Rainfall on unsealed roads causes a sediment flux that proceeds along the road,
through drains, and then down to river systems. Optimal placement of drains can minimise the
amount of sediment polluting the river. Two methods of minimising this sediment are considered
and compared. The methods provide excellent examples of mathematical modelling, the solution of
systems of non-linear equations, function minimisation and Taylor series.
Unsealed roads are common in natural parks and forest catchments, altering the natural catchment hydrology (Wemple et al, 2001). During rainfall events, runoff from the road (Reid & Dunne, 1984) can flow through to river and stream systems, threatening water quality and in-stream ecology (Croker et al, 1993; Richardson, 1985). Typically, runoff flows along the road before entering natural gullies or artificial drainage points. The runoff then flows naturally down hillslopes, depositing some sediment to the soil, until the remaining water and sediment reaches flowing water systems (Hairsine et al, 2002; Takken et al, 2006; 2008). This process is illustrated schematically in figure 1.
To minimise sediment flux to the river system, engineers place drains along the road, channelling the flow into the landscape. With well-designed drains, the sediment is deposited on the hillside before reaching the river system; the amount of sediment deposited is proportional to the distance to the river ((Hairsine et al, 2002). Thus a drain placed a long distance from a river is able to channel a large sediment flux without affecting river quality. There are also financial and physical constraints on the number of drains than can be placed along a road system. The goal is to place the minimum number of drains, in the optimal positions, so that sediment flux to the river is minimised or hopefully zero. This is complicated by the local geography, since the shortest flow distance from road to river will vary.
We will primarily consider the case of how to optimally find the position of a single drain, along a road segment bounded by two drains. This simplification allows techniques to be used that are appropriate for first- and second-year engineering mathematics students, while still retaining the essential system characteristics. The extension to multiple drains can be easily understood by many early undergraduate students, although the numerically-intense solution method required for multiple drains may only be appropriate for more advanced students, particularly those with good programming skills. However, the single drain solution still shows the fundamental properties of the solution.
We structure the paper by first detailing the modelling approach, before analysing two models separately. In each model, the general equations for multiple drains are given, followed by the solution method and then analysis for a single drain. The results give upper and lower bounds for the possible fluxes and show how the drain position varies with the river-to-road distance.
2 MODEL APPROACH
We describe here two modelling approaches. The first finds the number of drains needed (and their position) for a given rainfall so that no sediment flux reaches the river. The second method finds the position of a fixed number of drains that will minimise the sediment flux for a given rainfall.
[FIGURE 1 OMITTED]
Both models use the following notation and assumptions, which do not affect the generality of the model:
* The road runs along the x axis from x = a to x = b.
* Drains are placed at [x.sub.0], [x.sub.1], [x.sub.2], ..., [x.sub.n]. The locations [x.sub.0] = a and [x.sub.n] = b are known and fixed.
* The river is located in the y > 0, x > a quadrant.
* Water flows from left to right, in the positive x direction, picking up sediment, before exiting the road at a drain and flowing in the positive y direction towards the river. …