EFFECT OF PREFABRICATED VERTICAL DRAIN CLOGGING ON THE RATE OF CONSOLIDATION: ANUMERICAL STUDY

Transcription

1 Technical Paper by D. Basu and M.R. Madhav EFFECT OF PREFABRICATED VERTICAL DRAIN CLOGGING ON THE RATE OF CONSOLIDATION: ANUMERICAL STUDY ABSTRACT: Ground treatment of soft soil deposits by prefabricated vertical drains (PVDs) is a common practice in geotechnical engineering. The PVDs accelerate the consolidation process and help to rapidly increase the strength and stiffness of soil. Certain operational problems, such as clogging, are inherent to PVDs. The fine particles in the soil may become entrapped within the fibers of the filter, i.e. the geotextile, surrounding the PVD. If the pores of the filter sleeve are clogged, the discharge capacity of the PVD reduces and, consequently, the consolidation process is impeded. In this paper, a numerical study is carried out to investigate the effect of PVD geotextile filter clogging on the rate of soil consolidation. The rate of consolidation decreases with increasing clogging and is a function of the location of the clogged area in the PVD. In the parametric study, the PVD was sequentially clogged and its effect studied. The effect of smear was also included in the analysis. KEYWORDS: Clogging, Consolidation, Soft soil, Ground improvement, Prefabricated vertical drain. AUTHORS: D. Basu, Graduate Student, Department of Civil Engineering, Indian Institute of Technology, Kanpur , India, dibasu@iitk.ac.in, dipan52@yahoo.com; and M.R. Madhav, Professor, Department of Civil Engineering, Indian Institute of Technology, Kanpur , India, Telephone: 91/ , Telefax: 91/ , madhav@iitk.ac.in. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 1801 County Road B West, Roseville, Minnesota , USA, Telephone: 1/ , Telefax: 1/ Geosynthetics International is registered under ISSN DATES: Original manuscript received 27 January 2000, revised version received and accepted 19 June Discussion open until 1 March REFERENCE: Basu, D. and Madhav, M.R., 2000, Effect of Prefabricated Vertical Drain Clogging on the Rate of Consolidation: A Numerical Study, Geosynthetics International, Vol. 7, No. 3, pp

2 1 INTRODUCTION Soft soil deposits are a constant challenge in geotechnical engineering owing to their low strength and high compressibility. Ground treatment of these deposits includes heavy tamping (dynamic compaction), dynamic replacement (with or without mixing), vibratory densification (vibro-compaction and vibro-replacement), preloading without the addition of drains, and preloading with vertical drains. Preloading with vertical drains is successfully used to pre-empt large settlements in highly compressible deposits with very low permeability (Johnson 1970; Hansbo 1979; Forrester 1982; Jamiolkowski et al. 1983; Holtz 1987). This method of treatment ensures a significant increase in soil strength and stiffness. Vertical drains accelerate the consolidation process by reducing the drainage path and by reorienting the direction of flow into a more permeable (horizontal) direction. Their efficacy is at a maximum when there is a time constraint for ground improvement in soft soils. The earliest version of vertical drains was sand drains, which consisted of a borehole filled with sand. Dastidar et al. (1969) introduced sand-wicks, which had an advantage over sand drains regarding ease of construction and ensuring the continuity of the drain. Since the 1970s, band-shaped prefabricated vertical drains (abbreviated as PVD herein), originally developed by Kjellman (1948), have replaced sand-wicks. Various manufacturers throughout the world are producing a large variety of PVDs. There are two main types of PVDs (Akagi 1994). One common type consists of a thin geotextile filter sleeve (typically a nonwoven) surrounding a corrugated or studded plastic central core. The filter sleeve prevents fine soil particles from entering into the core, but allows easy entry of pore water. The central core acts as a drainage channel, while withstanding buckling and compressive stresses. The other type of PVD is a simple unit-sized strip of porous plastic having small continuous drainage holes inside, with sufficient strength and durability. The primary functions of these drains are (i) to filter the excess water from the consolidating soil (filtration) and (ii) to carry this water away from the compressible soil layers, by longitudinal flow (drainage). A typical PVD has a width of approximately 100 mm and thickness of 3 to 4 mm. They are usually installed in square or triangular arrays using a mandrel, which is often square, rectangular, or octagonal in shape. The popularity of PVDs can be attributed to the advantages they have over conventional sand drains. Installation of a PVD is fast and the material can be easily stored and transported. The tensile strength of PVDs maintains its continuity, and most importantly, PVDs are cheaper than sand drains (Hausmann 1990, pp ). However, the following are inherent problems encountered when using PVDs: (i) formation of a smear zone around the PVD; (ii) decrease in PVD discharge capacity due to the increase in lateral confining pressure with depth, folding, and kinking; and (iii) PVD clogging. During the installation of PVDs, the surrounding soil becomes disturbed. This disturbance decreases the horizontal permeability in that soil zone, thus, slowing down the consolidation process. This zone of disturbed soil is referred to as the smear zone, the extent of which depends on the PVD size and the method of installation. The effects of smear have been incorporated in analytical solutions (Barron 1948; Hansbo 1981). Many experimental studies have been reported that investigate the phenomenon of smear (Casagrande and Poulos 1969; Akagi 1977; Akai et al. 1981; Onoue 1991; Bergado et al. 1991). Madhav et al. (1993) performed a parametric study and concluded that 190

