Comparative Solution Project

Hi David,

Thank you for the detailed investigation! I was not aware of the fact that the script implements a simplified version. Can you let me know what exact simplification is done? I would also be interested in the data points of the plot so that I can do a comparison myself.

Kind regards

Bernd

Thanks for replying to my query.

I haven't checked all of the equations in the Spycher-Pruess papers and the Python code, but I can highlight one area of simplification, which relates to the calculation of fugacity coefficients. The Spycher-Pruess equations for fugacity require Redlich-Kwong a and b parameters a_CO2, a_H2O, b_CO2, b_CO2 and mixing rules (giving a_mix and b_mix). I have included below a couple of screen grabs of equations from the Spycher-Pruess paper:

·           Spycher, N., Pruess, K., and Ennis-King, J. 2003. CO₂-H₂O Mixtures in the Geological Sequestration of CO₂. I. Assessment and Calculation of Mutual Solubilities from 12 to 100°C and up to 600 Bar. Geochimica et Cosmochimica Acta 67 (16): 3015–3031.

In the Python script, b_CO2 is used instead of a b_mix term. In the case of the CO2 fugacity calculation, I have highlighted below in red where the full equations imply a b_mix term should be:

def fugacityCoefficientCO2(T, p, rhoCO2):

    molarMassCO2 = 44.01e-3 # [kg/mol]

    V = 1/(rhoCO2/molarMassCO2)*1e6 # molar volume [cm3/mol]

    p_bar = p/1e5 # phase pressure in bar

    a_CO2 = (7.54e7 - 4.13e4*T) # mixture parameter of Redlich-Kwong equation

    b_CO2 = 27.8 # mixture parameter of Redlich-Kwong equation

    R = 83.1446261815324 # universal gas constant [bar.cm3/mol.K]

    lnPhiCO2 = math.log(V/(V - b_CO2)) + b_CO2/(V - b_CO2) \

               - 2*a_CO2/(R*math.pow(T, 1.5)*b_CO2)*math.log((V + b_CO2)/V) \

               + a_CO2*b_CO2/(R*math.pow(T, 1.5)*b_CO2*b_CO2) \

                 *(math.log((V + b_CO2)/V) - b_CO2/(V + b_CO2)) \

               - math.log(p_bar*V/(R*T))

    return math.exp(lnPhiCO2)

def fugacityCoefficientH2O(T, p, rhoCO2):

    molarMassCO2 = 44.01e-3 # [kg/mol]

    V = 1/(rhoCO2/molarMassCO2)*1e6 # molar volume [cm3/mol]

    p_bar = p/1e5 # phase pressure in bar

    a_CO2 = (7.54e7 - 4.13e4*T) # mixture parameter of  Redlich-Kwong equation

    a_CO2_H2O = 7.89e7 # mixture parameter of Redlich-Kwong equation

    b_CO2 = 27.8 # mixture parameter of Redlich-Kwong equation

    b_H2O = 18.18 # mixture parameter of Redlich-Kwong equation

    R = 83.1446261815324 # universal gas constant [bar.cm3/mol.K]

    lnPhiH2O = math.log(V/(V - b_CO2)) + b_H2O/(V - b_CO2) \

               - 2*a_CO2_H2O/(R*math.pow(T, 1.5)*b_CO2)*math.log((V + b_CO2)/V) \

               + a_CO2*b_H2O/(R*math.pow(T, 1.5)*b_CO2*b_CO2) \

                 *(math.log((V + b_CO2)/V) - b_CO2/(V + b_CO2)) \

               - math.log(p_bar*V/(R*T))

    return math.exp(lnPhiH2O)

There may be other simplifications. As I noted in my earlier post, I do not know whether differences between Python-calculated solubilities and other solubilities are due to such simplifications.

You asked about the data points on the plot I shared.

·           The solubilities from simulators were obtained by from a 1d model initialised with a mixture of CO₂ and H₂O.

·           The web-based calculation of CO₂ solubility made use of the tool at https://www.zetaware.com/utilities/CO2/index.html (although other online CO₂ property calculators are available).

