Higher-order conservative interpolation between control-volume meshes: Application to advection and multiphase flow problems with dynamic mesh adaptivity

A general, higher-order, conservative and bounded interpolation for the dynamic and adaptive meshing of control-volume fields dual to continuous and discontinuous finite element representations is presented. Existing techniques such as node-wise interpolation are not conservative and do not readily generalise to discontinuous fields, whilst conservative methods such as Grandy interpolation are often too diffusive. The new method uses control-volume Galerkin projection to interpolate between control-volume fields. Bounded solutions are ensured by using a post-interpolation diffusive correction. Example applications of the method to interface capturing during advection and also to the modelling of multiphase porous media flow are presented to demonstrate the generality and robustness of the approach.

[1]  P. van Meurs,et al.  The Instability of Slow, Immiscible, Viscous Liquid-Liquid Displacements in Permeable Media , 1959 .

[2]  Hamdi A. Tchelepi,et al.  Linear stability analysis of immiscible two-phase flow in porous media with capillary dispersion and density variation , 2004 .

[3]  Oubay Hassan,et al.  Investigations into the applicability of adaptive finite element methods to two‐dimensional infinite Prandtl number thermal and thermochemical convection , 2007 .

[4]  Fred J. Hickernell,et al.  Linear stability of immiscible displacement in porous media , 1989 .

[5]  Matthew D. Piggott,et al.  Conservative interpolation between unstructured meshes via supermesh construction , 2009 .

[6]  R. H. Brooks,et al.  Hydraulic properties of porous media , 1963 .

[7]  Rainald Löhner.  Robust, Vectorized Search Algorithms for Interpolation on Unstructured Grids , 1995 .

[8]  J. Latham,et al.  Multiphase flow simulation through porous media with explicitly resolved fractures , 2015 .

[9]  Christopher C. Pain,et al.  A dynamic mesh approach for simulation of immiscible viscous fingering , 2015, ANSS 2015.

[10]  Jeffrey Grandy,et al.  Conservative Remapping and Region Overlays by Intersecting Arbitrary Polyhedra , 1999 .

[11]  Zhihua Xie,et al.  A Discontinuous Overlapping Control Volume Finite Element Method for Multi-Phase Porous Media Flow Using Dynamic Unstructured Mesh Optimization , 2015, ANSS 2015.

[12]  Igor Tsukerman.  Approximation of conservative fields and the element 'edge shape matrix' , 1998 .

[13]  Patrick E. Farrell,et al.  Galerkin projection of discrete fields via supermesh construction , 2009 .

[14]  Arthur Moncorge,et al.  Reservoir Simulation Prototyping Platform for High Performance Computing , 2014 .

[15]  Dimitrios Pavlidis,et al.  Compressive advection and multi‐component methods for interface‐capturing , 2016 .

[16]  Michael T. Heath,et al.  Common‐refinement‐based data transfer between non‐matching meshes in multiphysics simulations , 2004 .

[17]  J. Peiro,et al.  Adaptive remeshing for three-dimensional compressible flow computations , 1992 .

[18]  Dimitrios Pavlidis,et al.  Two- and three-phase horizontal slug flow simulations using an interface-capturing compositional approach , 2014 .

[19]  Martin Berzins,et al.  Mesh shape and anisotropic elements : theory and practice , 2000 .

[20]  Dimitrios Pavlidis,et al.  Control volume finite element modelling of segregation of sand and granular flows in fluidized beds , 2014 .

[21]  Patrick E. Farrell,et al.  Conservative interpolation between volume meshes by local Galerkin projection , 2011 .

[22]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[23]  C. Carstensen.  Clément Interpolation and Its Role in Adaptive Finite Element Error Control , 2006 .

[24]  G. Taylor,et al.  The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[25]  Paul-Louis George,et al.  Delaunay triangulation and meshing : application to finite elements , 1998 .

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

[27]  C.R.E. de Oliveira,et al.  Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations , 2001 .

[28]  George M. Homsy,et al.  Viscous fingering in porous media , 1987 .

[29]  Dimitrios Pavlidis,et al.  Anisotropic Mesh Adaptivity and Control Volume Finite Element Methods for Numerical Simulation of Multiphase Flow in Porous Media , 2015, Mathematical Geosciences.

[30]  Yannis C. Yortsos,et al.  Linear-stability analysis of immiscible displacement: Part 1-Simple basic flow profiles , 1986 .

[31]  M. Blunt,et al.  Reservoir Modeling for Flow Simulation by Use of Surfaces, Adaptive Unstructured Meshes, and an Overlapping-Control-Volume Finite-Element Method , 2015 .

[32]  Hamdi A. Tchelepi,et al.  Numerical simulation of immiscible two-phase flow in porous media , 2006 .