Space-time fractional diffusion: transient flow to a line source

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 exponents α and β are 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.