Directory UMM :Data Elmu:jurnal:A:Agricultural Water Management:Vol42.Issue3.Jan2000:
Agricultural Water Management 42 (2000) 291±312
A geostatistical approach to optimize the determination
of saturated hydraulic conductivity for large-scale
subsurface drainage design in Egypt
Mahmoud M. Moustafa1
Faculty of Environmental Science and Technology, Okayama University, 2-1-1,
Tsushima-naka, 700-8530, Okayama, Japan
Accepted 5 March 1999
Abstract
Measurements of saturated hydraulic conductivity (Ks) in the field are costly, time-consuming, and
relatively cumbersome, chiefly as hydraulic conductivity exhibits a large spatial variability, so that it
becomes difficult to find accurate representative values to correctly predict soil-water flow and
design irrigation and drainage systems. Ks was measured in seven different soils in Egypt to evaluate
its spatial variability and to develop a model for estimating its representative value for a large-scale
subsurface drainage design. Published data from East Delta was also used. Results showed that the
spatial structure of Ks is characterized by a high nugget effect with a correlation range varying from
1600 to 2700 m and is fairly correlated with the agricultural practices and geologic nature of field
soils. Based on the concepts of geostatistics, a simple correlation model was developed for estimating
reliable and rapid representative values of Ks. The validity of that model was tested statistically and
on field data of one Nile Delta soil and one Nile Valley soil. The results indicated that the model will
be practically valuable for estimating the representative value of Ks that could be used in the drainage
design of small blocks or large areas. The model was applied to the design drain spacings used in
Egypt and to estimate the minimum sample size required for estimating a mean value of Ks at a given
precision level taking into account the spatial variability of Ks. The results showed that neglecting
spatial variability of Ks may affect the design drain spacing by ÿ27% to 3%, and overestimate the
required sample size by about 76%. Such a model may be regarded as a helpful tool for drainage
design oriented professionals without prior knowledge of geostatistics procedures. It is necessary to
adequately characterize large areas to which hydrologic models, which require Ks, are to be applied.
Furthermore, the magnitude of spatial dependence of Ks presented in this paper may be of great help
for a better understanding and modeling of water and solutes movement in, and through, the
agricultural clay soils in Egypt. # 2000 Elsevier Science B.V. All rights reserved.
Keywords: Saturated hydraulic conductivity; Subsurface drainage; Spatial variability; Correlation model
1
Present address. Egyptian Public Authority for Drainage Projects, 13 Giza St., Giza, Egypt
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 4 2 - 6
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
1. Necessity of the study
One of the main end products of any rational drainage design is a drain spacing. While
using the drain spacing formulae one encounters the problem that the natural conditions
seldom lend themselves to idealization. The saturated hydraulic conductivity (Ks) is of
utmost importance to drainage design and affects the economic and technical feasibility
of large-scale subsurface drainage projects. However, it is one of the most difficult factors
to evaluate in any drain spacing equation (Schwab et al., 1996). Its variability is often so
wide that it becomes difficult to find representative values to use in drain spacing
calculations and/or soil management (Van Schilfgaarde, 1970; Bouwer and Jackson,
1974; Topp et al., 1980; Puckett et al., 1985; Dorsey et al., 1990; Gupta et al., 1993;
Mohanty et al., 1994). Large number of measurements may, therefore, be required to
account for this variability, so that a reliable estimate of Ks might be obtained. These
measurements are not only costly but also time-consuming and relatively cumbersome.
However, the designer must have some confidence in the design value of Ks before he can
have confidence in the drainage design. The most effective way to guarantee this is to
calculate the Ks-value based on water-table measurements, where lateral drains are
already installed in the field (e.g. Skaggs, 1976; El-Mowelhi and van Schilfgaarde, 1982).
However, it is not practical to install lateral drains for the sole purpose of measuring Ks
chiefly on large-scale drainage projects.
To overcome such difficulties, it is prudent to develop simplified methods to provide
rapid and reliable representative values of Ks that are practically helpful, even though
somewhat approximate and semiempirical (Ahuja et al., 1984; Vereecken, 1995). In the
context of this paper, a reliable representative value means estimation of mean value at
acceptable precision level, that is at minimum estimation variance. The two most
complementary used approaches to achieve a rapid and less expensive hydraulic
conductivity characterization of soils have been: (i) development of simpler field methods
(e.g. Topp and Binns, 1976; Bouma and Dekker, 1981; Chong et al., 1981; Jones and
Wagenet, 1984; Reynolds et al., 1984), and (ii) estimation of Ks from other easily
obtainable soil properties (pedo-transfer functions) (e.g. Ahuja et al., 1989; Franzmeier,
1991; Jabro, 1992; Rawls et al., 1993; Vereecken, 1995). Albeit intuitive appeal and ease
of application, these methods are not sufficiently advanced to be practical for routine
measurements on a large scale, particularly the latter methods (see Tietje and Hennings,
1996). The spatial variability of Ks, and some other physical and chemical soil properties
as well, is still a great source of inaccuracy for estimating reliable representative Ksvalues obtained from these methods and an extensive data base describing such variability
may not be available in many practical cases. Tietje and Hennings (1996) showed that
pedo-transfer function is inaccurate, because of the spatial variability of Ks which should
be interpreted as a random variable.
Many attempts have been made to infer the spatial variability of Ks (e.g. Bakr et al.,
1978; Alemi et al., 1988; Mulla, 1988; Mohanty et al., 1991; Rogers et al., 1991;
Romano, 1993; Moustafa and Yomota, 1998) and only a few studies have incorporated
this spatial variability into subsurface drainage design (Prasher et al., 1984; Gallichand et
al., 1991, 1992). Nevertheless, none of them have been applied in practice as they require
large amounts of data which are seldom available in practical situations. In addition,
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
293
some of them require a prior knowledge of the geostatistics concepts at a level beyond
that of many designers. Therefore, the procedures of the studies cited above seem to be
difficult to use and not practically feasible chiefly with the routine design process of the
large-scale subsurface drainage projects in Egypt (69 000±72 000 ha/year). Rapid and
reliable methods for estimating representative Ks-values for soils in the field seem,
therefore, to be not only of a crucial importance for reliable drainage design on a large
scale but also for reliable practical applications of many simulation flow models to
environmental studies (Warrick and Nielsen, 1980; Vereecken, 1995). However, for a
method to be successful, it should be able to estimate representative Ks-value easily and
inexpensively, without sacrificing its accuracy. Bouma (1989) indicated that the major
challenge for soil science is to `translate' data we have to the data we need, if only
because there will not be funds available to obtain soil properties data on a large scale.
These requirements were the motivations for this study.
The purpose of this study was to develop a method, based on geostatistics concepts, for
estimating a rapid and reliable representative value of Ks, from limited in situ
measurements, suitable for the large-scale drainage design in Egypt taking into
consideration the spatial variability of Ks.
2. Procedures
2.1. Study areas and measurement technique
2.1.1. Study areas
The most important factor in selecting a technique for determining Ks of a soil is the
end application of the data. In this case, the focus is on subsurface drainage design. Field
measurements are, in principle, preferable to laboratory measurements as they reflect
better natural boundary conditions which govern flow processes in the field (Bouma,
1980). Ks was therefore measured in situ in five various soils (A1,. . .,A5) located in the
East (ED), Middle (MD), and West Delta (WD) of Egypt (Fig. 1). These measurements
along with the published data from East Delta (A6) (Gallichand et al., 1991) were used to
interpret the spatial variability of Ks in the Nile Delta and to develop the proposed
predictive model. In addition, to validate that model, Ks was also measured in another two
soils A7 and A8 located in Nile Delta and Middle Egypt (ME), respectively (Fig. 1).
Study areas A1 to A7 were chosen carefully to represent the most prevalent soil,
hydrological, and agricultural conditions in the Nile Delta, whereas study area A8 is
considered representative of the Nile Valley and was chosen to be outside the Nile Delta
to evaluate the general applicability of the model. In practice, physical constraints (e.g.
location of irrigation canals, roads, railways, villages, etc.) and natural boundaries of each
area determine its size and the number of measurements (Table 1).
With the year-round availability of water in Egypt, two or three crops a year could be
grown in the study areas based on a rotation system. In the areas of Nile Delta
(A1,. . .,A7), a typical three-year crop rotation includes rice, cotton, and maize in summer
and wheat and berseem (Egyptian clover) in winter. Excluding A4, which has no land is
cultivated with rice, and rice is planted on 20% (A5,A7), 40% (A6), and 50% (A1,A2,A3)
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Fig. 1. Location of the study areas.
of the land in the other areas (Table 1). In A8, sugar cane is the main crop, in addition to
maize in summer and wheat, berseem and vegetables in winter.
The geology of the Nile Delta and Nile Valley areas is broadly classified into two
geologic units: Nile River alluvium and undifferentiated basement rocks. The Nile River
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Table 1
Some characteristics of the study areas (for their locations, see Fig. 1)
Study
area
Location
Total area
(ha)
Number of
measurements
Rice land
(%)
Average depth of
clay-silt layer (m)
A1
A2
A3
A4
A5
A6
A7
A8
WDa
WDa
MDb
MDb
EDc
EDc
MDb
MEd
1533
1848
941
1260
1470
33500
1260
1260
61
71
36
63
61
3488
63
41
50
50
50
0
20
40
20
0
5
50
50
15
8
30
45
10
a
West Delta.
Middle Delta.
c
East Delta.
d
Middle Egypt.
b
alluvium consists of the Nile River sands and the clay-silt layer. The Nile River sands
consist predominantly of beds of coarse and fine sands and are mostly overlain by the
near surface clay-silt layer which acts as a cap to the aquifer. The thickness of the
alluvium in the Nile Valley ranges from about 20 m near Lake Nasser (Fig. 1) to about
300 m at A8, whereas its thickness in the Nile Delta ranges from about 200 to 300 m. The
average thickness of the clay-silt layer in the Nile Valley varies from about 4 m to about
14 m. Laterally, this thickness becomes thinner toward the fringes of the valley and may
be locally absent. The thickness of this layer in the Nile Delta gets thicker toward the
north, ranging from about 5 m near Cairo to >50 m near the Mediterranean Coast. The
thickness also decreases toward the Delta fringes. This clay-silt layer forms the fertile
agricultural lands of both, the Nile Delta and the Nile Valley with an average Ks varying
between 0.05 and 0.50 m/day. Its approximate average depth in the study areas is
presented in Table 1. The soils are of alluvial and alluvio-marine deposits, consisting of
loam in A8 and clay to clay-loam in the areas of the Nile Delta. The soils can be
characterized as moderate saline soils in A1 and A8 and non-saline soils in the other
study areas with a water-table depth ranging from 64 to 110 cm below soil surface. The
annual average rainfall in the Nile Delta is about 180 mm with a maximum value of
650 mm at A1 (1980±1994, St. 318 at Al-Nozhah, Alexandria). The rainfall intensity
sharply decreases toward the south to about 1.5 mm/year near A8.
