Lateral solute mixing in homogeneous and layered sand columns

(;E()I)EH~MA ELSEVIER

Geoderma63 (1994) 109-121

Lateral solute mixing in homogeneous and layered sand columns Sabine Koch*, Hannes Fltihler ETH Zlirich, Institute of Terrestrial Ecology, Soil Physics, Grabenstr. 3, CH-8952 Schlieren, Switzerland

Received November 2, 1992; accepted after revision September 2, 1993)

Abstract Lateral dispersion was investigated in water unsaturated columns (length 18 or 20 cm and diameter 6 or 11 cm, respectively) at constant flow (q = 28.5 mm h - ~). Columns were either packed homogeneously with fine sand and coarse sand, or in horizontal or inclinded layers. A conservative dye tracer was applied as a point source in the centre of the column or as a pulse over the entire surface. Lateral mixing of the point source was investigated visually by cutting the columns vertically into two halves immediately after the dye was detected in the outflow. In non-stratified, homogeneously packed columns lateral solute mixing could be modelled using a convection--dispersion model with lateral dispersion term. In stratified columns, processes at or just above the layer boundaries significantly modified lateral mixing. These processes depend strongly on the sequence of neighbouring layers. In a fine textured layer overlying coarse sand streamlines may converge and thereby induce preferential flow. In a coarse layer immediately above a fine textured one lateral exchange of flowing solutes may be enhanced and thereby smooth out a moving solute front.

1. Introduction Lateral mixing is an important characteristic of solute transport in porous media. It may be conceptualized based on the stochastic continuum approach. This concept may be illustrated by assuming that solutes are trapped in parallel stream tubes moving at different local flow velocities isolated from other tubes. Longitudinal dispersion of a solute plume results from the variance of the distribution of the local velocities (Jury and Roth, 1990, pp. 142). The convective--dispersive and the stochastic-convective model may be considered as two limiting cases for dispersion in terms of this concept (Jury and Roth, 1990; Jury and Fliihler, 1992). Lateral exchange of solute particles between stream tubes is possible as solute *Corresponding author. 0016-7061/94/$07.00 © 1994 Elsevier Science Publishers B.V. All fights reserved SSD10016-7061 ( 9 3 ) E 0 0 9 7 - F

I I0

S. I~m'h, tt. Fhihler/Gemterma 63 ~1994) 109.-121

particles diffuse laterally into adjoining stream tubes. If the probability of lateral mixing is small (e.g. when the solute has just entered the medium) then the transport process will be stochastic-convective. The longitudinal dispersion coefficient D increases with travel distance x, which is known as the dispersion scale effect. If the probability of lateral mixing is large (e,g. when the travel distance is long enough to allow the solute molecules to diffuse laterally into different stream tubes) then the transport process is convective-dispersive. The longitudinal dispersion coefficient D has reached a constant value. Little is known about lateral dispersion in porous beds. Harleman and Rumer (1963) were among the first to measure lateral dispersion coefficients in a water saturated homogeneous bed of spheres. They found a linear dependence of the lateral dispersion coefficient DT on pore velocity t, similar to the relation for the longitudinal dispersion coefficient, But the lateral dispersivity DT/V was only about one thirteenth of the longitudinal one. Yule and Gardner (1978) measured lateral dispersion coefficients in an unsaturated sandy soil, but they found no significant dependence of DT on v. The results of Han et al. (1985) were similar to those of Harleman and Rumer (1963). They investigated in addition the influence of particle size distribution on the lateral dispersion coefficient: lateral dispersivities were less sensitive to particle size than longitudinal dispersivities. Leroy et al. (1992) investigated lateral dispersion in a water saturated bed of different glass spheres with layers parallel to flow. They found that the influence of lateral mixing depends on flow velocity. Raats (1973) proved theoretically that streamlines were refracted at the interface between two layers, with different permeabilites. Miyazaki (1990) visualized this refraction phenomena in saturated and unsaturated beds of layered sands and glass beads. Refraction of streamlines may contribute to lateral mixing at textural boundaries, especially when the interfaces are wavy. To the best of our knowledge, no studies on lateral mixing with flow perpendicular to stratification have been reported. Several field studies have shown that stratification in soil affects solute dispersion significantly. Butters and Jury ( 1989) reported an abrupt decrease of the longitudinal dispersion coefficient where bulk density changed with depth. They concluded that lateral mixing increased at this boundary. Several investigators (Starr et al., 1978; Glass et al., 1989a, b; Baker and Hillel, 1990; Kung, 1990a, b) reported fingering at boundaries between layers. To learn more about the effect of boundaries between layers on the dispersion process in a uniform flow field we investigated the mechanisms of lateral mixing in a laboratory study. A dye was used as a point source tracer at the surface of a small column packed with sand to stain a flow path within the column. The shape of the dye pattern in homogeneous fine and coarse materials was compared with patterns where sands were packed in layers.

