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

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#ifndef DUMUX_FVVELOCITY2P_HH
#define DUMUX_FVVELOCITY2P_HH
 
#include <dune/common/float_cmp.hh>
#include <dune/grid/common/gridenums.hh>
#include <dumux/porousmediumflow/2p/sequential/diffusion/properties.hh>
 
#include <dumux/common/deprecated.hh>
 
namespace Dumux {
 
template<class TypeTag>
class FVVelocity2P
{
    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 Element = typename GridView::Traits::template Codim<0>::Entity;
    using Intersection = typename GridView::Intersection;
 
    using Geometry = typename Element::Geometry;
    using JacobianTransposed = typename Geometry::JacobianTransposed;
 
    enum
    {
        dim = GridView::dimension, dimWorld = GridView::dimensionworld
    };
 
    enum
    {
        pw = Indices::pressureW,
        pn = Indices::pressureNw,
        pGlobal = Indices::pressureGlobal,
        vw = Indices::velocityW,
        vn = Indices::velocityNw,
        vt = Indices::velocityTotal,
        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 GlobalPosition = typename Element::Geometry::GlobalCoordinate;
    using GravityVector = Dune::FieldVector<Scalar, dimWorld>;
    using DimMatrix = Dune::FieldMatrix<Scalar, dim, dim>;
 
public:
    FVVelocity2P(Problem& problem) :
    problem_(problem), gravity_(problem.gravity())
    {
        if (getPropValue<TypeTag, Properties::EnableCompressibility>() && velocityType_ == vt)
        {
            DUNE_THROW(Dune::NotImplemented,
                    "Total velocity - global pressure - model cannot be used with compressible fluids!");
        }
        if (velocityType_ != vw && velocityType_ != vn && velocityType_ != vt)
        {
            DUNE_THROW(Dune::NotImplemented, "Velocity type not supported!");
        }
 
        density_[wPhaseIdx] = 0.;
        density_[nPhaseIdx] = 0.;
        viscosity_[wPhaseIdx] = 0.;
        viscosity_[nPhaseIdx] = 0.;
 
        vtkOutputLevel_ = getParam<int>("Vtk.OutputLevel");
    }
 
    void 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);
        }
    }
 
    void calculateVelocity(const Intersection&, CellData&);
 
    void calculateVelocityOnBoundary(const Intersection&, CellData&);
 
    bool calculateVelocityInTransport()
    {
        return true;
    }
 
    template<class MultiWriter>
    void addOutputVtkFields(MultiWriter &writer)
    {
        if (vtkOutputLevel_ > 0)
        {
            Dune::BlockVector < Dune::FieldVector<Scalar, dim> > &velocity = *(writer.template allocateManagedBuffer<Scalar,
                    dim>(problem_.gridView().size(0)));
            Dune::BlockVector < Dune::FieldVector<Scalar, dim> > &velocitySecondPhase =
            *(writer.template allocateManagedBuffer<Scalar, dim>(problem_.gridView().size(0)));
 
            // compute update vector
            for (const auto& element : elements(problem_.gridView()))
            {
                // cell index
                int eIdxGlobal = problem_.variables().index(element);
 
                CellData& cellData = problem_.variables().cellData(eIdxGlobal);
 
                const typename Element::Geometry& geometry = element.geometry();
 
                // get corresponding reference element
                const auto refElement = referenceElement(geometry);
 
                const int numberOfFaces=refElement.size(1);
                std::vector<Scalar> fluxW(numberOfFaces,0);
                std::vector<Scalar> fluxNw(numberOfFaces,0);
 
                // run through all intersections with neighbors and boundary
                for (const auto& intersection : intersections(problem_.gridView(), element))
                {
                    int isIndex = intersection.indexInInside();
 
                    fluxW[isIndex] += intersection.geometry().volume()
                    * (intersection.centerUnitOuterNormal() * cellData.fluxData().velocity(wPhaseIdx, isIndex));
                    fluxNw[isIndex] += intersection.geometry().volume()
                    * (intersection.centerUnitOuterNormal() * cellData.fluxData().velocity(nPhaseIdx, isIndex));
                }
 
