# Overland flow

Originally LAPSUS was developed to simulate overland flow (either Hortonian or saturation overland flow) erosion and sedimentation in the Mediterranean region of Southern Spain (Schoorl, 2001).

Overland flow (or surface run-off) occurs in two cases:

1. when the intensity of precipitation that reaches the surface exceeds the infiltration capacity of the soil. This process is known as Hortonian overland flow.

2. when the combination of  precipitation intensity and duration (and run-on from higher areas) saturates the soil and raises the water table to the surface. This process is known as saturation overland flow.

## Technical Information

The water erosion and sedimentation module of the model has two fundamental assumptions: 1) the potential energy of surface water flow is the driving force for sediment transport and 2) the difference between sediment input and output of a grid cell is equal to the net increase in storage (continuity equation for sediment movement) (Schoorl et al., 2000). The process description is derived from early works of Kirkby (Kirkby, 1971) and Foster and Meyer (Foster and Meyer, 1972a; Foster and Meyer, 1972b), who use 2D formulas to calculate water erosion and sedimentation. For the LAPSUS model the formulas are adapted to be able to simulate spatial (3D) water erosion and sedimentation (Schoorl et al., 2000). The formulas discussed below are based on the 2D formulas of Kirkby and Foster and Meyer and with the accompanying units (Foster and Meyer, 1972a; Foster and Meyer, 1972b; Kirkby, 1971).

After calculating discharge Q the sediment transport capacity C (m2 time-1) in the grid cell can be calculated as function of discharge and slope following:

(1.1)    C = α·Qm·Λn

Whereby Λ is the slope gradient (∂z/∂x) (-) and m (-) and n (-) are constants giving an indication of the system studied: m = 0 and n = 1 suggests soil creep, while m = n = 3 suggests large rivers (Kirkby, 1971). No strict boundaries can be given as the effect of m and n depends on interactions with other model parameters. Dummy variable α  is used to correct the units.

The sediment transport rate S (m2 time-1) is calculated following the integrated continuity equation for sediment movement (Eq. 1.2 and 1.3). The composition of the used e-power term in the formula depends on the balance between the transport rate of sediment already in transport S0 (m2 time-1) (incoming sediment fluxes of all higher neighbours in the grid cell) and sediment transport capacity C: if S0 < C erosion results, while when S0 > C sedimentation results. When the grid cell is eroded the following formula for sediment transport rate S is used:

(1.2)    S = C + (S0 - C)·e -dx·D/C

When sediments are deposited in the grid cell the following formula for sediment transport rate S is used:

(1.3)    S = C + (S0 - C)·e -dx·T/C

Whereby the transport rate of sediment S over grid cell size dx (m) is calculated by comparing sediment transport capacity C with transport rate of sediment already in transport S0 (m2 time-1) minus sediment transport capacity C, reduced by an e-power resulting from grid cell size, detachment capacity D or settlement capacity T and sediment transport capacity C.

Detachment capacity D (m time-1), representing how easy sediment is eroded of the surface, is calculated as function of discharge and slope following:

(1.4)    D = Kes·Q·Λ

whereby Kes (m-1) is a lumped surface factor indicating erodibility of the surface. Settlement capacity T (m time-1), representing how easy sediment is deposited on the surface, is calculated following:

(1.5)    D = Pes·Q·Λ

whereby Pes (m-1) is a surface factor indicating lumped sedimentation characteristics.

By comparing the sediment transport rate S of the grid cell with the sediment already in transport S0 the change in sediment transport rate dS, and thus erosion or sedimentation, can be calculated following:

(1.6)    dS = S - S0

dS can be recalculated to erosion or sedimentation in meter by dividing it by the grid length dx (m) and multiplying it by the time step (time). The resulting value is used to correct the digital elevation model and the soil depth map for the following time step.
The comparison of the e-power determines how much of the difference between transport capacity C and transport rate of sediment S can be “satisfied” in the grid cell. Depending on the values of the variables involved, the resultant of the e-power varies between 0 and 1. In extreme situations when dx and D/T combined are much large than C, the e-power approaches zero and transport rate of sediment S is equal to sediment transport capacity C. Then maximum erosion or sedimentation is reached. However, in the other extreme when dx and D/T combined are much smaller than C, the e-power approaches 1 and transport rate of sediments S is equal to the transport rate of sediments already in transport S0 and no erosion or deposition occurs. In less extreme situations the model is likely to simulate sediment transport rate S close to transport capacity C. In case S0 > C more sediment is transported than would be allowed based on the grid cells above and less than the maximum sediment is deposited. The e-power therefore results in under-concentrated and super-concentrated flows in the model, smoothing erosion and deposition over the slope. Obviously the outcome of the e-power comparison is very influential for erosion and sedimentation. Comparing Eq. 1.1 and 1.4/1.5 it is clear that discharge and slope are involved in both the transport capacity C and in the calculations of detachment capacity D or settlement capacity T. This means that in a situation when m = n = 1, the term in the e-power reduces to dx ·Kes or dx ·Pes. As the grid cell size is a constant value, the erodibility Kes and sedimentability Pes are the most important variables in a situation with low m and n values. When m and n are larger, the effect of the transport capacity C on the e-power term increases. The outcome of the e-power term is in that situation harder to predict.