Fluid Stretching in Heterogeneous Porous Media as a Lévy Process
Résumé
Stretching and compression of material fluid elements is key for the understanding and quantification of the dispersion
and mixing dynamics in heterogeneous porous media flows, because they represent the support of a transported
solute. The elongation and compression of a material strip determine the mixing volume and mixing rate
and thus the concentration content of a heterogeneous mixture. While linear and exponential elongation dynamics
typical for shear and chaotic flows, respectively, are well understood, the mechanisms that lead toobserved powerlaw
elongation in heterogeneous porous media are in general unknown. We cast the fluid deformation problem in
streamline coordinates, which reveals that the principal elongation mechanism for non-helical steady flows is due
to shear deformation and velocity fluctuations along the streamline. The impact of this coupling on the elongation
dynamics is quantified within a continuous time random walk (CTRW) approach. The CTRW describes the movement
of fluid particles in porous media flows through a random in both space and time, in which the transition time
over a characteristic velocity length scale `c is coupled kinematically to streamline velocity vs as = `c=vc.
In this framework, the elongation process isidentified as a coupled CTRW in which the elongation increment is
related to the transition time through the velocity-shear coupling. For a broad distribution of transition, as found
in strongly heterogeneous porous media, the elongation is a Lévy process. These dynamics describe a broad range
of algebraic stretching behaviors with mean strip elongations h`(t)i / t with 1=2 < 2. These findings have
broad implications for the understanding and prediction of dilution and mixing in heterogeneous porous media
flows.