A fully subordinated linear flow model for hillslope subsurface stormflow

Abstract

Hillslope subsurface stormflow exhibits complex patterns when natural soils with multiscale heterogeneity impart a spatiotemporally nonlocal memory on flow dynamics. To efficiently quantify such nonlocal flow responses, this technical note proposes a fully subordinated flow (FSF) equation where the time‐ and flow‐subordination capture the temporal and spatial memory, respectively. Results show that the time‐subordination component of the FSF model captures a wide range of delayed flow response due to various degrees of soil heterogeneity (especially for low‐conductivity zones), while the model's flow‐subordination term accounts for the rapid flow responses along preferential flow paths. In the FSF model, parameters defining spatiotemporal memory functions may be related to soil properties, while other parameters such as scalar factors controlling the overall advection and diffusion are difficult to predict and can be estimated from subsurface stormflow hydrographs. These parameters can be constants at the hillslope scale because the spatiotemporal subordination, an upscaling technique, can capture the impact of system heterogeneity on flow dynamics, leading to a linear FSF model that might be applicable for various slopes. Valid scale, limitation and extension of the FSF model, and modification of the model for other complex hydrological dynamics are also discussed.

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

“…Hence, a multi-index stochastic model, such as the distributed-order fractional-derivative flow model developed in this study, should be used to capture the full time-range of water dynamics. The time-subordination approach, proposed first by Baeumer et al (2001) to model anomalous transport for pollutants, has been applied to capture heavy-tailed responses from heterogeneous hillslopes (Harman et al, 2010;Zhang et al, 2017). Here we extended the approach to capture saturated flow in heterogeneous aquifers.…”

“…Based on the time subordination described by Equation A1 and given any linear spatial operator defined on the right-hand side (RHS) of the continuity Equation A2, the time-subordinated flow has a hydraulic head that solves (Zhang et al, 2017):…”

“…We randomize the operational time that a water parcel spends in motion in porous media, similar to the random operational time assumed by Harman et al (2010) for a flow impulse spent in motion along a heterogeneous hillslope after precipitation. Following the definition in Harman et al (2010) and Zhang et al (2017), we also assume that the PDF of the operational time follows the  E -stable density, whose subordinator satisfies the following governing equation:…”

Upscaling Heat Flow in Porous Media With Periodic Surface Temperature Fluctuation Using a One‐Dimensional Subordinated Heat Transfer Equation

“…The time index α may be sensitive to the degree of medium heterogeneity (as shown in Supplementary Material), which can be partially represented by the spatial variation of the hydraulic conductivity K , the thermal conductivity K e , or the flow condition, a trend found by Zhang, Baeumer, et al. (2017) for hillslope subsurface stormflow. For heat transport observed in our laboratory, the best‐fit time index α (=0.78–0.90) was relatively large, implying a weak heat storage impact of the medium, probably due to the relatively weak heterogeneity of the sand column compared to real‐world soil.…”

“…As some heat components move faster while others move slower, the operational time is not a constant value, but instead a random variable. The governing equation for the operational time density, (where denotes the physical or clock time and denotes the operational time), then takes the form (Zhang, Baeumer, et al., 2017): with the initial condition (the Dirac delta function), for the subordinator corresponding to a ‐ stable density (Samorodnitsky & Taqqu, 1994). Here [ T α−1 ] is the scale factor, and the operator denotes the Caputo fractional derivative in time with index [dimensionless] ().…”