·           The published tables of CO2 solubility at 50°C were obtained from Mao, Zhang, Li and Liu (https://www.sciencedirect.com/science/article/abs/pii/S0009254113001241), although many other measured and calculated CO₂/H₂O solubility tables are available in the literature.

I have summarised solubility values in the attached Excel file, with values at 10, 30, 50 and 70°C.

Regards,

Hi David,

Thank you for the detailed investigation! I was not aware of the fact that the script implements a simplified version. Can you let me know what exact simplification is done? I would also be interested in the data points of the plot so that I can do a comparison myself.

Kind regards

Bernd

Thank you! My guess is that the simplification is made since the definition of b_mix contains the solubilities which are to be calculated. I will check how this can be resolved. I'd certainly appreciate any hint in this direction.

Thanks for replying to my query.

I haven't checked all of the equations in the Spycher-Pruess papers and the Python code, but I can highlight one area of simplification, which relates to the calculation of fugacity coefficients. The Spycher-Pruess equations for fugacity require Redlich-Kwong a and b parameters a_CO2, a_H2O, b_CO2, b_CO2 and mixing rules (giving a_mix and b_mix). I have included below a couple of screen grabs of equations from the Spycher-Pruess paper:

·           Spycher, N., Pruess, K., and Ennis-King, J. 2003. CO₂-H₂O Mixtures in the Geological Sequestration of CO₂. I. Assessment and Calculation of Mutual Solubilities from 12 to 100°C and up to 600 Bar. Geochimica et Cosmochimica Acta 67 (16): 3015–3031.

In the Python script, b_CO2 is used instead of a b_mix term. In the case of the CO2 fugacity calculation, I have highlighted below in red where the full equations imply a b_mix term should be:

def fugacityCoefficientCO2(T, p, rhoCO2):

    molarMassCO2 = 44.01e-3 # [kg/mol]

    V = 1/(rhoCO2/molarMassCO2)*1e6 # molar volume [cm3/mol]

    p_bar = p/1e5 # phase pressure in bar

    a_CO2 = (7.54e7 - 4.13e4*T) # mixture parameter of Redlich-Kwong equation

    b_CO2 = 27.8 # mixture parameter of Redlich-Kwong equation

    R = 83.1446261815324 # universal gas constant [bar.cm3/mol.K]

    lnPhiCO2 = math.log(V/(V - b_CO2)) + b_CO2/(V - b_CO2) \

               - 2*a_CO2/(R*math.pow(T, 1.5)*b_CO2)*math.log((V + b_CO2)/V) \

               + a_CO2*b_CO2/(R*math.pow(T, 1.5)*b_CO2*b_CO2) \

                 *(math.log((V + b_CO2)/V) - b_CO2/(V + b_CO2)) \

               - math.log(p_bar*V/(R*T))

    return math.exp(lnPhiCO2)

def fugacityCoefficientH2O(T, p, rhoCO2):

    molarMassCO2 = 44.01e-3 # [kg/mol]

    V = 1/(rhoCO2/molarMassCO2)*1e6 # molar volume [cm3/mol]

    p_bar = p/1e5 # phase pressure in bar

    a_CO2 = (7.54e7 - 4.13e4*T) # mixture parameter of  Redlich-Kwong equation

    a_CO2_H2O = 7.89e7 # mixture parameter of Redlich-Kwong equation

    b_CO2 = 27.8 # mixture parameter of Redlich-Kwong equation

    b_H2O = 18.18 # mixture parameter of Redlich-Kwong equation

    R = 83.1446261815324 # universal gas constant [bar.cm3/mol.K]

    lnPhiH2O = math.log(V/(V - b_CO2)) + b_H2O/(V - b_CO2) \

               - 2*a_CO2_H2O/(R*math.pow(T, 1.5)*b_CO2)*math.log((V + b_CO2)/V) \

               + a_CO2*b_H2O/(R*math.pow(T, 1.5)*b_CO2*b_CO2) \

                 *(math.log((V + b_CO2)/V) - b_CO2/(V + b_CO2)) \

               - math.log(p_bar*V/(R*T))

    return math.exp(lnPhiH2O)

There may be other simplifications. As I noted in my earlier post, I do not know whether differences between Python-calculated solubilities and other solubilities are due to such simplifications.

