Improved Soil-Water Characteristic Curves and Permeability Functions for Unsaturated Soils

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1 Ipovd SoilWat haactitic uv ad Pailit Fuctio fo Uatuatd Soil Shada H. Kihapillai, Nadaajah Ravichada ivil Egiig Dpatt lo Uivit, lo, S, USA ABSTRAT Soilwat chaactitic cuv (SW) which pt th oituuctio latio of oil i o of th ipotat cotitutiv odl dd to iulat th havio of uatuatd oil. A ffctiv SW odl hould capal of calculatig th oituuctio vaiatio fo th ti ag of dg of atuatio. Applicailit of popula SW odl uch a Book ad o, va Gucht, ad Fdlud ad Xig i liitd, pciall i low (<2%) dg of atuatio ag. I thi tud, all th odl a odifid, o that th odl ca ffctivl ud i ti ag of dg of atuatio. Th Fdlud t al (994) pailit fuctio i alo odifid ad o th odifid Fdlud ad Xig SW odl. Th applicailit of th ipovd odl i ivtigatd caliatig th SW of vaiou tp of oil ad ptd i thi pap. Bad o thi tud it ca cocludd that th odifid odl a flxil ough to fit th xpital data fo th ti ag of dg of atuatio. RÉSUMÉ olau cou caactéitiqu d qui pét l d'huidité t d'apiatio latio d t l'u d odèl ipotat écai pou iul l copott d ol o atué. U odèl fficac dvait êt capal d calcul la vaiatio d l'huidité d'apiatio da tout l ga dgé d atuatio. L'applicailité d olau cou caactéitiqu odèl tl qu Book t o (B), va Gucht (vg), t Fdlud t Xig (FX) t liité, paticuli da la fouchtt d dgé d atuatio fail. Da ctt étud, tou c odèl ot odifié ou aélioé, d ot qu c odèl puvt êt utilié tout l ga dgé d atuatio. L Fdlud t al foctio d péailité t égalt odifié u l aélioé odèl. L'applicailité d odèl aélioé d'u quêt appofodi t caliat l cou caactéitiqu olau d div tp d ol t pété da l pét l papi. Baé u ctt étud, o put coclu qu l odèl odifié ot uffiat oupl pou 'adapt aux doé xpéital pou da tout l ga du dgé d atuatio. INTRODUTION Th Soil Wat haactitic uv (SW) i a latiohip tw th aout of wat pt i th oil (oitu) ad th uctio chaactitic of th oil atix. Th aout of wat pt i th oil ca xpd i t of dg of atuatio (S), volutic wat cott (ϴ), o gavitic wat cott (u). Ma ach hav idtifid th facto which ifluc th hap of th SW ad ad o that, a athatical SW odl w dvlopd. Gad (956), Book ad o (964), va Gucht (9), Kougi (994), ad Fdlud ad Xig (994) a o of th odl foud i th litatu. All th odl cofi a iv popotioal latiohip tw S ad uctio (ψ). Thi ca xplaid with th fudatal icu tho a follow. Wh th S ica, th adiu (R ) of th icu will ica. Wh R ica, th pu diffc tw th po ai pu ad th po wat pu (uctio) will dca ( Eq. ). g l 2T = p p [] R wh ψ i th uctio, p g i po ga pu, p l i po liquid pu, ad T i ufac tio. Th ait uctio that i alo kow a ulig pu ad po iz ditiutio a two aic paat icopoatd i ot of th SW odl. I odl uch a Book ad o (B), va Gucht (vg), ad Fdlud ad Xig (FX), th two paat a ptd a ad, pctivl. Th Book ad o odl (Eq. 2) i o of th aic SW odl dvlopd with two paat. Thi odl do ot povid a cotiuou athatical fuctio fo th ti ag of S. if ψ a / a if ψ a wh a ad a th fittig paat. Th paat a i latd to th ait uctio of th oil ad th i latd to th po iz ditiutio of th oil. ψ i uctio, θ i volutic wat cott, θ i idual wat cott, ad θ i atuatd wat cott. Th vg odl (Eq. 3) povid a igl quatio fo th ti ag of S. Thi odl ha a additioal fittig paat, th akig thi odl o flxil copad to th B odl. [3] ( a ) wh th fittig paat i latd to idual wat cott. All th oth paat a a a i th B odl. Th FX odl i ptd i Eq. 4. Th ailit of thi odl to pdict th SW fo ti ag of S i [2]

