A Statistical Model for the Relative Hydraulic Conductivity of Water Phase in Unsaturated Soils

Transcription

1 Intenational Jounal of Geosciences, 2011, 2, doi: /ijg Published Online Novembe 2011 ( A Statistical Model fo the Relative Hdaulic Conductivit of Wate Phase in Unsatuated Soils Abstact Nadaajah Ravichandan, Shada Kishnapillai Civil Engineeing Depatment, Clemson Univesit, Clemson, USA. navic@clemson.edu Received June 7, 2011; evised August 9, 2011; accepted Septembe 14, 2011 Pemeabilit coefficients of fluids occuping the poe space of a poous medium have significant influence on the flow of these fluids though the poous medium. In the case of unsatuated soils, in addition to othe paametes such as void atio, void distibution, paticle size distibution and initial densit the degee of satuation also affects the pemeabilit coefficient of wate. The degee of satuation, in unsatuated soil, is diectl elated to the matic suction of the soil though soil wate chaacteistic cuve. Matic suction is one of the two stess state vaiables widel used to chaacteize the defomation behavio of unsatuated soils. Theefoe, it can be stated that both flow and defomation behavios of unsatuated soil ae affected b the pemeabilit coefficient of wate. Numeical modeling of coupled defomation-flow behavio of unsatuated soil equies a mathematical equation that elates the pemeabilit coefficient to the degee of satuation. Since the paametes that affect the pemeabilit coefficient of wate in unsatuated soil have simila diect o indiect effects on the soil wate chaacteistic cuve, pemeabilit can be effectivel pedicted using the soil wate chaacteistic cuve as done in statistical models. In this pape, a statistical model is poposed fo the pemeabilit of wate in unsatuated soil using soil wate chaacteistic cuve of the soil. The calibated paametes of the soil wate chaacteistic cuve ae diectl used in the pediction of pemeabilit with- out additional calibation using measued pemeabilit data. The pedictive capabilit of the new equation is veified b matching the measued data of eight diffeent soils found in the liteatue. Kewods: Unsatuated Soils, Pemeabilit Function, Relative Pemeabilit of Unsatuated Soils, Relative Pemeabilit Using Soil-wate Chaacteistic Cuve 1. Intoduction Unsatuated soil is a thee-phase media consisting of solid paticles, wate and ai. A wide ange of poblems in Hdolog, Soil Phsics, Geoenvionmental Engineeing and Geotechnical Engineeing ae associated with unsatuated soils. Axial and lateal load capacit of foundations, contaminant tanspot though soil, eath slope failue afte extended peiods of ainfall, seepage though eathen stuctues, and shinking and swelling of poblematic fine gained soils ae some of the examples. All of these poblems shae a single commonalit: movement (flow) of wate though the poe space. The abilit of wate to move though a given soil is measued b pemeabilit coefficient. Theefoe, accuate evaluation of the pemeabilit is impotant fo accuate modeling of flow and defomation poblems in unsatuated soils. The classical satuated soil mechanics theoies fall well shot of captuing phenomena associated with flow of wate in unsatuated soils. Theefoe, a geate unde- standing of flow though unsatuated soil equies the incopoation of unsatuated soil pinciples. In the case of satuated soil in which the void space is completel filled with wate, the coefficient of pemeabilit is coelated to the void atio and/o the paametes of the paticle size distibution cuve such as effective size, D 10 and unifomit coefficient, Cu [1,2] of the soil. On the othe hand, the void space in unsatuated soil is filled patl with wate and the est with ai. The pemeabilit of wate in unsatuated soil is affected not onl b the void atio, poe size distibution, voids distibution and d densit [3] but also b the degee of satuation [4]. Compaed to pue flow poblems, coupled defomation-flow poblems ae complex at the same time common in civil engineeing. In a defomation poblem, the volumetic defomation of the solid skeleton due to

