Anomalous transport in correlated velocity fields

We examine different types of heterogeneous hydraulic conductivity fields to ascertain the basic structural features that dominate the transport behavior. We contrast two approaches to the analysis, within the framework of the continuous time random walk (CTRW), considering recent simulations of particle transport in two correlated flow fields to discern these key features. These flow fields are the steady-state solutions of Darcy flow in systems with correlated distributions, $\mathcal{P}\mathbf{\left(}K\left(\mathbf{x}\right)\mathbf{\right)}$, of hydraulic conductivity values $K\left(\mathbf{x}\right)$. One approach uses the organizational structure of the Lagrangian velocities determined from simulations to derive correlated space-time distributions for particle tracking, which are used to fit simulated breakthrough curve (BTC) data. These fits emphasize the ability to account for both early arrival times and late-time long tailing. The other approach, in this paper, treats the simulated BTCs as “measurements” and uses a truncated power-law form of $\psi \left(t\right)$, the probability density function (pdf) of local transit times, in a partial differential equation form of CTRW. Excellent fits to both data sets are obtained with a single value of $\beta$, the key parameter that characterizes the nature of the dispersive transport. The value of $\beta$ is derivable from the high $\xi$ behavior of the pdf histogram $\Phi \left(\xi \right)$ (where $\xi$ is the inverse velocity) of the Darcy field, which determines the late-time tail within $\psi \left(t\right)$. The quality of the two fits obtained herein with a physically derived parameter set is a probe of how heterogeneous hydraulic conductivity fields with different types of correlation can affect the larger-scale transport behavior. The features that give rise to a power-law tail of local transition times and a limit of the time range for non-Fickian behavior dominate the transport. The correlation structures of the different $\mathcal{P}\mathbf{\left(}K\left(\mathbf{x}\right)\mathbf{\right)}$ play a secondary role compared to the spectrum of less frequent events (e.g., low velocity regions) that have a large effect on the aggregate of median time transitions.

Figure 1

(Color online) Example realization of a multilognormal hydraulic conductivity $\left(K\right)$ field [9]. The scale to the right shows ${\text{log}}_{10}K$.

Figure 2

(Color online) Example realization of a connected hydraulic conductivity $\left(K\right)$ field [9]. The scale to the right shows ${\text{log}}_{10}K$.

Figure 3

First passage time distribution at a distance of 100 elements from the inlet multilognormal field, based on numerical simulations of particle tracking [10], and a fit with the CTRW using a TPL $\psi \left(t\right)$. Solid curve, numerical simulations; dashed curve, CTRW fit. The simulation curve here shows the ensemble averaged transport behavior based on 100 realizations of the underlying conductivity field. Parameters of the CTRW-TPL fit: $v_{\psi }=2.78$, $D_{\psi }=0.49$, $\beta =1.12$, ${\text{log}}_{10}\left(t_1\right)=0.15$, ${\text{log}}_{10}\left(t_2\right)=3.6$. Arbitrary units on $v\left[\text{L}/\text{T}\right]$, $D\left[{\text{L}}^2/\text{T}\right]$, $t_1\left[\text{T}\right]$, and $t_2\left[\text{T}\right]$.

Figure 4

First passage time distribution at a distance of 100 elements from the inlet connected field, based on numerical simulations of particle tracking [10], and a fit with the CTRW using a TPL $\psi \left(t\right)$. Solid curve, numerical simulations; dashed curve, CTRW fit. The simulation curve here shows the ensemble averaged transport behavior based on 100 realizations of the underlying conductivity field. Parameters of the CTRW-TPL fit: $v_{\psi }=7.4$, $D_{\psi }=1.8$, $\beta =1.12$, ${\text{log}}_{10}\left(t_1\right)=0.15$, ${\text{log}}_{10}\left(t_2\right)=3.7$. Arbitrary units on $v\left[\text{L}/\text{T}\right]$, $D\left[{\text{L}}^2/\text{T}\right]$, $t_1\left[\text{T}\right]$, and $t_2\left[\text{T}\right]$.