Laminar and non-laminar flow in geosynthetic and granular drains

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1 Geosythetics Iteatioal, 2012, 19, No. 2 Lamia ad o-lamia flow i geosythetic ad gaula dais J. P. Gioud 1, J. P. Gouc 2 ad E. Kavazajia, J. 1 Cosultig Egiee, JP Gioud, Ic., 587 Noth Ocea Blvd, Ocea Ridge, FL 45, USA, Telephoe: , Telefax: , jpg@ jpgioud.com 2 Pofesso, LTHE, Uivesity Geoble 1, BP 5, 8041 Geoble Cedex 1, Face, Telephoe , gouc@ujf-geoble.f Pofesso, School of Sustaiable Egieeig ad the Built Eviomet, Aizoa State Uivesity, Tempe, Aizoa , USA, Telephoe: , Telefax: , edkavy@asu.edu Received 6 July 2011, evised 5 Febuay 2012, accepted 5 Febuay 2012 ABSTRACT: Hydaulic tasmissivity tests o commo geosythetic ad gaula daiage mateials (e.g. geoets ad gavel) show that the hydaulic tasmissivity of these mateials ofte depeds heavily o the hydaulic gadiet, which idicates that the flow is o-lamia. Despite the o-lamia atue of flow i these mateials, Dacy s equatio ad equatios deived fom Dacy s equatio ae extesively used fo the desig of geosythetic ad gaula daiage systems, eve though these equatios ae stictly valid oly fo lamia flow. Theefoe it is impotat to idetify the daiage mateials ad flow coditios fo which the flow is lamia i ode to evaluate the applicability of Dacy s equatio. I classical hydodyamics, the coditios fo lamia flow ae geeally descibed i tems of a limitig Reyolds umbe. This pape povides guidace fo Reyolds umbe calculatio i geosythetic ad gaula dais, ad pesets a methodology to establish the coditios fo lamia flow as a fuctio of the Reyolds umbe. Numeical applicatios of the methodology show that, fo typical hydaulic gadiets used i hydaulic tasmissivity tests i the laboatoy ad ecouteed i daiage layes i the field, flow is geeally lamia i eedle-puched owove geotextiles ad sad, wheeas it is geeally olamia i geoets ad gavel. Howeve, i the case of geoets adjacet to geotextiles (such as i geocomposites), the flow becomes close to lamia coditios as the geotextile pogessively itudes ito the geoet chaels ude iceasig values of the applied omal stesses. Pactical ecommedatios ae give fo the use of Dacy s equatio ad equatios deived fom Dacy s equatio to obtai appoximate solutios whe flow is ot lamia. KEYWORDS: Geosythetics, Daiage, Flow, Geoets, Geosythetic dais, Geotextiles, Gaula dais, Lamia, Reyolds umbe, Specific suface aea, Tubulet REFERENCE: Gioud, J. P., Gouc, J. P. & Kavazajia, E. J. (2012). Lamia ad o-lamia flow i geosythetic ad gaula dais. Geosythetics Iteatioal, 19, No. 2, [ 1. INTRODUCTION # 2012 Thomas Telfod Ltd 1.1. Backgoud ad pupose of the pape I the fist pape pesetig a compehesive testig pogam o hydaulic tasmissivity of geoets, Williams et al. (1984) oted that liquid flow i geoets is geeally ot lamia. Futhemoe, umeous studies cited by Cedege (1989) have show that flow i coase gaula dais (e.g. gavel ad ockfill) is ofte ot lamia. The temiology o-lamia is used heei to desigate flow that is ot lamia. I the cotext of this pape, it is ot ecessay to distiguish betwee diffeet types of olamia flow, such as ietial lamia flow, whee the positio of fluid molecules is theoetically pedictable, ad tubulet flow, fo highe hydaulic gadiets, whee the positio of fluid molecules is ot theoetically pedictable. Dacy s equatio ad equatios deived fom Dacy s equatio ae outiely used to pefom hydaulic calculatios elated to daiage mateials whee flow is likely to be o-lamia, e.g. geoets ad gavel, eve though these equatios ae stictly valid oly fo lamia flow. Use of equatios fo lamia flow i desig of daiage layes i which flow is ot lamia ca lead to eos i estimatig the capacity of the daiage laye. Theefoe it is impo- 160

2 Lamia ad o-lamia flow i geosythetic ad gaula dais 161 tat to kow i which daiage mateials ad ude which coditios flow is lamia o o-lamia, ad it is impotat to povide desig egiees with pactical guidace fo the appopiate use of Dacy s equatio ad equatios deived fom Dacy s equatio i the case of o-lamia flow. The pupose of this pape is to povide useful ifomatio to desig egiees who deal with a situatio whee flow i commo geosythetic ad gaula daiage mateials is ot lamia ad wish to use Dacy s equatio ad equatios deived fom Dacy s equatio, as these equatios ae simple ad allow fo staightfowad calculatios, eve though they ae ot stictly applicable i this case Assumptios I the aalyses peseted i this pape, the liquid is assumed to be wate o a liquid, such as a aqueous solutio, with hydaulic chaacteistics simila to those of wate. I paticula, the aalyses peseted i this pape ae applicable to ladfill leachate if the cosideed leachate has desity ad viscosity close to those of wate (Stoltz et al. 2010), sice these two popeties have the most sigificat ifluece o the hydaulic chaacteistics of daiage mateials. Futhemoe, all of the hydaulic chaacteistics efeed to i this pape ae fo satuated flow. 1.. Mateials coceed with this study The study peseted heei is applicable to all poous mateials that ca be used to fom daiage layes. These mateials iclude: geosythetic mateials such as geoets, geomats, cuspated sheets ad eedle-puched owove geotextiles; ad gaula mateials such as ockfill, gavel ad sad Ogaizatio of the pape Relevat physical chaacteistics of geosythetic ad gaula daiage mateials ae peseted i Sectio 2 ad the hydaulic chaacteistics of these mateials ae discussed i Sectio. A methodology to detemie the coditios fo lamia flow i geosythetic ad gaula daiage layes is peseted i Sectio 4. Numeical applicatios of the methodology peseted i Sectio 5 show that lamia flow pevails i eedle-puched owove geotextiles ad sad, wheeas o-lamia flow pevails i geoets, gavel ad ockfill. Recommedatios that esult fom the aalyses peseted i the pape ae made i Sectio 6. Coclusios ae peseted i Sectio PHYSICAL CHARACTERISTICS OF DRAINAGE MATERIALS I this sectio, two key physical chaacteistics of poous media used as daiage mateials ae defied ad discussed: the specific suface aea ad the aveage flow-path diamete. These two physical chaacteistics (which ae liked) gove the hydaulic chaacteistics defied i Sectio. Theefoe they play a impotat ole i the methodology developed i Sectio 4. Two of the othe impotat chaacteistics of daiage mateials, the poosity ad the void atio, ae ot defied i Sectio 2, because they ae assumed to be well kow. Howeve, a commet elated to poosity ad void atio is peseted i Sectio Specific suface aea Defiitio of specific suface aea The specific suface aea, S s, of a solid is the atio of the suface aea to the volume of the solid. Whe a mateial cosists of discete elemets (such as fibes, ibs o paticles), the specific suface aea of the mateial is the volumetically weighted aveage of the specific suface aeas of its costituets (Gioud et al. 2002). If all costituets ae idetical (which is fequet i geosythetics), the specific suface aea of the mateial is equal to the specific suface aea of ay oe of its costituets Specific suface aea of geosythetic dai I the case of a eedle-puched owove geotextile, if the fibes ae all idetical ad ae assumed to have a uifom cicula coss-sectio, the specific suface aea is give by the followig classical equatio: S s ¼ 4 d f (1) whee d f is the fibe diamete. I the case of a geoet, Gioud et al. (2012) have show that the cotibutio of the mateials adjacet to the geoet to the specific suface aea that is elevat to liquid flow caot be igoed. I othe wods, the elevat specific suface aea is ot the itisic specific suface aea of the geoet, but a appaet specific suface aea that takes ito accout the impact o flow of the adjacet mateials. Gioud et al. (2012) poposed the followig equatio fo the appaet specific suface aea of a geoet betwee two igid mateials with smooth sufaces (e.g. betwee two geomembaes that ae stiff ad smooth): S s ¼ 4 2 þ (2) d ð1 Þt D whee d is the diamete of a cicle with a aea equal to the aveage coss-sectioal aea of the ibs of the geoet, is the poosity of the geoet, ad t D is the thickess of the daiage laye (i.e. the thickess of the geoet i the case of Equatio 2). Regadig d, two types of commecially available geoet ae cosideed heei: biplaa geoets (i.e. geoets that cosist of two layes of idetical ibs o top of each othe) ad tiplaa geoets (i.e. geoets that cosist of oe laye of paallel ibs havig a lage coss-sectio aea located betwee two layes of ibs havig a much smalle coss-sectioal aea). Gioud et al. (2012) have idicated that d is appoximately equal to t D /2 i the case of a typical biplaa geoet ad appoximately equal to t D i the case of a typical tiplaa geoet (because, i the commecially available tiplaa geoets cosideed heei, the thickess of the two layes of smalle ibs is egligible compaed with the thickess of the laye of lage ibs). Hece, fom Equatio 2: Geosythetics Iteatioal, 2012, 19, No. 2

3 162 Gioud, Gouc ad Kavazajia S s ¼ t D 1 fo biplaa geoets, ad S s ¼ 2 2 (4) t D 1 fo tiplaa geoets. It is impotat to ote that Equatios ad 4 ae fo a geoet betwee two boudaies that ae both igid ad smooth. No simple expessio has bee developed fo the appaet specific suface aea of geoets adjacet to ough sufaces such as textued geomembaes, o adjacet to compliat mateials such as geotextiles. I these cases, it ca be expected that the specific suface aea will be geate tha the value give by Equatio o Equatio 4. As show by Gioud et al. (2012) usig Equatios 1, ad 4 fo typical geoets havig poosities of 0.8, the pesece of boudaies that ae both igid ad smooth esults i a appaet specific suface aea that exceeds the itisic specific suface aea by 125% fo a typical biplaa geoet ad by 250% fo a typical tiplaa geoet. Based o these calculatios, it is clea that hydaulic tasmissivity tests pefomed o geoets betwee two plates that ae both igid ad smooth do ot give the itisic hydaulic tasmissivity of the tested geoets. I fact, the measued hydaulic tasmissivity is a fuctio of both the hydaulic tasmissivity of the geoet ad the iteactio of the flow ad the geoet boudaies Specific suface aea of gaula dai I the case of a spheical paticle, the specific suface aea is give by the followig classical equatio: () S s ¼ 6 d s (5) whee d s is the diamete of the spheical paticle. Accodig to Cama (197), Fai ad Hatch (19) ad Rose (1945) as cited by Gouc (1982, p. 1), the specific suface aea of a o-spheical paticle ca be expessed as S s ¼ 6C f (6) d whee d is the paticle size (as detemied i a covetioal paticle size test, such as sievig), ad C f is a shape facto whose value is 1.0 fo spheical paticles, 1.25 (i.e. 7.5/6) fo ouded sad ad gavel, 1.5 (i.e. 9/6) fo agula sad, ad 1.8 (i.e. 11/6) fo vey agula gavel. To these values, the authos popose addig 1.75 (i.e. 8.25/6) fo sub-agula sad o gavel, 1.5 (i.e. 9/6) fo semi-agula gavel, ad 1.67 (i.e. 10/6) fo agula gavel, based upo itepolatio amog the above values. These values of the shape facto ae summaized i Table 1. I a actual gaula mateial, paticles have vaious sizes. Theefoe a aveage specific suface aea must be established. A appoximate value of the aveage specific suface aea ca be calculated usig a fictitious liea paticle-size distibutio cuve i the taditioal axes of pecetage passig by mass (o weight) as a fuctio of Table 1. Shape facto fo calculatio of the specific suface aea of gaula mateials Type of paticle Pefectly spheical 1.00 Shape facto, C f Sad Rouded 1.25 (7.5/6) Sub-agula 1.75 (8.25/6) Agula 1.50 (9.0/6) Gavel Rouded 1.25 (7.5/6) Sub-agula 1.75 (8.25/6) Semi-agula 1.50 (9.0/6) Agula 1.67 (10/6) Vey agula 1.8 (11/6) logaithm of paticle size. This fictitious liea cuve should be established as close as possible to the actual paticle-size distibutio cuve (Figue 1). The slope of the fictitious liea paticle-size distibutio cuve is defied by the liea coefficiet of uifomity of the gaula mateial (Gioud 1982, 1996), C9 u, which is equal to the atio of ay two paticle sizes sepaated by 50% by mass. Hece: Factio fie (%) Factio fie (%) Geosythetics Iteatioal, 2012, 19, No d 0 d 0 d d d d (a) (b) d d d 100 Paticle size d 100 Paticle size Figue 1. Use of staight lie (d9 0 d9 100 ), called fictitious liea paticle-size distibutio cuve, associated to the actual paticle-size distibutio cuve (solid cuve) to detemie the liea coefficiet of uifomity (Equatio 7): (a) the fictitious liea paticle-size distibutio cuve follows the cetal potio of the actual paticle-size distibutio cuve; (b) the fictitious liea paticle-size distibutio cuve has the same d 10 ad d 60 as the actual paticle-size distibutio cuve

