Experimental evidence of lateral flow in unsaturated homogeneous isotropic sloping soil due to rainfall



[1] This paper describes laboratory experimental evidence for lateral flow in the top layer of unsaturated sloping soil due to rainfall. Water was applied uniformly on horizontal and V-shaped surfaces of fine sand, at rates about 100 times smaller than the saturated hydraulic conductivity. Flow regimes near the surface and in the soil bulk were studied by using dyes. Streamlines and streak lines and wetting fronts were visually studied and photographed through a vertical glass wall. Near wetting fronts the flow direction was always perpendicular to the fronts owing to dominant matrix potential gradients. Thus, during early wetting of dry sloping sand, the flow direction is directed upslope. Far above a wetting front the flow was vertical due to the dominance of gravity. Downslope flow was observed during decreasing rainfall and dry periods. The lateral movement was largest near the soil surface and decayed with soil depth. Unstable downslope lateral flow close to the soil surface was attributed to non-Darcian flow due to variable temporal and spatial raindrop distributions. The experiments verify the theory that predicts unsaturated downslope lateral flow in sloping soil due to rainfall dynamics only, without apparent soil texture difference or anisotropy. This phenomenon could have significant implications for hillside hydrology, desert agriculture, irrigation management, etc., as well as for the basic mechanisms of surface runoff and erosion.

1. Introduction

[2] Most of the flow processes involved in the transformation of rainfall to surface and subsurface runoff occur in what is called the hillside portion of watersheds. Many early hydrological models assumed a Horton-type runoff process of excess water that cannot infiltrate into the soil profile. However, field observations by Hewlett and Hibbert [1963], Sinai et al. [1981], McCord and Stephens [1987], and Torres et al. [1998] indicated occurrence of downslope lateral flow in sloping soil even under unsaturated conditions. Silliman et al. [2002] and Thorenz et al. [2002] observed in laboratory experiments lateral unsaturated flow in the capillary fringe just above the slanted phreatic surface of the saturated zone. Nieber and Walter [1981] measured and simulated successfully unsaturated lateral flow in a sloping sand box.

[3] Philip [1991a, 1991b] suggested a theory that predicts upslope lateral flow in sloping, unsaturated, homogeneous, isotropic soil due to the dynamic nature of rainfall events. Earlier work by Zaslavsky and Rogowski [1969], Zaslavsky [1970], and Zaslavsky and Sinai [1981a, 1981b, 1981c, 1981d] paved the way for a theory of unsaturated lateral flow in sloping soil. Zaslavsky and Sinai [1981b] predicted lateral flow in sloping soil, as well as above phreatic saturated sloping aquifers. Zaslavsky [1970], Zaslavsky and Sinai [1981a, 1981c, 1981d], Wallach et al. [1989, 1991], McCord et al. [1991], Pan et al. [1997], and Warrick et al. [1997] considered, besides soil surface slope, soil anisotropy and layering as the major causes of lateral flow. McCord et al. [1991] adopted this approach and claimed unsaturated lateral flow to be the result of state-dependent anisotropy in the soil hydraulic conductivity. Torres et al. [1998] indicated an ambiguity in the proposed causes for the observed unsaturated lateral flow.

[4] Another source of confusion emerges from the definition of the terms downslope and upslope. Philip [1991a, 1993] defined flow downslope if its vector is directed downward from the normal to the surface slope, whereas Zaslavsky [1970], Zaslavsky and Sinai [1981b], McCord and Stephens [1987], and Jackson [1993] defined flow downslope if its vector is directed downward from the vertical (see Figure 10). Thus, when the flow vector is between the normal to the surface slope and the vertical, Philip [1991a, 1993] calls it downslope and the other authors call it upslope lateral flow. Philip [1991a, 1991b] attributed unsaturated hillslope lateral flow to the dynamics of rainstorms. For unknown reasons, however, he considered that only upslope lateral flow occurs early during the wetting phase of a rainfall event. Note that Zaslavsky's definition of downslope direction (Figure 10a) is opposed to that of Philip (Figure 10b). Jackson [1992, 1993] indicated the shortcomings of Philip's theoretical solution to predict downslope lateral flow. Philip's [1993] comment did not resolve this problem. This different definition was discussed in the present paper In nature, most downslope lateral flow events occur during the drying/drainage phase after a rainstorm [Jackson, 1992; Pan et al., 1997]. There is no need to assume anisotropy or layering as was pointed out by Zaslavsky and Rogowski [1969], Zaslavsky and Sinai [1981a, 1981c, 1981d], Bronstert and Plate [1997], Miyazaki [1988], Warrick et al. [1997], McCord et al. [1991], and Walter et al. [2004].

