This is a repository copy of Numerical analysis of seepage deformation in unsaturated soils.

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1 This is repository copy of Numericl nlysis of seepge deformtion in unsturted soils. White Rose Reserch Online URL for this pper: Version: Accepted Version Article: Liu, E, Yu, H-S, Deng, G et l. (2 more uthors) (2014) Numericl nlysis of seepge deformtion in unsturted soils. Act Geotechnic, 9 (6). pp ISSN Springer-Verlg Berlin Heidelberg This is post-peer-review, pre-copyedit version of n rticle published in Act Geotechnic. The finl uthenticted version is vilble online t: Uploded in ccordnce with the publisher's self-rchiving policy. Reuse Unless indicted otherwise, fulltext items re protected by copyright with ll rights reserved. The copyright exception in section 29 of the Copyright, Designs nd Ptents Act 1988 llows the mking of single copy solely for the purpose of non-commercil reserch or privte study within the limits of fir deling. The publisher or other rights-holder my llow further reproduction nd re-use of this version - refer to the White Rose Reserch Online record for this item. Where records identify the publisher s the copyright holder, users cn verify ny specific terms of use on the publisher s website. Tkedown If you consider content in White Rose Reserch Online to be in brech of UK lw, plese notify us by emiling eprints@whiterose.c.uk including the URL of the record nd the reson for the withdrwl request. eprints@whiterose.c.uk

2 1 2 Numericl Anlysis of the Seepge-Deformtion in Unsturted Soils Enlong Liu 1, Hi-Sui Yu 2, Gng Deng 3, Jinhi Zhng 1, Siming He 4 1 Stte Key Lbortory of Hydrulics nd Nturl River Engineering, College of Wter Resource nd Hydropower, Sichun University, Chengdu, P.R. Chin ; 2 Nottinghm Centre for Geomechnics, University of Nottinghm, NG7 2RD, UK; 3 Stte Key Lbortory of Simultion nd Regultion of Wter Cycle in River Bsin, Chin Institute of Wter Resources nd Hydropower Reserch, Beijing , P.R. Chin; 4 Institute of Mountin Hzrds nd Environment, CAS, Chengdu , P.R. Chin. Corresponding uthor: Enlong Liu, Emil: liuenlong@scu.edu.cn Abstrct: A coupled elstic-plstic finite element nlysis bsed on simplified consolidtion theory for unsturted soils is used to investigte the coupling processes of wter infiltrtion nd deformtion of unsturted soil. By introducing reduced suction nd n elstic-plstic constitutive eqution for the soil skeleton, the simplified consolidtion theory for unsturted soils is incorported into n in-house finite element method code. Using the numericl method proposed here, the genertion of pore wter pressure nd development of deformtion cn be simulted under evportion or rinfll infiltrtion conditions. Through prmetric study nd comprison with the test results, the proposed method is found to describe well the chrcteristics during wter evportion/infiltrtion into unsturted soils. Finlly, n unsturted soil slope with wter infiltrtion is nlyzed in detil to investigte the development of the displcement nd genertion of pore wter pressure. Keywords: unsturted soil, wter infiltrtion, seepge-deformtion coupled nlysis, finite element method. Introduction There re currently mny geotechnicl engineering problems tht re relted to unsturted soils, such s rinfll induced slope filure nd expnsive soils. For embnkment nd slope engineering, filures re usully induced by rinfll infiltrtion becuse wter infiltrtion will result in n increse in sturtion nd 1

3 chnge in the distribution of pore-wter pressure, which in turn leds to chnge in the stress nd deformtion of soil. The chnge of stress nd deformtion, contrrily, ffects the seepge process by modifying the hydrulic properties of porosity nd permebility of the soil. Bsed on these results, it is necessry to conduct coupled nlysis of wter infiltrtion nd deformtion to study the filure mechnism of unsturted soil engineering; such coupled nlysis is becoming more importnt [16]. Mny experimentl studies hve been conducted to investigte the coupling interctions of flow through unsturted soils with rinfll infiltrtion. Columns composed of sndy mterils hve been tested to investigte the lekge s result of downlod dringe from n initilly sturted stte [23], nd new soil column pprtus for lbortory infiltrtion study ws developed tht cn mesure ll the vribles (porewter pressures, wter contents, inflow rte nd outflow rte) instntneously nd utomticlly in n infiltrtion process by control of ll the boundry conditions [36]. Becuse more lyered soils re encountered thn uniform soils in geotechnicl engineering, infiltrtion in lyered soils hs drwn much ttention nd hs been studied by mny reserchers [30, 33]. The lbortory test results of verticl infiltrtion in two soil columns of finer over corse soils subjected to simulted rinflls under conditions of no-ponding t the surfce nd constnt hed t the bottom hve been presented [37], nd n experimentl device used to investigte the trnsient unsturted-sturted hydrulic response of sndgeotextile lyers under conditions of 1-D constnt hed surfce infiltrtion hve been discussed in detil [4]. In ddition to lbortory experiments, in-situ tests hve been performed to observe the flow through unsturted soil msses. For exmple, studies were performed under nturl nd simulted rinfll on residul soil slope in Singpore [28] tht ws instrumented with pore wter pressure, wter content, nd rinfll mesuring devices. Rinfll infiltrtion into unsturted soils hs lso been nlyzed by nlyticl solutions for simple initil nd boundry conditions. Ignoring the coupling effects of flow nd deformtion, Morel-Seytoux [25, 26] 51 obtined n nlyti Bsh 52 [2, 3] derived multidimensionl non-stedy solutions for domins with prescribed surfce flux boundry 53 conditions nd bottom boundry conditions by using on, nd Chen et l. [11] obtined series solution tht hs the merit of esy clcultion by using Fourier integrl trnsformtion. To improve the understnding of the influence of hydrulic properties nd rinfll conditions on rinfll infiltrtion 2

