Published online 4 August 2005
Published in Soil Sci Soc Am J 69:1440-1447 (2005)
DOI: 10.2136/sssaj2004.0309
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
Soil & Water Management & Conservation
Validation of WEPP Sediment Feedback Relationships using Spatially Distributed Rill Erosion Data
X.-C. Zhanga,*,
Z.-B. Lib and
W.-F. Dingc
a USDA-ARS, Grazinglands Research Lab., 7207 W. Cheyenne St., El Reno, OK 73036
b Xi'an Technology Univ., Xi'an, Shaanxi 710048, PRC, and Institute of Soil and Water Conservation, CAS & MWR, Yangling, Shaanxi 712100, PRC
c Institute of Soil and Water Conservation, Yangtse Academy of Sciences, Wuhan, Hubei 430010, PRC
* Corresponding author (jzhang{at}grl.ars.usda.gov)
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ABSTRACT
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Process-based soil erosion models have not been thoroughly evaluated due to the lack of spatially distributed, instantaneous rill erosion data. The objective of this study was to quantitatively evaluate the sediment feedback relationships of the Water Erosion Prediction Project (WEPP) model using distributed instantaneous rill erosion data derived with rare earth element (REE) tracers. Four REE oxide powders (Ce2O3, Nd2O3, Sm2O3, and Dy2O3) were separately mixed with a loessial silt loam soil, and each mix was packed in a 1-m segment of a 4-m flume. Each packed flume was subjected to flow scouring at a selected slope and inflow rate for 13 to 17 min, depending on slope and discharge. Three slopes (10.5, 15.8, and 20.2%) and five inflow rates (2.5, 3.5, 4.5, 5.5, and 6.5 L min–1) were used, and two replicates were made for each combination. Runoff and sediment samples were collected at 1- or 2-min intervals. Flow velocity and width were monitored. Sediment samples were analyzed for the REE composition by Instrumental Neutron Activation Analysis (INAA). The REE concentrations in each sample were used to estimate sediment deliveries from each 1-m tagged segment. Net rill detachment rates tended to decrease linearly as sediment loads increased in the downslope direction. The negative slope of linear regression (or rate of the decrease), which became more negative with inflow rates at the 10.5% slope but less negative at the 20.2% slope, substantiated the sediment feedback relationships assumed in the WEPP model. The WEPP-calculated and REE-measured rill detachment rates agreed reasonably well, with the model efficiency being 0.511. Overall, the results show that the assumed sediment feedback relationships used in the WEPP model are reasonable for simulating rill detachment.
Abbreviations: INAA, instrumental neutron activation analysis • ME, model efficiency • REE, rare earth element • USLE, Universal Soil Loss Equation • WEPP, Water Erosion Prediction Project
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INTRODUCTION
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SPATIALLY AND TEMPORALLY distributed erosion data are of great importance for better understanding soil erosion dynamics, evaluating process-based erosion models, and advancing process-based erosion prediction technology. Process-based erosion prediction models such as the WEPP model (Flanagan and Nearing, 1995) have been developed in the past two decades to predict spatial and temporal distributions of soil erosion. Because soil erosion processes constantly change in time and space, process-based models have advantages over spatially and temporally lumped empirical prediction models such as the Universal Soil Loss Equation (USLE) (Wischmeier and Smith, 1978) in simulating soil erosion. However, these advantages will not be fully recognized unless the spatial and temporal predictability of those process-based models are thoroughly evaluated.
In the WEPP model, upland soil erosion is conceptually divided into interrill and rill erosion. Rill erosion is mainly caused by concentrated overland flow. The rill erosion equations in WEPP are primarily based on the sediment feedback relationship (also known as detachment–transport coupling concept), which was initially proposed by Foster and Meyer (1972). This relationship was developed using the analogy of first-order chemical reaction kinetics and takes the form of:
 | [1] |
where Dr is the net rill detachment rate (kg s–1 m–2), 
is a rate control coefficient (m–1), Tc is the sediment transport capacity (kg s–1 m–1), and G is the sediment load (kg s–1 m–1). It simply states that net rill detachment rates are proportional to sediment deficits in the flow. This bedrock relationship has not been thoroughly evaluated due to the lack of authentic, spatially distributed rill erosion data.