2. Materials and methods

The experiments were done in columns packed with sand and under unsaturated conditions. The columns were made from plexiglas so that their walls were transparent and they consisted of two halves so that longitudinal cross-sections could be made readily at the end of the experiments. Two sizes of columns were used. The larger column was 11 cm in diameter and 20 cm long, the smaller column 6 cm wide and 18 cm long. The columns were supported by porous plates at their bases so that a suction could be applied.

S. Koch, H. Fl~ihler/ Geoderma 63 (1994) 109-121

coarse sand

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fine sand Z~

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data model

model

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Fig. 1. Retention curves, pore size distributions calculated from the retention curves and functions of hydraulic conductivity of coarse and fine sand. The retention curves were fitted with Van Genuchten's model and the hydraulic conductivity was estimated by Mualem's model. K, was measured with the constant head method. Arrows indicate the flow rate which was used.

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The columns were packed with a special filling pipe (Stauffer, 1977; Stauffer and Dracos, 1986) which allowed a dense, homogeneous and reproducible packing. We used air-dry fine and coarse sand of 0.1-0.3 mm and 1.0-1.5 mm grain diameter and porosities of 0.42 and 0.37, respectively. Fig. 1 shows the respective retention curves, pore size distributions and functions of hydraulic conductivity of the two sands. The pore size distributions were calculated from the retention curves. The retention curves were fitted with Van Genuchten's model (1980). The saturated hydraulic conductivity K~ was measured with the constant head method and the functions of hydraulic conductivity were estimated by Mualem's model (1976). For fine sand we found Ks = 0.26 mm s- 1 and for coarse sand Ks = 0.67 mm s - 1. The columns were packed either homogeneously or in layers. The solution was applied at a constant rate with a slowly rotating sprinkler* of the same diameter as the column. The sprinkler was equipped with several hypodermic needles evenly distributed over the entire area ( 1 needle/3 cm2). We used needles with a inner diameter of 0.6 mm. In the centre of the sprinkler one needle was connected to a separate pump through which dye or water could be applied at the same rate as through the other needles. Water was applied at the top of the columns at a constant rate of q = 28.5 mm h - i and a constant suction of Ap = 3 kPa was maintained at the bottom of the columns in all experiments. This resulted in a steady-state percolation regime at time invariant water content. The volumetric water content of the columns was monitored throughout the experiments using micro TDR-probes (Malicki and Skierucha, 1989; Malicki et al., 1992) installed at four depths. The columns were pre-wetted in two different ways: (i) from underneath by a rising water table which ensured an even wetting, and (ii) from above by the sprinkler, so that air could be entrapped because the columns were closed at their bases by the saturated and therefore air-impervious porous plate. Dye was applied at the centre of the columns as a point source as described above or manually distributed with a syringe over the entire surface as a narrow pulse. We used the blue food dye Brilliant Blue FCF (colour index 42090) at a concentration of 5 g/l. The dye adsorbs only slightly on sandy material that contains little organic carbon (Flury and Reber, 1992). The experiments were stopped when the dye had reached the base of the columns. Each column was then carefully cut in half and the dye pattern was photographed and analysed visually.

3. Results and discussion 3.1. N o n - s t r a t i f i e d c o l u m n s

Fig. 2 shows schematic drawings of dye patterns (shaded area) along vertical profiles through the centres of two homogeneously packed columns, one with coarse sand (Fig. 2, right-hand side) and the other with fine sand (Fig. 2, left-hand side). The dye was applied *A detaileddrawing of the constructionis availableon request.