2.1.2. Measurement technique
The measurements in all study areas were performed on a regular 500 m square grid in
8 cm diameter and 2 m deep holes using the auger hole method (Van Beers, 1983). An
auger hole is made and after 24 h, during which the water table in the hole is allowed to
reach an equilibrium with the natural groundwater of the soil, the test is carried out. The
depth of the water table is recorded, the water is then removed from the hole using a
bailer and the velocity by which the water flows back into the hole is observed. This
method gives an average value of Ks for a soil profile and it is considered to be the most
effective method for use in subsurface drainage applications in clay soils (FAO, 1976;
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Table 2
Statistics summary of saturated hydraulic conductivity measurements (m/day) of the study areas
Study area
Minimum
Maximum
Arithmetic mean
Variance
CV (%)
A1
A2
A3
A4
A5
A6a
A7
A8
0.028
0.178
0.171
0.150
0.010
0.001
0.020
0.006
2.755
2.726
2.144
1.920
0.650
3.740
1.900
1.585
1.029
0.928
0.766
0.643
0.107
0.302
0.506
0.295
0.609
0.227
0.235
0.097
0.018
0.123
0.197
0.210
76
51
63
49
126
116
88
155
a
Published data.
Bouwer, 1969) owing to the large volume of soil involved in Ks-measurement which
tends to reduce the variability in the data sets.
The data sets were subjected to standard exploratory data analysis (Table 2) to verify
the nature of the frequency distribution of Ks in the study areas. Data sets were tested for
normality at 95% significance level using the procedure of D'Agostino et al. (1990) and
their distributions were found to be lognormally distributed, which is in agreement with
the general thinking that the measurements of many natural phenomena tend to have a
lognormal distribution. Consequently, the geometric mean was justified for determining
the best representative value of Ks (Bouwer, 1969; Tietje and Hennings, 1996).
Since field sampling with the auger hole method is done in a shallow or upper aquifer
clay-silt layer, the variability is dominated by the heterogeneous in this layer. Reasons for
this behavior may be due to uneven breaking of soil structure due to swelling and
shrinkage process and tillage system at this shallow layer. However, Camp (1977) and
Dorsey et al. (1990) showed that the auger hole method gave Ks values, in clay soils of
alluvial nature, similar to those calculated from field observations of water table
drawdown and drain outflow, pumping-test and velocity permeameter methods. Reynolds
and Zebchuk (1996) found that the auger hole method and the Guelph permeameter
methods yielded similar geometric mean values as well as similar spatial variability
structures in a fine textured soil. This may suggest that the scale effects on Ks values and
spatial variability parameters might be neglected. This assertion may moreover be
confirmed by the results of Butler and Healey (1998a, b) who showed that the difference
between pumping-test and slug-test parameters in a two-dimensional flow system is much
more likely to be an artifact than a scale dependence in hydraulic conductivity. Further, in
a rigorous numerical investigation, Sanchez-Vila et al. (1996) indicated that there will be
no scale dependence in the two-dimensional flow systems when hydraulic conductivity
can be represented as a lognormal random field. Bouwer and Jackson (1974) evaluated
several techniques for determining Ks and concluded that the variability of the soil is a
much greater source of variation in determining Ks of an area than the variation related to
the measurement method. In addition, the 2-m sampling depth of auger holes was deeper
than the design drain depth used in Egypt (1.25±1.50 m) in a range of 0.50±0.75 m,
whereas Rogers and Carter (1987) recommended a sampling depth of 0.3 m below the
design drain depth in a similar soil of alluvial nature. This may suggest that the sampling
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
297
method used in this study may give information on Ks at the scale of the practical
problem.
Nevertheless, sampling dimensions, such as diameter and depth of auger hole and
depth of the water pumped from the hole, may have some effects on the measured Ksvalues (Rogers, 1986), and thus, all holes were augered to similar depths and diameters
and were bailed to similar depths. Moreover, in clay soils, serious errors may result if the
true water table level is not established for determinations with the auger hole method
(Kirkham, 1965). These problems in the study fields are unlikely to have occurred as all
tests were conducted one day after the holes were augered.
The water-table depth is controlled by Ks of the soil between two adjacent lateral
drains. Bouwer (1969), using electrical analog simulations, showed that the representative
Ks-value is best represented by the geometric mean of point hydraulic conductivities
within the flow domain. Using a perturbation method, Matheron (1967) proved that, for
the particular case of two-dimensional flow and the lognormal distribution of Ks, the
representative Ks-value is equal to the geometric mean of Ks-measurements. Since the
auger-hole method samples a relatively small volume of soil compared with the soil
volume between two lateral drains, it can be considered as a point measurement method.
Consequently, for each study area, the geometric mean of all the measured values was
assigned as the representative Ks-value within the flow domain of each area.
According to Van Beers (1983), the flow system around the auger hole was solved
numerically with the relaxation technique under specific dimensional conditions. With
water-table depths in the range of 64 to 110 cm in the study areas and 8 cm diameter and
2 m deep auger holes, these dimensional conditions were found to be valid in this study.
Since the design requirements are often not accurately known and other uncertainties,
such as the lower boundary of the flow system and the entrance resistance of the drains,
may exist, the dimensionality and scale of measurement technique of Ks were considered
sufficiently reasonable in this study. Further, the hydraulic conductivity data sets are
expected to be homogenous and consistent as the sampling method and density were the
same in all areas of study.
2.2. Regionalized variable theory (RVT)
2.2.1. Spatial measure and kriging technique
Geostatistics approach utilizes the fact that variations of soil properties are not always
random, but have some spatial structures. It is based on the regionalized variable theory
(RVT) which takes into account both, the random and structured characteristics of
spatially distributed variables to provide quantitative tools, which are the most commonly
used methods in analysis of soil variability, for their description and optimal-unbiased
estimation.
Consider a field of area A, for which a set of n values were measured [z(xi), i 1,n], in
which each xi identifies a coordinate position in the space. Each z(xk) can be considered a
particular realization of a certain random variable, Z(xk), for a particular fixed point, xk.
The regionalized variable Z(xi), for all xi inside A, can be considered a realization of the
set of random variables [Z(xi), for all xi inside A]. This set of random variables is called a
random function (Journel and Huijbregts, 1978). Application of RVT assumes that the
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
covariance between any two locations in A depends only on the distance and direction of
separation between the two locations and not on their geographic location. Based on this
assumption, the average covariance for each lag distance can be estimated for a given
volume of three-dimensional space. Among others, the covariance variogram (referred to,
hereafter, as `variogram') is a measure for the correlation between measurements of a
regionalized variable and may be written as (Moustafa and Yomota, 1998):
v h C 0 ÿ C h
(1)
N h
C h
1 X
xi hz xi ÿ m1 m2
N h i1
(2)
where
v h is the variogram at a separation distance h for a sampling volume v, C(0) is
the sample variance, and C(h) the covariance function at a separation vector h, m1 and m2
denote the means of the data values z(xi h) and z(xi), respectively, and they are equal
under the assumption of second-order stationarity. N(h) is the number of pairs of
measurements [z(xi), z(xi h)]. As for any vectorial function,
v h depends on both the
magnitude and direction of h. When the value of the variogram depends upon the
direction of h, anisotropic conditions exist. For such cases, functions describing the
experimental anisotropic variogram are often submitted to a transformation to yield
isotropic behavior.
The main advantages in using variogram of Eq. (1) for interpreting spatial structure of
Ks and performing kriging are (Moustafa and Yomota, 1998):
(i) it incorporates possible non-stationarity in the data set and, hence, reveals better the
character of its spatial structure comparing to the traditional spatial function (i.e. semivariogram);
(ii) it squeezes the effect of preferential sampling of data set;
(iii) its spatial modeling has a lower standard error compared with the traditional
spatial function; and
(iv) it yields more robust solution of the kriging system and savings in terms of
computer time compared with the traditional function.
A positive definite continuous function (Journel and Huijbregts, 1978),
1 h, must be
fitted to the experimental variogram, computed using Eq. (1), to characterize the spatial
structure of the regionalized variable studied and to assure the mathematical consistency
required for kriging estimations. Kriging is a technique for calculating optimal and
unbiased linear estimation of a soil property at an unsampled location, z*(x0), with
minimum estimation variance:
z x0
N
X
i z xi
(3)
i1
where N is the number of neighboring sampled points used for estimation, and i is the
weight applied to the neighboring sample z(xi). The weights are chosen so that z*(x0) is an
unbiased estimate:
Ez x0 ÿ z x0 0
(4)
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
299
and the variance between z*(x0) and the true value of the soil property at point x0 is
minimized:
var z x0 ÿ z x0 minimum
(5)
The weights placed on each neighboring sample sum to unity, and their unique
combination for which estimation variance is minimized yields the kriging system:
N
X
j C xi ; xj ÿ C xi ; x0
for
i 1; 2; . . . N;
j1
X
j 1
(6)
and the minimum estimation variance or kriging variance, 2k , is given by:
2k x0 C 0 ÿ
N
X
i C xi ; x0
(7)
i1
The values C(xi,xj) and C(xi,x0) are the covariance functions between observed
locations xi and xj and between the observed location xi and the interpolated location x0,
respectively. This system consists of (N 1) linear equations and (N 1) unknowns (N
weights, i and one Lagrangian multiplier, ).
The validation of kriging estimates can then be done by the cross-validation method
(Kitanidis, 1993). The criteria for validation depends on the values of reduced mean error
and reduced variance which must be close to zero (no systematic error) and one
(consistency between the kriging variance and the squared error), respectively. Standard
deviation of the estimation error (e) and the percentage of errors lying within 2e can
also be calculated to assess the minimum variance condition of the kriged estimates. This
requires that the percentage of errors should be 95 or more under the assumption of
normality (Journel and Huijbregts, 1978).