Huang et al. (1996) conducted a field rill scouring study at four slope lengths from 2 to 9 m. Sediment deliveries from different rill lengths were used as "spatially distributed" data to evaluate the detachment-transport coupling concept. They concluded that the coupling concept was inadequate, and to the contrary, the decoupled approach as proposed by Meyer and Wischmeier (1969) explained their data better. Lei and colleagues (2002) performed a laboratory flume study by introducing inflow at eight different slope positions, starting near the flume outlet and sequentially moving up the flume in 1-m increments. The differences in sediment loads between the two consecutive slope lengths were treated as net rill detachment from the last segment near the end plate under the assumption that soil detachment behavior was the same along the rill under steady flow (Lei et al., 2002). Their results substantiated the principle of the sediment feedback relationship (i.e., net detachment decreased linearly with increasing sediment load in the flow).
Zhang et al. (2001) experimentally showed that REE oxide powders when directly mixed with soil were bound with soil particles or aggregates, and were uniformly incorporated into various-size soil aggregates. Considering the other attributes of REE tracers such as their low mobility, low background concentration in soils, accurate measurement, negligible environmental hazard, and availability of multiple tracers (similar in chemical properties but distinct in signature), they concluded that the REE oxide powders are ideal tracers for studying soil erosion and aggregation. Zhang et al. (2003) further demonstrated in a laboratory rainfall simulation study that the REE-tracing technique is capable of producing both spatially and temporally distributed soil erosion data, and therefore provides a significant opportunity for studying soil erosion dynamics and processes.
The objective of this study was to quantitatively evaluate the sediment feedback relationships of the WEPP model using spatially and temporally distributed rill erosion data measured with REE tracers.
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WEPP RILL EROSION EQUATION FORMULATION
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The WEPP model uses a steady-state mass balance equation for routing sediment along a slope (Nearing et al., 1989):
 | [2] |
where x is the downslope distance (m), and Di is the interrill erosion rate, which is zero for this particular application. Rill net detachment (Dr) is estimated by:
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 | [3b] |
Equation [3a] is identical to Eq. [1] when is set to Dc/Tc. Dc is the detachment capacity under clear water inflow conditions, and is expressed as:
 | [4] |
where Kr is the rill soil erodibility (s m–1), 
s is the flow shear stress acting on soil (N m–2), and c is the critical shear stress of the soil (N m–2). Flow shear stress is calculated by:
 | [5] |
where 
is the specific weight of water (kg m–2 s–2), S is the slope gradient, and R is the hydraulic radius (m). Flow shear stress () is partitioned into the part acting on soil (s) and other parts acting on plants, residue, or other objects, if any; and only the portion acting on soil is assumed to cause soil erosion in the WEPP model. Sediment transport capacity is estimated in WEPP as:
 | [6] |
where Kt is a transport coefficient and is calibrated with the transport capacity at the end of the slope using the Yalin equation (Finkner et al., 1989). The sediment deposition equation is not reviewed here because it is irrelevant to this particular application. Due to short slope length, high slope gradient, and relatively narrow rill width, sediment deposition is negligible in this study.
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MATERIALS AND METHODS
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Rare earth elements of Ce, Nd, Sm, and Dy were used for the study, and the target concentration for Element i (Ci) in µg g–1 was estimated by:
 | [7] |
where Bi is the background concentration of Element i in µg g–1, Ri is the expected fractional contribution of sediment from the segment tagged with Element i, and k is an assurance coefficient. The assurance coefficient is largely dependent on the detection limit and cost of the element. A value of k between one and three was recommended to ensure reliable results (Institute of Soil and Water Conservation, 1997; Tian et al., 1994).
A loessial silt loam soil was used. The soil was composed of soil particles of which approximately 36% were <0.001 mm, 20% were 0.001 to 0.01 mm, 41% were 0.01 to 0.05 mm, and 3% were >0.05 mm. Soil was sieved through a 10-mm sieve before mixing with tracers. The predetermined amounts of soil and REE oxide powder (Ce2O3, Nd2O3, Sm2O3, Dy2O3) were thoroughly mixed by serial dilutions to obtained the target concentration as determined from Eq. [7].
A 5-m long by 0.3-m wide by 0.5-m deep flumes was first filled with a 0.2-m layer of river sand, and then packed with a 0.2-m layer of tagged soil. Bulk density of the packed soil was between 1.25 and 1.3 Mg m–3. The soil was prewetted to saturation with a sprinkler before the introduction of inflow to achieve a uniform initial soil moisture and surface condition. The placement of four REE tracers in each 1-m segment is shown in Fig. 1. The upper most 1-m segment was lined with nonerodible materials and used for achieving steady flow velocity and better inflow distribution across the flume before entering the soil section.
