3.2.1 Inclinometers

Inclinometers are used to monitor the lateral (transverse) movements of natural slopes or embankments. An inclinometer casing has a grooved metal or plastic pipe that is placed into a borehole (Dunnicliff, 1988). The space between the wall of the borehole and the casing is backfilled with a sand or gravel grout. The bottom of the pipe must rest on a firm base to achieve a stable point of fixity. To monitor embankment performance, inclinometers are normally placed at or near the toe of the embankment where excessive lateral movement is usually of some concern.

3.2.2 Settlement indicators

Settlement plates or points are commonly installed where significant settlement is predicted (Dunnicliff, 1988) to record the magnitude and rate of settlement under a load. Therefore, they should be placed immediately after installing the vertical drains. In the simplest form, this instrument is a settlement plate consisting of a steel plate placed on the ground before construction of the embankment. Surface settlement points measure vertical displacement with depth, for example, along an embankment centerline. Typically, a reference rod and protecting pipe are attached to the settlement-monitoring platform. Settlement is often evaluated periodically until the surcharge embankment is completed, then at a reduced frequency, measuring the elevation of the top of the reference rod. Benchmarks used for reference data must be stable and remote from all other possible vertical movements. Further information about settlement points is given elsewhere (Dunnicliff, 1988).

3.2.3 Piezometers

A detailed description and analysis of various types of piezometers to measure in situ pore water pressure are presented by Hanna (1985) and Dunnicliff (1988). Piezometers should be installed at the bottom of the sand blanket, at various intermediate depths within the compressible layer. A dummy piezometer is usually installed a sufficient distance away from the embankment to record natural groundwater level, and excess pore water pressure at a given location is determined by comparison with the “dummy” level.

As shown in Fig. 3.4, the equivalent diameter of the influence zone (D_e) can be found in terms of the drain spacing (S) as follows (Hansbo, 1981):

$\begin{array}{cc}\hfill D_e=1.13S\hfill & \hfill \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{drains}\mathrm{installed}\mathrm{in}\mathrm{a}\mathrm{square}\mathrm{pattern}\hfill \end{array}$

and

$\begin{array}{cc}\hfill D_e=1.05S\hfill & \hfill \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{drains}\mathrm{installed}\mathrm{in}\mathrm{a}\mathrm{triangular}\mathrm{pattern}\hfill \end{array}$

Drains in a square pattern may be easier to lay out and control during installation in the field but a triangular pattern usually provides a more uniform consolidation between them.

3.3.2 Equivalent drain diameter of band-shaped vertical drain

Most prefabricated drains have a rectangular cross section (band-shaped, Fig. 3.5), but for design purposes, the rectangular (width a, thickness b) section has to be converted into an equivalent circle with a diameter of d_w, because the conventional theory of radial consolidation assumes that drains are circular.

The following typical equation is used to determine the equivalent drain diameter:

$\begin{array}{cc}\hfill d_w=2\left(\mathrm{a}+\mathrm{b}\right)/\pi \hfill & \hfill \left(\mathrm{Hansbo},1979\right)\hfill \end{array}$

Atkinson and Eldred (1981) proposed that a reduction factor of π/4 should be applied to Eq. (3.3) to take account of the corner effect, where the flow lines rapidly converge. From the finite element studies, Rixner et al. (1986) proposed that

$d_w=\left(a+b\right)/2$

Pradhan et al. (1993) suggested that the equivalent diameter of band-shaped drains should be estimated by considering the flow net around the soil cylinder of diameter d_e (Fig. 3.6). The mean-square distance of their flow net is calculated as

$s^{-2}=\frac{1}{4}{d}_{\mathit{e}}^2+\frac{1}{12}{a}^2-\frac{2\mathrm{a}}{{\pi }^2}{d}_{\mathit{e}}$

$\mathrm{Then},d_w=d_{\mathit{e}}-2\sqrt{\left(s^{-2}\right)}+b$

More recently, Long and Covo (1994) found that the equivalent diameter d_w could be computed using an electrical analog field plotter:

$d_w=0.5a+0.7b$

3.3.3 Discharge capacity

The discharge capacity is probably the most important parameter that controls the performance of prefabricated vertical drains. According to Holtz et al. (1991), the discharge capacity depends primarily on the following factors (Fig. 3.7): (1) the area of the drain core available for flow, (2) the effect of lateral earth pressure, (3) possible folding, bending, and crimping of the drain, and (4) infiltration of fine particles into the drain filter.

