Search

Dissolution and precipitation processes in reactive transport in porous media are ubiquitous in a multitude of contexts within the field of Earth sciences. In particular, the dynamic coupling between the reactive precipitation / dissolution processes and the solute transport is of interest, as it is capable of giving rise to the emergence of preferential flow paths in the porous host matrix. This coupling is critical to a variety of Earth science scenarios, although the approaches to its characterization remain disputed. It has been shown that the emergence of preferential flow paths can be considered a manifestation of transport self-organization in porous media, as they create spatial gradients that distance the system from the state of perfect mixing and allow for a faster and more efficient fluid transport through the host matrix. To investigate the dynamic feedback between the transport and the reactive process in the field and its influence on the emergence of transport self-organization, we consider a two-dimensional Darcy-scale formulation of a reactive transport setup, where the precipitation and dissolution of the host matrix are driven by the injection of an acid compound, establishing local equilibrium with the resident fluid and an initially homogeneous porous matrix, composed of a calcite mineral. The coupled reactive process is simulated in a series of computational analyses employing the Lagrangian particle tracking (LPT) approach, capable of capturing the subtleties of the multiple scale heterogeneity phenomena. As the reactive process evolves, the dissolution / precipitation reactions are allowed to affect the local values of porosity and hydraulic conductivity in the host matrix, thus creating a dynamic feedback in the system. Subsequently, we employ computational non-reactive tracer tests to obtain the solute concentration data in the field, used to calculate the Shannon entropy of the transport. We employ the Shannon entropy to quantify the emergence of self-organization in the field, which we define as a relative reduction in entropy, compared to its maximum value. Our findings show that transport self-organization in an initially homogeneous field increases with time, along with the emergence of the field heterogeneity due to the interplay between transport and reactive process. By studying the influence of the transport Peclet number on the reactive process, we arrive at a conclusion that self-organization is more pronounced in diffusion-dominated flows, characterized by small Peclet values. The explanation for this lies in the fact that in a completely homogeneous field, the dominant mechanism to drive reactive components out of equilibrium is the stochasticity of diffusion. The self-organization of the breakthrough curve exhibits the opposite tendencies, that are explained from the thermodynamic perspective. The hydraulic power, required to maintain the driving head pressure difference between inlet and outlet, increases with the increasing variance of the hydraulic conductivity in the field due to the evolution of the reactive process in the field. This increase in power, supplied to the fluid traveling in the porous medium, results in an increase in the level of transport self-organization in the medium.

Our study investigates interplays between dissolution, precipitation, and transport processes taking place across randomly heterogeneous conductivity domains and the ensuing spatial distribution of preferential pathways. We do so by relying on a collection of computational analyses of reactive transport performed in two-dimensional systems where the (natural) logarithm of conductivity is characterized by various degrees of spatial heterogeneity. Our results document that precipitation and dissolution jointly take place in the system, with the latter mainly occurring along preferential flow paths associated with the conductivity field and the former being observed at locations close to and clearly separated from these. High conductivity values associated with the preferential flow paths tend to further increase in time, giving rise to a self-sustained feedback between transport and reaction processes. The clear separation between regions where dissolution or precipitation takes place is imprinted onto the sample distributions of conductivity which tend to become visibly left skewed with time (with the appearance of a bimodal behavior at some times). The link between conductivity changes and reaction-driven processes promotes the emergence of non-Fickian effective transport features. The latter can be captured through a continuous-time random-walk model where solute travel times are approximated with a truncated power law probability distribution. The parameters of such a model shift towards values associated with increasingly high non-Fickian effective transport behavior as time progresses.

Conventional concepts for transport in porous media assume that the heterogeneous distribution of hydraulic conductivities is the source for the contaminant temporal and spatial heavy tail. This tailing, known as anomalous or non-Fickian transport, can be captured by the β parameter in the continuous-time random walk framework. This study shows that with the increase in spatial correlation length between these heterogeneous distributions of hydraulic conductivities, the transport’s anomaly reduces; yet, the β value is unchanged, suggesting a topological component of the conductivity field, captured by the β. This finding is verified by an analysis of the solute transport, showing that the changing conductivity values have a moderate effect on the transport shape.

