# A Discontinuous Galerkin Finite Element Framework for the Direct Numerical Simulation of Flow on High-Resolution Pore-Scale Images

- Authors
- Florian Frank (Rice University) | Chen Liu (Rice University) | Faruk O. Alpak (Shell International Exploration and Production Inc.) | Mauricio Araya-Polo (Shell International Exploration and Production Inc.) | Beatrice Riviere (Rice University)
- DOI
- https://doi.org/10.2118/182607-MS
- Document ID
- SPE-182607-MS
- Publisher
- Society of Petroleum Engineers
- Source
- SPE Reservoir Simulation Conference, 20-22 February, Montgomery, Texas, USA
- Publication Date
- 2017

- Document Type
- Conference Paper
- Language
- English
- ISBN
- 978-1-61399-483-2
- Copyright
- 2017. Society of Petroleum Engineers
- Disciplines
- 5 Reservoir Desciption & Dynamics, 5.3 Reservoir Fluid Dynamics, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation
- Keywords
- multigrid, Discontinuous Galerkin, direct pore-scale numerical simulation, iterative linear solvers
- Downloads
- 1 in the last 30 days
- 250 since 2007

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Advances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow on pore structures feasible. In this paper, we describe a two-phase flow simulator that solves mass and momentum balance equations valid at the pore scale, i.e. at scales where the Darcy velocity homogenization starts to break down. The simulator is one of the key components of a molecule-to-reservoir truly multiscale modeling workflow.

A Helmholtz free-energy driven, thermodynamically based diffuse-interface method is used for the effective simulation of a large number of advecting interfaces, while honoring the interfacial tension. The advective Cahn–Hilliard (mass balance) and Navier–Stokes (momentum balance) equations are coupled to each other within the phase-field framework. Wettability on rock-fluid interfaces is accounted for via an energy-penalty based wetting (contact-angle) boundary condition. Individual balance equations are discretized by use of a flexible discontinuous Galerkin (DG) method. The discretization of the mass balance equation is semi-implicit in time; momentum balance equation is discretized with a fully-implicit scheme, while both equations are coupled via an iterative operator splitting approach.

We discuss the mathematical model, DG discretization, and briefly introduce nonlinear and linear solution strategies. Numerical validation tests show optimal convergence rates for the DG discretization indicating the correctness of the numerical scheme. Physical validation tests demonstrate the consistency of the mass distribution and velocity fields simulated within our framework. Finally, two-phase flow simulations on two real pore-scale images demonstrate the utility of the pore-scale simulator. The direct pore-scale numerical simulation method overcomes the limitations of pore network models by rigorously taking into account the flow physics and by directly acting on pore-scale images of rocks without requiring a network abstraction step or remeshing. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.

File Size | 2 MB | Number of Pages | 22 |

### Supporting information

- SUPPLEMENTARY/SPE-182607-SUP.pdf

Andra, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E. H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A. and Zhan, X. 2013. Digital rock physics benchmarks &- Part I: Imaging and segmentation. Computers & Geosciences 50: 25&-32.

Andra, H., Combaret, N., Dvorkin, J., Glatt, E., Han, J., Kabel, M., Keehm, Y., Krzikalla, F., Lee, M., Madonna, C., Marsh, M., Mukerji, T., Saenger, E. H., Sain, R., Saxena, N., Ricker, S., Wiegmann, A. and Zhan, X. 2013. Digital rock physics benchmarks &- Part II: Computing effective properties. Computers & Geosciences 50: 33&-43.

Berg, S., Armstrong, R., Ott, H., Georgiadis, A. H., Klapp, S. A., Schwing, A., Neiteler, R., Brussee, N., Makurat, A., Leu, L., Enzmann, F., Schwarz, J.-O., Wolf, M., Khan, F., Kersten, M., Irvine, S., and Stampanoni, M. 2014. Multiphase flow in porous rock imaged under dynamic flow conditions with fast x-ray computed microtomography. Petrophysics 55: 304&-312.

Boek, E. S., Zacharoudiou, I., Gray, F., Shah, S. M., Crawshaw, J. P., and Yang, J. 2014. Multiphase Flow and Reactive Transport at the Pore Scale Using Lattice-Boltzmann Computer Simulations. Paper SPE-170941-MS presented at the SPE Annual Technical Conference and Exhibition, 27-29 October, Amsterdam, The Netherlands.

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