Anomalous transport on regular fracture networks: Impact of conductivity heterogeneity and mixing at fracture intersections

We investigate transport on regular fracture networks that are characterized by heterogeneity in hydraulic conductivity. We discuss the impact of conductivity heterogeneity and mixing within fracture intersections on particle spreading. We show the emergence of non-Fickian transport due to the interplay between the network conductivity heterogeneity and the degree of mixing at nodes. Specifically, lack of mixing at fracture intersections leads to subdiffusive scaling of transverse spreading but has negligible impact on longitudinal spreading. An increase in network conductivity heterogeneity enhances both longitudinal and transverse spreading and leads to non-Fickian transport in longitudinal direction. Based on the observed Lagrangian velocity statistics, we develop an effective stochastic model that incorporates the interplay between Lagrangian velocity correlation and velocity distribution. The model is parameterized with a few physical parameters and is able to capture the full particle transition dynamics.

Figure 1

(a) Schematic of the fracture network studied here, with two sets of links with orientation $\pm \alpha =\pm \pi /4$ and uniform spacing $l$. The conductivity values are reflected in the link thickness. We study log-normal conductivity distributions with three different conductivity variance values: ${\sigma }_{lnK}^2=0.1,1,5$. (b) Map of the spatially uncorrelated conductivity field with ${\sigma }_{lnK}^2=5$ shown in a log-scale color scheme. No flow boundary conditions on the top and bottom and constant hydraulic head on the left and right boundaries ensure a uniform mean flow.

Figure 2

(a) Normalized flow field ($|u_{ij}|/\overline{u}$) for log-normal conductivity distribution with variance $0.1$. (b) Normalized flow field for log-normal conductivity distribution with variance 1. (c) Normalized flow field for log-normal conductivity distribution with variance 5. Even though the underlying conductivity field is uncorrelated, the combined effect of network heterogeneity and the mass conservation constraint at nodes leads to a correlated flow field with preferential flow paths.

Figure 3

Schematic for the two different mixing rules for the case in which nodes at the two incoming and the two outgoing links have equal fluxes. (a) The streamline routing rule makes all the particles transit to the adjacent link because particles cannot switch between streamlines. (b) The complete mixing rule makes half of the particles move upward and the other half downward, following flux-weighted probabilities.

Figure 4

Particle distribution at $t=15\overline{t_l}$ for a given realization after the instantaneous release of particles at the origin (red star). $\overline{t_l}$ is the mean advective time along one link. (a) Low heterogeneity (${\sigma }_{\text{ln}K}^2=0.1$) with streamline routing at nodes. (b) Low heterogeneity (${\sigma }_{\text{ln}K}^2=0.1$) with complete mixing at nodes. (c) High heterogeneity (${\sigma }_{\text{ln}K}^2=5$) with streamline routing at nodes. (d) High heterogeneity (${\sigma }_{\text{ln}K}^2=5$) with complete mixing at nodes. For low heterogeneity, complete mixing significantly enhances transverse spreading. An increase in heterogeneity significantly enhances longitudinal spreading.

Figure 5

Time evolution of MSDs for complete mixing (solid line) and streamline routing (dashed line). (a) Longitudinal MSD for ${\sigma }_{lnK}^2=0.1$. (b) Transverse MSD for ${\sigma }_{lnK}^2=0.1$. (c) Longitudinal MSD for ${\sigma }_{lnK}^2=1$. (d) Transverse MSD for ${\sigma }_{lnK}^2=1$. Inset: Change in the time evolution of transverse MSD for complete mixing with increasing variance. (e) Longitudinal MSD with ${\sigma }_{lnK}^2=5$. Inset: Change in the time evolution of longitudinal MSD for complete mixing with increasing variance. (f) Transverse MSD with ${\sigma }_{lnK}^2=5$. Inset: Change in the time evolution of transverse MSD for streamline routing with increasing conductivity variance.

Figure 6

First passage time distribution $f_{\chi }\left(\tau \right)$ for ${\sigma }_{lnK}^2=0.1,1,5$ and different mixing rules. Conductivity heterogeneity has a major impact on particle breakthrough curves, in contrast to the mixing rules.

Figure 7

Transverse breakthrough positions distribution (TBPD) at the outlet plane. Comparison between the two different mixing rules for (a) ${\sigma }_{lnK}^2=0.1$, (b) ${\sigma }_{lnK}^2=1$, and (c) ${\sigma }_{lnK}^2=5$. The impact of the mixing rule on transverse spreading diminishes as the network heterogeneity increases.