2.2.2. Regularization effect
Measurements are based on samples with a certain support size. The scale at which
these measurements are made is called the support scale. According to Journel and
Huijbregts (1978), such support affects both, the sill and the range of the variogram, and
as a result, the kriging estimates might not be accurate. The process of measuring a
regionalized variable with a certain support size is called regularization. It is imperative,
therefore, to take into considerations such regularization effect on the variogram
modeling.
For a practical approximation, the relation between the regularized variogram,
v h, and the point variogram,
*(h), may be written as (Journel and Huijbregts,
1978):
v h
h ÿ
v; v
for h v
(8)
where
(v,v) is a constant term related to the dimensions and geometry of the sample
volume, v, of the regularization. Although it seems imperative to consider this
regularization effect on the variogram modeling, hence on the kriging estimates, it is
expected that this effect within the sampled data sets in this study may not affect the
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
variogram parameters, owing largely to the small sample volume used to measure
hydraulic conductivity (0.01 m3), which included less of the overall spatial variability
than larger volumes. Moreover, the sampling dimensions in the horizontal plane
(500 500 m2) used to estimate and model the variograms are much larger (250-fold)
than those of the vertical direction. As a check, the procedures suggested by Journel and
Huijbregts (1978) were carried out for consideration of such an effect on variogram
modeling, hence on kriging estimates.
2.2.3. Problem statement
Since the field-measured Ks in the Nile Delta was found spatially variable (Gallichand
et al., 1991; Moustafa and Yomota, 1998), the question becomes the following: given the
measured Ks-values which are spatially variable, how do we rapidly arrive at a reliable
representative value to use in drain-spacing calculations of a region or at a correction
factor (Cf) which should be introduced to any steady or unsteady-state drain-spacing
equation to account for the effect of spatial variability of measured Ks-values on the
calculated drain spacing?
One way to provide this information is through the application of RVT with the use of
kriging technique, which gives an optimal and unbiased estimation of Ks at unsampled
locations with minimum estimation variance. Hence, the interpolated values at unsampled
locations may be considered as the most accurate values that can be obtained from limited
in situ measurements and those that can be used with known confidence. Estimating
representative Ks-values for soils, accordingly, can be improved using the kriging
technique. Gallichand et al. (1991) indicated that the geometric mean of kriged Ks-values
(Krk) is the most accurate representative Ks-value for use in subsurface drainage design.
This study aims, therefore, to calculate the magnitude of spatial correlation of the
measured Ks-values of different study areas, from which a way for direct and rapid
calculation of Krk might be developed.
2.3. Developing predictive model for Krk
2.3.1. Model verification
In order to develop a predictive model for Krk, experimental variograms of study areas
(A1,. . .,A5) were calculated and theoretical spatial dependence functions were fitted to
them chiefly to explore the behavior of the variograms between the origin and the range
of dependency. Then, combining the calculated spatial dependence and the weighted
influence of nearby points, the measured Ks-values were interpolated at 50 m intervals by
the ordinary kriging technique. Fifty-meter intervals were selected to represent, as far as
possible, the Ks-values within the flow domain, since the practical design lateral drain
spacing in Egypt ranges from 30 to 100 m.
Representative kriged Ks-values (Krk1,. . .,Krk5) of study areas were then estimated by
the geometric mean of all the kriged values within each area and the best relationship
between them and their measured values (Krm1,. . .,Krm5) were verified. After establishing
such a relationship, a predictive model was developed using regression procedure for a
direct and rapid estimation of Krk as the most reliable representative Ks-value from the
limited in situ measurements.
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
301
2.3.2. Model validation
Several statistical evaluation procedures were employed to quantify the power of the
performance of the model. These included: correlation coefficient (R), which measures
the degree of association between kriged and their predictive values; the mean difference
between kriged and their predictive values (Md), which measure the degree of
coincidence; the relative error (Er); and comparison of the statistics for the kriged and
predicted results (mean, standard deviation).
In order to assess the validation of the model on field data, RVT procedures were
performed for another two areas (A7,A8), whose data were not used in the model
prediction. Representative kriged Ks-values of A7 and A8 (Krk7 and Krk8) were then
calculated and compared with their predicted values from the model.
3. Results and discussion
3.1. Spatial dependence of saturated hydraulic conductivity
The experimental variograms for Ks-measurements of the study areas were found to be
represented by a spherical spatial function:
"
#
3 h
1 h 3
ÿ
0ha
1 h c0 c1
2 a
2 a
1 h c0 c1
h>a
(9)
where
1 h is the estimated variogram for lag h, c0 is the nugget, c1 is the structural
component, c0 c1 is the sill (cs), h is the distance between measurement points (lag),
and a is the correlation range. Published data by Gallichand et al. (1991) showed that the
frequency distribution of Ks in A6 is a lognormal distribution, and hence the experimental
variogram was calculated by the semi-variogram of log-transforms of Ks-measurements
and was fitted to an exponential spatial function:
ÿh
1 h c0 c1 1 ÿ exp
(10)
a0
where a0 is a range parameter (approximately one-third of the apparent range, a). The
calculated spatial parameters of this function were c0 0.886, c1 0.413, and
a 5599 m.
Whether a distribution is normal or lognormal, has no particular significance on the
variogram estimation, except that it is often more difficult to interpret variograms of
highly skewed distributions, such as Ks data sets. In a recent study, Moustafa and Yomota
(1998) showed that the variogram of Eq. (1) reveals better the character of spatial
structure of highly skewed distributions of Ks data sets in the Nile Delta, using the
original data set, than the semi-variogram of log-transforms of the data. Hence, in this
study, all the experimental variograms were estimated using the variogram of Eq. (1) with
the original data sets. Variography for Ks-measurements of study areas is shown in Figs. 2
and 3 and all parameters of spherical function are presented in Table 3.
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Fig. 2. Experimental variograms and the fitted spherical spatial functions of the study areas used for model
prediction.
Table 3
Spatial parameters of spherical function fitted to experimental variograms of the study areas
Study area
c0
c1
cs
A1
A2
A3
A4
A5
A7a
A8a
0.250
0.158
0.143
0.035
0.014
0.146
0.001
0.400
0.080
0.124
0.086
0.004
0.064
0.210
0.650
0.238
0.267
0.121
0.018
0.210
0.211
a
Used for model validation.
a (m)
2000
1700
1600
2500
2700
2000
1800
c0/cs (%)
38
66
54
29
78
70
0.5
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
303
Fig. 3. Experimental variograms and the fitted spherical spatial functions of the study areas used for model
validation.
All these variograms exhibit similar behavior showing nugget effects (c0) and approach
the sill (cs) at different values. The nugget effect quantifies the amount of covariance not
explained as spatial correlation, due chiefly to measurement errors and variations that
occur over distances smaller than the sampling distance of 500 m. The spatial structure of
Ks in the Nile Delta (A1,. . .,A5,A7) was characterized by a high nugget effect with a
relative value (c0/cs) being on average equal to about 56%. In contrast, the spatial
structure of Ks in the Nile Valley (A8) has a low nugget value (Table 3). This is not
surprising, since it may be interpreted in terms of the influence of weathering conditions
and agricultural practices in both the regions.
The variogram range (a) represents the distance beyond which values of the
regionalized variable are no longer autocorrelated, and therefore the measurements can be
assumed to be randomly distributed. A degree of autocorrelation exists between Ksmeasurements at a range varying from 1600 to 2700 m, confirming that Ks in the Nile
Delta is spatially variable with spatial structures characterized by a high nugget effect.
Values of the variance (Table 2) and the sill, cs, (Table 3) for different study areas,
moreover, are comparable, thereby practically confirming absence of trends in the data
sets. A visual examination of the experimental variograms at the azimuths 08, 458, 908
and 1358, with angular regularization in each direction of 458, indicated that anisotropy of
Ks was not present. This was in agreement with the results obtained by Gallichand et al.
(1991) and Moustafa and Yomota (1998).
A perusal of the information presented in Tables 1 and 3 showed that non-rice areas
(A4,A8) have the lower relative nugget effect (c0/cs) values compared with those of rice
areas. In rice areas, a non-uniform increase of relative nugget effect and correlation range
(a) with the decrease of rice land was also observed. On the other hand, the structured
component (c1) was found increasing with the increase of rice lands, with no correlation
to the depth of the near surface clay-silt layer. c0/cs can be generally decreased with the
decrease of clay-silt layer depth as indicated by c0/cs-values of A1, A4, and A8, compared
to those of the other areas. However, different scale of variation may also be found as in
the case of A5. In the Nile Delta, the correlation range non-uniformly increases with the
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
decrease of clay-silt layer depth. These findings may conclude that the spatial variability
of Ks is fairly correlated with the geologic nature and agricultural practices in the study
areas. The spatial variation of Ks can, moreover, be compounded by the heterogeneous
nature of field soils, and thus different spatial structures and scale of variability might be
obtained.
3.2. Regularization
In order to assess the sample volume effect on the variogram modeling, the
deregularization procedure (see Journel and Huijbregts, 1978) was carried out to deduce
the parameters of the point spatial theoretical models from the calculated regularized
models. The results showed that the deregularized parameters were almost the same as
the previous calculated theoretical model parameters given in Table 3. The differences
are insignificant (0.03±0.08%). These results confirm, therefore, the previous assertion
that the regularization effect on the modeling of variogram could be neglected, and thus
the auger hole method can be considered as a point measurement method. This may
reflect that the sampling technique used in this study is highly reasonable with RVT
procedure to estimate accurate kriged estimates which, in turn, assure accurate estimates
of representative Ks-values for different study areas.
3.3. Kriging estimates and model development
Number of points used in kriging estimation were determined by the search radius
(Vieira et al., 1981). The determined search radii of study areas are presented in Table 4.
Using the fitted theoretical spatial functions to the experimental variograms and the
search radii, kriged estimates were validated by the cross-validation technique. The mean
reduced error was 0, whereas the mean reduced variance was almost equal to unity
(Table 4), suggesting that the variogram is appropriate and the kriging process is
performed consistently. The hypothesis of normally distributed kriging errors are tested
Table 4
Search radii and results of cross-validation of kriging estimates
Study
area
Search
radius (m)
Reduced
meana
Reduced
varianceb
Percentage
errors within 2e
A1
A2
A3
A4
A5
A6
A7
A8
2000
1000
1000
1750
1500
1200
1250
1000
ÿ0.0003
ÿ0.007
0.001
0.006
ÿ0.009
ÿ0.0004
0.004
ÿ0.008
0.998
1.005
1.001
1.007
1.005
1.012
1.007
1.013
94
95
98
98
96
Ðc
97
96
a
Average of standardized residuals.
