Mobile-immobile Mass Transfer Research Articles

Bimolecular reactive transport in a filled single fracture-matrix system considering the nonequilibrium sorption

Comprehending the transport mechanism of bimolecular reactive solutes in fracture-matrix systems is crucial for improving the efficiency of collaborative remediation using hydraulic fracturing and in-situ chemical oxidation in low-permeability contaminated sites. Previous studies have neglected the effects of physical nonequilibrium (PNE) caused by sedimentation and chemical nonequilibrium (SNE) caused by rete-limited sorption. In this study, a two-dimensional numerical model that considers mobile-immobile mass transfer in the fracture and two-site nonequilibrium sorption in both fracture and matrix is established to investigate the behavior of bimolecular reactive solutes in a filled fracture-matrix system. The results indicate that the PNE and SNE lead to two-stage characteristics and severe tailing effects in contaminant remediation. High tailing occurs when the equilibrium sorption fraction and fracture-mobile/fracture-immobile porosity ratio are relatively small and the longitudinal hydrodynamic dispersion coefficient is relatively large, requiring consideration of multi-nonequilibrium processes. Additionally, the solute concentration in the matrix is most sensitive to the matrix porosity (θk), less sensitive to fracture aperture (2b), and reagent bulk diffusivity (DOA). The remediation area (Sr) in the matrix has an exponential relationship with θk and a linear relationship with 2b and DOA. The quantitative relationship between Sr and key parameters facilitates the prediction of remediation effectiveness.

Transient dispersion regimes in heterogeneous porous media: On the impact of spatial heterogeneity in permeability and exchange kinetics in mobile–immobile transport

Transport processes in the subsurface are coupled with the heterogeneity of the porous structure. Obtaining an accurate description of mass transport in such media at all time scales remains a crucial task, as the subsurface is strongly impacted by human activity. Spreading of pollutants in the transient but also in the asymptotic regime, as well as the time to reach a critical location (e.g. aquifer, well) significantly depend on the underlying heterogeneity of the permeability field. Moreover, in the case of solute retention, transport is also characterized by local exchange kinetics that depend on the local aquifer properties. Consequently, exchange (retention) times are expected to be spatially heterogeneous. In this work we focus on the influence of spatially heterogeneous permeability and exchange times on the transient and asymptotic transport regimes and provide a parametric study. To this goal, we simulate in a first part the transport in a two-dimensional heterogeneous medium under spatially varying permeability and mobile–immobile mass transfer parameters. Equations are solved using a Lattice-Boltzmann two-relaxation-time (TRT) algorithm. We assume the following relation between the local permeability K and the local exchange time: τ∝Kγ. Taking into account this relation, we investigate the impact of the Damköhler number (Da, ratio of the advection and exchange time scales), the disorder of the permeability field and the value of the exponent of the coupling function (γ) on the spatial evolution of the concentration field and breakthrough curve. We show that, depending on the parameters (Da, γ, etc.), we can observe transient, non-Fickian dispersion, which is characterized by non-exponential tails of the solute breakthrough curves and a non-linear evolution of the spatial variance of the solute distribution.In a second part, we present a new continuous time random walk (CTRW) model to upscale these transport behaviors. The model is based on a spatial Markov model for particle velocities that couples advective–dispersive transport and heterogeneous mass transfer through a compound Poisson process. The upscaled model can be fully parameterized by the statistics of permeability and the hydraulic gradient (with no fitting parameter), that is, in terms of medium and flow characteristics. The results of the CTRW model fully capture the non-Fickian transient transport regimes both for the breakthrough curves and spatial concentration variance. In the longtime limit, as expected from the central limit theorem, the CTRW model predicts normal, Fickian behavior. Finally we show, that the time to reach the asymptotic regimes in heterogeneous media (e.g. heterogeneous exchange times) is parameter dependent and in average is two orders of magnitude larger than for the respective homogeneous case. To summarize, the coupling between the heterogeneous permeability field and the local mass transfer properties can strongly influence transient and asymptotic transport regimes and potentially explain experimentally observed non-Gaussian behaviors.

