NUPUS_2007-01_online.pdf - Universität Stuttgart International Research Training Group GRK 1398/1 “Non-linearities and upscaling in porous media“

# NUPUS_2007-01_online.pdf - Universität Stuttgart...

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Universität Stuttgart International Research Training Group GRK 1398/1 “Non-linearities and upscaling in porous media“ Yufei Cao Birgitte Eikemo Rainer Helmig Fractional Flow Formulation for Two-phase Flow in Porous Media Preprint 2007/1

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GRK 1398/1 - Geschäftsstelle - Pfaffenwaldring 61 70569 Stuttgart Telefon: 0711/685-60399 Telefax: 0711/685-60430 E-Mail: [email protected]
Fractional Flow Formulation for Two-phase Flow in Porous Media Yufei Cao School of Mathematics and System Science, Shandong University, Jinan 250100, China IWS, Department of Hydromechanics and Modeling of Hydrosystems, Universit¨at Stuttgart, Germany Birgitte Eikemo Dept. Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway Rainer Helmig IWS, Department of Hydromechanics and Modeling of Hydrosystems, Universit¨at Stuttgart, Germany Abstract: This paper focuses on the fractional flow formulation for two-phase flow in porous media and its applications. The IMPES (IMplicit-Pressure- Explicit-Saturation) concept is introduced for the fractional flow formulation using a finite volume element (FVE) method. Based on the differences between the fractional flow (FF) and fully coupled (FC) formulation, more adaptive pos- sibilities are proposed for the FF system. Numerical results and CPU times for both formulations are given and compared to prove that the FF formula- tion is more efficient for the solution of advection-dominated two-phase flow problems, and the numerical simulation for one of the adaptive possibilities is also shown to investigate its feasibility. Furthermore, the FF formulation is com- bined with time-of-flight (TOF) calculation using a discontinuous Galerkin (dG) method which gives another application of the FF formulation. Numerical sim- ulations for two real-life flow problems (single-phase models) are presented. Keywords: fractional flow formulation, fully coupled formulation, two-phase flow, finite volume element method, adaptive method, discontinuous Galerkin method, time-of-flight 2007

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Contents Notation IV 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Goal and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Physical background and mathematical modeling 4 2.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 Matrix and fluid parameters . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Constitutive relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Mathematical formulations for two-phase flow . . . . . . . . . . . . . . . . . . 9 2.2.1 Fully coupled formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Fractional flow formulation . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Adaptive methods 16 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Adaptive possibilities for fractional flow formulation . . . . . . . . . . . . . . 17 4 Numerical Examples 22 4.1 1D example: Buckley-Leverett problem . . . . . . . . . . . . . . . . . . . . . . 22 4.1.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.1.2 Comparative study of the results . . . . . . . . . . . . . . . . . . . . . . 24 4.2 2D example: five-spot waterflood problem . . . . . . . . . . . . . . . . . . . . 28 4.2.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2.2 Comparative study of the results . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Final remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 Application to time-of-flight simulation 32 5.1 Time-of-flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Discretization method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3.1 Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.3.2 Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.4 Final remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6 Summary 43

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List of Figures 2.1 From microscale to macroscale . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Contact angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 capillary pressure-saturation ( p c - S w ) relation after Brooks-Corey . . . . . . . . 8 2.4
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