Analysis of one-dimensional consolidation of soft soils with non-darcian flow caused by non-newtonian liquid

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1 Joural of Rock Mechacs ad Geotechcal Egeerg., 4 (3): 5 57 Aalyss of oe-dmesoal cosoldato of soft sols wth o-darca flow caused by o-newtoa lqud Kaghe Xe, Chuaxu L, *, Xgwag Lu 3, Yul Wag Isttute of Geotechcal Egeerg, Zheag Uersty, agzhou, 358, Cha School of Cl Egeerg ad Mechacs, Jagsu Uersty, Zheag, 3, Cha 3 Zheag Proce Isttute of Archtectural Desg ad Research, agzhou, 36, Cha Receed 6 February ; receed resed form August ; accepted September Abstract: Based o o-darca flow caused by o-newtoa lqud, the theory of oe-dmesoal (D) cosoldato was modfed to cosder arato the total ertcal stress wth depth ad tme. The fte dfferece method (FDM) was adopted to obta umercal solutos for excess pore water pressure ad aerage degree of cosoldato. Whe o-darca flow s degeerated to Darca flow, a comparso betwee umercal solutos ad aalytcal solutos was made to erfy relablty of fte dfferece solutos. Fally, takg to accout the ramp tme-depedet loadg, cosoldato behaors wth o-darca flow uder arous parameters were aalyzed. Thus, a comprehese aalyss of D cosoldato combed wth o-darca flow caused by o-newtoa lqud was coducted ths paper. Key words: oe-dmesoal (D) cosoldato; o-darca flow caused by o-newtoa lqud; o-uform dstrbuto of total ertcal stress; tme-depedet loadg Itroducto A lear relatoshp betwee flow elocty ad hydraulc gradet,.e. Darcy s law for flow, has bee wdely used the theory of cosoldato, whch ca forecast the settlemet rates ad dsspato rates of excess pore water pressure. I fact, Darcy s flow law was fouded o permeablty test of sady sol. For fe graed sol, both expermets ad feld obseratos dcate that water flow soft sol may deate from Darcy s law uder low hydraulc gradets. I 898, Kg (898) reported that, at low hydraulc gradets, the flow elocty porous meda was ot proportoally lear to the hydraulc gradet. I addto, may researchers (asbo, 96, 997, 3; Swartzedruber, 96; Mller ad Low, 963; Olse, 965, 985; Elaggak ad Krzek, 973; Dub ad Moul, 985) also obsered the deato of water flow from Darcy s law by expermets or feld Do:.374/SP.J * Correspodg Author. Tel: ; E-mal: lchuaxu@yeah.et Supported by the Natoal Natural Scece Foudato of Cha (599, 58789). obseratos. Kutĺek (97) summarzed exstg o-darca flow relatos, ad cosdered that water flow sols could be classfed to two categores. Oe was Newtoa lqud, ad the other was o-newtoa lqud. Thus, the deato of water flow from Darcy s law may be caused by the o-newtoa lqud sols. Elaggak ad Krzek (973) cosdered that water sols may be ether free water, Newtoa flud, or bod water. No- Newtoa flud ad a o-darca flow sols caused by o-newtoa flud may exst uder low hydraulc gradets. I order to geerally descrbe the o-darca flow caused by o-newtoa flud, Elaggak ad Krzek (973) proposed a emprcal relatoshp betwee flow elocty ad hydraulc gradet, whch ca descrbe all the expermetal data reported. As show Fg., ths relatoshp may be goered by k( a) exp () where s the flow elocty, s the hydraulc gradet, k s the slope of the asymptote Fg., s the zeroelocty tercept of ths asymptote, ad a ad θ are emprcal parameters.

