Directory UMM :Data Elmu:jurnal:A:Agricultural Water Management:Vol45.Issue2.Jul2000:

Agricultural Water Management 45 (2000) 169±184

An analytical solution for bi-level drainage design
in the presence of evapotranspiration
A. Upadhyayaa,*, H.S. Chauhanb
b

a
Water Technology Centre for Eastern Region, Bhubaneswar 751 023, India
G.B. Pant University of Agriculture and Technology, Pantnagar 263 145, India

Accepted 14 September 1999

Abstract
The linearized Boussinesq equation incorporating the effect of evapotranspiration with
appropriate initial and boundary conditions was solved analytically to predict a fall in the water
table in a bi-level drainage system. It was assumed that the evapotranspiration rate decreases
linearly with a reduction in the elevation of the water table above the drains. A special case of this
solution (i.e. with no evapotranspiration) was veri®ed by comparison with an independent and
accepted analytical method, and almost identical values of spatial and temporal distribution of
water table heights were obtained. The effects of various parameters like evapotranspiration rate

and depth-dependent reduction factor on spacing between bi-level and level drains were also
studied. Results suggested that inclusion of evapotranspiration in the analytical solution for bi-level
drainage design in arid and semi-arid regions is useful and makes a signi®cant difference in spacing.
We conclude that in a bi-level drainage system spacing between two drains can be increased by
9.61±13.75% for soils having a hydraulic conductivity of 3 m/day, if the contribution of
evapotranspiration at a rate of 8 mm/day in lowering the water table is taken into account. # 2000
Elsevier Science B.V. All rights reserved.
Keywords: Evapotranspiration; Subsurface drainage; Bi-level drainage; Boussinesq equation; Analytical
solution; Drain discharge

1. Introduction
One of the main constraints in the installation of subsurface drainage is high initial
investment. So any contribution in reducing the installation cost of drains, either in
*

Corresponding author. Tel.: ‡91-674-440016; fax: ‡91-674-441651.

0378-3774/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 3 7 7 4 ( 9 9 ) 0 0 0 7 2 - 4

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A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

materials used, method of installation or improvement of design of subsurface system,
would be economically advantageous. Installation of deep drains in arid regions generally
require the costly construction of a trench, whereas shallow drains don't require this
and are therefore more economical. De Boer and Chu (1975) suggested the use of a bilevel drainage system because, by having alternating deep and shallow drain lines,
excavation will be reduced substantially and, in arid and semi-arid regions, will prove
more economical compared to a level drainage system. They provided a theoretical
analysis for steady state design of subsurface drainage using Dupuit-Forchheimer
assumptions. They also developed an unsteady state theory using the Bouwer and Van
Schilfgaarde (1963) approach. Chu and DeBoer (1976) validated their theory using
laboratory experiments on a Hele-Shaw viscous flow model. They found a close
relationship in predicted fall of the maximum water table elevations with the
experimental results of the Hele-Shaw model. They also compared their theory with
field data of bi-level drainage experiments conducted at the James Valley Agricultural
Research and Extension Centre, Brookings, USA. They found good agreement between
predicted fall of maximum water tables and measured field values. Later, Sabti (1989)
obtained an analytical solution of linearized and a numerical solution of the nonlinear

unsteady state Boussinesq equation to describe spatial and temporal variation of water
tables between two drains in a bi-level subsurface drainage system. It was observed that
the numerical solution predicted higher values of water table elevations than the
analytical solution in the region of the deep drain, whereas heights of the water table
predicted analytically were higher than the numerical solution, near the shallow drain.
The numerical solution yielded higher values of water table elevations at the mid-point
between the drains, but only at later stages of the drainage cycle. The difference in the
values of water table elevations obtained from these two methods was a result of
linearization (approximating the average depth of flow). Sabti (1989) also concluded that
the difference between the two methods increased with an increase in the region of flow
or in the average depth of flow. Verma et al. (1998) also obtained an analytical solution to
the linearized Boussinesq boundary value problem involving Laplace transformation.
They validated their method with the existing solution and field data reported by Chu and
DeBoer (1976).
Reduction in the cost of installation of subsurface drains using a bi-level drainage
approach has been suggested by all the researchers mentioned above. Another way to
reduce the installation cost is to consider the effect of evapotranspiration (ET) from the
fields. ET plays a significant role in arid and semi-arid regions not only in the decline of
water tables, but also in the rise of salts. Consideration of ET would increase the spacing
and thus reduce drainage costs. With this in mind, a number of empirical relationships

describing variation of ET with depth to water table have been proposed by Averianov
(1956), Gardner and Fireman (1958), Singh et al. (1996), and Sharma and Prihar (1973).
A commonly recognized simplified model proposed by Grismer and Gates (1988) which
provides a linear relationship between ET from cropped lands and depth to water table
may be expressed as
Ed ˆ E0 ÿ b…h0 ÿ h†;

0 < …h0 ÿ h† < dc ;

(1)

where Ed is rate of ET [LTÿ1] at any depth d in the soil, E0 the rate of ET [LTÿ1] at the

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

171

land surface, b the depth-dependent reduction factor [Tÿ1], h0 the initially ¯at water table
height [L] near the soil surface, and h is the water table height [L] as a function of space
and time. At the Sampla drainage site it was observed that after 1.2 m depth (dc, the

threshold value of depth) in the soil pro®le the effect of ET becomes zero. Since a deep
drain is assumed to be located at 1.8 m depth and Ed becomes zero at 1.20 m below the
soil surface or 0.60 m above the deep drain level, for E0 ˆ 0.008 m/day Eq. (1) yields a
value of b of 0.00667 per day.
In prolonged wet periods with negligible ET, the values of E0 and b used in the
proposed solution may result in an under-designed drainage system. In a hydrological
situation where mixed conditions of aridity and humidity occur at different times of year,
commonsense will have to be used to select the appropriate values of E0 and b so that the
drainage system can take care of the cropping system prevalent in that region.
A number of investigators, such as Skaggs (1975), Pandey and Gupta (1990), Nikam
et al. (1992), and Singh et al. (1996), have studied the effect of ET on water table
fluctuation in the conventional level subsurface drainage system. All except Skaggs
(1975), who assumed a constant rate of ET, have used Eq. (1) to study the effect of ET
on lowering of the water table between two drains located at the same elevation. They
have also shown that consideration of constant or depth-dependent ET leads to an
increase in drain spacing, thus providing more economic drainage design. No methods
which incorporate the effect of ET to predict transient fall of water table for design
of drain spacing in a bi-level drainage system are presently available. The objective
of the present paper is to obtain an analytical solution of the linearized Boussinesq
equation which includes a term of depth-dependent ET. The special case of the

proposed solution is verified with data from Verma et al. (1998). In consideration
of zero ET, constant ET and depth-dependent ET, the variation in spacing, fall of
water table between bi-level drains, and decrease in discharge of drains have all
been discussed to demonstrate the importance of incorporating ET in bi-level drainage
design.

2. Theory
2.1. Problem de®nition
An illustration of the bi-level and conventional level drainage problem is shown in
Fig. 1. Initially, the horizontal water table is assumed to be close to the land surface
at an elevation of h0 above the deep drain. The shallow drain is located at an elevation of
h1 above the deep drain. The impervious layer is at a distance d1 below the deep drain.
The two drains are separated by a distance L. The aquifer is assumed to be homogeneous
and isotropic resting on a horizontal impermeable barrier. Considering the deep drain
as the origin, the coordinate axis are taken parallel and perpendicular to the impervious
layer.
The one-dimensional Boussinesq equation, having a sink term for ET along with
appropriate initial and boundary conditions, describes the above physical problem in
mathematical terms. The boundary value problem in analyzing the fall of water table in a

172

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

Fig. 1. Graphic illustration of a bi-level and a conventional level drainage problem.

bi-level drainage system is given as

 

@
@h
E0 ÿ b…h0 ÿ h†
f @h
…d1 ‡ h†
;
ÿ
ˆ
@X
@X

K
K @t
h…X; 0† ˆ h0 ;
h…0; t† ˆ 0;
h…L; t† ˆ h1 ;
@h…L; t†
ˆ 0;
@X

0 < X < L;

(2)
(3a)

t > 0;

(3b)

0 < t  t0 ;

(3c)

t > t0 :

(3d)

