Theories, Computations, and Design Procedures Involving Vertical Drains

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Chapter 2
Theories, Computations, and Design Procedures Involving Vertical Drains



The analyses in this chapter are based on the following established equations of simple consolidation theories:

where ue is the excess pore water pressure, cv = kv/mvγw = coefficient of consolidation for vertical flow, ch = kh/mvγw = coefficient of consolidation for horizontal flow, kv and kh are coefficients of permeability in the vertical and horizontal directions, mv the coefficient of volume compressibility, and γw the unit weight of water.

Emphasis is on preliminary analyses using charts and simple programs.

Owing to the necessary idealizations in formulations, parametric uncertainties, and imperfect information about geological material, good agreement cannot always be expected between theoretical predictions and actual monitored performance. Mitchell (1993) noted that in many cases the simple consolidation theory did not

produce satisfactory predictions of the settlement magnitude and the settlement rate; the possible causes were discussed, including soil behavior factors, modeling factors, effects of sample disturbance, and secondary compression and time effects. A nonlinear consolidation theory was also described. The limitations of simple consolidation theory were further assessed in Terzaghi et al. (1996); possible refinements and reformulations were suggested to account for nonhomogeneous consolidating layer, nonconstant coefficients of permeability kv and kh during consolidation, and finite strain formulations, for example. Terzaghi et al. (1996) also noted that, in some situations, the assumptions underlying the simple theories could be reasonable and might lead to realistic predictions of the rate of settlement; but in other cases (especially excess pore pressure predictions), reliable predictions would require more realistic assumptions about the permeability and compressibility of the consolidating layer. Other theoretical aspects of consolidation analysis can be found in Scott (1963), and Lambe and Whitman (1969).

The above paragraph serves to keep in proper perspective the potential limitations and possible inadequacies of the simple analysis and procedures of this chapter. Nevertheless, the usefulness and economy of simple and approximate analysis for preliminary design and sensitivity studies should not be underestimated.


2.2.1 Equations and Charts of One-Dimensional Consolidation

Equation (2.1) is Terzaghi's small-strain theory for one-dimensional consolidation of a saturated and homogeneous soil that obeys Darcy's law and has linear stress–strain relation. For uniform initial excess pore pressure u0, the solution is (e.g., Lambe and Whitman 1969; Terzaghi et al., 1996):



and where z is measured from the top of the consolidating layer and Hd is the maximum vertical drainage distance.

In terms of consolidation ratio Uz, Eq. (2.3) can be expressed as (e.g., Lambe and Whitman, 1969):

that is,

Equation (2.7) expresses the degree of consolidation Uz as a function of depth ratio z/Hd and time factor Tv. It can be plotted as shown in Fig. 2.1. For program-based computations in later sections, it is Eq. (2.7) that is more useful than the chart in Fig. 2.1.

The average degree of consolidation Ūv for the entire thickness of the consolidating layer is (Lambe and Whitman, 1969):

The subscript v in Ūv denotes that the average degree of consolidation is based on excess pore pressure dissipation due to vertical flow only, to distinguish it from the average degree of consolidation Ūh (discussed later) assuming excess pore pressure dissipation due to horizontal flow. A user-defined function code can be created in the Visual Basic programming environment of Microsoft Excel to allow easy computation of Ūv according to Eq. (2.8). The following VBA

(Visual Basic for Applications) function code achieves this objective:

Function UvConsol(Tv) As Double

sumMTv = 0

If Tv < 0.0001

Then Tv = 0.0001

For m = 0 To 10000

Prev = sumMTv

CapM = 1.5708 * (2 * m+1)

sumMTv = sumMTv + 2/CapMˆ2 * Exp(-CapM^2 * Tv)

If Abs(sumMTv - Prev) < 0.0001 Then Exit For

Next m

UvConsol = 1 - sumMTv

End Function

Once created, the function is used as one uses SQRT(x) or SIN(y), namely, by entering the equation ‘= UvConsol(Tv)’ in a cell, where Tv is the argument of the function.

Equation (2.8) as computed by the above function code is plotted in Fig. 2.2. The chart is convenient for manual calculation. However, for computer-based analysis and design, it is Eq. (2.8) (or equations approximating it) that will be more useful than the chart in Fig. 2.2. Among the various closed form approximations of Eq. (2.8), the following fitted equation (Sivaram and Swamee, 1977) is convenient:

Direct comparison will show that Eqs. (2.8) and (2.9) produce almost the same results for 0 ≤ Ūv≤ 0.997, or 0 ≤ Tv ≤ 6. The value of Ūv by Eq. (2.9) decreases slightly at large values of Tv

(e.g., Ūv=0.994 when Tv = 100) but such inconsistency is of little practical significance.

Eq. (2.8)0.1130.3570.5040.6980.8160.8870.9310.9940.9991.0001.0001.000
Eq. (2.9)0.1130.3570.5030.6960.8160.8870.9290.9860.9940.9960.9970.997

Sivaram and Swamee (1977) also suggested the following curve-fitted equation

which can be used to estimate the time required for a desired average degree of consolidation Uv.

2.2.2 Implicit Finite Difference Solution of One-Dimensional Consolidation

Equations (2.7) and (2.8) (and associated charts in Figs. 2.1 and 2.2) are applicable for uniform initial excess pore pressure of a consolidating layer with either single drainage boundary (half-closed layer) or double drainage boundary (open layer). They are also valid for nonuniform but linear initial excess pore pressure, provided both the top and bottom boundaries of the consolidating layer are permeable. When these conditions are not satisfied, a more general approach is to use the finite difference method to solve the one-dimensional consolidation equation (Eq. (2.1)). The principles of the Crank-Nicholson implicit finite difference scheme and its difference from the less robust explicit scheme are shown in Fig. 2.3. Understanding this figure will facilitate understanding of the more lengthy implicit finite difference formulations for combined vertical and radial flow shown later in Fig. 2.11. A simple and transparent implementation of the procedure in a spreadsheet is shown in Fig. 2.4, which uses the automatic iteration feature (‘Circular reference’) of Microsoft Excel. The implicit finite difference equation (of u2 in Fig. 2.3) is entered

only once in cell D10of Fig. 2.4, followed by autofilling downwards to cell D18 and rightwards to cell U18. The case in Fig. 2.4, with uniform initial excess pore pressure, has known closed form solutions

(Eq. (2.3)). The case in Fig. 2.5, with nonuniform initial excess pore pressure and impermeable bottom boundary, can be solved only by numerical methods. It is interesting to note that because the 10-m thick clay layer in Fig. 2.5 is half-closed, excess pore pressure at the bottom impermeable boundary increases at first flowing to flow from the upper higher-pressure layer, but subsequently dissipates and approaches zero after a long time.