Average of squared standardized residuals.
c
Not available.
b
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Table 5
Representative measured (Krm) and kriged (Krk) saturated hydraulic conductivity (m/day) used for model
prediction
Study area
Krma
Krkb
A1
A2
A3
A4
A5
A6
0.731
0.797
0.627
0.578
0.066
0.250c
0.800
0.819
0.691
0.531
0.097
0.236c
a
b
c
Geometric mean of the measured values.
Geometric mean of kriged estimates.
Average of 25 blocks ranging in size between 80 and 210 ha.
and the results are also presented in Table 4. This was verified by the percentage (95%)
of estimation errors lying within 2e. These results indicate that kriging estimates are
truly values with minimum variances, and thus they may represent truly representative
values of Ks at reasonable and acceptable accuracy.
For study areas (A1,. . .,A5), Ks-measurements were interpolated at 50-m intervals by
the ordinary kriging technique. The representative measured (Krm1,. . .,Krm5) and kriged
(Krk1,. . .,Krk5) values were then determined by the geometric mean of all the measured
and kriged values, respectively. The results are presented in Table 5 along with the
published data of A6.
The calculated Krk-values were plotted against Krm-values to establish the relationship
between them (Fig. 4). To deduce a model that can be applied for small blocks (drawing
areas) and/or large areas, the data of 25 blocks (80±210 ha) of A6 were used (Fig. 4)
instead of only one average value for the whole area. Using the least-squares procedure,
the best estimates (Krkp) for Krk were found to be given by a predictive model in a form of
a linear relationship as:
p
0:925 Krm 0:016
Krk
(11)
The coefficient of determination (R2) of this model was highly significant (R2 0.924,
F 0.0004). Having tested the adequacy of kriging estimates, this equation would be
useful for estimating, with reasonable accuracy, a rapid and reliable representative
saturated hydraulic conductivity directly for an area from limited field measurements for
use in a subsurface drainage design and/or in the analysis of any saturated-soil water-flow
system.
3.4. Model validation
The representative measured Ks-values (Krm) were used to estimate their predicted
kriged values (Krkp) using the developed model of Eq. (11). The actual kriged values
(Krk) were then compared with those predicted from the model (Krkp). The mean and
standard deviation of Krkp-values were 0.295 and 0.191 m/day, respectively, whereas they
were 0.295 and 0.199 m/day for Krk-values, indicating excellent agreement between the
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Fig. 4. Relation between representative measured (Krm) and kriged (Krk) values of saturated hydraulic
conductivity.
values of Krkp and Krk with a very high correlation (R 0.961) between them (Fig. 5).
The mean (Md) and standard deviation of the difference between the values of Krkp and
Krk were 0.0001 and 0.055, respectively, whereas their mean relative error (Er) was
ÿ0.62%.
Fig. 5. Relation between representative kriged (Krk) values of saturated hydraulic conductivity and those
predicted (Krkp) from the developed model of Eq. (11).
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
307
Table 6
Validation results of the predictive model on field data
Study area
A7
A8
Measured values (Krm)
0.335
0.096
Kriged values (Krk)
0.401
0.131
Predicted kriged values (Krkp)
0.326
0.105
The foregoing statistical tests have established that a relation exists between the
representative measured and kriged values of Ks in the Nile Delta and the actual kriged
values are comparable to those predicted from the developed model, indicating that the
model is performed well.
In order to assess the validity of the model on field data, kriging procedures were
performed for another two areas, A7 and A8. Their representative kriged values
(Krk7,Krk8) were then calculated and compared with their predictive kriged values
(Krk7p,Krk8p) from the model. Results are presented in Table 6 and are very satisfactory.
Based on these results, and from a practical point of view, it can be concluded that the
developed model might be used to determine directly and rapidly the representative value
of Ks in a region with reasonable accuracy. It is encouraging that one equation in the form
of Eq. (11) may be applied not only in the areas of the Nile Delta but also in areas having
different spatial structures in the Nile Valley. Therefore, the designers may find this
equation helpful to determine an accurate estimate of representative saturated hydraulic
conductivity required for a subsurface drainage design without extensive field
measurements. Furthermore, the advantage of this equation is that an estimate of
accurate representative Ks can be obtained, from limited in situ measurements, simpler
and quicker than by obtaining through the application of RVT procedures for every
drainage area which will be cumbersome and not practically feasible with the large scale
of on-going drainage projects in Egypt.
The developed model was built based on in situ measurements and quantitative random
and structured variation information of Ks in areas of different sizes (80±1848 ha). As a
result, it is expected to estimate a representative value which captures, with sufficiently
and acceptable accuracy, the hydraulic function of the drainage system. Tietje and
Hennings (1996) indicated that Ks is increasingly accepted to estimate as a random
variable depending on method and scale of the measurements and on the spatial
variability.
3.5. Practical application
The Hooghoudt's steady-state equation has been used in Egypt to calculate drain
spacing. This equation disregards radial flow to the drains and uses instead an equivalent
horizontal flow with a barrier at a reduced depth called the `equivalent depth' (de). The
equation is relatively simple and has been widely applied with reasonable and acceptable
accuracy (Moustafa, 1997) and has the form:
L2 frm ; q; de ; h
(12)
where L is the drain spacing (m), q is the steady outflow rate of the system (m/day), h is
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
the water-table height midway between drains (m), and Krm is the geometric mean of the
measured values of Ks. This equation in its present form does not take into consideration
the spatial variability of Ks; therefore, RVT procedures should be applied to the Ksmeasurements to yield a reliable representative value taking into account such variability
of measurements. In such case, the developed model of Eq. (11) is a helpful tool to
provide this information (Krkp) quickly with a reasonable accuracy and the resulting drain
spacing (L1) can be written as:
p
; q; de ; h
L21 f Krk
(13)
Another alternative way is to introduce a correction factor (Cf) directly to Eq. (12) to
account for spatial variability of Ks-measurements as:
L21 Cf L2
(14)
where Cf can be calculated from the developed model as:
Cf 0:925 0:016 Krm ÿ1
(15)
Fig. 6 illustrates the relation between Cf and the design range of the representative
measured saturated hydraulic conductivity (Krm), based on the Egyptian conditions.
Results reveal that neglecting of spatial variability of saturated hydraulic conductivity in
the drainage design affects the design drain spacing in a range varying from ÿ27% to 3%,
whereas its incorporation in the design assures an accurate drainage design with less costs
which can be considered to be important economically, chiefly with the large scale of
subsurface drainage design in Egypt. Based on these results, Cf was found to vary
between 1.62 and 0.96.
Fig. 6. Correction factor (Cf) for large-scale drain spacing design to account for spatial variability of fieldmeasured saturated hydraulic conductivity.
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Table 7
Minimum sample size of Ks in the study area A1 to obtain sample mean within 5%, 10%, and 15% of the
true mean
Probability
level (%)
95
90
Using measured
values (Krm)
Using kriged
values (Krk)
Using predicted kriged
values (Krkp)
5%
10%
15%
5%
10%
15%
5%
10%
15%
881
621
220
155
98
69
236
166
59
42
26
19
213
150
53
37
24
17
In summary, based on the developed model, Eqs. (13) and (14) are two equivalent
effective tools to give a rapid and accurate drainage design on a large scale owing to the
fact that they incorporate the random and structured variation of in situ saturated
hydraulic conductivity measurements into the calculated drain spacing.
Another example is given in Table 7 to realize the practical relevance of the developed
model in estimating minimum sample size of Ks in the study area A1. The minimum
sample size was calculated at 95% and 90% probability levels and allowing 5%, 10%,
and 15% errors around the true mean using Student's t-test. The sample size was lower
with the actual kriged values of Ks (Krk) (ÿ73%) and their predicted values from the
model (Krkp) (ÿ76%) than that associated with the original measured values (Krm) at any
given probability and precision levels. The sample size decreased with decreasing
precision and probability levels. The minimum sample size required using Krm was higher
than the present sample size of 61 in A1 at all the probability and precision levels,
whereas it was lower than the present sample size using Krk and Krkp at the precision
levels of 10% and 15%. At the highest precision level (5%), their minimum sample
sizes were higher than the present sample size. Based on these results, it has shown that
the developed model is a very useful tool for predicting reliable estimates of Ks at
minimum costs. This is an important issue for the large-scale subsurface drainage design
in Egypt.
4. Conclusion
This study presented a method, based on the concepts of geostatistics, for developing a
model, from which a rapid and reliable representative value of saturated hydraulic
conductivity from in situ measurements can be estimated and be used in the large-scale
subsurface drainage design of Egypt. The model was validated statistically and on field
data of two different soils. The results were encouraging. The obvious advantage of using
such a predictive model arises from the fact that a large number of field measurements of
saturated hydraulic conductivity is costly, time consuming, and cumbersome, whereas the
model provides a means for predicting reliably and rapidly the best estimate possible of
the representative value from limited in situ measurements. Notwithstanding, further
validation tests may be needed to further confirm the reliability of predictions chiefly in
areas outside the Nile Delta region.
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Acknowledgements
The author is grateful to engineers of the Egyptian Public Authority for Drainage
Projects (EPADP) who did the field work of this study. He also would like to thank the
Editor-in-Chief and two anonymous reviewers for providing comments and suggestions
that significantly improved the quality of this paper.
References
Ahuja, L.R., Naney, J.W., Green, R.E., Nielsen, D.R., 1984. Macroporosity to characterize spatial variability of
hydraulic conductivity and effects of land management. Soil Sci. Soc. Am. J. 48, 699±702.
Ahuja, L.R., Cassel, D.K., Bruce, R.R., Barnes, B.B., 1989. Evaluation of spatial distribution of hydraulic
conductivity using effective porosity data. Soil Sci. 148(6), 404±411.
Alemi, M.H., Shahriari, M.R., Nielsen, D.R., 1988. Kriging and cokriging of soil water properties. Soil Techn. 1,
117±132.
Bakr, A.A., Gelhar, L.W., Gutjahr, A.L., MacMillan, J.R., 1978. Stochastic analysis of spatial variability
in subsurface flows. 1. Comparison of one- and three-dimensional flows. Water Resour. Res. 14(2),
263±271.
