PTFlash.hpp

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// -*- mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
// vi: set et ts=4 sw=4 sts=4:
/*
  Copyright 2022 NORCE.
  Copyright 2022 SINTEF Digital, Mathematics and Cybernetics.
 
  This file is part of the Open Porous Media project (OPM).
 
  OPM is free software: you can redistribute it and/or modify
  it under the terms of the GNU General Public License as published by
  the Free Software Foundation, either version 2 of the License, or
  (at your option) any later version.
  OPM is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  GNU General Public License for more details.
 
  You should have received a copy of the GNU General Public License
  along with OPM.  If not, see <http://www.gnu.org/licenses/>.
 
  Consult the COPYING file in the top-level source directory of this
  module for the precise wording of the license and the list of
  copyright holders.
*/
#ifndef OPM_CHI_FLASH_HPP
#define OPM_CHI_FLASH_HPP
 
#include <opm/material/fluidmatrixinteractions/NullMaterial.hpp>
#include <opm/material/fluidmatrixinteractions/MaterialTraits.hpp>
#include <opm/material/fluidstates/CompositionalFluidState.hpp>
#include <opm/material/densead/Evaluation.hpp>
#include <opm/material/densead/Math.hpp>
#include <opm/material/common/MathToolbox.hpp>
#include <opm/material/common/Valgrind.hpp>
#include <opm/material/Constants.hpp>
#include <opm/material/eos/CubicEOS.hpp>
 
#include <opm/input/eclipse/EclipseState/Compositional/CompositionalConfig.hpp>
 
#include <dune/common/fvector.hh>
#include <dune/common/fmatrix.hh>
#include <dune/common/classname.hh>
 
#include <limits>
#include <iostream>
#include <iomanip>
#include <stdexcept>
#include <type_traits>
 
#include <fmt/format.h>
 
namespace Opm {

template <class Scalar, class FluidSystem, bool isThermal = false>
class PTFlash
{
    static constexpr int numPhases = FluidSystem::numPhases;
    static constexpr int numComponents = FluidSystem::numComponents;
    enum { oilPhaseIdx = FluidSystem::oilPhaseIdx};
    enum { gasPhaseIdx = FluidSystem::gasPhaseIdx};
    enum { waterPhaseIdx = FluidSystem::waterPhaseIdx};
    static constexpr int numMiscibleComponents = FluidSystem::numMiscibleComponents;
    static constexpr int numMisciblePhases = FluidSystem::numMisciblePhases; //oil, gas
    static constexpr int numEq = numMisciblePhases + numMisciblePhases * numMiscibleComponents;
 
    using EOSType = CompositionalConfig::EOSType;
 
public:
    template <class FluidState>
    static bool solve(FluidState& fluid_state,
                      const std::string& twoPhaseMethod,
                      Scalar flash_tolerance,
                      const EOSType& eos_type,
                      int verbosity = 0)
    {
        using ScalarFluidState = CompositionalFluidState<Scalar, FluidSystem>;
        ScalarFluidState fluid_state_scalar;
 
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            fluid_state_scalar.setKvalue(compIdx, Opm::getValue(fluid_state.K(compIdx) ) );
            fluid_state_scalar.setMoleFraction(compIdx, Opm::getValue(fluid_state.moleFraction(compIdx) ) );
        }
 
        fluid_state_scalar.setLvalue(Opm::getValue(fluid_state.L()));
        // other values need to be Scalar, but I guess the fluidstate does not support it yet.
        fluid_state_scalar.setPressure(FluidSystem::oilPhaseIdx,
                                       Opm::getValue(fluid_state.pressure(FluidSystem::oilPhaseIdx)));
        fluid_state_scalar.setPressure(FluidSystem::gasPhaseIdx,
                                       Opm::getValue(fluid_state.pressure(FluidSystem::gasPhaseIdx)));
 
        fluid_state_scalar.setTemperature(Opm::getValue(fluid_state.temperature(0)));
 
        const auto is_single_phase = flash_solve_scalar_(fluid_state_scalar, twoPhaseMethod, flash_tolerance, eos_type, verbosity);
 
        // the flash solution process were performed in scalar form, after the flash calculation finishes,
        // ensure that things in fluid_state_scalar is transformed to fluid_state
        for (int compIdx=0; compIdx<numComponents; ++compIdx){
                const auto x_i = fluid_state_scalar.moleFraction(oilPhaseIdx, compIdx);
                fluid_state.setMoleFraction(oilPhaseIdx, compIdx, x_i);
                const auto y_i = fluid_state_scalar.moleFraction(gasPhaseIdx, compIdx);
                fluid_state.setMoleFraction(gasPhaseIdx, compIdx, y_i);
        }
 
        // we update the derivatives in fluid_state
        updateDerivatives_(fluid_state_scalar, fluid_state, eos_type, is_single_phase);
 
        return is_single_phase;
    } //end solve

    template <class FluidState, class ComponentVector>
    static void solve(FluidState& fluid_state,
                      const ComponentVector& globalMolarities,
                      Scalar tolerance = 0.0)
    {
        using MaterialTraits = NullMaterialTraits<Scalar, numPhases>;
        using MaterialLaw = NullMaterial<MaterialTraits>;
        using MaterialLawParams = typename MaterialLaw::Params;
 
        MaterialLawParams matParams;
        solve<MaterialLaw>(fluid_state, matParams, globalMolarities, tolerance);
    }
 
    template <class Vector>
    static typename Vector::field_type solveRachfordRice_g_(const Vector& K, const Vector& z, int verbosity)
    {
        // Find min and max K. Have to do a laborious for loop to avoid water component (where K=0)
        // TODO: Replace loop with Dune::min_value() and Dune::max_value() when water component is properly handled
        using field_type = typename Vector::field_type;
        constexpr field_type tol = 1e-12;
        constexpr int itmax = 10000;
        field_type Kmin = K[0];
        field_type Kmax = K[0];
        for (int compIdx = 1; compIdx < numComponents; ++compIdx){
            if (K[compIdx] < Kmin)
                Kmin = K[compIdx];
            else if (K[compIdx] >= Kmax)
                Kmax = K[compIdx];
        }
        // Lower and upper bound for solution
        auto Vmin = 1 / (1 - Kmax);
        auto Vmax = 1 / (1 - Kmin);
        // Initial guess
        auto V = (Vmin + Vmax)/2;
        // Print initial guess and header
        if (verbosity == 3 || verbosity == 4) {
            std::cout << "Initial guess " << numComponents << "c : V = " << V << " and [Vmin, Vmax] = [" << Vmin << ", " << Vmax << "]" << std::endl;
            std::cout << std::setw(10) << "Iteration" << std::setw(16) << "abs(step)" << std::setw(16) << "V" << std::endl;
        }
        // Newton-Raphson loop
        for (int iteration = 1; iteration < itmax; ++iteration) {
            // Calculate function and derivative values
            field_type denum = 0.0;
            field_type r = 0.0;
            for (int compIdx = 0; compIdx < numComponents; ++compIdx){
                auto dK = K[compIdx] - 1.0;
                auto a = z[compIdx] * dK;
                auto b = (1 + V * dK);
                r += a/b;
                denum += z[compIdx] * (dK*dK) / (b*b);
            }
            auto delta = r / denum;
            V += delta;
 