Here h is height of water table at any time t above the deep drain, K the average hydraulic
conductivity [LTÿ1], and f is the drainable porosity of aquifer (dimensionless); t0 is the
time when slope of water table at X ˆ L becomes zero. The boundary condition (3d)
shows that for t > t0 the discharge in the shallow drain becomes zero and the water table
will fall below the shallow drain. For time t > t0, the shallow drain becomes
nonoperational. In a bi-level drainage system, the time in which the shallow drain is

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

173

operational is important, therefore the following solution of this boundary value problem
is valid only for t  t0.
Since Eq. (2) is a nonlinear partial differential equation which cannot be solved

analytically without linearization, it is linearized by neglecting (qh/qX)2 and replacing
(d1 ‡ h) with a constant D, average depth of flow, reducing Eq. (2) to the following
form:


@2h
E0 ÿ b…h0 ÿ h†
f @h
:
(4)
ˆ
ÿ
2
@X
KD
KD @t
According to van Schilfgaarde (1974) this method of linearization can be assumed if
variation in h is very small compared to that in qh/qX, i.e. change in depth of the
unconfined aquifer is small compared to the average depth. This restricts the application
of the linearized equation to cases where Dh  KD and also KD  L; these conditions

are also needed to satisfy Dupuit-Forchheimer assumptions. Here Dh is the change in
water table height with time and space, and L is the spacing between two drains in the
flow region. D, the average depth of flow, used in Eq. (4) is equal to the sum of depth of
the impermeable barrier below the drain and one half the initial water table height above
the drains. Taking into account the convergence of flow lines near the drain pipes, the
depth of the impermeable barrier below the deep drain is replaced by an equivalent depth,
de using the relationship of Hooghoudt (1940), such that D ˆ de ‡ h0/2 as considered by
Verma et al. (1998).
2.2. Analytical solution
To obtain an analytical solution of Eq. (4) with initial and boundary conditions given
by Eqs. (3a) ± (3c) a suitable transformation was devised. Eq. (4) is then transformed to a
heat transfer boundary value problem for which analytical solutions are available. The
transformation is:
h ˆ Veÿ…bt=f † ÿ

E0
‡ h0 :
b

(5)

Applying this transformation in Eq. (4) and Eqs. (3a) ± (3c) one gets the following
boundary value problem:
@ 2 V 1 @V
;
ˆ
@X 2 a @t
E0
ˆ f …X†; 0 < X < L;
V…X; 0† ˆ
b


E0
V…0; t† ˆ
ÿ h0 ebt=f ˆ f1 …t†; t > 0;
b


E0
‡ h1 ÿ h0 ebt=f ˆ f2 …t†; 0 < t < t0 :
V…L; t† ˆ
b

(6)
(7a)
(7b)
(7c)

Here a ˆ KD/f. Ozisik (1980) obtained the generalized solution to such a boundary value

174

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

problem using Duhamal's theorem. His solution is:


Z L
1
2
2X
X
X
0
0
0
f …X †sinbn X dX ‡ 1 ÿ
eÿabn t sinbn X
f1 …t† ‡ f2 …t†
V … X; t† ˆ
L nˆ1
L
L
0


Z
1
t
2
2
2 X sinbn X
f1 …0†eÿabn t ‡
eÿabn …tÿt† df1 …t†
ÿ
L nˆ1 bn
0


Z t
1
2X
n sinbn X
ÿab2n …tÿt†
ÿab2n t
e
df2 …t† :
…ÿ1†
f2 …0†e
‡
‡
L nˆ1
bn
0

(8)

Here bn ˆ np/L.
Substituting the expressions for f(X), f1(t), f2(t), f1(0), f2(0), df1(t), and df2(t) in Eq. (8)
and after some simplification one gets the following expression for V(X, t):
"
!#
1
bt=f
ÿab2n t
2
2X
1
1
e
ÿ
e
sinbn X …ÿh0 †eÿabn t ‡ …E0 ÿ bh0 †
‰1 ÿ …ÿ1†n Š
V…X; t† ˆ ÿ
p nˆ1 n
f
…b=f † ‡ ab2n
"
!# 

