Continuous elliptic and multi-dimensional hyperbolic Darcy-flux finite-volume methods Michael G. Edwards a,⇑ , Hongwen Zheng b , Sadok Lamine c , Mayur Pal c a Civil and Computational Engineering Centre, Swansea University, Swansea SA2 8PP, UK b Computer Modelling Group, #200, 3512 - 33 Street NW, Calgary, Canada c Shell Exploration and Production, Kessler Park 1, 2288GS Rijswijk, The Netherlands article info Article history: Received 26 May 2010 Received in revised form 12 December 2010 Accepted 22 December 2010 Available online 30 December 2010 Keywords: M-matrix Discrete maximum principle (DMP) CVD MPFA Anisotropy Multi-dimensional upwind methods abstract Elliptic and hyperbolic Darcy-flux approximations are presented. Families of flux-continuous finite-vol- ume methods are investigated for the elliptic full-tensor pressure equation with general discontinuous coefficients. Full pressure continuity across control-volume interfaces is built into the methods leading to an important distinction from the earlier pointwise continuous methods. The families of quasi-positive methods significantly reduce spurious oscillations (induced by earlier schemes) in discrete pressure solu- tions for strongly anisotropic full-tensor fields. Anisotropy favoring triangulation and non-linear flux splitting are also shown to be effective for computing solutions free of spurious oscillations. Multi-dimensional upwind schemes that reduce cross-wind numerical diffusion induced by the stan- dard upwind scheme are also presented for hyperbolic Darcy-flux approximation. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Elliptic and hyperbolic Darcy-flux approximations are pre- sented. Approximation of the elliptic pressure equation resulting from Darcy’s law and mass conservation requires that key physical constraints of continuity in normal flux and pressure be imposed at control-volume interfaces, across which strong discontinuities in permeability can occur. Families of pointwise flux-continuous methods that embody the generalization of the harmonic mean for full-tensor coefficients are presented in [1,2,4]. The methods are control-volume distributed (CVD) where rock and flow vari- ables are distributed to control-volumes and rely upon a single de- gree of freedom per control-volume per flow equation. Schemes of this type have also become known as multi-point flux approxima- tion schemes (MPFA), [7–9]. While proving effective for lower (full- tensor) anisotropy ratios the methods failure to satisfy a discrete maximum principle has lead to pressure solutions with severe spu- rious oscillations at higher full-tensor anisotropy ratios. Four methods for alleviating elliptic maximum principle viola- tion are considered. Families of locally conservative continuous Darcy-flux finite-volume schemes are presented in [13,14] for solv- ing the general tensor pressure equation. These schemes have full pressure continuity imposed across control-volume faces, in con- trast to the earlier families of schemes with pointwise continuity in pressure and flux. As a result the methods permit a full quadra- ture range and are further explored here in terms of their proper- ties and performance. In particular two of the methods based on specific quadrature rules are investigated. Two further methods based on anisotropy favoring triangulation [11] and non-linear flux splitting [12] respectively are also investigated. All four methods are shown to be effective for computing solutions free of spurious oscillations. Further methods for the pressure equation are pre- sented in [16,17]. Alternative approaches based on upscaling are presented in [18,19]. Three dimensional extensions of the methods considered here are presented in [15,10,6]. A summary of recent multi-dimensional upwind scheme devel- opment for convective flow is also presented. The well known sin- gle-point upstream weighting (first order upwind) scheme suffers from excessive smearing of steep fronts due to both coordinate- line diffusion and cross-wind diffusion that is inherent in the scheme. In contrast, multi-dimensional upwinding in the correct physical wave direction, reduces cross-wind diffusion and can pro- vide significant enhancement in resolution of discontinuities that travel across the mesh. Results are free of spurious oscillations in most cases and the new first order schemes require a minimal in- crease in support. The multidimensional schemes are further en- hanced by the development of a higher order multidimensional formulation, leading to a family of higher order multidimensional 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2010.12.029 ⇑ Corresponding author. E-mail address: m.g.edwards@swansea.ac.uk (M.G. Edwards). Computers & Fluids 46 (2011) 12–22 Contents lists available at ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid