Vertical drain consolidation with non-darcian flow and void ratio dependent compressibility and permeability

Transcription

1 University of Wollongong Research Online Faculty of Engineering Information Sciences - Papers Faculty of Engineering Information Sciences Vertical drain consolidation with non-darcian flow void ratio dependent compressibility permeability R Walker University of Wollongong, rohanw@uow.edu.au B Indraratna University of Wollongong, indra@uow.edu.au C Rujikiatkamjorn University of Wollongong, cholacha@uow.edu.au Publication Details Walker, R., Indraratna, B. & Rujikiatkamjorn, C. (). Vertical drain consolidation with non-darcian flow void ratio dependent compressibility permeability. Geotechnique: international journal of soil mechanics, 6 (), Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au

2 Vertical drain consolidation with non-darcian flow void ratio dependent compressibility permeability Abstract Vertical drains increase the rate of consolidation in soft soils by facilitating faster dissipation of excess pore water pressure through short, horizontal drainage paths. This paper presents an analytical solution for nonlinear radial consolidation under equal-strain conditions incorporating smear but ignoring well resistance. Three aspects of non-linearity are considered: (a) non-darcian flow, (b) a log-linear void-ratio-stress relationship; (b) a log-linear void-ratio-permeability relationship. The analytical solution to non-linear radial consolidation can explicitly capture the behaviour of both overconsolidated normally consolidated soils. For non-linear material properties, consolidation may be faster or slower when compared with the cases with constant material properties. The difference depends on the compressibility/ permeability ratios (Cc/Ck Cr/Ck), the preconsolidation pressure the stress increase. If Cc/Ck < or Cr/Ck < then the coefficient of consolidation increases as excess pore pressures dissipate, the corresponding rate of consolidation is greater. Keywords dependent, compressibility, permeability, drain, flow, vertical, darcian, non, consolidation, void, ratio Publication Details Walker, R., Indraratna, B. & Rujikiatkamjorn, C. (). Vertical drain consolidation with non-darcian flow void ratio dependent compressibility permeability. Geotechnique: international journal of soil mechanics, 6 (), This journal article is available at Research Online:

3 Walker, R. et al. (). Géotechnique 6, No., [ Vertical drain consolidation with non-darcian flow void-ratiodependent compressibility permeability R. WALKER,B.INDRARATNA C. RUJIKIATKAMJORN Vertical drains increase the rate of consolidation in soft soils by facilitating faster dissipation of excess pore water pressure through short, horizontal drainage paths. This paper presents an analytical solution for non-linear radial consolidation under equal-strain conditions incorporating smear but ignoring well resistance. Three aspects of non-linearity are considered: (a) non-darcian flow, (b) a log-linear void-ratio stress relationship; (b) a log-linear void-ratio permeability relationship. The analytical solution to non-linear radial consolidation can explicitly capture the behaviour of both overconsolidated normally consolidated soils. For non-linear material properties, consolidation may be faster or slower when compared with the cases with constant material properties. The difference depends on the compressibility/ permeability ratios (C c /C k C r /C k ), the preconsolidation pressure the stress increase. If C c /C k < or C r /C k < then the coefficient of consolidation increases as excess pore pressures dissipate, the corresponding rate of consolidation is greater. KEYWORDS: case history; consolidation; pore pressures; settlement Les drains verticaux augmentent la vitesse de consolidation dans les sols meubles, en facilitant une dissipation plus rapide de la pression d eau interstitielle à travers des canaux de drainage horizontaux courts. Cette communication présente une solution analytique pour la consolidation radiale non linéaire dans des conditions de contrainte égales, comprenant l effet smear, mais ne tenant pas compte de la résistance de puits. On examine trois aspects de non linéarité suivants: (i) écoulement non Darcien, (ii) rapports ratio de vide/contrainte log-linéaire ; et (iii) rapport ratio de vide/perméabilité log-linéaire. La solution analytique pour la consolidation radiale non linéaire est en mesure de capturer de façon explicite le comportement de sols surconsolidésetàconsolidation normale. Pour les propriétés de matières non linéaires, la consolidation peut être plus rapide ou plus lente par rapport aux cas à propriétés matérielles constantes. La différence est tributaire des ratios de compressibilité/ perméabilité (C c /C k et C r /C k ), de la pression de préconsolidation, et de l augmentation des contraintes. Si C c /C k < on C r /C k <, le coefficient de consolidation augmente, les pressions interstitielles excessives se dissipant et les ratios de consolidation correspondants étant supérieurs. INTRODUCTION Vertical drains increase the rate of consolidation in soft soils by facilitating faster dissipation of excess pore water pressure through short, horizontal drainage paths. Usually, the simplest analytical methods used to analyse vertical drain consolidation problems simulate a single soil cylinder with constant soil properties over a given stress range (Barron, 948; Hansbo, 98). When material behaviour is non-linear, the simple models are often inadequate. This paper presents analytical solutions for non-linear radial consolidation under equalstrain conditions incorporating smear but ignoring well resistance. For most commonly installed drain lengths in Australia Southeast Asia (less than 8 m), the well resistance is not significant compared with the smear effects (Indraratna & Redana, ). Three aspects of non-linearity are considered: (a) non-darcian flow, (b) log-linear void-ratio stress relationship, (c) log-linear void-ratio permeability relationship. In non-darcian flow, the velocity of flow, v, is related to the hydraulic gradient, i, by the power law v ¼ ~ ki n () where ~ k is the coefficient of permeability under non-darcian conditions, n is the non-darcian flow exponent. Void Manuscript received August ; revised manuscript accepted March. Published online ahead of print 7 August. Discussion on this paper closes on April 3, for further details see p. ii. Centre for Geomechanics Railway Engineering, Faculty of Engineering, University of Wollongong, Australia. ratio is related to effective stress permeability by the following relationships. e ¼ e C c log ó 9 () ó9 e ¼ e þ C k log ~! k (3) ~k where e is the void ratio; ó9 is the effective stress; C c is the compressibility index; C k is the permeability index; e, ó 9 ~ k are the initial values of void ratio, effective stress permeability respectively before the application of preloading. A review of the current literature reveals various attempts to model the corresponding problem with Darcian flow (e.g. Basak & Madhav, 978; Lekha et al., 998; Indraratna et al., 5). Lekha et al. (998) proposed the average excess pore pressure, u, for radial consolidation considering loglinear void-ratio stress log-linear void-ratio permeability relationships as u ¼ ó 9 T h â ó9 ì C c =C k " 3 â þ #) (4) Cc =C k â C c=c k ó 9 â ¼ þ ó 9 where is the change in total stress, T h is the time factor (5) 985

