Effective dispersivities for a two-dimensional periodic fracture network by a continuous time random walk analysis of single-intersection simulations

Fracture networks are of major importance for transport problems in hydrology. Their inclusion in models requires assumptions about both the single fracture and the network structure. This paper focuses on the macroscopic effect of microscopic mixing conditions in intersections within a periodic fracture network. Mixing is characterized by the transition probabilities for particles to move from one fracture to another or to stay in the same fracture. A periodic network is used, which excludes stochastic effects and also single-fracture dispersion. The periodic network can be solved analytically and is applicable to a variety of network, structures. We find that for this type of network, effective dispersivities in the sense of asymptotic values for long times and large scales do usually exist (i.e., the asymptotic dispersion is Gaussian). For the particular case of a quadratic network, particle transition probabilities at an intersection have been obtained by microscopic lattice Boltzmann simulations of flow and transport in addition to the assumptions of complete mixing and stream routing. The resulting dispersivities show that complete mixing is not a low Peclet number limit in the presence of regions of immobile water volumes. Dispersivities depend strongly on the direction between flow and network and on the mixing model; dispersivities are large for high Peclet numbers and flow parallel to the network. Especially for these cases, we also find a scale dependence up to some orders of magnitude, which has serious implications for the interpretation of simulations and measurements. Since these results also appear in stochastic network simulations, they are therefore not specific to the periodic structure of the fracture network.