Investigation of Several Interpolation Functions for Unstructured Meshes in Conjunction with Compositional Reservoir Simulation

One of the key parameters in numerical simulation of fluid flow problems is the interpolation function used for adjusting the numerical value from the position where it is evaluated to the interface of the control volumes; for instance, if the finite-volume method is employed. In this article, we present an investigation of several interpolation functions in conjunction with the element based finite-volume method (EbFVM) using unstructured triangular meshes. We investigate the mass weighted upwind (MWU) and a modified version of this method, the upwind scheme, a streamline based upwind scheme, and a limited second order upwind scheme. These interpolation functions are implemented in a compositional simulator using vertex cell unstructured grids. The accuracy of these interpolation functions are evaluated for several case studies. For some of them, we also compare the results with the available analytical solutions.

[1]  Bantwal R. Baliga,et al.  CO-LOCATED EQUAL-ORDER CONTROL-VOLUME FINITE-ELEMENT METHOD FOR MULTIDIMENSIONAL, INCOMPRESSIBLE, FLUID FLOW—PART II: VERIFICATION , 1994 .

[2]  Christopher William Stuteville Bruner,et al.  Parallelization of the Euler Equations on Unstructured Grids , 1997 .

[3]  S. Lassue,et al.  A SKEW UPWINDING SCHEME FOR NUMERICAL RADIATIVE TRANSFER , 2008 .

[4]  B. R. Baliga,et al.  A NEW FINITE-ELEMENT FORMULATION FOR CONVECTION-DIFFUSION PROBLEMS , 1980 .

[5]  R. F. Warming,et al.  Upwind Second-Order Difference Schemes and Applications in Aerodynamic Flows , 1976 .

[6]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

[7]  A. Harten.  High Resolution Schemes for Hyperbolic Conservation Laws , 2017 .

[8]  Barry Koren,et al.  A robust upwind discretization method for advection, diffusion and source terms , 1993 .

[9]  Lian-xia Li,et al.  An improved r-factor algorithm for TVD schemes , 2008 .

[10]  Gerry E. Schneider,et al.  A SKEWED, POSITIVE INFLUENCE COEFFICIENT UPWINDING PROCEDURE FOR CONTROL-VOLUME-BASED FINITE-ELEMENT CONVECTION-DIFFUSION COMPUTATION , 1986 .

[11]  With Invariant Submanifolds,et al.  Systems of Conservation Laws , 2009 .

[12]  Brett F. Sanders,et al.  Impact of Limiters on Accuracy of High-Resolution Flow and Transport Models , 2006 .

[13]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[14]  M. Berger,et al.  Analysis of Slope Limiters on Irregular Grids , 2005 .

[15]  Jingming Hou,et al.  A new TVD method for advection simulation on 2D unstructured grids , 2013 .

[16]  M. Darwish,et al.  TVD schemes for unstructured grids , 2003 .

[17]  Vaughan R Voller,et al.  STREAMLINE UPWIND SCHEME FOR CONTROL-VOLUME FINITE ELEMENTS, PART I. FORMULATIONS , 1992 .

[18]  Francisco Marcondes,et al.  An element-based finite-volume method approach for heterogeneous and anisotropic compositional reservoir simulation , 2010 .

[19]  Jingming Hou,et al.  Improved total variation diminishing schemes for advection simulation on arbitrary grids , 2012 .

[20]  Vaughan R Voller,et al.  Streamline Upwind Scheme for Control-Volume Finite Elements, Part II. Implementation and Comparison with the Supg Finite-Element Scheme , 1992 .

[21]  Clarence O. E. Burg,et al.  Higher Order Variable Extrapolation For Unstructured Finite Volume RANS Flow Solvers , 2005 .

[22]  Eva Farkas,et al.  General Purpose Compositional Model , 1985 .

[23]  G. D. Raithby,et al.  A critical evaluation of upstream differencing applied to problems involving fluid flow , 1976 .

[24]  G. Pope,et al.  A higher-order finite-difference compositional simulator , 1990 .

[25]  G. D. Raithby,et al.  Skew upstream differencing schemes for problems involving fluid flow , 1976 .

[26]  L. J. Lee,et al.  Simulation of Liquid Composite Molding Based on Control-Volume Finite Element Method , 2002 .

[27]  Christian Masson,et al.  A stable second‐order mass‐weighted upwind scheme for unstructured meshes , 2006 .

[28]  Christian Masson,et al.  Co‐located equal‐order control‐volume finite element method for two‐dimensional axisymmetric incompressible fluid flow , 1994 .

[29]  P. Roe.  CHARACTERISTIC-BASED SCHEMES FOR THE EULER EQUATIONS , 1986 .

[30]  S. Osher,et al.  High resolution applications of the Osher upwind scheme for the Euler equations , 1983 .

[31]  Yassin A. Hassan,et al.  a Stable Mass-Flow Two-Dimensional Skew Upwind Scheme , 1983 .

[32]  C. Prakash,et al.  AN IMPROVED CONTROL VOLUME FINITE-ELEMENT METHOD IFOR HEAT AND MASS TRANSFER, AND FOR FLUID FLOW USING EQUAL-ORDER VELOCITY-PRESSURE INTERPOLATION , 1986 .

[33]  Fernando S. V. Hurtado,et al.  An Element Based Conservative Scheme Using Unstructured Grids For Reservoir Simulation , 2005 .

[34]  C. Rhie,et al.  Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation , 1983 .

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

[36]  W. E. Brigham,et al.  Analysis Of Well-to-Well Tracer Flow To Determine Reservoir Layering , 1984 .

[37]  Yassin A. Hassan,et al.  A Stable Mass-Flow-Weighted Two-Dimensional Skew Upwind Scheme , 1983 .