MULTIPHASE FLOW SIMULATION WITH DYNAMIC ADAPTIVE DYADIC GRIDS Cl´ audio G. S. Cardoso Anamaria Gomide Jorge Stolfi ra014852,anamaria,stolfi @ic.unicamp.br Instituto de Computac ¸˜ ao Maria Cristina Cunha Instituto de Matem´ atica, Estat´ ıstica e Computac ¸˜ ao Cient´ ıfica cunha@ime.unicamp.br Denis J. Schiozer denis@fem.unicamp.br Faculdade de Engenharia Mecˆ anica Universidade Estadual de Campinas Caixa Postal 6176 13084-971 Campinas, SP Abstract. We address the problem of efficient simulation of multiphase flow in porous media — in particular, flow in natural oil reservoirs under advanced exploitation regimes such as water injection. We use a finite-element approach on a dynamic adaptive dyadic grid – a hierarchic mesh where a cell at level is partitioned into two equal children at level by a hyperplane perpendicular to coordinate axis . The topologyof dyadic grids is much simpler than that of arbitrary tetrahedral meshes, leading to very light data structures and efficient mesh manipulation algorithms. These advantages are expected to overwhelm their limitations, especially in problems with moving fronts. According to the finite-element paradigm, all relevant variables are represented as linear combinations of splines polynomial splines with small support, defined on a variable-resolution dyadic grid. Specifically, we consider the space of all functions that, within any leaf cell of , coincide with a multivariate polynomial of maximum degree in each coordinate, and are continuous to order . We describe here the theory of dyadic grids and dyadic splines, including efficient and entirely discrete algorithms to construct finite-element bases for the space . Finally we show how the equations for two-phase oil flow simulation are formulated in this model. Keywords: Finite elements, dyadic grids, adaptative grids, numerical integration, petroleum reservoir simulation.