A Modified Philip Dunne Infiltrometer for Measuring the Field-Saturated Hydraulic Conductivity of Surface Soil

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1 Published October 14, 2014 Origial Research A ew fallig-head ifiltrometer method is preseted for estimatig field-saturated hydraulic coductivity ad Gree Ampt wettig frot suctio for soil surfaces. The method was verified usig refereced laboratory colums, ad fallig-head data were geerated sythetically usig Richards equatio solutios for axisymmetric flow. A Modified Philip Due Ifiltrometer for Measurig the Field-Saturated Hydraulic Coductivity of Surface Soil F. Ahmed, R. Nestige, J.L. Nieber,* J.S. Gulliver, ad R.M. Hozalski There is a curret ad expadig eed to measure surface ifiltratio rate parameters for stormwater ifiltratio practices used to mitigate the detrimetal effects of lad developmet activities o watershed hydrology. We have developed a fallig-head soil surface ifiltrometer, termed the modified Philip Due (MPD) ifiltrometer, that is iexpesive to costruct, easy to use, ad requires a miimal water volume per test. Because of these characteristics, may MPD devices ca be deployed simultaeously to obtai ifiltratio rate data at multiple locatios withi a give ifiltratio practice. Gree Ampt theory was used to derive the expressios eeded for aalyzig the fallig-head data to solve for the field-saturated hydraulic coductivity (K fs ) ad Gree Ampt wettig-frot suctio (y). The accuracy of the aalysis was determied usig umerical experimets i which fallig-head data were geerated from a computatioal solutio of the axisymmetric form of the three-dimesioal Richards equatio for homogeeous ad isotropic porous media with specified iput parameters. The fallig-head data were the aalyzed usig a quasi-aalytical procedure, ad the resultig values of K fs ad y were compared with the iput values. The accuracy of K fs ad y derived from data acquired usig the MPD device was the assessed usig physical experimets ivolvig three large barrels packed with differet types of sad. The K fs values obtaied for the media i the barrels usig a MPD ifiltrometer were, o average, 82% of the values obtaied from whole barrel fallig-head tests. The resultig ucertaity i K fs values from the MPD ifiltrometer is cosidered to be small compared with the orders of magitude of variability commoly observed for K fs values i the field. Abbreviatios: MPD, modified Philip Due. F. Ahmed, J.S. Gulliver, ad R.M. Hozalski, Dep. of Civil, Evirometal ad Geo-egieerig, Uiv. of Miesota, Mieapolis, MN 55414; R. Nestige, Short Eliott Hedrickso, Ic., 3535 Vadais Ceter Drive, St. Paul, MN 55110; ad J.L. Nieber, Dep. of Bioproduct ad Biosystem Egieerig, Uiv. of Miesota, St. Paul, MN *Correspodig author (ieber@um.edu). Vadose Zoe J. doi: /vzj Received 30 Ja Supplemetal material olie. Soil Sciece Society of America 5585 Guilford Rd., Madiso, WI USA. All rights reserved. No part of this periodical may be reproduced or trasmitted i ay form or by ay meas, electroic or mechaical, icludig photocopyig, recordig, or ay iformatio storage ad retrieval system, without permissio i writig from the publisher. Ifiltratio basis, rai gardes, swales, ad other ifiltratio practices are stormwater cotrol measures that reduce ruoff volume through meas of ifiltratio ad evapotraspiratio, of which ifiltratio is the most sigificat. If ifiltratio is ot occurrig at a sufficiet rate, the capacity to reduce ruoff volume is decreased, potetially leadig to icreased pollutat discharge, icreased degradatio of stream chaels, ad icreased potetial for floodig of dowstream areas. The surfaces of ifiltratio practices are proe to compactio (Olso et al., 2013) due to foot ad equipmet traffic as well as cloggig from the fie particles typically trasported i stormwater ruoff. To evaluate the impacts of compactio ad particle accumulatio, it is importat to measure the field-saturated hydraulic coductivity of the media at the surface because it is the mai determiat of the ifiltratio capacity of a particular soil. Field-saturated hydraulic coductivity (K fs ) is the most importat soil property that cotrols water ifiltratio ad cosequetly surface ruoff. Methods to determie this soil property ca be placed ito two mai categories: methods based o steady-state ifiltratio ad methods based o usteady ifiltratio. Withi these categories there are subcategories that relate to the type of geometric ad pressure boudary coditios imposed. The Vadose Zoe Joural

2 commo types of geometries iclude boreholes, surface disks, ad surface rigs. Pressures applied at the soil boudary ca be positive, zero, or egative. Of iterest i this study is a type of measuremet that will yield the hydraulic properties of the soil surface. For this, several methods have bee developed, with water applicatio geometries that iclude disk ifiltrometers, sigle-rig ifiltrometers, ad double-rig ifiltrometers (ASTM, 2003) applied for steady-state flow or for usteady flow (Parr ad Bertrad, 1960; Reyolds ad Elrick, 1990; Bagarello et al., 2004; Lassabatere et al., 2006; Reyolds, 2008; Nimmo et al., 2009). It has bee foud that, both for field soils ad for ifiltratio practices, the K fs of a medium ca vary spatially up to two orders of magitude (Asleso et al., 2009; Olso et al., 2010, 2013). To capture the spatial variability of K fs ad to determie a represetative ifiltratio capacity, a large umber of measuremets are required to represet field coditios. This situatio calls for measuremet methods that are quick, require the least amout of water, ad are relatively easy to set up. To meet time costraits o projects, it is also desirable if the measuremets ca be made simultaeously at multiple sites by a sigle practitioer. This almost certaily requires that the methods ivolve measuremet of short-term usteady ifiltratio. To meet this data collectio requiremet for our ow eeds of data collectio, we developed ad tested a modified versio of the Philip Due borehole permeameter (referred to as the modified Philip Due ifiltrometer or MPD ifiltrometer). The device is simple, iexpesive to costruct, ad has low water volume requiremets per test; hece, 20 or more devices ca be deployed to measure the soil hydraulic parameters at as may differet locatios simultaeously. The MPD ifiltrometer requires?0.003 m 3 of water per test, while the most commoly used device, the double-rig ifiltrometer, requires?0.028 m 3 of water (Ahmed et