                // calculate velocity on reference element as the Raviart-Thomas-0
                // interpolant of the fluxes
                Dune::FieldVector<Scalar, dim> refVelocity;
                // simplices
                if (refElement.type().isSimplex()) {
                    for (int dimIdx = 0; dimIdx < dim; dimIdx++)
                    {
                        refVelocity[dimIdx] = -fluxW[dim - 1 - dimIdx];
                        for (int fIdx = 0; fIdx < dim + 1; fIdx++)
                        {
                            refVelocity[dimIdx] += fluxW[fIdx]/(dim + 1);
                        }
                    }
                }
                // cubes
                else if (refElement.type().isCube()){
                    for (int i = 0; i < dim; i++)
                        refVelocity[i] = 0.5 * (fluxW[2*i + 1] - fluxW[2*i]);
                }
                // 3D prism and pyramids
                else {
                    DUNE_THROW(Dune::NotImplemented, "velocity output for prism/pyramid not implemented");
                }
 
                const Dune::FieldVector<Scalar, dim> localPos =
                        refElement.position(0, 0);
 
                // get the transposed Jacobian of the element mapping
                const JacobianTransposed jacobianT =
                        geometry.jacobianTransposed(localPos);
 
                // calculate the element velocity by the Piola transformation
                Dune::FieldVector < Scalar, dim > elementVelocity(0);
                jacobianT.umtv(refVelocity, elementVelocity);
                elementVelocity /= geometry.integrationElement(localPos);
 
                velocity[eIdxGlobal] = elementVelocity;
 
                // calculate velocity on reference element as the Raviart-Thomas-0
                // interpolant of the fluxes
                // simplices
                if (refElement.type().isSimplex()) {
                    for (int dimIdx = 0; dimIdx < dim; dimIdx++)
                    {
                        refVelocity[dimIdx] = -fluxNw[dim - 1 - dimIdx];
                        for (int fIdx = 0; fIdx < dim + 1; fIdx++)
                        {
                            refVelocity[dimIdx] += fluxNw[fIdx]/(dim + 1);
                        }
                    }
                }
                // cubes
                else if (refElement.type().isCube()){
                    for (int i = 0; i < dim; i++)
                        refVelocity[i] = 0.5 * (fluxNw[2*i + 1] - fluxNw[2*i]);
                }
                // 3D prism and pyramids
                else {
                    DUNE_THROW(Dune::NotImplemented, "velocity output for prism/pyramid not implemented");
                }
 
                // calculate the element velocity by the Piola transformation
                elementVelocity = 0;
                jacobianT.umtv(refVelocity, elementVelocity);
                elementVelocity /= geometry.integrationElement(localPos);
 
                velocitySecondPhase[eIdxGlobal] = elementVelocity;
            }
 
            // switch velocities
            if (velocityType_ == vt)
            {
                 writer.attachCellData(velocity, "total velocity", dim);
            }
            else
            {
                writer.attachCellData(velocity, "wetting-velocity", dim);
                writer.attachCellData(velocitySecondPhase, "nonwetting-velocity", dim);
            }
        }
 
        return;
    }
 
private:
    Problem& problem_;
    const GravityVector& gravity_; 
    Scalar density_[numPhases];
    Scalar viscosity_[numPhases];
 
    int vtkOutputLevel_;
 
    static const int velocityType_ = getPropValue<TypeTag, Properties::VelocityFormulation>();
    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 FVVelocity2P<TypeTag>::calculateVelocity(const Intersection& intersection, CellData& cellData)
{
    auto elementI = intersection.inside();
    auto elementJ = intersection.outside();
 
    int eIdxGlobalJ = problem_.variables().index(elementJ);
 
    CellData& cellDataJ = problem_.variables().cellData(eIdxGlobalJ);
 
    // 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 index
    int isIndexI = intersection.indexInInside();
    int isIndexJ = intersection.indexInOutside();
 
    // get face normal
    const Dune::FieldVector<Scalar, dim>& unitOuterNormal = intersection.centerUnitOuterNormal();
 
    // 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);
 
