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Keywords:

  • unsaturated flow;
  • hillslope hydrology;
  • lateral flow;
  • transient flow

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[1] The distribution of soil moisture in a homogeneous and isotropic hillslope is a transient, variably saturated physical process controlled by rainfall characteristics, hillslope geometry, and the hydrological properties of the hillslope materials. The major driving mechanisms for moisture movement are gravity and gradients in matric potential. The latter is solely controlled by gradients of moisture content. In a homogeneous and isotropic saturated hillslope, absent a gradient in moisture content and under the driving force of gravity with a constant pressure boundary at the slope surface, flow is always in the lateral downslope direction, under either transient or steady state conditions. However, under variably saturated conditions, both gravity and moisture content gradients drive fluid motion, leading to complex flow patterns. In general, the flow field near the ground surface is variably saturated and transient, and the direction of flow could be laterally downslope, laterally upslope, or vertically downward. Previous work has suggested that prevailing rainfall conditions are sufficient to completely control these flow regimes. This work, however, shows that under time-varying rainfall conditions, vertical, downslope, and upslope lateral flow can concurrently occur at different depths and locations within the hillslope. More importantly, we show that the state of wetting or drying in a hillslope defines the temporal and spatial regimes of flow and when and where laterally downslope and/or laterally upslope flow occurs.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[2] Infiltration and fluid flow in hillslopes are transient, variably saturated processes. The temporal and spatial distribution of moisture content in the shallow subsurface, or flow regime, is vital to biological, chemical, and geomorphologic processes. For example, if rainfall-induced unsaturated flow has a significant lateral component parallel to the slope surface under wetting conditions, it could mechanically destabilize the slope leading to surface erosion or landsliding. On the other hand, if the rainfall-induced transient unsaturated flow is in the vertical direction under the drying conditions, it could stabilize the slope [Iverson and Reid, 1992; Reid and Iverson, 1992].

[3] In general, the flow regime in a homogeneous and isotropic hillslope is governed by rainfall characteristics, hillslope geometry, and the hydrological properties of the hillslope materials [e.g., Hewlett and Hibbert, 1963]. The major driving mechanisms for moisture movement are gravity and gradients in matric potential or moisture content [e.g., Sinai et al., 1981; McCord and Stephens, 1987]. Flow in a saturated homogeneous and isotropic hillslope under the driving force of gravity and a constant pressure boundary at the slope surface is, although not uniform, always in the laterally downslope direction under both transient and steady state conditions. [e.g.,Tóth, 1963; Freeze and Cherry, 1979; Reid, 1997].

[4] However, under variably saturated conditions, both gravity and moisture content gradients (or matric potential gradients as matric potential and moisture content are constitutively related) drive fluid motion leading to complex flow patterns [Sinai et al., 1981; Torres et al., 1998; Silliman et al., 2002; Thorenz et al., 2002]. In general, the flow field near the ground surface is variably saturated and transient, and the direction of flow could be laterally downslope, laterally upslope, or vertical. The division between downslope and upslope can be defined either as the direction normal to the slope (Figure 1a) or the vertical (Figure 1b).

image

Figure 1. Two commonly used definitions of downslope and upslope flow: (a) using the direction normal to the slope surface as the divide [e.g., Philip, 1991] and (b) using the vertical direction as the divide [e.g., Harr, 1977; Zaslavsky and Sinai, 1981b; Jackson, 1992]. According to Figure 1a, flux shown in the shaded area would be called downslope flow, whereas according to Figure 1b, it would be called upslope flow. In this paper, we adopt the definition presented in Figure 1b.

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[5] The exact conditions under which fluid flow is upslope, vertical, or downslope have been a subject of research over the past 3 decades. It is well known that layering and heterogeneity can promote strong lateral flow [e.g., Zaslavsky and Sinai; 1981a; Miyazaki, 1988; Ross, 1990; Warrick et al., 1997; Reid, 1997]. The highly heterogeneous and sometimes fractured nature of shallow subsurface environments, patterns of rainfall, and interactions with vegetation [e.g., Redding and Devito, 2008] can also induce non-Darcy and nonequilibrium unsaturated water flow [e.g., Swartzendruber, 1963; Hassanizadeh and Gray, 1987; Ritsema et al., 1993; Sinai and Dirksen, 2006]. For a homogeneous and isotropic hillslope with an initially uniform moisture distribution, the possible existence of lateral flow has been studied by Zaslavsky and Sinai [1981b] and McCord and Stephens [1987], who found that shallow unsaturated flow moves nearly parallel to the slope surface and converges in topographically concave areas in natural slopes. For a hillslope of initially constant moisture content that is subsequently subject to a constant but greater moisture content at the slope surface, Phillip [1991] arrived at a seminal analytical solution for the transient moisture content field as a function of two-dimensional space (planar) and time. He found that there is a time-independent component of horizontal flow into the slope and a time-dependent downslope (parallel to the slope surface) flow component. However, Phillip's [1991] solution does not apply to flow in hillslopes following the cessation of rainfall. Jackson [1992] solved Richards equation numerically to simulate flow both during and following precipitation in a homogeneous and isotropic hillslope. He concluded that the lateral downslope flow is largely a drainage phenomenon driven by the change from flux to no-flow boundary conditions at the surface. He showed that infiltration is nearly vertical during rainfall but when rainfall ceases, unsaturated flow near the surface becomes predominantly parallel to the slope surface.

