brineco2.hh

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#ifndef DUMUX_BRINE_CO2_FLUID_SYSTEM_HH
#define DUMUX_BRINE_CO2_FLUID_SYSTEM_HH
 
#include <type_traits>
 
#include <dune/common/exceptions.hh>
 
#include <dumux/common/parameters.hh>
#include <dumux/material/idealgas.hh>
#include <dumux/material/fluidsystems/base.hh>
#include <dumux/material/fluidsystems/brine.hh>
#include <dumux/material/fluidstates/adapter.hh>
 
#include <dumux/material/components/brine.hh>
#include <dumux/material/components/co2.hh>
#include <dumux/material/components/co2tablereader.hh>
#include <dumux/material/components/tabulatedcomponent.hh>
 
#include <dumux/material/binarycoefficients/brine_co2.hh>
 
#include <dumux/io/name.hh>
 
namespace Dumux {
 
// include the default tables for CO2
#ifndef DOXYGEN // hide tables from doxygen
#include <dumux/material/components/co2tables.inc>
#endif
 
namespace FluidSystems {
namespace Detail {
 
    template<bool useConstantSalinity>
    struct BrineCO2Indices;
 
    template<>
    struct BrineCO2Indices<true>
    {
        static constexpr int BrineIdx = 0;
    };
 
    template<>
    struct BrineCO2Indices<false>
    {
        static constexpr int H2OIdx = 0;
        static constexpr int NaClIdx = 2;
        static constexpr int comp2Idx = 2;
    };
} // end namespace Detail
 
template<bool salinityIsConstant, bool fastButSimplifiedRelations = false>
struct BrineCO2DefaultPolicy
{
    static constexpr bool useConstantSalinity() { return salinityIsConstant; }
    static constexpr bool useCO2GasDensityAsGasMixtureDensity() { return fastButSimplifiedRelations; }
};
 
template< class Scalar,
          class CO2Table,
          class H2OType = Components::TabulatedComponent<Components::H2O<Scalar>>,
          class Policy = BrineCO2DefaultPolicy</*constantSalinity?*/true> >
class BrineCO2
: public Base<Scalar, BrineCO2<Scalar, CO2Table, H2OType, Policy>>
, public Detail::BrineCO2Indices<Policy::useConstantSalinity()>
{
    using ThisType = BrineCO2<Scalar, CO2Table, H2OType, Policy>;
    using Base = Dumux::FluidSystems::Base<Scalar, ThisType>;
 
    // binary coefficients
    using Brine_CO2 = BinaryCoeff::Brine_CO2<Scalar, CO2Table>;
 
    // use constant salinity brine?
    static constexpr bool useConstantSalinity = Policy::useConstantSalinity();
 
    // The possible brine types
    using VariableSalinityBrine = Dumux::FluidSystems::Brine<Scalar, H2OType>;
    using ConstantSalinityBrine = Dumux::Components::Brine<Scalar, H2OType>;
    using BrineType = typename std::conditional_t< useConstantSalinity,
                                                   ConstantSalinityBrine,
                                                   VariableSalinityBrine >;
 
    static constexpr int BrineOrH2OIdx = 0;
    static constexpr int NaClIdx = 2;
 
public:
    using ParameterCache = NullParameterCache;
 
    using H2O = H2OType;
    using Brine = BrineType;
    using CO2 = Dumux::Components::CO2<Scalar, CO2Table>;
 
    static constexpr int numComponents = useConstantSalinity ? 2 : 3;
    static constexpr int numPhases = 2;
 
    static constexpr int liquidPhaseIdx = 0; 
    static constexpr int gasPhaseIdx = 1;    
    static constexpr int phase0Idx = liquidPhaseIdx; 
    static constexpr int phase1Idx = gasPhaseIdx;    
 
    static constexpr int comp0Idx = 0;
    static constexpr int comp1Idx = 1;
 
    // CO2 is always the second component
    static constexpr int CO2Idx = comp1Idx;
 
private:
 