al., 2011). The fallig-head Philip Due permeameter is iserted ito a borehole to a give depth ad is used to obtai the K fs ad Gree Ampt wettig-frot suctio (y) of the soil at that depth. It caot be used to measure K fs ad y at the soil surface of a ifiltratio practice. I cotrast, the MPD ifiltrometer is ot iserted ito a borehole but is drive ito the soil surface to a specified depth without removig ay soil. Because this modificatio chages the boudary coditios applied to the ifiltratig flow compared with the coditios associated with the Philip Due borehole permeameter, it is ot possible to use the approximate borehole ifiltratio aalysis of Philip (1993) to derive the hydraulic properties of the soil. Philip s aalysis, however, ca be modified to arrive at a similar approach for aalyzig the head vs. time data collected from the MPD ifiltrometer. Three fairly recet studies that describe alterative approaches to the oe we preset here are those by Bagarello et al. (2004), Lassabatere et al. (2006), ad Nimmo et al. (2009). The approach of Bagarello et al. (2004) applies the oe-dimesioal Gree Ampt formulatio of Philip (1992) to aalyze ifiltratio ito a iserted rig ifiltrometer. The approach of Lassabatere et al. (2006) fits a two-term ifiltratio formula to cumulative ifiltratio measuremets to derive soil hydraulic property scale parameters ad also a pedotrasfer fuctio approach with particle size distributio to derive shape parameters to describe the porous medium water retetio ad usaturated hydraulic coductivity fuctios. The approach of Nimmo et al. (2009) uses a sigle-rig ifiltrometer iserted ito the soil ad measures the time required for poded water to ifiltrate ito the soil. The data aalysis with this approach ivolves a sigle simple formula for field-saturated hydraulic coductivity. The formula cotais a capillarity parameter, but this is assumed to be kow or ca be estimated for a particular soil type. Of these three methods the oe of Nimmo et al. (2009) appears to be the closest, i terms of simplicity ad ease of use, to the techique preseted here. The accuracy of the MPD ifiltrometer, which was used to measure K fs ad y, was verified usig umerical experimets ad physical laboratory experimets. 6Theory 6 I our applicatio, the MPD ifiltrometer was a 0.1-m-diameter cylider that is drive 0.05 m ito the soil, although these dimesios are free to be selected by the user. The iitial moisture cotet of the soil ear the surface is measured, ad this iitial moisture cotet is assumed to represet the iitial moisture cotet of the uderlyig soil profile. The ifiltrometer is the filled with water to a specified level, ad the water level i the cylider is moitored with time. The test cotiues util sufficiet measuremets of water surface elevatio vs. time have bee take to estimate K fs, usually util the water has completely emptied out of the ifiltrometer cylider. The aalysis of Philip (1993) was based o the assumptios of the Gree Ampt model. The soil was assumed to be a isotropic ad homogeeous porous medium ad the wettig frot to be sharp ad to represet the three-dimesioal flow that was assumed as a ideal spherical geometry for the wettig frot by cosiderig symmetrical pressure-capillarity flow ad superimposig a symmetrical gravity flow. He foud that havig a spherical source geometry had little ifluece o estimates of K fs ad y, which was of primary cocer i our aalysis. A similar approach was take for the aalysis of the MPD ifiltrometer. However, due to the applicatio of the device at the surface rather tha i a borehole, the o-flow boudary at the soil surface outside of the cylider was take ito accout by represetig the wetted soil as a capped sphere, as illustrated i Fig. 1. I additio to modifyig the geometry of flow, the pressure loss alog the soil ecased withi the iserted portio of the device eeded to be added ito the aalysis. Vadose Zoe Joural p. 2 of 14

3 Fig. 1. Compariso of assumed wetted cross-sectios for the Philip Due permeameter ad the modified Philip Due ifiltrometer. Equatio Modificatios The derivatio of the goverig equatios for the MPD ifiltrometer was based o the work of Nestige (2007). The otatio used i the derivatio is illustrated i Fig. 2. Usig similar assumptios as Philip (1993), the equatio for cumulative ifiltratio, i(t), is expressed by a equatio usig the geometry of the wettig frot, which is a capped sphere with a radius of R(t) ad cetroid at a vertical distace of L max from the soil surface at the ceter of the cylider R(t) + L max. The soil withi the capped sphere has iitial ad field saturated moisture cotets of q i ad q s, respectively. The total volume of the wetted soil matrix is calculated by subtractig the volume of the equivalet spherical source (4/3)pr o 3, where r o = r 1 /2 is the radius of a equivalet spherical source, from the volume of the capped sphere defied by the advacig wettig frot. The volume of the capped sphere bouded by the wettig frot is (p/3)[r(t) + L max ] 2 {3R 2 (t) [R(t) + L max ]}, where R(t) is the sphere radius to the wettig frot ad L max is the distace that the ifiltrometer peetrates the soil. Thus the equatio for temporal cumulative ifiltratio is i( t) = p ( qs -q ) é i 2R 3 ( t) + 3R 2 ( t) Lmax -L 3 max -4ro 3ù 3 êë úû A mass balace of the water remaiig i the ifiltrometer ad the water that has ifiltrated ito the soil at a give time is used to compute R(t) as a fuctio of H(t) for use i the aalysis, replacig i(t) with [H i H(t)]pr 1 2, where H i is the iitial height ad H(t) is the depth of water i the cylider above the soil surface with time. With this we ca ow rewrite Eq. [1] to establish a relatio betwee H(t) ad R(t): é 2 Hi H( t) ù ë - ûp r1 = p ( qs -q i ) é2r ( t) + 3R ( t) Lmax -Lmax -4r ù o 3 êë úû [1] [2] Fig. 2. Importat parameters of the modified Philip Due ifiltrometer, icludig the iitial height of water (H i ), the height of water at time t [H(t)], the depth of isertio ito the soil (L max ), the equivalet source radius (r 0 ), the radius of the cylider (r 1 ), ay radius withi the wetted frot (r), ad the radius to the sharp wetted frot at time t [R(t)]. Equatios [1] ad [2] are applicable oly after R(t) is greater tha the value Ö(r L max 2 ). The head vs. time data before the wetted zoe reaches this miimum radius are eglected from the aalysis because this is a poit where the geometric shape for the problem becomes costat for the remaider of the experimet. Up to that poit, the geometry has chaged from oe-dimesioal flow alog the ecased cylider of soil, ad the to a sphere that grew util the top part of the sphere itersected with the soil surface ad the geometry became that of a capped sphere. It would be possible to accout for these itermediate chagig geometries withi the aalysis if the equatio set were expaded, but for the preset the aalysis we have limited ourselves to the formulatio usig Eq. [2]. Followig the aalysis procedure of Philip (1993), which ivolves differetiatig Eq. [2] with respect to time, the separatig the velocity ito two compoets, a pressure-capillary-drive flow ad a gravity-drive flow, the pressure-capillary flow velocity compoet, v c (r) at r betwee r o ad R(t) becomes æ 2 2r ö o vc() r = v co 2 ç èr + rl max ø 2 dr 2 = {( q ) ( ) ( ) } s -q é i R t RtL ù ê + max -2rK ë úû dt o fs 1 2 r + rl max [3] Vadose Zoe Joural p. 3 of 14