[6] Because of the difference in defining the flow components in a hillslope compared to that in a flat environment, confusion about the meaning of “downslope” and “upslope” has persisted over the years. On the basis of the two definitions shown in Figure 1, Jackson [1992] noted that Phillip's [1991] analytical solution under infiltration conditions can produce downslope flow using the definition in Figure 1a but can never predict downslope flow using the definition in Figure 1b because the resultant of gravity (vertical) and gradient of moisture content (inward normal to the slope surface) is in the upslope direction depicted in Figure 1b. Philip [1993] could not accept the definition of upslope and downslope defined in Figure 1b and was unable to reconcile the contention of no downslope flow under infiltration made by Jackson [1992]. In the reply to Philip's [1993] comments, Jackson [1993, p. 4169] correctly pointed out “This controversy boils down to an issue of semantics.” On the basis of the definition by Philip [1991], shown in the shaded area in Figure 1a, flow in the region between vertical (z) and the slope-normal directions (z*) is called downslope flow. This is impractical and inconvenient in studying hillslope hydrology, as unsaturated flow under such constraint will always reach the water table before ever moving toward the toe of the slope. The definition shown in Figure 1b provides a clear way to identify the “drainage” mechanism (i.e., downslope flow upon cessation of rainfall) identified numerically by Jackson [1992] and confirmed experimentally by Sinai and Dirksen [2006]. Throughout this paper we use upslope and downslope flow relative to the vertical direction as shown in Figure 1b. Upslope flow is flow in the direction from vertically downward toward the direction normal to the slope and downslope flow is the direction from vertically downward toward the slope surface.

[7] Experimental work by Sinai and Dirksen [2006] showed that cessation of rainfall is not a general condition for the occurrence of downslope unsaturated lateral flow. On the basis of their experiments, they concluded that “the necessary condition for downslope lateral flow to occur is not zero-flow at the slope surface, but decreasing rain intensity” [Sinai and Dirksen, 2006, p. 11]. In general, the movement of soil moisture in a homogeneous and isotropic hillslope is driven by both gravity and the gradient of moisture content (or matric potential). Thus, the flow direction may not be directly and concurrently correlated to the changes in the boundary conditions. While all previous work regarding unsaturated lateral flow defines flow regimes by the boundary conditions, i.e., constant rainfall, increasing rainfall intensity, decreasing rainfall intensity, or cessation of rainfall [e.g. Jackson, 1992; Sinai and Dirksen, 2006], we contend that more general conditions that clearly delineate unsaturated lateral flow can be established by identifying appropriate hydrologic conditions within a hillslope.

[8] Our hypothesis is that at any point within a homogeneous and isotropic hillslope, downslope lateral unsaturated flow will occur if that point is in the state of drying, while upslope lateral unsaturated flow will occur if that point is in the state of wetting (Figure 1b). This hypothesis provides a quantitative criterion in the time domain. Thus, together with the spatial distribution of the state of wetting or drying, flow regimes in a hillslope can be completely defined. The states of wetting, steady moisture conditions, and drying can be uniquely determined by the moisture content at a point. Specifically, we define wetting as equation image, drying as equation image, and steady as equation image, with moisture content equation image and time t. One of the practical implications is that the dynamic lateral flow direction in a hillslope can be inferred from time series data of moisture content collected in the field. This criterion also becomes instructional in any numerical modeling and experimental testing of hillslope hydrology.

[9] In the following, we test the above hypothesis using a well-calibrated two-dimensional numerical model. The extensive experimental data collected by Sinai and Dirksen [2006] under increasing and decreasing rainfall intensity are used to calibrate the model, reconfirm their conditions necessary for unsaturated lateral flow, and test the new hypothesis. The numerical model is then used to quantify a more complex rainfall scenario qualitatively studied by Sinai and Dirksen [2006] to further validate the conceptual model.

2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[10] Although all previous conceptualizations of flow regimes in hillslopes consider drying or wetting at the slope surface, they overlook a simple fact that wetting and drying are dynamic processes, such that the domain of wetting and drying can concurrently occur and change even when the boundary condition can be defined by a single state of wetting or drying. Strictly speaking, the conditions identified in previous work for upslope lateral flow under wetting and downslope lateral flow under decreasing rainfall intensity [e.g., Jackson, 1992] are only correct in the region immediately below the slope surface. Regions away from the slope surface could be in any state, i.e., wetting, steady, and drying, that is governed by the unsaturated flow process under the competing driving mechanisms of gravity and the gradient of moisture content. This leads to the concept of time-and-space regimes of flow in hillslope environments and our conceptual model that at any point within a homogeneous and isotropic hillslope, downslope lateral unsaturated flow will occur at a point if that point is in the state of drying and upslope lateral unsaturated flow will occur at the point if that point is in the state of wetting. The physical reasoning for such a model is illustrated in Figure 2 and is described in the next paragraph.

image

Figure 2. Conceptual illustration of flow regimes in a hillslope: (a) steady state, (b) wetting state, and (c) drying and wetting state. Dashed lines with arrows show the hypothetical path of a particle in the flow field. The water table is assumed to be far below the slope surface such that it has no effect on flow direction near the slope surface.

