**Spatial Markov Models of Anomalous Transport in Fracture Networks**

**Peter K. Kang¹, Marco Dentz², Tanguy Le Borgne³, and Ruben Juanes¹**

*¹Massachusetts Institute of Technology, Cambridge, Massachusetts, USA
²Spanish National Research Council (IDAEA-CSIC), Barcelona, Spain
³Université́ de Rennes 1, CNRS, Rennes, France
*

Anomalous transport, understood as the nonlinear scaling with time of the mean square displacement of transported particles, is observed in many physical processes, including contaminant transport through porous and fractured geologic media [1], animal foraging patterns [2], freely diffusing molecules in tissue [3], tracer diffusion in suspensions of swimming microorganisms [4], and biased transport in complex networks [5].

Anomalous transport often leads to a broad-ranged particle distribution density, both in space and time [6–8]. Understanding the origin of the slow-decaying tails in the probability density is essential, because they determine the likelihood of high-impact, “low-probability” events and therefore exert dominant control over the predictability of a system [9]. This becomes especially important when human health is at risk, such as in epidemic spreading through transportation systems [10] or radionuclide transport in the subsurface [11].

Past studies have shown that a high variability in the flow properties leads to anomalous transport [1,7]. Depending on the nature of the underlying disorder, distribution anomalous behavior can be transient or persist to asymptotic scales [12,13]. The continuous time random-walk (CTRW) formalism [14,15] offers an attractive framework to understand and model anomalous transport through disordered media and networks [1,5,16]. The CTRW model is intrinsically an annealed model because the disorder configuration changes at each random-walk step. A particle that returns to the same position experiences different velocity properties. The validity of the CTRW approach for average transport in quenched random environments has been studied for purely diffusive transport (e.g., [7]) and biased diffusion (e.g., [9,17–19]). Most studies that employ the CTRW approach assume that transition times associated with particle displacements are independent of each other, therefore neglecting velocity correlation between successive jumps [20]. Indeed, a recent study of transport on a lattice network has shown that CTRW with independent transition times emerges as an exact macroscopic transport model when velocities are uncorrelated [9].

However, a detailed analysis of particle transport simulations demonstrates conclusively that particle velocities in mass-conservative flow fields exhibit correlation along their spatial trajectory [17,21,22]. Mass conservation induces correlation in the Eulerian velocity field because fluxes must satisfy the divergence-free constraint at each intersection. This, in turn, induces a correlation in the velocity sequence along a particle trajectory. To take into account velocity correlation, Lagrangian models based on temporal [22,23] and spatial [17,21] Markovian processes have recently been proposed. These models successfully capture many important aspects of the Lagrangian velocity statistics and the particle transport behavior. In particular, the study of Le Borgne et al. [17] shows that introducing correlation in the Lagrangian velocity through a Markov process in space yields an accurate representation of the first and second moments of the particle density. The model is restricted, however, to particle trajectories projected onto the direction of the mean flow, and the study leaves open the question of whether spatial Markov processes can describe multidimensional features of transport.

Here, we investigate average transport in divergence-free flow through a quenched random lattice from the CTRW point of view. We show that the divergence-free condition arising from mass conservation is the source of strong and nontrivial correlation in the Lagrangian velocity, even when the underlying conductivity field is completely uncorrelated. Accounting for such correlation in the velocity is important to obtain quantitative agreement for the mean particle density and the first passage time (FPT) distribution. We propose and validate a spatial Markov model of transport on a lattice network that explicitly captures the multidimensional effects associated with changes in direction along the particle trajectory. This study opens the door to understanding the interplay between two sources of velocity correlation: the divergence-free condition and the spatial correlation in the permeability field. We suspect that correlation in the Lagrangian velocity exerts an even more dominant control over mixing (understood as the decay of the variance of the particle density [24–26]) than it does on spreading. This remains an exciting open question.

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*AAPG Search and Discovery Article #120034©2012 AAPG Hedberg Conference Fundamental Controls on Flow in Carbonates, Saint-Cyr Sur Mer, Provence, France, July 8-13, 2012*