Build up Wash off method

Build-up of pollution on the catchment

During dry weather periods, pollution or sediments accumulate on the surface of urban catchments. The most common formulation of this process is to assume that the build-up is a linear or an exponential function of time. The choice between the two formulations is not straightforward, due to insufficient experimental results.

The linear build up function is given as

\[ {\frac{dM}{dt}}~=~A_{c}\qquad\mathrm{for}\qquad M<M_{m a x} \]

Equation: Linear accumulation rate of surface pollutant mass until maximum storage capacity.

\[ {\frac{\mathrm{d}M}{\mathrm{d}t}} 0= \mathrm{for} M\geq M_{m a x} \]

Equation: Pollutant mass buildup rate equals zero when maximum storage capacity is reached.

and the exponential build-up function is given as

\[ \frac{dM}{dt} = A_c - D_{rem} M \text{ with } M_{max} = \frac{A_c}{D_{rem}} \]

Equation: Differential equation for exponential build-up and removal of pollutant mass.

Here:

Table: Variables in the linear and exponential build-up equations.

t	=	time in days
M	=	accumulated mass per area of pollutants at time t, (kg/ha)
Mmax	=	maximum accumulated mass per area of pollutants on the catchment, (kg/ha
Ac	=	daily accumulation rate, (kg/ha/day)
Drem	=	removal coefficient, (1/ha/day)

t

The coefficient D_rem represents the removal of pollution from the surface by various mechanisms (for example by wind, traffic, street sweeping, biological and chemical degradation) - all except wash-off. The accumulated mass M will increase until a limit M_max =A~c _/Drem~.

The initial value of pollutant mass on catchment surface is determined by building-up mass from 0 for ADWP_ini time (duration of antecedent dry weather period before the first event in the simulation).

An example of the two build-up formulations is shown in an example of the linear and the exponential build-up function.

When the sediments are being built up in the model, it is possible to attach pollutants to these sediments. The mass of the pollutant M_i,Pollutant is related to the mass of the sediment M_Sediment by a simple fraction f_i,PS

\[ M_{i,~P o l l u t a n t}=f_{i,~P S}M_{S e d i m e n t}\qquad\mathrm{with}\qquad{\sum_{i}}f_{i,~P S}\!\leq1 \]

Equation: Pollutant mass on surface proportional to sediment mass through partition coefficient.

The difference between pollutants and sediments from the modelling perspective is that sediments are used as an input to the Sediment Transport (ST) module and pollutants are used as an input to the Advection Dispersion (AD) module.

Figure: An example of the linear and the exponential build-up function. The values of parameters are: A_c=40 kg/ha/day, M_max =100 kg, ADWP_ini =0 h.

Wash-off of pollution by rainfall

Wash-off of pollution during a rain event can be divided into two processes: erosion by raindrops and erosion by overland flow. Only the erosion by raindrops is considered in this model. Erosion by raindrops is governed by several parameters. The most important are: rainfall intensity, rainfall depth, rainfall duration, raindrop size, catchment topography, particle characteristics and vegetation. However, parameters such as raindrop size and rainfall depth are rarely available, so a simpler approximation has been adopted in the model. This approximation states that the raindrop erosion is a function of the rain intensity and a detachment rate. The equation for the detachment by rainfall can be written as (ref ¹):

\[ \frac{dM_{rem}}{dt} = D_r\left(\frac{i_r}{i_d}\right)^n \]

Equation: Rainfall-driven erosion and detachment rate proportional to rainfall intensity.

where

Table: Variables in the rainfall wash-off equation.

dMrem/dt	=	pollutant mass washed off by the rain per unit of time, (kg/ha/day)
Dr	=	detachment coefficient for rainfall, (kg/ha/day)
ir	=	rainfall intensity, (mm/h)
id	=	rain intensity constant, (25.4 mm/h)
n	=	exponent (default value 2)

The default value n=2 assumes that kinetic energy of raindrops is causing the detachment of the pollutant. Kinetic energy is assumed to be quadratically related to rainfall intensity. The rainfall intensity has dimension of velocity, and it is known that kinetic energy E of water is proportional to square of flow velocity v (i.e., E a v²).

Only the dissolved pollution is routed together with the runoff. The routing happens based on a catchment model specified. If there are water losses in the catchment model, it is assumed that the pollution is not lost and is extracted back to the catchment surface.

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1. Svensson, G., 1987. Modelling of Solids and Metal Transport from small Urban Watersheds. Chalmers University of Technology, Göteborg, Sweden, 1987. ↩