Difference between revisions of "Bioretention: Sizing and modeling"

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Before beginning the sizing calculations most of the following parameters must be known or estimated.  
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This article describes recommended design approaches when available space for the practice is constrained.<br>
The exceptions are the depth (''d'') and Permeable area (''P''), as only one of these is required to find the other.  
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<br>
Note that some of these parameters are limited:
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Before beginning the sizing calculations certain parameters must be known or estimated. See [[Bioretention: Sizing]] for parameter descriptions and conceptual diagram illustrating key components of bioretention practices. Note that some of these parameters are limited:
 
#The ''maximum'' total depth will be limited by construction practices i.e. not usually > 2 m.
 
#The ''maximum'' total depth will be limited by construction practices i.e. not usually > 2 m.
 
#The ''maximum'' total depth may be limited by the [[Infiltration| conditions underground]] e.g. the groundwater or underlying geology/infrastructure.
 
#The ''maximum'' total depth may be limited by the [[Infiltration| conditions underground]] e.g. the groundwater or underlying geology/infrastructure.
#The minimum total depth may be limited by the need to support vegetation i.e. not < 0.6 m.
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#The minimum total depth will be limited by the need to support vegetation (e.g not less than 0.6 m to support deep rooting perennials and shrubs).
#[[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.
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#[[Bioretention]] has a maximum recommended catchment impervious area to practice permeable (footprint) area ratio, R (or I/P ratio) of 20.
#[[Infiltration trenches]], [[Infiltration chambers| chambers]] and [[bioretention cells]] have a maximum recommended I/P ratio of 20.
 
  
==Determine the effective infiltration coefficient==
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==Size a bioretention cell receiving flows directly to the storage reservoir for a constrained depth==
Determine the I/P ratio:
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If there is a constraint to the depth (''d<sub>T</sub>'') of the practice, calculate the required storage reservoir footprint area (''A<sub>r</sub>''), as:
:<math>R=\frac{A_{c}}{A_{p}}</math>
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<math>A_{r}=\frac{i\times D\times A_i}{(d_{r}\times n')+(f'\times D)}</math>
{{Plainlist|1= Where:
 
*''R'' = Ratio of catchment area (''A<sub>c</sub>'') to BMP footprint area (A<sub>p</sub>) syn. I/P ratio. 
 
*''A<sub>p</sub>'' = Area of the infiltration practice in m<sup>2</sup>
 
*''A<sub>c</sub>'' = Catchment area in m<sup>2</sup>}}
 
----
 
Apply an appropriate safety correction factor to the measured infiltration rate:
 
:<math>q'=\frac{q}{SCF}</math>
 
 
{{Plainlist|1=Where:
 
{{Plainlist|1=Where:
*''q''' = Infiltration coefficient in mm/hr (accounting for SCF)
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*''A<sub>r</sub>'' = Area of the infiltration practice storage reservoir (m<sup>2</sup>)
*''SCF'' = Safety correction factor}}
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*''A<sub>i</sub>'' = Catchment impervious area (m<sup>2</sup>)
----
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*''D'' = Duration of design storm (h)
 +
*''i'' = Intensity of design storm (mm/h)
 +
*''f''' = [[design infiltration rate]] (m/h)
 +
*''n''' = Effective porosity of the fill material in the storage reservoir of the practice
 +
*''d<sub>r</sub>'' = Storage reservoir depth, based on depth available between the elevation of the invert of the underdrain perforated pipe and one (1) metre above the seasonally high water table or top of bedrock  (m) or other value determined to be suitable through groundwater mounding analysis.}}<br>
 +
If R is greater than 20, consider decreasing catchment impervious area (A<sub>i</sub>) by draining less area to the practice.
  
 +
==Size a bioretention cell where drainage area and practice area are fixed==
 +
If the land area is limited, determine the I/P ratio, which is the ratio of catchment impervious area (A<sub>i</sub>) to practice pervious footprint area (A<sub>p</sub>):
 +
:<math>R=\frac{A_{i}}{A_{p}}</math>
  
The following equations assume that infiltration occurs primarily through the base of the facility.
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{{Plainlist|1= Where:
They may be easily applied for any shape and size of infiltration facility, in which the reservoir storage is mostly in an aggregate. 
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*''R'' = Ratio of catchment impervious area to practice pervious footprint area, also referred to as I/P ratio
 