4 986 WALKER, INDRARATNA AND RUJIKIATKAMJORN for horizontal consolidation, ì is a smear zone parameter for Darcian flow. Equation (4) is a linear function of the time factor T h, whereby at large values of T h unrealistic negative values of excess pore water pressure are predicted. The excess pore water pressure should decay to zero: thus equation (4) is unsuitable for estimating u as primary consolidation nears conclusion. Also, equation (4) is undefined if C c /C k ¼. Basak & Madhav (978) presented a more useful solution with equation (), but using a linear relationship between permeability effective stress. Indraratna et al. (5) expressed the excess pore pressure under instantaneous loading as u ¼ exp P av 8T h ì (6a) where " P av ¼ þ þ # Cc=C k (6b) ó 9 Equation (6) is very similar to the linear solution given by Hansbo (98); the main difference is in the parameter P av : P av represents the average between the beginning end values of the consolidation coefficient. This averaging of consolidation coefficient over the applied stress increment is implied in Hansbo s (98) solution. Thus equation (6) simply provides the best choice of consolidation coefficient for use with Hansbo s (98) equations. Although Indraratna et al. (5) recommend using the log-linear void-ratio stress relationship (equation ()) for settlement calculations, the solution expressed by equation (6) does not reflect the non-linear processes involved with pore pressure dissipation. The proposed model presented herein removes these simplifying assumptions. Hansbo s () equal-strain solution for radial drainage with non-darcian flow is extended to include the non-linear material properties expressed in equations () (3). A series solution to the resulting nonlinear partial differential equation is found, explicitly capturing the variation of permeability compressibility in the consolidation of normally overconsolidated soil. These solutions can also be used for benchmarking numerical analysis. CONSTITUTIVE RELATIONSHIP The horizontal coefficient of consolidation under non- Darcian flow, ~c h, can be written as ~c h ¼ ~ k h m v ª w (7) where m v is the volume compressibility, ª w is the unit weight of water. When considering material non-linearity, the coefficient of consolidation becomes dependent on the effective stress (under equal strain, the effective stress does not vary radially). The effective stress can be written as ó9 ¼ ó 9 þ W (8) where is the instantaneous change in total stress, W is a normalised pore pressure given by W ¼ u (9) The compressibility of soils previously subjected to higher effective stresses (overconsolidated) may increase markedly when the preconsolidation pressure, ó p 9, is exceeded. With reference to Fig., the compressibility relationships in the recompression zone (ó 9<ó p 9) the compression zone (ó 9.ó p 9) are described, respectively, by e ¼ e C r log ó9 (a) ó 9 e ¼ e C r log ó p 9 ó 9 C c log ó9 ó p 9 (b) The consolidation coefficient involves permeability volume compressibility. Volume compressibility is defined by the relationship m v () þ 9 Differentiating equation () to then substituting into equation () yields m v ¼ :434C r ó 9ð þ e Þ ; ó 9<ó p 9 (a) m v ¼ :434C c ó 9ð þ e Þ ; ó 9.ó p 9 (b) The relative change in volume compressibility with effective stress is thus expressed as m v ¼ ó9 ; ó 9<ó p 9 (3a) m v ó 9 m v ¼ C r ó 9 ; ó 9.ó p 9 (3b) m v C c ó 9 e e e C r C c C k e f log( σ ) log( σ p ) log( σ Δσ) log( σ ) log( k ~ f ) log( k ~ ) log( k ~ ) Fig.. Void-ratio-dependent compressibility permeability

5 Substituting equation () into equation (3) gives the relative change in permeability with effective stress as ~k ¼ ó 9 Cr =C k ; ó 9<ó p 9 (4a) ~k ó 9 VERTICAL DRAIN CONSOLIDATION 987 ~k ¼ ó 9 ð Cc C r Þ=C k Cc=C p ó 9 k ; ~k ó 9 ó9 ó 9.ó p 9 (4b) r w Combining equations (7), (8), (3) (4) results in the following expressions for the stress dependence of ~c h in the recompression compression zones k ~ h k ~ s r s Cr =C k ; ó9 <ó 9 p (5a) ~c h ¼ þ W ~c h ó 9 ó 9 ~c h ~c h ¼ C r C c ó 9 p ó 9 ð Cc C r Þ=C k 3 þ W (5b) Cc =C k ; ó 9.ó p 9 ó9 ó 9 r e The value of consolidation coefficient at the start of analysis is denoted ~c h : DEVELOPMENT OF GOVERNING EQUATION FOR NON-DARCIAN FLOW The consolidation coefficient is a function of the coefficient of volume compressibility, mv, of the soil permeability, which vary according to the effective stress. The stress dependence of the consolidation coefficient is treated implicitly incorporated explicitly in the following. Vertical drains, installed in a square or triangular pattern, are usually modelled analytically by considering an equivalent axisymmetric system. Pore water flows from a soil cylinder to a single central vertical drain with simplified boundary conditions. Moreover, this mathematical analysis considers only the lateral flow, because, for long vertical drains the contribution from the vertical consolidation coefficient (c v ) is insignificant. Therefore the question of anisotropy does not affect the mathematical derivations directly. For simplicity, the authors have assumed that the compressibility parameter m v is the same for both smear the undisturbed zone. Fig. shows a unit cell with external radius r e, drain radius r w smear zone radius r s : Depending on the soil stiffness, the size shape of the mrel, the installation method, the extent of the smear zone (r s ) can usually be estimated as r s ¼ ( to 3)r m, where r m is the equivalent radius of the mrel (Hansbo, 98). Gheharioon et al. () showed that the driving of the mrel does follow the theory of lateral cavity expansion in a nonlinear manner. Weber et al. () showed that a very clearly structured smear zone was observed from environmental scanning electron microscope photographs. The soil permeability compressibility parameters decrease nonuniformly with the radius within just outside the smear zone, as vividly discussed elsewhere by Indraratna & Redana (997, 998a) Walker (6), based on measurements from large-scale consolidation tests. However, in order to simplify the mathematical formulation of already complex equations, reduced lateral soil permeability was assumed as an equivalent average (constant) across the smear zone annulus. The lateral coefficient of consolidation (c h ) is therefore changed proportionately too. This assumption may lead to some overestimation of the settlement. The velocity of Fig.. Axisymmetric unit cell d e pore water flow in the smear undisturbed zones are respectively given by (Hansbo, ) v ¼ n s k s ; n > (6a) ª r v ¼ n k h ; ª r n > (6b) where ~ k h is the undisturbed horizontal permeability for non- Darcian flow; ~ k s is the horizontal permeability in the smear zone; u is the excess pore water pressure in the undisturbed zone; u s is the excess pore water pressure in the smear zone. The rate of fluid flow through the internal face of the hollow cylindrical slice with internal radius r is ðrv (7) The rate of volume change in the hollow cylindrical slice with internal radius r outer radius r e ð r e r t

