An Upwind Generalized Finite Difference Method (GFDM) for Meshless Analysis of Heat and Mass Transfer in Porous Media

TL;DR

This paper introduces an upwind generalized finite difference method (GFDM) for meshless simulation of heat and mass transfer in porous media, demonstrating stability, accuracy, and potential for complex geometries.

Contribution

The paper develops an upwind GFDM integrating a first-order upstream scheme, enabling stable, meshless simulation of coupled heat and mass transfer in porous media.

Findings

The upwind GFDM provides stable solutions for convection-diffusion equations.

Numerical examples show good accuracy and convergence.

Increasing influence domain radius raises temperature profile errors.

Abstract

In this paper, an upwind GFDM is developed for the coupled heat and mass transfer problems in porous media. GFDM is a meshless method that can obtain the difference schemes of spatial derivatives by using Taylor expansion in local node influence domains and the weighted least squares method. The first-order single-point upstream scheme in the FDM/FVM-based reservoir simulator is introduced to GFDM to form the upwind GFDM, based on which, a sequential coupled discrete scheme of the pressure diffusion equation and the heat convection-conduction equation is solved to obtain pressure and temperature profiles. This paper demonstrates that this method can be used to obtain the meshless solution of the convection-diffusion equation with a stable upwind effect. For porous flow problems, the upwind GFDM is more practical and stable than the method of manually adjusting the influence domain based on the prior information of the flow field to achieve the upwind effect. Two types of calculation errors are analyzed, and three numerical examples are implemented to illustrate the good calculation accuracy and convergence of the upwind GFDM for heat and mass transfer problems in porous media, and indicate the increase of the radius of the node influence domain will increase the calculation error of temperature profiles. Overall, the upwind GFDM discretizes the computational domain using only a point cloud that is generated with much less topological constraints than the generated mesh, but achieves good computational performance as the mesh-based approaches, and therefore has great potential to be developed as a general-purpose numerical simulator for various porous flow problems in domains with complex geometry.