Modeling the dynamics of soil erosion and size-selective sediment transport over nonuniform topography in flume-scale experiments

Authors


Abstract

[1] Soil erosion and the associated nutrient fluxes can lead to severe degradation of surface waters. Given that both sediment transport and nutrient sorption are size selective, it is important to predict the particle size distribution (PSD) as well as the total amount of sediment being eroded. In this paper, a finite volume implementation of the Hairsine-Rose soil erosion model is used to simulate flume-scale experiments with detailed observations of soil erosion and sediment transport dynamics. The numerical implementation allows us to account for the effects of soil surface microtopography (measured using close range photogrammetry) on soil erosion. An in-depth discussion of the model parameters and the constraints is presented. The model reproduces the dynamics of sediment concentration and PSD well, although some discrepancies can be observed. The calibrated parameters are also consistent with independent data in the literature and physical reason. Spatial variations in the suspended and deposited sediment and an analysis of model sensitivity highlight the value of collecting distributed data for a more robust validation of the model and to enhance parametric determinacy. The related issues of spatial resolution and scale in erosion prediction are briefly discussed.

1. Introduction

[2] Soil erosion and the subsequent transport of sediment by overland flow can have adverse effects on surface water quality. It is well known that clay and silt particles are preferentially transported by overland flow [Parsons et al., 1991; Sutherland et al., 1996; Leguédois and Le Bissonnais, 2004]. These particles carry with them high concentrations of sorbed nutrients and other potentially deleterious elements [Sharpley, 1980; McDowell et al., 1989; Zhang et al., 2003; Quinton and Catt, 2007]. Thus, it is important, in trying to model the effects of soil erosion on surface water quality, to predict not just total sediment fluxes but also the particle size distribution (PSD) of the eroded sediment.

[3] There are a number of size-selective erosion models in the literature [e.g., Woolhiser et al., 1990; Hairsine and Rose, 1991, 1992a, 1992b; Flanagan and Nearing, 1995; De Roo et al., 1996]. With the exception of the Hairsine-Rose (H-R) model, all adopt the transport capacity concept, which has been shown to be deficient under certain conditions [Huang et al., 1999; Sander et al., 2007]. The H-R model treats erosion and deposition processes independently, with the net outcome being the difference between these two process groups. It also models the development of a deposited layer that differs from the original soil in its cohesion and PSD. The feedback on erosion processes due to an evolving surface soil is thus accounted for; this allows us to model dynamical changes in erodibility and the PSD of the eroded soil.

[4] The H-R model has been applied with some success to various scenarios of soil erosion [e.g., Sander et al., 1996; Hairsine et al., 1999; Rose et al., 2003; Van Oost et al., 2004]. These applications, however, have been limited in various respects, e.g., using simplified approximations of the model, neglecting spatial or temporal variability, restricting to net erosion or net deposition conditions, and using either rainfall or runoff erosion processes. Moreover, the calibrated model parameters have not always been physically reasonable.

[5] Recently, Heng et al. [2009] presented a finite volume implementation of the full H-R model coupled with the Saint-Venant equations for overland flow. This allows us to model complex scenarios of soil erosion, where the processes of erosion by raindrop impact and surface runoff are concurrently active. We can also simulate the wetting up period, surface ponding, and the ensuing flow over nonuniform microtopography. In this paper, we apply the implementation to a series of detailed flume-scale experiments [Armstrong et al., 2010] to test the model's ability to reproduce the observed dynamics in terms of the PSD as well as the total concentration of sediment in the runoff. Brief descriptions of the model and the experimental study are presented in sections 2 and 3, respectively. We account for the topographical variation between experimental runs as well as the spatial heterogeneity per run by using photogrammetry-derived surface measurements in the model in section 4. Section 5 contains an in-depth discussion of the H-R model parameters (including the constraints) and our treatment of the preponding period. The results are presented in section 6, with a discussion of the calibrated parameters and a sensitivity analysis. We conclude with a brief discussion of spatial resolution and scale in erosion prediction.

2. Model Description

[6] We use the Saint-Venant equations to describe overland flow:

equation image
equation image

where h is the flow depth, q is the unit discharge, P is the rainfall intensity, f(t) is the rate of infiltration, and z is the bed elevation. The friction slope Sf can be expressed as

equation image

where n is Manning's roughness coefficient.

