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black oil system: use the UniformXTabulated2DFunction class for the oil formation factor

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this means that undersaturated oil is now properly supported.
This commit is contained in:
Andreas Lauser
2014-07-09 18:50:45 +02:00
parent d408ed8cc8
commit ec0902c0f3
2 changed files with 263 additions and 166 deletions
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12
opm/material/UniformXTabulated2DFunction.hpp
View File

@@ -252,8 +252,10 @@ public:
/*!
* \brief Set the x-position of a vertical line.
*
* Returns the i index of that line.
*/
void appendXPos(Scalar nextX)
size_t appendXPos(Scalar nextX)
{
#ifndef NDEBUG
if (xPos_.size())
@@ -262,17 +264,23 @@ public:
xPos_.push_back(nextX);
samples_.resize(xPos_.size());
return xPos_.size() - 1;
}
/*!
* \brief Append a sample point.
*
* Returns the i index of that line.
*/
void appendSamplePoint(int i, Scalar y, Scalar value)
size_t appendSamplePoint(int i, Scalar y, Scalar value)
{
assert(0 <= i && i < numX());
Scalar x = iToX(i);
samples_[i].push_back(SamplePoint(x, y, value));
return samples_[i].size() - 1;
}
/*!

417
opm/material/fluidsystems/BlackOilFluidSystem.hpp
View File

@@ -32,6 +32,7 @@
#include <opm/material/components/H2O.hpp>
#include <opm/material/fluidsystems/BaseFluidSystem.hpp>
#include <opm/material/fluidsystems/NullParameterCache.hpp>
#include <opm/material/UniformXTabulated2DFunction.hpp>
#include <opm/parser/eclipse/Utility/PvtoTable.hpp>
#include <opm/parser/eclipse/Utility/PvtwTable.hpp>
@@ -39,7 +40,6 @@
#include <array>
#include <vector>
#include <iostream>
namespace Opm {
namespace FluidSystems {
@@ -55,6 +55,8 @@ class BlackOil
typedef Opm::Spline<Scalar> Spline;
typedef std::vector<std::pair<Scalar, Scalar> > SplineSamplingPoints;
typedef Opm::UniformXTabulated2DFunction<Scalar> TabulatedFunction;
public:
//! \copydoc BaseFluidSystem::ParameterCache
typedef Opm::NullParameterCache ParameterCache;
@@ -112,29 +114,48 @@ public:
* \brief Sets the pressure-dependent oil viscosity, density and
* gas content using a table stemming from the Eclipse PVTO
* keyword.
*
* When calling this method, the surface densities of the fluids _must_ have already
* been set!
*/
static void setPvtoTable(const PvtoTable &pvtoTable)
{
const auto saturatedTable = pvtoTable.getOuterTable();
int numSamples = saturatedTable->numRows();
assert(numSamples > 1);
assert(saturatedTable->numRows() > 1);
gasDissolutionFactorSpline_.setXYArrays(numSamples,
gasDissolutionFactorSpline_.setXYArrays(saturatedTable->numRows(),
saturatedTable->getPressureColumn(),
saturatedTable->getGasSolubilityColumn(),
/*type=*/Spline::Monotonic);
oilFormationVolumeFactorSpline_.setXYArrays(numSamples,
saturatedTable->getPressureColumn(),
saturatedTable->getOilFormationFactorColumn(),
/*type=*/Spline::Monotonic);
oilViscositySpline_.setXYArrays(numSamples,
saturatedTable->getPressureColumn(),
saturatedTable->getOilViscosityColumn(),
/*type=*/Spline::Monotonic);
Scalar rhooRef = surfaceDensity(oilPhaseIdx);
Scalar rhogRef = surfaceDensity(gasPhaseIdx);
// we don't properly deal with undersaturated oil yet
// extract the table for the oil formation factor
for (int outerIdx = 0; outerIdx < saturatedTable->numRows(); ++ outerIdx) {
Scalar Rs = saturatedTable->getGasSolubilityColumn()[outerIdx];
Scalar XoG = Rs/(rhooRef/rhogRef + Rs);
oilFormationVolumeFactor_.appendXPos(XoG);
oilViscosity_.appendXPos(XoG);
assert(oilFormationVolumeFactor_.numX() == outerIdx + 1);
assert(oilViscosity_.numX() == outerIdx + 1);
const auto underSaturatedTable = pvtoTable.getInnerTable(outerIdx);
for (int innerIdx = underSaturatedTable->numRows() - 1; innerIdx >= 0; -- innerIdx) {
Scalar po = underSaturatedTable->getPressureColumn()[innerIdx];
Scalar Bo = underSaturatedTable->getOilFormationFactorColumn()[innerIdx];
Scalar muo = underSaturatedTable->getOilViscosityColumn()[innerIdx];
oilFormationVolumeFactor_.appendSamplePoint(outerIdx, po, Bo);
oilViscosity_.appendSamplePoint(outerIdx, po, muo);
}
}
referenceFormationVolumeFactor_[oilPhaseIdx] =
oilFormationVolumeFactor_.eval(/*XoG=*/0, surfacePressure_, /*extrapolate=*/true);
}
/*!
