Zhichao Shen ,Siau Chen Chian ,Siew Ann Tan ,Chun Fai Leung

a School of Marine Science and Engineering,South China University of Technology,Guangzhou,China

b Department of Civil and Environmental Engineering,National University of Singapore,Singapore

Keywords: Consolidation Vertical drain Smear effect Pore pressure Soil improvement

ABSTRACT Vertical drains are used to accelerate consolidation of clays in ground improvement projects.Smear zones exist around these drains,where permeability is reduced due to soil disturbance caused by the installation process.Hansbo solution is widely used in practice to consider the effects of drain discharge capacity and smear on the consolidation process.In this study,a computationally efficient diameter reduction method (DRM) obtained from the Hansbo solution is proposed to consider the smear effect without the need to model the smear zone physically.Validated by analytical and numerical results,a diameter reduction factor is analytically derived to reduce the diameter of the drain,while achieving similar solutions of pore pressure dissipation profile as the classical full model of the smear zone and drain.With the DRM,the excess pore pressure u obtained from the reduced drain in the original undisturbed soil zone is accurate enough for practical applications in numerical models.Such performance of DRM is independent of soil material property.Results also show equally accurate performance of DRM under conditions of multi-layered soils and coupled radial-vertical groundwater flow.

1.Introduction

Ground improvement is widely employed to increase the strength and stiffness of soft soils to provide sufficient support for infrastructures.Among ground improvement techniques for soft soils,the combination of prefabricated vertical drains (PVDs) and preloading using fill surcharge or vacuum pressure is a common choice to accelerate the consolidation process.With the application of closely spaced drains,the dominant direction of flow of pore water would be altered from vertical(without drains)to horizontal(with drains),which dramatically reduces the drainage path,and accelerate the consolidation process.

In practice,band-shaped PVDs are typically installed in triangular and square patterns to provide a drainage path for pore water from the surrounding known as the‘unit cell’.For the convenience of analysis,the PVD and unit cell are commonly simplified as circles of equivalent diameters to obtain the axisymmetric feature,as shown in Fig.1.The installation of the PVD creates a disturbed soil zone,known as the smear zone,around the PVD.Within the smear zone,the soil hydraulic conductivitykand compressibility are changed in comparison to those in the surrounding undisturbed zone,which results in a reduction in the rate of consolidation.This smear effect can be considered by either using a reduced diameter of the drain or by assuming a smear zone of reduced permeability around the drain(Gabr et al.,1997).The former strategy attracted a little attention.Several researchers evaluated the commonly used equivalent diameter formulations for a PVD using equal equipotentials and equal flow rate approach (Welker et al.,2000;Welker and Herdin,2003;Abuel-Naga and Bouazza,2009).However,the equivalent diameter formulations solely depend on the geometry of a PVD (i.e.width and thickness) without consideration of the property of undisturbed and smeared soils.Various analytical solutions were proposed for the single drain unit cell analysis using the latter strategy.Hansbo (1981) assumed a smear zone with a constant horizontal hydraulic conductivity around the drain and developed radial consolidation equations without consideration of vertical flow.Following Hansbo(1981)work,analytical solutions to excess pore water pressure have been deduced for various boundary conditions of the unit cell(Indraratna and Redana,1997;Indraratna et al.,2005;Basu et al.,2006,2010;Walker and Indraratna,2006;Walker et al.,2012;Liu et al.,2014;Lu et al.,2015,2016;Rujikiatkamjorn and Indraratna,2015;Huang et al.,2016;Deb and Behera,2017;Nguyen and Kim,2019),such as varying horizontal hydraulic conductivity and compressibility in the smear zone,assumption of a transition zone between smear and undisturbed zones,non-Darcian flow,time-variable loading and multi-layered soils.

Fig.1.The geometry of a unit cell.