            // Check if V is within the bounds, and if not, we apply bisection method
            if (V < Vmin || V > Vmax)
                {
                    // Print info
                    if (verbosity == 3 || verbosity == 4) {
                        std::cout << "V = " << V << " is not within the the range [Vmin, Vmax], solve using Bisection method!" << std::endl;
                    }
 
                    // Run bisection
                    // TODO: This is required for some cases. Not clear why
                    // since the objective function should be monotone with a
                    // single zero between the Lmin/Lmax interval defined by
                    // K-values.
                    decltype(Vmax) Lmin = 1.0;
                    decltype(Vmin) Lmax = 0.0;
                    auto L = bisection_g_(K, Lmin, Lmax, z, verbosity);
 
                    // Print final result
                    if (verbosity >= 1) {
                        std::cout << "Rachford-Rice (Bisection) converged to final solution L = " << L << std::endl;
                    }
                    return L;
                }
 
            // Print iteration info
            if (verbosity == 3 || verbosity == 4) {
                std::cout << std::setw(10) << iteration << std::setw(16) << Opm::abs(delta) << std::setw(16) << V << std::endl;
            }
            // Check for convergence
            if ( Opm::abs(r) < tol ) {
                auto L = 1 - V;
                // Should we make sure the range of L is within (0, 1)?
 
                // Print final result
                if (verbosity >= 1) {
                    std::cout << "Rachford-Rice converged to final solution L = " << L << std::endl;
                }
                return L;
            }
        }
 
        // Throw error if Rachford-Rice fails
        throw std::runtime_error(" Rachford-Rice did not converge within maximum number of iterations" );
    }
 
    // performing the flash calculation, which is done with Scalar without touching derivatives
    template <typename FluidState>
    static bool flash_solve_scalar_(FluidState& fluid_state,
                                    const std::string& twoPhaseMethod,
                                    const Scalar flash_tolerance,
                                    const EOSType& eos_type,
                                    const int verbosity = 0)
    {
        // Do a stability test to check if cell is is_single_phase-phase (do for all cells the first time).
        bool is_stable = false;
        auto L_scalar = fluid_state.L();
        using ScalarVector = Dune::FieldVector<Scalar, numComponents>;
        ScalarVector K_scalar, z_scalar;
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            K_scalar[compIdx] = fluid_state.K(compIdx);
            z_scalar[compIdx] = fluid_state.moleFraction(compIdx);
        }
 
        if ( L_scalar <= 0 || L_scalar == 1 ) {
            if (verbosity >= 1) {
                std::cout << "Perform stability test (L <= 0 or L == 1)!" << std::endl;
            }
            phaseStabilityTest_(is_stable, K_scalar, fluid_state, z_scalar, eos_type, verbosity);
        }
        if (verbosity >= 1) {
            std::cout << "Inputs after stability test are K = [" << K_scalar << "], L = [" << L_scalar << "], z = [" << z_scalar << "], P = " << fluid_state.pressure(0) << ", and T = " << fluid_state.temperature(0) << std::endl;
        }
        // TODO: we do not need two variables is_stable and is_single_hase, while lacking a good name
        // TODO: from the later code, is good if we knows whether single_phase_gas or single_phase_oil here
        const bool is_single_phase = is_stable;
 
        // Update the composition if cell is two-phase
        if ( !is_single_phase ) {
            // Rachford Rice equation to get initial L for composition solver
            L_scalar = solveRachfordRice_g_(K_scalar, z_scalar, verbosity);
            flash_2ph(z_scalar, twoPhaseMethod, K_scalar, L_scalar, fluid_state, flash_tolerance, eos_type, verbosity);
        } else {
            // Cell is one-phase. Use Li's phase labeling method to see if it's liquid or vapor
            L_scalar = li_single_phase_label_(fluid_state, z_scalar, verbosity);
        }
        fluid_state.setLvalue(L_scalar);
        return is_single_phase;
    }
 
    template <class Vector>
    static typename Vector::field_type bisection_g_(const Vector& K, typename Vector::field_type Lmin,
                                                    typename Vector::field_type Lmax, const Vector& z, int verbosity)
    {
        // Calculate for g(Lmin) for first comparison with gMid = g(L)
        typename Vector::field_type gLmin = rachfordRice_g_(K, Lmin, z);
 
        // Print new header
        if (verbosity >= 3) {
                std::cout << std::setw(10) << "Iteration" << std::setw(16) << "g(Lmid)" << std::setw(16) << "L" << std::endl;
        }
 
        constexpr int max_it = 10000;
 
        auto closeLmaxLmin = [](double max_v, double min_v) {
            return Opm::abs(max_v - min_v) / 2. < 1e-10;
            // what if max_v < min_v?
        };
 
        // Bisection loop
        if (closeLmaxLmin(Lmax, Lmin) ){
            throw std::runtime_error(fmt::format("Strange bisection with Lmax {} and Lmin {}?", Lmax, Lmin));
        }
        for (int iteration = 0; iteration < max_it; ++iteration){
            // New midpoint
            auto L = (Lmin + Lmax) / 2;
            auto gMid = rachfordRice_g_(K, L, z);
            // std::cout << ">>> Lmin = " << Lmin << "g(Lmin) = " << gLmin << " L = " << L << " g(L) = " << gMid << std::endl;
            if (verbosity == 3 || verbosity == 4) {
                std::cout << std::setw(10) << iteration << std::setw(16) << gMid << std::setw(16) << L << std::endl;
            }
 
            // Check if midpoint fulfills g=0 or L - Lmin is sufficiently small
            if (Opm::abs(gMid) < 1e-16 || closeLmaxLmin(Lmax, Lmin)){
                return L;
            }
            // Else we repeat with midpoint being either Lmin og Lmax (depending on the signs).
            else if (Dune::sign(gMid) != Dune::sign(gLmin)) {
                // gMid has different sign as gLmin, so we set L as the new Lmax
                Lmax = L;
            }
            else {
                // gMid and gLmin have same sign so we set L as the new Lmin
                Lmin = L;
                gLmin = gMid;
            }
        }
        throw std::runtime_error(" Rachford-Rice bisection failed with " + std::to_string(max_it) + " iterations!");
    }
 
    template <class Vector, class FlashFluidState>
    static typename Vector::field_type li_single_phase_label_(const FlashFluidState& fluid_state, const Vector& z, int verbosity)
    {
        // Calculate intermediate sum
        typename Vector::field_type sumVz = 0.0;
        for (int compIdx=0; compIdx<numComponents; ++compIdx){
            // Get component information
            const auto& V_crit = FluidSystem::criticalVolume(compIdx);
 