@@ -176,6 +197,8 @@ public:
pvdgTable.getFormationFactorColumn(),
/*type=*/Spline::Monotonic);
referenceFormationVolumeFactor_[gasPhaseIdx] =
gasFormationVolumeFactorSpline_.eval(surfacePressure_, /*extrapolate=*/true);
}
/*!
@@ -203,42 +226,177 @@ public:
{
gasDissolutionFactorSpline_.setContainerOfTuples(samplePoints, /*type=*/Spline::Monotonic);
assert(gasDissolutionFactorSpline_.monotonic());
// create the spline representing saturation pressure
// depending of the mass fraction in gas
int n = gasDissolutionFactorSpline_.numSamples()*5;
int delta =
(gasDissolutionFactorSpline_.xMax() - gasDissolutionFactorSpline_.xMin())/(n + 1);
SplineSamplingPoints pSatSamplePoints;
Scalar X_oG = 0;
for (int i=0; i <= n; ++ i) {
Scalar pSat = gasDissolutionFactorSpline_.xMin() + i*delta;
X_oG = saturatedOilGasMassFraction(pSat);
std::pair<Scalar, Scalar> val(X_oG, pSat);
pSatSamplePoints.push_back(val);
}
saturationPressureSpline_.setContainerOfTuples(pSatSamplePoints, /*type=*/Spline::Monotonic);
}
/*!
* \brief Initialize the spline for the oil formation volume factor
*
* \param samplePoints A container of (x,y) values which is suitable to be passed to a spline.
* The oil density/volume factor is a function of (p_o, X_o^G), but this method here
* only requires the volume factor of gas-saturated oil (which only depends on
* pressure) while the dependence on the gas mass fraction is estimated...
*/
static void setSaturatedOilFormationVolumeFactor(const SplineSamplingPoints &samplePoints)
{
oilFormationVolumeFactorSpline_.setContainerOfTuples(samplePoints, /*type=*/Spline::Monotonic);
assert(oilFormationVolumeFactorSpline_.monotonic());
Scalar XoGMin = 0.0;
Scalar RsMax = gasDissolutionFactorSpline_.eval(gasDissolutionFactorSpline_.xMax());
Scalar XoGMax = RsMax*surfaceDensity(gasPhaseIdx)/surfaceDensity(oilPhaseIdx);
Scalar poMin = samplePoints.front().first;
Scalar poMax = samplePoints.back().first;
size_t nX = 20;
size_t nP = samplePoints.size()*2;
Spline oilFormationVolumeFactorSpline;
oilFormationVolumeFactorSpline.setContainerOfTuples(samplePoints, /*type=*/Spline::Monotonic);
// calculate a table of estimated densities depending on pressure and gas mass
// fraction
for (size_t XIdx = 0; XIdx < nX; ++XIdx) {
Scalar XoG = XoGMin + (XoGMax - XoGMin)*XIdx/nX;
oilFormationVolumeFactor_.appendXPos(XoG);
for (size_t pIdx = 0; pIdx < nP; ++pIdx) {
Scalar po = poMin + (poMax - poMin)*pIdx/nP;
// Estimate the oil density. We use the method outlined in:
//
// A. Lauser: "Theory and Numerical Applications of Compositional
// Multi-Phase Flow in Porous Media", Ph.D. Thesis, University of
// Stuttgart, 2014
Scalar poSat = oilSaturationPressure(XoG);
// retrieve the gas formation factor and the oil formation volume factor
Scalar RsSat = gasDissolutionFactorSpline_.eval(poSat, /*extrapolate=*/true);
Scalar BoSat = oilFormationVolumeFactorSpline.eval(poSat, /*extrapolate=*/true);
// retrieve the derivatives of the oil formation volume
// factor and the gas formation factor regarding pressure
Scalar dBoSat_dp = oilFormationVolumeFactorSpline.evalDerivative(poSat, /*extrapolate=*/true);
