A reliable algorithm to determine the pollution transport within underground reservoirs: implementation of an efficient collocation meshless method based on the moving Kriging interpolation

The pollution propagation within the underground water reservoirs is a challenging and important phenomenon. In the current work, the numerical simulation of pollution transport in an underground channel is performed using the meshless method. To account the anomalous dispersion in a general case, the variable order fractional mass transfer equation is utilized for a rectangular channel. The clean fluid stream enters the channel and due to several phenomenon including the leakage of pollution from the channel walls, the internal pollution source, and the occurrence of the chemical reactions, the pollution content is affected. The non-dimensional form of the governing equation is derived to introduce the dominant dimensionless group numbers. The numerical solution of the obtained equation is established based on the meshless local Petrov–Galerkin method using the moving Kriging interpolation. The Dirac delta function is used as a test function over the local sub-domains. To discretize the present formulation in space variables, we apply the moving Kriging shape functions. Also, to estimate the fractional-order versus the time, finite difference relation is utilized. Using Kronecker’s delta property of moving Kriging interpolation shape functions the boundary conditions in the final system are imposed automatically. The main aim of this technique is to investigate a global estimation for the model, which consequently decrease such problems to those of solving a system of algebraic equations. To determine the accuracy and efficiency of the present method on regular and irregular domains, an example is given in various domains and with regular and irregular distributed points. Also, the effect of major parameters including the fractional order exponent, leakage velocity, chemical reaction rate constant, diffusion coefficient in addition to the stationary/moving pollution source is also examined. It will be shown that, by enhancement of the diffusivity from 0.1 to 20, the outlet concentration reduces by 25.1%, while diffusivity increase from 20 to 50 affects the exiting pollution by merely 7.0%.