            // Sum calculation
            sumVz += (V_crit * z[compIdx]);
        }
 
        // Calculate approximate (pseudo) critical temperature using Li's method
        typename Vector::field_type Tc_est = 0.0;
        for (int compIdx=0; compIdx<numComponents; ++compIdx){
            // Get component information
            const auto& V_crit = FluidSystem::criticalVolume(compIdx);
            const auto& T_crit = FluidSystem::criticalTemperature(compIdx);
 
            // Sum calculation
            Tc_est += (V_crit * T_crit * z[compIdx] / sumVz);
        }
 
        // Get temperature
        const auto& T = fluid_state.temperature(0);
 
        // If temperature is below estimated critical temperature --> phase = liquid; else vapor
        typename Vector::field_type L;
        if (T < Tc_est) {
            // Liquid
            L = 1.0;
 
            // Print
            if (verbosity >= 1) {
                std::cout << "Cell is single-phase, liquid (L = 1.0) due to Li's phase labeling method giving T < Tc_est (" << T << " < " << Tc_est << ")!" << std::endl;
            }
        }
        else {
            // Vapor
            L = 0.0;
 
            // Print
            if (verbosity >= 1) {
                std::cout << "Cell is single-phase, vapor (L = 0.0) due to Li's phase labeling method giving T >= Tc_est (" << T << " >= " << Tc_est << ")!" << std::endl;
            }
        }
 
 
        return L;
    }
 
    template <class FlashFluidState, class ComponentVector>
    static void phaseStabilityTest_(bool& isStable, ComponentVector& K, FlashFluidState& fluid_state, const ComponentVector& z, const EOSType& eos_type, int verbosity)
    {
        // Declarations
        bool isTrivialL, isTrivialV;
        ComponentVector x, y;
        typename FlashFluidState::ValueType S_l, S_v;
        ComponentVector K0 = K;
        ComponentVector K1 = K;
 
        // Check for vapour instable phase
        if (verbosity == 3 || verbosity == 4) {
            std::cout << "Stability test for vapor phase:" << std::endl;
        }
        checkStability_(fluid_state, isTrivialV, K0, y, S_v, z, /*isGas=*/true, eos_type, verbosity);
        bool V_unstable = (S_v < (1.0 + 1e-5)) || isTrivialV;
 
        // Check for liquids stable phase
        if (verbosity == 3 || verbosity == 4) {
            std::cout << "Stability test for liquid phase:" << std::endl;
        }
        checkStability_(fluid_state, isTrivialL, K1, x, S_l, z, /*isGas=*/false, eos_type, verbosity);
        bool L_stable = (S_l < (1.0 + 1e-5)) || isTrivialL;
 
        // L-stable means success in making liquid, V-unstable means no success in making vapour
        isStable = L_stable && V_unstable;
        if (isStable) {
            // Single phase, i.e. phase composition is equivalent to the global composition
            // Update fluid_state with mole fraction
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                fluid_state.setMoleFraction(gasPhaseIdx, compIdx, z[compIdx]);
                fluid_state.setMoleFraction(oilPhaseIdx, compIdx, z[compIdx]);
            }
        }
        // If not stable: use the mole fractions from Michelsen's test to update K
        else {
            for (int compIdx = 0; compIdx<numComponents; ++compIdx) {
                K[compIdx] = y[compIdx] / x[compIdx];
            }
        }
    }
 
protected:
 
    template <class FlashFluidState>
    static typename FlashFluidState::ValueType wilsonK_(const FlashFluidState& fluid_state, int compIdx)
    {
        const auto& acf = FluidSystem::acentricFactor(compIdx);
        const auto& T_crit = FluidSystem::criticalTemperature(compIdx);
        const auto& T = fluid_state.temperature(0);
        const auto& p_crit = FluidSystem::criticalPressure(compIdx);
        const auto& p = fluid_state.pressure(0); //for now assume no capillary pressure
 
        const auto& tmp = Opm::exp(5.3727 * (1+acf) * (1-T_crit/T)) * (p_crit/p);
        return tmp;
    }
 
    template <class Vector>
    static typename Vector::field_type rachfordRice_g_(const Vector& K, typename Vector::field_type L, const Vector& z)
    {
        typename Vector::field_type g=0;
        for (int compIdx=0; compIdx<numComponents; ++compIdx){
            g += (z[compIdx]*(K[compIdx]-1))/(K[compIdx]-L*(K[compIdx]-1));
        }
        return g;
    }
 
    template <class Vector>
    static typename Vector::field_type rachfordRice_dg_dL_(const Vector& K, const typename Vector::field_type L, const Vector& z)
    {
        typename Vector::field_type dg=0;
        for (int compIdx=0; compIdx<numComponents; ++compIdx){
            dg += (z[compIdx]*(K[compIdx]-1)*(K[compIdx]-1))/((K[compIdx]-L*(K[compIdx]-1))*(K[compIdx]-L*(K[compIdx]-1)));
        }
        return dg;
    }
 
    template <class FlashFluidState, class ComponentVector>
    static void checkStability_(const FlashFluidState& fluid_state, bool& isTrivial, ComponentVector& K, ComponentVector& xy_loc,
                                typename FlashFluidState::ValueType& S_loc, const ComponentVector& z, bool isGas, const EOSType& eos_type,
                                int verbosity)
    {
        using FlashEval = typename FlashFluidState::ValueType;
        using CubicEOS = typename Opm::CubicEOS<Scalar, FluidSystem>;
 
        // Declarations
        FlashFluidState fluid_state_fake = fluid_state;
        FlashFluidState fluid_state_global = fluid_state;
 
        // Setup output
        if (verbosity >= 3) {
            std::cout << std::setw(10) << "Iteration" << std::setw(16) << "K-Norm" << std::setw(16) << "R-Norm" << std::endl;
        }
 
        // Michelsens stability test.
        // Make two fake phases "inside" one phase and check for positive volume
        for (int i = 0; i < 20000; ++i) {
            S_loc = 0.0;
            if (isGas) {
                for (int compIdx=0; compIdx<numComponents; ++compIdx){
                    xy_loc[compIdx] = K[compIdx] * z[compIdx];
                    S_loc += xy_loc[compIdx];
                }
                for (int compIdx=0; compIdx<numComponents; ++compIdx){
                    xy_loc[compIdx] /= S_loc;
                    fluid_state_fake.setMoleFraction(gasPhaseIdx, compIdx, xy_loc[compIdx]);
                }
            }
            else {
                for (int compIdx=0; compIdx<numComponents; ++compIdx){
                    xy_loc[compIdx] = z[compIdx]/K[compIdx];
                    S_loc += xy_loc[compIdx];
                }
                for (int compIdx=0; compIdx<numComponents; ++compIdx){
                    xy_loc[compIdx] /= S_loc;
                    fluid_state_fake.setMoleFraction(oilPhaseIdx, compIdx, xy_loc[compIdx]);
                }
            }
 
            int phaseIdx = (isGas ? static_cast<int>(gasPhaseIdx) : static_cast<int>(oilPhaseIdx));
            int phaseIdx2 = (isGas ? static_cast<int>(oilPhaseIdx) : static_cast<int>(gasPhaseIdx));
            // TODO: not sure the following makes sense
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                fluid_state_global.setMoleFraction(phaseIdx2, compIdx, z[compIdx]);
            }
 
            typename FluidSystem::template ParameterCache<FlashEval> paramCache_fake(eos_type);
            paramCache_fake.updatePhase(fluid_state_fake, phaseIdx);
 
            typename FluidSystem::template ParameterCache<FlashEval> paramCache_global(eos_type);
            paramCache_global.updatePhase(fluid_state_global, phaseIdx2);
 
