Fuzzy Estimators of Drain spacing in Subsoil Drainage using Fuzzy
Logic and Possibility Theories
CHRISTOS TZIMOPOULOS, GEORGE PAPAEVANGELOU
Department of Rural and Surveying Engineering
Aristotle University of Thessaloniki, Greece
124, Egnatias Street, 54124, Thessaloniki
GREECE
Abstract: - In the permanent flow of subsoil drainage, a lot of equations are used, most of them based on the
Dupuit assumption. All related mathematical models present uncertainties and fuzziness, which create problems
in the design of drainage networks. Fuzzy Logic deals with this problem and allows the management of uncertain
information. This paper presents the solution of the Hooghout equation based on Fuzzy Logic and Possibility
theories, using the Reduced Transformation Method for the related numerical calculations. This results in a fuzzy
estimator for the drain spacing, whose α-cuts, provide, according to Possibility Theory, the confidence intervals
of the drain spacing with a certain strong probability. Results on subsoil drainage in the case of soils with parallel
drains located at any position from the impermeable bottom are presented. The possibility theory application
enables the engineers and designers of irrigation, drainage, and water resources projects to gain knowledge of
hydraulic properties (e.g., water level, outflow volume) and make the right decision for rational and productive
engineering studies.
Key-Words: Drainage Networks; Hooghout equation; fuzzy logic; possibility theory; transformation method.
Received: February 14, 2023. Revised: December 16, 2023. Accepted: February 11, 2024. Published: March 19, 2024.
1 Introduction
In underground drainage, drains are used to
control the water level by draining the excess water.
In practice, parallel drainage conduits are used which
are either ditches or tubular drains. The mathematical
description of the underground flow to the drains is
achieved by using the following assumptions: a)
Two-dimensional flow. This assumption is true for
long drains. b) Uniform distribution of rainfall
intensity, natural or artificial, c) Homogeneous and
isotropic soils.
Many drainage equations are reduced to one-
dimensional, accepting parallel and horizontal flow
lines. This assumption is valid if the impermeable
subsoil stratum is close to the drains. Hooghoudt [1,
2] first used Dupuit [3] assumptions in 1940 and
extracted an analytical form also known as the
Donnan equation [4]. He then considered the Dupuit
assumptions for the horizontal flow section beyond a
short distance from the drains and the radial curve
assumption for the area near the drains to be valid,
and derived a new equation based on equivalent
depth. He also presented tables for the determination
of the equivalent depth.
Many related solutions consider the case of
stratified soil with two permeable layers, and the
parallel drains to be located at any position from the
interface of the two layers (Toksöz and Kirkham [5,
6], Ernst [7, 8], Wesseling [9], Terzidis and
Karamouzis [10], Terzidis [11], Tzimopoulos [12],
Kirkham [13]. All these researchers presented a
series-type solution, based on two-dimensional flow
and potential theory, by solving the Laplace equation.
Ernst [7] provided an approximate method of solving
the problem, which is an extension of Hooghoudt’s
[1, 2] method for drainage of homogeneous soils,
which is mainly suitable for stratified soils with
certain limitations. Dagan [14] considered a radial
flow near the drains and a horizontal flow quite far
from them and presented an approximate solution.
The Toksöz-Kirkham [5, 6] method is an extension
of Kirkham’s method for draining homogeneous
soils, considering the flow to be two-dimensional,
and solving the corresponding Laplace equation.
Walczak et al. [15] presented an algorithm based on
the Kirkham equation. Van der Molen and Wesseling
[16] presented a closed form of the Hooghoudt
equivalent depth equation with great accuracy.
According to Lovel and Youngs [17], and Ritzema
[18], from the above-mentioned solutions, the
Hooghoudt equation in combination with the
simplistic solution of Van der Molen & Wesseling
gives the best results without any restrictions. More
recently, Mishra and Singh [19], modified
Hooghoudt's method and improved the free surface
in the area near the drains. Afruzi et al [20] have also
presented a solution of the two-dimensional Laplace
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DOI: 10.37394/232033.2024.2.8
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E-ISSN: 2945-1159
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equation for the flow into the drains using the
Schwarz – Christoffel transform in conformal
representation.
There have been more recent contributions to the
problem of subsoil drainage: Vlotman et al. [21],
provide a resource book for envelope design and
research, and collected related data from all over the
world. Rimidis and Dierickx [22] proposed a second-
degree polynomial, similar to the well-known
equation of Hooghoudt, in order to express the
relationship between hydraulic head loss and
discharge for each of the plots during each of the
measuring seasons. Skaggs et al. [23], developed
design drainage rates for use in the Hooghoudt
equation to estimate required drain depth and spacing
in the eastern United States. Castanheira and Santos
[24], use a two dimensional saturated - unsaturated
Galerkin finite element numerical model to predict
water table height between parallel drains. The
results obtained with this model agree well with
Khirkam’s and Hooghoudt analytical solution for the
distribution of total head in ideal drains and for the
total head calculations midway between drains.
Ali [25] describes the hydraulic design of
subsurface drains applied in Bangladesh. For steady
state problems, Hooghoudt’s equation is proposed,
based on the Dupuit-Forcheimer assumptions. He
also describes the DRAINMOD hydrologic model,
the Colorado State University Irrigation and
Drainage (CSUID) model, and the EnDrain model.