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[11] Consider two adjacent points A and B in a homogeneous and isotropic hillslope with the same elevation near the surface of the hillslope (see Figure 2). If the two points reach a steady state (i.e., the moisture content becomes invariant with time) and assuming the rainfall intensity is less than the saturated hydraulic conductivity of the slope materials, then the water flux at these two points should be equal to the constant vertical infiltration rate at the surface, leading to no gradient in matric potential or moisture content between the two points. Thus, the resulting unsaturated flow is predominantly vertical because of gravity, as shown in Figure 2a. If the rainfall intensity increases to another value (still less than the saturated hydraulic conductivity) at some time to, soil at the surface becomes wetter leading to an additional gradient of moisture content in the direction normal to and inward from the slope surface. This will result immediately in upslope lateral flow in the soil adjacent to the surface but not in the region of points A and B. Flow at points A and B will remain vertical under gravity at time to. In time, the region of upslope lateral flow will propagate into the slope directly normal and inward from the surface as shown in Figure 2b. The additional wetting front will arrive earlier at point B than at point A as the slope-normal distance from the surface to point B is shorter than that to point A, yielding a gradient in matric potential between points A and B. When this happens, upslope lateral flow occurs. Using the same logic, if the rainfall intensity decreases or the rainfall ceases, the opposite will occur as point B will drain earlier than point A, leading to downslope lateral flow as shown in Figure 2c. At the same time, on the slope-parallel horizon between points C and D shown in Figure 2c, the wetting front has just arrived, leading to lateral upslope flow by the same mechanism shown in Figure 2b under the wetting state. Because rainfall intensity varies and moisture movement in a hillslope is a dynamic process, we contend that defining upslope or downslope lateral flow is better accomplished at the local scale rather than by the simple hydrologic conditions at the slope surface. This conceptual model, that the state of wetting and drying of any point in homogeneous and isotropic hillslopes completely defines the upslope or downslope lateral flow at the point, is tested and demonstrated in sections 3 and 4.

3. Calibration of a 2-D Model With Previous Sandbox Experiments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[12] We used HYDRUS-2D, a finite element based software that solves the Richards equation for flow in variably saturated porous media [Šimůnek et al., 1999], to clarify and test the conceptual model on the lateral unsaturated flow regimes in hillslopes. The Richards equation, in terms of volumetric moisture content equation image, can be expressed as [e.g., Lu and Likos, 2004]

  • equation image

where xj are coordinate directions with j = 1 and 2 being horizontal and j = 3 being vertical, h is the matric suction head, equation image is called the soil water retention curve, equation image is the hydraulic conductivity function, and equation image is the Kronecker's delta with nonzero value when j = 3. The first term in the bracket of the left-hand side of equation (1) represents the flux driven by the gradient of moisture content, and the second term represents the flux driven by the gravity. We first used the results of sandbox experiments conducted by Sinai and Dirksen [2006] to calibrate our numerical model.

[13] Sinai and Dirksen [2006] performed a series of laboratory sand tank experiments to capture unsaturated lateral flow in homogeneous isotropic sloping soils. In their experiments, one and two V-shaped trenches were carved into originally horizontal sand surfaces to represent hillslopes. The dimensions of the sandbox were 0.81 m in height, 1.12 m in length, and 0.05 m in width (Figure 3a). Special effort was taken to pack the sandbox with fine sand (i.e., particle sizes <0.25 mm) to create as homogeneous conditions as possible. They controlled the infiltration rate using a rainfall simulator composed of 100 hypodermic needles on the top of the sandbox. To prevent water accumulation at the bottom of the sandbox and in order to conduct series of experiments without repacking the sand, ceramic porous filter tubes were placed on the bottom of the sandbox, and a constant suction head of 2 m was maintained. To analyze flow regimes visually, food dye was used as a tracer. The dye was injected into the soil pack using hypodermic needles through small holes drilled through the sides of the sandbox. The locations of injection points for the experiment are roughly shown in Figure 3a, and those used in our modeling analyses are clearly shown in Figure 3b.

image

Figure 3. (a) Experimental setup of Sinai and Dirksen [2006], (b) our numerical simulation domain, boundary conditions, locations of dye (particle) injection used in numerical modeling (black dots), and hydrologic properties, and (c) soil water characteristic curves with the original sand reported by Dirksen [1978] and with the calibrated parameters.

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[14] The dimensions of our model domain shown in Figure 3b were the same as that of the sandbox used in Sinai and Dirksen's [2006] experiments. We used the geometry of a single 56 cm wide, 14 cm deep V trench (26.6° slope) with a flat surface at the left of the sandbox as a reference. The model domain contained 21,181 elements, with finer mesh at the top and coarsening toward the bottom boundary to increase computational accuracy and to focus on the upper part of the domain near the sloping surface (Figure 3b). Because complete information on the hydrologic properties of the fine sand used was not available from Sinai and Dirksen [2006], we used the soil water characteristic curve and hydraulic conductivity function reported by Dirksen [1978] for another similar set of sandbox experiments. RETC [van Genuchten et al., 1991] was used to calculate the hydrologic parameters for the van Genuchten [1980] model, i.e.,

  • equation image

where equation image is the porosity, equation image is the unit weight of water, equation image is the residual water content, equation image is the inverse of the air entry pressure for soil, and n is pore size parameter.

[15] Some adjustments were made to the hydrologic parameters to reproduce the timing of the wetting front movement and the pathlines in the experiment. The original [Dirksen, 1978] and the final calibrated hydrologic parameters used in the numerical model are also shown in Figure 3b. The effect of hysteresis was not accounted for in the analysis, as it does not appear to have a dominant effect on the direction of flow. Full treatment of the effect of hysteresis on the direction of unsaturated flow is beyond the scope of the paper, but preliminary numerical modeling results using the empirical hysteresis model of Scott et al. [1983] indicate hysteresis has a small (few percent) influence on the magnitude of horizontal flux but has no effect on the direction of flow.