The current recommended values are given in Table 3.1 and the discharge capacities of various types of drains are shown in Fig. 3.8 as a function of lateral confining pressure.

In the field, vertical drains are installed using a steel mandrel, which is pushed into ground statically or dynamically then withdrawn, leaving the drain in the subsoil. This process causes significant remolding of the subsoil, especially in the immediate vicinity of the mandrel. The resulting smear zone will have reduced lateral permeability, which adversely affects consolidation process.

The combined effect of permeability and compressibility within the smear zone causes a different behavior from the undisturbed soil. Predicting soil behavior surrounding the drain requires an accurate estimation of the smear zone properties. In many classical solutions (Barron, 1948; Hansbo, 1981; Indraratna et al., 1997), the influence of the smear zone is considered with an idealized two-zone model.

Two parameters are necessary to characterize the smear effect, namely, the diameter of the smear zone (d_s) and the permeability ratio (k_h/k_s), that is, the value in the undisturbed zone (k_h) over the smear zone (k_s). Both the diameter of the smear zone and its permeability are difficult to quantify and determine from laboratory tests, and, so far, there is no comprehensive or standard method of measuring them. The extent of the smear zone and its permeability vary with the installation procedure, size, and shape of the mandrel, and the type and sensitivity of soil (macro fabric). Field and laboratory observations (Indraratna and Redana, 1998) indicated a continuous variation of soil permeability with the radial distance away from the drain centerline. Also, the smear zone diameter (d_s) has been the subject of much discussion in literature dealing with PVD.

Investigations by Holtz and Holm (1973) and Akagi (1977) indicate that

$d_s=2d_{\mathit{m}}$

where d_m is the diameter of the circle with an area equal to the cross-sectional area of the mandrel. Jamiolkowski and Lancellotta (1981) proposed that

$d_s=\left(2.5-3.0\right)d_{\mathit{m}}$

Hansbo (1981, 1997) proposed a different relationship:

$d_s=\left(1.5-3.0\right)d_w$

Based on laboratory study and back-analysis, Bergado et al. (1991) proposed that the following relation could be assumed:

$d_s=2d_w$

Indraratna and Redana (1998) proposed that the estimated smear zone could be as large as (4 − 5)d_w. This proposed relationship was verified using a specially designed large-scale consolidometer (Indraratna and Redana, 1995). Figure 3.9 shows the variation of the k_h/k_v ratio along the radial distance from the central drain. According to Hansbo (1987) and Bergado et al. (1991), the k_h/k_v ratio was found to be close to unity in the smear zone. This agrees with the study by Indraratna and Redana (1998). More recently, the primary author and his coworkers at the University of Wollongong attempted to estimate the extent of the smear zone caused by mandrel installation using the cylindrical cavity expansion theory. They used the modified Cam-clay (MCC) model and verified that the extent of the smear zone proposed by Indraratna and Redana (1998) was reasonable. Most recent results indicate that for most soft clays the extent of the smear zone is between 4d_w and 6d_w. The recommended smear zone parameters by different researchers are listed in Table 3.2.

3.4.2 Effect of a sand mat

Part or all water ingress to drains will flow to the ground first and then drain out through the sand mat. Because the hydraulic conductivity of sand is considerably higher than clay, it can usually be assumed that there is no hydraulic resistance in the sand mat. However, in some cases, depending on local materials, lower quality clayey sand may be used as a sand mat. In these instances, the hydraulic resistance in the sand mat may influence the rate of consolidation of the clay subsoil, the amount of which is a function of the hydraulic conductivity of the sand as well as the embankment geometry.

