Upscaling of Transmissibility for Field Scale Flow Simulation in Heterogeneous Media
Authors: Ding Y. , Urgelli D. 
Published: December 2007

Abstract
Techniques of upscaling absolute permeability have been discussed for a long time in the literature. To improve the accuracy for flow simulation, the upscaling technique must be integrated in the discretized numerical scheme. Now, more and more papers discuss the upscaling of transmissibility. In this paper, the technique of transmissibility upscaling for a control-volume type scheme is studied. The upscaling procedures are considered in near and far well regions for a problem that needs a full permeability tensor.

Transmissibility Upscaling
A control-volume numerical scheme is developed to discretize the flow equations with a full tensor permeability. A technique of transmissibility upscaling corresponding to this numerical scheme is presented. For a field scale simulation, a "radial" type upscaling technique must be considered to suit to high pressure gradient in the near well region. A technique for the transmissibility upscaling in the vicinity of wells is also presented.
Finite-Volume Numerical Scheme. Applying the control-volume technique to the flow equations, the elliptic operator can be discretized as a sum of flow terms at the interfaces (Fig. 1) with:(1)
where represents the fluid flow through the interface by a pressure gradient in x-direction and the fluid flow by a pressure gradient in y-direction, representing the cross flow. These two terms are approximated by:
(2)
(3)
where are the transmissibilities to be upscaled. These transmissibilities are upscaled on a shifted block (Fig. 2).
Upscaling on a Shifted Block. Lots of block permeability upscaling procedures are proposed in the literature, and all of them can be used to determine the equivalent transmissibility on the shifted block. The transmissibility in the above numerical scheme can be easily calculated using the permeability upscaling on the shifted block. It is not our objective to discuss the advantages of these upscaling methods. Our goal is to compare the transmissibility and permeability upscaling impact on the flow simulation. Without loss of generality, periodic boundary conditions are used in this paper for the transmissibility upscaling on the shifted blocks.
"Radial" Upscaling Procedure. A "radial" upscaling technique was proposed for a 5-point finite-difference scheme. In this paper, this method is extended to the control-volume scheme with a full tensor permeability. Using a tensor gives us one more degree to identify the equivalent transmissibility for an accurate flow calculation. Keeping the values of cross transmissibility obtained from an upscaling procedure on a shifted block, the transmissibility in the principle direction will be modified to suit the "radial" flow:
(4)
where is the total flow through a coarse grid interface, p is the coarse grid pressure. These parameters can be determined by a fine grid simulation in the near well region. The cross flow is calculated from Eq. (3).
The "radial" upscaling is particularly important for the imposed well pressure condition problem.

Numerical Example
A stratified field is considered. The fine grid has 99×99 gridblocks, and the coarse grid has 11x11 gridblocks (Fig. 3).
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