For most consolidation problems consisting of combinations of time-variable loading,multi-layered soils,multiple drains and plastic soil model,engineers and researchers typically carry out numerical analysis in view of complex boundary conditions and adopt a constant hydraulic conductivity within the smear zone(Indraratna and Redana,2000;Lin and Chang,2009;Lam et al.,2015;Liu and Rowe,2015;Oliveira et al.,2015;Yildiz and Uysal,2015).This is because the adoption of a variable hydraulic conductivity is outside the capacities of the most commonly used commercial software,such as Plaxis (2020).Modelling each PVD with surrounding smear zone in a multiple drains threedimensional (3D) numerical analysis usually complicates the creation of the numerical model and consumes extensive computation time due to the small size of PVD and smear zone compared to the whole model.In order to reduce the complexity of modelling drains and computational effort,the multiple drains 3D numerical model is usually simplified(Borges,2004;Chen et al.,2016).For example,Borges (2004) conducted a 3D analysis of embankments on soft soils incorporating vertical drains without considering the smear effect.Chen et al.(2016) applied the reduced soil hydraulic conductivity in the drainage zone of 3D model to consider smear effect without modelling smear zones physically.As noted by Chen et al.(2016),this assumption may not be representative of the conditions at the site.It is hence desirable to propose a practical rigorous method to simplify the modelling of vertical drains with consideration of the smear effect.

In this paper,a diameter reduction method(DRM)is proposed based on Hansbo solution (Hansbo,1981).This method considers the effect of smear by reducing diameters of vertical drains,which enables consideration of smear effect without having to model a smear zone physically or alter the soil properties in the numerical model.According to the DRM,the diameter of vertical drain is reduced based on the property of the drain,undisturbed and smeared soils,which is different from the equivalent diameter formulations that solely depend on the geometry of a PVD.Although the DRM is proposed based on Hansbo solution,it also permits easy implementation to layered soil profiles with coupled radial-vertical flow when multiple drains are involved.Under the condition where a constant hydraulic conductivity within the smear zone can be reasonably assumed,the DRM is valid and especially useful for a 3D numerical analysis including multiple drains.The arrangement of this paper is outlined as follows.First,Hansbo solution is extended to consider the excess pore pressureufor arbitrary locations in the unit cell rather than the average excess pore pressureuavein the original solution.This is more appropriate for engineers and researchers to validate their numerical models.Next,the development of the DRM would be described based on the extended Hansbo solution.Lastly,practical considerations of the DRM(especially practical conditions beyond the assumption of Hansbo solution and the DRM,such as multilayered soils with coupled radial-vertical flow and the change in compressibility in the smear zone) would be discussed and recommendations provided.

2.Finite element model and validation

2.1.Geometry and mesh

In this study,finite element simulations were conducted using the software Plaxis 2D(Plaxis,2020).An axisymmetric model with 15-noded elements was used to represent the cylinder of the unit cell,as shown in Fig.2.The drainage line element was employed to simulate the ideal drain without consideration of discharge capacity(or well resistance)of the vertical drain.The actual drain was simulated with the solid element type as soils.The groundwater flow boundary conditions of the unit cell model were set according to flow directions.For the vertical flow,only the flow boundary condition on the top side of the unit cell model was set toSeepageto allow for a permeable boundary and the other three sides(i.e.left,right and bottom) were set toClosedto consider an impermeable boundary.Fig.2 shows flow boundary conditions of the radial flow for ideal and actual drains,respectively.For the ideal drain,the drainage line element on the left side represents a permeable boundary.For the actual drain,only the boundary on the top of the drain is permeable(Fig.2b).For the coupled radial-vertical flow,the permeable boundaries are the combination of permeable boundaries of vertical and radial flows for both ideal and actual drains.The left and right sides of the unit cell were constrained to prevent horizontal displacement and the bottom side was constrained to prevent vertical displacement.

Fig.2.Model geometry and generated mesh:(a)Ideal drain without smear zone(free-strain case,radial flow),(b)Actual drain with smear zone(equal-strain case,radial flow),and(c) Reduced drain based on DRM.