Figure 8

(a) Lagrangian transition time distributions for ${\sigma }_{lnK}^2=0.1,1,5$ and complete mixing at the nodes. Inset: Lagrangian velocity distributions for the three different values of ${\sigma }_{lnK}^2$. As the network conductivity becomes more heterogeneous, both the transition time distribution and the velocity distribution become broader. (b) Velocity autocorrelation function in space. Error bars represent the coefficient of variation. An increase in network heterogeneity leads to stronger correlation. Inset: Comparison between the velocity autocorrelation in space and in time for ${\sigma }_{\text{ln}K}^2=5$. Velocity autocorrelation in time is normalized with the mean advective time along one link, and velocity autocorrelation in space is normalized with the link length.

Figure 9

Schematic of the velocity transition matrix for $d=2$ dimensional networks. The transition matrix considers all 16 possible transitions to capture the full particle transport dynamics. The matrix has information about the one-step correlation, directionality, and velocity heterogeneity.

Figure 10

(a) Velocity transition matrix with linear equiprobable binning for ${\sigma }_{lnK}^2=0.1$ and complete mixing at nodes. Of 16 transitions, only the four that have forward-forward movement in longitudinal direction (E, M, G, O) are possible. Note that the probability for each possible transition is almost identical. (b) Velocity transition matrix with linear equiprobable binning for ${\sigma }_{lnK}^2=0.1$ with streamline routing. Again, only the four transitions that have forward-forward movement in the longitudinal direction (E, M, G, O) are possible. Also, note that the probability for M and G transitions (0.89) is significantly higher than E and O transitions (0.11).

Figure 11

(a) Velocity transition matrix with linear equiprobable binning for ${\sigma }_{lnK}^2=5$ and complete mixing at the nodes. Due to strong heterogeneity, 12 different transitions, including backward movements, are possible. Also, note that up-up and down-down transitions (A, F, K, P) have triangular matrices. This indicates that velocity magnitudes mostly increase when a particle changes direction from the $-x$ direction to the $+x$ direction and vice versa. (b) Velocity transition matrix with linear equiprobable binning for ${\sigma }_{lnK}^2=5$ and streamline routing. Since strong heterogeneity dictates particle transitions, there is no significant difference between complete mixing and streamline routing.

Figure 12

Time evolution of MSDs obtained from Monte Carlo simulations (solid lines) and the model predictions from the correlated CTRW model (dashed lines). The developed correlated CTRW model is able to accurately capture the time evolution of the MSDs for all levels of heterogeneity strength and mixing rules. (a) ${\sigma }_{lnK}^2=0.1$, (b) ${\sigma }_{lnK}^2=1$, and (c) ${\sigma }_{lnK}^2=5$ with complete mixing, where the red line is longitudinal direction and the blue line is transverse direction. (d) ${\sigma }_{lnK}^2=0.1$, (e) ${\sigma }_{lnK}^2=1$, and (f) ${\sigma }_{lnK}^2=5$ with streamline routing, where the black line is longitudinal and the green line is transverse direction.

Figure 13

Probability distributions of transverse particle breakthrough position for Monte Carlo simulations and model predictions. (a) Correlated CTRW for ${\sigma }_{lnK}^2=0.1$. (b) Correlated CTRW for ${\sigma }_{lnK}^2=1$. (c) Correlated CTRW for ${\sigma }_{lnK}^2=5$. (d) Uncorrelated CTRW for ${\sigma }_{lnK}^2=0.1$. (e) Uncorrelated CTRW for ${\sigma }_{lnK}^2=1$. (f) Uncorrelated CTRW for ${\sigma }_{lnK}^2=5$. Uncorrelated CTRW does not have a capability of distinguishing complete mixing and streamline routing cases.

Figure 14

Particle breakthrough curves from Monte Carlo simulations and model predictions from (a) correlated CTRW and (b) uncorrelated CTRW.

Figure 15

Comparison between the time evolution of the MSDs obtained from the Monte Carlo simulations (solid lines) and the model predictions from the parametric correlated CTRW model (dashed lines). The proposed model is able to accurately capture the time evolution of the MSDs for all levels of heterogeneity and mixing rules under consideration. (a) ${\sigma }_{lnK}^2=0.1$, (b) ${\sigma }_{lnK}^2=1$, and (c) ${\sigma }_{lnK}^2=5$ with complete mixing, where the red line is longitudinal direction and the blue line is transverse direction. (d) ${\sigma }_{lnK}^2=0.1$, (e) ${\sigma }_{lnK}^2=1$, and (f) ${\sigma }_{lnK}^2=5$ with streamline routing, where the black line is longitudinal and the green line is transverse direction.

Figure 16

Probability distributions of the transverse particle breakthrough positions obtained from Monte Carlo simulations, and predictions from the parametric correlated CTRW model. (a) ${\sigma }_{lnK}^2=0.1$. (b) ${\sigma }_{lnK}^2=1$. (c) ${\sigma }_{lnK}^2=5$. Inset: Particle breakthrough curves from Monte Carlo simulations and model predictions from the parametric correlated CTRW model.