Technical Note

Storm-Water Management through Infiltration TrenchesBhagu R. Chahar1; Didier Graillot2; and Shishir Gaur3

Abstract:With urbanization, the permeable soil surface area through which recharge by infiltration can occur is reducing. This is resulting inmuch less ground-water recharge and greatly increased surface run-off. Infiltration devices, which redirect run-off waters from the surface tothe sub-surface environments, are commonly adopted to mitigate the negative hydrologic effects associated with urbanization. An infiltrationtrench alone or in combination with other storm water management practice is a key element in present day sustainable urban drainagesystems. A solution for the infiltration rate from an infiltration trench and, consequently, the time required to empty the trench is pre-sented. The solution is in the form of integrals of complicated functions and requires numerical computation. The solution is useful inquantifying infiltration rate and/or artificial recharge of ground-water through infiltration trenches and the drain time of the trench, whichis a key parameter in operation of storm water management practice. The solution has been applied to a case study area in Lyon, France.MATLAB programming has been used in the solution. DOI: 10.1061/(ASCE)IR.1943-4774.0000408. © 2012 American Society of CivilEngineers.

CE Database subject headings: Drainage; Infiltration; Stormwater management; Groundwater; Aquifers; Seepage; Artificial recharge;Best management practice.

Author keywords: Drainage trench; Infiltration trench; Urban drainage; Storm water; Infiltration; Groundwater; Aquifer; Seepage;Artificial recharge; Best management practice.

Introduction

Approximately half of the world’s population is living in urbanareas. Land-use modifications associated with urbanization, suchas the removal of vegetation, replacement of previously perviousareas with impervious surfaces, and drainage channel modificationsinvariably result in changes to the characteristics of the surfacerun-off hydrograph. The hydrologic changes that urban catchmentscommonly exhibit are, increased run-off peak, run-off volume,and reduced time to peak (Schneider 1975). Consequently, urbanareas are more susceptible to flooding affecting all land-use activ-ities (Hammer 1972). Urbanization also has a profound influenceon the quality of storm water run-off (Hall 1984). Kibler and Aron(1980) reviewed basic elements in urban run-off management. Thediversity of an urban catchment makes managing storm water verycomplicated (Jones and Macdonald 2007). Safe disposal of stormwater through traditional sewer systems is usually very expensive(Schluter and Jefferies 2002; Scholz 2006). The strengths andweaknesses of state and local storm water management programswere explored, with conclusions and recommendations to correctdeficiencies by Howells and Grigg (1981). Zoppou (2001)reviewed the diversity of approaches and parameters that are con-sidered in urban storm water models.

Storm water management in urban areas is becoming increas-ingly oriented to the use of low impact development (LID), sustain-able urban drainage systems (SUDS), water sensitive urban design(WSUD), best management practices (BMP), or low impact urbandesign and development (LIUDD) for countering the effects of ur-ban growth, wherein the storm water is controlled at its sourcethrough detention, retention, infiltration, storage, and retardation(Charlesworth et al. 2003; Elliott and Trowsdale 2007; Kirby2005; Martin et al. 2007). These methods include structural mea-sures, such as wetlands, ponds, swales, soakaways, infiltrationtrenches, roof storage systems, detention/retention basins, infiltra-tion basins, bioretention devices, vegetated filter strips, filter strips,and pervious pavements. The primary objective of these measuresis to replicate the pre-urbanization run-off hydrograph. Under ap-propriate conditions, these structural measures have proven to beeffective (Goonetillekea et al. 2005). Bioretention usage will growas design guidance matures as a result of continued research andapplication (Davis et al. 2009). The application of source controloptions in storm water management will improve the ecologicalintegrity of rivers and streams, reduce flooding in the city andin downstream areas, reduce sediment transport, and mitigate ero-sion; consequently, urban storm water can become a true resourceinstead of a nuisance (Niemczynowicz 1999; Braden and Johnston2004). Permeable pavement systems (PPS), which are sustainableand cost effective processes (Andersen et al. 1999), are suitable fora wide variety of residential, commercial, and industrial applica-tions (Scholz and Grabowiecki 2007). The general principle ofPPS is simply to collect, treat, and infiltrate freely any surfacerun-off to support ground-water recharge. The characteristic featureof sustainable urban drainage is that aesthetics, multiple use, andpublic acceptance of the drainage facilities play a very importantrole in the planning (Stahre 2005). Martin et al. (2007) conducted anational survey in France to collect feedback from BMP users ontheir experiences and found that retention processes were usedmore frequently than infiltration processes (68% versus 32%),and most of the organizations used BMPs for flood prevention

1Associate Professor, Dept. of Civil Eng., Indian Institute of Technol-ogy, New Delhi–110 016, India (corresponding author). E-mail: chahar@civil.iitd.ac.in

2Professor, SITE, Ecole nationale Supérieure des mines, St Etienne,France.