6 988 WALKER, INDRARATNA AND RUJIKIATKAMJORN where is the one-dimensional strain rate. For instantaneous loading the strain rate can be t ¼ v where u is the average excess pore pressure, m v is the volume compressibility of the soil. For continuity, the volume changes in equations (7) (8) can be equated; rearranging the resulting expression using equations (6) (9), the pore pressure gradient in the smear undisturbed zones is represented by " # y ð r wª w Þ =n r ~k h y ~c ~k s N y =n y ð r wª w Þ =n r y y =n ~c N (a) (b) where N ¼ r e /r w, a change of variable has been made such that y ¼ r/r w : (The derivation of equation () is given in the Appendix.) By using the binomial expansion, the terms involving y on the right-h side of equation () can be represented by a series as y N =n y =n X =n ¼ y j¼ f =n j! g j y j () N where {x} m, sometimes called the Pochhammer symbol or rising factorial, is defined by fg x m ¼ xxþ ð Þðx þ Þ... ðx þ m Þ, fg x ¼ () The series expansion performed in equation () others below are best obtained using computer algebra systems (CAS). For example, entering the text (-(x^)/(n^))^(/ n) x^(-/n) into the Wolfram Alpha LLC. () website yields, after simplification, equation (). Substituting equation () into equation () integrating (with the boundary conditions u s ¼ aty¼, u ¼ u s at y ¼ s, where s ¼ r s /r w ) yields the following pore pressure expressions =n u s ðyþ ¼ ðr w ª w Þ =n r ~c =n h ~c t ~c h 3 ~! =n (3a) k h [g(y) g()]; < y, s ~k s =n uy ð Þ ¼ ðr w ª w Þ =n r ~c =n h ~c t ~c h 8 3 gðyþ gs ðþþ ~! 9 < =n = k h : [g(s) g()] ~k ; ; < y, N s (3b) where g(y) is the integral of equation () with respect to y, given by X =n gðyþ ¼ ny j¼ f =ng j j! ½ðj þ Þn Š y j (4a) N (The derivation of g(y) is given in the Appendix.) Equation (4a) can be formulated using a recurrence relationship as gðyþ ¼ X a j ðyþ (4b) j¼ a ¼ ny =n n y ðjn n Þðjn n Þ a j ¼ a j N jnðjn þ n Þ (4c) The average excess pore pressure satisfies the algebraic expression ð r e ð rs ð re r w u ¼ ð ru s ðþdr r þ ð ruðþdr r (5a) r w r s or, in the transformed coordinate system " u ¼ ð s ð # N N yu s ðyþdy þ yuðyþdy s (5b) Substituting equation (3) into equation (5) performing the appropriate integrations gives the excess pore pressure ( governing equation) as u ¼ ðr w ª w Þ =n r ~c t =n ~c =n h â (6) ~c h with ( ) â ¼ gðnþ k =n [gðþþ gðþðn Þ] N þðk =n Þ[gðÞN sð s Þþ gðþ] s (7a) where k ¼ ~ k h = ~ k s g(y) is the product of y equation (4a), integrated with respect to y to give gðyþ¼ n X f =ng 3 =n j y j y j! ½ðj þ Þn Š½ðj þ 3Þn Š N j¼ (7b) Equation (7b) can be formulated using a recurrence relationship as gðyþ ¼ X j¼ a j ðyþ n y 3 =n a ¼ ðn Þð3n Þ y ðjn n Þðjn n Þ a j ¼ a j N jnðjn þ 3n Þ (7c) (7d) The pore pressure at any point in the soil can now be related to the average excess pore water pressure by substituting equation (6) into equation (3). The resulting expressions for pore water pressure in the smear undisturbed zones are! =n u s ðyþ ¼ u ~k h i h gy ðþ gðþ (8a) â ~ ks