4 Lamia ad o-lamia flow i geosythetic ad gaula dais 16 C9 u ¼ d9 50 d9 0 ¼ d9 60 d9 10 ¼ d9 70 d9 20 ¼ d9 80 d9 0 ¼ d9 90 d9 40 ¼ d9 100 d9 50 ¼ sffiffiffiffiffiffiffiffi d9 100 d9 0 whee d9 x is the fictitious paticle size that coespods to the odiate x% o the fictitious liea paticle-size distibutio cuve, d9 0 is the size of the smallest fictitious paticle o the fictitious liea paticle-size distibutio cuve, d9 50 is the size of the media fictitious paticle o the fictitious liea paticle-size distibutio cuve, ad d9 100 is the size of the lagest fictitious paticle o the fictitious liea paticle-size distibutio cuve (Figue 1). As metioed by Gioud (2010) fo filte desig, the fictitious liea cuve ca be selected by followig the cetal potio of the actual paticle-size distibutio cuve, as show i Figue 1a. Thee ae may othe possibilities fo selectig the fictitious liea cuve. Fo example, oe may use a staight lie that passes though the poits of the actual paticle-size distibutio cuve that coespod to d 10 ad d 60 (hece d9 10 ¼ d 10 ad d9 60 ¼ d 60 ) i ode to have C9 u equal to the classical C u (Figue 1b). I geeal, the cetal pats of the two cuves ae vey close. Theefoe, at least appoximately, the followig elatioship is applicable: d9 50 ¼ d 50 (8) whee d 50 is the actual media paticle size, i.e. the paticle size such that 50% by mass of the paticles ae smalle tha d 50 : The fictitious liea paticle-size distibutio cuve is defied by two paametes, fo istace d9 0 ad C9 u : Usig these two paametes, a appoximate value of the aveage specific suface aea of the gaula mateial ca be calculated usig the followig equatio (Gioud 1996, Gioud et al. 2002): " S s ¼ d9 0 l ðc9 u Þ 1 1 # ðc9 u Þ 2 (9) Equatio 9 was developed fo spheical paticles. This equatio ca be exteded to ay paticle shape by usig the shape facto fom Equatio 6. If it is assumed that the shape facto is the same fo all paticles of the cosideed mateial, the exteded equatio is S s ¼ C " f d9 0 l ðc9 u Þ 1 1 # ðc9 u Þ 2 (10) A limited seies expasio of Equatio 9 shows that, fo C9 u ¼ 1, this equatio teds towad Equatio 6 with d9 0 ¼ d. Similaly, a limited seies expasio of Equatio 8 shows that it teds towad Equatio 5 with d9 0 ¼ d s : It is ot coveiet to use d9 0 i equatios fo the specific suface aea (as i Equatio 10), because it is a fictitious paticle size liked to a fictitious liea paticlesize distibutio cuve. It is pefeable to use a paticle size fom the actual paticle-size distibutio cuve, such as d 50 : This ca be achieved by combiig Equatios 7, 8 ad 10, which gives the followig equatio fo the specific suface aea of a gaula medium: (7) S s ¼ C " f C9 u d 50 l ðc9 u Þ 1 1 # ðc9 u Þ 2 C f ¼ C9 u 1 d 50 l ðc9 u Þ C9 u (11) 2.2. Aveage flow-path diamete Defiitio of aveage flow-path diamete The aveage flow-path diamete of a poous medium, d FP, is the aveage diamete of flow chaels that ca be used i theoetical aalyses of flow though the cosideed poous medium. The aveage flow-path diamete of a poous medium ca be expessed by the followig equatio (Gioud et al. 2012): d FP ¼ 4 (12) S s Aveage flow-path diamete of geosythetic dai Combiig Equatios 1 ad 12 gives the followig expessio fo the aveage flow-path diamete of a eedlepuched owove geotextile: d FP ¼ d f (1) 1 Equatio 1 shows that, fo a poosity of (which is typical fo a eedle-puched owove geotextile if thee is o applied compessive stess; Gioud 1996, p. 582), the aveage flow-path diamete of a eedlepuched owove geotextile is appoximately 10 times the fibe diamete: hece d FP 00 ìm ¼ 0. mm fo a typical fibe diamete of 0 ìm. Combiig Equatios (o 4) ad 12 gives the followig expessios fo the aveage flow-path diamete of a geoet betwee two boudaies that ae both igid ad smooth: d FP ¼ 2t D fo a biplaa geoet, ad (14) 5 4 d FP ¼ 2t D fo a tiplaa geoet: (15) 2 Numeical calculatios usig Equatios 14 ad 15 ad typical geoet poosities (which age betwee 0.75 ad 0.85) show that the aveage flow-path diamete is betwee 0.75 ad 1.05 times the geoet thickess fo biplaa geoets ad betwee 1.0 ad 1. times the geoet thickess fo tiplaa geoets (Gioud et al. 2012). Theefoe it may be cocluded that the aveage flow-path diamete i a geoet is geeally of the ode of the geoet thickess Aveage flow-path diamete of gaula dai Combiig Equatios 6 ad 12 gives the followig expessio fo the aveage flow-path diamete of a gaula mateial with uifom paticle size, d: d FP ¼ 2 C f 1 Geosythetics Iteatioal, 2012, 19, No. 2 d (16)

5 164 Gioud, Gouc ad Kavazajia Equatio 16 shows that, fo typical values of poosity ad shape facto fo gaula daiage mateials, such as ¼ 1/ ad C f betwee 1.25 ad 1.8, d FP is of the ode of 0.2d. Combiig Equatios 11 ad 12 gives the followig expessio fo the aveage flow-path diamete of a gaula mateial with o-uifom paticle size: d FP ¼ 4 C f l ð C9u 1 Þ C9 u ð1=c9 u Þ d 50 (17) A limited seies expasio of Equatio 17 shows that, fo C9 u ¼ 1, this equatio teds towad Equatio 16 with d 50 ¼ d. Fo typical values of the shape facto ad coefficiet of uifomity fo gaula daiage mateials, such as C f betwee 1.25 ad 1.8 ad C9 u betwee 2 ad 10, umeical calculatios pefomed with Equatio 17 show that d FP is of the ode of d 50 fo ¼ 0.15 ad d 50 fo ¼ 0.0. Based upo these values, the followig appoximate equatio ca be used to obtai a typical value fo the aveage flow-path diamete of a gaula mateial: d FP d 50 (18) 2 I most hydaulic calculatios elated to gaula daiage mateials, it is ot ecessay to use a value of the aveage flow-path diamete moe accuate tha the value povided by Equatio 18. The above aalysis shows that the aveage flow-path diamete of gaula daiage media is sigificatly smalle tha the media paticle size, d 50. Howeve, it has sometimes bee assumed i the liteatue that the aveage flow-path diamete is equal to the media paticle size (e.g. Bea 1972, p. 125). The above aalysis suggests that the assumptio of a aveage flow-path diamete equal to the media paticle size sometimes foud i the liteatue is abitay ad ufouded. 2.. Poosity ad void atio All the equatios peseted i this pape that iclude the poosity,, ca be expessed i tems of the void atio, e, usig the elatioship ¼ e (19) 1 þ e Fom Equatio 19, it is possible to deive the followig expessios, which will be used late i this pape: e ¼ (20) 1 ð1 Þ 2 ¼ e 1 þ e. HYDRAULIC CHARACTERISTICS OF DRAINAGE MATERIALS (21).1. Geeal commets o hydaulic chaacteistics As idicated i Sectio 1., this pape is elated to liquid flow i daiage layes (i.e. gaula layes ad geosythetics). Theefoe liquid flow cosideed heei is withi Geosythetics Iteatioal, 2012, 19, No. 2 the plae of the daiage layes. I othe wods, thee is o flow compoet pepedicula to the plae of the daiage laye. I this sectio, two impotat hydaulic chaacteistics ae defied: the hydaulic coductivity ad the hydaulic tasmissivity. These two hydaulic chaacteistics (which ae liked) ae used i equatios fo the desig of daiage layes. Howeve, befoe these two hydaulic chaacteistics ae peseted, it is ecessay to give basic defiitios of flow ate ad velocity. Eve though most of the cocepts peseted i Sectio ae well kow, this sectio is ecessay to povide i a ogaized mae the equatios that will be used fo subsequet deivatios, ad to esue that these deivatios ae