinityToMolFrac_(Scalar salinity) {
const Scalar Mw = H2O::molarMass(); /* molecular weight of water [kg/mol] */
const Scalar Ms = 58.8e-3; /* molecular weight of NaCl [kg/mol] */
const Scalar X_NaCl = salinity;
/* salinity: conversion from mass fraction to mol fraction */
const Scalar x_NaCl = -Mw * X_NaCl / ((Ms - Mw) * X_NaCl - Ms);
return x_NaCl;
}
/*!
* \brief Returns the molality of NaCl (mol NaCl / kg water) for a given mole fraction (mol NaCl / mol solution)
*
* \param x_NaCl mole fraction of NaCL in brine [mol/mol]
*/
static Scalar molFracToMolality_(Scalar x_NaCl) {
// conversion from mol fraction to molality (dissolved CO2 neglected)
const Scalar mol_NaCl = 55.508 * x_NaCl / (1 - x_NaCl);
return mol_NaCl;
}
/*!
* \brief Returns the equilibrium molality of CO2 (mol CO2 / kg water) for a
* CO2-water mixture at a given pressure and temperature
*
* \param T the temperature [K]
* \param pg the gas phase pressure [Pa]
*/
static Scalar molalityCO2inPureWater_(Scalar temperature, Scalar pg) {
Scalar A = computeA_(temperature, pg); // according to Spycher, Pruess and Ennis-King (2003)
Scalar B = computeB_(temperature, pg); // according to Spycher, Pruess and Ennis-King (2003)
Scalar yH2OinGas = (1 - B) / (1. / A - B); // equilibrium mol fraction of H2O in the gas phase
Scalar xCO2inWater = B * (1 - yH2OinGas); // equilibrium mol fraction of CO2 in the water phase
Scalar molalityCO2 = (xCO2inWater * 55.508) / (1 - xCO2inWater); // CO2 molality
return molalityCO2;
}
/*!
* \brief Returns the activity coefficient of CO2 in brine for a
* molal description. According to "Duan and Sun 2003"
* given in "Spycher and Pruess 2005"
*
* \param temperature the temperature [K]
* \param pg the gas phase pressure [Pa]
* \param molalityNaCl molality of NaCl (mol NaCl / kg water)
*/
static Scalar activityCoefficient_(Scalar temperature,
Scalar pg,
Scalar molalityNaCl)
{
Scalar lambda = computeLambda_(temperature, pg); // lambda_{CO2-Na+}
Scalar xi = computeXi_(temperature, pg); // Xi_{CO2-Na+-Cl-}
Scalar lnGammaStar = 2 * lambda * molalityNaCl + xi * molalityNaCl
* molalityNaCl;
Scalar gammaStar = exp(lnGammaStar);
return gammaStar; // molal activity coefficient of CO2 in brine
}
/*!
* \brief Returns the paramater A for the calculation of
* them mutual solubility in the water-CO2 system.
* Given in Spycher, Pruess and Ennis-King (2003)
*
* \param T the temperature [K]
* \param pg the gas phase pressure [Pa]
*/
static Scalar computeA_(Scalar T, Scalar pg)
{
Scalar deltaP = pg / 1e5 - 1; // pressure range [bar] from p0 = 1bar to pg[bar]
const Scalar v_av_H2O = 18.1; // average partial molar volume of H2O [cm^3/mol]
const Scalar R = IdealGas::R * 10;
Scalar k0_H2O = equilibriumConstantH2O_(T); // equilibrium constant for H2O at 1 bar
Scalar phi_H2O = fugacityCoefficientH2O(T, pg); // fugacity coefficient of H2O for the water-CO2 system
Scalar pg_bar = pg / 1.e5;
Scalar A = k0_H2O / (phi_H2O * pg_bar) * exp(deltaP * v_av_H2O / (R * T));
return A;
}
/*!
* \brief Returns the paramater B for the calculation of
* the mutual solubility in the water-CO2 system.