Although the implicit finite difference scheme is numerically more stable than the explicit scheme, it is still necessary to control the time stepping so that the dimensionless ratio λ = cv∆t/(∆z)2 is small.


The consolidation of thick clay layers can often be accelerated by the use of vertical drains. The parameters required for design include the coefficients of consolidation for vertical drainage and for horizontal drainage, cv and ch respectively, the length of vertical drainage path, the length of horizontal drainage path, and the ratio of drain spacing to drain diameter. In many cases, there is much uncertainty about the in-situ values of cv and ch flowing to our imperfect knowledge of the subsoil profile at any given site, compounded by the possible smear effect accompanying installation of vertical drains. The length of the vertical drainage path may also be difficult to ascertain; it depends on whether continuous natural drainage layers exist within the clay stratum. In practice, field trials are often conducted to check whether vertical drains will significantly accelerate the consolidation at a given site, and to help in estimating the values of the design parameters.

The difficulties in selecting realistic values of the design parameters are discussed in the other chapters. In this section, the computational aspects of vertical drains are considered. The usual design charts are reviewed. An alternative chart is also shown, which facilitates the calculation of the average degree of consolidation for simultaneous vertical and radial drainage. A convenient option using spreadsheet is also described.

2.3.1 Barron's Solution and Carillo's Equation

When vertical drains are used, excess pore pressure dissipates by simultaneous radial and vertical flow, the former mode being more expeditious by virtue of the typically shorter radial drainage path compared to the longer vertical drainage distance. The simplest partial differential equation that governs excess pore pressure dissipation due to combined radial and vertical flow in a unit soil cylinder is Eq. (2.2). In practice, the excess pore pressure ratios ue/uo are computed separately on the basis of vertical flow alone (Eq. (2.1)) and of radial flow alone (Eq. (2.2) without the cv2ue/∂z2 component), and then combined using Carillo's equation (as summarized in Mitchell and Kathi, 1981, for example):

Or, equivalently,


Ūv = Average degree of consolidation for vertical drainage only

Ūh = Average degree of consolidation for horizontal drainage only

The subscripts v and h denote vertical and horizontal (in this case radial) flow, respectively. In Eq. (2.13), Uz is the degree of consolidation at depth z due to vertical flow only, and Uh is the degree of consolidation due to radial flow only. The value of Uh varies with radial distance from the drain and with time.

The consolidation of a clay layer installed with vertical drains is analyzed at the unit cell level on the basis of Eq. (2.2). The unit cell is idealized to be an equivalent soil cylinder of diameter de,having an impermeable outside surface and an inner cylindrical drain. If the drain is band-shaped, the equivalent diameter dw is that of a cylindrical drain having the same circumference (Fig. 2.6), that is,

where a and b = width and thickness, respectively, of the band-shaped drain.

According to Barron's equal-strain solution, the average degree of consolidation with respect to radial flow alone can be estimated from:

where n = the drain spacing ratio = (de/dw)

Equation (2.16) is plotted in the upper chart of Fig. 2.7 as a function of the dimensionless drain spacing ratio η and time factor Th. The broken curve in the same chart shows the average Ūv as a function of Tv, the same as the curve in Fig. 2.2. The values of Ūv and Ūh obtained from the chart are then used in Carillo's equation (Eq. (2.14)) to estimate theŪvh for combined vertical and horizontal drainage. In many cases, it is found that the contribution of Ūv to Ūvh is far less than that of Ūh to Ūvh. A trial-and-error procedure is often necessary when computing the time required to achieve a desired Ūvh.

Note that Terzaghi et al. (1996) defines the dimensionless time factor for radial flow in terms of the radius re of the equivalent clay cylinder rather than of diameter de:

Thus defined, the degree of consolidation is:

Equations (2.16) and (2.20) are consistent with their respective time factors, (2.18) and (2.19), and will give identical results.

The diameter of the equivalent clay cylinder tributary to a vertical drain is based on equivalent cross-sectional area. If the vertical drains are installed in a square grid pattern, the equivalent drainage diameter is obtained as follows:

If the vertical drains are spaced in a triangular grid pattern, the equivalent drainage diameter is:

2.3.2 Design Charts for Combined Vertical and Radial Drainage

The two separate charts (for determining and Ūv and Ūh)mentioned in the preceding paragraphs can be combined into a single convenient chart using Carillo's equation. From Eqs. (2.14) and (2.16), one obtains:


assuming that the time for consolidation is the same for both vertical flow and horizontal flow (i.e., no time-lag between the start of vertical flow and the start of radial flow).

Since the Ūv in Eq. (2.23) is a function of Tv, according to Eq. (2.8) (or its approximation Eq. (2.9)),Ūvh as expressed by Eq. (2.23) is a function of Tv and the dimensionless group parameter represented by λ.

Equation (2.23) has been plotted in Fig 2.8. The sample problem given in Fig. 2.8 can be solved readily once the λ value has been computed. Had the two separate charts for determining Ūv and Ūh been used, a number of iterations would have been necessary to

obtain the time t which satisfies Eq. (2.14) (which is the basis of Eq. (2.23)). For hand computations, Fig. 2.8 is also more efficient than Fig. 2.7 when a drain spacing is to be designed to achieve a target Ūvh for a given consolidation duration t.

A perhaps more obvious feature of the chart in Fig. 2.8 is that the sensitivity of Ūvh to the λ values and hence to the ratio (Eq. (2.23a)) is readily perceivable. The chart is therefore convenient for assessing how deviations from the assumed values of the parameters will affect the calculated average degree of consolidation for simultaneous vertical and radial consolidation. Note that the uppermost curve, with λ = 0, is the case with no vertical drains, the same as Fig. 2.2 and the broken curve in Fig. 2.7.

