A high-order, high-efficiency adaptive time filter algorithm for shale reservoir model based on coupled fluid flow with porous media flow

Abstract

In this paper, a third-order time adaptive algorithm with less computation, low complexity is provided for shale reservoir model based on coupled fluid flow with porous media flow. This algorithm combines a method of three-step linear time filters for simple post-processing and a second-order backward differential formula (BDF2), is third-order accurate in time, and provides no extra computational complexity. At the same time, the time filter method can also be used to damp non-physical oscillations inherent in the BDF2 method, ensuring stability. We prove the algorithm’s stability of the constant stepsize second-order backward differential formula plus time filter (BDF2-TF) and the third-order convergence properties of the fluid velocity $u$ and hydraulic head $\varphi$ in the $L^2$ norm. In numerical experiments, this adaptive algorithm automatically adjusts a time step in response to the varying characteristics of different models, ensuring that errors are maintained within acceptable limits. The algorithm addresses the issue that high-order algorithms may select inappropriate time steps, resulting in instability or reduced accuracy of a numerical solution, and thereby it enhances calculation accuracy and efficiency. We perform three-dimensional numerical tests to examine the BDF2-TF algorithm’s effectiveness, stability, and third-order convergence. Simultaneously, a simplified model is employed to simulate the process of shale oil extraction from reservoirs, further demonstrating the algorithm’s practical applicability.