Transmissivity estimation for highly heterogeneous aquifers: comparison of three methods applied to the Edwards Aquifer, Texas, USA

Abstract

Obtaining reliable hydrological input parameters is a key challenge in groundwater modeling. Although many quantitative characterization techniques exist, experience applying these techniques to highly heterogeneous real-world aquifers is limited. Three geostatistical characterization techniques are applied to the Edwards Aquifer, a limestone aquifer in south-central Texas, USA, for the purposes of quantifying the performance in an 88,000-cell groundwater model. The first method is a simple kriging of existing hydraulic conductivity data developed primarily from single-well tests. The second method involves numerical upscaling to the grid-block scale, followed by cokriging the grid-block conductivity. In the third method, the results of the second method are used to establish the prior distribution for a Bayesian updating calculation. Results of kriging alone are biased towards low values and fail to reproduce hydraulic heads or spring flows. The upscaling/cokriging approach removes most of the systematic bias. The Bayesian update reduced the mean residual by more than a factor of 10, to 6 m, approximately 2.5% of the total head variation in the aquifer. This agreement demonstrates the utility of automatic calibration techniques based on formal statistical approaches and lends further support for using the Bayesian updating approach for highly heterogeneous aquifers.

Résumé

Disposer de paramètres hydrogéologiques d’entrée fiables est une des clés de la modélisation hydrogéologique. Bien que de nombreuses techniques de caractérisation quantitative existent, leur application à des aquifères réels fortement hétérogènes est limitée. Trois techniques géostatistiques de caractérisation sont appliquées à l’aquifère Edwards, un aquifère calcaire situé au sud du centre du Texas, USA, afin d’évaluer leur performance dans un modèle hydrogéologique de 88 000 cellules. La première méthode est un krigeage simple des données disponibles de conductivité hydraulique obtenues à partir d’essais de puits. La seconde méthode fait intervenir une mise à l’échelle au niveau de celle du maillage puis un co-krigeage de la conductivité à cette échelle bloc. Dans la troisième méthode, les résultats obtenus à la deuxième méthode sont utilisés comme distribution préalable à une approche bayésienne. Les résultats du krigeage seul sont biaisés pour les faibles valeurs et ne permettent pas de reproduire les niveaux piézomètriques ou les écoulements des sources. L’approche changement d’échelle/co-krigeage supprime une grande partie du biais systématique. L’approche bayésienne a réduit le résidu moyen d’un facteur supérieur à 10, jusqu’à 6 m, environ 2.5% de la variation piézomètrique totale dans l’aquifère. Ceci démontre l’utilité des techniques de calibration automatique basées sur des approches statistiques formelles et accrédite davantage l’utilisation de l’approche bayésienne dans le cas d’aquifères fortement hétérogènes.

Resumen

La obtención de parámetros hidrológicos de entrada confiables es un desafío clave en la elaboración de modelos de aguas subterráneas. Aunque existen muchas técnicas cuantitativas de caracterización se cuenta con experiencia limitada para aplicar estas técnicas a acuíferos del mundo real altamente heterogéneos. Se han aplicado tres técnicas de caracterización geoestadística al acuífero Edwards, un acuífero de calizas en la parte sur central de Texas, Estados Unidos, con el propósito de cuantificar el desempeño en un modelo de agua subterránea de 88,000 celdas. El primer método es una interpolación kriging simple de los datos existentes de conductividad hidráulica obtenidos principalmente de pruebas en un solo pozo. El segundo método involucra escaleo ascendente numérico a la escala de bloques de la malla seguida por cokriging en la conductividad de los bloques de la malla. En el tercer método, se utilizan los resultados del segundo método para establecer la distribución anterior para un cálculo de actualización Bayesiano. Los resultados únicos de kriging están sesgados hacia valores bajos y fallan en reproducir presiones hidráulicas o flujos de manantial. El enfoque de escaleo superior/cokriging remueve casi todo el sesgo sistemático. La actualización Bayesiana reduce el residual medio en más de un factor de 10, a 6m, aproximadamente 2.5 por ciento de la variación total de presión en el acuífero. Esta consistencia demuestra la utilidad de las técnicas de calibración automática basadas en enfoques estadísticos formales y confiere apoyo adicional para el uso del enfoque de actualización Bayesiano en acuíferos altamente heterogéneos.

Appendix

Geostatistical analysis

Geostatistical analyses of the local scale hydraulic conductivity are needed to establish models for the spatial correlation. The confined and outcrop sections of the aquifer were treated as separate populations. A more detailed geostatistical analysis was undertaken for the confined region to better model spatial correlation in the extreme values of hydraulic conductivity.

