3 - Spatial numerical discretization methods for nanoparticles transport in porous media

Abstract

Nowadays, computational methods are becoming increasingly more of a third science research means, parallel with experimental and theoretical methods. Especially in oil engineering and the research of groundwater flow and transport phenomena, numerical simulation is turning the most essential method, due to the fast development of computers. Generally, when adopting numerical simulation to research problems, the first step is establishing a mathematical model according to some physical laws of the problems. The second procedure is discretizing the mathematical model, followed by the third step, which is to code and run it on the computer to get the results. Finally, we can understand the original problem through simulation results. In this chapter, we introduce numerical methods that will be used for spatial discretization in nanoparticle transport in porous media. This chapter starts with mesh generation using MATLAB, including uniform and nonuniform 1D/2D/3D grids. After that, we introduce the cell-centered finite difference method (CCFD), including the discretization of the pressure equation, Darcy's law, and how to treat the boundary conditions. Then, the vectored implementation (shifting-matrix) of the CCFD method was presented. Therefore, the harmonic mean of permeability and transmissibility matrices has been listed. Moreover, the finite element method (FEM) is discussed by highlighting its discretization and weak formulation. The theoretical foundation of the FEM requires presenting Raviart–Thomas space. Also, the mixed FEM has been discussed with some numerical examples as it is essential in solving partial differential equations (PDEs) that govern the flow and transport in porous media.