This paper describes a new three-dimensional (3D), three-phase, multipurpose reservoir simulator. An extensive description is given of the dual-porosity facility . Three user-selected options available for computing the material transfers between matrix and fractures are developed and discussed. The most sophisticated option accurately accounts for the capillary, gravity, and viscous forces. A simplified thermodynamical package is also presented and examples are given.

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Our previous paper [1] described and analyzed the performance and some of the advantages obtained when a reservoir simulator is upgraded with: the possibility of flow between any given grid blocks, especially from one to several blocks;a multiple time step calculation. These improvements were appreciated by modelling engineers. The upgraded simulator was used for the simulation of actual reservoirs with complex geometry (such as faulted reservoirs), and for its possibility of having a composite discretization grid (i.e. a grid with several nested levels of local sub-gridding). This practical need justified further extensive study of the flow behaviour near a sub-gridding boundary in a composite grid. Composite grids possess some features which do not exist in regular classical grids. In a regular grid system, when two blocks are neighbours in a certain direction, their grid centers are always drawn up in that direction. In a composite grid, this is not always the case for neighbouring blocks belonging to different levels of local sub-gridding. As a consequence, in a composite grid, approximation of material exchange between two blocks by the classical two point finite difference scheme may be less accurate than in a regular grid. Tests have shown that a significant grid effect may be induced by computing flows between neighbouring blocks belonging to different sub-gridding levels with this scheme. To eliminate these effects, a more sophisticated numerical scheme, involving more than two points, must be used for flow computation. This paper reports an analysis of this phenomenon. It also gives a description of some schemes especially designed for this purpose. These schemes may be viewed as being the classical two point scheme with one of the two points being a grid point but not a grid block center. Reservoir data are then approximated at this new grid point by interpolating data values at the grid block centers. More or less sophisticated interpolations are considered. It also describes some numerical examples performed to compare these schemes when they are used for the simulation of multiphase flows with a composite grid. One of the schemes is selected for its accuracy, its reliability, and its simplicity. On numerical examples, it is shown to nearly reproduce, with a composite grid, results obtained with a fine regular grid having a much larger number of blocks. Moreover, this scheme is implemented on a black oil simulator using a Sequential Solution Method and designed for the simulation of three dimensional three phase flows in large reservoirs with composite grids and several thousands of blocks.

Abstract. The simulation of sedimentary basins aims at reconstructing its historical evolution in order to provide quantitative predictions about phenomena leading to hydrocarbon accumulations. The kernel of this simulation is the numerical solution of a complex system of non-linear partial differential equations (PDE) of mixed parabolic-hyperbolic type in 3D. A discretisation and linearisation of this system leads to very large, ill-conditioned, non-symmetric systems of linear equations with three unknowns per mesh cell, i.e. pressure, geostatic load, and oil saturation. This article describes the parallel version of a preconditioner for these systems, presented in its sequential form in [7]. It consists of three steps: in the first step a local decoupling of the pressure and saturation unknowns aims at concentrating in the "pressure block" the elliptic part of the system which is then, in the second step, preconditioned by AMG. The third step finally consists in recoupling the equations. Each step is efficiently parallelised using a partitioning of the domain into vertical layers along the y-axis and a distributed memory model within the PETSc library (Argonne National Laboratory, IL). The main new ingredient in the parallel version is a parallel AMG preconditioner for the pressure block, for which we use the BoomerAMG implementation in the hypre library [4]. Numerical results on real case studies, exhibit (i) a significant reduction of CPU times, up to a factor 5 with respect to a block Jacobi preconditioner with an ILU(0) factorisation of each block, (ii) robustness with respect to heterogeneities, anisotropies and high migration ratios, and (iii) a speedup of up to 4 on 8 processors.

Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms. are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete two-dimensional convolutions with a minimum number of operations. Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms.