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