Macroscopic Dispersion Coefficient Research Articles

Determination of Nanoparticle Macrotransport Coefficients from Pore Scale Processes

Nanoparticle transport in porous media is modeled using a hierarchical set of differential equations corresponding to pore scale and macroscale. At the pore scale, movement and interaction of a single particle with a solid matrix is modeled using the advection–dispersion–sorption equation. A single nanoparticle entering the space encounters viscous, diffusion and surface forces. Surface forces (electrostatic and van der Waals forces) between nanoparticles and mineral grains appear as sorption propensity on solid matrix boundary condition. These local events are then transformed into a macroscale continuum by imposing periodic boundary conditions for contiguous unit cells representing porous media and using a scheme of moment analysis. At the macroscale, propagation and retention of particles are characterized by three position-independent coefficients: mean nanoparticle velocity vector $${\bar{\mathbf{U}}}^*$$ , macroscopic dispersion coefficient $${\bar{\mathbf{D}}}^*$$ , and mean nanoparticle retention rate constant $${\bar{K}}^*$$ . The modeling results are validated with a set of nanoparticle transport tests in porous microchips. We also present simulations of realistic porous media, where an actual image of sandstone samples is processed into binary tones. The representative unit cells are constructed from the resulting binary image by searching for areas within the sample with maximum similarities to the whole sample in terms of porosity and specific surface area, which are found to show strong correlations with the resulting $${\bar{\mathbf{U}}}^*$$ and $${\bar{K}}^*$$ , respectively.

Critical Role of the Immobile Zone in Non-Fickian Two-Phase Transport: A New Paradigm.

Using a visualization setup, we characterized the solute transport in a micromodel filled with two fluid phases using direct, real-time imaging. By processing the time series of images of solute transport (dispersion) in a two fluid-phase filled micromodel, we directly delineated the change of transport hydrodynamics as a result of fluid-phase occupancy. We found that, in the water saturation range of 0.6-0.8, the macroscopic dispersion coefficient reaches its maximum value and the coefficient was 1 order of magnitude larger than that in single fluid-phase flow in the same micromodel. The experimental results indicate that this non-monotonic, non-Fickian transport is saturation- and flow-rate-dependent. Using real-time visualization of the resident concentration (averaged concentration over a representative elementary volume of the pore network), we directly estimated the hydrodynamically stagnant (immobile) zones and the mass transfer between mobile and immobile zones. We identified (a) the nonlinear contribution of the immobile zones to the non-Fickian transport under transient transport conditions and (b) the non-monotonic fate of immobile zones with respect to saturation under single and two fluid-phase conditions in a micromodel. These two findings highlight the serious flaws in the assumptions of the conventional mobile-immobile model (MIM), which is commonly used to characterize the transport under two fluid-phase conditions.

Pore‐scale modeling of transverse dispersion in porous media

A physically based description is provided for the transverse dispersion coefficient in porous media as a function of Péclet number, Pe. We represent the porous medium as lattices of bonds with square cross section whose radius distribution is the same as computed for Berea sandstone and describe flow (Stokes equation) and diffusion (random walk method) at the pore scale (∼μm) to compute the transverse dispersion coefficient at a larger scale (∼cm to ∼m). We show that the transverse dispersion coefficient DT ∼ Pe for all Pe ≫ 1. A comprehensive comparative study of transverse dispersion with experiment indicates that the model can successfully predict the trends for the asymptotic macroscopic dispersion coefficient over a broad range of Péclet numbers, 0 < Pe < 105. We discuss the relation between transverse and longitudinal dispersion coefficient and show that unless one studies solute transport in the advection dominated regime, it is not appropriate to take DT to be 1 order of magnitude less than DL.

Persistence of anomalous dispersion in uniform porous media demonstrated by pore‐scale simulations

The anomalous dispersion observed in solute transport through porous media is often attributed to media heterogeneity at various scales, while chemical movement in homogenous media is traditionally thought to be Fickian. There has been increasing evidence over the past few years that even in carefully packed homogenous media, solute movement may not be Fickian as previously thought. On the basis of a pore‐scale modeling, we demonstrate that in a simple uniform porous medium where all pores are well hydraulically connected, the movement of passive tracers appears to be anomalous. It is found that the macroscopic dispersion coefficient increases with time approximately in a power law prior to reaching an asymptotic value, but the spatial distribution of the plume remains non‐Gaussian and cannot be described adequately by the advection‐dispersion equation. In particular, the spatial distribution of the plume has a persistent tail that does not decay with the distance from the peak in c ∝ exp(−bx2) as predicted by the advection‐dispersion equation but in c ∝ exp(−a∣x∣) with a depending on the Peclet number. The concentration in the plume front, on the other hand, decays faster than predicted by the advection‐dispersion equation. Comparing the results to the continuous time random walk (CTRW) reveals that the solute movement is well described by CTRW with a modified exponentially distributed transition time. The plume shows a transition from anomalous to normal distributions but at a slow rate. Given the time it takes for such a transition to complete in such a uniform medium, it is anticipated that in natural media where multiple‐scale heterogeneity exists, a transition from anomalous to normal transports may never complete.

