Fluid Stretching in Heterogeneous Porous Media as a Lévy Process

International audienceStretching and compression of material fluid elements is key for the understanding and quantification of the dispersionand mixing dynamics in heterogeneous porous media flows, because they represent the support of a transportedsolute. The elongation and compression of a material strip determine the mixing volume and mixing rateand thus the concentration content of a heterogeneous mixture. While linear and exponential elongation dynamicstypical for shear and chaotic flows, respectively, are well understood, the mechanisms that lead toobserved powerlawelongation in heterogeneous porous media are in general unknown. We cast the fluid deformation problem instreamline coordinates, which reveals that the principal elongation mechanism for non-helical steady flows is dueto shear deformation and velocity fluctuations along the streamline. The impact of this coupling on the elongationdynamics is quantified within a continuous time random walk (CTRW) approach. The CTRW describes the movementof 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 isrelated to the transition time through the velocity-shear coupling. For a broad distribution of transition, as foundin strongly heterogeneous porous media, the elongation is a Lévy process. These dynamics describe a broad rangeof algebraic stretching behaviors with mean strip elongations h`(t)i / t with 1=2 < 2. These findings havebroad implications for the understanding and prediction of dilution and mixing in heterogeneous porous mediaflows