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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 
33 
34 namespace Opm {
35 namespace BinaryCoeff {
36 
41 template<class Scalar, class H2O, class CO2, bool verbose = true>
42 class Brine_CO2 {
43  typedef ::Opm::IdealGas<Scalar> IdealGas;
44  static const int liquidPhaseIdx = 0; // index of the liquid phase
45  static const int gasPhaseIdx = 1; // index of the gas phase
46 
47 public:
55  template <class Evaluation>
56  static Evaluation gasDiffCoeff(const Evaluation& temperature, const Evaluation& pressure, bool extrapolate = false)
57  {
58  //Diffusion coefficient of water in the CO2 phase
59  Scalar k = 1.3806504e-23; // Boltzmann constant
60  Scalar c = 4; // slip parameter, can vary between 4 (slip condition) and 6 (stick condition)
61  Scalar R_h = 1.72e-10; // hydrodynamic radius of the solute
62  const Evaluation& mu = CO2::gasViscosity(temperature, pressure, extrapolate); // CO2 viscosity
63  return k / (c * M_PI * R_h) * (temperature / mu);
64  }
65 
72  template <class Evaluation>
73  static Evaluation liquidDiffCoeff(const Evaluation& /*temperature*/, const Evaluation& /*pressure*/)
74  {
75  //Diffusion coefficient of CO2 in the brine phase
76  return 2e-9;
77  }
78 
96  template <class Evaluation>
97  static void calculateMoleFractions(const Evaluation& temperature,
98  const Evaluation& pg,
99  Scalar salinity,
100  const int knownPhaseIdx,
101  Evaluation& xlCO2,
102  Evaluation& ygH2O,
103  bool extrapolate = false)
104  {
105  Evaluation A = computeA_(temperature, pg, extrapolate);
106 
107  /* salinity: conversion from mass fraction to mol fraction */
108  Scalar x_NaCl = salinityToMolFrac_(salinity);
109 
110  // if both phases are present the mole fractions in each phase can be calculate
111  // with the mutual solubility function
112  if (knownPhaseIdx < 0) {
113  Scalar molalityNaCl = moleFracToMolality_(x_NaCl); // molality of NaCl //CHANGED
114  Evaluation m0_CO2 = molalityCO2inPureWater_(temperature, pg, extrapolate); // molality of CO2 in pure water
115  Evaluation gammaStar = activityCoefficient_(temperature, pg, molalityNaCl);// activity coefficient of CO2 in brine
116  Evaluation m_CO2 = m0_CO2 / gammaStar; // molality of CO2 in brine
117  xlCO2 = m_CO2 / (molalityNaCl + 55.508 + m_CO2); // mole fraction of CO2 in brine
118  ygH2O = A * (1 - xlCO2 - x_NaCl); // mole fraction of water in the gas phase
119  }
120 
121  // if only liquid phase is present the mole fraction of CO2 in brine is given and
122  // and the virtual equilibrium mole fraction of water in the non-existing gas phase can be estimated
123  // with the mutual solubility function
124  if (knownPhaseIdx == liquidPhaseIdx)
125  ygH2O = A * (1 - xlCO2 - x_NaCl);
126 
127  // if only gas phase is present the mole fraction of water in the gas phase is given and
128  // and the virtual equilibrium mole fraction of CO2 in the non-existing liquid phase can be estimated
129  // with the mutual solubility function
130  if (knownPhaseIdx == gasPhaseIdx)
131  //y_H2o = fluidstate.