[5] Bodhinayaka et al. [2004] measured hydraulic conductivity of the topsoil of a sloping soil profile with the double-ring-infiltrometer method. Their field observations were compared with three-dimensional (3-D) numerical solutions of the Richards equation. Their results did not indicate an effect of soil slope on the measured and simulated hydraulic conductivity in the range from 0 to 20% water content. It implies absence of downhill lateral flow in their experiments. These surprising results can be explained, however, by the dynamics of their infiltration process, steady state infiltration in a double-ring infiltrometer. At steady state, lateral flow does not occur in homogeneous sloping soil, and so their findings did not conflict with the lateral flow observed as a result of dynamic changes in infiltration rate.

[6] In 1978, G. Sinai presented Zaslavsky's [1970] theory at the Soil and Environmental Department of the University of California, Riverside (UCR) and at the U.S. Salinity Laboratory. As a result, the authors conducted in 1978 relatively simple sandbox laboratory experiments at the U.S. Salinity Laboratory, aimed at clarifying the then already existing ambiguity concerning the process of lateral flow in unsaturated sloping soil. The experiments were designed to test whether lateral flow can occur in unsaturated homogeneous isotropic sloping soil as a result of rainfall intensities much smaller than the infiltration capacity of the soil. A box was packed with fine sand, rainfall was applied by a rainulator, and dyes were injected through a glass wall forming streamlines, streak lines, and pathlines. Lateral flow in unsaturated sloping soil was observed and Zaslavsky's theory was proven correct in these relatively simple experiments. In the wetting phase, the flow was upslope, during steady state it was vertical, and during drying/drainage, it was downslope. (Note that unless stated otherwise, upslope and downslope are defined throughout this paper according to Figure 10a in the present paper and by Zaslavsky and Sinai [1981b].) The dominance of the drainage phase in the generation of lateral flow in unsaturated isotropic homogeneous soil was apparent. Unfortunately, these findings were never published. Recently, by reading some of the new literature concerning unsaturated hillslope lateral flow reviewed above, we realized that the ambiguity as to the reasons for unsaturated lateral flow in hillslopes, as reflected, e.g., by Torres et al. [1998], still exists. We therefore decided to publish the experimental results obtained in 1978 to help resolve some of the ambiguity regarding the causes for unsaturated lateral flow in homogeneous sloping soil due to rainfall events.

2. Experimental Setup

2.1. Sandbox

[7] Two frames belonging to the front wall of the two-dimensional automatic gamma ray attenuation scanning apparatus described by Dirksen [1978] and Dirksen and Huber [1978] were mounted together. Each frame consisted of two 9.13-mm-thick glass plates mounted inside steel frames. The glass walls permitted visual observations of soil packing, wetting fronts, and dye streamlines. The overall dimensions of this sand box were height, 0.81 m, length, 1.12 m, and width, 0.05 m (Figure 1a). Ceramic porous filter tubes at the bottom of the sandbox were maintained at a constant suction of 2 m water by a vacuum pump. This drainage system was essential to prevent water accumulation at the bottom and facilitate new experiments without repacking.

Figure 1.

Experimental set up: (a) sandbox and (b) dimensions of one V-trench and location of dye injection holes.

2.2. Rainulator

[8] Water was applied on top of the sandbox by a rainfall simulator (rainulator). One hundred hypodermic needles (0.15 mm diameter) were mounted in the bottom of a 1.12 × 0.06 × 0.06 m pressure-equalizing box. Horizontal density of these needles was approximately one needle per square inch (25.4 × 25.4 mm). The rainulator was moved simultaneously in two horizontally perpendicular directions at different speeds such that every needle scanned precisely an area of 25.4 mm × 25.4 mm. A variable speed (Masterflex) tubing pump supplied water at a controlled constant rate to the rainulator. The spatial distribution of the application rate was tested and found fairly uniform (about 10% variation). Water application rates (rain intensities) were in the range of 25–250 mm/h.