The process of a fluid replacing a separate miscible fluid in a porous medium is present in many industrial and natural systems, such as enhanced oil recovery, CO2 sequestration and salt-fresh water interfaces in the ground. While the replacement can be approximated with the Darcy law, the mechanisms of the miscible phases mixing to the displacement remain unclear, specifically as the heterogeneity of the domain increases. As this mixing influences the reaction pattern between the fluids, it is important to estimate it using indirect measurements that are available, such as pressure and flux measurements. We propose a set of experiments that allow us to observe and measure the displacement and mixing process in high resolution and with the use of image analysis we can distinguish between the mechanisms. We can clearly see how the heterogeneous rate of the pore structure influences the mixing pattern, rate, and duration. Surprisingly, we found a clear and typical “mark” of the mechanisms on the flow rate, under constant pressure, which allows us to relate the heterogeneity level of the structure to the ratio of displacement to mixing.

Fluid injections into the underground occurs in many industrial processes as hydraulic fracturing for oil and gas recovery, wastewater disposal, enhanced geothermal energy systems (EGS) and Carbon storage technologies. Often, the increase in pore pressure due to the fluid injections lead to the activation of a preexisting underground shear fractures (named faults), forming unanticipated local earthquakes.While studying the mechanism of injection induced earthquakes, the rock deformation due to the fluid injection is unknown. Understanding the rock deformation coupling with the pressure change, requires detailed experiments linking the global and local deformation with the pressure change during flow, which ultimately influence the earthquake triggering.In this study we present a novel experiment on transparent plastic rocks, that offers a detailed analysis of the artificial rocks’ deformation due to pressurized flow. In these experiments, we inject a fluid through the artificial rocks and analyze the internal deformation by capturing the displacement of fluorescent microspheres embedded in the artificial rock structure. Our analysis allows a straight-forward correlation between the deformation of rocks, the pressure change and the fluid flow. The study points a similarity between the material deformation due to internal pressure induced by the fluid injection and material deformation due to an external pulling.

In recent years, Darcy scale transport in porous media was characterized to be Fickian or non-Fickian due to the homogeneity or heterogeneity of the porous medium conductivity layout. Yet, evidence shows that preferential flows that funnel the transport occur in heterogeneous and homogenous cases. We model the Darcy scale transport using a 2D conductivity field ranging from homogenous to heterogeneous and find that these preferential flows bifurcate, leaving voids where particles do not invade while forming a tortuous path. The fraction of bifurcations decreases downflow and reaches an asymptotical value, which scales as a power-law with the heterogeneity level. We show that the same power-law scaling of the bifurcations to heterogeneity level appears for the void fraction, tortuosity, and fractal dimension analysis with the same heterogeneity level. We conclude that the scaling with the heterogeneity is the dominant feature in the preferential flow geometry, which will lead to variations in weighting times for the transport and eventually to anomalous transport.

This study proposes an alternative framework to quantify and explain the enigmatic emergence of preferential flow and transport in heterogeneous saturated porous media, using concepts from thermodynamics and information theory. We examined simulations of two-dimensional fluid flow and solute transport based on the methods of Edery et al. (2014) at total head differences of 100 and 10 characterized the discrete probability distribution of solute particles to cross a distinct transversal position in a plane normal to the direction of the mean flow by means of the Shannon entropy. In general, we found a declining entropy with increasing downstream transport distance, reflecting a growing downstream self-organization due to the increasing concentration of particles in preferential flow paths. Strikingly, preferential flow patterns with lower entropies emerged when analyzing simulations in media with larger variances in hydraulic conductivity, and this enhanced self-organization appeared even stronger for simulations at lower head differences. This implies that macro-states of higher order are established, despite the higher randomness of ln(K) for a range of Peclet numbers representing strong and intermediate importance of advective transport. The key to explain this almost paradoxical behavior is the finding that power in the vertical flow components grows with the variance of the hydraulic conductivity field. Due to this larger energy input, the vertical/transversal flow component may perform more work to increase the order in the flow path distribution, through steepening transversal concentration gradients as reflected in lower entropies. The emergence of spatially organized preferential transport and the related decline in flow path entropy essentially requires that the entropy is exported from the system. Consistently, we found that a declining entropy in lateral distribution of flow paths goes along with a rising entropy in the associated breakthrough curve. This space-time asymmetry in entropy implies that perfect organization and certainty about both flow paths and travel times can never simultaneously occur. This required consummation of entropy and thus violation of the second law of thermodynamics. We thus propose that the combined use of free energy and entropy holds the key to characterize, quantify and predict the self-organized emergence of preferential flow phenomena and to explain the underlying cause of their emergence.References:Edery, Y., Guadagnini, A., Scher, H., and Berkowitz, B.: Origins of anomalous transport in disordered media: Structural and dynamic controls, Water Resour. Res., 50, 1490-1505, doi:10.1002/2013WR015111, 2014.

