Fractional dynamics of tracer transport in fractured media from local to regional scales

Abstract

Abstract:Tracer transport through fractured media exhibits concurrent direction-dependent super-diffusive spreading along high-permeability fractures and sub-diffusion caused by mass transfer between fractures and the rock matrix. The resultant complex dynamics challenge the applicability of conventional physical models based on Fick's law. This study proposes a multi-scaling tempered fractional-derivative (TFD) model to explore fractional dynamics for tracer transport in fractured media. Applications show that the TFD model can capture anomalous transport observed in small-scale single fractures, intermediate-scale fractured aquifers, and two-dimensional large-scale discrete fracture networks. Tracer transport in fractured media from local (0.255-meter long) to regional (400-meter long) scales therefore can be quantified by a general fractional-derivative model. Fractional dynamics in fractured media can be scale dependent, owning to 1) the finite length of fractures that constrains the large displacement of tracers, and 2) the increasing mass exchange capacity along the travel path that enhances sub-diffusion. PACS

Modeling mixed retention and early arrivals in multidimensional heterogeneous media using an explicit Lagrangian scheme

“…Anomalous transport with retention or early arrivals have been well documented in many disciplines, motivating the development of nonlocal physical models such as the fractional advection‐dispersion equation (fADE) [ Metzler and Klafter , ; Kilbas et al ., ; Herrmann , ]. In hydrology, anomalous diffusion due to retention has been observed for tracers transport in soils [ Pachepsky et al ., ; Levy and Berkowitz , ; Bromly and Hinz , ; Cortis and Berkowitz , ; Delay et al ., ; Hunt et al ., ], rocks [ LaBolle and Fogg , ; Kosakowski , ; Pedretti et al ., ; Ronayne , ], and streams [ Czernuszenko et al ., ; González‐Pinzón et al ., ], while anomalous diffusion with early arrivals has been found for conservative tracer transport in soils [ Zhang et al ., ], streams [ Deng et al ., , ; Kim and Kavvas , ], bed load or suspended sediment transport in rivers [ Stark et al ., ; Houssais and Lajeunesse , , among many others], and solutes moving along fractured media at various scales [ Benson et al ., ; Reeves et al ., , b; Zhang et al ., ]. The fADE is found to be superior to the second‐order advection‐dispersion equation (ADE) in quantifying the observed non‐Fickian transport behavior [ Benson et al ., , 2004; Schumer et al ., ; Zhang et al ., ].…”

A fully subordinated linear flow model for hillslope subsurface stormflow

“…The fractional‐derivative model (14) should be valid at any spatial and temporal scale as long as the control volume provides well‐defined statistics for flow dynamics, since mathematically, the tempered stable density is infinitely divisible (i.e., formally valid at all scales) [ Baeumer and Meerschaert , ]. Previous applications using real‐world data also demonstrated that the spatiotemporal fractional‐derivative models can capture anomalous dynamics of conservative tracers transport in complex fractured media varying from centimeter scales to multikilometer scales [ Zhang et al ., ] and quantify fluvial bed sediment transport across all scales [ Zhang et al ., ]. The FSF model (14), similar to other stochastic hydrologic models, is applicable for stationary media such as soil which is sometimes assumed to be fractal with scale‐invariant properties.…”

A New Compact Numerical Scheme for Solving Time Fractional Mobile-Immobile Advection-Dispersion Model

“…Fractional calculus has recently proven to be useful in mimicking aberrant behaviours that can occur in a range of scientific and engineering fields. Many mathematical models are developed using the fractional derivatives like the nonlinear behaviour of earthquake [6], continuum fluid-dynamics [7], porous media [8], hydrology [18], fractured media at regional scales [40], complex dynamics in biological tissues [17], non-Fickian transport [37,38], optimization of radiotherapy cancer treatments [5], SIQ mathematical model of Corona virus disease [23].…”