2 coidd a th ajo advatag of thi odl. Th F X odl au a axiu uctio of,, kpa at d coditio, whil th B ad th vg odl au ifiit valu of axiu uctio. Th FX odl i ath iila to th vg odl oth tha th coctio facto ( ) ad lt i th quatio. Th lt i v ffctiv i kpig th SW without achig zo oalizd wat cott i low uctio ag, pciall fo ad oil. Fdlud ad Xig (994) hav alo uggtd aoth fo of th odl (Eq. 5) which ca ud if a idual wat cott i kow. ( ) l ( ψ / a) = l ( ψ / a) ( ψ) l ( ψ ) 6 l ( ) wh i a paat latd to idual wat cott ad oth paat a a a i th vg odl. Th popti which affct th oil wat chaactitic cuv alo affct th pailit cofficit of po fluid i uatuatd oil. Thfo, SW ca ffctivl ud to calculat pailituctio latio, which i cool fd a pailit fuctio. Bad o FX SW odl, a pailit fuctio (Eq. 6) i popod Fdlud t al. (994). K l l av wh ψ i uctio, K (ψ) i th lativ pailit at uctio ψ, ψ av i th ait uctio, i a du vaial of itgatio, = l(l,,), θ i volutic wat cott giv i Eq. 4 ad θ i it divativ. a,, ad a fittig paat of th FX odl (Eq. 4). Th B, vg, ad FX odl a ig widl ud to calculat th oituuctio latio of uatuatd oil. Fo th B ad vg odl, a idual wat cott valu ha to pcifid. Howv th two odl calculat ualitic uctio wh th oalizd wat cott i zo o l, i.. wat cott of th oil i l tha o qual to th idual wat cott. I th FX odl, th axiu uctio i aud to,, kpa. Although th a thodaic cocpt to ack up thi axiu uctio, it i a coc to u a fixd valu fo all tp of oil. I additio, wh d d [4] [5] [6] th actual axiu uctio i low, uag of uch lag axiu uctio valu ight ov pdict ha tgth i uical iulatio. Siila to th B ad vg odl, th cod fo of th FX odl (Eq. 5) alo calculat a ualitic uctio wh th oalizd wat cott i zo o l. Thfo, to avoid a ualitic uctio valu at zo oalizd wat cott, th axiu uctio valu hould pcifid v with a idual wat cott pcifid. I additio, th fouth odl paat i th FX odl i cho fo a wid ag ( to,, kpa) ad it cat difficulti i achivig a uiqu t of caliatd odl paat. Alo, th affct th iitial potio of th cuv wh th valu of i lativl low ad it i coidd a aoth diadvatag (Log ad Rahadjo, 997). Th pia ojctiv of thi tud i to ica th flxiilit of th B ad vg odl o that th odl ca pdict alitic high uctio i low dg of atuatio without cauig uical itailiti i fiit lt iulatio. It i v challgig to odl th oil havio fo a full d coditio to a full atuatd coditio uig a igl full coupld fiit lt coput cod. Th cut tat of th at uggt that th a th ajo difficulti i dvlopig uicall tal iulatio capailit. Th a: difficulti i dalig with ultipl odal/lt vaial i fiit lt foulatio of poou dia at th xt coditio, difficulti i dvlopig ttai havio