# Reactive Transport Modelling of Dolomitisation Using the New CSMP++GEM Coupled Code: Governing Equations, Solution Method and Benchmarking Results

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## Abstract

Reactive transport modelling (RTM) is a powerful tool for understanding subsurface systems where fluid flow and chemical reactions occur simultaneously. RTM has been widely used to understand the formation of dolomite by replacement of calcite, which can be an important control on carbonate reservoir quality. Dolomitisation is a reactive transport process governed by slow dolomite precipitation and cannot be correctly simulated without a kinetic rate model. The new CSMP++GEM coupled RTM code uses the GEMS3K kernel for solving geochemical equilibria by the Gibbs energy minimisation method with the CSMP++ framework that implements a hybrid finite element–finite volume method to solve partial differential equations. The unique feature of the new coupling is the mineral reaction kinetics, implemented via additional metastability constraints. CSMP++GEM is able to simulate single-phase flow and solute transport in porous media together with chemical reactions at different pressure, temperature, and water salinity conditions. This RTM assures mass conservation which is crucial when simulating transport of solutes with low concentrations over geological time. A full feedback of mineral dissolution/precipitation on the fluid flow is provided via corresponding porosity/permeability evolution and two source terms in the pressure equation. First, the mass source term accounts for the mass of solutes released during mineral dissolution or taken from the solution by mineral precipitation. The second source term attributes to the fact that the solution density is affected by mineral dissolution/precipitation, too. This effect is included through the equivalent water salinity, which is calculated from the total amount of dissolved solutes and is used to update the properties of saline water from the H\(_2\)O–NaCl equation of state. This paper puts emphasis on the thorough mathematical derivation of the governing equations and a detailed description of the numerical solution procedure. Two sets of benchmarking results are presented. The first benchmark is a well-known 1D model of dolomitisation by MgCl\(_2\) solution with thermodynamic reactions. In the second benchmark, CSMP++GEM is compared with TOUGHREACT on a 1D model of dolomitisation by sea water taking into account mineral reaction kinetics. The results presented in this paper demonstrate the ability of the CSMP++GEM code to correctly reproduce dolomitisation effects.

## Keywords

Reactive transport modelling Gibbs energy minimisation Finite element finite volume method Sequential non-iterative approach Dolomitisation## List of Symbols

- \(\beta _f\)
Fluid compressibility (Pa\(^{-1}\))

- \(\beta _t\)
Total system compressibility (Pa\(^{-1}\))

- \(\beta _{\phi }\)
Pore compressibility (Pa\(^{-1}\))

- \(\varDelta t\)
Time step

- \(\gamma _i\)
Activity coefficient of

*j*-th DC- \(\kappa \)
Rate constant (mol/m\(^2\)/s)

- \(\kappa ^o\)
Rate constant at reference temperature \(25\,^{\circ }\mathrm {C}\) (mol/m\(^2\)/s)

- \(\varLambda \)
Arrhenius parameter

- \({\mathbf {g}}\)
Gravitational acceleration vector (m/s\(^2\))

- \({\mathbf {J}}\)
Component flux (mol/l m/s)

- \({\mathbf {v}}\)
Darcy velocity (m/s)

- \(\mu _f\)
Fluid dynamic viscosity (Pa s)

- \(\mu _j\)
Normalised chemical potential of the

*j*-th DC- \(\varOmega _k\)
Stability index of the

*k*-th phase- \({\overline{n}}_j^{(x)}\)
Upper AMR for the

*j*-th DC- \(\phi \)
Porosity \((-)\)

- \(\rho _f\)
Fluid density (kg/m\(^3\))

- \(\rho _{r}\)
Rock density (kg/m\(^3\))

- \({\underline{n}}_j^{(x)}\)
Lower AMR for the

*j*-th DC- \({\widehat{n}}\)
GEM problem primal solution vector

- \({\widehat{q}}\)
Vector of Lagrange multipliers

- \({\widehat{u}}\)
GEM problem dual solution vector

- \(A_{k,t}\)
Surface area of the

*k*-th solid phase at time*t*(m\(^2\))- \(A_{S,k}\)
Specific surface area of the

*k*-th solid phase (m\(^2\)/kg)- \(c_i\)
Aqueous concentration of the

*i*-th IC (mol/l)- \(c_{pf}\)
Fluid-specific heat capacity (J/kg K)

