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Brine_CO2.hpp
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28 #ifndef OPM_BINARY_COEFF_BRINE_CO2_HPP
29 #define OPM_BINARY_COEFF_BRINE_CO2_HPP
30 
32 
33 namespace Opm {
34 namespace BinaryCoeff {
35 
40 template<class Scalar, class H2O, class CO2, bool verbose = true>
41 class Brine_CO2 {
42  typedef ::Opm::IdealGas<Scalar> IdealGas;
43  static const int liquidPhaseIdx = 0; // index of the liquid phase
44  static const int gasPhaseIdx = 1; // index of the gas phase
45 
46 public:
54  template <class Evaluation>
55  static Evaluation gasDiffCoeff(const Evaluation& temperature, const Evaluation& pressure, bool extrapolate = false)
56  {
57  //Diffusion coefficient of water in the CO2 phase
58  Scalar k = 1.3806504e-23; // Boltzmann constant
59  Scalar c = 4; // slip parameter, can vary between 4 (slip condition) and 6 (stick condition)
60  Scalar R_h = 1.72e-10; // hydrodynamic radius of the solute
61  const Evaluation& mu = CO2::gasViscosity(temperature, pressure, extrapolate); // CO2 viscosity
62  return k / (c * M_PI * R_h) * (temperature / mu);
63  }
64 
71  template <class Evaluation>
72  static Evaluation liquidDiffCoeff(const Evaluation& /*temperature*/, const Evaluation& /*pressure*/)
73  {
74  //Diffusion coefficient of CO2 in the brine phase
75  return 2e-9;
76  }
77 
95  template <class Evaluation>
96  static void calculateMoleFractions(const Evaluation& temperature,
97  const Evaluation& pg,
98  Scalar salinity,
99  const int knownPhaseIdx,
100  Evaluation& xlCO2,
101  Evaluation& ygH2O,
102  bool extrapolate = false)
103  {
104  Evaluation A = computeA_(temperature, pg, extrapolate);
105 
106  /* salinity: conversion from mass fraction to mol fraction */
107  Scalar x_NaCl = salinityToMolFrac_(salinity);
108 
109  // if both phases are present the mole fractions in each phase can be calculate
110  // with the mutual solubility function
111  if (knownPhaseIdx < 0) {
112  Scalar molalityNaCl = moleFracToMolality_(x_NaCl); // molality of NaCl //CHANGED
113  Evaluation m0_CO2 = molalityCO2inPureWater_(temperature, pg, extrapolate); // molality of CO2 in pure water
114  Evaluation gammaStar = activityCoefficient_(temperature, pg, molalityNaCl);// activity coefficient of CO2 in brine
115  Evaluation m_CO2 = m0_CO2 / gammaStar; // molality of CO2 in brine
116  xlCO2 = m_CO2 / (molalityNaCl + 55.508 + m_CO2); // mole fraction of CO2 in brine
117  ygH2O = A * (1 - xlCO2 - x_NaCl); // mole fraction of water in the gas phase
118  }
119 
120  // if only liquid phase is present the mole fraction of CO2 in brine is given and
121  // and the virtual equilibrium mole fraction of water in the non-existing gas phase can be estimated
122  // with the mutual solubility function
123  if (knownPhaseIdx == liquidPhaseIdx)
124  ygH2O = A * (1 - xlCO2 - x_NaCl);
125 
126  // if only gas phase is present the mole fraction of water in the gas phase is given and
127  // and the virtual equilibrium mole fraction of CO2 in the non-existing liquid phase can be estimated
128  // with the mutual solubility function
129  if (knownPhaseIdx == gasPhaseIdx)
130  //y_H2o = fluidstate.