            //fugacity for fake phases each component
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                auto phiFake = CubicEOS::computeFugacityCoefficient(fluid_state_fake, paramCache_fake, phaseIdx, compIdx);
                auto phiGlobal = CubicEOS::computeFugacityCoefficient(fluid_state_global, paramCache_global, phaseIdx2, compIdx);
 
                fluid_state_fake.setFugacityCoefficient(phaseIdx, compIdx, phiFake);
                fluid_state_global.setFugacityCoefficient(phaseIdx2, compIdx, phiGlobal);
            }
 
 
            ComponentVector R;
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                if (isGas){
                    auto fug_fake = fluid_state_fake.fugacity(phaseIdx, compIdx);
                    auto fug_global = fluid_state_global.fugacity(phaseIdx2, compIdx);
                    auto fug_ratio = fug_global / fug_fake;
                    R[compIdx] = fug_ratio / S_loc;
                }
                else{
                    auto fug_fake = fluid_state_fake.fugacity(phaseIdx, compIdx);
                    auto fug_global = fluid_state_global.fugacity(phaseIdx2, compIdx);
                    auto fug_ratio = fug_fake / fug_global;
                    R[compIdx] = fug_ratio * S_loc;
                }
            }
 
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                K[compIdx] *= R[compIdx];
            }
            Scalar R_norm = 0.0;
            Scalar K_norm = 0.0;
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                auto a = Opm::getValue(R[compIdx]) - 1.0;
                auto b = Opm::log(Opm::getValue(K[compIdx]));
                R_norm += a*a;
                K_norm += b*b;
            }
 
            // Print iteration info
            if (verbosity >= 3) {
                std::cout << std::setw(10) << i << std::setw(16) << K_norm << std::setw(16) << R_norm << std::endl;
            }
 
            // Check convergence
            isTrivial = (K_norm < 1e-5);
            if (isTrivial || R_norm < 1e-10)
                return;
            //todo: make sure that no mole fraction is smaller than 1e-8 ?
            //todo: take care of water!
        }
        throw std::runtime_error(" Stability test did not converge");
    }//end checkStability
 
    // TODO: basically FlashFluidState and ComponentVector are both depending on the one Scalar type
    template <class FlashFluidState, class ComponentVector>
    static void computeLiquidVapor_(FlashFluidState& fluid_state, typename FlashFluidState::ValueType& L, ComponentVector& K, const ComponentVector& z)
    {
        // Calculate x and y, and normalize
        ComponentVector x;
        ComponentVector y;
        typename FlashFluidState::ValueType sumx=0;
        typename FlashFluidState::ValueType sumy=0;
        for (int compIdx=0; compIdx<numComponents; ++compIdx){
            x[compIdx] = z[compIdx]/(L + (1-L)*K[compIdx]);
            sumx += x[compIdx];
            y[compIdx] = (K[compIdx]*z[compIdx])/(L + (1-L)*K[compIdx]);
            sumy += y[compIdx];
        }
        x /= sumx;
        y /= sumy;
 
        for (int compIdx=0; compIdx<numComponents; ++compIdx){
            fluid_state.setMoleFraction(oilPhaseIdx, compIdx, x[compIdx]);
            fluid_state.setMoleFraction(gasPhaseIdx, compIdx, y[compIdx]);
        }
    }
 
    template <class FluidState, class ComponentVector>
    static void flash_2ph(const ComponentVector& z_scalar,
                          const std::string& flash_2p_method,
                          ComponentVector& K_scalar,
                          typename FluidState::ValueType& L_scalar,
                          FluidState& fluid_state_scalar,
                          const Scalar flash_tolerance,
                          const EOSType& eos_type,
                          int verbosity = 0) {
        if (verbosity >= 1) {
            std::cout << "Cell is two-phase! Solve Rachford-Rice with initial K = [" << K_scalar << "]" << std::endl;
        }
 
        // Calculate composition using nonlinear solver
        // Newton
        bool converged = false;
        if (flash_2p_method == "newton") {
            if (verbosity >= 1) {
                std::cout << "Calculate composition using Newton." << std::endl;
            }
            converged = newtonComposition_(K_scalar, L_scalar, fluid_state_scalar, z_scalar, flash_tolerance, eos_type, verbosity);
        } else if (flash_2p_method == "ssi") {
            // Successive substitution
            if (verbosity >= 1) {
                std::cout << "Calculate composition using Succcessive Substitution." << std::endl;
            }
            converged = successiveSubstitutionComposition_(K_scalar, L_scalar, fluid_state_scalar, z_scalar, false, flash_tolerance, eos_type, verbosity);
        } else if (flash_2p_method == "ssi+newton") {
            converged = successiveSubstitutionComposition_(K_scalar, L_scalar, fluid_state_scalar, z_scalar, true, flash_tolerance, eos_type, verbosity);
            if (!converged) {
                converged = newtonComposition_(K_scalar, L_scalar, fluid_state_scalar, z_scalar, flash_tolerance, eos_type, verbosity);
            }
        } else {
            throw std::logic_error("unknown two phase flash method " + flash_2p_method + " is specified");
        }
 
        if (!converged) {
            throw std::runtime_error("flash calculation did not get converged with " + flash_2p_method);
        }
    }
 
    template <class FlashFluidState, class ComponentVector>
    static bool newtonComposition_(ComponentVector& K,
                                   typename FlashFluidState::ValueType& L,
                                   FlashFluidState& fluid_state,
                                   const ComponentVector& z,
                                   const Scalar tolerance,
                                   const EOSType& eos_type,
                                   int verbosity)
    {
        // Note: due to the need for inverse flash update for derivatives, the following two can be different
        // Looking for a good way to organize them
        constexpr size_t num_equations = numMisciblePhases * numMiscibleComponents + 1;
        constexpr size_t num_primary_variables = numMisciblePhases * numMiscibleComponents + 1;
        using NewtonVector = Dune::FieldVector<Scalar, num_equations>;
        using NewtonMatrix = Dune::FieldMatrix<Scalar, num_equations, num_primary_variables>;
 
        NewtonVector soln(0.);
        NewtonVector res(0.);
        NewtonMatrix jac (0.);
 
        // Compute x and y from K, L and Z
        computeLiquidVapor_(fluid_state, L, K, z);
 