established o a igoous basis. It should be oted that the hydaulic coductivity ad the hydaulic tasmissivity used i this pape ae i fact the satuated hydaulic coductivity ad the satuated hydaulic tasmissivity, i.e. the hydaulic coductivity ad the hydaulic tasmissivity elated to a poous medium satuated with wate o simila liquid as defied i Sectio 1.2. Fo the sake of simplicity, the tem satuated is omitted i the emaide of this pape..2. Flow ate ad velocity.2.1. Flow ate ad appaet velocity The appaet velocity of liquid flow i a poous medium, also called the dischage velocity o the specific dischage, is defied by the equatio v app ¼ Q (22) A whee v app is the appaet velocity of liquid flow, Q is the flow ate, A is the coss-sectioal aea of the medium whee the flow takes place, ad Q/A is the flow ate pe uit aea. Equatio 22 ca be used with ay set of coheet uits. The elevat basic SI uits ae: v app (m/s), Q (m /s), A (m 2 ), ad Q/A (m/s). Whe a daiage laye with a ectagula coss-sectio (i.e. a daiage laye havig a uifom thickess) is flowig full, the coss-sectioal aea of the medium whee the flow takes place is A ¼ Bt D (2) whee B is the width of the daiage laye ad t D is the thickess of the daiage laye (peviously defied fo geoets, but applicable to all types of daiage laye). Combiig Equatios 22 ad 2 gives the followig equatio, applicable whe a daiage laye with a ectagula coss-sectio is flowig full (i.e. whe Q ¼ Q full ): v app ¼ Q full ¼ Q full=b (24) Bt D t D whee Q full is the flow capacity (i.e. the flow ate whe the daiage laye is full), ad Q full /B is the flow capacity pe uit width. Equatio 24 ca be used with ay set of coheet uits. The elevat basic SI uits ae: v app (m/s), Q full (m /s), Q full /B (m 2 /s), B (m), ad t D (m). The appaet velocity (which is i fact the flow ate pe uit aea, Q/A) is ot the actual velocity of the liquid. It is

6 Lamia ad o-lamia flow i geosythetic ad gaula dais 165 a fictitious velocity, cosistet with the cotiuum appoach, which cosists i assumig that the liquid occupies the etie volume (i.e. voids ad solids) ad ot oly the volume of voids Aveage pojected velocity The aveage pojected velocity of liquid flow i a poous medium is defied by the followig classical equatio: v avg ¼ Q A ¼ Q full ¼ v app (25) Bt D Equatio 25 ca be used with ay set of coheet uits. The elevat basic SI uits ae: v avg (m/s), Q (m /s), A (m 2 ), Q full (m /s), B (m), t D (m), ad v app (m/s); is dimesioless. The aveage pojected velocity, v avg, sometimes called the seepage velocity, is the pojectio o the diectio of the gadiet, i.e. o the geeal diectio of the flow, of the aveage liquid velocity i the chaels of the poous medium. Like the appaet velocity, this pojected velocity is also a fictitious velocity: it is based upo the assumptio that the fluid tavels betwee ay two poits A ad B, followig a smooth (but ot ecessaily staight) path i the diectio of the gadiet, i.e. a fictitious o-totuous flow path. It is ot the aveage velocity of the liquid i the chaels of the poous medium, as ofte stated iappopiately. The aveage pojected velocity is also sometimes called the tavel velocity, because it is the atio of the distace taveled by liquid betwee the two poits, A ad B, to the time equied fo tavelig: v avg ¼ AB (26) t AB whee AB is the distace taveled by liquid fom poit A to poit B (measued i the diectio of the gadiet, i.e. alog the geeal diectio of the flow, ot alog the totuous chaels), ad t AB is the time equied fo the wate to tavel fom poit A to poit B. Equatio 26 ca be used with ay set of coheet uits. The elevat basic SI uits ae: v avg (m/s), AB (m), ad t AB (s)... Hydaulic coductivity..1. Appaet hydaulic coductivity I a poous medium, the appaet velocity of the liquid (o flow ate pe uit aea) iceases as the hydaulic gadiet iceases: v app ¼ Q A ¼ k (i)i (27) The atio of the appaet velocity (o flow ate pe uit aea) i a poous medium to the hydaulic gadiet is called heei the appaet hydaulic coductivity, k (i). Combiig Equatios 25 ad 27 gives k (i) ¼ Q full Bt D i ¼ v avg i (28)..2. Dacy s law ad hydaulic coductivity The appaet hydaulic coductivity geeally depeds o Geosythetics Iteatioal, 2012, 19, No. 2 the flow velocity, ad theefoe depeds o the hydaulic gadiet (hece the symbol k (i) ). Obsevatios of fluid flow i poous media show that, whe the flow velocity is small, the flow is lamia (i.e. thee is o tubulece, ad the liquid molecules flow paallel to each othe), ad the atio of the appaet velocity to the hydaulic gadiet is costat. This atio is the called the hydaulic coductivity, k. This is kow as Dacy s law, accodig to which the appaet velocity (o flow ate pe uit aea) is popotioal to the hydaulic gadiet. Dacy s law is usually expessed by the followig equatio, kow as Dacy s equatio: v app ¼ Q A ¼ Q full Bt D ¼ ki (29) Equatio 29 is simila to Equatio 27, but with k costat. Dacy s equatio is valid oly fo lamia flow. Howeve, Dacy s equatio ad equatios based o Dacy s law (i.e. equatios that wee developed with the assumptio that Dacy s law is applicable) ae outiely used whe flow is ot lamia to itepet test esults ad pefom desig calculatios elated to flow i both geosythetic ad gaula dais. Theefoe it is impotat to kow whee the limit betwee lamia ad o-lamia flow is. I the case of o-lamia flow, Equatio 29 (Dacy s law) ad equatios deived fom Equatio 29 should be witte with k (i) (the gadiet-depedet hydaulic coductivity, as i Equatio 27), wheeas i the case of lamia flow the equatios ca be witte with k (the costat hydaulic coductivity fo lamia flow).... Relatioship betwee hydaulic coductivity ad physical chaacteistics I the case of lamia flow, the hydaulic coductivity of a poous medium ca be expessed as follows by the classical Kozey Cama equatio (Kozey 1927a, 1927b; Cama 197): k ¼ â k g! 1 ð1 Þ 2 S 2 s ¼ â k g! (0) e 1 1 þ e S 2 s whee â k is a totuosity facto, is the desity of liquid, is the viscosity of liquid (also efeed to as its dyamic viscosity), ad g is the acceleatio of gavity. Gioud (1996, pp ) pesets a deivatio of the Kozey Cama equatio, ad ecommeds a value â k ¼ 0.1 fo all poous media. Equatio 0 ca be used with ay set of coheet uits. The elevat basic SI uits ae: k (m/s), (kg/m ), g (m/s 2 ), (kg/(m s)), ad S s (m 1 ); â k ad ae dimesioless. I Equatio 0, S s ca be eplaced by its expessio give by Equatio 1 fo eedle-puched owove geotextiles, Equatio fo biplaa geoets, Equatio 4 fo tiplaa geoets, Equatio 5 fo uifomly gaded spheical paticles, Equatio 6 fo uifomly gaded gaula daiage mateials, ad Equatio 11 fo o-uifomly gaded gaula daiage mateials.

