A rational method for estimating erodibility and critical shear stress of an eroding rill

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1 Available online at Geoderma 144 (2008) A rational method for estimating erodibility and critical shear stress of an eroding rill T.W. Lei a,b,, Q.W. Zhang a,c, L.J. Yan b, J. Zhao a, Y.H. Pan a a State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, Institute of Soil and Water Conservation, CAS and MWR, Yangling, Shaanxi, , PR China b Laboratory of Modern Precision Agriculture Integration Research, College of Hydraulic and Civil Engineering, China Agricultural University, Beijing, , PR China c Institute of Environment and Sustainable Development in Agriculture, Chinese Academy of Agricultural Sciences, Beijing , PR China Received 23 March 2007; received in revised form 17 January 2008; accepted 21 January 2008 Available online 10 March 2008 Abstract Soil erodibility and critical shear stress are two of the most important parameters for physically-based soil erosion modeling. To aid in future soil erosion modeling, a rational method for determining the soil erodibility and critical shear stress of rill erosion under concentrated flow is advanced in this paper. The method suggests that a well-defined rill be used for shear stress estimation while infinite short rill lengths be used for determination of detachment capacity. The derivative of the functional relationship between sediment yield and rill length at the inlet of rill flow, as opposed to average detachment rate of a long rill, was used for the determination of detachment capacity. Soil erodibility and critical shear stress were then regressively estimated with detachment capacity data under different flow regimes. Laboratory data of rill erosion under well defined rill channels from a loess soil was used to estimate the soil erodibility and critical shear stress. The results showed that no significant change in soil erodibility (K r ) was observed for different slope gradients ranging from 5 to 25 while critical shear stress increased slightly with the slope gradient. Soil erodibility of the loess soil was ±0.001 s m 1. The soil erodibility and critical shear stress calculations were then compared with data from other resources to verify the feasibility of the method. Data comparison showed that the method advanced is a physically logical and feasible method to calculate the soil erodibility and critical shear stress for physically-based soil erosion models Published by Elsevier B.V. Keywords: Soil erodibility; Critical shear stress; Soil erosion prediction model; Rational method; Loess soil 1. Introduction Soil erosion is a serious environmental problem threatening the future development of agriculture and society. It is not only a major factor responsible for the long-term degradation of land quality, but also a major source of non-point water pollution. Increased attention to these concerns has led to improved measures for erosion control and a superior comprehension in soil erosion mechanics and soil loss prediction. Abbreviations: D r, rill detachment rate; D rmax, potential detachment rate; T c, transport capacity of the flowing water; WEPP, Water Erosion Prediction Project. Corresponding author. China Agricultural University, Qinghua Donglu, Beijing, , PR China. Tel.: address: ddragon@public3.bta.net.cn (T.W. Lei). A process-based model for soil erosion prediction is a group of mathematical functions based on soil erosion processes. Scientists have developed and worked with these process-based erosion models, such as the Water Erosion Prediction Project (WEPP) (Flanagan and Nearing, 1995), since the 1980 s. Soil erodibility (K r ) and critical shear stress (τ c ) are two important indices of soil properties and are used as essential parameters in WEPP (Nearing et al., 1989a) and are described in the following relationship: D r ¼ K r ðs s c Þ 1 qc ð1þ T c where D r is rill detachment rate, kg m 2 s 1 ; τ is the shear stress of flowing water, N m 2 ; τ c is the critical shear stress of soil, N m 2 ; q is unit flow rate, m 3 s 1 m 1 ; and c is sediment /$ - see front matter 2008 Published by Elsevier B.V. doi: /j.geoderma