4 where the gravity-drive flow term is give by 2r 2 o K fs. Applyig Darcy s law, the pressure-capillarity potetial drop, DP, from the Rt () spherical source to the wetted frot is give by DP = b ò r [v o c (r)/ K fs ]dr, which ca be evaluated to give Rt () vc() r D P=bò d r ro K fs é 2 R( t) + RtL ( ) max dr ù =b ( s i) B G q -q - Kfs dt êë úû where b is a coefficiet that takes ito accout the hydraulic iefficiecy of the actual flow path of ifiltrated water ito the soil, B = (1/ L max )(l{r(t)(r o + L max )/r o [R(t) + L max ]}), ad G = 2r o 2 B is the term arisig from the gravity-drive compoet of the flow. Through a exploratory aalysis, Philip (1993) estimated the b coefficiet to be p 2 /8, which is used here. The use of G for the gravity term follows from the aalysis of Reyolds (2011), who showed that, at t = 0 for the Philip Due borehole permeameter, the gravity term is equal to zero, while at large time (if the volume of water i the permeameter tube is ulimited), the gravity term coverges o r o. For the MPD ifiltrometer, the gravity term is zero at t = 0. while at large time the term coverges o 2r o 2 /L max l(1 + L max /r o ). To calculate the pressure, P o (t), at the surface of the spherical source, it is ecessary to accout for the pressure loss due to flow i the cylider of soil ecased withi the ifiltrometer. This loss is represeted with Darcy s law, for which the flux alog the cylider of soil is df ædp ö q=- Kfs =- Kfsç + 1 dz çèdz ø [4] [5a] where F is the total potetial withi the cylider of soil, ad z is the coordiate alog the legth of the cylider, positive upward with the origi at the soil surface. This flux is also kow from the rate of drop i the water level i the ifiltrometer reservoir, that is, dh( t) q =- [5b] dt Equatig these two ad itegratig with z, the pressure at the spherical source is potetial drop from the spherical source to the wetted frot ca thus be described by D Lmax dh( t) P =y- H( t) - Lmax + K dt [6] fs where y is the Gree Ampt wettig-frot suctio for the usaturated soil. The Gree Ampt wettig-frot suctio is defied as 0 K( h) y=ò dh [7] hi K fs where h i is the iitial pressure head ad K(h) is the usaturated hydraulic coductivity, which is a fuctio of the pressure head h. By equatig Eq. [4] ad [6], we get the followig two equatios that ca be used to simulate the temporal variatios i the podig depth i the ifiltrometer for a give combiatio of the ifiltrometer geometry ad soil properties (K fs ad y): 2 K ì é fs R( t) + RtL ( ) max dr ù dh = ï íb ( qs-qi) B-G L max Kfs dt ïî êë úû -y + H( t) + L } dt dt = max { } ( ) ( ) é 2 b q ( ) ( ) ù ( ) é ù s -q i êr t+ RtL ú -ê D ë max K û fslmax BdR L ë max Kfs Hú û [ 9] y- ( )- + é 2 H t L b ù max ê ë ( 2ro Lmax ) B ú û Equatios [8] ad [9] are just differet rearragemets of the equatio resultig from equatig Eq. [4] ad [6], ad each ca be used i a umerical scheme to optimize for K fs ad y give a time series of podig depth measuremets. Equatios [8] ad [9] are discretized i a implicit formulatio alog H ad t, respectively, to facilitate the calculatio of measuremets of head at give times or the calculatio of time at give heads. The discretized forms of the equatios are H -1 - H = æ 2 ( ) ( ) 1 K ì ér t ù ü R t L - fs + max ær R ö ê ( s i) ë úû - bï í q -q B -Gï Lmax K ý [10] fs èç Dt ø ç è ïî ïþ -1 -y + H + L Dt max ) [8] Lmax dh( t) P( t) = H( t) + Lmax - K dt [5c] fs Equatio [5c] is altered from Philip s aalysis to accout for the oe-dimesioal movemet of water through the distace L max of the MPD ifiltrometer ad to accout for the geometry of the capped sphere of wetted soil. The total pressure-capillarity -1 t - t = 2-1 { } ( ) æ ( ) ( ) ö b q ( ) s -q i ç êr t ú + R t L ë û max KfsLmax è B ø R -R y- - + é 2 H L b ù max ê ë ( 2ro Lmax ) B ú û ( Lmax Kfs ) DH - y- - + é 2 H L b ù max ê ë ( 2ro Lmax ) B ú û [11] Vadose Zoe Joural p. 4 of 14

5 where ad 1 represet the preset ad previous time steps, respectively, Dt is the time icremet, ad DH is the differece betwee the previous ad preset water levels. I the computatioal procedure, R(t) i Eq. [10] ad [11] is computed usig Eq. [2] ad the measured head i the ifiltrometer vs. time data. Equatios [10] ad [11] ca each be applied to simulate observed time series of podig depth vs. time data ad do this i a iterative maer to fid the optimum combiatio of K fs ad y that yields the best agreemet betwee the observed ad simulated podig depth time series. The formulatio leadig to Eq. [8] ad [9] is based o the approach preseted by Philip (1993) i which it was assumed that the flow from the ifiltrometer is drive by a combiatio of pressure (H) ad capillarity (y), perturbed by a symmetric gravity flow. The symmetry of the gravity flow does ot mea that the gravity compoet is zero. It is zero at the begiig but the approaches the full dowward gravity flow described above [2br o 2 l(1 + L max /r o )]. This type of behavior is well kow from ifiltratio theory (Philip, 1969). With that explaatio, we poit out that Cheg et al. (2011) preseted a formulatio of the MPD method, which they referred to as the modified Nestige (MN) method because they derived their aalysis startig from the work of Nestige (2007). I derivig their cumulative ifiltratio equatio, they did ot use the Philip (1993) formulatio with the perturbed flow, but rather they accouted for gravity flow by addig costat flow give by K fs (t t o ), where t o is the time at which the wettig frot reaches the base of the ifiltrometer tube ad the three-dimesioal flow begis. The additio of this term is ot cosistet with the origial Philip (1993) formulatio based o the Gree Ampt equatio. The effect of icorporatig gravity i such a ad hoc maer o the accuracy of their resultig formulatio remais to be determied. They also stated that the mass balace equatio for the Nestige formulatio eglected the volume of water ifiltrated ito the ecased cylidrical soil sectio, give by L max Dq. This claim is ot correct, though, because that volume of water is fully accouted for i our equatio for the capped sphere. 