    // calculate potential gradients
    Scalar potentialDiffW = cellData.potential(wPhaseIdx) - cellDataJ.potential(wPhaseIdx);
    Scalar 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)) ? 0.5 * (cellData.density(wPhaseIdx) + cellDataJ.density(wPhaseIdx)) :
                        density_[wPhaseIdx];
        density_[nPhaseIdx] =
                (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? 0.5 * (cellData.density(nPhaseIdx) + cellDataJ.density(nPhaseIdx)) :
                        density_[nPhaseIdx];
 
        potentialDiffW = (cellData.pressure(wPhaseIdx) - cellDataJ.pressure(wPhaseIdx));
        potentialDiffNw = (cellData.pressure(nPhaseIdx) - cellDataJ.pressure(nPhaseIdx));
 
        potentialDiffW += density_[wPhaseIdx] * (distVec * gravity_); //delta z/delta x in unitOuterNormal[z]
        potentialDiffNw += density_[nPhaseIdx] * (distVec * gravity_);
    }
 
    // store potentials for further calculations (velocity, saturation, ...)
    cellData.fluxData().setUpwindPotential(wPhaseIdx, isIndexI, potentialDiffW);
    cellData.fluxData().setUpwindPotential(nPhaseIdx, isIndexI, potentialDiffNw);
 
    cellDataJ.fluxData().setUpwindPotential(wPhaseIdx, isIndexJ, -potentialDiffW);
    cellDataJ.fluxData().setUpwindPotential(nPhaseIdx, isIndexJ, -potentialDiffNw);
 
    // 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)) ? 0.5 * (cellData.density(wPhaseIdx) + cellDataJ.density(wPhaseIdx)) :
                        density_[wPhaseIdx];
        density_[nPhaseIdx] =
                (Dune::FloatCmp::eq<Scalar, Dune::FloatCmp::absolute>(potentialDiffNw, 0.0, 1.0e-30)) ? 0.5 * (cellData.density(nPhaseIdx) + cellDataJ.density(nPhaseIdx)) :
                        density_[nPhaseIdx];
    }
 
    Scalar scalarPerm = permeability.two_norm();
 
    // calculate the gravity term
    Dune::FieldVector<Scalar, dimWorld> velocityW(unitOuterNormal);
    Dune::FieldVector<Scalar, dimWorld> velocityNw(unitOuterNormal);
 
    // 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)
    Scalar gravityTermW = (gravity_ * distVec) * density_[wPhaseIdx] * areaScaling;
    Scalar gravityTermNw = (gravity_ * distVec) * density_[nPhaseIdx] * areaScaling;
 
    // calculate velocity depending on the pressure used -> use pc = pn - pw
    switch (pressureType_)
    {
    case pw:
    {
        velocityW *= lambdaW * scalarPerm
                * ((cellData.pressure(wPhaseIdx) - cellDataJ.pressure(wPhaseIdx)) / dist + gravityTermW);
        velocityNw *= lambdaNw * scalarPerm
                * ((cellData.pressure(wPhaseIdx) - cellDataJ.pressure(wPhaseIdx)) / dist + gravityTermNw)
                + 0.5 * (lambdaNwI + lambdaNwJ) * scalarPerm * (pcI - pcJ) / dist;
        break;
    }
    case pn:
    {
        velocityW *= lambdaW * scalarPerm
                * ((cellData.pressure(nPhaseIdx) - cellDataJ.pressure(nPhaseIdx)) / dist + gravityTermW)
                - 0.5 * (lambdaWI + lambdaWJ) * scalarPerm * (pcI - pcJ) / dist;
        velocityNw *= lambdaNw * scalarPerm
                * ((cellData.pressure(nPhaseIdx) - cellDataJ.pressure(nPhaseIdx)) / dist + gravityTermNw);
        break;
    }
    case pGlobal:
    {
        velocityW *= (lambdaW + lambdaNw) * scalarPerm * (cellData.globalPressure() - cellDataJ.globalPressure()) / dist
                + scalarPerm * (lambdaW * gravityTermW + lambdaNw * gravityTermNw);
        velocityNw = 0;
        break;
    }
    }
 
    // store velocities
    cellData.fluxData().setVelocity(wPhaseIdx, isIndexI, velocityW);
    cellData.fluxData().setVelocity(nPhaseIdx, isIndexI, velocityNw);
    cellData.fluxData().setVelocityMarker(isIndexI);
 
    cellDataJ.fluxData().setVelocity(wPhaseIdx, isIndexJ, velocityW);
    cellDataJ.fluxData().setVelocity(nPhaseIdx, isIndexJ, velocityNw);
    cellDataJ.fluxData().setVelocityMarker(isIndexJ);
 
