Record-breaking statistics for random walks in the presence of measurement error and noise

We address the question of distance record-setting by a random walker in the presence of measurement error, $\delta$, and additive noise, $\gamma$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $\sim n^{1/2}$ for large $n$ for all jump distributions, including L\'evy flights, and for all $\delta$ and $\gamma$. In contrast to the universal growth exponent of 1/2, the pace of record setting, measured by the pre-factor of $n^{1/2}$, depends on $\delta$ and $\gamma$. In the absence of noise ($\gamma=0$), the pre-factor $S(\delta)$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $\delta$ whereas, in case of perfect measurement $(\delta=0)$, the corresponding pre-factor $T(\gamma)$ increases with $\gamma$. Our analytical results are supported by extensive numerical simulations and qualitatively similar results are found in two and three dimensions

Record setting during dispersive transport in porous media

How often does a contaminant ‘particle’ migrating in a porous medium set a distance record, i.e., advance farther from the origin than at all previous time steps? This question is of fundamental importance in characterizing the nature of the leading edge of a contaminant plume as it is transported through an aquifer. It was proven theoretically by Majumdar and Ziff (2008) that, in the 1d case for pure diffusion, record setting of a random walker scales with n1/2, where n is the number of steps, regardless of the length and time distribution of steps. Here, we use numerical simulations, benchmarked against the 1d analytical solution, to extend this result also for pure diffusion in 2d and 3d domains. We then consider transport in the presence of a drift (i.e., advective‐dispersive transport), and show that the record‐setting pace of random walkers changes abruptly from ∝ n1/2 to ∝ n1. We explore the dependence of the prefactor on the distribution of step length and number of spatial dimensions. The key implication is that when, after a brief transitional period, the scaling regime commences, the maximum distance reached by the leading edge of a migrating contaminant plume scales linearly with n, regardless of the drift magnitude

Preferential pathways for fluid and solutes in heterogeneous groundwater systems: Self-organization, entropy, work

Patterns of distinct preferential pathways for fluid flow and solute transport are ubiquitous in heterogeneous, saturated and partially saturated porous media. Yet, the underlying reasons for their emergence, and their characterization and quantification, remain enigmatic. Here we analyze simulations of steady-state fluid flow and solute transport in two-dimensional, heterogeneous saturated porous media with a relatively short correlation length. We demonstrate that the downstream concentration of solutes in preferential pathways implies a downstream declining entropy in the transverse distribution of solute transport pathways. This reflects the associated formation and downstream steepening of a concentration gradient transversal to the main flow direction. With an increasing variance of the hydraulic conductivity field, stronger transversal concentration gradients emerge, which is reflected in an even smaller entropy of the transversal distribution of transport pathways. By defining “self-organization” through a reduction in entropy (compared to its maximum), our findings suggest that a higher variance and thus randomness of the hydraulic conductivity coincides with stronger macroscale self-organization of transport pathways. Simulations at lower driving head differences revealed an even stronger self-organization with increasing variance. While these findings appear at first sight striking, they can be explained by recognizing that emergence of spatial self-organization requires, in light of the second law of thermodynamics, that work be performed to establish transversal concentration gradients. The emergence of steeper concentration gradients requires that even more work be performed, with an even higher energy input into an open system. Consistently, we find that the energy input necessary to sustain steady-state fluid flow and tracer transport grows with the variance of the hydraulic conductivity field as well. Solute particles prefer to move through pathways of very high power in the transversal flow component, and these pathways emerge in the vicinity of bottlenecks of low hydraulic conductivity. This is because power depends on the squared spatial head gradient, which is in these simulations largest in regions of low hydraulic conductivity

Investigating the Permeability Evolution of Artificial Rock During Ductile and Brittle Deformation Under Pressurized Flow

The drilling of geothermal energy, CO2 sequestration, and wastewater injection all involve the pressurized flow of fluids through porous rock, which can cause deformation and fracture of the material. Despite the widespread use of these industrial methods, there is a lack of experimental data on the connection between the pore pressure rise, the deformation and permeability changes in real rock. In order to address this gap in the literature, this study developed an artificial rock material that can be deformed and fractured at low pressures. By controlling the porosity, permeability, and strength of the material during the sintering process, it is possible to mimic various types of rock. The artificial rock was designed to accommodate radial flow and deformation, allowing for the tracking of deformation by monitoring the flux and driving pressure and thus calculating the permeability changes under various pressure conditions. The study was able to examine the impact of both ductile and brittle deformation on the permeability during pressurized flow, which were captured by two models that were adjusted to this scenario. This study provides a link between pressurized flow, rock formation permeability and ductile to brittle deformation, that can constrain risk assessment to geothermal energy and CO

Multimodel framework for characterization of transport in porous media

We consider modeling approaches to characterize solute transport in porous media, integrating them into a unique theoretical and experimental framework for model evaluation and data interpretation. To date, development of (conservative and reactive chemical) transport models and formulation of model calibration methods grounded on sensitivity-based collection of measurements have been pursued in parallel. Key questions that remain include: For a given set of measurements, which conceptual picture of the transport processes, as embodied in a mathematical model or models, is most appropriate? What are the most valuable space-time locations for solute concentration measurements, depending on the model selected? How is model parameter uncertainty propagated to model output, and how does this propagation affect model calibration? We address these questions by merging parallel streams of research – model formulation, reduction, calibration, sensitivity analysis, and discrimination – offering our view on an emerging framework that guides (i) selection of an appropriate number and location of time-dependent concentration measurements given a transport model; and (ii) assessment (through discrimination criteria) of the relative benefit of applying any particular model from a set of several models. Our strategy is to employ metrics to quantify the relative contribution of each uncertain model parameter to the variability of the model output. We evaluate these metrics through construction of a surrogate (or "meta") transport model that has the additional benefit of enabling sensitivity analysis and model calibration at a highly reduced computational cost. We demonstrate the applicability of this framework, focusing on transport of reactive chemicals in laboratory-scale porous media

Characterization of Bimolecular Reactive Transport in Heterogeneous Porous Media

We characterize the role of preferential pathways in controlling the dynamics of bimolecular reactive transport in a representative model of a heterogeneous porous medium. We examine a suite of numerical simulations that quantifies the irreversible bimolecular reaction (Formula presented.), in a two-dimensional heterogeneous domain (with log-conductivity, Y), wherein solute A is injected along an inlet boundary to displace the resident solute B under uniform (in the mean) flow conditions. We explore the feedback between the reactive process and (a) the degree of system heterogeneity, as quantified by the unconditional variance of Y, (Formula presented.), representing moderately to strongly heterogeneous media, and (b) the relative strengths of advective and diffusive mechanisms, as quantified by a grid Péclet number, (Formula presented.). Our analysis is based on the identification of particle preferential pathways, focusing on particle residence time within cells employed to discretize the flow domain. These preferential pathways are formed mainly by high conductivity cells and generally contain an important component of (sometimes isolated and a relatively small number of) lower conductivity values. A key finding of our analysis is that while the former dominate the behavior, the latter are shown to provide a non-negligible contribution to the global number of reactions taking place in the domain for strongly heterogeneous media, i.e., for the largest investigated values of (Formula presented.). Reactions are detected across the complete simulation time window (of about 5.5 pore volumes) for the strongly advective case. When diffusion plays an important role, the reactive process essentially stops after the injection of a limited amount ((Formula presented.)2.5) of pore volumes