Valipour [26] carried out a comparison between
horizontal and vertical drainage in anisotropic soils.
He determined this purpose, using EnDrainWin and
WellDrain softwares drain spacing and well spacing,
respectively. The results showed that in the same
situation, horizontal drainage systems due to the
higher spacing between drains were better than
vertical drainage systems. Valipour [27] has
investigated the effect of drainage parameters change
on drain discharge, which is essential in subsurface
drainage systems. For this purpose, he used to change
all the drainage parameters by EnDrain software and
investigated changes of drains’ discharge in
subsurface drainage systems. Skaggs [28] introduces
three coefficients, namely: a) the subsurface drainage
coefficient, calling it Kirkham Coefficient (KC). b)
The Drainage intensity coefficient (DI), and c) The
drainage coefficient (DC), which quantifies the
hydraulic capacity of the system and is estimated by
the Hooghoudt equation. Inclusion of these three
coefficients in the research and design projects would
facilitate comparison of results from different soils
and drainage systems, and generally, the meta-
analysis of data pertaining to drainage studies. The
KC, DI, and DC coefficients represent the minimum
information needed to characterize a drainage site.
Recently, several authors provided useful insights
concerning subsurface drainage system solutions
(Kacimov and Obnosov, [29], Chahar and Vadoria,
[30], Emikh, [31], Baru and Alan, [32], Sarmah and
Tiwari, [33], Ren et al, [34], Bao et al., [35], Zhang
et al., [36]).
In all above-mentioned models, the variables are
the hydraulic conductivity K, the rainfall intensity R
which is entirely infiltrated into the subsoil, the water
height in the middle distance between the drains h,
and the distance of the bottom of the drains from the
impermeable subsoil D. The K, R, D variables are
measurable, and contain inaccuracies and fuzziness
due to human error, due to measurement apparatus,
due to the inhomogeneity and the anisotropy of the
soils, etc. The above-mentioned uncertainties heavily
influence the reduction of precise conclusions and do
not allow engineers to take the right decisions in the
design of a drainage network.
In classical Logic, the mathematical models must
be extremely accurate, avoiding and dismissing
inaccuracies. However, the inaccuracy and the
fuzziness are very interesting because they contain
information concerning real processes especially in
cases where the inaccuracy becomes not acceptable.
According to Goguen [37], fuzziness is the rule rather
than the exception in problems of engineering, and
usually there is no well-defined perfect solution. This
weakness is covered by Fuzzy Logic which was
developed in 1965 from Zadeh [38]. Fuzzy logic
introduces fuzzy numbers, and any inaccuracy or
fuzziness is represented numerically. At the same
time, the models provide fuzzy numerical
calculations based on the theory of fuzzy sets, which
allows the management of fuzzy information.
According to Goualles [39], uncertainty and
fuzziness have received acceptance in scientific
research and in the scientific consideration of the
world in general.
For the realization of the fuzzy numerical
calculations, the following procedure is followed:
•Fuzzification of the variables Κ, S, d (d =
equivalent depth). Usually, symmetrical fuzzy
numbers of triagonal form are used.
•A Fuzzy model is chosen.
• The method of numerical calculations is chosen.
Because many calculations are involved in the
models, they are executed as interval calculations of
the cuts of the triangular numbers. To avoid
overestimation in the calculation of the intervals
when the number of variables is high, the VERTEX
method is used (Dong and Shah, [40]), or the
corresponding reduced transformation method
(Hanss, [41, 42]).
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DOI: 10.37394/232033.2024.2.8
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E-ISSN: 2945-1159
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• A fuzzy number that represents the length of the
drain spacing L is obtained, after an iteration
process, as the final result.
•The α-cuts of this fuzzy number, represent the
confidence intervals of L with probability ,
according to the Possibility Theory (Dubois and
Prade [43, 44], Dubois et al. [45, 46, 47], Mylonas
[48]).
In the present paper, we present the fuzzification
of the solution of Hooghoudt in the case of the
equivalent length d, to show the difference in the
drain spacing, and to estimate the related fuzziness.
The fuzzy estimation of variables K, S, R, was
obtained using the theory of non-asymptotic fuzzy
estimators (Sfiris and, Papadopoulos, [49]), while d
is estimated with the closed solution of Van der
Molen and Wesseling [16]. The choice of this model
was made because it was considered by Ritzema [50]
to provide the best results, without restrictions. As a
result, we obtain the estimation of drain spacing L as
a fuzzy number whose α-cuts according to the
possibility theory define the strong probability of the
confidence intervals of L. Consequently, the
possibility theory application, enables the engineers
and designers of irrigation, drainage, and water
resources projects to gain knowledge of hydraulic
properties and take the right decision for rational and
productive engineering studies.
2 Problem Formulation
In the present section, definitions of Fuzzy numbers,
Fuzzy sets and Possibility Theory are provided, in
paragraph 2.1. The presentation of the classic
Hooghout equation is reminded in paragraph 2.2. The
Fuzzy form of the equation is presented in paragraph
2.3, and the transformation method is presented in
paragraph 2.4, more specifically the decomposition
of the fuzzy numbers of the variables involved, and
the transformation of their intervals.
