Upscaling of Stochastic Models for Reservoir Simulation - An Integrated Approach
Authors: Lozano J.A. , Costa L.P. , Alves F.B. , Silva A.C. 
Published: December 2007

Upscaling of Stochastic Models for Reservoir Simulation - An Integrated Approach.

Abstract
Large oil reservoirs are seldom characterised by a high degree of homogeneity and a well defined layer structure. Even some of those initially classified within this category ended up by showing a significant level of heterogeneity after more thorough studies were carried out. Modelling based on geostatistical methods (stochastic modelling) is a very important technique, being the most appropriate for an increasing number of reservoirs. Naturally, stochastic modelling generates extensive and very detailed property maps. This, in turn, implies the development of an adequate upscaling methodology in order to make reservoir simulation models time efficient and less hardware consuming.
In this paper an upscaling process is proposed, which develops in three steps. The first deals with the optimisation of grid size and orientation based on well locations. The second step deals with the selection and application of known permeability upscaling techniques. Furthermore, a procedure is developed to generate coarse scale transmissibilities. It uses refined sector simulation models under stabilised conditions to obtain directional transmissibilities for the upscaled blocks based on the fine grid permeability distribution. Finally, the third step contemplates the aggregation of neighbouring blocks with similar properties, thus reducing the overall number of coarse grid blocks.

Introduction
Stochastic models are now frequently used as a means of characterising heterogeneous hydrocarbon reservoirs. Geostatistical techniques are adequate to handle the fine scale heterogeneities and are capable of providing alternative high resolution images of the reservoir at the core scale.
Unfortunately, this level of detail is far too high to be accommodated by a dynamic reservoir simulator and in order to perform flow simulation, reservoir properties must be "averaged" to a more suitable scale. The averaging procedure (upscaling), however, has to occur in such a way that the fine scale effects on reservoir displacement processes are captured and preserved as much as possible.
Having this as a fundamental principle, the coarse scale models should also accomplish the purpose of maximising computational efficiency by minimising the number of grid blocks required for simulation of reservoir performance. In this paper it is shown that a proper choice of the grid areal orientation relative to the well locations is a sensible step to achieve this purpose, allowing the optimisation of blocks size and number.
The next step would be the selection (or development) of good performance upscaling methods in order to convert the fine scale properties into coarse scale values. Permeability is our main concern in this paper. The other properties such as effective porosity or saturation seem properly handled when volume or pore volume weighted averages are used.
Durlofsky et al. distinguish two fundamental approaches in the scale up of detailed geological descriptions. One of them is supported by the application of pseudo relative permeability and pseudo capillary pressure curves. Although adequate for some problems, this approach is highly process dependent in the sense that pseudos are only valid for flow conditions similar to the ones used on their computation. The other approach relies on the calculation of effective or coarse grid block equivalent properties, using either analytical or numerical methods.
In this work some of the most robust analytical methods used in the calculation of effective permeability will be tested for screening purposes. Numerical methods are also dealt with and an upscaling procedure based on directional transmissibilities is proposed. This studies are performed in two dimensions with an extension to three dimensional problems suggested at the end.
Finally, there is a third step, in which the coarsened grid is post-processed. Neighbouring blocks with similar permeabilities are aggregated in order to further reduce the overall number of grid blocks.
P. 328