# American Institue of Mathematical Sciences

2009, 4(4): 755-788. doi: 10.3934/nhm.2009.4.755

## A mathematical model relevant for weakening of chalk reservoirs due to chemical reactions

 1 International Research Institute of Stavanger (IRIS), Prof. Olav Hanssensvei 15, NO-4068 Stavanger, Norway 2 University of Stavanger (UiS), 4036 Stavanger, Norway, Norway

Received  April 2009 Revised  August 2009 Published  October 2009

In this work a mathematical model is proposed for modeling of coupled dissolution/precipitation and transport processes relevant for the study of chalk weakening effects in carbonate reservoirs. The model is composed of a number of convection-diffusion-reaction equations, representing various ions in the water phase, coupled to some stiff ordinary differential equations (ODEs) representing species in the solid phase. More precisely, the model includes the three minerals $\text{CaCO}_3$ (calcite), $\text{CaSO}_4$ (anhydrite), and $\text{MgCO}_3$ (magnesite) in the solid phase (i.e., the rock) together with a number of ions contained in the water phase and essential for describing the dissolution/precipitation processes. Modeling of kinetics is included for the dissolution/precipitation processes, whereas thermodynamical equilibrium is assumed for the aqueous chemistry. A numerical discretization of the full model is presented. An operator splitting approach is employed where the transport effects (convection and diffusion) and chemical reactions (dissolution/precipitation) are solved in separate steps. This amounts to switching between solving a system of convection-diffusion equations and a system of ODEs. Characteristic features of the model is then explored. In particular, a first evaluation of the model is included where comparison with experimental behavior is made. For that purpose we consider a simplified system where a mixture of water and $\text{MgCl}_2$ (magnesium chloride) is injected with a constant rate in a core plug that initially is filled with pure water at a temperature of $T=130^{\circ}$ Celsius. The main characteristics of the resulting process, as predicted by the model, is precipitation of $\text{MgCO}_3$ and a corresponding dissolution of $\text{CaCO}_3$. The injection rate and the molecular diffusion coefficients are chosen in good agreement with the experimental setup, whereas the reaction rate constants are treated as parameters. In particular, by a suitable choice of reaction rate constants, the model produces results that agree well with experimental profiles for measured ion concentrations at the outlet. Thus, the model seems to offer a sound basis for further systematic investigations of more complicated precipitation/dissolution processes relevant for increased insight into chalk weakening effects in carbonate reservoirs.
Citation: Steinar Evje, Aksel Hiorth, Merete V. Madland, Reidar I. Korsnes. A mathematical model relevant for weakening of chalk reservoirs due to chemical reactions. Networks & Heterogeneous Media, 2009, 4 (4) : 755-788. doi: 10.3934/nhm.2009.4.755
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