2 N. RAVICHANDRAN ET AL. 485 extenal load can change both the void atio and degee of satuation of the soil. Fo example, a negative volumetic stain will incease the volumetic wate content in a epesentative element esulting in incease in pemeabilit coefficient. It is obseved that the pemeabilit coefficient of unsatuated soil vaies b an ode of magnitude of 10 when the degee of satuation of the soil vaies fom ve low to ve high [5]. Because the pemeabilit of unsatuated soil is uniquel influenced b the degee of satuation, the soil wate chaacteistic cuve (SWCC) of the soil can be used to pedict the pemeabilit coefficient. The SWCC is a unique constitutive equation in unsatuated soil that elates the degee of satuation to the matic suction and it incopoates the basic soil popeties associated with flow such as void atio, poe size distibution, void distibution, paticle size distibution and initial densit. The majo advantage of using the SWCC is that the moistuesuction elationship can be easil obtained expeimentall than the moistue-pemeabilit elationship. In this stud, a new statistical model fo the elative pemeabilit of wate in unsatuated soils has been developed using the SWCC of the same soil. The model paametes used in the SWCC ae used in the elative pemeabilit equation and these paametes ae calibated using SWCC data onl. The pedictive capabilit is veified using expeimental data of eight diffeent soils found in the liteatue. As veification, the pedictions ae compaed with that of the widel used Fedlund et al. s model [5]. The pedictions and the compaisons show that the poposed model accuatel pedicts the measued pemeabilit data ove a wide ange of degee of satuation. 2. Relative Pemeabilit Model fo Wate Phase in Unsatuated Soils 2.1. Existing Models and Modeling Techniques It is common pactice to expess the pemeabilit coefficient of wate phase in unsatuated soil (k us ) as a scala poduct of satuated pemeabilit tenso (k s ) and elative pemeabilit (k ) i.e., k us = k * k s. The modeling techniques of all of the elative pemeabilit functions available in the liteatue can be classified into thee goups: 1) empiical models, 2) macoscopic models and 3) statistical models. The empiical technique is puel a datadiven method. Hee, the unsatuated pemeabilit is expessed as a function of satuated pemeabilit and cetain fitting paametes of an equation. The fitting paametes depend upon the shape of the expeimental cuve [6-12] and ae adjusted to match the expeimental cuve with the empiical equation. It is woth noting that most of the existing unsatuated pemeabilit functions fit the expeimental data well in the mid to high ange of degee of satuation and exhibit a significant deviation in low degee of satuation ange. Analses of poblems that involve wide ange of degee of satuation change (d to full satuated condition) equie models that accuatel pedict the pemeabilit fom low degee of satuation to full satuated condition. Howeve, obtaining of the equisite amount of expeimental data especiall at a low degee of satuation is a difficult task. This is mainl because of change in fabic and the stuctue of cetain soils at low degee of satuation. The macoscopic models ae being developed b aveaging the micoscopic flow behavio ove a epesenttative element volume. The shape and the dimensions of the poe space and the flow channels in a epesentative element volume ae simplified to ease the calculation and integated to obtain the macoscopic esponse. The epesentative element size is selected so that the volume o the chaacteistic length is lage enough to include a sufficient numbe of poes and paticles to educe the micoscopic inhomogeneit at the same time small enough to educe the macoscopic inhomogeneit due to cacks etc. The model poposed b Mualem [13] is one of the ealiest models that not onl takes into account the micoscopic popeties but also models the hsteetic behavio due to wetting and ding phases. Although the macoscopic models ae developed based on fundamental phsical laws, the inabilit of scaling the micoscopic popeties to the macoscopic level and incopoating the poe size distibution index [9], makes it difficult to develop advanced models that eplicate actual soil sstems. The statistical models ae developed based upon the assumption that the soil poes consists of a netwok of inteconnected poes. When a fluid occupies a