Unstationary numerical scheme for multiphase multicomponent flows in sedimentary basins

A sedimentary basin is a large porous medium (several hundred kilometers in length and width and five kilometers in depth) which evolves in the course of time with the sedimentation and compaction effects. During this evolution, hydrocarbons appear and flow in this basin. We establish a model which allows for the simulation of a sedimentary basin evolution (compaction, sedimentation) and the hydrocarbon flows generation, migration and trapping. These phenomena occur during millions of years. Consequently, we mainly study time discretization of these equations. In order to solve the corresponding system of strongly coupled equations, we use an explicit scheme in time which is known in the petroleum world as IMPES (Implicit Pressure Explicit Saturations) and the 5 points Finite Volume method for the space discretization. As we concentrate our attention on the time discretization, we use a cartesian grid to mesh the rectangular domain. Due to the explicit scheme used and the nonlinear equations, a C.F.L. condition appears. Therefore, we develop an empirical time strategy which is based on flux throughout the discretization cells. This strategy permits to reduce CPU-time. Nevertheless, we remark that we lose computational time due to local phenomena, so we exhibit a time local refinement and time adaptive strategy. Adequate aposteriori estimators are obtained for the Finite Volume method developed in the basin simulator. Finally, we end this talk by showing a space adaptive mesh strategy which uses the previously developed aposteriori estimators. The main idea of this strategy consists to distinguish the regions where the solution is well computed and those where an improvement of the accuracy is necessary. In the latter, the accuracy is improved by means of a time step refinement and a new co mputation of all the quantities. Nethertheless, the distinction between good and bad areas constitues a very serious difficulty. We overcome this issue using adequate a posteriori estimators for which we obtain several theorical and numerical results in the case of linear parabolic equations. These a posteriori estimators are obtained for the Finite Volume method developped in our basin simulator. Finally, we end this talk by showing a space adaptive mesh strategy which uses the previously developed aposteriori estimators.