The excess pore water pressureuwas generated with the application of uniform load on the top of the soil cylinder.There are two fundamental cases according to the nature of the ground surface,i.e.(1)the free-strain case for a flexible areal load,and(2)the equal-strain case for a rigid areal load.For free-strain case,the uniform load was directly applied to the ground surface.For equalstrain case,a plate with high stiffness (e.g.bending stiffnessEI=1015kN m2)was placed at the ground surface and the uniform load was applied to the plate to keep the equal-strain deformation condition of the ground surface.

TheFine Meshoption was selected for theElement distributionwith the generated mesh illustrated in Fig.2.

2.2.Materials

The soil was regarded as a linear elastic material and under undrained condition during the application of loads.A Poisson’s ratio of ν′=0 was assumed to prohibit transverse strain.The only difference in material properties among actual drain,soils in the smear and undisturbed zones was the hydraulic conductivity.For the simplicity of the validation of the numerical model,the hydraulic conductivity of the unit cell in horizontal and vertical directions was set to be identical.It should be noted that the DRM is derived based on Hansbo solution,which is only related to horizontal hydraulic conductivitykh,and therefore the vertical hydraulic conductivitykvdoes not influence the derivation of DRM.The hydraulic conductivity of smeared soilskswas determined accordingly to the hydraulic conductivity ratio of undisturbed and smeared soils η.The hydraulic conductivity of drainkdwas calculated with the discharge capacity of the drainqd.

2.3.Calculations

TheK0procedure was used in the Initial Phase to generate the initial stresses.In Phase 1,aPlasticanalysis (i.e.elastoplastic undrained analysis)was conducted to generate the initial excess pore pressureu0,in which the uniform load was activated.In the following multiple Phases,Consolidationanalyses were performed for different time intervals.In Plaxis,the error limits of the numerical solution from the exact solution are linked closely to the specified value ofTolerated error.Within each step,the calculation continues to carry out iterations until the calculated errors are smaller than the specified value.In general,the default value of 0.01 is suitable for most calculation (Plaxis,2020).For higher accuracy,theTolerated erroris set to 0.001 in this study.

2.4.Validation

The numerical model of the unit cell with an ideal drain(Fig.2a)was validated with analytical solutions step by step.The vertical,radial and coupled radial-vertical flows were simulated,respectively,for the same model geometry.Note that Fig.2a only shows the free-strain case of radial flow among these three flows.Under the vertical flow condition,the numerical solution was validated with Terzaghi solution (Terzaghi,1925).The Barron solution(Barron,1948) was employed to validate the numerical solutions under the radial flow condition.According to Carrillo theory(Carrillo,1942),analytical solutions for the coupled radial-vertical flow can be derived with the product of Terzaghi and Barron solutions,which was used to validate the numerical solution for the coupled radial-vertical flow.The details of Terzaghi,Barron and Carrillo solutions are provided in Table 1.

Table 1Analytical solutions for validation of the numerical model.

Table 2 summarises inputs of the numerical model for validation.It should be noted that values adopted in Table 2 are only for the purpose of validation though some of them are proposed based on the field practice as illustrated in the note of Table 2.Fig.3 compares numerical results with analytical solutions under the vertical,radial and coupled radial-vertical flow conditions,respectively.It can be seen from Fig.3 that numerical results agree very well with analytical solutions at varying locations and consolidation times under all flow conditions,which validates the accuracy of the numerical model in this study.

Table 2Inputs of the numerical model for validation (ideal drain).

Fig.3.Validation of numerical models with analytical solutions (ideal drain):(a)vertical flow,(b)radial flow(free-strain case),(c)radial flow(equal-strain case),and (d)coupled radial-vertical flow (free-strain case).

Fig.4 compares the consolidation process between free-strain and equal-strain cases according to analytical and numerical results.Numerical results for these two cases agree very well with analytical solutions for the free-strain case at all times modelled in Figs.3 and 4.Good agreement in numerical results between freestrain and equal-strain case reflects that the difference between these two cases is minor,consistent with findings in previous studies (Barron,1948;Hansbo,1981).