7 166 Gioud, Gouc ad Kavazajia Combiig Equatio 12 fo the aveage flow-path diamete ad Equatio 0 gives k ¼ â k g ð 16 d FPÞ 2 (1) Equatio 1 (which is valid fo ay poous medium i the case of lamia flow) shows that the hydaulic coductivity is popotioal to the squae of the aveage flow-path diamete. Equatio 1 is cosistet with elatioships foud i some soil mechaics textbooks (e.g. Taylo 1948, p. 112). Recallig that the aveage flow-path diamete is popotioal to paticle size (Equatios 17 ad 18), Equatio 1 is also cosistet with the classical Haze s fomula (Haze 1911) elatig the hydaulic coductivity to the squae of a chaacteistic paticle size..4. Hydaulic tasmissivity.4.1. Appaet hydaulic tasmissivity The atio of the flow capacity pe uit width of a daiage laye with a ectagula coss-sectio (i.e. a daiage laye havig a uifom thickess) to the hydaulic gadiet is called heei the appaet hydaulic tasmissivity, Ł (i) : Ł (i) ¼ Q full=b (2) i Equatio 2 ca be used with ay set of coheet uits. The elevat basic SI uits ae: Q full (m /s), B (m), Q full /B (m 2 /s), ad Ł (i) (m 2 /s); i is dimesioless. Equatio 2 is valid whethe the daiage laye is composed of a sigle mateial o ot. If the daiage laye is composed of a sigle mateial, it is possible to combie Equatios 28 ad 2, which gives Ł (i) ¼ k (i) t D () Equatio ca be used with ay set of coheet uits. The elevat basic SI uits ae: Ł (i) (m 2 /s), k (i) (m/s) ad t D (m). Equatio shows that, i the case of a daiage laye composed of a sigle mateial, the appaet hydaulic tasmissivity is the poduct of the appaet hydaulic coductivity of the daiage mateial ad the thickess of the daiage laye Hydaulic tasmissivity The appaet hydaulic tasmissivity geeally depeds o the hydaulic gadiet (hece the symbol Ł ( i) ). Whe the hydaulic gadiet is small, the flow is lamia, ad the atio betwee the flow capacity ad the hydaulic gadiet is costat. This atio is the called the hydaulic tasmissivity. Equatios 2 ad the become, espectively, Ł ¼ Q full=b (4) i Ł ¼ kt D (5) whee Ł is the hydaulic tasmissivity. Equatios 4 ad 5 ae valid oly fo lamia flow. 4. CONDITIONS FOR LAMINAR FLOW 4.1. Reyolds umbe Defiitio of Reyolds umbe The Reyolds umbe quatifies the elative magitude of: (1) the ietia foces esultig fom gadiets i the fluid velocity field; ad (2) the foces imposed o the solid costituets ad boudaies of the daiage medium due to fictio esultig fom fluid viscosity. Classical expeimets o flow i pipes have show that the Reyolds umbe chaacteizes the tasitio betwee lamia flow ad o-lamia flow. Evetually, the Reyolds umbe cocept was geealized to poous media. Flow is lamia if the Reyolds umbe, Re, that chaacteizes the flow of the cosideed fluid i the cosideed medium is equal to o less tha a cetai limitig value, Re lim : Re < Re lim (6) The Reyolds umbe ca be calculated usig the followig classical equatio: Re ¼ Geosythetics Iteatioal, 2012, 19, No. 2 d c v (7) whee v is a chaacteistic velocity of the liquid, ad d c is a chaacteistic dimesio of the flow chaels. Equatio 7 ca be used with ay set of coheet uits. The elevat basic SI uits ae: v (m/s), (kg/m ), d c (m), ad (kg/ (m s)); Re is dimesioless. It should be oted that seveal Reyolds umbes ca be calculated fo a give daiage mateial, depedig o the selectio of the chaacteistic dimesio of the flow chaels ad the chaacteistic liquid velocity I the case of pipes, the iside diamete of the pipe is atually used fo the chaacteistic dimesio of the flow chael. Fo the sake of cosistecy with the pactice fo pipes, the aveage flow-path diamete, d FP, will be used heei as the chaacteistic dimesio fo poous media. I a pipe, the liquid velocity vaies pogessively fom the cete to the walls of the pipe i lamia flow, ad vaies adomly i tubulet flow. As the liquid velocity vaies fom oe poit to aothe, the Reyolds umbe fo pipes is calculated usig, fo the chaacteistic liquid velocity, the aveage liquid velocity (which, i the case of a pipe, is the same as the aveage pojected velocity). I the case of a poous medium, the followig liquid velocities ca be used i Reyolds umbe calculatios fo the chaacteistic liquid velocity: the appaet flow velocity (Equatio 22), the aveage pojected velocity (Equatio 25), ad the actual aveage liquid velocity i the flow chaels (which is pactically impossible to kow, owig to chael totuosity). Heei, the aveage pojected velocity (Equatio 25) will be used as the chaacteistic velocity fo the sake of cosistecy with the pactice fo pipes. Based o the above discussio, the Reyolds umbe fo poous media ca be calculated usig Equatio 7, witte as follows with v avg ad d FP :

8 Lamia ad o-lamia flow i geosythetic ad gaula dais 167 Re ¼ d FP v avg (8) Combiig Equatios 25 ad 8 gives the Reyolds umbe fo poous media, as follows: Re ¼ v app (9) dfp Data o limitig value of Reyolds umbe It is well kow that, i the case of pipes, the flow is lamia if the Reyolds umbe is less tha 2000 (i.e. Re lim ¼ 2000 fo pipes). Howeve, this well-kow limitig value does ot apply to media whee flow is totuous, such as poous media. Expeimetal evidece egadig the limit betwee lamia flow ad o-lamia flow i gaula media has bee eviewed by Bea (1972, p. 126), who dew the followig coclusio: Dacy s law is valid as log as the Reyolds umbe based o aveage gai diamete does ot exceed some value betwee 1 ad 10. Based o the cotext of the discussio by Bea (1972, p. 125), the efeece to aveage gai diamete implies that Bea used d 50 fo the diamete of the flow path (i.e. fo d FP i Equatios 8 ad 9). This assumptio is ot cosistet with the aalysis peseted i Sectio 2.2., which shows that d FP is much smalle tha d 50 (i.e. d FP is of the ode of d 50 fo ¼ 0.15 ad d 50 fo ¼ 0.0). Also, it should be oted that Bea (1972, p. 125) calculates the Reyolds umbe usig the appaet velocity fo the chaacteistic liquid velocity. I othe wods, Bea uses Equatio 8 with v app athe tha v avg, which is equivalet to usig Equatio 9 without. Based o the above data, the limitig values of the Reyolds umbe fo lamia flow i gaula media idicated by Bea (1972) should be multiplied by the facto (d FP /d 50 )/ to establish the limitig value of the Reyolds