advection.hh

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#ifndef DUMUX_FLUX_PNM_ADVECTION_HH
#define DUMUX_FLUX_PNM_ADVECTION_HH
 
#include <array>
#include <dumux/common/parameters.hh>
 
namespace Dumux::PoreNetwork::Detail {
 
template<class... TransmissibilityLawTypes>
struct Transmissibility : public TransmissibilityLawTypes... {};
 
} // end namespace Dumux::PoreNetwork::Detail
 
namespace Dumux::PoreNetwork {
 
template<class ScalarT, class... TransmissibilityLawTypes>
class CreepingFlow
{
 
public:
    using Scalar = ScalarT;
 
    using Transmissibility = Detail::Transmissibility<TransmissibilityLawTypes...>;
 
    template<class Problem, class Element, class FVElementGeometry,
             class ElementVolumeVariables, class SubControlVolumeFace, class ElemFluxVarsCache>
    static Scalar flux(const Problem& problem,
                       const Element& element,
                       const FVElementGeometry& fvGeometry,
                       const ElementVolumeVariables& elemVolVars,
                       const SubControlVolumeFace& scvf,
                       const int phaseIdx,
                       const ElemFluxVarsCache& elemFluxVarsCache)
    {
        const auto& fluxVarsCache = elemFluxVarsCache[scvf];
 
        // Get the inside and outside volume variables
        const auto& insideScv = fvGeometry.scv(scvf.insideScvIdx());
        const auto& outsideScv = fvGeometry.scv(scvf.outsideScvIdx());
        const auto& insideVolVars = elemVolVars[insideScv];
        const auto& outsideVolVars = elemVolVars[outsideScv];
 
        // calculate the pressure difference
        const Scalar deltaP = insideVolVars.pressure(phaseIdx) - outsideVolVars.pressure(phaseIdx);
        const Scalar transmissibility = fluxVarsCache.transmissibility(phaseIdx);
        using std::isfinite;    
        assert(isfinite(transmissibility));
 
        Scalar volumeFlow = transmissibility*deltaP;
 
        // add gravity term
        static const bool enableGravity = getParamFromGroup<bool>(problem.paramGroup(), "Problem.EnableGravity");
        if (enableGravity)
        {
            const Scalar rho = 0.5*insideVolVars.density(phaseIdx) + 0.5*outsideVolVars.density(phaseIdx);
            const Scalar g = problem.spatialParams().gravity(scvf.center()) * scvf.unitOuterNormal();
 
            // The transmissibility is with respect to the effective throat length (potentially dropping the pore body radii).
            // For gravity, we need to consider the total throat length (i.e., the cell-center to cell-center distance).
            // This might cause some inconsistencies TODO: is there a better way?
            volumeFlow += transmissibility * fluxVarsCache.poreToPoreDistance() * rho * g;
        }
 
        return volumeFlow;
    }
 
    template<class Problem, class Element, class FVElementGeometry, class ElementVolumeVariables, class FluxVariablesCache>
    static Scalar calculateTransmissibility(const Problem& problem,
                                            const Element& element,
                                            const FVElementGeometry& fvGeometry,
                                            const typename FVElementGeometry::SubControlVolumeFace& scvf,
                                            const ElementVolumeVariables& elemVolVars,
                                            const FluxVariablesCache& fluxVarsCache,
                                            const int phaseIdx)
    {
        if constexpr (ElementVolumeVariables::VolumeVariables::numFluidPhases() == 1)
            return Transmissibility::singlePhaseTransmissibility(problem, element, fvGeometry, scvf, elemVolVars, fluxVarsCache, phaseIdx);
        else
        {
            static_assert(ElementVolumeVariables::VolumeVariables::numFluidPhases() == 2);
 
            const auto& spatialParams = problem.spatialParams();
            using FluidSystem = typename ElementVolumeVariables::VolumeVariables::FluidSystem;
            const int wPhaseIdx = spatialParams.template wettingPhase<FluidSystem>(element, elemVolVars);
            const bool invaded = fluxVarsCache.invaded();
 
            if (phaseIdx == wPhaseIdx)
            {
                return invaded ? Transmissibility::wettingLayerTransmissibility(element, fvGeometry, scvf, fluxVarsCache)
                               : Transmissibility::singlePhaseTransmissibility(problem, element, fvGeometry, scvf, elemVolVars, fluxVarsCache, phaseIdx);
            }
            else // non-wetting phase
            {
                return invaded ? Transmissibility::nonWettingPhaseTransmissibility(element, fvGeometry, scvf, fluxVarsCache)
                               : 0.0;
            }
        }
    }
 