        // Print initial condition
        if (verbosity >= 1) {
            std::cout << " the current L is " << Opm::getValue(L) << std::endl;
            std::cout << "Initial guess: x = [";
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                if (compIdx < numComponents - 1)
                    std::cout << fluid_state.moleFraction(oilPhaseIdx, compIdx) << " ";
                else
                    std::cout << fluid_state.moleFraction(oilPhaseIdx, compIdx);
            }
            std::cout << "], y = [";
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                if (compIdx < numComponents - 1)
                    std::cout << fluid_state.moleFraction(gasPhaseIdx, compIdx) << " ";
                else
                    std::cout << fluid_state.moleFraction(gasPhaseIdx, compIdx);
            }
            std::cout << "], and " << "L = " << L << std::endl;
        }
 
        // Print header
        if (verbosity == 2 || verbosity == 4) {
            std::cout << std::setw(10) << "Iteration" << std::setw(16) << "Norm2(step)" << std::setw(16) << "Norm2(Residual)" << std::endl;
        }
 
        // AD type
        using Eval = DenseAd::Evaluation<Scalar, num_primary_variables>;
        // TODO: we might need to use numMiscibleComponents here
        std::vector<Eval> x(numComponents), y(numComponents);
        Eval l;
 
        // TODO: I might not need to set soln anything here.
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            x[compIdx] = Eval(fluid_state.moleFraction(oilPhaseIdx, compIdx), compIdx);
            const unsigned idx = compIdx + numComponents;
            y[compIdx] = Eval(fluid_state.moleFraction(gasPhaseIdx, compIdx), idx);
        }
        l = Eval(L, num_primary_variables - 1);
 
        // it is created for the AD calculation for the flash calculation
        CompositionalFluidState<Eval, FluidSystem> flash_fluid_state;
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            flash_fluid_state.setMoleFraction(FluidSystem::oilPhaseIdx, compIdx, x[compIdx]);
            flash_fluid_state.setMoleFraction(FluidSystem::gasPhaseIdx, compIdx, y[compIdx]);
            // TODO: should we use wilsonK_ here?
            flash_fluid_state.setKvalue(compIdx, y[compIdx] / x[compIdx]);
        }
        flash_fluid_state.setLvalue(l);
        // other values need to be Scalar, but I guess the fluid_state does not support it yet.
        flash_fluid_state.setPressure(FluidSystem::oilPhaseIdx,
                                      fluid_state.pressure(FluidSystem::oilPhaseIdx));
        flash_fluid_state.setPressure(FluidSystem::gasPhaseIdx,
                                      fluid_state.pressure(FluidSystem::gasPhaseIdx));
 
        // TODO: not sure whether we need to set the saturations
        flash_fluid_state.setSaturation(FluidSystem::gasPhaseIdx,
                                        fluid_state.saturation(FluidSystem::gasPhaseIdx));
        flash_fluid_state.setSaturation(FluidSystem::oilPhaseIdx,
                                        fluid_state.saturation(FluidSystem::oilPhaseIdx));
        flash_fluid_state.setTemperature(fluid_state.temperature(0));
 
        using ParamCache = typename FluidSystem::template ParameterCache<typename CompositionalFluidState<Eval, FluidSystem>::ValueType>;
        ParamCache paramCache(eos_type);
 
        for (unsigned phaseIdx = 0; phaseIdx < numMisciblePhases; ++phaseIdx) {
            paramCache.updatePhase(flash_fluid_state, phaseIdx);
            for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
                // TODO: will phi here carry the correct derivatives?
                Eval phi = FluidSystem::fugacityCoefficient(flash_fluid_state, paramCache, phaseIdx, compIdx);
                flash_fluid_state.setFugacityCoefficient(phaseIdx, compIdx, phi);
            }
        }
        bool converged = false;
        unsigned iter = 0;
        constexpr unsigned max_iter = 1000;
        while (iter < max_iter) {
            assembleNewton_<CompositionalFluidState<Eval, FluidSystem>, ComponentVector, num_primary_variables, num_equations>
                    (flash_fluid_state, z, jac, res);
            if (verbosity >= 1) {
                std::cout << " newton residual is " << res.two_norm() << std::endl;
            }
            converged = res.two_norm() < tolerance;
            if (converged) {
                break;
            }
 
            jac.solve(soln, res);
            constexpr Scalar damping_factor = 1.0;
            // updating x and y
            for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
                x[compIdx] -= soln[compIdx] * damping_factor;
                y[compIdx] -= soln[compIdx + numComponents] * damping_factor;
            }
            l -= soln[num_equations - 1] * damping_factor;
 
            for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
                flash_fluid_state.setMoleFraction(FluidSystem::oilPhaseIdx, compIdx, x[compIdx]);
                flash_fluid_state.setMoleFraction(FluidSystem::gasPhaseIdx, compIdx, y[compIdx]);
                // TODO: should we use wilsonK_ here?
                flash_fluid_state.setKvalue(compIdx, y[compIdx] / x[compIdx]);
            }
            flash_fluid_state.setLvalue(l);
 
            for (unsigned phaseIdx = 0; phaseIdx < numMisciblePhases; ++phaseIdx) {
                paramCache.updatePhase(flash_fluid_state, phaseIdx);
                for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
                    Eval phi = FluidSystem::fugacityCoefficient(flash_fluid_state, paramCache, phaseIdx, compIdx);
                    flash_fluid_state.setFugacityCoefficient(phaseIdx, compIdx, phi);
                }
            }
            ++iter;
        }
        if (verbosity >= 1) {
            for (unsigned i = 0; i < num_equations; ++i) {
                for (unsigned j = 0; j < num_primary_variables; ++j) {
                    std::cout << " " << jac[i][j];
                }
                std::cout << std::endl;
            }
            std::cout << std::endl;
        }
        if (!converged) {
            throw std::runtime_error(" Newton composition update did not converge within maxIterations " + std::to_string(max_iter));
        }
 
        // fluid_state is scalar, we need to update all the values for fluid_state here
        for (unsigned idx = 0; idx < numComponents; ++idx) {
            const auto x_i = Opm::getValue(flash_fluid_state.moleFraction(oilPhaseIdx, idx));
            fluid_state.setMoleFraction(FluidSystem::oilPhaseIdx, idx, x_i);
            const auto y_i = Opm::getValue(flash_fluid_state.moleFraction(gasPhaseIdx, idx));
            fluid_state.setMoleFraction(FluidSystem::gasPhaseIdx, idx, y_i);
            const auto K_i = Opm::getValue(flash_fluid_state.K(idx));
            fluid_state.setKvalue(idx, K_i);
            // TODO: not sure we need K and L here, because they are in the flash_fluid_state anyway.
            K[idx] = K_i;
        }
        L = Opm::getValue(l);
        fluid_state.setLvalue(L);
        return converged;
    }
 
