https://orcid.org/0000-0003-4530-5484
Figures
Abstract
Conventional designs of infiltration swales either neglect infiltration while the swale is filling or approximate the flow in the swale as being normal instead of gradually varied. The adequacy of these approximations are elucidated, and two common design configurations for infiltration swales are considered. For swales designed to store the water-quality volume behind check dams, the retained volume can be on the order of twice the design water-quality volume depending on the magnitude of the inflow rate normalized by the infiltration rate. In a second configuration, the swale is designed to infiltrate the water-quality flow, where the limiting assumption is that the flow is normal along the infiltration length. The actual required infiltration length can be expressed as a function of the normalized bottom width, and the required infiltration length can be up to 30% longer than derived using the conventional design. Graphical relations are developed that can be used to either quantify the factor of safety of conventional designs or provide credit for in the flood-control function of infiltration swales.
Citation: Chin DA (2024) Design of infiltration swales. PLOS Water 3(3): e0000228. https://doi.org/10.1371/journal.pwat.0000228
Editor: Tarun Kumar Lohani, Arba Minch Water Technology Institute: Arba Minch University, ETHIOPIA
Received: September 11, 2023; Accepted: January 17, 2024; Published: March 6, 2024
Copyright: © 2024 David A. Chin. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The results reported in this paper are all based on computer simulations following the flowcharts contained within the paper. The input data used in the simulations are all contained within the paper.
Funding: The author received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
Introduction
An infiltration swale is a shallow open channel in which infiltration is a substantial abstraction process. Infiltration swales are a subset within the conventional categories of grassed swales and dry swales, which are widely used as stormwater control measures in green infrastructure. Grassed and dry swales used for water-quality control typically rely on either biofiltration or retention and subsequent infiltration behind check dams. To facilitate infiltration, swales are sometimes underlain by engineered soils along with a subsurface drain embedded in a gravel layer. A common shortcoming of conventional design approaches for infiltration swales is that they do not take into account the infiltration that occurs when the swale is filling, which would potentially lead to different designs at sites with different infiltration capacities. In swales that are designed to be of sufficient length to infiltrate the water-quality flow, the conventional assumption is that the flow is normal along the infiltration length. However, this assumption is generally unrealistic since the flow depth must necessarily go to zero at the end of the infiltration length. The quantitative impacts of the aforementioned design assumptions on infiltration-swale designs are the foci of this paper.
Background
Swales are shallow vegetated open channels with small longitudinal and side slopes that transport runoff from adjacent land areas. Swales differ from regular grass-lined open channels in that they are shallow and have small side slopes, with an operational definition of a swale being a channel that has a depth of 45 cm (1.5 ft) or less and side slopes of 3:1 (H:V) or less [1]. Infiltration swales are typically designed to provide water-quality control by either storing and infiltrating the water-quality volume (WQV) or by treating the water-quality flow (WQF) as it passes through the swale. When storing and infiltrating the WQV, check dams are usually sized and spaced to store the WQV behind the dams, and this stored water is subsequently infiltrated within a specified time limit. When designed to treat the WQF, infiltration swales are sometimes designed to be of sufficient length that the volumetric infiltration rate over a design treatment length is equal to the WQF. Such swales are only feasible on sites where the required length of the swale can be accommodated within the available space. Inflow to infiltration swales can be either from the upstream end or distributed along the swale, commonly referred to as longitudinal and lateral inflows, respectively, and can influence the swale design.
Design guidelines
Authoritative guidelines for designing swales for water-quality control are co-published by the American Society of Civil Engineers and the Water Environment Federation [2]. Regulatory guidelines and associated requirements are provided in design manuals of state agencies [3–5], and design guidelines are also provided textbooks that incorporate findings and recommendations from the peer-reviewed technical literature [6]. The design guidelines for infiltration swales with check dams are similar to those used for bioretention cells [7]. Consensus conventional guidelines for designing infiltration swales derived from the aforementioned references are summarized below.
Site conditions
For infiltration swales to be effective, a typical minimum infiltration rate of the underlying soil is around 10 mm/h, which usually corresponds to soils in hydrologic soil groups A and B. For lesser native infiltration capacities, engineered soil with a subsurface drain can be a viable alternative, and these types of swales are commonly called enhanced dry swales or bioswales. The effectiveness of grassed swales can be enhanced by using planted grass instead of sod, since the soil binder in the sod usually has high silt and clay content [8]).
