# Rill Erosion

## Rill Initiation

Rills are initiated at a critical distance downslope, where overland flow becomes channelled. Overland flow breaks up into small channells of microrills (Moss et al., 1982)[1]. In addition to the main flow path downslope, secondary flow paths develop with a lateral component. Where these converge, the increase in discharge intensifies particle movement and small channels or trenches are cut by scouring.

In terms of hydraulic characteristics of flow (Reynolds, and Froude number) the change from interrill overland to rill flow passes through four stages (Merritt, 1984)[2]:

• Unconcentrated overland flow,
• Overland flow with concentrated flow paths,

At the point of rill initiation, flow conditions change from subcritical to supercritical (Savat, 1979)[3]. The overall change in flow conditions through the four stages takes place smoothly as the Froude number F increases from about 0.8 to 1.2, rather than occurring when a distinct threshold value is reached (Torri et al., 1987b[4]; Slattery and Bryan, 1992[5]). For this reason, attempts to explain the onset of rilling through the exceedance of a critical Froude number have not been successful and additional factors - such as particle size of the material (Savat, 1979)[3] and sediment concentration in the flow (Boon and Savat, 1981)[6] have had to be included when defining its value.

Greater success has been achieved relating rill initiation to the exceedance of a critical ratio between hydraulic shear stress ${\displaystyle \tau }$ exerted by the flow and the shear strength ${\displaystyle \tau _{s}}$ of the soil. When ${\displaystyle \tau /\tau _{s}}$ > 0.0001 − 0.0005, rills will form (Torri et al., 1987a)[7].

Using the Darcy-Weisbach uniform flow equation and assuming that discharge varies linearly with the downslope distance ${\displaystyle x}$, Nearing et al. (1989)[8] describe ${\displaystyle \tau }$ as function of the specific weight of water ${\displaystyle \rho _{w}}$, of peak runoff rate ${\displaystyle Q_{p}}$, of the Chezy discharge coefficient ${\displaystyle C}$ and of local slope gradient ${\displaystyle s}$:

${\displaystyle \tau =\rho _{w}{\Bigl (}{\frac {Q_{p}}{C}}xs{\Bigr )}^{\tfrac {2}{3}}}$   (1.23)

where ${\displaystyle C}$ is related to gravity ${\displaystyle g}$ and to a total rill friction factor ${\displaystyle f_{t}}$:

${\displaystyle C={\frac {8g^{\tfrac {1}{2}}}{f_{t}}}}$   (1.24)

The critical shear velocity (eqn. 1.10, cf. section → Soil Detachment by Interrill Overland Flow) for rill initiation ${\displaystyle u_{*crit}}$ is linearly related to the shear strength ${\displaystyle \tau _{s}}$ of the soil as measured at saturation using a Torvane device (Rauws and Govers, 1988)[9]:

${\displaystyle u_{*crit}=0.89+0.56\tau _{s}}$   (1.25)

when shear stress or shear velocity is applied wholly to the soil particles, both should strictly be known as grain shear stress or grain shear velocity.

Once rills have formed, their extension upslope occurs by the retreat of the headcut at the top of the channel. The rate of retreat is controlled by the cohesiveness of the soil, the height and angle of the headwall, the discharge and the velocity of the flow (De Ploey, 1989)[10]. Downslope extension of the rill is controlled by the shear stress exerted by the flow and the strength of the soil (Savat, 1979)[3].

## Soil Detachment in Rills

Soil detachment in rills occurs when hydraulic shear stress ${\displaystyle \tau }$ acting on the soil exceeds the critical shear stress ${\displaystyle \tau _{c}}$ of the soil and when sediment load is less than sediment transport capacity. For the case of rill erosion the detachment rate ${\displaystyle D_{f,r}}$ is determined by the detachment capacity of flow ${\displaystyle D_{c}}$, the sediment load of the flow ${\displaystyle G}$ and the sediment transport capacity in the rill ${\displaystyle T_{f}}$ (Nearing et al., 1989)[8]:

${\displaystyle D_{f,r}=D_{c}{\Bigl (}1-{\frac {G}{T_{f}}}{\Bigr )}}$   (1.26)

Shear stress also determines the detachment capacity of soil particles by flow within a rill (Foster, 1982)[11] as critical shear stress is exceeded:

${\displaystyle D_{c}=K_{r}(\tau -\tau _{c})}$   (1.27)

where ${\displaystyle K_{r}}$ is a measure of the detachability of the soil and ${\displaystyle \tau _{c}}$ is the critical shear stress of the soil. Rill detachment is considered zero when actual shear is less than critical shear of the soil.