2.3. Streamlines, Streak Lines, and Pathlines

[9] Flow directions can be derived from three types of flow lines: (1) A streamline is a curve that is everywhere tangent to the flux vector q, and thus it indicates the momentary direction of flow at every point in the flow domain; (2) a pathline is the curve followed by a hypothetical particle carried by the flow from an injection point; and (3) a streak line is the path followed by a stain of dye which was injected in a pulse at a given point in the flow domain. Under steady state conditions, all three types of lines merge into a single line.

2.4. Tracer Injection

[10] Tracers are a traditional experimental tool for visually analyzing flow regimes. Streamlines and streak lines were depicted in the sandbox by using green and red food dyes as tracers. The dyes were injected into the soil pack via hypodermic needles mounted with silicon rubber in small holes drilled through the glass (Figure 1b).

[11] The three types of flow lines were obtained as follows: (1) Streamlines were observed by photographs which were taken just after the injection of dyes; (2) pathlines were observed by following the path of dye, which was formed as a result of continuous injection of dye; and (3) streak lines were observed by following the motion of a stain of dye formed by an injected pulse of dye. Timing of simultaneous injection was important, particularly for the streamline study. For this, the needles were connected by tubes to an injection system and were pushed simultaneously through a number of holes (see Figure 12a). Progress of dyes and wetting fronts was observed visually and monitored by means of photographs.

2.5. Tests for Homogeneity and Isotropy of Sand Packs

[12] The sandbox was packed manually with fine sand (<0.25 mm) in layers of about 30 mm thick. Previous experience [Dirksen, 1978] indicated that special effort was needed during packing to avoid heterogeneity and anisotropy. The sand packing procedure was first tested for homogeneity and isotropy. In the first test, rain was applied on a horizontal soil surface for 48 hours at a constant rate of 100 mm/h, which was 100 times smaller than the saturated hydraulic conductivity. Significant anisotropic behavior was observed in that the streamlines were bent (Figure 2a). We suspected this to be due to a small horizontal crack formed during the packing. The sand box was repacked; this time each additional amount of sand was carefully stirred and mixed. The same test was repeated and now the streamlines were vertical, indicating a homogeneous isotropic sand pack (Figure 2b). Next, a V-trench was formed in one half of the same horizontal surface by carefully removing sand. Constant rainfall of 100 mm/h for 48 hours again yielded vertical streamlines and streak lines, indicating vertical flow also in the sloping surface section (Figure 2c). The improved packing procedure was followed for all following experiments.

Figure 2.

Preliminary tests for sand homogeneity and isotropy with steady rainfall: (a) bending of stream and pathlines with V-trench due to a horizontal crack; (b) vertical flow lines with flat sand surface; (c) vertical flow lines with flat and sloping soil surface.

3. Results and Discussion

3.1. Lateral Flow due to Surface Slope and Rainfall Events

[13] The influence of the surface slope on lateral flow was studied with one and two V-trenches carved into originally horizontal sand surfaces; a partially flat surface served as control. Spatially uniform rainfall was applied by the rainulator. Dyes were injected, and flow lines and wetting fronts were visually observed and marked on the glass wall, and then monitored by means of photographs. Three flow phases were examined: (1) wetting phase; (2) steady state phase; (3) redistribution and drainage phases following the termination of rainfall.

3.1.1. Wetting

[14] The wetting process with one and two V-trenches can be followed from the photographs in Figures 3 and 4, respectively, taken at the indicated times after rainfall was started. During the initial phase, the wetting front was more or less parallel to the soil surface, both in the horizontal and the sloping section (Figures 3a and 3b). As wetting proceeded and the fronts moved deeper, the effect of soil surface microtopography became weaker and the wetting front approached horizontal orientation (Figures 3c and 3d).

Figure 3.

Streamlines and streak lines with one V-trench and a flat surface section after (a) 30 min, (b) 1 hour, (c) 3.5 hours, and (d) 4.5 hours of rainfall. In addition, the wetting front is traced on the glass wall at 0.5-hour intervals.

Figure 4.