Swale channels
Infiltration swales typically have trapezoidal cross-sections, with side slopes not greater than 3:1 (H:V). However, limiting side slopes less than 3:1 are sometimes dictated by maintenance (mowing) concerns, and 4:1 side slopes or less are not uncommon for such requirements. For example, roadside swales in Florida commonly have recommended side slopes of 6:1 [9]. In cases of lateral inflow, triangular swales with side slopes of 6:1 (H:V) are sometimes preferred due to the sides acting as filter strips [10, 11]. Bottom widths of trapezoidal swales are typically in the range of 0.6–3.0 m, with the lower limit required for practical construction and adequate maintenance, and the upper limit required to prevent sub-channelization of the flow within the swale. Longitudinal slopes are typically in the range of 0.5%–2%, with constructibility being the main consideration for the lower limit, and erosion control being the main consideration for the upper limit. Some regulations limit the maximum slope to 4% [3–5], and a minimum slope of 0.5% is sometimes required [12]). For greater slopes, check dams are commonly used to reduce the effective slope.
Design with check dams
When check dams are used, the height of the check dam is typically limited to 30–45 cm, which corresponds to the maximum ponded depth within the swale. A minimum spacing of 15–30 m between check dams is usually preferred, although minimum spacings as low as 8 m are sometimes allowed. All ponded water behind check dams should typically be infiltrated within 24 h, although infiltration times up to 72 h are sometimes allowed. When check dams overflow, they are assumed to perform as broad-crested weirs. Placement of multiple check dams within a swale can create maintenance issues, so it is usually preferable to locate the check dam at the downstream end of the swale, or at the downstream drop inlet into the storm-sewer system.
Design based on the water-quality flow (WQF)
There is not a standard method of calculating the WQF. In some jurisdictions, the WQF is calculated using the TR-55 method [13], and in other jurisdictions the WQF is calculated using the rational method with a specified water-quality rainfall intensity [14]. In addition to these conventional approaches, more advanced methods for calculating the WQF have also been proposed [15]. In using the WQF to design a swale for water-quality control, two alternative approaches are commonly used. In one approach, the swale is designed as a biofiltration swale in which the flow depth is below the height of the grass, typically taken as 10–15 cm, and a minimum hydraulic residence time, minimum swale length, and maximum velocity are imposed. A minimum residence time of 5–9 min, a minimum swale length of 15–30 m, and a maximum velocity of around 30 cm/s are typical [16]. In a second approach, the water-quality control length of the swale is sized such that the WQF is completely infiltrated over the water-quality control length. The second approach is more consistent with the function of an infiltration swale, which is the focus of the present study.
Other considerations
A minimum freeboard of 15–30 cm is typically provided in swales for both storage and conveyance flow, with the lower freeboard typically associated with a lower flow depth [17]. The bottom of infiltration swales are typically required to be at least 60–90 cm above the seasonal high water table. Catchment areas for water-quality swales are typically less than 2 ha, while conveyance swales can have catchment areas up to around 20 ha. Infiltration swales typically cover 3%–10% of the catchment area depending primarily on the imperviousness of the catchment. When an infiltration swale is also used for conveyance, it is typically expected to be stable at 2-yr and 10-yr return-period flows, with a capacity to accommodate the 10-yr flow with a freeboard of around 15 cm.
Limitations of conventional approaches
A core limitation of conventional infiltration-swale design approaches relate to incorporation of the infiltration capacity into the swale design. Infiltration swales can generally be assumed to have infiltration capacities of at least 10 mm/h, and sometimes much higher, yet the infiltration capacity is usually not considered in calculating the required retention volume behind check dams. Also, when designing an infiltration swale for the WQF, accurate estimation of the swale length required to completely infiltrate the WQF usually assumes normal flow along the swale, which is inconsistent with continuous infiltration which ends with a flow depth of zero. It is sometimes assumed that infiltration swales primarily provide conveyance for flood-control flows, however, the potential for these swales to infiltrate a substantial portion of flood-control flows is often overlooked. The goals of the investigation reported in this paper were to provide quantitative guidance on accounting for infiltration when swales are filling behind retention dams and on accounting for nonuniform flow when runoff is infiltrating along swales.
Design based on WQV
The design of an infiltration swale to retain the WQV usually consists of sizing the storage volume(s) behind check dam(s) to store the WQV. The conventional approach neglects the infiltration that occurs during the filling of the storage volume. Whereas neglecting infiltration during filling might be appropriate for low infiltration capacities, for swales with higher infiltration capacities this approach could result in a substantial overdesign for both water-quality and flood-control functions of the swale. Whereas overdesign might be regarded as incorporating a safety factor, the extent of overdesign would generally be of interest. Accounting for the infiltration of the ponded runoff behind check dams while the swale is filling requires estimation of an inflow hydrograph and a storage routing calculation.