2.3.3 Accounting for Smear Effect by Reducing ch

Drain installation disturbs the soil to some degree depending on the soil's sensitivity and macrofabric. The effect of soil disturbance was included in the analyses by Barron (1948) and Hansbo (1979, 1981) by assuming an annulus of smeared clay around the drain. Within this annulus of diameter ds, the remolded soil has a coefficient of permeability ks which is lower than the kh of the undisturbed clay. The new boundary condition between the undisturbed zone and the smeared annulus affects the above solution for Ūh by changing the factor F (n) which becomes:

or, equivalently,

so that Eq. (2.16) becomes

The two additional parameters, s and kh/ks, are difficult to estimate.

If smear effect is ignored, then s =1 and kh = ks, and Eqs. (2.24) and (2.25) reduce to Eqs. (2.17) and (2.16), respectively.

In the presence of smear effect, charts prepared without smear effect in mind (e.g., the upper chart in Fig. 2.7, and Fig. 2.8) can still be used, provided either a modified drain spacing ratio (n*)or a modified coefficient of consolidation is adopted. The rationale for the modified parameters is straightforward:

  • By the equivalent reduced drain diameter approach, it is required that the n in Eq. (2.17) be modified to n*so that Eq. (2.17) yields the same value as Eq. (2.24), that is:

    This is the approach underlying Fig. 21 of NAVFAC (1982) DM7.1-252.
  • By the equivalent reduced ch approach, it is required that the original ch (in Eqs. (2.18) and (2.16)) be modified to so that Eq. (2.16) yields the same value as Eq. (2.25). This leads to the following equation:

Physically, the equivalent reduced coefficient may be regarded as the weighted average of the disturbed ch in the smear zone and the original undisturbed ch outside the smeared annulus. Since already incorporates the smear effect, it can be used with charts prepared with no smear in mind, that is with Figs. 2.7 or 2.8. Note that if s =1 and kh = ks, meaning no smear effect, then F (n)= Fs(n), and equivalent ch = undisturbed ch.

Equation (2.27a) is plotted in the lower chart of Fig. 2.7 for smear diameter ratio s = 2 and various values of kh/ks. The equivalent reduced coefficient () is to be used in computing Th according to Eq. (2.18), which is then used in the upper chart of Fig. 2.7 which

assumes no smear. The resulting Ūh value will of course be the same as that obtained by Eqs. (2.24a) and (2.25) directly.

The equivalent reduced ch concept was applied in Low et al. (1995) to the land reclamation test fill of Chek Lap Kok in Hong Kong. The main test area, located about 200 m offshore and 100 m square in plan, was divided into quadrants: one was a control area, with no treatment of the marine clay, and the remaining three quadrants were installed with vertical drains at different spacing through about 7 m thick mud. The offshore geotechnical investigations, the test fill, and the instrumentation program were described by Foott et al. (1987). Two test fill quadrants were installed with prefabricated band drains at 1.5 m and 3 m spacing, and the third quadrant was installed with sand drains at 3 m spacing.

The original undisturbed ch was estimated to be 3.7m2/year. With smear diameter ratio s =2 and kh/ks =4.63 (based on data from Chek Lap Kok), the equivalent ch for the three vertical drain quadrants can be calculated using either Eq. (2.27a) or the plot in Fig. 2.7. The results are tabulated below:

Band drain spacingnF(n)Fs(n)Equivalent ch(m2/year)
1.5 m band drain23.62.414.931.81
3 m band drain473.105.622.04
3 m sand-drain6.31.093.611.12

The three values of equivalent ch may be compared with the value of undisturbed ch(= 3.7m2/year). It is interesting to note the different degrees of reduction indicated by the different equivalent ch values in the three quadrants. It is also possible in principle to use Eq. (2.27) in reverse, namely to assess the radius of the smear zone or the ratio kh/ks (of Eq. 2.24) using the value of ch back analyzed from the measured rate of settlement.

2.3.4 Spreadsheet Template with Automatic Iteration

The charts in Figs. 2.7 and 2.8 are useful in giving an overall perspective of the parametric relationships which affect the degree

of consolidation. However, for parametric and what-if analysis, the spreadsheet template illustrated in Fig. 2.9 will be more time-efficient. It uses the Microsoft Excel's built-in GoalSeek routine to automatically obtain the time required to achieve a desired average

degree of consolidation Ūvh (e.g., 90% in Fig. 2.9) for given input values of drain spacing, square or triangular grid layout, cv, ch,and smear parameters. The example in Fig. 2.10automatically obtains the drain spacing required to achieve the desired 90% of average degree of consolidation within the specified time frame of 1.25 years. This example is tedious to solve manually using the charts of Fig. 2.7, because the drain spacing has to be determined by trial and error. Another convenient feature of the spreadsheet platform is that solutions of multiple scenarios (including those of Figs. 2.9 and 2.10) can be stored for easy recall using the Scenario Manager feature of the spreadsheet. This means that only one single template (Fig. 2.9) can be used for analyzing many parametric combinations, and all the cases are remembered for easy recall.

If a cost function is expressed in terms of drain spacing, consolidation time, and other relevant parameters, one can perform cost optimization using the Solver program in Microsoft Excel. The Solver routine is much more versatile than the GoalSeek routine, because it automatically performs cell-object oriented constrained optimization by changing user-specified input cell values (which can be over one hundred cells), subject to numerous user-specified constraints. Illustrations of how Solver and VBA programming can be harnessed to good effect, with much enhanced computational economy and conceptual clarity, can be found in Low and Tang (1997), Low et al. (1998) and Low (2001).

2.3.5 Implicit Finite Difference Solution for Combined Vertical and Radial Flow

In its simplest form, the excess pore pressure dissipation in a unit soil cylinder that has been installed with a central vertical drain is governed by Eq. (2.2). The solution can be obtained using the finite difference method in spreadsheet. Figure 2.11 shows the implicit finite difference scheme based on the Crank-Nicholson method, which assigns equal weights to the previous time step and the current time step. The implicit finite difference scheme is more robust and accurate than the simpler explicit finite difference scheme. By virtue of symmetry, only half the vertical cross-section of the soil

cylinder needs be analyzed, as shown in Fig. 2.12. Figure 2.13 shows the stages of excess pore pressure dissipation for a soil cylinder of radius 1.5 m and maximum vertical drainage distance Hd =4m. The vertical drain installed at the center of the cylinder had a radius rw =0.1 m. The coefficients of consolidation in the vertical and horizontal directions were cv =2 m2/year and ch =3 m2/year, respectively. Smear effects were not considered. The initial excess pore pressure was 100 kPa and uniform with depth. Each band in Fig. 2.13 represents 10kPa of excess pore pressure. The average degrees of consolidation Ūv shown in Fig. 2.13 were calculated from integrating radially and with depth. The values Ūvh = 0.26, 0.45, 0.65, and 0.85 at times 0.1, 0.25, 0.5, and 1.0 year, respectively, are in reasonable agreement with Ūvh =0.24, 0.43, 0.64, and 0.85 computed using Eq. (2.23).