3Post Doctoral Fellow, SITE, Ecole nationale Supérieure des mines,St Etienne, France.

Note. This manuscript was submitted on December 27, 2010; approvedon July 8, 2011; published online on August 17, 2011. Discussion periodopen until August 1, 2012; separate discussions must be submitted for in-dividual papers. This technical note is part of the Journal of Irrigation andDrainage Engineering, Vol. 138, No. 3, March 1, 2012. ©ASCE, ISSN0733-9437/2012/3-274–281/$25.00.

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(78.2%) rather than storm water pollution prevention (27.6%). Asper the survey, surface detention basins are, in general, the mostwidely used BMPs, followed by belowground storage tanks, sur-face retention ponds, roads, and car parks, along with reservoirstructures, swales, soakaways, infiltration trenches and lastly roofstorage systems. Detention basins are a common feature of stormwater management programs in urban areas and vast literature isavailable for design of detention basins (Akan 1990; Baker1977; Donahue et al. 1981; Froehlich 2009; Jones and Jones1984; McEnroe 1992; Mein 1980). Barrett (2008) explored the per-formance and relative pollutant removal of several common bestmanagement practices by using data contained in the InternationalStorm water BMP Database.

Infiltration supports ground-water recharge (Bouwer et al.1999), decreases ground-water salinity, allows smaller diametersfor sewers (resulting in cost reduction), and improves the waterquality of receiving waters. Therefore, BMPs based on infiltrationare the foundation of many low impact development and green in-frastructure practices. Various investigators (Emerson and Traver2008; Zheng et al. 2006; Guo 1999; Guo 2001; Guo and Hughes2001; Raimbault et al. 2002; Sample and Heaney 2006) haveundertaken studies on infiltration basins. Infiltration of storm waterthrough detention and retention basins may increase the risk ofground-water contamination, especially in areas where the soilis sandy and the water table is shallow, and contaminants maynot have a chance to degrade or sorb onto soil particles beforereaching the saturated zone (Fischer et al. 2003; Brattebo andBooth 2003). The ‘first flush’ is more polluted than the remainderbecause of the washout of deposited pollutants by rainfall (Deletic1998; Bertrand-Krajewski et al. 1998). This has to be consideredin the management and treatment of urban storm water run-offespecially through detention/retention basins (Goonetillekea et al.2005). Similarly, all run-off from manufacturing industrial areasshould be diverted away from infiltration devices because of theirrelatively high concentrations of soluble toxicants (Pitt et al. 1999).All other run-off should include pretreatment using sedimentationprocesses before infiltration to both minimize ground-water con-tamination and to prolong the life of the infiltration device (ifneeded). This pretreatment can take the form of grass filters, sedi-ment sumps, and wet detention ponds depending on the run-offvolume to be treated and other site-specific factors (Pitt et al. 1999).

A full-scale physical model of a modified infiltration trench wasconstructed by Barber et al. (2003) to test a new storm water bestmanagement practice called an ecology ditch. The ditch was con-structed by using compost, sand, and gravel, and a perforated drainpipe. A series of 14 tests were conducted on the physical model.For larger storms, the ecology ditch managed a peak reduction inthe range of 10 to 50%. A grass swale-perforated pipe system re-sults in a pleasant curbless design, which may replace open ditchsystems in low density residential areas. Abida and Sabourin(2006) studied a grass swale underlain by a section of perforatedpipe enclosed in an infiltration trench. They conducted field tests tomeasure the infiltration rates of typical grass swales and existingpipe trenches. The total seasonal discharge for a properly designedperforated pipe system was found to be 13 times smaller than thatfor a conventional storm water system.

Martin et al. (2007) applied a multicriteria approach to evaluatedifferent BMPs for the decision-making process. The analysisshowed that for local government with primary consideration ofcost minimisation, the rankings were infiltration trenches, soak-aways, porous pavements, roof storage, swales, surface wet reten-tion ponds, belowground storage tanks, and dry detention basins. Inthe case of regional planning (planning improvements), the orderwas infiltration trenches, surface dry detention and wet retention

basins, swales and porous pavements, roof storage and soakaways,with storage tanks winding up in the lowest position. For residentsassociation level (environmental protection), infiltration trenches,soakways, porous pavements, swales, surface dry detention andwet retention basins, roof storage and belowground storage tankswere the top-to-bottom ranking. Thus, the infiltration trenches areplaced first in all three levels. Their use remain relatively infre-quent, probably attributed to the fact that BMP users are more in-clined to choose classical storm water source control solutions,such as basins and ponds.