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coarse sand

fine sand

6 cm

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3-i

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r=0

D T = 5 . 3 . 1 0 -4 m m2 s"1

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DT = 1 . 5 . 1 0 -3 mn~ s1

D T = 2 . 3 . 1 0 -4 m m2 s1

DT = 3.0- 10 -3 mrn2 s1

D T = 4 . 6 . 1 0 4 mrrf2 s 1

Fig. 2. Schematic drawing of dye distributions in coarse sand and in fine sand along a vertical cross section through the centres of columns. Dye was applied as a point source. Lines indicate lateral dispersion estimated with Eq. (3). The lines are 0.001 g/l isoconcentration lines.

as a point source. The v o l u m e t r i c water contents (0) increased slightly f r o m top to bottom o f the c o l u m n with 01 = 0.15 m 3 m - 3 at x~ = - 3 . 5 cm, 02 = 0.18 m 3 m - 3 at x2 = - 7 . 3 cm, 0 3 = 0 . 2 3 m 3 m - 3 atx3 -- - 10.8 era, 0 4 = 0 . 2 6 m 3 m - 3 atx~ = - 14.3 c m for fine sand and 01 = 0 . 1 2 m 3 m - 3 0 2 = 0 . 1 4 m 3 m -3, 0 3 = 0 . 1 4 m 3 m -3, and 0 4 = 0 . 1 3 m 3 m - 3 for coarse sand at the s a m e depths. In both c o l u m n s the stained flow paths spread laterally with increasing depth. In the c o l u m n p a c k e d with fine sand (Fig. 2, left-hand side) the shaded

S. Koch, t4. Fliihler / Geoderma 63 (1994) 109 12/

114

area constricts near the outflow. Probably the experiment was stopped too early and the distribution of dye was not yet at steady state. Lateral mixing is more intense in coarse sand than in fine sand. For coarse sand the dye reached the column walls several centimetres above the bottom of the column. We calculated concentration profiles for different values of the lateral dispersion coefficient Dv using an analytical solution similar to that of Harleman and Rumer (1963). For a steady-state distribution of concentration longitudinal dispersion may be ignored and the transport equation becomes: OC(x,

v

r)

Ox

1 0

DT

r Or

(l)

Or

where C(x,r) is the solute concentration, v = q~ 0 is the mean pore velocity in the direction of x where 0 is the mean value of the four water contents measured along the column, and r is the radial coordinate. The initial and boundary conditions are:

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Vx, r

C(x,r)=O

forx=0

and

r>O

att>0

(2b)

C(x,r)=Co

forx=0

and

r=O

att>0

(2c)

for t < 0

(2a)

The solution is:

C(x, r) = QCo 1 1 - r2 ~ - 2,tr 2x(D~v) eXp ( ~ )

(3)

where Co is the input concentration of dye and Q is the volumetric flow rate. Harleman and Rumer (1963) gave an empirical relation for the lateral dispersion coefficient DT in homogeneous and water saturated porous media:

DT =cRe m

(4)

where c is a constant depending on particle shape and size distribution, Re is the particle Reynolds number depending on the pore velocity, mean grain diameter and kinematic viscosity of the solute, and the exponent m is a function of the particle size distribution. Their approximation yields Dv = 1.12 X 10 - 4 rnlTl 2 S- 1 for fine sand and DT = 5.3 X 10 - 4 mm 2 s - J for coarse sand using our experimental conditions but neglecting that our porous materials were unsaturated. The lines shown in Fig. 2 are isolines connecting all points of C(x,r)=O.O01 g/1 for different values of Dr. At this concentration the dye is no longer visible in the sand. Experimentally the dye could not be applied in a real point as assumed by Eq. (2c), but rather over a small area at the surface of the column. Therefore the analytically derived concentration profiles do not coincide with the dye pattern. The values estimated by the relation of Harleman and Rumer (1963) were used to calculate the profiles shown by the bold lines in Fig. 2. For the profiles shown by the plain and dashed lines we increased these values three and six times for coarse sand and two and four times for fine sand. The spreading of the dye in the experiment with coarse sand (Fig. 2a) is only partly simulated by the model. The dye spread more intensely than predicted by the model. This may be explained

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by effects from the wall of the column which are not accounted for in the model which assumes r--,oo. For fine sand (Fig. 2b) the model describes fairly well the dye trace, especially for the largest value of Dr. Hence the lateral dispersion coefficients for unsaturated sand lie in the same order of magnitude as estimated by the relation of Harleman and Rumer for saturated beds of spheres. 3.2. Stratified c o l u m n s

Fig. 3a shows a typical dye pattern in columns packed with three layers. The dye was added at a point as before. Fig. 3b shows the distribution of the volumetric water content as well as air and matrix volumes of the layers. Note that the coarse layer contains very little water ( 0 = 0.07 m 3 m - 3). The dye pattern shows a distinct funnelling when approaching the upper layer boundary. At this interface the stained flow path is confined to a small area when it enters into the coarse layer. This effect cannot be explained by lateral dispersion. It is possible that streamlines are diverted towards a region at the interface where the

matrix air water

Fig. 3. (a) Typical dye pattern for a column packed in 3 layers (fine-coarse-fine). Column size: 20 cm long, 11 cm diameter. The upper interface forces streamlines to converge,the lowerboundaryhas no visible influence. (b) Distribution of water, air, and matrix volume in the 3 layers.