Bouma, J., 1980. Field measurement of soil hydraulic properties characterizing water movement through
swelling clay soils. J. Hydrol. 45, 149±158.
Bouma, J., 1989. Using soil survey data for quantitative land evaluation. Adv. Soil Sci. 9, 177±213.
Bouma, J., Dekker, L.W., 1981. A method of measuring the vertical and horizontal Ksat of clay soils with
macropores. Soil Sci. Soc. Am. J. 45, 662±663.
Bouwer, H., 1969. Planning and interpreting soil permeability measurements. J. Irrig. Drain. Div., ASCE 95(3),
391±402.
Bouwer, H., Jackson, R.D., 1974. Determining soil properties. In: van Schilfgaarde, J. (Ed.), Drainage for
Agriculture. American Soc. of Agronomy, Madison, WI, pp. 611±674.
Butler, J.J. Jr., Healey, J.M., 1998a. Relationship between pumping-test and slug-test parameters: Scale effect or
artifacts?. Ground Water, 3
A geostatistical approach to optimize the determination
of saturated hydraulic conductivity for large-scale
subsurface drainage design in Egypt
Mahmoud M. Moustafa1
Faculty of Environmental Science and Technology, Okayama University, 2-1-1,
Tsushima-naka, 700-8530, Okayama, Japan
Accepted 5 March 1999
Abstract
Measurements of saturated hydraulic conductivity (Ks) in the field are costly, time-consuming, and
relatively cumbersome, chiefly as hydraulic conductivity exhibits a large spatial variability, so that it
becomes difficult to find accurate representative values to correctly predict soil-water flow and
design irrigation and drainage systems. Ks was measured in seven different soils in Egypt to evaluate
its spatial variability and to develop a model for estimating its representative value for a large-scale
subsurface drainage design. Published data from East Delta was also used. Results showed that the
spatial structure of Ks is characterized by a high nugget effect with a correlation range varying from
1600 to 2700 m and is fairly correlated with the agricultural practices and geologic nature of field
soils. Based on the concepts of geostatistics, a simple correlation model was developed for estimating
reliable and rapid representative values of Ks. The validity of that model was tested statistically and
on field data of one Nile Delta soil and one Nile Valley soil. The results indicated that the model will
be practically valuable for estimating the representative value of Ks that could be used in the drainage
design of small blocks or large areas. The model was applied to the design drain spacings used in
Egypt and to estimate the minimum sample size required for estimating a mean value of Ks at a given
precision level taking into account the spatial variability of Ks. The results showed that neglecting
spatial variability of Ks may affect the design drain spacing by ÿ27% to 3%, and overestimate the
required sample size by about 76%. Such a model may be regarded as a helpful tool for drainage
design oriented professionals without prior knowledge of geostatistics procedures. It is necessary to
adequately characterize large areas to which hydrologic models, which require Ks, are to be applied.
Furthermore, the magnitude of spatial dependence of Ks presented in this paper may be of great help
for a better understanding and modeling of water and solutes movement in, and through, the
agricultural clay soils in Egypt. # 2000 Elsevier Science B.V. All rights reserved.
Keywords: Saturated hydraulic conductivity; Subsurface drainage; Spatial variability; Correlation model
1
Present address. Egyptian Public Authority for Drainage Projects, 13 Giza St., Giza, Egypt
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 4 2 - 6
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
1. Necessity of the study
One of the main end products of any rational drainage design is a drain spacing. While
using the drain spacing formulae one encounters the problem that the natural conditions
seldom lend themselves to idealization. The saturated hydraulic conductivity (Ks) is of
utmost importance to drainage design and affects the economic and technical feasibility
of large-scale subsurface drainage projects. However, it is one of the most difficult factors
to evaluate in any drain spacing equation (Schwab et al., 1996). Its variability is often so
wide that it becomes difficult to find representative values to use in drain spacing
calculations and/or soil management (Van Schilfgaarde, 1970; Bouwer and Jackson,
1974; Topp et al., 1980; Puckett et al., 1985; Dorsey et al., 1990; Gupta et al., 1993;
Mohanty et al., 1994). Large number of measurements may, therefore, be required to
account for this variability, so that a reliable estimate of Ks might be obtained. These
measurements are not only costly but also time-consuming and relatively cumbersome.
However, the designer must have some confidence in the design value of Ks before he can
have confidence in the drainage design. The most effective way to guarantee this is to
calculate the Ks-value based on water-table measurements, where lateral drains are
already installed in the field (e.g. Skaggs, 1976; El-Mowelhi and van Schilfgaarde, 1982).
However, it is not practical to install lateral drains for the sole purpose of measuring Ks
chiefly on large-scale drainage projects.
To overcome such difficulties, it is prudent to develop simplified methods to provide
rapid and reliable representative values of Ks that are practically helpful, even though
somewhat approximate and semiempirical (Ahuja et al., 1984; Vereecken, 1995). In the
context of this paper, a reliable representative value means estimation of mean value at
acceptable precision level, that is at minimum estimation variance. The two most
complementary used approaches to achieve a rapid and less expensive hydraulic
conductivity characterization of soils have been: (i) development of simpler field methods
(e.g. Topp and Binns, 1976; Bouma and Dekker, 1981; Chong et al., 1981; Jones and
Wagenet, 1984; Reynolds et al., 1984), and (ii) estimation of Ks from other easily
obtainable soil properties (pedo-transfer functions) (e.g. Ahuja et al., 1989; Franzmeier,
1991; Jabro, 1992; Rawls et al., 1993; Vereecken, 1995). Albeit intuitive appeal and ease
of application, these methods are not sufficiently advanced to be practical for routine
measurements on a large scale, particularly the latter methods (see Tietje and Hennings,
1996). The spatial variability of Ks, and some other physical and chemical soil properties
as well, is still a great source of inaccuracy for estimating reliable representative Ksvalues obtained from these methods and an extensive data base describing such variability
may not be available in many practical cases. Tietje and Hennings (1996) showed that
pedo-transfer function is inaccurate, because of the spatial variability of Ks which should
be interpreted as a random variable.
Many attempts have been made to infer the spatial variability of Ks (e.g. Bakr et al.,
1978; Alemi et al., 1988; Mulla, 1988; Mohanty et al., 1991; Rogers et al., 1991;
Romano, 1993; Moustafa and Yomota, 1998) and only a few studies have incorporated
this spatial variability into subsurface drainage design (Prasher et al., 1984; Gallichand et
al., 1991, 1992). Nevertheless, none of them have been applied in practice as they require
large amounts of data which are seldom available in practical situations. In addition,
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
293
some of them require a prior knowledge of the geostatistics concepts at a level beyond
that of many designers. Therefore, the procedures of the studies cited above seem to be
difficult to use and not practically feasible chiefly with the routine design process of the
large-scale subsurface drainage projects in Egypt (69 000±72 000 ha/year). Rapid and
reliable methods for estimating representative Ks-values for soils in the field seem,
therefore, to be not only of a crucial importance for reliable drainage design on a large
scale but also for reliable practical applications of many simulation flow models to
environmental studies (Warrick and Nielsen, 1980; Vereecken, 1995). However, for a
method to be successful, it should be able to estimate representative Ks-value easily and
inexpensively, without sacrificing its accuracy. Bouma (1989) indicated that the major
challenge for soil science is to `translate' data we have to the data we need, if only
because there will not be funds available to obtain soil properties data on a large scale.
These requirements were the motivations for this study.
The purpose of this study was to develop a method, based on geostatistics concepts, for
estimating a rapid and reliable representative value of Ks, from limited in situ
measurements, suitable for the large-scale drainage design in Egypt taking into
consideration the spatial variability of Ks.
2. Procedures
2.1. Study areas and measurement technique
2.1.1. Study areas
The most important factor in selecting a technique for determining Ks of a soil is the
end application of the data. In this case, the focus is on subsurface drainage design. Field
measurements are, in principle, preferable to laboratory measurements as they reflect
better natural boundary conditions which govern flow processes in the field (Bouma,
1980). Ks was therefore measured in situ in five various soils (A1,. . .,A5) located in the
East (ED), Middle (MD), and West Delta (WD) of Egypt (Fig. 1). These measurements
along with the published data from East Delta (A6) (Gallichand et al., 1991) were used to
interpret the spatial variability of Ks in the Nile Delta and to develop the proposed
predictive model. In addition, to validate that model, Ks was also measured in another two
soils A7 and A8 located in Nile Delta and Middle Egypt (ME), respectively (Fig. 1).
Study areas A1 to A7 were chosen carefully to represent the most prevalent soil,
hydrological, and agricultural conditions in the Nile Delta, whereas study area A8 is
considered representative of the Nile Valley and was chosen to be outside the Nile Delta
to evaluate the general applicability of the model. In practice, physical constraints (e.g.
location of irrigation canals, roads, railways, villages, etc.) and natural boundaries of each
area determine its size and the number of measurements (Table 1).
With the year-round availability of water in Egypt, two or three crops a year could be
grown in the study areas based on a rotation system. In the areas of Nile Delta
(A1,. . .,A7), a typical three-year crop rotation includes rice, cotton, and maize in summer
and wheat and berseem (Egyptian clover) in winter. Excluding A4, which has no land is
cultivated with rice, and rice is planted on 20% (A5,A7), 40% (A6), and 50% (A1,A2,A3)
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Fig. 1. Location of the study areas.
of the land in the other areas (Table 1). In A8, sugar cane is the main crop, in addition to
maize in summer and wheat, berseem and vegetables in winter.
The geology of the Nile Delta and Nile Valley areas is broadly classified into two
geologic units: Nile River alluvium and undifferentiated basement rocks. The Nile River
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Table 1
Some characteristics of the study areas (for their locations, see Fig. 1)
Study
area
Location
Total area
(ha)
Number of
measurements
Rice land
(%)
Average depth of
clay-silt layer (m)
A1
A2
A3
A4
A5
A6
A7
A8
WDa
WDa
MDb
MDb
EDc
EDc
MDb
MEd
1533
1848
941
1260
1470
33500
1260
1260
61
71
36
63
61
3488
63
41
50
50
50
0
20
40
20
0
5
50
50
15
8
30
45
10
a
West Delta.
