1. “Consolidation Settlement with Sand Drains – Analytical
and Numerical Approaches”
Department of Civil Engineering, IIT Kanpur
CE 632
By –
Kundan Tripathi (10327365)
Rajeev Verma (10572)
Saurav Shekhar (10660)
Shashank Kumar (10327670)
Umed Paliwal (10327774)
Dated: 5th
April, 2014

2. Abstract & Objective
Sand drains are often used in important construction projects in order to accelerate the process of
consolidation settlement for the construction of some structures. Sand drains increase the rate of
consolidation such that the settlement that would occur in years can be hastened to occur in
months. When a surcharge is applied at ground surface, the pore water pressure in the clay will
increase, and there will be drainage in the vertical and horizontal directions. The horizontal
drainage is induced by the sand drains. Hence the process of dissipation of excess pore water
pressure created by the loading (and hence the settlement) is accelerated.
The objectives of this study are two-fold. Analytical and numerical approaches have been studied
herein. The analytical part includes a review of the existing literature and presents useful extracts
in regards to settlement, structure, installation and monitoring of sand drains. Popular subjects
such as free strain and equal strain cases with and without smear have been glanced at. The
numerical part is the result of finite element analysis of a drain unit cell using Plaxis 2d version
8.2. It addresses –
1. Reduction in time of consolidation by use of sand drains and also the changes in this
reduction as the loading is changed.
2. Relationship between ultimate settlement and loading.
3. Relationship between ultimate settlement and drain diameter.
*****

3. Table of Contents
S.No Topic Page No.
1. Part 1 - Analytical Approach 4
1.1 Popular Theory 4
1.2 Recent Research 9
2. Part 2 – Numerical Approach 12
2.1 Objective 12
2.2 General Settings 12
2.3 Soil Properties 12
2.4 Boundary Conditions 15
2.5 Initial Conditions 15
2.6 Calculations 15
2.7 Observations 15
2.8 Results & Discussion 17
2.9 Conclusions 23
3. References 24
*****

4. Part 1. Settlement of Foundations built on Sand
Drains - Analytical Approach"
The consolidation settlement of soft clay subsoil creates a lot of problems in foundation and
infrastructure engineering. Because of the very low clay permeability, the primary consolidation
takes a long time to complete. To shorten this consolidation time, sand drains can be used. Sand
drains are constructed by driving down casings or hollow mandrels into the soil. The holes are
then filled with sand, after which the casings are pulled out. When a surcharge is applied at
ground surface, the pore water pressure in the clay will increase, and there will be drainage in the
vertical and horizontal directions. Hence the process of dissipation of excess pore water pressure
created by the loading (and hence the settlement) is accelerated. The basic theory of sand drains
was presented by Rendulic (1935) and Barron (1948) and later summarized by Richart (1959)
and in the development of these theories; it is assumed that drainage takes place only in the
radial direction, i.e., no dissipation of excess pore water pressure in the vertical direction. In the
study of sand drains, two fundamental cases:
1. Free-strain case- When the surcharge applied at the ground surface is of a flexible nature, there
will be equal distribution of surface load. This will result in an uneven settlement at the surface.
2. Equal-strain case.-When the surcharge applied at the ground surface is rigid, the surface
settlement will be the same all over. However, this will result in an unequal distribution of stress.
1.1 Adapted from advanced soil mechanics from B.M Das
1.1.1Free-strain consolidation with no smear
Figure 6.37bshows the general pattern of the layout of sand drains. For triangular spacing of the
sand drains, the zone of influence of each drain is hexagonal in plan. This hexagon can be
approximated as an equivalent circle of diameter de. Other notations used in this section are as
follows:
1. re = radius of the equivalent circle = de/2.
2. rw= radius of the sand drain well.
3. rs= radial distance from the centerline of the drain well to the farthest point of the smear zone.
Note that, in the no-smear case, rw= rs.
The basic differential equation of Terzaghi’s consolidation theory for flow in the vertical
direction is given in Eq. (1).