Modeling Hydrologic Phenomena [Free opinion

With the aim of suggesting some practical rules for the use of hydrological models, G. De MARSILY in his "free opinion" (Rev. Sci. Eau 1994, 7(3): 219-234) proposes a classification of hydrologic models into two categories: - models built on data (observable phenomena) and ; - models without any available observations (unobservable phenomena). He claims that for the former group of observable phenomena, models developed through a learning process as well as those based on the underlying physical laws are of the black box type. For the latter group of unobservable phenomena, he suggests that physically-based hydrologic models be developed. Physically-based hydrologic models should introduce to the phenomenological laws the correct empirical coefficients, which correspond to the proper time and space scales (GANOULIS, 1986). Well-known examples are Darcy's permeability coefficient on the macroscopic scale as derived from the Navier-Stokes equations on the local scale and the macroscopic dispersion coefficients in comparison with the local Fickian diffusion coefficients. Misuse of these models by confusing the proper time and space scales and determining the coefficients by calibration is not a sufficient reason to consider them as belonging to the black box type. Black box type hydrologic models, although very useful when data are available, remain formally empirical. They fail to give correct answers when serious constraints of unity in place, time and action are not fulfilled. Concerning the second class of models, we may notice that purely unobservable phenomena without any available data do not really exist in hydrology. In the case of very rare events and complex systems, such as radioactivity impacts and forecasting of changes on a large scale, physically-based models with adequate parameters may be used to integrate scarce information from experiments and expert opinions in a Bayesian probabilistic framework (APOSTOLAKIS, 1990). The most important feature of hydrologic models capable of describing real hydrologic phenomena, is the possibility of handling imprecision and natural variabilities. Uncertainties may be seen in two categories: aleatory or noncognitive, and epistemic or cognitive. Probabilistic hydrologic models are more suitable for dealing with aleatory uncertainties. Fuzzy logic-based models may quantify epistemic uncertainties (GANOULIS et al., 1996). The stochastic and fuzzy modeling approaches are briefly explained in this free opinion as compared to the deterministic physically-based hydrologic modeling.

Pore‐scale modeling of longitudinal dispersion

We study macroscopic (centimeter scale) dispersion using pore‐scale network simulation. A Lagrangian‐based transport model incorporating flow and diffusion is applied in a diamond lattice of throats with square cross section whose radius distribution is the same as computed for Berea sandstone. We use physically consistent rules using a combination of stream‐tube routing and ideal mixing to transport particles across pore junctions. The influence of both heterogeneity and high Peclet numbers results in asymptotic behavior only being seen after movement through many throats. A comprehensive comparative study of longitudinal dispersion with experiments in consolidated and unconsolidated media indicates that the model can quantitatively predict the asymptotic macroscopic dispersion coefficient over a broad range of Peclet numbers, 0 < Pe < 105. In the low Peclet number region, molecular diffusion is more restricted for consolidated media as compared with unconsolidated media. The first effects of advection on dispersion are observed at Pe ∼ 0.1. In the advection‐dominated regions the longitudinal dispersion coefficient follows a weak nonlinear dependence on Peclet number (DL ∼ Pe1.19) followed by a linear dependence DL ∼ Pe for Pe > 400.

Early breakthrough of colloids and bacteriophage MS2 in a water‐saturated sand column

We conducted column‐scale experiments to observe the effect of transport velocity and colloid size on early breakthrough of free moving colloids, to relate previous observations at the pore scale to a larger scale. The colloids used in these experiments were bacteriophage MS2 (0.025 μm), and 0.05‐ and 3‐μm spherical polystyrene beads, and were compared with a conservative nonsorbing tracer (KCl). The results show that early breakthrough of colloids increases with colloid size and water velocity, compared with the tracer. These results are in line with our previous observations at the pore scale that indicated that larger colloids are restricted by the size exclusion effect from sampling all paths, and therefore they tend to disperse less and move in the faster streamlines, if they are not filtered out. The measured macroscopic dispersion coefficient decreases with colloid size due to the preferential flow paths, as observed at the pore scale. Dispersivity, typically considered only a property of the medium, is in this case also a function of colloid size, in particular at low Peclet numbers due to the size exclusion effect. Other parameters for colloid transport, such as collector efficiency and colloid filtration rates, were also estimated from the experimental breakthrough curve using a numerical fitting routine. In general, we found that the estimated filtration parameters follow the clean bed filtration model, although with a lower filtration efficiency overall.