Single region substitute models for dual-porosity advection–dispersion migration process based on contaminant distribution

Dual-porosity (DP) model is recommended for describing the complex non-Fickian contaminant transport in soil, while advection–dispersion-equation (ADE) model is the most popular transport model due to its simplicity and convenience. In this study, single region substitute approaches for the DP model were established based on the contaminant distribution in the mobile and immobile phases. A time dependent parameter Ω was used to determine the participation degree of the immobile phase in contaminant transport. The single region substitute (SRS) model was modified by combining the mobile phase retardation factor with the adsorbed concentration of immobile phase. Furthermore, dispersion-modified SRS (DSRS) model was established for calculating the equivalent dispersion coefficient of the coupled mobile-immobile mass transfer and advection via Fourier transformation. Correlation analyses of the modified models shown that the SRS model fits DP well when advection velocity or the mobile-immobile mass transfer rate is relatively slow under both Cauchy and Dirichlet inlet boundary, whereas that the DSRS model is applicable when mobile-immobile mass transfer rate is relatively higher. Both SRS and DSRS models can provide better understanding to the contaminant transport process of DP model.

Experimental Support for a Simplified Approach to CTRW Transport Models and Exploration of Parameter Interpretation

AbstractWe empirically test our earlier theoretical arguments about simplification of continuous‐time random walk (CTRW) solute transport models, namely that without loss of generality the velocity‐like term, vψ, may be set to mean groundwater velocity, the dispersion‐like term, Dψ, defined by a classical, velocity‐independent dispersivity, and the so‐called time constant, τ, to unity. We also argue that for small‐scale heterogeneous advection (HA) and mobile‐immobile mass transfer (MIMT) CTRW transition time distributions, ψ(t), are unaffected by mean flow velocity. To experimentally test these claims, we re‐analyze two bench‐scale transport experiments—one for HA, one for MIMT—each performed at multiple flow rates in otherwise identical conditions, and show it is possible to simultaneously explain all breakthrough curves in each, subject to the above constraints. We compare our calibrations with earlier efforts for the same data sets. In the HA calibration we identify a ψ(t) of the same functional form as previous authors, and which yielded breakthrough predictions essentially identical to theirs, but with greatly differing parameters. This illustrates how values of individual CTRW parameters may not map one‐to‐one onto underlying physics. We recommend reporting complete model descriptions, discuss how the simplified approach assists in this and other theoretical considerations.

Transport under advective trapping

Abstract

Characterization and upscaling of hydrodynamic transport in heterogeneous dual porosity media

We study the upscaling of pore-scale transport of passive solute in a carbonate rock sample. It is characterized by microporous regions displaying heterogeneous porosity distribution that are accessible due to diffusion only, and a strongly heterogeneous mobile pore space, characterized by a broad distribution of flow velocities. We observe breakthrough curves that are characterized by strong tailing, which can be attributed to velocity variability in the flowing medium portion, and solute retention in the microporous space. Using accurate numerical flow and transport simulations, we separate these two mechanisms by analyzing the statistics of residence times in the mobile phase, and the trapping and residence time statistics in the mmobile phase. We employ a continuous time random walk framework in order to upscale transport using a particle based implementation of mobile-immobile mass transfer, and heterogeneous advection. This approach is based on the statistics of the characteristic mobile and immobile residence times, and mass transfer rates between the two continua. While classical mobile-immobile approaches model mass transfer as a constant rate process, we find that the trapping rate increases with increasing mobile residence times until it reaches a constant asymptotic value. Based on these findings and the statistical characteristics of travel and retention times, we derive an upscaled Lagrangian transport model that separates the processes of heterogeneous advection and diffusion in the immobile microporous space, and provides accurate descriptions of the observed non-Fickian breakthrough curves. These results shed light on transport upscaling in highly complex dual-porosity rocks for which mobile-immobile mass transfer are controlled by a dual multirate process controlled by the heterogeneity of both the flow field in the connected porosity and the diffusion in the no-flow regions.