6 Materials ad Methods To determie the accuracy of the derived equatios for the MPD ifiltrometer, fallig-head data were geerated by solvig the axisymmetric form of the Richards equatio for a series of simulated ifiltratio experimets for differet soils. The MPD ifiltrometer Eq. [2], [10], ad [11] were the applied to estimate the hydraulic properties of the soils from the simulated fallig-head data. Compariso of the soil hydraulic parameters used i the Richards equatio simulatios with the optimized parameters from the fittig with Eq. [2], [10], ad [11] was the used to assess the accuracy of the MPD aalysis. The validity of the MPD device ad associated aalysis was assessed usig laboratory experimets ivolvig three large barrels packed with differet types of sad. Numerical Simulatios A umerical solutio of Richards equatio was used to provide the fallig-head data for evaluatio of the aalytical method described above. Because the hydraulic parameters are iputs to the umerical solutio, there is a exact way of assessig the accuracy of the approximate method. I this study, Richards equatio was solved with the commercial fiite elemet equatio solver cotaied withi the COMSOL Multiphysics software package (COMSOL, 2013). Oe compariso of the MPD ifiltrometer with the Philip Due borehole permeameter was coducted to illustrate some differeces i respose betwee the two, but most of the aalysis focused o the MPD cofiguratio. Boudary Coditios Costraits o the solutio of Richards equatio are the iitial coditio, which is iitial pressure or iitial saturatio, ad boudary coditios, either specified pressure or specified flux. For the borehole (Philip Due borehole permeameter) domai illustrated i Fig. 3a, the iitial coditio is oe of uiform iitial pressure, ad the boudary coditios for the idividual boudary segmets are give by the followig: AB: h = water depth(t) iside the permeameter BC, CD: (h + z)/ DE: h i = iitial soil water pressure EF: (h + z)/ = 1 AF = (h + z)/ r = 0 due to axial symmetry where is the uit ormal vector to the boudary, ad r ad z represet radial ad vertical directio, respectively. For the ifiltrometer (MPD ifiltrometer) domai illustrated i Fig. 3b, the iitial coditios are the same as for the borehole Fig. 3. Illustratios of the axisymmetric domai for (a) the Philip Due permeameter ad (b) the modified Philip Due (MPD) ifiltrometer. Vadose Zoe Joural p. 5 of 14

6 domai. The boudary coditios for the ifiltrometer domai are the followig: AB: h = water depth(t) iside the ifiltrometer BB² = B²C = CD: (h + z)/ = 0 DE: h i = iitial soil water pressure EF: (h + z)/ = 1 AF: (h + z)/ r = 0 due to axial symmetry For the two domais, the coditio for Boudary EF is for the assumptio that a uit hydraulic gradiet exists at the bottom boudary. The coditios for Boudaries DE ad EF are sufficiet as log as the wettig frot does ot reach the boudaries durig the ifiltratio evet. Details cocerig the solutio domai ad the COMSOL solver are preseted i the supplemetal material. Descriptio of Iput Parameters For the umerical simulatios, five types of soil that are amog the most commo types foud i ifiltratio practices were chose. The soil types are loamy sad, sady clay loam, silt loam, sady clay, ad silty clay. The hydraulic properties of the soils were defied i terms of va Geuchte (1980) parameters. The value of H i was kept costat at 0.43 m for all umerical simulatios, while the hydraulic properties of the soil icluded the field saturated volumetric soil moisture cotet q s, saturated hydraulic coductivity K sat, ad the va Geuchte parameters a ad were withi oe stadard deviatio of their respective mea values for the five soil types as described by Carsel ad Parrish (1988). I these simulatios, the iitial soil water pressure was set to yield moderate to full wettig frot potetial, as calculated from Eq. [7], for each type of soil, while Dq was limited to relatively small for the fiest soil (silty clay) because, i the umerical simulatios for fie soil, a very low soil water pressure eeds to be set for high Dq (dry coditio), which sometimes ca lead to makig the umerical simulatios difficult to complete due to ocovergece of solutios. Aother set of simulatios was also performed with the solutio to the Richards equatio for the case where was kept costat at 4.0 while a ad K sat were varied usig liear scalig theory (Vogel et al., 1991). Accordig to this theory, the value of a varies i direct proportio to the scalig factor g: a = a ref g, while the value of K sat varies with g 2 : K sat = K sat_ref g 2, where a ref = 4 ad K sat_ref = m/s are the referece values of a ad K sat, respectively. The iitial water depth iside the ifiltrometer was the same as i the previous simulatios, but the iitial moisture cotet ad saturated moisture cotet were set to be ad 0.375, respectively, thus makig Dq equal to 0.32 for all cases. The procedure for each simulatio was as follows for all of the soils described above. A set of values for q s, q r, a, K sat, ad h i was defied (Tables 1 ad 2) for use as iputs to the COMSOL-MP solver. Richards equatio (Richards, 1931) was the solved for the period whe the water level i the MPD ifiltrometer was above zero. Usig the head vs. time curve produced by the simulatio, the defied chage i moisture, ad the geometry of the ifiltrometer, the modified aalytical equatios were tested usig the MPD aalysis procedure described above to produce values of K fs ad y. These parameters were the compared with the values used as iputs for the simulatio. The K sat values used as iput for the simulatio ad the K fs values determied from the MPD aalysis are termed K eff ad K MPD, respectively. The value of y determied usig the selected values for the va Geuchte parameters i Eq. [7] ad determied from the MPD aalysis are termed y eff ad MPD, respectively; y eff is determied by applyig Eq. [7]. Aalysis Procedure A computatioal spreadsheet procedure i MS Excel with the Solver add-i ad Visual Basic applicatio was developed to fid solutios to Eq. [2], [10], ad [11] ad obtai optimal values of K fs ad y. The geeral procedure for fidig values of K fs ad y from the head vs. time data is as follows: 1. Iput all variables, icludig iitial moisture cotet, field saturated moisture cotet, iitial height, ad the head vs. time curve. 