3 an extremely disturbed inner smear zone adhering to the PVD significantly hinders the consolidation process. The problem of reduction of discharge capacity, q w (defined as q w = Q/i,whereQ is the discharge velocity and i is the hydraulic gradient), or well resistance, has been investigated by several researchers, including Lawrence and Koerner (1988), Ali (1991), Miura et al. (1993), and Kamon et al. (1994). Pore water reaching the PVD core travels vertically upward through the PVD. The term well resistance refers to the impedance of vertical flow of water through the PVD. According to Mesri et al. (1994), well resistance refers to the finite permeability of the vertical drain with respect to that of the soil. The lateral confining stress squeezes the PVD, thereby, constricting the drainage channel and increasing the well resistance. Moreover, as the soil consolidates, the ground settles and pulls the PVD down with it. This results in the development of compressive and bending stresses in the PVD, which further reduce flow. However, slightly flexible PVD cores with three-dimensional, interconnected flow channels resist this reduction in discharge (Ali 1991). If the settlement is large (which is often the case), the PVD buckles, folds, or forms kinks, further reducing the discharge. For most PVDs, the discharge capacity is less affected by bending, but decreases considerably due to kinking (Miura et al. 1993). The problems discussed in the preceding paragraph arise only when the pore water enters the interior of the PVD; but, the very process of ingress of pore water into the PVD core through the filter sleeve is hindered due to clogging of the filter sleeve by fine soil particles. While it is desirable that some fines should pass through the filter to permit the formation of a natural soil filter around it, too many fines could reduce the PVD discharge capacity due to clogging, sedimentation, and blockage, thus increasing the filter resistance (Miura et al. 1993). In addition, the fines entering the PVD core could clog the vertical channel reducing the discharge capacity. Miura et al. (1993) and Mesri et al. (1994) have reported this problem, but its effect on PVD performance has neither been experimentally quantified, nor theoretically predicted. This paper investigates the effect of clogging on the performance of PVDs. A parametric study is carried out using the finite difference technique to study the variation of the average degree of consolidation obtained using a PVD for different degrees of clogging. In the analysis, the shape of the PVD and the area of influence are maintained as constants. This differs from most other theoretical studies performed to date, in which both the shape and area of influence are transformed into equivalent circular shapes. 2 PROBLEM DEFINITION AND FORMULATION 2.1 Problem Definition The present work involves the study of the variation of the average degree of consolidation, U, as obtained using a PVD, with the time factor T, for different amounts of PVD clogging. A square arrangement (Figure 1) of PVDs, having a spacing of 1 m 1m (s x s y ), was considered in the analysis (PVD width, b = 100 mm, and thickness, w = 4 mm). It is reasonable to assume that water within the area shown in Figure 1 flows only into the PVD and there is no cross-boundary flow. Thus, the area can be treated 191

4 s x =1m Zone of smear (300 mm 200 mm) y x s y =1m PVD (100 mm 4 mm) Figure 1. Area of influence of a single PVD. as a unit cell with impervious boundaries, typically representing the entire area in which the PVDs are installed. Finite difference was chosen as the numerical technique and an implicit scheme (Euler Backward) was used for the time increments. For the differential equation used in the study, as described in Section 2.2, the implicit scheme is infinitely stable and, hence, no time-step value restriction was required. The scheme requires the solution of a set of simultaneous equations by using, for example, the Gauss Siedel technique. The effect of smear around the PVD was included in the present analysis. 2.2 Governing and Corresponding Finite Difference Equations In this study, the Terzaghi-Rendulic theory of consolidation was assumed to be valid, which is a common assumption (Das 1983, pp ). The differential equation governing the consolidation phenomenon is: u t = C h 2 u x + 2 u 2 y 2 where: u = excess pore water pressure; t = time; x and y = distance coordinates; and C h = coefficient of consolidation for flow in the horizontal direction, considering the medium to be isotropic. This equation was approximated by the finite difference technique using an implicit (Euler Backward) scheme. The resulting algebraic equation is: u t+ t u i, j t i, j = C t h ut+ t i+1, j 2ut+ t + u i, j t+ t i 1, j x 2 + ut+ t i, j+1 2ut+ t i, j + u t+ t i, j 1 y 2 In Equation 2, i and j are required for space discretisation and represent a point (i, j) (Figure 2), where: u i,j = excess pore water pressure; t = index for time step; x and y = step lengths in the x and y directions, respectively; and t = time increment. (1) (2) 192

5 (i, j +1) y x (i --- 1, j) (i, j) (i, j --- 1) (i +1,j) Figure 2. Convention followed for the finite difference discretisation. 2.3 Domain of Analysis, Discretisation, and Boundary and Initial Conditions The boundaries of the unit cell were assumed to be impervious, as discussed in Section 2.1, and a Neumann boundary condition ( u/ ñ = 0, ñ is the direction of the unit normal vector on any boundary) was imposed on the boundaries. The boundary along the PVD is pervious and a Dirichlet boundary condition (u specified) is required. Since the flow of water in the four quadrants of the unit cell is symmetrical, only one quadrant was considered as the domain for analysis (Figure 3). A Neumann boundary condition was also imposed along the quadrant boundaries because there is no cross-boundary flow. The PVD thickness (4 mm) was neglected in the numerical analysis for the sake of simplicity. To study the PVD clogging effect, a very fine discretisation near the PVD was necessary. The use of such a fine mesh over the entire domain would require a very large computation time; hence, a coarser mesh was used in the domain away from the PVD. The discretisation of the domain is shown in Figure 4. Four zones, Zones A, B, C, and D, four inter-zone boundaries, Boundaries a, b, c, and d, and four peripheral boundaries, Boundaries α, β, ψ, andδ are used. Although the discretisations are different in each s x /2 = 0.5 m y x u/ y =0 u/ x =0 u/ y =0 PVD s y /2 = 0.5 m b/2 = 50 mm Figure 3. Domain of analysis (without the smear zone). 193