[16] To replicate the experiments with a numerical model, we used an atmospheric flux boundary condition at the top, no flow boundaries on the sides, and constant pressure head boundary of −2 m at the bottom of the sandbox (Figure 3b). Throughout the simulations we used the various rainfall intensities applied in the original experiments. Residual moisture content of the fine sand equation image was prescribed as the initial condition for all simulations after calibration. Other calibrated hydraulic properties used in the modeling are shown in Figure 3b. The soil water characteristic curves with the original sand reported by Dirksen [1978] and with the calibrated parameters are plotted in Figure 3c for comparison.

[17] We generated pathlines during the simulations to visually compare the traces of dyes in the experiments with our numerical results. To capture the trends of the pathlines formed by the dye, the starting locations of the water “particles” tracked by the simulation were selected as close as possible to the reported locations and times of dye injection. The locations of injection points for the experiment and starting locations of tracer particles that we tracked in the simulations are shown as solid dots in Figures 3a and 3b.

[18] Four rainfall conditions were examined using the numerical simulations and compared with the experimental results quantitatively and qualitatively: (1) initial change to a constant rainfall intensity, (2) prolonged rainfall of a constant intensity, (3) decrease to zero rainfall intensity, and (4) varying rainfall intensity. The quantitative comparisons are presented in this section for the first three cases, and a qualitative comparison under varying rainfall intensity is presented in section 4.

3.1. Initial Change to a Constant Rainfall Intensity

[19] For the first set of simulations we applied a rainfall intensity of 50 mm/h to the top atmospheric boundary and simulated the movement of the wetting front and water particles as function of time. In the original experiments, both pathlines and streak lines were employed to track the flow field. A pathline is the trajectory of dye particles carried by the flow from an initial injection point, whereas a streak line is the trajectory of a pulse of dye particles injected at a given point in the flow domain during the experiment. Pathlines were used to visualize fluid motion from the top boundary, and streak lines were used to visualize flow direction over a shorter period of time near the point of interest. Because no information was available on the release times for the dye injected for the streak lines, we used only pathlines in our simulations for comparisons and for visualization. The traced particles were released at the beginning of the simulation from the top surface. At early times, less than 0.5 h after the 50 mm/h rainfall was introduced, the wetting front was approximately parallel to the slope surface (Figure 4a). As the wetting front proceeded, the effect of the slope gradually diminished (Figures 4b and 4c), and the wetting front approached horizontal (Figure 4d). Sinai and Dirksen [2006] traced the wetting front on the glass wall at 0.5 h intervals, and the simulated movement of the wetting fronts accords well with the experimental observations. Note that the comparisons of pathlines in the later times (Figures 4c and 4d) do not appear to be as good as those in the earlier times (Figures 4a and 4b). We attribute this to the use of pulse injections of dye for the streak lines, which would tend to smear and alter the pathlines (see the difference in pathlines near the slope surface in the middle of the slope shown in Figure 4d).

image

Figure 4. Comparison of flow field subject to a constant rainfall intensity of 50 mm/h between (left) experimental results and (right) numerical simulations at (a) 30 min after rainfall, (b) 1 h after rainfall, (c) 3.5 h after rainfall, and (d) 4.5 h after rainfall. Black lines in experimental results are streak lines. Black lines in numerical simulations are pathlines. See text for explanation of the differences between streak lines and pathlines. The saturated moisture content equation image for the simulations was 0.37.

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[20] Pathlines near the wetting fronts are almost perpendicular to them at all times in both experiments and simulations, and as the wetting fronts proceed, they become nearly vertical through time. This can be explained with our conceptual model presented in Figure 2, where the gradient in moisture content is diminished as infiltration progresses and gravity becomes the dominant driving mechanism for fluid flow. During earlier times, as rainfall infiltrates through the slope, the moisture content of the soil near point B in the downslope region increases before the moisture content in the upslope region near point A. Therefore moisture content will be lower in the upslope area such that water moves in the upslope direction under the gradient in moisture content. As the wetting front and particles move away from the slope surface, the gradient in moisture content is reduced, and the effect of the slope diminishes. Under this condition, gravitational force dominates over the gradient in moisture content, and the particles tend to move vertically. As can be seen from Figure 4, the particle movement lags behind the wetting front, and this lag becomes more obvious in time. This phenomenon is pertinent to unsaturated flow. The reason for this lag is the difference in the calculation of the velocities for the wetting front and the particle movement. The velocity of wetting front is calculated by equation image, where q is the Darcian flux and equation image and equation image are the initial and final water contents, respectively, meaning that the water partly or completely replaces the initial air phase. However, the velocity of the particle is calculated by equation image, which means that the particle moves in all of the water in the soil regardless if it is initial or added water (J. Šimůnek, personal communication, 2009).

