# Soil detachment by interrill overland flow

Interrill erosion

Interrill overland flow on a soil surface is a mass of anastomosing or braided water courses with no pronounced channels rather than a sheet of water of uniform depth. The flow is broken up by stones and cobbles and by the vegetation cover, often swirling around tufts of grass and small shrubs.

## Shear Stress of Flow

The important factor in the basic hydraulic relationships of Reynolds, and Froude numbers is the flow velocity, commonly expressed by the Manning equation:

${\displaystyle v={\frac {r^{\frac {2}{3}}s^{\frac {1}{2}}}{n}}}$    (1.7)

where ${\displaystyle r}$  is the hydraulic radius, ${\displaystyle s}$  is the slope and ${\displaystyle n}$  is Manning’s coefficient of roughness. The equation assumes fully turbulet flow moving over a rough surface.

Because of an inherent resistance of the soil, velocity must attain a threshold value before erosion commences. Basically, the detachment of an individual soil particle from the soil mass occurs when the forces exerted by the flo exceed the forces keeping the particle at rest. The processes involved and the forces at work when particle movement over relatively gentle slopes in rivers is initiated are controlled by critical conditions of the dimensionless shear stress of the flow ${\displaystyle \Theta }$ , and the particle roughness Reynolds number ${\displaystyle Re}$ , respectively, defined by:

${\displaystyle \Theta ={\frac {\rho _{w}u_{*}^{2}}{g(\rho _{s}-\rho _{w})D}}}$    (1.8)

${\displaystyle Re^{*}={\frac {u_{*}D}{v}}}$    (1.9)

where ${\displaystyle \Theta }$  is the Shields (1936) number[1], ${\displaystyle \rho _{w}}$  and ${\displaystyle \rho _{s}}$  are the densities of water and soil respectively, ${\displaystyle g}$  is the acceleration of gravity, ${\displaystyle D}$  is the diameter of the respective soil particle and ${\displaystyle u_{*}}$  is the shear velocity of the flow, expressed as:

${\displaystyle u_{*}=(grs)^{\frac {1}{2}}}$    (1.10)

When the value of ${\displaystyle Re^{*}}$  is greater than 40 (turbulent flow), the critical value of ${\displaystyle \Theta }$  for particle movement assumes a constant value of 0.05. Unfortunately, this value does not hold when the particles are not fully submerged or the flow has Reynolds numbers in the laminar range, as it is the case with overland flow. Studies with rock fragments in shallow flows suggest that ${\displaystyle \Theta }$  is about 0.01 in value (Poesen, 1987[2]; Torri and Poesen, 1988[3]).

Figure 1.2: Critical shear velocity for soil particle detachment in turbulent water flow as function of particle size[4]

Other research (Govers, 1987[5]; Guy and Dickinson, 1990[6]; Torri and Borselli, 1991[7]) indicates that Shields number consistently overpredicts the hydraulic requirements of particle movement. This implies that the initiation of particle movement is not solely a phenomenon of fluid shear stress but is enhanced by further factors:

• Effects of raindrop impact on the flow,
• Angle of repose of the particle in relation to ground slope,
• Strong influence of gravity as slope steepness increases,
• Cohesion of the soil,
• Changes in density of the fluid as sediment concentration in the flow increases,
• Abrasion between particles moving in the flow and the soil beneath.

Since the above approach has not proved satisfactory, empirical procedures have been adopted by Savat (1982)[8], based on a critical value of the flow’s shear velocity for initiating particle movement (fig. 1.2). Once the critical conditions for particle movement are exceeded, soil partilces may be detached from the soil mass at a rate that is dependent on the shear velocity of the flow and the unit discharge (Govers and Rauws, 1986)[9].

However this is only valid, if the shear velocity is exerted solely on the soil particles, which implies that the resistance to the flow wold entirely be due to grain resistance. This situation is only true for completely smooth bare soil surfaces. In practice, resistance due to the microtopographic form of the soil surface and the plant cover is usually more important and grain resistance may be as little as 5 % of the total resistance offered to the flow (Abrahams et al., 1992)[10].

## Soil Detachment

Since it is difficult to determine the level of grain resistance, only very generalized relationships can be developed for describing detachment rate ${\displaystyle D_{f}}$ , depending on a simplified relationship between detachment and flow velocity. Integrating the continuity and Manning’s velocity equations, Meyer (1965)[11] showed that:

${\displaystyle v=s^{\frac {1}{2}}Q^{\frac {1}{3}}}$    (1.11)

for constant roughness conditions, where ${\displaystyle Q}$  is discharge or flow rate. Assuming that the detachment rate ${\displaystyle D_{f}}$  varies with the square of the velocity Meyer and Wishmeier (1969)[12] showed that:

${\displaystyle D_{f}\propto Q^{f}s^{j}}$    (1.12)

with ${\displaystyle f={\tfrac {2}{3}}}$ , and ${\displaystyle j={\tfrac {2}{3}}}$ . Quansah (1985)[13] obtained more accurate exponent values of f = 1.5 and j = 1.44 experimentally for a range of soil types from clay to sand. Both versions relate only to the action of water flow over the soil surface. When flow is accompanied by rainfall, he found that the exponents decreased in value to f = 1.12 and j = 0.64 indicating that raindrop impact inhibits the ability of flow to detach soil particles.

However, detachment depends on the amount of sediment already suspended in the flow (Meyer and Monke, 1965)[14], ${\displaystyle D_{f}}$  in eqn. 1.12 applies to the detachment capacity that only occurs when the flow is clear. Unter other conditions, ${\displaystyle D_{f}}$  depends on the difference between the actual sediment concentration in the flow ${\displaystyle C}$  and the maximum concentration that the flow can hold ${\displaystyle C_{max}}$  (Foster and Meyer, 1972)[15]:

${\displaystyle D_{f}\propto (C_{max}-C)}$    (1.13)

This implies that in theory the detachment rate declines as sediment concentration in the flow increases and that when the maximum sediment concentration is reached, the detachment rate becomes zero.

Flanagan and Nearing (1995)[16] describe the interrill detachment rate ${\displaystyle D_{f,i}}$  as explicit function of baseline interrill erodibility ${\displaystyle K_{i}}$ , effective rainfall intensity ${\displaystyle I_{e}}$ , interrill runoff rate ${\displaystyle Q_{i}}$ , canopy, ground cover and interrill slope adjustment factors ${\displaystyle C_{c}}$ , ${\displaystyle C_{g}}$  and ${\displaystyle C_{s}}$ , spacing and width of rills ${\displaystyle R_{s}}$ and ${\displaystyle w}$ , and a sediment delivery ratio ${\displaystyle {SDR}}$ :

${\displaystyle D_{f,i}=K_{i}I_{e}Q_{i}C_{c}C_{g}C_{s}{\frac {R_{s}}{w}}{SDR}}$    (1.14)

with

${\displaystyle C_{c}=1-F_{c}e^{-0.34h_{c}}}$    (1.15)

${\displaystyle C_{g}=e^{-2.5f_{g}}}$    (1.16)

${\displaystyle C_{s}=1.05-0.85e^{4\sin \alpha }}$    (1.17)

where ${\displaystyle f_{c}}$  is the fraction of the soil protected by canopy cover, ${\displaystyle h_{c}}$  is the effective canopy height, ${\displaystyle f_{g}}$  is the fraction of interrill surface covered by plant residue, and ${\displaystyle \alpha }$  is the interrill slope angle. According to Foster (1982)[17] ${\displaystyle {SDR}}$  may be estimated as function of soil surface random roughness, interrill slope and interrill sediment particle size distribution.

## Bibliography

1. Shields, A. (1936). Anwendung der Aehnlichkeitsmechanik und der Turbulenzfoschung auf die Geschiebebewegung, volume 26 of Mitteilungen der Preussischen Anstalt Wasserbau und Schiffbau.
2. Poesen, J. (1987). Transport of rock fragments by rill flow: a field study. Catena Supplement, 8:35–54.
3. Torri, D. and Poesen, J. (1988). Incipient motion conditions for single rock fragments in simulated rill flow. Earth Surface Processes and Landforms, 13:225–237.
4. Savat, J. (1982). Common and uncommon selectivity in the process of fluid transportation: field observations and laboratory experiments on bare surfaces. Catena Supplement, 1:139–160.
5. Govers, G. (1987). Initiation of motion in overland flow. Sedimentology, 34:1157–1164.
6. Guy, B. and Dickinson, W. (1990). Inception of sediemnt transport in shallow overland flow. Catena Supplement, 17:91–109.
7. Torri, D. and Borselli, L. (1991). Overland flow and soil erosion: some processes and their interactions. Catena Supplement, 19:129–137.
8. Savat, J. (1979). Laboratory experiments on erosion and deposition of loess by laminar sheet flow and turbulent rill flow. In Vogt, H. and Vogt, T., editors, Colloque sur l’erosion agricole des sols en milieu tempere non Mediterraneen, pages 139–143. L’Universite Lois Pasteur, Strasbourg.
9. Govers, G. and Rauws, G. (1986). Transporting capacity of overladn flow on a plane and on irregular beds. Earth Surface Processes and Landforms, 11:515–524.
10. Abrahams, A., Parsons, A., and Hirsch, P. (1992). Field and laboratory studies of resistance to interrill overland flow on semi-arid hillslopes, southern arizona. In Parsons, A. and Abrahams, A., editors, Overland flow: hydraulics and erosion mechanics, pages 1–23. UCL Press, London.
11. Meyer, L. (1965). Mathematical relationships governing soil erosion by water. Journal of Soil and Water Conservation, 20:149–150.
12. Meyer, L. and Wishmeier, W. (1969). Mathematical simulation of the process of soil erosion by water. Transactions of the American Society of Agricultural Engineers, 12:754–758,762.
13. Quansah, C. (1985). Rate of soil detached by overland flow, with and without rain, and its relationship with discharge, slope steepness and soil type. In El-Swaify, S., Moldenhauer, W., and Lo, A., editors, Soil erosion and conservation, pages 406–423. Soil Conservation Society of America, Ankeny, IA.
14. Meyer, L. and Monke, E. (1965). Mechanics of soil erosion by rainfall and overland flow. Transactions of the American Society of Agricultural Engineers, 8:572–577.
15. Foster, R. and Meyer, L. (1972). A closed-form soil erosion equation for upland areas. In Shen, H., editor, Sedimentation. Department of Civil Engineering, Colorado State University, Cort Collins.
16. Flanagan, D. and Nearing, M. (1995). USDA-Water Erosion Prediction Project (WEPP: Technical Documentation, volume 10 of NSERL Report. National Soil Erosion Research Laboratory, West Lafayette, IN.
17. Foster, G. (1982). Modelling the soil erosion process. In Haan, C., Johnson, H., and Brakensiek, D., editors, Hydrologic modelling of small watersheds, volume 5, pages 940–947. American Society of Agricultural Engineers Monograph.