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

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. The algorithm combines the three-step linear time filters method for simple post-processing and the second-order backward differential formula (BDF2), is third-order accurate and provides, at 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 proves the variable time stepsize second-order backward differential formula plus time filter (BDF2-TF) algorithm's stability and the convergence properties of the fluid velocity u and hydraulic head $\phi$ in the $L^2$ norm with an order of $O(k_{n+1}^3 + h^3)$. In the experiments, the adaptive algorithm automatically adjusts the time step in response to the varying characteristics of different models, ensuring that errors are maintained within acceptable limits. This algorithm addresses the issue that high-order algorithms may select inappropriate time steps, resulting in instability or reduced precision of the numerical solution, thereby enhancing calculation accuracy and efficiency. We perform three-dimensional numerical experiments to verify 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.