- \(c_{pr}\)
Rock-specific heat capacity (J/kg K)

*D*Diffusion–dispersion coefficient (m\(^2\)/s)

- \(E_a\)
Activation energy (J/mol)

*G*Total Gibbs energy of the system (J)

- \(g^o_j\)
Standard chemical potential (Gibbs energy per mole of

*j*-th DC) (J/mol)*K*Thermal conductivity (W/m K)

*k*Permeability (m\(^{2}\))

- \(M_i\)
Molar mass of the

*i*-th IC (kg/mol)- \(m_j\)
Molality (moles per kilogram of water–solvent of the

*j*-th species) (mol/kg\(_w\))- \(n^{(\phi )}\)
Vector of equilibrium phase amounts

- \(n^{(b)}\)
Bulk composition vector,

*n*(*N*) components- \(n^{(x)}\)
Equilibrium speciation vector,

*n*(*L*) components- \(n_{k,t}\)
Mineral mole amount of the

*k*-th mineral at time*t*(mol)*p*Pressure (Pa)

*Q*Source/sink term for the mass exchange between the aqueous and solid phases (kg/m\(^3\)/s)

- \(q_i\)
Source/sink term, accounting for mineral dissolution/precipitation of the

*i*-th IC (mol/l/s)- \(q_{TX}\)
Source/sink term, accounting for the temperature and salinity-induced solution density change at constant pressure (kg/m\(^3\)/s)

- \(R_{k,t}\)
Net kinetic rate of the

*k*-th mineral at time*t*(mol/m\(^2\)/s)*T*Temperature (K)

*t*Time (s)

*X*Solution salinity (−)

- \(x_j\)
Mole fraction of the

*j*-th species- AMR
Additional metastability restrictions

- CFL
Courant–Friedrichs–Lewy condition

- DC
Dependent component

- Eh
Reduction potential (redox potential) (V)

- GEM
Gibbs energy minimisation

- IC
Independent component

- KKT
Karush–Kuhn–Tucker conditions

- LMA
Law of mass action

- pe
Negative logarithm of electron concentration in a solution (−)

- pH
Decimal logarithm of the reciprocal of the hydrogen ion activity in a solution (−)

- RTM
Reactive transport modelling

- SIA
Sequential iterative approach

- SNIA
Sequential non-iterative approach

## Mathematics Subject Classification

80A32 76M10 76M12 76S05## Notes

### Acknowledgements

We thank BG Group, Chevron, Petrobras, Saudi Aramco, and Wintershall for sponsorship of the University of Bristol ITF (Industry Technology Facilitator) project IRT-MODE. We are grateful to two anonymous reviewers and also Peter C. Lichtner for valuable comments that helped to improve the manuscript.

