A mathematical model for dynamic wettability alteration controlled by water-rock chemistry
Steinar Evje Aksel Hiorth
Previous experimental studies of spontaneous imbibition on chalk core plugs have shown that seawater may change the wettability in the direction of more water-wet conditions in chalk reservoirs. One possible explanation for this wettability alteration is that various ions in the water phase (sulphate, calcium, magnesium, etc.) enter the formation water due to molecular diffusion. This creates a non-equilibrium state in the pore space that results in chemical reactions in the aqueous phase as well as possible water-rock interaction in terms of dissolution/precipitation of minerals and/or changes in surface charge. In turn, this paves the way for changes in the wetting state of the porous media in question. The purpose of this paper is to put together a novel mathematical model that allows for systematic investigations, relevant for laboratory experiments, of the interplay between (i) two-phase water-oil flow (pressure driven and/or capillary driven); (ii) aqueous chemistry and water-rock interaction; (iii) dynamic wettability alteration due to water-rock interaction.
   In particular, we explore in detail a 1D version of the model relevant for spontaneous imbibition experiments where wettability alteration has been linked to dissolution of calcite. Dynamic wettability alteration is built into the model by defining relative permeability and capillary pressure curves as an interpolation of two sets of end point curves corresponding to mixed-wet and water-wet conditions. This interpolation depends on the dissolution of calcite in such a way that when no dissolution has taken place, mixed-wet conditions prevail. However, gradually there is a shift towards more water-wet conditions at the places in the core where dissolution of calcite takes place. A striking feature reflected by the experimental data found in the literature is that the steady state level of oil recovery, for a fixed temperature, depends directly on the brine composition. We demonstrate that the proposed model naturally can explain this behavior by relating the wettability change to changes in the mineral composition due to dissolution/precipitation. Special attention is paid to the effect of varying, respectively, the concentration of $\text{SO}_4^{2-}$ ions and $\text{Mg}^{2+}$ ions in seawater like brines. The effect of changing the temperature is also demonstrated and evaluated in view of observed experimental behavior.
keywords: aqueous chemistry precipitation water-rock interaction countercurrent flow wettability alteration. spontaneous imbibition dissolution
A mathematical model relevant for weakening of chalk reservoirs due to chemical reactions
Steinar Evje Aksel Hiorth Merete V. Madland Reidar I. Korsnes
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.
keywords: dissolution porous media flow upscaling. precipitation physico-chemical processes convection-diffusion-reaction equations damage mechanics water-rock interaction

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