Flow detachment and deposition
The ability of flowing water to erode its bed is assumed independent of the amount of material it carries and is only a function of the energy expended by the flow. Deposition takes place at a rate equal to wCVs , where w is the width of flow (m), C is the sediment concentration in the flow (kg.m-3) and Vs is the settling velocity of the particles (m.s-1). The concentration at transport capacity represents the sediment concentration at which the rate of erosion by the flow and accompanying rate of deposition are in balance. In this condition the net rate of erosion is zero and Df equals the deposition rate (wCTVs).
Tc = ds c(w-wc) d
in which Tc is the volumetric transport capacity (kg.m-3), Ds is the material density (2650 kg.m-3), w is the stream power (calculated as flow velocity*energy slope) and w c is the critical stream power defined by Govers (1990) for a fairly wide range of materials to be approximately 0.4 cm.s-1, c and d are experimental coefficients depending on the median texture (d50) of the material. This equation describes the transport capacity in rills. In LISEM equation 9 becomes (with V in m/s and the sine of the slope S):
Tc = 2650 c(VS100 – 0.4) d
Note that currently LISEM does not distinguish in its algorithms between interrill and rill erosion. In fact all erosion is assumed to be rill erosion, although there are no rills defined as such. Water simply flows from cell to cell following a drainage network (see below). Currently the definition of a separate rill network is under development and then rill and interrill transport capacity can be differentiated. LISEM uses the median of the texture (d50) as input to represent spatial variability of grain sizes. The model calculates the corresponding coefficients a and b based on work of Govers (1990):
c = [(D50 + 5)/0.32] -0.6
d = [(D50 + 5)/300] 0.25
NOTE: these equations are valid for materials with a grain size diameter larger than 32 m m.
Df = Y(Tc-C)Vs w dx
in which Df is the flow detachment (kg.s-1), Tc is the transport capacity of the flow (kg.m-3) and Y is a dimensionless efficiency factor. The latter is included to account for the fact that the detachment will be limited by the cohesion of the soil material: the pick-up rate for cohesive soil therefore needs to be reduced by a coefficient whenever C is less than Tc. Note that dx is added because the unit length of a spatial element in a raster environment becomes the grid cell size. Y is calculated as (Rauws and Govers, 1988):
Y = umin/uc = 1/(0.89+0.56Coh)
in which uc and umin are the critical shear velocity and the minimum critical shear velocity (cm.s-1), and Coh is the cohesion of the wet soil determined with a Torvane (kPa). Cohesion by plant roots can be accounted for in LISEM by a second cohesion map that is added to the soil cohesion.
Deposition DP (in kg.s-1) is calculated in a similar way except that an exponential decay function is added to simulate the effect that the settling of the particles will slow down when they reach the bottom of the water layer on the surface. Deposition takes place only when C is larger than TC, so when the transport capacity is exceeded. By definition the efficiency Y then equals 1 as there is no effect of soil strength on deposition:
DP = (Tc-C)(1-exp(dt Vs/d)) d w dx
where d is the water depth (m), and thus d*w*dx is the volume of water in the grid cell.
NOTE: in LISEM version 2.02 and earlier the deposition is calculated as a linear function of the settling velocity, similar to the flow detachment:
DP = (Tc-C)Vs w dx
The net sediment in suspension is transported between gridcells with the kinematic wave.
There are two checks carried out in LISEM to avoid the calculation of an incorrect Df or DP.The amount of erosion or deposition in a timestep depends on the settling velocity Vs. If the timestep is too large it may happen that all sediment has already settled before the end of the timestep. Therefore the deposition can never be more than the amount of transport capacity surplus (note that DP is a negative value):
DP = max(min(0, TC-C), DP)
NOTE: this used to be the sediment in suspension (LISEM version 2.02 and earlier):
DP = max(-sediment, DP)
However this caused sometimes a drop of all the sediment from the flow and a drop below the transport capacity is more correct. In the case of detachment, the amount of detached soil cannot be more than the remaining carrying capacity of the flow (Q in m3/s):
Df = (Tc – C) Q dt
Furthermore as of version 2.03 a minimum water height of 0.01 mm is included to prevent the occurence of extreme sediment concentrations with very low water depth at the surface. So the sediment concentration C, as well as Df and Dp
C: if (h < 0.0001) then C=0
Df: if (h < 0.0001) then Df=0
Dp: if (h < 0.0001) then Dp = all sediment in suspension