Origins of anomalous transport in heterogeneous media: Structural and dynamic controls

Abstract

Anomalous (or ''non-Fickian'') transport is ubiquitous in the context of tracer migration in geological formations. We quantitatively identify the origin of anomalous transport in a representative model of a heterogeneous porous medium under uniform (in the mean) flow conditions; we focus on anomalous transport which arises in the complex flow patterns of lognormally distributed hydraulic conductivity (K) fields, with several decades of K values. Transport in the domains is determined by a particle tracking technique and characterized by breakthrough curves (BTCs). The BTC averaged over multiple realizations demonstrates anomalous transport in all cases, which is accounted for entirely by a power law distribution $t 212b of local transition times. The latter is contained in the probability density function w(t) of transition times, embedded in the framework of a continuous time random walk (CTRW). A unique feature of our analysis is the derivation of w(t) as a function of parameters quantifying the heterogeneity of the domain. In this context, we first establish the dominance of preferential pathways across each domain, and characterize the statistics of these pathways by forming a particle-visitation weighted histogram, H w ðKÞ, of the hydraulic conductivity. By converting the ln(K) dependence of H w ðKÞ into time, we demonstrate the equivalence of H w ðKÞ and w(t), and delineate the region of H w ðKÞ that forms the power law of w(t). This thus defines the origin of anomalous transport. Analysis of the preferential pathways clearly demonstrates the limitations of critical path analysis and percolation theory as a basis for determining the origin of anomalous transport. Furthermore, we derive an expression defining the power law exponent b in terms of the H w ðKÞ parameters. The equivalence between H w ðKÞ and w(t) is a remarkable result, particularly given the nature of the K heterogeneity, the complexity of the flow field within each realization, and the statistics of the particle transitions.

Adaptive POD model reduction for solute transport in heterogeneous porous media

“…Figure 5 depicts the temporal evolution of the relative error (21), suggesting that, while the overall quality of the reduced model solution deteriorates with time, it does so to a lesser extent than the corresponding solution associated with the homogeneous set-up (see also Figure 2). This result might be related to the observation that the solute plume tends to follow the high conductivity paths in the system [17] and the effect of these paths close to the source tend to be influential to the plume behavior over time [5]. As such, capturing these features by the POD techniques enables the reduced solution to mimic the full model for longer time than in the homogeneous setting, where no preferential conductivity path is present.…”

Mechanisms of anomalous dispersion in flow through heterogeneous porous media

“…Edery et al [26] proposed to fit a power-law to the low-k end of p k (k) corresponding to the time regime, for which a prediction is desired. The resulting power-law approximation may then be used to make an approximation on the tailing behavior of the first-passage time distribution.…”

“…The spatial jumps and waiting times may be independent or correlated random variables [23]. The transition times in the CTRW are given in terms of the particle velocities v n , whose statistics have typically been estimated by using particle tracking simulations in the detailed heterogeneous flow [23][24][25][26]. The multirate mass transfer (MRMT) framework models the interplay of fast channels and slow advection by a mobile-immobile approach.…”

“…In fact, it has been shown [30][31][32] that both models are under certain conditions mathematically equivalent. For both modeling approaches, the relation between the (statistical) medium and flow properties and large scale transport is of central importance [2,26,[33][34][35]. Oftentimes, the distribution of transition times (CTRW) and the memory function (MRMT) are estimated on the basis of coupled flow and transport simulations or from experimental data, for example breakthrough curves.…”

Mathematical Modeling of Non-Fickian Diffusional Mass Exchange of Radioactive Contaminants in Geological Disposal Formations

“…Numerous studies indicate that the real nature of solute transport in geological formations exhibits anomalous behavior [1][2][3][4]. Multiscale subsurface systems often produce power-law tails in breakthrough curves [5][6][7][8], as well as in a nuclear waste repository site [9].…”