Fig.4.Comparison of free-strain and equal-strain cases under radial flow condition (z/h=0.5): (a) Curve point location r/rc=0.095,(b)Curve point location r/rc=0.476,and (c)Curve point location r/rc=0.952.

It is noteworthy that there exists a contrast in approach between the analytical equal-strain solution and other solutions(i.e.analytical free-strain,numerical equal-strain and free-strain) at the beginning of consolidation.For the analytical equal-strain case,initial excess pore water pressureu0is not spatially constant such thatu0is not necessarily identical to average initial excess pore water pressureu0,ave(i.e.u0≠u0,ave) at all locations,which can be calculated through the analytical equation in Table 1.In contrast,other solutions (analytical free-strain,numerical equal-strain and free-strain) assume thatu0is spatially constant such thatu0=u0,aveis satisfied at any location.For the numerical equal-strain and free-strain cases,the initial excess pore water pressureu0is generated by the application of a uniform load in an undrained analysis step and therefore is spatially constant in the unit cell model.Hence,differences in results between these two groups of solutions exist particularly at the beginning of consolidation whenTr≤0.05.

3.Extension of Hansbo solution and diameter reduction method

3.1.Review of Hansbo solution

Hansbo solution (Hansbo,1981) that considers the effect of discharge capacity and smear is based on Kjellman solution(Kjellman,1948) for the ideal drain.During the consolidation process,the horizontal sections of the unit cell are assumed to remain horizontal and without horizontal strains (i.e.equal strain hypothesis).Constant compressibility throughout the soil cylinder and the validity of Darcy’s law were assumed in the derivation of Hansbo solution.Solutions of average excess pore water pressureuaveand relationships betweenu(i.e.excess pore water pressure)and ?ε/?t(i.e.the rate of vertical strain of the unit cell) based on Hansbo solution are summarised in Tables 3 and 4,respectively,for different cases including ideal and actual drains and smear effect.The ideal and actual drains can be deemed as a drain with unlimited and limited discharge capacityqd,respectively.Hence,the parameter μdsin Table 3 can be adjusted to μswithqd→+∞,while μsconsiders the smear effect for the ideal drain case.

Table 3Hansbo solution for average excess pore water pressure uave.

3.2.Extension of Hansbo solution and validation

3.2.1.Extension of Hansbo solution

Hansbo(1981)proposed solutions for average excess pore water pressureuaverather than excess pore water pressureu.However,in some cases,it is more convenient to use the solutionu,such as calculation of excess pore pressure at a specific location and validation of numerical models.In Hansbo (1981),the relationship between excess pore pressureuand ?ε/?twas provided,as summarised in Table 4,but the relationship between average excess pore pressureuaveand ?ε/?twas not explicitly provided.If the relationship betweenuand ?ε/?tand the relationship betweenuaveand ?ε/?tare both known,the relationship betweenuanduaveshould be known.With the combination of this relationship and Hansbo solution,excess pore pressureuat an arbitrary location in the unit cell can be calculated.

Table 4Relationship between excess pore pressure u and ?ε/?t based on Hansbo solution.

The average excess pore pressureuavethroughout the soil cylinder can be expressed as

wherercis the equivalent radius of the unit cell,rdis the equivalent radius of the drain,rsis the equivalent radius of smear zone,ris the radial coordinate,uis the excess pore water pressure in the undisturbed zone,andusis the excess pore water pressure in the smear zone.

Substitutinguandusin Table 4 into Eq.(1) yields the relationship betweenuaveand ?ε/?t,which is summarised in Table 5.The extension of Hansbo solution for the excess pore pressureucan be obtained from the combination of Tables 3-5,as summarised in Table 6.

Table 5Relationship between average excess pore pressure uave and ?ε/?t based on Hansbo solution.

3.2.2.Validation

The extension of analytical Hansbo solution is validated with numerical results using the finite element model shown in Fig.2b which considers the discharge capacity and smear effect.Input parameters for the validation are provided in Tables 2 and 7.In this study,η=3 andm=3 were assumed for the purpose of validation.The influence of discharge capacity on the consolidation process increases with increasing length of the drain (Hansbo,1981).A small discharge capacity ofqd=1 × 10-4m3/d together with a small height of unit cellh=1 m were assumed to achieve a comparable effect of discharge capacity and smear on consolidation,which aims to clearly show individual contribution in their combined effect on consolidation.