Carbonate dissolution and precipitation in coastal environments: Laboratory analysis and theoretical consideration

We have conducted laboratory experiments to examine CaCO3 dissolution and precipitation in saltwater‐freshwater mixing zones, with a view to understanding and predicting porosity changes in coastal environments. Mixing of seawater or saline subsurface water with fresh water can be of major importance in the chemical diagenesis of carbonate rocks and sediments. We used artificial seawater and NaCl solutions of different concentrations under different CO2 partial pressures and with different mixing ratios. Two‐dimensional flow cells filled with glass beads and crushed calcium carbonate rock were used to measure calcium carbonate precipitation and dissolution, respectively. An important feature of these experiments is that the results are shown to agree well with a relatively simple transport theory describing mineral precipitation/dissolution that results from the nonlinear dependence of CaCO3 saturation upon electrolyte concentration. The theory demonstrates that the rate of dissolution or precipitation depends on the curvature (and sign) of the solubility as a function of salinity, the square of the salinity gradient, and the macroscopic dispersion coefficient. The theory is largely scale independent and depends upon field parameters that can be determined. Analysis of data from three field sites (Yucatan peninsula, Bermuda, and Mallorca) demonstrates excellent agreement between field observations and theory.

Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport

We study the time behavior of solute transport in a heterogeneous medium. We consider a spatially biased continuous time random walk (CTRW) governed by ψ( s ,t) , the joint probability density for an event–displacement s with an event–time t. In this effective transport framework the concentration distribution of a solute is given by a generalized master equation (GME). We present results detailing the time dependence for the resident and flux concentrations, the center of mass velocity and the macroscopic dispersion coefficients of the solute plume. The Laplace transform of the GME is converted to a spatial differential equation resembling the advection dispersion equation (ADE), with Laplace space dependent coefficients though, and can then be solved explicitly in Laplace space. The transport behavior of the solute is then determined by accurate numerical inverse Laplace transforms. The confirmation of the accuracy of our methods is demonstrated by the excellent agreement with efficient random walk simulations based on the same ψ( s ,t) . The ψ( s ,t) is given by the product of a Gaussian distribution for s and a truncated power-law distribution for t. This particular choice allows for a systematic study of the time regimes of anomalous and normal transport behavior and the transition from normal to anomalous behavior. The presented results show new aspects for the modeling of solute transport in heterogeneous media, in particular the effect of the system “memory” on plume patterns at asymptotically long times. In a specific application we solve for the contaminant flux entering a stream from a point injection of tracer in a catchment. The results are discussed as an independent test of a model of fractal stream chemistry in catchments.

STUDY ON THE MICROSCOPIC AND MACROSCOPIC DISPERSIONS FOR THE TRACER TRANSPORT IN THE NON-UNIFORM POROUS MEDIA

An evaluation of the dispersion coefficient or the dispersivity is important for the solute transport in groundwater flow. In this paper, an evaluation of the dispersion coefficient by the numerical simulation for the non-uniform porous media is attempted. The non-uniform porous media is consisted of 39×19 blocks with the packed glass beads of six different diameters. For each block, microscopic dispersivity, permeability and porosity are known. The forty cases for evaluating the macroscopic dispersion are studied. Specifically, the relations between the integral scale of the log permeability and macroscopic dispersivity is examined. It is suggested that the macroscopic dispersion depends on the integral scale and that the sufficient observation scale is necessary for obtaining the macroscopic dispersion coefficient.

