2p/sequential/diffusion/cellcentered/pressure.hh

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#ifndef DUMUX_FVPRESSURE2P_HH
#define DUMUX_FVPRESSURE2P_HH
 
#include <dune/common/float_cmp.hh>
 
// dumux environment
#include <dumux/porousmediumflow/sequential/cellcentered/pressure.hh>
#include <dumux/porousmediumflow/2p/sequential/diffusion/properties.hh>
 
namespace Dumux {
template<class TypeTag> class FVPressure2P: public FVPressure<TypeTag>
{
    using ParentType = FVPressure<TypeTag>;
 
    //the model implementation
    using Implementation = GetPropType<TypeTag, Properties::PressureModel>;
 
    using GridView = typename GetPropType<TypeTag, Properties::GridGeometry>::GridView;
    using Scalar = GetPropType<TypeTag, Properties::Scalar>;
    using Problem = GetPropType<TypeTag, Properties::Problem>;
 
    using SpatialParams = GetPropType<TypeTag, Properties::SpatialParams>;
 
    using Indices = typename GetPropType<TypeTag, Properties::ModelTraits>::Indices;
 
    using FluidSystem = GetPropType<TypeTag, Properties::FluidSystem>;
    using FluidState = GetPropType<TypeTag, Properties::FluidState>;
 
    using BoundaryTypes = GetPropType<TypeTag, Properties::SequentialBoundaryTypes>;
    using SolutionTypes = GetProp<TypeTag, Properties::SolutionTypes>;
    using PrimaryVariables = typename SolutionTypes::PrimaryVariables;
    using CellData = GetPropType<TypeTag, Properties::CellData>;
    using PressureSolutionVector = GetPropType<TypeTag, Properties::PressureSolutionVector>;
 
    using ScalarSolutionType = typename SolutionTypes::ScalarSolution;
 
    enum
    {
        dim = GridView::dimension, dimWorld = GridView::dimensionworld
    };
    enum
    {
        pw = Indices::pressureW,
        pn = Indices::pressureNw,
        pGlobal = Indices::pressureGlobal,
        sw = Indices::saturationW,
        sn = Indices::saturationNw,
        pressureIdx = Indices::pressureIdx,
        saturationIdx = Indices::saturationIdx,
        eqIdxPress = Indices::pressureEqIdx,
        eqIdxSat = Indices::satEqIdx
    };
    enum
    {
        wPhaseIdx = Indices::wPhaseIdx, nPhaseIdx = Indices::nPhaseIdx, numPhases = getPropValue<TypeTag, Properties::NumPhases>()
    };
 
    using Element = typename GridView::Traits::template Codim<0>::Entity;
    using Intersection = typename GridView::Intersection;
 
    using GlobalPosition = typename Element::Geometry::GlobalCoordinate;
    using GravityVector = Dune::FieldVector<Scalar, dimWorld>;
    using DimMatrix = Dune::FieldMatrix<Scalar, dim, dim>;
 
protected:
    using EntryType = typename ParentType::EntryType;
    enum
    {
        rhs = ParentType::rhs, matrix = ParentType::matrix
    };
 
public:
    void getSource(EntryType& entry, const Element& element, const CellData& cellData, const bool first);
 
    void getStorage(EntryType& entry, const Element& element, const CellData& cellData, const bool first);
 
    void getFlux(EntryType& entry, const Intersection& intersection, const CellData& cellData, const bool first);
 
    void getFluxOnBoundary(EntryType& entry,
    const Intersection& intersection, const CellData& cellData, const bool first);
 
    void updateMaterialLaws();
 
    void initialize(bool solveTwice = true)
    {
        ParentType::initialize();
 
        if (!compressibility_)
        {
            const auto element = *problem_.gridView().template begin<0>();
            FluidState fluidState;
            fluidState.setPressure(wPhaseIdx, problem_.referencePressure(element));
            fluidState.setPressure(nPhaseIdx, problem_.referencePressure(element));
            fluidState.setTemperature(problem_.temperature(element));
            fluidState.setSaturation(wPhaseIdx, 1.);
            fluidState.setSaturation(nPhaseIdx, 0.);
            density_[wPhaseIdx] = FluidSystem::density(fluidState, wPhaseIdx);
            density_[nPhaseIdx] = FluidSystem::density(fluidState, nPhaseIdx);
            viscosity_[wPhaseIdx] = FluidSystem::viscosity(fluidState, wPhaseIdx);
            viscosity_[nPhaseIdx] = FluidSystem::viscosity(fluidState, nPhaseIdx);
        }
 
        storePressureSolution();
        updateMaterialLaws();
 
        this->assemble(true);
        this->solve();
        if (solveTwice)
        {
            PressureSolutionVector pressureOld(this->pressure());
 
            this->assemble(false);
            this->solve();
 
