Time‐Fractional Flow Equations (t‐FFEs) to Upscale Transient Groundwater Flow Characterized by Temporally Non‐Darcian Flow Due to Medium Heterogeneity

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.