* Given in Spycher, Pruess and Ennis-King (2003)
*
* \param T the temperature [K]
* \param pg the gas phase pressure [Pa]
*/
static Scalar computeB_(Scalar T, Scalar pg) {
Scalar deltaP = pg / 1e5 - 1; // pressure range [bar] from p0 = 1bar to pg[bar]
const Scalar v_av_CO2 = 32.6; // average partial molar volume of CO2 [cm^3/mol]
const Scalar R = IdealGas::R * 10;
Scalar k0_CO2 = equilibriumConstantCO2_(T); // equilibrium constant for CO2 at 1 bar
Scalar phi_CO2 = fugacityCoefficientCO2(T, pg); // fugacity coefficient of CO2 for the water-CO2 system
Scalar pg_bar = pg / 1.e5;
Scalar B = phi_CO2 * pg_bar / (55.508 * k0_CO2) * exp(-(deltaP
* v_av_CO2) / (R * T));
return B;
}
/*!
* \brief Returns the parameter lambda, which is needed for the
* calculation of the CO2 activity coefficient in the brine-CO2 system.
* Given in Spycher and Pruess (2005)
* \param T the temperature [K]
* \param pg the gas phase pressure [Pa]
*/
static Scalar computeLambda_(Scalar T, Scalar pg)
{
Scalar lambda;
static const Scalar c[6] = { -0.411370585, 6.07632013E-4, 97.5347708,
-0.0237622469, 0.0170656236, 1.41335834E-5 };
Scalar pg_bar = pg / 1.0E5; /* conversion from Pa to bar */
lambda = c[0] + c[1] * T + c[2] / T + c[3] * pg_bar / T + c[4] * pg_bar
/ (630.0 - T) + c[5] * T * std::log(pg_bar);
return lambda;
}
/*!
* \brief Returns the parameter xi, which is needed for the
* calculation of the CO2 activity coefficient in the brine-CO2 system.
* Given in Spycher and Pruess (2005)
* \param T the temperature [K]
* \param pg the gas phase pressure [Pa]
*/
static Scalar computeXi_(Scalar T, Scalar pg)
{
Scalar xi;
static const Scalar c[4] = { 3.36389723E-4, -1.98298980E-5,
2.12220830E-3, -5.24873303E-3 };
Scalar pg_bar = pg / 1.0E5; /* conversion from Pa to bar */
xi = c[0] + c[1] * T + c[2] * pg_bar / T + c[3] * pg_bar / (630.0 - T);
return xi;
}
/*!
* \brief Returns the equilibrium constant for CO2, which is needed for the
* calculation of the mutual solubility in the water-CO2 system
* Given in Spycher, Pruess and Ennis-King (2003)
* \param T the temperature [K]
*/
static Scalar equilibriumConstantCO2_(Scalar T)
{
Scalar TinC = T - 273.15; //temperature in °C
static const Scalar c[3] = { 1.189, 1.304e-2, -5.446e-5 };
Scalar logk0_CO2 = c[0] + c[1] * TinC + c[2] * TinC * TinC;
Scalar k0_CO2 = pow(10, logk0_CO2);
return k0_CO2;
}
/*!
* \brief Returns the equilibrium constant for H2O, which is needed for the
* calculation of the mutual solubility in the water-CO2 system
* Given in Spycher, Pruess and Ennis-King (2003)
* \param T the temperature [K]
*/
static Scalar equilibriumConstantH2O_(Scalar T)
{
Scalar TinC = T - 273.15; //temperature in °C
static const Scalar c[4] = { -2.209, 3.097e-2, -1.098e-4, 2.048e-7 };
Scalar logk0_H2O = c[0] + c[1] * TinC + c[2] * TinC * TinC + c[3]
* TinC * TinC * TinC;
Scalar k0_H2O = pow(10, logk0_H2O);
return k0_H2O;
}
};
} // end namepace BinaryCoeff
} // end namespace Opm
#endif
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