132  xlCO2 = 1 - x_NaCl - ygH2O / A;
133  }
134 
138  template <class Evaluation>
139  static Evaluation henry(const Evaluation& temperature, bool extrapolate = false)
140  { return fugacityCoefficientCO2(temperature, /*pressure=*/1e5, extrapolate)*1e5; }
141 
150  template <class Evaluation>
151  static Evaluation fugacityCoefficientCO2(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
152  {
153  Valgrind::CheckDefined(temperature);
154  Valgrind::CheckDefined(pg);
155 
156  Evaluation V = 1 / (CO2::gasDensity(temperature, pg, extrapolate) / CO2::molarMass()) * 1.e6; // molar volume in cm^3/mol
157  Evaluation pg_bar = pg / 1.e5; // gas phase pressure in bar
158  Evaluation a_CO2 = (7.54e7 - 4.13e4 * temperature); // mixture parameter of Redlich-Kwong equation
159  Scalar b_CO2 = 27.8; // mixture parameter of Redlich-Kwong equation
160  Scalar R = IdealGas::R * 10.; // ideal gas constant with unit bar cm^3 /(K mol)
161  Evaluation lnPhiCO2;
162 
163  lnPhiCO2 = log(V / (V - b_CO2));
164  lnPhiCO2 += b_CO2 / (V - b_CO2);
165  lnPhiCO2 -= 2 * a_CO2 / (R * pow(temperature, 1.5) * b_CO2) * log((V + b_CO2) / V);
166  lnPhiCO2 +=
167  a_CO2 * b_CO2
168  / (R
169  * pow(temperature, 1.5)
170  * b_CO2
171  * b_CO2)
172  * (log((V + b_CO2) / V)
173  - b_CO2 / (V + b_CO2));
174  lnPhiCO2 -= log(pg_bar * V / (R * temperature));
175 
176  return exp(lnPhiCO2); // fugacity coefficient of CO2
177  }
178 
187  template <class Evaluation>
188  static Evaluation fugacityCoefficientH2O(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
189  {
190  const Evaluation& V = 1 / (CO2::gasDensity(temperature, pg, extrapolate) / CO2::molarMass()) * 1.e6; // molar volume in cm^3/mol
191  const Evaluation& pg_bar = pg / 1.e5; // gas phase pressure in bar
192  const Evaluation& a_CO2 = (7.54e7 - 4.13e4 * temperature);// mixture parameter of Redlich-Kwong equation
193  Scalar a_CO2_H2O = 7.89e7;// mixture parameter of Redlich-Kwong equation
194  Scalar b_CO2 = 27.8;// mixture parameter of Redlich-Kwong equation
195  Scalar b_H2O = 18.18;// mixture parameter of Redlich-Kwong equation
196  Scalar R = IdealGas::R * 10.; // ideal gas constant with unit bar cm^3 /(K mol)
197  Evaluation lnPhiH2O;
198 
199  lnPhiH2O =
200  log(V/(V - b_CO2))
201  + b_H2O/(V - b_CO2) - 2*a_CO2_H2O
202  / (R*pow(temperature, 1.5)*b_CO2)*log((V + b_CO2)/V)
203  + a_CO2*b_H2O/(R*pow(temperature, 1.5)*b_CO2*b_CO2)
204  *(log((V + b_CO2)/V) - b_CO2/(V + b_CO2))
205  - log(pg_bar*V/(R*temperature));
206  return exp(lnPhiH2O); // fugacity coefficient of H2O
207  }
208 
209 private:
215  static Scalar salinityToMolFrac_(Scalar salinity) {
216 
217  const Scalar Mw = H2O::molarMass(); /* molecular weight of water [kg/mol] */
218  const Scalar Ms = 58.8e-3; /* molecular weight of NaCl [kg/mol] */
219 
220  const Scalar X_NaCl = salinity;
221  /* salinity: conversion from mass fraction to mol fraction */
222  const Scalar x_NaCl = -Mw * X_NaCl / ((Ms - Mw) * X_NaCl - Ms);
223  return x_NaCl;
224  }
225 
231  static Scalar moleFracToMolality_(Scalar x_NaCl)
232  {
233  // conversion from mol fraction to molality (dissolved CO2 neglected)
234  return 55.508 * x_NaCl / (1 - x_NaCl);
235  }
236 
244  template <class Evaluation>
245  static Evaluation molalityCO2inPureWater_(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
246  {
247  const Evaluation& A = computeA_(temperature, pg, extrapolate); // according to Spycher, Pruess and Ennis-King (2003)
248  const Evaluation& B = computeB_(temperature, pg, extrapolate); // according to Spycher, Pruess and Ennis-King (2003)
249  const Evaluation& yH2OinGas = (1 - B) / (1. / A - B); // equilibrium mol fraction of H2O in the gas phase
250  const Evaluation& xCO2inWater = B * (1 - yH2OinGas); // equilibrium mol fraction of CO2 in the water phase
251  return (xCO2inWater * 55.508) / (1 - xCO2inWater); // CO2 molality
252  }
253 
263  template <class Evaluation>
264  static Evaluation activityCoefficient_(const Evaluation& temperature,
265  const Evaluation& pg,
266  Scalar molalityNaCl)
267  {
268  const Evaluation& lambda = computeLambda_(temperature, pg); // lambda_{CO2-Na+}
269  const Evaluation& xi = computeXi_(temperature, pg); // Xi_{CO2-Na+-Cl-}