Three slope gradients (10.5, 15.8, and 21.2%) and five inflow rates (2.5, 3.5, 4.5, 5.5, and 6.5 L min–1) were used. Rainfall simulation was not used during the scouring runs. Runoff and sediment were collected every minute for the first 3 min, and then collected every 2 min. Rill flow velocity (dye method) and flow width were measured for each segment during each collection interval within a run. Each run lasted 13 to 17 min depending on slope and inflow rate, and two replicates were made for each combination of slope and inflow rate. After each run, the soil remaining in the flume was thrown away, and the flume was repacked. In total more than 30 packed flumes were used in the study.
Runoff and sediment were collected in a pail during each collection interval and weighed to determine runoff volumes. Sediment and runoff were later separated by settling and siphoning. Sediment samples were air dried, weighed, and then well mixed. About 20 g of sediment was sieved through a 0.15-mm (100 mesh) sieve, and a 50-mg subsample was taken for measuring the REE composition by INAA (Institute of Soil and Water Conservation, 1997; Tian et al., 1994). For each sediment sample, sediment delivery (g) from Segment i or Tracer i (Li) was calculated as:
 | [8] |
where Lt is the total mass of the sediment sample in g, and Si is the concentration (µg g–1) of Tracer i in that sample. Using Eq. [8], each sediment sample (Lt) was partitioned into sediment deliveries from each of the four segments (Li, i = 1 ... 4). The relative error for each sediment sample was estimated as (
Li – Lt)/Lt. The overall relative error averaged over all sediment samples within each run was <23% in this experiment. Readers are referred to the article of Zhang et al. (2003) for detailed counts on REE analysis, sediment partitioning, and error estimation.
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RESULTS AND DISCUSSION
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Sediment delivery from the lowest segment of the flume (Dy segment) was not used in the analysis because of the effects of the end plate on rill flow hydraulics and erosion. Rill incision often initiated immediately above the end plate and reached more than 0.15 m by the end of most runs. Such alteration of the local slope gradient near the end plate could have considerable impacts on rill erosion in the segment.
Based on topographic evolution of rills, rill formation process could be divided into three stages in this experiment. In the rill initiation stage (0–3 min) laminar flow predominated (Reynolds Number [Re] ranged from 260 to 1000 for all inflow rates and slopes), flow was relatively broad and rapid (Fig. 2b), and scouring holes occurred along the flow path. In the rill development stage (3–13 min) flow was predominantly turbulent (Re ranged from 1000–4500) due to increased rill roughness and flow depth resulting from rigorous downward scouring and rill head cutting, as indicated by narrower flow width in Fig. 2a. In the rill maturity stage (to the end of run) downward incision subsided and lateral expansion by sidewall slumping increased. Conceptually, the rill erosion equation formulation in WEPP is primarily based on simulating the downward scouring by concentrated flow, and rill sidewall slump and rill head cutting are not explicitly modeled. Therefore, to evaluate WEPP's prediction, sediment data after the 13-min mark was not used in the analysis. Also, due to relatively short slope length, high slope gradients, and large inflow rates, deposition in rills was negligible, as evidenced by the presence of deeply incised rills after most runs.
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Fig. 2. Changes of average flow velocity and flow width along the slope profile at the 10.5% slope with scouring time and inflow rates.
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Figure 2 shows the temporal changes of average flow width and flow velocity at the 10.5% slope as an example. In general, average flow width along the entire profile narrowed in time for all inflow rates and slopes due to downward incision, and the final rill width was wider at larger inflow rates. The latter trend was less obvious at the 15.8 and 21.2% slopes than at the 10.5% slope (data not shown). Average flow velocity decreased in the first 3 min and gradually leveled off afterward for all inflow rates and slopes, as rill roughness increased and flow changed from laminar to turbulent flow (Fig. 2b). The steady flow velocities near the end of each run were faster at larger inflow rates; however, the trend was less clear at steeper slopes.