131  xlCO2 = 1 - x_NaCl - ygH2O / A;
132  }
133 
137  template <class Evaluation>
138  static Evaluation henry(const Evaluation& temperature, bool extrapolate = false)
139  { return fugacityCoefficientCO2(temperature, /*pressure=*/1e5, extrapolate)*1e5; }
140 
149  template <class Evaluation>
150  static Evaluation fugacityCoefficientCO2(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
151  {
152  Valgrind::CheckDefined(temperature);
153  Valgrind::CheckDefined(pg);
154 
155  Evaluation V = 1 / (CO2::gasDensity(temperature, pg, extrapolate) / CO2::molarMass()) * 1.e6; // molar volume in cm^3/mol
156  Evaluation pg_bar = pg / 1.e5; // gas phase pressure in bar
157  Evaluation a_CO2 = (7.54e7 - 4.13e4 * temperature); // mixture parameter of Redlich-Kwong equation
158  Scalar b_CO2 = 27.8; // mixture parameter of Redlich-Kwong equation
159  Scalar R = IdealGas::R * 10.; // ideal gas constant with unit bar cm^3 /(K mol)
160  Evaluation lnPhiCO2;
161 
162  lnPhiCO2 = log(V / (V - b_CO2));
163  lnPhiCO2 += b_CO2 / (V - b_CO2);
164  lnPhiCO2 -= 2 * a_CO2 / (R * pow(temperature, 1.5) * b_CO2) * log((V + b_CO2) / V);
165  lnPhiCO2 +=
166  a_CO2 * b_CO2
167  / (R
168  * pow(temperature, 1.5)
169  * b_CO2
170  * b_CO2)
171  * (log((V + b_CO2) / V)
172  - b_CO2 / (V + b_CO2));
173  lnPhiCO2 -= log(pg_bar * V / (R * temperature));
174 
175  return exp(lnPhiCO2); // fugacity coefficient of CO2
176  }
177 
186  template <class Evaluation>
187  static Evaluation fugacityCoefficientH2O(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
188  {
189  const Evaluation& V = 1 / (CO2::gasDensity(temperature, pg, extrapolate) / CO2::molarMass()) * 1.e6; // molar volume in cm^3/mol
190  const Evaluation& pg_bar = pg / 1.e5; // gas phase pressure in bar
191  const Evaluation& a_CO2 = (7.54e7 - 4.13e4 * temperature);// mixture parameter of Redlich-Kwong equation
192  Scalar a_CO2_H2O = 7.89e7;// mixture parameter of Redlich-Kwong equation
193  Scalar b_CO2 = 27.8;// mixture parameter of Redlich-Kwong equation
194  Scalar b_H2O = 18.18;// mixture parameter of Redlich-Kwong equation
195  Scalar R = IdealGas::R * 10.; // ideal gas constant with unit bar cm^3 /(K mol)
196  Evaluation lnPhiH2O;
197 
198  lnPhiH2O =
199  log(V/(V - b_CO2))
200  + b_H2O/(V - b_CO2) - 2*a_CO2_H2O
201  / (R*pow(temperature, 1.5)*b_CO2)*log((V + b_CO2)/V)
202  + a_CO2*b_H2O/(R*pow(temperature, 1.5)*b_CO2*b_CO2)
203  *(log((V + b_CO2)/V) - b_CO2/(V + b_CO2))
204  - log(pg_bar*V/(R*temperature));
205  return exp(lnPhiH2O); // fugacity coefficient of H2O
206  }
207 
208 private:
214  static Scalar salinityToMolFrac_(Scalar salinity) {
215 
216  const Scalar Mw = H2O::molarMass(); /* molecular weight of water [kg/mol] */
217  const Scalar Ms = 58.8e-3; /* molecular weight of NaCl [kg/mol] */
218 
219  const Scalar X_NaCl = salinity;
220  /* salinity: conversion from mass fraction to mol fraction */
221  const Scalar x_NaCl = -Mw * X_NaCl / ((Ms - Mw) * X_NaCl - Ms);
222  return x_NaCl;
223  }
224 
230  static Scalar moleFracToMolality_(Scalar x_NaCl)
231  {
232  // conversion from mol fraction to molality (dissolved CO2 neglected)
233  return 55.508 * x_NaCl / (1 - x_NaCl);
234  }
235 
243  template <class Evaluation>