    // Adapter policy for the fluid state corresponding to the brine fluid system
    struct BrineAdapterPolicy
    {
        using FluidSystem = VariableSalinityBrine;
 
        static constexpr int phaseIdx(int brinePhaseIdx) { return liquidPhaseIdx; }
        static constexpr int compIdx(int brineCompIdx)
        {
            switch (brineCompIdx)
            {
                assert(brineCompIdx == VariableSalinityBrine::H2OIdx || brineCompIdx == VariableSalinityBrine::NaClIdx);
                case VariableSalinityBrine::H2OIdx: return BrineOrH2OIdx;
                case VariableSalinityBrine::NaClIdx: return NaClIdx;
                default: return 0; // this will never be reached, only needed to suppress compiler warning
            }
        }
    };
 
    template<class FluidState>
    using BrineAdapter = FluidStateAdapter<FluidState, BrineAdapterPolicy>;
 
public:
 
    static std::string phaseName(int phaseIdx)
    {
        switch (phaseIdx)
        {
            case liquidPhaseIdx: return IOName::liquidPhase();
            case gasPhaseIdx: return IOName::gaseousPhase();
        }
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index " << phaseIdx);
    }
 
    static constexpr bool isMiscible()
    { return true; }
 
    static constexpr bool isGas(int phaseIdx)
    {
        assert(0 <= phaseIdx && phaseIdx < numPhases);
 
        return phaseIdx == gasPhaseIdx;
    }
 
    static constexpr bool isIdealGas(int phaseIdx)
    {
        assert(0 <= phaseIdx && phaseIdx < numPhases);
        // let the fluids decide
        if (phaseIdx == gasPhaseIdx)
            return useConstantSalinity ? (ConstantSalinityBrine::gasIsIdeal() && CO2::gasIsIdeal())
                                       : (H2O::gasIsIdeal() && CO2::gasIsIdeal());
        return false; // not a gas
    }
 
    static bool isIdealMixture(int phaseIdx)
    {
        assert(0 <= phaseIdx && phaseIdx < numPhases);
        if (phaseIdx == liquidPhaseIdx)
            return false;
        return true;
    }
 
    static constexpr bool isCompressible(int phaseIdx)
    {
        assert(0 <= phaseIdx && phaseIdx < numPhases);
        if (phaseIdx == liquidPhaseIdx)
            return useConstantSalinity ? ConstantSalinityBrine::liquidIsCompressible()
                                       : VariableSalinityBrine::isCompressible(VariableSalinityBrine::liquidPhaseIdx);
        return true;
    }
 
    static std::string componentName(int compIdx)
    {
        assert(0 <= compIdx && compIdx < numComponents);
        if (useConstantSalinity)
        {
            static std::string name[] = { ConstantSalinityBrine::name(), CO2::name() };
            return name[compIdx];
        }
        else
        {
            static std::string name[] = { VariableSalinityBrine::componentName(VariableSalinityBrine::H2OIdx),
                                          CO2::name(),
                                          VariableSalinityBrine::NaCl::name() };
            return name[compIdx];
        }
    }
 
    static Scalar molarMass(int compIdx)
    {
        assert(0 <= compIdx && compIdx < numComponents);
        if (useConstantSalinity)
        {
            static const Scalar M[] = { ConstantSalinityBrine::molarMass(), CO2::molarMass() };
            return M[compIdx];
        }
        else
        {
            static const Scalar M[] = { VariableSalinityBrine::molarMass(VariableSalinityBrine::H2OIdx),
                                        CO2::molarMass(),
                                        VariableSalinityBrine::molarMass(VariableSalinityBrine::NaClIdx) };
            return M[compIdx];
        }
    }
 
    /****************************************
     * thermodynamic relations
     ****************************************/
 
    // Initializing with salinity and default tables
    static void init()
    {
        init(/*startTemp=*/273.15, /*endTemp=*/623.15, /*tempSteps=*/100,
             /*startPressure=*/1e4, /*endPressure=*/40e6, /*pressureSteps=*/200);
    }
 