The effects of sediment loading on downslope rill detachment throughout each scouring run at each inflow rate are presented in Fig. 3 through 5 for the 10.5, 15.8, and 21.2% slopes, respectively. Averages of the two replicates were used in the plots. Note that the rill detachment rate and sediment-loading rate were expressed for the whole rill, therefore the dimension of per unit rill width was dropped. Symbols from left to right in each curve represent flume segments A, B, and C, respectively (see Fig. 1 for the layout). The rill detachment rates are the averages over the 1-m segment of interest, and the sediment loading rates are the total loads immediately above that segment. The overall trends show that sediment loads do have a negative impact on rill detachment rates, and that the linear relationship of Eq. [3] approximately holds. The deviation from linearity in some plots may be a result of the formation of scouring holes, which is largely random in nature and often leads to severe rill head cutting. The non-uniform formation of scouring holes and rill head cutting along the slope profile (across segments) would undoubtedly distort the linearity of Eq. [3]. Based on Eq. [4] and [6], detachment capacity, and transport capacity are functions of soil or sediment properties and flow hydraulics (particularly flow shear stress). Because of nonuniform downward incision, flow depth as well as flow shear stress as calculated by Eq. [5] varied along a flow path. As a result, the slope coefficient of Dc/Tc in Eq. [3a] was not a constant even at a given slope gradient and inflow rate, and it could even take the opposite sign on rare occasions. The overall results seem to indicate that Eq. [3] should hold if soil and flow hydraulics were homogeneous along a flow path.
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Fig. 3. Effects of sediment loading rates on downslope rill detachment during scouring tests at the 10.5% slope and five inflow rates.
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Fig. 5. Effects of sediment loading rates on downslope rill detachment during scouring tests at the 21.2% slope and five inflow rates.
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Fig. 4. Effects of sediment loading rates on downslope rill detachment during scouring tests at the 15.8% slope and five inflow rates.
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There was a tendency for the slopes of the plots in Fig. 3 to increase as flow rates increased while for the slopes in Fig. 5 to decrease as flow rates increased. To evaluate this observation, linear regression of rill detachment rate versus sediment loading rate, Eq. [3a], was performed for each curve in Fig. 3 and 5. The averaged regression slope coefficients after removal of the largest and smallest values for each inflow rate are shown in Table 1. The average regression slopes decreased (became more negative) as inflow rates increased for the 10.5% slope, while the opposite was true for the 21.2% slope. This result apparently agrees with the erosion theory assumed in WEPP. Detachment capacity (Dc) is a linear function of shear stress (Eq. [4]), and sediment transport capacity (Tc) is a power function of shear stress (Eq. [6]) in WEPP. Mathematically, when shear stress is relatively small, the increase of Dc exceeds the increase of Tc, therefore, the absolute ratio of Dc/Tc (the regression slope) increases. Conversely, when shear stress is relative large, the increase of Tc exceeds the increase of Dc, and thus the absolute ratio of Dc/Tc decreases. Further explanation and discussion are given below using measured data from this study.
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Table 1. Temporally averaged regression slope coefficients for the five inflow rates at the 10.5 and 21.2% slopes.
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As stated earlier, detachment capacity (Dc) is defined as the detachment rate under clear water inflow conditions. Thus, measured sediment and hydraulic data from Segment A were used to evaluate Eq. [4]. Time-averaged flow width and flow velocity of Segment A along with inflow rates were used to compute event-averaged flow depth. Assuming a rectangular cross-section, rill flow depth instead of hydraulic radius (R) was used in Eq. [5] to calculate event-averaged shear stress (s). The event-averaged rill detachment rates from Segment A were taken as Dc and were regressed against event-averaged shear stress over all inflow rates and slopes. Results are presented in Fig. 6. The fitted rill erodibility (Kr) was 0.0169 s m–1; critical shear stress (c) was 0.355 N m–2; and r2 was 0.74. The results indicate that the linear relationship of Eq. [4] is approximately valid at least under this experimental condition.
In WEPP, Eq. [6] is calibrated using Yalin's sediment transport capacity at the end of the slope (Finkner et al., 1989). In this study, Yalin's Tc was calculated for each event-averaged shear stress as stated above, assuming a D50 of 0.2 mm and an average particle density of 2.0 Mg m–3 for sediment including aggregates (note that D50 is a particle size at which 50% of particles by weight are finer than that value). Calculated Tc was regressed against event-averaged shear stress to a 1.5 power 
without an intercept. The regression coefficient (Kt) was 0.0475, and r2 was 0.99 (Fig. 7a). For comparison, the regression line of Eq. [4] is also shown in Fig. 7a, and the ratio of Dc/Tc (, rate control coefficient) and its reciprocal (critical slope length) are shown in Fig. 7b. The rate control coefficient initially increased with shear stress and then decreased steadily due to the faster increase in Tc than in Dc. Conversely, the critical slope length, defined as downslope distance needed to reach Tc at a constant flow rate, initially decreased with shear stress and then steadily increased. The relatively long critical slope length near zero shear stress occurred because rill detachment only occurs when flow shear stress is greater than critical shear stress and Dc is relatively low when flow shear stress barely exceeds critical shear stress. The steady increase in critical slope length when shear stress is relatively large is attributed to the faster increase in Tc than in Dc. This result is consistent with the field experiment data of Huang et al. (1996), who reported that critical slope length increased as transport capacity or shear stress increased.