2
1
2X
1
bh1 ebt=f ÿ eÿabn t
E0
n
ÿab2n t
bt=f
…ÿ1† sinbn X h1 e
ÿ
h
‡
‡
‡
0 e
p nˆ1 n
f
b
…b=f † ‡ ab2n
X
(9)
‡ h1 ebt=f ;
L
substituting Eq. (9) into Eq. (5) one gets the following expression for h(X, t):
"
!#
2
1
2 ÿ…bt=f † X
1
1
ebt=f ÿ eÿabn t
ÿab2n t
sinbn X …ÿh0 †e
‡ …E0 ÿ bh0 †
h…X; t† ˆ ÿ e
p
n
f
…b=f † ‡ ab2n
nˆ1
"
!#
1
bt=f
ÿab2n t
2
2 ÿ…bt=f † X
1
bh
e
ÿ
e
1
n
n
…ÿ1† sinbn X h1 eÿabn t ‡
‰1 ÿ …ÿ1† Š ‡ e
f
p
n
…b=f † ‡ ab2n
nˆ1
‡

X
h1 :
L

(10)

2.3. Special cases of the solution
1. If it is assumed that E0, a constant value of ET, is in¯uencing water table in the soil
pro®le, the value of b may be taken as 0 in Eq. (10) which gives:
"
!#
2
1
2X
1
E0 1 ÿ eÿabn t
ÿab2n t
‡
sinbn X …ÿh0 †e
‰1 ÿ …ÿ1†n Š
h…X; t† ˆ ÿ
f
p nˆ1 n
ab2n
1
h
i X
2
2X
1
…ÿ1†n sinbn X h1 eÿabn t ‡ h1 :
‡
p nˆ1 n
L

(11)

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

175

2. If it is assumed that ET does not in¯uence the water table then both E0 and b, may be
taken as 0 in Eq. (10) to obtain the following solution:
h…X; t† ˆ

1
1
X
2
2
2
1
2h0 X
1
X
…h1 ÿ h0 †
…ÿ1†n sinbn Xeÿabn t ‡
sinbn Xeÿabn t ‡ h1 :
p nˆ1 n
p
n
L
nˆ1

(12)

3. If drains are at the same level and ET does not in¯uence the water table, then h1 ˆ 0,
E0 ˆ 0 and b ˆ 0 in Eq. (10) and the solution takes the form as:
h…X; t† ˆ

1
2
2h0 X
1
sinbn Xeÿabn t ‰1 ÿ …ÿ1†n Š:
p nˆ1 n

(13)

4. If drains are at the same level and a constant ET, E0 in¯uences the water table, then
h1 ˆ 0 and b ˆ 0 in Eq. (10) which gives:
"
!#
2
1
2X
1
E0 1 ÿ eÿabn t
ÿab2n t
sinbn X …ÿh0 †e
‡
‰1 ÿ …ÿ1†n Š:
(14)
h…X; t† ˆ ÿ
p nˆ1 n
f
ab2n
5. If drains are at the same level and a depth-dependent ET in¯uences the water table,
then h1 ˆ 0 in Eq. (10) to obtain the following solution:
"
!#
2
1
2 ÿ…bt=f † X
1
1
ebt=f ÿeÿabn t
ÿab2n t
sinbn X …ÿh0 †e
‡ …E0 ÿbh0 †
‰1ÿ…ÿ1†n Š:
h…X; t† ˆ ÿ e
2
p
n
f
…b=f
†‡ab
n
nˆ1

(15)

The nondimensionalized analytical solution obtained by Sabti (1989), assuming the initial
water table height as a fourth degree parabola, was dimensionalized. If an initially ¯at
water table is assumed, the expression is similar to that of Eq. (12), which is a special
case of the proposed solution. This partially validates the correctness of the proposed
solution.
2.4. Discharge equation for bi-level drains
Discharge of the drains can be computed using Darcy's Law, which states that the
quantity of water passing a unit cross-section of soil is proportional to the gradient of
hydraulic head. The expression for the gradient of hydraulic head is obtained by
differentiating Eq. (10) with respect to X. Discharge per unit length of the drains is
obtained by multiplying the first derivative of h(X, t) at X ˆ 0 and at X ˆ L with ÿKD.
Final expressions for discharge of deep and shallow drains are as follows:
(
!)
"
2
1
h1 2 ÿ…bt=f † X
1
ebt=f ÿ eÿabn t
ÿab2n t
ÿ e
ÿh0 e
‡ …E0 ÿ bh0 †
Qd ˆ ÿKD
f1 ÿ …ÿ1†n g
2
L L
f
…b=f
†
‡
ab
n
nˆ1
(
!)#
2
1
bt=f
ÿab
t
X
2
2
bh1 e ÿ e n
;
(16)
‡ eÿ…bt=f † …ÿ1†n h1 eÿabn t ‡
f
L
…b=f † ‡ ab2n
nˆ1

