Two-grid reduced-dimension method and applications for the nonlinear single phase flow model coupled with ground stress in porous median

Abstract

This article mainly discusses the two grid model dimension reduction about the coupling problem between the non-linear single phase fluid model and the ground stress including the unknown multi physical quantities such as the natural gas motion speed and pressure, and ground stress, poriness, and structure displacement. To address this issue, a mixed variational form, a temporal semi discrete Crank-Nicolson algorithm with 2nd-order temporal precision, and a two-grid Crank-Nicolson mixed finite element (TGCNMFE) algorithm for the non-linear problem are established, and the existence, stationarity, and error estimations of the temporal semi discrete solutions and the TGCNMFE solutions are stated. However, the ultimate goal of this article is to reduce the dimension for the unknown TGCNMFE solution coefficient vectors and establish the TGCNMFE reduced-dimension (TGCNMFERD) model by using proper orthogonal decomposition, and attest the existence, unconditional stationarity, and error estimates for the TGCNMFERD solutions by means of matrix analysis. Moreover, through some numerical tests to the changes in displacement, speed, pressure, and stress near the oil well, the rightness for the obtained theory results and the effectiveness for the TGCNMFERD method are verified. The results show that the TGCNMFERD model can greatly reduce unknowns, alleviate calculating load, and improve computational efficiency.