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hydraulic conductivity is slightly larger. This causes preferential entries into the underlying coarse material. It is well known (Starr et al., 1978; Glass et al., 1989a, b; Baker and Hillel. 1990; Kung 1990a, b) that a wetting front can break up into fingers when passing a boundary from a fine textured top layer to an underlying coarse textured layer. Hillel and Baker (1988) gave a convincing hypothesis to explain this phenomenon. They suppose that initially parallel streamlines converge and hence form preferential flow paths when the flux density increases along the direction of flow. Although they restrict their analysis to infiltration and redistribution their findings can be applied to our experiments. We use steady flow (q =const.). Comparison with the hydraulic conductivity in Fig. 1 shows that if the coarse layer were to conduct water throughout its entire cross-sectional area at the given flux density of q = 28.5 mm h ~ , this would be only possible at a mean water content of 0 > 0.32 m 3 m-3. But the mean water content which was measured by the TDR probes is 0= 0.07 m 3 m -3. Locally the water content in the coarse layer is greater where the streamlines converge and this enables a locally higher flow rate. The water content and flow rate in this zone of convergence could not be measured. We can estimate them, however, by the following approximation according to Hillel and Baker (1988). Conserving mass we can write for Q, the volumetric flow rate:

Q=

qAdy c =

qinterAinter,

(5)

where A a y e is the cross-sectional area of the stained flow path before it converges and Ainter is its cross-sectional area at the interface between fine and coarse layer and qin,~r is the local flux-density of the flow path at this interface. The cross-sectional areas can be estimated from Fig. 3a by measuring the diameter of the dyed flow path in the fine sand before converging and at the interface. From Eq. (5) we find with the so measured values for qioter= 65.7 mm h - ~. Hence the local flux density of the flow path at the interface is more than twice that of the mean flow q. Using the hydraulic conductivity functions of Fig. 1 and assuming unit gradient flow with qi,~ = K (0inter) w e find that the local water content of the flow path when entering the coarse layer is about 0=0.33 m 3 m -3. Lateral mixing is again intense in the coarse layer which becomes better visible immediately after entering the underlying fine layer. This second interface is passed without visible discontinuities. In the underlying fine layer two zones are visible, a darkly stained one in the centre and a lightly stained one at the edges of the flow path. This is an optical effect. The concentration spreads radially by lateral dispersion like a normal distribution (i.e. Eq. 3). The lightly stained area is the "tail" of the horizontal distribution. This dye pattern was observed also in other experiments (e.g. Fig. 4b) when the interface had no additional effect on lateral mixing. Fig. 4a shows a typical dye pattern for a layered column when air is entrapped. The dye was added here too as a point source. An air bubble is confined at one side in the coarse layer (lighter area at inner column walls) directing the streamlines towards the opposite side of the column, where the water content and hence the hydraulic conductivity are greater. Consequently, the stained flow path is diverted towards this side already in the upper fine layer. Distinct funnelling is visible as streamlines converge like in Fig. 3a. Again this can be explained by the increasing local flux density when approaching the coarse layer. Lateral

