Correct Anisotropic and Heterogeneous Transmissibilities Calculation in Corner-Point Grids

Abstract

The transmissibility coefficient, present in commercial codes in petroleum reservoir simulation, was firstly defined considering an orthogonal grid in an isotropic and homogeneous medium, applying the traditional cell center control volume method. This type of problem results in a five point stencil for two dimensional domains, and the mass flux over a control volume face is, in this particular case, evaluated correctly with two pressure points. Afterwards, Hegre and Dalen presented a useful calculation method, widely used in commercial reservoir simulators, to any distorted corner point grid, still resulting in a five point stencil for a heterogeneous and anisotropic medium. Recently, some works showed in distorted and two dimensional grids and/or anisotropic medium that the mass flux over a control volume face cannot be accurately evaluated with only two pressure points, i.e., a five point stencil. This can introduce significant errors in the problem solution and strong grid orientation effects. This work presents an extension of the traditional control volume method, based on Hegre’s transmissibilities, considering the correct mass flux. The resultant stencil involves nine pressure points in a two dimensional case and twenty-three in a three dimensional case. The transmissibilities calculation proposed is designed to be applied in anisotropic (full tensor) and/or heterogeneous medium. The results of the tests performed are in good agreement with the correct nine point reference solution that considers the correct mass flux. In addition, the transmissibility method is straightforward to be extended in a computational code based on Hegre’s transmissibility.