[7] It may be argued that the Saint-Venant equations are not valid where the roughness is of the same order of magnitude as the flow depth. Nevertheless, they are an accepted approximation of overland flow [Zhang and Cundy, 1989; Tayfur et al., 1993; Esteves et al., 2000]. The specific form of the equations adopted in this work also assumes that slopes are gentle and that flow momentum is not significantly affected by infiltration. These simplifications are appropriate under most occurrences of shallow overland flow. The effect of rainfall on flow momentum is accounted for in Manning's n (see section 5.1).

[8] Surface sealing is a common phenomenon with many soils under heavy rainfall (see Assouline [2004] for a review). It reduces the infiltration capacity of a soil, eventually leading to overland flow. Attempts have been made to model the physical process and the consequent effect on infiltration, but the parameters required can be onerous. We have adopted, as a first attempt, the simple Hortonian model

equation image

where f0 and fc are the initial and steady state infiltration capacities, respectively, and k is a decay constant, as a phenomenological description of the process.

[9] The H-R equations governing soil erosion are

equation image
equation image

where ci is the concentration in mass per unit volume and mi is the deposited mass per unit area of sediment in the ith sediment class. The terms on the right-hand side of (3) and (4) are the rates of sediment entrainment (by surface runoff), reentrainment (of the deposited sediment), detachment (by raindrop impact), redetachment, and deposition, given by

equation image
equation image
equation image
equation image
equation image

respectively. Here equation image, where mT = ∑mi and equation image is the mass of deposited (cohesionless) sediment required to shield the original cohesive soil from further erosion, pi is the fraction of class i particles in the original soil, equation image (W m−2) is the effective stream power, J (J kg−1) is the energy expended in entraining a unit mass of cohesive sediment, equation image and equation image are the densities of sediment and water, respectively, a and ad (kg m−3) are the detachabilities of the original soil and the deposited sediment, respectively, and vi (m s−1) is the settling velocity of class i sediment in still water. Implicit in (9) is the assumption that the suspended sediment is uniformly mixed in the water column.

[10] The system of equations (1)(4) can be solved using the finite volume method [Heng et al., 2009]. We apply the model to a flume-scale experimental study described in section 3.

3. Experimental Study

[11] The experimental study is detailed by Armstrong et al. [2010] but is summarized here for completeness. A 3.9 m × 1.4 m flume was used in this study. To minimize boundary effects, the collection trough at the end of the flume was divided into three sections; surface runoff flowing into the two outer sections (each 0.15 m wide) was disregarded.

[12] The rainfall simulator is similar to that described by Strauss et al. [2000] and produces raindrops with a mean diameter of 2 mm that impact the soil surface at approximately 5.5 m s−1. The nozzles are opened and closed intermittently to regulate the rainfall intensity; the sampling period was designed to minimize biases due to this intermittency [Armstrong and Quinton, 2009]. While the mean rainfall intensity was fixed at 47 mm h−1 for this study, there was some spatial variation, particularly in the longitudinal direction (Figure 2), that can be attributed to the overlapping of nozzle spray areas. The numerical model allows us to specify the rainfall intensity in each finite volume cell, so we were able to take this spatial variation into account.

[13] Silt loam (4.6% clay, 49.9% silt, and 45.5% sand) was placed and packed in the flume to various slope gradients (3%, 6%, and 9%), with triplicate runs (R1, R2, and R3) for each gradient. We determined the PSD of the silt loam and divided soil particles into 10 equal proportion size classes. The class representative particle sizes and settling velocities (determined using Cheng's [1997] formula, assuming a sediment density of 2000 kg m−3) are shown in Table 1.

Table 1. Class Representative Soil Particle Sizes and Corresponding Settling Velocitiesa
ClassSize (μm)Settling Velocity (×10−3 m s −1)
  • a

    The soil fraction in each class is 10%.