## Sediment Deposition

Deposition of transported soil particles occurs when sediment load ${\displaystyle G}$ is greater than the sediment transport capacity ${\displaystyle \tau _{f}}$. In this case, sediment detachment rate ${\displaystyle D_{f,r}}$ becomes negative and may be expressed as function of effective settling velocity for the sediment particles ${\displaystyle v_{s}}$, flow discharge ${\displaystyle Q}$ per unit width (Nearing et al., 1989)[8]:

${\displaystyle D_{f,r}={\frac {v_{s}}{Q}}{\Bigl (}T_{f}-G{\Bigr )}}$   (1.28)

The transport capacity of rill flow ${\displaystyle \tau _{f}}$ is approximately represented by eqns. 1.19, 1.21 or 1.20 (cf. section → Transport of Soil Particles by Flow).

Govers (1992)[12] experimentally related flow velocity in rills is related to the rill flow discharge ${\displaystyle Q}$:

${\displaystyle v=3.52Q^{0.294}}$   (1.29)

which gave better predictions than the Manning equation (eqn. 1.7, cf. section → Soil Detachment by Interrill Overland Flow). This was because over a range from 2 to 8° slope had no effect on flow velocity, neither did the grain roughness or the soil surface form. Govers (1992)[12] therefore modified eqn. 1.21 (cf. Section → Transport of Soil Particles by Flow) by replacing the velocity term in eqn. 1.29:

${\displaystyle C_{max}=a(3.52Q^{0.294}-\omega _{c})Q}$   (1.30)

where ${\displaystyle a}$ is dependent upon the grain size of the sediment and ${\displaystyle \omega _{c}}$ is the critical value of unit stream power, which amounts at 0.0074. The equation expresses the maximum sediment concentration that can be carried by runoff in a rill.

The actual sediment concentration - and therefore the erosion - may vary considerably from this. This is because the supply of sediment to the rill is not solely dependent upon the detachment of soil particles by the flow. The rill has to adjust continually for pulse influxes of sediment

• due to wash-in from interrill flow,
• due to erosion and collapse of the head wall,
• due to collapse of side walls.

Mass failure of side walls can contribute more than half of the sediment removed in rills (Govers and Poesen, 1988)[13].

Since rill flow is non-selective in particle size it can carry, large grains, and even rock fragments up to 9 cm in diameter (Poesen, 1987)[14], can be moved. Rill erosion accounts for the bulk of the sediment removed from a hillside, depending, besides the soil properties, on the spacing of the rills and the extent of the area affected. Material transported in rills may account for 54-78 % of the total erosion (Govers and Poesen, 1988)[13].

## Bibliography

1. Moss, A., Green, P., and Hutka, J. (1982). Small channels: their formation, nature and significance. Earth Surface Processes and Landforms, 7:401–415.
2. Merritt, E. (1984). The identification of four stages during microrill development. Earth Surface Processes and Landforms, 9:493–496.
3. a b c 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.
4. Torri, D., Sfalanga, M., and Del Sette, M. (1987b). Splash detachment: Runoff depth and soil cohesion. Catena, 14:149–155.
5. Slattery, M. and Bryan, R. (1992). Hydraulic conditions for rill incision under simulated rainfall: a laboratory experiment. Earth Surface Processes and Landforms, 17:127–146.
6. Boon, W. and Savat, J. (1981). A nomogram for the prediction of rill erosion. In Morgan, R., editor, Soil conservation: problems and prospects, pages 303–319. Wiley, Chichester.
7. Torri, D., Sfalanga, M., and Chisci, G. (1987a). Threshold conditions for incipient rilling. Catena Supplement, 8:97–105.
8. a b c Nearing, M., Foster, G., Lane, L., and Finkner, S. (1989). A process-based soil erosion model for usda-water erosion prediction project technology. Transactions of the American Society of Agricultural Engineers, 32(5):1587–1593.
9. Rauws, G. and Govers, G. (1988). Hydraulic and soil mechanical aspects of rill generation on agricultural soils. Journal of Soil Science, 39:111–124.
10. De Ploey, J. (1989). A model for headcut retreat in rills and gullies. Catena Supplement, 14:81–86.
11. 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.
12. a b Govers, G. (1992). Relationship between discharge, velocity and flow area for rills eroding loose, non-layered materials. Earth Surface Processes and Landforms, 17:515–528.
13. a b Govers, G. and Poesen, J. (1988). Assessment of the interrill and rill contributions to total soil loss from an upland field plot. Geomorphology, 1:343–354.
14. Poesen, J. (1987). Transport of rock fragments by rill flow: a field study. Catena Supplement, 8:35–54.