There are several methods for establishing the two-point spatial correlation, each with its advantages and disadvantages in a given situation (e.g., Journel and Huijbregts 1978; Deutsch and Journel 1998). The most familiar approach is the sample semi-variogram, which is the sample semi-variance as a function of lag (separation) distance. An alternative approach is to calculate the sample nonergodic covariance

$$ \Gamma {\left( h \right)} = \frac{1} {{2N{\left( h \right)}}}{\sum\limits_{i = 1}^{N{\left( h \right)}} {{\rm Z}{\left( {u_{i} } \right)}{\rm Z}{\left( {u_{i} + h} \right)}} } - m^{2}_{h} $$

or the closely related nonergodic correlogram

$$ \rho {\left( h \right)} = {\Gamma {\left( h \right)}} \mathord{\left/ {\vphantom {{\Gamma {\left( h \right)}} {\sigma ^{2}_{h} }}} \right. \kern-\nulldelimiterspace} {\sigma ^{2}_{h} } $$

where Z(u_i) is a sample value (of the random field Z) at location u_i, and h is the lag (separation) distance. The quantities m _h and $\sigma ^{2}_{{\text{h}}}$ are the sample mean and sample variance of the N(h) data pairs, which need not be the same as the mean and variance of the entire population. The correlogram ranges between 0 and 1 with the value 1 corresponding to perfect correlation; the value zero corresponds to the uncorrelated situation. The semi-variogram is the most widely used measure of spatial correlation. However, it is particularly sensitive to two statistical features that often occur in combination in hydrological data: heteroscedasticity and clustering of extreme values. The term heteroscedasticity refers to the situation when the local variability of the data is related to the local mean, or more generally, changes over the study area. When this occurs in combination with clustering of data, spatial correlation, as measured by the semi-variogram, is often masked. The nonergodic correlogram is preferred in such situations, as it is known to be robust against these statistical artifacts (Deutsch and Journel 1998).

It is advantageous to apply these measures of spatial correlation not to the original K data, but to some transforms of K. For example, it is common to calculate the semi-variogram or correlogram for the logarithmic transforms. Another powerful technique is to apply the semi-variogram or correlogram to indicator transforms. The indicator transform at a given cutoff value is obtained by replacing those values above the cutoff with a 1 and those below with a 0. By repeating this process for several cutoff values and calculating the variogram or correlogram for each, it is possible to develop a nonparametric model for the spatial correlation. Such an indicator model contains much more information than a traditional semi-variogram model, and is preferred if enough data are available. In particular, an indicator model is better in reproducing spatial correlation in the tails of the distribution, which controls upscaling.

Omnidirectional indicator correlograms for the confined region K are shown in Fig. 12 for five cutoff values corresponding to the 1st, 2nd, 5th, 8th and 9th decile. The fact that these correlograms are nonzero for nonzero lag distances means that significant spatial correlation exists for all cutoff values. For these data, the indicator correlograms for the 1st decile are larger than the corresponding ones for the median or upper cutoffs. This suggests that low values of hydraulic conductivity are better correlated spatially than the large values. The fact that all of these correlograms are significantly less than 1 as lags approach 0 is a manifestation of the nugget effect. A nonzero nugget value means that hydraulic conductivity measured in two closely spaced wells would be imperfectly correlated due to the effects of very small-scale variability or measurement errors.

Fig. 12

Results of the geostatistical analysis. The solid curves are fitted correlogram models

The solid curves in Fig. 12 are fitted correlogram models. The indicator correlogram for each cutoff is well fitted by a nested exponential model of the form Eq. (1). For each cutoff, the two correlation structures contribute roughly equally to the composite correlation structure. The short correlation range is 2.1–3.6 km, depending on the cutoff value, and the long correlation range is 15 km except for the first decile, which has a practical correlation range of 21 km.

The indicator semi-variogram for the 9th decile is shown for comparison purposes in Fig. 13. The semi-variogram is constant for all lags except for random fluctuations. This suggests no spatial correlation in the upper 10% of the hydraulic conductivity distribution, as opposed to the correlogram in Fig. 12, which shows significant spatial correlation. Careful review of the data suggests that this apparent discrepancy is caused by m _h and $\sigma ^{2}_{{\text{h}}}$ being different for small and large lags, a statistical feature that is known to mask spatial correlation. There are two possible causes for m _h and $\sigma ^{2}_{{\text{h}}}$ being different for small and large lags: heteroscedasticity and sampling bias. Sampling bias is likely in this dataset, as a landowner that drills an unproductive well is likely to try again on the same property. This type of biased sampling may alter the statistics of closely spaced wells compared with the general population. Whatever the cause, the correlogram is robust to this statistical feature and picks up spatial correlation that would have been overlooked using the traditional semi-variogram.

Fig. 13

Semi-variogram for the 9th decile. Unlike the non-ergodic correlogram of Fig. 12, the semi-variogram reveals no significant spatial correlation