            PressureSolutionVector pressureDiff(pressureOld);
            pressureDiff -= this->pressure();
            pressureOld = this->pressure();
            Scalar pressureNorm = pressureDiff.infinity_norm();
            pressureNorm /= pressureOld.infinity_norm();
            int numIter = 0;
            while (pressureNorm > 1e-5 && numIter < 10)
            {
                updateMaterialLaws();
                this->assemble(false);
                this->solve();
 
                pressureDiff = pressureOld;
                pressureDiff -= this->pressure();
                pressureNorm = pressureDiff.infinity_norm();
                pressureOld = this->pressure();
                pressureNorm /= pressureOld.infinity_norm();
 
                numIter++;
            }
            // std::cout<<"Pressure defect = "<<pressureNorm<<"; "<<
            //        numIter<<" Iterations needed for initial pressure field"<<std::endl;
        }
 
        storePressureSolution();
    }
 
    void update()
    {
        timeStep_ = problem_.timeManager().timeStepSize();
        //error bounds for error term for incompressible models
        //to correct unphysical saturation over/undershoots due to saturation transport
        if (!compressibility_)
        {
            maxError_ = 0.0;
            int size = problem_.gridView().size(0);
            for (int i = 0; i < size; i++)
            {
                Scalar sat = 0;
                switch (saturationType_)
                {
                case sw:
                    sat = problem_.variables().cellData(i).saturation(wPhaseIdx);
                    break;
                case sn:
                    sat = problem_.variables().cellData(i).saturation(nPhaseIdx);
                    break;
                }
                if (sat > 1.0)
                {
                    using std::max;
                    maxError_ = max(maxError_, (sat - 1.0) / timeStep_);
                }
                if (sat < 0.0)
                {
                    using std::max;
                    maxError_ = max(maxError_, (-sat) / timeStep_);
                }
            }
        }
 
        ParentType::update();
 
        storePressureSolution();
    }
 
    void updateVelocity()
    {
        updateMaterialLaws();
 
        //reset velocities
        int size = problem_.gridView().size(0);
        for (int i = 0; i < size; i++)
        {
            CellData& cellData = problem_.variables().cellData(i);
            cellData.fluxData().resetVelocity();
        }
    }
 
    void storePressureSolution()
    {
        // iterate through leaf grid
        for (const auto& element : elements(problem_.gridView()))
        {
            asImp_().storePressureSolution(element);
        }
    }
 
    void storePressureSolution(const Element& element)
    {
        int eIdxGlobal = problem_.variables().index(element);
        CellData& cellData = problem_.variables().cellData(eIdxGlobal);
 
        if (compressibility_)
        {
            density_[wPhaseIdx] = cellData.density(wPhaseIdx);
            density_[nPhaseIdx] = cellData.density(nPhaseIdx);
        }
 
        switch (pressureType_)
        {
        case pw:
        {
            Scalar pressW = this->pressure()[eIdxGlobal];
            Scalar pc = cellData.capillaryPressure();
 
            cellData.setPressure(wPhaseIdx, pressW);
            cellData.setPressure(nPhaseIdx, pressW + pc);
 
            Scalar gravityDiff = (problem_.bBoxMax() - element.geometry().center()) * gravity_;
            Scalar potW = pressW + gravityDiff * density_[wPhaseIdx];
            Scalar potNw = pressW + pc + gravityDiff * density_[nPhaseIdx];
 
            cellData.setPotential(wPhaseIdx, potW);
            cellData.setPotential(nPhaseIdx, potNw);
 
            break;
        }
        case pn:
        {
            Scalar pressNw = this->pressure()[eIdxGlobal];
            Scalar pc = cellData.capillaryPressure();
 
            cellData.setPressure(nPhaseIdx, pressNw);
            cellData.setPressure(wPhaseIdx, pressNw - pc);
 
            Scalar gravityDiff = (problem_.bBoxMax() - element.geometry().center()) * gravity_;
            Scalar potW = pressNw - pc + gravityDiff * density_[wPhaseIdx];
            Scalar potNw = pressNw + gravityDiff * density_[nPhaseIdx];
 
            cellData.setPotential(wPhaseIdx, potW);
            cellData.setPotential(nPhaseIdx, potNw);
 
            break;
        }
        case pGlobal:
        {
            Scalar press = this->pressure()[eIdxGlobal];
            cellData.setGlobalPressure(press);
 
            Scalar pc = cellData.capillaryPressure();
            Scalar gravityDiff = (problem_.bBoxMax() - element.geometry().center()) * gravity_;
 