It may be observed that the excess pore pressure isochrones are near horizontal midway between vertical drains (i.e., at the exterior cylindrical shell of the idealized unit soil cylinder), and descend sharply in the vicinity of the central vertical drain. The approximately inverted-L shape of the isochrones is the consequence of the permeable boundaries at the left and at the top. Had the top surface been impermeable, the isochrones would all be coaxial cylindrical shells and appear as vertical lines in cross-sectional view.

Hansbo (1994) noted that a mandrel inserted into the soil may deviate considerably from the vertical, especially in deep installations, and that such a deviation from the vertical could have an unpredictable effect on the consolidation process at great depth.

The inevitable deviation from verticality of the drains and of the pore-pressure monitoring device also means that there is some uncertainty about the horizontal position of the device relative to the drain. In Fig. 2.13, pore-pressure measuring devices at A, B, and C will register 70, 60, and 40 kPa of excess pore water pressure, respectively, based on the uncoupled simple model of Eq. (2.2) and assuming regular cylindrical unit cell (itself a questionable assumption if the drains are long and out of plumb). Since it cannot always be assumed that the pore-pressure monitoring device is located exactly midway between supposedly vertical drains, trying to assess

the degree of consolidation from measured excess pore pressure at depth may not be as straightforward as one would wish.


2.4.1 Program VDrainSt and Excel Spreadsheet Template

This section describes an approximate procedure for calculating the rate of consolidation settlement. The iterative computations are coded in the VBA programming environment of Microsoft Excel. The resulting VBA function is named VDrainSt, short for VDrain-Settlement. It takes into account: (1) staged loading; (2) load reduction due to fill submergence; (3) delayed vertical drain installation; (4) changes in length of vertical drainage path with time; and (5) variation of soil stress history with depth.

A practical algorithm for prediction of rate of settlement and design of vertical drains was adopted because, even in the relatively simple approach adopted here, 15 or more values of individual input parameters were required. The uncertainties associated with some of these parameters will limit the accuracy of prediction even if sophisticated models are used. The computational algorithm in VDrainSt is not fully rigorous because idealizations and approximations have been made. Nevertheless, limited comparisons (including analysis of the Chek Lap Kok test fill) made by the author suggest that the degree of accuracy achieved is adequate for most cases. Note that in any use of the program for actual design, it is the sole responsibility of the user to check whether the output is correct or is correctly interpreted or whether the problem is correctly modeled.

Program VDrainSt is a revised version of VDrain95 which was written by B.K. Low for the analyses of the Chek Lap Kok test embankments. The current version (VDrainSt) is coded in the Visual Basic for Applications (or VBA) programming language of the spreadsheet software Microsoft Excel. Values of 15 parameters are required, together with three sets of data defining the staged loading,

the initial effective vertical stress profile, and the maximum past pressure profile, as shown in Fig. 2.14. VDrainSt is meant for preliminary analysis and sensitivity studies, where one wants to assess the effect of varying one or more parameters on the rate of settlement, keeping all other values of parameters the same. Parameters required for analysis

Figure 2.14 shows example input data for analysis using program VDrainSt. Any system of units can be used, provided consistency is observed. This means that if length (e.g., mud thickness, elevation, drain spacing, drain diameter) is in meter and time in day, then the units for cv and ch have to be in m2/day. In Fig. 2.14 the units of the input data are meter, m2/day, day, kN/m3, kN/m2, as appropriate.

The function VDrainSt (codes shown in Fig. 2.17) needs only to be entered in the first cell (cell A25) of Fig. 2.14, below the time series, followed by autofilling to the right. To run a different problem, the user need only modify the input data of Fig. 2.14 to reflect the case in hand. Entering VDrainSt in cell A25 is most conveniently done in the Excel window shown in Fig. 2.15, invoked by Insert/Function/Category: User-defined/VDrainSt. Such windows are automatically created by Excel for any VBA functions created by the user. As several input data may need to be varied from one case to another, the syntax ‘= if($H$3, VDrainSt(…), 0)’ in cells A25:N25 of Fig. 2.14 allows user to suspend program calculation during input mode by entering ‘false’ in cell H3.

The first 15 parameters in the top half of Fig. 2.14 are mostly self-explanatory, and only brief comments will be made:

Cell H10: Elevation of sea level has to be equal to or higher than original mud-top elevation (Cell H9). For projects on land, the elevation of sea level can be set equal to the mud-top elevation.

Cell H12: Single drainage (j = 1) means that the top surface of the mud is open (i.e., a drainage boundary) and the bottom surface closed (i.e., impervious), or vice versa. Double drainage (j =0.5) means that both the top and bottom surfaces are open. Program VDrainSt computes

the length of drainage path in the vertical direction as equal to (j× thickness of mud). The value is needed in the computation of the time factor Tv for vertical drainage.

Cell H13: The original undisturbed value of the coefficient of consolidation for horizontal flow should be used. Smear effect is accounted for separately in the program code when smear diameter ratio and smear k ratio (cells H16 and H17) are both greater than 1.

Cell H14: The equivalent drain spacing is the diameter of an equivalent cylinder of soil around the vertical drain. It is equal to 1.05s for vertical drains on triangular grids, and 1.13s for vertical drains on square grids, where s is the actual spacing of the drains.

Cell H15: The equivalent drain diameter , for a band-shaped drain of width b and thickness t, can be taken to be

2(b + t)/π. It is equal to the actual diameter of the drain if the drain is round as in sand-drains.

Cell H16: The smear diameter ratio is ds/dw =(smear zone diameter/drain diameter).

Cell H17: The smear k ratio is kh/ks, that is, the ratio of the coefficients of permeability for horizontal flow in the undisturbed zone and in the smeared zone.

