Abstract
Nonlocal diffusion to a line source well is addressed by space-time fractional diffusion to model transients governed by both long-range connectivity and distorted flow paths that result in interruptions in the geological medium as a consequence of intercalations, dead ends, etc.The former, superdiffusion, results in long-distance runs and the latter, subdiffusion, in pauses. Both phenomena are quantified through fractional constitutive laws, and two exponentsαandβare used to model subdiffusion and superdiffusion, respectively. Consequently, we employ both time and space fractional derivatives. The spatiotemporal evolution of transients in 2D is evaluated numerically and insights on the structure of solutions described through asymptotic solutions are confirmed numerically. Pressure distributions may be classified through two situations (i) wherein 2α = β + 1 in which case solutions may be grouped on the basis of the classical Theis solution, and (ii) wherein 2α ≠ β + 1 in which case conventional expectations do not hold; regardless, at long enough times for the combined case, power-law responses are similar to those for pure subdiffusive flows. Pure superdiffusion on the other hand, although we consider a system that is infinite in its areal extent, interestingly, results in behaviors similar to steady-state flow. To our knowledge, documented behaviors are yet to be reported.
Highlights
The ubiquity of heterogeneity in geologic media hardly needs an introduction – every occasional gardener knows firsthand
One immediate advantage of this work is that it provides an understanding of long term behaviors of the flow of fluids produced through complex wellbores, for pseudoradial flow will prevail in all situations
Studies of this type provide a richer understanding of transient diffusion in heterogeneous porous solids and may be extended to other similar applications
Summary
The ubiquity of heterogeneity in geologic media hardly needs an introduction – every occasional gardener knows firsthand. Many field evaluations of well responses in recent times (Bernard et al, 2006; Chu et al, 2019a, b; Cloot and Botha, 2006; Doe et al, 2014; Meerschaert et al, 2008; Raghavan and Chen, 2019; Scott et al, 2015, Sinkov et al, 2021; Thomas et al, 2005) note that the proper assessment of responses in systems producing with complex geology requires going beyond the classical transient diffusion; that is, beyond the premises of the Theis (1935), solution They have proposed incorporation of the internal topology of the complex porous network along with actual paths where distant locations may be connected rather well whereas nearby locations are isolated because of localized barriers and dead ends which lead to mean square displacements of the form h(Dx)2i / ta with a \1⁄4[ 1. Both the exponents a and b are
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