heavyoil.hh

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#ifndef DUMUX_HEAVYOIL_HH
#define DUMUX_HEAVYOIL_HH
 
#include <dumux/material/idealgas.hh>
#include <dumux/material/constants.hh>
 
#include <dumux/material/components/base.hh>
#include <dumux/material/components/liquid.hh>
#include <dumux/material/components/gas.hh>
 
namespace Dumux {
namespace Components {

template <class Scalar>
class HeavyOil
: public Components::Base<Scalar, HeavyOil<Scalar> >
, public Components::Liquid<Scalar, HeavyOil<Scalar> >
, public Components::Gas<Scalar, HeavyOil<Scalar> >
{
    using Consts = Dumux::Constants<Scalar>;
    using IdealGas = Dumux::IdealGas<Scalar>;
public:
    static std::string name()
    { return "heavyoil"; }

    constexpr static Scalar molarMass()
    { return .350; }

    constexpr static Scalar refComponentMolecularWeight()
    { return .400; }

 
    constexpr static Scalar molecularWeight()
    { return .350; }

    constexpr static Scalar specificGravity()
    { return 0.91; }

    static Scalar tripleTemperature()
    {
        DUNE_THROW(Dune::NotImplemented, "tripleTemperature for heavyoil");
    }

    static Scalar triplePressure()
    {
        DUNE_THROW(Dune::NotImplemented, "triplePressure for heavyoil");
    }
 
    static Scalar refComponentSpecificGravity()
    {
        const Scalar A = 0.83;
        const Scalar B = 89.9513;
        const Scalar C = 139.6612;
        const Scalar D = 3.2033;
        const Scalar E = 1.0564;
 
        const Scalar mW = refComponentMolecularWeight() *1000. ;  // in [g/mol];
 
        using std::pow;
        return A+(B/mW)-(C/pow((mW+D),E));
    }
 
    static Scalar perbutationFactorBoilingTemperature()
    {
        const Scalar A = -7.4120e-2;    //All factors for 1 atm / 101325 pascals [760 mmHg]
        const Scalar B = -7.5041e-3;
        const Scalar C = -2.6031;
        const Scalar D = 9.0180e-2;
        const Scalar E = -1.0482;
 
        using std::log;
        Scalar deltaSpecificGravity = log(refComponentSpecificGravity()/specificGravity());
        Scalar deltaMolecularWeight = log(refComponentMolecularWeight()/molecularWeight());
 
        using std::pow;
        return A*pow(deltaSpecificGravity,2) + B*deltaSpecificGravity + C*pow(deltaMolecularWeight,2) + D*deltaMolecularWeight
                + E*deltaSpecificGravity*deltaMolecularWeight;
    }
 
    static Scalar perbutationFactorCriticalTemperature()
    {
        const Scalar A = -6.1294e-2;
        const Scalar B = -7.0862e-2;
        const Scalar C = 6.1976e-1;
        const Scalar D = -5.7090e-2;
        const Scalar E = -8.4583e-2;
 
        using std::log;
        Scalar deltaSpecificGravity = log(refComponentSpecificGravity()/specificGravity());
        Scalar deltaMolecularWeight = log(refComponentMolecularWeight()/molecularWeight());
 
        using std::pow;
        return A*pow(deltaSpecificGravity,2) + B*deltaSpecificGravity + C*pow(deltaMolecularWeight,2) + D*deltaMolecularWeight
                + E*deltaSpecificGravity*deltaMolecularWeight;
    }
 
    static Scalar perbutationFactorCriticalPressure()
    {
        const Scalar A = 1.8270e-1;
        const Scalar B = -2.4864e-1;
        const Scalar C = 8.3611;
        const Scalar D = -2.2389e-1;
        const Scalar E = 2.6984;
 
        using std::log;
        Scalar deltaSpecificGravity = log(refComponentSpecificGravity()/specificGravity());
        Scalar deltaMolecularWeight = log(refComponentMolecularWeight()/molecularWeight());
 
        using std::pow;
        return A*pow(deltaSpecificGravity,2) + B*deltaSpecificGravity + C*pow(deltaMolecularWeight,2) + D*deltaMolecularWeight
                + E*deltaSpecificGravity*deltaMolecularWeight;
    }
 
    static Scalar refComponentBoilingTemperature()
    {
        const Scalar A = 477.63;    //All factors for 1 atm /  101325 pascals [760 mmHg]
        const Scalar B = 88.51;
        const Scalar C = 1007;
        const Scalar D = 1214.40;
 
