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A benchmark study on reactive two-phase flow in porous media: Part I - model description

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Abstract

This paper proposes a benchmark study for reactive multiphase multicomponent flow in porous media. Modeling such problem leads to a highly nonlinear coupled system of partial differential equations, ordinary differential equations and algebraic constraints, which requires special numerical treatment. The benchmark consists of five test problems in total (both in 1D and in 2D), with varying degrees of difficulty, designed to verify the algorithms and the codes dedicated to simulating coupled isothermal Hydro-Chemical processes during injection and storage of CO2 in the subsurface. It is intended to be used as a basis for comparing codes in order to better understand different couplings such as chemical reactions with two-phase flow, phase behavior with equilibrium reactions, dissolution and precipitation.

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Data availability statement

The original statement of the benchmark, as well as a set of files containing the data given in Tables 2 to 9 and Tables 10 to 15 are available from https://doi.org/10.5281/zenodo.8319570.

Additionally, all the data necessary to run the benchmark with OpenDarts [24] are contained in a Python script that is available at https://gitlab.com/open-darts/darts-models/-/tree/development/publications/23_chemical_benchmark, and also as a Jupyter Notebook at https://gitlab.com/open-darts/darts-workshop/-/blob/main/4.Chemical_benchmark.ipynb.

Abbreviations

A :

Exponent for porosity–permeability dependence

\(a_{cw}\) :

Activity of species c

C :

Number of fluid species

\(C_{\alpha }\) :

Phase compressibility of phase \(\alpha \)

\(d_{c\alpha }\) :

Diffusion coefficient

\(f_{c\alpha }\) :

Fugacity of species c in phase \(\alpha \)

g :

Gravity constant

h :

vertical coordinate

\(\mathbb {K}\) :

Permeability

\(\mathbb {K}_0\) :

Initial permeability

\(K_q\) :

Equilibrium constant for reaction q

\(K_{c\alpha }\) :

Phase partition coefficient for species c in phase \(\alpha \)

\(k_{r\alpha }\) :

Relative permeabilty function for phase \(\alpha \)

\(k_{r\alpha }^e\) :

End-point for relative permeability for phase \(\alpha \)

\(l_c\) :

Total flux of fluid species c

M :

Number of mineral phases

\(M_w\) :

Number of moles of water per kg of water

\({\mathcal M}_m\) :

Molar mass of mineral species m

\(n_c\) :

Number of moles of fluid species c

\(n_K\) :

Number of kinetic reactions

\(n_m\) :

Number of moles of mineral species m

\(n_Q\) :

Number of equilibrium reactions

\(n_{\alpha }\) :

Brooks-Corey exponent for phase \(\alpha \)

\(N_{x,y,z}\) :

Number of control volumes

P :

Number of fluid phases

p :

Pressure

\(p_0\) :

Reference pressure

\(P_{\text {ini}}\) :

Initial pressure

\(P_{\text {out}}\) :

Outflow pressure

\(q_c\) :

Total well flow-rate for fluid species c

\(Q_q\) :

Reaction quotient for equilibrium reaction q

\(Q_{\text {inj}}\) :

Injection rate

\(r_k^K\) :

Reaction rate for kinetic reaction k

\(r_q^Q\) :

Reaction rate for equilibrium reaction q

\(\textbf{r}^K\) :

Vector of kinetic reaction rates

\(\textbf{r}^Q\) :

Vector of equilibrium reaction rates

\(s_{\alpha }\) :

Saturation of phase \(\alpha \)

\(s_{r\alpha }\) :

Residual saturation for phase \(\alpha \)

T :

Simulation time

\(\textbf{u}_{\alpha }\) :

Darcy velocity of phase \(\alpha \)

\(v_{ik}\) :

Stoichiometric coefficient for species i in reaction k

\(\textbf{v}^K\) :

Stoichiometrix matrix for kinetic reactions

\(\textbf{v}^Q\) :

Stoichiometric matrix for equilibrium reactions

\(x_{c\alpha }\) :

Molar fraction of species c in phase \(\alpha \)

\(z_c\) :

Overall mole fraction of component c

\(\Delta \{x, y, z\}\) :

Control volume dimension

\(\phi \) :

Porosity

\(\phi _0\) :

Initial porosity

\(\phi _{c\alpha }\) :

Fugacity coefficient for species c in phase \(\alpha \)