2. For each measuremet of head, use the relatioship i Eq. [2] to fid the correspodig distace of the sharp wettig frot (ote: Solver i Microsoft Excel 2010 ad a macro were used to automate this step). 3. Estimate the chage i head with respect to time ad the chage i wettig frot distace with respect to time by usig the forward fiite differece method for all values of R(t) equal to or greater tha the distace Ö(r L max 2 ). 4. Make iitial guesses for the values of K fs ad y. By default, the iitial guess for K fs ad y are set as cm/s ad 100 cm, respectively. We foud that for a fier soil that has a K fs value of < cm/s, the iitial guess of y might eed to be chaged to improve the covergece rate. 5. Solve Eq. [10] ad [11] for DH ad Dt at each icremetal value of R(t). 6. Miimize the absolute differeces betwee DH foud i Step 5 ad chage i the measured head ad betwee Dt ad the measured time iterval by iteratig the values of K fs ad y. Betwee these two optimizatio procedures (DH ad Dt), the oe with the miimum RMSE betwee measured ad estimated data was chose to calculate K fs ad y. Occasioally, oe of the two does ot coverge, which is the primary reaso that the dual fittig procedure is recommeded. Experimets Experimet Setup Three barrels with a diameter of 0.56 m, height of 0.91 m, ad volume of m 3 (Greif, Ic.) were chose as vessels for the calibratio media. Each barrel was fitted with a threaded polyviyl chloride valve alog the side ear the bottom that allowed the water to drai. A thi coatig of sad was attached to the Vadose Zoe Joural p. 6 of 14

7 Table 1. Compariso of field-saturated hydraulic coductivity (K fs ) ad Gree Ampt wettig-frot suctio (y) values determied from the modified Philip Due (MPD) aalysis of the simulated fallig-head data with the values used as iputs to the COMSOL simulatios for the case of variable (K eff ad y eff, respectively) ad the iitial soil water pressure i the umerical simulatios (h i ), residual ad iitial soil moisture cotet (q r ad q i, respectively), ad va Geuchte parameters a ad. The pressure parameter h s i the Vogel et al. (2000) modified va Geuchte equatios was set to 0.04 m for all simulatios show here. Soil type a h i q r q i K MPD K eff K MPD / K eff y MPD y eff y MPD /y eff m 1 m % m /s m Loamy sad Sady clay loam Silt loam Sady clay Silty clay Table 2. Compariso of field saturated hydraulic coductivity (K fs ) ad Gree Ampt wettig-frot suctio (y) values determied from the modified Philip Due (MPD) aalysis of the simulated fallig head data with the values used as iputs to the COMSOL simulatios for the case of costat (K eff ad y eff, respectively) ad the iitial soil water pressure i the umerical simulatios (h i ), residual ad iitial soil moisture cotet (q r ad q i, respectively), ad va Geuchte parameters a ad. The parameters a ad K eff were derived from liear scalig theory (Vogel et al., 1991), with Case 2 beig the referece coditio. Case a h i q r q i K MPD K eff K MPD / K eff y MPD y eff y MPD /y eff m 1 m % m /s m ier walls of the barrels with a spray adhesive to roughe the surface ad miimize the potetial for preferetial flow of water alog the walls. A m layer of pea gravel (media diameter = m) was placed at the bottom of the barrel ad covered with a coarse filter fabric to isolate the gravel from the medium above. Sad media of three particle size distributios were added to the three barrels over the filter fabric to a height of 0.51 m, stoppig (roughly) every 0.05 m to tamp dow the sad to prevet large voids ad o-uiform compactio. The three sad media used i the experimetal testig were: (i) 100% ASTM C-33 sad (Barrel 1), (ii) 80% (w/w) ASTM C-33 sad with 20% US Silica F110 sad (Barrel 2), ad (iii) 100% US Silica F110 sad (Barrel 3). The media were selected to represet a rage of relatively high permeability egieered soils used i bioretetio facilities ad other ifiltratio practices. For example, Wiogradoff (2002) recommeds usig 50 to 60% clea ASTM C-33 costructio sad with 20 to 30% sady loam or loamy sad ad 20 to 30% leaf compost material for a soil medium. Other mauals recommed similar mixes. The compost was omitted from our media to achieve homogeeous mixtures that would ot chage with time due to dissolutio or degradatio of the orgaic material. The sad media were mixed i a portable mortar mixer before additio to the barrels. The particle size distributio for each sad mixture was determied by a sieve aalysis (ASTM, 2006) ad is give i Fig. 4. Modified Philip Due Ifiltrometer Tests The MPD ifiltrometer was iserted 5 cm ito the surface of the soil ear the ceter of the barrel. Iitial soil moisture measuremets were made from five locatios aroud the outside edge of the ifiltrometer at the soil surface. The iitial soil moisture cotet was assumed to be uiform for the whole media. These measuremets were made either gravimetrically (Garder, 1986; ASTM, 2000, 2005) or with a calibrated moisture probe (Theta Probe ML2x). The temperature of the water used to fill the ifiltrometer was measured. The MPD ifiltrometer was the filled to a height of 0.43 m with the water. The head of water with time Vadose Zoe Joural p. 7 of 14