Cell H18: In Fig. 2.14 the time of V-Drain installation is 0 (day), whereas the time of the first increment of fill thickness (to 0.5 m) is –40(days). This means that the first increment of fill thickness was done 40days before the installation of the vertical drains.

Cell H19: Compression ratio of mud: This parameter is dimensionless and is equal to Cc/(1 + e), where Cc is the compression index and e the initial void ratio.

Cell H22: The buoyant unit weight of fill is equal to the difference between the saturated unit weight of the fill and the unit weight of water. This parameter is used to account for the reduction in fill load due both to initial submergence and to subsequent submergence induced by consolidation of the underlying soft mud. (For projects onshore where no submergence effect need be accounted for, the buoyant unit weight can be set equal to the total unit weight of fill above water table (cell H21)).

The next three sets of data in Fig. 2.14 define staged loading, initial effective overburden pressure profile, and maximum past pressure profile. Brief comments on the three sets of data are given next.

Data set #1: Step increments of fill thickness with time

The user specifies the staged loading information (up to a maximum of 12 points) to define the total fill thickness, versus time. Four points are shown in Fig. 2.14, each point being defined by two values representing the time and the total thickness, respectively. The fill thickness before the time defined by the first point is taken to be zero. For the duration between two successive points, the fill thickness is constant and equal to the thickness defined by the earlier point. Thus, for the four points given in data set #1, the

staged loading scenario is as depicted in the sketch at the bottom left of Fig. 2.14. Note that since unloading (with possible swelling) is not considered by program VDrainSt, data points depicting decreasing thickness with time are not acceptable.

Data set #2: Initial effective overburden stress of mud

The user can enter up to 12 points to define the variation of initial effective overburden pressure with depth. Each point consists of two values: the elevation and the stress. Linear variation of stress with depth is assumed between two adjacent points. Note that the stress profile for the entire original mud thickness must be defined. It very frequently happens that, during sensitivity studies, one changes the original mud thickness (cell H8) without remembering to redefine the stress profile corresponding to the changed thickness. The following formula in cell K3 safeguards this: ‘= IF(H8 = MAX(K8:K21)-MIN(K8:K21), IF(H9 = K8,TRUE,FALSE), FALSE)’, which checks to ensure that the first and last points in data set #2 have elevations equal to the original mud-top and mud-bottom elevations, respectively. If this condition is not satisfied, cell K3 will display ‘FALSE’.

Data set #3: Maximum past pressure profile of mud

The number of points (maximum 12) can be greater or smaller than that in data set #2. As for data set #2, linear variation of maximum past pressure with depth is assumed between two adjacent data points. The following formula is entered in cell N3: ‘= IF(H8 = MAX(N8:N21)-MIN(N8:N21), IF(H9 = N8,TRUE,FALSE), FALSE)’, which checks whether the variation of the maximum past pressure with depth has been defined for the entire original mud thickness.

Program VDrainSt divides the soft mud into 21 sublayers, and calculates for each sublayer both the effective overburden stress and the maximum past pressure. Such calculations are based on the information given in data sets #2 and #3.

Output of settlement with time needs to be set up only once using standard Excel charting techniques. The display will be updated in

subsequent parametric or case studies, in accordance with changes in cells A24:N25.

2.4.2 Example VDrainSt Output

In Fig. 2.14 the vertical drain is installed at time 0 (day). The time in row 24 means the number of days since the installation of the vertical drains. Had time zero been used to specify the time of the first increment of the fill thickness in data set #1 of Fig. 2.14, the time of the vertical drain installation would be 40days (cell H18); the time values in row 24 would then denote the number of days since the time of the first increment of the fill thickness.

The time values in row 24 are input by user. The settlement versus time curve will exhibit kinks at time values corresponding to increase in fill thickness. It is therefore desirable to specify these time values (50 and 260 days for the case in Fig. 2.14) and shortly after (55 and 265 days). The other time values can be arbitrarily spaced, or spaced at regular intervals, the guideline being to obtain a relatively smooth curve of settlement versus time plot.

The cv and ch values of 0.0055 and 0.0082 m2/day in Fig. 2.14 correspond to about 2.0and 3.0m2/year, respectively. The same template of Fig. 2.14 can be used to obtain the rate of settlement for the

case with no vertical drains. This is done by setting the value of ch to zero; the other vertical-drain related parameters (cells H14:H17) can be left at their typical values and will have no effect on computed results when ch is set to zero. Figure 2.16 compares the rate of settlement with and without vertical drains. Except for the values of ch, all other parameters are as shown in Fig. 2.14.

If the project is on land and has no submergence effect (e.g., a highway embankment constructed on soft clay), it is only necessary to input the buoyant unit weight of fill as equal to the unit weight of fill above water level.

2.4.3 Computation Procedure in Program VDrainSt

The computation rationale in VDrainSt is pragmatic and aims at simplicity of model and algorithm. Approximations are involved. Nevertheless, the program is able to capture the salient features of the rate of consolidation settlement involving staged loading, load reduction due to fill submergence, delayed vertical drain installation, changes in length of vertical drainage path with time, and variation of soil stress history with depth.

Program VDrainSt combines vertical and horizontal (radial) drainage using Carillo's equation. The depth-dependent degree of consolidation due to vertical flow (Uz) and the equal-strain solution of the degree of consolidation due to radial flow (Ūh)are used. VDrainSt requires the user to input the thickness of fill versus time. For each incremental thickness, the degrees of consolidation for vertical and radial drainage, Uz and Ūh respectively, are computed from Eqs. (2.7) and (2.16), and then combined to obtain the average degree of consolidation Uvh at the particular depth z. In computing Ūh, the time t in the term Tv of Eq. (2.7) is equal to the duration between the current time and the time the particular thickness increment took place. In computing h, the time t in the term Th of Eq. (2.16) is equal to the duration between the current time and either the time the particular thickness increment took place, or the time the vertical drains were installed, whichever gives the smaller duration. Also, in computing Tv, the layer thickness of the soft mud is taken to be the average of the thickness at the beginning of the

duration t (the time the fill thickness increment took place) and the thickness at the end of the duration t (the current time).

One value of the average degree of consolidation Uvh will be computed for each fill thickness increment that has already taken place. Each value of Uvh is multiplied by the incremental fill load (equal to incremental fill thickness × fill unit weight) to obtain the increase

in effective vertical stress in the clay due to the particular fill increment. The total increase in effective vertical stress (up to the time in question) due to all the fill increments which have taken place is obtained by superposition:

where the summation index i goes from 1 to the number of fill increments up to the time in question.