244  static Evaluation molalityCO2inPureWater_(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
245  {
246  const Evaluation& A = computeA_(temperature, pg, extrapolate); // according to Spycher, Pruess and Ennis-King (2003)
247  const Evaluation& B = computeB_(temperature, pg, extrapolate); // according to Spycher, Pruess and Ennis-King (2003)
248  const Evaluation& yH2OinGas = (1 - B) / (1. / A - B); // equilibrium mol fraction of H2O in the gas phase
249  const Evaluation& xCO2inWater = B * (1 - yH2OinGas); // equilibrium mol fraction of CO2 in the water phase
250  return (xCO2inWater * 55.508) / (1 - xCO2inWater); // CO2 molality
251  }
252 
262  template <class Evaluation>
263  static Evaluation activityCoefficient_(const Evaluation& temperature,
264  const Evaluation& pg,
265  Scalar molalityNaCl)
266  {
267  const Evaluation& lambda = computeLambda_(temperature, pg); // lambda_{CO2-Na+}
268  const Evaluation& xi = computeXi_(temperature, pg); // Xi_{CO2-Na+-Cl-}
269  const Evaluation& lnGammaStar =
270  2*molalityNaCl*lambda + xi*molalityNaCl*molalityNaCl;
271  return exp(lnGammaStar);
272  }
273 
282  template <class Evaluation>
283  static Evaluation computeA_(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
284  {
285  const Evaluation& deltaP = pg / 1e5 - 1; // pressure range [bar] from p0 = 1bar to pg[bar]
286  Scalar v_av_H2O = 18.1; // average partial molar volume of H2O [cm^3/mol]
287  Scalar R = IdealGas::R * 10;
288  const Evaluation& k0_H2O = equilibriumConstantH2O_(temperature); // equilibrium constant for H2O at 1 bar
289  const Evaluation& phi_H2O = fugacityCoefficientH2O(temperature, pg, extrapolate); // fugacity coefficient of H2O for the water-CO2 system
290  const Evaluation& pg_bar = pg / 1.e5;
291  return k0_H2O/(phi_H2O*pg_bar)*exp(deltaP*v_av_H2O/(R*temperature));
292  }
293 
302  template <class Evaluation>
303  static Evaluation computeB_(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
304  {
305  const Evaluation& deltaP = pg / 1e5 - 1; // pressure range [bar] from p0 = 1bar to pg[bar]
306  const Scalar v_av_CO2 = 32.6; // average partial molar volume of CO2 [cm^3/mol]
307  const Scalar R = IdealGas::R * 10;
308  const Evaluation& k0_CO2 = equilibriumConstantCO2_(temperature); // equilibrium constant for CO2 at 1 bar
309  const Evaluation& phi_CO2 = fugacityCoefficientCO2(temperature, pg, extrapolate); // fugacity coefficient of CO2 for the water-CO2 system
310  const Evaluation& pg_bar = pg / 1.e5;
311  return phi_CO2*pg_bar/(55.508*k0_CO2)*exp(-(deltaP*v_av_CO2)/(R*temperature));
312  }
313 
321  template <class Evaluation>
322  static Evaluation computeLambda_(const Evaluation& temperature, const Evaluation& pg)
323  {
324  static const Scalar c[6] =
325  { -0.411370585, 6.07632013E-4, 97.5347708, -0.0237622469, 0.0170656236, 1.41335834E-5 };
326 
327  Evaluation pg_bar = pg / 1.0E5; /* conversion from Pa to bar */
328  return
329  c[0]
330  + c[1]*temperature
331  + c[2]/temperature
332  + c[3]*pg_bar/temperature
333  + c[4]*pg_bar/(630.0 - temperature)
334  + c[5]*temperature*log(pg_bar);
335  }
336 
344  template <class Evaluation>
345  static Evaluation computeXi_(const Evaluation& temperature, const Evaluation& pg)
346  {
347  static const Scalar c[4] =
