# SPE Journal Volume 17, Number 1, March 2012, pp. 198-211

SPE-119183-PA

### Multiscale Finite-Volume Formulation for Saturation Equations

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DOI  10.2118/119183-PA http://dx.doi.org/10.2118/119183-PA

### Citation

• Zhou, H., Lee, S.H., and Tchelepi, H.A. 2012. Multiscale Finite-Volume Formulation for Saturation Equations. SPE J. 17 (1): 198-211. SPE-119183-PA. http://dx.doi.org/10.2118/119183-PA.

### Discipline Categories

• 6.5 Reservoir Simulation
• 6.5.1 Simulator Development

### Keywords

• reservoir simulation, multiscale method, finite-volume, saturation equations

### Summary

Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator-based framework proposed by Zhou and Tchelepi (SPE Journal, June 2008, pages 267-273) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators--namely, restriction and prolongation--are used to construct the multiscale saturation solution. The restriction operator is defined as the sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexities, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but at a much lower computational cost.

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### History

• Original manuscript received: 4 November 2009
• Meeting paper published: 2 February 2009
• Revised manuscript received: 2 March 2011
• Manuscript approved: 8 March 2011
• Published online: 12 December 2011
• Version of record: 13 March 2012