//                        printvector(std::cout, cellData.fluxData().velocity(), "velocity", "row", 4, 1, 3);
    return;
}
 
template<class TypeTag>
void FVVelocity2P<TypeTag>::calculateVelocityOnBoundary(const Intersection& intersection, CellData& cellData)
{
    auto element = intersection.inside();
 
    // get face index
    int isIndex = intersection.indexInInside();
 
    // get face normal
    const Dune::FieldVector<Scalar, dim>& unitOuterNormal = intersection.centerUnitOuterNormal();
 
    BoundaryTypes bcType;
    //get boundary type
    problem_.boundaryTypes(bcType, intersection);
    PrimaryVariables boundValues(0.0);
 
    if (bcType.isDirichlet(eqIdxPress))
    {
        problem_.dirichlet(boundValues, intersection);
 
        // 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();
 
        // distance vector between barycenters
        GlobalPosition distVec = globalPosJ - globalPosI;
 
        // compute distance between cell centers
        Scalar dist = distVec.two_norm();
 
        // 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 pressBound = boundValues[pressureIdx];
 
        // old material law interface is deprecated: Replace this by
        // const auto& fluidMatrixInteraction = spatialParams.fluidMatrixInteractionAtPos(element.geometry().center());
        // after the release of 3.3, when the deprecated interface is no longer supported
        const auto fluidMatrixInteraction = Deprecated::makePcKrSw(Scalar{}, problem_.spatialParams(), element);
 
        const Scalar pcBound = fluidMatrixInteraction.pc(satW);
 
        // determine phase pressures from primary pressure variable
        Scalar pressWBound = 0;
        Scalar pressNwBound = 0;
        if (pressureType_ == pw)
        {
            pressWBound = pressBound;
            pressNwBound = pressBound + pcBound;
        }
        else if (pressureType_ == pn)
        {
            pressWBound = pressBound - pcBound;
            pressNwBound = pressBound;
        }
 
        // get temperature at current position
        const Scalar temperature = problem_.temperature(element);
 
        Scalar densityWBound = density_[wPhaseIdx];
        Scalar densityNwBound = density_[nPhaseIdx];
        Scalar viscosityWBound = viscosity_[wPhaseIdx];
        Scalar viscosityNwBound = viscosity_[nPhaseIdx];
 
        if (compressibility_)
        {
            FluidState fluidState;
            fluidState.setSaturation(wPhaseIdx, satW);
            fluidState.setSaturation(nPhaseIdx, satNw);
            fluidState.setTemperature(temperature);
            fluidState.setPressure(wPhaseIdx, pressWBound);
            fluidState.setPressure(nPhaseIdx, pressNwBound);
 
            densityWBound = FluidSystem::density(fluidState, wPhaseIdx);
            densityNwBound = FluidSystem::density(fluidState, nPhaseIdx);
            viscosityWBound = FluidSystem::viscosity(fluidState, wPhaseIdx) / densityWBound;
            viscosityNwBound = FluidSystem::viscosity(fluidState, nPhaseIdx) / densityNwBound;
        }
 
        Scalar lambdaWBound = fluidMatrixInteraction.krw(satW)
                / viscosityWBound;
        Scalar lambdaNwBound = fluidMatrixInteraction.krn(satW)
                / viscosityNwBound;
 
        Scalar potentialDiffW = cellData.fluxData().upwindPotential(wPhaseIdx, isIndex);
        Scalar potentialDiffNw = cellData.fluxData().upwindPotential(nPhaseIdx, isIndex);
 
        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)) ? 0.5 * (cellData.density(wPhaseIdx) + densityWBound) : 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)) ? 0.5 * (cellData.density(nPhaseIdx) + densityNwBound) : density_[nPhaseIdx];
        }
 
        // calculate potential gradient
        if (pressureType_ == pGlobal)
        {
            potentialDiffW = (cellData.globalPressure() - pressBound - fractionalNwI * (pcI - pcBound));
            potentialDiffNw = (cellData.globalPressure() - pressBound + fractionalWI * (pcI - pcBound));
        }
        else
        {
            potentialDiffW = (cellData.pressure(wPhaseIdx) - pressWBound);
            potentialDiffNw = (cellData.pressure(nPhaseIdx) - pressNwBound);
        }
 
        potentialDiffW += density_[wPhaseIdx] * (distVec * gravity_);
        potentialDiffNw += density_[nPhaseIdx] * (distVec * gravity_);
 