    // TODO: the interface will need to refactor for later usage
    template<typename FlashFluidState, typename ComponentVector, size_t num_primary, size_t num_equation >
    static void assembleNewton_(const FlashFluidState& fluid_state,
                                const ComponentVector& global_composition,
                                Dune::FieldMatrix<double, num_equation, num_primary>& jac,
                                Dune::FieldVector<double, num_equation>& res)
    {
        using Eval = DenseAd::Evaluation<double, num_primary>;
        std::vector<Eval> x(numComponents), y(numComponents);
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            x[compIdx] = fluid_state.moleFraction(oilPhaseIdx, compIdx);
            y[compIdx] = fluid_state.moleFraction(gasPhaseIdx, compIdx);
        }
        const Eval& l = fluid_state.L();
        // TODO: clearing zero whether necessary?
        jac = 0.;
        res = 0.;
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            {
                // z - L*x - (1-L) * y
                auto local_res = -global_composition[compIdx] + l * x[compIdx] + (1 - l) * y[compIdx];
                res[compIdx] = Opm::getValue(local_res);
                for (unsigned i = 0; i < num_primary; ++i) {
                    jac[compIdx][i] = local_res.derivative(i);
                }
            }
 
            {
                // f_liquid - f_vapor = 0
                auto local_res = (fluid_state.fugacity(oilPhaseIdx, compIdx) -
                                  fluid_state.fugacity(gasPhaseIdx, compIdx));
                res[compIdx + numComponents] = Opm::getValue(local_res);
                for (unsigned i = 0; i < num_primary; ++i) {
                    jac[compIdx + numComponents][i] = local_res.derivative(i);
                }
            }
        }
        // sum(x) - sum(y) = 0
        Eval sumx = 0.;
        Eval sumy = 0.;
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            sumx += x[compIdx];
            sumy += y[compIdx];
        }
        auto local_res = sumx - sumy;
        res[num_equation - 1] = Opm::getValue(local_res);
        for (unsigned i = 0; i < num_primary; ++i) {
            jac[num_equation - 1][i] = local_res.derivative(i);
        }
    }
 
    template <typename FlashFluidStateScalar, typename FluidState>
    static void updateDerivatives_(const FlashFluidStateScalar& fluid_state_scalar,
                                   FluidState& fluid_state,
                                   const EOSType& eos_type,
                                   bool is_single_phase)
    {
        if(!is_single_phase)
            updateDerivativesTwoPhase_(fluid_state_scalar, fluid_state, eos_type);
        else
            updateDerivativesSinglePhase_(fluid_state_scalar, fluid_state);
 
    }
 
    template <typename FlashFluidStateScalar, typename FluidState>
    static void updateDerivativesTwoPhase_(const FlashFluidStateScalar& fluid_state_scalar,
                                           FluidState& fluid_state,
                                           const EOSType& eos_type)
    {
        using InputEval = typename FluidState::ValueType;
        using ComponentVector = Dune::FieldVector<InputEval, numComponents>;
        ComponentVector z;
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            z[compIdx] = fluid_state.moleFraction(compIdx);
        }
 
        // getting the secondary Jocobian matrix
        constexpr size_t num_equations = numMisciblePhases * numMiscibleComponents + 1;
        constexpr size_t secondary_num_pv = isThermal ? numComponents + 2 : numComponents + 1;
        // secondary variables: pressure, [temperature if thermal], z for all the components
        using SecondaryEval = Opm::DenseAd::Evaluation<double, secondary_num_pv>;
        using SecondaryComponentVector = Dune::FieldVector<SecondaryEval, numComponents>;
        using SecondaryFlashFluidState = Opm::CompositionalFluidState<SecondaryEval, FluidSystem>;
 
        SecondaryFlashFluidState secondary_fluid_state;
        SecondaryComponentVector secondary_z;
        // p and z are the primary variables here
        // pressure
        const SecondaryEval sec_p = SecondaryEval(fluid_state_scalar.pressure(FluidSystem::oilPhaseIdx), 0);
        secondary_fluid_state.setPressure(FluidSystem::oilPhaseIdx, sec_p);
        secondary_fluid_state.setPressure(FluidSystem::gasPhaseIdx, sec_p);
 
        if constexpr (isThermal) {
            // set the temperature with derivatives
            const SecondaryEval sec_T = SecondaryEval(fluid_state_scalar.temperature(0), 1);
            secondary_fluid_state.setTemperature(sec_T);
        } else {
            // set the temperature as a scalar
            secondary_fluid_state.setTemperature(Opm::getValue(fluid_state_scalar.temperature(0)));
        }
 
        // composition variable offset: 2 for thermal (pressure=0, temperature=1, z starts at 2)
        //                               1 for non-thermal (pressure=0, z starts at 1)
        constexpr unsigned z_offset = isThermal ? 2 : 1;
        for (unsigned idx = 0; idx < numComponents; ++idx) {
            secondary_z[idx] = SecondaryEval(Opm::getValue(z[idx]), idx + z_offset);
        }
        // set up the mole fractions
        for (unsigned idx = 0; idx < numComponents; ++idx) {
            // TODO: double checking that fluid_state_scalar returns a scalar here
            const auto x_i = fluid_state_scalar.moleFraction(oilPhaseIdx, idx);
            secondary_fluid_state.setMoleFraction(FluidSystem::oilPhaseIdx, idx, x_i);
            const auto y_i = fluid_state_scalar.moleFraction(gasPhaseIdx, idx);
            secondary_fluid_state.setMoleFraction(FluidSystem::gasPhaseIdx, idx, y_i);
            // TODO: double checking make sure those are consistent
            const auto K_i = fluid_state_scalar.K(idx);
            secondary_fluid_state.setKvalue(idx, K_i);
        }
        const auto L = fluid_state_scalar.L();
        secondary_fluid_state.setLvalue(L);
        // TODO: Do we need to update the saturations?
        // compositions
        // TODO: we probably can simplify SecondaryFlashFluidState::ValueType
        using SecondaryParamCache = typename FluidSystem::template ParameterCache<typename SecondaryFlashFluidState::ValueType>;
        SecondaryParamCache secondary_param_cache(eos_type);
        for (unsigned phase_idx = 0; phase_idx < numMisciblePhases; ++phase_idx) {
            secondary_param_cache.updatePhase(secondary_fluid_state, phase_idx);
            for (unsigned comp_idx = 0; comp_idx < numComponents; ++comp_idx) {
                SecondaryEval phi = FluidSystem::fugacityCoefficient(secondary_fluid_state, secondary_param_cache, phase_idx, comp_idx);
                secondary_fluid_state.setFugacityCoefficient(phase_idx, comp_idx, phi);
            }
        }
 
        using SecondaryNewtonVector = Dune::FieldVector<Scalar, num_equations>;
        using SecondaryNewtonMatrix = Dune::FieldMatrix<Scalar, num_equations, secondary_num_pv>;
        SecondaryNewtonMatrix sec_jac;
        SecondaryNewtonVector sec_res;
 
 
        //use the regular equations
        assembleNewton_<SecondaryFlashFluidState, SecondaryComponentVector, secondary_num_pv, num_equations>
            (secondary_fluid_state, secondary_z, sec_jac, sec_res);
 