6 (i) β (ii) A B Point 1 a Point 2 α ψ (iv) d (v) b Point 4 D c (vi) PVD Point 3 δ C (iii) Figure 4. Finite difference discretisation. zone, x and y were taken to be equal in each zone. Hence, if x = y= s, Equation 2 becomes: u t+ t u i, j t i, j = C t h ut+ t + i+1, j ut+ t + i 1, j ut+ t + i, j+1 ut+ t i, j 1 4ut+ t i, j For points in Zones A, B, and C, s was taken to be s e = 25 mm and the working equation for these zones, as derived from Equation 3, is: s 2 (3) u t+ t i, j = β e u t+ t i+1, j + u t+ t i 1, j + u t+ t i, j+1 + u t+ t i, j u t i, j (4) where β e = C h t s 2 e (5) For points in Zone D, Equation 4 is valid with β e being replaced by β i where: β i = C h t s 2 i (6) with s i = 5 mm. 194

7 For points on Boundaries α, β, ψ, andδ, a Neumann boundary condition was imposed by modifying Equation 3. The equations for points lying on the inter-zone boundaries, Boundaries c and d, were mathematically more involved because of the unequal step lengths on either side of the points along the x and y directions. The details of these equations are given in the Appendix. There were further problems for points along Boundaries c and d because the mesh was coarser on one side of these boundaries. Therefore, finite difference equations could not be applied to the points on Boundaries c and d between the nodes of the coarser mesh, due to an absence of nodes along the x and y directions, respectively. To solve this problem, imaginary nodes were assumed along the grid lines next to Boundaries c and d as shown in Figure 5. The values of u at these imaginary nodes were calculated using linear interpolation after excess pore water pressure values, u, were determined at the real nodes at every time step. For points on the PVD, a Dirichlet boundary condition was initially imposed. This boundary condition implied the free flow of water through the PVD (i.e. no clogging) and the value of u was set to zero since the excess pore water pressure completely dissipates at the PVD. Since Equation 1 is a time-dependent equation (initial value problem), initial values of u at all of the nodes were arbitrarily set to 100, except at the nodes on the PVD boundary where u = 0 because the excess pore water pressure dissipates completely at the PVD boundary immediately after PVD installation (when there is no clogging). Imaginary nodes Coarse grid Inter-zone boundary, d Interpolation required for these points Imaginary nodes Fine resolution grid PVD Inter-zone boundary, c Figure 5. Imaginary nodes used for the linear interpolation. 195

8 2.4 Model for PVD Clogging The model described thus far does not take into account PVD clogging. Clogging was taken into account by imposing a Neumann boundary condition for points on the PVD boundary. When the gradient at a point on the PVD boundary is made equal to zero it implies that no flow occurs across that point. Since the entire PVD is represented by a set of discrete points (nodes), a clogged node implies that the section of the PVD lying within half the step length on both sides of the node is clogged (Figure 6). Thus, if a single node is clogged (except the end nodes), it implies 10% PVD clogging for the discretisation scheme adopted (Figure 4) in this study. The initial condition specified at the clogged points is u = 100, similar to the interior points. 2.5 Degree of Consolidation and Time Factor At every time step, the average degree of consolidation, U, was calculated such that it could be plotted against the corresponding time factor, T. The expression for U is: U = 1 A u(x, y, T)dx dy A u ini dx dy where u ini and u(x,y,t) are the excess pore water pressures at any point in the entire domain initially and at any time factor T, respectively. Integration over the entire domain A can be performed using the trapezoidal or Simpson s rule for a two-dimensional case. The time factor, T, is defined as: (7) T = C h t d 2 e (8) where: t = time; and d e = equivalent diameter of the unit cell, calculated by equating the area of the real domain and its equivalent circle. The expression for d e is: d e = 4 s x s y π (9) Clogged node Section of the PVD represented by the node Figure 6. Numerical modeling of PVD clogging. 196

9 where s x and s y are the PVD spacings in the x and y directions, respectively. The average degree of consolidation, U, can thus be expressed in terms of T. The basic algorithm followed in the study is illustrated in Figure Analysis Considering the Smear Zone The extent of the smear zone surrounding a PVD is not uniquely known. Casagrande and Poulos (1969), McDonald (1985), and Aboshi and Inoue (1986) recommend that the area of the smear zone be taken as equal to that of the mandrel, while Holtz and Holm (1973), Akagi (1977), and Hansbo (1981) assume the diameter of the smear zone to be twice the equivalent diameter of the mandrel. Based on the back-calculated values from the observed time-settlement relationships obtained from a laboratory-scale consolidation test, Bergado et al. (1991) support the results of Hansbo (1981). Madhav et al. (1993) studied the actual variation of the horizontal coefficient of permeability and Input d e, T initial, T maximum, and T Set u(i, j) values of unclogged PVD nodes to 0 Set remaining u(i, j) values to 100 (initial condition) Increase T from T initial to T maximum in steps of T At every step, form u(i, j) equations using values from previous time step Solve set of simultaneous equations by Gauss Siedel iterative technique Calculate U Store u(i, j) values for use in next step If U 99.99%, exit loop End loop for T End Figure 7. Basic algorithm used for the numerical study. 197