5. ……………….(1)
For radial drainage, this equation can be written as
Where
u=excess pore water pressure
r=radial distance measured from center of drain well
Cvr =coefficient of consolidation in radial direction

6. For solution of above Eq., the following boundary conditions are used:
1. At time t=0, u=ui
2. At time t>0, u=0 at r=rw.
3. At r=re, du=dr.
With the above boundary conditions, above Eq. yields the solution for excess pore water pressure
at any time t and radial distance r:
where
J0=Bessel function of first kind of zero order
J1=Bessel function of first kind of first order
Y0=Bessel function of second kind of zero order
Y1=Bessel function of second kind of first order
Where kh is the coefficient of permeability in the horizontal direction. The average pore water
pressure uav throughout the soil mass may now be obtained from as

7. 1.1.2 Equal-strain consolidation with no smear
The problem of equal-strain consolidation with no smear (rw=rs) was solved by Barron (1948).
The excess pore water pressure at any time t and radial distance r is given by
Uav=average value of pore water pressure throughout clay layer.
The average degree of consolidation due to radial drainage is
For re/rw>5 the free-strain and equal-strain solutions give approximately the same results for the
average degree of consolidation.
Olson (1977) gave a solution for the average degree of consolidation Ur for time-dependent
loading (ramp load) similar to that for vertical drainage. The surcharge increases from zero at
time t=0 and q at time t=tc. For t ≥ tc, the surcharge is equal to q. For this case

8. 1.1.3 Effect of smear zone on radial consolidation
Barron (1948) also extended the analysis of equal-strain consolidation by sand drains to account
for the smear zone. The analysis is based on the assumption that the clay in the smear zone will
have one boundary with zero excess pore water pressure and the other boundary with an excess
pore water pressure that will be time dependent. Based on this assumption.

9. 1.2 Recent Research
Recently several analytical and experimental studies have reported on sand drain consolidation
of clayey soils, some of them are listed below:
1.2.1 TOYOAKI NAGOMI, AND MAOXIN LI, (2003) CONSOLIDATION OF CLAY
WITH A SYSTEM OF VERTICAL AND HORIZONTAL DRAINS -
Consolidation behavior with the drain system is formulated using the transfer matrix method.
Special care is given to formulation of thin pervious layers for efficient computation. The
developed formulation is verified using available numerical and field information. Parametric
studies are conducted to study the consolidation characteristics of clay with the drain system.
Based on the findings, a design method for an optimum system of horizontal and vertical drains
is proposed and design charts are presented for such a design. The consolidation behavior of clay
with a system of horizontal drains and vertical cylindrical drains is formulated using the transfer
matrix approach. The developed formulation can handle the inhomogeneous profile in clay and
multiple horizontal drains made of either thin sand layers or geotexstile sheets. The number of
terms used in series is five terms in the r direction, and five to ten terms in the z direction
depending on the behavior in the series expression. As Terzaghi’s consolidation solution, only
one or two terms in the expansion in the z direction are sufficient to compute the consolidation
behavior in the later stage of consolidation but the upper-side number of terms is required in the
early stage of consolidation. The formulation is found to be very efficient and convenient for
computation.
1.2.2. K.R.LEKHA, N.R.KRISHNASWAMY, AND P.BASAK, (1998) CONSOLIDATION
OF CLAY BY SAND DRAIN UNDERTIME-DEPENDENT LOADING -
The literature contains a nonlinear theory of sand drain consolidation under time-dependent
loading that can take into account any effective stress/void ratio/permeability variations. A
generalized governing equation, capable of yielding a large class of analytical solutions for these
variations is derived in this paper. Closed-form solutions are presented for the variation of pore
water pressure with a time factor and load increment ratio under time-dependent loading. The
analytical formulation is validated by comparing the solution with the standard results available
in the literature for instantaneous loading, constant permeability, and constant compressibility.
Governing equation for equal strains and drain problems in time-dependent loading is given in its