Estimation of inflow hydrograph
In conventional designs, swale inflows are characterized by peak inflow rates commonly calculated using the TR-55 method. The TR-55 method uses the Natural Resources Conservation Service (NRCS) unit hydrograph along with incremental rainfall excesses derived from one of the four NRCS Type rainfall distributions to estimate runoff hydrograph and hence the peak runoff rate [18]. Two key technical drawbacks of this approach are: (1) the Type rainfall distributions are known to not be representative of actual local rainfall distributions [19, 20], and (2) the TR-55 method uses incremental application of the curve number method to determine the incremental rainfall excesses within a storm event, which is known to be an inappropriate application of the curve-number method [21]. In this study, appropriate modifications of the TR-55 method were made to address the aforementioned limitations. The steps followed to estimate the runoff hydrograph, which is also taken as being the swale inflow hydrograph, are described below.
Calculation procedure.
The given quantities in a typical design are the total rainfall depth, which is usually taken to be equal to the water-quality rainfall, WQR, the local intensity-duration function, IDF, the catchment curve number, CN, and the time of concentration, tc, of the catchment.
- Step 1:. Calculate the 24-h rainfall-intensity distribution from the given IDF function, which is assumed to have the following form:
(1) where i is the average rainfall intensity over a duration t, and α, β, and c are empirical constants. Using Eq 1, the cumulative rainfall depth, Pt, over any duration t (in hours) can be expressed in terms of the 24-h rainfall depth, P24, as
(2) where t24 = 24 h, and in the present context, P24 is equal to WQR. The 24-h rainfall-intensity distribution is constructed using the NRCS method for constructing a hyetograph with a nested cumulative distribution function [22], where the cumulative distribution is given by Eq 2, and the time increment is taken as 0.1 h. The constructed rainfall-intensity distribution is represented as in, where n ∈ [1, N] and N = 240.
- Step 2:. Calculate the water-quality depth, WQD, corresponding to the WQR and CN of the catchment using the conventional curve-number method:
(3) where WQD, WQR, and S are each expressed in centimeters.
- Step 3:. Using the rainfall-intensity distribution calculated in Step 1, determine the constant infiltration capacity, ϕ, of the catchment that corresponds to CN and yields the WQD. This requires that ϕ satisfy the following relation:
(4) where N is the number of increments in the rainfall-intensity distribution determined in Step 1.
- Step 4:. Use the value of ϕ determined in Step 3 to calculate the rainfall excess, Qe,n, in each increment as
(5) This rainfall-excess distribution corresponds to a total runoff depth equal to WQD.
- Step 5:. Combine the rainfall excess, Qe,n, in each increment with the NRCS unit hydrograph [23] to generate the inflow hydrograph to the swale, Qin(t), using the following relations:
(6) where Qp is the peak of the unit hydrograph occurring at t = Tp, and Qp and Tp are estimated using the following conventional relations:
(7) where Qp is in (m3/s)/cm, A is the catchment area in km2, and Tp, Δt, and tc are expressed in hours. Eq 7 uses the standard peak rate factor (PRF) of 2.08, however, site-specific values of PRF can vary widely from the typical value of 2.08. For example, PRF values calculated for 26 catchments in New Jersey were found to be in the range of 0.73–4.19 [24].
Storage routing through an infiltration swale
An infiltration swale should be sized such that the depth of the swale at any section is at least equal to the maximum ponded depth plus the freeboard. In response to a rainfall event, a swale fills according to the continuity equation, which requires that the inflow rate minus the infiltration rate and outflow rate be equal to the rate of change of stored volume within the swale. This relation can be expressed as
(8)
where V is the volume of runoff stored in the swale, Qin is the volumetric inflow rate, F is the volumetric infiltration rate, and Qout is the volumetric outflow rate. To facilitate computation of the swale storage, V, as a function of time, Eq 8 can be written in finite-difference form as
(9)
where j is the time-step index. Eq 9 can be applied with Qout = 0 to determine the maximum value of V corresponding to the water-quality runoff event.
Stage-storage function.