    template<class Problem, class Element, class FVElementGeometry, class ElementVolumeVariables, class FluxVariablesCache>
    static std::array<Scalar, 2> calculateTransmissibilities(const Problem& problem,
                                                             const Element& element,
                                                             const FVElementGeometry& fvGeometry,
                                                             const ElementVolumeVariables& elemVolVars,
                                                             const typename FVElementGeometry::SubControlVolumeFace& scvf,
                                                             const FluxVariablesCache& fluxVarsCache)
    {
        static_assert(ElementVolumeVariables::VolumeVariables::numFluidPhases() == 1);
        const Scalar t = calculateTransmissibility(problem, element, fvGeometry, scvf, elemVolVars, fluxVarsCache, 0);
        return {t, -t};
    }
};
 
template <class ScalarT, class... TransmissibilityLawTypes>
class NonCreepingFlow
{
public:
    using Scalar = ScalarT;
 
    using TransmissibilityCreepingFlow = Detail::Transmissibility<TransmissibilityLawTypes...>;
 
    class Transmissibility: public TransmissibilityCreepingFlow
    {
    public:
        class SinglePhaseCache : public TransmissibilityCreepingFlow::SinglePhaseCache
        {
            using ParentType = typename TransmissibilityCreepingFlow::SinglePhaseCache;
        public:
            using Scalar = ScalarT;
 
            template<class Problem, class Element, class FVElementGeometry, class ElementVolumeVariables, class FluxVariablesCache>
            void fill(const Problem& problem,
                    const Element& element,
                    const FVElementGeometry& fvGeometry,
                    const typename FVElementGeometry::SubControlVolumeFace& scvf,
                    const ElementVolumeVariables& elemVolVars,
                    const FluxVariablesCache& fluxVarsCache,
                    const int phaseIdx)
            {
                ParentType::fill(problem, element, fvGeometry,scvf, elemVolVars, fluxVarsCache, phaseIdx);
                const auto elemSol = elementSolution(element, elemVolVars, fvGeometry);
 
                for (const auto& scv : scvs(fvGeometry))
                {
                    const auto localIdx = scv.indexInElement();
                    const Scalar throatToPoreAreaRatio = fluxVarsCache.throatCrossSectionalArea() / problem.spatialParams().poreCrossSectionalArea(element, scv, elemSol);
 
                    // dimensionless momentum coefficient
                    const Scalar momentumCoefficient = problem.spatialParams().momentumCoefficient(element, scv, elemSol);
 
                    // dimensionless kinetik-energy coefficient
                    const Scalar kineticEnergyCoefficient = problem.spatialParams().kineticEnergyCoefficient(element, scv, elemSol);
 
                    expansionCoefficient_[localIdx] = getExpansionCoefficient_(throatToPoreAreaRatio, momentumCoefficient, kineticEnergyCoefficient);
                    contractionCoefficient_[localIdx] = getContractionCoefficient_(throatToPoreAreaRatio, momentumCoefficient, kineticEnergyCoefficient);
                }
            }
 
            Scalar expansionCoefficient(int downstreamIdx) const
            { return expansionCoefficient_[downstreamIdx]; }
 
            Scalar contractionCoefficient(int upstreamIdx) const
            { return contractionCoefficient_[upstreamIdx]; }
 
        private:
 
            Scalar getExpansionCoefficient_(const Scalar throatToPoreAreaRatio, const Scalar momentumCoefficient, const Scalar kineticEnergyCoefficient) const
            {
                Scalar expansionCoefficient = throatToPoreAreaRatio * throatToPoreAreaRatio * (2 * momentumCoefficient - kineticEnergyCoefficient)
                                              + kineticEnergyCoefficient - 2 * momentumCoefficient * throatToPoreAreaRatio
                                              - (1 - throatToPoreAreaRatio * throatToPoreAreaRatio);
 
                return expansionCoefficient;
            }
 
            Scalar getContractionCoefficient_(const Scalar throatToPoreAreaRatio, const Scalar momentumCoefficient, const Scalar kineticEnergyCoefficient) const
            {
                const Scalar contractionAreaRatio = getContractionAreaRatio_(throatToPoreAreaRatio);
                Scalar contractionCoefficient = (1 - (throatToPoreAreaRatio * throatToPoreAreaRatio * kineticEnergyCoefficient - 2 * momentumCoefficient /*+1-throatToPoreAreaRatio*throatToPoreAreaRatio*/)
                                                     * contractionAreaRatio * contractionAreaRatio - 2 * contractionAreaRatio) / (contractionAreaRatio * contractionAreaRatio);
 
                return contractionCoefficient;
            }
 
            Scalar getContractionAreaRatio_(const Scalar throatToPoreAreaRatio) const
            {
                return 1.0-(1.0-throatToPoreAreaRatio)/(2.08*(1.0-throatToPoreAreaRatio)+0.5371);
            }
 
            std::array<Scalar, 2> expansionCoefficient_;
            std::array<Scalar, 2> contractionCoefficient_;
        };
    };
 
    template<class Problem, class Element, class FVElementGeometry,
             class ElementVolumeVariables, class SubControlVolumeFace, class ElemFluxVarsCache>
    static Scalar flux(const Problem& problem,
                       const Element& element,
                       const FVElementGeometry& fvGeometry,
                       const ElementVolumeVariables& elemVolVars,
                       const SubControlVolumeFace& scvf,
                       const int phaseIdx,
                       const ElemFluxVarsCache& elemFluxVarsCache)
    {
        const auto& fluxVarsCache = elemFluxVarsCache[scvf];
 