To account for the variation of the stress history (overburden effective vertical stress and maximum past pressure) of the soft mud with depth, program VDrainSt divides the soft mud into 21 sublayers of equal thickness. For each sub-layer the overburden effective vertical stress and the maximum past pressure are interpolated from the soil stress history profile provided by the user. The number of points defining variation of overburden effective vertical stress and maximum past pressure with depth can be as many as justified by the certainty of knowledge concerning them. The induced consolidation settlement will differ from one sublayer to the next if the overburden effective vertical stress and the maximum past pressure vary from one sublayer to the next, which is typical. The consolidation settlement up to the time in question is obtained by superposing the compression of all the sublayers:

where subscript j refers to sublayer j of the soft mud, and Ho is the original mud thickness.

The reduction in fill load due to submergence can be accounted for by using buoyant unit weight for the submerged portion of the fill. The submerged thickness of the fill affects the magnitude of the soft mud settlement, which in turn determines the thickness of the fill which is submerged. Hence an iterative procedure is used by program VDrainSt. Settlement is first calculated based on current known thicknesses of fill above and below water. The settlement calculation is then repeated, using updated thicknesses of fill above and below water in the light of the previously calculated settlement. The iteration in settlement calculation continues until the difference between two successive calculations is smaller than 0.005 of the value of settlement computed in the current iteration.

As mentioned earlier, if there is no submergence effect, the buoyant unit weight of fill (cell H22 in Fig. 2.14) should be set equal to the total unit weight of the fill above water level.

2.4.4 Limitations of Program VDrainSt

Program VDrainSt is meant for analysis of a single layer where the values of the coefficients of consolidation (in the vertical and radial directions) and the compression ratio are assumed to be independent of the magnitude of the stress increment. Unloading is not considered. Also, the coefficients of consolidation in the recompression range are assumed to be the same as those in the virgin compression range. For the soft mud in the Chek Lap Kok project, where the difference between maximum past pressure and overburden effective vertical stress is small compared to the magnitude of the applied stress, little error is introduced by the assumption. For other situations, the higher values of the coefficients of consolidation in re-compression than in virgin compression may need to be explicitly accounted for if the prediction of settlement rate is to be realistic.


2.5.1 The Vertical Band Drain Quadrants of the Chek Lap Kok Test Embankment

The Chek Lap Kok test reclamation fill on marine clay installed with vertical drains was reported in Foott et al. (1987), and studied

in Choa et al. (1990) and Low et al. (1995). In this section the measured settlement record of the marine clay are compared with computed results from Program VDrainSt.

The following values of coefficients of consolidation are used in VDrainSt computations:

Undisturbed cv = 1 m2/yearBased on studies of Chek Lap Kok data.
Undisturbed ch = 3.7 m2/yearASCE JGT 2/1987, p. 93, Fig. 2 piezocone dissipation.
Remolded ch = 0.8 m2/yearBased on studies of Chek Lap Kok data.

The original cv and ch values of 1m2/year and 3.7m2/year are divided by 365 days to obtain 0.00274 and 0.0101 m2/day, respectively, for use with program VDrainSt. The effect of smear is accounted for using smear diameter ratio s = 2 and smear permeability ratio (kh/ks)= 3.7/0.8=4.63 as part of the input for VDrainSt. Basis for the values of other parameters used in VDrainSt computations

Based on Choa et al. (1990)'s studies of the available soft clay data, a compression ratio CR of 0.36 will be used. The recompression ratio CRR is about 5–10% of the compression ratio. For analysis 7% is assumed. Note that for the soft mud layer the recompression settlement is negligible when compared to the settlement due to virgin compression.

As mentioned earlier, a cv value of 1 m2/year is used in computation. Although there is evidence that in the recompression range cv could be much higher, the assumption of the same value of cv for virgin compression as well as for recompression will introduce little error because recompression settlement is negligible, and also the difference between overburden effective vertical stress and maximum past pressure is small compared with the applied stress.

Another factor to be resolved is the length of the drainage path in the vertical direction. For double drainage condition (top and bottom surface of soft clay open), it is equal to half the layer thickness,

corresponding to j =0.5 in program VDrainSt. For single drainage condition (top surface open but bottom surface closed), it is equal to the entire soft clay thickness, or j = 1 in program VDrainSt. For analysis using VDrainSt, it is not certain which condition prevails, hence both conditions will be analyzed. Fortunately, in the presence of simultaneous radial drainage to the vertical drains, the rate of consolidation settlement is not very sensitive to the values of the length of drainage path in the vertical direction or of the coefficient of consolidation (cv) in the vertical direction.

The above paragraphs have provided the basis for the values of the main input parameters for analysis using program VDrainSt. Other necessary information such as thickness of soft mud and variation of fill thickness with time are based on Hong Kong's Geotechnical Control Office GCO.4/90Drawings #1-10, made available to Choa et al. in their 1990studies. The soil stress profile (initial effective stress and maximum past pressure) is based on Choa et al. (1990)'s studies of available data. The measured range of settlement for each test embankment quadrant is based on GCO.4/90Drawing #1.

2.5.2 Comparing VDrainSt Computations with Measured Settlements

Results from analyses using program VDrainSt are plotted in Figs. 2.18 and 2.19, corresponding to the 1.5 m (spacing) band-drain and 3 m band-drain quadrants, respectively. The lists of input values for each quadrant are also shown in the respective figures. For comparison, the measured range of settlements are also plotted. Note that there are two computed curves for each quadrant: the upper (slower) curve (with ‘o’ markers) represents single drainage condition (j = 1), while the lower (faster) curve (with ‘*’markers) represents double drainage condition (j =0.5).

The following may be observed in 2.18:

  1. Whether single drainage or double drainage is assumed has negligible influence on the computed rate of settlement. This is because for the 1.5 m band-drain spacing, consolidation due to radial drainage far outweighs that due to vertical drainage.

  2. The shape of the computed curves is similar to the shape of the measured range, both reflecting the staged loading. The computed rate of settlement is only slightly slower than the measured rate.
  3. The computed final consolidation settlement (2.7 m) is slightly smaller than the average of the measured range of settlement, by about 0.2 m.