            //This is only an estimation!!! -> only used for uwind directions and time-step estimation!!!
            Scalar potW = press - cellData.fracFlowFunc(nPhaseIdx) * pc + gravityDiff * density_[wPhaseIdx];
            Scalar potNw = press - cellData.fracFlowFunc(wPhaseIdx) * pc + gravityDiff * density_[nPhaseIdx];
 
            cellData.setPotential(wPhaseIdx, potW);
            cellData.setPotential(nPhaseIdx, potNw);
 
            break;
        }
        }
        cellData.fluxData().resetVelocity();
    }
 
    template<class MultiWriter>
    void addOutputVtkFields(MultiWriter &writer)
    {
        int size = problem_.gridView().size(0);
        ScalarSolutionType *pressure = writer.allocateManagedBuffer(size);
        ScalarSolutionType *pressureSecond = 0;
        ScalarSolutionType *pc = 0;
        ScalarSolutionType *potentialW = 0;
        ScalarSolutionType *potentialNw = 0;
        ScalarSolutionType *mobilityW = 0;
        ScalarSolutionType *mobilityNW = 0;
 
        if (vtkOutputLevel_ > 0)
        {
            pressureSecond = writer.allocateManagedBuffer(size);
            pc = writer.allocateManagedBuffer(size);
            mobilityW = writer.allocateManagedBuffer(size);
            mobilityNW = writer.allocateManagedBuffer(size);
        }
        if (vtkOutputLevel_ > 1)
        {
            potentialW = writer.allocateManagedBuffer(size);
            potentialNw = writer.allocateManagedBuffer(size);
        }
 
        for (int i = 0; i < size; i++)
        {
            CellData& cellData = problem_.variables().cellData(i);
 
            if (pressureType_ == pw)
            {
                (*pressure)[i] = cellData.pressure(wPhaseIdx);
                if (vtkOutputLevel_ > 0)
                {
                    (*pressureSecond)[i] = cellData.pressure(nPhaseIdx);
                }
            }
            else if (pressureType_ == pn)
            {
                (*pressure)[i] = cellData.pressure(nPhaseIdx);
                if (vtkOutputLevel_ > 0)
                {
                    (*pressureSecond)[i] = cellData.pressure(wPhaseIdx);
                }
            }
            else if (pressureType_ == pGlobal)
            {
                (*pressure)[i] = cellData.globalPressure();
            }
            if (vtkOutputLevel_ > 0)
            {
                (*pc)[i] = cellData.capillaryPressure();
                (*mobilityW)[i]  = cellData.mobility(wPhaseIdx);
                (*mobilityNW)[i]  = cellData.mobility(nPhaseIdx);
                if (compressibility_)
                {
                    (*mobilityW)[i] = (*mobilityW)[i]/cellData.density(wPhaseIdx);
                    (*mobilityNW)[i] = (*mobilityNW)[i]/cellData.density(nPhaseIdx);
                }
            }
            if (vtkOutputLevel_ > 1)
            {
                (*potentialW)[i]  = cellData.potential(wPhaseIdx);
                (*potentialNw)[i]  = cellData.potential(nPhaseIdx);
            }
        }
 
        if (pressureType_ == pw)
        {
            writer.attachCellData(*pressure, "wetting pressure");
            if (vtkOutputLevel_ > 0)
            {
                writer.attachCellData(*pressureSecond, "nonwetting pressure");
            }
        }
        else if (pressureType_ == pn)
        {
            writer.attachCellData(*pressure, "nonwetting pressure");
            if (vtkOutputLevel_ > 0)
            {
                writer.attachCellData(*pressureSecond, "wetting pressure");
            }
        }
        if (pressureType_ == pGlobal)
        {
 
            writer.attachCellData(*pressure, "global pressure");
        }
 
        if (vtkOutputLevel_ > 0)
        {
            writer.attachCellData(*pc, "capillary pressure");
            writer.attachCellData(*mobilityW, "wetting mobility");
            writer.attachCellData(*mobilityNW, "nonwetting mobility");
        }
        if (vtkOutputLevel_ > 1)
        {
            writer.attachCellData(*potentialW, "wetting potential");
            writer.attachCellData(*potentialNw, "nonwetting potential");
        }
 
        if (compressibility_)
        {
            if (vtkOutputLevel_ > 0)
            {
                ScalarSolutionType *densityWetting = writer.allocateManagedBuffer(size);
                ScalarSolutionType *densityNonwetting = writer.allocateManagedBuffer(size);
                ScalarSolutionType *viscosityWetting = writer.allocateManagedBuffer(size);
                ScalarSolutionType *viscosityNonwetting = writer.allocateManagedBuffer(size);
 
                for (int i = 0; i < size; i++)
                {
                    CellData& cellData = problem_.variables().cellData(i);
                    (*densityWetting)[i] = cellData.density(wPhaseIdx);
                    (*densityNonwetting)[i] = cellData.density(nPhaseIdx);
                    (*viscosityWetting)[i] = cellData.viscosity(wPhaseIdx);
                    (*viscosityNonwetting)[i] = cellData.viscosity(nPhaseIdx);
                }
 
                writer.attachCellData(*densityWetting, "wetting density");
                writer.attachCellData(*densityNonwetting, "nonwetting density");
                writer.attachCellData(*viscosityWetting, "wetting viscosity");
                writer.attachCellData(*viscosityNonwetting, "nonwetting viscosity");
            }
        }
    }
 