Fig.5 compares analytical solutions to numerical results under three conditions,i.e.(1)discharge capacity,(2)smear effect and(3) discharge capacity and smear effect.It can be seen from Fig.5a-c that analytical solutions reasonably agree with numerical results under all conditions,which validates the extension of Hansbo solution.The relatively large difference in the case of discharge capacity is attributed to the simplification in the derivation of Hansbo solution.Under the condition of ideal drain,the direction of water flow is radial without a vertical component.However,under the condition of actual drain,the flow direction is mainly radial and with a minor vertical component.Hansbo solution was proposed based on the assumption of pure radial flow in the soil zone.Therefore,compared to Fig.5(a)for actual drain,Figs.3 and 5b for ideal drain and pure radial flow show better agreement between analytical and numerical results.The flow rate of pore water in the vertical direction vv=kviv=（kv/γw）（?u/?z）can be obtained,whereivis the hydraulic gradient in the vertical direction.For an identical initial pore pressureu0,a larger height of unit cellhresults in smallerivandvv,which reduces the error caused by the assumption of pure radial flow for actual drain.Comparison between Fig.5a forh=1 m and Fig.5d forh=20 m illustrates a better agreement between analytical and numerical results with a largerh.

Fig.5.Validation of the extension of Hansbo solution:(a)Discharge capacity,(b)Smear effect,(c)Discharge capacity and smear effect,and(d)Discharge capacity(qd=0.03 m3/d,h=20 m, z/h=0.5).

3.3.Diameter reduction method

In this paper,DRM aims to consider the smear effect by replacing the drain with a smear zone using a diameter-reduced drain without a smear zone,which enables consideration of smear effect without modelling a smear zone physically.According to Hansbo solution,this method can ensure the accuracy of excess pore pressureuin the undisturbed soil zone,which occupies most of the volume in the unit cell.It should be noted that for a constant discharge capacityqd,the reduced diameter leads to an increase in the hydraulic conductivity of the drain.The DRM is explained as follows.

Consider a drain of radiusrdand discharge capacityqdand the corresponding smear zone of radiusrsand hydraulic conductivityks.The excess pore pressure solution for the case of actual drain and smear effect in Table 6 can be rewritten as

Omitting the insignificant term（m2-1）/（2n2）,Eq.(2a) can be simplified to

Now consider a reduced drain of radiusr′dand discharge capacityqdwithout the smear zone.According to excess pore pressure solution for the case of actual drain in Table 6,the excess pore pressureu′can be expressed as follows:

Eqs.(3) and (4) can be rewritten as

Omitting the insignificant term,Eq.(5) can be simplified to

Comparisons betweenuin Eq.(2b)andu′in Eq.(7)and between μdsin Table 3 and μ′din Eq.(6) reveal that the equation ln（rd/）=（η-1）lnmensures relationships ofu=u′and μds=μ′dforrs≤r≤rc.Therefore,a reduction factorRFcan be used to determine the diameter of the reduced drain to consider the smear effect,analytically:

Note that the discharge capacities of actual and reduced drains are identical,i.e.qd=.Hence,the diameter of reduced drainD′dand hydraulic conductivity in the reduced draink′dcan be calculated with the original diameterDdand hydraulic conductivitykdas follows:

In the deduction of DRM,the discharge capacity is considered.Based on the previous analysis,DRM is also suitable for an ideal drain with unlimited discharge capacity.The application of DRM to the numerical simulation is illustrated by the comparison of Fig.2b and c.