potion of the poe space, a fluid-filled tube foms and the flow of that paticula fluid occus onl though the flow tubes. In addition to the size and the distibution of these tubes, the degee of satuation also affects the flow of a given liquid. Fo example, at highe degees of satuation, the flow tubes will be bigge in coss sections that will esult in a lage flow. The statistical method is used to quantif the size and the distibution of these flow tubes. It should be noted that the distibution of the poes and poe sizes affect the suction at a given degee of satuation. Theefoe, the suction-degee of satuation elationship can be indiectl used to develop the pemeabilit function fo unsatuated soils [5,13-16] i.e., a calibated SWCC model can be used to pedict the pemeabilit of unsatuated soil at vaious degee of satuation. Of the man pemeabilit functions, to ou knowledge, the model poposed b. [5], shown in Equation (1) is commonl used in the finite element

3 486 N. RAVICHANDRAN ET AL. simulations of coupled defomation-flow poblems in unsatuated soil. The model uses the SWCC poposed b Fedlund and Xing [6]. Since the esidual wate content is assumed to be zeo in the Fedlund and Xing model, the nomalized wate content and the degee of satuation ae equal. Theefoe, this pemeabilit function can be utilized with eithe volumetic o gavimetic wate content o with the degee of satuation. and K b ln ln e e e d b e s e e d aev (1) The functions and C ae given b s C( ψ) (2) m n ln e ψ a ln 1 ψ C C 1 ln C whee is the soil suction, K is the elative pemeabilit at suction, aev is the ai-ent value of the soil unde consideation, is a dumm vaiable of integation epesenting a suction, b = ln (l,000,000), θ is the volumetic wate content and θ is the deivative of θ. C is a paamete elated to esidual wate content, and a, n and m ae the fitting paametes fo the SWCC. The paamete a epesents the ai-ent suction, the paamete n epesents the poe size distibution of the soil, and paamete m elates to the asmmet of the soil wate chaacteistic cuve. Based on ou expeience the. [5] model involves a complicated integation pocedue [5] fo calculating the pemeabilit using the coesponding SWCC. It also exhibits a significant deviation at low degee of satuation (high suction) ange. Leong and Rahadjo [16] suggested anothe pemeabilit function incopoating the soil suction and a fitting paamete p that vaies with soil tpe. This method was futhe studied b Fedlund et al. [18] using almost 300 sets of pemeabilit data to obtain tpical values fo p fo common tpes of soils. This method is effective fo couse-gained soils but it is not suitable fo fine-gained soils [18,19] New Pemeabilit Function The poe-size distibution is an impotant popet in unsatuated soils, because it diectl influences the soil suction and pemeabilit. In most of the popula soilwate chaacteistic cuves (SWCCs), a fitting paamete n which is elated to the poe-size distibution is used to elate the soil suction to the degee of satuation. The (3) pemeabilit of wate in unsatuated soils is govened not onl b the poe-size distibution but also b the volumetic wate content (θ/θ s ) o the degee of satuation. Thee ae man available pemeabilit models which elate the pemeabilit of the unsatuated soils to the SWCC model paametes [5,15]. Othe paamete that affects the pemeabilit coefficient of wate is the matic suction. The effect of suction is significant in low degee of satuation ange because the stong adhesion between paticles and the wate film at the cones of the paticles. Theefoe, in geneal, the pemeabilit functions can be expessed as a function of volumetic wate content, poe-size distibution index, and soil suction as shown in Equation (4). K f, n, (4) The new shown in Equation (5) [20] is used to pedict the pemeabilit coefficient in this pape. A detail compaison stud of this SWCC with existing models and its pefomance in the finite element simulation of unsatuated soil ae pesented in Kishnapillai and Ravichandan [20]. N = s 1 n 1m ln1 ψ a ai m The functions N is given b N N 1 (6) N 1 max whee a, n and m ae the fitting paametes; a is elated to the ai-ent suction, n is elated to the poe-size distibution of the soil, m is elated to asmmet of