10 found the extent of the disturbed zone to be up to a distance of 700 mm from the center of the mandrel. Madhav et al. (1993) observed that within the smear zone there exists two distinct sub-zones, namely, (i) an inner highly disturbed zone around the PVD where the coefficient of horizontal permeability was approximately 0.2 times that of the undisturbed zone and (ii) a transition zone in which the horizontal permeability increased with distance away from the PVD until it reaches its maximum value in the undisturbed zone. Madhav et al. (1993) also showed that the transition zone has a less significant effect on the consolidation process. Based on all of these observations, a single smear zone, with a coefficient of horizontal permeability, k hs, equal to the value suggested by Madhav et al. (1993) for the inner smear zone and extending up to a distance of 100 mm from the PVD surface, was considered to be a reasonable assumption in the current analysis (Figure 8). The equations used were derived following the procedure given by Das (1983, pp ). The soil was assumed to be isotropic and its compressibility unaffected by disturbance, thus, the coefficients of consolidation of both zones are proportional to the respective coefficients of permeability. The details of the equations are given in the Appendix. 3 COMPARISON WITH THE ANALYTICAL SOLUTION 3.1 Comparison Without the Smear Zone The numerical solution for the no clogging condition was compared to the analytical solution available for a circular drain (axi-symmetric case) with equal strain case (Barron 1948). The analytical solution for U is: U = 1 e 8T F(n) (10) where s x /2 = 0.5 m y PVD x p 150 mm k hs q r k h 100 mm s y /2 = 0.5 m Zone of smear b/2 = 50 mm Figure 8. Domain of analysis including the smear zone. 198

11 F(n) = n 2 n 2 1 ln n 3n2 1 4n 2 (11) with n = d e d w (12) where: n = ratio of the equivalent diameter of the unit cell to that of the PVD; F = function of n; d e = equivalent diameter of the unit cell (Equation 9); and d w = equivalent PVD diameter obtained by equating the perimeters of the real drain to an equivalent circular drain (Hansbo 1981). The equivalent diameter d w is given by the following: d w = 2 π (b + w) (13) where b and w are the PVD width and thickness, respectively. 3.2 Comparison With the Smear Zone The numerical result, including the smear effect for the no-clogging condition, was also compared with the analytical solution proposed by Hansbo (1981). The expression for U is the same as Equation 10 with F(n) being modified as follows: F(n) = ln n m + k h k hs ln m 3 4 (14) In Equation 14, the new term m is the ratio of the equivalent diameter, d s,ofthe smear zone to that of the drain, d w,andk h and k hs are the coefficients of horizontal permeability for the undisturbed and smear zones, respectively. The equivalent diameter of the smear zone was calculated in the same way as the equivalent diameter of the unit cell. 4 RESULTS AND DISCUSSION 4.1 Test of Convergence In order to minimize the numerical error, a test of convergence was performed. The discretisation of the outer domain (Zones A, B, and C) was successively reduced until the difference in results between two successive discretisations was within tolerable limits (0.76%). In each case, the new discretisation was a subset of its preceding one; however, the discretisation of the inner domain (Zone D) was not changed, it being understood that such a fine discretisation would yield acceptable results. The details of the adopted discretisation scheme were discussed in Section 2.3. Although there was no restriction on the magnitude of the time increment, a test on the time increment was performed and it was found that aberrations were negligible for time increments up to 0.1T. Hence, in the analysis, the increment in time was varied from T to 0.1T. 199

12 4.2 Comparison of Numerical and Analytical Solutions for an Unclogged PVD The numerical solution is compared with the analytical solution for the no PVD clogging condition in Figure 9. The curves closely match each other. The maximum difference in U is approximately 6% at T=0.5. This difference results from the fact that, in the analytical solution, the domain and PVD are circular whereas in the present case they are rectangular. For lesser values of T, the consolidation for the numerical case is faster than the analytical case because the PVD in the numerical case has a straight surface through which water travels orthogonally and readily drains. However, complete consolidation should occur faster in the analytical solution than that of the numerical solution because, in the latter case, water has to travel a relatively longer distance from the corners as can be seen in Figure Clogging from the Tip of the PVD To study the effect of clogging on the value of U, the node at the tip of the PVD was clogged (Case A), implying 5% clogging of the PVD area. The amount of clogging was then increased from 5 to 95% by progressively clogging additional nodes toward the center of the PVD (Figure 10). As expected, the rate of consolidation decreases with increased clogging. For example, the value of T required for 90% consolidation to take place for the no PVD clogging condition is 0.75, while those for 15, 35, 55, 75, and 95% clogging are 0.85, 0.92, 1.04, 1.25, and 1.8, respectively. Thus, if the PVD is 95% clogged, the time required for 90% consolidation increases to more than double the time 100 Degree of consolidation, U (%) 50 s x /s y =1.0 s x /b = 10.0 n = Analytical Analytical Numerical PVD Unit cell Time factor, T Figure 9. Comparison of numerical and analytical results for the case of no PVD clogging. 200

13 Degree of consolidation, U (%) s x /s y =1.0 s x /b = 10.0 PVD Direction of progressive clogging 0% 15% 35% Clogged section 95% 75% 55% Time factor, T Figure 10. Clogging from the tip of the PVD (Case A). requiredbyanuncloggedpvd. Again, fora particularvalue oft, say 1.0, the value of U obtained is 95.2% for the unclogged PVD whereas those for 15, 35, 55, 75, and 95% clogging are 93.4, 91.5, 89, 84.8, and 73.3%, respectively. Thus, the decrease in the rate of consolidation with increasing clogging is substantial. 4.4 Clogging from the Center of the PVD The effect of clogging from the center of the PVD was also studied (Figure 11). In this case (Case B), the PVD was progressively clogged from the center toward the tip of the PVD; however, it was found that the effect of clogging on consolidation is not as pronounced as it was for Case A. For Case B, the T values required for 90% consolidation are 0.79, 1.06 and 1.17 for 55, 75, and 95% clogging and are much less compared to those in Case A. Again, for T = 1.0, the degree of consolidation obtained for 55, 75, and 95% clogging are 94.5, 93.7, and 92.3%, respectively. Thus, the effect of clogging from the center is less critical compared to the effect clogging from the tip of the PVD. The probable reason for this difference is that, for Case A, the effective width of the PVD through which water could escape was reduced and water in the unit cell has to travel a longer distance leading to an increased time for consolidation (Figure 12a). However, for Case B, the PVD and, hence, the unit cell are bifurcated into two zones. The unclogged PVD section is inside and closer to the area to be drained (Figure 12b) and as a result the consolidation is faster. 201