2.1 Fuzzy numbers
To facilitate the readers not familiar with the fuzzy
theory, some definitions are provided here,
concerning some preliminaries in Fuzzy Logic theory
and Possibility theory.
Definition 1. A fuzzy number is a fuzzy set :
R1→I=[0,1] with the following properties: (i) u is
upper semicontinuous, (ii) u (x)=0 outside of some
interval [c, d], (iii) there are real numbers a and b,
c≤a≤b≤d such that u is increasing (non-decreasing)
on [c, a], decreasing (non-increasing) on [b, d] and
u (x)=1 for each x [a,b], (iv) u (λx+(1+λ)x)≥min{
u (λx),u ((1+λ)x)},λ[0,1], u is convex, (v) This
fuzzy number has a membership function, denoting
the degree of set membership. The membership
function of a fuzzy set u is denoted by
or by
u (x).
Definition 2. Define by:
where
denotes the closure of the support of .
Then it is easily established that is a fuzzy number
if and only if: (i) is a closed and bounded
interval for each , and (ii)
The is called level set of .
Definition 3. Let K(X) the family of all nonempty
compact convex subsets of a Banach space. A fuzzy
set on X is called compact if
. The
space of all compact and convex fuzzy sets on X is
denoted as Ƒ (X).
Definition 4. Let Ƒ (R). The α-cuts of , are
¨. According to representation
theorem of Negoita and Ralescu [51] and the theorem
of Goetschel and Voxman [52], the membership
function and the α-cut form of a fuzzy number , are
equivalent and in particular the α-cuts uniquely
represent, provided that the two functions are
monotonic (
increasing,
decreasing) and
The arithmetic operations of α-cuts are the same as
for a set of classical interval numbers (Moore [53],
Moore et al. [54]). The arithmetic operations for
interval numbers have the properties of associativity
and commutativity. However, distributivity does not
always hold, which implies that after distributing, the
interval would probably be widened. The cause of
failure of distributivity is due to the treatment of two
occurrences of identical interval numbers as two
independent interval numbers. To prevent the
widening, additional methods can be used: a) Vertex
method (Dong and Shah, [40]), or the Reduced
Transformation Method (Hanss, [42, 55]).
Definition 5. Possibility theory (Dubois et al. [45]).
Let be a fuzzy subset of referential set Ω, viewed as
the set of admissible, mutually exclusive values of a
variable x. Let
be another subset of Ω; one may
evaluate to what extent
intersects (possibility of
event
) and to what extent
contains certainty of
event
):
(i) Possibility of
: Π
α
α
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(ii) Necessity (certainty) of
:
α
α
The above relation (ii) means that
Π
i.e. the certainty of
reflects the impossibility
of its complement
.
Definition 6. A degree of necessity NessX on a set X
(e.g., a set of reels) is characterized by the non‐
possibility (one minus possibility) of A complement
(AC)
Definition 7. A probability distribution p and a
possibility distribution π are said to be consistent only
if (Dubois et al., [45], Mylonas,
[48]).
Definition 8. Two possibility distributions �x, �΄x are
consistent with the probability distribution px. The �x
distribution is more specific than �΄x, if it is �x< �΄x.
A possibility distribution �΄x consistent with the
probability distribution px is called maximal
specificity, if it is more specific that each other
possibility distribution:
�΄x (x) < �x (x), ∀�
Definition 9. For a number Y with a known and
continuous probability distribution function p, the
fuzzy number
, which has a possibility measure
is the fuzzy estimator of Y and has an α‐
cut of
This fuzzy number satisfies the
consistency principle and verifies
, so that the probability of the
possibility α‐cut is equal to 1−α. The α‐cuts
α
are the confidence intervals of P, and the confidence
level is α.
Definition 10. Conjecture (Mylonas [48]). For a
function Y = Y (X1,X2,....Xn) with unknown
probability distribution function, a fuzzy number
may be constructed
and the
α‐cut is equal to the following:
α
α
α
α.
In this case, the fuzzy number
is the fuzzy
estimator of Y and verifies the following:
α
αα,
so that the probability of the possibility α‐cut is
greater than 1-α.
2.2 Hooghoudt classic function
Hooghoudt [1] presented a formula for the drain
spacing accepting parallel and horizontal streamlines,
and Hooghoudt [2] considered more practical to have
a formula accounting for the extra resistance caused
by the radial flow, and he introduced a reduction of
the depth D to a smaller equivalent depth d. Finally,
his formula is:
(1)
where: Κ =the hydraulic conductivity (m/d),
R=recharge rate per unit surface area (m/d), h=height
above the drain level, midway between two drains
(m), D = the actual thickness of the aquifer between
the drains and the impervious bottom (m), d = the
equivalent depth (m). Hooghoudt [2], presented
tables with values for the equivalent depth d, for
different values of L (5 to 250 m), D (0.5 to 60m),
and radius drain r0=0.1m.
Moody [56], proposed the following iterative
formula for the equivalent depth d, which is quite
accurate:
(2)
(3)
Van der Molen and Wesseling [16], proposed the
following formula for the equivalent depth d:
. (4)
Equation 4 is a combination between the Hooghout’s
[2] and Kirkham’s [13] equations.