270  const Evaluation& lnGammaStar =
271  2*molalityNaCl*lambda + xi*molalityNaCl*molalityNaCl;
272  return exp(lnGammaStar);
273  }
274 
283  template <class Evaluation>
284  static Evaluation computeA_(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
285  {
286  const Evaluation& deltaP = pg / 1e5 - 1; // pressure range [bar] from p0 = 1bar to pg[bar]
287  Scalar v_av_H2O = 18.1; // average partial molar volume of H2O [cm^3/mol]
288  Scalar R = IdealGas::R * 10;
289  const Evaluation& k0_H2O = equilibriumConstantH2O_(temperature); // equilibrium constant for H2O at 1 bar
290  const Evaluation& phi_H2O = fugacityCoefficientH2O(temperature, pg, extrapolate); // fugacity coefficient of H2O for the water-CO2 system
291  const Evaluation& pg_bar = pg / 1.e5;
292  return k0_H2O/(phi_H2O*pg_bar)*exp(deltaP*v_av_H2O/(R*temperature));
293  }
294 
303  template <class Evaluation>
304  static Evaluation computeB_(const Evaluation& temperature, const Evaluation& pg, bool extrapolate = false)
305  {
306  const Evaluation& deltaP = pg / 1e5 - 1; // pressure range [bar] from p0 = 1bar to pg[bar]
307  const Scalar v_av_CO2 = 32.6; // average partial molar volume of CO2 [cm^3/mol]
308  const Scalar R = IdealGas::R * 10;
309  const Evaluation& k0_CO2 = equilibriumConstantCO2_(temperature); // equilibrium constant for CO2 at 1 bar
310  const Evaluation& phi_CO2 = fugacityCoefficientCO2(temperature, pg, extrapolate); // fugacity coefficient of CO2 for the water-CO2 system
311  const Evaluation& pg_bar = pg / 1.e5;
312  return phi_CO2*pg_bar/(55.508*k0_CO2)*exp(-(deltaP*v_av_CO2)/(R*temperature));
313  }
314 
322  template <class Evaluation>
323  static Evaluation computeLambda_(const Evaluation& temperature, const Evaluation& pg)
324  {
325  static const Scalar c[6] =
326  { -0.411370585, 6.07632013E-4, 97.5347708, -0.0237622469, 0.0170656236, 1.41335834E-5 };
327 
328  Evaluation pg_bar = pg / 1.0E5; /* conversion from Pa to bar */
329  return
330  c[0]
331  + c[1]*temperature
332  + c[2]/temperature
333  + c[3]*pg_bar/temperature
334  + c[4]*pg_bar/(630.0 - temperature)
335  + c[5]*temperature*log(pg_bar);
336  }
337 
345  template <class Evaluation>
346  static Evaluation computeXi_(const Evaluation& temperature, const Evaluation& pg)
347  {
348  static const Scalar c[4] =
349  { 3.36389723E-4, -1.98298980E-5, 2.12220830E-3, -5.24873303E-3 };
350 
351  Evaluation pg_bar = pg / 1.0E5; /* conversion from Pa to bar */
352  return c[0] + c[1]*temperature + c[2]*pg_bar/temperature + c[3]*pg_bar/(630.0 - temperature);
353  }
354 
361  template <class Evaluation>
362  static Evaluation equilibriumConstantCO2_(const Evaluation& temperature)
363  {
364  Evaluation temperatureCelcius = temperature - 273.15;
365  static const Scalar c[3] = { 1.189, 1.304e-2, -5.446e-5 };
366  Evaluation logk0_CO2 = c[0] + temperatureCelcius*(c[1] + temperatureCelcius*c[2]);
367  Evaluation k0_CO2 = pow(10.0, logk0_CO2);
368  return k0_CO2;
369  }
370 
377  template <class Evaluation>
378  static Evaluation equilibriumConstantH2O_(const Evaluation& temperature)
379  {
380  Evaluation temperatureCelcius = temperature - 273.15;
381  static const Scalar c[4] = { -2.209, 3.097e-2, -1.098e-4, 2.048e-7 };
382  Evaluation logk0_H2O =
383  c[0] + temperatureCelcius*(c[1] + temperatureCelcius*(c[2] + temperatureCelcius*c[3]));
384  return pow(10.0, logk0_H2O);
385  }
386 
387 };
388 
389 } // namespace BinaryCoeff
390 } // namespace Opm
391 
392 #endif
Relations valid for an ideal gas.
Some templates to wrap the valgrind client request macros.
Binary coefficients for brine and CO2.
Definition: Brine_CO2.hpp:42
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:188
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:97
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:56
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:151
static Evaluation liquidDiffCoeff(const Evaluation &, const Evaluation &)
Binary diffusion coefficent [m^2/s] of CO2 in the brine phase.
Definition: Brine_CO2.hpp:73
static Evaluation henry(const Evaluation &temperature, bool extrapolate=false)
Henry coefficent for CO2 in brine.
Definition: Brine_CO2.hpp:139
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