According to parameters provided in Table 7,the reduction factorRF=9.Fig.6a compares results obtained from the actual drain with smear zone and reduced drain without smear zone for the undisturbed zone ofrs≤r≤rc.Good agreement of analytical and numerical results between the actual drain and reduced drain validates the proposed DRM.Fig.6b illustrates the error caused by DRM in the smear zone ofrd≤r≤rsthrough the comparison of analytical results.The error decreases with the increasing radial distancerin the smear zone and becomes insignificant atr=rs.For consolidation timeTr,the error decreases with increasing consolidation time.Fig.6b also reveals that the DRM is associated with a corresponding decrease in hydraulic gradient within the smear zone.A smaller pressure difference for the case of reduced drain indicates a smaller hydraulic gradient within the smear zone in comparison to the case of actual drain.

Table 7Inputs parameters for discharge capacity and smear effect.

Fig.6.Validation of the diameter reduction method (DRM): (a) Undisturbed zone (rs ≤r ≤rc),and (b) Smear zone (rd ≤r ≤rs).

4.Practical considerations

The DRM is derived based on the extension of Hansbo solution for the radial flow and is validated with elastic soil material.In practice,consolidation problems are more complex,such as multilayered soils with plastic nature and coupled radial-vertical groundwater flow.Here,the DRM is discussed in detail with regard to these practical considerations.

4.1.Plastic soil materials

The coefficient of consolidationcris dependent on the coefficient of volume compressibilitymv,which is identical to the reciprocal of oedometer modulusEoed(i.e.mv=1/Eoed).For elastic soil materials,Eoedis a function of effective Young’s modulusE′and effective Poisson’s ratio ν′,and can be expressed asEoed=(1-ν′)E′/[(1+ν′) (1-2ν′)],which is taken as a constant value during the consolidation process.In contrast,for plastic soil models (e.g.theHardening Soilmodel andSoft Soilmodel in Plaxis),Eoedvaries during the consolidation process due to the stress dependency of soil stiffness in these models.Therefore,the main difference between elastic and plastic soil materials in their influence on the dissipation of pore water is the variation ofcrduring the consolidation process.

However,the derivation of DRM is independent of the value ofcr.Hence,DRM is also suitable for plastic soil materials.Fig.7 compares the dissipation of excess pore pressureufor an actual drain and that for the reduced drain at two representative locations ofr=rsandr=rcfor a plastic soil model under the condition of radial flow.Good agreement of results between the actual and reduced drains validates the above analysis.

Fig.7.Application of DRM in plastic soil materials (Hardening Soil model): (a) Curve point: r= rs, z=0.5h;and (b) Curve point: r= rc, z=0.5h.

4.2.Coupled radial-vertical groundwater flow

Typically in soil improvement,the vertical dimension of a unit cell is much larger than that in the radial dimension (i.e.h＞＞rc),resulting in a much shorter drainage path in the radial direction.Hence,the radial flow is dominant in most zones of the unit cell,where the DRM should offer good performance.However,for the zone near the ground surface,the vertical flow and radial flow may be comparable,reflecting a significant feature of coupled radialvertical flow.

The vertical and radial dimensions of the unit cell model shown in Fig.2b are nearly identical,which can be used to simulate the coupled radial-vertical flow.The groundwater flow boundary conditions at the ground surface are set toSeepageto allow for an open boundary to accommodate coupled radial-vertical flow during the consolidation process.The elastic soil material given in Table 2 is employed since the performance of DRM is independent of soil materials.

Fig.8 compares numerical results obtained from actual and reduced drain simulations for a group of points in a horizontal line and a group of points in a vertical line,respectively.The results based on the reduced drain agree very well with that based on the actual drain,validating the DRM under the coupled radial-vertical flow conditions.The strong performance of DRM under the coupled radial-vertical flow conditions is expected since the hydraulic conductivity in the limited zone of 0 ≤r＜rsis adjusted according to DRM,while the zone ofrs≤r＜rccovering the majority of the flow remains unchanged.

Fig.8.Application of DRM under a coupled radial-vertical flow condition: (a) Points in a horizontal line at z=0.5h,and (b) Points in a vertical line at r= rc.