Scalar dRsSat_dp = gasDissolutionFactorSpline_.evalDerivative(poSat, /*extrapolate=*/true);
// oil compressibility of the oil from SPE-9 [kg/m^3 / Pa)]
Scalar oilCompressibility = (1.1200 - 1.1189)/((5000 - 4000)*6894.76);
// define the derivatives of oil regarding oil component
// mass fraction and pressure
Scalar drhoo_dXoO =
surfaceDensity_[oilPhaseIdx]
* (1 + oilCompressibility)*(po - 1.0135e5);
Scalar drhoo_dp = oilCompressibility;
// Calculate the derivative of the density of saturated
// oil regarding pressure
Scalar drhoosat_dp = - surfaceDensity_[oilPhaseIdx]*dBoSat_dp / (BoSat * BoSat);
// calculate the derivative of the gas component mass
// fraction regarding pressure in saturated oil
Scalar dXoOsat_dp =
- surfaceDensity_[gasPhaseIdx]/surfaceDensity_[oilPhaseIdx]
*(BoSat * dRsSat_dp + RsSat * dBoSat_dp);
// Using the previous derivatives, define a derivative
// for the oil density in regard to the gas mass fraction.
Scalar drhoo_dXoG =
drhoo_dXoO + (drhoo_dp - drhoosat_dp) / dXoOsat_dp;
// calculate the composition of saturated oil.
Scalar XoGsat = saturatedOilGasMassFraction(po);
Scalar XoOsat = 1.0 - XoGsat;
Scalar rhoo =
surfaceDensity_[oilPhaseIdx]/BoSat*(1 + drhoo_dp*(po - poSat))
+ (XoOsat - (1 - XoG))*drhoo_dXoO
+ (XoGsat - XoG)*drhoo_dXoG;
Scalar Bo = surfaceDensity(oilPhaseIdx)/rhoo;
oilFormationVolumeFactor_.appendSamplePoint(XIdx, po, Bo);
}
}
referenceFormationVolumeFactor_[oilPhaseIdx] =
oilFormationVolumeFactor_.eval(/*XoG=*/0, surfacePressure_, /*extrapolate=*/true);
}
/*!
* \brief Initialize the spline for the oil formation volume factor
*
* This is a function of (Rs, po)...
*/
static void setOilFormationVolumeFactor(const TabulatedFunction &Bo)
{
oilFormationVolumeFactor_ = Bo;
referenceFormationVolumeFactor_[oilPhaseIdx] =
oilFormationVolumeFactor_.eval(/*XoG=*/0, surfacePressure_, /*extrapolate=*/true);
}
/*!
* \brief Initialize the spline for the viscosity of gas-saturated
* oil.
*
* \param samplePoints A container of (x,y) values which is suitable to be passed to a spline.
* This is a function of (Rs, po)...
*/
static void setOilViscosity(const TabulatedFunction &muo)
{ oilViscosity_ = muo; }
/*!
* \brief Initialize the oil viscosity
*
* The oil viscosity is a function of (p_o, X_o^G), but this method here only requires
* the viscosity of gas-saturated oil (which only depends on pressure) while the
* dependence on the gas mass fraction is assumed to be zero...
*/
static void setSaturatedOilViscosity(const SplineSamplingPoints &samplePoints)
{
oilViscositySpline_.setContainerOfTuples(samplePoints, /*type=*/Spline::Monotonic);
assert(oilViscositySpline_.monotonic());
}
Scalar XoGMin = 0.0;
Scalar RsMax = gasDissolutionFactorSpline_.eval(gasDissolutionFactorSpline_.xMax());
Scalar XoGMax = RsMax*surfaceDensity(gasPhaseIdx)/surfaceDensity(oilPhaseIdx);
/*!
* \brief Set the volume factor of a phase at reference pressure.
*
* This should usually be 1, but some codes like ECLiPSE also
* allow different values for unknown reasons.
*
* \param phaseIdx The index of the fluid phase.
* \param val The value of the reference volume factor.
