**Once there is water at the surface which surpasses the surface storage, it is routed downhill towards the catchment outlet with a kinematic wave function. The water flows over a predefined flow network, called Local Drainage Direction.**

A gridcell can have more than one type of surface: e.g. a road (smaller than DX) is impermeable, a wheeltrack is compacted and a field surface may be ponded or not. The infiltration characteristics vary according to the surface and the infiltration is calculated for each type. An average water height is than calculated for each gridcell, resulting in an average hydraulic radius with which the velocity is calculated, The field surface has a certain roughness and therefore only a part of the water will move.

*Figure 1. Calculation of average water height and flowwidth*

The velocity V (m/s) is calculated with Manning’s formula:

*V = R ^{2/3} * sqrt(S) / n*

In which: R = hydraulic radius (m), calculated with the flowwidth and average water height(see figure 5); S = sine of the slope (fraction); n = Manning’s n (dimensionless)

The discharge Q (m^{3} /s) per cell is then calculated with (Chow et al., 88)

In which:

*b = 0.6*

* A = wet cross section (m ^{2})*

There is evidence that alpha is constant for small rills and independent of slope and resistance (Govers, 1997), because the flow will find a new equilibrium and uses its energy to form new rill dimensions, while the velocity does not change. Research is currently being done to see if this should be incorporated in LISEM.

For the distributed overland and channel flow routing, a four-point finite-difference solution of the kinematic wave is used together with Manning’s equation. Procedures of the numerical solution can be found in Chow et al. (1988) and Moore & Foster (1990). The kinematic wave is done over the Local Drain Directions map that forms a network which connects cells in 8 directions. The kinematic wave equation is as follows:

Where Q and A are defined as above and q and i are external in and out fluxes such as rainfall and infiltration in the case of overland flow. For practical reasons LISEM handles this slightly different: the rainfall (minus interception) is added to the water layer on the surface and the infiltration is subtracted from it. The new water depth is then used to calculate velocity and discharge, which is routed with the kinematic wave.

*For LISEM 2.12 and later the following procedure is added*: In order to still have the possibility of infiltration *inside *the kinematic wave module, surplus infiltration is calculated and used instead of the term “q-i”. The surplus infiltration is calculated as the potential infiltration in a timestep minus the actual infiltration that already took place. *Thus there is generally more infiltration in LISEM 2.12 then in earlier versions.*

Some cells may have a channel (ditch, gully) for which a separate kinematic wave is done (see fig 6). The cells that have a channel receive a part of the overland flow, depending on the velocity. Thus the velocity is considered the average velocity existing in the cell. The channel is considered to be in the center of the cell so that the distance from the edge to the channel is 0.5*(DX-channelwidth). The part of the water that flows into the channel is therefore:

*f = dt .V/(0.5*(dx-cw))*

The channelwidth can increase during the run if the channel is not rectangular and the water level in it rises. In that case it cannot become larger than 0.9 of the cell width dx.

Figure 6. Overland and channel flow with LDD structure