Spatial Markov Model of Anomalous Transport Through Random Lattice Networks

Flow through lattice networks with quenched disorder exhibits a strong correlation in the velocity field, even if the link transmissivities are uncorrelated. This feature, which is a consequence of the divergence-free constraint, induces anomalous transport of passive particles carried by the flow. We propose a Lagrangian statistical model that takes the form of a continuous time random walk with correlated velocities derived from a genuinely multidimensional Markov process in space. The model captures the anomalous (non-Fickian) longitudinal and transverse spreading, and the tail of the mean first-passage time observed in the Monte Carlo simulations of particle transport. We show that reproducing these fundamental aspects of transport in disordered systems requires honoring the correlation in the Lagrangian velocity.

Figure 1

(a) Schematic of the lattice network considered here, with two sets of links with orientation $\pm \alpha =\pm \pi /4$ and spacing $l=1$. (b) Particle distribution at nodes (represented by circles of different sizes) at $t=30$ for a single realization after injection at the origin at $t=0$.

Figure 2

(a) Aggregate transition matrix for $N=100$ velocity classes distributed with logarithmic scale. (b) Transition probabilities after $m=5$ steps from direct Monte Carlo computation (blue solid line) and calculated from the Markov assumption (green symbols). Shown are probability densities for two initial velocity classes: a low velocity class ($j=5,\circ$), and a high velocity class ($j=90,*$). Inset: Probability of returning to the same initial velocity class as a function of the number of steps for a high initial velocity (class $j=90$).

Figure 3

Contour plot of the mean particle density at $t=5\times {10}^2$, computed from direct Monte Carlo simulation (blue solid line), correlated CTRW model (green solid line), and uncorrelated CTRW model (red solid line).

Figure 4

(a) Time evolution of the longitudinal MSD. Inset: Transverse MSD. (b) Cumulative FPT distribution.