An interior penalty discontinuous Galerkin approach for 3D incompressible Navier-Stokes equation for permeability estimation of porous media

Abstract Permeability estimation of porous media from directly solving the Navier–Stokes equations has a wide spectrum of applications in petroleum industry. In this paper, we utilize a pressure-correction projection algorithm in conjunction with the interior penalty discontinuous Galerkin scheme for space discretization to build an incompressible Navier–Stokes simulator and to use this simulator to calculate permeability of real rock samples. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.

[1]  Beatrice Riviere,et al.  A finite volume / discontinuous Galerkin method for the advective Cahn–Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging , 2016, Computational Geosciences.

[2]  John Lekner,et al.  Viscous flow through pipes of various cross-sections , 2007 .

[4]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[5]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[6]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[7]  Pavel B. Bochev,et al.  On the Finite Element Solution of the Pure Neumann Problem , 2005, SIAM Rev..

[8]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[9]  Khosro Shahbazi,et al.  Short Note: An explicit expression for the penalty parameter of the interior penalty method , 2005 .

[10]  C. Ross Ethier,et al.  Exact fully 3D Navier–Stokes solutions for benchmarking , 1994 .

[11]  Qian Fang,et al.  Relevance of computational rock physics , 2011 .

[12]  Faruk O. Alpak,et al.  References and benchmarks for pore-scale flow simulated using micro-CT images of porous media and digital rocks , 2017 .

[13]  Mary F. Wheeler,et al.  A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems , 2004, Math. Comput..

[14]  Mine Akbas,et al.  An analogue of grad-div stabilization in nonconforming methods for incompressible flows , 2017 .

[15]  A. Chorin.  Numerical Solution of the Navier-Stokes Equations* , 1989 .

[16]  Charles-Henri Bruneau,et al.  BOUNDARY CONDITIONS ON ARTIFICIAL FRONTIERS FOR INCOMPRESSIBLE AND COMPRESSIBLE NAVIER-STOKES EQUATIONS , 2000 .

[17]  Ramon Codina,et al.  Error estimates for an operator-splitting method for incompressible flows , 2004 .

[18]  Martin Kronbichler,et al.  A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow , 2016, J. Comput. Phys..

[19]  Steffen Müthing,et al.  A stable and high-order accurate discontinuous Galerkin based splitting method for the incompressible Navier-Stokes equations , 2016, J. Comput. Phys..

[20]  Rolf Rannacher,et al.  ARTIFICIAL BOUNDARIES AND FLUX AND PRESSURE CONDITIONS FOR THE INCOMPRESSIBLE NAVIER–STOKES EQUATIONS , 1996 .

[21]  Mine Akbas,et al.  The analogue of grad–div stabilization in DG methods for incompressible flows: Limiting behavior and extension to tensor-product meshes , 2018, Computer Methods in Applied Mechanics and Engineering.

[22]  Tapan Mukerji,et al.  Digital rock physics benchmarks - part II: Computing effective properties , 2013, Comput. Geosci..

[23]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[24]  B. Rivière,et al.  Estimation of penalty parameters for symmetric interior penalty Galerkin methods , 2007 .

[25]  Rolf Rannacher,et al.  Finite Element Methods for the Incompressible Navier-Stokes Equations , 2000 .