7 8 uðyþ ¼ u < gðyþ gs ðþþ â : VERTICAL DRAIN CONSOLIDATION 989 ~! 9 =n = k h ~a gs ðþ gðþ ¼ W n Ł n ~k ; n s ðj nþðj Æ=C (8b) k Þ ~a j ¼ ~a j þ ó 9 W þ j n þ j n Ł jð þ j nþ W j n Ł j n (33c) Usually the expressions for excess pore water pressure would involve explicit functions of time (Hansbo, 98, ); however, as shown below, the solution of equation (6) gives consolidation time as a non-invertible function of u: The corresponding time is determined when a suitable value for u is specified. For the special case of Darcian flow (i.e. derive equation (3) with n ¼ ) the recurrence relationships become ~a ¼ ln W Ł ANALYTICAL SOLUTION OF THE GOVERNING EQUATION Equation (6) can be rewritten ~T W n ~c h ~c h (9) where ~T is a modified time factor defined by ~T ¼ 8 T h n (3a) â n r w ª w T h ¼ ~c h t 4r (3b) e Using a Taylor power series, expansion of equation (5a) about the point W ¼ gives (in the recompression zone) ~c h W n ~c h ¼ þ ó9 3 X j¼ ð Cr=C k Þ f C r =C k j! g j þ ó 9 (3) ðwþ j n Substituting equation (3) into equation (9), integrating with the initial condition W ¼ at t ¼, results in the following time factor-normalised pore pressure relationship in the recompression zone ~T ¼ þ ó X j¼ ð Cr =C k Þ f C r =C k g j j! ðj n þ Þ þ ó 9 j ðw j nþ 3 Þ5 (3) To avoid writing such large expressions as the right-h side of equation (3), a shorth notation is used, whereby a function F, depending on parameters Æ, Ł W, is described by FW, ½ Æ, ŁŠ ¼ þ ó9 3 4 X j¼ f Æ=C k g j j! ðj n þ Þ ð Æ=Ck Þ þ ó 9 j ðw j nþ Ł j nþ Þ5 (33a) Equation (33a) can be formulated using a recurrence relationship as FW, ½ Æ, ŁŠ ¼ þ ð Æ=Ck ÞX ~a j (33b) ó 9 l¼ 3 ~a ¼ Æ C k þ ó 9 ðw ŁÞ ðj Þðj Æ=C k Þ ~a j ¼ ~a j j þ ó 9 W j Ł j W j Ł j (33d) Equation (3) can now be written in the concise notation as ~T ¼ FW, ½ C r,š (34) Equation (34) is valid during recompression: that is, when W > W p, where W p corresponds to the preconsolidation pressure. W p is calculated from W p ¼ ó9 ó9 p ó9 (35) The time factor required to reach the preconsolidation pressure, ~T p, is determined by substituting W p into equation (34): hence ~T p ¼ FW ½ p, C r,š (36) Now the power series representation of equation (5b) is substituted into equation (9) solved with the initial condition W ¼ W p at ~T ¼ ~T p : The resulting expression for consolidation in the compression phase is ~T ¼ FW, ½ C c, W p Š C c C r ó9 p ó9 Cc C r ð Þ=C k þ FW ½ p, C r,š (37) Equation (37) can be used for normally consolidated soils by putting C r ¼ C c W p ¼. The m v value in ~T should always be calculated using the recompression index, C r : Substituting a given normalised excess pore pressure into equations (34) (37), the resulting value of ~T can be found. The time required to reach the specified degree of consolidation is then found from equation (3). Equation (3) is undefined for integer values of n; however, values very close to integer values give the appropriate consolidation times (e.g. use n ¼., not n ¼ ). When C r /C k ¼ C c /C k ¼ (i.e. ~c h does not change during consolidation), equation (34) is numerically equivalent to Hansbo s () non-darcian radial consolidation equation. For the case of Darcian flow n ¼, the corresponding value of ~T Darcy is given by ~T Darcy ¼ 8T h ì (38) The ì parameter can then be any of those derived for Darcian flow conditions. The following expressions for ì were derived for

8 99 WALKER, INDRARATNA AND RUJIKIATKAMJORN (a) (b) (c) a smear zone of constant reduced permeability (Hansbo, 98) ì ¼ ln N þ ðkþln ðþ s :75 (39a) s similar to (a), but plane-strain conditions (Indraratna & Redana, 997) " ì ¼ k 3 # N ðn sþ3 3ðN ÞN þ ðn sþ3 3ðN ÞN (39b) a smear zone with parabolic variation of permeability in the smear zone (Walker & Indraratna, 6) ì ¼ ln N 3 s 4 þ kðs Þ s ks þ k ln s p ffiffiffi k ss ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi! (39c) kk ð Þ k þ k ln ðs pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ks þ kþ k k where k is the ratio of undisturbed permeability to the permeability at the drain/soil interface. Equation (39c) was simplified by ignoring insignificant terms, can be used when n. ; otherwise the full expression presented by Walker & Indraratna (6) is more appropriate, usually for smaller drain spacing. When n ¼ C r /C k ¼ C c /C k ¼, equation (34) is numerically equivalent to Hansbo s (98) radial consolidation equation. EFFECT OF C c /C k ON NORMALLY CONSOLIDATED SOIL For normally consolidated soils, if C c /C k,, then c h increases over time, the consolidation rate is faster. If C c /C k. then c h decreases over time, the consolidation rate is slower. The degree to which consolidation is slowed or hastened is controlled by the normalised factor =ó 9: Taken together, C c /C k =ó 9 give an indication of the total change in ~c h over the consolidation period, in accordance with equation (5). In practice, C c /C k ranges between.5. (Berry & Wilkinson, 969; Mesri & Rokhsar, 974), with C k taken from the empirical relation C k ¼. 5e : Figure 3 gives consolidation curves based on the ratio between the final initial consolidation coefficients (c hf c h respectively) for Darcian flow. Although the same change in c h can be obtained with different C c /C k =ó 9 values, the resulting difference in degree of consolidation is minimal. The actual values of C c /C k =ó 9 used in producing Fig. 3 are given in Table. Using different parameter values from those in Table will result in slightly faster or slower consolidation. Faster consolidation would occur if c h /c h was more than unity; otherwise, slower Degree of consolidation: % Chf/ Cho T ho /μ Fig. 3. Consolidation curves for normally consolidated soil under Darcian flow conditions Table. Parameters used to produce consolidation curves shown in Fig. 3 C c /C k =ó 9 c hf /c h consolidation would occur. Analysis including the non-linear soil properties is particularly relevant for soils where relatively high values of =ó 9 will lead to large changes in c h, thus potentially large deviations in degree of consolidation compared with calculations based on constant soil properties. For c hf /c h values close to unity, each % increase in c hf /c h decreases the time to reach 9% consolidation by approximately 5% (compared with the ideal case of c hf /c h ¼ ). Each % decrease in c hf /c h increases the time to reach 9% consolidation by approximately %. SETTLEMENT Primary consolidation settlements, r, result from changes in effective stress. Once excess pore water pressures are found (i.e. based on W ¼ u= ), then the settlements can be calculated using the following expressions. If the initial final effective stresses (ó 9 ó 9 þ ) fall in the recompression zone then settlements are calculated with r ¼ HC r log þ ð WÞ (4a) þ e ó 9 where H is the depth of soil. If the initial final effective stresses fall in the compression zone then settlements are given by r ¼ HC c log þ ð WÞ (4b) þ e ó 9 If the initial effective stress is less than the preconsolidation pressure, ó p 9, the final effective stress is greater than ó p 9 then settlements in the recompression zone are the same as in equation (4a); settlements in the compression zone are then expressed as r ¼ H ðc r C c ÞlogðOCRÞþ C c log þ ð WÞ þ e ó 9 (4c) where OCR is the overconsolidation ratio, defined by OCR ¼ ó9 p (4) ó 9 The total primary settlement, r, can be calculated by putting W ¼ in the above equations. DEGREE OF CONSOLIDATION BASED ON SETTLEMENT AND PORE PRESSURE The degree of consolidation, as determined by pore pressure dissipation, is simply given by