Similar observations may be made for Fig. 2.19, except that the difference between single drainage condition and double drainage condition is somewhat larger than in Fig. 2.18. This is due to the increasing role of drainage in the vertical direction in this quadrant where the band-drains are spaced further apart (3 m) than in Fig. 2.18.

The discrepancies in the computed and measured magnitude of final settlement for the band-drain quadrants may be attributed to undrained immediate settlement and secondary compression settlement, both of which are not accounted for in program VDrainSt, and to uncertainties (probably small) associated with the instrumented results. As for the rate of settlement, comparison of the computed and measured curves seems to support double drainage conditions (i.e., j = 0.5) for the control quadrant and the 3 m band-drain quadrant. The computations also indicate the importance of taking the smear effect into account.

In both Figs. 2.18 and 2.19, the discrepancies between the computed curves and the measured ranges are partly attributable to the underestimation of final consolidation settlement by VDrainSt. The final consolidation settlements are functions of CR, CRR,stress history, and applied loadings, but are not affected by the values of the rate parameters cv and ch, nor by the idealizations and approximations related to excess pore pressure dissipation.

The VDrainSt results will be even closer to the observed settlement curves if different values of ch and compression index are used, but such speculations belong to the realm of back analysis and hindsight and will not be pursued here. Instead, something will be said about sensitivity.

2.5.3 Sensitivity Studies

The accuracy of design and prediction is often limited not so much by the lack of rigor in the analytical model as by the uncertainties associated with the input parameters used in the analysis. This is particularly so when the number of parameters is large, each with its flown degree of uncertainty. In these circumstances, parametric and sensitivity studies are useful for identifying situations in which uncertainty in a parameter will have significant influence on the calculated results and situations in which the uncertainty is of little consequence. For instance, whether to use lab cv or field cv and what vertical drainage length to assume are certainly much more important questions for the control quadrant (without vertical drains) than for the 1.5 m band drain quadrant. In principle, it should be possible to incorporate the uncertainty of the various parameters into the computational procedure, so that relationships/charts/equations can be developed to give the likely range of the settlement rate, in addition to the rate based on mean parametric values.

Figure 2.18 shows that the rate of settlement is not sensitive to whether drainage occurs at both the top and bottom boundaries of the mud (i.e., i =0.5), or at the top boundary only (i.e., i =1), for the two sets of points are practically the same, reflecting the predominant role of radial drainage in the quadrant with band drains spaced at 1.5 m in triangular grids. The sensitivity of the settlement rate to cv can also be inferred from the same figure. One notes that the time factor for vertical flow depends on cv/(i × mud thickness)2; therefore if the values of cv and i change such that the ratio cv/i2 remains the same, the settlement rate will not change. This means the results for (i =1, cv =4 m2/year) will be identical to the (i = 0.5, cv =1 m2/year) curve (marked ‘*’) in Fig. 2.18; but the latter is very close to the (i =1, cv =1m2/year) points (marked ‘o’). It follows that (i =1, cv =4 m2/year) ≈ (i =1, cv =1m2/year). Hence VDrainSt settlement curve is not sensitive to the uncertainty associated with the value of cv in this quadrant, where the soft mud is about 7 m thick and the band drains are spaced at 1.5 m on triangular grids.

Similarly, if the values of ch, equivalent drain spacing de, and drain diameter dw all change but in such a way that the ratio remains constant, then the computed settlement versus time curves will be the same. These anticipations can be verified by program VDrainSt.

The computed settlement rate for the 3 m band drain quadrant, Fig. 2.19, is by comparison more sensitive to variations in the ratio cv/i2. This is readily comprehensible because the value for this quadrant is only 22% of the value for the 1.5 m band drain quadrant; the role of radial drainage is correspondingly less dominant in the 3 m band drain quadrant. (Both quadrants have about the same thickness of mud.)

It follows that the sensitivities of the parameters which affect the rate of settlement (cv, ch, length of vertical drainage path Hd, equivalent drain spacing de, and drain diameter dw) can be studied via two terms: , and , where n = de/dw.

One can, for example, present a plot showing time to 90% (or some other %) of theoretical final consolidation settlement versus , for different values of . This chart can be used to assess the sensitivity of the time required for 90% consolidation to changes in any of the five parameters affecting the rate of settlement. The complete settlement versus time curves can also be plotted on the basis of lower bound and the upper bound values of and , in addition to those based on their most likely values.

Yet another option is to perform reliability analysis and reliability-based design accounting explicitly for the estimated uncertainties of the parameters. The classical method of computing the Hasofer-Lind reliability index is intricate when correlation, autocorrelation, and nonnormal distributions are involved and when the performance function is complicated or implicit. These hitherto tedious problems can be solved, with relative ease and transparency, using spreadsheet-automated constrained optimization and the expanding ellipsoid perspective (Low, 1996; Low and Tang, 1997). By this perspective, the quadratic form defining the Hasofer-Lind index is visualized as a tilted multidimensional ellipsoid (centered at the mean m or equivalent mean mN) in the original space of the

random variables; there is no need to diagonalize the covariance or correlation matrix. The Rackwitz-Fiessler equivalent normal transformation was used when dealing with nonnormals (Low and Phoon, 2002). The concepts of coordinate transformation and frame-of-reference rotation were not required. Iterative searching and partial derivatives were automatic using constrained optimization in the ubiquitous spreadsheet platform. The versatility of the spreadsheet constrained optimization approach is enhanced when used in combination with user-defined functions coded in the programming environment of a spreadsheet, for example the Visual Basic (VBA) programming environment of the Microsoft Excel spreadsheet software. This means that the performance function can be implicit, iterative, and based on numerical methods (e.g., Low and Tang, 1997; Low et al. 1998; Low, 2001). Another probabilistic approach was to use the response surface method, as described, for example, in Li (2000), which studied the Changi land reclamation settlement and pore pressure data and conducted both deterministic and probabilistic analysis (including Monte Carlo simulations) involving the use of vertical drains in land reclamation.


The general considerations summarized in this section are gleaned from a survey of other researchers' and practitioners' published works as listed in the References and Bibliography section.