10. most general form, which can conveniently account for any effective stress/void
ratio/permeability variation. This equation is linear and requires evaluation of only one integral
to yield the solution for a large class of problems in time-dependent loading with variable
permeability and compressibility. The theory is an extension of the solution by Basakand
Madhav(1970), for the case of instantaneous loading and variation of compressibility and
permeability. The analytical formulation is validated by comparing the solution with the standard
results available for instantaneous loading, constant permeability, and constant compressibility.
The results are presented for the variation of pore water pressure with a time factor and load
increment ratio.
1.2.3. IEW-ANN TAN, (1993) ULTIMATE SETTLEMENT BY HYPERBOLIC PLOT
FOR CLAYS WITH VERTICAL DRAINS
The rectangular hyperbola method (Tv/U versus Tv) is extended to the case of drains and
surcharge by considering the hyperbolic plots for combined vertical and radial flow
consolidation in clays of varying thickness and drain spacing ratio for typical soil
properties of Cv of 1-5 m2
/yr. The results indicate that the hyperbolic plots are linear
between U50% and U90%. For the lines radiating from the origin to U50% point , the slope is
(1/0.5 = 2.0), and to the U90% point, the slope is (1/0.9 = 1.11). Thus, the ratio of the
slopes of these radiating lines to the slope of the linear portion of the hyperbolic plots
identifies the U50% and U90% for any settlement record using drains and surcharge. It is found
that the estimate of ultimate settlement from the U50% and U90% is more accurate than the
conventional inverse slope .approach of the hyperbolic method, especially for data between
the 50% and 90% consolidation points. The hyperbolic method of settlement analysis can be
extended to the practical case of vertical drains and surcharge. The use of the inverse of
the slope of the first straight-line portion of the hyperbolic plot of settlement data tends
to overestimate the amount of ultimate primary compression. To some extent, this
overestimation compensates for the effects of secondary compression, but in field
applications the amount of secondary compression is uncertain. However, when the
hyperbolic method is used to obtain the 50% and 90% points of the settlement record,
the ultimate compression obtained from these points agrees reasonably well with long-term
compression data of the Skh-Edeby test fill. Therefore, this method can provide a useful
and practical check on the progress of consolidation in field applications using vertical
drains and surcharge, especially in the absence of reliable soil properties data.

11. 1.2.4. CHIN JIAN LEO, (2004) EQUAL STRAIN CONSOLIDATION BY VERTICAL
DRAINS-
Closed-form analytic solutions of equal strain consolidation by a vertical drain with smear and
well resistance have been developed in the present paper. Solutions in this paper, however, have
been derived for coupled radial and vertical drainage and covered a step-loading or a ramp-
loading situation. Comparisons made with the corresponding analytic solutions of Hansbo and
Barron showed that the differences between the solutions of the present paper and the solutions
of Hansbo and/or Barron are generally quite small. In keeping with Barron (1948), consolidation
is considered in the undisturbed soil mass only, not in the vertical drain or the smeared zone, and
only radial drainage is assumed in smeared zone.
1.2.5 BUDDHIMA INDRARATNA, ALA AlIJORANY ANS CHOLACHAT
RUJIKATKARNOM, ANALYTICAL AND NUMERICAL MODELLING OF
CONSOLIDATION BY VERTICAL DRAIN BENEATH A CIRCULAR EMBANKMENT
While analyzing the axisymmetric problems, it is tried that aspects of geometry, material
properties, and loading characteristics are either maintained as constants or represented by
continuous functions in the circumferential direction. In the case of radial consolidation beneath
a circular embankment by vertical drains i.e., circular oil tanks or silos, the discrete system of
vertical drains can be substituted by continuous concentric rings of equivalent drain walls. An
equivalent value for the coefficient of permeability of the soil is obtained by matching the degree
of consolidation of a unit cell model. A rigorous solution to the continuity equation of radial
drainage towards cylindrical drain walls is presented and verified by comparing its results with
the existing unit cell model. The proposed model is then adopted to analyze the consolidation
process by vertical drains at the Skå-Edeby circular test embankment. The calculated values of
settlement, lateral displacement, and excess pore-water pressure indicate good agreement with
the field measurements.
1.2.6. TUNG WEN SHU and HUI_JYE LU, (2013) CONSOLIDATION FOR RADIAL
DRAINAGE UNDER TIME-DEPENDENT LOADING
It represents the details of consolidation for radial drainage under linear time-dependent
loading with varying loading dependent coefficients of radial consolidation by using a visco
elastic approach. By extending Barron’s solution for radial consolidation of small strain
sustained constant load, the convolution integral with time as the variable was used to analyze
the consolidation under time-dependent loading. Four different loading rates were applied in the
consolidation tests on three types of remolded clay with various plasticity indices to study the
behavior of radial consolidation. The findings indicate that the predicted consolidation
settlements accounting for the loading rate-dependent Cr values more closely match the
experimental results than the predictions using an assumed constant Cr.