Solving Diffusion Time in Heterogeneous Microscale Rock Matrix by 3D Computations: Non-Fickian Dispersion Observed in Porous Media

Understanding and simulating the underlying microscopic physics of the rock matrix is very useful for determining macroscopic physical properties such as permeability. Matrix diffusion is an important transport parameter controlling the late-time behaviour of breakthrough curves (BTCs). We compute the memory function, implemented in the sink/source term of Mobile-immobile mass transfer by solving the matrix diffusion using a time diffusion random-walk approach. The diffusion is controlled by different parameters like the porosity, tortuosity, mobile-immobile interface and immobile domain cluster shapes. All these properties are investigated by X-ray microtomography that captures the main characteristics of matrix diffusion at three dimensions. We compare the memory function deduced from the field-scale tracer tests well with the computed memory function. Simulation results of the memory function appeared to be coherent with that measured from the tracer test for a large tortuosity value. Probably, the diffusion paths are longer, and they are controlled by the properties mentioned above. From a representative elementary volume of natural reservoirs studied here, we conclude that, microscale diffusion process in the immobile domain play a crucial role to better understand the non-Fickian dispersion measured from the tracer test.

Local Equilibrium and Retardation Revisited.

In modeling solute transport with mobile-immobile mass transfer (MIMT), it is common to use an advection-dispersion equation (ADE) with a retardation factor, or retarded ADE. This is commonly referred to as making the local equilibrium assumption (LEA). Assuming local equilibrium, Eulerian textbook treatments derive the retarded ADE, ostensibly exactly. However, other authors have presented rigorous mathematical derivations of the dispersive effect of MIMT, applicable even in the case of arbitrarily fast mass transfer. We resolve the apparent contradiction between these seemingly exact derivations by adopting a Lagrangian point of view. We show that local equilibrium constrains the expected time immobile, whereas the retarded ADE actually embeds a stronger, nonphysical, constraint: that all particles spend the same amount of every time increment immobile. Eulerian derivations of the retarded ADE thus silently commit the gambler's fallacy, leading them to ignore dispersion due to mass transfer that is correctly modeled by other approaches. We then present a particle tracking simulation illustrating how poor an approximation the retarded ADE may be, even when mobile and immobile plumes are continually near local equilibrium. We note that classic "LEA" (actually, retarded ADE validity) criteria test for insignificance of MIMT-driven dispersion relative to hydrodynamic dispersion, rather than for local equilibrium.

Transient modeling of non-Fickian transport and first-order reaction using continuous time random walk

Contaminants in groundwater may experience a broad spectrum of velocities and multiple rates of mass transfer between mobile and immobile zones during transport. These conditions may lead to non-Fickian plume evolution which is not well described by the advection–dispersion equation (ADE). Simultaneously, many groundwater contaminants are degraded by processes that may be modeled as first-order decay. It is now known that non-Fickian transport and reaction are intimately coupled, with reaction affecting the transport operator. However, closed-form solutions for these important scenarios have not been published for use in applications. In this paper, we present four new Green's function analytic solutions in the uncoupled, uncorrelated continuous time random walk (CTRW) framework for reactive non-Fickian transport, corresponding to the quartet of conservative tracer solutions presented by Kreft and Zuber (1978) for Fickian transport. These consider pulse injection for both resident and flux concentration combined with detection in both resident and flux concentration. A pair of solutions for resident concentration temporal pulses with detection in both flux and resident concentration is also presented. We also derive the relationship between flux and resident concentration for non-Fickian transport with first-order reaction for this CTRW formulation. An explicit discussion of employment of the new solutions to model transport with arbitrary upgradient boundary conditions as well as mobile–immobile mass transfer is then presented. Using the new solutions, we show that first-order reaction has no effect on the anomalous spatial spreading rate of concentration profiles, but produces breakthrough curves at fixed locations that appear to have been generated by Fickian transport. Under the assumption of a Pareto CTRW transition distribution, we present a variety of numerical simulations including results showing coherence of our analytic solutions and CTRW particle tracking.

Push‐pull tracer tests: Their information content and use for characterizing non‐Fickian, mobile‐immobile behavior