14 Degree of consolidation, U (%) s x /s y =1.0 s x /b = 10.0 PVD Clogged section Direction of progressive clogging 0% 55% 95% 75% Time factor, T Figure 11. Clogging from the center of the PVD (Case B). (a) Flow path of water after clogging (b) Influence area gets divided into two parts Flow path of water before clogging Clogged PVD section Clogged PVD section (c) (d) Clogged PVD sections Clogged PVD sections (alternate with pervious sections) Figure 12. Different methods of PVD clogging: (a) clogging from the tip of the PVD (Case A); (b) clogging from the center of the PVD (Case B); (c) clogging at the middle of each half of the PVD (Case C); (d) discontinuous clogging (Case D). 202

15 4.5 Clogging at the Middle of Each PVD Half and Discontinuous Clogging From Section 4.4, it can be inferred that the effect of clogging on the degree of consolidation is dependent on the location of the clogged area in the PVD. To investigate the effect of clogging location, two more cases (Cases C and D) were considered. For Case C, 50% of the PVD was clogged at the middle in each half of the PVD (Figure 12c). Thus, the PVD was divided into three sections. For Case D, 50% of the PVD was clogged such that the clogged zones alternate with the pervious zones (Figure 12d). The results for Cases C and D are presented in Figure 13 along with those for Cases A and B. The decrease in the rate of consolidation is minimal for Case D and slightly more for Case C. In these two cases, the decrease in consolidation is less than that for Case B. Thus, for the same amount of PVD clogging, the more scattered the clogged sections over the PVD width, the less the effect it has on the rate of consolidation. This result follows from the discussion in Section 4.4 because, for Case C, water travels less distance than for Case B. For Case D, water travels an even smaller distance. However, this is true when the PVD is not clogged at its tip. In the latter case, the effect is most pronounced as has already been discussed in Sections 4.3 and 4.4. Since the deviations in values of U for Cases C and D are negligible and less critical compared to that for Case B, Cases C and D are not considered further. 100 Degree of consolidation, U (%) No clogging Case D Case C Case B 50 Case A Case A Enlarged s x /s y =1.0 s x /b = Time factor, T Figure 13. Variation of the degree of consolidation, U, with the time factor, T, for 50% PVD clogging at different positions. 203

16 4.6 Comparison of Numerical and Analytical Solutions for a Clogged PVD When the PVD gets clogged from its tip (Case A), its effective width is reduced (Figure 12a). This type of clogging is equivalent to a reduction of the PVD size. Thus, the value of n (Equation 12) changes for the PVD. Analytical solutions were obtained for different values of n calculated from the reduced PVD size corresponding to 55, 75, and 95% clogging. The analytical and numerical results are compared in Figures 14, 15, and 16. The curves for corresponding percentages of clogging for Case B and for the noclogging condition of the PVD are also plotted in Figures 14, 15, and 16. As discussed in Section 4.2, the rate of consolidation is slower for the analytical case than that for Case A at lesser time durations and vice versa at greater time durations. However, with an increase in n, the rates of consolidation for the two cases become more or less equal at lesser times, while the difference increases at greater times. In all of these cases, the rates of consolidation for the analytical solution are faster than that for Case B. 4.7 Variation of Excess Pore Water Pressure To investigate the reason for the differences in the average degree of consolidation values, U, for Cases A and B, the variations of excess pore water pressure (as a percentage of the initial excess pore water pressure) with time at four points (selected at the center of Zones A, B, C, and D, namely Points 1, 2, 3, and 4 in Figure 4) were studied (Figure 17). The rate of dissipation of pore water pressure is almost the same for Points 1 and 3. Being the closest to the PVD, the rate of dissipation for Point 4 is strikingly fast while that for Point 2 is the slowest, it being located farthest from the PVD. Figure 18 shows the change in the rate of dissipation of pore water pressure for Points 1, 2, and Degree of consolidation, U (%) s x /s y =1.0 s x /b = 10.0 No clogging n = Numerical (Case B) Numerical (Case A) Analytical Figure Time factor, T Comparison of analytical and numerical solutions for 55% PVD clogging. 204

17 100 Degree of consolidation, U (%) 50 s x /s y =1.0 s x /b = 10.0 n = No clogging Numerical (Case B) Numerical (Case A) Analytical Time factor, T Figure 15. Comparison of the analytical and numerical solutions for 75% PVD clogging. 100 s x /s y =1.0 Degree of consolidation, U (%) 50 s x /b = 10.0 No clogging n = Numerical (Case B) Numerical (Case A) Analytical Time factor, T Figure 16. Comparison of analytical with numerical solutions for 95% clogging. 205

