On the evaluation of dispersion coefficients from visualization experiments in artificial porous media

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Abstract

High-resolution single source–solute transport experiments in glass-etched pore networks presented in a previous publication are used for quantifying the hydrodynamic dispersion as a function of Peclet number (Pe). This type of artificial porous media is ideal for understanding the effects of pore-scale phenomena on solute dispersion thanks to its well characterized topology and pore geometry. The hydrodynamic dispersion coefficients are estimated by matching the spatial and temporal distributions of the solute concentration over various regions of the network with the numerical solution of the advection–dispersion equation and using a parameter space analysis to ensure well conditioning of the parameters. The source release function and assumptions on boundary conditions are found to affect significantly the reliability of the dispersion coefficient estimates. The estimated longitudinal dispersion coefficients are in close quantitative agreement with literature data for dispersion in porous media. The extracted transverse dispersivity (αT) values indicate an apparent decrease of αT with increasing flow velocity, which is at variance of constant dispersivity approximation adopted commonly in soil hydrology models and is attributed to incomplete diffusive mixing at the pore intersections. Calculated values of an effective mixing ratio at pore junctions are in agreement with published theoretical estimates and reveal a transition from partial mixing to streamline routing as Pe increases within the range investigated here.

Introduction

Dispersion plays a critical role in numerous processes and practical applications, including contaminant transport in groundwater, filtration, chromatography, fluid–solid catalytic and non-catalytic reactions, etc. Hydrodynamic dispersion in a porous medium occurs as a consequence of two different processes: (i) molecular diffusion, which originates from the random molecular motion of solute molecules, and (ii) mechanical dispersion, which is caused by non-uniform velocities and flow path distribution. Molecular diffusion and mechanical dispersion cannot be separated in a flow regime (Bear, 1979). In diffusion models, the effects of molecular diffusion and mechanical dispersion are usually added together in a single diffusive flux term where the value of the hydrodynamic dispersion coefficient equals the sum of the molecular diffusion and mechanical dispersion coefficients.

Experimental and numerical studies, in the last three decades, indicate that the hydrodynamic dispersion coefficients are, in general, non-linear functions of the Peclet number (Pe). Regarding the longitudinal dispersion, five dispersion regimes have been distinguished depending on the prevailing transport mechanism (Fried and Comparnous, 1971, Sahimi, 1995): (i) the diffusion regime (Pe<0.3), (ii) the transition regime (0.3<Pe<5) where the contribution of mechanical dispersion becomes appreciable, (iii) the power law regime (5<Pe<300) where the contribution of mechanical dispersion is predominant, (iv) the pure advection or mechanical dispersion regime (300<Pe<105), and (v) the turbulent dispersion regime (Pe>105) where the flow regime is out of the domain of validity of Darcy's law. Regarding the transverse dispersion, Fried and Comparnous (1971) reported four dispersion regimes equivalent to those of the longitudinal dispersion, however, shifted to higher Peclet values. Furthermore, experimental (field and laboratory) data in porous media and theoretical evidence demonstrate that there is also a scale-dependence of hydrodynamic dispersion (e.g. Gelhar et al., 1992, Birovljev et al., 1994).

Field scale dispersion of solutes has been observed in field applications to differ from the hydrodynamic dispersion measured in laboratory experiments. The behavior of field scale dispersion (macrodispersion) has been studied extensively over the past 25 years. Several works (e.g. Dagan, 1982, Gelhar and Axness, 1983, Neuman et al., 1987) have demonstrated that, at the field scale, the mean concentration of solute in a steady-state velocity field can be adequately represented at large times by the advection–dispersion equation with constant (asymptotic) dispersivities, which are determined by the degree of heterogeneity of the media and, in the case of transverse dispersivities, are strongly affected by the temporal variations of the flow field. These macrodispersivities are orders of magnitude larger than the dispersivity values observed at the laboratory scale, which results in diminishing the effect of local (laboratory) scale dispersion in such applications. However, recent studies (e.g. Kapoor and Gelhar, 1994a, Kapoor and Gelhar, 1994b; Kapoor and Kitanidis, 1998, Pannone and Kitanidis, 1999) have shown that local dispersion (especially the transverse component) plays a very important role in determining the behavior of the concentration variance, in the dilution of conservative solutes, and in the mixing of reacting compounds. For example, a model that assumes perfect mixing between two reactants within the pores or at the pore intersections of a porous medium may overpredict the amount of reaction products that actually form (Gramling et al., 2002). Such errors in model prediction can significantly affect our understanding of reactive transport processes, and the estimates of the timescales and efficacy of groundwater remediation schemes.

