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# Upscaling Immiscible Two-Phase Flows in an Adaptive Frame

## Citation

Strinopoulos, Theofilos (2006) Upscaling Immiscible Two-Phase Flows in an Adaptive Frame. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/AP0P-8S11. https://resolver.caltech.edu/CaltechETD:etd-02192006-165348

## Abstract

We derive the two-scale limit of a linear or nonlinear saturation equation with a flow-based coordinate transformation. This transformation consists of the pressure and the streamfunction. In this framework the saturation equation is decoupled to a family of one-dimensional nonconservative transport equations along streamlines. This simplifies the derivation of the two-scale limit. Moreover it allows us to obtain the convergence independent of the assumptions of periodicity and scale separation. We provide a rigorous estimate on the convergence rate. We combine the two-scale limit with Tartar's method to complete the homogenization.

To design an efficient numerical method, we use an averaging approach across the streamlines on the two-scale limit equations. The resulting numerical method for the saturation has all the advantages in terms of adaptivity that methods have. We couple it with a moving mesh along the streamlines to resolve the shock more efficiently. We use the multiscale finite element method to upscale the pressure equation because it gives access to the fine scale velocity, which enters in the saturation equation, through the basis functions. We propose to solve the pressure equation in the coordinate frame of the initial pressure and saturation, which is similar to the modified multiscale finite element method.

We test our numerical method in realistic permeability fields, such as the Tenth SPE Comparative Solution Project permeabilities, for accuracy and computational cost.

Item Type: Thesis (Dissertation (Ph.D.)) adaptive; cell problem; convection-dominated; homogenization; hyperbolic equations; immiscible; incompressible; macrodispersion; porous media; pressure streamline; two-phase flow; upscaling California Institute of Technology Engineering and Applied Science Applied And Computational Mathematics Public (worldwide access) Hou, Thomas Y. Hou, Thomas Y. (chair)Pullin, Dale IanPierce, Niles A.Bruno, Oscar P. 30 November 2005 theofilos13 (AT) yahoo.com CaltechETD:etd-02192006-165348 https://resolver.caltech.edu/CaltechETD:etd-02192006-165348 10.7907/AP0P-8S11 No commercial reproduction, distribution, display or performance rights in this work are provided. 680 CaltechTHESIS Imported from ETD-db 27 Feb 2006 06 May 2020 21:01

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