You asked about the data points on the plot I shared.

·           The solubilities from simulators were obtained by from a 1d model initialised with a mixture of CO₂ and H₂O.

·           The web-based calculation of CO₂ solubility made use of the tool at https://www.zetaware.com/utilities/CO2/index.html (although other online CO₂ property calculators are available).

·           The published tables of CO2 solubility at 50°C were obtained from Mao, Zhang, Li and Liu (https://www.sciencedirect.com/science/article/abs/pii/S0009254113001241), although many other measured and calculated CO₂/H₂O solubility tables are available in the literature.

I have summarised solubility values in the attached Excel file, with values at 10, 30, 50 and 70°C.

Regards,

Hi David,

Thank you for the detailed investigation! I was not aware of the fact that the script implements a simplified version. Can you let me know what exact simplification is done? I would also be interested in the data points of the plot so that I can do a comparison myself.

Kind regards

Bernd

Actually, the issue is addressed in the paragraph below (23) (in my case (B7)) in the paper, where they also suggest the simplification.

Thank you! My guess is that the simplification is made since the definition of b_mix contains the solubilities which are to be calculated. I will check how this can be resolved. I'd certainly appreciate any hint in this direction.

Thanks for replying to my query.

I haven't checked all of the equations in the Spycher-Pruess papers and the Python code, but I can highlight one area of simplification, which relates to the calculation of fugacity coefficients. The Spycher-Pruess equations for fugacity require Redlich-Kwong a and b parameters a_CO2, a_H2O, b_CO2, b_CO2 and mixing rules (giving a_mix and b_mix). I have included below a couple of screen grabs of equations from the Spycher-Pruess paper:

·           Spycher, N., Pruess, K., and Ennis-King, J. 2003. CO₂-H₂O Mixtures in the Geological Sequestration of CO₂. I. Assessment and Calculation of Mutual Solubilities from 12 to 100°C and up to 600 Bar. Geochimica et Cosmochimica Acta 67 (16): 3015–3031.

In the Python script, b_CO2 is used instead of a b_mix term. In the case of the CO2 fugacity calculation, I have highlighted below in red where the full equations imply a b_mix term should be:

def fugacityCoefficientCO2(T, p, rhoCO2):

    molarMassCO2 = 44.01e-3 # [kg/mol]

    V = 1/(rhoCO2/molarMassCO2)*1e6 # molar volume [cm3/mol]

    p_bar = p/1e5 # phase pressure in bar

    a_CO2 = (7.54e7 - 4.13e4*T) # mixture parameter of Redlich-Kwong equation

    b_CO2 = 27.8 # mixture parameter of Redlich-Kwong equation

    R = 83.1446261815324 # universal gas constant [bar.cm3/mol.K]

    lnPhiCO2 = math.log(V/(V - b_CO2)) + b_CO2/(V - b_CO2) \

               - 2*a_CO2/(R*math.pow(T, 1.5)*b_CO2)*math.log((V + b_CO2)/V) \

               + a_CO2*b_CO2/(R*math.pow(T, 1.5)*b_CO2*b_CO2) \

                 *(math.log((V + b_CO2)/V) - b_CO2/(V + b_CO2)) \

               - math.log(p_bar*V/(R*T))

    return math.exp(lnPhiCO2)

def fugacityCoefficientH2O(T, p, rhoCO2):

    molarMassCO2 = 44.01e-3 # [kg/mol]