        // store potential gradients for further calculations
        cellData.fluxData().setUpwindPotential(wPhaseIdx, isIndex, potentialDiffW);
        cellData.fluxData().setUpwindPotential(nPhaseIdx, isIndex, potentialDiffNw);
 
        // 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)) ? 0.5 * (cellData.density(wPhaseIdx) + densityWBound) : 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)) ? 0.5 * (cellData.density(nPhaseIdx) + densityNwBound) : density_[nPhaseIdx];
        }
 
        const Scalar scalarPerm = permeability.two_norm();
 
        // calculate the gravity term
        Dune::FieldVector<Scalar, dimWorld> velocityW(unitOuterNormal);
        Dune::FieldVector<Scalar, dimWorld> velocityNw(unitOuterNormal);
 
        // 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)
        Scalar gravityTermW = (gravity_ * distVec) * density_[wPhaseIdx] * areaScaling;
        Scalar gravityTermNw = (gravity_ * distVec) * density_[nPhaseIdx] * areaScaling;
 
        // calculate velocity depending on the pressure used -> use pc = pn - pw
        switch (pressureType_)
        {
        case pw:
        {
            velocityW *= lambdaW * scalarPerm * ((cellData.pressure(wPhaseIdx) - pressBound) / dist + gravityTermW);
            velocityNw *= lambdaNw * scalarPerm * ((cellData.pressure(wPhaseIdx) - pressBound) / dist + gravityTermNw)
                    + 0.5 * (lambdaNwI + lambdaNwBound) * scalarPerm * (pcI - pcBound) / dist;
            break;
        }
        case pn:
        {
            velocityW *= lambdaW * scalarPerm * ((cellData.pressure(nPhaseIdx) - pressBound) / dist + gravityTermW)
                    - 0.5 * (lambdaWI + lambdaWBound) * scalarPerm * (pcI - pcBound) / dist;
            velocityNw *= lambdaNw * scalarPerm * ((cellData.pressure(nPhaseIdx) - pressBound) / dist + gravityTermNw);
            break;
        }
        case pGlobal:
        {
            velocityW *= (lambdaW + lambdaNw) * scalarPerm * (cellData.globalPressure() - pressBound) / dist
                    + scalarPerm * (lambdaW * gravityTermW + lambdaNw * gravityTermNw);
            velocityNw = 0;
            break;
        }
        }
 
        // store velocities
        cellData.fluxData().setVelocity(wPhaseIdx, isIndex, velocityW);
        cellData.fluxData().setVelocity(nPhaseIdx, isIndex, velocityNw);
        cellData.fluxData().setVelocityMarker(isIndex);
 
    } // end Dirichlet boundary
 
    else if (bcType.isNeumann(eqIdxPress))
    {
        problem_.neumann(boundValues, intersection);
 
        Dune::FieldVector<Scalar, dimWorld> velocityW(unitOuterNormal);
        Dune::FieldVector<Scalar, dimWorld> velocityNw(unitOuterNormal);
 
        velocityW *= boundValues[wPhaseIdx];
        velocityNw *= boundValues[nPhaseIdx];
 
        if (!compressibility_)
        {
            velocityW /= density_[wPhaseIdx];
            velocityNw /= density_[nPhaseIdx];
        }
 
        // store potential gradients for further calculations
        cellData.fluxData().setUpwindPotential(wPhaseIdx, isIndex, boundValues[wPhaseIdx]);
        cellData.fluxData().setUpwindPotential(nPhaseIdx, isIndex, boundValues[nPhaseIdx]);
 
        cellData.fluxData().setVelocity(wPhaseIdx, isIndex, velocityW);
        cellData.fluxData().setVelocity(nPhaseIdx, isIndex, velocityNw);
        cellData.fluxData().setVelocityMarker(isIndex);
    } // end Neumann boundary
    else
    {
        DUNE_THROW(Dune::NotImplemented, "No valid boundary condition type defined for pressure equation!");
    }
 
//                        printvector(std::cout, cellData.fluxData().velocity(), "velocity", "row", 4, 1, 3);
    return;
}
} // end namespace Dumux
#endif