An elevation view of a typical segment of an infiltration-swale is illustrated in Fig 1. If y0 is the depth at the downstream end of a swale segment of length L and the cross section of the swale is trapezoidal, then the volume, V1, between the upstream end of the segment (where the water surface intersects the bottom of the swale) and the toe of the downstream check can be calculated as
(10)
where b is the bottom width of the swale, y is the ponded depth at any given section, m is the inverse side slope, and x is the distance along the channel measured from the upstream end. Since y = S0x and y0 = S0L, where S0 is the longitudinal slope of the swale and y0 is the ponded depth at the downstream end of the swale, Eq 10 can be evaluated as follows:
(11)
The storage volume, V2, at the downstream check dam can be calculated as
(12)
where A0 is the cross-sectional area of the swale at the toe of the check dam, and m1 is the inverse slope of the face of the check dam. Combining Eqs 11 and 12 yields the following relation for storage, V, in the swale segment as a function of the depth y0 at the downstream end of the segment:
(13)
Infiltration function.
The volumetric infiltration rate, F, is equal to the ponded infiltration capacity, fp, multiplied by the wetted area in the swale. For a swale segment of length L, the volumetric infiltration rate, F1, between the upstream end of the segment and the toe of the downstream check can be calculated as follows:
(14)
The infiltration rate, F2, at the downstream check dam can be calculated as
(15)
where P0 is the perimeter of the swale at the toe of the check dam. Combining Eqs 14 and 15 yields the following relation for the infiltration rate, F, in the swale segment as a function of the depth y0 at the downstream end of the segment:
(16)
Outflow function.
For most outflow structures, the rate of flow through the structure can be expressed as a function of the storage depth at the structure. Hence, the outflow rate, Qout, can be related to the depth, y at the structure. For a grate outlet, Qout depends on the depth of flow just upstream of the grate location, and for a weir outlet Qout depends on the head over the weir crest, which requires specification of the height of the weir crest, which is usually equal to the height of the check dam.
System configuration
In the conventional WQV-based design procedure, a swale is designed to store the WQV behind one or more check dams. The key design parameters are the WQV, the maximum height of the check dam(s), and the stage-storage function of the swale segment(s). These data are sufficient to calculate the required number of swale segments, the length of each segment, and the required height of each check dam. The conventional design procedure neglects any infiltration that occurs when the volume behind the check dams is being filled. The focus of this investigation is to estimate the additional runoff volume, beyond the WQV, that can be accommodated behind the check dams if infiltration during inflow is taken into account.
Dimensional analysis
Let the actual volume of runoff that can be stored and infiltrated in a segment of an infiltration swale be represented by Vs, the design storage volume by V0, the peak runoff rate by Qp, the ponded infiltration capacity by fp, and the length, bottom width, inverse side slope, and longitudinal slope of the trapezoidal swale be represented by L, b, m, and S0, respectively. The design storage volume, V0, is equal to the volume of runoff corresponding to the water-quality depth, WQD. Due to infiltration when the swale is filling, the storage capacity, Vs, will be more than the design storage volume, V0, and the objective of this study is to determine the relative magnitudes of Vs and V0 for given values of the other variables. From the listed relevant variables we can assert the existence of the following functional relation:
(17)
where ψ represents an unknown function. Eq 17 must necessarily satisfy the boundary condition that Vs = V0 when fp = 0. If V0, b, and m are fixed, then L can be determined for any given value of S0, so S0 can be removed as an independent variable in Eq 17. Further, for a trapezoidal swale, the wetted perimeter at the end of the swale, P0, is determined by V0, L, b, and m. Given these interdependencies, Eq 17 can be expressed as
(18)
where P0 or LP0 can replace L for convenience; it is noteworthy that LP0 has physical meaning as approximately equal to the surface area over which infiltration occurs within the swale, as assumed in the conventional design approach. Eq 18 can be expressed in a more fundamental form as
(19)
where ds is the actual depth of runoff stored in the swale, d0 is the design runoff depth stored in the swale, Ac is the area of the catchment, and qp0 is the unit peak runoff rate determined using the NRCS unit-hydrograph method for a given rainfall distribution and catchment time of concentration. In terms of the listed variables in Eq 19, qp0Ac depends only on d0Ac, and with this interdependency, Eq 19 can be expressed as
(20)
This functional relation involves 6 variables in 2 dimensions (length and time) and so, according to the Buckingham Pi theorem can be expressed as a functional relation between 4 dimensionless groups, such as
(21)
Note that the mapping of d0 to qp0 that is implicit in Eq 21 depends on the magnitude of d0, since a higher d0 requires a higher rainfall amount, which maps nonlinearly into the runoff hydrograph. In the subsequent analysis, qp0Ac will be represented as Qp0, in which case Eq 21 can be represented as
(22)
where it is noted that, for a given d0, the variable Qp0 varies depending on the area of the catchment.
