Infiltration: Sizing and modeling

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To calculate the required depth of an infiltration facility in a specified footprint area...
To calculate the required footprint area of an infiltration facility with a known depth constraint....
To calculate the drainage time of ponded water on the surface of a facility footprint...
To calculate the drainage time of an underground infiltration facility...

The sizing calculations require that most of the following parameters be known or estimated. The exceptions are the depth (d) and the permeable footprint area of the practice (Ap), as only one of these is required to find the other. Note that some of these parameters can be limited by site conditions and factors influencing constructability:

The maximum total depth will be limited by construction practices i.e. usually ≤ 2 m to avoid the need for benching and shoring open cut excavations.
The maximum total depth may be limited by the conditions underground (e.g. water table or underlying geology/infrastructure).
The maximum total depth may be limited by the desire to install the practice below the maximum frost penetration depth in the proposed location.
The maximum total depth may be limited by the desire to support vegetation cover over it (e.g. at least 30 cm of planting soil backfill over the BMP to support grasses)
Infiltration trenches, chambers and bioretention have a maximum recommended I/P ratio of 20.

Inputs
Symbol	Units	Parameter
D	h	Duration of design storm
i	mm/h	Intensity of design storm
f'	mm/h	Design infiltration rate of the underlying native soil, calculated from measured infiltration rate and applied safety factor
n	-	Porosity of the aggregate or other void-forming fill material(s) in the storage reservoir of the practice. *Note: For systems that have significant storage in open chambers surrounded by clear stone aggregate, an effective porosity value (n') may be estimated for the whole installation and used in the calculations below. Effective porosity will vary according to the geometry of the storage chambers, so advice should be sought from product manufacturers. Permit applications should include the basis for n' estimates.
A_i	m²	Impermeable area within the catchment
d_r	m	Depth of the storage reservoir of the practice
d_T	m	Total depth of the practice
A_p	m²	Permeable footprint area of the practice
x	m	Perimeter of the practice
K_f	mm/hr	Minimum saturated hydraulic conductivity of the filter media or planting soil used in the practice

This spreadsheet tool has been set up to perform all of the infiltration practice sizing calculations shown below
Download the infiltration practice sizing tool

To calculate the required storage reservoir depth where the area of the facility is constrained (1D drainage)[edit]

In some very constrained sites, the surface area of the practice may be limited, in this case the required active storage reservoir depth, d_r of the bioretention cell or infiltration trench can be calculated. Note that in most cases the results of this calculation will be very similar to those from the equation below assuming 3 dimensional drainage.

d{r}={\frac {D\left[\left({\frac {Ai}{Ap}}\right)i-f'\right]}{n}}

To calculate the require facility area or footprint where the depth is constrained (1D drainage)[edit]

In many locations throughout Ontario, there may be limited depth of soil available into which stormwater may be infiltrated. In this case the required storage needs to be distributed more widely across the landscape. The area of the storage reservoir, A_r can be calculated: $Ar={\frac {iDAi}{ndr+f'D}}$

Time required to drain surface ponded water[edit]

The following equation assumes one dimensional drainage over the surface ponding area. It is best applied to calculate the maximum duration of ponding on the surface of bioretention cells, and upstream of check dams of bioswales and enhanced grass swales to ensure all surface ponding drains within 48 hours. To calculate the time (t) to fully drain surface ponded water through the filter media or planting soil: $t={\frac {dp'}{Kf}}$ Where
d_p' is the effective or mean surface ponding depth (mm).
K_f is the minimum acceptable saturated hydraulic conductivity of the filter media or planting soil when compacted to 85% maximum dry density (mm/h).

Time to drain the storage reservoir[edit]

The target drainage time for the active storage reservoir depth of an infiltration facility is typically between 48 and 72 hours or based on the average inter-event period for the location. Contact the local municipality or conservation authority for criteria. See Drainage time for more information about how inter-event periods vary across Ontario and to help select what is suitable for the site.

To calculate the required depth, where the area of the facility is constrained (3D drainage)[edit]

In some very constrained sites, the surface area of the BMP may be limited, in this case the required depth of cell or trench can be calculated:

d=a[e^{\left(-bD\right)}-1]

Where $a={\frac {Ap}{x}}-{\frac {iAi}{xf'}}$ and $b={\frac {xf'}{nAp}}$

(The rearrangement to calculate the required footprint area of the facility for a given depth using three dimensional drainage is not available at this time. Elegant submissions are invited.)

Try the Darcy drainage calculator tool for estimating drainage time assuming either one or three-dimensional drainage from the practice:
Download the Darcy drainage time calculator

Three footprint areas of 9 m².
From left to right x = 12 m, x = 20 m

For some geometries (e.g. particularly deep facilities or linear facilities), it preferable to account for lateral infiltration. The 3D equation make use of the hydraulic radius (A_p/x), where x is the perimeter (m) of the facility.
Maximizing the perimeter of the facility directs designers towards longer, linear shapes such as infiltration trenches and bioswales.

To calculate the time (t) to fully drain the facility: $t={\frac {nAp}{f'x}}ln\left[{\frac {\left(d+{\frac {Ap}{x}}\right)}{\left({\frac {Ap}{x}}\right)}}\right]$ Where "ln" means natural logarithm of the term in square brackets