## References

- Al-Helal, A., Whitaker, F., Xiao, Y.: Reactive transport modeling of brine reflux: dolomitization, anhydrite precipitation, and porosity evolution. J. Sediment. Res.
**82**, 196–215 (2012)CrossRefGoogle Scholar - Arvidson, R., Mackenzie, F.: The dolomite problem; control of precipitation kinetics by temperature and saturation state. Am. J. Sci.
**299**(4), 257–288 (1999)CrossRefGoogle Scholar - Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, Amsterdam (1972)Google Scholar
- Bejan, A., Kraus, A.: Heat Transfer Handbook. Wiley, London (2003)Google Scholar
- Bethke, C.: Geochemical and Biogeochemical Reaction Modeling. Cambridge University Press, Cambridge (2008)Google Scholar
- Blanc, P., Lassin, A., Piantone, P., Azaroual, M., Jacquemet, N., Fabbri, A., Gaucher, E.: Thermoddem: a geochemical database focused on low temperature water/rock interactions and waste materials. Appl. Geochem.
**27**(10), 2107–2116 (2012)CrossRefGoogle Scholar - Consonni, A., Ronchi, P., Geloni, C., Battistelli, A., Grigo, D., Biagi, S., Gherardi, F., Gianelli, G.: Application of numerical modelling to a case of compaction-driven dolomitization: a jurassic palaeohigh in the po plain, Italy. Sedimentology
**57**(1), 209–231 (2010)CrossRefGoogle Scholar - Corbella, M., Gomez-Rivas, E., Martn, J., Stafford, S., Teixell, A., Griera, A., Trav, A., Cardellach, E., Salas, R.: Insights to controls on dolomitization by means of reactive transport models applied to the benicssim case study (maestrat basin, eastern Spain). Petrol. Geosci.
**20**(1), 41–54 (2014)CrossRefGoogle Scholar - de Dieuleveult, C., Erhel, J., Kern, M.: A global strategy for solving reactive transport equations. J. Comput. Phys.
**228**, 6395–6410 (2009)CrossRefGoogle Scholar - Driesner, T.: The system H\(_2\)O–NaCl. Part II: correlations for molar volume, enthalpy, and isobaric heat capacity from 0 to \(1000^{\circ }\), 1 to 5000 bar, and 0 to 1 XNaCl. Geochim. Cosmochim. Ac.
**71**, 4902–4919 (2007)CrossRefGoogle Scholar - Ehrenberg, S.N.: Porosity and permeability in miocene carbonate platforms of the marion plateau, offshore ne Australia: relationships to stratigraphy, facies and dolomitization. In: Braithwaite, C., Rizzi, G., Darke, G. (eds.) The Geometry and Petrogenesis of Dolomite Hydrocarbon Reservoirs, vol. 235, pp. 233–253. Geological Society (London) Special Publication, London (2004)Google Scholar
- Ehrenberg, S.N.: Porositypermeability relationships in miocene carbonate platforms and slopes seaward of the great barrier reef, Australia (odp leg 194, Marion Plateau). Sedimentology
**53**, 1289–1318 (2006)CrossRefGoogle Scholar - Fabricius, I.L.: Chalk: composition, diagenesis and physical properties. Geol. Soc. Den. Bull.
**55**, 97–128 (2007)Google Scholar - Fowler, S., Kosakowski, G., Driesner, T., Kulik, D., Wagner, T., Wilhelm, S., Masset, O.: Numerical simulation of reactive fluid flow on unstructured meshes. Transp. Porous Med.
**112**, 283–312 (2016)CrossRefGoogle Scholar - Gabellone, T., Whitaker, F.: Secular variations in seawater chemistry controlling dolomitisation in shallow reflux systems: insights from reactive transport modelling. Sedimentology
**63**, 1233–1259 (2015)CrossRefGoogle Scholar - Geiger, S., Driesner, T., Matthai, S., Heinrich, C.A.: Multiphase thermohaline convection in the Earth’s crust: I. A novel finite element–finite volume solution technique combined with a new equation of state for NaCl–H\(_2\)O. Transp. Porous Med.
**63**(3), 399–434 (2006)CrossRefGoogle Scholar - Geiger, S., Roberts, S., Matthai, S., Zoppou, C., Burri, A.: Combining finite element and finite volume methods for efficient multiphase flow simulations in highly heterogeneous and structurally complex geologic media. Geofluids
**4**, 284–299 (2004)CrossRefGoogle Scholar - Gregg, J.: Basin fluid flow, base metal mineralization, and the development of dolomite petroleum reservoirs. In: Braithwaite, C., Rizzi, G., Darke, G. (eds.) The Geometry and Petrogenesis of Dolomite Hydrocarbon Reservoirs, vol. 235, pp. 157–175. Geological Society (London) Special Publication, London (2004)Google Scholar
- Hammond, G.E., Lichtner, P.C.: Field-scale model for the natural attenuation of uranium at the hanford 300 area using high-performance computing. Water Resour. Res.
