A simulation study of reactive flow in 2-D involving dissolution and precipitation in sedimentary rocks
Introduction
The study of flow, dissolution and precipitation in rocks is of fundamental interest in many areas of scientific, industrial and engineering processes as it involves an understanding of reaction kinetics and transport phenomena. Such studies play a crucial role in several important applications like stimulation of petroleum reservoirs (Daccord et al., 1993), environmental contaminant transport, mineral mining, geologic sequestration of carbon dioxide (Xu et al., 2003, Xu et al., 2004), chemical weathering, diagenesis, concrete degradation, bioremediation, and hydrothermal processes. Apart from involving multiple processes like advection, diffusion and reaction, the temporal evolution of the pore space further complicates matters. The changes in the pore space of the rock may significantly affect the hydrological properties like porosity and permeability which in turn, modify the transport properties of the fluid flow through such media. Accounting for both physical and chemical heterogeneity of the rocks at different scales is a challenge to be addressed. The chemical heterogeneity present at the level of grain scale size will have different reaction rates. This is accounted for in our proposed model, whereby the dissolution and precipitation rates vary at this length scale. This leads to the dynamic evolution of the pore space which in turn affects the physical property of transport at a larger scale. Our model can recognize the heterogeneity at the grain level to predict changes in physical properties at a larger scale. However the heterogeneities may contribute to scale dependence in mineral dissolution rates and thereby lead to discrepancies between the laboratory and field rates (Molins et al., 2012, Li et al., 2008, Malmstrom et al., 2004). The definition of a pertinent support volume in the frame of the continuum approach has been studied by Whitaker, 1999, Hornung, 1997 for instance. Common approaches to pore scale modelling applied to study of reactive transport include pore network models (Li et al., 2006, Li et al., 2007, Algive et al., 2010, Varloteaux et al., 2013), Lattice Boltzmann method (Kang et al., 2002, Kang et al., 2003, Kang et al., 2005, Kang, 2010), particle methods (Tartakovsky et al., 2007, Tartakovsky et al., 2007, Tartakovsky et al., 2008) and direct pore-scale simulations (Flukiger and Bernard, 2009). Several studies have considered the up scaling of mass transfer processes in heterogeneous media (Edwards et al., 1993, Litchner and Tartakovsky, 2003, Meile and Tuncay, 2006) using different analytical methods. Few theoretical works have established conditions for accurate prediction of upscaling from pore scale reactive transport processes to the continuum scale (Kechagia et al., 2002, Battiato and Tartakovsky, 2011). However physical and chemical mechanisms have often been considered separately. Experimental data on the effects of reactive fluid flow through porous rocks e.g. (Carroll et al., 2013, Luquot and Gouze, 2009, Noiriel et al., 2009, Izgec et al., 2008, Saldi et al., 2009, Saldi et al., 2010, Saldi et al., 2012, Salehikhoo and Li, 2013) emphasized the complexity of the mechanisms controlling both porosity and permeability considering both subsurface and deep environments. The main identified parameters governing porosity and permeability relationships are the pore structure and its heterogeneity that controls the flow distribution and the chemical heterogeneity that controls the fluid–rock mass transfer rate. Yet, experimental investigations implement complex reactions even in simple cases such as pure calcite rocks (Gouze and Luquot, 2011).
In this study we present a simulation study of reactive flow through a simplified natural rock, eg. pure forsterite . The process under consideration may be taken to be the carbonation of this rock triggered by the injection of -rich fluid envisaged for underground mineralization storage or naturally occurring at some oceanic ridges. However, possible feedback effects between carbonation reactions and changes in the reservoir porosity and permeability must be studied for assessing the efficiency and sustainability of this process. The carbonation efficiency is controlled by the local renewal of the reactants, for example, the olivine dissolution and carbonate precipitation (differential) kinetics and the heterogeneity of the pore structure. Andreani et al. (2009) observed that (i) the mass transfers were highly variable at pore scale and controlled by the local Peclet and Damkohler number and (ii) high flow rates will decrease the carbonation efficiency of the reservoir while low flow rates may reduce the permeability irreversibly. They concluded that moderate injection rates will ensure a partial carbonation of the rock and maintain the reservoir permeability. Any reservoir-scale model used for predicative calculations will require a detailed information of the controlling parameters including the flow rate or the pressure gradient, the concentration of the reactants and the kinetic reaction coefficient for both the dissolution and the precipitation.
