Vertical Drain Consolidation with Parabolic Distribution of Permeability in Smear Zone

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1 Vertical Drain Consolidation with Parabolic Distribution of Permeability in Smear Zone Rohan Walker and Buddhima Indraratna, F.ASCE Downloaded from ascelibrary.org by ESCOLA POLITECNICA- USP on 07/5/3. Coyright ASCE. For ersonal use only; all rights reserved. Abstract: A vertical drain radial consolidation equation based on a arabolic reduction in ermeability toward the drain is resented. The roosed equation, based on Hansbo s equal strain theory, is comared with settlement data from a laboratory test in a large scale consolidometer. DOI: 0.06/ASCE :7937 CE Database subject headings: Drainage; Consolidation; Laboratory tests; Pore water ressure; Permeability. Introduction Vertical drains are used to imrove soft soil by roviding a horizontal drainage ath along which excess ore water ressures, caused by a surcharge, can dissiate faster than by a vertical drainage ath alone. Installation of vertical drains results in disturbance of the soil zone adjacent to the drain. A number of researchers Chai and Miura 999; Sharma and Xiao 000; Hawlader et al. 00 have noted that the disturbance in this smear zone increases toward the drain. However, to date, most analytical models e.g., Hansbo 98; Zhu and Yin 004 have included smear effects by incororating a reduced horizontal ermeability that is held constant throughout the smear zone. Tyically, the size of smear zone is assumed as a function of mandrel or drain size with the smear zone ermeability set equal to the undisturbed vertical ermeability. The extent of smearing deends on the mandrel size and soil tye Lo 998; Eriksson et al Assumed radii for a constant ermeability smear zone range in size from.6 to 4 times the equivalent drain or mandrel radius Hansbo 98; Indraratna and Redana 998b. If sufficient measurements are available, then the smear zone roerties do not have to be assumed. Fig. shows that for laboratory work on reconstituted clays Onoue et al. 99; Indraratna and Redana 998b a arabolic decay in horizontal ermeability towards the drain is aroriate. The model roosed in this technical note is an extension of Hansbo 98 theory incororating a arabolic ermeability distribution. The model redictions are comared with observed settlement data using a large scale consolidation cell. Doctoral Student, School of Civil Engineering, Faculty of Engineering, Univ. of Wollongong, Wollongong City, NSW 5, Australia. Professor, School of Civil Engineering, Faculty of Engineering, Univ. of Wollongong, Wollongong City, NSW 5, Australia. Note. Discussion oen until December, 006. Searate discussions must be submitted for individual aers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscrit for this technical note was submitted for review and ossible ublication on January 3, 005; aroved on January 3, 006. This technical note is art of the Journal of Geotechnical and Geoenvironmental Engineering, Vol. 3, No. 7, July, 006. ASCE, ISSN /006/ /$5.00. Analytical Solution Vertical drains, installed in a square or triangular attern, are usually modeled analytically by considering an equivalent axisymmetric system. Pore water flows from a soil cylinder to a single central vertical drain with simlified boundary conditions. Fig. shows a unit cell with an external radius r e, and an initial drainage ath length l. The radius of the vertical drain and smear zone are r w and r s, resectively. According to Hansbo 98, for axisymmetric flow, the average degree of consolidation, Ū h, on a horizontal lane at a deth z and at time t is Ū h = ex 8T h / hs a where the value of hs for smear effect, assuming no well resistance, is given by hs =lnn/s k h /klns h 0.75 Including smear and well resistance hs =lnn/s k h /klns h 0.75 zl zk h /q w Neglecting both smear and well resistance b c h =lnn 0.75 d In the receding, n=r e /r w and s=r s /r w see Fig. ; q w = drain discharge caacity; k h = horizontal coefficient of ermeability; and k h = horizontal coefficient of ermeability in the smear zone, which is assumed constant throughout the smear zone in the Hansbo 98 theory. Parabolic Smear Zone Permeability This section resents a revised radial consolidation equation based on a arabolic decay in ermeability towards the drain. The calculation stes in Hansbo s 98 axisymmetric analysis are followed with modification, in order to incororate the decay of ermeability in the smear zone. Real soil roerties may change during consolidation but in this analysis, for simlicity, the coefficient of consolidation is assumed constant horizontal ermeability varies satially but not with time. The velocity of water flow in the undisturbed zone Darcy s law is given by JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JULY 006 / 937

