Effect of velocity fluctuations on pore scale stretching kinematics in 3D porous media

Fluid stretching plays an important role in controlling mixing dynamics in porous media. Recent advances have shown that stretching at pore scale in 3D porous media is chaotic, leading to exponential elongation of mixing interfaces [e.g. 1,2]. Yet, it is not known how the associated stretching rate depend on the pore scale velocity heterogeneity. In this study, we perform particle tracking simulations in a periodic flow fields to investigate how flow heterogeneity control the transient evolution of the stretching rate as well as the asymptotic stretching rate (Lyapunov exponent). Our results reveal that rare low velocity events have a significant impact on the Lyapunov exponent, while these regions are numerically more difficult to treat and thus sometimes excluded from statistics [1]. Moreover, we discuss conceptual difficulties associated to velocity pdf with heavy tails towards low velocities: Ensemble averages of the deformation gradient tensor do not converge under these conditions when taken them at equal advective distances, as opposed to at equal times. As a consequence, the meaning of the steady state stretching rate must be discussed in the context of long memories. Using Continuous Time Random Walks (CTRW) we derive analytical expressions for the averages and discuss low velocity cutoffs to guarantee convergence. We further discuss the nature of the pre-asymptotic stretching kinematics, which can have a dominant effect on mixing processes. We show that the strength of the transient is controlled by the typical shear rate while the duration is determined by the Lyapunov exponent. Weaker chaotic stretching are associated to a longer lasting transient regime.

[1]: Turuban et al. (2019) In: JFM 
[2]: Heyman et al. (2020) In: PNAS