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.

翻译：本文针对基于流体与多孔介质流动耦合的页岩储层模型，提出了一种计算量小、复杂度低的三阶时间自适应算法。该算法结合了用于简单后处理的三步线性时间滤波方法与二阶向后微分公式（BDF2），在无额外计算复杂度的情况下实现了三阶精度。同时，时间滤波方法还能抑制BDF2方法固有的非物理振荡，确保稳定性。我们证明了变时间步长的二阶向后微分公式加时间滤波（BDF2-TF）算法的稳定性，以及流体速度 u 和水头 $\phi$ 在 $L^2$ 范数下的收敛性，其收敛阶为 $O(k_{n+1}^3 + h^3)$。在实验中，自适应算法能根据不同模型的变化特性自动调整时间步长，确保误差维持在可接受的范围内。该算法解决了高阶算法可能因选择不当的时间步长而导致数值解不稳定或精度下降的问题，从而提高了计算精度与效率。我们进行了三维数值实验，验证了BDF2-TF算法的有效性、稳定性及三阶收敛性。同时，采用一个简化模型模拟页岩油从储层中开采的过程，进一步证明了该算法的实际适用性。