    FVPressure2P(Problem& problem) :
            ParentType(problem), problem_(problem), gravity_(problem.gravity()), maxError_(0.), timeStep_(1.)
    {
        if (pressureType_ != pw && pressureType_ != pn && pressureType_ != pGlobal)
        {
            DUNE_THROW(Dune::NotImplemented, "Pressure type not supported!");
        }
        if (pressureType_ == pGlobal && compressibility_)
        {
            DUNE_THROW(Dune::NotImplemented, "Compressibility not supported for global pressure!");
        }
        if (saturationType_ != sw && saturationType_ != sn)
        {
            DUNE_THROW(Dune::NotImplemented, "Saturation type not supported!");
        }
 
        ErrorTermFactor_ = getParam<Scalar>("Impet.ErrorTermFactor");
        ErrorTermLowerBound_ = getParam<Scalar>("Impet.ErrorTermLowerBound");
        ErrorTermUpperBound_ = getParam<Scalar>("Impet.ErrorTermUpperBound");
 
        density_[wPhaseIdx] = 0.;
        density_[nPhaseIdx] = 0.;
        viscosity_[wPhaseIdx] = 0.;
        viscosity_[nPhaseIdx] = 0.;
 
        vtkOutputLevel_ = getParam<int>("Vtk.OutputLevel");
    }
 
private:
    Implementation &asImp_()
    { return *static_cast<Implementation *>(this); }
 
    const Implementation &asImp_() const
    { return *static_cast<const Implementation *>(this); }
 
    Problem& problem_;
    const GravityVector& gravity_; 
 
    Scalar maxError_;
    Scalar timeStep_;
    Scalar ErrorTermFactor_; 
    Scalar ErrorTermLowerBound_; 
    Scalar ErrorTermUpperBound_; 
 
    Scalar density_[numPhases];
    Scalar viscosity_[numPhases];
 
    int vtkOutputLevel_;
 
    static const bool compressibility_ = getPropValue<TypeTag, Properties::EnableCompressibility>();
    static const int pressureType_ = getPropValue<TypeTag, Properties::PressureFormulation>();
    static const int saturationType_ = getPropValue<TypeTag, Properties::SaturationFormulation>();
};
 
template<class TypeTag>
void FVPressure2P<TypeTag>::getSource(EntryType& entry, const Element& element
        , const CellData& cellData, const bool first)
{
    // cell volume, assume linear map here
    Scalar volume = element.geometry().volume();
 
    // get sources from problem
    PrimaryVariables sourcePhase(0.0);
    problem_.source(sourcePhase, element);
 
    if (!compressibility_)
    {
        sourcePhase[wPhaseIdx] /= density_[wPhaseIdx];
        sourcePhase[nPhaseIdx] /= density_[nPhaseIdx];
    }
 
    entry[rhs] = volume * (sourcePhase[wPhaseIdx] + sourcePhase[nPhaseIdx]);
}
 
template<class TypeTag>
void FVPressure2P<TypeTag>::getStorage(EntryType& entry, const Element& element
        , const CellData& cellData, const bool first)
{
    //volume correction due to density differences
    if (compressibility_ && !first)
    {
        // cell volume, assume linear map here
        Scalar volume = element.geometry().volume();
 
        Scalar porosity = problem_.spatialParams().porosity(element);
 
        switch (saturationType_)
        {
        case sw:
        {
            entry[rhs] = -(cellData.volumeCorrection()/timeStep_ * porosity * volume
                    * (cellData.density(wPhaseIdx) - cellData.density(nPhaseIdx)));
            break;
        }
        case sn:
        {
            entry[rhs] = -(cellData.volumeCorrection()/timeStep_ * porosity * volume
                    * (cellData.density(nPhaseIdx) - cellData.density(wPhaseIdx)));
            break;
        }
        }
    }
    else if (!compressibility_ && !first)
    {
        // error term for incompressible models to correct non-physical saturation over/undershoots due to saturation transport
        // error reduction routine: volumetric error is damped and inserted to right hand side
        Scalar sat = 0;
        switch (saturationType_)
        {
        case sw:
            sat = cellData.saturation(wPhaseIdx);
            break;
        case sn:
            sat = cellData.saturation(nPhaseIdx);
            break;
        }
 
        Scalar volume = element.geometry().volume();
 
        Scalar error = (sat > 1.0) ? sat - 1.0 : 0.0;
        if (sat < 0.0) {error =  sat;}
        error /= timeStep_;
 
        using std::abs;
        Scalar errorAbs = abs(error);
 
        if ((errorAbs*timeStep_ > 1e-6) && (errorAbs > ErrorTermLowerBound_ * maxError_) && (!problem_.timeManager().willBeFinished()))
        {
            entry[rhs] = ErrorTermFactor_ * error * volume;
        }
    }
}
 
template<class TypeTag>
void FVPressure2P<TypeTag>::getFlux(EntryType& entry, const Intersection& intersection
        , const CellData& cellData, const bool first)
{
    auto elementI = intersection.inside();
    auto elementJ = intersection.outside();
 
    const CellData& cellDataJ = problem_.variables().cellData(problem_.variables().index(elementJ));
 