∞
,
(5),
The above series converges rapidly for x> 0.5 with a
mean reduced error less than 0.15 %. In this case it
takes the form:
(6)
For x << 0.5 convergence is slow, but for this case,
Van der Molen and Wesseling compared it with
Dagan’s formula and proposed the following closed
approximation:
(7)
Equation 7 converges conveniently with the above
series with a mean reduced error equal to 0.000129
for . Figure 1 illustrates the series with a line
and the closed solution with black squares.
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Figure 1. F(x) series and F(x) exact formula.
2.3 Hooghoudt fuzzy function
For the case where the variables
are fuzzy
numbers of triangular form, Equation 1 becomes:
ℎ
ℎ
(8)
The fuzzy variables
are triangular form fuzzy
estimators of K, R and is a triangular form fuzzy
estimator of d with his form:
(9)
For practical reasons it is posed:
: ,
and Equations 8 and 9 become:
ℎℎ
(10)
According to Nguyen theorem [57], if:
(11)
then, a sufficient and necessary condition for
obtaining the following equality,
(12)
is that the function is continuous, and the
following relation is achieved:
(13)
Applying Equation 10, each fuzzy number is
decomposed into as set of (m+1) intervals X(j)
(j=0,1, 2, … m)
(14)
with
(15)
The arithmetic operations of α-cuts are made
applying the Reduced Transformation Form,
and finally the following fuzzy number arises
(Fig.2):
Figure 2. Fuzzy number and fuzzy intervals using
possibility theory.
The fuzzy number is an estimator of the crisp
number F1 and according to the Possibility theory
(Dubois and Prade [44], Dubois et al. [46], Dubois et
al. [47], Mylonas [48]), there is a Possibility
distribution function, and the following relationship
is valid:
(16)
Additionally, for the α-cut=0.01, it is valid:
(17)
2.4 Transformation Method
2.4.1 Decomposition of fuzzy numbers
The Transformation method can be divided into two
forms: a General and a Reduced Form, Hanss ([41],
[42], [55]). The Reduced Form is used for cases in
which there is a function with n independent
parameters, assumed to be uncertain. In addition, the
function is monotonic with respect to each variable,
without local extrema. In the opposite case, the
general transformation method can be applied for
0
5
10
15
20
25
0 0,5 1 1,5 2 2,5
...)5,3,1=n(
)e1(n
e4
=)x(F ∑
∞
1n
nx2
nx2
π2
x
ln+
x4
π
=)x(F
2
F(x)
x
0
0.2
0.4
0.6
0.8
1
α=0.01
α=0
α=0.1
α=0.2
α=0.3
α=0.4
α=0.5
α=0.6
α=0.7
α=0.8
α=0.9
α=1.0
%99]}b,a[]F
~
Pr{[ 01.0α01.0α01.0α1
a
b
1
F
~
1
F
~
μ
1
F
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Volume 2, 2024
complex, non-monotonic problems. In this article the
Reduced Transformation Form is used and the
function (Equation 10) is monotonic,
nonlinear and has three fuzzy variables.
The fuzzy number can be decomposed into m several
intervals, j=0, 1, ..., m, given by the α-cuts at the α-
levels μj
(18)
Figure 3. Decomposition of the fuzzy number
into intervals.
The fuzzy numbers of Equation 12 (n=3 in this case)
can be decomposed into a set of (m + 1) intervals,
j=0, 1, …, m, of the form (decomposition principle,
Zadeh [38])
(19)
with
ℓℓ
(20)
The set ℓ can be referred to as the
worst-case interval (Hanss [42]).
Notation: All the fuzzy parameters
can be seen as the coordinates
of points on the n-dimensional hypersurfaces
, nested according to their level of
membership. In the case of the Reduced Transformation
Form, only the 2n vertex points of the n-dimensional
cuboids are considered for the evaluation of the problem.
Fig. 4 illustrates the case n=3, which is a cube with 23
vertices and α=0, 1/3, 2/3, 1. The cuboid for the
membership level μ=1, is degenerated to one single point.
2.4.2 Transformation of the intervals
For the case of the Reduced Transformation
Form, the intervals are
transformed into arrays of the following form:
(21)
22)
Figure 4. Geometric interpretation of the
transformation scheme for n=3.
In the present case for, (n=3 variables) it is:
(23)
where is the index of the α-level.
In case for (n=2 variables) it is:
(24)
Notation: Between the matrices of the Reduced
Transformation Form (RTF) and the VERTEX method
there is the following relation:
ℎ
ℎ,
i.e. the columns of the Reduced Transformation Form
are the rows of the VERTEX method. Besides, the
columns are the coordinates of n- dimensional
hyperface vertices.