        using std::log;
        return A*log((1000.*refComponentMolecularWeight() + B)/(1000.*refComponentMolecularWeight()+C)) + D;
    }
 
    static Scalar refComponentCriticalTemperature()
    {
        const Scalar A = 226.50;
        const Scalar B = 6.78;
        const Scalar C = 1.282e6;
        const Scalar D = 2668;
 
        using std::log;
        return A*log((1000.*refComponentMolecularWeight() + B)/(1000.*refComponentMolecularWeight()+C)) + D ;
    }
 
    static Scalar refComponentCriticalPressure()
    {
        const Scalar A = 141.20;
        const Scalar B = 45.66e-2;
        const Scalar C = 16.59e-3;
        const Scalar D = 2.19;
 
        using std::pow;
        return (A*1000.*molecularWeight())/(pow(B + (C*1000.*molecularWeight()),D)) ;
    }

    static Scalar boilingTemperature()
    {
        using std::pow;
        return refComponentBoilingTemperature() * pow((1 + 2*perbutationFactorBoilingTemperature())/(1 - 2*perbutationFactorBoilingTemperature()),2);
    }

    static Scalar criticalTemperature()
    {
        using std::pow;
        return refComponentCriticalTemperature() * pow((1 + 2*perbutationFactorCriticalTemperature())/(1 - 2*perbutationFactorCriticalTemperature()),2);
    }

    static Scalar criticalPressure()
    {
        using std::pow;
        return refComponentCriticalPressure() * pow((1 + 2*perbutationFactorCriticalPressure())/(1 - 2*perbutationFactorCriticalPressure()),2);
    }

    static Scalar vaporPressure(Scalar temperature)
    {
        const Scalar A = 8.25990;
        const Scalar B = 2830.065;
        const Scalar C = 42.95101;
 
        Scalar T = temperature - 273.15;
 
        using std::pow;
        return 100*1.334*pow(10.0, (A - (B/(T + C))));  // in [Pa]
    }
 
    static Scalar vaporTemperature(Scalar pressure)
    {
        const Scalar A = 8.25990;
        const Scalar B = 2830.065;
        const Scalar C = 42.95101;
 
        const Scalar P = pressure;
 
        using std::log10;
        return  Scalar ((B/(A-log10(P/100*1.334)))-C);
    }

    static Scalar liquidEnthalpy(const Scalar temperature,
                                 const Scalar pressure)
    {
        // Gauss quadrature rule:
        // Interval: [0K; temperature (K)]
        // Gauss-Legendre-Integration with variable transformation:
        // \int_a^b f(T) dT  \approx (b-a)/2 \sum_i=1^n \alpha_i f( (b-a)/2 x_i + (a+b)/2 )
        // with: n=2, legendre -> x_i = +/- \sqrt(1/3), \apha_i=1
        // here: a=273.15K, b=actual temperature in Kelvin
        // \leadsto h(T) = \int_273.15^T c_p(T) dT
        //              \approx 0.5 (T-273.15) * (cp( 0.5(temperature-273.15)sqrt(1/3) ) + cp(0.5(temperature-273.15)(-1)sqrt(1/3))
 
        // Enthalpy may have arbitrary reference state, but the empirical/fitted heatCapacity function needs Kelvin as input and is
        // fit over a certain temperature range. This suggests choosing an interval of integration being in the actual fit range.
        // I.e. choosing T=273.15K  as reference point for liquid enthalpy.
        using std::sqrt;
        const Scalar sqrt1over3 = sqrt(1./3.);
        const Scalar TEval1 = 0.5*(temperature-273.15)*        sqrt1over3 + 0.5*(273.15+temperature)  ; // evaluation points according to Gauss-Legendre integration
        const Scalar TEval2 = 0.5*(temperature-273.15)* (-1)*  sqrt1over3 + 0.5*(273.15+temperature)  ; // evaluation points according to Gauss-Legendre integration
 
        const Scalar h_n = 0.5 * (temperature-273.15) * ( liquidHeatCapacity(TEval1, pressure) + liquidHeatCapacity(TEval2, pressure) ) ;
 
        return h_n;
    }

    static Scalar heatVap(Scalar temperature,
                   const  Scalar pressure)
    {
        using std::min;
        using std::max;
        temperature = min(temperature, criticalTemperature()); // regularization
        temperature = max(temperature, 0.0); // regularization
 