Results
The form of the functional relation given by Eq 22 was determined for the configuration shown in Fig 1. The values of the parameters used in this determination, the computation sequence, and the results are described below.
Parameter values.
Various parameters are necessary to characterize the rainfall, catchment, runoff hydrograph, swale geometry, and infiltration properties. The relevant variables were assigned realistic values, and the magnitudes of some of these variables were changed to obtain the functional relation. The parameter values used in this study are given in Table 1. For all cases, b and m were fixed at typical values of 1.5 m and 3, respectively, and two typical runoff depths were considered, d0 = 2.54 cm (1 in) and d0 = 5.08 cm (2 in). The infiltration capacity, fp, of the swale was varied in the typical range of 10–100 mm/h, and the maximum allowable check dam height was set at 46 cm, with the actual height being that required to store the runoff volume equal to d0Ac. The rainfall characteristics were taken to be typical of Miami, Florida, where the rainfall distribution is described by Eq 2 with β = 9.8 min and c = 0.74. To generate the WQDs of d0 = 2.54 cm and d0 = 5.08 cm using the curve number equation, corresponding WQRs of 6.75 cm and 10.03 cm and a CN value of 80 were set. The time of concentration of the catchment was set at the typical value of 10 min.
Computations.
The sequence of computations to determine the functional relation given by Eq 22 is shown in Fig 2. The values assigned to the fixed parameters are shown in Table 1. The variable parameters are the water-quality rainfall depth, WQR, longitudinal slope, S0, infiltration capacity of the swale, fp, and catchment area, Ac. With the curve number, CN, of the catchment set at 80, WQR values of 6.75 cm and 10.03 cm yield values of the water-quality depth, d0, equal to 2.54 cm and 5.08 cm, respectively. For each value of d0, values of S0, fp, and Ac were varied in the loops shown in Fig 2. For each set of these variable parameters, the runoff hydrograph, Qin(t) and associated peak runoff rate, Qp0 were calculated using Eq 6. For the runoff volume, V = d0Ac, the water depth at the downstream end of the swale, y0, was calculated using Eq 13, and the corresponding wetted perimeter, P0, of the trapezoidal channel was calculated using the relation
(23)
and hence the dimensionless parameter Qp0/(fpLP0) was then calculated. Finally, the adjusted value of the water-quality depth, ds, was varied so as to cause a proportional increase in the runoff hydrograph which was then routed through the swale using Eq 9. The value of ds that produced a maximum water level in the swale equal to y0, was determined, and the calculated value of ds/d0 was plotted versus Qp0/(fpLP0) in Figs 3–5.
Storage factor versus normalized peak flow for d0 = 2.54 cm: (a) slope = 0.5%; (b) slope = 1.5%; and (c) slopes = 0.5%, 1.5%.
Storage factor versus normalized peak flow for d0 = 5.08 cm: (a) slope = 0.5%; (b) slope = 1.5%; and (c) slopes = 0.5%, 1.5%.
Results for d0 = 2.54 cm.
For d0 equal to 2.54 cm (1 in), values of Qp0 were varied by changing the catchment area in the range of 202–2020 m2, fp was varied in the range of 10–100 mm/h, the longitudinal slope, S0, was fixed at 0.5%, and the other environmental parameters were fixed at the values shown in Table 1. Using these values, the functional relation between ds/d0 and Qp0/(fpP0L) is shown in Fig 3a. These results show that the relative capacity of the swale to store runoff (as measured by ds/d0) increases substantially for smaller values of Qp0/(fpP0L), with a 50% increase in capacity at Qp0/(fpP0L) ≈ 2.5, and a 100% increase in capacity at Qp0/(fpP0L) ≈ 1.2. To verify the asserted insensitivity of the results to the longitudinal slope of the swale, S0, this analysis was repeated by changing the slope from 0.5% to 1.5% and these results are shown separately in Fig 3b, and combined with the 0.5% results in Fig 3c. It is apparent that the ds/d0 versus Qp0/(fpP0L) relation is insensitive to S0, which supports this assertion in the dimensional analysis.
Results for d0 = 5.08 cm.