Streamlines and streak lines with two V-trenches after (a) 10 min, (b) 30 min, (c) 1 hour, and (d) 2 hours of rainfall.

[15] Streamlines near and above the wetting front were perpendicular to the front at all wetting stages (Figure 3). Pathlines from the soil surface to the wetting front were marked at regular time intervals; they became vertical after prolonged wetting. As the wetting front moved deeper, the flow in the upper soil layer became more vertical regardless of the soil surface slope. Thus upslope lateral flow was significant close to the sloping surface during initial wetting, but as the wetting proceeded the flow direction gradually became vertical. A second experiment with two V- trenches showed the same regime (Figure 4).

3.1.2. Steady State

[16] After 48 hours of continuous rainfall the flow regime approached that of steady infiltration. The suction drainage through the filter tubes was started when the wetting front approached the bottom of the box. The effect of surface slope had diminished long before (Figures 3d and 4d) and the pathlines and streamlines were nearly vertical. Flow lines under steady state rainfall conditions are shown in Figures 1a, 2b, and 2c.

3.1.3. Redistribution and Drainage

[17] This phase occurred after rainfall had been terminated. Flow direction changed and downward lateral flow was observed (Figures 5 and 6) . The angle of the streamline vectors changed gradually with depth. Near the soil surface, the direction of the streamlines was almost parallel to the soil surface, and at 15 cm depth the flow became nearly vertical. We define α as the angle between the soil surface and the horizontal direction (α = 23°) and β as the angle between the direction of the streamline and the vertical. Then, β varied from about 60° near the soil surface (βmax = 90°-α = 67°) to 0° at 15 cm depth. Figure 7 shows this relationship for both sides of the V-trench. These findings indicate significant downslope lateral flow beneath unsaturated sloping soil during the drainage phase.

Figure 5.

Streamlines and streak lines during drainage phase (a) 30 min, (b) 1 hour, (c) 2 hours after 100 mm/h rainfall ceased, and (d) 30 min after 250 mm/h rainfall ceased.

Figure 6.

Enlargement of the V-trench zone (a) 30 min and (b) 11 hours after 250 mm/h rainfall ceased.

Figure 7.

Shift in the direction of infiltration vector with soil depth of the two slopes of the V-trench during the drainage phase shown in Figure 5.

[18] Streak lines were also observed during the drainage phase. It was hard to take photographs of the streak lines so we illustrate their behavior in Figures 8 and 9. The shaded areas indicate where the sand was colored by dye injected through the holes marked by the black points. The little arrows in Figures 8b, 9a, and 9c indicate approximately the trajectory of the dye stains, which again indicate downslope lateral flow. The stained area grew with time from the beginning of the drainage phase.

Figure 8.

Illustration of streamlines and stained areas (a) 15 min, (b) 30 min, (c) 1 hour, and (d) 2 hours after 100 mm/h rainfall ceased (see Figure 5).

Figure 9.

Illustration of streamlines and stained areas (a) 30 min and (b) 3 hours after 250 mm/h rainfall ceased. (c) Difference between 100 and 250 mm/h rainfall after 30 min of drainage.

3.1.4. Discussion

[19] First, lateral flow must be defined in combination with the coordinate system selected for a long planar hillslope of homogeneous isotropic soil. The two key systems suggested by Zaslavsky [1970] and Philip [1991a] are depicted in Figure 10, where the arrows indicate the directions which are taken positive. Note the difference in the positive direction, both vertically and normal to the slope. As a result, positive infiltration fluxes are also defined differently, as depicted in the bottom panels of Figure 10. Observe that v and ud are not orthogonal, and hence they do not belong to the same Cartesian coordinate system. These differences in definitions have caused confusion in hillslope flow analysis, as Philip [1991a] pointed out. Another source of confusion arose from the term “downslope infiltration.” Philip [1991a, 1993] defined infiltration vectors directed upslope from the vertical as downslope lateral flow. Jackson [1993] stated that “from a hillslope hydrologist's viewpoint…unsaturated infiltration should be described as downslope only if the resultant points downslope of vertical….”

Figure 10.

The coordinate systems according to (a) Zaslavsky and Sinai [1981b] and (b) Philip [1991a]. (top) Positive coordinate axes of space and sloping soil and (bottom) the nomenclature of the corresponding flow vectors.