4 56 mechnism nd hence on the pore-wter pressure distributions in single- nd two-lyer unsturted soil 57 systems, n nlyticl prmetric study ws performed [38]. Bsed on -liner elstic constitutive model for unsturted soil, the nlyticl solution to one-dimensionl coupled wter infiltrtion nd deformtion problem by Fourier integrl trnsform ws obtined; the results indicted tht the volume chnge due to chnge in soil suction nd the rtio of rinfll intensity to sturted permebility hs significnt effect on the distribution of the negtive pore wter pressure nd deformtions long the soil profile [35]. By ssuming tht the verticl loding vries exponentilly with time, n nlyticl solution to one-dimensionl consolidtion in unsturted soils with finite thickness under confinement in the lterl direction ws lso presented [29]. Recently, to ddress complicted initil nd boundry conditions, numericl solutions hve been used to nlyze the soil engineering relted to unsturted soils. Using the modified Mohr-Coulomb filure criterion [12], the process of infiltrtion into slope due to rinfll nd its effect on soil slope behvior were exmined using two-dimensionl finite element flow-deformtion coupled nlysis progrm. The finite element nlysis of trnsient wter flow through unsturted-sturted soils [10] ws used to investigte the effects of the hydrulic chrcteristics, the initil reltive degree of sturtion, the methods used to consider boundry conditions, nd the rinfll intensity nd durtion on the wter pressure in the slopes. The elsto-plstic finite element model [18] in conjunction with novel nlyticl formultion for the suction stress bove the wter tble were used to nlyze the unsturted slope stbility. A seepge deformtion coupled pproch [15] including the effective pore-pressure nd the effective stress concept ws used to nlyze the deformtion nd the locliztion of strins on unsturted soils due to seepge flow. The net stress concept [1] ws used to compute the deformtions nd the vrition of the sfety fctor with time of n unstble slope in profile of wethered overconsolidted cly using n unsturted coupled hydromechnicl model. Bsed on the theory of porous medi [17], multiphse coupled elstoviscoplstic finite-element nlysis formultion ws used to describe the rinfll infiltrtion process into one-dimensionl soil column, which demonstrtes tht the genertion of pore wter pressure nd volumetric strin is minly controlled by mteril prmeters tht describe the soil wter chrcteristic curve. Bsed on the existing hydro-mechnicl model for non-expnsive unsturted soil, n elstoplstic constitutive model ws developed for predicting the hydrulic nd mechnicl behvior of unsturted 3

5 expnsive soils [34]. To consider the sptil vribility of the properties of the mterils of the soil deposits, stochstic finite element model of unsturted seepge through flood defense embnkment with rndomly heterogeneous mteril ws presented [22]. The stochstic finite element model ws lso used to simulte the contminnt trnsport through soils, focusing on the incorportion of the effects of soil heterogeneity [27]. Recently, Borj et l. proposed physicl-bsed hydro-mechnicl continuum model nd employed finite element method to study the series of properties of unsturted soils, which included the following: (i) the stresses, pore pressures, nd deformtion within slope were generted, nd then the fctor of sfety ws determined with limit-equilibrium solution [6]; (ii) the 3-D multi-physicl spects triggering lndslides were quntified by ccounting for the loss of sediment strength due to incresed sturtion s well s the frictionl drg exerted by the moving fluid [7]; (iii) the effect of the sptilly vrying degree of sturtion on triggering sher bnd in grnulr mterils ws investigted with new vritionl formultions for porous solids nd two criticl stte formultions [8]; nd (iv) mthemticl frmework for coupled solid-deformtion/fluid-diffusion model in unsturted porous mteril considering geometric nonlinerity in the solid mtrix ws developed tht relies on the continuum principle of thermodynmics to identify n effective or constitutive stress for the solid mtrix, long with wter-retention lw tht highlights the interdependence of the degree of sturtion, suction, nd porosity of the mteril [32]. From the relevnt works mentioned bove, we know tht the coupling processes of wter infiltrtion nd deformtion of unsturted soils is n interesting topic tht hs not yet been fully studied. Solving the problems of unsturted soil engineering requires slightly simplifying the generl consolidtion theory for unsturted soil, which hs so mny unknowns [13, 20] nd hs been implemented using numericl methods by only few reserchers. By introducing reduced suction nd n elstic-plstic constitutive eqution for the soil skeleton, coupled elstic-plstic finite element nlysis bsed on simplified consolidtion theory for unsturted soils [14, 31], compred with the generl or stndrd consolidtion theory for unsturted soils, is used here to investigte the coupled processes of wter infiltrtion nd deformtion of unsturted soil. The genertion of pore wter pressure nd deformtion is simulted under rinfll infiltrtion conditions nd then compred with the experimentl results, which enbles the detiled nlysis of n unsturted soil slope with wter infiltrtion to investigte the development of displcement nd genertion of pore wter pressure. 4

6 Simplified consolidtion theory for unsturted soils Effective stress The formultion is bsed firmly within the modified for unsturted soils by Bishop [5] in the form: u ( u u w), (1) in which nd re the effective nd totl stress, respectively, u is the ir pressure in the voids, nd u is the pore wter pressure within the unsturted soil mtrix. The term s ( u u ) is clled the w mtrix suction. We define the reduced suction s by the following expression [31]: s ( u u ). (2) w So we hve s / s, clled the coefficient of reduced suction. As shown in Fig. 1, the reduced suction cn be determined by compring the results of the compression curve of sturted soils with those of the drying shrinkge curve of the unsturted soils. Assuming p nd s re the pressure sustined by sturted soils nd mtrix suction induced in the unsturted soils with the sme void rtio, respectively, we hve w 125 s pby the equivlent strin theory, so the coefficient of reduced suction cn be described by the 126 expression: p/ s. Under the function of the suction s. When the suction is smller thn the ir entry vlue s b, the coefficient of reduced suction is equl to unity; when suction is smller thn the ir entry vlue s b, the coefficient of reduced suction vries with the mtrix suction in the mnner proposed here s follows: s s m2 ( ), in which sb ( u u w) b nd m 2 is mterils constnt. Pore ir pressure b (3) To signify the ir content (ir mss) in the voids of soil element within unit volume, we define the rtio of the pore ir of unit soil element, n, s follows (see ppendix I): in which n is the porosity, h n [1 (1 ch) Sr], n (4) c is the Henry coefficient of solubility nd S r is the degree of sturtion of the 137 cn be obtined: 5