3.4.3 Well resistance

Well resistance refers to the finite permeability of the vertical drain with respect to the soil. Head loss occurs when water flows along the drain and delays radial consolidation. Well resistance is controlled not only by the discharge capacity of the drain q_w, but also by the permeability of the soil k_h, the maximum discharge length l_m, and any geometric deficiencies (bending, kinks, etc.) on the drains.

Mesri and Lo (1991) proposed the governing equation for vertical flow within the vertical drain in terms of excess pore water pressure at soil–drain interface. Based on Mesri’s equation, a well-resistance factor (R) is defined as

$R=\pi \left(k_w/k_{\mathit{h}}\right){\left(r_w/l_m\right)}^2=q_{\mathit{w}}/\left(k_{\mathit{h}}{l}_m^2\right)$

Analysis of the field performance of vertical drains indicated that the well resistance is negligible when R > 5. In other words, the minimum discharge capacity q_w(min) of vertical drains required for negligible well resistance may be determined from

$q_{\mathit{w}\left(\min \right)}=5k_{\mathit{h}}{l}_m^2$

This relationship is illustrated in Fig. 3.10 for most typical values of k_h and l_m.

Table 3.3 summarizes the well-resistance indices proposed by various investigators to evaluate the influence of finite discharge capacity of vertical drains. Note that the proposed indices are also transformed to the well-resistance factor (R) proposed by Mesri and Lo (1991). These indices depend on R, except for in the expression proposed by Aboshi and Yoshikuni (1967) and Stamatopoulos and Kotzias (1985), in which the drain spacing is also included.

Table 3.3

Summary of proposed well-resistance indices

Source	Index of well resistance
Aboshi and Yoshikuni (1967)	$R_i=\frac{\left(n^2-1\right)}{4F\left(n\right)n^2}\frac{k_{\mathit{h}}}{k_w}{\left(\frac{l_m}{r_w}\right)}^2=\frac{\pi \left(n^2-1\right)}{4F\left(n\right)n^2}\frac{1}{R}$
Yoshikuni and Nakanodo (1974), Onoue (1988)	$L=\frac{8}{{\pi }^2}\frac{k_{\mathit{h}}}{k_w}{\left(\frac{l_m}{r_w}\right)}^2=\frac{8}{\pi }\frac{1}{R}$
Hansbo (1981)	$W=2\frac{k_{\mathit{h}}}{k_w}{\left(\frac{l_m}{r_w}\right)}^2=2\pi \frac{1}{R}$
Stamatopoulos and Kotzias (1985)	$R_i=\frac{1}{F\left(n\right)}\frac{k_{\mathit{h}}}{k_w}{\left(\frac{l_m}{r_w}\right)}^2=\frac{\pi }{F\left(n\right)}\frac{1}{R}$
Zeng and Xie (1989)	$G=\frac{1}{4}\frac{k_{\mathit{h}}}{k_w}{\left(\frac{l_m}{r_w}\right)}^2=\frac{\pi }{4}\frac{1}{R}$
Mesri and Lo (1991)	$R=\pi \frac{k_w}{k_{\mathit{h}}}{\left(\frac{r_w}{l_m}\right)}^2=\frac{q_{\mathit{w}}}{k_{\mathit{h}}{l}_m^2}$

n = D_e/d_w is the spacing ratio.

In general, laboratory and field data indicate that the discharge capacities of most commercial PVDs have little influence on the consolidation rate of clay, especially for drains that are not too long (Indraratna et al., 1994). For values of q_w > 100–150 m³/year (in the field) and where drains are shorter than 30 m, there should be no significant increase in the consolidation time. Given these circumstances, it may be claimed that for commercial PVDs, well resistance is usually negligible in most practical situations unless the drains are excessively long and geometric deficiencies occur during installation (bending, kinks, etc.). In most soft clays, well resistance can be ignored for PVD < 15 m long.

