Darcy’s Law defines a simple relationship that relates the instantaneous discharge rate through a porous medium to the local hydraulic gradient (change in hydraulic head over a distance) and the hydraulic conductivity (k) at that point. It is one of the basic relationships of hydrogeology.
For the purposes of these guidelines, Darcy’s law is applied in the absence of a known groundwater level, therefore:
- Q is the total discharge (e.g. m3/s)
- k is the hydraulic conductivity based on the properties of the soils on site following compaction (e.g. m/s)
- A is the cross-sectional area to flow (e.g. m2)
- ha /is the level of the pond surface (e.g. m)
- hb / is the level of the base of the impermeable layer (e.g. m)
- L is the distance between the level of the pond surface and the level of the base of the impermeable layer (m) i.e. depth of water + depth of impermeable layer
- x is the thickness of the liner (impermeable layer) (e.g. m)
The parameters are schematically demonstrated below:
Normal application of Darcy’s law (and measurement of k) uses the full cross-sectional area of the soil because it is readily measurable. The velocity (v) obtained then is only a notional velocity as it is calculated as if flow is occurring over the whole cross section:
Notional velocity v = Q / A
In reality the movement of water is restricted through the voids that are only a fraction of the total cross section. This means that to pass the flow rate, actual seepage velocities through the pores are faster than the notional velocity—effectively by the ratio of total cross-sectional area to the area of void or the inverse of porosity.
A saturation moisture content of 18 per cent corresponds to a porosity of approximately 33 per cent, so for such a soil the seepage velocity (or the advance of the pond water) would be three times the velocity obtained from Darcy’s law.
Based on this, a modification of Darcy’s Law should be used to evaluate the required combination of liner thickness and liner permeability to ensure compliance with the risk-based level of permeability (see Section 6.7) –
Actual velocity vt = v / 0.33
The time taken for water to pass through the liner is therefore:
Time to pass through liner: T = x / vt
The steps in this process may be combined to the following final equation:
- T = 1000 x2 / 3 k L (assumes porosity of 33 per cent) where the parameters and units of measurement are:
- T is the time to pass through liner (days)
- x is the depth of the impermeable layer (m)
- k is the predicted saturated hydraulic conductivity (mm/day)
- L is the distance between the level of the pond surface and the level of the base of theimpermeable layer (m) i.e. depth of water + depth of impermeable layer
The volume leaked during the period of ponding may be calculated using the following equation:
V = Q ta 86.4
Where V is the volume leaked during the period the pond is filled (ML/ha),
- Q is the flow (m3/s),
- ta / is the number of days that leakage may occur (the maximum likely filled period of the containment structure (days) minus the time to pass through the liner (T)), and 86.4 is a unit conversion factor from m3/s to ML/day.
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