18 Pore water pressure (%) 100 Point 3 Point 1 A B 1 2 D Point C 50 Domain of analysis s x /s y =1.0 Point 2 s x /b = Time factor, T Figure 17. condition. Variation of pore water pressures at Points 1, 2, 3, and 4 for no PVD clogging Pore water pressure (%) Point 1 (no clogging) Point 4 (Case A) Point 2 (no clogging) Point 2 (Case A) Point 1 (Case A) C L Point 4 (no clogging) Point 4 0 (Case B) Time factor, T Point 1 (Case B) Point 2 (Case B) Case A Case B s x /s y =1.0 s x /b = 10.0 Clogged PVD sections Figure 18. Variation of pore water pressures at Points 1, 2, and 4 for 55% PVD clogging (Cases A and B). 206

19 4 for 55% clogging for Cases A and B. The curves for Point 3 are omitted since the response of Points 1 and 3 are very similar. For all of the points, the dissipation of pore water pressure is faster for Case B than that for Case A. At Point 4, this difference is the maximum. For Case A clogging, the accumulation of water in Zone D is the maximum and, as a result, the rate of consolidation is slower. 4.8 Sequential Clogging The clogging considered thus far occurs from the beginning of consolidation; however, it is likely that clogging occurs after the start of consolidation and during flow into the PVD. To study this phenomena, 55% PVD clogging from the tip (Case A) at T = 0.01, 0.05, and 0.1, respectively, were simulated and the curves plotted in Figure 19 along with the curves for 55% clogging from T = 0 and for the no PVD clogging condition. The effect of clogging is immediately reflected in the U values at different times; however, all of the clogging curves merge into one at large time factors. For example, the value of T for 95% consolidation is approximately 1.4 for all of the curves. Thus, irrespective of the time when a PVD becomes clogged, for the same amount of clogging, the time required for 90% consolidation or more to occur is practically the same. 4.9 Effect of PVD Spacing The PVD spacing considered thus far is 1 m 1m(s x s y ). To study the effect of PVD spacing, three additional PVD spacings were considered (1.25 m 1m,1.5m Degree of consolidation, U (%) s x /s y =1.0 s x /b = 10.0 No clogging 55% clogging at T =0.1 55% clogging at T = % clogging at T =0.05 Initial 55% clogging Time factor, T Figure 19. Study of sequential PVD clogging (55%) for Case A. 207

20 1m,and2m 1 m) for the case of 55% PVD clogging from the tip (Case A). Figure 20 shows the effect of PVD spacing on U for no clogging and 55% clogging. The curves for no clogging and 55% clogging for all of the PVD spacings form two bands. As s x increases, the rate of consolidation decreases for both the no clogging and 55% PVD clogging conditions; however, it is interesting to note that the decrease in U values from no clogging to 55% clogging is practically the same for all PVD spacings. For example, at T = 1.0, the decrease in U values are 6.2, 6.2, 6.4, and 6.6% for PVD spacings of 1 m 1m,1.25m 1m,1.5m 1m,and2m 1 m, respectively. For T = 0.3, the decrease inuvaluesare 10.0, 11.2, 11.0, and9.8%, respectively, forthe PVD spacingsconsidered. Thus, the clogging effect is practically the same for the PVD spacings considered Effect of Smear (Without and With Clogging) The numerical and analytical results for the no clogging condition considering smear are compared in Figure 21. The curves are in good agreement, and the minor deviations are the result of differences in the shapes of zones of influence, as discussed in Section 4.2. Figure 22 shows the effect of smear and clogging on the average degree of consolidation-time factor (U -T) relationships. As before, 55% clogging from the tip of the PVD (Case A) and a spacing of 1 m 1 m were considered. The difference in U values between no clogging and 55% clogging for the case with smear (16.0% at T = 1.0) is greater than that without smear (6.2% at T = 1.0). Thus, the presence of a smear zone further decreases the rate of consolidation. 100 Degree of consolidation, U (%) 50 s x /s y =2.0 s x /s y =1.0 s x /s y =1.25 s x /s y =1.50 No clogging 55% clogging (Case A) Time factor, T Figure 20. Variation of the degree of consolidation, U, with PVD spacing. 208

21 100 s x /s y =1.0 Degree of consolidation, U (%) 50 s x /b = 10.0 n = m = k hs /k h =0.2 Analytical Numerical Time factor, T Figure 21. zone. Comparison of analytical with numerical solutions considering the smear Degree of consolidation, U (%) s x /s y =1.0 s x /b = 10.0 k hs /k h =0.2 55% clogging without smear (Case A) No clogging without smear 55% clogging with smear (Case A) No clogging with smear Time factor, T Figure 22. Effect of smear on PVD clogging. 209