the model, ψ is the soil suction, θ is the volumetic wate content, θ s is the satuated wate content, θ is the esidual wate content, ψ max is the maximum suction o suction at d condition, and N is a numbe elated to esidual wate content. This equation can be used eithe with maximum suction o esidual wate content concepts. Fo the maximum suction concept (at zeo volumetic wate content), the esidual wate content is set to zeo (θ = 0) and fo the esidual wate content the paamete N is set to zeo (N = 0). Although the existing pemeabilit functions pedict the measued data well, significant deviation is obseved in low degee of satuation ange because the actual mechanics of unsatuated soil behavio at low degee of satuation ange is complex because of fabic and stuctue change especiall in clae soils. Howeve, in the statistical appoach, if the SWCC is flexible enough to fit the expeimental data well in the low degee of satuation ange, then the pemeabilit function will also be able to s (5)

4 N. RAVICHANDRAN ET AL. 487 fit the measued data well in the low degee of satuation ange. The SWCC used fo pedicting the pemeabilit function is flexible enough to fit the measued data in low suction ange (Kishnapillai and Ravichandan, 2011). The poposed statistical model is given in Equation (7). It should be noted that the poposed equation is obtained b tial and eo pocedue knowing that the pemeabilit is invesel popotional to the matic suction. Afte calibating the model paametes using the SWCC data, the numbes in the equations wee adjusted until the poposed model fits the measued pemeabilit data. It woth noting hee that the model paametes wee not calibated using measued pemeabilit data but calibated using measued SWCC data. K F 1 1 The function F is given b 1 (1.25 ) 2n s n 3.5 n F 1.5 (8) 1 ψ a ai s whee K is the elative pemeabilit at suction. The pemeabilit is the scala poduct of elative pemeablit and the satuated pemeabilit s n 1 3. Calibation and Validation of the Poposed Relative Pemeabilit Function The pedictive capabilit of the new model is investigated using expeimental esults of eight diffeent tpe of soils found in the liteatue. Soil ae chosen based upon the availabilit of both moistue-suction and moistueemeabilit elationships. The dataset includes sands, silts and clas. The available popeties of these soils and coesponding efeences ae listed in Table 1. The SWCC model paametes ae fist calibated b matching the measued moistue-suction data. It should be noted that the expeimental pemeabilit values ae not matchb adjusting the model paametes; the calibated SWCC model paametes ae, instead, diectl used to pedict the elative pemeabilit Calibation of SWCC Model Paametes The calibations of the Kishnapillai and Ravichandan (2011) SWCC model paametes fo these eight soils ae shown in Figue 1(a) though (h). Figues 1(a) and (b) show the calibation of SWCC model paametes fo Supestition sand (data fom [21]) and Lakeland sand (data fom [22]), espectivel. Figues 1(c) and (d) show the calibation of SWCC model paametes fo Colum- (7) bia Sand loam (data fom [9]) and Touchet silt loam (data fom [9]), espectivel. The Figue 1(e) is fo Silt loam (data fom [22]) and (f) is fo Guelph loam (data fom [23]). The Figues 1(g) and (h) ae fo Yolo light cla (data fom [24]) and Speswhite Kaolin (data fom [25]), espectivel. As seen in these figues, the measued moistue-suction data fo these soils ae unavailable fo the full ange (0% - 100%) of degee of satuation. Fo the Supestition sand and Lakeland sand, the available expeimental data show an appoximate satuation ange between 30 to 100% degees (see Figue 1); fo the Columbia sand loam between 50 to 100%; fo Touchet silt loam between 20 to 100%; fo silt loam between 50 to 100%; fo Guelph loam between 45 to 100%; fo Yolo light cla between 45 to 100%; and fo the Speswhite kaolin between 55 to 100%. Fo each soil, the SWCC model paametes wee adjusted to match the expeimental data. Fom the Figues 1(a)-(h), it can be seen that the Shada and Ravichandan (2010) SWCC model closel matches the expeimental data. Howeve, pedicting the suction beond the available expeimental data ange, i.e., in the low degee of satuation ange fo all soils, is