 12.90.0034
 25.60.013
 38.70.031
 4130.065
 5180.13
 6260.27
 7380.60
 8591.4
 9953.4
 101709.6

4. Photogrammetric Soil Surface Measurement

[14] Stereoimages of the flume were captured before and after each run using two high-resolution digital cameras (Nikon D80, 10 MP) mounted above the flume. These were processed using Leica Photogrammetry Suite 9.0 to generate digital elevation models (DEMs) of the soil surface at 1cm resolution. The DEMs revealed irregularities, such as depressions and confluences on the soil surfaces, as well as differences in topography between the experimental replicates for each slope gradient [Heng et al., 2010]. These small-scale topographic features and variations may partly account for the observed runoff and erosion responses. The numerical model allows us to test this hypothesis.

[15] To make use of the topographic data in the one-dimensional numerical model, the following procedure was adopted.

[16] 1. Extract slope profiles from each DEM along three longitudinal transects spaced 0.3 m apart.

[17] 2. Average the slope profiles to obtain a one-dimensional approximation of the topography.

[18] 3. Downsample the data to 10 cm resolution.

[19] 4. Construct a cubic spline on the basis of the downsampled data.

[20] 5. Discretize the cubic spline at 2 cm resolution to obtain the input mesh for the model.

[21] Figure 1 compares the averaged DEM profile (after step 2) with the final model input mesh for run 1 on the 3% slope.

Figure 1.

Comparing the model input topography with the digital elevation model (DEM) profile (averaged over three transects) for run 1 on the 3% slope. An enlargement of one section of the slope is shown in the inset.

[22] This procedure essentially masks the grain- and aggregate-scale microtopography to satisfy the condition of a gradually varying flow (as assumed in the Saint-Venant equations). At the same time, sufficient detail is preserved at a practical resolution to allow differentiation between slopes (Figure 2).

Figure 2.

Rainfall and topographical inputs to the model.

5. Model Application

5.1. Surface Runoff

[23] The infiltration parameters f0, fc, and k were calibrated for each run on the basis of the time to runoff, the shape of the discharge curve, and its asymptotic value. The time to runoff (Table 2) is the time period between the start of rainfall and the collection of the first sample. Where there are significant depressions in the soil surface, the time to runoff can be much greater than the time to ponding, which occurs when P = f(t). For overland flow, we assume n = 0.05, after Engman [1986]. This friction coefficient includes the retarding effect of raindrop impact.

Table 2. Observed Times to Runoff
Slope (%)ReplicateTime to Runoff (min)
3114
 211
 34
615
 24
 32
916
 25
 33

5.2. Soil Erosion

[24] The H-R model has a number of parameters that are difficult to measure accurately. Nonetheless, since every parameter has physical meaning, we may restrict the parameter space within reasonable bounds. Moreover, constraints may be derived that relate one parameter to another, effectively reducing the number of parameters.

[25] Soil detachment by raindrop impact is known to vary with flow depth [e.g., Mutchler and McGregor, 1983; Schultz et al., 1985; Torri et al., 1987; Proffitt et al., 1991]. Schultz et al. [1985] and Torri et al. [1987] showed that splash transport declined exponentially with the depth of ponding, becoming negligible for h > 2 mm. On the other hand, Mutchler and McGregor [1983] and Proffitt et al. [1991] showed that soil loss in surface runoff continued to be significant up to h = 20 mm. Clearly, sediment detachment and suspension due to raindrop-induced turbulence remain important in ponded water even as direct transport by raindrop splash becomes negligible.

[26] Mutchler and McGregor [1983] proposed an exponential relation between near-steady soil loss (SL, in kg h−1) and h:

equation image

[27] This makes SL (due to rainfall) tend to a nonzero value as h → ∞, which is not reasonable. The power law suggested by Proffitt et al. [1991] seems more appropriate:

equation image
equation image

where a0 and ad0 are the detachabilities of cohesive and loose sediment, respectively, for flow depths below a threshold h0, and b is a positive constant. Since SL = qcT, where cT is the total sediment concentration in the runoff and cTa at steady state [Hairsine and Rose, 1991], a power law for a implies the same for SL. The best fit power law for the data of Mutchler and McGregor [1983] is SL = 133.4h−1.13, with R2 = 0.90. In other words, for the silt loam used in their experiments, b = 1.13. For comparison, Proffitt et al. [1991] found b ≈ 0.66 for the two clayey soils in their study. We conveniently take b = 1 for the silt loam in this study. Following Mutchler and Hansen [1970], we set h0 = 0.33 DR, where DR is the mean raindrop size, 2 mm in our case.