Middle Delta.
c
East Delta.
d
Middle Egypt.
b
alluvium consists of the Nile River sands and the clay-silt layer. The Nile River sands
consist predominantly of beds of coarse and fine sands and are mostly overlain by the
near surface clay-silt layer which acts as a cap to the aquifer. The thickness of the
alluvium in the Nile Valley ranges from about 20 m near Lake Nasser (Fig. 1) to about
300 m at A8, whereas its thickness in the Nile Delta ranges from about 200 to 300 m. The
average thickness of the clay-silt layer in the Nile Valley varies from about 4 m to about
14 m. Laterally, this thickness becomes thinner toward the fringes of the valley and may
be locally absent. The thickness of this layer in the Nile Delta gets thicker toward the
north, ranging from about 5 m near Cairo to >50 m near the Mediterranean Coast. The
thickness also decreases toward the Delta fringes. This clay-silt layer forms the fertile
agricultural lands of both, the Nile Delta and the Nile Valley with an average Ks varying
between 0.05 and 0.50 m/day. Its approximate average depth in the study areas is
presented in Table 1. The soils are of alluvial and alluvio-marine deposits, consisting of
loam in A8 and clay to clay-loam in the areas of the Nile Delta. The soils can be
characterized as moderate saline soils in A1 and A8 and non-saline soils in the other
study areas with a water-table depth ranging from 64 to 110 cm below soil surface. The
annual average rainfall in the Nile Delta is about 180 mm with a maximum value of
650 mm at A1 (1980±1994, St. 318 at Al-Nozhah, Alexandria). The rainfall intensity
sharply decreases toward the south to about 1.5 mm/year near A8.
2.1.2. Measurement technique
The measurements in all study areas were performed on a regular 500 m square grid in
8 cm diameter and 2 m deep holes using the auger hole method (Van Beers, 1983). An
auger hole is made and after 24 h, during which the water table in the hole is allowed to
reach an equilibrium with the natural groundwater of the soil, the test is carried out. The
depth of the water table is recorded, the water is then removed from the hole using a
bailer and the velocity by which the water flows back into the hole is observed. This
method gives an average value of Ks for a soil profile and it is considered to be the most
effective method for use in subsurface drainage applications in clay soils (FAO, 1976;
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Table 2
Statistics summary of saturated hydraulic conductivity measurements (m/day) of the study areas
Study area
Minimum
Maximum
Arithmetic mean
Variance
CV (%)
A1
A2
A3
A4
A5
A6a
A7
A8
0.028
0.178
0.171
0.150
0.010
0.001
0.020
0.006
2.755
2.726
2.144
1.920
0.650
3.740
1.900
1.585
1.029
0.928
0.766
0.643
0.107
0.302
0.506
0.295
0.609
0.227
0.235
0.097
0.018
0.123
0.197
0.210
76
51
63
49
126
116
88
155
a
Published data.
Bouwer, 1969) owing to the large volume of soil involved in Ks-measurement which
tends to reduce the variability in the data sets.
The data sets were subjected to standard exploratory data analysis (Table 2) to verify
the nature of the frequency distribution of Ks in the study areas. Data sets were tested for
normality at 95% significance level using the procedure of D'Agostino et al. (1990) and
their distributions were found to be lognormally distributed, which is in agreement with
the general thinking that the measurements of many natural phenomena tend to have a
lognormal distribution. Consequently, the geometric mean was justified for determining
the best representative value of Ks (Bouwer, 1969; Tietje and Hennings, 1996).
Since field sampling with the auger hole method is done in a shallow or upper aquifer
clay-silt layer, the variability is dominated by the heterogeneous in this layer. Reasons for
this behavior may be due to uneven breaking of soil structure due to swelling and
shrinkage process and tillage system at this shallow layer. However, Camp (1977) and
Dorsey et al. (1990) showed that the auger hole method gave Ks values, in clay soils of
alluvial nature, similar to those calculated from field observations of water table
drawdown and drain outflow, pumping-test and velocity permeameter methods. Reynolds
and Zebchuk (1996) found that the auger hole method and the Guelph permeameter
methods yielded similar geometric mean values as well as similar spatial variability
structures in a fine textured soil. This may suggest that the scale effects on Ks values and
spatial variability parameters might be neglected. This assertion may moreover be
confirmed by the results of Butler and Healey (1998a, b) who showed that the difference
between pumping-test and slug-test parameters in a two-dimensional flow system is much
more likely to be an artifact than a scale dependence in hydraulic conductivity. Further, in
a rigorous numerical investigation, Sanchez-Vila et al. (1996) indicated that there will be
no scale dependence in the two-dimensional flow systems when hydraulic conductivity
can be represented as a lognormal random field. Bouwer and Jackson (1974) evaluated
several techniques for determining Ks and concluded that the variability of the soil is a
much greater source of variation in determining Ks of an area than the variation related to
the measurement method. In addition, the 2-m sampling depth of auger holes was deeper
than the design drain depth used in Egypt (1.25±1.50 m) in a range of 0.50±0.75 m,
whereas Rogers and Carter (1987) recommended a sampling depth of 0.3 m below the
design drain depth in a similar soil of alluvial nature. This may suggest that the sampling
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
297
method used in this study may give information on Ks at the scale of the practical
problem.
Nevertheless, sampling dimensions, such as diameter and depth of auger hole and
depth of the water pumped from the hole, may have some effects on the measured Ksvalues (Rogers, 1986), and thus, all holes were augered to similar depths and diameters
and were bailed to similar depths. Moreover, in clay soils, serious errors may result if the
true water table level is not established for determinations with the auger hole method
(Kirkham, 1965). These problems in the study fields are unlikely to have occurred as all
tests were conducted one day after the holes were augered.
The water-table depth is controlled by Ks of the soil between two adjacent lateral
drains. Bouwer (1969), using electrical analog simulations, showed that the representative
Ks-value is best represented by the geometric mean of point hydraulic conductivities
within the flow domain. Using a perturbation method, Matheron (1967) proved that, for
the particular case of two-dimensional flow and the lognormal distribution of Ks, the
representative Ks-value is equal to the geometric mean of Ks-measurements. Since the
auger-hole method samples a relatively small volume of soil compared with the soil
volume between two lateral drains, it can be considered as a point measurement method.
Consequently, for each study area, the geometric mean of all the measured values was
assigned as the representative Ks-value within the flow domain of each area.
According to Van Beers (1983), the flow system around the auger hole was solved
numerically with the relaxation technique under specific dimensional conditions. With
water-table depths in the range of 64 to 110 cm in the study areas and 8 cm diameter and
2 m deep auger holes, these dimensional conditions were found to be valid in this study.
Since the design requirements are often not accurately known and other uncertainties,
such as the lower boundary of the flow system and the entrance resistance of the drains,
may exist, the dimensionality and scale of measurement technique of Ks were considered
sufficiently reasonable in this study. Further, the hydraulic conductivity data sets are
expected to be homogenous and consistent as the sampling method and density were the
same in all areas of study.
2.2. Regionalized variable theory (RVT)
2.2.1. Spatial measure and kriging technique
Geostatistics approach utilizes the fact that variations of soil properties are not always
random, but have some spatial structures. It is based on the regionalized variable theory
(RVT) which takes into account both, the random and structured characteristics of
spatially distributed variables to provide quantitative tools, which are the most commonly
used methods in analysis of soil variability, for their description and optimal-unbiased
estimation.
Consider a field of area A, for which a set of n values were measured [z(xi), i 1,n], in
which each xi identifies a coordinate position in the space. Each z(xk) can be considered a
particular realization of a certain random variable, Z(xk), for a particular fixed point, xk.
The regionalized variable Z(xi), for all xi inside A, can be considered a realization of the
set of random variables [Z(xi), for all xi inside A]. This set of random variables is called a
random function (Journel and Huijbregts, 1978). Application of RVT assumes that the
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covariance between any two locations in A depends only on the distance and direction of
separation between the two locations and not on their geographic location. Based on this
assumption, the average covariance for each lag distance can be estimated for a given
volume of three-dimensional space. Among others, the covariance variogram (referred to,
hereafter, as `variogram') is a measure for the correlation between measurements of a
regionalized variable and may be written as (Moustafa and Yomota, 1998):
v h C 0 ÿ C h
(1)
N h
C h
1 X
xi hz xi ÿ m1 m2
N h i1
(2)
where
v h is the variogram at a separation distance h for a sampling volume v, C(0) is
the sample variance, and C(h) the covariance function at a separation vector h, m1 and m2
denote the means of the data values z(xi h) and z(xi), respectively, and they are equal
under the assumption of second-order stationarity. N(h) is the number of pairs of
measurements [z(xi), z(xi h)]. As for any vectorial function,
v h depends on both the
magnitude and direction of h. When the value of the variogram depends upon the
direction of h, anisotropic conditions exist. For such cases, functions describing the
experimental anisotropic variogram are often submitted to a transformation to yield
isotropic behavior.
The main advantages in using variogram of Eq. (1) for interpreting spatial structure of
Ks and performing kriging are (Moustafa and Yomota, 1998):
(i) it incorporates possible non-stationarity in the data set and, hence, reveals better the
character of its spatial structure comparing to the traditional spatial function (i.e. semivariogram);
(ii) it squeezes the effect of preferential sampling of data set;
(iii) its spatial modeling has a lower standard error compared with the traditional
spatial function; and
(iv) it yields more robust solution of the kriging system and savings in terms of
computer time compared with the traditional function.
A positive definite continuous function (Journel and Huijbregts, 1978),
1 h, must be
fitted to the experimental variogram, computed using Eq. (1), to characterize the spatial
structure of the regionalized variable studied and to assure the mathematical consistency
required for kriging estimations. Kriging is a technique for calculating optimal and
unbiased linear estimation of a soil property at an unsampled location, z*(x0), with
minimum estimation variance:
z x0
N
X
i z xi
(3)
i1
where N is the number of neighboring sampled points used for estimation, and i is the
weight applied to the neighboring sample z(xi). The weights are chosen so that z*(x0) is an
unbiased estimate:
Ez x0 ÿ z x0 0
(4)
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
299
and the variance between z*(x0) and the true value of the soil property at point x0 is
minimized:
var z x0 ÿ z x0 minimum
(5)
The weights placed on each neighboring sample sum to unity, and their unique
combination for which estimation variance is minimized yields the kriging system:
N
X
j C xi ; xj ÿ C xi ; x0
for
i 1; 2; . . . N;
j1
X
j 1
(6)
and the minimum estimation variance or kriging variance, 2k , is given by:
2k x0 C 0 ÿ
N
X
i C xi ; x0
(7)
i1
The values C(xi,xj) and C(xi,x0) are the covariance functions between observed
locations xi and xj and between the observed location xi and the interpolated location x0,
respectively. This system consists of (N 1) linear equations and (N 1) unknowns (N
weights, i and one Lagrangian multiplier, ).
