Abstract
Since the use of the fractional-differential mathematical model of anomalous geomigration process based on the MIM (mobile–immoble media) approach in engineering practice significantly complicates simulations, a corresponding simplified mathematical model is constructed. For this model, we state a two-dimensional initial-boundary value problem of convective diffusion of soluble substances under the conditions of vertical steady-state filtration of groundwater with free surface from a reservoir to a coastal drain. To simplify the domain of simulation, we use the technique of transition into the domain of complex flow potential through a conformal mapping. In the case of averaging filtration velocity over the domain of complex flow potential, an analytical solution of the considered problem is obtained. In the general case of a variable filtration velocity, an algorithm has been developed to obtain numerical solutions. The results of process simulation using the presented algorithm shows that the constructed mathematical model can be efficiently used to simplify and accelerate modeling process.
Highlights
We study the problem of mathematical modeling of anomalous dynamics of soluble substances’
In the general case of variable filtration velocity, we present an algorithm for obtaining its numerical solution and the results of simulations
Numerical modeling of the dynamics of the considered migration process using the original MIM model and the proposed simplified mathematical model, as well as a comparative analysis of simulation results according to both of these models, were performed with respect to dimensionless variables determined by the relations in Equation (14) under the condition C0(φ, ψ) ≡ 0
Summary
We study the problem of mathematical modeling of anomalous dynamics of soluble substances’. One of the effective approaches to accelerate and simplify estimative engineering calculations when studying migration dynamics under the conditions of geofiltration consists in the simplification of the original mathematical model by an appropriate approximation of fractional derivatives and transition to a new, simplified mathematical model based on the classical convective diffusion equation. We obtain from Equation (4) the equation of a simplified mathematical model of the considered process of convective diffusion with particles’ immobilization in the form aCt(x, y, t) = L(C) − b (C(x, y, t) − C(x, y, 0)). Within the framework of the simplified mathematical model based on Equation (6), we state a two-dimensional initial-boundary value problem of convective diffusion of soluble substances under the conditions of vertical steady-state filtration of groundwater with a free surface from a reservoir into a coastal drain. In the general case of variable filtration velocity, we present an algorithm for obtaining its numerical solution and the results of simulations
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