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For the special case of a stratified porous medium with flow parallel to the bedding it is shown that the transport of solute cannot, in general, be represented by the usual convection-diffusion equation, even for large time. The necessary c6nditions for the appearance of a Fickian diffusive process are discussed and compared with previous work done by Gelhar et al. (1979) and Marie et al. (1967). It is shown, however, that when the flow is not exactly parallel to the stratification, diffusive behavior is much more likely to appear. The need for further work on the mechanism of transport in porous media is then emphasized. assumption holds even in a real porous medium as long as the tracer has not reached the boundaries.Solute transport is assumed to be controlled by convection and dispersion with the following properties:1. Convective transport has the local seepage velocity u(z) at each elevation z of the medium. Dispersive transport has a constant dispersion tensor: the dispersion coefficients DL andDr in the longitudinal x and transversal z directions are assumed to be constant, i.e., inde-901 902 MATHERON AND DE MARSILY: SOLUTE TRANSPORT IN GROUNDWATER

Heterogeneity can be dealt with by defining homogeneous equivalent properties, known as averaging, or by trying to describe the spatial variability of the rock properties from geologic observations and local measurements. The techniques available for these descriptions are mostly continuous Geostatistical models, or discontinuous facies models such as the Boolean, Indicator or Gaussian-Threshold models and the Markov chain model. These facies models are better suited to treating issues of rock strata connectivity, e.g. buried high permeability channels or low permeability barriers, which greatly affect flow and, above all, transport in aquifers. Genetic models provide new ways to incorporate more geology into the facies description, an approach that has been well developed in the oil industry, but not enough in hydrogeology. The conclusion is that future work should be focused on improving the facies models, comparing them, and designing new in situ testing procedures (including geophysics) that would help identify the facies geometry and properties. A world-wide catalog of aquifer facies geometry and properties, which could combine site genesis and description with methods used to assess the system, would be of great value for practical applications.RØsumØ On peut aborder le problme de l'hØtØrogØnØitØ en s'efforçant de dØfinir une permØabilitØ Øquivalente homogne, par prise de moyenne, ou au contraire en dØ-crivant la variation dans l'espace des propriØtØs des roches à partir des observations gØologiques et des mesures locales. Les techniques disponibles pour une telle description sont soit continues, comme l'approche GØosta-tistique, soit discontinues, comme les modles de facis, BoolØens, ou bien par Indicatrices ou Gaussiennes SeuillØes, ou enfin Markoviens. Ces modles de facis sont mieux capables de prendre en compte la connectivitØ des strates gØologiques, telles que les chenaux enfouis à forte permØabilitØ, ou au contraire les facis fins de barrires de permØabilitØ, qui ont une influence importante sur les Øcoulement, et, plus encore, sur le transport. Les modles gØnØtiques rØcemment apparus ont la capacitØ de mieux incorporer dans les modles de facis les observations gØologiques, chose courante dans l'industrie pØ-trolire, mais insuffisamment dØveloppØe en hydrogØo-logie. On conclut que les travaux de recherche ultØrieurs devraient s'attacher à dØvelopper les modles de facis, à les comparer entre eux, et à mettre au point de nouvelles mØthodes d'essais in situ, comprenant les mØthodes gØo-physiques, capables de reconnaître la gØomØtrie et les propriØtØs des facis. La constitution d'un catalogue mondial de la gØomØtrie et des propriØtØs des facis aquifres, ainsi que des mØthodes de reconnaissance utilisØes pour arriver à la dØtermination de ces systmes, serait d'une grande importance pratique pour les applications.Resumen La heterogeneidad se puede manejar por medio de la definición de características homogØneas equivalentes, conocidas como promediar o tratando de describir la variabilidad espacia...

