Exact Conditioning to Linear Constraints in Kriging and Simulation

Abstract

A recurrent problem in reservoir characterization is the need to generate realizations of an attribute conditioned not only to core data measurements, but constrained also to attribute averages, defined on a larger support. A related problem is that of downscaling, in which there is a need to generate realizations at a scale smaller than the available data, yet preserving some type of average relation between the data and the downscaled realization. Typical larger support data are those coming from well tests, production data, or geophysical surveys. A solution is proposed to approach these two problems for the case in which the large support data can be expressed as linear functions of the original, smaller support, attribute values, or of some local transform of them. The proposed approach considers two random functions, one for the point (small support) data, and one for the block (large support) data. The algorithm is based on the full specification of the point to block and block to block covariances from the point to point covariance. Once all direct-and cross-covariances are specified, co-kriging or co-simulation can be used to either produce estimation or simulation maps. Similar approaches have been attempted, but this approach distinguishes itself because it is exact, in the sense that the constraints are exactly honored in the final maps. Although the theoretical basis for this constrained estimation or simulation is reasonably straightforward, its implementation is not. In particular, the building of the co-kriging systems and the concept of search neighborhood presents some non-negligible challenges, which have been efficiently solved, even for the non-trivial case of overlapping supports of the constraining data.