In the present case, the dimensions of the
matrices are and for each column
corresponding to a vertex , are created 23=8
0
1
j
1
j
Δμ
)j(
i
a
)j(
i
b
i
~
)x(
i
~
i
(a1a2a3)0
(a1a2b3)0
(a1b2b3)0
(a1b2a3)0
(b1a2b3)
0
(b1b2b3)0
(b1a2a3)0
(b1b2a3)0
x1
x2
x3
(a1)0
(a2)0
(b1)0
(a3)0
(b3)0
α=0
α=0.33
α=0.6
α=1
α=0.66
(b2)0
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values of the function ()
and 22=4 for function ℓℓ. It
is examined now the following relation:
ℓℓℓℓ
(ℓ =the number of the matrix (23) column)
ℓℓℓℓℓℓ
ℓℓ
or:
ℓ
ℓℓ
ℓ (25)
3 Results
In the present section a soil sample was used with the
following parameters: A hydraulic conductivity K
with mean value and standard deviation
, a recharge rate R per unit surface area
with mean value and standard
deviation , a distance D above the
impervious floor to the drains level with mean value
and standard deviation . All
these parameters follow a normal distribution law.
The above (K, R, D) random sample is of size N=40
observations. The drainpipes with a radius of r=0.1
m, are placed at a depth of 1.8 m below the soil
surface and the height h above the drain level,
midway between the two drains is 0.6 m.
Figure 5. Definition sketch.
3.1 Fuzzy estimators
We apply the theory of non-asymptotic fuzzy
estimators (NAFE) (Sfiris and Papadopoulos, [59]).
A non-asymptotic fuzzy estimator is a complete
triangular form fuzzy number, whose a-cuts are as
follows:
ℎ
ℎ
where
and
.
Φ denotes the cumulative distribution function of the
standard normal distribution. In our case γ=0.01 and
.
(a)
(b)
Figure 5. Fuzzy estimators: (a) Hydraulic
Conductivity K, (b) Rainfall Intensity R.
1.8m
D=5.0m
h=0.6m
R(m/d)
Impermeable layer
0
0.2
0.4
0.6
0.8
1
1.3 1.4 1.5 1.6 1.7 1.8 1.9
Κ
~
Κ
(m/d)
Κ
~
α1
α2
α3
α4
α5
α6
0
0.2
0.4
0.6
0.8
1
0.0016 0.0017 0.0018 0.0019 0.002 0.0021 0.0022 0.0023 0.0024
R
~
R
~
)d/m(R
α1
α2
α3
α4
α5
α6
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DOI: 10.37394/232033.2024.2.8
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E-ISSN: 2945-1159
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Volume 2, 2024
Figure 6. Fuzzy estimator of D.
3.2 Fuzzy Estimation of equivalent depth
and drain spacing.
To find the drain spacing an iterative process has
started in two steps:
1rst step. With the aid of equation 10(a) a first value
of drain spacing estimator is calculated. This value
was given by the transformation method and six α-
cuts (α1=0.05, α2=0.2, α3=0.4, α4=0.6, α5=0.8 and α6=1)
were used for each variable
2nd step. This value was used to estimate the
variable estimator (first iteration) using the
equation 10(b).
The iterative process was stopped when the absolute
difference of two successive values was negligible.
Figures 6 and 7 illustrate the estimators of and
and the successive forms in every iteration.
Figure 6. Successive forms of drain spacing
estimators.
Figure 7. Successive forms of equivalent depth
estimators.
After a cycle of four iterations, the iterative process
stopped because the absolute difference ε between
the two last iterations was negligible.
.
Tables 1 and 2 present the absolute difference ε
between iterations 3 and 4 for every αi (i=1,2,…6)
and the mean value of ε is 9.96 E-04 for and 2.16
E-04 for
Table 1. Reduced absolute difference between two
iterations for
Table 2. Reduced absolute difference between two
iterations for
4 Discussion
In classical logic in mathematical models, there are
imprecisions and fuzziness, which are rejected as
being instability factors. However, the higher the
precision achieved, the higher the fuzziness becomes.
This problem is covered by the Fuzzy Logic and
Possibility theories, which introduce the notion of
fuzzy numbers, providing fuzzy numerical operations
on the base of the Fuzzy Sets Theory.
