Evaluation of Dynamic Pseudo Functions for Reservoir Simulation
Authors: Guzman R.E. , Giordano D. , Fayers F.J. , Godi A. , Aziz K. 
Published: December 2007

Abstract
This paper presents a detailed investigation on the reliability of the dynamic pseudo functions used to up-scale flow properties in reservoir simulation. A theoretical study and a real field application are used to evaluate Kyte and Berry (1975). Stone (1991), and a new flux weighted potential (FWP) method. A derivation of Stone's method is presented and shown to use an inconsistent set of equations. Stone's analytical example was used to illustrate how pseudo relative permeabilities that exhibit non-physical behavior may still give acceptable results, but this success can disappear with changes in boundary conditions. The pseudo functions from the field application were not able to match the 2D simulations from which they were calculated, even when a different pseudo function was used for each coarse grid block. Improvements were obtained when directional pseudo functions were used, but still the results were not satisfactory. Similar results were found when comparing fine and coarse grid 3D simulations for a quarter of a five-spot pattern in this field. The results presented in this paper suggest that dynamic pseudo functions, as applied here and as commonly used in industry, may not be an adequate approach to up-scaling. The possibility of large errors and the difficulty in predicting when they may occur make the use of pseudo functions unreliable.

Introduction
Oil and gas reservoirs are very complex systems in which rock and flow properties vary at all scales (pore to reservoir scale). Rock properties (e.g. porosity and absolute permeability) and flow properties (e.g. relative permeability and capillary pressure) show variations that can be significant to oil recovery at scales below the size of common simulation grid blocks. One of the most important problems in reservoir simulation is that of accurately accounting for such small scale variations. In addition to the low resolution, coarse grid solutions can be strongly affected by numerical diffusion. Several pseudo function techniques have been proposed to reproduce fine grid results (including detailed descriptions of heterogeneity and with minimum numerical diffusion) using the typical coarse grids of field simulations.
The dynamic pseudo functions of Jacks et al. (1973) and Kyte and Berry (1975) were designed with the purpose of reducing the dimensionality of a problem (e.g. including 3D effects within a 2D model), and in general, reducing numerical diffusion in coarse gridded models. Simple rules to average absolute permeability were also included in these methods. More recently, up-scaling due to heterogeneities has become the main issue. It was realized that heterogeneities not included in coarse grid simulation could have a major impact on oil recovery predictions. New methods were presented by Hewett and Berhens (1991), Stone (1991), Beier (1992) for dynamic pseudo functions, and by King (1989) and King et al. (1993) for single and two-phase renormalization.
The main problem with renormalization is the systematic error that is introduced due to the artificial boundary conditions (e.g. no flow or constant pressure) imposed on each block at each renormalization step. Hewett and Berhens (1993) and Yamada and Hewett (1995) have explained the origin of this error as the forcing of incorrect flow paths or streamlines through the rescaled grid blocks. They have shown that the magnitude of the error can be large and yield incorrect up-scaling (also see Malick and Hewett (1995) for a quantification of the error). The errors in multiphase renormalization are larger than in single-phase since, in addition to incorrect boundary conditions in pressure, multiphase flow renormalization imposes incorrect boundary conditions for fractional flow.
The method of Durlofsky et al. (1994) tries to remedy the problems with the renormalization technique. In this method, a non-uniform coarse grid is defined using the flow velocity from a single-phase flow solution (i.e., trying to honor the streamlines), and up-scaling using periodic instead of no flow boundary conditions. Significant errors are still introduced by the artificial boundary conditions (Yamada 1995), and only single-phase flow was up-scaled properly.
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