*/
static void setReferenceVolumeFactor(int phaseIdx, Scalar val)
{ refFormationVolumeFactor_[phaseIdx] = val; }
Scalar poMin = samplePoints.front().first;
Scalar poMax = samplePoints.back().first;
size_t nX = 20;
size_t nP = samplePoints.size()*2;
Spline muoSpline;
muoSpline.setContainerOfTuples(samplePoints, /*type=*/Spline::Monotonic);
// calculate a table of estimated densities depending on pressure and gas mass
// fraction
for (size_t XIdx = 0; XIdx < nX; ++XIdx) {
Scalar XoG = XoGMin + (XoGMax - XoGMin)*XIdx/nX;
oilViscosity_.appendXPos(XoG);
for (size_t pIdx = 0; pIdx < nP; ++pIdx) {
Scalar po = poMin + (poMax - poMin)*pIdx/nP;
Scalar muo = muoSpline.eval(po);
oilViscosity_.appendSamplePoint(XIdx, po, muo);
}
}
}
/*!
* \brief Initialize the spline for the formation volume factor of dry gas
@@ -249,6 +407,9 @@ public:
{
gasFormationVolumeFactorSpline_.setContainerOfTuples(samplePoints, /*type=*/Spline::Monotonic);
assert(gasFormationVolumeFactorSpline_.monotonic());
referenceFormationVolumeFactor_[gasPhaseIdx] =
gasFormationVolumeFactorSpline_.eval(surfacePressure_, /*extrapolate=*/true);
}
/*!
@@ -298,23 +459,6 @@ public:
// finally, for oil phase, we take the molar mass from the
// spe9 paper
molarMass_[oilCompIdx] = 175e-3; // kg/mol
// create the spline representing saturation pressure
// depending of the mass fraction in gas
int n = gasDissolutionFactorSpline_.numSamples()*5;
int delta =
(gasDissolutionFactorSpline_.xMax() - gasDissolutionFactorSpline_.xMin())/(n + 1);
SplineSamplingPoints pSatSamplePoints;
Scalar X_oG = 0;
for (int i=0; i <= n; ++ i) {
Scalar pSat = gasDissolutionFactorSpline_.xMin() + i*delta;
X_oG = saturatedOilGasMassFraction(pSat);
std::pair<Scalar, Scalar> val(X_oG, pSat);
pSatSamplePoints.push_back(val);
}
saturationPressureSpline_.setContainerOfTuples(pSatSamplePoints, /*type=*/Spline::Monotonic);
}
//! \copydoc BaseFluidSystem::phaseName
@@ -429,7 +573,11 @@ public:
Scalar p = fluidState.pressure(phaseIdx);
switch (phaseIdx) {
case oilPhaseIdx: return oilViscositySpline_.eval(p, /*extrapolate=*/true);
case oilPhaseIdx: {
Scalar XoG = fluidState.massFraction(oilPhaseIdx, gasCompIdx);
// ATTENTION: XoG is represented by the _first_ axis!
return oilViscosity_.eval(XoG, p);
}
case waterPhaseIdx: return waterViscosity_(p);
case gasPhaseIdx: return gasViscosity_(p);
}
@@ -446,29 +594,30 @@ public:
{ return surfaceDensity_[phaseIdx]; }
/*!
* \brief Returns the oil formation volume factor \f$B_o\f$ for a given pressure
* \brief Returns the oil formation volume factor \f$B_o\f$ of saturated oil for a given pressure
*
* \param pressure The pressure of interest [Pa]
*/
static Scalar oilFormationVolumeFactor(Scalar pressure)
static Scalar saturatedOilFormationVolumeFactor(Scalar pressure)
{
Valgrind::CheckDefined(pressure);
return oilFormationVolumeFactorSpline_.eval(pressure,
/*extrapolate=*/true);
// calculate the mass fractions of gas and oil
Scalar XoG = saturatedOilGasMassFraction(pressure);
// ATTENTION: XoG is represented by the _first_ axis!
return oilFormationVolumeFactor_.eval(XoG, pressure);
}
/*!
* \brief Returns the gas formation volume factor \f$B_g\f$ for a given pressure
* \brief Return the normalized formation volume factor of gas.
*
* \param pressure The pressure of interest [Pa]
* "Normalized" means "1 at surface pressure".
