Measurement and analysis of anomalous diffusion in porous rock

Molecular diffusion, or more specifically, diffusion-controlled transport, of tracers, contaminants, and chemical species --- in soil and rock formations, and in river, lake, and marine sediments --- plays a critical role in many dynamic processes that affect water chemistry and properties of the host domain. The spreading of dissolved ionic species via Brownian motion is generally described by a Gaussian law for the probability density function, with diffusion (embodying Fick’s second law) then being described by the classical diffusion equation. Solution of this equation shows that the spreading pattern of chemical species is characterized by a mean squared displacement that scales linearly with time. However, in other porous domains like biological tissues and cells, dense liquids, and gels, diffusion behavior often deviates from Fickian, instead exhibiting anomalous (or non-Fickian) diffusion. More specifically, tracer movements in these “crowded environments” exhibit a spreading pattern wherein the mean squared displacement scales as a power law. Somewhat surprisingly, in studies involving water-saturated porous rock, diffusion of chemical species is generally assumed to follow Fick’s second law, ignoring the possible occurrence of anomalous diffusion. To test this assumption, we measure molecular diffusion in five chalk and dolomite rock samples using a specially designed diffusion cell. The set-up enables high-resolution measurement of extended, long-time tailing at the measurement plane, which is required to distinguish between Fickian and anomalous diffusion behavior. In all of the rock samples, the diffusion behavior is demonstrated to be significantly different than Fickian, with extreme long-time tailing of tracer advance relative to conventional Fickian diffusion. The measured breakthrough curves are then analyzed using a continuous time random walk framework that describes anomalous diffusion in heterogeneous porous materials. The analysis (i) provides a framework to distinguish between Fickian and anomalous diffusion, and (ii) demonstrates that anomalous diffusion in geological formations is likely ubiquitous and implies that diffusion-controlled transport processes should be analyzed using tools that account for such behavior.