Block upscaling

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Rubin and Gdme -Hemdndez (1990) and Indelman and Dagan, (1993) have developed analytical expressions fotyupscaling both the mean and the variance of spatially autocorrelated lnOfc fdatcL Since macroscale dispersion can be related to the variance of ln( sat), it may be possible to apply their expressions for predicting variance as a function of block size to the problem of upscaling estimates of K determined in the laboratory to the field scale. Unfortunately, numerical simulations based upon this approach are likely to be computationally expensive. 

For the solution at the large scale, an upscaling procedure is applied over the small-scale model to determine the best conductivity value to be assigned to each square block of 10 by 10 cells. This procedure results in a conductivity tensor that... 

Figure J. Set of four prescribed head conditions applied around a skin centered on the block being upscaled...

The following procedure is taken to define an upscaling rule that will not only reproduce total flows at the 50 m by 50 m block scale, but also at... 

Figure 4. Diagonal components of the logiohydraulic conductivity tensor (logm m/s) after upscaling a smalt scale realization. The blocks are 50 m by 50 m in size.

The block of 10 by 10 cells to be upscaled is isolated along with a skin. The flow equation is solved using the boundary conditions shown in the Figure. These boundary conditions force flow in four different directions. Then, a simple least-square procedure is used to determine the single conductivity tensor that is able to reproduce best the flows crossing the block for all four boundary conditions. The result is a full hydraulic conductivity tensor with arbitrary principal directions, not necessarily parallel to the Cartesian axes. In Figure 4, the diagonal components of the... 

Similar upscaling techniques, motivated by the need to reduce grid block number, are important in practice. But the equivalent permeabilities within any reservoir will change if the reservoir is produced by different arrangements or patterns of wells, because the parallel and serial nature of the flow has changed. Upscaled quantities are not properties of the formation but are also related to the production method. However, several simulators compute fixed upscaled properties and use them in contrasting production scenarios. 

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