Upscaling Immiscible Gas Displacements: Quantitative Use of Fine-Grid Flow Data in Grid-Coarsening Schemes
- N.H. Darman (Petronas Research & Scientific Services Sdn. Bhd.) | L.J. Durlofsky (Chevron Petroleum Technology Co. and Stanford U.) | K.S. Sorbie (Heriot-Watt U.) | G.E. Pickup (Heriot-Watt U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2001
- Document Type
- Journal Paper
- 47 - 56
- 2001. Society of Petroleum Engineers
- 5.2.1 Phase Behavior and PVT Measurements, 5.5.3 Scaling Methods, 5.2 Reservoir Fluid Dynamics, 5.4.2 Gas Injection Methods, 5.1.5 Geologic Modeling, 5.7.2 Recovery Factors, 5.4 Enhanced Recovery, 5.3.2 Multiphase Flow, 5.3.1 Flow in Porous Media, 5.5 Reservoir Simulation, 4.1.4 Gas Processing, 4.3.4 Scale, 7.2.2 Risk Management Systems, 5.1 Reservoir Characterisation
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This paper presents and assesses grid-coarsening schemes based on the quantitative use of fine-scale two-phase-flow information. The basic approach is motivated from a volume-average analysis of the fine-scale saturation equation including gravitational effects. Extensive results for layered systems are presented. We show that coarse-grid simulation error correlates closely with specific subgrid quantities involving higher moments of fine-grid variables, which can be computed from the fine-scale simulations. With formation of a coarse grid that minimizes the appropriate subgrid property, optimal coarse-scale descriptions can be generated. The overall approach is shown to be applicable to coarse-scale descriptions with either rock or pseudorelative permeability curves. The accuracy of the coarse-grid calculations is, however, significantly better when pseudofunctions are used. The method is applied to determine the optimal number and configuration of coarse-grid layers in more general cases and it is shown that coarse-grid results do not always improve as the number of coarse layers is increased.
In modern reservoir characterization, the spatial resolution that may be incorporated into geological models often significantly exceeds the computational capabilities of fluid-flow simulators. Therefore, some level of upscaling must be applied to the fine-scale geological models before they can be used for practical flow calculations. This upscaling may be a simple block averaging of the single-phase permeability or may involve the application of a complex upscaling procedure. When the degree of upscaling is very large, the use of a dynamic technique, which may involve generation of upscaled or pseudorelative permeabilities, is generally required.
Several such upscaling methods have been developed and described in the literature: for example, the Kyte and Berry1 (KB), Stone,2 vertical-equilibrium (VE),3 and transmissibility-weighted (TW)4 methods. [The TW method originally was referred to as the transmissibility-potential-weighted (TPW) method. However, we now use transmissibility weighting only because the potential is nonunique. Numerically, the TW method gives results virtually identical to those in Ref. 4.] Hewett and Archer5 and Hewett and Yamada6 have also suggested approaches based on streamline methods. In general, all these dynamic methods (except VE) involve some procedure for using the fine-grid flows to generate modified pseudorelative permeability and pseudocapillary pressure curves at the coarse-block scale.
When successfully applied, these pseudofunctions accurately incorporate the interaction between small-scale multi-phase-fluid flow and heterogeneity and also correct for the numerical dispersion in the coarse-grid models. The principal metric is that the upscaling method provides a coarse-scale flow model that accurately reproduces the results (recovery profiles, breakthrough times, and other such factors) computed with the fine-grid model. However, pseudofunctions do have costs and limitations associated with them that must be considered when such an approach is applied. Pseudorelative permeabilities may be subject to so-called process dependence, meaning that the coarse-scale pseudofunctions vary with varying global boundary conditions. This in turn can result in a lack of robustness in the coarse-scale model. In addition, when pseudofunctions are generated through simulation of a global fine-scale flow problem, the computational requirements can be excessive in some cases.
Recently, Darman et al.4 described a new pseudogeneration scheme called the TW method. They found it to be particularly suitable for upscaling immiscible gas processes where large adverse mobility ratios and high gravity numbers are commonly encountered. They also found that the error in the coarse-grid-model predictions (relative to the reference fine-grid results) correlated closely with the subgrid variability in the gas saturation. This variability was expressed as the coefficient of variation in the gas saturation, the Cv in Sg [this is computed as the pore-volume average of the Cv in each of the coarse gridblock over all the calculated timesteps (Cv is equal to the standard deviation divided by the mean)]. Because of the close correlation between the coarse-grid error and the Cv in Sg, Darman et al.4 could approximately minimize the coarse-grid simulation error by choosing the coarse grid that minimizes the Cv in Sg. This result forms the basis for a new grid-coarsening scheme.
This method differs from conventional coarsening schemes in that it takes into account not only the static properties of the fine-grid models but also the dynamic properties of the fine-grid simulation runs. This in turn can lead to more accurate predictions of important quantities when pseudofunctions are applied, such as recovery factor and gas/oil ratio. The idea of this coarsening method is to identify regions of the fine-grid models where there is low variability of gas saturation and to take such regions as the corresponding coarse gridblock. As a result, the final composite coarse-grid model may include both finely and coarsely gridded regions.
This criterion (minimizing the Cv in Sg), is not the only one that might be applied in developing the grid-coarsening scheme. Durlofsky7,8 developed volume-averaged saturation equations for viscous-dominated immiscible displacements (i.e., gravity and capillary pressure effects were absent). He showed that the coarse-grid volume-averaged equations contain terms involving higher moments of certain fine-grid quantities. The specific higher moments that appear are the variance of saturation, s2S, and the velocity-saturation covariance, svS. (Note that Durlofsky used Sw instead of Sg. Also note that the difference in calculating s2S and the Cv in Sg is that, for the Cv in Sg, one takes the square root of s2S and then divides the number by the averaged saturation value for that particular coarse-grid block.) Because these higher moments are not explicitly modeled in coarse-grid simulations, it might be expected that coarse-grid errors could be reduced by minimizing the terms containing these higher moments. This could be accomplished by forming the coarse grid so that s2S and svS are minimized. This expectation is in fact quite consistent with previous results achieved with a nonuniform coarsening procedure for viscous-dominated displacements.7
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