The validation of kriging estimates can then be done by the cross-validation method
(Kitanidis, 1993). The criteria for validation depends on the values of reduced mean error
and reduced variance which must be close to zero (no systematic error) and one
(consistency between the kriging variance and the squared error), respectively. Standard
deviation of the estimation error (e) and the percentage of errors lying within 2e can
also be calculated to assess the minimum variance condition of the kriged estimates. This
requires that the percentage of errors should be 95 or more under the assumption of
normality (Journel and Huijbregts, 1978).
2.2.2. Regularization effect
Measurements are based on samples with a certain support size. The scale at which
these measurements are made is called the support scale. According to Journel and
Huijbregts (1978), such support affects both, the sill and the range of the variogram, and
as a result, the kriging estimates might not be accurate. The process of measuring a
regionalized variable with a certain support size is called regularization. It is imperative,
therefore, to take into considerations such regularization effect on the variogram
modeling.
For a practical approximation, the relation between the regularized variogram,
v h, and the point variogram,
*(h), may be written as (Journel and Huijbregts,
1978):
v h
h ÿ
v; v
for h v
(8)
where
(v,v) is a constant term related to the dimensions and geometry of the sample
volume, v, of the regularization. Although it seems imperative to consider this
regularization effect on the variogram modeling, hence on the kriging estimates, it is
expected that this effect within the sampled data sets in this study may not affect the
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
variogram parameters, owing largely to the small sample volume used to measure
hydraulic conductivity (0.01 m3), which included less of the overall spatial variability
than larger volumes. Moreover, the sampling dimensions in the horizontal plane
(500 500 m2) used to estimate and model the variograms are much larger (250-fold)
than those of the vertical direction. As a check, the procedures suggested by Journel and
Huijbregts (1978) were carried out for consideration of such an effect on variogram
modeling, hence on kriging estimates.
2.2.3. Problem statement
Since the field-measured Ks in the Nile Delta was found spatially variable (Gallichand
et al., 1991; Moustafa and Yomota, 1998), the question becomes the following: given the
measured Ks-values which are spatially variable, how do we rapidly arrive at a reliable
representative value to use in drain-spacing calculations of a region or at a correction
factor (Cf) which should be introduced to any steady or unsteady-state drain-spacing
equation to account for the effect of spatial variability of measured Ks-values on the
calculated drain spacing?
One way to provide this information is through the application of RVT with the use of
kriging technique, which gives an optimal and unbiased estimation of Ks at unsampled
locations with minimum estimation variance. Hence, the interpolated values at unsampled
locations may be considered as the most accurate values that can be obtained from limited
in situ measurements and those that can be used with known confidence. Estimating
representative Ks-values for soils, accordingly, can be improved using the kriging
technique. Gallichand et al. (1991) indicated that the geometric mean of kriged Ks-values
(Krk) is the most accurate representative Ks-value for use in subsurface drainage design.
This study aims, therefore, to calculate the magnitude of spatial correlation of the
measured Ks-values of different study areas, from which a way for direct and rapid
calculation of Krk might be developed.
2.3. Developing predictive model for Krk
2.3.1. Model verification
In order to develop a predictive model for Krk, experimental variograms of study areas
(A1,. . .,A5) were calculated and theoretical spatial dependence functions were fitted to
them chiefly to explore the behavior of the variograms between the origin and the range
of dependency. Then, combining the calculated spatial dependence and the weighted
influence of nearby points, the measured Ks-values were interpolated at 50 m intervals by
the ordinary kriging technique. Fifty-meter intervals were selected to represent, as far as
possible, the Ks-values within the flow domain, since the practical design lateral drain
spacing in Egypt ranges from 30 to 100 m.
Representative kriged Ks-values (Krk1,. . .,Krk5) of study areas were then estimated by
the geometric mean of all the kriged values within each area and the best relationship
between them and their measured values (Krm1,. . .,Krm5) were verified. After establishing
such a relationship, a predictive model was developed using regression procedure for a
direct and rapid estimation of Krk as the most reliable representative Ks-value from the
limited in situ measurements.
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
301
2.3.2. Model validation
Several statistical evaluation procedures were employed to quantify the power of the
performance of the model. These included: correlation coefficient (R), which measures
the degree of association between kriged and their predictive values; the mean difference
between kriged and their predictive values (Md), which measure the degree of
coincidence; the relative error (Er); and comparison of the statistics for the kriged and
predicted results (mean, standard deviation).
In order to assess the validation of the model on field data, RVT procedures were
performed for another two areas (A7,A8), whose data were not used in the model
prediction. Representative kriged Ks-values of A7 and A8 (Krk7 and Krk8) were then
calculated and compared with their predicted values from the model.
3. Results and discussion
3.1. Spatial dependence of saturated hydraulic conductivity
The experimental variograms for Ks-measurements of the study areas were found to be
represented by a spherical spatial function:
"
#
3 h
1 h 3
ÿ
0ha
1 h c0 c1
2 a
2 a
1 h c0 c1
h>a
(9)
where
1 h is the estimated variogram for lag h, c0 is the nugget, c1 is the structural
component, c0 c1 is the sill (cs), h is the distance between measurement points (lag),
and a is the correlation range. Published data by Gallichand et al. (1991) showed that the
frequency distribution of Ks in A6 is a lognormal distribution, and hence the experimental
variogram was calculated by the semi-variogram of log-transforms of Ks-measurements
and was fitted to an exponential spatial function:
ÿh
1 h c0 c1 1 ÿ exp
(10)
a0
where a0 is a range parameter (approximately one-third of the apparent range, a). The
calculated spatial parameters of this function were c0 0.886, c1 0.413, and
a 5599 m.
Whether a distribution is normal or lognormal, has no particular significance on the
variogram estimation, except that it is often more difficult to interpret variograms of
highly skewed distributions, such as Ks data sets. In a recent study, Moustafa and Yomota
(1998) showed that the variogram of Eq. (1) reveals better the character of spatial
structure of highly skewed distributions of Ks data sets in the Nile Delta, using the
original data set, than the semi-variogram of log-transforms of the data. Hence, in this
study, all the experimental variograms were estimated using the variogram of Eq. (1) with
the original data sets. Variography for Ks-measurements of study areas is shown in Figs. 2
and 3 and all parameters of spherical function are presented in Table 3.
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Fig. 2. Experimental variograms and the fitted spherical spatial functions of the study areas used for model
prediction.
Table 3
Spatial parameters of spherical function fitted to experimental variograms of the study areas
Study area
c0
c1
cs
A1
A2
A3
A4
A5
A7a
A8a
0.250
0.158
0.143
0.035
0.014
0.146
0.001
0.400
0.080
0.124
0.086
0.004
0.064
0.210
0.650
0.238
0.267
0.121
0.018
0.210
0.211
a
Used for model validation.
a (m)
2000
1700
1600
2500
2700
2000
1800
c0/cs (%)
38
66
54
29
78
70
0.5
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
303
Fig. 3. Experimental variograms and the fitted spherical spatial functions of the study areas used for model
validation.
All these variograms exhibit similar behavior showing nugget effects (c0) and approach
the sill (cs) at different values. The nugget effect quantifies the amount of covariance not
explained as spatial correlation, due chiefly to measurement errors and variations that
occur over distances smaller than the sampling distance of 500 m. The spatial structure of
Ks in the Nile Delta (A1,. . .,A5,A7) was characterized by a high nugget effect with a
relative value (c0/cs) being on average equal to about 56%. In contrast, the spatial
structure of Ks in the Nile Valley (A8) has a low nugget value (Table 3). This is not
surprising, since it may be interpreted in terms of the influence of weathering conditions
and agricultural practices in both the regions.
The variogram range (a) represents the distance beyond which values of the
regionalized variable are no longer autocorrelated, and therefore the measurements can be
assumed to be randomly distributed. A degree of autocorrelation exists between Ksmeasurements at a range varying from 1600 to 2700 m, confirming that Ks in the Nile
Delta is spatially variable with spatial structures characterized by a high nugget effect.
Values of the variance (Table 2) and the sill, cs, (Table 3) for different study areas,
moreover, are comparable, thereby practically confirming absence of trends in the data
sets. A visual examination of the experimental variograms at the azimuths 08, 458, 908
and 1358, with angular regularization in each direction of 458, indicated that anisotropy of
Ks was not present. This was in agreement with the results obtained by Gallichand et al.
(1991) and Moustafa and Yomota (1998).
A perusal of the information presented in Tables 1 and 3 showed that non-rice areas
(A4,A8) have the lower relative nugget effect (c0/cs) values compared with those of rice
areas. In rice areas, a non-uniform increase of relative nugget effect and correlation range
(a) with the decrease of rice land was also observed. On the other hand, the structured
component (c1) was found increasing with the increase of rice lands, with no correlation
to the depth of the near surface clay-silt layer. c0/cs can be generally decreased with the
decrease of clay-silt layer depth as indicated by c0/cs-values of A1, A4, and A8, compared
to those of the other areas. However, different scale of variation may also be found as in
the case of A5. In the Nile Delta, the correlation range non-uniformly increases with the
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
decrease of clay-silt layer depth. These findings may conclude that the spatial variability
of Ks is fairly correlated with the geologic nature and agricultural practices in the study
areas. The spatial variation of Ks can, moreover, be compounded by the heterogeneous
nature of field soils, and thus different spatial structures and scale of variability might be
obtained.
3.2. Regularization
In order to assess the sample volume effect on the variogram modeling, the
deregularization procedure (see Journel and Huijbregts, 1978) was carried out to deduce
the parameters of the point spatial theoretical models from the calculated regularized
models. The results showed that the deregularized parameters were almost the same as
the previous calculated theoretical model parameters given in Table 3. The differences
are insignificant (0.03±0.08%). These results confirm, therefore, the previous assertion
that the regularization effect on the modeling of variogram could be neglected, and thus
the auger hole method can be considered as a point measurement method. This may
reflect that the sampling technique used in this study is highly reasonable with RVT
procedure to estimate accurate kriged estimates which, in turn, assure accurate estimates
of representative Ks-values for different study areas.
