Reservoir Flow Simulation by Adaptive ADER Schemes

In this chapter, an extension of ADER schemes is presented in order to solve both linear and nonlinear scalar conservation laws on unstructured triangulations. The proposed scheme is conservative and belongs to the class of finite volume schemes. It combines high order reconstruction techniques with a high order flux evaluation method to update cell average values through fluxes across cell interfaces. The ADER approach results in an explicit, one-step scheme based on the solution of generalized Riemann problems across cell interfaces. Moreover, the triangulation is adaptively modified during the simulation to effectively combine high order accuracy with locally refined meshes and therefore reduce the computational costs. The required adaption rules for the refinement and coarsening of the triangular mesh rely on a customized error indicator. Numerical experiments confirm the expected orders of accuracy and show the good performance of the proposed scheme for linear and nonlinear problems. Finally, the adaptive ADER schemes are applied to a test case from oil industry, which plays an important role in the modelling of fluid flow in petroleum reservoirs.

[1]  Eleuterio F. Toro,et al.  TVD Fluxes for the High-Order ADER Schemes , 2005, J. Sci. Comput..

[2]  O. Friedrich,et al.  Weighted Essentially Non-Oscillatory Schemes for the Interpolation of Mean Values on Unstructured Grids , 1998 .

[3]  Walter Schempp,et al.  Constructive Theory of Functions of Several Variables: Proceedings of a Conference Held at Oberwolfach, Germany, April 25 - May 1, 1976 , 1977, Constructive Theory of Functions of Several Variables.

[4]  C. Ollivier-Gooch.  Quasi-ENO Schemes for Unstructured Meshes Based on Unlimited Data-Dependent Least-Squares Reconstruction , 1997 .

[5]  Armin Iske,et al.  Effective node adaption for grid-free semi-Lagrangian advection , 2001 .

[6]  Michael Griebel,et al.  Meshfree Methods for Partial Differential Equations , 2002 .

[7]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[8]  Rémi Abgrall,et al.  On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation , 1994 .

[9]  Jean Duchon,et al.  Splines minimizing rotation-invariant semi-norms in Sobolev spaces , 1976, Constructive Theory of Functions of Several Variables.

[10]  D. Kröner.  Numerical Schemes for Conservation Laws , 1997 .

[11]  Steven J. Ruuth,et al.  Two Barriers on Strong-Stability-Preserving Time Discretization Methods , 2002, J. Sci. Comput..

[12]  P. Frederickson,et al.  Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction , 1990 .

[13]  G. R. Cowper,et al.  Gaussian quadrature formulas for triangles , 1973 .

[14]  Ewald Quak,et al.  Tutorials on Multiresolution in Geometric Modelling, Summer School Lecture Notes , 2002 .

[15]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[16]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

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

[18]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[19]  Carl Ollivier-Gooch,et al.  A computational study of the effect of unstructured mesh quality on solution efficiency , 1997 .

[20]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[21]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[22]  Martin Käser,et al.  A comparative study of explicit differential operators on arbitrary grids , 2001 .

[23]  Eleuterio F. Toro,et al.  Towards Very High Order Godunov Schemes , 2001 .

[24]  K. Aziz,et al.  Petroleum Reservoir Simulation , 1979 .

[25]  Charles L. Lawson,et al.  Solving least squares problems , 1976, Classics in applied mathematics.

[26]  Armin Iske,et al.  Scattered Data Modelling Using Radial Basis Functions , 2002, Tutorials on Multiresolution in Geometric Modelling.

[27]  Armin Iske,et al.  Detection of discontinuities in scattered data approximation , 1997, Numerical Algorithms.

[28]  Eleuterio F. Toro,et al.  ARBITRARILY ACCURATE NON-OSCILLATORY SCHEMES FOR A NONLINEAR CONSERVATION LAW , 2002 .

[29]  Thomas Sonar,et al.  On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations : polynomial recovery, accuracy and stencil selection , 1997 .

[30]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[31]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[32]  P. Lax,et al.  Systems of conservation laws , 1960 .

[33]  Armin Iske,et al.  Conservative semi‐Lagrangian advection on adaptive unstructured meshes , 2004 .

[34]  S. E. Buckley,et al.  Mechanism of Fluid Displacement in Sands , 1942 .

[35]  Claus-Dieter Munz,et al.  ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D , 2002, J. Sci. Comput..

[36]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[37]  Armin Iske,et al.  Adaptive Meshfree Method of Backward Characteristics for Nonlinear Transport Equations , 2003 .

[38]  A. Harten,et al.  Multi-Dimensional ENO Schemes for General Geometries , 1991 .

[39]  E. Toro,et al.  Solution of the generalized Riemann problem for advection–reaction equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[40]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .