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A geostatistical approach is proposed for characterizing the uncertainty about the transmissivity field of an aquifer and analyzing its effect on predicted head values. A new methodology is developed, which couples conditional simulation and groundwater flow modeling. Conditional simulation is used for generating different two-dimensional transmissivity fields that all have the same spatial variability as the true field and are consistent with the measured T values at well locations. Two case studies are presented in order to illustrate the method, and conclusions are drawn for future investigation. INTRODUCTION A new subdivision of hydrogeology has recently evolved: the so-called 'stochastic hydrogeology.' Its object is to deal with uncertainty in groundwater modeling. Of course, at least in a non-Bayesian approach an underground reservoir cannot really be random: it is unique. But uncertainty about it exists, which can be accounted for by spatial variability and fragmentary sampling of the flow parameters. The complex variability of transmissivity T (and other parameters) within a given aquifer has induced research into the use of the language and mathematical tools of the theory of random functions. Several approaches have been used in previous studies. The work by Freeze [1975] applied a Monte Carlo technique to one-dimensional flows but ignored the spatial correlation structure. Later approaches made use of perturbation theory, either using space domain spectral analysis [Gelhar, 1976] or not [Tang and Pinder, 1977]. The purpose of the present paper is to show how the newly developed method of 'conditional simulations' (CS)can be applied to this analysis of the uncertainty resulting from the spatial variability of transmissivity. Basically, the CS method is a technique of the Monte Carlo type (here simulation is to be understood in a statistical sense) for generating two-or threedimensional fields. Two particular features must be emphasized. First, the simulated values of Thave the same autocorrelation structure as the true field. Second, at pumping test locations they are consistent with the observed values. Thus different conditional simulations of a reservoir can be considered possible versions of reality. Using them as different sets of parameters for a digital model, one can analyze the resulting statistical distribution of the various predicted head values.The proposed method is part of G. Matheron's topoprobabilistic approach and is closely related to kriging. Therefore the paper starts by reviewing some important points about estimation and simulation of regionalized variables (ReV). After exposing the principles of CS, two examples--a rather academic exercise and a real-world case study--are included to illustrate the method. Limitations and implications for future work are discussed. A REVIEW OF GEOSTATISTICAL CONCEPTSGeostatistics are now more than 15 years old. The word was created at the beginning of the 1960's by Matheron [1962, 1963] and initially stood for his own approach: to solve practic...

We review the main stages of the evolution of ideas and methods for solving the inverse problem in hydrogeology; i.e., the identification of the transmissivity field in single-phase flow from piezometric data, in mainly steady-state and, occasionally, transient flow conditions. We first define the data needed to solve an inverse problem in hydrogeology, then describe the numerous approaches that have been developed over the past 40 years to solve it, emphasizing the major contributions made by Shlomo P. Neuman. Finally, we briefly discuss fitting processes that start by defining the unknown field as geological images (generated by Boolean or geostatistical methods).The early attempts at solving the inverse problem were direct, i.e., the transmissivity field was directly determined by using stream lines of the flow and inverting the flow equation along these lines. Faced with the poor results obtained in this manner, hydrogeologists have tried many different ways of minimizing the balance error representing an integral of the mass-balance error for each mesh for a given transmissivity field. These attempts were accompanied by constraints imposed on the transmissivity field in order to avoid instabilities.The idea then emerged that the unknown field should reproduce the local observations of the pressure at the measurement points instead of minimizing a balance error. Second, it should also satisfy a condition of plausibility, which means that the transmissivity field obtained through the inverse solution should not deviate too far from an a priori estimate of the real transmissivity field. This a priori notion led to the inclusion of a Bayesian approach resulting in the search for an optimal solution by maximum likelihood, as expounded later.Simultaneously, the existence of locally measured values in the transmissivity field (obtained by pumping tests) allowed geostatistical methods to be used in the formulation of the problem; the result of this innovation was that three major approaches came into being: (1) the definition of the a priori transmissivity field by kriging; (2) the method of cokriging;(3) the pilot point method. Furthermore, geostatistics made it possible to pose the inverse problem in a stochastic framework and to solve an ensemble of possible and equally probable fields, each of them equally acceptable as a solution.

On the assumption that the behavior of an aquifer is linear, it is shown that recharge from an ephemeral stream can be computed by deconvolution of the fluctuations of the piezometers in the vicinity of the stream. First, the 'impulse response' of the piezometer has to be determined, either from an exceptional flood event or with the help of a digital model of the aquifer. Application of this method to the plain of Kairouan (Tunisia) proved to give estimates of recharge during a 5-year period within the same order of magnitude as direct volume estimates obtained by an automatic contouring technique of the recovery of the aquifer after infiltration (kriging). An empirical runoff-recharge relationship was then determined to extend the estimation of recharge to a 23-year period.

Kriging is of particular interest in network design because of its ability to estimate streamflow values using existing stations. Another possibility offered by kriging is the estimation of variance reduction gained by addition of fictitious stations in regions of high variance. In this article we give a brief description of kriging theory as developed at the Ecole des Mines de Paris. In order to improve and optimize the Quebec streamflow recording network design we kriged specific streamflows with a given return period over the Quebec province, using the data observed at existing stations. For the evaluation of the given return period flow and its sampling variance at each gauged site we use the log Pearson type 3 distribution model. Kriging is an optimal estimation technique, in terms of minimum variance, and contrary to other methods it gives an estimation variance for any point in the kriged domain, which is essential in network design.

In 1999, an oilfield experiment was initiated to test the application of electrical measurement technologies to permanent reservoir monitoring. The principal objective of the experiment was to demonstrate the feasibility of monitoring water movement between an injection and an observation well. This paper describes the interpretation of the data provided by the resistivity arrays and discusses the data quality and reliability of the measurements.Two wells were drilled into the Mansfield sandstone reservoir in Indiana, U.S.A. The D-8 injector well was located in the center of four development wells. The OB-1 monitoring well was offset 233 ft to the southwest in a location midway between the D-8 injector and the No. 3 production well. The injector was instrumented with a 16-electrode resistivity array that was run on the outside of insulated casing and cemented into the annulus of the well. A similar array was cemented into the annulus of the monitoring well.In March 1999, the D-8 well was perforated and acidized. A surface gauge was used to monitor injection rates and pressures. Initially, injection proceeded at a rate of approximately 20 B/D, increasing to 90 B/D after fracture stimulation. The D-8 array records responses to wellbore operations and injection. It clearly distinguishes the movement of the waterfront in different zones. The OB-1 electrical array clearly indicates early water breakthrough by means of an induced fracture. The data show good signal-to-noise ratio and high reciprocity.The experiment has demonstrated the viability of using permanently installed resistivity arrays to monitor the movement of oil/ water contacts and salinity fronts that are some tens of feet away from the wellbore. Results demonstrate the feasibility of using such arrays to monitor oil/water contact movements remote from injection, monitoring, and production wells.

In very thinly bedded sand-shale sequences, sand counts and sand/total ratios can be determined from high resolution electrical images of the borehole wall. This technique is illustrated here by a case study ~·n turbidites with sand laminae down to 1 em in thickness. The numerical modeling of the tool response and a comparison with fullbore core data help assess the accuracy of bed thicknesses derived from electrical images for this type of formations.

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