Experimental tests of truncated diffusion in fault damage zones

Abstract

Fault zones affect the flow paths of fluids in groundwater aquifers and geological reservoirs. Fault‐related fracture damage decreases to background levels with increasing distance from the fault core according to a power law. This study investigated mass transport in such a fault‐related structure using nonlocal models. A column flow experiment is conducted to create a permeability distribution that varies with distance from a main conduit. The experimental tracer response curve is preasymptotic and implies subdiffusive transport, which is slower than the normal Fickian diffusion. If the surrounding area is a finite domain, an upper truncated behavior in tracer response (i.e., exponential decline at late times) is observed. The tempered anomalous diffusion (TAD) model captures the transition from subdiffusive to Fickian transport, which is characterized by a smooth transition from power‐law to an exponential decline in the late‐time breakthrough curves.

Subdiffusive flow in a composite medium with a communicating (absorbing) interface

“…For subdiffusive flows, fractional flux laws of the form proposed here have been used for many years in a variety of contexts concerning diffusion: in biophysics (Magin et al, 2013), fractal structures (Dassas and Duby, 1995;Metzler et al, 1994), flow in porous media (Albinali et al, 2016;Lẽ Mehautẽ and Crepy, 1983;Nigmatullin, 1984a, b;Płociniczak, 2015;Raghavan, 2011;Raghavan and Chen, 2017;Su, 2014;Su et al, 2015), contaminant transport (Fomin et al, 2011;Kim et al, 2015;Molz et al, 2002;Suzuki et al, 2016), heat transmission (Angulo et al, 2000;Gurtin and Pipkin, 1968;Moodie and Tait, 1983;Norwood, 1972). Experimental aspects are discussed in Tao et al (2016), Strizhak (2018, 2019) and Yanga et al (2019).…”

On the maximum principle for a time-fractional diffusion equation

“…The time-fractional transport equation (1.2) was already employed for modeling various anomalous transport processes including the mass and heat transfer for characterizing geothermal reservoirs ( [22,23,24,26]). However, compared to the comprehensive results already obtained for the time-fractional diffusion equation of type (1.1), it may be a surprise that until now only few theoretical publications were devoted to the fractional transport equations, for instance, to the problem of unique existence of solution to the initial-boundaryvalue problem (1.2)- (1.4).…”

A study in fractional diffusion: Fractured rocks produced through horizontal wells with multiple, hydraulic fractures

“…For subdiffusive flows, fractionalflux laws were proposed over 30 years ago; see Le Mẽhautẽ and Crepy (1983) and Nigmatullin (1984Nigmatullin ( , 1986. As discussed in Raghavan (2011) the need for a fractional fluxlaw is implied in the development in Metzler et al (1994) which improves upon the work of O'Shaughnessy and Procaccia (1985a) for transient diffusion in fractal structures; see also Albinali et al (2016) and Suzuki et al (2016). As noted earlier, ideas based on Continuous Time Random Walks (Hilfer and Anton, 1995;Kenkre et al, 1973;Montroll and Weiss, 1965;Noetinger and Estebenet, 2000) and waiting-time solutions (Henry et al, 2010) also yield such laws.…”

“…The earth-science literature abounds with examples of situations where subdiffusive models apply, namely in situations wherein cracks, fissures and cervices abound (Caine et al, 1996;Evans, 1988;Mitchell and Faulkner, 2009;Savage and Brodsky, 2011;Scholz et al, 1993). Mathematical models depicting flows in porous rocks may be found in Jourde et al (2002), Cortis and Knudby (2006) and Suzuki et al (2016). In combined situations, we consider o a t p ¼ o bþ1…”