The analyses performed for d0 = 2.54 cm were repeated for d0 = 5.08 cm, and the results are shown in Fig 4. These results show that the ds/d0 versus Qp0/(fpP0L) relation for d0 = 5.08 cm is still insensitive to S0. More importantly, these results also show that the ds/d0 versus Qp0/(fpP0L) relation is not the same for d0 values of 2.54 cm and 5.08 cm, which is an expected outcome based on the dimensional analysis. The functional relations for d0 equal to 2.54 cm and 5.08 cm are superimposed in Fig 5, which show that for any given value of Qp0/(fpP0L) a higher proportional increase in swale capacity can be expected if the swale is designed for a water-quality depth of 5.08 cm versus 2.54 cm.
Assessment of results
The results presented here demonstrate that the increased capacity of an infiltration swale relative to its design capacity can be assessed using the parameter Qp0/(fpP0L). For the typical catchment conditions considered here, it is shown that the actual available storage capacity can be double the design water-quality storage for lower values of Qp0/(fpP0L). For infiltration swales designed to retain the water-quality volume, this additional storage capacity could be credited to flood control, since it will further reduce the swale discharge in the flood-control design. The significance of this additional storage capacity will depend on the flood-control design parameters, however, the results developed in this study provide essential quantitative values for such an analysis.
Design based on WQF
The design of biofiltration swales typically uses the Manning equation to calculate the uniform-flow characteristics, and then ensures that the flow depth is less than the grass height for a defined detention time, where the detention time is equal to the length of the swale divided by the flow velocity (ASCE, 2012). The design of an infiltration swale is based on ensuring that the length of the swale is sufficient that the infiltration rate over the wetted area is equal to the WQF. However, for infiltration swales in which the infiltration capacity is substantial, the usual design assumption of uniform flow over the infiltration length is questionable. Consider the case where the swale inflow rate is equal to the WQF and L is the swale length required to completely infiltrate the incoming flow rate as shown in Fig 6. It is apparent that the wetted perimeter and incremental infiltration surface varies along the length of the swale, which requires calculation of the water-surface profile under the WQF inflow condition.
Infiltration length in swale: (a) profile view; and (b) section A-A.
Conventional formulation
The scenario shown in Fig 6 is in contrast to the conventional design approach of determining L by assuming that the normal depth of flow occurs over L [25], which is not realistic. Conventional analytic design equations used in determining the length, L, under infiltrating conditions are generally based on the formulation:
(24)
where Ln is the required infiltration length under the normal-flow condition, Qp is the peak runoff rate into the swale, fp is the ponded infiltration capacity, and Pn is the wetted perimeter under the normal-flow condition. In the present context, Qp is equated to the WQF. A useful analytic relation can be derived from Eq 24 for the best hydraulic trapezoidal section in which the wetted perimeter is given by (Chin, 2021)
(25)
where yn is the normal flow depth and m is the inverse side slope. The best hydraulic section is one which minimizes the flow area for a given flow rate. Combining Eqs 24 and 25 with the Manning equation for a trapezoidal channel yields
(26)
where b is the bottom width, n is the Manning roughness coefficient, and S0 is the longitudinal slope.
Water-surface profile
The actual water-surface profile when the WQF enters the swale at the upstream end can be estimated by incremental application of the energy equation, taking into account the flow loss due to infiltration along the swale. In this context, the energy equation applied to an incremental control volume is given by:
(27)
where Δx is the incremental distance along the swale, y1 and y2 are the upstream and downstream flow depths, respectively, within each increment, Q2 is the flow rate at the downstream section,
and
are the average wetted perimeter and friction slope, respectively, between the upstream and downstream sections, A1 and A2 are the flow areas corresponding to y1 and y2, respectively, and g is the gravity constant. Note that the flow at the upstream section, Q1, is related to the flow at the downstream section by
. The average wetted perimeter,
, and average friction slope,
, are calculated using the following relations:
(28)
where P1 and P2 are the wetted perimeters at the upstream and downstream sections, respectively, and n1 and n2 are Manning’s n at the upstream and downstream sections, respectively. Manning’s n generally depends on the flow conditions, which change along the length of the swale. For grass-lined infiltration swales, Manning’s n at any section, i, can be estimated by [26]
(29)
where Cn is the grass-roughness coefficient, γ is the specific weight of water (= 9879 N/m3 at 20°C), and Ri and Sfi are the hydraulic radius and friction slope, respectively, at the given section. For full infiltration of the incoming flow, the upstream and downstream boundary conditions that must be satisfied by Eq 27 are
(30)
where x = 0 and x = L correspond to the upstream and downstream ends of the swale, respectively. The water-surface profile in the swale under the design condition can be solved by specifying y = 0 at a section along the swale, and then applying Eq 27 incrementally along the channel to calculate the flow condition at various distances upstream of the starting section. This gives swale-inflow rates and associated distances over which full infiltration occurs.