Traditional concepts for flow in porous media assume that the heterogenous distribution of hydraulic conductivity is the source for the contaminant temporal and spatial heavy tail, a process known as anomalous or non-Fickian transport; this anomalous transport behavior can be captured by the β parameter in the continues time random walk (CTRW) framework. In previous studies we showed that there is a functional form relating the β parameter to the permeability variance1 and fracture alignment in fracture fields2. Moreover, we showed that this variance is strongly influencing the reaction pattern during transport3. This study shows that as the spatial correlation length, between these heterogenous distribution of hydraulic conductivities, increase, the anomaly of the flow reduces, yet the β value is unchanged suggesting that there is a topological component to the flow field, captured by the β4. This finding is verified by an analysis on the flow field, showing that the changes in the conductivity values have little effect on the flow field morphology, which points to the topological component in the flow.Y. Edery, A. Guadagnini, H. Scher and B. Berkowitz, Water Resources Research 50 (2), 1490-1505 (2014). Y. Edery, S. Geiger and B. Berkowitz, Water Resources Research 52 (7), 5634-5643 (2016). Y. Edery, G. M. Porta, A. Guadagnini, H. Scher and B. Berkowitz, Transport in Porous Media 115 (2), 291-310 (2016). Y. Edery, arXiv preprint arXiv:1906.07061 (2019).

Anomalous transport is ubiquitous in a wide range of disordered systems, notably in fractured porous formations. We quantitatively identify the structural controls on anomalous tracer transport in a model of a real fractured geological formation that was mapped in an outcrop. The transport, determined by a continuum scale mathematical model, is characterized by breakthrough curves (BTCs) that document anomalous (or “non-Fickian”) transport, which is accounted for by a power-law distribution of local transition times ψ(t) within the framework of a continuous time random walk (CTRW). We show that the determination of ψ(t) is related to fractures aligned approximately with the macroscopic direction of flow. We establish the dominant role of fracture alignment, and assess the statistics of these fractures by determining a concentration-visitation weighted residence time histogram. We then convert the histogram to a probability density function (pdf) that coincides with the CTRW ψ(t) and hence anomalous transport. We show that the permeability of the geological formation hosting the fracture network has a limited effect on the anomalous nature of the transport; rather, it is the fractures transverse to the flow direction that play the major role in forming the long BTC tail associated with anomalous transport. This is a remarkable result, given the complexity of the flow field statistics as captured by concentration transitions. This article is protected by copyright. All rights reserved.

We present quasi-3D visualization and analysis of engineered nanoparticle (ENP) transport behavior in an experimental setup that uses a transmitted light imaging technique. A flow cell was packed with specially adapted, water-transparent, spherical polyacrylamide beads, which carry a negative surface charge representative of many natural environments. Ubiquitous, oppositely-charged ENPs - Au and Ag NPs - were synthesized and introduced into a flow cell subjected to a macroscopically uniform flow field via point source pulse injection, at three different flow rates. The negatively-charged ENPs behaved like a conservative tracer, in terms of spatio-temporal plume evolution. The positive AgNPs, however, displayed a decrease in their initially strong tendency to attach to the oppositely-charged porous medium. As a result, immobilization of the positive AgNPs was spatially and temporally limited to the vicinity of the point of injection; beyond this region, the AgNPs were mobile and effluent contained AgNPs with hydrodynamic diameters significantly larger than those of the injected AgNPs. This behavior is understood by dynamic light scattering and ζ potential measurements, which showed aggregation processes and inversion in particle surface charge to occur during transport of the positive ENPs. These findings have broad implications for ENP mobility and reactivity in the environment.

Nickel migration measured in laboratory-scale, natural soil column experiments is shown to display anomalous (non-Fickian) transport, nonequilibrium adsorption and desorption patterns, and precipitation/dissolution. Similar experiments using a conservative tracer also exhibit anomalous behavior. The occurrence of ion exchange of nickel, mainly with calcium (but also with other soil components), is measured in both batch and flow-through column experiments; adsorption and desorption isotherms demonstrate hysteresis. Strong retention of nickel during transport in soil columns leads to delayed initial breakthrough (∼40 pore volumes), slow increase in concentration, and extended concentration tailing at long times. We describe the mechanisms of transport and retention in terms of a continuous time random walk (CTRW) model, and use a particle tracking formulation to simulate nickel migration in the column. This approach allows us to capture the non-Fickian transport and the subtle local effects of adsorption/desorption and precipitation/dissolution. Consideration also of preferential pathways accounts for the evolution of the measured breakthrough curve and measured spatial concentration profiles. The model uses non-Fickian transport parameters estimated from the conservative tracer and, as a starting point, adsorption/desorption parameters based on batch experiments and a precipitation parameter based on Ksp values. The batch parameters are found to underestimate the actual amount of adsorption. We suggest that the sorption and precipitation/dissolution dynamics, and resulting breakthrough curves, are influenced strongly by preferential pathways; such pathways significantly alter the availability of sorption sites and ion availability for precipitation. Analysis of these results provides further understanding of the interaction and dynamics among transport, precipitation, and sorption mechanisms in natural soil.