    // get global coordinates of cell centers
    const GlobalPosition& globalPosI = elementI.geometry().center();
    const GlobalPosition& globalPosJ = elementJ.geometry().center();
 
    // get mobilities and fractional flow factors
    Scalar lambdaWI = cellData.mobility(wPhaseIdx);
    Scalar lambdaNwI = cellData.mobility(nPhaseIdx);
    Scalar lambdaWJ = cellDataJ.mobility(wPhaseIdx);
    Scalar lambdaNwJ = cellDataJ.mobility(nPhaseIdx);
 
    // get capillary pressure
    Scalar pcI = cellData.capillaryPressure();
    Scalar pcJ = cellDataJ.capillaryPressure();
 
    // get face normal
    const Dune::FieldVector<Scalar, dim>& unitOuterNormal = intersection.centerUnitOuterNormal();
 
    // get face area
    Scalar faceArea = intersection.geometry().volume();
 
    // distance vector between barycenters
    GlobalPosition distVec = globalPosJ - globalPosI;
 
    // compute distance between cell centers
    Scalar dist = distVec.two_norm();
 
    // compute vectorized permeabilities
    DimMatrix meanPermeability(0);
 
    problem_.spatialParams().meanK(meanPermeability, problem_.spatialParams().intrinsicPermeability(elementI),
            problem_.spatialParams().intrinsicPermeability(elementJ));
 
    Dune::FieldVector<Scalar, dim> permeability(0);
    meanPermeability.mv(unitOuterNormal, permeability);
 
    Scalar rhoMeanW = 0;
    Scalar rhoMeanNw = 0;
    if (compressibility_)
    {
        rhoMeanW = 0.5 * (cellData.density(wPhaseIdx) + cellDataJ.density(wPhaseIdx));
        rhoMeanNw = 0.5 * (cellData.density(nPhaseIdx) + cellDataJ.density(nPhaseIdx));
    }
 
    // calculate potential gradients
    Scalar potentialDiffW = 0;
    Scalar potentialDiffNw = 0;
 
    // if we are at the very first iteration we can't calculate phase potentials
    if (!first)
    {
        potentialDiffW = cellData.potential(wPhaseIdx) - cellDataJ.potential(wPhaseIdx);
        potentialDiffNw = cellData.potential(nPhaseIdx) - cellDataJ.potential(nPhaseIdx);
 
        if (compressibility_)
        {
            density_[wPhaseIdx] = (potentialDiffW > 0.) ? cellData.density(wPhaseIdx) : cellDataJ.density(wPhaseIdx);
            density_[nPhaseIdx] = (potentialDiffNw > 0.) ? cellData.density(nPhaseIdx) : cellDataJ.density(nPhaseIdx);
 
            density_[wPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffW, 0.0, 1.0e-30)) ? rhoMeanW : density_[wPhaseIdx];
            density_[nPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? rhoMeanNw : density_[nPhaseIdx];
 
            potentialDiffW = cellData.pressure(wPhaseIdx) - cellDataJ.pressure(wPhaseIdx);
            potentialDiffNw = cellData.pressure(nPhaseIdx) - cellDataJ.pressure(nPhaseIdx);
 
            potentialDiffW += density_[wPhaseIdx] * (distVec * gravity_);
            potentialDiffNw += density_[nPhaseIdx] * (distVec * gravity_);
        }
    }
 
    // do the upwinding of the mobility depending on the phase potentials
    Scalar lambdaW = (potentialDiffW > 0.) ? lambdaWI : lambdaWJ;
    lambdaW = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffW, 0.0, 1.0e-30)) ? 0.5 * (lambdaWI + lambdaWJ) : lambdaW;
    Scalar lambdaNw = (potentialDiffNw > 0) ? lambdaNwI : lambdaNwJ;
    lambdaNw = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? 0.5 * (lambdaNwI + lambdaNwJ) : lambdaNw;
 
    if (compressibility_)
    {
        density_[wPhaseIdx] = (potentialDiffW > 0.) ? cellData.density(wPhaseIdx) : cellDataJ.density(wPhaseIdx);
        density_[nPhaseIdx] = (potentialDiffNw > 0.) ? cellData.density(nPhaseIdx) : cellDataJ.density(nPhaseIdx);
 
        density_[wPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffW, 0.0, 1.0e-30)) ? rhoMeanW : density_[wPhaseIdx];
        density_[nPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? rhoMeanNw : density_[nPhaseIdx];
    }
 
    Scalar scalarPerm = permeability.two_norm();
    // calculate current matrix entry
    entry[matrix] = (lambdaW + lambdaNw) * scalarPerm / dist * faceArea;
 
