# Infiltration: Sizing and modeling

This is an image map; clicking on components will load the appropriate article.
• 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:

1. 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.
2. The maximum total depth may be limited by the conditions underground (e.g. water table or underlying geology/infrastructure).
3. The maximum total depth may be limited by the desire to install the practice below the maximum frost penetration depth in the proposed location.
4. 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)
5. 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 m/h Intensity of design storm
f' m/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.
Ai m2 Catchment impervious area
dr m Water storage reservoir depth. For practices without an underdrain (i.e. full infiltration design), dr is the total depth of the practice (i.e. includes surface ponding and filter media depths). For practices with an underdrain (i.e. partial infiltration design), dr is the depth below the invert of the underdrain perforated pipe or riser pipe upturn.
Ap m2 Permeable footprint area of the practice. For practices where runoff is directed to a surface ponding area, Ap is the surface ponding area. For practices where runoff is directed to a subsurface water storage reservoir, Ap is the footprint area of the internal water storage reservoir, Ar.
x m Perimeter of the practice
Kf m/h Minimum acceptable saturated hydraulic conductivity of the filter media or growing medium (i.e., planting soil) used in the practice, when compacted to 85% maximum dry density

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

## To calculate the required storage reservoir footprint area where the depth is fixed or constrained (1D drainage)

To ensure that the water storage capacity of the facility is available at the onset of a storm event, it is recommended to size the storage reservoir despth, dr, based on the depth of water that will drain via infiltration between storm events. So dr can be calculated as:
${\displaystyle d_{r}={\frac {f'\times t}{n}}}$ Where
t = drainage time, based on local criteria or long-term average inter-event period for the location (e.g. 72 hr in southern Ontario).
n = Porosity of the stone bed aggregate material (typically 0.4 for 50 mm dia. clear stone)
In many locations there may be a limited depth of soil available above the seasonally high water table or top of bedrock elevation into which stormwater may be infiltrated. In such cases the required storage needs to be distributed more widely across the landscape.
Where the storage reservoir depth is fixed or constrained the footprint area of the water storage reservoir, Ar can be calculated: ${\displaystyle A_{r}={\frac {i\times D\times Ai}{(n\times d_{r})+f'D}}}$

## To calculate the required storage reservoir depth where the area is fixed or constrained (1D drainage)

On densely developed sites, the surface area available for the practice may be constrained. In such cases the required storage reservoir depth, dr of the bioretention cell or infiltration trench can be calculated based on available surface area, Ap:

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

Note that in most cases the results of this calculation will be very similar to those from the equation below assuming three-dimensional drainage.

## To calculate the required storage reservoir depth, where the area is fixed or constrained (3D drainage)

On densely developed sites, the surface area available for the facility may be constrained. In such cases the required water storage reservoir depth of the bioretention cell or infiltration trench, d, can be calculated assuming three-dimensional drainage as follows

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

Where

${\displaystyle a={\frac {A_{r}}{x}}-{\frac {i\times A_{i}}{x\times f'}}}$

and

${\displaystyle b={\frac {x\times f'}{n\times A_{r}}}}$

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

## Time required to drain surface ponded water (1D drainage)

The following equations assume 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 stormwater tree trenches, and upstream of check dams of bioswales and enhanced grass swales to ensure all surface ponding drains within 24 hours.
To calculate the time (t) to fully drain surface ponded water through the filter media or growing medium: ${\displaystyle t={\frac {d_{p}'}{K_{f}}}}$
Where
dp' is the effective or mean surface ponding depth (mm).
Kf is the minimum acceptable saturated hydraulic conductivity of the filter media and growing medium when compacted to 85% maximum dry density (mm/h); minimum of 25 mm/hr is recommended for bioretention filter media; minimum of 15 mm/hr is recommended for enhanced grass swale and stormwater tree trench growing medium.

For full infiltration design practices that do not feature an underdrain, once the internal water storage capacity has been filled, the length of time required to fully drain surface ponded water is limited by the saturated hydraulic conductivity of the underlying in-situ (native) subsoil. To calculate the time (t) to fully drain surface ponded water once filled to capacity: ${\displaystyle t={\frac {d_{p}'}{f'}}}$
Where
dp' is the effective or mean surface ponding depth (mm).

## Time to drain internal water storage reservoir

The target drainage time for the internal water storage reservoir portion 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.

Try the Darcy drainage time calculator tool for estimating the time required to fully drain the water storage reservoir of the facility assuming either one or three-dimensional drainage:

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

For some geometries, particularly deep or linear facilities, it desirable to account for lateral drainage, out the sides of the storage reservoir. The following equation makes use of the hydraulic radius (Ar/x), where Ar is the area of the reservoir and x is the perimeter (m) of the reservoir.

Maximizing the perimeter of the water storage reservoir of the facility will enhance drainage performance and directs designers towards longer, linear shapes such as infiltration trenches and bioswales. See illustration for an example.
To calculate the time (t) to fully drain the facility assuming three-dimensional drainage: ${\displaystyle t={\frac {n\times A_{r}}{f'\times x}}ln\left[{\frac {\left(d_{r}+{\frac {A_{r}}{x}}\right)}{\left({\frac {A_{r}}{x}}\right)}}\right]}$ Where "ln" means natural logarithm of the term in square brackets
Adapted from CIRIA, The SUDS Manual C753 (2015).