2 T.W. Lei et al. / Geoderma 144 (2008) concentration, kg m 3 ; T c is transport capacity of the flowing water, kg s 1 m 1 ; and K r is the erodibility of soil, s m 1. Gilley et al. (1993) suggested a method to estimate the soil erodibility and critical shear stress and is as follows. Field experiments of rain simulation with runoff plots of 0.46 by 9 m were run. Detachment rates under different flow rates were computed with collected runoff samples, and the shear stresses of the water flow were determined by the flow rates and slope gradients. These detachment rates and shear stresses were used to estimate the erodibility and critical shear stress as parameters in the following regression model: D rmax ¼ K r ðs s c Þ ð2þ where D rmax is potential detachment rate. This method has been used to estimate soil erodibility and critical shear stress throughout the US (Elliot et al., 1989). Importantly, Eq. (2), as compared with Eq. (1), indicates that the potential detachment rate can only be estimated with clean water, in which c=0. Under steady concentrated flow into a well defined (uniform slope with no variation in width) rill channel, the relationship between sediment concentration distribution in flow water and rill length was conceptually, numerically, and experimentally illustrated by Huang et al. (1996), Lei et al. (1998) and Lei et al. (2001), respectively as: c ¼ A 1 e bx ð3þ where, A and β are regression coefficients, varying with soil and hydraulic conditions; x is downslope rill distance, m. This functional relationship indicates that sediment concentration increases with rill length and approaches a maximum value of A, the sediment concentration at transport capacity. The increase in sediment concentration, however, decreases exponentially with rill length. This incremental reduction in the rate of increase in sediment concentration is due to a decrease in soil detachment rate. This phenomenon of detachment rate reduction of sediment loaded flow has been experimentally demonstrated by Lei et al. (2002) and indicates that long rills give a very poor estimation of potential detachment rates. Based on this recognition, Huang et al. (1996) mentioned that in a rill detachment/transport model, a short channel is needed for the (potential) detachment rate. As a result, experiments on their small soil samples (12.7 cm in diameter) (Nearing et al., 1991; Nearing and Parker, 1994) gave much higher detachment rates than did the rill erosion experiments conducted by Laflen et al. (1991) and Nearing et al. (1999) because of the influences of sediment presence in the flowing water, as discussed by Cochrane and Flanagan (1997) and Merten et al. (2001). The objectives of this study then were to: 1) develop a method to estimate detachment capacity of steady rill flow, based on the sedimentation process of rill erosion and the rational estimation method of detachment rate as a function of rill length; 2) outline a rational method for the determination of soil erodibility and critical shear stress; and 3) estimate the soil erodibility and critical shear stress of a loess soil, with the newly suggested method. 2. Methodology When the sediment concentration in rill channel flow is zero, the detachment rate approaches its maximum value, or reaches its potential detachment rate or detachment capacity. Under this condition, Eq. (1) is reduced to Eq. (2), which is also equivalent to: D rmax ¼ K r s K r s c Sediment content in the rill flow comes from soil detachment in the rill bed by flowing water. The detachment rate is defined as the amount (kg) of soil detached from a unit area (m 2 )ina unit time (s). Based on the mass balance, Rose et al. (1983) proposed the sediment continuity equation. When the rill length approaches zero at the upend where clear water is introduced into the rill, we assume that the sediment concentration in clear water of rill channel flow is zero, thus the sediment continuity equation based on mass balance is derived as, AðcqÞ Ax þ AðchÞ ¼ 0 ð5þ At where, c is sediment concentration, kg m 3 ; q is flow rate per unit rill width, m 2 s 1 ; x is down slope distance, m; h is flow depth, m; and t is time, s. For the unit area, AðchÞ At is taken as the detachment rate (D r ). An analytic method for determining detachment rate of concentrated steady flow in eroding rills was advanced by Lei et al. (2002) under a given initial and boundary condition: D r ¼ lim Dc Dx Q w ¼ dqc ð Þ dx where, Q is inflow rate, m 3 s 1 ; and w is rill width, m. This equation means that the rill detachment rate under steady flow is the change rate of the sediment concentration in the flowing water, with respect to rill length, times the flow rate of unit rill width. Or equivalently, the soil detachment rate is the change in sediment yield with respect to rill length. Eq. (6) requires that the unit flow rate be uniform along the rill. The sedimentation processes, as related to rill length, or the relationship between sediment concentration and rill length is given in Eq. (3). Once the sedimentation process and the relationship between sediment concentration and the rill length are determined, the detachment rate can be easily obtained. It is done by substituting Eq. (3) into Eq. (6), yielding: D r ¼ qbae bx ¼ T c be bx where, T c is transport capacity of flowing water, kg m 1 s 1. The detachment rate as expressed in Eq. (7) decreases exponentially with rill length. Depending on the actual value of β, the reduction in detachment rate at a given rill length varies. At a rill length of x=3/β, the detachment rate is reduced to 5% (exp ( 3)=0.05) of the potential. Therefore, the average detachment of a long rill as a potential detachment rate vastly underestimates the detachment capacity. This explains why the small soil samples (12.7 cm in diameter) (Nearing et al., 1991; ð4þ ð6þ ð7þ