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.

[1] We apply a general strategy based on Global Sensitivity Analysis (GSA) and model discrimination criteria to (a) calibrate the parameters embedded in competing models employed to interpret laboratory-scale tracer experiments, (b) rank these models, and (c) estimate the relative degree of likelihood of each model through a posterior probability weight. We consider a conservative transport experiment in a uniform porous medium. We apply GSA to three transport models, based on: the classical advection-dispersion equation (ADE), a dual-porosity (DP) formulation with mass transfer between mobile and immobile regions, and the Continuous Time Random Walk (CTRW) approach. GSA is performed through Polynomial Chaos Expansion of the governing equations, treating key model parameters as independent random variables. We show how this approach allows identification of (a) the relative importance of model-dependent parameters, and (b) the space-time locations, where the models are most sensitive to these parameters. GSA is then employed to assist parameter estimates within a Maximum Likelihood framework. Finally, formal model identification criteria are employed to (a) rank the alternative models, and (b) associate each model with a posterior probability weight for the specific case study. The GSA-based calibration of each model returns an acceptable approximation (remarkably accurate in the case of the CTRW model) of all available concentration data, with calibration being performed using minimum sets of observations corresponding to the most sensitive (space-time) locations.

We review the analysis of the dynamics of reactive transport in disordered media, emphasizing the nature of the chemical reactions and the role of small-scale fluctuations induced by the structure of the porous medium. We are motivated by results and interpretations of laboratory-scale experiments, for which detailed characterization of the system is possible. Modeling approaches based on continuum and particle tracking (PT) schemes are examined critically, highlighting how fluctuations are incorporated. The continuum approach spans a large literature. Traditional formats of reactive transport equations, such as the advection–dispersion–reaction equation (ADRE), are based on a series of assumptions related mainly to scale separation and relative magnitude of time scales involved in the reactive transport setting. These assumptions as well as further developments are assessed in depth. PT methods offer an alternative means of accounting for pore-scale dynamics, wherein space–time transitions are drawn from appropriate probability distributions that have been tested to account for anomalous transport. While PT methods have been employed for many years to describe conservative transport, their application to laboratory-scale reactive transport problems in the context of both Fickian and non-Fickian regimes is relatively recent. We concentrate on experimental observations of different types of reactions in disordered media: (1) the dynamics of a bimolecular reactive transport (A + B → C) in passive (non-reactive) media, and (2) a multi-step chemical reaction, as exemplified in the process of dedolomitization involving both dissolution and precipitation. The fluctuations in a number of the key variables controlling the processes prove to have a dominant role; elucidation of this role forms the basis of the present study and the comparison of methods.

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−1−β of local transition times. The latter is contained in the probability density function ψ(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 ψ(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, Hw(K), of the hydraulic conductivity. By converting the ln(K) dependence of Hw(K) into time, we demonstrate the equivalence of Hw(K) and ψ(t), and delineate the region of Hw(K) that forms the power law of ψ(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 β in terms of the Hw(K) parameters. The equivalence between Hw(K) and ψ(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.

Mixing zone dynamics of a reaction product $C$ during diffusion of two species ($A$ and $B$) are examined, using a two-dimensional particle tracking model (PT) for the reaction $A+B\ensuremath{\rightarrow}C$, allowing for both Fickian and non-Fickian transitions. The form of PT we use is equivalent to a continuous time random walk, which is a widely used model for anomalous transport and diffusion. It is shown that the basic patterns of the $C$ dynamics---the temporal evolution of the spatial profile and the temporal $C$ production---are similar for both modes of diffusion. However, the distinctive time scale for the non-Fickian case is very much larger even when the median transition steps are matched with the Fickian case. For immobile $C$, the spatial profile pattern is a broadening (Gaussian) reaction front evolving to a concentration-fluctuation dominated (Lorentzian) shape. The temporal $C$ production is fit well by a stretched exponential for both diffusion types. In analyzing experiments, the appearance of a Gaussian $C$ profile does not prove that the diffusion process is Fickian.