7 (1 u / p ), (5) in which, 0 is the initil density of ir in the void, u is the excess ir pressure bove the reference dtum p, which is the tmospheric pressure tht is equl to P, nd u p is the bsolute ir pressure. Under the conditions tht the ir in the voids cnnot be dischrged, we hve (see Appendix II): in which, 0 h r0 0 u n 0 ( 1) p, n n [1 (1 c ) S ] n, S r0 is the initil degree of sturtion, nd n 0 is the initil porosity. (6) If the pore ir within the soil element cn be dischrged freely (tht is, the excess pore ir could be completely dischrged), the pore ir pressure generted will finlly dissipte nd is equl to the tmospheric pressure ( u 0 density of the pore ir will remin constnt. Under these conditions, to dissipte the pore ir pressure completely, the chnge of the volume of the pore ir within unit soil element per unit time is mss of the pore ir dischrged within unit soil element per unit time is n. n, so the 150 Under the conditions tht the ir in the voids is dischrged prtilly, it is ssumed tht the mss of ir 151 dischrged per unit time is q, so we define the dischrge speed of pore ir,, which is the rtio of the mss of the pore ir under dischrged prtilly per unit time to tht of the pore ir dischrged completely per unit time, s follows: q n So we cn obtin the expression of the increment of pore-ir pressure s follows (see Appendix III):. (7) u p u n (1 ) n. If is constnt, by integrting eqution (8), we obtin the expression of u s follows (see Appendix IV): u n 0 (1 ) [( ) 1] p. n When is equl to 0, the eqution (9) cn be reduced to eqution (6); when is equl to 1, u 0, which corresponds to the conditions tht the ir is dischrged completely. Governing equtions (8) (9) 6

8 Ignoring the flow of dissolved ir in the pore-wter, the vpor in the pore-ir nd the influence of temperture, the consolidtion equtions for unsturted soils hve the following expressions [31]: (10) Equilibrium equtions in incrementl form Continuous equtions of pore-wter [ L] d db 0, (11) Continuous equtions of pore-ir uw ( Srn) div kwgrd z, t g w (12) The reltionship of the effective stress-displcement t u (1 S ) n c S n div k grd, g r h r 174 T d D L du, (13) The reltionship of the sturtion-mtrix suction The coefficient of permebility of the pore-wter The coefficient of permebility of the pore-ir S f ( s ), (14) r r k ( ), w fw s (15) in which, k f ( s ), (16) 7

9 0 0 0 x y z 183 L y x z, { d } nd { d } re the increments of the totl stress nd z y x the effective stress, respectively, { db} is the increment of the body force, g is the grvittionl ccelertion, [D] is the mtrix of stress-strin nd { du } is the increment of displcement. 186 Simplified equtions in 2-Dimensions In this mnuscript, we use eqution (9) to describe the chnge of pore ir pressure upon loding, so eqution (16) will not be used nd the displcements nd pore wter pressure re solved simultneously. n n Sr s w n v Sr From eqution (4), we hve n ( u u ) nd v n for soil element, so the increment of totl stress from eqution (1) cn be described s follows: A in which 1 s 1 n Sr P s Sr s n S, 2 r P S s r A A u A (17) 1 w 2 v, n ( s 1) P s n, P (1 )( p u)/ n, n Sr 1 P S s r n / Sr (1 ch) n nd n / n 1 (1 c ) S. Substituting the bove equtions into equtions (10), we obtin: h r ux ux ux ( d11 A2 ) ( d 2 14 d41) d44 2 x x z z uz uz uz uw d14 ( d 2 12 d44 A2 ) d42 A 2 1 Fx, x x z z x ux ux ux d41 ( d 2 21 d44 A2 ) d24 2 x x z z uz uz uz uw d44 ( d 2 24 d42 ) ( d22 A2 ) A 2 1 Fz, x x z z z (18) (18b) 8

10 197 in which ux nd uz re the increments of the horizontl (or x) nd verticl (or z) displcements, respectively, d 11, d 12 re the elements of the elstic-plstic mtrix of stress-strin, Fx nd increments of lods in the horizontl (or x) nd verticl (or z) directions, respectively. From eqution (11), the continuous eqution of pore-wter cn be formulted s follows, Fz re the in which, / u k h k h n S t x x z z t h u g z, kwx nd kwz w w verticl directions, respectively, nd / w wx wz v r r, (19) re the coefficients of permebility in the horizontl nd S u. Constitutive equtions for unsturted soils w 205 Soil-wter chrcteristic curve The soil-wter chrcteristic curve is divided into two sections ccording to the vlues of the mtrix suction s nd the ir entry suction s b. When the vlue of mtrix suction s is smller thn tht of the ir entry suction s b, the soil cn be ssumed to be qusi-sturted nd the degree of sturtion of qusi-sturted soil is ssumed to be S r1 (the vlue of S r1 pproches 1.0, such s 0.96), so the degree of sturtion cn be expressed s follows using Hilf formultion [19]: p ( uw sb) Sr Sr1 ( s s b), (20) p (1 c ) S ( u s ) h r1 w b in which S r1 is the degree of sturtion of soil msses when the mtrix suction is equl to the vlue of ir entry suction degree of sturtion s b. When the vlue of mtrix suction s is greter thn tht of the ir entry suction Sr is computed s follows [9]: s S S S S s s m1 r ro ( r1 ro)( ) ( b), sb s b, the in which Sr0 nd m 1 re soil constnts. During the process of drying shrinkge or bsorbing wter, the (20b) 217 prmeters S r 0, r1 S nd s b my be different. By the derivtion of eqution (20b), we hve: 218 S S m 1 m1 ( r1 ro) ( ). s sb s (21) 9

