Diffusion in sparsely connected pore spaces: Temporal and spatial scaling

Abstract

[1] Diffusion both disperses and retards mass movement of contaminants through porous media. The retardation generally involves mass transfer of contaminants from faster to slower flow paths, making prediction more difficult and remediation slower. In ordinary (Fickian) diffusion, root-mean-square displacement of a population of diffusing molecules increases with the square root of the elapsed time, and diffusivity (D, L 2 T À1 ) of the porous medium is constant in time. The diffusivity can be measured using time-dependent methods such as pulsed field gradient (PFG) magnetic resonance (MR), or steady state diffusion experiments. However, it is known that diffusion behaves anomalously in sparsely connected pore spaces at the percolation threshold. We conducted a series of Monte Carlo simulations using random walks, designed to mimic measurement of D using both time-dependent and steady state methods. In the time-dependent simulations, diffusivity scaled as D $ T À0.48 at the percolation threshold. Above the percolation threshold, D decreased at early times, then became constant, with the time required for constant D being greater when the medium was less interconnected. In the steady state simulations we examined the effect of sample size on apparent diffusivity. Above the percolation threshold, diffusivity decreased, then became constant as sample size increased, but at the percolation threshold, diffusivity scaled as D $ L À 1.83 . The drop in diffusivity with sample size was caused by both a decrease in effective porosity and an increase in tortuosity, each of which also followed scaling laws. A single measurement will not suffice to determine whether a sample's pore space is at the percolation threshold, necessitating measurement of diffusivity at more than one scale in time or space. Using a scale-dependent measurement to make predictions at a different scale can result in serious overestimation or underestimation of diffusive mass transfer and consequent retardation.

Using X-ray computed tomography in pore structure characterization for a Berea sandstone: Resolution effect

“…Comparison in Table 1 of accessible fractions shows clearly that XCT based results are larger than MIP-based results. This can be caused by two reasons: (1) the low resolution XCT overestimates the pore size, consequently, it may overestimate the pore connectivity by taking two disconnected smaller pores as one larger pore; and (2) scale effect exists for pore connectivity, as indicated by Ewing and Horton (2002) that the pore connectivity was higher at smaller scale.…”

Low nanopore connectivity limits gas production in Barnett formation

“…Fluid flow and mass transport through rock is controlled by the rock pores' geometry (e.g., pore size distribution) [ Bear , ] and topology (i.e., pore connectivity) [ Dullien , ]. Especially when pore connectivity is low, topological factors outweigh the better‐known geometrical factors [ Ewing and Horton , ; Hu et al , ; Hunt et al , ]. Low pore connectivity in rocks such as tight shales has only recently been documented [ Hu et al , ; Hu and Ewing , ].…”

“…Low pore connectivity in rocks such as tight shales has only recently been documented [ Hu et al , ; Hu and Ewing , ]. However, it is known that diffusion in sparsely connected pore spaces is not of classical Fickian type; rather, diffusion will be anomalous as described by percolation theory [ Stauffer and Aharony , ; Ewing and Horton , ; Ewing et al , ].…”

Pore structure and tracer migration behavior of typical American and Chinese shales

“…Imbibition tests, which are much faster than diffusion tests, involve exposing one face of a rock sample to liquid (for example, water, API brine or n-decane), and monitoring the fluid mass uptake over time (e.g., Hu et al 2001;Schembre and Kovscek 2006). Using the network modeling results of Ewing and Horton (2002), we can probe pore connectivity, as indicated by the slope of log (imbibed liquid mass) versus log (time). The imbibition behavior-whether the imbibition slope is , changing to , or -conveniently classifies a rock's pore connectivity .…”

“…Especially when pore connectivity is low, topological factors outweigh the better known geometrical factors (Ewing and Horton 2002;Hunt et al 2014). However, the prevalence of low pore connectivity in tight shales, and its impacts on fluid flow and chemical transport, is poorly documented and understood.…”