A fractional diffusion model for single-well simulation in geological media

A fractional Darcian model‐based analytical solution for non‐Darcian flow toward a fully penetrating well in a confined aquifer

“…55 Obembe established a fractional diffusion model for single-well simulation in hydrocarbon reservoirs using a modified fractional Darcy flux expression to exhibit distorted flow paths and near power-law behaviors. 56 Zhou et al proposed a fractional diffusion model of diffusive transport for modeling the non-Darcian flow and solute transport in porous media by using the Caputo-Fabrizio fractional derivative. 57 Although the behavior of non-Darcian flow toward a well in various aquifer systems is commonly described by the Forchheimer and Izbash equations, both of these two equations inevitably introduce problems such as more or less empirical nature, and dimensional unbalance.…”

“…presented a variable‐order fractional diffusion equation to depict the diffusion process of chloride in concrete 55 . Obembe established a fractional diffusion model for single‐well simulation in hydrocarbon reservoirs using a modified fractional Darcy flux expression to exhibit distorted flow paths and near power‐law behaviors 56 . Zhou et al.…”

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

“…We extend the fractional engine (Metzler and Klafter, 2000, 2004), a promising tool originally proposed for solute transport (Zhang et al., 2009), to efficiently upscale long‐term groundwater flow. The s‐FFE is one of the main fractional‐derivative models, and it has been proposed by various researchers to replace Darcy's law‐based flow equations to quantify non‐Darcy flow (meaning that the groundwater velocity is a non‐linear function of the hydraulic gradient) (Cloot & Botha, 2006; Mehdinejadiani et al., 2013; Obember, 2020; among others). He (1998) was the first to describe non‐Darcy flow using a space fractional derivative (see Equation 41 in He (1998)): …”

“…Recent studies showed that pumping or injection can cause groundwater to flow differently from that described by the Darcy's law‐based flow models, motivating the upscaling of groundwater flow using parsimonious flow models as a logical extension to the classical, stochastic flow models (Dagan, 1989; Gelhar, 1993). Examples of the flow‐upscaling method include the continuous‐time random walk theory‐based diffusion model (Cortis & Knudby, 2006), the multi‐rate mass transfer (MRMT) flow model (Municchi & Icardi, 2020; Silva et al., 2009), and the space‐fractional flow equation (s‐FFE) assuming non‐Darcian flow (shown below) (Cloot & Botha, 2006; Obember, 2020). A detailed debate of stochastic hydrological approaches including the models mentioned above (especially for pollutant transport) can be seen in Cirpka and Valocchi (2016), Fiori et al.…”

“…Examples of the flow-upscaling method include the continuous-time random walk theory-based diffusion model (Cortis & Knudby, 2006), the multi-rate mass transfer (MRMT) flow model (Municchi & Icardi, 2020;Silva et al, 2009), and the space-fractional flow equation (s-FFE) assuming non-Darcian flow (shown below) (Cloot & Botha, 2006;Obember, 2020). A detailed debate of stochastic hydrological approaches including the models mentioned above (especially for pollutant transport) can be seen in Cirpka and Valocchi (2016), Fiori et al (2016), Fogg and Zhang (2016), Rajaram (2016), andFernàndez-Garcia (2016).…”