[28] Since the mass of deposited sediment required to shield the original soil decreases with decreasing rainfall erosivity, equation image is implicitly a function of h. In fact, all else being equal, equation image is a constant [Lisle et al., 1998]. Thus, analogously to ad,

equation image

[29] Another way of looking at a0 (kg m−3) is that it is the ratio of the kinetic energy per unit rainfall (J m−3) to the energy required to detach a unit mass of the original soil (J kg−1). Thus, we can write

equation image

where vR is the rainfall impact velocity. Given vR = 5.5 m s−1, we have J ≈ 15,000/a0 as another parametric constraint.

[30] Hairsine and Rose [1992a] originally formulated equation image as equation image, where equation image, equation image is the threshold stream power below which there is no entrainment or reentrainment, and F is the effective fraction of excess stream power. We can rewrite the original formulation as

equation image

by making the substitutions equation image and equation image.

[31] equation image can be thought of as the stream power expended on form roughness. Its form implies that form resistance increases with stream power or, equivalently for uniform flow, discharge. This is consistent with observations that form resistance increases with the Reynolds number [Rauws, 1988] or, perhaps more accurately, flow depth [Lawrence, 1997] as the form roughness elements are gradually inundated. A reasonable and parsimonious approximation would be to assume that equation image is constant, implying that equation image varies linearly with equation image.

[32] The last term in equation (13) is simply the critical stream power in connection with grain roughness alone [Abrahams et al., 1998; Ferguson, 2005], which we will denote as equation image. With some algebraic manipulation, the formulation proposed by Abrahams et al. [1998] becomes

equation image

where equation image is the critical Shields parameter for incipient motion, D50 is the median particle size, and equation image. We note that the formulation of Ferguson [2005] is virtually identical, if we neglect relative size effects. For equation image, equation image kg m−3, and D50 = 20 μm, the above formulation gives critical stream powers of 1.9 × 10−4, 1.5 × 10−4, and 1.2 × 10−4 W m−2 for slopes of 3%, 6%, and 9%, respectively.

[33] The above rationalization reduces the number of independent parameters in the H-R model to four: a0, ad0, equation image, and equation image. The detachability of the original soil a0 may be sensitive to initial conditions that affect cohesion and soil strength [Le Bissonnais et al., 1995; Römkens et al., 2001; Mamedov et al., 2006] and so may vary between experimental runs. On the other hand, we may assume ad0 to be a constant for the soil used in this study. Obviously, ad0a0 is a parametric constraint. Furthermore, we deduce from Abrahams and Parsons [1994] that equation image is larger than 0.9.

[34] On the basis of bulk density measurements [Armstrong et al., 2010], a 1 mm thick layer of deposited sediment has a mass of approximately 1.4 kg m−2. The deposited layer required to shield the original soil completely cannot be much thinner than that. Thus, a physically reasonable lower bound for equation image is 1 kg m−2.

[35] The four independent parameters were calibrated to fit the experimental data in terms of PSD as well as total sediment concentration. Given the computational cost of solving the model, calibration was performed manually, with the goodness of fit evaluated visually. The effect of uncertainties in the parametric constraints and assumptions are examined in section 6.5.

5.3. Initial and Boundary Conditions

[36] We used the same initial conditions for all model runs: h = 0, q = 0, and mi = 0. The periods before ponding were explicitly modeled. Sediment detached by raindrop impact in the preponding period is not transported off-site (assuming that the net sediment transport due to rain splash is negligible on gentle slopes), so there is a buildup of loose sediment on the soil surface. Since ad is typically much larger than a, the presence of predetached sediment on the soil surface results in a more rapid initial rise in suspended sediment concentration as well as a higher peak concentration than would otherwise be the case. It is important therefore to model this buildup of loose sediment prior to runoff generation.