3.5.1 Rendulic and Carillo diffusion theory

Rendulic (1936) formulated and solved the differential equation for one-dimensional vertical compression by radial flows

$\frac{\partial u}{\partial t}=c_{\mathit{h}}\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}\right)$

where r is the coordinate and c_h the horizontal coefficient of consolidation (k_h/γ_wm_v).

Carillo (1942) showed that for radial drainage and associated 1D consolidation, the excess pore water pressure u_r,z is given by

$\frac{\partial u}{\partial t}=c_{\mathit{h}}\left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}\right)+c_{\mathit{v}}\frac{{\partial }^{2}u}{\partial z^2}$

$u_{r,z}=u_r{u}_z/u_0$

where, u_r and u_z are the excess pore water pressure due to radial flow and vertical flow only, and u₀ is the initial pore water pressure. By substituting the average excess pore water pressure into Eq. (3.16), the average degree of consolidation of the compressible stratum can be obtained by combining U_z and U_r, thus

$\left(1-\overline{U}\right)=\left(1-{\overline{U}}_z\right)\left(1-{\overline{U}}_r\right)$

where Ū is the average degree of consolidation of the clay at time t for combined vertical and radial flow, and Ū_z and Ū_r are the average degree of consolidation at time t for vertical and radial flow, respectively. It should be noted that both Rendulic and Carillo’s solutions are for “ideal” drains only (infinite discharge capacity with no smear zone).

3.5.2 Barron—equal strain rigorous solution

Barron (1948) addressed the smear and well-resistance effects that can decrease the performance of vertical drains. He presented closed-form solutions for two extreme cases for radial drainage-induced consolidation, namely, free strain and equal strain, and showed that the average consolidation obtained in both cases is almost the same.

The free strain hypothesis assumes that the load is uniform over a circular zone of influence for each vertical drain. The differential settlements occurring over this zone do not affect the redistribution of stresses caused by the fill load arching. In contrast, the equal strain assumes that arching occurs in the upper layer during consolidation without any differential settlement in the clay layer; its simplicity is why it is commonly used by researchers.

Figure 3.12 shows the schematic illustration of a soil cylinder with a central vertical drain, where r_w is the radius of the drain, r_s is the radius of smear zone, R is the radius of soil cylinder, and l is the length of the drain installed into soft ground. The coefficient of permeability in the vertical and horizontal directions are k_v and k_h, respectively, and k_h′ is the coefficient permeability in the smear zone. Based on equal strain, Barron (1948) proposed a solution to Eq. (3.14) taking the smear effect into account as

$u_r=\overline{u}\frac{1}{v}\left[ln\left(\frac{r}{r_{\mathit{s}}}\right)-\frac{\left(r^2-r_{\mathit{s}}^2\right)}{2R^2}+\frac{k_{\mathit{h}}}{k_{\mathit{h}}^{\prime }}\left(\frac{n^2-s^2}{n^2}\right)ln\left(s\right)\right]$

where the smear factor ν is given by

$v=F\left(n,s,k_{\mathit{h}},k_{\mathit{h}}^{\prime }\right)=\left[\frac{n^2}{n^2-s^2}ln\left(\frac{n}{s}\right)-\frac{3}{4}+\frac{s^2}{4n^2}+\frac{k_h}{k_{\mathit{h}}^{\prime }}\left(\frac{n^2-s^2}{n^2}\right)ln\left(s\right)\right]$

and

$\overline{u}=u_{0}exp\left(-8T_{\mathrm{h}}/v\right)$

In Equations 3.18 through 3.20, s = r_s/r_w, n = R/r_w, T_h is the time factor given by T_h = c_ht/4R², ū is the average excess pore water pressure, and u₀ is the initial excess pore water pressure.