The modified rational method was applied by Akan (2002a) tosize infiltration basins and trenches to control storm water run-off,and the same method was used by Froehlich (1994) to size smallstorm water pump stations. The critical storm duration producingthe maximum run-off volume depends on characteristics of thecatchment and rainfall-intensity-duration relation. Although themaximum inflow rate to a detention basin will result from a stormof duration equal to time-of-concentration, the maximum volumewill be produced by a storm that lasts significantly longer than thetime-of-concentration of the catchment. Akan (2002b) presented adesign aid for sizing storm water infiltration trenches. The proposedprocedure is based on the hydrological storage equation for an in-filtration structure coupled with the Green and Ampt infiltrationequation. For the filling process, the two equations were solvedsimultaneously by using a numerical method. For the emptyingprocess, the governing equations were integrated analytically re-sulting in an algebraic equation that can be solved for the emptyingtime explicitly. De Souza et al. (2002) presented an experimentalstudy on two infiltration trenches at IPH-UFRGS, in Porto Alegre,Brazil. Both trenches were able to control excessive run-off vol-umes that ultimately infiltrated into the soil. The Bouwer Model(1965) was selected to represent the hydraulic functioning of thetrenches, taking into account the typical characteristics of theregional soil (with high percentage of clay).

The literature review shows that urbanization of a watershedwith its associated effect on the quantity and quality of storm waterrun-off has resulted in the implementation of a number of alterna-tives for storm water management. Infiltration trenches are one ofthem. An infiltration trench is an underground-storage zone filledwith clean gravel or stone (Fig. 1). Infiltration trenches are con-structed to temporarily store storm run-off and let it percolate intothe underlying soil. Such trenches are used for small drainage areas.They are typically used for control of run-off from residential lots,commercial areas, parking lots, and open spaces such as ring roads.Also, they are relatively easy to construct in the perimeters andother unutilized areas of a development site. Moreover, they can beused below the porous pavements or with grass swales (Fig. 1) andcan be combined with detention basins. Furthermore, they can beprovided below pavements, walkways, pedestrian, or cycle tracksso no additional area is required such as other storm water man-agement practices. Infiltration trench emptying time is important

Fig. 1. Trapezoidal infiltration trench below a porous pavement or withgrass cover

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to operate and manage storm water. If the time between two suc-cessive storms is less than the trench emptying time then the excessstorm water should be diverted to another detention basin or to thestorm sewer. Unlike detention basins (Emerson and Traver 2008;Zheng et al. 2006; Guo 1999; Guo 2001; Guo and Hughes 2001;Sample and Heaney 2006), widely accepted design standards andprocedures for infiltration trenches do not exist. The present studyfinds a solution for the infiltration rate from a trapezoidal trench andtime required to empty the infiltration trench.

Analytical Solution

Infiltration trenches are generally long, moderately wide, and shal-low in dimensions. They are filled with coarse gravel to providestorage; they collect run-off from adjacent paved areas and infiltratethe water into the aquifer beneath. The coarse gravel fill material inthe trench is usually much more permeable than the underlying soil,so negligible resistance to flow occurs within the trench, and theperimeter of the trench is an equipotential surface. Let the aquiferbe composed of a multilayer porous medium, such that the upperlayer has hydraulic conductivity less than the lower layers. If thewater table in the aquifer is lower than the bottom of the top layer,then the wetting front of infiltrating water from the trench willadvance all around and may saturate the low permeable top layerbut seepage flow in lower more pervious layers will be unsaturated.For example, when seepage from a lined canal takes place, andliner conductivity is much less than that of the underlying soilmedium, the soil medium remains unsaturated (Polubarinova-Kochina 1962). In such situations the lower unsaturated layersact as drainage layer to the top saturated layer and ultimately re-charge the aquifer. The position of the water table in the aquifer isgoverned by horizontal or vertical controls with regards to river,stream, or pumping wells present within the aquifer boundary.Let a trapezoidal trench (as shown in Fig. 2) of bed width b (m),depth of water y (m), and side slope m (1 Vertical: m Horizontal) isconstructed in such an aquifer and the saturated hydraulic conduc-tivity of the top layer is k (m=s). Also, assume the thickness of thetop layer below the bed of the trench is d (m). As the length of thetrench is very large, seepage flow can be considered 2D in the ver-tical plane. Initially, the top layer is unsaturated and seepage fromthe trench is unsteady, but after some time the layer will get satu-rated and steady seepage will establish.