AbstractPath reversibility and radial symmetry are often assumed in push‐pull tracer test analysis. In reality, heterogeneous flow fields mean that both assumptions are idealizations. To understand their impact, we perform a parametric study which quantifies the scattering effects of ambient flow, local‐scale dispersion, and velocity field heterogeneity on push‐pull breakthrough curves and compares them to the effects of mobile‐immobile mass transfer (MIMT) processes including sorption and diffusion into secondary porosity. We identify specific circumstances in which MIMT overwhelmingly determines the breakthrough curve, which may then be considered uninformative about drift and local‐scale dispersion. Assuming path reversibility, we develop a continuous‐time‐random‐walk‐based interpretation framework which is flow‐field‐agnostic and well suited to quantifying MIMT. Adopting this perspective, we show that the radial flow assumption is often harmless: to the extent that solute paths are reversible, the breakthrough curve is uninformative about velocity field heterogeneity. Our interpretation method determines a mapping function (i.e., subordinator) from travel time in the absence of MIMT to travel time in its presence. A mathematical theory allowing this function to be directly “plugged into” an existing Laplace‐domain transport model to incorporate MIMT is presented and demonstrated. Algorithms implementing the calibration are presented and applied to interpretation of data from a push‐pull test performed in a heterogeneous environment. A successful four‐parameter fit is obtained, of comparable fidelity to one obtained using a million‐node 3‐D numerical model. Finally, we demonstrate analytically and numerically how push‐pull tests quantifying MIMT are sensitive to remobilization, but not immobilization, kinetics.

Non-Fickian Transport Under Heterogeneous Advection and Mobile-Immobile Mass Transfer

We study the combined impact of heterogeneous advection and mobile–immobile mass transfer on non-Fickian transport using the continuous-time random walk (CTRW) approach. The CTRW models solute transport in heterogeneous media as a random walk in space and time. Our study is based on a d-dimensional CTRW model that accounts for both heterogeneous advection and mass transfer between mobile and immobile zones, to which we also refer as solute trapping. The flow heterogeneity is mapped into the distribution of advective transition times over a characteristic heterogeneity scale. Mass transfer into immobile zones is quantified by a trapping rate and the distribution of particle return times. The total particle transition time over a characteristic heterogeneity scale then is given by the advective time and the sum of trapping times over the number of trapping events. We establish explicit integro-partial differential equations for the evolution of the concentration and discuss the relation to the multirate mass transfer approach, specifically the relation between the trapping time distribution and the memory function. We then analyze the signatures of anomalous transport due to advective heterogeneity and trapping in terms of spatial moments and first passage times or breakthrough curves. The behaviors for different disorder scenarios are analyzed analytically and through random walk particle tracking simulations. Assuming that advective mass transfer is faster than diffusive, we identify three regimes of distinct transport behaviors, which are separated by the characteristic trapping rate and trapping times. (1) At early times, we identify a pre-asymptotic time regime that is fully determined by advective heterogeneity and which is characterized by superlinear growth of longitudinal dispersion. (2) For longitudinal dispersion, we identify an intermediate regime of strong superlinear diffusion. This regime is determined by the combined effect of advective heterogeneity and trapping. (3) At larger time, the asymptotic trapping-driven regime shows the signatures of diffusion in immobile zones, which leads to both sub- and superlinear dispersion. These results shed some new light on the mechanism of non-Fickian transport and their manifestation in spatial and temporal solute distributions.

Continuous time random walks for non-local radial solute transport

This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile–immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection–dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer tests.

Modeling the effectiveness of U(VI) biomineralization in dual-porosity porous media

Recent attention has focused on the in situ immobilization of uranium by stimulation of dissimilatory metal-reducing bacteria (DMRB). The objective of this work was to investigate the effectiveness of this approach in heterogeneous and structured porous media, since such media may significantly affect the geochemical and microbial processes taking place in contaminated sites, impacting remediation efficiency during biostimulation. A biogeochemical reactive transport model was developed for uranium remediation by immobile-region-resident DMRB in two-region porous media. Simulations were used to investigate the parameter sensitivities of the system over wide ranging geochemical, microbial and groundwater transport conditions. The results suggest that optimal biomineralization is generally likely to occur when the regional mass transfer timescale is less than one thirtieth the value of the volumetric flux timescale, and/or the organic carbon fermentation timescale is less than one thirtieth the value of the advective timescale, and/or the mobile region porosity ranges between equal to and four times the immobile region porosity. Simulations including U(VI) surface complexation to Fe oxides additionally suggest that, while systems exhibiting U(VI) surface complexation may be successfully remediated, they are likely to display different degrees of remediation efficiency over varying microbial efficiency, mobile-immobile mass transfer, and porosity ratios. Such information may aid experimental and field designs, allowing for optimized remediation in dual-porosity (two-region) biostimulated DMRB U(VI) remediation schemes.