9 U h ¼ u ¼ W (4) The degree of consolidation, based on settlement, is written as U hs ¼ r (43) r The relationship between U hs W depends on the stress history of the soil. In equation (4), when the stress range is either completely in the recompression zone or in the compression zone, the term U hs is then related to W by U hs ¼ log þ ð =ó 9 Þð WÞ (44a) log½ þ ð =ó 9ÞŠ For other cases, if ó 9<ó p 9 then log þ ð =ó 9Þð WÞ U hs ¼ ð C c =C r ÞlogðOCRÞþ C c =C r log½ þ ð =ó 9ÞŠ (44b) VERTICAL DRAIN CONSOLIDATION 99 _ U hs : % _ 6 8 U h : % (a) 8 Δσ/ σ 8 4 If ó 9.ó9 p then ð C c =C r ÞlogðOCRÞ þc c =C r log þ ð =ó 9Þð WÞ U hs ¼ ð C c =C r ÞlogðOCRÞ þc c =C r log½ þ ð =ó 9ÞŠ (44c) The degrees of consolidation based on pore pressure settlement are different (Leroueil et al., 99; Indraratna et al., 5). For equation (44a) U h lags behind U hs depending on =ó 9, as shown in Fig. 4. When determining the degree of consolidation for normally consolidated soils by settlement data, it is important to note that the effective stress in the soil will be less than expected, particularly during the middle stages of consolidation, if U h is assumed to be equal to U hs : A large variety of behaviour can occur for overconsolidated soil, as shown in Fig. 5, depending on the value of C c /C r, OCR =ó 9 : Dissipation of excess pore pressure is fast during the recompression stage, because C r is low relative to C c, while the resulting settlements are relatively small compared with the ultimate settlement. This means that U h will be greater than U hs during recompression, but U hs may or may not become greater than U h during the compression stage. The many parameters involved, the subsequent large variety of behaviour relating U hs to U h for _ U hs : % C c / C r OCR Δσ/ σ _ 6 8 U h : % Fig. 4. Comparison between degree of consolidation based on settlement pore pressure for normally consolidated soil _ U hs : % 6 4 Δσ/ σ _ 6 8 U h : % (b) Fig. 5. Comparison between degree of consolidation based on settlement pore pressure for overconsolidated soil (OCR.5): (a) C c /C r ; (b) C c /C r overconsolidated soil, emphasises the need for accurate determination of the soil stress history, the care needed when specifying construction milestones based on degree of consolidation. APPROXIMATION FOR ARBITRARY LOADING The consolidation behaviour expressed by equation (37) is valid only for instantaneous loading. Unfortunately, including arbitrary time-dependent loading such as Conte & Troncone (9), or even a simple ramp load (Walker & Indraratna, 9), will make equation (9) non-separable, to the authors knowledge, analytically unsolvable. However, arbitrary loading can be simulated by subdividing a continuous loading function into a finite number of instantaneous step loads (see Fig. 6). The only restriction is that average excess pore pressure cannot become negative (u. ). For example, slightly decreasing loads associated with submergence of fill can be modelled, but the swelling associated with preload removal (caused by dissipation of negative pore pressure) cannot be modelled with the consolidation equations presented. For the mth loading stage, as shown in Fig. 6, at ~T m the load increases from m to m : The loading stage ends at ~T mþ : The excess pore water pressure at the end of the last increment, u m, is known (e.g. in the first loading step u m ¼ ). The pore pressure after the load application, uþ m,is given by

10 99 WALKER, INDRARATNA AND RUJIKIATKAMJORN Δσ Δσ m Δσ m Actual load Piecewise load (d) Step 4: calculate ~T in small increments of W, with equation (47), until ~T. ~T mþ : (e) Step 5: calculate u mþ : Either use the last W value obtained in step 4, or interpolate between the last two values of W found in step 4. ( f) Step 6: repeat steps 3 to 5 for each loading stage. (g) Step 7: convert the time factor values obtained in above steps to time values using equation (3). u Actual load ~ T m (a) ~ T m T ~ In the steps just described, changes, in the mth loading stage is equal to m : For cases where soil properties vary with depth, the soil profile can be divided into sublayers the above process performed for each sublayer. The above steps have been implemented in a VBA/spreadsheet program that can be downloaded from edu.au/eng/research/geotechnical/software/index.html. u m u m u þ m ¼ u m þ m m (45) The normalised pore pressure at the beginning of the load increment, W þ m, is then described by W þ m ¼ uþ m m (46) The initial conditions for solution of equation (9) have now been found: W ¼ W þ m at ~T ¼ ~T m : There are three cases to consider in the solution of equation (9). If the preconsolidation pressure has been exceeded in previous loading steps, then the normalised pore pressure in the mth loading step is governed by ~T ¼ ~T m þ F W, C c, W þ m ~ T m T m C c C r ó9 p ó9 Cc C r ð Þ=C k (47a) If the preconsolidation pressure has not been exceeded in previous load steps, then the expressions for normalised pore pressure in the recompression compression zones are ~T ¼ ~T m þ F W, C r, W þ m (47b) ~T ¼ ~T m þ FW, ½ C c, W p Š C ð Cc C c ó p 9 r Þ=C k þ ~T C r ó 9 ¼ ~T m þ F W p, C r, W þ m ~ Fig. 6. Schematic diagram of piecewise loading in terms of loading rate combined with resulting excess pore water pressure (47c) The consolidation behaviour can now be described at any point. The following list of steps describes the process of constructing a graph of pore pressure against time. The process is best automated in a computer program. (a) Step : approximate the arbitrary loading by a finite (b) number of instantaneous step loads. Step : convert the loading times to time factor values, ~T using equation (3). (c) Step 3: calculate W þ m using equations (45) (46). (b) ILLUSTRATIVE EXAMPLE A pore pressure settlement analysis has been performed on a soil/drain system with the following properties: r w ¼ r s ¼. 7 m; r e ¼. 4m; C c ¼. 7; C r ¼. 75; C k ¼. 875; ó 9 ¼ kpa; ó9 p ¼,, 3 kpa; n ¼.,. 3; ª w ¼ kn/m 3 ; e ¼ ; H ¼ m. No smear zone is modelled. The resulting average pore pressure settlement plots are shown in Fig. 7. There are a few salient points to note from this figure. For the two overconsolidated soils (ó 9 p ¼, 3 kpa), the change from recompression to compression can be observed during the first ramp loading stage. At the preconsolidation pressure, the slope of the excess pore pressure plot sharply increases, reflecting the slower rate of consolidation during compression. Just prior to the preconsolidation pressure being reached, the dissipation of pore pressure in the highly overconsolidated soil (ó 9 p ¼ 3 kpa) actually exceeds that generated by the load application; the pore pressure reduces despite load still being applied. This occurs because C r /C k,, resulting in an increasingly rapid consolidation reaching a maximum just prior to the preconsolidation pressure. The difference between the Darcian flow of Fig. 7(a) the non-darcian flow of Fig. 7(b) is small for the highly overconsolidated soils. For the normally consolidated soil (ó 9 p ¼ kpa), the difference is illustrated during the first ramp loading stage. For non-darcian flow the pore pressure curve is slightly flatter, because, owing to the power law flow relationship, the higher pore pressures result in faster flow. The settlement plots illustrate the importance of minimising the disturbance caused by vertical drain installation, which can to some extent destroy any existing structure in the soil. Hence the preconsolidation pressure may be lowered, which, as shown in Fig. 7, causes greater settlements for the same pressure increase. This example shows that, with the equations presented above, almost any primary consolidation problem involving radial drainage can be modelled (provided the effective stress increases with time). When more than one soil layer is present, the analysis can simply be repeated with different soil properties (Walker & Indraratna, 9). LABORATORY VERIFICATION The large-scale consolidometer at the University of Wollongong is a steel cylinder 45 mm in diameter 95 mm high, with the ability to monitor the pore pressure settlement response of soils under load. Since 995, the consolidometer has been used by University of Wollongong researchers to study various aspects of vertical drain consolidation (Indraratna & Redana, 995, 998a, 998b; Indraratna et al.,, 4). Fig. 8 shows a schematic diagram of the large-scale consolidometer. The cylindrical