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A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

(
!)
2
1
h1 2 ÿ…bt=f † X
1
ebt=f ÿ eÿabn t
ÿab2n t
ÿ e
Qs ˆ ÿKD
‡ …E0 ÿ bh0 †
ÿh0 e
f…ÿ1†n ÿ 1g
2
L L
f
…b=f
†
‡
ab
n
nˆ1
(
!)#
2
1
X
2 ÿ…bt=f †
bh1 ebt=f ÿ eÿabn t
ÿab2n t
h1 e
‡
:
(17)
‡ e
f
L
…b=f † ‡ ab2n
nˆ1
"

Here Qd and Qs denote discharge from deep and shallow drains. Total discharge will be
the algebraic sum of absolute values of discharge from both the bi-level drains.

3. Results and discussion
To validate the proposed analytical solution the field data of the conventional level
drainage system at the Sampla (India) drainage site, as reported by Verma et al. (1998),
were utilized (Table 1). Nikam et al. (1992) considered values of ET and reduction factor
b, varying from 0.0 to 0.008 m/day and 0.0±0.0088 per day, respectively, while studying
the effect of ET on time taken to lower the water table by 30 cm in a conventional level
drainage system at the same site. These values of ET and b were also used to study their
effect on spacing between two drains based on the drainage criteria of lowering the water
table by 30 cm in 2 days from the initially assumed water table at the soil surface.
3.1. Water table elevations above the deep drain level computed from the proposed
solution for bi-level drainage system and comparison with the Verma et al. (1998)
solution
Water table elevations above the deep drain level (1.8 m below the soil surface)
between two bi-level drains separated by a distance of 50 m were computed by the
proposed analytical solution for zero ET, constant ET and depth-dependent ET at
X ˆ 15.0 and 35.0 m away from the deep drain (Table 2). In the proposed solution, the
values of water table elevation were found to converge for number of terms, (n ˆ 5). So
only five terms were retained in the solution. The values of water table elevation
predicted by Eq. (12), and a special case of Eq. (10) were compared with the water table
elevations obtained from the solution of Verma et al. (1998). It may be observed from
Table 2 that the temporal distribution of water tables at X ˆ 15.0 and 35.0 m, computed
from both analytical solutions, is almost identical.

Table 1
Parameters of the conventional level drainage system at the Sampla (India) drainage site
Parameter

Value

Hydraulic conductivity (m/day)
Drainable porosity (dimensionless)
Thickness of the envelope material (m)
Tile radius (m)

3.00
0.14
0.10
0.05

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

177

Table 2
Comparison of water table elevations (m) above the deep drain level predicted by Verma et al. (1998) and
proposed solutionsa
Time (days)

Solution Verma
et al. (1998)

Proposed solution
Without ET

With constant ET

With depth dependent ET

X ˆ 15.0 m
1
2
3
4
5
6
7
8
9
10
11
12
13

1.60
1.33
1.12
0.96
0.84
0.73
0.64
0.57
0.51
0.45
0.41
0.37
0.34

1.60
1.33
1.12
0.96
0.83
0.73
0.64
0.57
0.50
0.45
0.41
0.37
0.34

1.55
1.23
0.98
0.79
0.64
0.51
0.40
0.31
0.24
0.17
0.12
0.07
0.04

1.55
1.25
1.04
0.88
0.75
0.65
0.64
0.57
0.50
0.45
0.41
0.37
0.34

X ˆ 35.0 m
1
2
3
4
5
6
7
8
9
10
11
12
13

1.67
1.48
1.32
1.18
1.06
0.96
0.88
0.81
0.74
0.69
0.65
0.61
0.58

1.67
1.48
1.32
1.18
1.06
0.96
0.88
0.80
0.74
0.69
0.65
0.61
0.58

1.61
1.37
1.18
1.01
0.87
0.74
0.64
0.55
0.47
0.41
0.36
0.31
0.28

1.62
1.40
1.22
1.08
0.96
0.87
0.79
0.73
0.68
0.64
0.61
0.61
0.58

a

Spacing between bi-level drains ˆ 50 m; depth of deep and shallow drains ˆ 1.8 and 1.2 m.

The water table elevations above the deep drain computed from the proposed analytical
solution, considering a constant value of ET as 0.008 m/day and a linearly decreasing ET,
show that there is a sharp decline of water table when the value of ET is assumed as
constant compared to the case in which a linearly decreasing depth-dependent ET is
considered. It is also observed from Table 2 that for the case of depth-dependent ET the
values of water table elevations less than about 0.6 m above the deep drain are not
influenced by ET. This may be due to the fact that after about 1.2 m depth in the soil
profile the ET becomes zero as observed at the Sampla drainage site. Eq. (1) is valid only
up to 1.2 m depth below the soil surface or 0.6 m above the deep drain, because if in this
equation the values of h0, h, E0 and b are substituted as 1.8, 0.6 m, 0.008 m/day and
0.00667, respectively, it yields Ed ˆ 0.0 m/day.

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A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

3.2. Computation of water table elevations for the case of conventional level drainage
and comparison with the Kumar et al. (1994) solution
The spatial distribution of water table elevations after 5 days without ET, with constant
and depth-dependent ET for the proposed solution of the conventional level drainage
system (drains located at 1.8 m below the soil surface and separated by a distance of
50 m) was computed (Table 3). Comparison of water table elevations above the deep
drain (obtained from the proposed solution without considering ET) with those obtained
from the Kumar et al. (1994) solution show that all the values are identical. This indicates
that the proposed solution is more general and the solution for level drainage may be
obtained as a special case. In a conventional level drainage system the decline of the
water table was relatively faster when constant ET was assumed to occur throughout the
soil profile as compared to the case of depth-dependent ET. The declining trend in a
conventional level drainage system was similar to the trend in the bi-level drainage
system. It is also observed that after 1.20 m depth from the soil surface, the depthdependent ET did not influence the water table.
3.3. Spatial distribution of water table elevations
Spatial distribution of water table elevations with depth-dependent ET and without ET
between two bi-level drains spaced at 50 m distance considering initial water table,
h0 ˆ 1.8 m and shallow drain located at h1 ˆ 0.6 m (all heights measured above the deep
drain level) for 2, 6 and 12 days is shown in Fig. 2. With time the position of the
maximum water table shifts from the mid-point towards the shallow drain, as was also
observed by Verma et al. (1998). The water table elevations obtained from the proposed
solution for 2 and 6 days considering depth-dependent ET is lower than that obtained
without considering ET. After 12 days ET had no effect on the functioning of the drainage
Table 3
Comparison of water table elevations (m) above the drain level between two level drains at various distances
after 5 days of drainage operation with and without ETa
X (m)

0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
45.00
50.00
a

Kumar et al. (1994)

0.00
0.30
0.57
0.78
0.92
0.96
0.92
0.78
0.57
0.30
0.00

Proposed solution
Without ET

With constant ET

With depth dependent ET

0.00
0.30
0.57
0.78
0.92
0.96
0.92
0.78
0.57
0.30
0.00

0.00
0.21
0.41
0.58
0.69
0.73
0.69
0.58
0.41
0.21
0.00

0.00
0.30
0.57
0.70
0.82
0.86
0.82
0.70
0.57
0.30
0.00

Spacing between two drains ˆ 50 m; drain depth ˆ 1.8 m.

