Upscaling groundwater flow is a fundamental challenge in hydrogeology. This study proposed time-fractional flow equations (t-FFEs) for upscaling long-term, transient groundwater flow and propagation of pressure heads in heterogeneous media. Monte Carlo simulations showed that, with increasing variance and correlation of the hydraulic conductivity (K), flow dynamics gradually deviated from Darcian flow and exhibit sub-diffusive, time-dependent evolution which can be separated into three major stages. At the early stage, the interconnected high-K zones dominated flow, while at intermediate times, the transverse flow due to mixed high- and low-K zones caused delayed rise of the piezometric head. At late times when flow in the relatively high-K domains reached stability, cells with very low-K continued to block the entry of water and generate “islands” with low piezometric head, significantly extending the temporal evolution of the piezometric head. The elongated water breakthrough curve cannot be quantified by the flow equation with an effective K, the space-fractional flow equation, or the multi-rate mass transfer (MRMT) flow model with a few rates, motivating the development of t-FFEs assuming temporally non-Darcian flow. Model applications showed that both the early and intermediate stages of flow dynamics can be captured by a single-index t-FFE (whose index is the exponent of the power-law probability density function of the random operational time for water parcels), but the overall evolution of flow dynamics, especially the enhanced retention of flow at later times, required a distributed-order t-FFE with variable indexes for different flow phases that can dominate flow dynamics at different stages. Therefore, transient groundwater flow in aquifers with spatially stationary heterogeneity can be temporally non-Darcian and non-stationary, due to the time-sensitive, combined effects of interconnected high-K channels and isolated low-K deposits on flow dynamics (which is the hydrogeological mechanism for the temporally non-Darcian flow and sub-diffusive pressure propagation), whose long-term behavior can be quantified by multi-index stochastic models.

中文翻译：

---

时间分数流方程 (t-FFE) 到以时间非达西流为特征的高档瞬态地下水流，由于中等异质性

增加地下水流量是水文地质学的一项基本挑战。本研究提出了时间分数流方程 (t-FFE)，用于放大非均质介质中的长期、瞬态地下水流和压力水头的传播。Monte Carlo 模拟表明，随着水力传导率 ( K ) 的方差和相关性的增加，流动动力学逐渐偏离达西流，并表现出亚扩散、时间相关的演变，可分为三个主要阶段。在早期，相互连接的高K区主导流动，而在中间时期，由于高、低K区混合导致的横向流动导致测压头延迟上升。在相对较高的K 值流动的后期域达到稳定，具有非常低K 的细胞继续阻止水的进入并产生具有低测压头的“岛”，显着延长了测压头的时间演变。延长的水突破曲线无法通过具有有效K的流量方程量化、空间分数流动方程或具有少量速率的多速率传质 (MRMT) 流动模型，推动了 t-FFE 的发展，假设时间上非达西流动。模型应用表明，流动力学的早期和中期都可以通过单指数 t-FFE（其指数是水块随机运行时间的幂律概率密度函数的指数）来捕获，但是流动动力学的整体演变，尤其是后期流动保持力的增强，需要一个分布式阶次 t-FFE，其具有不同流动阶段的可变指数，可以支配不同阶段的流动动力学。因此，具有空间平稳异质性的含水层中的瞬态地下水流在时间上可能是非达西和非平稳的，因为时间敏感，K通道和孤立的低K沉积物流动动力学（这是时间非达西流和亚扩散压力传播的水文地质机制），其长期行为可以通过多指数随机模型量化。