3 630 T.W. Lei et al. / Geoderma 144 (2008) Nearing and Parker, 1994) gave much higher detachment rates than did the rill erosion experiments conducted by Nearing et al. (1989b, 1999). Therefore, the use of long rills are not desirable for detachment capacity estimation and any estimation of soil erodibility using long rills will be much lower than its true value. Obviously then, to estimate the potential detachment, short rills are needed, but how short is short enough for detachment rate determination? Mathematically, it should be infinitely short. Eq. (7) clearly shows that D r has its maximum value or detachment capacity when x approaches 0, which is: D rmax ¼ qba Combining Eqs. (4) and (8) gives: qba ¼ K r s K r s c The soil erosion process is the combination of a series of interaction processes between soil erodibility and erosive force. For example, overland flow has a certain velocity and energy as it flows across the surface of the land. The flowing water imposes a shear stress on the soil surface. Soil detachment occurs only (not necessarily) when shear stress of water flow exceeds the critical shear stress of the soil. Provided that the velocity of flow at an infinitesimal cross section of da is u and the flow rate across da is: dq ¼ uda ð8þ ð9þ ð10þ and the average velocity of a section is denoted as u then the flow rate at cross section A can be expressed as follows: Z Z Q ¼ dq ¼ uda ¼ ua ð11þ Q A According to the principle of hydrodynamics, shear stress of water flow is equal to the gravitational component along the flow direction. The actual derivation of flow shear stress on an inclined surface is the product of the weight density of water times the flow depth times the sine of the slope angle, and the figures in the paper used the computation of shear stress utilizing the sine of the slope angle (Wu, 1997). Thus the shear stress can be expressed as: s ¼ gsh ¼ gs Q ð12þ uw where γ is specific gravity of water, 9800 N m 3 ; s is hydraulic slope, Sin(α), α is slope in degrees; u is averaged velocity, m s 1. To compute the shear stress needed by Eqs. (9) and (12), the rill needs to be well defined in terms of both rill slope and rill width. Lei et al. (1998), Lei and Nearing (1999) and Lei et al. (2001) conducted a series of laboratory rill erosion experiments with a sandy loamy soil (Cecil), with a flume that was 0.5 m wide by 8 m long. The inflow rates were 3.8, 7.6, 11.4 and 15.2 L min 1 (1, 2, 3, 4 gpm), and the rill bed slope gradients were 1, 3, 5, and 7%. It was found that the rill width alternated periodically at a length downslope of 3 to 4.5 m, depending on rill bed slope and inflow rate, and the rill width changed from 35 mm to about 300 mm. These results indicate that a rill is very poorly defined in terms of width when using wide flume experiments because the fluctuation is a random process. Therefore, the shear stress needed in Eq. (12) cannot be estimated with data from these kinds of experiments. Moreover, the increase in rill width means that the energy dissipation per unit width in the flowing water is decreased, and so are the hydraulic shear and the transport capacity, and deposition may occur. All of these combined factors result in a lower sediment concentration in the runoff at the outlet of the rill. As a result, the average detachment rate estimated with data from this kind of experiment is much lower than the potential. Therefore, to estimate soil erodibility and critical shear stress, the following are suggested: Establish a sedimentation function with well defined rills (Lei et al., 2001) under different hydraulic conditions, such as different slope gradients and/or different inflow rates; Estimate the potential detachment rate with Eq. (8), based upon the sedimentation processes; Estimate the shear stress of the well defined rill, with Eq. (12); Regress potential detachment rates with Eq. (9) to get soil erodibility and critical shear stress. 