S. Koch, H. Fl(ihle r / Geoderma 63 (1994) 109-121

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mixing in the coarse layer looks different from that in Fig. 3a. The flow path is tortuous and it splits into several channels instead of spreading continuously. This pattern is caused not by lateral dispersion but by non-vertical flow. In the coarse layer entrapped air directs the streamlines towards regions of higher water content and hence higher hydraulic conductivity. The underlying interface causes an intense lateral spreading. It seems that at this interface water must be distributed laterally to various smaller pores of the fine layer. Only few pores in the coarse sand are filled with water, but they conduct water at a high local pore velocity. Because the average flow is continuous at the interface water escapes laterally and enhances lateral mixing. In the sublayer of fine sand the dye is evenly distributed, no lighter coloured region at the edge of the stained area is visible like in Fig. 3a. Hence the spreading at the interface is not caused by lateral dispersion but by diverging streamlines. The local flux density decelerates at this boundary and streamlines diverge. This is the reverse process which is observed at the upper interface. Roth et al. (1991), who investigated transport of chloride through an unsaturated field soil, found similar results. The tracer pulse they observed split into a slowly moving main front and a succession of fast pulses which moved rapidly. These fast pulses could be measured at a textural interface in the soil where they became diluted by increasing transverse flow of the solute flux. Laterally diverging movement of tracers was also found by Ritsema et al. (1993) in a field experiment. A water repellent horizon just at the soil's surface induced fingering. Below this water repellent zone water and solutes diverged laterally out of the preferential flow paths and from there the tracers were transported with fronts parallel to soil's surface. This effect of lateral flow at the interface cannot be detected in Fig. 4b where the lower boundary is inclined. Water passes this interface easily because of the inclination, and streamlines are already diverted towards the inclination to the right in the overlaying coarse layer. As a consequence lateral mixing in the coarse layer is more intense at the right-hand side of the stained flow path. Fig. 5 shows typical dye patterns in layered columns when dye was applied at the entire column's surface as a pulse. The experiments were stopped when the dye had passed the interface between the layers. In Fig. 5a the pulse passed an interface from coarse to fine sand and in Fig. 5b an interface from fine to coarse sand. In Fig. 5a the front of the pulse is sharp and slightly wavy. The dye pattern in Fig. 5b looks quite different. The front breaks up to form a preferential flow path at that point where the dye first reaches the boundary. To test the influence of the layer boundary we repeated the experiment with non-stratified columns and stopped them when the pulse had passed more than one half of the length of the column. The dye pattern, not shown here, had a sharp front slightly diluting at the upper end and retarding towards the walls of the column. We could not detect any preferential flow paths. It seems that breakthrough curves measured at the outflow end of the stratified columns would look completely different for different sequences of layers and this contrasts with the results of Selim et al. (1977) and Leij and Dane ( 1991 ), who assumed that breakthrough curves are not influenced by the sequence of layers.

ye pattern with an inclined lower boundary. Column size: 20 cm long, 11 cm diameter.

ix. 4. (a) Typicul dye pattern for a layered column with entrapped air in the coarse layer. Column size: 18 cm long, 6 cm diameter. The upper interface forces strca inlmu, converge. At the lower interface water is redistributed laterally (diverging streamlines! Tortuous channelling in the coarse layer is forced by the entrapped] ,~*r = h

az -z.: g

Fig. 5. Typical dye pattern for layered columns when dye is applied as a pulse over the entire surface. Column size: 20 cm long, 11 cm diameter. (a) Sequence coarsefine, (b) sequence fine-coarse.

~D

.z:

12!!

~' K.c/i. tt 1717hler/ ( ; e ~ d e r m a 63 ( 1t)941 109- 12 /

4. Conclusions Experiments on unsaturated flow in small columns show that lateral dispersion of solute is influenced by the grain size, or equivalently pore size distribution, of a porous material. Lateral mixing is more intense in coarse than in fine material for steady-state flow. Unsaturated beds tend to have a higher lateral dispersion coefficient than saturated ones. Flow patterns near textural boundaries are not determined by lateral dispersion but by non-vertical flow. A flow path passing an interface from afine to a coarser material may be forced to converge and form a preferential flow path. The local water content and flow density increase in the zone of convergence. In contrast when a flow path passes a textural boundary from coarse tofine material an intense lateral spreading of this flow path may be caused by the interface. Streamlines tend to diverge at such a textural boundary. Thus the sequence of layers influences breakthrough time and spreading of solutes under unsaturated flow conditions. The effects shown here at the laboratory scale also occur in the field. Most soils are stratified, and many have textural boundaries where sand lenses are surrounded by finer materials. Pore discontinuities, where large pores meet many finer ones or vice versa, are very common in coarsely aggregated soils. It is not clear from our investigation whether solute transport in homogeneously packed sands should be described by a convective-dispersive or stochastic-convective model. Several investigators (e.g. Khan and Jury, 1990) showed that transport in homogeneously packed columns is convective-dispersive. An ultimate validation with different column lengths would show that for our materials. But the effects on lateral mixing in stratified columns, indicate that longitudinal dispersion is distinctly influenced by stratification transverse to flow. Experiments monitoring breakthrough curves at several depths inside a column will be conducted to test whether our common modelling tools can describe solute transport across boundaries between contrasting layers.

Acknowledgements We thank Dr P. Reichert, E A W A G , for his help to find the analytical solution ofEq. ( 1 ). These experiments were presented as a poster at the scientific colloquium "Porous or Fractured Unsaturated Media: Transport and Behaviour" at Monte Verita, Centro Stefano Francini - ETH Ztirich, Ascona, October 5-9, 1992. Intense discussions with several participants brought valuable ideas which were included in this paper.