with appopiat t tat vaial at th xt coditio, ad difficulti i accuatl calculatig th uctio ov th ti ag of dg of atuatio. Th odifid odl ca icopoatd i fiit lt iulatio without itoducig uical itailiti ai fo SW. I thi tud, th B ad vg odl a odifid icopoatig coctio facto. Alo, th coctio facto i th FX odl i odifid to avoid th ffct of additioal fittig paat. Icopoatig th axiu uctio a pat of th odl icad it flxiilit i fittig aud data of vaiou oil ov th full ag of S. All th odl a ipovd with th fatu to pcif oth idual wat cott ad axiu uctio valu. Th capailit of th ipovd odl i vifid atchig with th xpital data ad pdictio of oigial odl. Bad o th ipovd FX odl, th pailit fuctio popod Fdlud t al. (994) i odifid ad ptd. 2 IMPROVED SW MODELS AND OMPARISONS Although th a uou SW odl availal i th litatu, thi tud i itdd to ipov th popula B, vg, ad FX odl. Th B ad vg odl a odifid piail to ak u that th odl o log calculat high uctio wh th oalizd wat cott i zo o l. Ad alo th odifid odl hav th fatu to pcif oth idual wat cott ad axiu uctio valu. 2. Th Ipovd Book ad o (IB) Modl

3 Dg of atuatio (%) Dg of atuatio (%) Dg of atuatio (%) Dg of atuatio (%) Th ipovd Book ad o (IB) odl i giv i Equatio 7. To pv th advatag of th B odl, o additioal fittig paat i itoducd. Ev though th axiu uctio ψ ax i icopoatd i th quatio, it caot coidd to a fittig paat, a th hap of th SW caot chagd adjutig th ψ ax. Th IB odl do ot povid a cotiuou athatical fuctio fo th ti ag of S. / a if if ψ ψ a ax wh ψ ax i axiu uctio ad oth paat a a a i th B odl. 2.. opaio of th B ad th IB Modl apailit of th ipovd B (IB) odl i pdictig th oituuctio latio i ivtigatd ad copad with th B odl fo fou difft oil. Th copaio of B ad IB Modl fo oluia ad loa (data Book & o 964) i how i Figu. Th Figu 2 ad 3 how th copaio fo Madid cla ad ad Aligto oil, pctivl. Th Figu 4 how th copaio fo Idia had till (data Vaapalli t al. 999). It hould otd that th xpital SW data a ot availal fo th full ag of S (%). Bad o th xpital data, th axiu uctio of,, kpa i cho fo all fou oil. Th idual wat cott i aud to zo fo all fou oil. A how i th figu, th IB odl i capal of calculatig th oituuctio latio fo full ag of S, wha th B odl i ot ffctiv. Th B, IB odl a ot ffctiv fo ad oil ad it i vidtl how i Figu a th odl faild to kp th SW without achig zo oalizd wat cott i low uctio ag a B IB oluia ad loa B (a = 5 kpa, =.8) IB (a = 5 kpa, =.8) [7] Figu. B ad IB SW fo oluia ad loa 4 2 B IB B (a = 32 kpa, =.3) IB (a = 32 kpa, =.3) Madid cla ad Figu 2. B ad IB SW fo Madid cla ad 4 2 B IB B (a = 68 kpa, =.5) IB (a = 68 kpa, =.45) Aligto oil Figu 3. B ad IB SW fo Aligto oil 4 2 B IB B (a = 23 kpa, =.23) IB (a = 22 kpa, =.25) Idia had till Figu 4. B ad IB SW fo Idia had till 2.2 Th Ipovd va Gucht (IvG) Modl Th ipovd va Gucht (IvG) odl i giv i Equatio 8. Sic th paat a i latd to th ait uctio, th odl i vid o that th paat a ha th uit of uctio. Th IvG odl i dvlopd with th fatu to pcif oth idual wat cott ad axiu uctio valu with o additioal fittig paat.

