The Modeling of a Three-Dimensional Reservoir with a Two-Dimensional Reservoir Simulator-The Use of Dynamic Pseudo Functions
Authors: Jacks Hugh H. , Smith Owen J.E. , Mattax C.C. 
Published: December 2007

Abstract
Dynamic pseudo-relative permeabilities derived from cross-section models can be used to simulate three-dimensional flow accurately in a two-dimensional areal model of a reservoir Techniques are presented for deriving and using dynamic pseudos that are applicable over a wide range of rates and initial fluid saturations. Their validity is demonstrated by showing calculated results from comparable runs in a vertical cross-section model and in a one-dimensional areal model using the dynamic pseudo-relative permeabilities and vertical equilibrium (VE) pseudo-capillary pressures. Further substantiation is provided by showing the close agreement in calculated performance for a three-dimensional model and corresponding two-dimensional areal model representing a typical pattern on the flanks of a large reservoir. The areal pattern on the flanks of a large reservoir. The areal model gave comparable accuracy with a substantial savings in computing and manpower costs.

Introduction
Meaningful studies can be made for almost all reservoirs now that relatively efficient three-dimensional reservoir simulators are available. In many instances, however, less expensive two-dimensional areal (x-y) models can be used to solve the engineering problem adequately, provided the nonuniform distribution and flow of fluids in the implied third, or vertical, dimension of the areal model is properly described. This is accomplished through the use of special saturation-dependent functions that have been labeled pseudo-relative permeability (k ) and pseudo-capillary pressure permeability (k ) and pseudo-capillary pressure (P ) or, for simplicity "pseudo functions", to distinguish them from the conventional laboratory measured values that are used in their derivation. Two types of reservoir models have been suggested in the past to derive pseudo functions: the vertical equilibrium (VE) model of Coats et al., which is based on gravity-capillary equilibrium in the vertical direction; and the stratified model of Hearn, which assumes that viscous forces dominate vertical fluid distribution. Neither of these models accounts for the effects of large changes in flow rate that take place as a field is developed, approaches and place as a field is developed, approaches and maintains its peak rate, and then falls into decline. This paper presents an alternative method for developing pseudo functions that are applicable over a wide range of flow rates and over the complete range of initial fluid saturations. The functions may be both space and time dependent and, again for clarity and convenience in nomenclature, we have labeled them "dynamic pseudo functions".

Description Of Pseudo-Relative Permeability Functions Permeability Functions
Methods for developing pseudo functions have been presented in the literature. The distinction between our method and those used by others lies in the technique for deriving the vertical saturation distribution upon which the pseudo-relative permeabilities are based. In our approach, the permeabilities are based. In our approach, the vertical saturation distribution is developed through detailed simulation of the fluid displacement in a vertical cross-section (x-z) model of the reservoir. The simulation is run under conditions that are representative of those to be expected during the period to be covered in the areal model simulations. period to be covered in the areal model simulations. Results of the cross-section simulation are then processed to give depth-averaged fluid saturations processed to give depth-averaged fluid saturations (S ) and "dynamic" pseudo-relative permeability values (k ) for each column of blocks in the cross-section model at each output time. The above approach can result in a different set of dynamic pseudo functions for each column of blocks due to differences in initial saturation, rate of displacement, reservoir stratification, and location. However, differences between columns are frequently minor or they can be accounted for by correlation of the data. In this and several other reservoir studies, it was possible to reduce the complexity of the pseudo function sets through correlations with initial fluid saturations and fluid velocities.