0
0.2
0.4
0.6
0.8
1
4.7 4.8 4.9 5 5.1 5.2 5.3
α1
α2
α3
α4
α5
α6
D
~
D
~
)m(D
0
0.2
0.4
0.6
0.8
1
105 115 125 135 145 155
L1
L2
L3
L4
α1=0.05
α2
α3
α4
α5
α6
115.83
139.72
%95]72.13983.115Pr[
127.25
1
L
~
2
L
~
3
L
~
4
L
~
L
~
μ
L
0
0.2
0.4
0.6
0.8
1
3.7 3.8 3.9 4 4.1 4.2
α1=0.05
α2
α3
α4
α5
α6
3.76
4.08
%95]08.476.3Pr[
1
d
~
2
d
~
3
d
~
4
d
~
d
~
μ
)m(d
ε ε
Iteration 3 Iteration 4 Iteration 3 Iteration4
α1115.96 115.83 1.12E-03 139.84 139.72 8.84E-04
α2119.63 119.50 1.07E-03 135.58 135.46 9.19E-04
α3122.22 122.09 1.05E-03 132.73 132.61 9.44E-04
α4124.13 124.01 1.03E-03 130.69 130.57 9.63E-04
α5125.80 125.67 1.01E-03 128.97 128.84 9.79E-04
α6127.37 127.25 9.94E-04 127.37 127.25 9.94E-04
α-
α+
εε
Iteration 3 Iteration 4 Iteration 3 Iteration4
α13.7560 3.7550 2.52E-04 4.0793 4.0785 1.84E-04
α23.8089 3.8080 2.39E-04 4.0251 4.0244 1.94E-04
α33.8455 3.8446 2.31E-04 3.9881 3.9873 2.01E-04
α43.8722 3.8713 2.25E-04 3.9611 3.9603 2.06E-04
α53.8951 3.8942 2.20E-04 3.9381 3.9373 2.11E-04
α63.9166 3.9157 2.16E-04 3.9166 3.9157 2.16E-04
α-
α+
International Journal of Environmental Engineering and Development
DOI: 10.37394/232033.2024.2.8
Christos Tzimopoulos, George Papaevangelou
E-ISSN: 2945-1159
95
Volume 2, 2024
The present paper, presents the solution of the
Hooghout equation based on Fuzzy Logic and
Possibility theories, using the Reduced
Transformation Method for the related numerical
calculations. This results in a fuzzy estimator for the
drain spacing, whose α-cuts, provide, according to
Possibility Theory, the confidence intervals of the
drain spacing with probability greater than the α-
level. Results are presented in Figures 6 and 7 after
an iterative process, witch after four iterations
attained a value of estimator L (drains spacing) and
d (equivalent depth) very close to real values.
According to Figure 6 presented above, the drain
spacing on the base of classical logic, is 127.25m,
while based on Fuzzy Logic and Possibility theories
the drain spacing is the interval [115.53,139.73] with
a probability greater than 95%.
5 Conclusion
From the above, it can be concluded that with the
application of the Fuzzy Logic and Possibility
theories to the problem of the design of drainage
networks, the designer has obtained confidence
intervals for the spacing of the drains for any
probability level. Consequently, the Fuzzy Logic and
Possibility theories application, enables the engineers
and designers of irrigation, drainage, and water
resources projects to gain knowledge of hydraulic
properties and take the right decision for rational and
productive engineering studies and design the
networks based on these values.
References
[1] Hooghoudt, S.B., Bepaling van den
doorlaatfaktor van den grond met behulp van
pompproeven (z.g. boorgatenmethode).
Verslagen Landbouwkundige Onderzoekingen,
42B, 1936, 449-541.
[2] Hooghoudt, S.B., Bijdragen tot de kennis van
enige natuurkundige grootheden van de grond,
Versl. Landbook Onderzoek, 46 (14B), 1940,
515-707.
[3] Dupuit, J., Etudes Theoriques et Pratiques sur le
Mouvement des Eaux, dans les Canaux
Découverts et a travers les Terrains
Perméables, Dunod: Paris, France, pp.338,
1863.
[4] Donnan, W. W., Model tests of a tile-spacing
formula. Soil Science Society of America
Journal, 11(C), 1946, 131-136.
[5] Toksöz, S.; Kirkham, D., Steady Drainage of
Layered Soils: I. Theory. Journal of Irrigation
and Drainage. Division of the American Society
of Civil Engineering Proceedings, 97(IR1),
1971a 1-18.
https://doi.org/10.1061/JRCEA4.0000768
[6] Toksöz, S.; Kirkham, D., Steady Drainage of
Layered Soils: II. Nomographs. Journal of
Irrigation and Drainage. Division of the
American Society of Civil Engineering
Proceedings, 97 (IR1), 1971b, 19-37.
[7] Ernst, L.F., Calculation of the steady flow of
groundwater in vertical cross sections. Neth. J.
Agric. Sci., 4, 1956, 126-131.
https://doi.org/10.18174/njas.v4i1.17793.
[8] Ernst, L.F., Grondwaterstromingen in de
verzadigde zone en hun berekening bij
aanwezigheid van horizontale evenwijdige open
leidingen. Versl. Landbouwk. Onderz 15.
Pudoc, Wageningen. 189 p, 1956.
[9] Wesseling, J., Drainage Principles and
Applications. II Theories of field drainage and
watershed runoff, Int. Inst. for Land
Reclamation and Improvement, Publ. 16, Vol.
II, Wageningen, 1973.
[10] Terzidis, G. and Karamouzis D., Drainage
Principles, Ziti Eds, Thessaloniki, Greece, 1956.
[11] Terzidis, G., Simple and accurate method of
layered soils steady drainage, In: Proceedins of
the 3rd Hellenic Conference of Agriculture,
2003, 505-513.
[12] Tzimopoulos, C., Drainage Principles. Ziti Eds,
Thessaloniki, Greece, 1982.
[13] Kirkham, D., Seepage of steady rainfall through
soil into drains. Trans. Am. Geophys. Union, 39
(IR2), 1958, 892-908.
[14] Dagan, G., Spacings of drains by an
approximate method. Journal of the Irrigation
and Drainage Division ASCE, 90, 1964, 41-46.
https://doi.org/10.1061/JRCEA4.0000297.
[15] Walczak, R.T, van der Ploeg R.R., Kirkham D.,
An algorithm for the calculation of drain spacing
for layered soils, Soil Science Society of
America Journal, 52, 1988, 336-340.
https://doi.org/10.2136/sssaj1988.03615995005
200020006x.