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

179

Fig. 2. Variation of water table with increase in horizontal distance from deep drain at different values of time.

system because all the water table values are less than or equal to 0.6 m (the threshold
value of water table above the deep drain level at which Ed becomes zero).
3.4. Temporal variation of maximum height of the water table
Temporal variation of maximum height of the water table for both bi-level and the level
drainage systems (drains separated by a distance of 50 m) considering zero ET and depthdependent ET is shown in Fig. 3. There was a slower rate of decline of the water table in
the bi-level drainage system compared to the level drainage system with the same drainspacing. The difference in the water table elevations was initially smaller, but gradually
increased with time. This trend was also observed by Verma et al. (1998). If depthdependent ET is taken into account in both drainage systems, the decline of the water
table is relatively faster, and once the water table above the deep drain reaches a value of
about 0.60 m the effect of depth-dependent ET becomes zero.
3.5. Effect of parameter E0 on spacing
Parameter E0, which represents rate of ET at the soil surface, affects the drain-spacing
significantly. This rate varies from season to season and also within a season. To study the
effect of varying rates of ET prevailing in the area on drain-spacing, in both the cases of
bi-level and level drains, various values of ET, varying from 0.002 to 0.008 m/day, were
considered. Spacings that satisfy the drainage criteria (lowering of water table by 30 cm
in 2 days from the initially flat water table near the soil surface) were computed. The
values of spacing obtained for two bi-level drainage cases and for a conventional level
case are given in Table 4. It may be noted from Table 4 that drain-spacing may be

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A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

Fig. 3. Variation of maximum hydraulic head with time for level and bi-level drainage system considering zero
ET and depth-dependent ET.

increased by 0.55±0.64% for E0 ˆ 0.002 m/day and 9.61±10.79% for E0 ˆ 0.008 m/day.
Here, a constant value of b ˆ 0.00667 was considered for computation.
3.6. Effect of parameter b on spacing
The parameter b represents the rate at which ET decreases with increase in depth to the
water table. If constant ET throughout the soil profile is assumed then b becomes zero.
Similarly E0 and b become zero at the depth where the effect of ET on water table
vanishes. The value of b also depends on type of soils. To study its effect, values of b
were varied from 0.0 to 0.0088 with a constant value of E0 (0.008 m/day) and results are
given in Table 5. The results show that with increase in the value of b drain-spacing is
Table 4
Effect of various rates of E0 (m/day) on spacing for bi-level and level drainage system
System

Bi-level drainage
h0 ˆ 1.8, h1 ˆ 0.6 (%)
Bi-level drainage
h0 ˆ 1.5, h1 ˆ 0.3 (%)
Level drainage
h0 ˆ 1.8, h1 ˆ 0.0 (%)

Computed spacing (m)
E0 ˆ 0.00,
b ˆ 0.00

E0 ˆ 0.002,
b ˆ 0.00667

E0 ˆ 0.004,
b ˆ 0.00667

E0 ˆ 0.006,
b ˆ 0.00667

E0 ˆ 0.008,
b ˆ 0.00667

42.94

43.18
0.56
40.49
0.64
46.12
0.55

44.41
3.42
41.71
3.68
47.36
3.25

45.77
6.59
43.06
7.03
48.74
6.26

47.29
10.13
44.57
10.79
50.28
9.61

40.23
45.87

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A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184
Table 5
Effect of parameter b on spacing for bi-level and level drainage system, E ˆ 0.008 m/day
System

Bi-level drainage system
h0 ˆ 1.8, h1 ˆ 0.6 (%)
Bi-level drainage system
h0 ˆ 1.5, h1 ˆ 0.3 (%)
Level drainage system
h0 ˆ 1.8, h1 ˆ 0.0 (%)

Computed spacing (m)
b ˆ 0.0

b ˆ 0.0055

b ˆ 0.00667

b ˆ 0.00778

b ˆ 0.0088

48.51
12.97
45.76
13.75
51.50
12.27

47.50
10.62
44.78
11.31
50.49
10.07

47.29
10.13
44.57
10.79
50.28
9.61

47.09
9.66
44.37
10.29
50.07
9.16

46.90
9.22
44.20
9.87
49.89
8.76

found to decrease in the range of 13.75±8.76% with respect to the drain-spacing
computed without considering ET.
3.7. Effect of ET on spacing
The effect of considering constant ET and depth-dependent ET on drain-spacing for the
two cases of bi-level and for a single case of level drainage system (compared to without
ET) are given in Table 6. The results show that for a uniformly constant value of ET as
0.008 m/day the computed drain spacing, as compared to zero ET, increased in the range
of 12.27±13.75%. In the case of depth dependent ET, the drain spacing, compared to zero
ET, increased in the range of 9.61±10.79%. Since depth dependent ET is a more realistic
value compared to constant ET, an increase of 9.61±10.79% in spacing is probably the
best estimate.
3.8. Variation of discharge with time
Temporal variations of discharge in deep and shallow drains of bi-level drainage
system with depth dependent ET and without ET are shown in Fig. 4. It can be observed
that discharge of the deep drain is always greater than the discharge of the shallow drain.
Therefore, for the shallow drain, a pipe of smaller diameter than the deep drain pipe may
be required. It can also be seen that the effect of depth dependent ET is more pronounced
Table 6
Effect of ET on spacing for bi-level and level drainage system
System