    V = 1/(rhoCO2/molarMassCO2)*1e6 # molar volume [cm3/mol]

    p_bar = p/1e5 # phase pressure in bar

    a_CO2 = (7.54e7 - 4.13e4*T) # mixture parameter of  Redlich-Kwong equation

    a_CO2_H2O = 7.89e7 # mixture parameter of Redlich-Kwong equation

    b_CO2 = 27.8 # mixture parameter of Redlich-Kwong equation

    b_H2O = 18.18 # mixture parameter of Redlich-Kwong equation

    R = 83.1446261815324 # universal gas constant [bar.cm3/mol.K]

    lnPhiH2O = math.log(V/(V - b_CO2)) + b_H2O/(V - b_CO2) \

               - 2*a_CO2_H2O/(R*math.pow(T, 1.5)*b_CO2)*math.log((V + b_CO2)/V) \

               + a_CO2*b_H2O/(R*math.pow(T, 1.5)*b_CO2*b_CO2) \

                 *(math.log((V + b_CO2)/V) - b_CO2/(V + b_CO2)) \

               - math.log(p_bar*V/(R*T))

    return math.exp(lnPhiH2O)

There may be other simplifications. As I noted in my earlier post, I do not know whether differences between Python-calculated solubilities and other solubilities are due to such simplifications.

You asked about the data points on the plot I shared.

·           The solubilities from simulators were obtained by from a 1d model initialised with a mixture of CO₂ and H₂O.

·           The web-based calculation of CO₂ solubility made use of the tool at https://www.zetaware.com/utilities/CO2/index.html (although other online CO₂ property calculators are available).

·           The published tables of CO2 solubility at 50°C were obtained from Mao, Zhang, Li and Liu (https://www.sciencedirect.com/science/article/abs/pii/S0009254113001241), although many other measured and calculated CO₂/H₂O solubility tables are available in the literature.

I have summarised solubility values in the attached Excel file, with values at 10, 30, 50 and 70°C.

Regards,

Hi David,

Thank you for the detailed investigation! I was not aware of the fact that the script implements a simplified version. Can you let me know what exact simplification is done? I would also be interested in the data points of the plot so that I can do a comparison myself.

Kind regards

Bernd

I have just noticed that someone else independently brought this up in an issue on the GitHub repository back in April, but has not received any response: https://github.com/Simulation-Benchmarks/11thSPE-CSP/issues/28

I have just noticed that someone else independently brought this up in an issue on the GitHub repository back in April, but has not received any response: https://github.com/Simulation-Benchmarks/11thSPE-CSP/issues/28

------------------------------
Adam Turner
Reservoir Engineer
RPS Energy
------------------------------

Thanks Adam & David for noticing this. Giacomo Rivolta and I  submitted lthe issue (CO2 solubility behavior at high pressures · Issue #28 · Simulation-Benchmarks/11thSPE-CSP

) long time ago, and I also wrote to Olav Moyner about.

This picture compare SPE11 python script output with experimental data:

Original Message:
Sent: 07-26-2024 08:19 AM
From: Adam Turner
Subject: Calculation of CO2 solubility

I have just noticed that someone else independently brought this up in an issue on the GitHub repository back in April, but has not received any response: https://github.com/Simulation-Benchmarks/11thSPE-CSP/issues/28

------------------------------
Adam Turner
Reservoir Engineer
RPS Energy

I double-checked with the data you provided and they coincide. I also came to the conclusion that there is no inconsistency in using Spycher for the solubilities and NIST for density and enthalpy, as required by the CSP description. The script now returns the Spycher values only.

Thanks Adam & David for noticing this. Giacomo Rivolta and I  submitted lthe issue (CO2 solubility behavior at high pressures · Issue #28 · Simulation-Benchmarks/11thSPE-CSP

) long time ago, and I also wrote to Olav Moyner about.

This picture compare SPE11 python script output with experimental data:

Original Message:
Sent: 07-26-2024 08:19 AM
From: Adam Turner
Subject: Calculation of CO2 solubility

I have just noticed that someone else independently brought this up in an issue on the GitHub repository back in April, but has not received any response: https://github.com/Simulation-Benchmarks/11thSPE-CSP/issues/28

------------------------------
Adam Turner
Reservoir Engineer
RPS Energy

I double-checked with the data you provided and they coincide. I also came to the conclusion that there is no inconsistency in using Spycher for the solubilities and NIST for density and enthalpy, as required by the CSP description. The script now returns the Spycher values only.