    // Initializing with custom tables
    static void init(Scalar startTemp, Scalar endTemp, int tempSteps,
                     Scalar startPressure, Scalar endPressure, int pressureSteps)
    {
        std::cout << "The Brine-CO2 fluid system was configured with the following policy:\n";
        std::cout << " - use constant salinity: " << std::boolalpha << Policy::useConstantSalinity() << "\n";
        std::cout << " - use CO2 gas density as gas mixture density: " << std::boolalpha << Policy::useCO2GasDensityAsGasMixtureDensity() << std::endl;
 
        if (H2O::isTabulated)
            H2O::init(startTemp, endTemp, tempSteps, startPressure, endPressure, pressureSteps);
    }
 
    using Base::density;
    template <class FluidState>
    static Scalar density(const FluidState& fluidState, int phaseIdx)
    {
        Scalar T = fluidState.temperature(phaseIdx);
        if (phaseIdx == liquidPhaseIdx)
            return liquidDensityMixture_(fluidState);
 
        else if (phaseIdx == gasPhaseIdx)
        {
            if (Policy::useCO2GasDensityAsGasMixtureDensity())
                // use the CO2 gas density only and neglect compositional effects
                return CO2::gasDensity(fluidState.temperature(phaseIdx), fluidState.pressure(phaseIdx));
            else
            {
                // assume ideal mixture: steam and CO2 don't "see" each other
                Scalar rho_gH2O = H2O::gasDensity(T, fluidState.partialPressure(gasPhaseIdx, BrineOrH2OIdx));
                Scalar rho_gCO2 = CO2::gasDensity(T, fluidState.partialPressure(gasPhaseIdx, CO2Idx));
                return (rho_gH2O + rho_gCO2);
            }
        }
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
    using Base::molarDensity;
    template <class FluidState>
    static Scalar molarDensity(const FluidState& fluidState, int phaseIdx)
    {
        Scalar T = fluidState.temperature(phaseIdx);
        if (phaseIdx == liquidPhaseIdx)
            return density(fluidState, phaseIdx)/fluidState.averageMolarMass(phaseIdx);
        else if (phaseIdx == gasPhaseIdx)
        {
            if (Policy::useCO2GasDensityAsGasMixtureDensity())
                return CO2::gasMolarDensity(fluidState.temperature(phaseIdx), fluidState.pressure(phaseIdx));
            else
            {
                // assume ideal mixture: steam and CO2 don't "see" each other
                Scalar rhoMolar_gH2O = H2O::gasMolarDensity(T, fluidState.partialPressure(gasPhaseIdx, BrineOrH2OIdx));
                Scalar rhoMolar_gCO2 = CO2::gasMolarDensity(T, fluidState.partialPressure(gasPhaseIdx, CO2Idx));
                return rhoMolar_gH2O + rhoMolar_gCO2;
            }
        }
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
    using Base::viscosity;
    template <class FluidState>
    static Scalar viscosity(const FluidState& fluidState, int phaseIdx)
    {
        Scalar T = fluidState.temperature(phaseIdx);
        Scalar p = fluidState.pressure(phaseIdx);
 
        if (phaseIdx == liquidPhaseIdx)
            return useConstantSalinity ? ConstantSalinityBrine::liquidViscosity(T, p)
                                       : VariableSalinityBrine::viscosity( BrineAdapter<FluidState>(fluidState),
                                                                           VariableSalinityBrine::liquidPhaseIdx );
        else if (phaseIdx == gasPhaseIdx)
            return CO2::gasViscosity(T, p);
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
    using Base::fugacityCoefficient;
    template <class FluidState>
    static Scalar fugacityCoefficient(const FluidState& fluidState,
                                      int phaseIdx,
                                      int compIdx)
    {
        assert(0 <= compIdx && compIdx < numComponents);
 
        if (phaseIdx == gasPhaseIdx)
            // use the fugacity coefficients of an ideal gas. the
            // actual value of the fugacity is not relevant, as long
            // as the relative fluid compositions are observed,
            return 1.0;
 
        else if (phaseIdx == liquidPhaseIdx)
        {
            Scalar T = fluidState.temperature(phaseIdx);
            Scalar pl = fluidState.pressure(liquidPhaseIdx);
            Scalar pg = fluidState.pressure(gasPhaseIdx);
 
            assert(T > 0);
            assert(pl > 0 && pg > 0);
 
            // calulate the equilibrium composition for given T & p
            Scalar xlH2O, xgH2O;
            Scalar xlCO2, xgCO2;
            const Scalar salinity = useConstantSalinity ? ConstantSalinityBrine::salinity()
                                                        : fluidState.massFraction(liquidPhaseIdx, NaClIdx);
            Brine_CO2::calculateMoleFractions(T, pl, salinity, /*knownGasPhaseIdx=*/-1, xlCO2, xgH2O);
 
