Characterizing the impact of particle behavior at fracture intersections in three-dimensional discrete fracture networks

We characterize the influence of different intersection mixing rules for particle tracking simulations on transport properties through three-dimensional discrete fracture networks. It is too computationally burdensome to explicitly resolve all fluid dynamics within a large three-dimensional fracture network. In discrete fracture network (DFN) models, mass transport at fracture intersections is modeled as a subgrid scale process based on a local Péclet number. The two most common mass transfer mixing rules are (1) complete mixing, where diffusion dominates mass transfer, and (2) streamline routing, where mass follows pathlines through an intersection. Although it is accepted that mixing rules impact local mass transfer through single intersections, the effect of the mixing rule on transport at the fracture network scale is still unresolved. Through the use of explicit particle tracking simulations, we study transport through a quasi-two-dimensional lattice network and a three-dimensional network whose fracture radii follow a truncated power-law distribution. We find that the impact of the mixing rule is a function of the initial particle injection condition, the heterogeneity of the velocity field, and the geometry of the network. Furthermore, our particle tracking simulations show that the mixing rule can particularly impact concentrations on secondary flow pathways. We relate these local differences in concentration to reactive transport and show that streamline routing increases the average mixing rate in DFN simulations.

Figure 1

A fracture intersection with two inflow branches and two outflow branches. All branches have equivalent discharge magnitudes. In complete mixing (a) mass from both inlets (red and blue) mix at the intersection and mass is distributed equally to each outlet (purple). In streamline routing (b) incoming red mass from the bottom inlet and blue mass from the top inlet are forced to their respective adjacent fractures and do not mix.

Figure 2

One realization of a DFN with fracture lengths drawn from a power-law distribution. Fractures are colored by their permeability, which is positively correlated with fracture radius.

Figure 3

Cumulative distribution of first passage arrival times (breakthrough curves) for point injection (a) and flux-weighted (b) initial condition. Thick lines are median breakthrough curves for 25 realizations and transparent lines correspond to single realizations. For each realization, solid lines indicate streamline routing and dashed lines indicate complete mixing. Colors correspond to different aperture variances; ${\sigma }_{\text{ln}\left(b\right)}^2=0.1,0.5,1$ are depicted with blue (steepest slope), red, and green (least steep slope), respectively.

Figure 4

The transverse distribution at sequential control planes through a lattice network with a point injection initial condition for one realization. All lattices have the same mean aperture size. Variance of aperture size is selected from a lognormal distribution and increases from the left to right column. Simulations are completed for complete mixing (row 1) and streamline routing (row 2) intersection rules for the same network realization. Row 3 gives the ratio of streamline routing to complete mixing transverse breakthrough position concentrations. Colorbars show log probabilities (rows 1 and 2) and absolute ratio values (row 3). Complete mixing enhances particle spreading. As velocity field heterogeneity increases, the impact of the mixing rule on particle spreading decreases. The transverse direction $z$ is normalized by half the length the of domain $z^{*}$.

Figure 5

The transverse breakthrough position distribution at each control plane for one realization of the lattice with a flux-weighted injection. Simulations are completed for complete mixing (a) and streamline routing (b) intersection rules. (c) gives the ratio of streamline routing to complete mixing transverse breakthrough position concentrations. Colorbars show log probabilities [(a) and (b)] and the absolute ratio (c). ${\sigma }_{\text{ln}\left(b\right)}^2=0.5$ is the only aperture variance shown because results do not significantly change for different velocity field heterogeneities. The mixing rule has less impact on particle spreading with a flux-weighted initial condition. The transverse direction $z$ is normalized by half the length of domain $z^{*}$.

Figure 6

Particles within the quasi-two-dimensional lattice network. In (a) and (b), particles injected from the same point source on the left boundary and driven right by a pressure gradient. In (a), particles adhere to a complete mixing rule (red) and in (b) they follow streamline routing (blue). The choice of a complete mixing rule in combination with a point injection leads to higher transverse dispersion than if streamline routing is used under the same initial conditions. (c) shows particles injected using flux weighting adhering to both mixing rules (red: complete mixing/blue: streamline routing). Here, no significant difference between the distribution of particle locations between the two rules is observed.

Figure 7

The mean MSD in the lattice network is calculated at each control plane by averaging over all particles and all 25 realizations. Solid lines indicate streamline routing and stars indicate complete mixing. Plotted is MSD for both the point (a) and flux-weighted (b) initial conditions. Colors correspond to different aperture variances; ${\sigma }_{\text{ln}\left(b\right)}^2=0.1,0.5,1$ are depicted with blue [upper curve in (b)], red, and green [lower curve in (b)], respectively. As velocity field heterogeneity increases the mixing rule's impact on MSD decreases. A flux-weighted injection results in less spatial variability of MSD.

Figure 8

Median breakthrough curves of ten TPL realizations for complete mixing (green stars) and streamline routing (thick orange lines). Solid lines correspond to point injections and dashed lines correspond to a flux-weighted initial condition. Each breakthrough curve realization is plotted with a transparent line. The mixing rule has a negligible impact of the distribution of arrival times. A flux-weighted initial condition homogenizes the spread of breakthrough curves.

Figure 9

The transverse breakthrough $z$-position distribution for one realization of a TPL network for a point (left column) and flux-weighted (right column) initial injection mode. Simulations are completed for complete mixing [row 1: (a) and (d)] and streamline routing [row 2: (b) and (e)] intersection rules. Row 3 gives the ratio of streamline routing to complete mixing transverse breakthrough position concentrations. Colorbars show log probabilities [rows 1 and 2: (a), (b), (d), and (e)] and the absolute ratio value [row 3: (c) and (f)]. Streamline routing increases channelization of particles to secondary fractures, shown as areas of yellow in the ratio figures. The transverse direction $z$ is normalized by half the length of domain $z^{*}$.

Figure 10

The MSD is averaged over all particles in ten realizations of stochastically generated TPL networks for complete mixing (green) and streamline routing (orange) with a point (solid lines) and flux-weighted (dashed lines) injection. The MSD for the flux-weighted injections has less spatial variability. The network structure drives particle spreading and so the impact of the mixing rule on MSD is negligible.

Figure 11

The ratio streamline routing to complete mixing local mixing ratios is shown for one TPL network realization for a point (a) and flux-weighted (b) initial injection. The front right face is the plane of injection and the back left face is the domain outlet. Colorbars plot the absolute ratio value. Near the inlet plane, the ratio between mixing rules is 1 because transport has yet to encounter fracture intersections. Near the outlet plane, we observe streaks where local mixing rates differ by a factor of 100. Differences in mixing rate occur from differences in channelization of particles due to the mixing rule.

Figure 12

The mean mixing rate averaged over ten TPL realizations at each control plane is compared for complete mixing (green) and streamline routing (orange) with a point injection (a) and flux-weighted (b) initial injection. In each realization, the mixing rate is normalized by maximum mixing rate observed for streamline routing. Streamline routing increases the mean mixing rate. At distances greater than $l^{*}$ from the particle source the difference between mixing rules is greater because particles have encountered at least one fracture intersection.