By means of the previously stated assumptions, the seepagefrom the infiltration trench becomes identical to the steady seepagedischarge per unit length of channel qs (m2=s) from a trapezoidalchannel analyzed by Chahar (2007). In that work, an exact analyti-cal solution for the quantity of seepage from a trapezoidal channelunderlain by a drainage layer at a shallow depth was obtained by

using an inverse hodograph and Schwarz-Christoffel transforma-tion, the solution is

qs ¼ 2kðd þ yÞKðffiffiffiffiffiffiffiffiγ=β

pÞ=Kð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=β

pÞ ð1Þ

where β and γ = transformation variables; and Kð ffiffiffiffiffiffiffiffiγ=β

p Þ andKð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=βp Þ = complete elliptical integrals of the first kindwith a modulus ð ffiffiffiffiffiffiffiffi

γ=βp Þ and ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=βp Þ, respectively (Byrd

and Friedman 1971). This involves two transformation parametersβ and γ that can be determined by solving the following twoequations:

d þ yy

¼ 2Kðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=β

pÞBð1=2; σÞ

. ffiffiffiβ

p Zβ

γ

Bτ ð1=2;σÞdτffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðτ � γÞp

ð2Þ

by¼ 2

Z1

β

ðBð1=2;σÞ � Bτ ð1=2;σÞÞdτffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðτ � βÞðτ � γÞp

�Zβ

γ

Bτ ð1=2;σÞdτffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðτ � γÞp

ð3Þwhere πσ ¼ cot�1m; τ = dummy variable; Bð1=2; σÞ = completeBeta function (Abramowitz and Stegun 1972); and Bτ ð1=2; σÞ =incomplete Beta function (Abramowitz and Stegun 1972) definedas

Bτ ð1=2;σÞ ¼ 2ffiffiffiτ

p2F1ð1=2; 1� σ; 3=2; τÞ ð4Þ

in which 2F1 is a Gauss-Hypergeometric series (Abramowitz andStegun 1972) given by

2F1ða; b; c; τÞ ¼ 1þ a · bc

τ þ aðaþ 1Þ · bðbþ 1Þc · ðcþ 1Þ · 1 · 2 τ2

þ aðaþ 1Þðaþ 2Þbðbþ 1Þðbþ 2Þc · ðcþ 1Þ · ðcþ 2Þ · 1 · 2 · 3 τ3 þ… ð5Þ

The range of transformation parameters is 0 ≤ γ ≤ β ≤ 1. Theparameter γ represents the effect of the drainage layer such thatγ → 0 as d=y → ∞ and γ → β as d=y → 0; whereas, the parameterβ represents the effect of the water depth in the trench such thatβ → 0 as b=y → ∞ and β → 1 as b=y → 0. It is evident fromEqs. (1)–(3) that the infiltration from a trench depends on trenchdimensions, depth of water in the trench, hydraulic conductivityof the porous medium, and depth of the drainage layer (i.e., lowerunsaturated medium of higher hydraulic conductivity), and locationof the ground-water table.

The trenches are designed to store and to infiltrate a capturedvolume of run-off that is generated from its contributing area duringa specific-design storm. The rational formula is used for calculatingrun-off from small catchments, particularly in urban areas where a

Fig. 2. Trapezoidal infiltration trenches underlain by an unsaturated porous medium

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large portion of the land surface is impervious. The correspondingtimes of concentration and, thus, the critical rainfall durations willalso be small, typically much less than one h. The filling processbegins when the run-off first reaches the trench. The filling processcan end and the emptying process can begin while the trench is stillreceiving run-off if the rate of infiltration from the trench exceedsthe inflow rate. However, run-off rates are normally much morethan the infiltration rates during a design storm event. Therefore,it is reasonable to assume that the filling process will continue untilthe entire captured run-off has entered the trench. The run-off cap-tured during the filling process is stored partly in the trench andpartly within the wetted zone of the soil. The emptying processstarts when the run-off into the trench ceases. The emptying timeof the infiltration trench is a function of the initial volume of waterin trench and the rate of infiltration from it. To find the time toempty the infiltration trench, let the initial water depth in the trenchbe y, and the steady infiltration rate be qs, and then the water level inthe trench will be lowered by dy in small time interval dt attributedto steady infiltration. Equating the volume of water in this strip tothe infiltration volume in the time interval dt:

qsdt ¼ ηðbþ 2myÞdy ð6Þ

where η = porosity of refilled material in the trench. This equationcan be integrated, after substituting qs from Eq. (1), to determinethe time taken in lowering the water level in the trench from y1 to y2as

Δt ¼ η2k

Zy1

y2

ðbþ 2myÞKð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=βp Þðd þ yÞKð ffiffiffiffiffiffiffiffi