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*''A<sub>p</sub>'' = Practice pervious footprint area in m<sup>2</sup>
To calculate the required depth, where the area of the facility is constrained:
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*''A<sub>i</sub>'' = Catchment impervious area in m<sup>2</sup>}}
<math>d=\frac{D\left[Ri-q'\right]}{V_{R}}</math>
 
  
To calculate the require facility area or footprint where the depth is constrained:
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Then calculate the required storage reservoir depth (''d<sub>r</sub>''), as:
<math>A_{p}=\frac{IiD}{V_{R}d+\left ( \frac{q}{SCF} \right )D}</math>
+
<math>d_{r}=\frac{D \left[ (R\times i)-f'\right]}{n'}</math>
  
 
{{Plainlist|1=Where:
 
{{Plainlist|1=Where:
*''D'' = Duration of design storm in hrs
+
*''D'' = Duration of design storm (h)
*''i'' = Intensity of design storm in mm/hr
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*''i'' = Intensity of design storm (m/h)
*''q''' = Infiltration coefficient in mm/hr (accounting for SCF)
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*''f''' = Design infiltration rate (m/h)  
*''SCF'' = Safety correction factor
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*''n''' = Effective porosity of the storage reservoir fill material}}
*''V<sub>R</sub>'' = Void ratio (porosity), as measured (or default to 0.35 for all aggregates)*
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These equations assume that infiltration occurs primarily through the base of the facility.<br>  
*''R'' = Ratio of catchment area (''A<sub>c</sub>'') to BMP footprint area (A<sub>p</sub>) syn. I/P ratio. 
+
<br>
*''A<sub>p</sub>'' = Area of the infiltration practice in m<sup>2</sup>
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This spreadsheet tool has been set up to perform all of the infiltration practice sizing calculations shown above.<br>
*''A<sub>c</sub>'' = Catchment area in m<sup>2</sup>
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{{Clickable button|[[Media:Infiltration Sizing 20220617 locked (1).xlsx|Download the infiltration practice sizing tool]]}}
*''d'' = Depth of infiltration practice in m.}}
 
  
*Note: For systems that have significant storage in clear open chambers, an effective porosity value (''V<sub>R</sub><nowiki>'</nowiki>'') 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 ''V<sub>R</sub><nowiki>'</nowiki>'' estimates.
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==Calculate drawdown time==
----
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[[file:Hydraulic radius.png|thumb|Two footprint areas of 9 m<sup>2</sup>.<br>
 +
Perimeter = 12 m (left) Perimeter = 20 m (right)]]
  
 +
{{Clickable button|[[Media:Darcy drainage_20200528_locked.xlsx|Download the Darcy drainage time calculator tool]]}}
  
This spreadsheet tool has been set up to perform either of the above calculations.<br>
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In some situations, it may be possible to reduce the size of the bioretention required, by accounting for rapid drainage. Typically, this is only worth exploring over sandy soils with rapid infiltration.
<strong>[[Media:Infiltration Sizing.xlsx|Download .xlsx calculation tool]]</strong>
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Note that narrow, linear bioretention features drain faster than round or blocky footprint geometries.  
 
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*Begin the drainage time calculation by dividing the storage reservoir area of the practice (''A<sub>r</sub>'') by the perimeter (''x'').
To calculate the time (''t'') to fully drain the facility:  
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*Use the following equation to estimate the time (''t'') to fully drain the facility:  
<math>t=\frac{nd}{q}</math>
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:<math>t=\frac{nA_{r}}{f'x}ln\left [ \frac{\left (d_{r}+ \frac{A_{r}}{x} \right )}{\left(\frac{A_{r}}{x}\right)}\right]</math>
 
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{{Plainlist|1=Where:
==Accounting for lateral infiltration==
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*''n'' is the porosity of the storage reservoir fill material
[[file:Hydraulic radius.png|thumb|Three footprint areas of 9 m<sup>2</sup>.<br>
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*''A<sub>r</sub>'' is the storage reservoir footprint area (m<sup>2</sup>),
From left to right x = 12 m, x = 14 m, and x = 16 m]]
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*''f''' is the design infiltration rate of the native soil (mm/h),  
For some geometries (e.g. particularly deep facilities or linear facilities), it may be preferred to also account for lateral infiltration.
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*''x'' is the perimeter of the practice (m), and
The 3 dimensional equations make use of the hydraulic radius (''P''/''x''), where ''x'' is the perimeter (m) of the facility. <br>
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*''d<sub>r</sub>'' is the depth of the storage reservoir (m).}}
Maximizing the perimeter of the facility directs designers towards longer, linear shapes such as [[infiltration trenches]] and [[bioswales]]. 
 