 
        // assembly the major matrix here
        // primary variables are x, y and L
        constexpr size_t primary_num_pv = numMisciblePhases * numMiscibleComponents + 1;
        using PrimaryEval = Opm::DenseAd::Evaluation<double, primary_num_pv>;
        using PrimaryComponentVector = Dune::FieldVector<double, numComponents>;
        using PrimaryFlashFluidState = Opm::CompositionalFluidState<PrimaryEval, FluidSystem>;
 
        PrimaryFlashFluidState primary_fluid_state;
        // primary_z is not needed, because we use z will be okay here
        PrimaryComponentVector primary_z;
        for (unsigned  comp_idx = 0; comp_idx < numComponents; ++comp_idx) {
            primary_z[comp_idx] = Opm::getValue(z[comp_idx]);
        }
        for (unsigned comp_idx = 0; comp_idx < numComponents; ++comp_idx) {
            const auto x_ii = PrimaryEval(fluid_state_scalar.moleFraction(oilPhaseIdx, comp_idx), comp_idx);
            primary_fluid_state.setMoleFraction(oilPhaseIdx, comp_idx, x_ii);
            const unsigned idx = comp_idx + numComponents;
            const auto y_ii = PrimaryEval(fluid_state_scalar.moleFraction(gasPhaseIdx, comp_idx), idx);
            primary_fluid_state.setMoleFraction(gasPhaseIdx, comp_idx, y_ii);
            primary_fluid_state.setKvalue(comp_idx, y_ii / x_ii);
        }
        PrimaryEval l;
        l = PrimaryEval(fluid_state_scalar.L(), primary_num_pv - 1);
        primary_fluid_state.setLvalue(l);
        primary_fluid_state.setPressure(oilPhaseIdx, fluid_state_scalar.pressure(oilPhaseIdx));
        primary_fluid_state.setPressure(gasPhaseIdx, fluid_state_scalar.pressure(gasPhaseIdx));
        primary_fluid_state.setTemperature(fluid_state_scalar.temperature(0));
 
        // TODO: is PrimaryFlashFluidState::ValueType> PrimaryEval here?
        using PrimaryParamCache = typename FluidSystem::template ParameterCache<typename PrimaryFlashFluidState::ValueType>;
        PrimaryParamCache primary_param_cache(eos_type);
        for (unsigned phase_idx = 0; phase_idx < numMisciblePhases; ++phase_idx) {
            primary_param_cache.updatePhase(primary_fluid_state, phase_idx);
            for (unsigned comp_idx = 0; comp_idx < numComponents; ++comp_idx) {
                PrimaryEval phi = FluidSystem::fugacityCoefficient(primary_fluid_state, primary_param_cache, phase_idx, comp_idx);
                primary_fluid_state.setFugacityCoefficient(phase_idx, comp_idx, phi);
            }
        }
 
        using PrimaryNewtonVector = Dune::FieldVector<Scalar, num_equations>;
        using PrimaryNewtonMatrix = Dune::FieldMatrix<Scalar, num_equations, primary_num_pv>;
        PrimaryNewtonVector pri_res;
        PrimaryNewtonMatrix pri_jac;
 
        //use the regular equations
        assembleNewton_<PrimaryFlashFluidState, PrimaryComponentVector, primary_num_pv, num_equations>
            (primary_fluid_state, primary_z, pri_jac, pri_res);
 
        // the following code does not compile with DUNE2.6
        // SecondaryNewtonMatrix xx;
        // pri_jac.solve(xx, sec_jac);
        pri_jac.invert();
        sec_jac.template leftmultiply<PrimaryNewtonMatrix>(pri_jac);
 
        ComponentVector x(numComponents), y(numComponents);
        InputEval L_eval = L;
 
        // use the chainrule (and using partial instead of total
        // derivatives, DF / Dp = dF / dp +  dF / ds * ds/dp.
        // where p is the primary variables and s the secondary variables. We then obtain
        // ds / dp = -inv(dF / ds)*(DF / Dp)
 
        const auto p_l = fluid_state.pressure(FluidSystem::oilPhaseIdx);
        const auto p_v = fluid_state.pressure(FluidSystem::gasPhaseIdx);
        // at the moment, the temperature is the same for both phases
        // T_input is not used for non-thermal case
        [[maybe_unused]] const auto& T_input = fluid_state.temperature(0);
 
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            x[compIdx] = fluid_state_scalar.moleFraction(FluidSystem::oilPhaseIdx,compIdx);//;z[compIdx] * 1. / (L + (1 - L) * K[compIdx]);
            y[compIdx] = fluid_state_scalar.moleFraction(FluidSystem::gasPhaseIdx,compIdx);//;x[compIdx] * K[compIdx];
        }
 
        // then we try to set the derivatives for x, y and L against P, [T] and z.
        // p_l and p_v are the same here, in the future, there might be slightly more complicated scenarios when capillary
        // pressure joins
 
        constexpr size_t num_deri = InputEval::numVars;
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            std::vector<double> deri(num_deri, 0.);
            // derivatives from P
            for (unsigned idx = 0; idx < num_deri; ++idx) {
                deri[idx] = -sec_jac[compIdx][0] * p_l.derivative(idx);
            }
            // derivatives from T (only for thermal case)
            if constexpr (isThermal) {
                for (unsigned idx = 0; idx < num_deri; ++idx) {
                    deri[idx] += -sec_jac[compIdx][1] * T_input.derivative(idx);
                }
            }
 
            for (unsigned cIdx = 0; cIdx < numComponents; ++cIdx) {
                const double pz = -sec_jac[compIdx][cIdx + z_offset];
                const auto& zi = z[cIdx];
                for (unsigned idx = 0; idx < num_deri; ++idx) {
                    deri[idx] += pz * zi.derivative(idx);
                }
            }
            for (unsigned idx = 0; idx < num_deri; ++idx) {
                x[compIdx].setDerivative(idx, deri[idx]);
            }
            // handling y
            for (unsigned idx = 0; idx < num_deri; ++idx) {
                deri[idx] = -sec_jac[compIdx + numComponents][0] * p_v.derivative(idx);
            }
            // derivatives from T (only for thermal case)
            if constexpr (isThermal) {
                for (unsigned idx = 0; idx < num_deri; ++idx) {
                    deri[idx] += -sec_jac[compIdx + numComponents][1] * T_input.derivative(idx);
                }
            }
            for (unsigned cIdx = 0; cIdx < numComponents; ++cIdx) {
                const double pz = -sec_jac[compIdx + numComponents][cIdx + z_offset];
                const auto& zi = z[cIdx];
                for (unsigned idx = 0; idx < num_deri; ++idx) {
                    deri[idx] += pz * zi.derivative(idx);
                }
            }
            for (unsigned idx = 0; idx < num_deri; ++idx) {
                y[compIdx].setDerivative(idx, deri[idx]);
            }
 
