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