umbe as defied by Equatio 8 o 9. This facto is appoximately 0.5, as show by Equatio 18. Theefoe, to accout fo Bea s use of a icoect value of the aveage flow-path diamete ad of the appaet velocity athe tha the aveage pojected velocity, the age of 1 to 10 poposed by Bea (1972) fo the limitig Reyolds umbe fo lamia flow i gaula mateials metioed by Bea (1972) should i fact be 0.5 to 5 fo the Reyolds umbe as defied by Equatio 8 o 9. Taylo (1948, p. 122) calculates the Reyolds umbe usig the appaet velocity fo the chaacteistic liquid velocity, ad the aveage paticle size, d avg, fo the flowpath diamete. I othe wods, Taylo uses Equatio 8 with v app athe tha v avg ad d avg athe tha d FP. Based o wok by Fache et al. (19), he idicates that the flow i a poous medium is lamia if the Reyolds umbe thus calculated is less tha 1. Theefoe the citeio fo lamia flow accodig to Taylo is d avg v app < 1 (40) Accodig to Gioud (1996, p. 569), the aveage paticle size of a give soil ca be defied as the paticle size of a uifomly gaded soil with the same specific suface aea. Theefoe the followig equatio is obtaied by aalogy with Equatio 6: d avg ¼ 6C f (41) S s Elimiatig S s betwee Equatios 11 ad 41 gives d avg ¼ 2l ð C9 uþ C9 u ð1=c9 u Þ d 50 (42) Combiig Equatios 17 ad 42 gives d FP ¼ 2 C f 1 d avg (4) It should be oted that Equatio 4 could have bee obtaied diectly fom Equatios 12 ad 41. Howeve, the above deivatio povides a oppotuity to itoduce Equatio 42 fo the aveage paticle size. Combiig Equatios 40 ad 4 gives the followig expessio fo the citeio fo lamia flow i gaula daiage mateials poposed by Taylo: C f 2 1 Geosythetics Iteatioal, 2012, 19, No. 2 d FP v app < 1 (44) Combiig Equatios 6, 9 ad 44 gives the followig equatio, which expesses the limitig Reyolds umbe poposed by Taylo i tems of the Reyolds umbe defiitio adopted i this pape (see Equatio 9): 2 Re lim ¼ (45) C f ð1 Þ Fo a typical age of values of the poosity fo gaula mateials,, of 0.15 to 0. ad typical values of the shape facto, C f, fom 1.25 to 1.8 (see Sectio 2.1. ad Table 1), Equatio 45 shows that the Reyolds umbe should be less tha appoximately 0.4 to 0.8 to esue lamia flow. I othe wods, the value of 1 fo the limitig Reyolds umbe fo lamia flow i gaula mateials metioed by Taylo (1948) should i fact be 0.4 to 0.8 fo the Reyolds umbe as defied by Equatio 9. This is appoximately cosistet with the age of values (0.5 to 5) deived fom the age poposed by Bea (see above). Gouc (1982) has peseted a detailed study of the limitig Reyolds umbe fo poous media. Based o published expeimetal data o flow though poous media made of cylides (modelig fibes) ad glass beads (modelig gaula mateials), ad o upublished tests he pefomed o geotextiles, Gouc (1982) deived limitig Reyolds umbes fo lamia flow. The paametes selected by Gouc (1982, p. 14) to deive the Reyolds umbe ae the followig: (1) the aveage pojected velocity fo the chaacteistic liquid velocity; ad (2) the aveage flow-path diamete fo the chaacteistic dimesio of the flow chaels. Theefoe the paametes used by Gouc (1982) ae cosistet with the paametes ecommeded i this pape ad used i Equatio 8. Gouc (1982, pp. 15 ad 2) cocluded that flow is lamia if the Reyolds umbe is less tha 1 fo sad (which is cosistet with the fidigs of Taylo ad of

9 168 Gioud, Gouc ad Kavazajia Bea metioed ealie i this sectio) ad 5 fo eedlepuched owove geotextiles Adopted limitig values of Reyolds umbe Based o the above discussio, the followig values ae adopted heei fo the limitig Reyolds umbe fo lamia flow: 1 fo gaula mateials, 5 fo eedlepuched owove geotextiles, ad somewhee betwee 1 ad 10 fo geoets (as accuate ifomatio o Reyolds umbe fo geoets is lackig). Compaig the limitig Reyolds umbe value of 2000 fo lamia flow i pipes, o oe had, to the value of 1 to 10 fo poous media, o the othe had, shows that flow ceases to be lamia at a much lowe velocity i a poous medium tha i a pipe. This is a impotat fact, because thee is a tedecy to believe that the well-kow limitig Reyolds umbe of 2000 fo pipes is a uivesal value applicable to all daiage mateials, fom pipes to poous media. It is impotat to emembe that the above values of the Reyolds umbe ae applicable oly if the Reyolds umbe is calculated usig Equatio 8 o 9 with the aveage flow-path diamete defied i Sectio Limit betwee lamia ad o-lamia flow o lamia flow coditio Geeal expessio fo Reyolds umbe fo poous media Combiig Equatios 24 ad 9 gives Re ¼ dfp Qfull (46) t D B Combiig Equatios 28 ad 46 gives the followig expessio fo the Reyolds umbe: Re ¼ dfp k (i) i (47) Combiig Equatios 2 ad 46 gives the followig expessio fo the Reyolds umbe: Re ¼ dfp Ł (i) i (48) t D Equatios 47 ad 48 ae equivalet, ad ae actually applicable to both gaula ad geosythetic daiage mateials. Howeve, Equatio 47 is moe coveiet fo gaula daiage mateials, ad Equatio 48 is moe coveiet fo geosythetic daiage mateials. Whe flow is lamia, k (i) i Equatio 47 ad Ł (i) i Equatio 48 ae to be eplaced by k ad Ł espectively Expessio of limit betwee lamia ad olamia flow Equatio 6 states that the flow is lamia if the Reyolds umbe that chaacteizes the flow of the cosideed fluid i the cosideed medium is equal to o less tha a cetai limitig value, Re lim : Fom the limitig value of the Reyolds umbe, it is possible to deive a limitig hydaulic gadiet by combiig Equatios 29, 5, 6 ad 9: i < Relim d FP k ¼ td Relim d FP Ł ¼ i lim (49) Equatio 49 shows that the flow is lamia if the hydaulic gadiet is equal to o less tha a limitig hydaulic gadiet, ad it povides two expessios fo this limitig hydaulic gadiet: oe with the hydaulic coductivity of the daiage medium, the othe with its hydaulic tasmissivity. It appeas fom Equatio 49 that the limitig hydaulic gadiet, i lim, depeds o the poous medium ad the fluid, as k ad Ł ae costat fo a give geosythetic subject to lamia flow (i.e. fo i < i lim ). It should be oted that, whe flow is lamia, Equatio 47 (with k ( i) ¼ k) o Equatio 48 (with Ł ( i) ¼ Ł) combied with Equatio 49 give Re ¼ i if i < i lim : (50) Re lim i lim Whe the flow is lamia, i.e. whe the coditio expessed by Equatio 49 (which is equivalet to