SPE Journal
Volume 13, Number 4, December 2008, pp. 400-411

Summary

Upscaling is often needed in reservoir simulation to coarsen highly detailed geological descriptions. Most existing upscaling procedures aim to reproduce fine-scale results for a particular geological model (realization). In this work, we develop and test a new approach, ensemble-level upscaling, for efficiently generating upscaled two-phase flow parameters (e.g., upscaled relative permeabilities) for multiple geological realizations. The ensemble-level upscaling approach aims to achieve agreement between the fine- and coarse-scale flow models at the ensemble level, rather than realization-by-realization agreement, as is the intent of existing upscaling techniques. For this purpose, flow-based upscaling calculations are combined with a statistical procedure based on a cluster analysis. This approach allows us to compute numerically the upscaled two-phase flow functions for only a small fraction of the coarse blocks. For the majority of blocks, these functions are estimated statistically on the basis of single-phase velocity information (attributes), determined when the upscaled single-phase parameters are calculated. The procedure is designed to maintain close correspondence between the cumulative distribution functions (CDFs) for the numerically computed and statistically estimated two-phase flow functions. We apply the method to 2D synthetic models of multiple realizations for uncertainty quantification. Models with different geological heterogeneity and fluid-mobility ratios are considered. It is shown that the method consistently corrects the biases evident in primitive coarse-scale predictions and can capture the ensemble statistics (e.g., P50, P10, P90) of the fine-scale results almost as accurately as the full flow-based upscaling procedures but with much less computational effort. The overall approach is flexible and can be used with any combination of upscaling procedures.

Introduction

In recent years, a wide variety of upscaling procedures has been developed and applied. These techniques generally take as their starting point a fine-scale geological model of the subsurface. The intent is then to generate a coarser model, which retains the geological realism of the underlying fine-scale description, for use in flow simulation. Though model sizes can vary substantially depending on the application, typical fine-scale geocellular models may contain 10⁷ to 10⁸ cells, while typical simulation models may contain 10⁴ to 10⁶ blocks.

Recent reviews and assessments (e.g., Barker and Thibeau 1997; Barker and Dupouy 1999; Farmer 2002; Darman et al. 2002; Gerritsen and Durlofsky 2005; Chen 2005) describe and apply a variety of upscaling techniques. These procedures can be categorized in different ways. One important distinction is in terms of the coarse-scale parameters that are computed by a particular method. Specifically, a technique that generates only upscaled single-phase parameters (permeability or transmissibility) can be classified as a single-phase upscaling procedure even though it may be applied to two- or three-phase flow problems. A method that additionally generates upscaled relative permeability functions is termed a two-phase upscaling procedure. Another way to distinguish upscaling procedures is according to the problem solved to determine the coarse-scale parameters. In particular, methods may be classified as local, extended local, quasiglobal, or global in order of increasing computational effort, depending on the problem solved in the upscaling computations. In general, two-phase upscaling methods are more computationally expensive than single-phase upscaling procedures, as a time-dependent two-phase flow problem must be solved in this case.

The appropriate upscaling procedure for any particular problem depends on the required level of accuracy and the degree of coarsening. For example, for permeability fields characterized by two-point geostatistics (variogram-based models), with only a moderate degree of coarsening, the use of local single-phase upscaling procedures, possibly coupled with nonuniform gridding, may provide acceptable coarse models. For more challenging cases, however, such as channelized systems characterized by multipoint geostatistics and high degrees of upscaling, extended local or (quasi) global single-phase upscaling coupled with two-phase upscaling may be necessary.

In recent work (Chen and Durlofsky 2006b), we introduced an upscaling procedure that combines quasiglobal single-phase upscaling, which was accomplished through a local-global procedure, with a specialized two-phase upscaling. The technique was shown to provide reasonable degrees of accuracy for challenging problems, though it was observed that the speedups between fine-grid simulation and the upscaling plus coarse-scale simulations were not that dramatic (e.g., approximately a factor of 4 to 10). Speedups will be much more substantial if the model is simulated many times, because the computation time required for the two-phase upscaling calculations is large compared to the coarse-grid simulations. It would, however, still be useful to accelerate these upscaling computations. This is particularly desirable in cases with substantial uncertainty in the underlying geological model, in which case many realizations (or scenarios) are to be simulated. In such cases, realization-by-realization agreement between fine and coarse models is less essential. Rather, what is required in this case is agreement of a statistical nature, such as agreement in the CDFs (e.g., the P10, P50, P90 predictions) for relevant production quantities such as cumulative oil recovered or net present value. The required level of accuracy of the upscaling, on the realization-by-realization basis, could be slightly less for such cases, though the method should be unbiased.

The intent of this paper is to develop and test procedures for substantially accelerating two-phase upscaling procedures for cases in which many realizations are to be considered. Toward this goal, we couple upscaling with statistical estimation techniques. Several statistical techniques were considered, though the best performance was achieved using K-means clustering. Application of this approach allows us to compute upscaled two-phase functions through full-flow simulation for only a small fraction of the coarse-scale blocks. For the rest of the blocks, these functions are estimated statistically on the basis of velocity information (attributes) computed during the single-phase upscaling. The overall method can be used with any combination of single-phase and two-phase upscaling procedures and is shown to provide a high level of accuracy in the statistical sense described above for example cases involving different heterogeneity models.

There has been very little research reported on the development of upscaling procedures for multiple permeability realizations. Previous researchers considered related problems involving the handling of upscaled multiphase flow parameters (e.g., the grouping of pseudorelative permeabilities). Dupouy et al. (1998) applied a statistical procedure to group the numerically computed global pseudorelative permeabilities to reduce the number of pseudofunctions used in flow simulation. Their work did not involve the estimation of upscaled relative permeabilities, though they noted that such an approach would be useful in practice because it would reduce the number of pseudofunctions to be numerically computed. Christie and Clifford (1998) suggested an a priori approach to grouping upscaled parameters for compositional simulation. They used the concept of tracer-breakthrough curves to represent coarse-scale blocks, and applied K-means clustering analysis to group the upscaled functions. Neither of these studies, however, considered upscaling over multiple reservoir models and the associated assessment of uncertainty for fine-scale predictions.

Our work here is also related to previous studies on error modeling of coarse-scale simulation models (Omre and Lødøen 2004, Lødøen et al. 2004), though the approaches are quite different. In the error modeling studies, some fine-scale calibration runs were required to model upscaling error and correct the bias in the coarse-scale simulation results, while our approach here estimates the upscaled flow parameters directly. The statistical estimation procedure (based on cluster analysis) used here can be viewed as a proxy or surrogate method that avoids the need to numerically generate upscaled two-phase parameters. In this sense, any proxy can be applied in the procedure. Statistical clustering approaches are used in many applications and have been applied recently in reservoir engineering as proxies for simulations in genetic algorithm-based optimization (Artus et al. 2006).

The outline of this paper is as follows: We first provide the governing equations and a brief overview of the relevant upscaling procedures. Next, we describe and illustrate the ensemble-level upscaling approach based on clustering to estimate statistically the upscaled two-phase flow functions. This is followed by extensive numerical results for a variety of 2D systems. We conclude with a discussion and summary.

© 2008. Society of Petroleum Engineers

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History

• Original manuscript received: 4 December 2006
• Meeting paper published: 26 February 2007
• Revised manuscript received: 17 January 2008
• Manuscript approved: 5 February 2008
• Version of record: 15 December 2008