2 Downloaded from ascelibrary.org by ESCOLA POLITECNICA- USP on 07/5/3. Coyright ASCE. For ersonal use only; all rights reserved. Fig.. a Ratio of horizontal to vertical ermeability along radial distance from drain in large scale consolidometer original data from Indraratna and Redana 998b; b roosed horizontal ermeability distribution for isotroic soil original data from Onoue et al. 99 v r = k h w r u where w =unit weight of water; u=ore water ressure; and r = radial coordinate. A similar relation exists in the smear zone, hence v r = k h w u r where uore water ressure in the smear zone. Hansbo 98 used a constant value of k h. In the writers model, k h is a function of r determined by the conditions k h r w = k 0 3 4a Fig.. Axisymmetric unit cell The flow of ore water through the boundary of the cylinder with radius r is equal to the change in volume of the hollow cylinder with outer radius r e and inner radius r, which gives rv r = r e r 6 t where /t=deth averaged vertical strain rate. Substituting Eq. into Eq. 6 and rearranging gives the ore ressure gradient in the undisturbed zone r = u w k h t r r e r r s r r e 7 Similarly in the smear zone the ore ressure gradient is u = w r k h t r r e r r w r r s 8 For vertical flow in the drain, the change in flow from the entrance to the exit of the slice with thickness dz Fig. is given by dq = r wk w w ur w,z z dzdt 9 where k w =drain ermeability. The radial flow into the slice is determined from k h r s = k h 4b k h r s /r =0 4c The arabolic curve that satisfies the above conditions, shown schematically in Fig. 3, is given by k h r = k 0 A B Cr/r w A B Cr/r w where =k h /k 0 ; A=/ ; B=s/s ; and C=/s. 5 Fig. 3. Permeability distribution 938 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JULY 006

3 Downloaded from ascelibrary.org by ESCOLA POLITECNICA- USP on 07/5/3. Coyright ASCE. For ersonal use only; all rights reserved. dq = r wk 0 w ur w,zdzdt 0 r To satisfy continuity of flow dq = dq Assuming no sudden dro in ore ressure at the drain-soil boundary i.e., u=u at r=r w, substituting Eqs. 9 and 0 into Eq. yields the following: ur w,z r k w r k 0 ur w,z z =0 Substituting Eq. into Eq. 8 and integrating in the z direction subject to the boundary conditions: u=0 at z=0, u=0 at z=l, and u/z=0 at z=l, the ore ressure at the drain-soil boundary is determined as ur w,z = w k w tn lz z 3 With the substitution of Eq. 5, Eq. 8 is integrated in the r direction with the same boundary conditions as mentioned reviously. In the same manner, Eq. 7 is then integrated assuming u=u at r=r s. The resulting exressions for ore water ressure on either side of the smear zone boundary are where u = wr e k h tln r/r w s n r s r w A A A B lns ln BE n C ln BE A k h k w n lz z u = w r e k 0 t ln r A B A BF A BG A r w n AC A BF A BG k 0 k w n lz z E =ln A A Fr =ln A B Cr/r w A Gr =ln A B Cr/r w A 4 5 If ū is the average excess ore ressure in the soil cylinder at deth z, then ūr e r w =r w r s rudr r s r e rudr 6 Substituting Eqs. 4 and 5 into Eq. 6 and subsequent solution gives ū = wr e k h 7a t where = n n A n k zl z 7b q h n = A B s lns s and A ABE A B ln C B A B ln A n C4 Bln 3ABE 3B 7c =lns n 3 4 s s s n A 4n n A B ln s BE A ln BE n C A 7d As n is generally much greater than s in most cases, the insignificant terms can then be ignored, thus Eq. 7b can be reduced to =ln n s 3 4 s s s ln s ss s s ln k zl z q h 7e The final term involving q w is omitted when ignoring well resistance. As and s, Eq. 7 aroaches the ideal case in Eq. c. Errors using Eq. 7e rather than Eq. 7b become significant when n0. Eq. 7 may now be combined with the Terzaghi constitutive equation for one-dimensional comression t = m ū v = m t v 8 t where m v =coefficient of volume comressibility m v in smear and undisturbed zone assumed equal and =average effective stress. Combining Eqs. 8 and 7 with the initial condition ū=ū 0 at t=0 gives ū = ū 0 ex 8T h / 9 where T h =c h t/4r e =horizontal time factor and c h =k h /m v w = horizontal coefficient of consolidation. The average degree of consolidation in the radial direction at a articular deth with well resistance, as in Eq. a, is now given by Ū h = ū ū 0 = ex 8T h / 0 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JULY 006 / 939

