Estimation of the transmissmty of the Santiago aquifer, Chile, using different geostatistical methods

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1 Groundwater Management: Quantity and Quality (Proceedings of the Benidorm Symposium, October 1989). IAHS Publ. no. 188,1989. Estimation of the transmissmty of the Santiago aquifer, Chile, using different geostatistical methods J. F. MUNOZ-PARDO & R. GARCIA Escuela de Ingenieria, Pontificia Universidad Catolica de Chile, Casilla 6177, Santiago, Chile INTRODUCTION Abstract Three geostatistical methods to estimate the transmissmty of an aquifer using the information commonly available are presented. The three methods discussed are simple kriging, kriging combined with linear regression and co-kriging. A comparison between the different methods is presented in terms of the accuracy of the estimations. It was found that kriging with linear regression gives better estimators than the other two methods and hence, in this case, is a preferable appoach considering the information available. Estimation de la transmissivité de la nappe de Santiago (Chile) par différentes méthodes géostatistiques Résumé On présente trois méthodes d'estimation de la transmissivité d'un aquifère utilisant l'information généralement disponible. Les trois méthodes mises au point sont le krigeage simple, le krigeage associé à un régression linéaire et le co-krigeage. Ces trois méthodes sont comparées en termes de précision des estimations. On trouve que le krigeage associé à une regression linéaire fournit de meilleurs estimateurs que les deux autres méthodes. Cette approche est donc, dans ce cas, préférable compte-tenu de l'information disponible. Deterministic numerical models simulating the behaviour of groundwater are useful tools in the evaluation, planning and management stages of this resource. More sophisticated and elaborate models are continuously being proposed. These, bi- or tri-dimensional numerical models are usually solved by finite difference or finite element methods for different boundary conditions to account for various aquifer configurations. However, the information available is usually limited and subject to large sampling and interpretative errors. This restricts the development of more complex models which may be applicable to studies of this type. To overcome these difficulties, it then becomes necessary to introduce into these models the effects of quality, quantity and spatial distribution of the input data. 77

2 /. F. Munoz-Pardo & R. Garcia 78 Geostatistical analysis (Matheron, 1965; Delhomme, 1976) constitutes in this sense a powerful tool. It allows both a correct conceptual interpretation of the phenomenon under consideration and an evaluation of the uncertainty associated with the information available as well as with the results obtained by the application of the numerical model. The information available for estimating the transmissivity T of an aquifer consists in many instances of a large number of results of pumping tests with variable flow rate realized in all the existing wells (maximum rate, specific capacity etc.) and a few results of pumping tests at constant flow rate. The latter data are more difficult and more expensive to obtain, but they provide more reliable values of T. In this paper three geostatistical methods to estimate the transmissivity of an aquifer using the information available on a routine basis are presented together with a comparison between these different methods in terms of the accuracy of the results. The methods discussed are simple kriging (method no. 1), kriging combined with linear regression (method no. 2) and co-kriging (method no. 3). The study area corresponds to the valley of the city of Santiago,Chile, where the deposits form a phreatic aquifer of an average thickness of approximately 400 m. The phreatic surface is in general similar to the topographic configuration of the soil surface. Its depth in the city centre is about 70 m. There are about 1000 wells with depths varying from 30 to 120 m. For the purposes of this study 80 wells had information on both transmissivity and specific capacity and 533 wells had solely specific capacity values. Additionally 13 wells had only transmissivity values. DATA ANALYSIS Classical statistical analysis The basic statistics of the two groups of data (613 values of specific capacity, q, and 93 values of transmissivity, T) are given in Table 1. From this table it can be concluded that there is a better fitting of the two groups of data to a lognormal distribution than to a normal distribution. This is in complete agreement with other results obtained with the same variables for other aquifers (Delhomme, 1976; Ahmed & de Marsily, 1987). A linear regression analysis between the variables Z = log T and Table 1 Statistics of transmissivity T and specific capacity q Variable Mean Variance Skewness Kurtosis T (m 2 day' 1 ) q (l s m ) z = l ho T Y = log, nq x

3 79 Estimation of the transmissivity of the Santiago aquifer, Chile Y = log q for the 80 measured pairs shows the presence of a strong correlation. This analysis led to a correlation coefficient of R = and to the following regression model: Z = ay+b (1) where a = and b = These results suggested the use of the linear regression method to estimate the transmissivity at the points where only the specific capacity had been measured (520 points predicted in this case). The prediction of Z. at the point ; where Y. is available leads to estimators associated with a variance of estimation or variance of the prediction error a? given by Snedecor & Cochran (1980). Geostatistical analysis When hydrogeological properties are studied, for example the field of transmissivities of an aquifer, and when an experimental realization of NP measured values is considered, it is necessary to introduce the hypotheses of ergodicity as well as of stationarity which allow the estimation of the experimental variogram given by: 7(h) = ^ E ^ W + h) - Z{x)f (2) where N(h) is the number of pairs of values [Z(x), Z(x + h)] separated by a vector h. Furthermore the variogram of the two variables Z = log T and T = log q was calculated assuming the phenomenon is isotropic. The cross-variogram between the variables Z and Y was estimated by the following expression: 1 N(h) Jzyih) = 7YZ(h) = _ E. = \ \ZQC.) - Z(x. + h)]\yqc.) - YQc. + h)] A theoretical spherical model given by: (3) 7(h) = C 0 + (C 2 - C Q )[1.5(h/a) - 0.5(/z/a) 3 ] y(h) = C 2 h < a h > a (4) was adjusted to each of the two experimental variograms as well as to the experimental cross-variogram. In equation (4) C Q is the nugget effect corresponding to the percentage of variance unexplained by the sampling, and C 2 is the sill and represents the global variance of the phenomenon; a is the range or distance beyond which two experimental observations can be considered to be spatially independent. Table 2 gives the values of these three coefficients for the different

4 /. F. Munoz-Pardo & R. Garcia 80 Table 2 Coefficients of the theoretical variogram and cross-variogram functions Parameter Z = log i0 T Y = l 8io<l Z - Y c o c z a(km) theoretical adjusted functions. Figure 1 represents the associated experimental and fitted variograms or cross-variogram. The cross-variogram model thus fitted satisfies the conditions ascertaining that the co-kriging matrix formed is positive definite (Ahmed & de Marsily, 1987). GEOSTATISTICAL METHODS OF ESTIMATION In models simulating the behaviour of an aquifer it is often necessary to estimate the hydrogeological properties at the nodes of a gridded domain. In this work the three methods have been compared through simultaneous estimations of T. Method no. 1: kriging The estimation of a value in x Q by the kriging method is accomplished through the use of NP experimental values. The estimator has the form: z **^o) =r^i x «z^«) ( 5 ) The determination of the weights X ;. is achieved through a probabilistic interpretation of the phenomenon as well as a reasoning in terms of random functions. This leads to an unbiased estimator with minimum variance which takes into account the spatially dependent structure of the phenomenon through the variogram. The resolution of the so-called kriging-equation-system (Delhomme, 1976) yields the values of the weights X. and the variance of the estimation error which is given by: k = 4iV,o + V. (6) where IL is the Lagrangian multiplier which minimizes the variance and y iq is the value of the variogram for A = JC. - x Q. Moreover the kriging method presents the following advantage in the estimation of a natural phenomenon: the estimator Z k * and the variance of the error of estimation, or kriging variance, a^2, depend exclusively on the relative position of the experimental points between each other and on the spatially dependent structure of the phenomenon.

5 81 Estimation of the transmissivity of the Santiago aquifer, Chile h (km.) I.U 5(h)" C J 2^^ ^ ^ 0 ^^^^^ ^^^^^ m C0=0,3 b) C 2=0.57 a =57 km h(km.) h ( km.) Fig. 1 (a) Mean variogram for Z = log T; (b) mean variogram for Y = log q; and (c) mean cross-variogram of Z and Y. Method no. 2: kriging with linear regression The kriging method can be modified to use additional field data with a quality different from the rest of the experimental measurements. In the case of an aquifer, values of T predicted from a linear regression obtained from values of the specific capacity q (equation (1)) are used. The predicted values of T have an associated uncertainty given by the variance of the prediction error oh Assuming the corresponding errors of prediction e(x.j are: unsystematic (E[e(x t )] = 0, Vz); uncorrected between themselves (cov[e(x / ), e(x.)] = 0, Vz * ;'); uncorrected with the variable (cov[e(x j ),Z(x.)] = 0, Vz, Vc; the estimator associated with the kriging combined with linear regression is given by: NP Modifying the system of equations (Delhomme, 1976; Ahmed & de Marsily, 1987) leads to the same expression for the variance of the estimation error (7)

6 /. F. Munoz-Pardo & R. Garcia 82 that one obtained with simple kriging: 0 NP k R = h-i X^0 + M (8) where the weights X^ are now different for they are determined considering the quality or uncertainty of each measured or predicted point value. Method no. 3: co-kriging This geostatistical technique introduced by Matheron (1971) improves the precision of the estimation obtained by kriging by using the information of another variable spatially correlated with the variable to be estimated and generally better sampled. This fact is very important in the case of the transmissivity T of an aquifer which is highly correlated with specific capacity, values more easily measurable. The level of cross-spatial-correlation between two variables is given by the cross-variogram (equation (3)) and in this case the estimator given by the co-kriging method is (Journel & Huijbregts, 1978; Vauclin et al, 1983): This equation yields a co-kriging system (Vauclin, 1983; Ahmed & de Marsily, 1987) whose resolution gives the values of the weights \ 1;. and \ 2k. The variance of estimation is given by: where ^ivipv^i^pio + ^i y l iq is the value of the variogram of the variable Z for the pair (x f x Q )\ 7^Q is the value of the cross-variogram between Z and Y for the pair (x k3 x Q ); do) N 1 is the number of experimental measured values of Z; N 2 is the number of experimental measured values of Y; [L x is the Lagrangian multiplier associated with variable Z. This technique is only convenient when the two variables Z and Y are highly correlated and when a large amount of information on the variable Y contributes to the estimation of the variable Z. APPLICATION TO THE CASE OF THE SANTIAGO AQUIFER The three geostatistical estimation methods were used to estimate the transmissivity of the Santiago aquifer on a 76 point grid over an area of 30 x 45 km 2. The different data used for each case were the following: (a) Method no. 1: 93 values of T determined in the field from pumping tests at constant rate. The adjusted theoretical variogram of the variable Z = log T is presented in Fig. 1(a). For each estimation the values of the measured T at the 10 closest points were used. (b) Method no. 2: The same adjusted variogram of the variable Z was

7 83 Estimation of the transmissivity of the Santiago aquifer, Chile used. 533 values of T for the experimental points where only q had been measured were predicted through the linear regression equation (equation(l)) which itself is the result of a linear regression analysis between the logarithm of the variables calculated at the 80 locations where both types of information are available. For each estimation the 10 closest values of measured (93) or predicted (533) values of T were used. (c) Method no. 3: 93 values of r and 613 values of q were used. The different adjusted theoretical variograms' and cross-variogram are already defined. For each estimation the 10 closest values of T and the 20 closest values of q were used. The results obtained with the three methods are given in Table 3. It may be observed that the kriging combined with the linear regression gives the best results. The average variance of the estimation error is minimum for that case. Table 3 Statistics of the results obtained by different methods Method Estimated values Min Max Mean Var Estimation error Min Max Mean Var No. No. No The same comparison of the three methods was achieved while increasing the number of neighbour points considered in the estimation from 10 to 20. The results showed no important reduction in the variance of the estimations and moreover kriging combined with linear regression remained the best method. The reason that could possibly explain that in this particular case kriging with linear regression is better than co-kriging is that the residuals e (. = Z i - a.y'. - b showed no spatial structure. Ahmed & de Marsily (1987) also found this property to influence strongly the relative superiority of one method over the other when kriging with linear regression and co-kriging were compared. CONCLUSION From the above results it may be observed than taking into account the additional information contained in the specific capacity values usually available greatly improves the estimation of the transmissivity of an aquifer. In the study of the Santiago aquifer, Chile, it was observed that kriging combined with linear regression gave better estimates than the two other methods presented.

8 /. F. Munoz-Pardo & R. Garcia 84 Acknowledgements This study was undertaken within the project "Spatial Variability of Parameters Related to Irrigation" granted by the Research Department of the Universidad Catolica de Chile. The authors wish to thank Dr D. Marchand for valuable comments and suggestions. REFERENCES Ahmed, S. & de Marsily, G. (1987) Comparison of geostatistical methods for estimating transmissivity using data on transmissivity and specific capacity. Wat. Resour. Res. 23 (9), Delhomme, J. P. (1976) Application de la théorie des variables régionalisées dans les sciences de l'eau. Thèse de docteur-ingénieur, Université Pierre et Marie Curie, Paris, France. Journel, A. G. & Huijbregts, J. C. (1978) Mining Geostatistics. Academic Press, London. Matheron, G. (1965) Les Variables Régionalisées et Leur Estimation. Masson, Paris, France. Matheron, G. (1971) The theory of regionalized variables and its applications. In: Cahiers du Centre de Morphologie Matljématique, vol. 5. Ecoles des Mines de Paris, Fontainebleau, France. Snedecor, G. W. & Cochran, W. G. (1980) Statistics Methods. The Iowa State University Press, USA. Vauclin, M., Vieira, S. R., Vachaud, G. & Nielsen, D. D. R (1983) The use of cokriging with limited field soil observations. Soil Sci. Soc. Am. J. 47 (2),