Hydrodynamic disersion Disersive transort of contaminants results from the heterogeneous distribution of water flow velocities within and between different soil ores (Figure 1, left). Disersion can be derived from Newton s law of viscosity which states that velocities within a single caillary tube follow a arabolic distribution, with the largest velocity in the middle of the ore and zero velocities at the walls (Figure 1, left). For this reason contaminants in the middle of a ore will travel faster than contaminants that are farther from the center. Since the distribution of contaminant ions within a ore deends on their charge, as well as on the charge of ore walls, some contaminants may move considerably faster than others. In some circumstances (e.g., in fine-tetured soils), anion eclusion may occur. Since the anions in such soils are located redominantly in the faster moving liquid in the center of a ore away from the negatively charge solid surface, anionic contaminants may travel faster than the average velocity of water (e.g., Nielsen et al. 1986). Based on Poiseuille s law, one can further show that velocities in a caillary tube deend strongly on the radius of the tube, and that the average velocity increases with the radius to the second ower. Since soils consist of ores of many different diameters, contaminant flues in ores of different diameters will be significantly different, with some contaminants again traveling faster than others (Figure 1, right). Furthermore, contaminants may travel according to athways of different length. All these factors result in a bell-shaed distribution of velocities and thus of arrival times, tyical of a breakthrough curve. slow velocity fast velocity short athway long athway Figure 1: Distribution of velocities in a single ore (left) and distribution of velocities in a ore system (right) resulting in different arrival times of contaminants (modified from Šimůnek and van Genuchten, 006). he above ore-scale disersion rocesses lead to an overall (macroscoic) hydrodynamic disersion rocess that can be described mathematically in aroimately the same way as molecular diffusion using Fick s first law. Addition of the disersion and diffusion rocesses leads to the following eression for the contaminant mass flu in the liquid hase J h : c c Jh Dh ( Dm D) z z (Eq. 1) 1
where D h is the hydrodynamic disersion coefficient [ -1 ] that accounts for both molecular diffusion and mechanical disersion (Fetter, 1999), D m is the mechanical disersion coefficient [ -1 ], and D is the liquid hase diffusion coefficient [ -1 ]. he mechanical disersion coefficient in one-dimensional systems has been found to be aroimately roortional to the average ore-water velocity v (= q/) [ -1 ], with the roortionality constant generally referred to as the (longitudinal) disersivity λ (Biggar and Nielsen 1967). he discussion above holds for one-dimensional transort; multi-dimensional alications require the use of a more comlicated disersion tensor involving longitudinal and transverse disersivities (e.g., Bear 197). Disersivity is a transort arameter that is often obtained eerimentally by fitting measured breakthrough curves with analytical solutions of the advection-disersion equation. he disersivity often changes with the distance over which contaminants travel, the so-called disersion-scale effect. Values of the longitudinal disersivity usually range from about 1 cm for relatively short, acked laboratory columns, to about 5 or 10 cm for field soils. ongitudinal disersivities can be significantly larger (u to hundreds of meters) for regional groundwater transort roblems (Gelhar et al. 1985). If no other information is available, a good first aroimation is to use a value of one-tenth of the transort distance for the longitudinal disersivity (e.g, Anderson 1984), and a value of one-hundred of the transort distance for the transverse disersivity when multi-dimensional alications are considered. In many cases, esecially in groundwater sediments, flow is three-dimensional. In such case, a three-dimensional flow and transort equation should be used. Often, such flow systems can be simlified to a two-dimensional system. For a two-dimensional flow system, the advection-disersion equation for a sorbing chemical element undergoing radioactive decay is written as (Bear, 197): C C C R D ij qi / ne R µc t i j (Eq. 3) i where D ij is the hydrodynamic disersion tensor, q i is the i-th comonent of the Darcy flu, and i (i = 1,) is satial coordinate. he hydrodynamic disersion D ij now accounts for disersion in the direction of the main flow comonent, characterized by the longitudinal ( ) disersivity, and in the direction orthogonal to the main flow direction, characterized by the transversal ( ) disersivity, and is given by (Frind en Hokkanen, 1985): D D 11 v1 / v v / v D (Eq. 4) v1 / v v / v D (Eq. 5) D1 D1 ( ) v1 v / v (Eq. 6) where <v> = (v 1 + v ) 0.5. Usually the transverse disersivity is 10 to 0 times smaller than the longitudinal disersivity. he effect of different ratios for / is seen in Figure. As the ratio becomes smaller, the lume will develo a less elongated shae, indicating that more chemicals will move in the direction orthogonal to the main water flow direction. his henomenon is tyical for layered sediments.