3. Calculation and results analysis The simulated experiments of rill erosion were conducted in the laboratory to collect data for estimation of erodibility and critical shear stress. The experimental details were given by Lei et al. (2001) and the parameters required in Eq. (3) were estimated. Five slopes (5, 10, 15, 20, 25 ) and three flow rates (2 L min 1,4Lmin 1,8Lmin 1 ; i.e., 0.12 m 3 h 1,0.24m 3 h 1, 0.48 m 3 h 1 ) were used in the experiments. The soil was a silt loam (loess) soil, typical of the Loess Plateau, and composed of 12.6% sand particles (N 0.05 mm), 72.3% silt particles (0.05 to mm), and 15.1% clay particles (b mm). The maximum (potential) detachment rates D rmax were calculated with Eq. (8) and flow shear stress τ c was estimated with Eq. (12). The relationships between D rmax and flow shear stress τ c under different slope gradients are graphically shown in Fig. 1. When analyzing rill erosion data (Fig. 1), the slopes of the lines describing the relationship of detachment capacity and shear stress represent soil erodibilities (K r ) with a steeper slope meaning a bigger K r. The value of flow shear stress can be thought of as the critical shear stress of the soil τ c when the net detachment rate is 0, which is the intercept of a line with the shear stress axis. That also means the negative ratio of the intercept with the vertical axis to the erodibility (K r ) is critical shear stress τ c. Fig. 1 shows that the slopes of the lines are almost the same under different slope gradients while the intercepts for different slope gradients differ by only a little. No significant change in soil erodibility (K r ) was observed for different slope gradients ranging from 5 to 25 and the critical shear stresses vary a little in this study. The actual numbers for the estimated soil erodibilities and critical shear stresses are listed in Table 1.

4 T.W. Lei et al. / Geoderma 144 (2008) Fig. 1. Relationship between the potential detachment rates and shear stress. Table 1 shows numerically that K r values under different slope gradients are nearly the same. The average soil erodibility (K r ) was ±0.001 s m 1. The K r estimated from the experimental data indicated that the soil used is vulnerable to erosion and has a high susceptibility to erosive agents. The determination coefficients (R 2 ) for Eq. (9), under different slopes, were quite high (N0.74) which suggests that the method advanced is very effective. This erodibility value was much higher than those given by Gilley et al. (1993). Actually, it is about 10 times higher than their highest value. It is believed that soil texture was responsible for this difference, but the method of estimating the potential detachment rate of the soil made a significant contribution as well. Data reported by Zhang et al. (2002) produced a soil erodibility of s m 1 that is much closer to the value recorded here. The soil they used for the study was 17.2% sand, 58.5% silt and 24.3% clay and small soil samples of 10 cm in diameter were used for the detachment rate measurement. Therefore, the correlation between our value and their's obtained through the use of soil with higher clay content as well as a rather short soil sample of 10 cm adds credence to the values obtained in this paper. What should be noted is that soil erodibility is a physical characteristic of soil. The K r estimated with data from experiments under different slope gradients and flow rates should be the same under different conditions according to the theoretical analysis given by Liu et al. (1999). The data listed in Table 1 supports this opinion. From Table 1, the critical shear stresses τ c increased a little with the increasing slope gradients, namely 3.19 N m 2, 3.96 N m 2, 4.13 N m 2, 4.39 N m 2, 4.57 N m 2 under 5, 10, 15, 20, 25 respectively with the coefficients of determination (R 2 ) being above When the shear stress exceeds the critical shear, soil will begin to erode. Soil structure is not necessarily deformed as stated in the paper given by Fan et al. (1997). Shear stress on soil increases with an increase of slope gradient because the force along the slope increases with slope gradient (Eq. (12)). Further study, though, is needed in this regard. Comparison of soil erodibility and critical shear stress computed by using the rational method and the experimental data from short rills was made and is presented in Fig. 2. The lines in the figures are in a 1:1 scale and the closer the dots are to the lines, the better the correlation between the data set from Table 1 K r and τ c under different slope gradients Slope gradient K r (s m 1 ) τ c (N m 2 ) n Regressed function R D rmax =0.3209τ D rmax =0.3211τ D rmax =0.3210τ D rmax =0.3213τ D rmax =0.3214τ