11 Settlement: m Excess pore pressure: kpa T c t/4r h Load h (a) e VERTICAL DRAIN CONSOLIDATION 993 σ p : kpa Hoist Pressure gauge To data logger Pressure chamber Soil Pulley Line connecting to mrel LVDT To vacuum pump Loading piston Top plate To air jack compressor Movable plate Porous material (Two half cylinders) Steel cell mm Teflon sheet Vertical drain Piezometer points Excess pore pressure: kpa Settlement: m T c t/4r h Load cell consists of two stainless steel sections bolted together along the two joining flanges. Top bottom drainage can be facilitated by placing a geotextile on the cell base clay surface. If only radial drainage is required, the permeable geotextile is replaced by an impervious plastic sheet. The clay is thoroughly mixed with water to ensure full saturation, the resulting slurry is placed in the consolidation cell in layers approximately cm thick. It is not necessary to fill the entire consolidometer, since an internal riser can be used to transfer loads from the loading piston to the shortened sample. The ring friction expected with such a large height/diameter ratio (. 5 ) is almost eliminated by using an ultra-smooth Teflon membrane around the cell boundary (friction coefficient less than. 3). Surcharge loading with a maximum capacity of kn can be applied by an air jack compressor system by way of a rigid piston 5 mm thick. Vacuum loading with a maximum capacity of kpa can be applied through the central hole of the rigid piston. An LVDT (linear variable differential h (b) e σ p : kpa 3 Fig. 7. Non-linear radial consolidation for non-darcian flow exponent: (a) n.; (b) n Fig. 8. Schematic diagram of large-scale consolidation apparatus (Indraratna & Redana, 998a, with permission from ASCE) transformer) transducer is placed on top of the piston to monitor surface settlement. Pore water pressures are monitored by strain gauge type pore pressure transducers installed through small holes in the steel cell at various positions in the soil. Transducers are easily located on the cell periphery, or can be placed within the clay by using small-diameter stainless steel tubes. The LVDT pore pressure transducers are connected to a PC-based data logger. The soil is subjected to an initial preconsolidation pressure (usually ó p 9 ¼ kpa), until the settlement rate becomes negligible (strain rate, mm/day). The load is then removed, a single vertical drain is installed using a rectangular steel mrel that acts as a protector for the drain a guide during penetration. S compaction piles may be installed with a circular pipe mrel. Depending on the purpose of the test, vertical horizontal samples may be cored to investigate the effect of drain installation on soil properties. Once the drain has been installed, the required loading sequence is applied to the sample. The consolidation test described below was conducted by Rujikiatkamjorn (6) to verify the radial consolidation model of Indraratna et al. (5) expressed by equation (6). The model of Indraratna et al. (5) determines the best value of c h to use in Hansbo s (98) consolidation equation with reference to C c, C k the stress range. The soil consisted of reconstituted alluvial clay from the Moruya floodplain. The clay size particles (, ìm) accounted for about 4 5% of the soil specimen. For this Moruya clay, the water content plastic limits were 45% 7% respectively. The saturated unit weight was 7 kn/m 3 : In two separate tests, the soil was subjected to an initial preconsolidation pressure, ó9 p ¼ kpa 5 kpa for five days. The load was removed, a mm 3 4 mm b drain was centrally installed, the preconsolidation loads were reapplied. Drainage in the vertical direction was prevented. Once settlements became negligible, the load was increased by 3 kpa 5 kpa respectively (i.e. ¼ 3, 5 kpa). Settlements were monitored after this point, with the soil