    // calculate right hand side
    // calculate unit distVec
    distVec /= dist;
    Scalar areaScaling = (unitOuterNormal * distVec);
    // this treatment of g allows to account for gravity flux through faces where the face normal has no z component (e.g. parallelepiped grids)
    entry[rhs] = (lambdaW * density_[wPhaseIdx] + lambdaNw * density_[nPhaseIdx]) * scalarPerm * (gravity_ * distVec) * faceArea * areaScaling;
 
    if (pressureType_ == pw)
    {
        // add capillary pressure term to right hand side
        entry[rhs] += 0.5 * (lambdaNwI + lambdaNwJ) * scalarPerm * (pcI - pcJ) / dist * faceArea;
    }
    else if (pressureType_ == pn)
    {
        // add capillary pressure term to right hand side
        entry[rhs] -= 0.5 * (lambdaWI + lambdaWJ) * scalarPerm * (pcI - pcJ) / dist * faceArea;
    }
}
 
template<class TypeTag>
void FVPressure2P<TypeTag>::getFluxOnBoundary(EntryType& entry,
const Intersection& intersection, const CellData& cellData, const bool first)
{
    auto element = intersection.inside();
 
    // get global coordinates of cell centers
    const GlobalPosition& globalPosI = element.geometry().center();
 
    // center of face in global coordinates
    const GlobalPosition& globalPosJ = intersection.geometry().center();
 
    // get mobilities and fractional flow factors
    Scalar lambdaWI = cellData.mobility(wPhaseIdx);
    Scalar lambdaNwI = cellData.mobility(nPhaseIdx);
    Scalar fractionalWI = cellData.fracFlowFunc(wPhaseIdx);
    Scalar fractionalNwI = cellData.fracFlowFunc(nPhaseIdx);
 
    // get capillary pressure
    Scalar pcI = cellData.capillaryPressure();
 
    // get face index
    int isIndexI = intersection.indexInInside();
 
    // get face normal
    const Dune::FieldVector<Scalar, dim>& unitOuterNormal = intersection.centerUnitOuterNormal();
 
    // get face area
    Scalar faceArea = intersection.geometry().volume();
 
    // distance vector between barycenters
    GlobalPosition distVec = globalPosJ - globalPosI;
 
    // compute distance between cell centers
    Scalar dist = distVec.two_norm();
 
    BoundaryTypes bcType;
    problem_.boundaryTypes(bcType, intersection);
    PrimaryVariables boundValues(0.0);
 
    if (bcType.isDirichlet(eqIdxPress))
    {
        problem_.dirichlet(boundValues, intersection);
 
        //permeability vector at boundary
        // compute vectorized permeabilities
        DimMatrix meanPermeability(0);
 
        problem_.spatialParams().meanK(meanPermeability,
                problem_.spatialParams().intrinsicPermeability(element));
 
        Dune::FieldVector<Scalar, dim> permeability(0);
        meanPermeability.mv(unitOuterNormal, permeability);
 
        // determine saturation at the boundary -> if no saturation is known directly at the boundary use the cell saturation
        Scalar satW = 0;
        Scalar satNw = 0;
        if (bcType.isDirichlet(eqIdxSat))
        {
            switch (saturationType_)
            {
            case sw:
            {
                satW = boundValues[saturationIdx];
                satNw = 1 - boundValues[saturationIdx];
                break;
            }
            case sn:
            {
                satW = 1 - boundValues[saturationIdx];
                satNw = boundValues[saturationIdx];
                break;
            }
            }
        }
        else
        {
            satW = cellData.saturation(wPhaseIdx);
            satNw = cellData.saturation(nPhaseIdx);
        }
 
        const Scalar temperature = problem_.temperature(element);
 
        // get Dirichlet pressure boundary condition
        const Scalar pressBound = boundValues[pressureIdx];
 
        //calculate constitutive relations depending on the kind of saturation used
 
        const auto fluidMatrixInteraction = problem_.spatialParams().fluidMatrixInteractionAtPos(element.geometry().center());
        const Scalar pcBound = fluidMatrixInteraction.pc(satW);
 
        // determine phase pressures from primary pressure variable
        Scalar pressW = 0;
        Scalar pressNw = 0;
        if (pressureType_ == pw)
        {
            pressW = pressBound;
            pressNw = pressBound + pcBound;
        }
        else if (pressureType_ ==  pn)
        {
            pressW = pressBound - pcBound;
            pressNw = pressBound;
        }
 
        Scalar densityWBound = density_[wPhaseIdx];
        Scalar densityNwBound = density_[nPhaseIdx];
        Scalar viscosityWBound = viscosity_[wPhaseIdx];
        Scalar viscosityNwBound = viscosity_[nPhaseIdx];
        Scalar rhoMeanW = 0;
        Scalar rhoMeanNw = 0;
 
        if (compressibility_)
        {
            FluidState fluidState;
            fluidState.setSaturation(wPhaseIdx, satW);
            fluidState.setSaturation(nPhaseIdx, satNw);
            fluidState.setTemperature(temperature);
            fluidState.setPressure(wPhaseIdx, pressW);
            fluidState.setPressure(nPhaseIdx, pressNw);
 
            densityWBound = FluidSystem::density(fluidState, wPhaseIdx);
            densityNwBound = FluidSystem::density(fluidState, nPhaseIdx);
            viscosityWBound = FluidSystem::viscosity(fluidState, wPhaseIdx) / densityWBound;
            viscosityNwBound = FluidSystem::viscosity(fluidState, nPhaseIdx) / densityNwBound;
 
            rhoMeanW = 0.5 * (cellData.density(wPhaseIdx) + densityWBound);
            rhoMeanNw = 0.5 * (cellData.density(nPhaseIdx) + densityNwBound);
        }
 
