# 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.

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