12 994 WALKER, INDRARATNA AND RUJIKIATKAMJORN consolidating in the compression range. The properties of the soil drain system are given in Table, based on the smear zone with constant reduced permeability. Only for the purpose of analysis, Darcian flow was considered (i.e. n ¼. ). The degree of consolidation based on settlement was calculated compared with laboratory data, the proposed model, Indraratna et al. (5). The comparison of the three approaches is shown in Fig. 9. Figure 9 shows good agreement between the measured predicted values of degree of consolidation for both the proposed non-linear equations Indraratna et al. (5). As mentioned above, in this case the method of Indraratna et al. (5) will give the same results as Hansbo (98). The proposed equations give a slightly better match in the early stages of consolidation; the difference is less for Test than for Test. Both methods are expected to converge to % consolidation at longer times, as both are based on the same value of ultimate settlement. As the consolidation coefficient is increasing during consolidation (C c /C k, ), by using an average value of c h as in Indraratna et al. (5), settlement is expected to be overpredicted in the initial stages of consolidation, as in Fig. 9. The combination of C c /C k ¼.64 Table. Parameters used in analysis (Indraratna et al., 5) Parameter Test Test C c C k Influence radius, r e :m Equivalent drain radius, r w :m Smear zone radius, r s :m.. N ¼ r e /r w s ¼ r s /r w Initial horizontal permeability, k h : 3 m/s k h /k s.5.5 Initial void ratio, e.95 c h : 3 3 m /day..68 Initial height, H:m Preconsolidation pressure, ó p 9: kpa 5 Load, : kpa 3 5 _ U hs : % Test Test Calculated from laboratory settlement data Proposed model Indraratna et al. (5) =ó 9 ¼.5 produces ratios of final to initial consolidation coefficient of respectively, for Test Test. Fig. 3 showed that for such values of c hf /c h the difference between the non-linear equations the Hansbo (98) equation are small. This explains the small difference between the analysis based on the proposed relationship that of Indraratna et al. (5). Larger more significant discrepancies between the two approaches would be expected for combinations of C c /C k =ó 9 producing c hf /c h values much greater or much less than one. APPLICATION TO A CASE STUDY AND THE IMPLICATIONS OF PARAMETER n At the site of the Second Bangkok International Airport (SBIA), several embankments were constructed over soft clay stabilised by vertical drains vacuum pressure, combined with surcharge loading. The surface settlement excess pore water pressure at the middle of two embankments, TV, incorporating vacuum loading vertical drains are analysed here. The basic soil parameters were reported by Indraratna et al. (5). The initial in situ soil properties are assumed to be the same for both Darcian non-darcian flow analysis. A summary of the properties used for analysis is given in Table 3. Soil layers are divided into m thick sublayers (. 5 m for when required), with stresses pore pressure calculated at sublayer midpoints. Surcharge load reduces with depth according to a Boussinesq stress distribution under the centre of a trapezoidal embankment with 4 m base width 36 m crest width. Vacuum pressure load factor reduces from unity at surface to zero at 5 m depth. Linear load graphs in Fig. are discretised into small step loads every.744 days. Geometry/smear properties for all layers are: k h /k s ¼, r e ¼.5m, r w ¼.5 m, r s ¼. m. The analysis in detail can be found in the VBA/spreadsheet mentioned above. Fig. presents the loading history with both measured surface settlements at the embankment centreline pore water pressures 3 m below the embankment centreline. To study the effect of non-darcian flow, the predicted settlements pore pressures are calculated based on n values varying between. 5 (vacuum pressure is treated as an equivalent surcharge). For settlement predictions (Fig. (b)), when n ¼, a good agreement between the predicted measured results can be obtained. When n increases, the predicted settlements overestimate the measured data. The predicted measured pore pressures are shown in Fig. (c). As expected, the pore pressure dissipates rapidly when n increases. Within 6 days, the agreement between field observations predictions is better according to Darcian flow. However, beyond 6 days, the prediction with non-darcian flow (n ¼. 5) is more realistic. 4 6 Time: days Fig. 9. Comparison between non-linear model Indraratna et al. (5) MODEL LIMITATIONS AND FUTURE RECOMMENDATIONS Although the proposed analytical model is capable of determining degree of consolidation based on both pore Table 3. Parameters used in analysis for embankment TV Layer Unit weight: ó p 9: kpa C r C c C k e k h : kn/m 3 3 m/s.. m m m m Constant within a layer.

13 Loading: kpa Settlement: m Pore pressure: kpa pressure dissipation strain, it has some inherent limitations, either through the assumptions made, or through simplifications applied to facilitate the mathematical formulations solutions in this analytical approach. Some of these limitations are highlighted below. (a) (b) Surcharge Vacuum 4 8 Time: days (a) 4 8 Time: days (b) 4 8 Time: days (c) Field n n n 5 Fig.. Embankment TV: (a) loading history; (b) surface settlements at centreline; (c) pore pressure at 3 m depth below centreline The equations do not capture the strain-rate effects directly, as the model is a stress-based approach, does not incorporate creep effects. For a given time settlement curve, the rate of dissipation of excess pore pressure tends to be slower when considering creep that constitutes part of the total time-dependent settlement. However, extension of this work is currently being carried out at the University of Wollongong adopting a largestrain radial consolidation theory. The proposed model does not include the effects of vertical consolidation through c v, because, for long vertical drains (say exceeding m or so) with relatively close spacing (..5 m), the radial consolidation is predominantly governed by c h : Therefore the question of anisotropy (c h /c v or k h /k v ratio) is not a major concern in this analysis. However, for short vertical drains with larger drain spacing, the contribution from c v will become increasingly relevant, this is to be captured as future work where short prefabricated vertical drains VERTICAL DRAIN CONSOLIDATION 995 can still be used effectively, for example under rail tracks, as demonstrated by Indraratna et al. (). (c) Lateral soil displacements based on elliptical cavity expansion theory (ECET) have been recently analysed (Gheharioon et al., ); however, for simplicity, the mathematical formulations related to ECET have not been incorporated in the current analysis. (d) The non-darcian flow equation adopted in this study is similar to Hansbo () Sathananthan & Indraratna (6). However, the scope of possible flow is limited by (e) the value of n the type of algebraic expression. Well resistance is ignored in the equations, as the authors have shown elsewhere that unless the vertical drains exceed m or so in length, the well resistance is insignificant compared with the smear effects (Indraratna & Redana, ). ( f ) Indraratna & Redana (997) proved, using large-scale consolidation tests, that even within the smear zone, the lateral permeability k h hence the lateral coefficient of consolidation c h can vary. Subsequently, Walker & Indraratna (6) introduced a parabolic fit to this variation. However, for simplicity of the mathematical formulations, the authors have used a constant k h within the smear zone, but it is a reduced value (an equivalent (g) (h) average) compared with the higher k h in the outer undisturbed zone. Use of a parabolic fit may provide a more comprehensive model, but the associated cumbersome mathematical solutions may not necessarily give a much greater accuracy. The model assumes full saturation steady-state flow (both Darcian non-darcian). The unsaturated conditions at the time of installation of relatively dry drains the occurrence of air gaps (occluded air) at the drain/ soil interfaces (Indraratna et al., 4) have not been considered in this analysis. The current solutions will be further enhanced if the effects of soil/drain interfaces for partially saturated conditions are incorporated. In situ (undisturbed) soils usually possess a characteristic structure, whose behaviour can be significantly different from that of the same material in a reconstituted state, as explained below. (i) (ii) Soil structure creates a material that is initially quite stiff at relatively low stress levels. When a soil with structure is remoulded/reconstituted, usually there is a reduction in the voids ratio. (iii) During virgin yielding, structured soil is generally more compressible than the reconstituted soil (Liu & Carter, ). A short preconsolidation period in the laboratory may cause minimal effect on soil settlements excess pore pressure dissipation. The use of reconstituted soil parameters to predict field behaviour may result in the underestimation of soil settlement excess pore pressure. The selection of soil parameters (reconstituted or in situ soil) should be based on soil conditions to obtain accurate predictions of soil settlement excess pore pressure. CONCLUSION An analytical solution to non-linear radial consolidation problems has been presented verified against a largescale laboratory test. The evolution of pore pressure with time described by equation (37) is valid for both Darcian non-darcian flow, can capture the behaviour of overconsolidated normally consolidated soils. Consolidation may be faster or slower when compared with the cases with constant material properties for non-linear material properties. The difference depends on the compressibility/ permeability ratios (C c /C k C r /C k ), the preconsolidation