11 The effect of the degree of sturtion on the permebility of unsturted soils is ssumed s follows: where s sb kw kwsexp( ck ), p kws is the permebility for wter under sturted conditions nd c k is constnt. When (22) s s b, 222 k k. w ws 223 Elstic-plstic model for soil skeleton According to the principle of effective stress, i.e., eqution (1), the stress in the following equtions of this section refers to the effective stress. Ignoring the influence of temperture, the double hrdening elsticplstic model for sturted soils [24] is used to describe the mechnicl fetures of the soil skeleton of unsturted soils. 228 Let 1 m kk 3, 3 2 s ij ij ss, s ij ij kk ij, 2 ee, 3 s ij ij e 1 3 ij ij kk ij.the yield 229 function of the model is expressed s follows: in which, / s F(, p p, ) p v s p ( v ), (23) m 1 [ ] p ( ) s m, nd m is the prmeter of yield function. When m = 1.2, the shpe of the yield 232 surfce is close to n ellipse, s shown in Fig. 2. p nd re the two hrdening prmeters, which cn be 233 expressed s the functions of plstic volumetric strin p v nd plstic sher strin p s, respectively, s 234 follows, p p v p0 exp( ), c c nd m m 0 in which, ee, e 3 p p, p 2 p p v kk s ij ij curve nd rebound curve, respectively, nd 0 c e p ( )exp( s ), 1 3 p p p ij ij kk ij c, cc nd p p is the reference pressure with 0 (24) (25) c e re the slopes of compressionl v. Eqution (24) is in the sme form s the hrdening prmeter of originl Cm-cly model. In eqution (25), 10

12 m m ( 1 m )sin, is internl frictionl ngle, nd 0 nd c re two other prmeters tht cn be determined by unloding trixil compression test, in which the xil lod is kept constnt nd the confining pressure is reduced grdully. Assuming tht the flow rule is ssocited, the plstic strin increment cn be determined by use of elstic-plstic theory s follows, F d d (26) p ij ij Or (27) (27b) F d d, p v m p 3 F d s d, 2 s 250 where d is the plstic multiplier, which cn be derived from the consistency conditions, 251 F F F d d d p p ij p v p s ij v s (28) Substituting for the plstic volumetric strin increment p d s in equtions (27-) nd (27-b), d is obtined s follows, p d v nd the plstic sher strin increment 255 d F d ij H ij (29) in which the Hrdening modulus H cn be expressed s 258 H 3 F F F F 3 F F F p F. p p p p 2 2 p s s v m s s v m (30) Formultions of the finite-element equtions 11

13 Anlyticl solutions to one-dimensionl seepge-deformtion coupled problem in unsturted soil could be derived by considering homogeneous elstic mteril. Anlyticl solutions re simple nd esy to implement but cnnot ccount for the complex initil nd boundry conditions, the soil heterogeneities [8, 32], the nonliner stress-strin nd the hydrulic reltions involved in prcticl geotechnicl problems, wheres the numericl solutions re more prcticl due to their flexibility. In the following, the formultions of finite element equtions for the theory proposed here re presented. Isoprmetric elements re implemented with eight-node interpolting functions for the displcements (u x, u z ) nd four node interpolting functions for the pore wter pressures (u w ), which cn be expressed s follows, in which u xi, uzi nd u Nu, u Nu, u Nu, (31) x i xi z i zi w i wi i 1 i 1 i 1 uwi re the nodl vribles t nodl point I nd i N nd N i re interpolting functions for the displcements nd pore-wter pressure, respectively. Wek forms of equtions (18), (18b) nd (19) re discretized in spce nd solved by the finite element method [21]. The time domin is divided into number of elements or steps, nd then the integrtion is performed for ech step to obtin the chnges of the prmeters u xi, uzi nd u wi. The step-by-step integrtions my then be summed to determine the totl chnge of the prmeters. The bckwrd differentition method is used for the time discretiztion of the equtions (18), (18b) nd (19) s follows: 278 tk tk tk Gdt t 1 G G t G G (32) k tk tk tk k tk 279 where tk is the time increment, Gt k nd G tk tk re the corresponding function vlues t steps t k nd 280 t k t k, respectively, nd G is the incrementl function vlue nd is n integrl prmeter. When > 0.5, the difference formt is unconditionlly stble, nd thus, Therefore, the finite element equtions for node i (i = 1, = 2/3 is used here. N t ) cn be formulted s follows, 283 N t [ ij xj ij zj ij j] xi, j 1 k u k u k h F (33) 284 N t [ ij xj ij zj ij j ] zi, j 1 k u k u k h F (33b) 12

14 285 nd N t [ ij xj ij zj ij ( j 0 j ) ij j ] i, j 1 k u k u k h h s h Q (33c) in which N t is the totl number of nodl points;, Fxi Fzi nd i Q re the lod increments nd flux 288 increment t node i, respectively; h i0 nd h i re the initil vlue of the wter hed nd the increment of 289 the wter hed t node i in computtionl time step t k ; h i =z i + u wi w, in which, z i is the loction of the 290 wter hed t node i; w is the weight of wter. The elements of the coefficient mtrix in the finite element equtions (33) re given in the Appendix V. Numericl simultions Simultion of the experimentl results The proposed benchmrk is bsed on n experiment performed by Likopolos [23] on column of Del Monte snd nd instrumented to mesure the moisture tension t severl points long the column during its desturtion due to grvittionl effects. Before the strt of the experiment, wter ws continuously dded from the top nd llowed to drin freely t the bottom through filter until the uniform flow conditions were estblished. At the strt of the experiment, the wter supply cesed, nd the tensiometer redings were recorded, s well s the outflow nd outflow rte t the bottom. The test digrm is shown in Fig. 3(). The initil conditions re s follows: for t=0, s=0 for ll the nodes, which corresponds to stedy flow of wter through the snd column. Furthermore, stte of mechnicl equilibrium is ssumed for t=0. All the displcements re relted to these initil displcements, which correspond to the equilibrium stte. The boundry conditions re s follows. For the lterl surfce: there is no flow horizontlly, nd q w =0 nd u x =0. For the top surfce: t>0 nd u =p. For the bottom surfce: t>0, free wter outflow, u =p, s=0 for t>0, u x = u z =0. The computed prmeters used here re s follows: =2.0 g/cm 3, k o n o =0.2975, k ws =0.03 cm/s, c k =1.0, c c =0.006, c e =0.0001, S ro =0.1, S r1 =0.96, m 1 = m 2 =0.1, s b =25 kp, m=1.0, o=0.75, c =0.05 nd =0.6. Fig. 3(b) presents the simulted results nd the tested results of the development of the pore wter 13