[21] A more quantitative illustration of upslope flow during the wetting state can be seen in Figures 5a and 5b, where the deviation of the flow direction from vertical and the horizontal flow component as functions of time and space are shown for the profile in the middle of the slope (A-A′ in Figure 4a). When rainfall commences, the flow direction is normal to the slope surface, about 26.6° on average from vertical (Figure 5a), and the upslope component of flux (negative in our coordinate system) is the strongest (Figure 5b). In time, the upslope deviations from vertical diminish (Figure 5a). For example, after 10 min of rainfall, the flow in the upper 5 cm deviates upslope from vertical by between 23° and 32°. At 1 h, the deviations from vertical of the flow directions in the upper 15 cm have all decreased to around 15°. At 4.5 h, when the sandbox experiment was terminated [Sinai and Dirksen, 2006], the deviations from vertical in the upper 48 cm have all decreased to around 6°. Initially, the upslope flux (or the magnitude of horizontal flux) at the slope surface is at its highest and diminishes over time, whereas the vertical flux at the slope surface increases slightly in time and approaches the infiltration rate. The region of changes in both horizontal and vertical fluxes, as well as changes in total flux direction, coincide with the downward propagation of the wetting fronts in time, as shown in Figure 5. After 7 days a steady state (vertical flux equal to the infiltration rate and zero horizontal component) in the numerical simulations was reached. Three distinct features can be observed. First, compared to the arrival times of the wetting fronts shown in Figure 4, flow only changes direction in the domain behind the wetting fronts. Second, at a given time the angle of the upslope flow direction remains relatively constant with depth except near the wetting front, where it is drastically reduced to zero (i.e., no change in flow direction from vertical). Finally, the magnitude of the upslope flow direction is related to the moisture content. These three features indicate that it is the gradient of moisture content at the point of interest that controls the flow direction at that point. This is most pronounced and best illustrated at the moving wetting front, where gradient of moisture content (shown in Figure 5d) is the highest.

image

Figure 5. Profiles of simulated quantities in the middle of the slope (14 cm horizontally from the toe of the slope shown as A-A′ cross section in Figure 4a) at different times under constant rainfall intensity: (a) deviation of flux from the vertical direction, (b) horizontal flux, (c) vertical flux, and (d) moisture content.

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3.2. Prolonged Constant Rainfall Intensity

[22] For the second set of simulations, on the basis of the flow rates used by Sinai and Dirksen [2006], we applied continuous rainfall of 100 mm/h for 7 days until a steady state condition was reached. After 7 days of continuous rainfall the horizontal unsaturated flux was zero, and the vertical flux was 2.8 × 10−5 m/s (i.e., 100 mm/h). The simulation results confirm the experimental and theoretical findings that under steady state infiltration conditions unsaturated flow in hillslopes is vertically downward. This condition was used as the initial condition for the case of the cessation of rainfall shown in Figures 6 and 7, and discussed below.

image

Figure 6. Comparison of flow fields after cessation of steady rainfall intensity of 100 mm/h between (left) experimental results and (right) numerical simulations at (a) 30 min after rainfall ended, (b) 1 h after rainfall ended, and (c) 2 h after rainfall ended.

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image

Figure 7. Profiles of simulated quantities in the middle of the slope (14 cm from the toe of the slope shown as A-A′ cross section in Figure 4a) at different times after rainfall ended: (a) deviation of flux from vertical direction, (b) horizontal flux, (c) vertical flux, and (d) water content.

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3.3. Decreasing to Zero Rainfall Intensity

[23] For the third set of simulations we stopped the rainfall and observed the movement of particles through time. Before the cessation of rainfall, rainfall intensities of 100 mm/h were applied for 7 days until steady state conditions were reached. These steady state conditions were introduced as initial condition for the simulations of the drying state, and the results are shown in Figure 6 for flow fields at different times. After the rainfall ended, the pathlines simulated numerically change direction, indicating downslope lateral flow (shown in Figure 6, right), and compare favorably with the pathlines from the sandbox experiments (shown in Figure 6, left). Near the slope surface the pathlines were almost parallel to the sloping surface. This can be observed in both experiments and simulations. For the particles released farther away from the slope surface the pathlines become more vertical. Because the effect of hydrodynamic dispersion is evident in the experimental results, pathlines are difficult to quantify, making precise one-to-one comparison between the experiments and simulations impossible. However, the phenomenon of downslope flow and the overall flow patterns can be confidently observed in both the experiments and simulations. Our conceptual model illustrated in Figure 2 can be used to understand such phenomenon. Drainage in the downslope region near point B will be faster than that in the upslope region near point A because of the smaller volume of soil above point B than that above point A. This difference in the drainage rate leads to a downslope lateral gradient in moisture content and downslope lateral flow when the drying front arrives. As a particle moves away from the slope, the volume difference between that above point A and point B becomes less significant, and the lateral gradient in moisture content diminishes, leading to vertical water flow (Figure 6).

[24] The drying front is not as visually prominent as the wetting front in the sandbox experiments but can be clearly identified in simulations by plotting contours of water content shown in Figure 6 (right). It is also clear from the moisture content contours, the model scale, model materials (sand), and duration of infiltration that boundary effects strongly influence results in the area below the bottom of the valley.

[25] Further quantitative understanding of downslope lateral flow during the drying state can be gained by examining the total flux direction, the horizontal and vertical flow components, and moisture content as functions of time and space along a vertical profile located in the middle of the slope shown as A-A′ in Figure 4. Semiquantitative comparison with the sandbox experiments is also possible as Sinai and Dirksen [2006] recorded the ultimate change in the direction of the infiltration vector with soil depth during the drying experiments, as shown in Figure 7a. As in the wetting case, although the downslope horizontal flux is the strongest at the slope surface and diminishes with time, the vertical flux is the strongest at the slope surface and increases with time. The largest change in flow direction occurs at the slope surface and remains at about 62° down from vertical, indicating that the lateral downslope flow near the surface is nearly parallel to the slope surface (28°). The horizontal flow component (downslope is positive in our coordinate system shown in Figure 2) is the strongest at the cessation of rainfall and diminishes as time elapses. For example, 10 min after the rainfall stops, the deviation of the flow direction from vertical in the upper 10 cm varies from 0° at 10 cm depth to ∼60° at the slope surface (Figure 7a). Over time, the zone of downslope horizontal flow progresses deeper into the slope (Figure 7b). From the simulation, we can see that the drying process is much slower than that of wetting and that the sandbox experiment was terminated long before a new steady state was reached, as shown in Figures 7c and 7d. Even 4 months after the cessation of rainfall, water contents (Figures 7d) are well above the residual moisture content equation image of 0.02.