Equatio 6) is met, the flow capacity is less tha a limitig value, as show by the followig equatio, obtaied by combiig Equatios 6 ad 46: Q full B < td Re lim ¼ d FP Q full B lim (51) Equatio 51 idicates that the flow is lamia if Q full /B is ot geate tha the limitig flow capacity, (Q full / B) lim, which depeds o the poous medium ad the fluid. I othe wods, the limitig flow capacity, (Q full / B) lim, is a costat fo a give poous medium ad a give fluid. This is illustated i Figue 2 whee it is see that if Q full /B is geate tha (Q full /B) lim the flow capacity is less tha popotioal to the hydaulic gadiet. Fom Equatio 49, it is deived that fo lamia flow, i.e. if i < i lim, the hydaulic coductivity ad the hydaulic tasmissivity ae expessed by the followig two equatios, espectively: Flow capacity, Q / B Geosythetics Iteatioal, 2012, 19, No. 2 full Q full B lim Lamia flow η ρ t D Re d lim FP i lim Hydaulic gadiet No-lamia flow Figue 2. Typical cuve of flow capacity, Q full /B, agaist hydaulic gadiet, i

10 Lamia ad o-lamia flow i geosythetic ad gaula dais 169 k ¼ Relim d FP i lim Ł ¼ td Relim d FP i lim (52) (5) It is well kow that the hydaulic coductivity ad hydaulic tasmissivity values fo lamia flow ae always geate tha the values fo o-lamia flow. Theefoe, if i > i lim, the appaet hydaulic coductivity ad hydaulic tasmissivity fo o-lamia flow meet the followig coditios deived fom Equatios 52 ad 5: k (i) < Relim ¼ k fo i. i lim ; (54) d FP i lim Ł (i) < td Relim ¼ Ł fo i. i lim : (55) d FP i lim This is illustated i Figue. I this figue, the shape of the cuve of k (i) o Ł ( i) fo i. i lim is as demostated i a techical ote by Gouc et al. (2012) Gaphical epesetatio of limit betwee lamia ad o-lamia flow A dimesioless epesetatio of the limit betwee lamia ad o-lamia flow ca be obtaied by witig Equatios 52 ad 5 as follows: Y ¼ ¼ Re lim i lim dfp k ¼ dfp Ł t D (56) I a plot of Y agaist i lim, Equatio 56 (i.e. the limit betwee lamia ad o-lamia flow) is a hypebola, sice Re lim is a costat fo a give type of daiage mateial. Usig a hypebola is ot coveiet. It is pefeable to use a logaithmic scale, i which case the limit is a staight lie, as illustated i Figue 4. O a logaithmic scale, Equatio 56 becomes log Y ¼ log d FP k ¼ log d FPŁ (57) t D ¼ log Re lim log i lim The hypebola (o the staight lie) sepaates two aeas: hece the tem limitig cuve. O the left side of that limitig cuve the flow is lamia, ad theefoe, k, Ł ad Y ae costat (Figue 5). O the ight side of that limitig cuve the flow is ot lamia, ad k ( i), Ł (i) ad Y ( i) decease whe i iceases, with Y (i) defied by the followig equatio: Y Y (i) ¼ dfp k (i) ¼ (58) d FP Ł (i) t D Limitig cuve (hypebola) No-lamia flow aea Lamia flow aea log Y Limitig cuve (staight lie) No-lamia flow aea Lamia flow aea k () θ () i i Hydaulic coductivity, o hydaulic tasmissivity, Lamia flow i lim k o θ Hydaulic gadiet, i No-lamia flow (a) i lim (b) log i lim Figue 4. I a plot of the dimesioless paamete Y (defied by Equatio 56) agaist the hydaulic gadiet, i, limitig cuve sepaatig the aea whee flow is lamia fom the aea whee flow is ot lamia: (a) liea axes; (b) log-log axes Y Limitig cuve Yi ( lim ) Yi () log Y Limitig cuve Yi ( lim ) Yi () Figue. Typical cuve of appaet hydaulic coductivity, k ( i), o appaet hydaulic tasmissivity, Ł ( i), agaist hydaulic gadiet, i. Whe flow is lamia (i < i lim ), k ( i) k (costat) ad Ł ( i) Ł (costat) ad the cuve of hydaulic coductivity o tasmissivity is a hoizotal lie. (Note: No equatio has bee poposed fo the cuve beyod i i lim : but the shape of the cuve is deived fom the shape of the cuve i log-log axes, which is explaied afte Equatio 58 ad show i Figues 5, 7 ad 8.) Geosythetics Iteatioal, 2012, 19, No. 2 (a) i, i lim log i, log (b) Figue 5. Repesetatio of the dimesioless paamete Y ( i) (defied by Equatio 58) agaist the hydaulic gadiet, i: (a) liea axes; (b) log-log axes. The shape of the cuve is as explaied i Sectio 4.2., afte Equatio 58. The limitig cuve (dashed) sepaates the aea whee flow is lamia fom the aea whee flow is o-lamia (see Figue 4) i lim

11 170 Gioud, Gouc ad Kavazajia The shape (o, moe specifically, the cocavity) of the cuve of Y (i) o the ight side of the limitig cuve is as demostated by Gouc et al. (2012). The mai featues of the cuve of Y ( i) ae as follows. I a log-log plot, the slope of the cuve of Y ( i) is zeo (i.e. the cuve is hoizotal) fo i ¼ i lim : Theefoe thee is a smooth tasitio betwee the hoizotal staight lie fo lamia flow (i < i lim ) ad Y ( i) i a log-log plot. I a log-log plot, the slope of the cuve of Y ( i) is 0.5 whe i teds towad ifiity. This is half the slope of the limitig cuve, which is 1. Y Logaithmic scale A B Lamia flow aea Limit ( Re lim 1) betwee lamia ad o-lamia flow C D i lim (gavel) i lim (sad) Logaithmic scale Yi () fo a give gavel sample No-lamia flow aea Yi () fo a give sad sample The shape of the cuves of k (i) ad Ł (i) would be the same as the shape of the cuve of Y (i) show i Figue 5. Thee limitig cuves epeseted by staight lies i logaithmic scale ae show i Figue 6. These limitig cuves coespod to the values of the Reyolds umbe adopted i Sectio 4.1.: Figue 7. Schematic example of the dimesioless gaphical epesetatio of Y ( i) (defied by Equatio 58) fo two give samples of gaula dais (gavel ad sad). (Note: The shape of the cuves is as explaied i Sectio 4.2., afte Equatio 58.) The staight lie with Re lim ¼ 1 coespods to gaula dais. The staight lie with Re lim ¼ 5 coespods to eedle-puched owove geotextiles used as daiage layes. The staight lies with Re lim ¼ 1 ad Re lim ¼ 10 coespod to the two boudaies of the limit betwee lamia ad o-lamia flow fo geoets. Dimesioless paamete, Y Figue 7 illustates the use of the dimesioless gaph Uppe limit fo geoets Needle-puched owove geotextiles Gaula mateial ad lowe limit fo geoets 10 5 Limitig Reyolds umbe Limitig hydaulic gadiet, i lim Figue 6. Plot of the dimesioless paamete Y (defied by Equatio 56) agaist the limitig hydaulic gadiet, i lim, epesetig i logaithmic scale the limitig cuve coespodig to values of the limitig Reyolds umbe fo diffeet types of daiage mateials. (Dashed lies ae used fo cosistecy with Figues 4, 5, 7, 8 ad 1.) 