15 pressure t different depths. Compred with the test results nd the simulted results proposed previously, the new method proposed here cn model the drying shrinkge process of unsturted soils. Prmetric study () Effects of dischrge speed of pore ir The mesh, shown in Figure 4(), is used to simulte seepge-deformtion processes under infiltrtion 314 conditions in only the verticl direction (z direction) with different dischrge speeds of the pore ir. The soil mss is 100 cm in length nd 10 cm in width, with infiltrtion through the upper surfce. The initil suction distribution is shown in Fig. 4(b). The boundry conditions re s follows. For the lterl surfce: there is no flow horizontlly, nd q w =0 nd u x =0. For the top surfce: drined freely (wter nd ir). For the bottom surfce: s=0 (wter level locted t the bottom surfce) for t>0 nd u x = u z =0. The soil mss is infiltrted by rinfll with rte of mm/hours, nd the computed prmeters employed re s follows: =2.0 g/cm 3, k o n o =0.432, k ws = cm/s, c k =1.0, c c =0.06, c e =0.01, S ro =0.2, S r1 =0.998, m 1 = m 2 =0.1, s b =25kP, m=1.0, o=0.75 nd c =0.05. The dischrge speeds of 323 pore ir re ssumed to be 0.2, 0.4, 0.6, 0.8 nd 1.0, respectively. 324 Fig. 4 (c)-(e) show the vrition of suction during rinfll infiltrtion with different dischrge speed of 325 pore ir. It is obvious tht the dischrge speed of pore ir hs greter influence on the development of suction in the unsturted soils. In the process of infiltrtion, the mgnitude of the suction with smller is greter thn tht of the suction with lrger becuse the pore ir pressure will decrese with the 328 incresing vlue of. With rinfll infiltrtion, the mgnitude of the suction becomes smller for the sme 329 vlue of (b) Effects of the evportion intensity I The effects of evportion intensity I on the seepge-deformtion coupling processes of unsturted soils re investigted using the computtionl mesh nd boundry conditions shown in Fig. 5 () with different vlues of evportion intensity I. Before the strt of the evportion, the soils re fully sturted. The initil conditions re s follows: for t=0, s=0 for ll the nodes, which corresponds to stedy flow of 14

16 wter through the soil column. Furthermore, stte of mechnicl equilibrium is ssumed for t=0. All the displcements re relted to these initil displcements, which correspond to the equilibrium stte. The boundry conditions re s follows: For the lterl surfce: there is no flow horizontlly, nd q w =0 nd u x =0. For the top surfce: evportion, t>0, u =p nd drined boundry (wter nd ir). For the bottom surfce: u =p, s=0 for t>0, nd u x = u z =0. The computed prmeters used here re s follows: =2.0 g/cm 3, k o n o =0.2975, k ws =0.003 cm/s, c k =1.0, c c =0.006, c e =0.0001, S ro =0.1, S r1 =0.96, m 1 = m 2 =0.1, s b =25kP, m=1.0, o=0.75 nd c =0.05, =0.6. The evportion intensity I = cm/hrs, cm/hrs nd cm/hrs re used here to study the influence of evportion intensity on the fetures of the soil column. Fig. 5 (b)-(d) present the distributions of the pore wter pressure during the process of evportion with different vlues of evportion intensity, which demonstrtes tht the mgnitude of the evportion intensity hs gret influence on the development of pore wter pressure in the unsturted soils. During the process of evportion, the mgnitude of the pore wter pressure (compressive is positive) increses grdully with the sme depth t the sme time. With the increse of the evportion intensity, the mgnitude of the pore wter pressure lso increses grdully t the sme time with the sme depth of the soil column. When the mgnitude of the evportion intensity is greter on the upper surfce of the soil column, there is greter pore wter pressure with negtive vlue, thus resulting in equilibrium of the unsturted soil msses. (c) Effects of the sturted permebility k ws For the computtionl mesh nd boundry conditions shown in Fig. 5(), the effects of the permebility of unsturted soils k ws on the seepge-deformtion coupling processes of unsturted soils re investigted with different vlues of k ws. The initil nd boundry conditions re the sme s those of section (b) 357 discussed bove. The computed prmeters used here re s follows: =2.0 g/cm 3, k o 358 n o =0.2975, I=0.012 cm/hrs, c k =1.0, c c =0.006, c e =0.0001, S ro =0.1, S r1 =0.96, m 1 = m 2 =0.1, s b =25kP, m=1.0, 359 o=0.75, c =0.05, nd =0.6. The sturted permebility k ws =0.03 cm/s, cm/s nd cm/s re used here to investigte the influence of the sturted permebility on the fetures of the soil column. Figs. 6 ()-(c) present the distributions of the pore wter pressure during the process of evportion with different vlues of sturted permebility, which lso demonstrtes tht the mgnitude of the sturted 15