22 5 CONCLUSIONS Band-shaped prefabricated vertical drains (PVDs) can be successfully used, along with preloading, for improving soft soil deposits; however, there are operational problems when using PVDs, one of which is clogging. Fine soil particles may become entrapped within the fibers of the geotextile filter/sleeve thereby clogging the filter pores. As a result, the pore water has a decreased surface area in which to enter the PVD and the consolidation process is retarded. In this paper, the effect of clogging of the PVD filter sleeve on the consolidation process was numerically simulated and studied. The variation of the average degree of consolidation with time factors was plotted for different amounts of PVD clogging. The effects of smear and PVD spacing were also analysed. Based on the results of the study, the following conclusions can be made: 1. Clogging of PVDs hinders the consolidation process of soils. The larger the amount of clogging, the slower the degree of consolidation. 2. The consolidation process is increasingly retarded when the PVD is clogged at its tip than when it is clogged at any intermediate position along its width. Depending upon where the clogging has occurred, the same amount of clogging may lead to different rates of consolidation. 3. The time required to reach 90% or more of total consolidation settlement is practically the same whether clogging occurs at the commencement of consolidation or after some time. Thus, the total time required to reach 90% of consolidation or more is independent of the time when the PVD becomes clogged. 4. Due to clogging of the PVD geotextile filter/sleeve, the PVD spacing (for the spacings considered) has a negligible effect on retarding the rate of consolidation. 5. The presence of a smear zone coupled with the effect of clogging results in enhanced retardation of the rate of consolidation. REFERENCES Aboshi, H. and Inoue, T., 1986, Prediction of Consolidation Settlement of Clay Layer, Especially in the Case of Soil Stabilization by Sand Drains, Proceedings of IEM - Japanese Society of Soil Mechanics and Foundation Engineering, Joint Symposium on Geotechnical Problems, pp Akagi, T., 1977, Effect of Mandrel-Driven Sand Drains on Strength, Proceedings of the Ninth International Conference of Soil Mechanics and Foundation Engineering, Vol. 1, Tokyo, Japan, pp Akagi, T., 1994, Hydraulic Applications of Geosynthetics to Filtration and Drainage Problems with Special Reference to Prefabricated Band-Shaped Drains, Keynote Lecture, Proceedings of the Fifth International Conference on Geotextiles, Geomembranes and Related Products, Vol. 4, Singapore, pp Akai, H., Yano, T. and Hwang, F.Y., 1981, Laboratory Tests on the Consolidation Effect due to Installation of Sand Drains, Sixth Annual Meeting of the Japanese Society of Soil Mechanics and Foundation Engineering, pp (in Japanese) 210

23 Ali, F.H., 1991, The Flow Behavior of Deformed Prefabricated Vertical Drains, Geotextiles and Geomembranes, Vol. 10, No. 3, pp Barron, R.A., 1948, Consolidation of Fine Grained Soils by Drain Wells, Transactions of American Society for Civil Engineers, Vol. 113, No. 2346, pp Bergado, D.T., Asakami, H., Alfaro, M.C. and Balasubramaniam, A.S., 1991, Smear Effect of Vertical Drains on Soft Bangkok Clay, Journal of Geotechnical Engineering, Vol. 117, No. 10, pp Casagrande, L. and Poulos, S., 1969, On the Effectiveness of Sand Drains, Canadian Geotechnical Journal, Vol. 6, August 1969, pp Das, B.M., 1983, Advanced Soil Mechanics, McGraw-Hill, Inc., New York, New York,USA,511p. Dastidar, A.G., Gupta, S. and Ghosh, T.K., 1969, Application of Sandwick in a Housing Project, Proceedings of the Seventh International Conference of Soil Mechanics and Foundation Engineering, Vol. 2, Mexico, pp Forrester, K., 1982, Accelerating Settlement by Vertical Drains, Department of Main Roads, Sydney, New South Wales, Australia. Hansbo, S., 1979, Consolidation of Clay by Band-Shaped Prefabricated Drains, Ground Engineering, Vol. 12, No. 5, pp Hansbo, S., 1981, Consolidation of Fine Grained Soils by Prefabricated Drains, Proceedings of the Tenth International Conference of Soil Mechanics and Foundation Engineering, Balkema, Vol. 3, Stockholm, Sweden, June 1981, pp Hausmann, M.R., 1990, Engineering Principles of Ground Modification, McGraw- Hill, Inc., New York, New York, USA, 632 p. Holtz, R.D., 1987, Preloading with Prefabricated Vertical Strip Drains, Soft Soil Stabilization Using Geosynthetics, Koerner, R.M., Editor, Elsevier Applied Science, 1988, Proceedings of the First Geosynthetic Research Institute Seminar,held in Philadelphia, Pennsylvania, USA, October 1987, pp Holtz, R. and Holm, G., 1973, Excavation and Sampling Around Some Drains at Ska- Edeby, Sweden, Scandinavian Geotechnical Meeting, Trondheim, Norwegian Geotechnical Institute, pp Jamiolkowski, M., Lancellotta, R. and Wolski, W., 1983, Precompression and Speeding up Consolidation, General Report Speciality Session 6, Proceedings of the Eighth European Conference on Soil Mechanics and Foundation Engineering,Balkema, Vol. 3, Helsinki, Finland, May 1983, pp Johnson, S.J., 1970, Foundation Precompression with Vertical Sand Drains, Journal of Soil Mechanics and Foundation Division, Vol. 96, No. 1, pp Kamon, M., Pradhan, T.B.S., Suwa, S., Hongo, T., Akai, T. and Imanishi, H., 1994, The Evaluation of Discharge Capacity of Prefabricated Band-Shaped Drains, Proceedings of a Symposium on Geotextile Test Methods, Japanese Society of Soil Mechanics and Foundation Engineering, Tokyo, Japan, pp (in Japanese) Kjellman, W., 1948, Accelerating Consolidation of Fine-Grained Soils by Means of Cardboard Wicks, Proceedings of the Second International Conference of Soil Mechanics and Foundation Engineering, Vol. 2, London, United Kingdom, pp