2 Kaghe Xe et al. / J Rock Mech Geotech Eg., 4 (3): Fg. Cure of flow elocty,, ersus hydraulc gradet,. It ca be see from Fg. that permeablty creases wth creasg hydraulc gradet; k ad k m were termed as ultmate permeablty ad mmum permeablty, respectely. was called by Elaggak as apparet threshold gradet. If a s equal to oe, or θ becomes zero, or becomes zero, the o-darca flow descrbed by Eq. () wll degeerate to Darcy s law. Sce the permeablty creases wth creasg hydraulc gradet, water flow sols may deate from Darcy s law durg the progress of cosoldato, ad Eq. () ca bascally descrbe the relatoshp betwee flow elocty ad hydraulc gradet. It has a theoretcal sgfcace estgatg the fluece of ths water flow relatoshp o D cosoldato behaor of soft sol foudatos. I fact, may studes o cosoldato behaor wth o- Darca flow were coducted (Teh ad Ne, ; Xe et al., 7; Lu et al., 9; L et al., ), ad may terestg results hae bee obtaed. Elaggak ad Krzek (973) aalyzed D cosoldato wth ths water flow relatoshp, but exteral tme-depedet loadg a actual egeerg ad tal excess pore water pressure chagg wth depth were ot corporated. Accordg to Zhu ad Y (998), exteral loadg may cause a crease total ertcal stress whch chages wth depth ad tme durg the progress of cosoldato. Therefore, ths paper, a aalyss of D cosoldato wth o-darca flow relatoshp s made, cosderg a chage total ertcal stress combato wth tme ad depth. Assumptos ad deelopmet of goerg equatos. Assumptos I order to estgate cosoldato behaor wth o-darca flow descrbed by Eq. (), the followg assumptos are made, cosderg a chage ertcal total stress wth tme ad depth: () A deal homogeeous sol layer s saturated. () Deformato ad water flow oly ertcal drecto are cosdered. (3) The relatoshp betwee od rato ad effecte stress s lear, ad the coeffcet of compressblty keeps costat durg the progress of cosoldato. (4) The relatoshp betwee flow elocty ad hydraulc gradet s goered by Eq. (), ad all the parameters are costat durg the progress of cosoldato. (5) A casual tme-depedet loadg s appled to the surface of a sol layer, ad ertcal addtoal stress caused by exteral loadg ares learly wth depth. As show Fg., the ertcal total stress ca be wrtte as (z, t) (, ) (, ) o =(, t c ) =(, t c ) (,t) (, t) t c (a) Stress wth tme. o z= (, t) (z, t) z= (, t) z (b) Stress wth depth. Fg. Varato the ertcal total stress wth tme ad depth. (, t) z ( t tc) (, z t) () z ( t tc) where s the thckess of a sol layer; (z, t) s a t

3 5 Kaghe Xe et al. / J Rock Mech Geotech Eg., 4 (3): 5 57 fucto of total ertcal stress wth tme, t, ad depth, z; t c s the costructo tme; s the total ertcal stress crease at z = ad t = t c ; ad s the total ertcal stress crease at z = ad t = t c. It s assumed that the total ertcal stress the clayey layer ares learly wth depth ad remas uchaged after tme t c, ad t c = dcates that the exteral load s appled statly ad the total ertcal stress may chage wth depth. Equato = meas that the total ertcal stress becomes uform dstrbuto wth depth. Equatos t c = ad = mea that the arato patter of the total ertcal stress s reduced to the basc assumpto of Terzagh s theory of cosoldato. If exteral loadg s ramp loadg, Eq. () ca be wrtte as t z ( t tc) tc (, z t) z ( t tc) (3). Deelopmet of the goerg equatos A arbtrary ut cell was take out from a sol layer. Accordg to aforemetoed assumptos, the chage water quatty the ut cell should be equal to the olumetrc chage ths ut cell, thus ths cotuty codto ca be wrtte as u (, z t) m z t t (4) where u s the excess pore water pressure the z-drecto, ad m s the coeffcet of