Original Message:
Sent: 07-26-2024 09:07 AM
From: Alberto Cominelli
Subject: Calculation of CO2 solubility

Thanks Adam & David for noticing this. Giacomo Rivolta and I  submitted lthe issue (CO2 solubility behavior at high pressures · Issue #28 · Simulation-Benchmarks/11thSPE-CSP

) long time ago, and I also wrote to Olav Moyner about.

This picture compare SPE11 python script output with experimental data:

Original Message:
Sent: 07-26-2024 08:19 AM
From: Adam Turner
Subject: Calculation of CO2 solubility

I have just noticed that someone else independently brought this up in an issue on the GitHub repository back in April, but has not received any response: https://github.com/Simulation-Benchmarks/11thSPE-CSP/issues/28

------------------------------
Adam Turner
Reservoir Engineer
RPS Energy

Dear Prof. Flemisch

It seems that the new make_solubility_table.py script updated the solubility table. I want to confirm if we need to switch to using the new table for simulations to ensure that all participating teams are using the same solubility data?

Best regards

Chaojie Di

University of Calgary

Original Message:
Sent: 07-31-2024 04:34 AM
From: Bernd Flemisch
Subject: Calculation of CO2 solubility

I double-checked with the data you provided and they coincide. I also came to the conclusion that there is no inconsistency in using Spycher for the solubilities and NIST for density and enthalpy, as required by the CSP description. The script now returns the Spycher values only.

Original Message:
Sent: 07-26-2024 09:07 AM
From: Alberto Cominelli
Subject: Calculation of CO2 solubility

Thanks Adam & David for noticing this. Giacomo Rivolta and I  submitted lthe issue (CO2 solubility behavior at high pressures · Issue #28 · Simulation-Benchmarks/11thSPE-CSP

) long time ago, and I also wrote to Olav Moyner about.

This picture compare SPE11 python script output with experimental data:

Original Message:
Sent: 07-26-2024 08:19 AM
From: Adam Turner
Subject: Calculation of CO2 solubility

I have just noticed that someone else independently brought this up in an issue on the GitHub repository back in April, but has not received any response: https://github.com/Simulation-Benchmarks/11thSPE-CSP/issues/28

------------------------------
Adam Turner
Reservoir Engineer
RPS Energy

Hi Chaojie,

yes, this would be preferred.

Kind regards

Bernd

Dear Prof. Flemisch

It seems that the new make_solubility_table.py script updated the solubility table. I want to confirm if we need to switch to using the new table for simulations to ensure that all participating teams are using the same solubility data?

Best regards

Chaojie Di

University of Calgary

Original Message:
Sent: 07-31-2024 04:34 AM
From: Bernd Flemisch
Subject: Calculation of CO2 solubility

I double-checked with the data you provided and they coincide. I also came to the conclusion that there is no inconsistency in using Spycher for the solubilities and NIST for density and enthalpy, as required by the CSP description. The script now returns the Spycher values only.

Original Message:
Sent: 07-26-2024 09:07 AM
From: Alberto Cominelli
Subject: Calculation of CO2 solubility

Thanks Adam & David for noticing this. Giacomo Rivolta and I  submitted lthe issue (CO2 solubility behavior at high pressures · Issue #28 · Simulation-Benchmarks/11thSPE-CSP

) long time ago, and I also wrote to Olav Moyner about.

This picture compare SPE11 python script output with experimental data:

Original Message:
Sent: 07-26-2024 08:19 AM
From: Adam Turner
Subject: Calculation of CO2 solubility

I have just noticed that someone else independently brought this up in an issue on the GitHub repository back in April, but has not received any response: https://github.com/Simulation-Benchmarks/11thSPE-CSP/issues/28

------------------------------
Adam Turner
Reservoir Engineer
RPS Energy