Computer Simulation Based on Non-local Model of the Dynamics of Convective Diffusion of Soluble Substances in the Underground Filtration Flow under Mass Exchange Conditions

摘要

The paper deals with the problem of modeling the dynamics of locally nonequilibrium in time process of soluble substances convective diffusion under the conditions of flat-vertical steady-state groundwater filtration with free surface taking into account the presence of phase-to-phase mass transfer. The urgency of solving such problem is due, in particular, to the need for development of measures for soil flushing as well as desalination and purification of groundwater from pollutants. For mathematical modeling of the corresponding transfer process in media with a property of temporal nonlocality this paper used the apparatus of fractional-order integro-differentiation. The corresponding nonlinear fractional differential model of the migration process has been developed using Caputo-Katugampola generalized fractional order derivative of a function with respect to another function which allows us in a sense to control the modeling process. In this model the nonequilibrium convection-diffusion process in a porous medium is considered under conditions of mass exchange. For the proposed mathematical model the formulation of the corresponding boundary value problem was carried out and the technique for its numerical solution was developed. This technique is based on a preliminary transition using the conformal mapping method from the physical How domain to the domain of complex potential which is canonical. The algorithm for approximate solution of the considered boundary value problem in the domain of complex potential is based on a linearized version of the locally one-dimensional difference scheme of A.A. Samarsky. The results of computer simulations demonstrate that the value of the exponent in the Caputo-Katugampola derivative significantly affects the simulation results giving both sub-diffusion and super-diffusion patterns of concentration fields distribution. Computational experiments also show that when mass exchange phenomenon is taken into account while modeling pollution propagation from water bodies to soil media it leads to a delay in the concentration front development in a liquid phase. The paper has drawn the conclusions regarding the influence of the mathematical model parameters on the resulting picture of concentration fields formation.