5 632 T.W. Lei et al. / Geoderma 144 (2008) concentrated flow at the start point of the rill, i.e. at a rill length of 0. A rational method was then advanced to determine soil erodibility and critical shear stress with data thus obtained. Experimental data from well defined rill channels of loess type soil were used to compute the soil erodibility and critical shear stress. The output results were compared with data from other sources and were found to be reasonably good, which presents the validation of the method suggested. However, the rill erodibility was about ten times greater than those estimated with method used by WEPP. The results also revealed that soil erodibility that has been estimated with average detachment rates from a long rill may be considerably underestimated. In order to reduce the estimation error of rill erodibility, a welldefined rill channel with steady shallow flow and relatively narrow width was used to ensure the steady transport capacity and the flow shear stress along a rill. This point should be always taken into consideration when using this rational method. It is hoped that this rational and logical method for the determination of soil erodibility and critical shear stress will supply a solid base and a useful tool for quantifying the important parameters needed in soil erosion prediction models. Acknowledgments Fig. 2. Comparison of soil erodibility and critical shear stress estimated directly from experimental data of short rills and computed with the rational method. the rational method and the short rill experimental data. From Fig. 2, the correlation of soil erodibility (K r ) and critical shear stress is very good between the rational method and those estimated directly from experimental data of short rills. The correlative function of soil erodibility (K r ) from the rational method and those estimated with experimental data is Y=0.999 X. The correlative function of soil critical shear stress from the rational method and those estimated with experimental data is Y=0.998X. The coefficient of determination (R 2 )ofk r is about 1 under different hydraulic conditions and the coefficient of determination (R 2 ) for critical shear stress is more than Thus, the rational method results show extremely good prediction of the experimentally obtained erodibility and critical shear stress. 4. Conclusion Estimation of soil erodibility and critical shear stress involves the determination of potential detachment rates under different hydraulic regimes. The shear stress of rill flow is best estimated with a well defined rill channel, while random rill width alternating in a wide runoff plot produces unpredictable hydraulic conditions. The detachment capacity is ideally and mathematically estimated from the derivative value of the function between sediment concentration and rill length of rill erosion under This work was supported by the Natural Science Foundation of China under project No and supported by Program for Changjiang Scholars and Innovative Research Team in University, as well as supported by the CAS/SAFEA International Partnership Program for Creative Research Teams, Thanks to Mr. Baoji Wang and Ms Yongcui Yuan from China Agricultural University Library for their services of literature search. References Cochrane, T.A., Flanagan, D.C., Detachment in a simulated rill. Transactions of the ASAE 40 (1), Elliot, W.J., Liebenow, A.M., Laflen, J.M., Kohl, K.D., A compendium of soil erodibility data from WEPP cropland soil field erodibility experiments 1987 and 88. NSERL Rep., vol. 3. USDA-ARS, West Lafayette, IN, p Fan, X.K., Jiang, D.S., Zhao, H.L., Analysis on anti-shear strength of shallow original state soil in Loess Plateau. Journal of Soil Erosion and Soil and Water Conservation 3 (4), (in Chinese with English abstract). Flanagan, D.C., Nearing, M.A., USDA-Water erosion prediction project hillslope profile and watershed model documentation. NSERL Report, vol. 10. USDA-ARS Nation soil erosion research Laboratory, West Lafayette, IN, p Gilley, J.E., Elliot, W.J., Laflen, J.M., Simanton, J.R., Critical shear stress and critical flow rates for initiation of rilling. Journal of Hydrology 142, Huang, C.J., Bradford, M., Laflen, J., Evaluation of the detachmenttransport coupling concept in the WEPP rill erosion equation. Soil Science Society of America Journal 60, Laflen, J.M., Elliot, W.J., Simanton, R., Holzhey, S., Kohl, K.D., WEPP soil erodibility experiments for rangeland and cropland soils. Journal of Soil and Water Conservation 46 (1), Lei, T.W., Nearing, M.A., Flume experiments for determining rill hydraulics, erosion and rill patterns. 10th ISCO Int. Con., West Lafayette, IN, USA, May 21 28, Lei, T.W., Nearing, M.A., Haghighi, K., Bralts, V.F., Rill erosion and morphological evolution: a simulation model. Water Resource Research 34 (11),

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