Reservoir Modeling Using Multiple-Point Statistics

Two approaches are traditionally used to build numerical models for facies distributions within a reservoir. Pixel-based techniques aim at generating simulated realizations that honor the well data values, and reproduce a given variogram which models two-point spatial correlation. However, because the variogram cannot look at spatial continuity between more than two locations at a time, pixel-based algorithms give poor representations of the actual facies geometries. In contrast, object-based techniques allow reproducing crisp geometries, but the conditioning on well data requires iterative ``trial-anderror'' corrections, which can be time-consuming, particularly when the data are dense with regard to the average object size. This paper presents a new approach that combines the easy conditioning of pixel-based algorithms with the ability to reproduce ``shapes'' of object-based techniques, without being too time and memory demanding. In this new approach, the complex geological structures expected to be present in the reservoir are characterized by multiple-point statistics, which express joint variability at many more than two locations at a time. Such multiple-point statistics cannot be inferred from typically sparse well data but could be read from training images depicting the expected subsurface heterogeneities. A training image need not carry any locally accurate information on the reservoir; it need only reflect a prior stationary geological/structural concept. Thus training images can be generated by object-based algorithms freed of the constraint of data conditioning. The multiple-point statistics inferred from the training image(s) are then exported to the reservoir model, where they are anchored to the well data using a pixel-based sequential simulation algorithm. ∗ Now with Chevron Petroleum Technology Co. This algorithm is tested for the simulation of a turbidite system where flow is controlled by meandering channels with cross-bedding. The training image reflecting the channel patterns is an unconditional realization generated by an objectbased algorithm. The final simulated numerical models reproduce these channel patterns, and honor exactly all well data values at their locations. The methodology proposed appears to be practical, general, and fast. Introduction Most current stochastic reservoir simulation algorithms aim at reproducing statistics inferred from the well data solely. Since these data are typically sparse they can give, at best, only an estimate of the two-point correlation within the reservoir. Hence most traditional simulation techniques, such as sequential Gaussian simulation, are limited to the reproduction of a variogram model or covariance function inferred from a few sample two-point correlation moments. Identification of two-point statistics is, however, not sufficient to reproduce connected patterns over long ranges, nor clusters of similar values with a characteristic shape (e.g. meandering channels), features which are common occurrences in oil reservoirs. The modeling of such specific patterns calls for characterizing the spatial continuity at three or more locations at a time. Reproduction of multiple-point continuity in the reservoir model is critical, not so much to produce geologically realistic looking maps, but to provide accurate flow performance predictions. The quest for more realistic models is motivated by the fact that structural information much beyond the sample twopoint correlation calculated from the well data is actually available. Stratigraphic interpretation from well logs, seismic data analysis, and comparison with reservoir analogues do provide information about the type of geological bodies present in the reservoir. Variogram-based algorithms mostly ignore such valuable information. One straightforward way to reproduce geological bodies with crisp shapes consists in generating objects with similar shapes then distribute these objects over the reservoir model. There are, however, three major limitations to such object-based approach: • Each different geometric shape requires its own parameterization, which means that each object-based algorithm can handle only one type of reservoir. SPE 71324 Reservoir Modeling Using Multiple-Point Statistics Sebastien B. Strebelle, SPE, Stanford University, and Andre G. Journel, SPE, Stanford University 2 S.B. STREBELLE, A.G. JOURNEL SPE 71324 • The conditioning well data may not be all honored if they are dense with regard to the average object size. • It is difficult to constrain the reservoir model to spatial trends, as represented by a vertical facies proportion curve or an areal proportion map. We propose an alternative approach to reproduce reservoir specific facies geometries that uses multiple-point statistics, i.e. (cross) correlation moments at three or more locations at a time. To provide an analytical expression of these correlation moments, some new terminology is now introduced. Consider an attribute S, taking K possible states {sk, k=1.K}. S can be a categorical property, for example facies, or a continuous value such as porosity or permeability, with its interval of variability discretized into K classes by (K-1) threshold values. A data event dn of size n centered at location u is constituted by: • the data geometry defined by n vectors {hα,α=1.n} • the n data values {s(u+hα),α=1.n} A data template τn comprises only the previous data geometry. The data event dn is said to be “associated” with the geometric template τn. The categorical transform of the variable S at location u is defined as: ( )    = = not if 0 ) ( if 1 ; k s S k I u u The multiple-point statistics used in this paper are actually probabilities of occurrence of the data events ( ) { } n á s S d á k á n . 1 , = = = u , i.e. probabilities that the n values s(u1). s(uk) are jointly in the respective states n k k s s . 1 . For any data event dn, that probability is also the expected value of the product of the n corresponding indicator data: { } ( ) { } (     ∏ = = = =