[20] Zaslavsky and Sinai [1981c] derived a basic relationship between the horizontal component of lateral flow qx and the vertical hydraulic potential gradient ∂ψ/∂z in unsaturated soil sloping at angle α with the horizontal x direction:

equation image

where ψ is the hydraulic potential, z is the vertical, positive upward coordinate (Figure 10a), and K is soil hydraulic conductivity.

[21] Several conclusions emerge from equation (1), which are summarized in Table 1: (1) Early during a wetting phase, the wetting front is near the soil surface. The hydraulic potential decreases with depth, i.e., ∂ψ/∂z > 0, and thus qx < 0 is directed upslope; (2) during steady rainfall, the hydraulic potential is the same at any depth, i.e., ∂ψ/∂z = 0, and thus qx = 0 and no lateral flow occurs downslope or upslope; and (3) just after rainfall recession or cessation, the soil is draining and the hydraulic potential increases with depth, i.e., ∂ψ/∂z < 0, and thus qx > 0 is directed downslope.

Table 1. Effect of Rainstorm Dynamics on the Direction of Lateral Flow
Rainstorm DynamicsStatus of Topsoilψ/∂zqxInterpretation
Onsetwetting>0<0upslope flow
Steadyconstant moisture00vertical flow
Terminationdrying, drainage<0>0downslope flow

[22] Zaslavsky and Sinai [1981d] solved numerically by finite element method the unsaturated-saturated differential equation in 2-D unsteady state condition. They assumed constant, as well as time-variable flux conditions at a sinusoidal soil surface. The solution for a homogeneous isotropic soil profile in the form of lines of equal hydraulic potential was shown in Figure 13 of their paper for the three typical phases mentioned in Table 1. Since the simulation was for isotropic homogeneous soil, the infiltration vector was perpendicular to the equal hydraulic potential lines or parallel to the hydraulic potential gradient. Early during wetting, the flux was directed normal to the soil surface, implying upslope lateral flow. Similar results were obtained by Philip [1991a] with his analytic solution and by Jackson [1992], Pan et al. [1997], and Miyazaki [1988] with numerical solutions and laboratory experiments. During steady state, the flux was vertical as predicted by Zaslavsky and Sinai [1981a, 1981c] and confirmed by many studies. During drainage, the flux above the wetting front was directed downslope, as was observed in our laboratory experiments and supported by simulations by Jackson [1992] and Pan et al. [1997]. Torres et al. [1998] indicated that earlier finite difference solutions of Freeze [1972] also showed downslope lateral unsaturated flow during the drainage period. The solutions of Philip [1991a, 1991b] and Su [2002] were limited to the wetting period, so they showed only upslope lateral flow. To obtain solutions for the drainage period, they should have extended their solutions to the boundary condition at z = 0 and the initial conditions: θ = θo during t ≤ 0; θ = θs during 0 < t < τ; and θ = θo during τ < t < ∞, where τ is the time that rainfall ceased. A more realistic analytical solution for infiltration under rainfall conditions would be obtained with flux rather than the water content boundary conditions used by Philip [1991a] and Su [2002].

[23] Jackson [1992] simulated hillslope lateral flow by a numerical solution of the Richard equation for a case similar to the sand dune observations of McCord and Stephens [1987]. His simulation confirmed the numerical solutions of Zaslavsky and Sinai [1981d]. He also pointed to the dependence of the infiltration flux direction on the dynamics of the rainfall events, i.e., the strong effect of the flux boundary conditions. He emphasized the drainage stage when the infiltration vector turned downslope and was almost parallel to the slope. These findings confirmed Zaslavsky's theory and also our laboratory observations (Figures 59). Jackson concluded, however, that the reason for downslope lateral flow during the drainage phase is the change in boundary conditions when rainfall stops: “… the hillslope surface becomes a no-flow boundary…therefore, at the intersection of the head contours and the soil surface, the head contours rotate and become perpendicular to the soil surface. Shallow unsaturated flow moves nearly parallel to the surface as a consequence of this boundary condition.” However, we found the reason is not only the change to no-flow boundary conditions, but more generally the change in the vertical gradient ∂ψ/∂z (equation (1)) as a result of temporal changes due to recession in the rain intensity as will be shown later.