γ=βp Þ dy ð7Þ

Eq. (2) can be used to eliminate ðd þ yÞ. Let time zero denotewhen the trench first starts to empty, then the total time required toempty the trench is

t ¼ η4kBð1=2; σÞ

Zy

0

ffiffiffiβ

p

Kð ffiffiffiffiffiffiffiffiγ=β

p Þ

×Z

β

γ

Bτ ð1=2; σÞdτffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðτ � γÞp

�byþ 2m

�dy

ð8Þ

Simultaneous solution of Eqs. (2) and (3) for the given trenchdimensions (b and σ), the depth of the unsaturated layer (d), and thedepth of water in trench (y) at a particular instant, results in cor-responding parameters β and γ. Thus, integrand in Eqs. (7) or (8)for any y and corresponding β and γ for fixed values of b, σ, and dcan be computed. However, these steps involve complicated inte-grals with implicit transformation variables. These integrals (com-plete and incomplete beta functions, complete and incompleteelliptical integrals, and remaining improper integrals) can be evalu-ated by using numerical integration (Press et al. 1992) after con-verting the improper integrals into proper integrals (Chahar 2007).

If single trench is insufficient for a given storm run-off contrib-uting area, then an array of parallel trenches may be adopted. Theminimum center-to-center spacing S (m) between two adjacenttrenches (see Fig. 2) can be determined by using the followingrelation (Chahar 2007):

S ¼ ðd þ DÞ ffiffiffiβ

p

Kð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=βp ÞBð1=2;σÞ

Zγ

0

ðBð1=2;σÞ � Bτ ð1=2; σÞÞdτffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðγ� τÞp

ð9Þ

where D = full depth of trench (m); and β and γ are simultaneoussolutions of Eqs. (2) and (3) with y = D. There will be interference

between the infiltrations from adjacent trenches if the spacing iskept smaller than S.

Generally, excavating machinery digs a trench with verticalsides. If the soil can support vertical side slopes temporarily (untillrefilled with gravel), then rectangular trenches (rather than trapezoi-dal trenches) are more convenient, economical, and faster to con-struct. For a rectangular trench, the corresponding relations are

d þ yy

¼ πKðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=β

p� ffiffiffi

βp Z

β

γ

tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðτ � γÞp dτ

ð10Þ

by¼

Z1

β

π � 2tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðτ � βÞðτ � γÞp dτ

�Zβ

γ

tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðτ � γÞp dτ

ð11Þ

t ¼ ηb2k

Zy

0

Kð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=βp Þðd þ yÞKð ffiffiffiffiffiffiffiffi

γ=βp Þ dy

¼ η2πk

Zy

0

�by

ffiffiffiβ

p

Kð ffiffiffiffiffiffiffiffiγ=β

p Þ

Zβ

γ

tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðτ � γÞp dτ

�dy ð12Þ

S ¼ ðd þ DÞ ffiffiffiβ

p

πKð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � γÞ=βp Þ

Zγ

0

π � 2tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτðβ � τÞðγ� τÞp dτ ð13Þ

Many times, the top soil layer may extend up to large depth(d=D > b=Dþ 2mþ 5) and the water table may also lie at largedepth then the solution becomes independent of the location of themore previous lower layer. For this case γ → 0 since d=y → ∞.The corresponding relations for trapezoidal trenches with γ ¼ 0 are

by¼ 2

Z1

β

ðBð1=2; σÞ � Bτ ð1=2;σÞÞdττ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðτ � βÞp�Z

β

0

Bτ ð1=2;σÞdττ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � τÞpð14Þ

t ¼ η2πkBð1=2;σÞ

Zy

0

� ffiffiffiβ

p �2mþ b

y

�Zβ

0

Btð1=2; σÞdττ

ffiffiffiffiffiffiffiffiffiffiffiffiβ � τ

p�dy

ð15Þ

S ¼ 2πDBð1=2; σÞffiffiffiβ

p�Z

β

0

Btð1=2;σÞdττ

ffiffiffiffiffiffiffiffiffiffiffiffiβ � τ

p ð16Þ

For the similar condition, the following are the solution equa-tions for an array of rectangular trenches:

