ANOMALOUS TRANSPORT IN GEOLOGICAL FORMATIONS: THEORY AND OBSERVATIONS

Anomalous transport of chemical tracers has been observed at field and laboratory scales, in porous and fractured geological formations. Quantification of this widespread phenomenon has been a long-standing problem. These formations have multi-scale heterogeneity, and capturing the complexities of tracer plume migration patterns suggests that, contrary to current practice, small-scale heterogeneities cannot be "averaged out". A general theory developed within the continuous time random walk (CTRW) framework, based on a picture of transport as a sequence of particle transfer rates, provides an effective means to quantify this anomalous transport. In disordered systems, statistically rare, slow transition rates limit transport. Hence, the key step is to retain the entire range of these transitions with a pdf ψ(s,t), where s is a transition step displacement and t is the transfer time, instead of upscaling from mean local rates. Comparison to a variety of laboratory- and field-scale observations has generated a new level of confirmation and further development of the theory. The CTRW has been developed within the framework of partial differential equations (pde), and incorporates complex systems involving reactive tracers and interactions with "immobile states". These pde's are nonlocal in time as they contain a convolution of a memory function M(t), based on ψ(s,t), with an advection-dispersion operator. The Laplace space forms can be solved by both analytical and conventional numerical methods. The CTRW is shown to provide an overarching framework for quantitative modeling; the uncoupled form (ψ(s,t) = p(s)ψ(t)) encompasses a variety of multirate mass transport and fractional derivative equation approaches.