Dimensional analysis
For any given infiltration swale, the length L required to completely infiltrate the runoff entering at an inflow rate Q0 can be expressed by the following functional relation:
(31)
where fp is the ponded infiltration capacity, b is the bottom width of the swale, m is the inverse side slope, and S0 is the longitudinal slope. If the flow is normal at the entrance to the swale and Manning’s n is given by Eq 29, then the Manning equation requires that
(32)
where An and Pn are the flow area and wetted perimeter, respectively, under normal-flow conditions, which can be expressed as
(33)
where yn is the normal-flow depth. Eqs 32 and 33 can be combined and expressed in either of the following functional forms:
(34)
Combining Eqs 31 and 34 it is apparent that Pn can be used as an alternative variable to S0, which leads to
(35)
This is a relation between 6 variables in 2 dimensions (length and time), and so, according to the Buckingham Pi theorem can be expressed as a relation between 4 dimensionless groups as
(36)
The denominator on the left-hand side of Eq 36 is the infiltration length Ln under the condition that the wetted perimeter remains constant at Pn over the entire infiltration length, as given by Eq 24. Using this relation, Eq 36 can be conveniently expressed as
(37)
Since Ln is the conventional estimate of the required infiltration length, then L/Ln ≈ 1 indicates good agreement with the conventional design length. In general, L/Ln indicates the extent to which the conventional estimate, Ln, is an underestimate of the required infiltration length, L.
Results
The form of the functional relation given by Eq 37 was determined for the configuration shown in Fig 6. The values of the parameters, the computation sequence, and the generated results are described below.
Parameter values.
The current analysis only requires specification of parameters that characterize the swale. Values of m and fp were kept constant at the typical values of 3 and 100 mm/h, respectively, and the roughness characteristic of grass, Cn, used in Eq 29 was fixed at the typical value of 0.165. The bottom width of the swale, b, was varied between 0.5 m and 3 m, and the longitudinal slope, S0, was varied between 0.5% and 1.5%.
Computations.
The computation sequence to determine the functional relation given by Eq 37 is shown in Fig 7. For each (b, S0) set, the water-surface profile was computed using Eq 27 starting from a downstream location specified as x = 0 where the flow depth, y, and flow rate, Q, are both set to zero. The upstream location was incremented by Δx = 0.1 m and the corresponding upstream depth of flow, y, was calculated using Eq 27. The flow rate, Q, at the upstream location is equal to the known downstream flow rate plus the infiltration rate over Δx. The normal flow depth, yn, and wetted perimeter, Pn, corresponding to Q is calculated using the Manning equation. Considering the calculated values of Q and x as being at the upstream end of the swale, then the required infiltration length, L, is equal to x, and the corresponding conventional infiltration length, Ln, is equal to Q/(fpPn). The computed values of L/Ln and b/Pn give the quantitative relation between these variables that are expressed in functional form by Eq 37. The results obtained for specific values of S0 and b are described below.
Results for S0 = 0.5%.
The relation between L/Ln and b/Pn for S0 = 0.5% is shown in Fig 8a for b equal to 0.5 m and 3 m. It is apparent from Fig 8a that the L/Ln versus b/Pn relation is independent of the magnitude of b, except for b/Pn greater than around 0.85 in the case of b = 0.5 m. The third dimensionless group, Pn/Ln, was allowed to vary based on the computed values of Pn and Ln for the given values of b. For the cases considered, the values of Pn/Ln generally increase with increasing values b/Pn, and Pn/Ln < 0.1 for b/Pn less than 0.81 and 0.94 for b values of 0.5 m and 3 m, respectively. These results collectively indicate that as along as Pn/Ln ≪ 0.1 then the L/Ln versus b/Pn relation is independent of Pn/Ln. The results indicate that the magnitude of Pn/Ln is only important for Pn/Ln ∼ 1, in which case the wetted perimeter at the inflow section of the swale likely has a significant influence on the infiltration length, L. The primary result in Fig 8a is that decreasing values of b/Pn lead to increasing values of L/Ln, thus increasing required swale length, L, relative to swale length of the conventional design, Ln. For example, for b/Pn ≈ 0.2 the required swale length is 35% longer than required by the conventional design.
Normalized infiltration length versus normalized bottom width: (a) slope = 0.5%, (b) slope = 1.5%; and (c) slopes = 0.5% and 1.5%.
Results for S0 = 1.5%.