by¼

Z1

β

π � 2tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

τffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðτ � βÞp dτ

�Zβ

0

tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

τffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðβ � τÞp dτ

ð17Þ

t ¼ ηπ2k

Zy

0

�by

ffiffiffiβ

p Zβ

0

tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

τffiffiffiffiffiffiffiffiffiffiffiffiβ � τ

p dτ�dy ð18Þ

S ¼ π2D=ffiffiffiβ

p Zβ

0

tan�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiτ=ð1� τÞp

τffiffiffiffiffiffiffiffiffiffiffiffiβ � τ

p dτ ð19Þ

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Application on a Case Study

Lyon is the second largest city in France located on the RhoneRiver, which is the third largest river in France. The populationof the urban area in Lyon is 1.2 million and the city area is approx-imately 500 km:2 In the past, storm water was managed throughcombined a storm water sewer system in the old part of the cityand through a separate system in the new area and at the periphery.With increase in area and population, the traditional system becameinefficient and uneconomical. The authorities are now looking forother best management practices that are more efficient and moreadaptable to linear drainage areas such as ring roads. Thus, the localauthorities of Grand Lyon adopted alternative storm managementpractices, such as detention and infiltration basins. At present, 100devices (detention and infiltration trenches and basins) exist in thecity area that are managing 106 m3 of storm water. Studies and ex-periments on infiltration trenches (Chocat et al. 1997; Proton 2008)have demonstrated their performance to reduce storm water flows.Dechesne et al. (2005) studied long-term evaluation of cloggingand soil pollution in four infiltration basins in Lyon. These basinsare 10 to 21 years old and still have good infiltration capacities.Winiarski et al. (2006) investigated the effects of storm water onan aquifer medium of the Django-Reinhardt infiltration basin inLyon. Goutaland et al. (2007, 2008), a conducted hydrogeophysicalstudy of the same infiltration basin.

Lyon is on the banks of the Rhone River below which alluvialdeposits underlie. Types of alluvial deposits in Lyon are glacioflu-vial and fluvial deposits, and they form good aquifers. The vadosezone overlying these aquifers plays a dominant role in rechargingaquifers and in contaminant retention mechanisms. Sedimentarydeposits, constituting aquifers and vadose zones, are complex,three-dimensional, heterogeneous, and commonly anisotropic(Goutaland et al. 2008). Infiltration trenches have been built inreal-scale in completely controlled conditions adjacent to one ofthe main ring roads in northern Grand Lyon. The drainage basinis limited to a band along the road and is rather impervious.The contributing area is approximately 2.2 ha. Interception ofrun-off is achieved with a pipe of 100-mm diameter which is con-nected to the sewer under the road. The discharge into the trenchesis regulated with a gate so that they can be completely isolated fromthe sewer. In this case, the trenches are not connected to the sewerand are directly supplied with storm run-off. Before inlet into theinfiltration trenches, storm water is stored in a detention basin,which is lined with an impervious geomembrane. The volume ofthe detention basin is 60 m:3 Considering the average rainfall inthis region, the detention basin can be filled 30 times per year.A pumping system allows controlled supply to the trenches. Thewater levels into the trenches are measured with submerged pres-sure sensors. All this equipment has been described by Proton(2008). The shape of the observation trench is trapezoidal, andit has dimensions as follws: depth ¼ 1:0 m; bed width ¼ 0:8 m;and side slopes ¼ 0:45 (1 Vertical: 0.45 Horizontal). The lengthof the trench is 12.0 m and the refilled material in thetrench provides a porosity ¼ 0:25. The soil adjacent to the trenchhas varying hydraulic conductivity. The top soil layer is underlainby another highly pervious layer at a depth of 10.0 m, and the pre-vailing water table is approximately 18.0 m below the ground sur-face. Observations on water level versus time are available at threelocations (H1, H2, and H3) from 1986 to 1991 by Essai (Proton2008) and 2005 by Proton (2008). The starting water depth variedfrom 0.62 to 0.75, and the emptying time was 60 to 150 min.

The time Eq. (8) involves β and γ in the integrand. For β and γ,Eqs. (2) and (3) should be solved simultaneously. However, be-cause these equations are nonlinear and contain improper integrals,

an indirect method has been used to find the β and γ values. Themethod consists of the fsolve function of the MATLAB program(MATLAB 2010). The objective function has been constituted as

f ðβ; γÞ ¼�dyþ 1� f 1ðσ;β; γÞ

�2þ�by� f 2ðσ;β; γÞ

�2

ð20Þ

where f 1ðσ;β; γÞ and f 2ðσ;β; γÞ are right hand sides ofEqs. (2) and (3), respectively. Because minimum of this functionis zero, it can only be attained when both parts of the function reachzero values and, hence, satisfy Eqs. (2) and (3). After removingsingularities and, by using Gaussian quadratures (96 points forweights and abscissa for both inner and outer integrals) for numeri-cal integration (Abramowitz and Stegun 1972), the function hasbeen minimized for β and γ for a particular set of σ, b=y andd=y. To find the emptying time of the trench, the previous schemehas been incorporated in computation of Eq. (8) through theMATLAB program (MATLAB 2010).

