Computational Geosciences 6: 29–47, 2002.  2002 Kluwer Academic Publishers. Printed in the Netherlands. Numerical calculation of equivalent cell permeability tensors for general quadrilateral control volumes Chuanping He a , Michael G. Edwards a,b and Louis J. Durlofsky a,c a Department of Petroleum Engineering, Stanford University, Stanford, CA 94305-2220, USA E-mail: lou@pangea.stanford.edu b Department of Civil Engineering, University of Wales, Swansea, SA2 8PP, UK c ChevronTexaco Exploration and Production Technology Company, San Ramon, CA, USA Received 23 May 2001; accepted 20 September 2001 A new method for upscaling ﬁne scale permeability ﬁelds to general quadrilateral-shaped coarse cells is presented. The procedure, referred to as the conforming scale up method, ap- plies a triangle-based ﬁnite element technique, capable of accurately resolving both the coarse cell geometry and the subgrid heterogeneity, to the solution of the local ﬁne scale problem. An appropriate averaging of this solution provides the equivalent permeability tensor for the coarse scale quadrilateral cell. The general level of accuracy of the technique is demonstrated through application to a number of ﬂow problems. The real strength of the conforming scale up method is demonstrated when the method is applied in conjunction with a ﬂow-based grid- ding technique. In this case, the approach is shown to provide results that are signiﬁcantly more accurate than those obtained using standard techniques. Keywords: effective permeability, heterogeneity, irregular geometry, reservoir simulation, upscaling 1. Introduction Geologic models of oil and gas reservoirs typically contain a much higher level of detail than can be accommodated in ﬂow models. As a result, these geologic models must be coarsened, or upscaled, for use in the ﬂow simulator. The most fundamental quantity computed in the upscaling step is the coarse scale equivalent permeability ten- sor. This permeability is computed from the underlying ﬁne scale permeabilities in such a way that the average (local) ﬂow response computed using the coarse grid permeability replicates that obtained using the ﬁne scale permeability description. In the past, reservoir simulation models generally made use of gridding procedures in which the coarse scale simulation grid exactly overlaid the ﬁne grid; i.e., all of the vertices of the coarse grid cells corresponded to ﬁne grid cell vertices. The coarse grid is generated, in these cases, by simply removing grid lines from the ﬁne grid system [6]. Recent ﬂow-based grid generation schemes, discussed in more detail below, produce coarse grids in which the grid lines do not correspond to the ﬁne grid. These gridding