        Scalar T_crit = criticalTemperature();
        Scalar Tr1 = boilingTemperature()/criticalTemperature();
        Scalar p_crit = criticalPressure();
 
        //        Chen method, eq. 7-11.4 (at boiling)
        using std::log;
        const Scalar DH_v_boil = Consts::R * T_crit * Tr1
                                        * (3.978 * Tr1 - 3.958 + 1.555*log(p_crit * 1e-5 /*Pa->bar*/ ) )
                                        / (1.07 - Tr1); /* [J/mol] */
 
        /* Variation with temp according to Watson relation eq 7-12.1*/
        using std::pow;
        const Scalar Tr2 = temperature/criticalTemperature();
        const Scalar n = 0.375;
        const Scalar DH_vap = DH_v_boil * pow(((1.0 - Tr2)/(1.0 - Tr1)), n);
 
        return (DH_vap/molarMass());          // we need [J/kg]
    }
 

    static Scalar gasEnthalpy(Scalar temperature, Scalar pressure)
    {
        return liquidEnthalpy(temperature,pressure) + heatVap(temperature, pressure);
    }

    static Scalar gasDensity(Scalar temperature, Scalar pressure)
    {
        return IdealGas::density(molarMass(), temperature, pressure);
    }

    static Scalar gasMolarDensity(Scalar temperature, Scalar pressure)
    { return IdealGas::molarDensity(temperature, pressure); }

    static Scalar liquidDensity(Scalar temperature, Scalar pressure)
    {
 
        /* according to Lashanizadegan et al (2008) in Chemical Engineering Communications:  */
        /* Simultaneous Heat and Fluid Flow in Porous Media: Case Study: Steam Injection for Tertiary Oil Recovery */
        Scalar rhoReference = 906.; // [kg/m^3] at reference pressure and temperature
        Scalar compressCoeff = 1.e-8; // just a value without justification
        Scalar expansCoeff = 1.e-7; // also just a value
        Scalar rho = rhoReference * (1. + (pressure - 1.e5)*compressCoeff) * (1. - (temperature - 293.)*expansCoeff);
 
        return rho; // [kg/m^3]
    }

    static Scalar liquidMolarDensity(Scalar temperature, Scalar pressure)
    { return liquidDensity(temperature, pressure)/molarMass(); }

    static constexpr bool gasIsCompressible()
    { return true; }

    static constexpr bool gasIsIdeal()
    { return true; }

    static constexpr bool liquidIsCompressible()
    { return true; }

    static Scalar gasViscosity(Scalar temperature, Scalar pressure, bool regularize=true)
    {
        using std::min;
        using std::max;
        temperature = min(temperature, 500.0); // regularization
        temperature = max(temperature, 250.0);
 
        // reduced temperature
        Scalar Tr = temperature/criticalTemperature();
 
        using std::pow;
        using std::exp;
        Scalar Fp0 = 1.0;
        Scalar xi = 0.00474;
        Scalar eta_xi =
            Fp0*(0.807*pow(Tr,0.618)
                 - 0.357*exp(-0.449*Tr)
                 + 0.34*exp(-4.058*Tr)
                 + 0.018);
 
        return eta_xi/xi/1e7; // [Pa s]
    }

    static Scalar liquidViscosity(Scalar temperature, Scalar pressure)
    {
 
        /* according to Lashanizadegan et al (2008) in Chemical Engineering Communications:  */
        /* Simultaneous Heat and Fluid Flow in Porous Media: Case Study: Steam Injection for Tertiary Oil Recovery */
 
        //using std::exp;
        //return 1027919.422*exp(-0.04862*temperature); // [Pa s]
 
        //according to http://www.ecltechnology.com/subsur/reports/pvt_tgb.pdf [Page 10]
        Scalar temperatureFahrenheit = (9/5)*(temperature-273.15)+32;
        Scalar API = 9;
 
        using std::pow;
        return ((pow(10,0.10231*pow(API,2)-3.9464*API+46.5037))*(pow(temperatureFahrenheit,-0.04542*pow(API,2)+1.70405*API-19.18)))*0.001;
 
    }

    static Scalar liquidHeatCapacity(const Scalar temperature,
                                     const Scalar pressure)
    {
        return 618.; // J/(kg K)
    }

    static Scalar liquidThermalConductivity( Scalar temperature,  Scalar pressure)
    {
        return 0.127;
    }
};
 
} // end namespace Components
 
} // end namespace Dumux
 
#endif