olume costraed compressblty. By substtutg Eq. () to cotuty codto Eq. (4), the goerg equato of cosoldato ca be obtaed by k u u ( a) exp wm z w z u (, z t) (5) t t where w s the ut weght of water. Two commoly ecoutered boudary codtos of ether perous top ad perous bottom (PTPB), or perous top ad mperous bottom (PTIB), are estgated: u(, t) u (, t) (6a) u(, t) u z z (6b) Eq. (6a) meas that both sdes of the sol layer are freely draed. Eq. (6b) meas that oe sde of the sol layer s freely draed, but the other sde s mperous. The tal dstrbuto of excess pore water pressure wth depth s uz (, ) (, z ) (7) Eq. (5) s the goerg equato of D cosoldato wth o-darcy s law caused by o-newtoa lqud. If a equals, or θ equals, or equals, Eq. (5) ca be smplfed to Terzagh s lear equato of D cosoldato wth Darcy s law. For the complexty of secod-order olear partal dfferetal equato, aalytcal soluto for Eq. (5) ca hardly be obtaed. Thus, fte dfferece method s adopted ths paper to get the soluto for the goerg equato. 3 Solutos for aerage degree of cosoldato 3. Dmesoless form of the goerg equatos To smplfy the process of soluto, the followg dmesoless arables are troduced: z Z (8a) kt T wm (8b) ktc Tc wm (8c) u U (8d) qh w (8e) (8f) (8g) ( Z, T ) QZ (, T ) (8h) I terms of these dmesoless arables, Eqs. (5), (6a), (6b) ad (7) ca be expressed by

4 Kaghe Xe et al. / J Rock Mech Geotech Eg., 4 (3): U q U U Q h ( a) exp Z Z T T U(, T ) U(, T ) U(, T ) U Z Z ( Z,) U( Z, ) (9) (a) (b) () 3. Fte dfferece soluto for excess pore water pressure To obta the umercal solutos for aerage degree of cosoldato wth o-darca flow caused by o-newtoa lqud, the spatal ad tme domas are subdded to dfferece grds. The spatal doma Z s frstly dded to parts by equal legth Z /, ad spatal odal pots are deoted as Z ΔZ (,,, 3,, ) () Meawhle, a cremet alog the tme axs s take equal to Δ, ad the tme odal pots are deoted as T T (,,, 3, ) (3) Thus, a dfferece grd s obsered betwee the spatal doma ad the tme doma. I terms of Crak-Ncolso dfferece scheme, Eqs. (8a) (8h) ca be expressed as U ( ) U U U U U Q Q ( ) ( ) (,,, 3,, ;,, 3, ) (4) where U s the dmesoless alue of excess pore water pressure at Z Z ad, Q s the dmesoless alue of total ertcal stress at Z Z, T T, ad T /( Z). s decded by qh U U ( a) exp Z (5) The boudary ad tal codtos descrbed by Eqs. (a)(b) ad () may be rewrtte terms of the dscrete odal pots as U (,,, 3, ) (6a) U U U U (,,, 3, ) (6b) ( Z,) U (,,, 3,, ) (7) I terms of matrx, the Eq. (4) ca be expressed cosderg the boudary codtos ad tal codtos as AU B (8) For a lear parabolc partal dfferetal equato, Crak-Ncolso dfferece scheme s absolutely coerget. Ufortuately, there s o geeral theoretcal kowledge of stablty ad coergece crtera for olear parabolc partal dfferetal equato, thus the learzato techque may be employed some staces to obta dfferece soluto for olear partal dfferece equato. For example, cosderg arable coeffcet as the cremet of costats for a ge tme, the exact soluto ca be obtaed by terato. Therefore, the elemets of matrx A Eq. (8) ca be obtaed by replacg the dmesoless alue of excess pore water pressure at tme wth that at tme ( ). A approxmato soluto for matrx U ca be obtaed by terato. A mathematcal problem for Eq. (9) wll be ecoutered at the mperous surface where excess pore water pressure may keep costat. To aod ths mathematcal dffculty, o-darcy law ca be substtuted by Darcy s law at the mperous surface. Thus, the expresso for boudary codto (Eq. (6b)) terms of elemets of matrxes ca be wrtte as A ( ) A ( ) (9) B ( ) U U ( Q Q ) Boudary codto (Eq. (6a)) ca be expressed as A A ( ) () B I addto, f total ertcal stress keeps homogeeous dstrbuto wth depth, the mddle surface of a sol layer s mperous surface. O ths codto, Darcy s law should be adopted stead of o-darcy law at mperous surface. 3.3 Derato of aerage degree of cosoldato If the compressblty ad permeablty of homogeeous foudato keep costat the process of cosoldato, the aerage degree of cosoldato terms of deformato s equal to that terms of stress. The defto of aerage degree of