[16] Van der Molen W.H.; Wesseling J., A solution
in closed form and a series solution to replace
the tables for the thickness of the equivalent
layer in Hooghoudt’s drain spacing formula.
Agricultural Water Management, 19, 1991, 1-
16. https://doi.org/10.1016/0378-
3774(91)90058-Q.
[17] Lovell, C.J.; Youngs E.G., A comparison of
steady-state land drainage equations,
Agricultural Water Management, 9 (1), 1984, 1-
International Journal of Environmental Engineering and Development
DOI: 10.37394/232033.2024.2.8
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Volume 2, 2024
21. https://doi.org/10.1016/0378-
3774(84)90015-5
[18] Ritzema, H.P., Drainage Principles and
Applications, International Institute for Land
Reclamation and Improvement, Wageningen,
The Netherlands, pp. 371, 1973
https://edepot.wur.nl/262058.
[19] Μishra, G.C., Singh, V., A new drain spacing
formula, Hydrological Sciences, 52(2), 2007,
338-351. https://doi.org/10.1623/hysj.52.2.338
[20] Αfruzi, Α., Νazemi, Α.Η. and Sadraddini, A.A.
Steady-state subsurface drainage of ponded
fields by rectangular ditch drains, Irrigation and
Drainage, 63, 2014, 668–681.
https://doi.org/10.1002/ird.1857
[21] Vlotman, W. F., Willardson, L. S., Dierickx, W.,
Envelope design for subsurface drains,
International Institute for Land Reclamation and
Improvement, P.O. Box 45,6700 AA
Wageningen, The Netherlands, 2000.
[22] Rimidis, A., Dierickx, W., Evaluation of
subsurface drainage performance in Lithuania,
Agricultural Water Management 59, 2003, 15–
31.
[23] Skaggs, R. W., Youssef, M.A., Chescheir, G.M.,
Drainage design coefficients for eastern United
States, Agricultural Water Management, 86,
2006.
[24] Castanheira, P. J., Santos F. L., A simple
numerical analyses software for predicting
water table height in subsurface drainage, Irrig
Drainage Syst, 2009, 23:153–162, DOI
10.1007/s10795-009-9079-
[25] Ali M.H., Practices of Irrigation & On-farm
Water Management: Volume 2, Springer
Science+Business Media, LLC ISBN 978-1-
4419-7636-9, DOI 10.1007/978-1-4419-7637-6,
Springer New York Dordrecht Heidelberg
London, pp. 571, 2011.
[26] Valipour M. A., Comparison between
Horizontal and Vertical Drainage Systems
(Include Pipe Drainage, Open Ditch Drainage,
and Pumped Wells in Anisotropic Soils, IOSR
Journal of Mechanical and Civil Engineering
(IOSR-JMCE) ISSN: 2278-1684, 1, 2012, 7-12.
[27] Valipour M., Effect of Drainage Parameters
Change on Amount of Drain Discharge in
Subsurface Drainage Systems, IOSR Journal of
Agriculture and Veterinary Science (IOSR-
JAVS, ISSN: 2319-2380, ISBN: 2319-2372), 4,
2012, 10-18.
[28] Skaggs, R.W., Coefficients for Quantifying
Subsurface Drainage Rates, American Society
of Agricultural and Biological Engineers, 33(6),
2017, 793-799, ISSN0883-8542, https:
//doi.org/10.13031/aea.12302.
[29] Kacimov A.R., Obnosov Y.V., Analytical
determination of seeping soil slopes of a
constant exit gradient, Zeitschrift fur
Angewande Mathematic und Mechanic 82 (6),
2002, 363-376.
[30] Chahar B.R., Vadoria G.P., Steady sub-surface
drainage of homogeneous soils by ditches.
Proceedings of the ICE-Water Management 161
(6), 2008, 303-311.
[31] Emikh V.N., Mathematical models of ground
water flow with a horizontal drain. Water
resources 35 (2), 2008, 205-211.
[32] Baru G., Alan W., An analytical solution for
predicting transient seepage into ditch drains
from a ponded field. Advances in Water
Resources 52, 2013, 78-92.
[33] Sarmah R. and Tiwari S., A two-dimensional
transient analytical solution for a ponded ditch
drainage system under the influence of
source/sink, Journal of Hydrology, 558, 2018,
196–204.
[34] Ren X., Wang S., Yang P., Tao Y., Performance
Evaluation of Different Combined Drainage
Forms on Flooding and Waterlogging Removal,
Water, 13, 2021, 2968,
https://doi.org/10.3390/w13212968.
[35] Bao T., Zhang S., Liu C., Xu Q., Experimental
Study on the Effect of Hydraulic Deterioration
of Different Drainage Systems on Lining Water
Pressure, Processes 10, 2022, 1975.
https://doi.org/10.3390/pr10101975.
[36] Zhang C., Liu N., Chen K., Ren F.Z., Study on
drainage mode and anti‑clogging performance
of new waterproofing and drainage system in a
tunnel. Scientific Reports, 2023, 5354.Goguen,
J. A., L-fuzzy sets. Journal of Mathematical
Analysis and Applications, 18, 1967, 145-174.
https://doi.org/10.1016/0022-247X(67)90189-
8.
