Soil Mechanics and Erosion

Soil phase relationships

Phase relationships describe how the total volume and mass of a soil are divided among solids, water and air. In the three-phase diagram, the void volume is V_v = V_w + V_a, and the total volume is V = V_s + V_v.

Key definitions:

• Void ratio — e = V_v/V_s.
• Porosity — n = V_v/V = e/(1+e).
• Water content (mass basis) — w = M_w/M_s, always taken on the dry mass of solids.
• Degree of saturation — S = V_w/V_v = wG/e.
• Volumetric water content — \theta = V_w/V.

The wet density is \rho = (M_s + M_w)/V, and the dry density is \rho_d = \rho/(1+w). With G the specific gravity of solids and \rho_w = 1 g/cm^3, e = G\rho_w/\rho_d - 1. A given wet density and water content therefore fix the dry density, the void ratio and the porosity.

For a partly saturated soil, the bulk unit weight is \gamma = \dfrac{G + Se}{1+e}\gamma_w, which reduces to the saturated form when S = 1. The result carries the units of \gamma_w, so no further conversion is needed if \gamma_w is given in N/m^3.

When a sample is oven-dried without any shrinkage, the total volume stays fixed. The water mass lost equals the moist mass minus the oven-dry mass, and the water volume equals that mass divided by \rho_w. The solid volume V_s = V/(1+e) must agree with M_s/(G\rho_w), which serves as a useful check before computing S = V_w/V_v.

For a core sample, the depth of water per unit depth of soil equals the volumetric water content, \theta = (\text{water mass}/\rho_w)/V_{\text{core}}. Multiplying \theta by a soil depth gives the equivalent water depth.

Common traps:

• Water content is referred to dry mass, not wet mass.
• Void ratio and porosity are different quantities, and they are linked only by n = e/(1+e).
• The total sample mass does not change intensive properties such as density, porosity or saturation.

Wind erosion and windbreak protection

Wind erosion moves soil by three modes of transport:

• Saltation — sand-sized grains bounce along the surface after being lifted by the wind.
• Surface creep — coarse grains roll or slide along the ground.
• Suspension — the finest particles are carried in the air over long distances.

Saltation transports the largest share of wind-eroded soil, because grains falling back to the surface dislodge further particles. This is why it is the dominant mode in wind erosion problems.

Erosion begins when the wind shear at the surface exceeds the threshold velocity of the most erodible size fraction. In Bagnold's form, the transport rate grows with the cube of the excess of the actual wind velocity V over the threshold velocity V_t, that is q \propto (V - V_t)^3. The particle diameter enters through a square-root term, so the transport rate is governed mainly by the wind speed above threshold.

A windbreak reduces wind speed on its lee side. The sheltered zone extends downwind as a multiple of the barrier height H. The distance of full protection is found from an empirical shelter relation that links the ratio of threshold to actual wind velocity at 15 m height with the barrier height, so the distance is not simply a fixed multiple in every case.

When the wind does not blow perpendicular to the barrier, the effective barrier length normal to the wind is reduced to L_{eff} = L\cos\theta, where \theta is the deviation angle. The protected area is the effective length times the protected downwind distance, and it is usually reported in hectares.

Bench terrace design

A bench terrace is a level or nearly level step cut into a sloping field. Its layout is set by the vertical interval VI between successive benches and the land slope S. The horizontal interval is HI = VI/S, so the benches are farther apart on gentler land.

The batter (riser) slope is given as horizontal to vertical. A riser with a 1:1 slope projects horizontally by the same amount as its height. The working width of the bench is therefore WW = HI - (\text{horizontal projection of the batter}). The area lost for cultivation is the riser strip expressed as a percentage of the interval area. A steeper batter reduces the working width and the area lost.

The drainage direction of the bench depends on rainfall:

• Inward slope — used where heavy rainfall makes surface runoff the main flow, so water is caught in a drain at the inner edge and does not overtop the outer riser.
• Level top — suits moderate rainfall, allowing infiltration on the bench.
• Outward slope — used where conservation of water is the aim and runoff is low.

Earthwork per hectare is found by integrating the cut area across one horizontal interval and scaling to a hectare. The cut on the uphill part of the bench and the triangular cut in the riser both contribute. When cutting and filling are equal, the earthwork per unit area follows directly from the cut cross-section and the number of intervals per hectare.

USLE topographic factor (LS)

The Universal Soil Loss Equation is A = RKLSCP, where the topographic factor LS combines the effect of slope length L and slope gradient S. The standard form is

LS = \left(\dfrac{\lambda}{22.13}\right)^{m}\left(65.41\sin^2\theta + 4.56\sin\theta + 0.065\right)

where \lambda is the slope length in metres, 22.13 m is the standard unit plot length, \theta is the slope angle, and \sin\theta = S/\sqrt{1+S^2} when S is the gradient expressed as a fraction (for example, 0.08 for 8 per cent slope).

The exponent m depends on the slope. It is about 0.5 for slopes of 5 per cent or more and smaller on gentler land. Because the length term is a power law, halving the slope length multiplies the length factor by (0.5)^m. For m = 0.5 this gives 0.707, so the soil loss falls by about 29 per cent with all other factors unchanged.

Sediment delivery ratio

The sediment delivery ratio (SDR) is the fraction of gross erosion within a watershed that actually reaches the outlet:

SDR = \dfrac{\text{sediment yield at outlet}}{\text{gross erosion}}

Its value is always less than or equal to 1. Sediment is deposited on hillslopes, in channels and in depressions along the flow path, so a larger contributing area gives more chances for deposition. The SDR therefore decreases as the drainage area A increases. Empirical relations express this as an approximate negative power of area, so SDR varies inversely with a fractional power of A. The exponent is empirical and differs between published relations, so it must be taken from the formula given in the problem.