Bioretention: Sizing and modeling

Before beginning the sizing calculations most of the following parameters must be known or estimated. The exceptions are the depth (d) and Permeable area (P), as only one of these is required to find the other. Note that some of these parameters are limited:

1. The maximum total depth will be limited by construction practices i.e. not usually > 2 m.
2. The maximum total depth may be limited by the conditions underground e.g. the groundwater or underlying geology/infrastructure.
3. The minimum total depth may be limited by the need to support vegetation i.e. not < 0.6 m.
4. Green roofs, absorbent landscapes and permeable paving often receive very little flow from other surfaces, so that the I/P ratio is close to 1.
5. Infiltration trenches, chambers and bioretention cells have a maximum recommended I/P ratio of 20.

Inputs

Symbol	Units	Parameter
D	hrs	Duration of design storm (for MOECC volume based caclulations set to 1)
i	mm/hr	Intensity of design storm (for MOECC volume based calculations use whole storm depth (link to map))
q	mm/hr	Infiltration coefficient, calculated from measured infiltration rate and applied safety factor
VR	-	Porosity (void ratio), as measured (or default to 0.35 for all aggregates). *Note: For systems that have significant storage in clear open chambers, 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.
I	m2	Impermeable area i.e. catchment
d	m	depth of infiltration facility or BMP
P	m2	Permeable area i.e. footprint area of the facility or BMP

The following equations assume that infiltration occurs primarily through the base of the facility. They may be easily applied for any shape and size of infiltration facility, in which the reservoir storage is mostly in an aggregate.

To calculate the required depth, where the area of the facility is constrained: $d={\frac {D\left[\left(R\right)i-q\right]}{n}}$

To calculate the require facility area or footprint where the depth is constrained: $P={\frac {IiD}{nd+qD}}$

This spreadsheet tool has been set up to perform either of the above calculations.
Download .xlsx calculation tool

To calculate the time (t) to fully drain the facility: $t={\frac {nd}{q}}$

Accounting for lateral infiltration

Three footprint areas of 9 m².
From left to right x = 12 m, x = 14 m, and x = 16 m

For some geometries (e.g. particularly deep facilities or linear facilities), it may be preferred to also account for lateral infiltration. The 3 dimensional equations make use of the hydraulic radius (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 required depth:

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

Where $a={\frac {P}{x}}-{\frac {iI}{Pq}}$ and $b={\frac {xq}{nP}}$

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

To calculate the time (t) to fully drain the facility: $t={\frac {nP}{qx}}ln\left[{\frac {\left(d+{\frac {P}{x}}\right)}{\left({\frac {P}{x}}\right)}}\right]$

The ratio of the impervious catchment (drainage) area to the pervious (footprint) area of the receiving BMP.

The porosity (n) of a mixture is the ratio of the volume of void-space to the total or bulk volume of the mixture. It is closely related to the concept of void ratio (e) where void ratio is the ratio of the volume of void-space to the volume of solids. n = Volume of voids/Total volume of mixture = e/(1+e)

The void ratio (e) of a mixture is the ratio of the volume of void-space to the volume of solids. It is closely related to the concept of porosity (n) where porosity is the ratio of the volume of void-space to the total or bulk volume of the mixture. e = Volume of voids/Volume of solids = n/(1-n)

A broad category of particulate material used in construction, including sand, gravel, crushed stone, slag, recycled concrete and geosynthetic aggregates, and available in various particulate size gradations.

The porosity (n) of a mixture is the ratio of the volume of void-space to the total or bulk volume of the mixture. It is closely related to the concept of void ratio (e) where void ratio is the ratio of the volume of void-space to the volume of solids. n = Volume of voids/Total volume of mixture = e/(1+e)

The land draining to a single reference point (usually a structural BMP); similar to a subwatershed, but on a smaller scale.

Best management practice. State of the art methods or techniques used to manage the quantity and improve the quality of wet weather flow. BMPs include: source, conveyance and end-of-pipe controls.

A broad category of particulate material used in construction, including sand, gravel, crushed stone, slag, recycled concrete and geosynthetic aggregates, and available in various particulate size gradations.

The slow movement of water into or through a soil or drainage system.

Penetration of water through the ground surface.