[26] Three distinct features of the drying state, in contrast to the wetting state, can be identified. First, the downslope lateral flow direction is always most pronounced at the slope surface, invariant with time, and approximately parallel to the slope surface (about 62° downslope from vertical in this case), whereas in the wetting state, the lateral flow direction is initially constant behind the wetting front but decreases with time (Figure 5a). The occurrence of the maximum deviation of the flow direction from vertical at the slope surface and the diminishing deviation away from the slope surface compare favorably with the sandbox experiment results, as shown in Figure 7a. Second, the downward progress of the drying front is much slower than that of wetting front. For the slope with the same fine sand, 7 days is sufficient to reach a steady state under constant rainfall, whereas a strong drying front does not occur after the rainfall ceases. This is probably due to the fact that the gradient of moisture content during drying generally acts against gravity. As illustrated in Figures 7a and 7b, 4 months after the rainfall stops, the direction of flow becomes more downslope, although the magnitude of the flux is small. Finally, the angle of the flow direction is related to the gradient in moisture content, which is always at its maximum at the slope surface. These distinct features again strongly indicate that it is the gradient of moisture content at the point of interest that controls the flow direction at that point.

[27] From the simulations of wetting, steady, and drying states of the experiments reported by Sinai and Dirksen [2006], we show that our 2-D model can reproduce well the flow regimes controlled by the boundary conditions: upslope lateral flow when rainfall of a constant intensity begins, vertical downward flow when steady state is reached, and downslope lateral flow when rainfall ceases. We further identify some distinct features pertinent to the spatial and temporal variations of the wetting and drying fronts, supporting the new conceptual model that the wetting and drying states at a given point completely define the upslope or downslope lateral flow at that point in a homogeneous and isotropic hillslope. From these analyses, we also find that the dimension and the time scales involved in the original sandbox experiments caused significant boundary effects, i.e., strong downward flow beneath the bottom of the valley. Because these experiments were conducted under monotonic wetting or drying scenarios with steady state initial conditions, the lateral flow direction does not change during each of the experiments. Therefore, we used a numerical simulation with larger dimensions with a different sand with varying rainfall intensity to further test our conceptual model below.

4. Flow Regimes Under Varying Rainfall Intensity

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[28] Sinai and Dirksen [2006] reported only qualitative results on four sandbox experiments in which they varied the rainfall intensity. The four experiments involved (1) periodic alternation of the rainfall rate from 100 to 0 mm/h every 15 min, (2) periodic alternation of the rainfall rate from 100 to 0 mm/h every 30 min, (3) periodic alternation of the rainfall rate from 150 to 50 mm/h with no specified period, and (4) fluctuations around a rainfall rate of 100 mm/h with maximum and minimum intensities of 250 and 0 mm/h, respectively. From these experiments they found that upslope lateral flow always occurred after rainfall rate increases and downslope lateral flow always occurred when the rainfall rate decreases. The latter finding supersedes Jackson's [1992] previous zero-rainfall condition for downslope lateral flow. On the basis of a series of experiments, Sinai and Dirksen [2006] constructed a hypothetical 6 h variable rainfall rate experiment and qualitatively predicted the expected temporal changes in the direction of unsaturated lateral flow. In the following, we used a numerical model of a field-scale sandy hillslope subject to two different rainfall conditions: constant rainfall of 40 mm/h for 1 h and the same 6 h varying rainfall intensity used by Sinai and Dirksen [2006].

4.1. Flow Regimes Under a Step Increase in Rainfall Intensity Followed by Cessation of Rainfall

[29] This simulation illustrates quantitatively, using a numerical model of a field-scale sandy hillslope, that the general conditions determining upslope, vertical, or downslope flow are not the rainfall conditions at the slope surface, but rather the state of wetting or drying within the hillslope. The conceptual model that the state of wetting or drying is the sole controlling factor determining lateral flow direction can be quantitatively confirmed by comparing time series of horizontal flux and moisture content variation. The changes of flow direction completely coincide with the changes in moisture content.

[30] The simulation domain, boundary conditions, and material hydrologic properties are shown in Figure 8a. One hour of constant rainfall of 40 mm/h is imposed on the slope surface with initially unsaturated hydrostatic condition throughout the entire hillslope domain. The evolution of flow patterns in this sandy hillslope is illustrated in the combined water content contour and flow direction vector plots for the upper middle region of the hillslope as shown in Figures 8b8f at different times after the rainfall ceases.

image

Figure 8. Simulation results from a 40 mm/h step increase then cessation rainfall intensity boundary condition: (a) simulated domain; velocity direction and moisture content (b) at 10 min, (c) at 1 h, (d) at 1.5 h, and (e) at 6 h; and (f) velocity direction and equation image at 6 h.