            // handling derivatives of L
            std::vector<double> deriL(num_deri, 0.);
            for (unsigned idx = 0; idx < num_deri; ++idx) {
                deriL[idx] = -sec_jac[2 * numComponents][0] * p_v.derivative(idx);
            }
            // derivatives from T (only for thermal case)
            if constexpr (isThermal) {
                for (unsigned idx = 0; idx < num_deri; ++idx) {
                    deriL[idx] += -sec_jac[2 * numComponents][1] * T_input.derivative(idx);
                }
            }
            for (unsigned cIdx = 0; cIdx < numComponents; ++cIdx) {
                const double pz = -sec_jac[2 * numComponents][cIdx + z_offset];
                const auto& zi = z[cIdx];
                for (unsigned idx = 0; idx < num_deri; ++idx) {
                    deriL[idx] += pz * zi.derivative(idx);
                }
            }
 
            for (unsigned idx = 0; idx < num_deri; ++idx) {
                L_eval.setDerivative(idx, deriL[idx]);
            }
        }
 
        // set up the mole fractions
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            fluid_state.setMoleFraction(FluidSystem::oilPhaseIdx, compIdx, x[compIdx]);
            fluid_state.setMoleFraction(FluidSystem::gasPhaseIdx, compIdx, y[compIdx]);
        }
        fluid_state.setLvalue(L_eval);
    } //end updateDerivativesTwoPhase
 
    template <typename FlashFluidStateScalar, typename FluidState>
    static void updateDerivativesSinglePhase_(const FlashFluidStateScalar& fluid_state_scalar,
                                              FluidState& fluid_state)
    {
        using InputEval = typename FluidState::ValueType;
        // L_eval is converted from a scalar, so all derivatives are zero at this point
        InputEval L_eval = fluid_state_scalar.L();;
 
        // for single phase situation, x = y = z;
        // and L_eval have all zero derivatives
        for (unsigned compIdx = 0; compIdx < numComponents; ++compIdx) {
            fluid_state.setMoleFraction(FluidSystem::oilPhaseIdx, compIdx, fluid_state.moleFraction(compIdx) );
            fluid_state.setMoleFraction(FluidSystem::gasPhaseIdx, compIdx, fluid_state.moleFraction(compIdx) );
        }
        fluid_state.setLvalue(L_eval);
    } //end updateDerivativesSinglePhase
 
 
    // TODO: or use typename FlashFluidState::ValueType
    template <class FlashFluidState, class ComponentVector>
    static bool successiveSubstitutionComposition_(ComponentVector& K,
                                                   typename ComponentVector::field_type& L,
                                                   FlashFluidState& fluid_state,
                                                   const ComponentVector& z,
                                                   const bool newton_afterwards,
                                                   const Scalar flash_tolerance,
                                                   const EOSType& eos_type,
                                                   const int verbosity)
    {
        // Determine max. iterations based on if it will be used as a standalone flash or as a pre-process to Newton (or other) method.
        const int maxIterations = newton_afterwards ? 5 : 100;
 
        // Store cout format before manipulation
        std::ios_base::fmtflags f(std::cout.flags());
 
        // Print initial guess
        if (verbosity >= 1)
            std::cout << "Initial guess: K = [" << K << "] and L = " << L << std::endl;
 
        if (verbosity == 2 || verbosity == 4) {
            // Print header
            int fugWidth = (numComponents * 12)/2;
            int convWidth = fugWidth + 7;
            std::cout << std::setw(10) << "Iteration" << std::setw(fugWidth) << "fL/fV" << std::setw(convWidth) << "norm2(fL/fv-1)" << std::endl;
        }
        //
        // Successive substitution loop
        //
        for (int i=0; i < maxIterations; ++i){
            // Compute (normalized) liquid and vapor mole fractions
            computeLiquidVapor_(fluid_state, L, K, z);
 
            // Calculate fugacity coefficient
            using ParamCache = typename FluidSystem::template ParameterCache<typename FlashFluidState::ValueType>;
            ParamCache paramCache(eos_type);
            for (int phaseIdx=0; phaseIdx<numMisciblePhases; ++phaseIdx){
                paramCache.updatePhase(fluid_state, phaseIdx);
                for (int compIdx=0; compIdx<numComponents; ++compIdx){
                    auto phi = FluidSystem::fugacityCoefficient(fluid_state, paramCache, phaseIdx, compIdx);
                    fluid_state.setFugacityCoefficient(phaseIdx, compIdx, phi);
                }
            }
 
            // Calculate fugacity ratio
            ComponentVector newFugRatio;
            ComponentVector convFugRatio;
            for (int compIdx=0; compIdx<numComponents; ++compIdx){
                newFugRatio[compIdx] = fluid_state.fugacity(oilPhaseIdx, compIdx)/fluid_state.fugacity(gasPhaseIdx, compIdx);
                convFugRatio[compIdx] = newFugRatio[compIdx] - 1.0;
            }
 
            // Print iteration info
            if (verbosity >= 2) {
                int prec = 5;
                int fugWidth = (prec + 3);
                int convWidth = prec + 9;
                std::cout << std::defaultfloat;
                std::cout << std::fixed;
                std::cout << std::setw(5) << i;
                std::cout << std::setw(fugWidth);
                std::cout << std::setprecision(prec);
                std::cout << newFugRatio;
                std::cout << std::scientific;
                std::cout << std::setw(convWidth) << convFugRatio.two_norm() << std::endl;
            }
 
            // Check convergence
            if (convFugRatio.two_norm() < flash_tolerance) {
                // Restore cout format
                std::cout.flags(f);
 
                // Print info
                if (verbosity >= 1) {
                    std::cout << "Solution converged to the following result :" << std::endl;
                    std::cout << "x = [";
                    for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
                        if (compIdx < numComponents - 1)
                            std::cout << fluid_state.moleFraction(oilPhaseIdx, compIdx) << " ";
                        else
                            std::cout << fluid_state.moleFraction(oilPhaseIdx, compIdx);
                    }
                    std::cout << "]" << std::endl;
                    std::cout << "y = [";
                    for (int compIdx = 0; compIdx < numComponents; ++compIdx) {
                        if (compIdx < numComponents - 1)
                            std::cout << fluid_state.moleFraction(gasPhaseIdx, compIdx) << " ";
                        else
                            std::cout << fluid_state.moleFraction(gasPhaseIdx, compIdx);
                    }
                    std::cout << "]" << std::endl;
                    std::cout << "K = [" << K << "]" << std::endl;
                    std::cout << "L = " << L << std::endl;
                }
                // Restore cout format format
                return true;
            }
            //  If convergence is not met, K is updated in a successive substitution manner
            else {
                // Update K
                for (int compIdx=0; compIdx<numComponents; ++compIdx){
                    K[compIdx] *= newFugRatio[compIdx];
                }
 
                // Solve Rachford-Rice to get L from updated K
                L = solveRachfordRice_g_(K, z, 0);
            }
        }
        // did not get converged. check whether we will do more newton later afterward
        {
           const std::string msg = fmt::format("Successive substitution composition update did not converge within maxIterations {}.", maxIterations);
           if (!newton_afterwards) {
               throw std::runtime_error(msg);
           } else if (verbosity > 0) {
               std::cout << msg << std::endl;
           }
        }
 
        return false;
    }
 
};//end PTFlash
 
} // namespace Opm
 
#endif