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Fig. 7. Fitted detachment capacity (Dc) and sediment transport capacity (Tc) as well as calculated rate control coefficients (Dc/Tc) as functions of flow shear stress.
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In contrast, Lei et al. (2002) reported that (or Dc/Tc) increased as slope and discharge (i.e., shear stress) increased. In their flume experiment, they introduced inflow at eight slope positions, starting near the outlet and sequentially moving upward. The differences in sediment loads between the two adjacent slope lengths were treated as the net rill detachment from the last segment near the end plate under the assumption that soil detachment behavior was the same along the rill under steady flow. In this type of "time-space-substitution" approach, the preceding erosion event might have an undesirable effect on the succeeding event. For example, the last segment near the end plate was eroded eight times longer than the uppermost segment. The prolonged erosion in the lowest segment would alter rill geometry, especially the local slope gradient, as well as the sediment availability. Such alterations could intensify as slope and discharge increased. As a result, net detachment rates in the last segment would tend to decrease over time, and the rate of decrease would be greater at steeper slopes and for greater discharges. This bias in time would translate to a bias in space and would artificially magnify the effects of sediment loading or slope length on net rill detachment. The magnification would increase as slope and discharge increased.
Finally, WEPP's instantaneous rill detachment rates, calculated for each segment and each sample (time interval) using Eq. [3a], were compared with the REE-measured detachment rates. Shear stress was calculated for each segment and time interval using the corresponding flow velocity and flow width assuming a rectangular cross-section. Dc and Tc were then calculated for each segment and time interval using the corresponding shear stress and the regression equations in Fig. 7a. The WEPP-calculated instantaneous rill detachment rates for all slopes and inflow rates are plotted against the REE measured rill detachment rates in Fig. 8. Linear regression showed that r2 was 0.526 and the regression coefficient was 0.593, which was significantly different from one at P = 0.05. The coefficient indicated that the WEPP model tended to underestimate rill detachment rates, especially at higher values. A plausible explanation for this underestimation is that high detachment bursts are often caused by gravitational erosion such as severe rill head cutting and/or sidewall slumping, and these eroding mechanisms are not explicitly simulated in the model. Model efficiency (ME), which is a good measure of model prediction relative to measured data in reference to the 1:1 line (Nash and Sutcliffe, 1970), was also calculated. If ME = 1, the model produces the exact prediction for each data point. A zero value of ME implies that a single mean measured value is as good an overall predictor as the model. A value of 0.511 was found in this study, indicating that the WEPP prediction of rill detachment is reasonable. The dispersiveness in Fig. 8 may be the result of nonuniform hydraulic and erosion processes along the flow paths. Specifically, sidewall slumping and rigorous advance of rill heads could result in considerable variations in rill detachment rates both in space and time, which in essence are not explicitly modeled in WEPP.
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Fig. 8. The Water Erosion Prediction Project (WEPP) model-calculated rill detachment rates versus rare earth element (REE)-measured detachment rates at each measurement interval and segment at all slopes and inflow rates.
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CONCLUSIONS
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This study showed that the sediment feedback relationships in the WEPP model are reasonable for simulating rill detachment. For most cases, net rill detachment rates decreased linearly as sediment loads in flow increased. The absolute rate of change increased with inflow rates at the 10.5% slope, but decreased with inflow rates at the 21.2% slope. This trend is consistent with the theoretical pattern that the rate control coefficient () increases with shear stress (s) at low shear stresses and then decreases at high shear stresses, because Tc increases much faster with s than does Dc. Overall, WEPP-calculated instantaneous rill detachment rates agreed reasonably well with the REE-measured rill detachment rates for all slopes and inflow rates, with a model efficiency of 0.511. However, the WEPP model tended to underpredict rill detachment rates at high values, because nonuniform erosion processes such as sidewall slumping and rill head cutting, which often generate bursts of high sediment loads, are not explicitly simulated.
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ACKNOWLEDGMENTS
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This article is based on the work partially supported by the National Natural Science Foundation of China under Grant No. 40371075.
Received for publication September 17, 2004.
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REFERENCES
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ABSTRACT
INTRODUCTION