17 permebility ffects the development of pore wter pressure in the unsturted soils gretly. With the decrese of the sturted permebility, the mgnitude of the pore wter pressure increses grdully t the sme time with the sme depth of the soil column. In ddition, the mgnitude of pore wter pressure increses grdully with the sme depth t the sme time for the sme sturted permebility. The greter the sturted permebility is, the quicker the dissiption of pore wter. Thus, we obtined the computtionl results presented here. Simultion of the unsturted soil slope with rinfll infiltrtion The seepge-deformtion processes under evportion nd rinfll infiltrtion conditions re simulted for n unsturted soil slope tht is 20 m in depth nd 54 m in width. The computtionl mesh nd boundry conditions re shown in Figure 7(). For the lterl surfces, there re undrined nd constrined boundries (u x = u z =0); for the bottom surfce, constrined boundry, the pore wter pressure is zero ll the time with the surfce of the wter tble; for the upper surfces composed of three plnes, they drin freely, nd evportion/infiltrtion occurs on these surfces. When t=0, the soil slope is sturted nd is in equilibrium with the weight stress stte. During the first 1100 dys, the evportion rte of the soil slope is 0.3 mm per dy, nd during the subsequent 300 dys, the infiltrtion rte of rinfll is 0.5 mm per dy. The computed prmeters re s follows: =2.0 g/cm 3, k o e o =0.7, k ws = cm/s, c k =0.2, c c =0.0332, c e =0.0064, S ro =0.2, S r1 =0.96, m 1 = m 2 =0.1, s b =20 kp, m=1.0, o=0.75, =0.6 nd c =0.05. Figs. 7 (b) nd (c) present the distribution of the pore wter pressure t the end of the evportion nd rinfll infiltrtion, respectively. At the end of evportion, the pore wter with negtive vlue is highest ner the upper surfces of the soil slope, nd it reduces grdully with the wter infiltrtion into the unsturted soil slope. Figs. 7 (d) nd (e) present the distributions of the displcement t the end of evportion nd rinfll infiltrtion, respectively. From the simultion of the evportion nd rinfll infiltrtion for n unsturted soil slope, the method proposed here cn model the seepge-deformtion process of unsturted soils qulittively. Conclusions 16

18 A new FEM code ws developed on the bsis of simplified consolidtion theory for unsturted soils tht cn be used to nlyze the seepge-deformtion of unsturted soil slope under evportion/rinfll infiltrtion conditions. The numericl exmples demonstrte tht the reduced suction nd constnt dischrge speed of the pore ir cn be introduced to simplify the consolidtion equtions for unsturted soils, thereby mking it esy to progrm the consolidtion theory into the numericl nlysis code of the elstic-plstic finite element method. Through prmeter study nd comprison with the tested results, the results of this study demonstrted tht the proposed method cn describe well the fetures in the process of wter evportion/infiltrtion into unsturted soils. References 1. Alonso EE, Gens A, Delhye CH (2003) Influence of rinfll on the deformtion nd stbility of slope in overconsolidted clys: cse study. Hydrogeology Journl 11: Bsh HA (1999) Multidimensionl linerized non-stedy infiltrtion with prescribed boundry conditions t the soil surfce. Wter Resources Reserch 35(1): Bsh HA (2000) Multidimensionl linerized non-stedy infiltrtion towrd shllow wter tble. Wter Resources Reserch 36(9): Bthurst RJ, Ho AF, Siemens G (2007) A column pprtus for investigtion of 1-D unsturted-sturted response of snd-geotextile system. Geotechnicl Testing Journl 30(6): Bishop AW (1955) The principl of effective stress. Teknsik Ukebld 106: Borj RI, Liu X, White JA (2012) Multiphysics hillslope processes triggering lndslides. Act Geotechnic 7: Borj RI, White JA, Liu X, Wu W (2012) Fctor of sfety in prtilly sturted slope inferred from hydromechnicl continuum modeling. Interntionl Journl for Numericl nd Anlyticl Methods in Geomechnics 36(2): Borj RI, Song X, Wu W. (2013). Cm-cly plsticity, Prt VII: Triggering sher bnd in vribly sturted porous medi. Comput. Methods Appl. Mech. Engrg : Brooks RH, Corey AT (1964) Hydrulic properties of porous medium. Hydrology Pper No.3, Civil Engineering Deprtment, Colordo Stte University, Fort Collins, Co. 10. Ci F, Ugi K (2004) Numericl nlysis of rinfll effects on slope stbility. Interntionl Journl of Geomechnics 4(2):

19 Chen JM, Tn YC, Chen CH (2001) Multidimensionl infiltrtion with rbitrry surfce fluxes. Journl of Irrigtion nd Dringe Engineering 127(6): Cho SE, Lee SR (2001) Instbility of unsturted soil slopes due to infiltrtion. Computers nd Geotechnics 28: Conte E (2004) Consolidtion nlysis for unsturted soils. Cndin Geotechnicl Journl 41: Deng G, Shen ZJ (2006) Numericl simultion of crck formtion process in clys during drying nd wetting. Geomechnics nd Geoengineering: An Interntionl Journl 1(1): Ehlers W, Grf T, Ammnn M (2004) Deformtion nd locliztion nlysis of prtilly sturted soil. Compute Methods in Applied Mechnics nd Engineering 193: Fredlund DG (2006) Unsturted soil mechnics in engineering prctice. Journl of Geotechnicl nd Geoenvironmentl Engineering 132(3): Grci E, Ok F, Kimoto S (2011) Numericl nlysis of one-dimensionl infiltrtion problem in unsturted soil by seepge-deformtion coupled method. Interntionl Journl for Numericl nd Anlyticl Methods in Geomechnics 35(5): Griffiths DV, Lu N (2005) Unsturted slope stbility nlysis with stedy infiltrtion or evportion using elstoplstic finite elements. Interntionl Journl for Numericl nd Anlyticl Methods in Geomechnics 29: Hilf JW (1948) Estimting construction pore pressures in rolled erth dms. In Proc.2 nd Int.Conf. Soil Mech. Found. Eng. Rotterdm, The Netherlnds, vol.3: Khlili N, Vllippn S, Wn CF (1999) Consolidtion of fissured clys. Geotechnique 49: Lewis RW, Schrefler BA (1987).The finite element method in the deformtion nd consolidtion of porous medi. Wiley: New York. 22. Le TMH, Gllipoli D, Sánchez M, Wheeler SJ (2012). Stochstic nlysis of unsturted seepge through rndomly heterogeneous erth embnkments. Interntionl Journl of Numericl nd Anlyticl Method in Geomechnics 36: Likopoulos A (1964). Trnsient flow through unsturted porous medi. Ph.D. Thesis, University of Cliforni t Berkeley. 24. Liu EL, Xing, HL (2009). A double hrdening thermo-mechnicl constitutive model for overconsolidted clys. Act Geotechnic 4: Morel-Seytoux HJ (1978). Derivtion of equtions for vrible rinfll infiltrtion. Wter Resources Reserch 14(4):