Experimental measurement of dispersion processes at short times using a pulsed field gradient NMR technique

Dispersion at short times is studied using a PFG-NMR (pulsed field gradient NMR) technique inside a fixed bed of nonconsolidated spherical beads saturated with water flowing at a constant velocity. This allows measurement of the probability distribution of the displacement of water molecules along the magnetic field gradient during a preset measurement time Δ: the mean displacement of the water molecules is varied between 0.1 and 7.3 times the bead diameter by varying Δ between 20 and 100 ms and the bead diameter between 800 and 81 μm. At short times, the displacement of the molecules is small enough so that the local displacement is proportional to the local velocity component along the magnetic field gradient. At mean displacements larger than 5 bead diameters, the displacement distribution is Gaussian and centered about the mean displacement; the width of the distribution corresponds to the macroscopic dispersion coefficient as measured by other techniques. At intermediate displacements, this distribution displays two peaks corresponding to a combination of the two processes. The main features of this transition can be reproduced by a simple Monte-Carlo simulation modeling the porous medium as a set of finite length tubes with random orientations.

On the stochastic theory of solute transport by unsteady and steady groundwater flow in heterogeneous aquifers

In this study the exact and approximate deterministic partial differential equations of the time-space evolution of mean solute concentration, of solute concentration two-point moment and of solute concentration n-point moment for conservative solute transport by unsteady and steady groundwater flow in a heterogeneous aquifer were developed in the real time-space domain under second-order cumulant expansion. These derivations were performed by means of the cumulant expansion method, combined with the calculus for the time-ordered exponential and with the calculus for Lie operator. The mean solute transport equations which describe the time-space evolution of mean conservative solute concentration under transport by unsteady and steady groundwater flows in a heterogeneous aquifer have convective-dispersive forms whose convective and dispersive coefficients are defined precisely in terms of the mean and covariance functions of the pore flow velocity random field. However, owing to the spatial nonuniformity of groundwater flow in heterogeneous porous media a new convective coefficient, besides the standard mean velocity vector, appeared in the derived mean solute transport equations in the case of steady flows. In the case of transport by unsteady groundwater flow in a heterogeneous aquifer, the new convective coefficient (besides the standard mean velocity vector) is due to the non-zero divergence of the pore flow velocity. The fundamental reason for the convective-dispersive form of the mean transport equations is the second-order truncation of the formal cumulant expansion. In the derived mean solute transport equations the macroscopic dispersion coefficient emerges as the time integral of the covariance function of the pore flow velocity. However, both in the case of transport by unsteady groundwater flow and in the case of transport by steady spatially nonstationary-nonuniform groundwater flow the macroscopic dispersion coefficients and convective coefficients vary with spatial locations within the porous medium. Only in the case of transport by steady spatially stationary-nonuniform flow are both the macroscopic dispersion coefficient and the convective coefficient constant with respect to spatial location within the heterogenous aquifer. Therefore, only in the case of conservative solute transport by steady spatially stationary-nonuniform groundwater flow in a heterogeneous aquifer does the mean transport equation apply to all spatial locations within the aquifer with the same parameter values at each time instant.

The concept of the Dilution Index

In many applications, it is important to make the distinction between spreading and dilution of a plume in groundwater. Spreading is associated with the stretching and deformation of a contaminant plume, whereas dilution is associated with the increase in volume of the fluid occupied by the solute. The dilution and spreading of a Gaussian plume in a homogeneous porous medium with constant velocity are related in a simple fashion and are both characterized by the same parameters, the dispersion coefficients. However, the geological formations of interest in field applications are heterogeneous, and the plumes are irregular in shape. The dispersion coefficients that are deduced from tracer tests usually measure an overall rate at which a tracer plume spreads about its centroid and depend critically on the heterogeneity of the formation. These macroscopic dispersion coefficients are not reliable measures of the rate at which the maximum concentration is reduced because in heterogeneous formations the rates of dilution and spreading can be quite different. The main objective of this work is to introduce a new macroscopic measure of dilution, the dilution index E. Examples serve to demonstrate the usefulness of the measure. A general expression for the rate of dilution of a tracer plume is derived. The exact rate of increase of the dilution index under the idealized conditions of constant dispersion coefficients and a Gaussian plume is computed, and a lower bound is found to the same quantity for non‐Gaussian plumes. For the general heterogeneous case the analysis demonstrates that the instantaneous rate of increase of ln E is proportional to the small‐scale dispersion coefficients, everything else being the same. The rate of increase of ln E depends also on the degree of irregularity in the shape of the plume. Thus, in the long term, geologic heterogeneity should increase the rate of dilution because spatial variability in the flow velocity tends to deform plumes and make them less regular.

Dispersion and Convection in Periodic Porous Media

The problem of transport of a passive solute in a porous medium by convection and dispersion is analysed by the method of homogenization. Assuming that the geometry is periodic, the expressions for the macroscopic dispersion coefficients are derived. A few possible scalings are compared and we find that the most interesting one provides a local balance between drift and diffusion.