Abstract
A wide variety of multiscale methods have been proposed in the literature to reduce runtime and provide better scaling for the solution of Poisson-type equations modeling flow in porous media. The key idea of all these methods is to construct a set of prolongation operators (also called basis functions) that map between unknowns associated with cells in a fine-scale geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as in flow-based upscaling methods. The prolongation operators are then combined with a suitable restriction operator to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, multiscale basis functions have sub-scale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner and enables us to reconstruct an approximate fine-scale solution once the global coarse-scale system is inverted.
For the past ten years or so, my research group has worked on adapting and extending these methods so that they can be applied to simulate real petroleum reservoirs. In the first part of the talk, I will review key ideas that have been researched, and point out failures and successes. In the second part of the talk, I discuss what is currently the most versatile and robust multiscale formulation, the multiscale restriction-smoothed basis (MsRSB) method. This method has been implemented in the prototyping branch of a commercial simulator and has three main advantages: First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and very robust compared to existing multiscale methods. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. I end the talk by discussing several examples, including 3-phase black-oil models, polymer flooding with non-Newton fluid rheology, and embedded fracture models.
For the past ten years or so, my research group has worked on adapting and extending these methods so that they can be applied to simulate real petroleum reservoirs. In the first part of the talk, I will review key ideas that have been researched, and point out failures and successes. In the second part of the talk, I discuss what is currently the most versatile and robust multiscale formulation, the multiscale restriction-smoothed basis (MsRSB) method. This method has been implemented in the prototyping branch of a commercial simulator and has three main advantages: First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and very robust compared to existing multiscale methods. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. I end the talk by discussing several examples, including 3-phase black-oil models, polymer flooding with non-Newton fluid rheology, and embedded fracture models.