4 Downloaded from ascelibrary.org by ESCOLA POLITECNICA- USP on 07/5/3. Coyright ASCE. For ersonal use only; all rights reserved. Comarison with Laboratory Testing Soil roerties, testing rocedures, settlement, and ore ressure data for the laboratory test described here are described fully in Indraratna and Redana 998a,b. The relevant data summarized in the following from this test is reanalyzed here with the roosed consolidation equations. Predicted and measured settlement data are comared. Reconstituted alluvial clay from Moruya 40 50% clay sized articles m, 40% saturated water content, liquid limit of 70, lastic limit of 30, and 7 kn/m 3 saturated unit weight was thoroughly mixed and laced in the steel consolidation cell, which is a stainless steel cylinder height=950 mm and diameter=450 mm, where drainage is rovided at the to of the soil. The height of the samle can be shortened by using an internal riser. The ring friction exected with such a large height/diameter ratio.5 is almost eliminated by using an ultrasmooth Teflon membrane around the cell boundary friction coefficient less than The soil was subjected to an initial reconsolidation ressure, =35 kpa until the settlement rate became negligible. The load was then removed and a single vertical drain Flowdrain 75 mm4 mm was installed using a rectangular steel mandrel 80 mm0 mm. In order to measure the disturbance of the soil due to insertion of the mandrel, small horizontal and vertical secimens were cored from the consolidometer. These samles were subject to one-dimensional consolidation using conventional 50 mm diameter oedometers. The measured soil roerties are as follows: comression index C c =0.34, recomression index C r =0.4, vertical coefficient of consolidation c v =.50 8 m /s c v in smear and undisturbed zone assumed equal, vertical coefficient of ermeability k v =.50 0 m/s, and the horizontal ermeability distribution is shown in Fig. a. The high C r /C c ratio exceeding 0.4 is due to remolding, and a similar value for remolded Winieg clay was determined by Graham and Li 985. The equivalent radius of the band drain after Rixner et al. 986 is r w =754/4=0 mm. The fitted arabolic curve in Fig. a is described by k h /k 0 =.6 at r=r w, k 0 is assumed equal to k v, r e /r w =.5, and r s /r w =8.4. These arameters give k h = m/s, k 0 =.50 0 m/s, =.5, and c h =.40 8 m /s. Owing to the short vertical drainage length, the well resistance was ignored, and vertical degree of consolidation was considered by Terzaghi s one-dimensional equation Ū z = m=0 M ex M T z where M =m /; m=,,...; and T z =c v t/l =vertical time factor. Consolidation by vertical and horizontal drainage are combined with Carillo s 94 relationshi Ū = Ū z Ū h After drain installation an overconsolidated initial state 0 for the soil was first induced with the alication of 0 =0 kpa. When the settlement rate became negligible, the surcharge ressure was increased to 50, 00, and 00 kpa threestage loading. Using Eqs. 0, the average excess ore ressure was calculated using the arabolic ermeability distribution within the smear zone. For an initial void ratio of e 0 =0.95, the settlement,, was calculated using: Fig. 4. Predicted and measured settlements for large scale consolidometer = HCr e 0 log, 0 HC r e 0 log 0 HC c e 0 log 3, Settlement curves are shown in Fig. 4. Also shown in Fig. 4 are the corresonding settlement lots with constant ermeability throughout the smear zone Eq. b, k h /k 0 is the same as for the arabolic case, i.e.,.6 for the ideal drain no smear: r s /r w = and an assumed uer bound for maximum smear r s /r w =6. The writers solution with arabolic ermeability decay, r s /r w =8.4, and Hansbo 98 with constant ermeability are identical only in the case of Hansbo s r s /r w =.6. This shows that the extent of smearing is much greater than that assumed when considering a smear zone with constant reduced ermeability smear zone. Fig. 4 confirms that the effects of smear can be assessed by using existing assumtions about the size of a constant ermeability smear zone with a radius of.6 4 times the equivalent drain or mandrel radius Hansbo 98; Indraratna and Redana 998b. However, more meaningful interretations of the extent of smear can be made using the roosed arabolic change of lateral ermeability within the smear zone, justifiable based on laboratory observations. By