12. PART 2. Numerical Approach toward study of Sand
Drains
2.1 Objective:
1. Comparative analysis of consolidation times with and without sand drains.
2. Find variation of ultimate consolidation settlement with applied stress.
3. Find variation of consolidation time with sand drain diameter at constant applied stress.
The finite element software Plaxis 8.2 was used for the modeling of vertical sand drains through
a layer of saturated clay. The goal was to discover the qualitative relationship between the
ultimate consolidation settlement and time taken vs. the load applied and the diameter of sand
drains. The time required for consolidation with and without drains was compared. The geometry
of the unit cell modelled is given in figures 2.1. and 2.2.
2.2 General Settings:
The following general settings were used for the modeling-
Model: Axisymmetry
Elements: 15 noded
x-acceleration: 0
y-acceleration: 0
Earth gravity : 9.8 m/s^2
Length: metre
Force: kN
Time: day
2.3 Soil Properties:
Three material sets were created for this problem- Dense Sand, Stiff Clay and Soft Clay.
Sand Drain: The sand used for making the drain was accorded the following properties-
Material Set: Dense Sand
Unsaturated unit weight = 17 kN/m3

13. Saturated unit weight = 20 kN/m3
Permeability Kx = ky = 1 m/day
Cohesion = 1 kpa
Internal angle of friction = 35 degrees
Angle of dilatancy = 3 degrees
Young’s Modulus = 40000 Kpa
Poisson’s ratio = 0.30
Fig. 2.1 Sample geometry of unit
cell without sand drain. Plaxis
8.2
Height
of
the
unit
cell
=
15m
Clay
layer

14. Clay layer: Two different types of clay were studied-
Clay 1 material set: Stiff Clay
Unsaturated unit weight = 18 kN/m3
Saturated unit weight = 19 kN/m3
Permeability Kx = ky = 0.001 m/day
Cohesion = 50 kPa
Internal angle of friction = 0 degrees
Angle of dilatancy = 0 degrees
Young’s Modulus = 50000 Kpa
Fig. 2.2 Sample geometry of the unit cell with
sand drain. Plaxis 8.2
Height
of
clay
layer
=
15m
Clay
surrounding
the
sand
drain
in
a
unit
cell
Sand
Drain

15. Poisson’s ratio = 0.35
Clay 2 material set: Soft clay
Unsaturated unit weight = 15 kN/m3
Saturated unit weight = 17 kN/m3
Permeability Kx = ky = 0.01 m/day
Cohesion = 15 kpa
Internal angle of friction = 25 degrees
Angle of dilatancy = 0 degrees
Young’s Modulus = 10000 Kpa
Poisson’s ratio = 0.25
2.4 Boundary Conditions:
The two vertical sides of the unit cell and one bottom side were accorded the standard fixities
boundary condition (no displacements) and closed consolidation boundary condition. In short,
settlement and consolidation both were allowed only through the top surface.
2.5 Initial conditions:
The effective stresses and ground water pore pressure were generated in the standard way (K0
procedure and phreatic surface respectively).
2.6 Calculations:
Calculations were run till the excess pore pressure developed reached 1 kPa or below at all points
in the unit cell.
2.7 Observations:
The observations are summarized in the tables below-
2.7.1 U (settlement) vs. Δσ (Applied Stress)
Taking diameter of sand drain = 0.4 m