[37] Goguen, J. A., L-fuzzy sets. Journal of
Mathematical Analysis and Applications, 18,
1967, 145-174. https://doi.org/10.1016/0022-
247X(67)90189-8.
[38] Zadeh, L. A., Fuzzy Sets, Information and
Control, 8, 1965, 338—353.
https://doi.org/10.1016/S0019-9958(65)90241-
X.
[39] Goualles Th. E., Fuzzy system in uncertain
environments. PhD, ΕΜP, Athens-Greece,
2001, pp.225.
[40] Dong, W., Shah, H.C., Vertex method for
computing functions of fuzzy variables. Fuzzy
International Journal of Environmental Engineering and Development
DOI: 10.37394/232033.2024.2.8
Christos Tzimopoulos, George Papaevangelou
E-ISSN: 2945-1159
97
Volume 2, 2024
Sets and Systems, 24(1), 1987, 65-78.
https://doi.org/10.1016/0165-0114(87)90114-X
[41] Hanss M., The transformation method for the
simulation and analysis of systems with
uncertain parameters, Fuzzy sets and systems,
130(3), 2002, 277-289.
https://doi.org/10.1016/S0165-0114(02)00045-
3.
[42] Hanss M., The Extended Transformation
Method for The Simulation and Analysis Of
Fuzzy-Parameterized Models, International
Journal of Uncertainty Fuzziness and
Knowledge-Based Systems, 11(06), 2003, 711-
728, https://doi.org/10.1142/S0218488503002491.
[43] Dubois D. and Prade H., Fuzzy sets, probability
and measurement, European Journal of
Operational Research, 40, 1989, 135-154.
[44] Dubois D. and Prade H., When upper
probabilities are possibility Measures, Fuzzy
Sets and Systems, 49, 1992, 65-74.
https://doi.org/10.1016/0165-0114(92)90110-P
[45] Dubois D., Moral S., Prade H., A Semantics for
Possibility Theory Based on Likelihoods,
Journal of Mathematical Analysis and
Applications, 205, 1997, 359-380.
[46] Dubois D., Foulloy L., Mauris G., Prade H.,
Probability-possibility transformations,
triangular fuzzy sets, and probabilistic
inequalities. Reliab Comput, 10, 2004, 273–297.
DOI:10.1023/B:REOM.0000032115.22510.b5
[47] Dubois, D.; Foulloy, L.; Mauris, G.; Prade, H.,
Transformations, Triangular Fuzzy Sets, and
Probabilistic Inequalities, Reliable Computing,
10 (4), 2004, 273–297.
[48] Mylonas N., Applications in fuzzy statistic and
approximate reasoning, PhD thesis, Dimokritos
University of Thrace-Greece, pp. 164, 2022, (In
Greek).
[49] Sfiris D.S., Papadopoulos B.K., Non-asymptotic
fuzzy estimators based on confidence intervals,
Information Sciences, 279, 2014, 446-459.
[50] Ritzema, H.P., Subsurface Flow to Drains,
Drainage Principles and Applications,
International Institute for Land Reclamation,
and Improvement, Wageningen, The
Netherlands, 1994.
[51] Negoita, C.V., Ralescu, D.A., Representation
theorems for fuzzy concepts, Kybernetes, 4(3),
1975, 169 – 174.
https://doi.org/10.1108/eb005392.
[52] Goetschel, R., Voxman, W., Elementary fuzzy
calculus. Fuzzy Sets and Systems, 18, 1986, 31-
43. https://doi.org/10.1016/0165-
0114(86)90026-6.
[53] Moore, R.E., Interval Analysis. Prentice-Hall,
Englewood Cliffs, NJ, 1966, DOI:
10.1126/science.158.3799.365.
[54] Moore R. E., Kearfott R.B., Cloud M. J.,
Introduction to Interval Analysis, Society for
Industrial and Applied Mathematics,
Philadelphia, pp. 223, 2009.
https://doi.org/10.1137/1.9780898717716.
[55] Hanss M., Applied Fuzzy Arithmetic-An
Introduction with Engineering Applications,
Springer Berlin Heidelberg New York, pp.256,
2005. ISBN 3-540-24201-5.
[56] Moody, W.T., Nonlinear differential equation of
drain spacing, J. of Irrigation and Drainage
Engineering, ASCE, 92, IR2, 1966, 1-9.
https://doi.org/10.1061/JRCEA4.0000420.
[57] Nguyen H. T., A Note on the Extension
Principle for Fuzzy Sets, Journal of
Mathematical. Analysis and Applications, 64,
1978, 369-380. https://doi.org/10.1016/0022-
247X(78)90045-8.
Contribution of Authors
Christos Tzimopoulos: conceptualization,
methodology, software, validation, formal analysis,
resources, data curation, writing - original draft
preparation.
George Papaevangelou: investigation, visualization,
supervision, project administration, writing – review
and editing.
Sources of Funding
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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International Journal of Environmental Engineering and Development
DOI: 10.37394/232033.2024.2.8
Christos Tzimopoulos, George Papaevangelou
E-ISSN: 2945-1159
98
Volume 2, 2024