Flow d irec tion / = 0 / = 5 / = / = 1 Figure Effect of horizontal versus transverse disersivity on lume sreading. In case of a uniform two-dimensional velocity field with the direction of flow arallel to the -ais, Eq. (3) is simlified to (with v = v = 0): 3
R C t D C D C y q C / ne i R µc (Eq. 7) where D and D are the longitudinal and transversal disersion coefficient, resectively, equal to: D D v (Eq. 8) D D v For the determination of aroriate disersivities an etensive literature review was carried out. Details of the review and the data analysis are given by Mallants et al. (1998). Only the main results will be resented here. 1E+003 Neuman, 1990 1E+00 Ma. ongitudinal disersivity (m) 1E+001 1E+000 1E-001 A = 0.087 Reliability Min. Intermediate Porous Best estimate Xu & Eckstein, 1995 Fractured High Gelhar et al. [199] 1E-00 Intermediate 1E-003 A = 0.0089 High his study 1E-001 1E+000 1E+001 1E+00 1E+003 1E+004 1E+005 1E+006 Scale (m) Figure 3 ongitudinal disersivity versus scale: only data with high and intermediate reliability are shown. Dashed line indicates best estimate, i.e. = 0.067. Horizontal arrows indicate maimum (=90m), best estimate ( > 1000 m) = 6.7 m, and minimum ( = 5.5 m) disersivity in the range of travel distances between 1000 and 10000 m (from Mallants et al., 1998). A summary of the longitudinal disersivities,, whose reliability was estimated to be intermediate or high, is lotted in Figure 3. Also indicated on the grah are the best estimate 4
relationshi = 0.0067, where is travel distance, together with the 5% and 75% values for /. he best estimate reresents the 50% value of the / robability distribution. Note that these relationshis are only valid for travel distances smaller than 1000 m. For > 1000 m, a constant value of 6.7 m is roosed. Figure 3 clearly shows the increasing trend for with increasing travel distance. However, for travel distances larger than or equal to 1000 m, too few reliable data were available to derive a relationshi between and. We therefore assumed that remained constant with scale for distances larger than 1000 m. he best estimate value was equal to 6.7 m, which is based on the best estimate relationshi / = 0.067, with = 1000 m. his aroach may underestimate at very large distances, but this is conservative. At distances of 1000 m or more, the minimum and maimum value for were equal to the minimum and maimum observed values, resectively. In addition to the longitudinal disersivity, also estimates for the horizontal transverse, H, and the vertical transverse, V, disersivity were derived (in case transort is three dimensional, three disersivities aly,, H, and V, where H accounts for disersion in the y-direction and V in the z-direction). For these two arameters, H and V, best estimates were calculated directly from the relationshi between and H and from the relationshi between H and V, resectively. In other words, each samled value for will lead to a value for H by using H = /5.08 (Mallants et al., 1998). he value for H thus obtained should be within the minimum (= 0.001 m) and maimum (=1.5m) observed values for H. he calculated value for H will then be used to calculate a value for V, according to V = H /.7. his value should be within the following limits: 0.0004 V 0.345 m. A summary of the statistical arameters for each of the disersivity coefficients is given in able 1. able 1 Statistical arameters for distributions of, /, H, and V (source: Mallants et al., 1998). [m] / $ [-] ** H [m] *** V [m] 50th ercentile (median) 6.7 0.067 0.1000 0.00 minimum 5.5 0.0089 $ 0.0010 0.0004 # maimum 90 0.0870 * 1.500 0.3450 # log-uniform distribution ( 1000 m); $ log-triangular distribution ( < 1000 m) ** values calculated from H = /5.08; *** values calculated from V = H /.7 #, based on observed values; $ 5th ercentile; * 75th ercentile 5
References Anderson, M.P., Movement of contaminants in groundwater: Groundwater transort-advection and disersion.. 37-45. In: Groundwater Contamination, Studies in Geohysics, National Academy Press, Washington, DC., 1984. Bear, J., Dynamics of Fluids in Porous Media, Elsevier Science, NY., 197. Biggar, J. W., and D. R. Nielsen, Miscible dislacement and leaching henomena, Agronomy, 11, 54-74, 1967. Fetter,C.W., Contaminant hydrology. Uer Saddle River; Prentice Hall, 458. 1999. Frind, E.O., and Hokkanen, G.E., 1985. Simulation of the Borden lume using the alternating direction Galerkin technique. Water Resour. Res., 1(), 15-169. Gelhar,. W., A. Mantaglou, C. Welty, and K. R. Rehfeldt, A review of Field Scale Physical Solute ransort Processes in Unsaturated and Saturated Porous Media, EPRI oicl Reort EA-4190, Electric Power Research Institute, Palo Alto, CA, 1985. Mallants, D., Marivoet, J., and Volckaert, G., 1998. Review of recent literature on the disersivity arameter for saturated and fractured orous media. echnical Note 44, Det. W&D, SCK CEN, Mol, Belgium. Nielsen, D. R., van Genuchten, M. h., and Biggar, J. W., Water flow and solute transort rocesses in the unsaturated zone. Water Resour. Res., 18(9), 89S-108S, 1986. Šimůnek, J., and M. h. van Genuchten, Contaminant ransort in the Unsaturated Zone: heory and Modeling, Chater in he Handbook of Groundwater Engineering, Ed. Jacques Delleur, Second Edition, CRC Press,..1-.46, 006. 6