[37] We observe that the left side of equation (3) is zero before ponding, so dieri = ei. Substituting this into equation (4) and summing over all i, we then get equation image. Given mT = 0 at t = 0, the solution of the differential equation is

equation image

[38] Thus, the mass of loose sediment on the soil surface approaches the critical mass required to shield the original soil completely, and the rate at which equation image is approached increases with the ratio of aP to equation image. Because of the spatially varying topography, we model this process numerically. This is achieved by setting hci = 0 and incrementing mi by equation image at each time step until hhp, where hp is a small artificial threshold to distinguish between ponding and nonponding in the numerical model [Heng et al., 2009].

[39] The boundary conditions are a reflective (zero flux) boundary at the top of the slope and a transmissive one at the bottom.

5.4. Time Steps

[40] The allowable time step for the numerical solution tends to be governed by the condition [Heng et al., 2009]

equation image

where vI is the settling velocity of the largest particles. This severely restricts the time step since h → 0 at the top of the slope (where there is no inflow). We may relax this constraint by reasoning that only particles smaller than h can logically be suspended in the surface runoff. The preponding process described in section 5.3 continues to apply for larger particles until h > D, the size of the particles. The criterion then becomes

equation image

where hk is the flow depth in the kth cell and vj,k is the settling velocity of the jth sediment class such that Dj < hk < Dj+1. This modification of the criterion for the allowable time step improves computational efficiency by an order of magnitude.

6. Model Output

[41] The best fit model parameters are shown in Table 3, and the model output is plotted against the observed data in Figure 3.

Figure 3.

Model output (lines) versus observed data (points) for discharge, total suspended sediment concentration, and particle size distribution (PSD) evolution on the (a) 3%, (b) 6%, and (c) 9% slopes. D(0.9), D(0.5), and D(0.1) denote the 90th, 50th, and 10th percentile particle sizes, respectively.

Table 3. Calibrated Model Parameters
Slope (%)Replicatef0 (mm h−1)fc (mm h−1)k (×10−3 s−1)a0 (kg m−3)ad0 (kg m−3)equation image(kg m−2)equation image (%)
3145130.801719001.598.0
 27500.551119003.098.0
 34970.752119002.298.0
613940.711019002.598.0
 249170.941019002.098.5
 34600.931119002.598.5
916160.69919003.099.0
 25200.501019004.099.0
 34501.002719005.098.0

6.1. Surface Runoff

[42] The model generally reproduces the measured discharge well, although discrepancies can be observed. For R1 on the 3% slope, the one-dimensional model predicts a rapid initial rise in discharge associated with the overflow of the depression at the end of the slope. On the actual two-dimensional surface, the overflow could have been incremental, resulting in a more gradual rise.

[43] For R1 on the 9% slope, the modeled discharge curve matched the early and late time data well, but it was significantly higher for 8 ≥ t ≥ 60 min. The first plateau in the measured discharge suggests that the initial source area was small. The filling up of depression storage on the surface resulted in increased connectivity and hence a larger source area (corresponding to the final plateau). The sequence of images shown in Figure 4 lends support to this conjecture.

Figure 4.

Images captured during replicate 1 on the 9% slope at approximately (a) 10 min, (b) 20 min, and (c) 50 min after the start of rainfall, showing an increase in runoff connectivity as storage areas filled up.

[44] In R2 and R3 on the 9% slope, the measured discharge at steady state (or that suggested by the trend) exceeded the theoretical maximum based on a 3.9 m × 1.1 m source area and zero infiltration. The photogrammetric measurements [Heng et al., 2010] indicate that rainfall on the exclusion zones on either side of the flume (designed to eliminate boundary effects) very likely flowed into the main central zone, effectively increasing the source area.

[45] The one-dimensional model cannot properly account for the effects highlighted in the above discussion. Nonetheless, it is a valid first approximation insofar as the transverse flow component is an order of magnitude less significant than the longitudinal. The calibrated infiltration parameters (Table 3) reflect experimental variabilities such as the antecedent moisture content, the rate of surface sealing, and the characteristics of the surface seal.

6.2. Soil Erosion

[46] The magnitude and timing of the peaks in sediment concentration as well as the subsequent declines are well reproduced by the model. The only significant discrepancy appears for R1 on the 9% slope: the observed behavior is atypical and appears to be related to the irregularity in the discharge (highlighted in section 6.1).