An appropriate value of hydraulic conductivity was not known,so an average of observed time to drop water level from 0.6 to 0.2 mat three locations (H1, H2, and H3) for three years (1987, 1989, and1991) equal to 45 min has been used to get the equivalent hydraulicconductivity, which came out to be = 1:7809 × 10�5 m=s. Withk ¼ 1:7809 × 10�5 m=s; η ¼ 0:25; b ¼ 0:8 m; and m ¼ 0:45, theresulting graphs with different starting water depths (i.e., 0.9, 0.75,0.6, 0.45, and 0.3 m) are plotted in Fig. 3. The computed emptyingtimes have been 173.7, 159.9, 145.4, 130.0, and 113.1 min, respec-tively. The graphs are asymptote to the time axis and, thus, result ina large emptying time for the final small water depths in the trench.Had the time taken to empty the last one cm of water depth not beenconsidered, the empty times would have been 104.1, 90.3, 75.8,60.4, and 43.4 min, respectively. If multiple trenches were adopted,then Eqs. (9) or (19) would yield the required spacing betweentrenches = 4.59 m.

For the case of a rectangular trench, the corresponding graphsare shown in Fig. 4. The emptying times for the rectangular ditchare 147.8, 140.1, 131.2, 120.9, and 108.3 min, respectively, for thecomplete emptying case and 78.3, 70.6, 61.7, 51.3, and 38.7 min,respectively, when the last cm of water depth is not considered. Therequired spacing between trenches in this case = 4.15 m. Bothgraphs show that substantial time is taken to drain the last 1 cmof water depth. For more effective operation of the trenches, they

Starting Water Depth

Time (minutes)0 10 20 30 40 50 60 70 80 90 100 110 120

Wat

er D

epth

in T

renc

h (m

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.90 m0.75 m0.60 m0.45 m0.30 m

Fig. 3. Emptying time for different starting depths in a trapezoidalinfiltration trench

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may be refilled with water before complete emptying. This will alsoestablish early saturated flow in the next cycle.

Discussion

Eq. (1) assumes that ground-water flow is viscous and steady andfollows Darcy’s law, so the governing equation is the 2D Laplaceequation. It has also been assumed that soil around the trench issaturated. During the initial period, the medium is unsaturated,the flow is unsteady, and the infiltration rates are high. As the sat-uration of the soil around the trench increases, the infiltration ratedecreases exponentially with time. It may acquire a relatively con-stant rate (approaching to saturated hydraulic conductivity) within20–30 min (Duchene et al. 1994). The wetting front moves fast, andsaturated flow conditions exist within this front. If antecedent-soilmoisture is present, then the attainment of saturation and a constantrate of infiltration are even faster. Generally, the surface infiltrationrate is more than the hydraulic conductivity of the aquifer material,so the slow ground-water motion will cause saturation to the sur-rounding area of the trench. As a result, the operation of the trenchis controlled by the saturated seepage rate rather than the infiltrationrate at the surface. Under this condition, the trench designed witha high infiltration rate becomes undersized; hence, it is importantto consider saturated seepage rates (Guo 1998). Therefore, theassumption of saturated porous medium with constant hydraulicconductivity is realistic except for a limited initial phase of theoperation. Effects of these assumptions are an underestimationof the rate of infiltration from the trench and an overestimation ofthe drain time of the trench and, thus, these assumptions are on theconservative side.

Infiltration trenches are more effective where the soil has ad-equate hydraulic conductivity. In most alluvial deposits the soilis stratified. In many cases, highly permeable layers of sand andgravel underlie the top low permeable layer of finite depth. In thosecases, the high conductivity lower layer acts as a free drainage layerfor the top seepage layer because all the seeping water receivedby this layer is insufficient to saturate it. If the stratified mediumis composed of more than two layers and the top saturated layer hashydraulic conductivity less than that of the next layer, which isunsaturated, then Eq. (1) is still valid irrespective of hydraulic con-ductivities and saturation conditions in the remaining lower layers.Thus, the boundary condition assumed in Eq. (1) is likely to be

applicable in many field problems. The efficiency of an infiltrationtrench decreases with an increase in the depth of drainage layer (d).For the drainage layer and water table both at large depth,i.e., d=D > b=Dþ 2mþ 5, the special case solutions given byEqs. (14)–(19) are applicable. If the water table and/or bedrockare at a shallow depth, the infiltration trenches are ineffectiveand, hence, should not be used.