Modelling the Competitive Sorption Process of Multiple Solutes During their Transport in Porous Media

A general mathematical model to solve the advection–dispersion transport equation for multiple solutes was developed, where the dual porosity mobile–immobile mass transfer, the two-site non-equilibrium model and first-order transformation reactions were included. The two-site model was expressed with an equilibrium sorption term and a kinetic term. One of three kinetic models could be used: the non-linear, the bilinear and the pore diffusion model. The traditional Freundlich or Langmuir isotherms were employed to simulate no-interaction between the solutes, but with the extended Freundlich or extended Langmuir isotherms, a competitive sorption could be simulated. The transport equation was solved with the Moving Concentration Slope method. The mathematical model was tested and further simplified by using real data from soil column experiments, with 1,2-cis-dichloroethene and trichloroethene as model contaminants and silica gel and real soil samples as porous medium.

Colloid and Bromide Transport in Undisturbed Soil Columns: Application of Two‐Region Model

Bromide tracer breakthrough and natural soil colloid leaching curves for undisturbed soil columns were used to characterize dissolved and suspended matter transport at the field scale. Data from 33 columns of 20‐cm diameter and 20‐cm height were used. Columns were collected in a grid of 25 by 30 m at an agricultural field. A two‐region (mobile–immobile water phase, MIM) solute transport model was fitted to data. The model was used to estimate bromide and colloid transport parameters including mobile and immobile water contents (θm, θim), bromide and colloid advective velocities (vBr, vColl), and mobile–immobile mass transfer coefficients (αbr, αcoll). Both soil physical properties and transport parameters were highly variable across the sampling field. Comparison of bromide transport parameters with basic soil physical properties revealed that vBr was proportional to soil clay content and bulk density (ρb), but αBr was inversely proportional to these parameters. Colloid transport parameters, vColl and αColl, however, showed only a weak correlation with clay content and ρb. Also, vColl was typically three to four times higher than vBr. The colloid velocity was generally higher than the bromide velocity, implying size exclusion of colloids. The spatial distributions of soil physical properties, bromide and colloid transport parameters, and leached quantities of particles were compared. The results suggested that bromide and colloid mass transfer (diffusion) were not controlled by the same soil physical conditions, and that soil structure and macropore flow are more important than the quantity of dispersible colloids in controlling colloid leaching.

Longevity of granular iron in groundwater treatment processes: changes in solute transport properties over time

Although progress has been made toward understanding the surface chemistry of granular iron and the mechanisms through which it attenuates groundwater contaminants, potential long-term changes in the solute transport properties of granular iron media have until now received relatively little attention. As part of column investigations of alterations in the reactivity of granular iron, studies using tritiated water ( 3H 2O) as a conservative and non-partitioning tracer were periodically conducted to independently isolate transport-related effects on performance from those more directly related to surface reactivity. Hydraulic residence time distributions (HRTDs) within each of six 39-cm columns exposed to bicarbonate solutions were obtained over the course of 1100 days of operation. First moment analyses of the data revealed generally modest increases in mean pore water velocity ( v) over time, indicative of decreasing water-filled porosity. Gravimetric measurements provided independent estimates of water-filled porosity that were initially consistent with those obtained from 3H 2O tracer tests, although at later times, porosities derived from gravimetric measurements deviated from the tracer test results owing to mineral precipitation. The combination of gravimetric measurements and 3H 2O tracer studies furnished estimates of precipitated mineral mass; depending on the assumed identity of the predominant mineral phase(s), the porosity decrease associated with solute precipitation amounted to 6–24% of the initial porosity. The accumulation of mineral and gas phases led to the formation of regions of immobile water and increased spreading of the tracer pulse. Application of a dual-region transport model to the 3H 2O breakthrough curves revealed that the immobile water-filled region increased from initially negligible values to amounts ranging between 3% and 14% of the total porosity in later periods of operation. For the aged columns, mobile–immobile mass transfer coefficients ( k mt) were generally in the range of 0.1–1.0 day −1 and reflected a slow exchange of 3H 2O between the two regions. Additional model calculations incorporating sorption and reaction suggest that although changes in HRTD can have an appreciable effect on trichloroethylene (TCE) transformation, the effect is likely to be minor relative to that stemming from passivation of the granular iron surface.