1 Geosythetics Iteatioal, 2012, 19, No. 2 fo gaula daiage layes. The coditio fo lamia flow is epeseted by the dashed staight lie fo Re lim ¼ 1, the Reyolds umbe adopted fo gaula dais i Sectio Fou data poits (A to D), assumed to have bee obtaied i a laboatoy pemeability test o a give gavel sample, ae plotted o axes of Y ( i) agaist i, with Y (i) defied by Equatio 58. Data poits A ad B, assumed to be obtaied with hydaulic gadiets smalle tha the limitig hydaulic gadiet, coespod to lamia flow. Data poits C ad D, assumed to be obtaied with hydaulic gadiets geate tha the limitig hydaulic gadiet coespod, to olamia flow. If the hydaulic gadiet of the gavel has bee measued oly ude lamia flow coditios (data poits A o B) ad the gavel laye is used i the field with a hydaulic gadiet geate tha the limitig hydaulic gadiet (i.e. ude o-lamia flow coditios), the laboatoy-measued hydaulic coductivity is geate tha the hydaulic coductivity i the field, which is ucosevative. The same dimesioless gaph ca be used fo othe gaula daiage mateials. Fo example, the cuve fo a give sad sample is show i Figue 7 as a thi solid lie. This cuve is sigificatly lowe tha the cuve fo gavel, because both k ( i) ad d FP ae smalle fo sad tha fo gavel. The cuve of Y fo sad beig lowe tha the cuve of Y fo gavel, i lim is highe fo sad tha fo gavel. Theefoe, fo a give hydaulic gadiet, flow is moe likely to be lamia i sad tha i gavel. Figue 8 illustates the use of the dimesioless gaph fo geoets. The coditio fo lamia flow is epeseted by two dashed staight lies, oe fo Re lim ¼ 1 ad the othe fo Re lim ¼ 10, the lowe ad uppe boudaies fo the Reyolds umbe adopted fo geoet dais i Sectio Fou data poits (A to D), assumed to have bee obtaied i a laboatoy tasmissivity test o a geoet, ae plotted o axes of Y agaist i, with Y defied by Equatio 56. If the hydaulic tasmissivity of the geoet was meas-

12 Lamia ad o-lamia flow i geosythetic ad gaula dais 171 Logaithmic scale Y A B Lamia flow aea Uppe bouday of limit ( Re lim 10) betwee lamia ad o-lamia flow Rage of Lowe bouday of limit ( Re lim 1) betwee lamia ad o-lamia flow C Logaithmic scale D fo the cosideed geoet i lim Yi () fo a give geoet No-lamia flow aea Figue 8. Schematic example of the dimesioless gaphical epesetatio of Y ( i) (defied by Equatio 58) fo a geoet. Hee (cotay to the case of gaula mateials illustated i Figue 7) thee is some ucetaity egadig the exact locatio of the tasitio betwee lamia flow ad olamia flow: hece the uppe ad lowe boudaies. (Note: The shape of the cuve is as explaied i Sectio 4.2., afte Equatio 58.) i coditio is coceed, because the flow is still lamia at the limit betwee lamia flow ad o-lamia flow. May expessios ca be deived usig the appoach descibed above. Geeal examples give i the emaide of Sectio 4. ae expessios fo the two costats idetified i Sectio 4.2.2, (Q full /B) lim ad i lim, ad expessios fo the Reyolds umbe. Examples specific to cetai types of daiage mateials will be peseted i Sectio Ifluece of daiage mateial physical popeties o limitig flow capacity Combiig Equatios 12 ad 51 gives the followig expessio fo the limitig flow capacity: Q full B ¼ ð1 ÞtD S s Re lim (59) lim 4 This equatio makes it possible to evaluate the ifluece of the physical popeties of the daiage mateial (poosity, specific suface aea, ad laye thickess) o the limitig flow capacity. The value of the limitig Reyolds umbe to be used i this equatio is give i Sectio ued ude the same coditios but fo fou diffeet hydaulic gadiets, icludig data poits A ad B, with i A ad i B below i lim, ad data poits C ad D, with i C ad i D above i lim, the hydaulic tasmissivity values Ł A ad Ł B ae equal, as they coespod to lamia flow, wheeas Ł C ad Ł D ae smalle, as they coespod to o-lamia flow. Numeical applicatios peseted i Sectio 5 will show that i geeal, with geoets, i lim is vey small, ad as a esult the data poits ae geeally to the ight of the limit betwee lamia flow ad o-lamia flow. I othe wods, the hydaulic tasmissivity of geoets is geeally measued with hydaulic gadiets geate tha i lim : The gaphical epesetatio peseted i this sectio is mostly useful fo eseach puposes. It will be used i a slightly diffeet way to discuss hydaulic tasmissivity test esults fo geoets i Sectio Ifluece of physical popeties of daiage mateial o lamia flow coditio The appoach The ifluece of the physical popeties of the daiage mateial o the lamia flow coditio ca be evaluated by eplacig, i equatios givig the Reyolds umbe (Sectio 4.2.1) ad i expessios of the lamia flow coditio (see Equatio 49 i Sectio 4.2.2), d FP ad k by thei expessio as a fuctio of basic physical popeties of the daiage mateial, e.g. poosity, daiage laye thickess t D, ad specific suface aea S s. Expessios fo d FP as a fuctio of basic physical popeties ca be foud i Sectio 2, ad the expessio fo k as a fuctio of basic physical popeties is the classical Kozey Cama equatio (see Equatio 0 i Sectio..). The Kozey Cama equatio ca be used oly if the flow is lamia. This equiemet is satisfied as fa as the lamia flow 4... Ifluece of daiage mateial physical popeties o limitig hydaulic gadiet Combiig Equatios 12, 20, 0 ad 49 gives the followig expessio fo the limitig hydaulic gadiet: i lim ¼ Re lim 4â k g ¼ Re lim 4â k g e S s ðs s Þ ð Þ (60) This equatio makes it possible to evaluate the ifluece of the physical popeties of the daiage mateial (poosity, specific suface aea, ad laye thickess) o the limitig hydaulic gadiet. The value of the limitig Reyolds umbe to be used i this equatio is give i Sectio Combiig Equatios 12, 1 ad 49 (o combiig Equatios 12 ad 60) gives the followig expessio fo the limitig hydaulic gadiet: i lim ¼ Re lim â k g Geosythetics Iteatioal, 2012, 19, No ðd FP Þ (61) This equatio makes it possible to evaluate the ifluece of the aveage flow-path diamete o the lamia flow coditio Ifluece of daiage mateial physical popeties o Reyolds umbe I the case of lamia flow, the Reyolds umbe ca be calculated usig the followig equatio, obtaied by combiig Equatios 12, 0 ad 47 with k ( i) ¼ k: