A geostatistical approach to optimize the determination
of saturated hydraulic conductivity for large-scale
subsurface drainage design in Egypt
Mahmoud M. Moustafa
1Faculty 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.Kswas 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 ofKsis 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 ofKs. 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 ofKsthat 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 ofKsat a given precision level taking into account the spatial variability ofKs. The results showed that neglecting spatial variability ofKsmay affect the design drain spacing byÿ27% 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 requireKs, are to be applied. Furthermore, the magnitude of spatial dependence ofKspresented 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.
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 ofKs 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 ofKsbefore 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 measuringKs chiefly on large-scale drainage projects.
To overcome such difficulties, it is prudent to develop simplified methods to provide rapid and reliable representative values ofKs 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 ofKs, and some other physical and chemical soil properties as well, is still a great source of inaccuracy for estimating reliable representativeK s-values 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 ofKswhich should be interpreted as a random variable.
Many attempts have been made to infer the spatial variability ofKs(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,
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 representativeKs-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 ofKs.
2. Procedures
2.1. Study areas and measurement technique
2.1.1. Study areas
The most important factor in selecting a technique for determiningKsof 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).Kswas 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,Kswas 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)
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
Fig. 1. Location of the study areas.
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 averageKsvarying 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 ofKsfor a soil profile and it is considered to be the most effective method for use in subsurface drainage applications in clay soils (FAO, 1976; Table 1
Some characteristics of the study areas (for their locations, see Fig. 1) Study
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 ofKsin 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 ofKs (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 gaveKsvalues, 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 onKsvalues 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 determiningKsand concluded that the variability of the soil is a much greater source of variation in determiningKsof 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 Table 2
Statistics summary of saturated hydraulic conductivity measurements (m/day) of the study areas
Study area Minimum Maximum Arithmetic mean Variance CV (%)
A1 0.028 2.755 1.029 0.609 76
A2 0.178 2.726 0.928 0.227 51
A3 0.171 2.144 0.766 0.235 63
A4 0.150 1.920 0.643 0.097 49
A5 0.010 0.650 0.107 0.018 126
A6a 0.001 3.740 0.302 0.123 116
A7 0.020 1.900 0.506 0.197 88
A8 0.006 1.585 0.295 0.210 155
aPublished data.
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 measuredK s-values (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 representativeKs-value is equal to the geometric mean ofKs-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 representativeKs-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 ofKswere 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 areaA, for which a set ofnvalues were measured [z(xi),i1,n], in
which eachxiidentifies a coordinate position in the space. Eachz(xk) can be considered a
particular realization of a certain random variable,Z(xk), for a particular fixed point,xk.
The regionalized variableZ(xi), for allxiinsideA, can be considered a realization of the
set of random variables [Z(xi), for allxiinsideA]. This set of random variables is called a
random function (Journel and Huijbregts, 1978). Application of RVT assumes that the
covariance between any two locations inAdepends 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 ÿC h (1)
the sample variance, andC(h) the covariance function at a separation vectorh,m1andm2 denote the means of the data valuesz(xih) andz(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(xih)]. As for any vectorial function,v hdepends 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
Ksand 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. semi-variogram);
(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:
whereNis the number of neighboring sampled points used for estimation, andiis the
weight applied to the neighboring samplez(xi). The weights are chosen so thatz*(x0) is an
unbiased estimate:
and the variance between z*(x0) and the true value of the soil property at point x0 is minimized:
varz x0 ÿz 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:
XN
j1
jC xi;xj ÿC xi;x0 for i1;2;. . .N; X
j1 (6)
and the minimum estimation variance or kriging variance,2k, is given by:
2k x0 C 0 ÿ XN
i1
iC xi;x0 (7)
The values C(xi,xj) and C(xi,x0) are the covariance functions between observed
locationsxiandxjand between the observed locationxiand the interpolated locationx0,
respectively. This system consists of (N1) linear equations and (N1) unknowns (N
weights,iand 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 within2ecan 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 hv (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
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 (500500 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-measuredKsin the Nile Delta was found spatially variable (Gallichand et al., 1991; Moustafa and Yomota, 1998), the question becomes the following: given the measuredKs-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 ofKsat 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 krigedKs-values (Krk) is the most accurate representativeKs-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 ofKrk might be developed.
2.3. Developing predictive model for Krk
2.3.1. Model verification
In order to develop a predictive model forKrk, 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 measuredKs-values were interpolated at 50 m intervals by the ordinary kriging technique. Fifty-meter intervals were selected to represent, as far as possible, theKs-values within the flow domain, since the practical design lateral drain spacing in Egypt ranges from 30 to 100 m.
Representative krigedKs-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 ofKrk as the most reliable representativeKs-value from the limited in situ measurements.
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 forKs-measurements of the study areas were found to be represented by a spherical spatial function:
1 h c0c1 component,c0c1is the sill (cs),h is the distance between measurement points (lag), andais the correlation range. Published data by Gallichand et al. (1991) showed that the frequency distribution ofKsin A6 is a lognormal distribution, and hence the experimental variogram was calculated by the semi-variogram of log-transforms ofKs-measurements and was fitted to an exponential spatial function:
1 h c0c1 1ÿexp
wherea0 is a range parameter (approximately one-third of the apparent range,a). The calculated spatial parameters of this function were c00.886, c10.413, and
a5599 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 asKsdata 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 forKs-measurements of study areas is shown in Figs. 2 and 3 and all parameters of spherical function are presented in Table 3.
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 a(m) c0/cs(%)
A1 0.250 0.400 0.650 2000 38
A2 0.158 0.080 0.238 1700 66
A3 0.143 0.124 0.267 1600 54
A4 0.035 0.086 0.121 2500 29
A5 0.014 0.004 0.018 2700 78
A7a 0.146 0.064 0.210 2000 70
A8a 0.001 0.210 0.211 1800 0.5
aUsed 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 K s-measurements at a range varying from 1600 to 2700 m, confirming thatKs 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
Kswas 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/cscan be generally decreased with the decrease of clay-silt layer depth as indicated byc0/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 Fig. 3. Experimental variograms and the fitted spherical spatial functions of the study areas used for model validation.
decrease of clay-silt layer depth. These findings may conclude that the spatial variability ofKsis fairly correlated with the geologic nature and agricultural practices in the study areas. The spatial variation ofKs 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 representativeKs-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 bAverage of squared standardized residuals. cNot available.
and the results are also presented in Table 4. This was verified by the percentage (95%) of estimation errors lying within2e. These results indicate that kriging estimates are truly values with minimum variances, and thus they may represent truly representative values ofKsat 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 calculatedKrk-values were plotted againstKrm-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) forKrkwere found to be given by a predictive model in a form of a linear relationship as:
Krkp 0:925 Krm 0:016 (11)
The coefficient of determination (R2) of this model was highly significant (R20.924,
F0.0004). 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 ofKrk
p
-values were 0.295 and 0.191 m/day, respectively, whereas they were 0.295 and 0.199 m/day forKrk-values, indicating excellent agreement between the Table 5
Representative measured (Krm) and kriged (Krk) saturated hydraulic conductivity (m/day) used for model prediction
Study area Krma Krkb
A1 0.731 0.800
A2 0.797 0.819
A3 0.627 0.691
A4 0.578 0.531
A5 0.066 0.097
A6 0.250c 0.236c
aGeometric mean of the measured values. bGeometric mean of kriged estimates.
cAverage of 25 blocks ranging in size between 80 and 210 ha.
values ofKrkpandKrkwith a very high correlation (R0.961) between them (Fig. 5). The mean (Md) and standard deviation of the difference between the values ofKrkpand
Krk were 0.0001 and 0.055, respectively, whereas their mean relative error (Er) was ÿ0.62%.
Fig. 4. Relation between representative measured (Krm) and kriged (Krk) values of saturated hydraulic conductivity.
Fig. 5. Relation between representative kriged (Krk) values of saturated hydraulic conductivity and those predicted (Krkp) from the developed model of Eq. (11).
The foregoing statistical tests have established that a relation exists between the representative measured and kriged values ofKsin 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 ofKsin 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 representativeKscan 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 ofKsin 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)
whereLis the drain spacing (m),qis the steady outflow rate of the system (m/day),his Table 6
Validation results of the predictive model on field data
Study area Measured values (Krm) Kriged values (Krk) Predicted kriged values (Krkp)
A7 0.335 0.401 0.326
A8 0.096 0.131 0.105
the water-table height midway between drains (m), andKrmis the geometric mean of the measured values ofKs. This equation in its present form does not take into consideration the spatial variability of Ks; therefore, RVT procedures should be applied to the K s-measurements 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:
L21 f Krkp;q;de;h (13)
Another alternative way is to introduce a correction factor (Cf) directly to Eq. (12) to account for spatial variability ofKs-measurements as:
L21 CfL2 (14)
whereCfcan be calculated from the developed model as:
Cf 0:9250:016 Krmÿ1 (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 fromÿ27% 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 field-measured saturated hydraulic conductivity.
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 ofKs in the study area A1. The minimum sample size was calculated at 95% and 90% probability levels and allowing5%,10%, and15% errors around the true mean using Student'st-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 usingKrmwas 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 usingKrk and Krk
p
at the precision levels of10% and15%. 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.
Table 7
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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