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

Abstract

Groundwater flow upscaling is a longstanding topic in hydrogeology, where either the whole model domain or each model grid is homogenized to capture the impact of sub-domain or sub-grid heterogeneity on flow (

An application of a multiindex, time-fractional differential equation to evaluate heterogeneous, fractured rocks

“…t-FFEs represent models incorporating sub-diffusive propagation of the pressure fronts at the Theis scale and result in temporally non-Darcy, non-stationary flow to account for the complex circuitry. As these models are based on motions that depend on a Continuous Time Random Walks (CTRW) (or a waiting-time), such models yield power-law declines in flux (Raghavan and Chen, 2017a;Xia et al, 2021). Examples of subdiffusive behaviors are reported in a number of sources in the earth sciences (Bisdom et al, 2016;Caine et al, 1996;Chu et al, 2017Chu et al, , 2018Cortis and Knudby, 2006;Doe, 1991;Evans, 1988;Jourde et al, 2002;Knudby et al, 2002;Mitchell and Faulkner, 2009;Noetinger and Estebenet, 2000;Noetinger et al, 2001Noetinger et al, , 2016Raghavan and Chen, 2018;Savage and Brodsky, 2011;Scholz et al, 1993;Suzuki et al, 2016;Xia et al, 2021).…”

“…As these models are based on motions that depend on a Continuous Time Random Walks (CTRW) (or a waiting-time), such models yield power-law declines in flux (Raghavan and Chen, 2017a;Xia et al, 2021). Examples of subdiffusive behaviors are reported in a number of sources in the earth sciences (Bisdom et al, 2016;Caine et al, 1996;Chu et al, 2017Chu et al, , 2018Cortis and Knudby, 2006;Doe, 1991;Evans, 1988;Jourde et al, 2002;Knudby et al, 2002;Mitchell and Faulkner, 2009;Noetinger and Estebenet, 2000;Noetinger et al, 2001Noetinger et al, , 2016Raghavan and Chen, 2018;Savage and Brodsky, 2011;Scholz et al, 1993;Suzuki et al, 2016;Xia et al, 2021). This paper examines applications of distributive fractional differential equations in a manner similar to those discussed in Luchko (2011) in two-porosity rocks with high and low permeability along the lines in Barenblatt et al (1960).…”

“…This paper examines applications of distributive fractional differential equations in a manner similar to those discussed in Luchko (2011) in two-porosity rocks with high and low permeability along the lines in Barenblatt et al (1960). Our choice is dictated because of the model's ubiquitousness and its ability to apply to situations beyond fractured rocks such as the Mobile-IMmobile-zones (MIM) model of Silva et al (2009) which is similar to the Multi Rate Mass Transfer (MRMT) model discussed in Haggerty et al (2000); see Xia et al (2021).…”

“…Solutions applicable to the evaluation of single and multiple (interference, tomographic) well tests are presented. The motivation for this study is to evaluate the consequences of finite speeds of propagation of the transient in complex systems of the form where conventional representations of heterogeneity through log-normal representations of permeability are inadequate (Xia et al, 2021). Two exponents that reflect the heterogeneity in permeability and denoted by the symbol a j are used to study the rate of propagation of pressure fronts.…”

Fractional Advection Diffusion Models for Radionuclide Migration in Multiple Barriers System of Deep Geological Repository

“…The fractional dynamic approach is emerging as a novel description of anomalous transport processes [4][5][6]. Numerous researchers have applied the fractional derivative models to describe the turbulent flow [7], non-Darcian flow [8][9][10], transient flow [11], atmospheric pollutant dispersion [12,13], solute transport [14], and contaminant migration [15,16]. Moreover, fractional diffusion models, in terms of different definitions of a fractional derivative, also have been considered to depict the advective-dispersive transport in single porous media [17][18][19].…”