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Multiple realizations of facies, porosity, and permeability are used for better representation of reservoir heterogeneity for more accurate performance forecasting and uncertainty assessment. These geostatistical realizations must reproduce all available data to be reliable. The available well, seismic, and production data are at different scales and must be linked to the reservoir modeling scale. The most common geostatistical simulation algorithms call for a Gaussian or normal transformation. It is not possible to merge the data types after this non-linear transformation. This has led to development of "Direct" approaches for simulation and data integration. Considering the variables without transformation ensures reproduction of the different data and the prescribed covariance or variogram model; however, until now, the resulting global histogram of the simulated realizations is not reproduced. The problem is that there is no theory that specifies the shape of the conditional distributions. The mean and variance are determined from the well-known normal or simple kriging equations; however, theory has not existed to specify the shape of the conditional distributions. A number of ad-hoc solutions have been proposed, but they violate the data and artificially reduce the modeled space of uncertainty. This paper develops a theory for determination of the required distribution shapes and reproduction of the global histogram. The results have significant theoretical and practical consequences. Data from multiple sources and scales can be directly reproduced in reservoir models with no need for transformation. There is no need for ad-hoc post-processing transformation or correction schemes. Several applications with synthetic data are shown to illustrate the technique.

This paper introduces an algorithm for rule induction intended to provide new insights, improve the reliability and expedite the utilization of large petrophysical and geologic databases. Very large petrophysical, geophysical, and geological databases contain multiple data types, which must be interpreted for application in subsurface modeling. This paper presents a significant advance in discovering complex and even nontrivial data relationships from such databases. Geoscientists are often challenged to predict subsurface lithologies and properties from multivariate relationships within large databases of core, wireline, and seismic data. Many data analysis techniques are used including histograms, parametric and non-parametric regression, n-dimensional histograms, cluster analysis, discrimininant analysis, principal components analysis. This paper introduces a new algorithm that seeks to discover "rule-like" relationships within the data that can be used to make predictions. The method is loosely derived from a data mining technology of classification. Concepts of data attribute distinguishability and importance are introduced to assess the value of the data and the outcomes to predictability. The new theory, implementation details, and an application are presented. Current petrophysical, seismic, and geostatistical analysis benefit from the rule induction algorithm presented. Improved reservoir characterization and forecasting result.

Accurate prediction of reservoir performance depends on an accurate estimate of the subsurface structure, lithofacies, associated petrophysical properties, and fluid distribution. Often the reservoir heterogeneities that are controlled by stratigraphic architecture and sedimentological trends are difficult to predict, particularly at subseismic resolution and far from well control. An ongoing challenge in subsurface modeling has been the utilization of analogs of complex geology along with seismic and sparse well data to predict the natural geologic complexity in models of the subsurface. Time surfaces provide very important constraints on the geometric connectivity and continuity of facies and petrophysical properties in reservoirs. Such time surfaces are a suitable framework for facies and petrophysical properties modeling. Instead of modeling each reservoir later as a whole, the elementary sediment units are more easily modeled separately; the final composite model will show realistic heterogeneity patterns consistent with the underlying physics. This work presents a hybrid deterministic, rule-based, and stochastic technique to generate surface models. These surface models are utilized as a framework to preserve sediment trends and honor analog and well data. Petrophysical properties are modeled for each sediment unit to reproduce trends. Finally, the individual sediment units are assembled into a reservoir model. The surface model is created stochastically with parameterized surface templates. The shape, extent, height, orientation and regularity of the surfaces are controlled by user-specified distributions. The location of each surface in the reservoir is chosen on the basis of previous events. The addition of each surface is based on sedimentological rules. Conditional Gaussian simulation is used to ensure that the surfaces reflect realistic uncertainty through undulations and that well data intersections are honored. The surface model divides the reservoir layer into sediment units. From geology and well data, trends are parameterized with mathematical functions as trend templates. Residuals are characterized after removing the trend. For each sediment unit, a trend and a residual model are generated stochastically. The observed well logs serve as conditioning data to guide the deployment of trends and to condition the generation of residuals. The model of each sediment unit combines its trend plus residual. The final reservoir model is obtained by assembling the separate sediment units.