**46**(9), 1–31 (2010)CrossRefGoogle Scholar - Helgeson, H.C., Kirkham, D.H., Flowers, D.C.: Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: Iv. calculation of activity coefficients, osmotic coefficients, and apparent molal and standard and relative partial molal properties to \(600^{\circ }\) and 5 kb. Am. J. Sci.
**281**, 1249–1516 (1981)CrossRefGoogle Scholar - Jones, G., Xiao, Y.: Dolomitization, anhydrite cementation, and porosity evolution in a reflux system: insights from reactive transport models. AAPG Bull.
**89**(5), 577–601 (2005)CrossRefGoogle Scholar - Karpov, I., Chudnenko, K., Kulik, D.: Modeling chemical mass-transfer in geochemical processes: thermodynamic relations, conditions of equilibria and numerical algorithms. Am. J. Sci.
**297**, 767–806 (1997)CrossRefGoogle Scholar - Karpov, I., Chudnenko, K., Kulik, D., Avchenko, O., Bychinskii, V.: Minimization of gibbs free energy in geochemical systems by convex programming. Geochem. Int.
**39**, 1108–1119 (2001)Google Scholar - Kulik, D., Thien, B., Curti, E.: Partial-equilibrium concepts to model trace element uptake. In: Goldschmidt2012 Conference, Montreal (2012)Google Scholar
- Kulik, D., Wagner, T., Dmytrieva, S., Kosakowski, G., Hingerl, F., Chudnenko, K., Berner, U.: Gem-selektor geochemical modeling package: revised algorithm and gems3k numerical kernel for coupled simulation codes. Comput. Geosci.
**17**, 1–24 (2013)Google Scholar - Leal, A., Blunt, M., LaForce, T.: A chemical kinetics algorithm for geochemical modelling. Appl. Geochem.
**55**, 46–61 (2015)CrossRefGoogle Scholar - Lichtner, P.C.: Continuum model for simultaneous chemical reactions and mass transport in hydrothermal systems. Geochim. Cosmochim. Ac.
**49**, 779–800 (1985)CrossRefGoogle Scholar - Lu, P., Cantrell, D.: Reactive transport modelling of reflux dolomitization in the Arab-d reservoir, Ghawar field, Saudi Arabia. Sedimentology
**63**, 865–892 (2016)CrossRefGoogle Scholar - Machel, H.G.: Concepts and models of dolomitization, a critical reappraisal. In: Braithwaite, C., Rizzi, G., Darke, G. (eds.) The Geometry and Petrogenesis of Dolomite Hydrocarbon Reservoirs, vol. 235, pp. 7–63. Geological Society (London) Special Publication, London (2004)Google Scholar
- Marini, L., Ottonello, G., Canepa, M., Cipolli, F.: Water-rock interaction in the bisagno valley (Genoa, Italy): application of an inverse approach to model spring water chemistry. Geochim. Cosmochim. Ac.
**64**, 2617–2635 (2000)CrossRefGoogle Scholar - Matthai, S., Nick, H., Pain, C., Neuweiler, I.: Simulation of solute transport through fractured rock: a higher-order accurate finite-element finite-volume method permitting large time steps. Transp. Porous Med.
**83**(2), 289–318 (2009)CrossRefGoogle Scholar - Matthäi, S.K., Geiger, S., Roberts, S.G.: Complex Systems Platform: CSP3D3.0. user’s guide. ETH, Eidgenössische Technische Hochschule Zürich, Institut für Isotopengeologie und Mineralische Rohstoffe (2001). doi: 10.3929/ethz-a-004432279
- Mironenko, M., Zolotov, M.: Equilibrium–kinetic model of water–rock interaction. Geochem. Int.
