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The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

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

The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

Quantitative methods taking into account the sedimentological characteristics will answer the needs of reservoir engineers. We propose here a geostatistical method for the conditional modelling of the facies of a sedimentary fluvio-deltaic series. This model was elaborated jointly by I.F.P. and the Paris School of Mines, with the aim of modelling reservoir heterogeneities. From the sedimentological study contained in the paper by C. Ravenne et al., we present several simulations, conditioned by "drill-core" taken from the outcrop. The block permeabilities are then calculated from the values given to the facies. Introduction Reservoir engineers have, for a long time, been asking what type of models could be entered into reservoir simulators. The geological models that are normally used are essentially qualitative and so it is difficult to numerize them, except by correlating the drillholes facies, which is not always self evident. This often leads to models with too many constraints for simulating reservoirs. Their dynamic behaviour worsens considerably going from very continuous layers of sandstone to disseminated lenses. (Fig. 1). This raises the question of how to characterize the geometry of the sandstone given the drillhole data, the geologist's interpretation and also other measurements (e.g. seismic recordings), and how to model the reservoir levels to suit this shape. As well as this, the models must match the lithology along the drillholes. In this article, we present a method for conditionally modelling the lithology which is designed for sedimentary processes. This approach was tested using the geological section of a cliff-face in Yorkshire (England) which shows a fluvio-deltaic environment similar to some of the levels in the Brent formation in the North Sea. A detailed description of the geology of the cliff-face studied and of the approach used in this project have been presented by Ravenne et al. Here we shall only consider the problem of modelling random sets by using a probabilistic method for representing the spatial distribution of the facies (sandstone, shaly sandstone, shale) in a heterogeneous reservoir. Before presenting the method, we review the main procedures for modelling random sets, that are used in the petroleum industry. REVIEW OF THE EXISTING METHODS Boolean Sets A simple way is to consider a heterogeneous medium as consisting of sandstone lenses in a shale matrix (or vice versa). Boolean sets (Matheron, Serra, Jeulin) are a mathematical way for modelling this type of deposit, that has been used for many years in other fields (Fig. 2). This method consists of putting lenses of a predetermined shape (e.g. ellipses or rectangles) at random points in the domain under study (i.e. the points are statistically uniformly distributed in space). Lenses are not correlated. The advantage of this approach is that it is easy to use in 2D or 3D spaces. It only depends on a few parameters: the number of seed points per unit space (called density), the shape of the lenses (fixed or variable), their size and orientation. The model is very flexible. The parameters can be modified locally in order to reproduce the real phenomenon more accurately. Clearly the more complicated the model is, the more parameters there are to fit but this can be overcome by fitting them by trial and error. P. 591^

ABSTRACT. In this paper, a new procedure for non linear estimation is proposed: it is better than the usual best linear estimation, and necessitates less prerequisites than the conditional expectation. O. INTRODUCTION.In applied Geostatistics. we need more and more often to estimate certain variables which depend in a non linear way on the grades of an orebody. Simple examples are given by the estimation procedure of the "transfer functions". In such a case, it is generally no longer possible to use the classical linear kriging estimotor. which is inadequate for these prob lema and can be dangerous ly biased. On the other hand, one cannot resort to the conditional expectation technique, either because the available information is not sufficient, or simply because the computations are too expensive. Under the name of Dis junctive Kriging (O .K.) we propose in this paper an intermediate method, more powerful than the simple linear combinations, but less sophisticated' than the condition-The starting idea is the following. Let Z be a random variable (R.V.) to be estimated from a set of other~ R.V. Za' a· 1,2,

16. Random Sets and Integral Geometry. By G. Matheron. New York and London, Wiley, 1975. xxiii, 261 p. 914″. £9.10. (Wiley Series in Probability and Mathematical Statistics.)

No abstract