[24] McCord and Stephens [1987] observed about 2-m downslope displacement of a bromide tracer plume in an unsaturated sandy hillslope during drainage periods following rainfall. They showed time- and place-average values of measured bromide concentrations. McCord et al. [1991] attempted to explain these field observations by adopting the concept of variable state-dependent anisotropy in soil hydraulic conductivity. However, this is not the only possible explanation of the observed phenomena, because McCord and Stephens [1987] did not report the dynamics of their tracer plume propagation. The downslope flow they observed can also be explained on the basis of lateral flow during the drainage periods in the dry spells between rainstorms, as was successfully simulated by Jackson [1992].

3.2. Lateral Flow due to Variation in Rain Intensity

[25] The observed significant lateral flow downward during the drainage phase was intriguing and induced us to test whether lateral flow can occur also as a result of changes in rain intensity, as predicted theoretically by equation (1). Wetting is the change from R = 0 to R = R0 and drainage the change from R = R0 to R = 0, where R is rain intensity. Generally, events include continuous dynamic changes in rain intensity, i.e., R = R(t). Dynamic changes in rain intensity result in dynamic changes of ∂ψ/∂z, and therefore lateral flows can be anticipated (Table 1).

3.2.1. Experiments

[26] Since real rain intensities rarely remain steady for long periods, several experiments were conducted with variable rain intensity. The results are reported qualitatively.

[27] Experiment 1 contained periodic alterations of rainfall rate from R = 100 mm/h to R = 0 every 15 min.

[28] Experiment 2 was the same as experiment 1, but with alterations every 30 min. Significant lateral flow was observed in the upper soil zone. The downslope component in experiment 2 was greater than that of experiment 1.

[29] Experiment 3 contained periodic changes in rain intensity from 150 to 50 mm/h, and back to 150 mm/h. Downslope lateral flow was observed during changes from 150 to 50 mm/h and upslope flow during changes from 50 to 150 mm/h.

[30] Experiment 4 featured fluctuations in average rain intensity of 100 mm/h. The highest temporal intensity was 250 mm/h and the lowest was 0 mm/h. Significant lateral flow was observed under these fluctuating conditions, while under a steady 100-mm/h rain only vertical flow was observed, not downslope flow.

3.2.2. Discussion

[31] The observations made during the experiments with variable rain intensities are applied to a hypothetical 6-hour period of variable rain intensity, including a dry period of 1 hour and 15 min (Figure 11a). The expected temporal directions of unsaturated lateral flow are shown in Figure 11b. During the beginning of a rainstorm, upslope lateral flow qx < 0 occurs due to the dominance of the water content gradient at the wetting front, which is about parallel to the sloping soil surface. As the storm continues and the wetting front penetrates deeper into the soil profile, the direction of lateral flow changes slowly toward the vertical (qx → 0). The sharp decrease in rain intensity after 1 hour and 30 min causes an immediate change in the lateral flow direction from upslope (qx < 0) to downslope (qx > 0). The downslope lateral flow continues during the recession and the drainage period (no rain) until the second rain storm begins at 4 hours and 45 min. The magnitude of the downslope lateral flow decreases with time due to the continuous decrease of the soil water content and unsaturated hydraulic conductivity of the topsoil. This phenomenon was both observed and predicted theoretically by Zaslavsky and Sinai [1981d] and Jackson [1992].

Figure 11.

Lateral flow due to a hypothetical variable rain intensity: (a) variable rain intensity; (b) unsaturated lateral flow direction; (c) 15-min time derivative of rain intensity (dR/dt-15min); (d) event-average time derivative of rain intensity (dR/dt-event).

[32] The lateral flow regime in Figure 11b, representing qualitative experimental results, was also predicted based on Zaslavsky's theory and the time derivative of the rain intensity, dR/dt. The latter was calculated in two ways: the average rate of change per 15 min (dR/dt-15 min) and the event-average rate of change (dR/dt-event). The highest value of dR/dt-15 min was 40 mm/h2 and the minimum value was −40 mm/h2 (Figure 11c). The dR/dt-event was calculated for the wetting period, recession period, drainage period, etc. (Figure 11d). This continued to be negative until t = 4.75 hours, while dR/dt-15 min was equal to zero from 3.5 to 4.75 hours. These findings highlight the importance of the event-average dR/dt in the generation of downslope lateral flow.