**50**, 1–7 (2012)CrossRefGoogle Scholar - Nielsen, L., De Yoreo, J., DePaolo, D.: General model for calcite growth kinetics in the presence of impurity ions. Geochim. Cosmochim. Ac.
**115**, 100–114 (2013)CrossRefGoogle Scholar - Nordstrom, D., Plummer, L., Wigley, T., Wolery, T., Ball, J., Jenne, E., Bassett, R., Crerar, D., Florence, T., Fritz, B., Hoffman, M., Holdren, G., Lafon, G., Mattigod, S., McDuff, R., Morel, F., Reddy, M., Sposito, G., Thrailkill, J.: A comparison of computerized chemical models for equilibrium calculations in aqueous systems. In: Jenne, E. (ed.) Chemical Modeling in Aqueous Systems—Speciation, Sorption, Solubility, and Kinetics, vol. 93, pp. 857–892. American Chemical Society, Series, Washington (1979)CrossRefGoogle Scholar
- Palandri, J., Kharaka, Y.: A compilation of rate parameters of water–mineral interaction kinetics for application to geochemical modelling. Open File Report 2004–1068, U.S.G.S., Menlo Park CA (2004)Google Scholar
- Pruess, K.: Eos7, an equation-of-state module for the tough2 simulator for two-phase flow of saline water and air. Report LBL-31114, Lawrence Berkeley Laboratory, Berkeley CA (1991)Google Scholar
- Putnis, A.: Mineral replacement reactions: from macroscopic observations to microscopic mechanisms. Mineral. Mag.
**66**(5), 689–708 (2002)CrossRefGoogle Scholar - Reed, M.: Calculation of multicomponent chemical equilibria and reaction processes in systems involving minerals, gases and an aqueous phase. Geochim. Cosmochim. Ac.
**46**, 513–528 (1982)CrossRefGoogle Scholar - SAMG: User’s Manual, 25a1 edn. Fraunhofer Institute SCAI, St. Augustin (2010)Google Scholar
- Schott, J., Oelkers, E., Benezeth, P., Godderis, Y., Francois, L.: Can accurate kinetic laws be created to describe chemical weathering? C. R. Geosci.
**344**, 568–585 (2012)CrossRefGoogle Scholar - Scislewski, A., Zuddas, P.: Estimation of reactive mineral surface area during water-rock interaction using fluid chemical data. Geochim. Cosmochim. Ac.
**74**, 6996–7007 (2010)CrossRefGoogle Scholar - Shao, H., Dmytrieva, S., Kolditz, O., Kulik, D., Pfingsten, W., Kosakowski, G.: Modeling reactive transport in non-ideal aqueoussolid solution system. Appl. Geochem.
**24**(7), 1287–1300 (2009)CrossRefGoogle Scholar - Steefel, C., Lasaga, A.: A coupled model for transport of multiple chemical species and kinetic precipitation/dissolution reactions with application to reactive flow in single phase hydrothermal systems. Am. J. Sci.
**294**, 529–592 (1994)CrossRefGoogle Scholar - Steefel, C., MacQuarrie, K.: Approaches to modeling of reactive transport in porous media. Rev. Mineral. Geochem.
**34**, 85–129 (1996)Google Scholar - Thien, B., Kulik, D., Curti, E.: A unified approach to model uptake kinetics of trace elements in complex aqueous–solid solution systems. Appl. Geochem.
**41**, 135–150 (2014)CrossRefGoogle Scholar - Thoenen, T., Hummel, W., Berner, U., Curti, E.: The psi/nagra Chemical Thermodynamic Database 12/07. PSI Bericht 14-04, Paul Scherrer Institut, Villigen (2014)Google Scholar
- Wagner, T., Kulik, D., Hingerl, F., Dmytrieva, S.: Gem-selektor geochemical modeling package: Tsolmod library and data interface for multicomponent phase models. Can. Mineral.
**50**, 1173–1195 (2012)CrossRefGoogle Scholar - Whitaker, F., Xiao, Y.: Reactive transport modeling of early burial dolomitization of carbonate platforms by geothermal convection. AAPG Bull.
**94**(6), 889–917 (2010)CrossRefGoogle Scholar - Wilson, A., Sanford, W., Whitaker, F., Smart, P.: Spatial patterns of diagenesis during geothermal circulation in carbonate platforms. Am. J. Sci.
**301**, 727–752 (2001)CrossRefGoogle Scholar - Wolthers, M., Nehrke, G., Gustafsson, J.P., Van Cappellen, P.: Calcite growth kinetics: modeling the effect of solution stoichiometry. Geochim. Cosmochim. Ac.
**77**, 121–134 (2012)CrossRefGoogle Scholar - Xu, T., Spycher, N., Sonnenthal, E., Zheng, L., Pruess, K.: TOUGHREACT User’s Guide: A Simulation Program for Non-Isothermal Multiphase Reactive Geochemical Transport in Variably Saturated Geologic Media, Version 2.0. Lawrence Berkeley National Laboratory, Berkeley (2012)Google Scholar