using the measured arabolic ermeability distribution, which gives good agreement with the model analysis as shown in Fig. 4, the need for assuming a smear zone radius and the consequent uncertainty in the analysis are reduced. It is acknowledged that hs is easier to calculate than, but in soils where the rate of consolidation is deendent on the roerties of the smear zone, the current model rovides enhanced reliability, in site of the more rigorous comutational rocedure. Conclusion In this study, a modified vertical drain radial consolidation equation based on the arabolic reduction of ermeability towards a vertical drain is resented. The arabolic reduction in ermeability is based on laboratory evidence. Hansbo s 98 well known radial consolidation equations where a constant coefficient of 940 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JULY 006

5 Downloaded from ascelibrary.org by ESCOLA POLITECNICA- USP on 07/5/3. Coyright ASCE. For ersonal use only; all rights reserved. consolidation is assumed are modified, without increasing the number of variables. The validity of the method has been examined by comarison with settlement data using a large scale consolidometer. Greater confidence in redicted results is gained by basing the analysis on a measured ermeability distribution rather than the ast conventional aroach of assuming the size of smear zone with a constant ermeability. Notation The following symbols are used in this technical note: A,B,C,E,F,G arameters in ; C c comression index; C r recomression index; c h radial consolidation coefficient; c v radial consolidation coefficient; H deth of soil; k h horizontal soil ermeability in undisturbed zone; k h horizontal soil ermeability in smear zone; k w ermeability in drain; k 0 horizontal soil ermeability at r w ; l vertical drainage length; M summation term; m summation index; m v soil volume comressibility; n ratio of influence radius to drain radius; Q, ore water flow volumes; q w drain discharge caacity; r radial coordinate; r e radius of influence; r s radius of smear zone; r w radius of drain; s ratio of smear radius to drain radius; T h radial time factor; T z vertical time factor; t time; average degree of radial consolidation; Ū h Ū z average degree of vertical consolidation; u ore water ressure in undisturbed zone; u ore water ressure in smear zone; ū average excess ore water ressure; v r radial ore water velocity; z deth; w unit weight of water; strain; smear zone ermeability ratio; drain and soil arameter for constant k h ; drain and soil arameter for arabolic k; h arameters in ; and effective stress. hs, References Carrillo, N. 94. Simle two and three dimensional cases in the theory of consolidation of soils. J. Math. Phys.,, 5. Chai, J.-C., and Miura, N Investigation of factors affecting vertical drain behavior. J. Geotech. Geoenviron. Eng., 53, 6 6. Eriksson, U., Hansbo, S., and Torstensson, B.-A Soil imrovement at Stockholm-Arlanda Airort. Ground Imrovement, 4, Graham, J., and Li, E. C Comarison of natural and remolded lastic clay. J. Geotech. Eng., 7, Hansbo, S. 98. Consolidation of fine-grained soils by refabricated drains. Proc., 0th Int. Conf. Soil Mechanics and Foundations Engineering, Balkema, Stockholm, Hawlader, B. C., Imai, G., and Muhunthan, M. 00. Numerical study of the factors affecting the consolidation of clay with vertical drains. Geotext. Geomembr., 04, Indraratna, B., and Redana, I. W. 998a. Develoment of the smear zone around vertical band drains. Int. J. of Ground Imrovement, 4, Indraratna, B., and Redana, I. W. 998b. Laboratory determination of smear zone due to vertical drain installation. J. Geotech. Geoenviron. Eng., 4, Lo, D Vertical drain erformance: myths and facts. Transactions, Hong Kong Inst. Eng., 5, Onoue, A., Ting, N.-H., Germaine, J. T., and Whitman, R. V. 99. Permeability of disturbed zone around vertical drains. Proc., Geotechnical Engineering Congress, Boulder, Vol., Rixner, J. J., Kraemer, S. R., and Smith, A. D Prefabricated vertical drains. Engineering Guidelines, FHWA/RD-86/68, Federal Highway Administration, Washington, D.C. Sharma, J. S., and Xiao, D Characterization of a smear zone around vertical drains by large-scale laboratory tests. Can. Geotech. J., 376, Zhu, G., and Yin, J.-H Consolidation analysis of soil with vertical and horizontal drainage under ram loading considering smear effects. Geotext. Geomembr.,, JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING ASCE / JULY 006 / 94