Conventional diffusion equations for fluid flow through porous media do not consider the effects of the history of rock, fluid, and flow. This limitation can be overcome by the incorporation of “memory” in the model, using fractional-order derivatives. Inclusion of fractional-order derivatives in the diffusion equation, however, adds complexity to both the equation and its numerical approximation. Of particular importance is the choice of temporal mesh, which can dramatically affect the convergence of the scheme. In this article, we consider a memory-based radial diffusivity equation, discretized on either uniform or graded meshes. Numerical solutions obtained from these models are compared against analytical solutions, and it is found that the simulation using properly chosen graded meshes gives substantially smaller errors compared to that using uniform meshes. Experimental data from one-dimensional flow through a porous layer with constant pressure gradient are collected from the literature and used to fit the fractional order in the diffusivity equation considered here. A reasonable value of the fractional order is found to be 0.05; this is further validated by performing numerical simulations to match these experiments, demonstrating substantial improvement over the classical Darcy’s model.

流体通过多孔介质的传统扩散方程式没有考虑岩石，流体和流动历史的影响。通过使用分数阶导数在模型中合并“内存”可以克服此限制。但是，在扩散方程中包含分数阶导数会增加方程及其数值逼近的复杂度。临时网格的选择尤其重要，它可以极大地影响方案的收敛性。在本文中，我们考虑基于记忆的径向扩散系数方程，该方程在均匀或渐变网格上离散。从这些模型获得的数值解与解析解进行比较，并且发现，与使用均匀网格相比，使用正确选择的渐变网格进行的模拟产生的误差要小得多。从文献中收集了通过具有恒定压力梯度的多孔层的一维流动的实验数据，并将其用于拟合此处考虑的扩散率方程式中的分数阶。分数阶的合理值是0.05；通过执行数值模拟以匹配这些实验进一步验证了这一点，证明了对经典达西模型的实质性改进。分数阶的合理值是0.05；通过执行数值模拟以匹配这些实验进一步验证了这一点，证明了对经典达西模型的实质性改进。分数阶的合理值是0.05；通过执行数值模拟以匹配这些实验进一步验证了这一点，证明了对经典达西模型的实质性改进。