16. Table 2.4 Clay 1
Δσ (kPa) Total Consolidation
Settlement (m)
Total Time with Sand
drain, t1 (days)
Total Time without
Sand Drain, t2 (days)
100
200
300
400
500
600
700
800
900
1000
0.019
0.038
0.063
0.09
0.117
0.144
0.171
0.198
0.225
0.253
11.484
22.972
19.619
17.234
18.006
20.912
20.688
28.395
30.295
30.345
122.5
183.75
153.13
157.92
169.4
149.3
172.87
179.69
175.62
181.9
Table 2.5 Clay 2
Δσ (kPa) Total Consolidation
Settlement (m)
Total Time with Sand
drain, t1 (days)
Total Time without
Sand Drain, t2 (days)
100
200
300
400
500
600
700
800
900
1000
0.077
0.169
0.261
0.353
0.445
0.537
0.629
0.721
0.814
0.906
22.968
45.938
45.938
45.938
45.938
45.938
45.938
45.938
61.251
61.251
61.25
91.876
91.876
91.876
91.876
91.876
91.876
91.876
91.876
91.876
2.7.2 U (Settlement) vs. Diameter (of drain)
Keeping applied stress at 300 kPa
Table 2.6 Clay 1
d (m) Total Consolidation
Settlement (m)
Total Time (days)

17. 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.063
0.063
0.063
0.064
0.064
0.064
0.063
11.875
42.109
31.582
15.312
12.919
10.287
8.373
Table 2.7 Clay 2
d (m) Total Consolidation
Settlement (m)
Total Time (days)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.265
0.265
0.263
0.261
0.259
0.261
0.253
93.876
61.251
45.938
46.057
30.626
28.261
23.088
2.8 Results and Discussion:
2.8.1 Time vs. Time
From Tables 2.4 and 2.5, it is clear that sand drains effectively reduce the time taken for
consolidation of saturated clay for both stiff and soft clays. As expected, this reduction is much
more pronounced in case of stiff clays where the time taken reduces by about 6 to 11 times.
While in the case of soft clays, the reduction factor is 1.5 to 3 times.
It is noticeable that as the applied stress increases and the final settlement (U) and time taken (t1
and t2) increase with it, the reduction factor is seen to decrease. Sand drains become less and less
effective as the time of consolidation increases. All these results may be noticed in Tables 2.8
and 2.9.

18. Clay 1 Table 2.8
Time with sand drain, t1
(days)
Time without sand drain, t2
(days)
Reduction Ratio t2/t1
11.484
22.972
19.619
17.234
18.006
20.912
20.688
28.395
30.295
30.345
122.5
183.75
153.13
157.92
169.4
149.3
172.87
179.69
175.62
181.9
10.66701
7.998868
7.805189
9.163282
9.407975
7.139441
8.356052
6.328227
5.796996
5.994398
Clay 2 Table 2.9
Time with sand drain, t1
(days)
Time without sand drain, t2
(days)
Reduction Ratio t2/t1
22.968
45.938
45.938
45.938
45.938
45.938
45.938
45.938
61.251
61.251
61.25
91.876
91.876
91.876
91.876
91.876
91.876
91.876
91.876
91.876
2.666754
2
2
2
2
2
2
2
1.499992
1.499992
2.8.2 U vs Δσ
For both stiff and soft clays, settlement steadily increases with applied load. This is in
accordance with theory and intuition.