14 996 WALKER, INDRARATNA AND RUJIKIATKAMJORN pressure (ó 9 p ) the stress increase ( =ó 9 ). If C c /C k, or C r /C k,, then the coefficient of consolidation increases as excess pore pressures dissipate, the corresponding consolidation is faster. If C c /C k. orc r /C k., then the coefficient of consolidation decreases as excess pore pressures dissipate, the corresponding consolidation becomes slower. For the case where C c /C k ¼, the solution is identical to Hansbo (). The equations presented give an analytical solution to non-linear radial consolidation that can be used to verify purely numerical methods. Many vertical drain problems where vertical drainage is negligible effective stresses always increase can be analysed with the approximation for arbitrary loading. ACKNOWLEDGEMENTS The authors would like to express their gratitude to the Australian Research Council Coffey Geotechnics (Sydney Brisbane) for supporting this area of research over a long period of time. The first third authors PhD studies at the University of Wollongong (UOW) under the supervision of Professor Buddhima Indraratna (second author) were sponsored through UOW s Australian Postgraduate Award International Postgraduate Research Scholarship scheme respectively, with further financial support by Queensl Department of Main Roads during that time. APPENDIX Derivations of equations (a) (b) By equating equations (7) (8), the radial flow rate is assumed to be equal to the rate of volume change of soil mass in the vertical direction. ðrv ¼ ð r e r t Substituting equations (6) (9) @ r ¼ " ð # =n r e r mv ª w ; n > ðr ~ k " ð # =n r e r mv ª ðr ~ ; n > (49b) k Substituting N ¼ r e /r w y ¼ r/r w in equations (49a) (49b) yields equations (a) (b). Derivation of g(y) From equation (), g(y) can be expressed as ð X f =ng =n j y j gðyþ ¼ y (5) j! N j¼ ð f =n y j =n (5) gðyþ ¼ X j¼ g j j!n j gðyþ ¼ X f =ng j y j ð=nþþ j¼ j!n j j ð=nþþ Let a j Then a ðyþ ¼ X j¼ f =ng j =n ny ð Þ ðyþ ¼ n f =n ðyþ ¼ ðj a j y j =n j!n j j ð=nþþ ð Þþ g j y j Þ!N ðj Þ j ð Þ ð=nþþ ð Þ ð=nþþ (5) (53) (4b) (54) Dividing equation (53) by equation (54) gives a j a j a j a j a j a j a j a j ðyþ ðyþ ¼ f =ng j y j ð=nþþ j!n j j ð=nþþ f =n ðj Þ!N j g j y j ð Þ ðyþ ðyþ ¼ f =ng j ðj Þ! f =ng j j! ðyþ ðyþ 3 j =n y ð Þþ y ð j Þ =n ¼ =n þ j j ð Þ ð=nþþ ðj Þ ð=nþþ N ðj Þ N j ðj Þ ð=nþþ ð Þþ j ð=nþþ y jn n N jn þ n ðyþ ðyþ ¼ y ðjn n Þðjn n Þ N jnðjn þ n Þ (55) (56) (57) (58) NOTATION C c compression index C k permeability change index C r recompression index ~c h horizontal coefficient of consolidation for non-darcian flow c hf final value of horizontal consolidation coefficient ~c h initial horizontal coefficient of consolidation for non- Darcian flow d e diameter of influence area e void ratio e initial void ratio e f final void ratio F function g function H height of soil; drainage length i hydraulic gradient; integer j integer ~k non-darcian coefficient of consolidation ~k h undisturbed non-darcian horizontal permeability ~k s smear zone non-darcian horizontal permeability m integer m v coefficient of volume compressibility m v initial value of volume compressibility N ratio of influence radius to drain radius n non-darcian flow index OCR overconsolidation ratio P av averaging factor to account for changing coefficient of consolidation r radial coordinate r e radius of influence area r m equivalent radius of mrel r s radius of smear zone r w radius of drain s ratio of smear zone radius to drain radius T h time factor for horizontal consolidation T h horizontal time factor calculated from initial parameters ~T modified time factor ~T Darcy Darcian time factor ~T m modified time factor at application of mth instantaneous load ~T p modified time factor at preconsolidation pressure t time U degree of consolidation U h average degree of consolidation in horizontal direction U hs degree of consolidation calculated with settlement data u pore water pressure u m pore pressure immediately before application of mth instantaneous load u þ m pore pressure immediately after application of mth instantaneous load u s smear zone pore pressure u average excess pore pressure u initial excess pore pressure W normalised pore pressure normalised pore pressure at preconsolidation pressure W p