Various researchers and practitioners have suggested that, for proper functioning of prefabricated vertical drains, the filter sleeve surrounding the plastic core of prefabricated band drains should satisfy several criteria: (a) Its permeability should be greater than that of the surrounding soil; (b) Fine soil particles should be retained to prevent clogging of flow channels in the drain core; (c) The filter sleeve should be stiff enough so as not to be pressed into the core channels of the vertical band drains by high lateral soil pressures; (d) The filter sleeve should be strong enough so as not to be damaged during installation; (e) The filter should not undergo physical,

chemical, or biological deterioration during the intended life of the vertical drain.

It is also generally recognized that the geotechnical information needed to design vertical drain systems is likely to include the following:

  • Stress history of the deposit – preconsolidation stress or overconsolidation ratio.
  • Coefficients of consolidation for horizontal flow (ch)and vertical flow (cv)
  • Extent of the smeared zone, ds, and other drain installation effects.
  • Coefficient of permeability of the remolded clay within the smeared zone, ks.
  • Undrained shear strength, cu, and undrained Young's modulus, Eu.
  • Secondary compression index or secondary compression ratio.
  • Discharge capacity of the drain, qw, and its variation with total lateral stress and time.

A typical testing program for projects involving precompression and the use of vertical drains may aim at obtaining information on the type of soil and their index properties; the in-situ shear strength of the soil (for short-term stability evaluation); parameters determining the magnitude of settlement (Cc, eo,and at various depths, and Cα in the case of secondary compression); parameters for estimating shear strength gain with increase in effective consolidation pressure; parameters controlling the rate of settlement, namely the coefficients of consolidation due to vertical flow and horizontal flow, cv and ch respectively.

The greatest uncertainty and difficulty are in assessing the coefficients of consolidation ch and cv and the extent and properties of the smear zone.

All the data except ch can be obtained by routine tests either in the lab or the field.

Gambin (1987) suggested the following methods for ch:

  1. Measure cv in an oedometer test and from the geological data assume a possible ratio for ch/cv (generally between 1 and 10).
  2. Measure ch in a large Rowe cell with a central vertical drain. A major difficulty in laboratory testing for determination of ch is that very large samples are required in many cases if results representative of the true in-situ stratification are to be obtained.
  3. Measure cv and kv in an oedometer tests and measure kh in-situ, then derive ch from the equation ch/cv = kh/kv.
  4. Back analysis from settlement observations.

With respect to back analysis from measured settlements, the following have been noted in the literature:

  1. Estimation of final consolidation settlement from laboratory tests has an accuracy of ±20% at best. A more common approach is to extrapolate the observed field settlement curves to obtain a probable value of the final consolidation settlement, for example, using Asaoka's procedure or the hyperbolic settlement model.
  2. It is necessary to deduct from the observed ground surface settlement the settlements resulting from the compression of the layers above and below the one in which consolidation is accelerated by vertical drains. This requires reliable settlement measurements at different depths.
  3. The vertical displacement from the immediate settlement, si, should be subtracted from the settlement measured at a given time, t. It is difficult to estimate si because of the difficulty of estimating the undrained modulus Eu and Poisson's ratio ν reliably, and because of partial drainage occurring during construction. If granular layers are included in the soil profile, their contribution to si also has to be estimated. It may also be necessary to consider the effect on vertical settlement of large horizontal movements which occur because of overstressed zones near the edges of the embankment.
  4. With overconsolidated soil, it is advisable to subtract the re-compression settlement (up to the in-situ value of ) from the observed settlement.

Field instrumentation may include the following (more details in other chapters):

  • Open stand-pipes to record the actual level of the water table at and adjacent to the site.
  • Settlement gauges at the original ground surface or buried in the drainage blanket.
  • Settlement gauges at depth, located in each different layer of soil.
  • Piezometers, close circuit type or with a gas back-pressure circuit.
  • Load pressure cells to verify the actual pressure applied by the fill on the natural soil.
  • Alignment markers to assess horizontal movements of the surrounding ground surface.
  • Inclinometers to assess horizontal movements of the compressible soil at depth.

Many case histories indicate that excess pore pressure may seem to remain although settlement has come to a complete stop. Hence the most feasible way of checking the rate of consolidation is to measure the rate of settlement and, in addition, the increase in undrained shear strength and preconsolidation pressure. Pore pressure observation can complement these control methods.

2.6.1 Some Practical Considerations

If the compressible layer thickness H is large relative to the width of the loaded area B, then there will be an immediate settlement si. Its magnitude can be estimated using the elastic theory. The consolidation settlement in these cases may be smaller than that based on one-dimensional procedure.

Lateral drainage becomes important in accelerating the rate of consolidation when the value of H/B is large. The rate of settlement will be even greater if the horizontal permeability, and therefore ch, is greater than the vertical permeability and cv.

Sand drains, if used, should be installed carefully, preferably in pre-bored holes or by careful jetting. If large-diameter drains (> 0.3 m) are installed by driving or careless jetting, the disturbance

effects may be very pronounced, which can reduce the efficiency of the drains. In one case, driven displacement-type sand drains (0.5 m in diameter) had no effect on the consolidation process, probably due to necking. Displacement sand drains should be avoided when:

  • The clay sensitivity exceeds 4–6.
  • The soil deposit has a highly developed macrofabric, associated with high anisotropy of the permeability coefficient.
  • Stability is critical for design.
  • The load transmitted to the foundation is small and the soil is overconsolidated. (This applies to all types of vertical drains.)

2.6.2 Well Resistance

The following established equation can be used to account for the effect of well resistance:

where l=longest drainage path along vertical drain: half the drain length for ‘open-end’ drains, or entire length for ‘closed-end’ drains; qw=well discharge capacity=Awkw; Aw=cross-sectional area of the drain; kh= coefficient of permeability for horizontal flow; z=depth.

In the presence of well resistance, the degree of consolidation is no longer constant with depth, and consolidation is retarded in the lower part of the drain well. The influence of finite drain permeability on the consolidation rate increases as the drain length increases. Typical values of the well discharge capacity of many prefabricated drains are around 100 to 500 m3/year. From the theoretical point of view, the well resistance does not significantly affect the design unless the drains are very long and under high lateral stresses. In practice, well resistance can develop and increase for three reasons:

  • The deterioration of the drain filter may lead to a significant reduction of the cross-section.

  • Fine soil particles may pass through the filter and decrease the area available for flow (siltation).
  • Folding of the drain because of large settlements may result in a reduced discharge capacity.

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Theories, Computations, and Design Procedures Involving Vertical Drains

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