The average degree of consolidation, Ū_r, in the soil body is given by

${\overline{U}}_r=1-exp\left(-\frac{8T_{\mathit{h}}}{v}\right)$

3.5.3 Hansbo—analysis with smear and well resistance

Hansbo (1981) presented an approximate solution for vertical drain based on the equal strain by taking both smear and well resistance into consideration. The Ū_r, presented by Hansbo (1981), can be expressed as

${\overline{U}}_r=1-exp\left(-8T_{\mathit{h}}/\mu \right)$

where ignoring the insignificance terms, gives

$\mu =ln\left(\frac{n}{s}\right)+\left(\frac{k_h}{k_h^{\prime }}\right)ln\left(s\right)-0.75+\pi z\left(2l-z\right)\frac{k_h}{q_w}$

3.6.1 Shinsha et al.—permeability transformation

Shinsha et al. (1982) first proposed an acceptable matching criterion for converting the permeability coefficients. The equivalent coefficient of permeability was calculated on the assumption that the required time for a 50% degree of consolidation in both schemes was the same, giving

$k_{\mathit{pl}}/k_{\mathit{ax}}={\left(B/D_e\right)}^2{T}_{h50}/T_{r50}$

where T_h50 = 0.197 is a dimensionless time factor at 50% consolidation of laminar flow and T_r50 is the corresponding radial flow.

3.6.2 Hird et al.—geometry and permeability matching

By adapting Hansbo’s (1981) theory for the plane strain case, Hird et al. (1992) showed that the average degrees of consolidation, U, at any depth and time in the two unit cells were theoretically identical if well resistance was ignored:

$\frac{k_{\mathit{pl}}}{k_{\mathit{ax}}}=\frac{2B^2}{3R^2\left[ln\left(\frac{R}{r_{\mathit{s}}}\right)+\left(\frac{k_{\mathit{ax}}}{k_s}\right)ln\left(\frac{r_s}{r_w}\right)-\frac{3}{4}\right]}$

where subscripts “ax” and “pl” represent the axisymmetric and plane strain conditions, respectively. Note that the geometric matching is achieved by substituting k_pl = k_ax = k_h in Eq. (3.25), whereas the permeability matching is obtained by substituting B = R. For incorporating well resistance, the following dimensionless expression can be used:

$Q_w/q_w=2B/\pi R^2$

3.6.3 Bergado and Long—equal discharge concept

The converted permeability, including smear effect, is introduced based on the condition of the equal discharge rate (Bergado and Long, 1994) in both schemes and on the assumption that the coefficient of permeability is independent of the seepage state:

$\frac{k_{\mathit{pl}}}{k_{\mathit{ax}}}=\frac{\pi D\left(1-a_{\mathrm{s}}\right)}{2S\left[ln\left(\frac{\alpha D}{d_s}\right)+\left(\frac{k_{\mathit{ax}}}{k_{\mathit{s}}}\right)ln\left(\frac{d_s}{d_w}\right)\right]}$

where a_s = t/D, t is the thickness of the walls in 2D model, D and S are the row spacing and pile spacing of the actual case, respectively, and α = D_e/D, S = D, and α = 1.05 for square pattern, and S = 0.866D and α = 1.13 for triangular pattern.

3.6.4 Chai et al.—well resistance and clogging

Chai et al. (1995) successfully extended the analysis by Hird et al. (1992) to include the effect of well resistance and clogging. In this approach, the discharge capacity of the drain in plane strain (q_wp) for matching the average degree of horizontal consolidation is given by

$q_{wp}=\frac{4k_{\mathit{h}}{l}^2}{3B\left[ln\left(\frac{n}{s}\right)+\frac{k_{\mathit{h}}}{k_{\mathit{s}}}ln\left(s\right)-\frac{17}{12}+\frac{2l^2\pi k_{\mathit{h}}}{3q_{\mathit{w}\mathrm{a}}}\right]}$