20 Morel-Seytoux HJ (1981) Anlyticl results for prediction of vrible rinfll infiltrtion. Journl of Hydrology 59: Nezhd MM, Jvdi AA, Al-Tbb A, Abbsi F. (2013). Numericl study of soil heterogeneity effects on contminnt trnsport in unsturted soil (model development nd vlidtion). Interntionl Journl of Numericl nd Anlyticl Method in Geomechnics 37: Rhrdjo H, Lee TT, Leong EC, Rezur RB (2005) Response of residul soil slope to rinfll. Cndin Geotechnicl Engineering 42: Qin A, Sun D, Tn Y (2010) Anlyticl solution to one-dimensionl consolidtion in unsturted soils under loding vrying exponentilly with time. Computers nd Geotechnics 37: Serrno SE (1990) Modeling infiltrtion in hysteretic soils. Advnce in Wter Resources 13(1): Shen ZJ (2003) Simplified consolidtion theory for unsturted soils nd its ppliction. Hydro-science nd Engineering (4):1-6 (in Chinese with English bstrct). 32. Song X, Borj RI (2014). Mthemticl frmework for unsturted flow in the finite deformtion rnge. Int. J. Num. Meth. Engng 97: Stuffer F, Drcos T (1986) Experimentl nd numericl study of wter nd solute infiltrtion in lyered porous medi. Journl of Hydrologic Engineering 84: Sun W, Sun D (2012). Coupled modelling of hydro-mechnicl behvior of unsturted compcted expnsive soils. Interntionl Journl of Numericl nd Anlyticl Method in Geomechnics 36: Wu LZ, Zhng LM (2009) Anlyticl solution to 1D coupled wter infiltrtion nd deformtion in unsturted soils. Interntionl Journl for Numericl nd Anlyticl Methods in Geomechnics 33: Yng H, Rhrdjo H, Wibw B, Leong EC (2004) A soil column pprtus for lbortory infiltrtion study. Geotechnicl Testing Journl 27(4): Yng H, Rhrdjo H, Leong EC (2006) Behvior of unsturted lyered soil columns during infiltrtion. Journl of Hydrologic Engineering 11(4): Zhn LT, Ng CWW(2004) Anlyticl nlysis of rinfll infiltrtion mechnism in unsturted soils. Interntionl Journl of Geomechnics 4: Appendix I From the simplified phse digrms (see Fig. A1) for n unsturted soil, we hve V v = V + V w, nd V=V +V w + V s =1, S r = V w /V v, n=v v /V, 19

21 so we obtin V =V v - V w. However, for the unsturted soils, some ir is embedded in the wter, which cn be formulted by the simplified method s V =c h V w, so the totl ir content of unit volume of unsturted soil element cn be obtined: V = V v - V w + c h V w =nv[1- S r +c h S r ]= [1- S r +c h S r ] n. We define n =[1-S r +c h S r ]n=[1-(1-c h )S r ]n, where n is the rtio of the pore ir of unit soil element, which signifies the ir content in the voids of soil element within unit volume. Appendix II For unit volume soil element, the pore ir pressure u will be generted upon loding. Under the conditions tht the ir in the voids cnnot be dischrged, from B ( p u )( V cv ) ( p u )( V cv ), (II-1) h w 0 0 h w in which, p is the tmospheric pressure, u is the pore ir pressure, 0 V is the initil content of pore ir of unit volume soil element (the corresponding pore ir pressure u 0 0 ), nd V is the content of pore ir of unit volume soil element. For unit volume soil element (see Fig. A1), we hve Vw ns r nd V n(1 S ). Combining eqution (4), n [1 (1 c ) S ] n, nd eqution (II-1), we hve r h r ( p u ) n ( p u ) n, (II-2) 0 0 in which, n 0 is the initil rtio of pore ir of unit soil element, 0 [1 (1 h) r0] 0 degree of sturtion nd n0 is initil porosity. n c S n, Sr0 is initil Becuse the initil ir pressure is equl to the tmospheric pressure ( u 0 0 ), from eqution (II-2), we hve: Appendix III u n 0 ( 1) p. n (II-3) or (6) In the following, we ignore the flow of the dissolved ir in the pore-wter, the vpor in the pore-ir nd the influence of temperture. For unit volume soil element, the mss of pore ir dischrged per unit time hs the following eqution (mss conservtion eqution): (III-1) dq (1 Sr) n chsn r n, t t dt 20

22 in which q is the mss of pore ir dischrged within unit soil element. From eqution (III-1), we cn deduce the following eqution in incrementl form: 2) Differentiting eqution (5), 0(1 u / p ), we hve From eqution (7), we hve n n q. (III- 0 u / p. (III-3) q n, (III-4) Substituting equtions (III-3) nd (III-4) into eqution (III-2), we obtin n 0 u / p n n, (III-5) Substituting eqution (5), 0(1 u / p ), into eqution (III-5), we hve 516 u p u n (1 ) n. (III-6) or (8) 517 Appendix IV 518 The eqution (8) cn be expressed s: 519 du 1 p u n dn, (IV-1) Integrting the bove eqution, we obtin the following: ln( p u) (1 )ln( n) const., (IV-2) Substituting the initil conditions n 0 nd u 0 into eqution (IV-2), we hve Combining equtions (IV-2) nd (IV-3), we hve const. ln[( p u ) n ], (IV-3) (1 ) u n 0 (1 ) [( ) 1] p. n (IV-4) 526 Appendix V 21

