Dual Scale Simulations: A Tool for Pseudoisation
Authors: Thibeau S. 
Published: December 2007

Abstract
This paper deals with the generation of dynamical pseudo functions: pseudo relative permeabilities, pseudo capillary pressures and, for compositional simulations, -factors.
The generation of pseudo functions is complex due to the fact that they are flow dependent. It is therefore necessary to reproduce the flow conditions, for which they will be used, to compute these parameters. For a homogeneous areal reservoir, the oil and water pseudo relative permeabilities of a cell are shown to be dependent on the angle of the flood with the downstream interface.
Dual Scale Simulations are presented. They are simulations of models containing a limited number of cells, that enable to reproduce 1) the macroscopical flow in the reservoir and 2) the fine scale flow in a coarse cell. The results of these simulations can be post-processed into dynamical pseudo functions.
This method is applied to a 3D model of a lean gas injection scheme in a stratified reservoir containing near critical fluids. Its validation is obtained by comparison of the results with fine grid simulation results in terms of local reproduction of fine scale saturation and pressure patterns; computation of pseudo functions; accurate correction of the coarse model with the pseudo function.

Introduction
Nowadays, flow simulation grid cells are too large to take into account small scale heterogeneities (such as laminations or impermeable barriers) and to reproduce some small scale physical phenomena (as gravity tongues, fingering, fronts, high velocity zones close to the perforations…).
Pseudo functions are a way to transfer this otherwise lost small scale information into the flow simulation model (or 'coarse' model).
The generation and use of dynamical pseudo functions (such as pseudo relative permeabilities, pseudo capillary pressures) is a global problem, containing various steps:–the grouping of the cells into 'rock types', that will be characterized by specific pseudo functions;–the reproduction of the fine scale flow patterns in a cell to compute the local pseudo functions;–the post processing of this information into pseudo functions thanks to an 'upscaling method';–the knowledge of the dependence of the pseudo functions with flow conditions, in order not to have to recompute these functions each time the flow changes, during a sensibility study of any parameter of the model (well description, permeability array…).
This paper concentrates on the second item. Indeed, several authors have pointed out the fact that the pseudo functions are flow dependent (or 'boundary conditions' dependent), which means that the pseudo functions of a specific cell must be generated under the flow conditions proper to that cell.
To solve this dependence problem, we present here the Dual Scale Simulations. The principle of these simulations is to simulate simultaneously the coarse grid model and the finely gridded cell where the pseudo functions are to be computed. This way, the macroscopical flow is reproduced, and hence, the flow patterns in the coarse cell are obtained under correct flow conditions. The flow data can then be post-processed into accurate pseudo functions.
Why are the pseudo functions flow dependent? The dependency of the pseudo functions can be understood comparing the upscaling of absolute and relative permeabilities.

Upscaling the absolute permeability
A common way to compute 'effective' absolute permeability is to solve, with appropriate boundary conditions, the Laplace equation, standing for a one phase, incompressible fluid conservation equation:
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