19. 0
0.05
0.1
0.15
0.2
0.25
0.3
100
200
300
400
500
600
700
800
900
1000
Total
SeDlement
(m)
Applied
Stress
(kPa)
Fig.
2.3
SeDlement
vs
Loading
for
Clay
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
100
200
300
400
500
600
700
800
900
1000
Total
SeDlement
(m)
Applied
Stress
(kPa)
Fig.
2.4
SeDlement
vs
Loading
for
Clay
2

20. 2.8.3 Settlement vs. Diameter of Sand Drain
As expected, the final settlement did not vary with the diameter of sand drain. And the time of
consolidation steadily decreases with increase in the diameter of drains. This is also in
accordance with theory.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
100
200
300
400
500
600
700
800
900
1000
RaOo
Applied
Stress
(kPa)
Fig.
2.5
RaOo
of
SeDlement
SoP
Clay
to
SOff
Clay

21. 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Total
SeDlement
(m)
Diameter
of
Sand
Drain
(m)
Fig.
2.6
SeDlement
vs.
Drain
Dia
for
Clay
1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Total
SeDlement
(m)
Diameter
of
Sand
Drain
(m)
Fig.
2.6
SeDlement
vs.
Drain
Dia
for
Clay
2

22. 0
10
20
30
40
50
60
70
80
90
100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ConsolidaOon
Time
(days)
Diameter
of
Sand
Drain
(m)
Fig.
2.7
Total
Time
vs.
Drain
Diameter
for
Clay
1
0
10
20
30
40
50
60
70
80
90
100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ConsolidaOon
Time
(days)
Diameter
of
Sand
Drain
(m)
Fig.
2.7
Total
Time
vs.
Drain
Diameter
for
Clay
2

23. 2.9 Conclusions
1. Sand drains effectively reduce the time taken for consolidation of saturated clay for both
stiff and soft clays.
2. This reduction is much more pronounced in case of stiff clays where the time taken
reduces by about 6 to 11 times.
3. Sand drains become less and less effective as the time of consolidation increases.
4. For both stiff and soft clays, settlement steadily increases with applied load.
5. The settlement of soft clay was found to be 3 to 5 times more than that for stiff clay.
6. The final settlement does not vary with the diameter of sand drain.
7. And the time of consolidation steadily decreases with increase in the diameter of drains.
*****

24. References:
1. Leo, C. (2004). ”Equal Strain Consolidation by Vertical Drains.” J. Geotech. Geoenviron.
Eng., 130(3), 316–327.
2. Xiao, D., Yang, H., and Xi, N. (2011) Effect of Smear on Radial Consolidation with
Vertical Drains. Geo-Frontiers 2011: pp. 4339-4348. doi: 10.1061/41165(397)444
3. Hsu, T. and Liu, H. (2013). ”Consolidation for Radial Drainage under Time-Dependent
Loading.” J. Geotech. Geoenviron. Eng., 139(12), 2096–2103.
4. Indraratna, B., Aljorany, A., and Rujikiatkamjorn, C. (2008). ”Analytical and Numerical
Modeling of Consolidation by Vertical Drain beneath a Circular Embankment.” Int. J.
Geomech., 8(3), 199–206.
5. Nogami, T. and Li, M. (2003). ”Consolidation of Clay with a System of Vertical and
Horizontal Drains.” J. Geotech. Geoenviron. Eng.,129(9), 838–848.
6. Lekha, K., Krishnaswamy, N., and Basak, P. (1998). ”Consolidation of Clay by Sand
Drain under Time-Dependent Loading.” J. Geotech. Geoenviron. Eng., 124(1), 91–94.
7. Tan, S. (1993). ”Ultimate Settlement by Hyperbolic Plot for Clays with Vertical
Drains.” J. Geotech. Engrg., 119(5), 950–956.
8. Das, B. M. (2008). “Advanced Soil Mechanics”, 3rd Ed., Taylor and Francis, London and
New York.