348  { 3.36389723E-4, -1.98298980E-5, 2.12220830E-3, -5.24873303E-3 };
349 
350  Evaluation pg_bar = pg / 1.0E5; /* conversion from Pa to bar */
351  return c[0] + c[1]*temperature + c[2]*pg_bar/temperature + c[3]*pg_bar/(630.0 - temperature);
352  }
353 
360  template <class Evaluation>
361  static Evaluation equilibriumConstantCO2_(const Evaluation& temperature)
362  {
363  Evaluation temperatureCelcius = temperature - 273.15;
364  static const Scalar c[3] = { 1.189, 1.304e-2, -5.446e-5 };
365  Evaluation logk0_CO2 = c[0] + temperatureCelcius*(c[1] + temperatureCelcius*c[2]);
366  Evaluation k0_CO2 = pow(10.0, logk0_CO2);
367  return k0_CO2;
368  }
369 
376  template <class Evaluation>
377  static Evaluation equilibriumConstantH2O_(const Evaluation& temperature)
378  {
379  Evaluation temperatureCelcius = temperature - 273.15;
380  static const Scalar c[4] = { -2.209, 3.097e-2, -1.098e-4, 2.048e-7 };
381  Evaluation logk0_H2O =
382  c[0] + temperatureCelcius*(c[1] + temperatureCelcius*(c[2] + temperatureCelcius*c[3]));
383  return pow(10.0, logk0_H2O);
384  }
385 
386 };
387 
388 } // namespace BinaryCoeff
389 } // namespace Opm
390 
391 #endif
Relations valid for an ideal gas.
Binary coefficients for brine and CO2.
Definition: Brine_CO2.hpp:41
static Evaluation fugacityCoefficientH2O(const Evaluation &temperature, const Evaluation &pg, bool extrapolate=false)
Returns the fugacity coefficient of the H2O component in a water-CO2 mixture.
Definition: Brine_CO2.hpp:187
static void calculateMoleFractions(const Evaluation &temperature, const Evaluation &pg, Scalar salinity, const int knownPhaseIdx, Evaluation &xlCO2, Evaluation &ygH2O, bool extrapolate=false)
Returns the mol (!) fraction of CO2 in the liquid phase and the mol_ (!) fraction of H2O in the gas p...
Definition: Brine_CO2.hpp:96
static Evaluation gasDiffCoeff(const Evaluation &temperature, const Evaluation &pressure, bool extrapolate=false)
Binary diffusion coefficent [m^2/s] of water in the CO2 phase.
Definition: Brine_CO2.hpp:55
static Evaluation fugacityCoefficientCO2(const Evaluation &temperature, const Evaluation &pg, bool extrapolate=false)
Returns the fugacity coefficient of the CO2 component in a water-CO2 mixture.
Definition: Brine_CO2.hpp:150
static Evaluation liquidDiffCoeff(const Evaluation &, const Evaluation &)
Binary diffusion coefficent [m^2/s] of CO2 in the brine phase.
Definition: Brine_CO2.hpp:72
static Evaluation henry(const Evaluation &temperature, bool extrapolate=false)
Henry coefficent for CO2 in brine.
Definition: Brine_CO2.hpp:138
static Scalar molarMass()
The mass in [kg] of one mole of CO2.
Definition: CO2.hpp:66
static Evaluation gasViscosity(Evaluation temperature, const Evaluation &pressure, bool extrapolate=false)
The dynamic viscosity [Pa s] of CO2.
Definition: CO2.hpp:203
static Evaluation gasDensity(const Evaluation &temperature, const Evaluation &pressure, bool extrapolate=false)
The density of CO2 at a given pressure and temperature [kg/m^3].
Definition: CO2.hpp:189
static const Scalar molarMass()
The molar mass in of water.
Definition: H2O.hpp:80
Relations valid for an ideal gas.
Definition: IdealGas.hpp:38
static const Scalar R
The ideal gas constant .
Definition: IdealGas.hpp:41