[47] The model also reproduced the PSD of the sediment in the runoff and the coarsening trends through time. The general pattern is one of relatively rapid coarsening near the beginning of the erosion event, with a gradual decline in the rate of change as the system approaches steady state (Figure 3). The continuing upward trend in particle sizes toward the end of the experiments (predicted as well as observed) indicates that steady state had not been reached.

[48] Although the model accurately predicted the 90th percentile particle size (denoted by D(0.9) in Figure 3) and its evolution through time, model predictions for the median and 10th percentile particle sizes, denoted by D(0.5) and D(0.1), respectively, start to deviate from the observed values at 20–30 min after the start of rainfall. On average, the predicted D(0.5) and D(0.1) are 46% and 28% larger than the values observed toward the end of the experiments, respectively. This means there were higher proportions of fine particles in the runoff than predicted. This may be due in part to the breakdown of aggregates, which effectively changed the PSD of the soil being eroded. Thus, the observed D(0.5) and D(0.1) seem to approach steady state values that are smaller than predicted by the model.

[49] Difficulties in sampling may also account for the particle size discrepancies. The analysis of particle sizes by laser diffraction necessitated taking small subsamples from each sample [Armstrong et al., 2010]. Although care was taken to mix each sample thoroughly before taking subsamples, the subsamples may still be biased toward finer particles, as sand-size particles dropped out of suspension very quickly. This systematic bias is not constant across samples but becomes more pronounced with the coarsening of the sediment in the runoff; this may explain why differences between the predicted and observed D(0.5) and D(0.1) were larger toward the end than at the beginning of the experiments.

[50] In view of these uncertainties, the “errors” may not be fatal to the model's potential for predicting sediment and nutrient export. Further work would be necessary to identify the reasons and impact of this underprediction of fines content on water quality modeling.

6.3. Spatial Variations

[51] As an instance of the spatial variations in sediment concentration and the deposited sediment mass, the model output for R1 on the 6% slope at t = 100 min is shown in Figure 5. Comparing this with the microtopography of the slope shown in Figure 2, one can observe that the drops in cT (and corresponding jumps in mT) occur where the local gradient is gentle because of the relatively large flow depths (and hence small a, ad, and equation image). There is also a fining of the suspended sediment in these regions due to the rapid deposition of large particles. This can be observed in Figure 5 (bottom); equation image is the mean weight diameter of the sediment. A corresponding drop in the MWD of the deposited sediment is also evident.

Figure 5.

Spatial variation of the suspended and deposited sediment (from model simulations) for R1 on the 6% slope at t = 100 min. Here cT is the total suspended sediment concentration, mT is the total deposited mass per unit area, and MWD is the mean weight diameter of the deposited or suspended sediment.

[52] The spatial variations highlight the potential use of distributed measurements in the development, validation, and calibration of a soil erosion model [Jetten et al., 2003]. Such measurements are difficult to obtain during an erosion event. Instead, Figure 5 suggests that a distributed sampling and analysis of the surface layer after an erosion event may yield important data for comparison with the model output. A visual inspection of the surface after the event does lend some support to the model's prediction that the deposited layer is finer in local depressions, but a more detailed study is needed to obtain quantitative data.

6.4. Parametric Analysis

[53] The model predictions are meaningless without a consistent set of parameters. The calibrated H-R model parameters (mean ±1 standard deviation) are a0 = 140 ± 6.3 kg m−3, ad0 = 1900 kg m−3, equation image kg m−2, and equation image.

[54] The detachability parameters a0 and ad0 can be related to the interrill erodibility constant KIR (kg s m−4) used in the Water Erosion Prediction Project (WEPP) erosion model. Interrill sediment delivery (in terms of sediment mass per unit area per unit time) is described in the WEPP model by equation image, where fs = 1.05 − 0.85 exp[−4 sin (arctan S0)] is a slope factor and equation image (m s−1) is the interrill runoff rate [Flanagan and Nearing, 1995]. Suppose rills are spaced equation image apart; the unit discharge into each rill is then equation image. If the sediment concentration in the discharge is cT, then equation image. The H-R solution for cT at steady state, for hh0, is [Tromp-van Meerveld et al., 2008]

equation image

where equation image is the weighted mean settling velocity. Equating the two definitions for DIR, we get

equation image

[55] Substituting a0 = 14 kg m−3, ad0 = 1900 kg m−3, fc = 5.2 mm h−1 (mean value), equation image mm s−1, and S0 = 6% into 15, we get KIR = 1.57 × 106 kg s m−4. This is close to the erodibility constant of 1.45 × 106 kg s m−4 determined experimentally for the Caribou soil (which is of a similar texture to ours) in the WEPP database [Elliot et al., 1989]. Thus, the detachability values obtained in this study appear to be consistent with independently obtained data.