5 54 Kaghe Xe et al. / J Rock Mech Geotech Eg., 4 (3): 5 57 cosoldato s the rato of the cosoldato settlemet at ay tme to the fal cosoldato settlemet of the foudato,.e. St U () t S The settlemet of the sol layer at ay tme ca follow as S [ (, ) (, )] d t m z t u z t z () The fal cosoldato settlemet of the sol layer ca be wrtte as z S Combg Eqs. (), (), (3) ad (), the aerage degree of cosoldato ca be expressed as ( ) dz (3) Q(, T ) U( Z, T)d Z ( T Tc) (4) U( Z, T)d Z ( T Tc) For a partcular case of ramp loadg, the aerage degree of cosoldato ca be wrtte as T U( Z, T)d Z ( T Tc) T c (5) U( Z, T)d Z ( T Tc) Sce the aalytcal soluto of pore water pressure caot be obtaed, umercal tegrato must be adopted ad lear terpolato pots may use the pots of a fte dfferetal grd. I terms of umercal tegrato, the aerage degree of cosoldato ca be expressed as Q(, T) U U ( T T c) (6) U U ( T Tc) where U s the dmesoless alue of excess pore water pressure at Z Z. For ramp loadg, Eq. (6) ca be wrtte as T U U ( T Tc) Tc (7) U U ( T Tc) 3.4 Verfcato of the dfferece programmg If a equals, or θ equals, or equals, Eq. () degeerates to Darcy s law, ad the theory of cosoldato wth o-darca flow turs to the lear theory of cosoldato. Takg ramp loadg from tme-depedet loadg as a example, aalytcal soluto for the theory of cosoldato wth Darcy s law has bee acheed by Zhu ad Y (998). To erfy the relablty of the dfferece programmg, as show Table, a comparso s made betwee the results of aerage degree of cosoldato by the FDM ad that by the aalytcal method. Table Comparso of results obtaed by FDM ad aalytcal method (Δz=., =., a =, or θ =, or = ). Tme factor c =.5, =.5, PTIB Results by FDM Aerage degree of cosoldato Aalytcal soluto c =., =.5, PTPB Results by FDM Aalytcal soluto A good agreemet betwee the results by the preset umercal method ad those by the aalytcal method ca be obsered from Table. The maxmum absolute error of aerage degree of cosoldato Table s less tha.%, ad correspodg relate error s less tha.8%. Therefore, the results of the preset umercal method are relable computg D cosoldato wth Darcy s law. The dfferece of theory of cosoldato betwee Darcy s law ad o-darcy law the process of umercal calculato s the arat alue of. For the theory of cosoldato wth Darcy s law, =; otherwse, the alue of s decded by Eq. (5). If the accuracy of expresso for ca be esured ad the method of terato s adopted the dfferece programmg, the umercal results of cosoldato wth o-darca

6 Kaghe Xe et al. / J Rock Mech Geotech Eg., 4 (3): flow should be relable. 4 Parametrc aalyses of cosoldato behaors 4. Ifluece of parameters of flow law o cosoldato behaor As show Fg. 3, the parameters of water flow law, a, θ ad, sgfcatly fluece the rate of cosoldato. Fg. 3(a) shows that the rate of a=, PTPB 4 a=.5, PTPB a=.5, PTPB a=.75, PTPB 6 a=, PTPB a=, PTIB a=.5, PTIB 8 a=.5, PTIB a=.75, PTIB a=, PTIB... (a) (b) c =.5, =.5, =, q h =, =.5 c =.5, a=.5, =, q h =, =.5 =, PTPB 4 =.5, PTPB =.5, PTPB =.75, PTPB 6 =, PTPB =, PTIB =.5, PTIB 8 =.5, PTIB =.75, PTIB =, PTIB =.5, PTPB =5, PTPB =7.5, PTPB =, PTPB =.5, PTIB =5, PTIB =7.5, PTIB =, PTIB... c =.5, a=, =, q h =, =.5 (c) Fg. 3 The fluece of parameters of water flow law o aerage degree of cosoldato uder both boudary codtos. cosoldato creases wth the creasg alue of a the case of ether PTIB or PTPB. a= dcates that the o-darca flow descrbed by Eq. () s degeerated to Darca flow, ad the rate of cosoldato comes to the maxmum ths case. The fluece of θ o cosoldato behaor s show as Fg. 3(b), ad the rate of cosoldato s the fastest the case of θ=, furthermore, the greater the alue of θ s, the slower the rate of cosoldato uder both boudary codtos s. The fluece of o the rate of cosoldato ca be see from Fg. 3(c), ad the rate of cosoldato decreases wth the creasg alue of. 4. Ifluece of rato of equalet water head of exteral load to sol layer thckess o rate of cosoldato Fg. 4 shows that the fluece of parameter q h o aerage degree of cosoldato. The rate of cosoldato creases wth creasg alue of q h uder both boudary codtos. Accordg to Eq. (8e), the defto of parameter q h s the rato of equalet water head of aerage total ertcal stress a foudato to the thckess of a sol layer. That s, greater alue of aerage total ertcal stresses wll crease the rate of cosoldato,.e. the ther the thckess of a sol layer s, the faster the rate of cosoldato s. Therefore, the rate of cosoldato wth o-darca flow descrbed by Eq. () s depedet o the alue of aerage total ertcal stresses. I addto, the requred cosoldato tme of a sol layer s o loger proportoal to the square of draage dstace of a sol layer, because the aerage degree of cosoldato ares wth the alue of q h, ee f the tme factor s the same. Such cosoldato behaor ca be determed by Eq. (5), whch was also obsered by Elaggak ad Krzek (973) the codto that the exteral load s statly appled q h =., PTIB q h =.5, PTIB q h =, PTIB q h =5, PTIB q h =, PTIB c =, =, a=.5, =... Fg. 4 The fluece of parameter q h o aerage degree of cosoldato uder both boudary codtos. q h =., PTPB q h =.5, PTPB q h =, PTPB q h =5, PTPB q h =, PTPB

7 56 Kaghe Xe et al. / J Rock Mech Geotech Eg., 4 (3): Ifluece of o-uform dstrbuto of total ertcal stress o rate of cosoldato Fg. 5 dsplays the fluece of o-uform dstrbuto of total ertcal stress o the rate of cosoldato. = dcates that the dstrbuto of total ertcal stress s up-tragle, ad the rate of cosoldato s the lowest the case of PTIB. = meas that the dstrbuto of total ertcal stress s dow-tragle, ad the rate of cosoldato s the fastest the case of PTIB. The rate of cosoldato creases wth the creasg alue of the case of PTIB. oweer, Fg. 5, the cosoldato cures almost cocde well wth each other uder arous alues of, ad o-uform dstrbuto of total ertcal stress has o effect o the rate of cosoldato the case of PTPB. I addto, as show Fg. 6, the dstrbuto of total ertcal stress strogly flueces dsspato of excess pore water pressure the case of PTPB. Meawhle, the area betwee the cosoldato cures ad ordates axs s almost detcal, ad ths ca expla why the dstrbuto of total ertcal stress has o fluece o the rate of cosoldato the case of PTPB. 4 =, PTIB 6 =.5, PTIB =, PTIB 8 =.5, PTIB =, PTIB =,.5,,.5,, PTPB... Fg. 5 The fluece of parameter o aerage degree of cosoldato uder both boudary codtos. Z =, =, q h =, c =., a=.5 = =.5 = =.5 = =, =, q h =, c =., a=.5, =. U Fg. 6 The fluece of parameter o excess pore water pressure the case of PTPB. 4.4 Ifluece of rate of exteral loadg o rate of cosoldato The rate of exteral loadg has flueces o the rate of cosoldato, ad the less the costructo tme s, the faster the rate of