A new methodology for solution of the inverse problem in groundwater hydrology is proposed and applied to a site in southeastern New Mexico with extensive hydrogeologic data. The methodology addresses the issue of nonuniqueness of the inverse solutions by generating an ensemble of transmissivity fields considered to be equally likely, each of which is in agreement with the measured transmissivity and pressure data. It consists of generating a selected number of conditionally simulated transmissivity fields and then calibrating each of the fields to match the measured steady state or transient pressures, in a least squares sense. The calibration phase involves an iterative implementation of an automated pilot point approach coupled with conditional simulations. Pilot points are the parameters of calibration. They are synthetic transmissivity data which are added to the transmissivity database to produce a revised conditional simulation during calibration. Coupled kriging and adjoint sensitivity analysis is employed for the optimal location of pilot points, and gradient search methods are used to derive their optimal transmissivities. The pilot point methodology is well suited for characterizing the spatial variability of the transmissivity field in contrast to methods using zonation. Pilot points are located where their potential for minimizing the objective function is the highest. This minimizes the perturbations in the transmissivities which are optimally assigned to the pilot point and results in minimal changes to the covariance structure of the transmissivity field. The calibrated fields honor the transmissivity measurements at their locations, preserve the variogram, and match the measured pressures in a least squares sense. Since there are numerous options in the execution of this methodology, computational experiments have been conducted to identify the most efficient among them. The method has been applied to the Waste Isolation Pilot Plant (WIPP) site, in southeastern New Mexico, where the U.S. Department of Energy is conducting probabilistic system assessment for the permanent disposal of transuranic nuclear waste. The resulting calibrated transmissivity fields are input to a Monte Carlo analysis of the total system performance. The present paper, paper 1 of a two‐paper presentation, describes the methodology. Paper 2, a companion paper, presents the methodology's application to the WIPP site.

A large‐scale investigation of fracture flow was recently conducted in a granite uranium mine at Fanay‐Augères, France. Its aim was to develop a methodology for the investigation of possible nuclear waste repository sites in crystalline environments, and thus to determine what measurements to make and what models to use in order to predict the flow and transport properties of the medium, i.e., their average behaviors and spatial variabilities at different scales. Four types of data were collected: (1) geometry of the fracture network; (2) local hydraulic properties measured by injection tests in boreholes; (3) global hydraulic behavior from flow rate and piezometric head distribution at a 106 m3 scale; and (4) tracer tests performed at a scale of up to 40 m. A stochastic fracture network model assuming negligible matrix permeability was developed and calibrated essentially on data 1 and 2 above; this was then used to predict data 3 and 4 in an attempt to validate both the parameters and the structure of the model. In this first part, only the flow problem (data 1) is discussed.

Abstract. This paper describes the first major attempt to compare seven different inverse approaches for identifying aquifer transmissivity. The ultimate objective was to determine which of several geostatistical inverse techniques is better suited for making probabilistic forecasts of the potential transport of solutes in an aquifer where spatial variability and uncertainty in hydrogeologic properties are significant. Seven geostatistical methods (fast Fourier transform (FF), fractal simulation (FS), linearized cokriging (LC), linearized semianalytical (LS), maximum likelihood (ML), pilot point (PP), and sequential self-calibration (SS)) were compared on four synthetic data sets. Each data set had specific features meeting (or not) classical assumptions about stationarity, amenability to a geostatistical description, etc. The comparison of the outcome of the methods is based on the prediction of travel times and travel paths taken by conservative solutes migrating in the aquifer for a distance of 5 km. Four of the methods, LS, ML, PP, and SS, were identified as being approximately equivalent for the specific problems considered. The magnitude of the variance of the transmissivity fields, which went as high as 10 times the generally accepted range for linearized approaches, was not a problem for the linearized methods when applied to stationary fields; that is, their inverse solutions and travel time predictions were as accurate as those of the nonlinear methods. Nonstationarity of the "true" transmissivity field, or the presence of "anomalies" such as high-permeability fracture zones was, however, more of a problem for the linearized methods. The importance of the proper selection of the semivariogram of the 1og•0 (T) field (or the ability of the method to optimize this variogram iteratively) was found to have a significant impact on the accuracy and precision of the travel time predictions. Use of additional transient information from pumping tests did not result in major changes in the outcome. While the methods differ in their underlying theory, and the codes developed to implement the theories were limited to varying degrees, the most important factor for achieving a successful solution was the time and experience devoted by the user of the method. •2Stanford University, Stanford, California.•3Duke Engineering and Services, Inc., Austin, Texas.•4University of Arizona, Tucson.•Slnstitut Franqais du Pftrole, Rueil-Malmaison, France.•6University of California, Berkeley.Copyright 1998 by the American Geophysical Union. Paper number 98WR00003.0043-1397/98/98WR-00003509.00 tion, or performance assessment of planned waste disposal projects, it is no longer enough to determine the "best estimate" of the distribution in space of the aquifer parameters. A measure of the uncertainty associated with this estimation is also needed. Geostatistical techniques are ideally suited to filling this role. Basically, geostatistics fits a "structural model" to the data, reflecting their spatial variability. Then, both "best estim...