 
 
To calculate the required depth:
 
:<math>d=a[e^{\left ( -bD \right )} -1]</math>
 
Where
 
<math>a=\frac{P}{x}-\frac{i I}{P q}</math>
 
and
 
<math>b=\frac{xq}{nP}</math>
 
 
 
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:
+
This 3 dimensional equation makes use of the hydraulic radius (''A<sub>r</sub>''/''x''), where ''x'' is the perimeter (m) of the facility. <br>
<math>t=\frac{nP}{qx}ln\left [ \frac{\left (d+ \frac{P}{x} \right )}{\left(\frac{P}{x}\right)}\right]</math>
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Maximizing the perimeter of the facility directs designers towards longer, linear shapes such as [[bioswales]]
  
 
[[category: modeling]]
 
[[category: modeling]]
 
[[category: infiltration]]
 
[[category: infiltration]]

Latest revision as of 15:49, 27 June 2022

This article describes recommended design approaches when available space for the practice is constrained.

Before beginning the sizing calculations certain parameters must be known or estimated. See Bioretention: Sizing for parameter descriptions and conceptual diagram illustrating key components of bioretention practices. 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 will be limited by the need to support vegetation (e.g not less than 0.6 m to support deep rooting perennials and shrubs).
  4. Bioretention has a maximum recommended catchment impervious area to practice permeable (footprint) area ratio, R (or I/P ratio) of 20.

Size a bioretention cell receiving flows directly to the storage reservoir for a constrained depth[edit]

If there is a constraint to the depth (dT) of the practice, calculate the required storage reservoir footprint area (Ar), as:

Where:

  • Ar = Area of the infiltration practice storage reservoir (m2)
  • Ai = Catchment impervious area (m2)
  • D = Duration of design storm (h)
  • i = Intensity of design storm (mm/h)
  • f' = design infiltration rate (m/h)
  • n' = Effective porosity of the fill material in the storage reservoir of the practice
  • dr = Storage reservoir depth, based on depth available between the elevation of the invert of the underdrain perforated pipe and one (1) metre above the seasonally high water table or top of bedrock (m) or other value determined to be suitable through groundwater mounding analysis.


If R is greater than 20, consider decreasing catchment impervious area (Ai) by draining less area to the practice.

Size a bioretention cell where drainage area and practice area are fixed[edit]

If the land area is limited, determine the I/P ratio, which is the ratio of catchment impervious area (Ai) to practice pervious footprint area (Ap):

Where:

  • R = Ratio of catchment impervious area to practice pervious footprint area, also referred to as I/P ratio
  • Ap = Practice pervious footprint area in m2
  • Ai = Catchment impervious area in m2

Then calculate the required storage reservoir depth (dr), as:

Where:

  • D = Duration of design storm (h)
  • i = Intensity of design storm (m/h)
  • f' = Design infiltration rate (m/h)
  • n' = Effective porosity of the storage reservoir fill material

These equations assume that infiltration occurs primarily through the base of the facility.

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

Calculate drawdown time[edit]

Two footprint areas of 9 m2.
Perimeter = 12 m (left) Perimeter = 20 m (right)

Download the Darcy drainage time calculator tool

In some situations, it may be possible to reduce the size of the bioretention required, by accounting for rapid drainage. Typically, this is only worth exploring over sandy soils with rapid infiltration. Note that narrow, linear bioretention features drain faster than round or blocky footprint geometries.

  • Begin the drainage time calculation by dividing the storage reservoir area of the practice (Ar) by the perimeter (x).
  • Use the following equation to estimate the time (t) to fully drain the facility:

Where:

  • n is the porosity of the storage reservoir fill material
  • Ar is the storage reservoir footprint area (m2),
  • f' is the design infiltration rate of the native soil (mm/h),
  • x is the perimeter of the practice (m), and
  • dr is the depth of the storage reservoir (m).

This 3 dimensional equation makes use of the hydraulic radius (Ar/x), where x is the perimeter (m) of the facility.
Maximizing the perimeter of the facility directs designers towards longer, linear shapes such as bioswales.