[56] The presence of P and fc in 15 implies either (1) KIR is a complex function of P and fc, if a0 and ad0 are constants, or (2) a0 and ad0 are not constants, if KIR is a soil property. Supposing the latter to be true, 15 shows that ad0 can effectively be assumed constant for a given soil (one of our assumptions in calibrating the model) since fc is generally much smaller than equation image (3 orders of magnitude in our case).

[57] It is probable that all three erodibility “constants” are dependent on event size. If so, the assumption that they are invariant would result in prediction errors when the event is much larger or smaller than that for which the parameters were calibrated. In practice, we may calibrate a model on the basis of the more frequent medium-intensity events and expect to get acceptable predictions in most scenarios.

[58] The equation image values (i.e., the deposited sediment mass required for complete shielding when hh0) obtained in this study are physically realistic but appear to be much larger (1–2 two orders of magnitude) than those obtained in previous applications of the model [e.g., Sander et al., 1996; Rose et al., 2007; Tromp-van Meerveld et al., 2008]. Part of the discrepancy can be attributed to the assumption in previous studies that equation image is independent of h. In other words, the effect of flow depth was incorporated in the calibrated equation image. Given the flow depths were an order of magnitude larger than h0, we can deduce from 12 that the corresponding equation image are an order of magnitude larger than the calibrated equation image. Thus, the equation image values obtained in this study are consistent with the higher end values obtained in previous studies. The less realistic lower-end values in previous studies may be due to the various assumptions made (e.g., constant excess rainfall and steady flow) to simplify the governing equations for analytical solution.

[59] There is some variability in equation image between runs. In agreement with the results of Walker et al. [2007], there appears to be an inverse relationship between equation image and the rate of infiltration (Figure 6). Since infiltration changes with time, equation image is strictly not even constant for each run. Ideally, equation image should be defined as a function of f(t) and h; this one function can then be used for all runs. There is, however, a paucity of data on which to base this function. We have conducted additional experiments to investigate this dependence and will be discussing the results in a forthcoming paper.

Figure 6.

Variation of equation image (the deposited sediment mass per unit area required to shield the underlying soil completely) with fc (the steady state infiltration capacity).

[60] Finally, the calibrated values of equation image indicate that a large proportion of total stream power is expended on form resistance, in agreement with Abrahams and Parsons [1994]. There is a weak increasing trend with slope, possibly due to decreasing flow depths and hence relative submergence of roughness elements.

6.5. Sensitivity Analysis

[61] To better understand the behavior of the model, we performed a series of tests with R1 on the 6% slope as the baseline.

[62] In relating rainfall detachment to flow depth by 10 and 11, we had assumed, on the basis of previous studies, that b = 1 and h0 = 0.7 mm. Figure 7 a shows that the model is relatively insensitive to b. However, cT drops considerably with a smaller h0, the effect becoming more pronounced with distance downslope (Figure 8a), as h/h0 increases. The form of the power laws 10 and 11 implies that the effect of reducing h0 by a factor equation image can be offset by correspondingly increasing a0 and ad0 by a factor equation image. The PSD of the eroded sediment, in terms of the MWD, is insensitive to b and h0 (Figure 7a).

Figure 7.

Sensitivity of the model output in terms of cT and the MWD of the suspended sediment at x = L, with (a) varying b and h0, (b) varying equation image and a0, and (c) varying equation image and ad0. See text for further explanation.

Figure 8.

Sensitivity of the model output in terms of cT and mT at t = 100 min, with (a) varying b and h0, (b) varying equation image and a0, and (c) varying equation image and ad0. See text for further explanation.

