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We investigate tracer transport on random discrete fracture networks that arecharacterized by the statistics of the fracture geometry and hydraulicconductivity. While it is well known that tracer transport through fracturedmedia can be anomalous and particle injection modes can have major impact ondispersion, the incorporation of injection modes into effective transportmodelling has remained an open issue. The fundamental reason behind thischallenge is that---even if the Eulerian fluid velocity is steady---theLagrangian velocity distribution experienced by tracer particles evolves withtime from its initial distribution, which is dictated by the injection mode, toa stationary velocity distribution. We quantify this evolution by a Markovmodel for particle velocities that are equidistantly sampled alongtrajectories. This stochastic approach allows for the systematic incorporationof the initial velocity distribution and quantifies the interplay betweenvelocity distribution and spatial and temporal correlation. The proposedspatial Markov model is characterized by the initial velocity distribution,which is determined by the particle injection mode, the stationary Lagrangianvelocity distribution, which is derived from the Eulerian velocitydistribution, and the spatial velocity correlation length, which is related tothe characteristic fracture length. This effective model leads to a time-domainrandom walk for the evolution of particle positions and velocities, whose jointdistribution follows a Boltzmann equation. Finally, we demonstrate that theproposed model can successfully predict anomalous transport through discretefracture networks with different levels of heterogeneity and arbitrary tracerinjection modes.

Anomalous transport in disordered fracture networks: spatial Markov model for dispersion with variable injection modes