A System Approach to Mathematical and Computer Modeling of Geomigration Processes Using Freefem++ and Parallelization of
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A SYSTEM APPROACH TO MATHEMATICAL AND COMPUTER MODELING OF GEOMIGRATION PROCESSES USING FreeFem++ AND PARALLELIZATION OF COMPUTATIONS V. A. Herus,1† N. V. Ivanchuk,1‡ and P. M. Martyniuk1††
UDC 517.9:519.6
Abstract. A method is described for constructing mathematical models of interrelated processes in porous media that are complex multicomponent systems. The performance capabilities of the package FreeFem++ are shown as applied to solving corresponding free boundary-value problems for systems of quasilinear parabolic equations by the finite element method using the parallelization of computations. Keywords: mathematical and computer modeling, multicomponent porous medium, chemical and mechanical suffosion, filtration consolidation, heat-and-mass transfer, free movable boundary, system of quasilinear parabolic equations, finite element method, FreeFem++, parallel computing. INTRODUCTION This paper proposes a mathematical model of geoinformatics, which is used to mathematically describe the dynamics of the process of nonisothermal filtration consolidation of deformable porous media saturated with saline solutions. The investigation of consolidation processes in saturated deformable porous media is a topical problem, in particular, in connection with questions of assurance of the safety and reliability of engineering objects such as superficial reservoirs for domestic and industrial wastewaters, means of protection of underground waters from pollution by toxic contents of such reservoirs, etc. [1–4]. Quite often reservoirs are filled with industrial waste, i.e., concentrated chemical solutions. To estimate processes of consolidation of their bases, it is incorrect to use the classical theory of filtration consolidation based on the assumption that the filtrate in a massif is pure water. Moreover, natural porous media are multicomponent and the investigation of physicochemical processes in them requires the consideration of the mutual influence and interdependence of these processes and also the consideration of a porous medium as a complex multicomponent system [5]. Interrelated processes and, in particular, heat transfer in porous media are investigated in [6, 7]. In [1–4, 8, 9], interrelated processes in soils are studied one of which is the migration of salts and ions and the presence of electrolytes in soil pores (in general, mass transfer processes). Interrelated processes of filtering, migration of nanoparticles, and suffosion and colmatage in porous media are considered in [3, 10, 11]. When investigating several interrelated processes in multicomponent porous media, their mutual influences can be taken into account in the following two ways: (1) the direct interaction of flows of components (chemical osmosis, thermal osmosis, thermal diffusion of dissolved substances, etc. [4; 12, Ch. 9]); (2) the specification of the dependence of the parameters of a porous medium (porosity, porosity factor, filtration coefficient, etc.) on the determinative parameters of processes (temperature, concentration of chemical ag
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