For regional aquifer modeling it is often necessary to produce maps of the distribution of the transmissivity in the aquifer, for example, as initial input for the calibration phase of the model, either by automatic or by trial and error procedures. Such estimations must be based on all possible information available in the field; in many instances, direct transmissivity measurements from primping tests are scarce, whereas indirect estimations based on specific capacity data are more numerous. It is, however, possible to use jointly both types of data when a geostatistical estimation technique is used. Four such methods will be compared here: (1) kriging combined with linear regression, (2) cokriging, (3) kriging with an external drift, and (4) kriging with a guess field. This comparison is made both on a set of real field data and on a theoretical case, where the "true" solution is known. measurements of transmissivlties obtained in the wells where pumping tests had been carried out. In practice, however, such data are scarce, and in many wells only the specific capacity data are available.It is well known, however, that this specific capacity is strongly correlated to the transmissivity of the aquifer and should thus be incorporated in the estimation technique. Delhornroe [1974, 1976] suggested a method that combined kriging with linear regression which made this possible. Other methods are also available: (1) cokriging [Matheron, 1971; Joumel and Huijbregts, 1978;Myers, 1982Myers, , 1984Myers, , 1985 was recently used for aquifer transmissivity by Aboufirassi and Marino [1984]; (2) krlging with an external drift, which was used by Delhomme [1979b], Delfiner et al. [1983], Galli and Meunier [1987], and Moinard [1987] for other type s of data, Ahmed, S., Analyse g6ostatistique de quelques variables hydrog6ologiques, Rep. C.F.S.G. S-184, Centre de G6ostatistique, Ecole des Mines de Paris, Fontainebleau, France, 1985. Ahmed, S., G. De Marsily, and A. Talbot, Combined use of hydraulic and electrical properties of an aquifer in a geostatistical e•timation of transmissivity, Ground Water, 26(1), 1987. Binsariti, A. A., Statistical analysis and stochastic modelling of the Cortaro aquifer in southern Arizona, Ph.D. thesis, 243 pp., Univ. of Arizona, Tucson, 1980. Combes, P., Construction du module math•matique de la nappe du calcaire carbonif•re, Tech. Rep. LHM/RD/81/17, Ecole des Mines de Paris, Fontainebleau, France, 1981. Dagan, G., Models of groundwater flow in statistically homogeneous porous formations, Water Resout'. Res., 15(1), 47-63, 1979. Dagan, G., Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 1, Conditional simulation and direct problem, Water Resour. Res., 18(4), 813-833, !982. Darricau-Beucher, H., Approche g•ostatistique du passage des donn•es de terrain aux param•tres des modbles en hydrog•ologie, Ph.D. thesis, Ecole des Mines de Paris, Fontainebleau, France, 1981. Delfiner, P., J.P. Delhomme, and J. Pelissier-Combescure, Application of geostatistical analysi...

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