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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.

SUMMARYUnstructured mesh based discretization techniques can o er certain advantages relative to standard ÿnite di erence approaches which are commonly used in reservoir simulation, due to their exibility to model complex geological features and to their ability to easily incorporate mesh adaptation. Within such class of methods the most frequently used are the ÿnite element method (FEM) and the ÿnite volume method (FVM). The latter is particularly attractive in reservoir problems due to its local and global conservation properties. The main goal of the present paper is to describe an unstructured edgebased FVM formulation which is used to solve the partial di erential equations resulting from the modelling of two-phase uid ow of oil and water in homogeneous porous media, when an IMPES (IMplicit Pressure Explicit Saturation) approach is used. This ÿnite volume formulation is similar to the edge-based FEM formulation when linear triangular or tetrahedral elements are employed. Some model examples are presented in order to validate the presented formulation.

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The numerical simulation of elliptic type problems in strongly heterogeneous and anisotropic media represents a great challenge from mathematical and numerical point of views. The simulation of flows in non-homogeneous and non-isotropic porous media with full tensor diffusion coefficients, which is a common situation associated with the miscible displacement of contaminants in aquifers and the immiscible and incompressible two-phase flow of oil and water in petroleum reservoirs, involves the numerical solution of an elliptic type equation in which the diffusion coefficient can be discontinuous, varying orders of magnitude within short distances. In the present work, we present a vertex-centered edge-based finite volume method (EBFV) with median dual control volumes built over a primal mesh. This formulation is capable of handling the heterogeneous and anisotropic media using structured or unstructured, triangular or quadrilateral meshes. In the EBFV method, the discretization of the diffusion term is performed using a node-centered discretization implemented in two loops over the edges of the primary mesh. This formulation guarantees local conservation for problems with discontinuous coefficients, keeping second-order accuracy for smooth solutions on general triangular and orthogonal quadrilateral meshes. In order to show the convergence behavior of the proposed EBFV procedure, we solve three benchmark problems including full tensor, material heterogeneity and distributed source terms. For these three examples, numerical results compare favorably with others found in literature. A fourth problem, with highly non-smooth solution, has been included showing that the EBFV needs further improvement to formally guarantee monotonic solutions in such cases.

Purpose The purpose of this paper is to present a methodology for parallel simulation that employs the discrete element method (DEM) and improves the cache performance using Hilbert space filling curves (HSFC). Design/methodology/approach The methodology is well suited for large-scale engineering simulations and considers modelling restrictions due to memory limitations related to the problem size. An algorithm based on mapping indexes, which does not use excessive additional memory, is adopted to enable the contact search procedure for highly scattered domains. The parallel solution strategy uses the recursive coordinate bisection method in the dynamical load balancing procedure. The proposed memory access control aims to improve the data locality of a dynamic set of particles. The numerical simulations presented here contain up to 7.8 millions of particles, considering a visco-elastic model of contact and a rolling friction assumption. Findings A real landslide is adopted as reference to evaluate the numerical approach. Three-dimensional simulations are compared in terms of the deposition pattern of the Shum Wan Road landslide. The results show that the methodology permits the simulation of models with a good control of load balancing and memory access. The improvement in cache performance significantly reduces the processing time for large-scale models. Originality/value The proposed approach allows the application of DEM in several practical engineering problems of large scale. It also introduces the use of HSFC in the optimization of memory access for DEM simulations.

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