\(\gamma \) :

Activity coefficient

\(\mu _{\alpha }\) :

Dynamic viscosity of phase \(\alpha \)

\(\nu _{\alpha }\) :

Mass fraction of phase \(\alpha \)

\(\hat{\rho }_{\alpha }\) :

Mass density of phase \(\alpha \)

\(\rho _{\alpha ,0}\) :

Reference molar density of phase \(\alpha \)

\(\rho _{\alpha }\) :

Molar density of phase \(\alpha \)

c :

Fluid species

m :

Mineral species

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Acknowledgements

The work of E. Ahusborde and B. Amaziane has been partly supported by the Carnot ISIFoR Institute, and “la Région Nouvelle-Aquitaine”, France. These supports are gratefully acknowledged. The authors are grateful to E. Flauraud (IFPEN, France) for providing the initial values for the “Extended chemical model” in Section 4.2.2. We wish also to thank all of the teams who took part in the Benchmark presented herein for their very active participation, as well as for a careful reading of the paper: M. El Ossmani (UPPA, France), E. Flauraud (IFPEN, France), F. Hamon (TotalEnergies), A. Socié, K. U. Mayer, D. Su (University of British Columbia, Canada), M. Tóth (University of Heidelberg, Germany). The authors also thank the anonymous reviewers whose comments led to significant improvements to the paper.

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Authors and Affiliations

  1. Department of Geoscience & Engineering, T.U. Delft, Stevinweg 1, 2628 CN, Delft, Netherlands

    Stephan de Hoop & Denis Voskov

  2. Department for Energy Science and Engineering, School of Earth Sciences, Stanford University, 367 Panama Street, Stanford, CA, 94305, USA

    Denis Voskov

  3. Universite de Pau et des Pays de l’Adour, E2S UPPA, CNRS, LMAP, Pau, France

    Etienne Ahusborde & Brahim Amaziane

  4. Inria, Paris research Center, 2 rue Simone Iff, 75012, Paris, France

    Michel Kern

  5. CERMICS, ENPC, 77455, Marne-la-Vallée, France

    Michel Kern

Authors
  1. Stephan de Hoop
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  2. Denis Voskov
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  3. Etienne Ahusborde
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  4. Brahim Amaziane
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  5. Michel Kern
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Corresponding author

Correspondence to Michel Kern.

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Appendix: Molar composition for initial and injection conditions

Appendix: Molar composition for initial and injection conditions

Tables 10, 11, 12, 13, 14 and 15 give the initial and injection composition in terms of molar fraction of individual species and saturation of each phases.

Table 10 Properties based on 1D injection state: \([P, z_{\text {H}_2\text {O}}, z_{\text {CO}_2}, z_{\text {Ca}}, z_{\text {CO}_3}] =[165, 10^{-12}, 1, 10^{-12}, 10^{-12}]\)
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Table 11 Properties based on 1D initial state: \([P, z_{\text {H}_2\text {O}}, z_{\text {CO}_2}, z_{\text {Ca}}, z_{\text {CO}_3}] =[95, 0.15, 10^{-12}, 0.075, 0.075]\)
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Table 12 Properties based on 2D initial in \(\Omega _2\) state: \([P, z_{\text {H}_2\text {O}}, z_{\text {CO}_2}, z_{\text {Ions}} ] =[95, 0.15, 10^{-12}, 0.15]\)
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Table 13 Properties based on 2D initial in \(\Omega _1\) state: \([P, z_{\text {H}2\text {O}}, z_{\text {CO}_2}, z_{\text {Ions}} ] =[95, 0.4, 2.66\times 10^{-12}, 0.4]\)
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Table 14 Properties based on 2D injection (water stream) state: \([P, z_{\text {H}_2\text {O}}, z_{\text {CO}_2}, z_{\text {Ions}}] =[95, 1, 2\times 10^{-12}, 10^{-12}]\)
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Table 15 Properties based on 2D injection (gas stream) state: \([P, z_{\text {H}_2\text {O}}, z_{\text {CO}_2}, z_{\text {Ions}} ] =[95, 0, 1, 10^{-12}]\)
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de Hoop, S., Voskov, D., Ahusborde, E. et al. A benchmark study on reactive two-phase flow in porous media: Part I - model description. Comput Geosci 28, 175–189 (2024). https://doi.org/10.1007/s10596-024-10268-z

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