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[31] Qualitatively, the behavior of the flow regimes is consistent with our conceptual model; that is, the wetting or drying state at a particular point in hillslope controls the flow direction (Figure 8). When rainfall begins, upslope flow is most pronounced near the slope surface (see lateral flow direction at 10 min in Figure 8b). The upslope lateral flow propagates into the hillslope with the wetting front (Figures 8c8e). Behind the wetting front a quasi-steady zone develops (Figures 8c8e). This zone is characterized by dominantly vertical flow. Upon the cessation of rainfall, strong downslope lateral flow develops near the slope surface, as shown in Figure 8d. The downslope flow direction near the slope surface persists after the rainfall ceases (Figures 8d and 8e). On the basis of these observations we conclude that both upslope lateral flow and vertical flow can occur during rainfall and all three flow regimes; upslope lateral flow, vertical flow, and downslope lateral flow can occur after the rainfall ends (Figure 8f)

[32] The dynamic variation of the lateral flow directions and moisture content at different depths is quantitatively illustrated in Figures 9a and 9b, and profiles of flow direction, lateral and vertical flow magnitude, and moisture content in the middle of the hillslope are shown in Figures 9c and 9d. For given points in the hillslope at three different vertical depths (0.1, 0.5, and 1.0 m) from the slope surface, upslope lateral flow occurs when the water content at those points increases, nearly vertical flow occurs when water content is quasi-steady, and downslope lateral flow occurs when water content decreases. The onset of upslope lateral flow occurs with the change to a wetting state at the depths of 0.1, 0.5, and 1.0 m in Figure 9b. At the depth of 0.1 m the onset time is 3 min, at the depth of 0.5 m the onset time is about 40 min, and at the depth of 1.0 m the onset time is about 1 h and 45 min. A zone of nearly vertical flow behind the wetting front develops as it is reaches a steady state (e.g., depth between 0.0 and 0.6 m at t = 1 h in Figure 9c). Upon the cessation of rainfall the onset of downslope lateral flow coincides with the change to a drying state, as shown in Figure 9b for the 0.1, 0.5, and 1.0 m depths. At the depth of 0.1 m the onset of downslope flow is almost immediate, at the depth of 0.5 m the onset of downslope flow occurs after about 1.5 h, and at the depth of 1.0 m the onset of downslope flow is at about 4.5 h. While strong downslope lateral flow develops near the slope surface and advances downward as shown in Figures 8d8f, concurrently strong upslope flux continues near the wetting front (Figures 9b9d). As the wetting front continues advancing into the slope, three different flow regimes, upslope, vertical, and downslope, occur simultaneously and evolve dynamically, as is clearly shown in Figures 8c8f and Figures 9b9d.

image

Figure 9. Simulation results from a step increase then cessation rainfall intensity boundary condition: (a) moisture content as a function of time at different depths, (b) horizontal flux and rate of moisture content change as a function of time at different depths, (c) profiles of deviation of flow direction from vertical at the middle of the slope at different times, and (d) profiles of horizontal flux at the middle of the slope at different times.

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[33] The simulation also confirms the observations made in section 3. Upslope flow is most pronounced near the wetting front (Figures 8b8f, 9c, and 9d). Downslope lateral flow is mostly restricted to the region near the slope surface and can persist for a long time (Figures 8f and 9c). The general condition determining upslope, vertical, or downslope flow is not the rainfall condition at the slope surface; rather, it is the state of wetting or drying within the hillslope.

4.2. Flow Regimes Under Transient Rainfall Intensity

[34] The final numerical test of our conceptual model is to conduct a simulation under varying rainfall intensity rather than under constant rainfall conditions. For comparison and discussion we chose the rainfall history suggested by Sinai and Dirksen [2006, Figure 11]. They suggested that the necessary condition predicted by their hypothesis for changes in the direction of downslope lateral flow is decreasing rainfall intensity. The same sandy slope defined in section 4.1 (Figure 8a) was used with varying rainfall intensity shown in Figure 10a and the expected temporal directions of lateral flow by Sinai and Dirksen [2006] shown in Figure 10b.

image

Figure 10. Simulation results from varying rainfall intensity: (a) infiltration intensity versus time at the slope surface, (b) simulated lateral flux in the hillslope at the slope surface and anticipated flow direction from Sinai and Dirksen [2006], (c) water content at different depths as a function of time, and (d) horizontal flux and moisture content change at different depths as a function of time.

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[35] The 6 h varying rainfall history consists of five periods (Figure 10a): I, increasing rainfall intensity (from 0.5 to 1.5 h); II, decreasing rainfall intensity (1.5–3.5 h); III, no rainfall (from 3.5 to 4.75 h); IV, increasing rainfall (4.75–5.5 h); and V, constant rainfall (time >5.5 h). Sinai and Dirksen's [2006] hypothesis predicts that three flow regimes will occur, as shown in Figure 10b: upslope lateral flow from 0.5 to 1.5 h, downslope lateral flow from 1.5 to about 4.75 h, and upslope lateral flow for times greater than 4.75 h.

[36] Results using our conceptual model are shown in Figure 10b, where the lateral flow component at the slope surface follows a similar pattern as that proposed by Sinai and Dirksen [2006] but the arrival times of changes in lateral flow direction are determined by the wetting and drying states, not by the rainfall history at the slope boundary.

[37] As shown in Figure 10b, at the surface of slope the upslope lateral flux begins once the rainfall starts. However, when rainfall intensity starts to decline at 1.5 h, the lateral flow direction does not switch immediately to the downslope direction as predicted by Sinai and Dirksen [2006]; rather, the change in direction occurs at around 2 h and 20 min and persists until the end of the period III, when it switches to the upslope direction again in response to increasing rainfall intensity at the beginning of period IV. The simulation results shown in Figure 10b, however, confirm that vertically downward flow occurs when a steady state (i.e., for times >7 h) is reached.