4.3.Multi-layered soils

Following the above discussion,the finite element model shown in Fig.2b is divided into two layers to investigate the performance of DRM under conditions of multi-layered soils with different hydraulic conductivity and coupled radial-vertical flows.The interface of these two soil layers is located atz/h=0.5.The soil parameters provided in Tables 2 and 7 are used for the top layer.For simplicity,the only difference between these two layers is assumed that the hydraulic conductivity in the bottom soil layer is one-tenth of that in the top one.The reduction factorsRFof two layers are identical due to the identical radius ratiomand hydraulic conductivity ratio η.

Comparison of results from the actual and reduced drain is illustrated in Fig.9.The same two groups of points as that in the last section of coupled radial-vertical flow are selected for comparison.Points in Fig.9a are all located at the soil layer interface and those in Fig.9b are distributed in both two soil layers.Good agreement of results between actual and reduced drain in Fig.9 validates the strong performance of DRM under quite complex soil and flow conditions.

Fig.9.Application of DRM under multi-layered soil condition: (a) Points in a horizontal line at z=0.5h,and (b) Points in a vertical line at r= rc.

4.4.Case study

A case study of Changi East Reclamation Project in Singapore(Arulrajah et al.,2005) is provided herein to further validate the strong performance of DRM in soil improvement.The soil material properties are summarised in Table 8.For simplicity,parameters of geometry and hydraulic conductivity of drain,smear zone and unit cell(i.e.rc,rd,mand η)in Tables 2 and 7 are adopted.The anisotropy of hydraulic conductivity in the undisturbed soil zone is considered in this case study.The equal hydraulic conductivity in both vertical and horizontal directions is assumed in the smear zone (Bergado et al.,1991;Indraratna and Redana,1998).The discharge capacity of drainqdis assumed to be 0.822 m3/d.According to laboratory observation of soil disturbance caused by drain installation,Rujikiatkamjorn et al.(2013) adopted a lower compression indexCc=0.18 in the smear zone in comparison toCc=0.23 in the undisturbed zone.A difference of 22% is assumed between the compressibility in the smear zone and that in the undisturbed zone(Table 8),following Rujikiatkamjorn et al.(2013) study.

Table 8Soil parameters for the case study (Arulrajah et al.,2005).

Fig.10 shows the magnitudes of the surcharge loadings with time.Fig.11 compares the dissipation of excess pore water pressure in soil improvement between actual and reduced drains for two groups of points atr=rsandr=rc,respectively.Fig.12 compares corresponding settlement for the group of points atr=rc.Good agreement in both excess pore water pressure and settlement between actual and reduced drains confirm the accuracy and suitability of DRM in soil improvement.The anisotropy of hydraulic conductivity in the undisturbed zone and the change in compressibility in the smear zone have an insignificant influence on the performance of DRM.

Fig.10.The simplified surcharge for the case study.

Fig.11.Comparison of dissipation of excess pore water pressure: (a) Points in a vertical line at r= rs,and (b) Points in a vertical line at r= rc.

Fig.12.Comparison of settlement (points in a vertical line at r= rc).

5.Conclusions

Vertical drains are widely employed to accelerate the consolidation of soft soils in soil improvement.Installation of vertical drains unavoidably generates smear zones around these drains,in which the hydraulic conductivity is reduced due to soil disturbance.Modelling smear zones physically in a finite element model including multiple drains can be very computationally demanding and time-consuming,especially for a 3D simulation.A practical rigorous method based on Hansbo solution,the DRM,is proposed to consider the smear effect without the need to model the smear zone physically.The performance of DRM is discussed pertaining to some representative soil conditions in practice.

The classical Hansbo solution to average excess pore water pressureuaveis extended to consider the excess pore pressureuat every location in the soil domain,which is validated with numerical results.Based on the extension of Hansbo solution,the DRM is developed,and a diameter reduction factorRFis proposed.It has been validated by analytical and numerical results that the smear effect can be considered by reducing the diameter of the vertical drain.It should be noted that the solution of excess pore water pressureuby DRM is accurate enough in the undisturbed zone and the error ofucaused by DRM in the smear zone decreases with the increasing radial distancerand becomes insignificant at the outer boundary of smear zone (i.e.r=rs).The hydraulic conductivity in the reduced drain needs to be adjusted and improved to keep a constant discharge capacity due to the reduced area of the horizontal section of the drain.