In this paper we propose a model for the simulation of reactive flow in sedimentary rock samples generated by the ‘Relaxed Bidisperse Ballistic Deposition Model’ (RBBDM) (Dutta and Tarafdar, 2003, Sadhukhan et al., 2007) with the aim to study the effect of variation of different flow parameters like Peclet number, Damkholer number, concentration of reactive species and pressure on dissolution and precipitation reactions through their manifestation on rock properties like porosity and permeability. Our model of reactive flow is done on a 2-D porous rock whose porosity can be varied to match that of a real rock. We have solved the Stokes’ equation using the finite difference method to evaluate the pressure and velocity field. Though in this paper we have considered the fluid to have low viscosity, the model can be adapted to consider the flow of high viscous fluid flow too. Modelling of reactive flow is done through walkers which move through the pore channels with appropriate velocity, performing either drift or diffusion. The heterogeneity in the mineral composition of the sedimentary rock is factored in our model through the assignment of different reaction rates according to the mineral present. This in turn affects the rates of dissolution and precipitation at the mineral sites. We have varied the flow parameters mentioned above to study their effect on macroscopic properties like porosity and permeability. Although we have taken the example of sequestration on carbonate rocks to elucidate a point of discussion, being a simulation model, the study can be adapted to reactive flow in other porous media through suitable choice of parameters.
In the following sections, we discuss our basic algorithm for the generation of the porous rock and determination of velocity field of fluid flow solving the Stokes’ equation. Reactants in the fluid follow either advection or diffusion as determined from the Peclet number. In this study we focus on the effect of concentration of the reactive species, the Pe number and the ratio of reaction rate constants of dissolution and precipitation on the changes in porosity and permeability.
Section snippets
Theory
Fluid transport in a porous structure under a suitable pressure gradient is described by the Navier Stokes’ equationwhere , P and represent the velocity, pressure and external force per unit volume respectively, and are respectively the density and dynamic viscosity of the fluid. Assuming that the fluid is of low viscosity, we neglect the inertial term. Assuming further that no external forces are acting on the fluid, Eq. (1) simplifies towhere
Simulation
We generate a porous stochastic structure in 2-D and simulate flow of a single fluid through it using a numerical finite difference solution of the steady state Stokes’ Eq. (2). The details of the generation of the porous rock sample are given in Dutta and Tarafdar, 2003, Sadhukhan et al., 2007. Above the percolation threshold, the simulated structure has at least one connected pore structure. As we study transport through the rock structure, we simulate rocks above this threshold.
We give an
Results
The 2-dimensional rock structure was generated with and . The square grid mesh that was superposed on the system to enable calculation using finite difference method, was of size cm, s. is the same size as a particle. The average value of initial porosity obtained as the volume fraction of the number of voids in the sample was calculated over 150 configurations. Before the transport of the reactive fluid began, its value was 0.42. This falls in the range of
Conclusion
Reactive flow involving advection and diffusion, may cause dissolution and precipitation in a porous structure and changes in porosity and permeability. In the frame of storage by mineralization in peridotite, these changes of the hydrodynamic properties triggered by olivine dissolution and carbonate precipitation is a main issue for determining the sustainability of the process. Apart from the complex geometry of the pore space, these reactions are determined by the combination of several
Acknowledgment
The authors are grateful to Marco Dentz, Sujata Tarafdar and Madhumita Mukhopadhyay for their useful suggestions and discussions. We thank IFCPAR CEFIPRA for the Grant of Project No. 4409-1, which supported this research.
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