3.2.3. Anisotropy

[33] As mentioned in the section 1, many scientists dealt with infiltration into unsaturated sloping soils and attributed lateral flow to either macroscopic or microscopic anisotropy in the soil hydraulic conductivity. There is no need, however, to assume anisotropy or layering. Dynamic changes in the rain intensity are sufficient to cause downslope lateral flow as predicted by Zaslavsky and Sinai [1981d] and Jackson [1992]. Anisotropy or layering only increases downslope lateral flow, except at steady state. Theoretically, under steady rainfall conditions the downslope component qx is zero (i.e., vertical infiltration) in homogeneous sloping soil, but in an anisotropic or layered soil profile a downslope horizontal component exists (i.e., qx > 0). This behavior was predicted theoretically and observed [e.g., Pan et al., 1997; Miyazaki, 1988; Walter et al., 2004]. It misled Zaslavsky [1970], Zaslavsky and Sinai [1981a, 1981b, 1981c], McCord et al. [1991], Miyazaki [1988], and Warrick et al. [1997] to address primarly anisotropy and layering as the major causes of unsaturated downslope lateral flow. Except for Zaslavsky and Sinai [1981d], Jackson [1992], and Pan et al. [1997], downslope lateral flow due to temporal changes in boundary conditions was generally neglected. It is therefore emphasized in this paper. Pan et al. [1997] discussed the occurrence of downslope unsaturated lateral flow in sloping soils due to changes in rainfall rate. They stated,

[34] “This phenomenon cannot be observed from steady state analysis, because there is no wetting front under steady state. This is the reason that Wallach et al. [1989, 1991] observed that the surface elevation gradient by itself did not contribute to lateral transition of water in their perturbation analysis of a steady state infiltration problem with a general surface shape. Yet this phenomenon should occur quite often in the real world. In some sense, the sloping wetting front is a 'moving interface' between the wetter and dryer soil layers, each of which has a different hydraulic conductivity due to water content rather than textural difference.”

[35] The two approaches for explaining lateral flow presented in the discussion can be appreciated in view of our experiments and the statement by Pan et al. [1997] quoted above. The experiments showed clearly that lateral flow in unsaturated sloping soil can occur as a result of rainfall dynamics even in homogeneous isotropic soil. There is no need to assume anisotropy as a condition required to induce lateral flow. However, anisotropy magnifies the extent of lateral flow, and in addition explains why lateral downslope flow can occur during steady infiltration.

4. Non-Darcian Lateral Flow due to Variable Temporal and Spatial Raindrop Distributions

[36] We observed downslope lateral flow in the upper 70 mm during unsaturated steady infiltration. Because this contradicted the conventional assumption of vertical infiltration under steady rainfall in isotropic sloping soil, we conducted a special test. Dye was injected through hypodermic needles exactly at the soil surface, and steady rainfall was applied at a rate of 100 mm/h, which was about 2 orders of magnitude smaller than the saturated hydraulic conductivity of the sand. The streamline directions were indicated by the dye plume (Figure 12). The lateral downslope component was the largest near the soil surface and nearly parallel to it. It decreased slowly with depth, and at about 70 mm the flow direction became vertically downward. A closer look at that phenomenon revealed dynamic changes in flow direction with time just below the soil surface, creating a manifold of directions from parallel to the sloping soil surface to the vertical. The time-average direction of the dynamic unstable lateral flows was downslope.

Figure 12.

Non-Darcian lateral flow just beneath soil surface due to variable temporal and spatial raindrop distribution: (a) dye injection and plumes; (b) plumes of dye during steady rainfall indicating downslope lateral flow; (c) average streak lines issuing from dye injection holes at the soil surface, indicating downslope lateral flow (steady state).