cosoldato s the case of ether PTIB or PTPB (Fg. 7). Ths cosoldato behaor wth o-darca flow s the same as that wth Darca flow c =, PTIB c =.5, PTIB a=.5, =, =, q h =, = c =.5, PTIB c =., PTIB c =.5, PTIB a=.5, =, =, q h =, = c =, PTPB c =.5, PTPB c =., PTPB c =.5, PTPB (a) PTIB. (b) PTPB. Fg. 7 The fluece of parameter c o aerage degree of cosoldato the cases of PTIB ad PTPB. 5 Coclusos c c =.5, PTPB... Ge a aalyss of D cosoldato wth o-darca flow caused by o-newtoa lqud, the followg coclusos ca be draw cosderg a chage total ertcal stress wth tme ad depth: () The parameters of water flow law sgfcatly fluece the rate of cosoldato. The aerage degree of cosoldato creases wth the creasg alue of a, ad decreases wth the creasg alues of θ ad uder both boudary codtos. () The rato of the equalet water head of exteral loadg to the thckess of sol layer strogly flueces the rate of cosoldato. The greater ths

8 Kaghe Xe et al. / J Rock Mech Geotech Eg., 4 (3): rato s, the faster the rate of cosoldato uder both boudary codtos s. The smlarty relatoshp that the cosoldato tme s proportoal to the square of draage dstace of sol layer the classcal theory of cosoldato s ot true. (3) No-uform dstrbuto of total ertcal stress greatly flueces rate of cosoldato the case of PTIB, howeer, t has o effect o the rate of cosoldato the case of PTPB. The dsspato of excess pore water pressure s affected by o-uform dstrbuto of total ertcal stress the case of ether PTIB or PTPB. (4) The slower the loadg rate s, the slower the rate of cosoldato s. oweer, D cosoldato of layered sol wth o-darca flow caused by o-newtoa lqud s ot cosdered the paper, ad further study ca be apprecated. Refereces Dub B, Moul G. Ifluece of a crtcal gradet o the cosoldato of clays. I: Yog R N, Towsed F C ed. Cosoldato of Sols: Testg ad Ealuato (STP 89). West Coshohocke: ASTM, 985: Elaggak A, Krzek R J. Effect of o-darcy flow o tme rate of cosoldato. Joural of the Frakl Isttute, 973, 96 (5): asbo S. Cosoldato of clay wth specal referece to fluece of ertcal dras. I: Swedsh Geotechcal Isttute Proceedgs. Stockholm: Swedsh Geotechcal Isttute Press, 96: 455. asbo S. Aspects of ertcal dra desg: Darca or o-darca flow. Geotechque, 997, 47 (5): asbo S. Deato from Darcy s law obsered oe-dmesoal cosoldato. Geotechque, 3, 53 (6): 665. Kg F. Prcples ad codtos of moemet of groud water. Uted States Geologcal Surey: Neteeth Aual Report (part ), 898: Kutĺek M. No-Darca flow of water sols-lamar rego: a reew. Deelopmets Sol Scece, 97, : L Chuaxu, Xe Kaghe, Wag Ku. Aalyss of D cosoldato wth o-darca flow descrbed by expoet ad threshold gradet. Joural of Zheag Uersty (Scece A),, (9): Lu Zhogyu, Su Lyu, Yue Jchao, Ma Chogwu. D cosoldato theory of saturated clay based o o-darcy flow. Chese Joural of Rock Mechacs ad Egeerg, 9, 8 (5): ( Chese). Mller R J, Low P E. Threshold gradet for water flow clay systems. Sol Scece Socety of Amerca Joural, 963, 7 (6): Olse W. Deatos from Darcy s law saturated clays. Sol Scece Socety of Amerca Joural, 965, 9 (): 354. Olse W. Osmoss: a cause of apparet deato from Darcy s law. Caada Geotechcal Joural, 985, (): 384. Swartzedruber D. Modfcato of Darcy s law for the flow of water sols. Sol Scece, 96, 93 (): 9. Teh C I, Ne X Y. Coupled cosoldato theory wth o-darca flow. Computers ad Geotechcs,, 9 (3): 699. Xe ala, Wu Qag, Zhao Zegm, J Xaol, L Jua. Cosoldato computato of aqutard cosderg o-darcy flow. Rock ad Sol Mechacs, 7, 8 (5): 6 65 ( Chese). Zhu Guofu, Y Jahua. Cosoldato of sol uder depth-depedet ramp load. Caada Geotechcal Joural, 998, 35 ():