8 durig the test was recorded at a rate of six readigs per miute with a ultrasoic sesor (MassaSoic M-5000) mouted above the device. Immediately after the water had completely draied from the ifiltrometer tube, the ifiltrometer was removed from the barrel ad five fial moisture cotet measuremets were made. Eleve, 17, ad 19 idepedet tests were coducted with the MPD ifiltrometer o Barrels 1, 2, ad 3, respectively. Referece Fallig-Head Tests To perform a referece fallig-head test, the barrel was filled at a approximate flow rate of L/s from a hose coected to a valve opeed ito the pea gravel layer at the bottom of the barrel. This method of fillig the barrels from the bottom up at low flow was used to miimize the amout of etrapped air i the soil voids. The flow rate durig fillig was maitaied below that required to fluidize the sad so as ot to disturb the bed. Whe the water level was approximately 0.2 m above the sad surface, the valve was closed ad the hose was discoected. A ultrasoic sesor was the mouted to the top of the barrel. The valve at the bottom was opeed ad the head vs. time data were recorded. The aalysis for the referece fallig-head test was similar to the aalysis of a fallig-head laboratory permeameter. I the case of a fallig head, the flow ad hydraulic gradiet are both time depedet. Darcy s law (Klute ad Dirkse, 1986) is used to calculate the saturated hydraulic coductivity K sat accordig to K sat L æ Ti + L ö = D t çèt + L ø i+ 1 Fig. 4. Particle size distributios of the three media used for ifiltratio testig. [12] where U K_MPD is the ucertaity of the K fs value determied by the MPD aalysis for each medium, U K_ref is the ucertaity of the K sat value determied by referece fallig-head aalysis for each medium, K MPD is the mea value of K fs determied by the MPD aalysis for each medium, ad K is the mea value of K ref sat determied by the referece fallig-head aalysis for each medium. The values of U K_MPD ad U K_ref are determied by U K_MPD ad s = ts N K_MPD MPD where L is the legth of the soil colum, ad T i ad T i+1 are poded-head depths at the begiig ad ed of the time iterval Dt. The barrels were coditioed for these tests by performig fillig ad draiig i the same maer as described above approximately seve times before begiig the referece fallig-head tests. Twety-five, 20, ad 21 referece fallig-head tests were performed o Barrels 1, 2, ad 3, respectively. Referece Fallig-Head Tests vs. Modified Philip Due Ifiltrometer The mea K fs for each medium determied by the MPD ifiltrometer usig the MPD aalysis procedure ( K ) was compared MPD with the mea K sat of the referece fallig-head tests for the same medium ( K ). The ucertaity, U ref r, of the ratio K R = KMPD / K is defied by followig the ASME stadard techique (Aberethy et al., ref 1985): 2 2 æ K ö æ R K ö R R = K_MPD K_ref + ç è K MPD ø çè K ref ø U U U 2 2 æ 1 ö æk ö MPD U K_MPD U 2 K_ref ç èk ref ø çè K ref ø = + [13] U K_ref s = ts N K_ref ref where s K_MPD ad s K_ref represet the stadard deviatio of MPD measuremets ad referece fallig-head measuremets, respectively, t s is the Studet s t value, ad N MPD ad N ref represet the umber of measuremets for MPD aalysis ad referece fallig-head aalysis, respectively. 6 Results ad Discussio Richards Equatio Simulatios Borehole Permeameter vs. Modified Philip Due Ifiltrometer It is istructive to compare computer simulatios of ifiltratio ito a homogeeous ad isotropic soil for the two cases, oe for the Philip-Due permeameter ad oe for the MPD ifiltrometer, just to show the differeces i ifiltratio characteristics. For this, a soil with the followig soil moisture characteristics was used: q s = 0.375, q r = 0.05, a = 4 m 1, = 4, ad K sat = m/s. The iitial water pressure was set to 1.0 m, which gives a Vadose Zoe Joural p. 8 of 14

9 correspodig iitial moisture cotet of The simulatio result for a 0.05-m-deep, 0.05-m-radius borehole (Philip Due permeameter) is show i Fig. 5. The results for the MPD ifiltrometer with a 0.05-m radius peetrated to the 0.05-m depth are show i Fig. 6. Both plots show the moisture distributio i the soil surroudig the ifiltratio surface at the momet that the podig i the tube becomes zero. There are some similarities i the geometry of the flow for both cases, but there are also some differeces. Primarily, the water i the MPD ifiltrometer is forced to pass oe-dimesioally through the 0.05-m-log soil core before allowig for three-dimesioal flow i the soil beeath ad aroud the ifiltrometer. This flow costrait for the MPD ifiltrometer results i a reduced water pressure at the ed of the tube compared with a Philip Due permeameter due to the pressure loss withi the soil core, ad this the results i a loger time for the ifiltrometer tube to empty ad a more diffusive wettig frot tha for the permeameter. To illustrate the effect of borehole depth ad ifiltrometer peetratio depth o the time variatio of water height iside the permeameter ad ifiltrometer tubes, simulatios for a few differet peetratio depths were coducted usig Richards equatio for the same iitial coditio as described above; the results of these are preseted i Fig. 7. It ca be observed from the plots that the borehole depth for the permeameter does ot affect the rate of water height decrease for borehole depths i the rage betwee 0.15 ad 0.05 m. The depth of the borehole does begi to ifluece the ifiltratio rate at 0.02 m ad eve more so for surface applicatio. This ifluece of borehole depth results from the fact that whe the wettig frot reaches the soil surface, which it would do for the shallower boreholes, the rate of ifiltratio is restricted because there is less volume of soil to be ivaded by the advacig frot. Ifiltratio rates for the MPD ifiltrometer cofiguratio are much slower tha for the permeameter at equivalet borehole or peetratio depths. Also, it ca be observed that the time required for emptyig of the iitially filled volume for the case with a 0.05-m tube peetratio is more tha twice the time required for the case of surface applicatio ad about 30% more time tha for the case of 0.02-m peetratio. This results from the pressure loss that occurs i trasmittig the water through the ecased soil volume as opposed to a ope borehole. Simulatios of the Modified Philip Due Ifiltrometer for Various Soils The iput parameters for the soils where the value of the va Geuchte parameters (a ad ) were reported by Carsel ad Parrish (1988) for the five soil textures of loamy sad, sady clay loam, silt loam, sady clay, ad silty clay are preseted i Table 1, alog with the correspodig K fs ad y values estimated from the MPD aalysis procedure for each texture. The iput parameters for the soils with parameters derived from liear scalig theory are Fig. 5. Soil moisture cotet at 380 s for the Philip Due permeameter at the ed of the simulatio period whe the permeameter tube has emptied. Fig. 6. Soil moisture cotet at 1200 s for the modified Philip Due ifiltrometer at the ed of the simulatio period whe the ifiltrometer tube has emptied. Vadose Zoe Joural p. 9 of 14

