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SUMMARYThe simulation of two-phase ow of oil and water in inhomogeneous porous media represents a great challenge because rock properties such as porosity and permeability can change abruptly throughout the reservoir. This fact can produce velocities which vary several orders of magnitude within very short distances. The presence of complex geometrical features such as faults and deviated wells is quite common in reservoir modelling, and unstructured mesh procedures, such as ÿnite elements (FE) and ÿnite volume (FV) methods can o er advantages relative to standard ÿnite di erences (FD) due to their ability to deal with complex geometries and the easiness of incorporating mesh adaptation procedures. In uid ow problems FV formulations are particularly attractive as they are naturally conservative in a local basis. In this paper, we present an unstructured edge-based ÿnite volume formulation which is used to solve the partial di erential equations resulting from the modelling of the immiscible displacement of oil by water in inhomogeneous porous media. This FV formulation is similar to the edge-based ÿnite element formulation when linear triangular elements are employed. Flow equations are modelled using a fractional ux approach in a segregated manner through an IMplicit Pressure-Explicit Saturation (IMPES) procedure. The elliptic pressure equation is solved using a two-step approach and the hyperbolic saturation equation is approximated through an artiÿcial di usion method adapted for use on unstructured meshes. Some representative examples are shown in order to illustrate the potential of the method to solve uid ows in porous media with highly discontinuous properties.

SUMMARYThe Galerkin finite element method is used as the basis for the construction of schemes for the solution of the two-dimensional compressible Euler equations on unstructured triangular grids. The use of a side-based data structure readily allows for the construction of a local (structured) stencil and the incorporation of a high-resolution shock-capturing method formulated within the TVD concept. The essential features of the finite element side-based scheme and the 1D TVD approach are described and their numerical implementation is discussed. The choice of limiters and the support for their computation are analysed and the solutions of some inviscid flows, obtained by advancing explicitly in time, are presented. K E Y WORDS TVD Unstructured mesh algorithms Euler equations Hypersonic flows