            // normalize the phase compositions
            using std::min;
            using std::max;
            xlCO2 = max(0.0, min(1.0, xlCO2));
            xgH2O = max(0.0, min(1.0, xgH2O));
            xlH2O = 1.0 - xlCO2;
            xgCO2 = 1.0 - xgH2O;
 
            if (compIdx == BrineOrH2OIdx)
                return (xgH2O/xlH2O)*(pg/pl);
 
            else if (compIdx == CO2Idx)
                return (xgCO2/xlCO2)*(pg/pl);
 
            // NaCl is assumed to stay in the liquid!
            else if (!useConstantSalinity && compIdx == NaClIdx)
                return 0.0;
 
            DUNE_THROW(Dune::InvalidStateException, "Invalid component index.");
        }
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
    template <class FluidState>
    static Scalar equilibriumMoleFraction(const FluidState& fluidState,
                                          const ParameterCache& paramCache,
                                          int phaseIdx)
    {
        Scalar T = fluidState.temperature(phaseIdx);
        Scalar p = fluidState.pressure(phaseIdx);
 
        assert(T > 0);
        assert(p > 0);
 
        Scalar xgH2O;
        Scalar xlCO2;
 
        // calulate the equilibrium composition for given T & p
        const Scalar salinity = useConstantSalinity ? ConstantSalinityBrine::salinity()
                                                    : fluidState.massFraction(liquidPhaseIdx, NaClIdx);
        Brine_CO2::calculateMoleFractions(T, p, salinity, /*knowgasPhaseIdx=*/-1, xlCO2, xgH2O);
 
        if (phaseIdx == gasPhaseIdx)
            return xgH2O;
        else if (phaseIdx == liquidPhaseIdx)
            return xlCO2;
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
 
    using Base::diffusionCoefficient;
    template <class FluidState>
    static Scalar diffusionCoefficient(const FluidState& fluidState, int phaseIdx, int compIdx)
    { DUNE_THROW(Dune::NotImplemented, "Diffusion coefficients"); }
 
    using Base::binaryDiffusionCoefficient;
    template <class FluidState>
    static Scalar binaryDiffusionCoefficient(const FluidState& fluidState,
                                             int phaseIdx,
                                             int compIIdx,
                                             int compJIdx)
    {
        assert(0 <= compIIdx && compIIdx < numComponents);
        assert(0 <= compJIdx && compJIdx < numComponents);
 
        if (compIIdx > compJIdx)
        {
            using std::swap;
            swap(compIIdx, compJIdx);
        }
 
        Scalar T = fluidState.temperature(phaseIdx);
        Scalar p = fluidState.pressure(phaseIdx);
        if (phaseIdx == liquidPhaseIdx)
        {
            if (compIIdx == BrineOrH2OIdx && compJIdx == CO2Idx)
                return Brine_CO2::liquidDiffCoeff(T, p);
            if (!useConstantSalinity && compIIdx == BrineOrH2OIdx && compJIdx == NaClIdx)
                return VariableSalinityBrine::binaryDiffusionCoefficient( BrineAdapter<FluidState>(fluidState),
                                                                          VariableSalinityBrine::liquidPhaseIdx,
                                                                          VariableSalinityBrine::H2OIdx,
                                                                          VariableSalinityBrine::NaClIdx );
 
            DUNE_THROW(Dune::NotImplemented, "Binary diffusion coefficient of components " <<
                                             compIIdx << " and " << compJIdx  << " in phase " << phaseIdx);
        }
        else if (phaseIdx == gasPhaseIdx)
        {
            if (compIIdx == BrineOrH2OIdx && compJIdx == CO2Idx)
                return Brine_CO2::gasDiffCoeff(T, p);
 
            // NaCl is expected to never be present in the gas phase. we need to
            // return a diffusion coefficient that does not case numerical problems.
            // We choose a very small value here.
            else if (!useConstantSalinity && compIIdx == CO2Idx && compJIdx == NaClIdx)
                return 1e-12;
 