[63] The detachability of the original soil a0 has a significant impact on the mean and steady state sediment concentrations (also evident from 14), but its effect on the PSD of the sediment is negligible (Figure 7b). On the other hand, increasing/decreasing equation image (i.e., decreasing/increasing the proportion of total stream power acting on the sediment) significantly decreases/increases the rate of coarsening of the eroded sediment (the rate of approach to steady state), with relatively little impact on the steady state sediment concentration.

[64] As with equation image, increasing/decreasing ad0 (the detachability of loose sediment) increases/attenuates the peak sediment concentration and increases/decreases the sediment turnover rate so that it takes a shorter/longer time to reach steady state (Figure 7c). The converse is true for equation image (the deposited sediment mass required for complete shielding).

[65] The effect of parametric uncertainties on mT is most pronounced in local depressions (Figure 8; see Figure 2 for the microtopography), which act as sediment traps.

[66] The above sensitivity analysis illustrates how the H-R parameters may be calibrated. We can more or less determine a0 on the basis of the steady state cT, which is insensitive to the other parameters. It is harder to determine equation image, ad0, and equation image because they produce similar effects on model output at x = L. It helps to have multiple data sets from experiments under different conditions (as in this study) and to impose parametric constraints (such as ad0 is constant for all runs). Distributed data may also help to separate out the effects of the three parameters, as Figure 8 demonstrates. In lieu of this, the a posteriori check against an independent measure of erodibility (section 6.4) lends credibility to the calibrated values for a0 and ad0.

7. Discussion and Conclusions

[67] We have presented in this paper an application of the full H-R model to a detailed flume-scale experimental study. The numerical implementation of the model coupled with the Saint-Venant equations allows us to model soil erosion on complex topography and with realistic initial conditions. The model output matches fairly closely the observed data, including the dynamic evolution of the eroded sediment's PSD, which is requisite to predicting accurately the fluxes of nutrients sorbed onto the sediment. Importantly, the calibrated parameters are consistent with independent data in the literature, which is important if this is not to become a mere curve-fitting exercise.

[68] While the model represents faithfully the processes of soil erosion, the question inevitably arises as to its applicability in the field and at larger scales, where data on soil PSD, topography, etc., would be much sparser. Figures 9 and 10 show the effect of using a uniform averaged slope to represent the 6% R1 slope in conjunction with model parameters that were calibrated with respect to the nonuniform topography. The total sediment yield predicted on the basis of the uniform slope is 6.3% greater than that based on the varying topography. The corresponding increase in the MWD of the eroded sediment (Figure 9) suggests that the increase in sediment yield is due to a greater amount of coarse sediment in the runoff. Figure 10 shows that the uniform slope approximation results in an underprediction of the deposited sediment mass, particularly in areas of depression. It is thus evident that a larger number of coarse particles is transported out of the eroding area (instead of being trapped in depressions) when the slope is assumed to be uniform.

Figure 9.

Effect of assuming a uniform slope on the modeled sediment concentration cT and the sediment MWD at x = L for R1 on the 6% slope. The same model parameters calibrated with respect to the nonuniform topography were used for the uniform slope simulation. The observed experimental data are also plotted for comparison.

Figure 10.

Effect of assuming a uniform slope on the modeled sediment concentration cT and the deposited sediment mass mT at t = 100 min for R1 on the 6% slope. The same model parameters calibrated with respect to the nonuniform topography were used for the uniform slope simulation.

[69] In essence, this is a scale issue that has been explored by others [Jetten et al., 2003; Rojas et al., 2008] at larger scales. It seems clear that erosion parameters calibrated at one spatial resolution will produce poorer predictions at a different spatial resolution. The general lack of data at the temporal and spatial resolutions necessary for physics-based erosion prediction continues to hamper progress in this direction (although LIDAR and other remote sensing technologies show promise as possible solutions). Nonetheless, inasmuch as a scale-independent model must account for “those erosion products that are entrained as sediment load but are redeposited” [Osterkamp and Toy, 1997, p. 155], the H-R model is a step in the right direction. A necessary step forward would be to apply this model at field and catchment scales, which should yield some interesting results as well as raise further research questions.

Acknowledgments

[70] This work was supported by the Natural Environment Research Council, United Kingdom, under grant NE/E007015/1.