a challenging task since the patten of vaiation is unknown. In this stud, the SWCC model paametes ae adjusted not onl to match the measued data but also to each an assumed maximum suction fo each soil. Although some eseaches assumed infinit as the maximum possible suction [19],. [5] poved using themodnamic pinciples that maximum suction fo an soil is 10 6 kpa. It was shown in that the measued moistue-suction data wee fitted well with Soil Table 1. Popeties of the selected soils. Poosit Plasticit index (%) Lakeland sand Refeence Elzeftaw & Catwight 1981 Supestition sand Richads 1952 Columbia sand loam unknown Books & Coe 1964 Touchet silt loam Books & Coe 1964 Silt loam unknown Reisenaue 1963 Guelph loam Elick & Bowmann 1964 Yolo light cla Mooe 1939 Speswhite kaolin unknown Peoni et al. 2003

5 488 N. RAVICHANDRAN ET AL. (a) (b) (c) (d) (e) (f) (g) (h) Figue 1. Calibation of chandan SWCC model paametes fo vaious soils. model poposed b Kishnapillai and Ravichandan [20] with maximum suction less than the theoetical maximum compaed the Fedlund and Xing model with the maximum suction of 10 6 kpa. In this stud, maximum possible suctions of 10 5 kpa and 10 6 kpa ae assumed fo sand and clae soil, espectivel. The calibated SWCC

6 N. RAVICHANDRAN ET AL. 489 moel paametes fo the Kishnapillai and and Ravichandan [20] model and Fedlund and Xing [17] models ae listed in Table 2. The shape of the SWCCs fo the fist fou soils matches a tpical shape of sand soils (i.e. exhibiting a sudden dop in the vaiation of degee of satuation when the suction is appoximate to the ai-ent value). The calibated values of n fo these soils ae also elativel high (highe than 6). It is appaent that the Tuochet silt loam (Figue 1(d)) consists of consideable amount of sand, since its SWCC is analogous to the tpical shape of sand soil. Similal, the last thee figues (Figues 1(f)- (h)) show a tpical shape of clae soils (i.e. a unifom eduction the degee of satuation when the suction inceases and with a elativel small calibated value of n, less than 2). The shape of the SWCC of the Silt loam, shown in Figue 1(e), looks simila to a tpical SWCC of clae soil; it can thusl be assumed that the amount of cla in the Silt loam is moe than the amount of sands Pediction of Relative Pemeabilit The pemeabilit coefficients of the above mentioned eight soils wee pedicted using the poposed pemeabilit model that uses the same fitting paametes that wee calibated and match the expeimental SWCC. Figue 2 illustates the pediction of elative pemeabilit of Supestition sand, which is compaed with expeimental data (fom [21]) and pediction fom the. model [5] model. It should be noted that the poposed pemeabilit model paametes ae not calibated o adjusted to match the measued pemeabilit values. Instead, the model paametes ae calibated b matching the measued SWCC used to pedict the pemeabilit using the poposed model. The poposed model shows bette pediction while the. method shows small deviation at highe suction ange (at a low degee of satuation). The pedicted elative pemeabilit of Lakeland sand (expeimental data fom [22]), is shown in Figue 3. As illustated in the figue the poposed model shows a bette pediction compaed to the. model. The. pediction significantl diffes in the highe suction ange. When the suction is appoximatel 100 kpa (with a degee of satuation of 30%), the diffeence between the pedictions b. and the autho s poposed model is appoximatel one ode of magnitude. When the suction is appoximatel 1000 kpa (degee of satuation of 20%), the diffeence neal doubles to an appoximatel incease of neal two odes of magnitude. The pedicted elative pemeabilit of Columbia sand loam is shown in Figue 4. As shown thee, the new model and the. model pedict the ai = 2.25 kpa a = 1.35 n = 7.25 m = 1.0 N = 1 max Figue 2. Compaison of elative pemeabilit of wate fo Supestition sand (expeimental data Richads 1952) ai = 2 kpa a = 1.5 n = 7 m = N = 1 max Figue 3. Compaison of elative pemeabilit of wate fo Lakeland sand (expeimental data Elzeftaw and Catwight 1981). ai = 5 kpa a = 1.4 n = 8.5 m = 1 max Figue 4. Compaison of elative pemeabilit of wate fo Columbia sand loam (expeimental data Books & Coe 1964).