            DUNE_THROW(Dune::NotImplemented, "Binary diffusion coefficient of components " <<
                                             compIIdx << " and " << compJIdx  << " in phase " << phaseIdx);
        }
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
    using Base::enthalpy;
    template <class FluidState>
    static Scalar enthalpy(const FluidState& fluidState, int phaseIdx)
    {
        Scalar T = fluidState.temperature(phaseIdx);
        Scalar p = fluidState.pressure(phaseIdx);
 
        if (phaseIdx == liquidPhaseIdx)
        {
            // Convert J/kg to kJ/kg
            const Scalar h_ls1 = useConstantSalinity ? ConstantSalinityBrine::liquidEnthalpy(T, p)/1e3
                                                     : VariableSalinityBrine::enthalpy( BrineAdapter<FluidState>(fluidState),
                                                                                        VariableSalinityBrine::liquidPhaseIdx )/1e3;
 
            // mass fraction of CO2 in Brine
            const Scalar X_CO2_w = fluidState.massFraction(liquidPhaseIdx, CO2Idx);
 
            // heat of dissolution for CO2 according to Fig. 6 in Duan and Sun 2003. (kJ/kg)
            // In the relevant temperature ranges CO2 dissolution is exothermal
            const Scalar delta_hCO2 = (-57.4375 + T * 0.1325) * 1000/44;
 
            // enthalpy contribution of water and CO2 (kJ/kg)
            const Scalar hw = H2O::liquidEnthalpy(T, p)/1e3;
            const Scalar hg = CO2::liquidEnthalpy(T, p)/1e3 + delta_hCO2;
 
            // Enthalpy of brine with dissolved CO2 (kJ/kg)
            return (h_ls1 - X_CO2_w*hw + hg*X_CO2_w)*1e3;
        }
        else if (phaseIdx == gasPhaseIdx)
        {
            // we assume NaCl to not enter the gas phase, only consider H2O and CO2
            return H2O::gasEnthalpy(T, p)*fluidState.massFraction(gasPhaseIdx, BrineOrH2OIdx)
                   + CO2::gasEnthalpy(T, p) *fluidState.massFraction(gasPhaseIdx, CO2Idx);
        }
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
    template <class FluidState>
    static Scalar componentEnthalpy(const FluidState& fluidState, int phaseIdx, int componentIdx)
    {
        const Scalar T = fluidState.temperature(phaseIdx);
        const Scalar p = fluidState.pressure(phaseIdx);
 
        if (phaseIdx == liquidPhaseIdx)
        {
            if (componentIdx == BrineOrH2OIdx)
                return H2O::liquidEnthalpy(T, p);
            else if (componentIdx == CO2Idx)
                return CO2::liquidEnthalpy(T, p);
            else if (componentIdx == NaClIdx)
                DUNE_THROW(Dune::NotImplemented, "The component enthalpy for NaCl is not implemented.");
            DUNE_THROW(Dune::InvalidStateException, "Invalid component index " << componentIdx);
        }
        else if (phaseIdx == gasPhaseIdx)
        {
            if (componentIdx == BrineOrH2OIdx)
                return H2O::gasEnthalpy(T, p);
            else if (componentIdx == CO2Idx)
                return CO2::gasEnthalpy(T, p);
            else if (componentIdx == NaClIdx)
                DUNE_THROW(Dune::InvalidStateException, "Implementation assumes NaCl not to be present in gas phase");
            DUNE_THROW(Dune::InvalidStateException, "Invalid component index " << componentIdx);
        }
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index " << phaseIdx);
    }
 
    using Base::thermalConductivity;
    template <class FluidState>
    static Scalar thermalConductivity(const FluidState& fluidState, int phaseIdx)
    {
        if (phaseIdx == liquidPhaseIdx)
            return useConstantSalinity ? ConstantSalinityBrine::liquidThermalConductivity( fluidState.temperature(phaseIdx),
                                                                                           fluidState.pressure(phaseIdx) )
                                       : VariableSalinityBrine::thermalConductivity( BrineAdapter<FluidState>(fluidState),
                                                                                     VariableSalinityBrine::liquidPhaseIdx );
        else if (phaseIdx == gasPhaseIdx)
            return CO2::gasThermalConductivity(fluidState.temperature(phaseIdx), fluidState.pressure(phaseIdx));
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
    using Base::heatCapacity;
    template <class FluidState>
    static Scalar heatCapacity(const FluidState &fluidState,
                               int phaseIdx)
    {
        if(phaseIdx == liquidPhaseIdx)
            return useConstantSalinity ? ConstantSalinityBrine::liquidHeatCapacity( fluidState.temperature(phaseIdx),
                                                                                    fluidState.pressure(phaseIdx) )
                                       : VariableSalinityBrine::heatCapacity( BrineAdapter<FluidState>(fluidState),
                                                                              VariableSalinityBrine::liquidPhaseIdx );
        else if (phaseIdx == gasPhaseIdx)
            return CO2::liquidHeatCapacity(fluidState.temperature(phaseIdx),
                                           fluidState.pressure(phaseIdx));
 
