Use of Higher Moments for the Description of Upscaled, Process Independent Relative Permeabilities
- Louis J. Durlofsky
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 1997
- Document Type
- Journal Paper
- 474 - 484
- 1997. Society of Petroleum Engineers
- 4.3.4 Scale, 5.8.6 Naturally Fractured Reservoir, 4.6 Natural Gas, 5.5 Reservoir Simulation, 1.6.9 Coring, Fishing, 5.1.5 Geologic Modeling, 5.5.3 Scaling Methods, 5.3.1 Flow in Porous Media, 5.1 Reservoir Characterisation, 5.3.2 Multiphase Flow
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Traditional approaches for the scale up of highly detailed reservoir descriptions to the coarser scales more appropriate for reservoir simulation often rely on the use of pseudo relative permeabilities. Two of the major limitations in the use of pseudo relative permeabilities are the process dependency inherent in the resulting curves and the need for a different set of curves in each coarse scale grid block. In this paper, these limitations are illustrated and quantified through numerical simulations of viscous dominated displacements in fine scale, heterogeneous, two dimensional systems. It is concluded that the functional form of traditional pseudo relative permeability descriptions is too limited to capture a wide variety of flow behavior and that an 'extended description,' in which the upscaled relative permeabilities depend on variables in addition to the phase saturations, is required. The additional variables we propose are higher moments of the fine scale variables, specifically the variance of saturation, the variance of pressure and the velocity-saturation covariance. It is then shown that these new variables are capable of accurately correlating upscaled relative permeabilities for a variety of displacements in a process independent manner; i.e., the extended description can account for varying inlet conditions and permeability realizations. The practical use of this methodology in an actual reservoir simulation will, however, require the derivation and solution of coarse scale evolution equations for the moments.
The level of geologic detail built into reservoir descriptions continues to exceed the computational capabilities of reservoir simulation by a significant margin. Depending on the sophistication of the geologic model, the type of simulation to be performed and the computer resources available, the geologic description may contain one to several orders of magnitude more detail (grid blocks) than the simulation model can accommodate. This disparity has motivated the development of a variety of techniques to coarsen, or scale up, the geologic model for purposes of flow simulation. This scale up is complicated by the fact that very fine scale details in the geologic description can have a major impact on flow simulation results, and these must be accurately accounted for in the coarsened model.
Several approaches have been developed in recent years for the scale up of detailed geologic descriptions. The more traditional approaches generally involve the modification, or ‘pseudoization,' of the fine scale relative permeability curves (often taken to be the core scale relative permeability curves) to account for heterogeneities on various scales. Many of these approaches are discussed in Ref. 1. A key limitation of pseudoization methods is that the resulting relative permeability curves are generally dependent on the displacement process and history. This means that their applicability is restricted to flow fields, initial conditions, boundary conditions and parameter ranges ‘close' to those for which they were derived. More recent methods avoid some of these problems through the detailed calibration of the pseudo relative permeability curves and estimation of the relevant flow parameters, which provides these approaches with greater generality.2,3 Though these methods do not entirely eliminate the inherent limitations of the pseudo relative permeability approach,4 they provide a more robust coarse scale description of multiphase flow than do the traditional pseudoization approaches.
In some cases, the displacement process and flow field are well established, and the process dependency of the pseudo relative permeabilities does not represent a major limitation. In such instances, pseudo relative permeabilities can be used quite effectively for coarse grid reservoir modeling. For problems of this type, significant speedups can be obtained with only a slight loss of accuracy relative to the fine grid model.
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