Stochastic parameters representing geological uncertainties in reservoir modeling may be classified in 2 types: 1) Continuous stochastic variables (e.g., degree of communication through a fault); and 2) Discrete stochastic variables representing different geological interpretations (e.g., same/different channel observed in different wells) each with a given probability.A method for optimizing reservoir performance is presented, which may take into account both these types of uncertainties in a consistent and simple way by finding values of reservoir management variables that optimize the expected performance over the population of possible reservoirs. The method is based on response surfaces and experimental design and is implemented in a user-friendly computer program which may be used together with any reservoir simulator or analytical flow model.

The distribution of zeros of the grand partition function is calculated in the thermodynamic limit for a class of one-dimensional gas models in two ways: (1) from the equation of state and (2) directly from the partition function. In this way one obtains (for these cases) a verification of the assumptions we had to make in order to associate a unique distribution of zeros with a given equation of state. In the Appendix we present some numerical evidence for the validity of these assumptions also in the case of the van der Waals gas.

Summary This paper describes a multistep pseudofunction-generation process designed to incorporate several scales of process designed to incorporate several scales of heterogeneities into one set of final pseudofunctions to be used in large gridblocks for field simulations. A detailed description of the multistep scale-up process is provided. The calculational procedure and the results of an extensive numerical scaling-up experiment from Kyte and Berry pseudofunctions are described. Three geological descriptions involving pseudofunctions are described. Three geological descriptions involving random permeability variations are used. The effect of three scales (sizes) of heterogeneities on a standard oil/water relative permeability curve is determined from a three-step pseudofunction-generation process. Ranges of mobility ratios, viscosity/gravity ratios, and viscosity/capillary ratios are used in the displacements to provide a guide to the effect of various types and scales of heterogeneities on fluid flow in several regimes. Introduction Pseudorelative permeabilities and capillary pressures have been used Pseudorelative permeabilities and capillary pressures have been used for several years in reservoir simulations to reduce the number of dimensions of the flow field or to reduce the total number of gridblocks (to a computationally economical number). The process allows us to replace the geologically detailed fine grid of rock (laboratory) relative permeability and capillary pressure curves wita few large homogeneous gridblocks containing an effective permeability and pseudofunctions. When the displacement is permeability and pseudofunctions. When the displacement is simulated with the coarse grid of effective permeabilities and the generated pseudofunctions (relative permeability and capillary pressure), the pseudofunctions (relative permeability and capillary pressure), the fluid flow across the coarse-grid boundaries or into the production wells will be the same as in the fine-grid simulation. Thus, the effect of increased numerical dispersion and the decreased detail of the reservoir description in the coarse grid are overcome by the pseudofunctions. pseudofunctions. Recent trends in reservoir-description research have been directed at replacing a fine-grid reservoir containing a specific type of heterogeneity with a very coarse (homogeneous) grid. Davies and Haldorsen replaced stochastic discontinuous shale bodies with pseudorelative permeabilities. Pande et al. studied the pseudorelative permeabilities. Pande et al. studied the replacement of a noncommunicating layered system with a 1D description of the layered system by determining the appropriate pseudofunctions. Both works consider one scale of heterogeneity pseudofunctions. Both works consider one scale of heterogeneity e. g., the discontinuous shales or the noncommunicating layers. Their implicit assumption was that smaller scales of heterogeneities (e.g., variations in the rock permeability between the shales or within the layers) were not important and that the rock curves could be used in the fine gridblocks. Lasseter et al. extended the classic concept for the use of pseudofunctions in heterogeneous reservoir description. The idea water pseudofunctions in heterogeneous reservoir description. The idea water use pseudofunctions in a multistep process to account for several scales of rock heterogeneity and, at the same time, to control numerical dispersion. The process begins with laboratory rock relative permeabilities and capillary pressures at the small scale and permeabilities and capillary pressures at the small scale and ends up, after several steps, with final pseudofunctions for large gridblocks designed for use in field-scale simulations. These final pseudofunctions control numerical dispersion as do the standard pseudofunctions control numerical dispersion as do the standard one-step pseudofunctions, but they also account for the several scales of reservoir heterogeneities. A detailed explanation of the scalingup process follows in a later section. The research project associated with this paper (see Acknowledgment) is meant to expand the work of Lasseter et al. and to provide an understanding of the effect of various scales of heterogeneity provide an understanding of the effect of various scales of heterogeneity in different geological settings on the pseudofunctions. A scaling-up system was defined and displacement calculations were made over a range of mobility ratios, viscosity/capillary ratios, and vis-cosity/gravity ratios for three geological types. A comparison othe results shows the effect of the different heterogeneities on displacements in various fluid-flow regimes. Fluid-flow regimes for each geological description are given where the scaling-up process is not required to give a good approximation of the displacement in the coarse grid.

Stochastic parameters representing geological uncertainties in reservoir modeling may be classified in 2 types: 1) Continuous stochastic variables (e.g., degree of communication through a fault); and 2) Discrete stochastic variables representing different geological interpretations (e.g., same/different channel observed in different wells) each with a given probability.A method for optimizing reservoir performance is presented, which may take into account both these types of uncertainties in a consistent and simple way by finding values of reservoir management variables that optimize the expected performance over the population of possible reservoirs. The method is based on response surfaces and experimental design and is implemented in a user-friendly computer program which may be used together with any reservoir simulator or analytical flow model.

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