10 Fig. 7. Water depth i the Philip-Due (PD) permeameter or modified Philip Due (MPD) ifiltrometer vs. time. Iitial water depth = 0.43 m. preseted i Table 2, ad agai the K fs ad y values estimated from the MPD aalysis procedure correspodig to each soil are also give i Table 2. Note that all the soils with the scaled parameters had the parameter set equal to 4. The K sat values that were iput i the umerical simulatios are termed K eff for both Tables 1 ad 2. Accordig to the results i Table 1 ad Fig. 8, the MPD aalysis procedure overestimated K fs by 10% (silty clay, a = 0.4, = 1.1) to 36% (sady clay, a = 0.9, = 1.2) for the soil cases examied. The variatio of the K MPD /K eff ratio with a is show i Fig. 8. Part of the overestimatio of K fs is believed to be due to the distortio of the actual flow path lies caused by the o-flow boudaries of the ifiltrometer. This distortio effect should be accouted for i the value of b, ad the distortio should icrease with icreasig capillarity. It is ot clear that the costat value of b as assiged by Philip (1993) is correct or whether it would be better to assig a value of Fig. 8. Variatio of the ratio of the field-saturated hydraulic coductivity estimated from the modified Philip Due (MPD) aalysis procedure with the saturated hydraulic coductivity values iput i the umerical simulatios (K MPD /K eff ) for differet values of the va Geuchte parameters a ad (variable or costat). b that accouts for capillarity. This effect should be examied further. Reyolds (2011) used a ad hoc modified versio of HYDRUS-2D to show that usig b = p 2 /8 i the Philip Due permeameter aalysis leads to overestimatio of the K fs ragig from a few percetage poits to about 23% for the differet soils he examied. For the borehole permeameter, he suggested this coefficiet to be 1. I his aalysis usig b = 1, the measured K fs value was cosistetly more accurate ( 20% differece). A secod possible reaso for the overestimated K fs value is that the MPD aalysis assumes that the wettig frot is sharp. I the field, however, ad also i simulatios of Richards equatio, the wettig frot is ot sharp but will be diffuse to a degree determied by the capillarity of the soil. Accordig to the results i Table 2 ad Fig. 8, where was kept costat, the MPD aalysis procedure i some cases overestimated K fs (Case 9, 34%) ad i some cases uderestimated K fs (Case 5, 6%). Overall i these cases, however, the MPD aalysis predicted K fs better tha the cases where was variable (Table 1). The reaso is probably because a higher value, such as = 4, represets a arrower particle size distributio ad a less diffuse wettig frot, which correspods more closely to the assumptio of a sharp wettig frot made i the Gree Ampt aalysis. From Tables 1 ad 2 we ca also coclude that the MPD formulatio overestimated y i some cases (84% for loamy sad ad 57% for Case 1) ad uderestimated it i other cases (6% for sady clay ad 30% for Case 9). A similar kid of tred, goig from coarser textured materials to fier textured materials, was show by Reyolds (2011) for the borehole permeameter. He oted that this iaccuracy seems to be associated with the gravity term G i the Gree Ampt solutio formulatio, which was kept costat durig the etire period of ifiltratio i his aalysis ad i the aalysis of Philip (1993). The gravity term was kept costat i those two aalyses because i both cases the goverig equatio was itegrated aalytically with time, which required that the gravity term be costat. I cotrast, i our aalysis, the gravity term was ot kept costat because we itegrated the goverig equatio (Eq. [8] or [9]) umerically, allowig the flexibility of a time-variat gravity term. Reyolds (2011) foud that the detrimetal effect of a costat gravity term o the estimated y is greatest for coarser soils ad demostrated that a costat value of zero (rather tha r o ) leads to improved estimates of the wettig frot suctio for the etire rage of soil textures. Vadose Zoe Joural p. 10 of 14

11 Aother assumptio made i the MPD formulatio is that the wettig frot will be hemispherical i shape. I reality, however, because of the effect of gravity, the wettig frot caot be perfectly hemispherical after some period of ifiltratio. It will be more elogated i shape (i.e., bulged dowward). This elogatio may be aother cause of deviatio of parameter estimates from the iput values. To illustrate this effect, the momet of iertia about the axis of symmetry was calculated for each of the Richards equatio solutios. The momet of iertia (M) about the axis of symmetry is oe way to quatitatively represet the shape of the wettig frot. It was calculated as a fuctio of time usig t M( t) = ò 2pq( t) r 2 dd r z [14] o Simulatios of Richards equatio were performed with ad without the presece of gravity for each type of soil ad the values of M were calculated from Eq. [14] for the istat i time at which the ifiltrometer just emptied. Results for these itegratios are listed i Table 3, where the momet of iertia for cases without gravity (M C ) are give as a ratio with the momet of iertia for the cases with gravity (M C+G ). The amout of elogatio depeds o the ratio of capillary forces to gravitatioal forces. From Table 3, it ca be see that the coarser soils (i.e., loamy sad ad sady clay loam) Table 3. Compariso of the spread of the wettig frot at the ed of the simulatio period, whe the ifiltrometer tube has emptied, for differet types of soil, where K eff is the saturated hydraulic coductivity values that were iput i the umerical simulatio ad M C ad M C+G are the momets of iertia without ad with gravity, respectively. Soil type K eff M C /M C+G m/s Loamy sad Sady clay loam Silt loam Sady clay Silty clay have a larger ratio tha the fier soils, idicatig the importace of the gravitatioal compoet of flow for those soils. It also idicates that the wettig frot will ot be spherical as assumed i the Gree Ampt aalysis, ad this could be a cause for some reductio i accuracy of parameter estimates with the MPD aalysis. Illustratio of the elogatio of the wetted volume for the case of the loamy sad soil is preseted i Fig. 9. Figure 9a shows the Fig. 9. Illustratio of the effect of gravity o dowward distortio of the otherwise spherical shaped wetted domai. Loamy sad is compared (a) with gravity ad (b) without gravity. Vadose Zoe Joural p. 11 of 14