[38] We anticipate that the flow regime in regions away from the slope surface will follow far more complicated patterns but can be precisely predicted by the state of wetting or drying at each point in the hillslope. The strong correlation between variations in the lateral flow direction and moisture content change with time is illustrated in Figures 10cand 10d at different depths from the slope surface. For example, at the depth of 0.1 m the duration of upslope component of horizontal flux (from 0.75 to 2.3 h) corresponds to the period when moisture content is increasing, the duration of downslope lateral flow (from 2.3 to 5.0 h) coincides with the period when moisture content is decreasing, and the duration of upslope lateral flow (from 5.0 to 7.0 h) coincides with the period when moisture content is increasing. For regions away from the slope surface (e.g., at the depth of 0.5 m), increase in moisture content and consequent upslope lateral flow begin at about 2 h, i.e., 1 h and 30 min after the rainfall begins. Moisture content decreases and downslope lateral flow starts around 4 h, i.e., 30 min after rainfall ceases. Downslope lateral flow persists for another 1 h and 15 min until about 5 h and 45 min into the simulation, when moisture content increases again and upslope lateral flow occurs. For a region deeper beneath the slope surface (e.g., at the depth of 0.7 m), lateral flow is mostly upslope. For a brief period (5.25–6.0 h), flow is laterally downslope, and then the magnitude of the flux is small. This brief period of downslope lateral flow occurs concurrently with a small decrease in moisture content (Figures 10c and 10d). For a region even deeper from the slope surface (e.g., at the depth of 1.0 m), flow never changes direction and is always upslope during the entire simulation period of 12 h. This is again consistent with the change in moisture content at this point in that it only increases and over time reaches a constant value.

[39] To summarize the observations above, the state of wetting or drying controls the flow direction, and the rainfall history is inadequate to define the flow regime in a hillslope. Specifically, when a point in a hillslope is subject to rapid increases in moisture content, upslope lateral flow occurs; when moisture content at the point stays relatively steady or the rate of change is small, predominantly downward vertical flow occurs; and when moisture content rapidly decreases, downslope lateral flow occurs. This can be quantitatively confirmed by comparing the time series of horizontal flux and the rate of change in moisture content (Figure 10d). At all points within the hillslope the changes of flow direction completely coincide with the changes in the rate of moisture content.

5. Summary and Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[40] The direction of saturated groundwater flow directions in isotropic and homogeneous hillslopes is generally laterally downslope under the driving force of gravity and constant pressure boundary conditions along the slope surface, regardless of transient or steady rainfall conditions. However, under unsaturated conditions an additional mechanism, namely, the gradient of moisture content or matric potential, plays a significant role. Furthermore, the boundary condition along the slope surface is better described by time-dependent flux rather than time-dependent moisture content. Such a transient flux boundary condition and concurrent moisture mechanisms of gravity and gradient in moisture content yield spatially and temporally complex flow patterns in hillslopes: upslope lateral flow, downward vertical flow, and downslope lateral flow. In general, all three patterns can simultaneously exist within the variably saturated flow regime in an isotropic and homogenous hillslope subject to time-varying flux boundary conditions.

[41] Previous studies considered flow patterns to be solely controlled by rainfall conditions at the slope surface [Jackson, 1992; Sinai and Dirksen, 2006]. They found that the condition for upslope lateral flow is increasing rainfall intensity, the condition for vertical flow is steady state infiltration, and the condition for downslope lateral flow is decreasing or zero rainfall intensity. These conditions were confirmed using laboratory-scale hillslope infiltration tests [Sinai and Dirksen, 2006]. Using a two-dimensional numerical model calibrated to those laboratory tests, we demonstrate that the previous conclusions for flow regimes in hillslopes are only valid in the region immediately adjacent to the slope surface. We show that flow directions in isotropic and homogeneous unsaturated hillslopes could be simultaneously upslope, vertical, and downslope throughout the soil profile. We propose a conceptual model that at any point within a homogeneous and isotropic hillslope, downslope lateral unsaturated flow will occur at a point if that point is in a drying state and upslope lateral unsaturated flow will occur at that point if that point is in a wetting state. We found that a quasi-steady zone behind the wetting front will develop and is characterized by predominantly downward vertical flow. Numerical simulations of the previous laboratory experiments under constant rainfall conditions and of a field-scale sand hillslope under varying rainfall intensity conditions confirm our conceptual model. Thus, if the history of moisture content within a homogeneous and isotropic hillslope is known (e.g., from field measurements), the pattern of unsaturated flow in the hillslope can be unambiguously defined. For hillslopes with significant heterogeneity and anisotropy, the pattern of unsaturated flow could be complex; thus, the role of the moisture content gradient versus matric potential gradient in controlling flow direction needs to be further explored.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References

[42] The authors would like to thank Mark Reid, Brian Collins, the Associate Editor Tetsu Tokunaga, four anonymous referees, and Charles Jackson for their constructive comments. This research is partially supported by a grant from the National Science Foundation (NSF CMMI-0855783) to N.L. and J.W.G. and a grant from the U. S. Geological Survey (USGS G09AC00085) to N.L.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Conceptual Model for the “Wetting” and “Drying” State Hypothesis
  5. 3. Calibration of a 2-D Model With Previous Sandbox Experiments
  6. 4. Flow Regimes Under Varying Rainfall Intensity
  7. 5. Summary and Conclusions
  8. Acknowledgments
  9. References