3.6.5 Kim and Lee—time factor analysis

Kim and Lee (1997) assume that the time durations for the two systems (plane strain and axisymmetric) to achieve a 50% and 90% degree of consolidation are the same. Then, the following simple expression is obtained:

$\frac{k_{\mathit{pl}}}{k_{\mathit{ax}}}={\left(\frac{B}{R}\right)}^3\frac{T_{\mathit{r}50}}{T_{\mathit{h}50}}\frac{T_{\mathit{r}90}}{T_{\mathit{h}90}}\frac{S}{\pi d_w}$

3.6.6 Indraratna and Redana—rigorous solution for parallel drain wall

Indraratna and Redana (1997) converted the vertical drain system shown in Fig. 3.13 into an equivalent parallel drain wall by adjusting the coefficient of soil permeability. They assumed that the half-widths of unit cell B, of drains b_w, and of smear zone b_s are the same as their axisymmetric radii R, r_w, and r_s, respectively. The equivalent permeability of the model is then determined by

$k_{\mathit{hp}}=\frac{k_{\mathit{h}}\left[\alpha +\left(\beta \right)\frac{k_{\mathit{hp}}}{k_{\mathit{hp}}^{\prime }}+\left(\theta \right)\left(2lz-z^2\right)\right]}{\left[ln\left(\frac{n}{s}\right)+\left(\frac{k_{\mathit{h}}}{k_{\mathit{h}}^{\prime }}\right)ln\left(s\right)-0.75+\pi \left(2lz-z^2\right)\frac{k_{\mathit{h}}}{q_{\mathit{w}}}\right]}$

The associated geometric parameters α and β and the flow term θ are given by

$\alpha =\frac{2}{3}\frac{{\left(n-s\right)}^3}{\left(n-1\right)n^2}$

$\beta =\frac{2}{3}\frac{\left(s-1\right)}{\left(n-1\right)n^2}\left[3n\left(n-s-1\right)+\left(s^2+s+1\right)\right]$

and

$\theta =\frac{2k_{\mathit{hp}}}{B{\mathit{q}}_{\mathit{z}}}\left(1-\frac{1}{n}\right)$

where q_z = 2q_w/πB is the equivalent plane strain discharge capacity.

In Eq. (3.30), as k_hp appears on both sides of the equation the solution is obtained by iteration with an initially assumed k_hp/k_hp′ ratio.

To verify this model, a finite element analysis was undertaken for both axisymmetric and equivalent plane strain models. As an example, a unit drain was analyzed with a drain installed to a depth of 5 m below the ground surface at 1.2 m spacing. The model parameters and soil properties were r_w = 0.03 m, r_m = 0.05 m, k_h = 1 × 10^–8 m/s, and k_h′ = 5 × 10^–9 m/s, and the corresponding equivalent coefficients of plane strain permeability were k_hp′ = 5.02 × 10^–10 m/s and k_hp = 2.97 × 10^–9 m/s based on Eq. (3.30). The water table was assumed to be at the surface and r_s = 5r_m (based on experimental results). For the elasto-plastic finite element analysis, the MCC model was used as follows: λ = 0.2, κ = 0.04, M = 1.0, e_cs = 2, and Poisson’s ratio ν = 0.25, with a saturated unit weight of 18 kN/m³.

The results of both axisymmetric and plane strain analysis are plotted in Fig. 3.14. The average degree of radial consolidation U_h (%) is plotted against the time factor T_h for perfect drain conditions. As illustrated, the proposed plane strain analysis agree perfectly with the axisymmetric analysis, with the maximum deviation between the two methods being less than 5%.

Figures 3.15 and 3.16 illustrate the settlements and excess pore pressure variations over time, including smear plus well resistance for a single drain, and again the axisymmetric model and the equivalent plane strain model agreed.