In macroscopic modelling of fluid flow and solute transport in porous media, when the coordinate system is aligned with the mean velocity vector, the longitudinal (DL) and transverse (DT) hydrodynamic dispersion coefficients are typically related to the pore fluid velocity through the following relationshipsDL=αL|v|+DeDT=αT|v|+Dewhere De is the effective molecular diffusion coefficient of the solute in the porous medium, |v| is the magnitude of the velocity vector, and αL and αT are the longitudinal and transverse components of the dispersivity tensor, respectively. The longitudinal and transverse dispersivities are considered to be characteristic properties of a region of a porous medium (Gelhar et al., 1992, Domenico and Schwartz, 1990) and typically, they are conveniently treated as independent of the pore fluid velocity and Peclet number. In a recent publication, Klenk and Grathwohl (2002) presented data on transverse vertical dispersivity, which are in contrast to earlier studies. Their paper gives an overview on literature values of transverse vertical dispersivities measured at different flow velocities and compares them with their results from laboratory tank experiments on mass transfer of trichloroethene (TCE) across the capillary fringe. Their measurements and literature values indicated that transverse vertical dispersivity apparently declines with increasing flow velocity or Peclet number. The latter was attributed by the authors to incomplete diffusive mixing at the pore-scale (pore throats) for flow velocities greater than 4 md−1 or equivalently Pe>128. Park et al. (2001) examined the effects of mixing characteristics on solute transport behavior in fracture networks of a regular lattice. Their work also demonstrated a decreasing trend in the transverse dispersivity values as the Peclet number or flow velocity increases.

Theodoropoulou et al. (2003) describe the design, implementation and results of visualization experiments on the transport of tracer in transparent glass-etched pore networks, which have frequently served as simplified porous media (see, for instance, Tsakiroglou and Payatakes, 2000) or as fracture models (Tsakiroglou, 2002) for characterization and transport studies. The temporal and spatial evolution of solute concentration distribution throughout the model pore network as resulted from the continuous injection of a dense HCl solution through a single entrance position into a flowing dilute HCl solution, was measured with the help of an image analysis system. The longitudinal and transverse dispersion coefficients were estimated by fitting the experimentally measured transient solute concentration profiles to an analytic solution of the 2D advection–dispersion equation based on a specified flux boundary condition at the source (Theodoropoulou et al., 2003). Such an investigation is extremely useful for understanding the solute spreading process under flow conditions, thanks to the well defined topology of the medium and the accurately known geometry of the individual pores. However, both the estimated longitudinal and transverse dispersion coefficients were substantially overestimated (see Fig. 17; Theodoropoulou et al., 2003) as a result of the incorrect calculation of the solute flux and, hence, inaccurate conclusions could be drawn. Corrected values of the longitudinal and transverse dispersion coefficients vs. Peclet number are presented here that were obtained using the experimental data and an analytic solution of the 2D advection–dispersion equation.

In addition, a more complete model of the dispersion process coupled with a detailed description of boundary conditions is presented and solved numerically. An inverse method is employed for the direct evaluation of the dispersion coefficients using concentration profiles from the dispersion visualization experiments. The technique is applied to the particular case of the concentration profiles that were obtained in Theodoropoulou et al. (2003) and compared to experimental data and theoretical estimates from literature. A fruitful discussion on the effect of increasing the flow velocity on transverse dispersion is presented aiming at improved understanding of pore-scale phenomena and their effects on the macroscopic dispersion coefficient. Emphasis is placed at the range of relatively low Peclet number values, which presents a particular interest in major applications (for instance, soil pollution by organic mixtures) and becomes relevant in cases where the interplay of molecular diffusion and mechanical dispersion is expected to control the dispersion process.

Section snippets

Hydrodynamic dispersion experiments

A high-resolution visualization technique was developed by Theodoropoulou et al. (2003) for the measurement of the transient changes of the solute concentration distribution in the void space of a model porous medium. The porous medium was fabricated by etching mirror image patterns of a square network of capillaries on two glass plates with hydrofluoric acid, and sintering the pre-aligned etched plates in a programmable furnace. The cross-sections of the pores have an approximately elliptical

Simulation model

The reliability of parameter estimates by an inverse procedure is highly dependent on the correctness of the structure of the simulation model used to describe the true system (Gaganis and Smith, 2001). The observed fluctuations in the values of dispersion coefficients shown in Fig. 2, which are attributed to the simplified assumption of a semi-infinite domain in the analytical model, as well as other approximations made in the 2D description of the experimental set-up do not allow high

Results and discussion

The apparent longitudinal and transverse dispersion coefficients are estimated from the experimentally measured transient and steady-state solute concentration profiles (Theodoropoulou et al., 2003) at five Pe values, namely, 0.53, 1.06, 2.67, 5.3, and 10.6. The cases of best matching between model results and experimental steady-state concentrations for Pe values 0.53 and 2.67, as well as the corresponding actual contour images are shown in Fig. 5, Fig. 6, respectively.

Fig. 7 shows the

Concluding remarks

Solute concentration measurements as obtained from high-resolution visualization experiments in glass-etched pore networks, presented in a previous publication, can be used for the computation of the hydrodynamic dispersion coefficient as a function of Peclet number. The apparent longitudinal and transverse dispersion coefficients are estimated through an inverse analysis that involves matching the numerical simulation results to the experimentally measured transient and steady-state solute

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