*/
static Scalar gasFormationVolumeFactor(Scalar pressure)
{
if (pressure < gasFormationVolumeFactorSpline_.xMin()) {
return (pressure - Bg0_.first)*mBg0_ + Bg0_.second;
}
return gasFormationVolumeFactorSpline_.eval(pressure,
/*extrapolate=*/true);
Scalar BgRaw = gasFormationVolumeFactorSpline_.eval(pressure, /*extrapolate=*/true);
return BgRaw/referenceFormationVolumeFactor_[gasPhaseIdx];
}
/*!
@@ -477,10 +626,7 @@ public:
* \param pressure The pressure of interest [Pa]
*/
static Scalar gasDissolutionFactor(Scalar pressure)
{
return gasDissolutionFactorSpline_.eval(pressure,
/*extrapolate=*/true);
}
{ return gasDissolutionFactorSpline_.eval(pressure, /*extrapolate=*/true); }
/*!
* \brief Returns the fugacity coefficient of a given component in the water phase
@@ -552,14 +698,6 @@ public:
return phi_gG / x_oGf;
}
/*!
* \brief Return the oil phase compressibility at _constant_ composition
*/
static Scalar oilCompressibility()
{
return (1.1200 - 1.1189)/((5000 - 4000)*6894.76); // [kg/m^3 / Pa)]
}
/*!
* \brief Returns the saturation pressure of the oil phase [Pa]
* depending on its mass fraction of the gas component
@@ -623,70 +761,38 @@ public:
return xoG;
}
// density of gas-saturated oil
static Scalar saturatedOilDensity(Scalar pressure)
/*!
* \brief Return the normalized formation volume factor of (potentially)
* under-saturated oil.
*
* "Normalized" means "1 at surface pressure".
*/
static Scalar oilFormationVolumeFactor(Scalar oilPressure, Scalar XoG)
{
// oil formation volume factor at reservoir pressure
Scalar Bo = oilFormationVolumeFactor(pressure);
// oil formation volume factor at standard pressure
Scalar BoRef = refFormationVolumeFactor_[oilPhaseIdx];
// surface density of oil
Scalar rhoRef = surfaceDensity_[oilPhaseIdx];
// reservoir density is surface density scaled by the ratio of
// the volume formation factors
return rhoRef * BoRef/Bo;
// ATTENTION: XoG is represented by the _first_ axis!
Scalar BoRaw = oilFormationVolumeFactor_.eval(XoG, oilPressure);
return BoRaw/referenceFormationVolumeFactor_[oilPhaseIdx];
}
// density of gas (potentially) under-saturated oil
static Scalar oilDensity(Scalar oilPressure, Scalar massFractionGasInOil)
/*!
* \brief Return the density of (potentially) under-saturated oil.
*/
static Scalar oilDensity(Scalar oilPressure, Scalar XoG)
{
Scalar poSat = oilSaturationPressure(massFractionGasInOil);
return 650;
Scalar Bo = oilFormationVolumeFactor(oilPressure, XoG);
return surfaceDensity_[oilPhaseIdx]/Bo;
}
// retrieve the gas formation factor and the oil formation volume factor
Scalar Rs = gasDissolutionFactorSpline_.eval(poSat, /*extrapolate=*/true);
Scalar Bo = oilFormationVolumeFactor(poSat);
// retrieve the derivatives of the oil formation volume
// factor and the gas formation factor regarding pressure
Scalar dBo_dp = oilFormationVolumeFactorSpline_.evalDerivative(poSat, /*extrapolate=*/true);
Scalar dRs_dp = gasDissolutionFactorSpline_.evalDerivative(poSat, /*extrapolate=*/true);
// define the derivatives of oil regarding oil component
// mass fraction and pressure
Scalar drhoo_dXoO =
surfaceDensity_[oilPhaseIdx]
* (1 + oilCompressibility()*(oilPressure - 1.0135e5));
Scalar drhoo_dp = oilCompressibility();
// Calculate the derivative of the density of saturated
// oil regarding pressure
Scalar drhoosat_dp = - surfaceDensity_[oilPhaseIdx]*dBo_dp / (Bo * Bo);
// calculate the derivative of the gas component mass
// fraction regarding pressure in saturated oil
Scalar dXoOsat_dp =
- surfaceDensity_[gasPhaseIdx]/surfaceDensity_[oilPhaseIdx]
*(Bo * dRs_dp + Rs * dBo_dp);
// Using the previous derivatives, define a derivative
// for the oil density in regard to the gas mass fraction.
Scalar drhoo_dXoG =
drhoo_dXoO + (drhoo_dp - drhoosat_dp) / dXoOsat_dp;
// calculate the composition of saturated oil.