4 Dg of atuatio (%) Dg of atuatio (%) Dg of atuatio (%) Dg of atuatio (%) a [8] Aligto oil vg IvG ax wh ψ ax i axiu uctio ad oth paat a a a i th vg odl Pdictiv apailit of th IvG Modl apailit of th ipovd vg (I vg) odl i pdictig th oituuctio latio i ptd fo oluia ad loa, Madid cla ad, Aligto oil, ad Idia had till i figu 5 though 8, pctivl. Siila to th IB odl, axiu uctio of,, kpa ad idual wat cott of zo a ud fo all fou oil. A how i figu 5 though 8, th IvG odl i capal of calculatig th oituuctio latio fo full ag of S, wha th vg odl i ot ffctiv. A how i Figu 5, th vg, IvG odl a alo ot uital fo ad oil a th odl alo faild to kp th SW without achig zo oalizd wat cott i low uctio ag Figu 5. vg ad IvG SW fo oluia ad loa oluia ad loa vg IvG vg (a =.8 kpa, =, =.) vg IvG IvG (a = 6 kpa, =, =.) vg (a =.3 kpa, = 2.85, =.) IvG (a = 33 kpa, = 2.9, =.5) Madid cla ad Figu 6. vg ad IvG SW fo Madid cla ad 4 2 vg (a =.2 kpa, = 2., =.77) IvG (a = 85 kpa, = 2, =.77) Figu 7. vg ad IvG SW fo Aligto oil Idia had till vg IvG 4 vg (a =.3 kpa, =., =.23) 2 IvG (a = 37 kpa, =., =.2) Figu 8. vg ad IvG SW fo Idia had till 2.3 Th Ipovd Fdlud ad Xig (IFX) Modl Th ipovd Fdlud ad Xig (IFX) odl i giv i Equatio 9. Th IFX odl i dvlopd with th fatu to pcif oth idual wat cott ad axiu uctio valu without th paat, i.. with ol th fittig paat. Thfo, th ffct of i th iitial potio of th FX odl (Log ad Rahadjo, 997) i avoidd i th IFX odl. ( ) = l a ax wh all th paat a a a i th IvG odl Pdictiv apailit of th IFX Modl Th pdictiv capailit of th IFX odl i pdictig th oituuctio latio i ptd i figu 9 though 2. Siila to th IB, IvG odl,,, kpa axiu uctio ad zo idual wat cott.5 [9]

5 Dg of atuatio (%) Dg of atuatio (%) Dg of atuatio (%) Dg of atuatio (%) a ud. It ca otd that th IFX odl i alo ffctiv i full ag of S. Howv th IFX odl ca coidd tt a it ha ol th fittig paat, wha th FX odl ha fou. FX IFX FX (a = 6.2 kpa, = 8.5, 4 =.49, = 4 kpa) IFX (a = 6.2 kpa, = 8.5, 2 =.5) oluia ad loa Figu 9. FX ad IFX SW fo oluia ad loa 4 2 FX IFX Madid cla ad FX (a = kpa, =.8, =.64, = 4 kpa) IFX (a = 64 kpa, =.6, =.7) Figu. FX ad IFX SW fo Madid cla ad Aligto oil FX IFX 4 FX (a =.25 kpa, =.6, 2 =.3, = 4 kpa) IFX (a = kpa, =.3, =.43) Figu. FX ad IFX SW fo Aligto oil 4 Idia Had till FX (a = 3 kpa, =, =.46, = 3 kpa) FX IFX 2 IFX (a = 575 kpa, =.95, =.72) Figu 2. FX ad IFX SW fo Idia had till 3 MODIFIED PERMEABILITY FUNTION AND OMPARISONS Bad o FX SW odl, a pailit fuctio i popod Fdlud t al. (994) ad it i ig widl ud. Thfo, it i ipotat to odif th Fdlud t al pailit fuctio (FAll odl) ad o th IFX SW odl. Th FAll odl i odifid ad o th IFX SW odl, ad ptd a IFAll odl i Equatio. Th ol diffc tw th FAll ad I FAll odl, i th coctio facto ( ψ ). K l l av Th fuctio ad a giv ( ψ) l ( ψ / a) ax ad.5 d d [] wh ax i axiu uctio ad oth paat a a a i th FAll odl. 3. Pdictiv apailit of th IFAll Modl Th pailit cofficit of wat i fou difft oil a pdictd with FAll ad IFAll odl ad ptd i Figu 3 though 6. Figu 3 illutat th pdictio fo Suptitio ad ad th copaio with xpital data (fo Richad, 952). A how i Figu 3, th FAll ad IFAll odl how tt atch with th xpital data. Howv, cau of