The relation between L/Ln and b/Pn for S0 = 1.5% is shown in Fig 8b for values of b equal to 0.5 m and 3 m. The results are similar to those for S0 = 0.5%, and show that Pn/Ln < 0.1 for b/Pn less than 0.89 and 0.96 for b values of 0.5 m and 3 m, respectively, which is reflected in the deviation from the L/Ln versus b/Pn relation occurring at higher values of b/Pn.
Overall.
The results shown in Fig 8a and 8b are superimposed in Fig 8c which shows that the combined results are independent of S0, as asserted in the dimensional analysis. The results in Fig 8c support the assertion that infiltration-swale performance can be inferred from the combined L/Ln versus b/Pn relation in Fig 8c) as long as Pn/Ln < 0.1. Furthermore, and perhaps most importantly from a practical viewpoint, for lower values of b/Pn infiltration swales might need to be substantially longer than indicated by conventional designs in order to completely infiltrate the water-quality flow.
Design example
To illustrate practical application of the results presented here, consider the case of an infiltration swale that is to be designed to treat a WQD of 2.54 cm (1 in) from a 1011 m2 catchment in Miami, Florida. Application of the NRCS unit hydrograph yields a WQF of 0.015 m3/s. The infiltration swale is expected to have an infiltration capacity of 50 mm/h (= 1.39 × 10−5 m/s). The swale is to be on a slope of 0.5%, and have a trapezoidal cross section with a bottom width of 1.5 cm and an inverse side slope of 3. Two options are to be considered: (1) ponded infiltration with a maximum check dam height of 46 cm, and (2) an infiltration length sufficient to infiltrate the WQF.
Option 1
From the given data: d0 = 2.54 cm, Q0 = 0.015 m3/s, fp = 1.39 × 10−5 m/s, S0 = 0.005, b = 1.5 m, m = 3, and y0 = 0.46 m. Using these data yields
Using Qp0/(fpP0L) = 2.66 in Fig 5 gives ds/d0 = 1.45, which means that the 92-m long swale segment will store a runoff depth of 1.45(2.54 cm) = 3.74 cm before overflowing into a downstream conveyance channel, which is 45% more storage than is obtained by neglecting infiltration during runoff.
Option 2
To determine the infiltration length requires calculation of the normal-flow condition. Taking Cn = 0.165 and γ = 9789 N/m3 to calculate Manning’s n as given by Eq 32, the Manning equation gives yn = 0.0766 m and Pn = 1.984 m, which yields
Using b/Pn = 0.76 in Fig 8 gives L/Ln = 1.08, which means that the required infiltration length is 8% longer than that required using the uniform-flow assumption. In other words, the swale would need to be 44 m longer than the conventional design length of 544 m.
Assessment of options
The analyses presented here indicate that Option 1 is preferable in that it requires only a length of 92 m, versus a length of 544 m. The actual storage provided by Option 1 is 45% greater that the storage calculated using the conventional approach. This difference can either be associated with a factor of safety or credited toward the flood-control function of the swale.
Conclusions
Infiltration swales are commonly used for water-quality control in which either the water-quality volume or the water-quality flow is completely infiltrated within the swale. In cases where the water-quality volume is retained behind a check dam, it is common practice to size the storage volume to be equal to the water-quality volume, which neglects any infiltration while runoff is entering the swale. The results of this study show that the efficacy of this approach is dependent on the value of the inflow rate normalized by the infiltration rate, as quantified by Qp0/(fpP0L), where Qp0 is the peak inflow rate, fp is the ponded infiltration capacity, P0 is the wetter perimeter just upstream of the check dam, and L is the length of the infiltration swale. For lower values of Qp0/(fpP0L), the actual volume of retained runoff can be substantially greater than the design storage volume, and it is shown that under practical conditions the retained volume can be as high as twice the design storage volume. In the second configuration, where the swale is designed to infiltrate the water-quality flow, the limiting conventional assumption is that flow conditions are normal along the infiltration length, where in reality the flow depth decreases to zero along the infiltration length, resulting in a longer required infiltration length than obtained using the conventional design methodology. The results of this study show that the relative increase in the required infiltration length can be estimated based on the value of b/Pn, where b is the bottom width of the swale and Pn is the normal-flow wetted perimeter associated with the water-quality flow rate. As b/Pn decreases the required swale length increases relative to that required by the conventional design, and it is shown that the required infiltration length can be up to 30% longer than derived using the conventional design. Overall, the graphical relations presented in this study can be used to quantify the factor of safety of conventional designs, or to provide credit for the flood-control function of infiltration swales.
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