Scalar XoGsat = saturatedOilGasMassFraction(oilPressure);
Scalar XoOsat = 1.0 - XoGsat;
Scalar rhoo =
surfaceDensity_[oilPhaseIdx]/Bo*(1 + drhoo_dp*(oilPressure - poSat))
+ (XoOsat - (1 - massFractionGasInOil))*drhoo_dXoO
+ (XoGsat - massFractionGasInOil)*drhoo_dXoG;
return rhoo;
/*!
* \brief Return the density of gas-saturated oil.
*/
static Scalar saturatedOilDensity(Scalar pressure)
{
// mass fraction of gas-saturated oil
Scalar XoG = saturatedOilGasMassFraction(pressure);
return oilDensity(pressure, XoG);
}
private:
@@ -694,26 +800,14 @@ private:
{
// gas formation volume factor at reservoir pressure
Scalar Bg = gasFormationVolumeFactor(pressure);
// gas formation volume factor at standard pressure
Scalar BgRef = refFormationVolumeFactor_[gasPhaseIdx];
// surface density of gas
Scalar rhoRef = surfaceDensity_[gasPhaseIdx];
// reservoir density is surface density scaled by the ratio of
// the volume formation factors
return rhoRef * BgRef/Bg;
return surfaceDensity_[gasPhaseIdx]/Bg;
}
static Scalar waterDensity_(Scalar pressure)
{
// compressibility of water times standard density
Scalar rhoRef = surfaceDensity_[waterPhaseIdx];
return rhoRef *
(1 +
waterCompressibilityScalar_
* (pressure - 1.0135e5));
return rhoRef*(1 + waterCompressibilityScalar_*(pressure - 1.0135e5));
}
static Scalar gasViscosity_(Scalar pressure)
@@ -725,32 +819,30 @@ private:
static Scalar waterViscosity_(Scalar pressure)
{ return waterViscosityScalar_; }
static Spline oilFormationVolumeFactorSpline_;
static Spline oilViscositySpline_;
static TabulatedFunction oilFormationVolumeFactor_;
static TabulatedFunction oilViscosity_;
static Spline gasDissolutionFactorSpline_;
static Spline gasFormationVolumeFactorSpline_;
static Spline saturationPressureSpline_;
static Spline gasFormationVolumeFactorSpline_;
static Spline gasViscositySpline_;
static std::pair<Scalar, Scalar> Bg0_;
static Scalar mBg0_;
static Scalar waterCompressibilityScalar_;
static Scalar waterViscosityScalar_;
static Scalar refFormationVolumeFactor_[numPhases];
static const Scalar surfacePressure_;
static Scalar surfaceDensity_[numPhases];
static Scalar referenceFormationVolumeFactor_[numPhases];
static Scalar molarMass_[numComponents];
};
template <class Scalar>
typename BlackOil<Scalar>::Spline
BlackOil<Scalar>::oilFormationVolumeFactorSpline_;
typename BlackOil<Scalar>::TabulatedFunction
BlackOil<Scalar>::oilFormationVolumeFactor_;
template <class Scalar>
typename BlackOil<Scalar>::Spline
BlackOil<Scalar>::oilViscositySpline_;
typename BlackOil<Scalar>::TabulatedFunction
BlackOil<Scalar>::oilViscosity_;
template <class Scalar>
typename BlackOil<Scalar>::Spline
@@ -769,20 +861,17 @@ typename BlackOil<Scalar>::Spline
BlackOil<Scalar>::gasViscositySpline_;
template <class Scalar>
std::pair<Scalar, Scalar>
BlackOil<Scalar>::Bg0_;
template <class Scalar>
Scalar BlackOil<Scalar>::mBg0_;
template <class Scalar>
Scalar
BlackOil<Scalar>::refFormationVolumeFactor_[BlackOil<Scalar>::numPhases];
const Scalar
BlackOil<Scalar>::surfacePressure_ = 101325.0; // [Pa]
template <class Scalar>
Scalar
BlackOil<Scalar>::surfaceDensity_[BlackOil<Scalar>::numPhases];
template <class Scalar>
Scalar
BlackOil<Scalar>::referenceFormationVolumeFactor_[BlackOil<Scalar>::numPhases];
template <class Scalar>
Scalar
BlackOil<Scalar>::molarMass_[BlackOil<Scalar>::numComponents];