Based on the previous single drain analysis, Figures 3.14 through 3.16 provide concrete evidence that the equivalent (converted) plane strain model is an excellent substitute for the axisymmetric model. In finite element modeling, the 2D plane strain analysis is expected to reduce computational time considerably compared to the time taken by a 3D, axisymmetric model, especially in multidrain analysis.

3.8.1 Element types for soil and soil–drain interface

The types of elements used in consolidation analysis in the finite element code ABAQUS are shown in Fig. 3.17. The basic element type is a 4-node bilinear displacement and pore pressure element (CPE4P) consisting of 4 displacement and pore pressure nodes at the corners. The higher order of this element is a 20-node triquadratic displacement and trilinear pore pressure nodes with reduced integration (C3D20RP), which contain 20 displacement nodes and 8 pore pressure nodes. As explained by Hibbitt and Sorensen (2004), reduced integration elements use a lower order of integration to form element stiffness. ABAQUS recommends using reduced integration elements because it usually gives more accurate results and is less time consuming than full integration. The common element type used in the analysis presented here is the CPE8RP element, which contains 8 displacement nodes and 4 pore pressure nodes.

Interface elements are most appropriate to simulate soil–drain interaction. Because the thickness of PVD is relatively small compared to its spacing, the interface element is envisaged as the soil element having properties similar to the adjacent soil except for permeability. A three-node interface element (ASI3) is shown in Fig. 3.17, where there are two pore pressure nodes at the ends.

In finite element analysis, the pore pressure shape function is usually one order less than the displacement shape function. In most of the elements shown in Fig. 3.17, the pore pressure shape function is linear, while the displacement shape functions are either quadratic or cubic.

Figure 3.18 presents a typical discretized finite element mesh, which is used for numerical analysis, where only one-half of the embankment is considered by symmetry. A foundation depth of 15 m was considered sufficient for analysis, assuming the existence of a stiff clay layer beneath this depth. The mesh consists of more than 1000 CPE8RP elements and the vertical drains are modeled by an interface element (ASI3). A finer mesh was employed for the zone beneath the embankment, with a half-width of 20 m. The embankment loading is simulated by applying incremental vertical loads.

3.8.3 Influence of drain spacing

To investigate the effect of vertical drains on embankment stability, four different drain spacings were considered in the analysis: 1, 1.5, 2, and 3 m. The embankment is raised to a maximum height of 4 m, with two different construction rates: 0.1 m/week and 0.35 m/week. The slope of the embankment is assumed to be 2:1.

Figures 3.23 and 3.24 show the predicted surface settlement at the centerline and toe of the embankment, respectively. For a construction rate of 0.1 m/week, impending failure is not noticed for a small drain spacing of up to 2 m, which suggests that the higher dissipation of pore pressure and slower construction rate allow the soft clay foundation to gain sufficient strength to support a 4 m high embankment. If the construction rate is increased to 0.35 m/week, the foundation stabilized with PVD at 1 m spacing reaches its ultimate settlement within a shorter period (100 days) compared to a construction rate of 0.1 m/week (300 days). It is not possible to reach the final embankment height of 4 m if the drain spacing is 1.5, 2, or 3 m (Fig. 3.24b).

3.9.1 Embankment constructed to failure

An embankment built for catastrophic failure is shown in Fig. 3.25, located just north of embankment #1. The cross section of the embankment, showing the key instruments with subsoil variation and the discretized finite element mesh, is shown in Figures 3.26 and 3.27, respectively. The piezometers P5 and P6 and the inclinometers I3 and I4 are used to monitor embankment failure. The embankment was raised with a fill material of 20.5 kN/m³ bulk unit weight at a constant rate of 0.4 m/week (Indraratna et al., 1992). The MCC parameters and the in situ stresses are given in Tables 3.6 and 3.7, respectively.