24 Lawrence, C.A. and Koerner, R.M., 1988, Flow Behavior of Kinked Strip Drains, Geosynthetics for Soil Improvement, Holtz, R.D., Editor, ASCE Geotechnical Special Publication No. 18, Proceedings of a symposium held in Nashville, Tennessee, USA, May 1988, pp Madhav, M.R., Park, Y.M. and Miura, N., 1993, Modelling and Study of Smear Zones Around Band Shaped Drains, Soils and Foundations, Vol. 33, No. 4, pp McDonald, P., 1985, Settlement of Fills on Soft Clay with Vertical Drains, Proceedings of the Eleventh International Conference of Soil Mechanics and Foundation Engineering, Balkema, San Francisco, California, USA, August 1985, pp Mesri, G., Lo, D.O.K. and Feng, T.W., 1994, Settlement of Embankments on Soft Clays, Keynote Lecture, Predicted and Measured Behavior of Five Spread Footings on Sand, Briaud, J-L. and Gibbens, R., Editors, ASCE, proceedings of the Prediction Symposium during the Settlement 94 ASCE Conference at Texas A&M University, College Station, Texas, USA, June Miura, N., Park, Y. and Madhav, M.R., 1993, Fundamental Study on Drainage Performance of Plastic Board Drains, Journal of the Japanese Society of Civil Engineers, 483/III-25, pp (in Japanese) Onoue, A., 1991, Permeability of Disturbed Zone Around Vertical Drains, 26 th Annual Meeting of Japanese Society of Soil Mechanics and Foundation Engineering, pp (in Japanese) NOTATIONS Basic SI units are given in parentheses. b = width of PVD (m) C h = coefficient of consolidation for flow in horizontal direction (m 2 /s) d e = equivalent diameter of unit cell (m) d s = equivalent diameter of smear zone (m) d w = equivalent diameter of PVD (m) F = function of n (Equation 11) (m) i = hydraulic gradient (dimensionless) k h = coefficient of horizontal permeability for undisturbed zone (m/s) k hs = coefficient of horizontal permeability for smear zone (m/s) m = ratio of equivalent diameter of smear zone to PVD (dimensionless) n = ratio of equivalent diameter of unit cell to PVD (dimensionless) Q = discharge velocity (m/s) q w = discharge capacity = Q/i (m/s) s x, s y = PVD spacing along x and y directions, respectively (m) T = time factor (dimensionless) T initial = initial time factor, Figure 7 (dimensionless) 212

25 T maximum = maximum time factor, Figure 7 (dimensionless) t = time (s) U = average degree of consolidation (%) u = excess pore water pressure (N/m 2 ) u(x, y, T) = excess pore water pressure at any point in entire domain at any time factor, T (N/m 2 ) u ini = initial excess pore water pressure at any point in entire domain (N/m 2 ) w = thickness of PVD (m) x, y = distance coordinates (m) β e = constant for Zones A, B, and C (Equation 5) (dimensionless) β i = constant for Zone D (Equation 6) (dimensionless) s, s e, s i = step lengths in finite difference discretisation (m) Vector ñ = direction of unit normal vector on any boundary 213

26 APPENDIX The equations for Boundaries a, b, c, and d (Figure 4), obtained by imposing a Neumann boundary condition to Equation 4, respectively, are: u t+ t i, j = β e 2u t+ t i+1, j + u t+ t i, j+1 + u t+ t i, j u t i, j (A-1) u t+ t i, j = β e u t+ t i+1, j + u t+ t i 1, j + 2u t+ t i, j u t i, j (A-2) u t+ t i, j = β e 2u t+ t i 1, j + u t+ t i, j+1 + u t+ t i, j u t i, j (A-3) u t+ t i, j = β e u t+ t i+1, j + u t+ t i 1, j + 2u t+ t i, j u t i, j (A-4) For points lying on Boundary α within the interior of Zone D, Equation A-1 is applicable, but only by replacing β e with β i. For the corner points, Points (i), (ii), and (iii), the equations are, respectively: u t+ t i, j = β e 2u t+ t i+1, j + 2u t+ t i, j u t i, j (A-5) u t+ t i, j = β e 2u t+ t i 1, j + 2u t+ t i, j u t i, j (A-6) u t+ t i, j = β e 2u t+ t i 1, j + 2u t+ t i, j u t i, j (A-7) The equations for points lying on Boundary d, an inter-zone boundary, are: β u t+ t e i, j = βi u t+ t i+1, j + β 1 + 2β i + β i u t+ t i 1, j + β ea u t+ t i, j+1 + β ia u t+ t i, j 1 ei (A-8) β e + u t i, j 1 + 2β i + β ei and equations for Boundary c are obtained by interchanging i and j. For Points (iv), (v), and (vi), however, the equations are different. The equations for Points (iv) and (vi) can be obtained by imposing a Neumann boundary condition to Equation A-8 using the same method as for Equations A-1 to A-4. For Point (v): 214

27 u t+ t i, j = 1 βea u t+ t i+1, j + β 1 + 4β ia u t+ t i 1, j + β ea u t+ t i, j+1 + β ia u t+ t i, j 1 ei + 1 u t i, j (A-9) i In Equations A-8 and A-9, the terms β ei, β ea,andβ ia are given by β ei = C h t s e s i β ea = C h t s e s a β ia = C h t s i s a (A-10) (A-11) (A-12) where s a is given by: s a = s e + s i (A-13) 2 The equations for all other points are the same as Equation 4. When the PVD is clogged, Equation A-4 is used for the clogged points with β i replacing β e. To incorporate the effect of smear, the equations for the smear boundaries, Boundaries pq and qr (Figure 8), were modified to give the following equation for Boundary pq (except the end points): u t+ t i, j = β e 1 + 2β e 1 + k hs k h β e 1 + k hs k h ut i, j k hs k h u t+ t i+1, j k hs k h u t+ t i 1, j + ut+ t i, j+1 + k hs u t+ t k i, h j 1 (A-14) and the equation for Boundary qr can be obtained by interchanging i and j in Equation A-14. The equations for Points p and r can be obtained by imposing a Neumann boundary condition to Equation A-14 using the same method as for Equations A-1 to A-4. For the corner point, Point q, the equation is: u t+ t i, j = β e 1 + β e 3 + k hs k h ut+ t i+1, j k hs β e 3 + k hs k h ut i, j k h u t+ t i 1, j + ut+ t i, j k hs k h u t+ t i, j 1 (A-15) 215