6 Rlativ pailit Rlativ pailit Rlativ pailit Rlativ pailit th lack of xpital data, th accuac of th two odl i th high uctio ag could ot vifid. Th Figu 4 how th copaio of pdictd ult ad xpital data fo oluia ad loa (xpital data fo Book & o 964). Siila to th Suptitio ad, th pdictio of FAll ad IFAll odl atch wll with th xpital data i th low uctio ag. A how i Figu 5, iila pdictio a otaid fo th Toucht ilt loa (xpital data fo Book & o, 964). Figu 6 how th pdictio ad copaio fo Yolo light cla (data fo Moo 939). A how th, th diffc tw th xpital data ad th pdictio of FAll ad IFAll odl ica a th uctio ica. I additio, th pdictio of FAll odl lightl dviat fo th pdictio of IFAll odl at high uctio ag FAll IFAll FX (a = 2.6 kpa = 9 =.46 = 3 kpa) IFX (a = 2.6 kpa = 9 =.46) 2 3 Figu 3. FAll ad IFAll odl fo Suptitio ad FX (a = 6.2 kpa = 8.5 =.49 = 4 kpa) IFX (a = 6.2 kpa = 8.5 =.5) FAll IFAll 2 3 Figu 4. FAll ad IFAll odl fo oluia ad loa FX (a = 8.4 kpa = =.5 = 3 kpa) IFX (a = 8.4 kpa = =.5) FAll IFAll 2 3 Figu 5. FAll ad IFAll odl fo Toucht ilt loa FX (a = 2.9 kpa =.9 =.4 = kpa) IFX (a = 2.9 kpa =.9 =.4) FAll IFAll 2 3 Figu 6. FAll ad IFAll odl fo Yolo light cla 4 ONLUSION Th Book ad o, va Gucht, ad Fdlud & Xig odl a odifid to captu th high uctio at low dg of atuatio. Both axiu uctio ad idual wat cott ca ud a iput i th odifid odl. Sic th i o data availal to vif it capailit i high uctio ag, th flxiilit of th odifid odl ha vifid fittig xpital data fo fou difft oil i high uctio ag ad th pdictio fo oigial odl. REFERENES Book R.H., o A.T Hdaulic popti of poou dia. Hdolog Pap, oloado Stat Uivit. Fot olli, 27(3). Fdlud D.G., Xig A Equatio fo th oilwat chaactitic cuv. aadia Gotchical Joua, 3: Fdlud D.G., Xig A., Huag S Pdictig th pailit fuctio fo uatuatd oil uig th oilwat chaactitic cuv. aadia Gotchical Joual, 3: Kougi, K Th paat logoal ditiutio odl fo oil wat ttio. Wat Rouc Rach, 3: 899. Log, E.., ad Rahadjo, H Rviw of oilwat chaactitic cuv quatio. Joual of Gotchical ad Goviotal Egiig, 23:67. Moo R.E Wat coductio fo hallow wat tal. Hilgadia, 2: Richad L.A Wat coductig ad taiig popti of oil i latio to iigatio. Pocdig, Itatioal Spoiu o Dt Rach, Jual, va Gucht M.Th. 9. A clod fo quatio fo pdictig th hdaulic coductivit of uatuatd oil. Soil Scic Socit of Aica Joual, 44: Vaapalli, S. K., Pufahl, D. E., ad Fdlud, D. G Th ifluc of oil tuctu ad t hito o th oilwat chaactitic of a copactd till. Gotchiqu, 49(2): 4359.

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