The infiltration trenches may also experience clogging problemsresulting from the settlement of fine sand particles in the intersticesof the soil. To minimize this, the run-off should be passed throughwell-maintained sediment filters or detention basins before entryinto the infiltration trench. Further, Duchene et al. (1994) observedthat the effects of sediment clogging in the bottom of the trench islimited; hence, the effect of clogging has not been considered inthis study. The porous medium in the vicinity of the trench maynot be homogeneous and isotropic in the true sense; hence, the es-timation of equivalent hydraulic conductivity of the medium maybe difficult. Clogging resulting from the migration of sediments anddevelopment of microbial growth will further change the hydraulicconductivity of the medium. As per Eq. (8), the emptying time isinversely proportional to the hydraulic conductivity of the saturatedporous medium of the top layer. Therefore, any alteration in theconductivity value can easily be incorporated into the emptyingtime while other parameters remain unaffected. Also, the analysisis based on the assumption of seepage flow 2D in the vertical plane,which will happen for a very large length of a trench. For the finitelength of a trench, if its length to bed- width ratio is more than 10then the seepage flow will be 2D in the vertical plane except at theends; therefore, the present analysis is valid with negligible errorfor such trenches.

A risk of ground-water contamination exists if the volume ofcontaminants infiltrated is greater than the natural attenuationcapacity of the underlying soils. This can happen if contaminantsmove too rapidly through the soils of high hydraulic conductivityoverlying an aquifer. The type of soil underlying an infiltrationtrench and the distance to the water table are major determinantsof the potential of ground-water contamination. A minimum of1.25 m between the bottom of the trench and the ground-water tableshould be ensured (Guo 1998).

Emptying time is important to operate the infiltration trench forstorm water management. The captured volume of run-off is tem-porarily stored in the voids of the gravel and, subsequently, it willinfiltrate into the soil adjacent to the trench and down to the aquifer.After emptying time or design storage time, the trench will beempty and ready for the next run-off. Both the captured volumeand emptying time depend on the purpose of the infiltration struc-ture and the storm water management (Akan 2002b). The capturedvolume for infiltration trenches can be calculated as the volume of12.5 mm of run-off over the impervious portion of the contributingarea, and the storage time for water quality infiltration basins varyfrom 24 to 72 h for different agencies (Akan 2002a, b). The con-tributing drainage area to an infiltration trench is usually less than4 ha because of the storage requirements for peak-run-off control(Duchene et al. 1994). Partial storm water control is provided forstorms that produce more run-off than can be stored within thetrench. An overflow for the trench is necessary to the handle excessrun-off that is produced from storms larger than the design storm.On large sites, other storm water practices, such as detention basins,can be used in conjunction with trenches to provide the necessarypeak-run-off control. Moreover, if the time between two successivestorms is less than the trench emptying time, then the excess stormwater should be diverted to detention basins. Infiltration trenchesshould be designed to drain completely within 72 h after the designevent (Duchene et al. 1994). This allows the soils underlying a

Starting Water Depth

Time (minutes)0 10 20 30 40 50 60 70 80 90 100

Wat

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epth

in T

renc

h (m

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.90 m0.75 m0.60 m0.45 m0.30 m

Fig. 4. Emptying time for different starting depths in a rectangularinfiltration trench

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trench to drain and to maintain aerobic conditions, which improvesthe pollutant removal capability of the soil underlying the trench.Therefore, to manage a storm water generated by a particular catch-ment, trench dimensions can initially be fixed on the basis of therun-off volume and the porosity of the refilled material, and thenthe corresponding drain time can be computed. If the drain time ofthe trench is not within desired limits, the depth and width of thetrench may be adjusted to achieve it. Thus a trial-and-error methodmay be adopted to arrive at an appropriate design of an infiltrationtrench.

Conclusions

Infiltration trenches can control quality and quantity of storm waterfrom small urban catchment. The surface hydrology of the catch-ment (i.e., run-off volume) determines the size of an infiltrationtrench; whereas, while the hydraulic conductivity of the aquifergoverns the emptying time of the trench. Solutions derived forthe steady saturated seepage state can provide a guideline for de-termining the required size of trenches and their spacing/numbers.The proposed design is simple enough to obtain the first estimationof the required time to empty the trench. It overestimates emptyingtime and, hence, estimates are on the conservative side to provide amargin-of-safety. The presented results may assist a storm watermanagement engineer in the design of infiltration trenches.

Notation

The following symbols are used in this paper:B(.,.) = complete Beta function [dimensionless];Bt(.,.) = incomplete Beta function [dimensionless];

b = bed width of trench [m];D = full depth of trench [m];d = depth of unsaturated medium/aquifer below bed of

trench [m];K(.) = complete elliptical integral of the first kind

[dimensionless];k = hydraulic conductivity of top layer [m=s];m = side slope of trench (1 Vertical: m Horizontal)

[dimensionless];qs = seepage discharge per unit length of trench [m2=s];S = spacing between adjacent trenches [m];T = top width of trench at full depth [m];t = trench empty time [s];y = water depth in trench [m];

β, γ = transformation variables [dimensionless];σ = ð1=πÞcot�1m [dimensionless]; andτ = dummy variable [dimensionless].

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