Discontinuous approximation of viscous two-phase flow in heterogeneous porous media

Runge-Kutta Discontinuous Galerkin (RKDG) and Discontinuous Finite Volume Element (DFVE) methods are applied to a coupled flow-transport problem describing the immiscible displacement of a viscous incompressible fluid in a non-homogeneous porous medium. The model problem consists of nonlinear pressure-velocity equations (assuming Brinkman flow) coupled to a nonlinear hyperbolic equation governing the mass balance (saturation equation). The mass conservation properties inherent to finite volume-based methods motivate a DFVE scheme for the approximation of the Brinkman flow in combination with a RKDG method for the spatio-temporal discretization of the saturation equation. The stability of the uncoupled schemes for the flow and for the saturation equations is analyzed, and several numerical experiments illustrate the robustness of the numerical method.

[1]  Konstantinos T. Panourgias,et al.  A limiting approach for DG discretizations on mixed type meshes , 2015 .

[2]  Ricardo Ruiz-Baier,et al.  New fully-mixed finite element methods for the Stokes–Darcy coupling☆ , 2015 .

[3]  Raimund Bürger,et al.  Second-order schemes for conservation laws with discontinuous flux modelling clarifier–thickener units , 2010, Numerische Mathematik.

[4]  N. Risebro,et al.  Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media , 2014, Computational Geosciences.

[5]  Raimund Bürger,et al.  A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes , 2012, SIAM J. Sci. Comput..

[6]  Alessio Fumagalli,et al.  A numerical method for two-phase flow in fractured porous media with non-matching grids , 2013 .

[7]  Zhendong Luo,et al.  A fully discrete stabilized mixed finite volume element formulation for the non-stationary conduction–convection problem , 2013 .

[8]  Ivan Lunati,et al.  Mixed finite element -- primal finite volume element discretization of multicontinuum models , 2015 .

[9]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[10]  Amiya K. Pani,et al.  Discontinuous Galerkin finite volume element methods for second‐order linear elliptic problems , 2009 .

[11]  Zhiliang Xu,et al.  A new Runge-Kutta discontinuous Galerkin method with conservation constraint to improve CFL condition for solving conservation laws , 2014, J. Comput. Phys..

[12]  Willi Jäger,et al.  On The Interface Boundary Condition of Beavers, Joseph, and Saffman , 2000, SIAM J. Appl. Math..

[13]  Ricardo Ruiz-Baier,et al.  An augmented velocity–vorticity–pressure formulation for the Brinkman equations , 2015 .

[14]  J. Flaherty,et al.  Parallel, adaptive finite element methods for conservation laws , 1994 .

[15]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[16]  Sarvesh Kumar.  On the Approximation of incompressible miscible Displacement Problems in porous Media by mixed and Standard finite Volume element Methods , 2013, Int. J. Model. Simul. Sci. Comput..

[17]  M. Wheeler,et al.  An augmented-Lagrangian method for the phase-field approach for pressurized fractures , 2014 .

[18]  Chi-Wang Shu,et al.  Discontinuous Galerkin Method for Time-Dependent Problems: Survey and Recent Developments , 2014 .

[19]  Ricardo Ruiz-Baier,et al.  Equal Order Discontinuous Finite Volume Element Methods for the Stokes Problem , 2015, J. Sci. Comput..

[20]  Yalchin Efendiev,et al.  An adaptive local–global multiscale finite volume element method for two-phase flow simulations , 2007 .

[21]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[22]  Chunjia Bi,et al.  A discontinuous finite volume element method for second‐order elliptic problems , 2012 .

[23]  Peter Bastian,et al.  A fully-coupled discontinuous Galerkin method for two-phase flow in porous media with discontinuous capillary pressure , 2013, Computational Geosciences.

[24]  Michael Presho,et al.  Operator Splitting Multiscale Finite Volume Element Method for Two-Phase Flow with Capillary Pressure , 2011 .

[25]  P. Frolkovic,et al.  Numerical modelling of convection dominated transport coupled with density driven flow in porous media , 2000 .

[26]  Xiangxiong Zhang,et al.  Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes , 2011, Journal of Scientific Computing.

[27]  Rainer Helmig,et al.  Multiphase Processes in Porous Media , 2006 .

[28]  Sigal Gottlieb,et al.  On High Order Strong Stability Preserving Runge–Kutta and Multi Step Time Discretizations , 2005, J. Sci. Comput..

[29]  G. D. Veerappa Gowda,et al.  A finite volume method for a two-phase multicomponent polymer flooding , 2014, J. Comput. Phys..

[30]  Xiu Ye,et al.  A Discontinuous Finite Volume Method for the Stokes Problems , 2006, SIAM J. Numer. Anal..

[31]  Dmitri Kuzmin,et al.  Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis , 2013 .

[32]  Siam Staff,et al.  Godunov-Type Methods for Conservation Laws with a Flux Function Discontinuous in Space , 2004 .

[33]  Ricardo Ruiz-Baier,et al.  Numerical solution of a multidimensional sedimentation problem using finite volume-element methods , 2015 .

[34]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[35]  Xiangxiong Zhang,et al.  On maximum-principle-satisfying high order schemes for scalar conservation laws , 2010, J. Comput. Phys..

[36]  Ricardo Ruiz-Baier,et al.  A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem , 2016, Numerische Mathematik.

[37]  Kai Schneider,et al.  Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux , 2008, 0807.0403.

[38]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[39]  Jérôme Jaffré,et al.  Godunov-Type Methods for Conservation Laws with a Flux Function Discontinuous in Space , 2004, SIAM J. Numer. Anal..

[40]  Pierre Sagaut,et al.  A problem-independent limiter for high-order Runge—Kutta discontinuous Galerkin methods , 2001 .

[41]  Ricardo Ruiz-Baier,et al.  Stabilized mixed approximation of axisymmetric Brinkman flows , 2015 .

[42]  Raimund Bürger,et al.  Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation , 2015, J. Comput. Phys..

[43]  Stanimire Tomov,et al.  Adaptive finite volume element method for convection-diffusion-reaction problems in 3-D , 2001 .

[44]  David L. Darmofal,et al.  Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation , 2010, J. Comput. Phys..

[45]  Sarvesh Kumar.  A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media , 2012 .

[46]  Jérôme Jaffré,et al.  The DFLU flux for systems of conservation laws , 2013, J. Comput. Appl. Math..

[47]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..

[48]  R. Rannacher,et al.  Simple nonconforming quadrilateral Stokes element , 1992 .