23 The elements of the coefficient mtrix in the finite element eqution (33) re s follows, 11 N Nj N Nj N Nj Nj N k [( 11 2) i i 44 14( i i ij d A d d )] dxdz x x z z x z x z 12 N Nj N Nj N Nj Nj N k [( 12 2 ) i i i i ij d A d14 d24 d44 ] dxdz x z x x z z x z 21 N N i j N N i j N N i j N N i j kij [( d12 A2 ) d14 d24 d 44 )] dxdz z x x x z z x z 22 N N i j N N i j N Nj N i j Ni kij [( d22 A2 ) d44 d24( )] dxdz z z x x z x z x 33 N N i j N N i j kij [ kx kz ] dxdz x x z z 13 Ni kij wg A1 N jddz r 23 Ni kij wg A1 N jdxdz z N 31 j kij wg Sr Nidxdz x N 32 j kij wg Sr Nidxdz z, (V-1), (V-2), (V-3), (V-4), (V-5), (V-6), (V-7), (V-8), (V-9) sij wg csnin jdxdz, (V-10) where c / s n S r

24 List of figures Fig. 1. Determintion of the reduced suction. Fig. 2 Double hrdening yield surfces. Fig. 3 Tested nd simulted results. () The Likopoulos test problem. (b) Tested nd simulted results. 567 Fig. 4 Effects of the dischrge speed of pore ir,. () Computtionl mesh nd boundry conditions. (b) The initil suction distribution t t=0.0 hours. (c) The suction distribution t t=100 hours of infiltrtion. (d) The suction distribution t t=205 hours of infiltrtion. (e) The suction distribution t t=310 hours of infiltrtion. Fig. 5 Effects of the evportion intensity I. () Computtionl mesh nd boundry conditions. (b) The distribution of pore wter pressure with evportion intensity I=0.012 cm/hours. (c) The distribution of 23

25 pore wter pressure with evportion intensity I=0.024 cm/hours. (d) The distribution of pore wter pressure with n evportion intensity I=0.036 cm/hours. Fig. 6 Effects of sturted permebility k ws. () The distribution of pore wter pressure with k ws =0.03 cm/s. (b) The distribution of pore wter pressure with k ws =0.003 cm/s. (c) The distribution of pore wter pressure with k ws = cm/s. Fig. 7 Simultion of unsturted soil slope with rinfll infiltrtion. () Computtionl mesh nd boundry conditions. (b) Pore wter pressure distribution t the end of evportion (kp). (c) Pore wter pressure distribution t the end of infiltrtion (kp). (d) Displcement distribution t the end of evportion. (e) Displcement distribution t the end of rinfll infiltrtion. Fig. A1 Simplified three phse digrm

26 e Drying shrinkge curve e c s p Compression curve o p s p(s) Fig.1 Determintion of the reduced suction 1

27 s =0.9 =0.7 =0.5 =0.3 =0.1 p=1.0 m Fig.2 () p=const. s p=1.0 p=0.8 p=0.4 p=0.6 =0.5 p=0.2 Fig.2 (b) =const. m Fig.2 Double hrdening yield surfces 2

28 For t>0, u = p o X For t=0, mtrix suction s=0(full impervious nd constrined boundry u x =0 sturtion with wter) impervious constrined u x =0 nd boundry 1.0m 0.1m For t>0, free wter outflow, u = p ; constrined boundry u x = u z =0 Z Fig.3 ()The Likopoulos test problem 3

29 Fig.3 (b) Comprisons with lbortory test results of the Likopoulos test problem (Likopolos 1965) 4

30 Rinfll drined boundry; (wter nd ir) o X Undrined boundry; Constrined boundry u x =0 Undrined boundry; Constrined boundry u x =0 Szie:1.0m 0.1m; 20 elements (0.05 m 0.1m) Z t>0,s=0;constrined u x = u z =0 boundry Fig.4(). Computtionl mesh nd boundry conditions 5

31 Fig.4(b). The initil suction distribution t t=0.0hours 6

32 Fig.4(c). The suction distribution t t=100hours of infiltrtion 7

33 Fig.4(d). The suction distribution t t=205hours of infiltrtion 8

34 Fig.4(e). The suction distribution t t=310hours of infiltrtion 9

35 Evportion For t>0, u =p o X For t=0, suction s=0(full sturtion with wter) for ll the soil elements Undrined boundry; Constrined boundry u x =0 Undrined boundry; Constrined boundry u x =0 Size:1.0m 0.1m; 20 elements (0.05 m 0.1m) Z For t>0, s=0; constrined boundry u x = u z =0 Fig.5(). Computtionl mesh nd boundry conditions 10

36 Fig.5(b). The distribution of pore wter pressure with evportion intensity I=0.012cm/hours 11

37 Fig.5(c). The distribution of pore wter pressure with evportion intensity I=0.024cm/hours 12

38 Fig.5(d). The distribution of pore wter pressure with evportion intensity I=0.036cm/hours 13

39 Fig.6(). The distribution of pore wter pressure with k ws =0.03cm/s 14

40 Fig.6(b). The distribution of pore wter pressure with k ws =0.003cm/s 15

41 Fig.6(c). The distribution of pore wter pressure with k ws =0.0008cm/s 16

42 Mesh:54m in width 20m in depth x Evportion/infiltrtion surfce, drined freely Undrined constrined boundry z nd When t=0, the soil slope is sturted nd in equilibrium with weight stress stte. Undrined nd constrined boundry Wter level Constrined boundry u x=u z=0 Fig.7(). Computtionl mesh nd boundry conditions 17

43 Fig.7 (b)pore wter pressure distribution t the end of evportion (kp) 18

44 -20 Fig.7 (c) Pore wter pressure distribution t the end of infiltrtion (kp) 19

45 cm The biggest displcement is 13.10cm. Fig.7 (d) Displcement distribution t the end of evportion 20

46 cm The biggest displcement is 12.68cm. Fig.7 (e) Displcement distribution t the end of rinfll infiltrtion 21

47 Volume Volume Air V V Wter V w Soil solids V s Fig.A1 Simplified three phse digrm 22