The performance of DRM is independent of soil material properties.Numerical tests and a field case study validate the strong performance of DRM under conditions of multi-layered soils and coupled radial-vertical flow,which are common in the field.

The extension of Hansbo solution is very convenient analytically,to validate the dissipation of excess pore water pressure in the numerical unit cell model with a vertical drain.The DRM can be used to simplify a finite element model using reduced diameter drains without smear zone,to model multiple drains surrounded by smear zones.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors wish to acknowledge the generous financial support from the Singapore Maritime Institute (SMI) for this research within the project ‘Evaluation of In-situ Consolidation of Dredged and Excavated Materials at Reclaimed Next Generation Tuas Port’(Project ID:SMI-2018-MA-01).The first author is supported by“the Fundamental Research Funds for the Central Universities”in China.

List of symbols

CcCompression index

cv=kv/(mvγw) Vertical coefficient of consolidation

cr=kh/(mvγw) Horizontal coefficient of consolidation

c′refReference value of the cohesion

DdEquivalent diameter of drain

D′dEquivalent diameter of reduced drain

EYoung’s modulus

E′Effective Young’s modulus

EoedOedometer modulus

HHeight of unit cell

IMoment of inertia

ivHydraulic gradient in the vertical direction

J0Bessel function of first kind of zero order

J1Bessel function of first kind of first order

Coefficient of lateral earth pressure for a normally consolidated stress state

kdHydraulic conductivity of drain

k′dHydraulic conductivity of reduced drain

khHorizontal hydraulic conductivity of undisturbed soil

ksHydraulic conductivity of smeared soil

kvVertical hydraulic conductivity

m=rs/rdRadius ratio of smear zone and drain

mvCoefficient of volume compressibility

n=rc/rdRadius ratio of unit cell and drain

qd=πdDischarge capacity of drain

RFReduction factor

rRadial coordinate

rcEquivalent radius of unit cell

rdEquivalent radius of drain

r′dEquivalent radius of reduced drain

rsEquivalent radius of smear zone

Tr=crt/(4) Time factor for radial flow

Tv=cvt/h2Time factor for vertical flow

tTime

U0(αin)J0(αi)Y0(αi)-Y0(αin)J0(αi)

U0(αir/rd)J0(αir/rd)Y0(αi)-Y0(αir/rd)J0(αi)

U1(αi)J1(αi)Y0(αi)-Y1(αi)J0(αi)

uExcess pore water pressure

u′Excess pore water pressure for the case of reduced drain

uaveAverage excess pore water pressure

urSolution of excess pore pressure to radial consolidation

usExcess pore water pressure in the smear zone

uvSolution of excess pore pressure to vertical consolidation

uvrSolution of excess pore pressure to coupled radial-vertical consolidation

u0Initial excess pore water pressure

u0,aveAverage initial excess pore pressure

vvFlow rate of pore water in the vertical direction

Y0Bessel function of second kind of zero order

Y1Bessel function of second kind of first order

zVertical coordinate

αi(i=1,2,…) Roots of Bessel function that satisfyJ1(αin)Y0(αi)-Y1(αin)J0(αi)=0

ε Vertical strain of the unit cell

γsatUnit soil weight below water level

γunsatUnit soil weight above water level

γwUnit weight of water

η Horizontal hydraulic conductivity ratio of undisturbed and smeared soils

φ′Effective friction angle

k* Modified swelling index

λ* Modified compression index

μ Parameter for ideal drain

μdParameter for actual drain considering discharge capacity

μ′dParameter for actual reduced drain considering discharge capacity

μdsParameter for actual drain considering discharge capacity and smear effect

μsParameter for ideal drain considering smear effect

ν′Poisson’s ratio

ν′urEffective Poisson’s ratio for unloading-reloading

ψ Dilatancy angle of the soil material