7 490 N. RAVICHANDRAN ET AL. expeimental data (expeimental data fom [9]) well in the lowe suction ange (highe degee of satuation). Howeve, the accuac of these two models in the highe suction ange (lowe degee of satuation) could not be veified because the expeimental esults ae available onl fo the lowe suction anges (less than 12 kpa). A simila discepanc is obseved fo the Touchet silt loam as shown in Figue 5 (expeimental data fom [9]). The pediction and compaison fo the Silt loam ae shown in Figue 6 (expeimental data fom [23]). Of paticula inteest is the obsevation that the poposed model matches the expeimental data well while the Fedlund et al. model is shifted to the ight. Figues 7-9 show the pedictions and compaisons of the elative pemeabilit of Guelph loam (data fom [24]), Yolo light cla (data fom [25]), and Speswhite kaolin (data fom [26]), espectivel. Although the pedictions ai = 7 kpa a = 1.35 n = 7.5 m = 1.35 N = 1 max Figue 5. Compaison of elative pemeabilit of wate fo Touchet silt loam (GE3) (expeimental data Books & Coe 1964). ai = 10 kpa a = 4 n = 1.95 m = 2.8 max N = Figue 6. Compaison of elative pemeabilit of wate fo Silt loam (expeimental data Reisenaue 1963) ai = 3 kpa a = 3.65 n = 1.8 m = 0.9 N = 4 max = 10 6 kpa Figue 7. Compaison of elative pemeabilit of wate fo Guelph loam (expeimental data Elick & Bowmann 1964). ai = 1.5 kpa a = 3.75 n = 1.71 m = N = 4 max = 10 6 kpa Figue 8. Compaison of elative pemeabilit of wate fo Yolo light cla (expeimental data Mooe 1939). ai = 10 kpa a = 5.7 n = 2 m = max = 10 6 kpa N = Figue 9. Compaison of elative pemeabilit of wate fo Speswhite kaolin (expeimental data Peoni et al. 2003).

8 N. RAVICHANDRAN ET AL. 491 ae compaable fo Guelph loam, as shown in Figue 7, both models show slight deviations fom the measued data. In the case of Yolo light cla, the diffeence between the expeimental data and the. pediction inceases as the suction inceases (Figue 8) while the poposed model matches the expeimental data well. Because expeimental data fo the Speswhite kaolin is available fo onl a naow ange of suction (Figue 9), possible pedictive capabilit is not elucidated hee. Fom these obsevations, the poposed model pedicts the expeimental values well while the. model (one of the cuentl available popula models) shows significant diffeences in the highe suction ange. 4. Conclusions A new elative pemeabilit function fo wate in unsatuated soil was developed using the SWCC and the SWCC model paametes of the soil. The capabilit and the accuac of the new pemeabilit function wee veified b compaing the pedictions of the new pemeabilit function with both expeimental values and pedicttions of. s model fo eight diffeent soils. The compaisons show that the new model pedicts the expeimental data well ove a wide ange of suction (0-1,000,000 kpa) and the accuac of the new model in highe suction ange seems bette than the. model. The poposed elative pemeabilit equation must be used with the coesponding equation fo the soil wate chaacteistic cuve. Because the model paametes in these two equations wee identical, the model paametes can be obtained b calibating against the measued SWCC fo the soil instead of the pemeabilit coefficients. It should be noted, howeve, that measuing SWCC fo a soil ove the full ange of degee of satuation is easie than measuing the pemeabilit coefficient. This is a singula advantage of the autho s poposed model. Based on the autho s expeience, this new model is capable of pediction the pemeabilit of wate in unsatuated soils and can be used in finite element simulation of flow and defomation poblems in unsatuated soils. 