Numerical modeling of the dissolution of karstic cavities

摘要

The karstic cavity dissolution problems are often studied from a hierarchical point of view. Based on a discussion of the frequently adopted assumptions, a pore-scale model is first developed for a simple geochemistry scheme. The impact of implementing reactive or thermodynamic equilibrium boundary condition at the dissolving surface is discussed. Such a pore-scale model is subsequently used as a basis for developing models at higher scale levels. The first problem deals with transport from a heterogeneous and rough surface characterized by a mixed boundary condition. The resulting macro-scale model takes the form of an effective surface theory. In the homogenized model developed with the effective surface concept (denote ESCM), the original rough surface is replaced locally by a homogeneous and smooth surface, where effective boundary conditions are prescribed. To develop the concept of effective surface, a multi-domain decomposition approach is applied. In this framework the velocity, pressure and concentration are estimated at the micro-scale with an asymptotic expansion of deviation terms with respect to macro-scale velocity and concentration fields. Closure problems for the deviations are obtained and used to define the effective surface position and the corresponding boundary conditions. The evolution of some effective properties and the impact of surface geometry and some dimensionless numbers are investigated. A comparison between the numerical results obtained with this effective model and those from direct numerical simulations with the original rough surface shows good agreements. In the case corresponding to mass transport in porous media, upscaling is carried out with the method of volume averaging to develop a macro-scale porous medium model (denote PMM), starting from a pore-scale transport problem involving thermodynamic equilibrium or nonlinear reactive boundary conditions. A general expression to describe the macro-scale mass transport is obtained involving several effective parameters which are given by specific closure problems. The impact on the effective parameters of the fluid properties, in terms of pore-scale Péclet number (Pe), and the process chemical properties, in terms of pore-scale Damköhler number (Da) and reaction order (n), is studied for periodic stratified, 2D and 3D unit cells. An example of the application of the macro-scale model is presented with the emphasis on the potential impact of additional, non-traditional effective parameters appearing in the theoretical development on the improvement of the accuracy of the macro-scale model. The above developed PMM is also used as a Diffuse Interface Model (DIM) to describe the evolution of a gypsum cavity formation induced by dissolution. The method is based upon the assumption of a pseudo-component dissolving with a thermodynamic equilibrium boundary condition. A methodology is proposed in order to choose suitable parameters for the DIM model and hence predict the correct dissolution fluxes and surface recession velocity. Additional simulations are performed to check which type of momentum balance equation should be used. Calculations with a variable density and Boussinesq approximation were also performed to evaluate the potential for natural convection. The results showed that the impact of density driven flows were negligible in the cases under investigation. The potential of the methodology is illustrated on two large-scale configurations: one corresponding to a gypsum lens contained within a porous rock layer and the other to an isolated pillar in a flooded gypsum quarry. Geomechanical consequences of the dissolution in terms of mechanical stability is evaluated with the help of a simplified geomechanical model. A final case is also studied in which gypsum is replaced by salt to show the applicability of the proposed methodology to a rapidly dissolving material