We examine distance record setting by a random walker in the presence of a measurement error $\ensuremath{\delta}$ and additive noise $\ensuremath{\gamma}$ and show that the mean number of (upper) records up to $n$ steps still grows universally as $⟨{R}_{n}⟩\ensuremath{\sim}{n}^{1/2}$ for large $n$ for all jump densities, including L\'evy distributions, and for all $\ensuremath{\delta}$ and $\ensuremath{\gamma}$. In contrast, the pace of record setting, measured by the amplitude of the ${n}^{1/2}$ growth, depends on $\ensuremath{\delta}$ and $\ensuremath{\gamma}$. In the absence of noise ($\ensuremath{\gamma}=0$), the amplitude $S(\ensuremath{\delta})$ is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing $\ensuremath{\delta}$ whereas, in the case of perfect measurement ($\ensuremath{\delta}=0$), the corresponding amplitude $T(\ensuremath{\gamma})$ increases with $\ensuremath{\gamma}$. The exact results for $S(\ensuremath{\delta})$ offer a new perspective for characterizing instrumental precision by means of record counting. Our analytical results are supported by extensive numerical simulations.

[1] We simulate the processes of dedolomitization and calcium carbonate precipitation using particle tracking. The study is stimulated by the results of a laboratory experiment that examined reactive transport of injected CaCl2/HCl, into a column of sucrosic dolomite particles, with a constant flow field. The injected fluid supplies Ca2+ and H+. Dedolomitization is a protonation reaction yielding carbonic acid; a subsequent deprotonation reaction yields , and reaction with the abundant Ca2+ forms the precipitate CaCO3. The dedolomitization and precipitation processes involve multistep, multispecies chemical reactions, with both irreversible and reversible stages. The particle tracking is governed by spatial and temporal distributions within a continuous time random walk framework. This accounts for the effects of disorder of heterogeneous media (leading to non-Fickian transport) and includes the option of treating purely advective-dispersive (Fickian) transport. The dynamics of dedolomitization are examined for different flow conditions and reaction rates. The fluctuations in the local velocity distributions, due to porosity changes, create conditions for positive feedbacks leading to development of preferential pathways, large-scale nonlinearity, and precipitation banding. These features have been observed in the laboratory experiments and are now accounted for by the simulation results at similar time frames, velocities, and pH levels.

[1] We use a particle tracking approach to analyze the dynamics that control bimolecular reactive transport (A + B → C) in porous media. Particle transitions are governed by spatial and temporal distributions to account for the transport within a continuous time random walk framework. Particle tracking simulations are compared to measurements from a laboratory experiment of bimolecular reactive transport in a constant flow field. The simulations capture the experimental sequence of evolving C particle profiles using a marginally Fickian temporal distribution to quantify the particle transitions. The first profile is a fit with the model parameters, and subsequent ones are predictions. The rate of production of reaction product C over time is found to follow a power law. At early times after the injection of A particles into a uniform distribution of B particles, the strong contact and reaction between A and B particles induces the formation of a spatial void between the reactants. At longer times, the production of C is nearly constant and depends on the fluctuations of velocities of reactant particles that can surmount the void. We probe the behavioral dependence of the A, B, and C spatial profiles on the spectra of velocity fluctuations of the reactants. The latter are generated by different temporal distributions, namely, a decaying exponential distribution, which is equivalent to advective-dispersive (Fickian) transport, and the truncated power law with degrees of non-Fickian behavior, which is characteristic of transport in heterogeneous media. We demonstrate that the C profile exhibits subtle dynamics because of competition between the dispersion (spreading of the plumes) of A and B and the (power law) production rate.

[1] We quantitatively account for the measured concentration profile from a laboratory experiment of bimolecular (A + B → C) reactive transport in a porous medium with a particle tracking (PT) model. The PT results are in contrast to the analytical solution of the continuum scale advection-dispersion-reaction equation, which results in an excess quantity of reaction product (C). The approaches differ in the treatment of the mixing zone, the fluctuations due to the low reactant concentrations, and the localized nature of the reaction. The PT can accommodate a range of transport modes with different temporal distributions. An exponential temporal distribution is equivalent to Fickian transport, which we use for the comparison to the laboratory data; a truncated power-law (TPL) temporal distribution yields a non-Fickian transport characteristic of heterogeneous media. We study the influence of disorder on the mixing zone and the product concentration profiles via these contrasting transport modes.