        Scalar lambdaWBound = fluidMatrixInteraction.krw(satW)
                / viscosityWBound;
        Scalar lambdaNwBound = fluidMatrixInteraction.krn(satW)
                / viscosityNwBound;
 
        Scalar fractionalWBound = lambdaWBound / (lambdaWBound + lambdaNwBound);
        Scalar fractionalNwBound = lambdaNwBound / (lambdaWBound + lambdaNwBound);
 
        Scalar fMeanW = 0.5 * (fractionalWI + fractionalWBound);
        Scalar fMeanNw = 0.5 * (fractionalNwI + fractionalNwBound);
 
        Scalar potentialDiffW = 0;
        Scalar potentialDiffNw = 0;
 
        if (!first)
        {
            potentialDiffW = cellData.fluxData().upwindPotential(wPhaseIdx, isIndexI);
            potentialDiffNw = cellData.fluxData().upwindPotential(nPhaseIdx, isIndexI);
 
            if (compressibility_)
            {
                density_[wPhaseIdx] = (potentialDiffW > 0.) ? cellData.density(wPhaseIdx) : densityWBound;
                density_[nPhaseIdx] = (potentialDiffNw > 0.) ? cellData.density(nPhaseIdx) : densityNwBound;
 
                density_[wPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffW, 0.0, 1.0e-30)) ? rhoMeanW : density_[wPhaseIdx];
                density_[nPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? rhoMeanNw : density_[nPhaseIdx];
            }
 
            // calculate potential gradient
            switch (pressureType_)
            {
            case pw:
            {
                potentialDiffW = (cellData.pressure(wPhaseIdx) - pressBound);
                potentialDiffNw = (cellData.pressure(nPhaseIdx) - pressBound - pcBound);
                break;
            }
            case pn:
            {
                potentialDiffW = (cellData.pressure(wPhaseIdx) - pressBound + pcBound);
                potentialDiffNw = (cellData.pressure(nPhaseIdx) - pressBound);
                break;
            }
            case pGlobal:
            {
                potentialDiffW = (cellData.globalPressure() - pressBound - fMeanNw * (pcI - pcBound));
                potentialDiffNw = (cellData.globalPressure() - pressBound + fMeanW * (pcI - pcBound));
                break;
            }
            }
 
            potentialDiffW += density_[wPhaseIdx] * (distVec * gravity_);
            potentialDiffNw += density_[nPhaseIdx] * (distVec * gravity_);
        }
 
        // do the upwinding of the mobility depending on the phase potentials
        Scalar lambdaW = (potentialDiffW > 0.) ? lambdaWI : lambdaWBound;
        lambdaW = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffW, 0.0, 1.0e-30)) ? 0.5 * (lambdaWI + lambdaWBound) : lambdaW;
        Scalar lambdaNw = (potentialDiffNw > 0.) ? lambdaNwI : lambdaNwBound;
        lambdaNw = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? 0.5 * (lambdaNwI + lambdaNwBound) : lambdaNw;
 
        if (compressibility_)
        {
            density_[wPhaseIdx] = (potentialDiffW > 0.) ? cellData.density(wPhaseIdx) : densityWBound;
            density_[wPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffW, 0.0, 1.0e-30)) ? rhoMeanW : density_[wPhaseIdx];
            density_[nPhaseIdx] = (potentialDiffNw > 0.) ? cellData.density(nPhaseIdx) : densityNwBound;
            density_[nPhaseIdx] = (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? rhoMeanNw : density_[nPhaseIdx];
        }
 
        Scalar scalarPerm = permeability.two_norm();
        // calculate current matrix entry
        entry[matrix] = (lambdaW + lambdaNw) * scalarPerm / dist * faceArea;
        entry[rhs] = entry[matrix] * pressBound;
 