The excess pore pressure variation along the embankment centerline, the surface settlement, and the lateral displacement at 10 m from the centerline are plotted in Figures 3.28 through 3.30 for a fill height of 5 m. As expected, the lateral displacement is significantly reduced in the stiffer clay layer. In general, the MCC theory overestimates the lateral displacements. The reason for this discrepancy can be attributed to several factors, including the lateral variability of soil parameters, the use of a simplified associated flow rule, and the effect of the stiff surficial crust (Potts and Zdravkovic, 2001). It was found that lateral displacements are also sensitive to nominal changes of compression parameter λ (Indraratna et al., 1992).

3.9.2 Embankment stabilized with geosynthetic vertical drain

An embankment stabilized with a geosynthetic vertical drain is shown in Fig. 3.25 as embankment #14. The embankment cross section with key instrumentation and the associated subsoil profile is shown in Fig. 3.31. The equivalent drain radius based on Eq. (3.4) is estimated to be r_w = 0.03 m and the smear zone radius is taken as r_s = 0.15 m. The Cam-clay parameters and equivalent plane strain permeabilities based on Eq. (3.30) are given in Table 3.8. Table 3.9 tabulates the in situ stress distribution. Embankment construction was carried out in two loading stages; during the first 14 days the height was raised to 2.57 m (stage 1) and after a 90-day rest period the height was raised to 4.74 m in 24 days (stage 2).

The finite element mesh of the embankment is shown in Fig. 3.32 and the location of inclinometer (ID1, 23 m away from the centerline) and piezometers are conveniently defined at mesh nodes. The well resistance of the drain was included because it was 18 m long. The well resistance was simulated by considering the vertical permeability of the transformed drain wall, as previously shown in Eq. (3.31c). The equivalent coefficient of permeability of the drain was estimated as 0.0005 m/s by a single drain analysis.

The predicted and measured settlements at the centerline and along the surface are shown in Figures 3.33 and 3.34, respectively. Heave is also predicted beyond the toe of the embankment, that is, at about 45 m away from the centerline, but, regrettably, no field data were available for comparison. The predictions acceptably agree with the limited field measurements obtained near the centerline.

The evaluated and measured excess pore water pressure variations are shown in Fig. 3.35. The measured excess pore pressure does not indicate much dissipation during stage 2 due to the piezometer malfunctioning. Even though the prediction of excess pore pressure is made accurately in stage 1 by including the smear effect, the predicted postconstruction pore pressure only improved slightly by including both smear and well resistance. As expected, the “perfect drain” underestimates the measurements. Observed and predicted lateral deformations are plotted in Fig. 3.36. Acceptable agreement between the field data and predictions is obtained when both the smear and well resistance are considered. The perfect drain condition gives the smallest lateral deformation while maximizing vertical deformation.

3.9.3 Normalized deformation factors

The lateral displacement and settlement can be normalized with respect to the corresponding fill height to examine the effectiveness of the ground improvement techniques. Thus, the following “stability” indicators are defined (Indraratna et al., 1997): β₁, the ratio between lateral displacement and the corresponding fill height, β₂ the ratio between settlement and the corresponding fill height, and α = β₁/β₂.

Figure 3.37 shows variation of β₁ with depth and Fig. 3.38 shows the variation of α with depth. The normalized displacement of a PVD-stabilized embankment is considerably less than an embankment constructed to failure. These results clearly show that vertical drains effectively decrease lateral deformations and enable the critical height of an embankment to be increased. After 13 days, the untreated embankment fails with unacceptably large lateral displacement (Fig. 3.37). The PVD-stabilized foundation takes more than 7 years before lateral displacement becomes similar to the failed embankment.

The normalized deformation factors for a few trial embankments are also compared in Table 3.10. In comparison with the unstabilized embankment constructed to failure, the stabilized foundations are characterized by considerably smaller values for α and β₁, which elucidates their obvious implications on stability. The normalized settlement (β₂) on its own is not a proper indicator of instability, but is still a useful stability indicator when taken in conjunction with α and β₁. For example, the foundation having SCP gives the lowest values of β₁ and β₂, clearly suggesting the benefits of sand compaction piles over band drains.