On the Numerical Approximation of Mobile-Immobile Advection-Dispersion Model of Fractional Order Arising from Solute Transport in Porous Media

Abstract

The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.

Physics-Informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media

“…Solute transport in porous media is governed by three main mechanisms: advection, diffusion, and dispersion [39][40][41]. The mathematical formulation of the advection-diffusiondispersion equation is explained briefly here in terms of implementing the flow and transport factors that affect the transport process.…”

A Review on Process-Based Groundwater Vulnerability Assessment Methods

“…Many scholars [81,82] have found that the conventional CDE model falls short in explaining the occurrence of early breakthrough and tailing phenomena in the transport of solutes in heterogeneous media. Some scholars [83,84] have discovered that even in soil columns or troughs that are uniformly filled indoors, irregular solute transport phenomena may occur. Furthermore, the CDE model is limited in capturing variations in pollutant concentrations only within the upper layers of soil, and falls short in providing satisfactory simulation results for deeper layers [85].…”