        // calculate right hand side
        // calculate unit distVec
        distVec /= dist;
        Scalar areaScaling = (unitOuterNormal * distVec);
        // this treatment of g allows to account for gravity flux through faces where the face normal has no z component (e.g. parallelepiped grids)
        entry[rhs] -= (lambdaW * density_[wPhaseIdx] + lambdaNw * density_[nPhaseIdx]) * scalarPerm * (gravity_ * distVec)
                * faceArea * areaScaling;
 
        if (pressureType_ == pw)
        {
            //add capillary pressure term to right hand side
            entry[rhs] -= 0.5 * (lambdaNwI + lambdaNwBound) * scalarPerm * (pcI - pcBound) / dist * faceArea;
        }
        else if (pressureType_ ==  pn)
        {
            //add capillary pressure term to right hand side
            entry[rhs] += 0.5 * (lambdaWI + lambdaWBound) * scalarPerm * (pcI - pcBound) / dist * faceArea;
        }
    }
    // set Neumann boundary condition
    else if (bcType.isNeumann(eqIdxPress))
    {
        problem_.neumann(boundValues, intersection);
 
        if (!compressibility_)
        {
            boundValues[wPhaseIdx] /= density_[wPhaseIdx];
            boundValues[nPhaseIdx] /= density_[nPhaseIdx];
        }
        entry[rhs] = -(boundValues[wPhaseIdx] + boundValues[nPhaseIdx]) * faceArea;
    }
    else
    {
        DUNE_THROW(Dune::NotImplemented, "No valid boundary condition type defined for pressure equation!");
    }
}
 
template<class TypeTag>
void FVPressure2P<TypeTag>::updateMaterialLaws()
{
    for (const auto& element : elements(problem_.gridView()))
    {
        int eIdxGlobal = problem_.variables().index(element);
 
        CellData& cellData = problem_.variables().cellData(eIdxGlobal);
 
        const Scalar temperature = problem_.temperature(element);
 
        // determine phase saturations from primary saturation variable
        const Scalar satW = cellData.saturation(wPhaseIdx);
 
        const auto fluidMatrixInteraction = problem_.spatialParams().fluidMatrixInteractionAtPos(element.geometry().center());
        const Scalar pc = fluidMatrixInteraction.pc(satW);
 
        // determine phase pressures from primary pressure variable
        Scalar pressW = 0;
        Scalar pressNw = 0;
        if (pressureType_ == pw)
        {
            pressW = cellData.pressure(wPhaseIdx);
            pressNw = pressW + pc;
        }
        else if (pressureType_ == pn)
        {
            pressNw = cellData.pressure(nPhaseIdx);
            pressW = pressNw - pc;
        }
 
        if (compressibility_)
        {
            FluidState& fluidState = cellData.fluidState();
            fluidState.setTemperature(temperature);
 
            fluidState.setPressure(wPhaseIdx, pressW);
            fluidState.setPressure(nPhaseIdx, pressNw);
 
            density_[wPhaseIdx] = FluidSystem::density(fluidState, wPhaseIdx);
            density_[nPhaseIdx] = FluidSystem::density(fluidState, nPhaseIdx);
 
            viscosity_[wPhaseIdx]= FluidSystem::viscosity(fluidState, wPhaseIdx);
            viscosity_[nPhaseIdx] = FluidSystem::viscosity(fluidState, nPhaseIdx);
 
            //store density
            fluidState.setDensity(wPhaseIdx, density_[wPhaseIdx]);
            fluidState.setDensity(nPhaseIdx, density_[nPhaseIdx]);
 
            //store viscosity
            fluidState.setViscosity(wPhaseIdx, viscosity_[wPhaseIdx]);
            fluidState.setViscosity(nPhaseIdx, viscosity_[nPhaseIdx]);
        }
        else
        {
            cellData.setCapillaryPressure(pc);
 
            if (pressureType_ != pGlobal)
            {
                cellData.setPressure(wPhaseIdx, pressW);
                cellData.setPressure(nPhaseIdx, pressNw);
            }
        }
 
        // initialize mobilities
        Scalar mobilityW = fluidMatrixInteraction.krw(satW) / viscosity_[wPhaseIdx];
        Scalar mobilityNw = fluidMatrixInteraction.krn(satW) / viscosity_[nPhaseIdx];
 
        if (compressibility_)
        {
            mobilityW *= density_[wPhaseIdx];
            mobilityNw *= density_[nPhaseIdx];
        }
 
        // initialize mobilities
        cellData.setMobility(wPhaseIdx, mobilityW);
        cellData.setMobility(nPhaseIdx, mobilityNw);
 
        // initialize fractional flow functions
        cellData.setFracFlowFunc(wPhaseIdx, mobilityW / (mobilityW + mobilityNw));
        cellData.setFracFlowFunc(nPhaseIdx, mobilityNw / (mobilityW + mobilityNw));
 
        const Scalar gravityDiff = (problem_.bBoxMax() - element.geometry().center()) * gravity_;
 
        Scalar potW = pressW + gravityDiff * density_[wPhaseIdx];
        Scalar potNw = pressNw + gravityDiff * density_[nPhaseIdx];
 
        if (pressureType_ == pGlobal)
        {
            potW = cellData.globalPressure() - cellData.fracFlowFunc(nPhaseIdx) * pc + gravityDiff * density_[wPhaseIdx];
            potNw = cellData.globalPressure() - cellData.fracFlowFunc(wPhaseIdx) * pc + gravityDiff * density_[nPhaseIdx];
        }
 
        cellData.setPotential(wPhaseIdx, potW);
        cellData.setPotential(nPhaseIdx, potNw);
    }
}
 
} // end namespace Dumux
#endif