A linear, mass-conserving, multi-time-step compact block-centered finite difference method for incompressible miscible displacement problem in porous media

TL;DR

This paper introduces a novel high-order finite difference scheme for simulating incompressible miscible displacement in porous media, achieving mass conservation and high accuracy in both space and time.

Contribution

It develops a decoupled, linearized multi-time-step scheme with second-order temporal and fourth-order spatial accuracy, validated through theoretical proofs and numerical experiments.

Findings

Scheme is mass-conserving and highly accurate.

Numerical results confirm theoretical convergence and accuracy.

Successfully simulates viscous fingering phenomena.

Abstract

In this paper, a two-dimensional incompressible miscible displacement model is considered, and a novel decoupled and linearized high-order finite difference scheme is developed, by utilizing the multi-time-step strategy to treat the different time evolutions of concentration and velocity/pressure, and the compact block-centered finite difference approximation for spatial discretization. We show that the scheme is mass-conserving, and has second-order temporal accuracy and fourth-order spatial accuracy for the concentration, the velocity and the pressure simultaneously. The existence and uniqueness of the developed scheme under a rough time-step condition is also proved following the convergence results. Numerical experiments are presented to confirm the theoretical conclusions. Besides, some 'real' simulations are also tested to show good performance of the proposed scheme, in particular, the viscous fingering phenomenon is verified.