        // Get the inside and outside volume variables
        const auto& insideScv = fvGeometry.scv(scvf.insideScvIdx());
        const auto& outsideScv = fvGeometry.scv(scvf.outsideScvIdx());
        const auto& insideVolVars = elemVolVars[insideScv];
        const auto& outsideVolVars = elemVolVars[outsideScv];
 
        // calculate the pressure difference
        Scalar deltaP = insideVolVars.pressure(phaseIdx) - outsideVolVars.pressure(phaseIdx);
 
        // add gravity term
        static const bool enableGravity = getParamFromGroup<bool>(problem.paramGroup(), "Problem.EnableGravity");
        if (enableGravity)
        {
            const Scalar rho = 0.5*insideVolVars.density(phaseIdx) + 0.5*outsideVolVars.density(phaseIdx);
 
            // projected distance between pores in gravity field direction
            const auto poreToPoreVector = fluxVarsCache.poreToPoreDistance() * scvf.unitOuterNormal();
            const auto& gravityVector = problem.spatialParams().gravity(scvf.center());
 
            deltaP += rho * (gravityVector * poreToPoreVector);
        }
        const Scalar creepingFlowTransmissibility = fluxVarsCache.transmissibility(phaseIdx);
        const Scalar throatCrossSectionalArea = fluxVarsCache.throatCrossSectionalArea();
        using std::isfinite;
        assert(isfinite(creepingFlowTransmissibility));
 
        assert(scvf.insideScvIdx() == 0);
        assert(scvf.outsideScvIdx() == 1);
 
        // determine the flow direction to predict contraction and expansion
        const auto [upstreamIdx, downstreamIdx] = deltaP > 0 ? std::pair(0, 1) : std::pair(1, 0);
 
        const Scalar contractionCoefficient = fluxVarsCache.singlePhaseFlowVariables().contractionCoefficient(upstreamIdx);
        const Scalar expansionCoefficient = fluxVarsCache.singlePhaseFlowVariables().expansionCoefficient(downstreamIdx);
        const Scalar mu = elemVolVars[upstreamIdx].viscosity();
 
        const Scalar A = (contractionCoefficient * elemVolVars[upstreamIdx].density() + expansionCoefficient * elemVolVars[downstreamIdx].density())
                          / (2.0 * mu * mu * throatCrossSectionalArea * throatCrossSectionalArea);
        const Scalar B =  1.0/creepingFlowTransmissibility;
        using std::abs;
        const Scalar C = -abs(deltaP);
 
        using std::sqrt;
        const auto discriminant = B*B - 4*A*C;
        const auto q = (-B + sqrt(discriminant)) / (2*A);
 
        if (upstreamIdx == 0)
            return q;
        else
            return -q;
 
    }
 
    template<class Problem, class Element, class FVElementGeometry, class ElementVolumeVariables, class FluxVariablesCache>
    static Scalar calculateTransmissibility(const Problem& problem,
                                            const Element& element,
                                            const FVElementGeometry& fvGeometry,
                                            const typename FVElementGeometry::SubControlVolumeFace& scvf,
                                            const ElementVolumeVariables& elemVolVars,
                                            const FluxVariablesCache& fluxVarsCache,
                                            const int phaseIdx)
    {
        static_assert(ElementVolumeVariables::VolumeVariables::numFluidPhases() == 1);
        return Transmissibility::singlePhaseTransmissibility(problem, element, fvGeometry, scvf, elemVolVars, fluxVarsCache, phaseIdx);
    }
 
    template<class Problem, class Element, class FVElementGeometry, class ElementVolumeVariables, class FluxVariablesCache>
    static std::array<Scalar, 2> calculateTransmissibilities(const Problem& problem,
                                                             const Element& element,
                                                             const FVElementGeometry& fvGeometry,
                                                             const ElementVolumeVariables& elemVolVars,
                                                             const typename FVElementGeometry::SubControlVolumeFace& scvf,
                                                             const FluxVariablesCache& fluxVarsCache)
    {
        static_assert(ElementVolumeVariables::VolumeVariables::numFluidPhases() == 1);
        const Scalar t = calculateTransmissibility(problem, element, fvGeometry, scvf, elemVolVars, fluxVarsCache, 0);
        return {t, -t};
    }
 
};
 
} // end namespace Dumux
 
#endif