Scaling of Reactive Transport in Fracture Networks with Incomplete Mixing: A Metapopulation Network Approach

The study of networks as complex systems has revolutionized many disciplines in physics and the social and natural sciences. Recently, the focus of network science has shifted from the analysis of the network topology to the study of the dynamics of processes that take place on them. Here, we adopt a bosonic (metapopulation) network approach to characterize transport and reaction on fracture networks. In a bosonic network, nodes contain populations of particles, which may undertake contact processes within them. This coarse-grained conceptualization permits modeling of nonequilibrium phenomena such as incomplete mixing and kinetic reactions. Particles then move between connected nodes along links, with a rate that reflects the traffic patterns through the network. We generate fracture networks from realistic statistical properties of fracture density, orientation and aperture, and solve potential flow on the network for simple flow configurations. It is well known that the transport of passive particles with complete mixing at the nodes on a fracture network or a scale-free network is anomalous [1,2]. Here, we extend this analysis to account for incomplete mixing at the nodes by considering two types of particles, A and B, which come in contact at the nodes at a rate α (the mixing rate) to produce type-C particles: A+B-> 2C. In the limit of α -> ∞ we recover the instantaneous mixing case. Further, type-C particles decay into stable type-D particles at a rate λ : C→ D. As a result, we capture the interplay among the transport, mixing and reaction time scales on a fracture network. Our analysis demonstrates the strong dependence of global mixing and spreading on incomplete local mixing, and allows us to obtain robust scalings for the spatio-temporal distributions of particles.