Generalized Regression (Kriging - Cokriging)

Any unsampled value, a porosity for example, can be estimated by generalized regression from surrounding measurements of the same value once the statistical relationship between the unknown being estimated and the available sample values has been defined. This is exactly what the correlogram provides: a prior model of the statistical similarity between data values. This generalized regression can also include nearby measurements of some different variable—seismic travel time, for example, or a facies code. When using such secondary information, one also needs a prior model of the cross-correlogram, which provides information on the statistical similarity between different variables at different locations.

Integrating Data of Different Types

Multivariate regression, or "cokriging" as a geostatistician would tail it, is usually not a convenient framework for the integration of too widely different types of data such as qual-itative geological information, which is usually only indicative in nature, and direct laboratory measurements. At a particular location where one does not yet have a porosity measurement, a consideration of the lithofacies information might provide a reasonable interval within which the unknown value should fall. If we are certain that we are in a particular type of sandstone, for example, we might know that the porosity must be somewhere in the interval from 10% to 30%. If, in addition, we also have enough core plug measurements within that type of sandstone to build a histogram we could go further than simply stating the previous constrained interval. We could use that histogram to provide a probability distribution that might, for example, tell us that the unknown porosity is more likely to be on the low end of our 10% to 30% range than on the high end.

Stochastic Imaging of Reservoir Heterogeneities

The thrust of modern geostatistics is not least-squares spatial regression but the building of probability distributions that characterize the present uncertainty about a reservoir rock and fluid properties. These probability distributions should account for all relevant information through models of the spa-tial dependence between each piece of information and the unknown variable of interest. As more information is collected, the uncertainty about the unknown variable of interest lessens and the spread of the posterior probability distribution decreases. The "unknown" can be a single particular unsampled value, say porosity, at a single location, or it can be the unsampled values of a particular variable at many locations (the nodes of a regular grid, for example). It can even be the unsampled values of all relevant variables-porosity, permeability, satura-tions, pressures,...-at many locations. With a single variable at a single location, one has a l-variate problem; with a sin-gle variable at N locations, the problem becomes an N-variate problmn; with K interdependent variables at N locations, the problem becoms a (K*N)-variate problem.