5. Refeences [1] A. Hazen, Wate Suppl, Ameican Civil Enginees Handbook, Wile, New Yok, [2] R.P. Chapuis, Pedicting the Satuated Hdaulic Conductivit of Sand and gavel Using Effective Diamete and Void Ratio, Canadian Geotechnical Jounal, Vol. 41, No. 5, 2004, pp doi: /t [3] C. P. K. Gallage and T. Uchimua, Effects of D Densit and Gain Size Distibution on Soil-Wate Chaacteistic Cuves of Sand Soils, Soils and Foundations, Vol. 50, No. 1, 2010, pp doi: /sandf [4] A. Lloet and E. E. Alonso, Consolidation of Unsatuated Soils Including Swelling and Collapse Behavio, Géotechnique, Vol. 30, No. 4, 1980, pp doi: /geot [5] D. G. Fedlund, A. Xing, S. Huang, Pedicting the Pemeabilit Function fo Unsatuated Soils Using the Soil-Wate Chaacteistic Cuve, Canadian Geotechnical Jounal, Vol. 31, No. 4, 1994, pp doi: /t [6] D. G. Fedlund and A. Xing, Equations fo the Soil- Wate Chaacteistic Cuve, Canadian Geotechnical Jounal, Vol. 31, No. 3, 1994, pp doi: /t [7] G. P. Wind, Field Expeiment Concening Capilla Rise of Moistue in Heav Cla Soil, Nethelands Jounal of Agicultual Science, Vol. 3, 1955, pp [8] W. R. Gadne, Some Stead State Solutions of the Unsatuated Moistue Flow Equation with Application to Evapoation fom a Wate Table, Soil Science, Vol. 85, 1958, pp doi: / [9] R. H. Books and A. T. Coe, Hdaulic Popeties of Poous Media, Hdolog Pape, Coloado State Univesit, Fot Collins, [10] P. E. Rijtema, An Analsis of Actual Evapotanspiation, Agicultual Reseach Repots, Wageningen, 1965, p [11] J. M. Davidson, L. R. Stone, D. R. Nielsen and M. E. Laue, Field Measuement and Use of Soil-Wate Popeties, Wate Resouces Reseach, Vol. 5, 1969, pp doi: /wr005i006p01312 [12] J. D. Campbell, Poe pessues and volume changes in unsatuated soils, Ph.D. Thesis, Univesit of Illinois at Ubana-Champaign, Ubana, [13] Y. Mualem, Hsteetical Models fo Pediction of the Hdaulic Conductivit of Unsatuated Poous Media Media, Wate Resouces Reseach, Vol. 12, No. 6, 1976, pp doi: /wr012i006p01248 [14] Y. Mualem, A New Model fo Pedicting the Hdaulic Conductivit of Unsatuated Poous Media, Wate Resouces Reseach, Vol. 12, No. 3, 1976, pp doi: /wr012i003p00513 [15] M. Th. van Genuchten, A Closed Fom Equation fo Pedicting the Hdaulic Conductivit of Unsatuated Soils, Soil Science Societ of Ameica Jounal, Vol. 44, No. 5, 1980, pp doi: /sssaj x [16] E. C. Leong and H. Rahadjo, Pemeabilit functions fo unsatuated soils, Jounal of Geotechnical and Geoenvionmental Engineeing, Vol. 123, No. 12, 1997, pp doi: /(asce) (1997)123:12(1118) [17] D. G. Fedlund and A. Xing, Equations fo the Soil-

9 492 N. RAVICHANDRAN ET AL. Wate Chaacteistic Cuve, Canadian Geotechnical Jounal, Vol. 31, No. 4, 1994, pp doi: /t [18] D. G. Fedlund, M. D. Fedlund and N. Zakezadeh, Pedicting the Pemeabilit Function fo Unsatuated Soils, Intenational Confeence on Clas and Cla Mineolog, Shioukoza, Janua 2001, pp [19] J. P. Lobbezoo and S. K. Vanapalli, A Simple Technique fo Estimating the Coefficient of Pemeabilit of Unsatuated Soils, Poceedings of the 55th Canadian Geotechnical Confeence, Niaga Falls, [20] H. K. Shada and N. Ravichandan, New Soil-Wate Chaacteistic Cuve and Its Pefomance in Finite Element Simulation of Unsatuated Soils, Intenational Jounal of Geomechanics, accepted with mino evision, Octobe [21] L. A. Richads, Wate Conducting and Retaining Popeties of Soils in Relation to Iigation, Poceedings of Intenational Smposium on Deset Reseach, Jeusalem. 1952, pp [22] A. Elzeftaw and K. Catwight, Evaluating the Satuated and Unsatuated Hdaulic Conductivit of Soils, ASTM Standads and Engineeing Digital Liba, West Conshohocken, 1981, pp doi: /stp28323s [23] A. E. Reisenaue, Methods fo Solving Poblems of Multi-Dimensional Patiall Satuated Stead Flow in Soils, Jounal of Geophsical Reseach, Vol. 68, No. 20, 1963, pp [24] D. E. Elick and D. H. Bowman, Note on an impoved appaatus fo soil moistue flow measuements, Soil Science Societ of Ameica Poceedings, Vol. 28, 1964, pp doi: /sssaj x [25] R. E. Mooe, Wate Conduction fom Shallow Wate Tables, Univesit of Califonia, Bekele, 19377, pp [26] N. Peoni, E. Fatalocchi and A. Taantino, Wate Pemeabilit of Unsatuated Compacted Kaolin, Poceedings of the Intenational Confeence, Weima, Septembe 2003.