3.3. Kriging estimates and model development
Number of points used in kriging estimation were determined by the search radius
(Vieira et al., 1981). The determined search radii of study areas are presented in Table 4.
Using the fitted theoretical spatial functions to the experimental variograms and the
search radii, kriged estimates were validated by the cross-validation technique. The mean
reduced error was 0, whereas the mean reduced variance was almost equal to unity
(Table 4), suggesting that the variogram is appropriate and the kriging process is
performed consistently. The hypothesis of normally distributed kriging errors are tested
Table 4
Search radii and results of cross-validation of kriging estimates
Study
area
Search
radius (m)
Reduced
meana
Reduced
varianceb
Percentage
errors within 2e
A1
A2
A3
A4
A5
A6
A7
A8
2000
1000
1000
1750
1500
1200
1250
1000
ÿ0.0003
ÿ0.007
0.001
0.006
ÿ0.009
ÿ0.0004
0.004
ÿ0.008
0.998
1.005
1.001
1.007
1.005
1.012
1.007
1.013
94
95
98
98
96
Ðc
97
96
a
Average of standardized residuals.
Average of squared standardized residuals.
c
Not available.
b
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Table 5
Representative measured (Krm) and kriged (Krk) saturated hydraulic conductivity (m/day) used for model
prediction
Study area
Krma
Krkb
A1
A2
A3
A4
A5
A6
0.731
0.797
0.627
0.578
0.066
0.250c
0.800
0.819
0.691
0.531
0.097
0.236c
a
b
c
Geometric mean of the measured values.
Geometric mean of kriged estimates.
Average of 25 blocks ranging in size between 80 and 210 ha.
and the results are also presented in Table 4. This was verified by the percentage (95%)
of estimation errors lying within 2e. These results indicate that kriging estimates are
truly values with minimum variances, and thus they may represent truly representative
values of Ks at reasonable and acceptable accuracy.
For study areas (A1,. . .,A5), Ks-measurements were interpolated at 50-m intervals by
the ordinary kriging technique. The representative measured (Krm1,. . .,Krm5) and kriged
(Krk1,. . .,Krk5) values were then determined by the geometric mean of all the measured
and kriged values, respectively. The results are presented in Table 5 along with the
published data of A6.
The calculated Krk-values were plotted against Krm-values to establish the relationship
between them (Fig. 4). To deduce a model that can be applied for small blocks (drawing
areas) and/or large areas, the data of 25 blocks (80±210 ha) of A6 were used (Fig. 4)
instead of only one average value for the whole area. Using the least-squares procedure,
the best estimates (Krkp) for Krk were found to be given by a predictive model in a form of
a linear relationship as:
p
0:925 Krm 0:016
Krk
(11)
The coefficient of determination (R2) of this model was highly significant (R2 0.924,
F 0.0004). Having tested the adequacy of kriging estimates, this equation would be
useful for estimating, with reasonable accuracy, a rapid and reliable representative
saturated hydraulic conductivity directly for an area from limited field measurements for
use in a subsurface drainage design and/or in the analysis of any saturated-soil water-flow
system.
3.4. Model validation
The representative measured Ks-values (Krm) were used to estimate their predicted
kriged values (Krkp) using the developed model of Eq. (11). The actual kriged values
(Krk) were then compared with those predicted from the model (Krkp). The mean and
standard deviation of Krkp-values were 0.295 and 0.191 m/day, respectively, whereas they
were 0.295 and 0.199 m/day for Krk-values, indicating excellent agreement between the
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Fig. 4. Relation between representative measured (Krm) and kriged (Krk) values of saturated hydraulic
conductivity.
values of Krkp and Krk with a very high correlation (R 0.961) between them (Fig. 5).
The mean (Md) and standard deviation of the difference between the values of Krkp and
Krk were 0.0001 and 0.055, respectively, whereas their mean relative error (Er) was
ÿ0.62%.
Fig. 5. Relation between representative kriged (Krk) values of saturated hydraulic conductivity and those
predicted (Krkp) from the developed model of Eq. (11).
M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
307
Table 6
Validation results of the predictive model on field data
Study area
A7
A8
Measured values (Krm)
0.335
0.096
Kriged values (Krk)
0.401
0.131
Predicted kriged values (Krkp)
0.326
0.105
The foregoing statistical tests have established that a relation exists between the
representative measured and kriged values of Ks in the Nile Delta and the actual kriged
values are comparable to those predicted from the developed model, indicating that the
model is performed well.
In order to assess the validity of the model on field data, kriging procedures were
performed for another two areas, A7 and A8. Their representative kriged values
(Krk7,Krk8) were then calculated and compared with their predictive kriged values
(Krk7p,Krk8p) from the model. Results are presented in Table 6 and are very satisfactory.
Based on these results, and from a practical point of view, it can be concluded that the
developed model might be used to determine directly and rapidly the representative value
of Ks in a region with reasonable accuracy. It is encouraging that one equation in the form
of Eq. (11) may be applied not only in the areas of the Nile Delta but also in areas having
different spatial structures in the Nile Valley. Therefore, the designers may find this
equation helpful to determine an accurate estimate of representative saturated hydraulic
conductivity required for a subsurface drainage design without extensive field
measurements. Furthermore, the advantage of this equation is that an estimate of
accurate representative Ks can be obtained, from limited in situ measurements, simpler
and quicker than by obtaining through the application of RVT procedures for every
drainage area which will be cumbersome and not practically feasible with the large scale
of on-going drainage projects in Egypt.
The developed model was built based on in situ measurements and quantitative random
and structured variation information of Ks in areas of different sizes (80±1848 ha). As a
result, it is expected to estimate a representative value which captures, with sufficiently
and acceptable accuracy, the hydraulic function of the drainage system. Tietje and
Hennings (1996) indicated that Ks is increasingly accepted to estimate as a random
variable depending on method and scale of the measurements and on the spatial
variability.
3.5. Practical application
The Hooghoudt's steady-state equation has been used in Egypt to calculate drain
spacing. This equation disregards radial flow to the drains and uses instead an equivalent
horizontal flow with a barrier at a reduced depth called the `equivalent depth' (de). The
equation is relatively simple and has been widely applied with reasonable and acceptable
accuracy (Moustafa, 1997) and has the form:
L2 frm ; q; de ; h
(12)
where L is the drain spacing (m), q is the steady outflow rate of the system (m/day), h is
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
the water-table height midway between drains (m), and Krm is the geometric mean of the
measured values of Ks. This equation in its present form does not take into consideration
the spatial variability of Ks; therefore, RVT procedures should be applied to the Ksmeasurements to yield a reliable representative value taking into account such variability
of measurements. In such case, the developed model of Eq. (11) is a helpful tool to
provide this information (Krkp) quickly with a reasonable accuracy and the resulting drain
spacing (L1) can be written as:
p
; q; de ; h
L21 f Krk
(13)
Another alternative way is to introduce a correction factor (Cf) directly to Eq. (12) to
account for spatial variability of Ks-measurements as:
L21 Cf L2
(14)
where Cf can be calculated from the developed model as:
Cf 0:925 0:016 Krm ÿ1
(15)
Fig. 6 illustrates the relation between Cf and the design range of the representative
measured saturated hydraulic conductivity (Krm), based on the Egyptian conditions.
Results reveal that neglecting of spatial variability of saturated hydraulic conductivity in
the drainage design affects the design drain spacing in a range varying from ÿ27% to 3%,
whereas its incorporation in the design assures an accurate drainage design with less costs
which can be considered to be important economically, chiefly with the large scale of
subsurface drainage design in Egypt. Based on these results, Cf was found to vary
between 1.62 and 0.96.
Fig. 6. Correction factor (Cf) for large-scale drain spacing design to account for spatial variability of fieldmeasured saturated hydraulic conductivity.
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Table 7
Minimum sample size of Ks in the study area A1 to obtain sample mean within 5%, 10%, and 15% of the
true mean
Probability
level (%)
95
90
Using measured
values (Krm)
Using kriged
values (Krk)
Using predicted kriged
values (Krkp)
5%
10%
15%
5%
10%
15%
5%
10%
15%
881
621
220
155
98
69
236
166
59
42
26
19
213
150
53
37
24
17
In summary, based on the developed model, Eqs. (13) and (14) are two equivalent
effective tools to give a rapid and accurate drainage design on a large scale owing to the
fact that they incorporate the random and structured variation of in situ saturated
hydraulic conductivity measurements into the calculated drain spacing.
Another example is given in Table 7 to realize the practical relevance of the developed
model in estimating minimum sample size of Ks in the study area A1. The minimum
sample size was calculated at 95% and 90% probability levels and allowing 5%, 10%,
and 15% errors around the true mean using Student's t-test. The sample size was lower
with the actual kriged values of Ks (Krk) (ÿ73%) and their predicted values from the
model (Krkp) (ÿ76%) than that associated with the original measured values (Krm) at any
given probability and precision levels. The sample size decreased with decreasing
precision and probability levels. The minimum sample size required using Krm was higher
than the present sample size of 61 in A1 at all the probability and precision levels,
whereas it was lower than the present sample size using Krk and Krkp at the precision
levels of 10% and 15%. At the highest precision level (5%), their minimum sample
sizes were higher than the present sample size. Based on these results, it has shown that
the developed model is a very useful tool for predicting reliable estimates of Ks at
minimum costs. This is an important issue for the large-scale subsurface drainage design
in Egypt.
4. Conclusion
This study presented a method, based on the concepts of geostatistics, for developing a
model, from which a rapid and reliable representative value of saturated hydraulic
conductivity from in situ measurements can be estimated and be used in the large-scale
subsurface drainage design of Egypt. The model was validated statistically and on field
data of two different soils. The results were encouraging. The obvious advantage of using
such a predictive model arises from the fact that a large number of field measurements of
saturated hydraulic conductivity is costly, time consuming, and cumbersome, whereas the
model provides a means for predicting reliably and rapidly the best estimate possible of
the representative value from limited in situ measurements. Notwithstanding, further
validation tests may be needed to further confirm the reliability of predictions chiefly in
areas outside the Nile Delta region.
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M.M. Moustafa / Agricultural Water Management 42 (2000) 291±312
Acknowledgements
The author is grateful to engineers of the Egyptian Public Authority for Drainage
Projects (EPADP) who did the field work of this study. He also would like to thank the
Editor-in-Chief and two anonymous reviewers for providing comments and suggestions
that significantly improved the quality of this paper.
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