        DUNE_THROW(Dune::InvalidStateException, "Invalid phase index.");
    }
 
private:
 
    template<class FluidState>
    static Scalar liquidDensityMixture_(const FluidState& fluidState)
    {
        const auto T = fluidState.temperature(liquidPhaseIdx);
        const auto p = fluidState.pressure(liquidPhaseIdx);
 
        if (T < 273.15)
            DUNE_THROW(NumericalProblem, "Liquid density for Brine and CO2 is only "
                                         "defined above 273.15K (T = " << T << ")");
 
        if (p >= 2.5e8)
            DUNE_THROW(NumericalProblem, "Liquid density for Brine and CO2 is only "
                                         "defined below 250MPa (p = " << p << ")");
 
        // density of pure water
        Scalar rho_pure = H2O::liquidDensity(T, p);
 
        // density of water with dissolved CO2 (neglect NaCl)
        Scalar rho_lCO2;
        if (useConstantSalinity)
        {
            // use normalized composition for to calculate the density
            // (the relations don't seem to take non-normalized compositions too well...)
            // TODO Do we really need this normalization???
            using std::min;
            using std::max;
            Scalar xlBrine = min(1.0, max(0.0, fluidState.moleFraction(liquidPhaseIdx, BrineOrH2OIdx)));
            Scalar xlCO2 = min(1.0, max(0.0, fluidState.moleFraction(liquidPhaseIdx, CO2Idx)));
            Scalar sumx = xlBrine + xlCO2;
            xlBrine /= sumx;
            xlCO2 /= sumx;
 
            rho_lCO2 = liquidDensityWaterCO2_(T, p, xlBrine, xlCO2);
        }
        else
        {
            // rescale mole fractions
            auto xlH2O = fluidState.moleFraction(liquidPhaseIdx, BrineOrH2OIdx);
            auto xlCO2 = fluidState.moleFraction(liquidPhaseIdx, NaClIdx);
            const auto sumMoleFrac = xlH2O + xlCO2;
            xlH2O = xlH2O/sumMoleFrac;
            xlCO2 = xlCO2/sumMoleFrac;
            rho_lCO2 = liquidDensityWaterCO2_(T, p, xlH2O, xlCO2);
        }
 
        // density of brine (water with nacl)
        Scalar rho_brine = useConstantSalinity ? ConstantSalinityBrine::liquidDensity(T, p)
                                               : VariableSalinityBrine::density( BrineAdapter<FluidState>(fluidState),
                                                                                 VariableSalinityBrine::liquidPhaseIdx );
 
        // contribution of co2 to the density
        Scalar contribCO2 = rho_lCO2 - rho_pure;
 
        // Total brine density with dissolved CO2
        // rho_{b, CO2} = rho_pure + contribution(salt) + contribution(CO2)
        return rho_brine + contribCO2;
    }
 
 
    static Scalar liquidDensityWaterCO2_(Scalar temperature,
                                         Scalar pl,
                                         Scalar xlH2O,
                                         Scalar xlCO2)
    {
        const Scalar M_CO2 = CO2::molarMass();
        const Scalar M_H2O = H2O::molarMass();
 
        const Scalar tempC = temperature - 273.15;        /* tempC : temperature in °C */
        const Scalar rho_pure = H2O::liquidDensity(temperature, pl);
 
        // xlH2O is available, but in case of a pure gas phase
        // the value of M_T for the virtual liquid phase can become very large
        xlH2O = 1.0 - xlCO2;
        const Scalar M_T = M_H2O * xlH2O + M_CO2 * xlCO2;
        const Scalar V_phi = (37.51 +
                              tempC*(-9.585e-2 +
                                     tempC*(8.74e-4 -
                                            tempC*5.044e-7))) / 1.0e6;
        return 1/(xlCO2 * V_phi/M_T + M_H2O * xlH2O / (rho_pure * M_T));
    }
};
 
} // end namespace FluidSystems
} // end namespace Dumux
 
#endif