12 wetted profile for the case with gravity, ad Fig. 9b shows the case without gravity. There is a clear effect of gravity i the elogatio of the wettig profile i this case. Experimetal Results Treatmet of Outliers Although the flow rate durig fillig was maitaied below that required to fluidize the sad, i some cases the soil was still slightly fluidized. This led to some uusually high K sat values or outliers, which were ot observed whe the soil was freshly recompacted. These outliers were removed to obtai a data set free of erroeous measuremets caused by experimetal error such as equipmet malfuctios ad operatioal issues. Outliers were idetified (ad removed from further cosideratio) usig the media absolute differece method developed by Rousseeuw (1990). The techique icorporates a estimate of scale, S, ( ) S= media Ki - K [15] med where K i is the saturated hydraulic coductivity of the ith test ad K med is the media of the K i values. The a Z score for each data poit is determied from Ki -Kmed Zi = [16] S Ay data that have a Z score above a critical value is cosidered a outlier. If the distributio is Gaussia, critical Z scores of 2.5 ad 1.5 correspod to the iclusio of 98.5 ad 84% of the total data, respectively. A histogram of idividual K ref ad K MPD values (icludig outliers) of each medium type is show i Fig. 10. For Media 1 ad 2, there is a wider rage of K ref values tha of K MPD values, ad i Medium 3, the variatio of K MPD is wider. Because of this wide rage of K ref ad K MPD values, which leads to difficulty i obtaiig represetative fallig-head tests, a low Z value of 1.5 was used to treat the outliers. Approximately 16% of the data was idetified as outliers, idicatig that the measuremets varied from a Gaussia distributio at the extremes. Outlier distributio amog the differet devices ad barrels appeared to be radom. Compariso betwee Fallig-Head Test ad Modified Philip Due Test The ratio of the arithmetic mea of K MPD ad the arithmetic mea of K ref (excludig outliers) for the three sads measured with the MPD ifiltrometer are preseted i Fig. 11, with correspodig descriptive statistics i Table 4. The graph also shows the upper ad lower limits of the ratio (K MPD /K ref ) withi the 67% cofidece iterval aroud the ratio of the mea K MPD ad mea K ref. Equatio [13] was used to calculate the upper ad lower cofidece iterval of the ratio (K MPD /K ref ) for each barrel. Fig. 10. Histogram of the mea field-saturated hydraulic coductivity value for each medium determied by the modified Philip Due (MPD) ifiltrometer (K MPD ) ad the mea saturated hydraulic coductivity of the referece fallig-head tests (K ref ) before removig outliers for (a) Medium 1, (b) Medium 2, ad (c) Medium 3. Medium 1 has the highest K sat value ad Media 2 ad 3 have roughly equivalet K sat values, which are because of the fies preset that restrict the passage of water, ad the fies are similar i both media. The average iitial moisture cotet for Media 1, 2, ad 3 are 6, 10.5, ad 7.6%, respectively, for the MPD ifiltrometer test. The coefficiets of variatio (CVs) for all the tests are relatively low, ragig from 4.8 to 21.5%. Muñoz-Carpea et al. (2002) reported CV values of 39 to 101% whe comparig permeameter results from field measuremets. Asleso et al. (2009) reported CV values betwee 54 ad 178% i a ifiltratio assessmet of rai gardes with egieered soil. Lower CV values would be expected for a cotrolled laboratory compariso Vadose Zoe Joural p. 12 of 14

13 because the sad medium is homogeous i compariso to field soils. The relatively low skewess ad kurtosis values idicate that the data sets may be described as ormally distributed. This could also be a cosequece of the relatively homogeous sads used i the testig. Comparatively, it is typical for field-measured hydraulic coductivity to be represeted by a logormal distributio (Asleso et al., 2009). A Games ad Howell procedure (Toothaker, 1994) was used to compare the mea K sat values for each of the barrels with the K fs values estimated with the MPD ifiltrometer. The results showed that the mea K fs values from the MPD ifiltrometer measuremets were statistically differet at the 5% level from the mea referece fallig-head test K sat values for Barrels 1 ad 3 but ot for Barrel 2. I geeral, the K fs values from the MPD ifiltrometer tests were i good agreemet with the referece fallig-head tests. Nevertheless, there was a cosistet bias i that the mea K fs values were 80, 92, ad 78% of the mea referece fallig-head tests for Barrels 1, 2, ad 3, respectively. Oe possible reaso for this bias is the air etrapmet or ecapsulatio by the dowward advacig wettig frot durig the MPD measuremets. The MPD ifiltrometer measures the field-saturated hydraulic coductivity (K fs ), which will geerally be less tha that of fully saturated media (K sat ). Bouwer (2002) showed that because of etrapped air, the hydraulic coductivity i a wetted zoe durig ifiltratio is less tha the fully saturated value ad idicated that for sady soil this ca cause decreases up to a factor of 2 i the measured K fs value. Give that the barrels were wetted to saturatio from the bottom up before the fallig-head tests, - much less etrapped air could reasoably be expected i the sads durig the fallig-head tests ad hece a greater saturated hydraulic coductivity value. Fortuately, this amout of bias is relatively low i compariso to the amout of variability observed for MPD ifiltrometer measuremets at field sites (mea CV of 107%; Asleso et al., 2009). Fig. 11. Compariso of ratio of arithmetic mea of K MPD ad arithmetic mea of K ref for three differet porous media. Error bars represet 67% cofidece itervals. 6 Coclusios Ifiltratio rate parameters, icludig K fs, should be measured at the soil surface i stormwater ifiltratio practices because surface issues (e.g., compactio, particle accumulatio) ca severely limit ifiltratio ad the overall performace of these stormwater maagemet practices. We have developed a ovel ad iexpesive device for surface ifiltratio rate measuremets, called the modified Philip Due ifiltrometer, ad a associated approximate data aalysis method. Such a device could be used to determie whe ad where to perform maiteace i stormwater ifiltratio practices. I additio, this device could also be used for evaluatio of ladscape-alterig practices such as costructio activities. A MPD aalysis method was developed based o the Gree Ampt model, which assumes a sharp, spherical wettig frot. The Table 4. Descriptive statistics for the hydraulic coductivity of three barrels of soil media determied by the modified Philip Due ifiltrometer ad the fallig-head method. Statistic Medium 1 Medium 2 Medium 3 MPD Fallig head MPD Fallig head MPD Fallig head Mi., m/s Max., m/s Mea, m/s Media, m/s SD, m/s CV, % Skewess Kurtosis N Coefficiet of variatio. Sample size. Vadose Zoe Joural p. 13 of 14