[37] Theoretically, the infiltration flux into unsaturated uniform sloping soil under steady, spatially uniform rain intensity should be vertically downward. We suggest that the above reported deviation from this flow pattern just beneath the sloping soil surface may be due to non-Darcian flow as a result of nonuniformity of rain intensity on a small scale. Bear [1972a, 1972b] stated, “By employing the definition of REV (representative elementary volume), we have replaced the actual medium by a fictitious continuum in which we may assign values of any property to any mathematical point.” Darcy's law, in this case Richards equation, requires a continuous porous medium. Accordingly, non-Darcian microscopic effects may occur in domains smaller than the REV. The ratio between REV and typical pore size is often taken to be 100 or more. In our experiments, the typical pore size was about 0.25 mm and thus the REV about 25 mm, while the average raindrop diameter was about 1 mm. Thus the infiltration of single raindrops should be considered as a non-Darcian flow process and microscopic effects can be expected. On a field scale, a typical horizontal spacing between two adjacent raindrops that hit the ground simultaneously may be in the range of 25–500 mm. This makes the infiltration process similar to that of time- and space-variable point sources rather than that of spatially uniform rainfall, as is generally assumed. Flow patterns from fixed point sources at the soil surface have been studied intensively in relation to drip irrigation. At the initial stage of drip irrigation, infiltrating water is subjected to a sharp radial gradient of water content (pressure potential) around the point source. As the wetting proceeds, gravity forces gradually become dominant, changing the flow pattern to form the well-known “onion-shaped” wetting zone beneath a point source. Similar behavior occurs beneath point sources on a sloping soil surface, with one difference: Flow directions are asymmetrically skewed downslope. This asymmetry causes unstable downslope lateral flow that changes direction with time due to variable temporal and spatial raindrop distributions.

5. Conclusions

[38] Lateral flow in the unsaturated vadose zone is a physical process that was not yet fully understood. Disagreement and confusion among scientists emerged from the definition of downslope lateral flow in hillslope hydrology. Another source of confusion was the assumption as to what causes lateral flow: either anisotropy and layering of the soil profile, or dynamic changes of the boundary conditions at the soil surface.

[39] Laboratory experiments were conducted in a sandbox to try to investigate these issues. The effect of rainfall dynamics on the direction of the infiltration vector in sloping soil was clearly demonstrated. The infiltration vector was directed normal to the soil slope (upslope) during the wetting phase, changed to the vertical direction during steady rainfall, and moved on to downslope from the vertical during the drainage phase, precisely as was found by simulation earlier by Sinai and Zaslavsly and published by Zaslavsky and Sinai [1981d].

[40] The explanation given by Jackson [1992], that downslope unsaturated lateral flow occurs after rainfall ceases, was shown to be only partially correct. The necessary condition for downslope lateral flow to occur is not zero-flow conditions at the soil surface, but decreasing rain intensity. Under such conditions, significant downslope lateral flow was also observed. Table 2 summarizes qualitatively our findings regarding lateral flow. For four dynamic stages, the relevant directions of lateral flow are indicated. The hope is that this will clarify some of the confusion reflected in the literature regarding hillslope hydrology. Note the different flow behavior in isotropic and anisotropic soils, and the differences between the normal Darcian flow in the bulk soil and the non-Darcian flow in the upper soil layer.

Table 2. Direction of Lateral Flow due to Rainfall Dynamics in Isotropic Soil, Anisotropic Soil, and in the Upper Soil Layera
Rainfall DynamicsChange in Rainfall Rate ∂R/∂tDirection of Lateral Flow Component, qx
Darcian FlowNon-Darcian Flow due to Raindrop Distribution
Isotropic SoilAnisotropic or Layered Soil
  • a

    Variables are qx, horizontal component of lateral flow; equation imagex, time average of qx; qx > 0, lateral flow downslope from vertical; qx < 0, lateral flow upslope from vertical; qx = 0, vertical flow.

Wetting>0qx < 0qx < 0equation imagex > 0
Steady0qx = 0qx ≥ 0equation imagex > 0
Recession<0qx > 0qx > 0qx > 0
Drainage0qx > 0qx > 0equation imagex = 0


[41] The research was conducted at the USDA Salinity Laboratory, Riverside, California, in cooperation with the Soil and Environmental Department, University of California, Riverside. Thanks are extended to the staff of these institutions for enabling us to perform these experiments and to M. Huber for assisting us with these experiments. We also thank D. Zaslavsky, J. Morin, and E. Henkin, who encouraged us to conduct the research. Thanks are also due Ruth Adoni for editing, to Arieh Aines for the graphic works, and to David Nezlobin for assisting in the literature survey.