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Glass, R.J., Parlange, J.-Y. and Steenhuis, T.S., 1989a. Wetting front instability. 1. Theoretical discussion and dimensional analysis. Water Resour. Res., 25:1187-1194. Glass, P.J., Steenhuis, T.S. and Parlange, J.-Y., 1989b. Wetting front instability. 2. Experimental determination of relationships between system parameters and two-dimensional unstable flow field behavior initially dry porous media. Water Resour. Res., 25:1195-1207. Hart, N.-W., Bhakta, J. and Carbonell, R.G., 1985. Longitudinal and lateral dispersion in packed beds: Effect of column length and particle size distribution. Am. Ind. Chem. Eng. J., 31: 277-288. Harleman, D.R.F. and Rumer, R.R., 1963. Longitudinal and lateral dispersion in an isotropic porous medium. Fluid Mech., 16: 385-394. Hillel, D. and Baker, R.S., 1988. A descriptive theory of fingering during infiltration into layered soils. Soil Sci., 146: 51-56. Jury, W.A. and Fliihler, H., 1992. Transport of chemicals through soil: mechanisms, models and field application. Adv. Agron., 47: 141-201. Jury, W.A. and Roth, K., 1990. Transfer Functions and Solute Movement Through Soil: Theory and Application. Birkh~iuser, Basel, 226 pp. Khan, A.U.-H. and Jury, W.A., 1990. A laboratory study of the dispersion scale effect in column outflow experiments. J. Contain. Hydrol., 5:119-131. Kung, K.-J.S., 1990a. Preferential flow in a sandy vadose zone. 1. Field observations. Geoderma, 46: 51-58. Kung, K.-J.S., 1990b. Preferential flow in a sandy vadose zone. 2. Mechanism and implications. Geoderma, 46: 59-71. Leij, F.J. and Dane, J.H., 1991. Solute transport in a two-layer medium investigated with time moments. Soil Sci. Soc. Am. J., 55: 1529-1535. Leroy, C., Hulin J.P. and Lenormand, R., 1992. Tracer dispersion in stratified porous media: influence of transverse dispersion and gravity. J. Contam. Hydrol., 11: 51-68. Malicki, M.A. and Skierucha, W.M., 1989. A manually controlled TDR soil moisture meter operating with 300 ps rise-time needle pulse. Irrig. Sci., 10: 153-163. Malicki, M.A., Plagge, R., Renger, M. and Walczak, R., 1992. Application of time-domain reflectometry (TDR) soil moisture microprobe for the determination of unsaturated soil water characteristics from undisturbed soil cores. Irrig. Sci., 13: 65-72. Miyazaki, T., 1990. Visualization of refractional water flow in layered soils. Soil Sci., 149:317-319. Mualem, Y., 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res., 12: 513-522. Raats, P.A.C., 1973. Refraction of a fluid at an interface between two anisotropic porous media. J. Appl. Math. Phys., 24: 43-53. Ritsema, C.J., Dekker, L.W., Hendrickx, J.M.H. and Hamminga, W., 1993. Preferential flow in a water repellent sandy soil. Water Resour. Res,, 29:2183-2193. Roth, K., Jury, W.A., Fliihler, H. and Attinger, W., 1991. Transport of chloride through an unsaturated field soil. Water Resour. Res., 27: 2533-2541. Selim, H.M., Davidson, J.M. and Rao, P.S.C., 1977. Transport of reactive solutes through mnltilayered soils. Soil Sci. Soc. Am. J., 41: 3-10. Starr, J.L., DeRoo, H.C., Frink, C.R. and Parlange, J.-Y., 1978. Leaching characteristics of a layered field soil. Soil Sci. Soc. Am. J., 42: 386-391. Stauffer, F., 1977. Einfluss der kapillaren Zone anf instation~ire Drainagevorg~nge. Int. Rep. R13-77, Inst. f. Hydromechanik u. Wasserwirtschafl, ETH Ztirich, pp. 58-60, unpubl. Stauffer, F. and Dracos, T., 1986. Experimental and numerical study of water and solute infiltration in layered porous media. J. Hydrol., 84: 9-34. Van Genuchten, M.T., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soe. Am. J., 44: 892-898. Yule, D.F. and Gardner, W.R., 1978. Longitudinal and transverse dispersion coefficients in unsaturated Plainfield sand. Water Resour. Res.: 14, 582-588.