Bi-level drainage system
h0 ˆ 1.8, h1 ˆ 0.6 (%)
Bi-level drainage system
h0 ˆ 1.5, h1 ˆ 0.3 (%)
Level drainage system
h0 ˆ 1.8, h1 ˆ 0.0 (%)

Computed spacing (m)
Without ET

With constant ET

With depth dependent ET

42.94

48.51
12.97
45.76
13.75
51.50
12.27

47.29
10.13
44.57
10.79
50.28
9.61

40.23
45.87

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A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

Fig. 4. Variation in discharges of deep and shallow drains with time.

on the discharge of the shallow drain compared to the discharge of the deep drain and that
after 10 days the effect of depth dependent ET on discharges of both the drains vanishes.
The discharge of the shallow drain becomes zero on the 13th day and boundary condition
(3d) becomes effective. It indicates that the proposed solution for the bi-level drainage
system is valid only up to t0 ˆ 13 days.
3.9. Field application
The proposed analytical solution is a more generalized method which can be used for
designing bi-level or level subsurface drainage systems in arid and semi-arid regions. The
solution takes into account the effect of constant or depth dependent ET on the fall of the
water table. A fast fall results in increase of drain spacing and thus provides economy in
design. Therefore, this solution may be recommended for arid and semi-arid regions
where effect of ET is pronounced and cannot be neglected.

4. Conclusions
An analytical solution to the linearized Boussinesq equation for bi-level drainage
system design was developed in the presence of ET. A special case of the proposed
solution was compared with the Verma et al. (1998) solution and identical values for the
water table were obtained. The analytical solution was obtained after devising a simple
transformation through which the boundary value problem was transformed to a heat
transfer problem for which a solution was available. This technique is relatively easy and

A. Upadhyaya, H.S. Chauhan / Agricultural Water Management 45 (2000) 169±184

183

yields a simpler expression of h(X, t) than the one obtained after applying Laplace
transformation, as done by Verma et al. (1998).
The effect of parameter E0 and b on drain spacing is significant. Therefore values of
these parameters should be selected based on the soil and climatic conditions of the
region. However, it was observed that for a specific value of b, with increase in the value
of E0, the computed drain spacing increases, whereas for a particular value of E0, with
increase in the value of b, computed drain spacing decreases.
Consideration of constant or depth dependent ET in the linearized Boussinesq equation
results in a faster fall of the water table and thus wider drain spacing. In a bi-level drainage
system spacing between two drains can be increased by as much as 9.61±13.75% for soils
having a hydraulic conductivity of 3 m/day, if the contribution of evapotranspiration at
the rate of 8 mm/day in lowering the water table is taken into account. Through the
illustrative example, it was also observed that depth-dependent ET affected the water
table between two drains and the discharge of deep and shallow drains only up to 10 days.
After 10 days, the effect of depth-dependent ET on the water table and discharge of drains
vanished. Discharge of the shallow drain becomes zero on the 13th day. This indicates
that the proposed solution for a bi-level drainage system is valid only up to 13 days.
Therefore, we conclude that a bi-level drainage system can be profitably introduced in
arid and semi-arid regions of the developing countries as it provides controlled drainage
and economy of design, if the effect of ET is considered.

Acknowledgements
The first author acknowledges the financial assistance of CSIR, New Delhi, in carrying
out this study and in the preparation and submission of this manuscript. He also thanks
the Water Technology Centre for Eastern Region (ICAR), Bhubaneswar for sponsorship
in carrying out higher studies at G.B. Pant University of Agriculture and Technology,
Pantnagar. This paper is a part of the first author's Ph.D. Thesis at this university.

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