The recent progress in computer architecture lead to increased demand for simulations including more, and more realistic; physics, and the geosciences are no exception: nuclear waste simulations involve coupling between thermal, hydraulic, mechanical and chemical phenomena. The same is true for CO2 storage in deep saline aquifers, which requires full coupling between two-phase flow and chemical reactions. The goal of the DEDALES project is to contribute to the mathematical methods, and the simulation tools, that will be needed to enable simulation of these complex phenomena on the next generation Exascale computers. A characteristic of these architectures will certainly be the hierarchical nature of their computing elements. For applications requiring the solution of (systems of) partial differential equations, domain decomposition methods (in space and / or in time) have shown to lend themselves well to this architecture: in a DD method, an outer algorithm iterates over subdomain solves, and these can be arranged to be independent, and solved in parallel (this is a coarse grain type of parallelism). In turn, each of the subdomain solves is itself the solution of a problem of the same type (or simpler), and can be parallelized using techniques for the parallel solution of PDEs. Such an approach is moreover well suited to local time stepping in the subdomains. Linear solvers have always been, and will continue to be, at the center of simulation codes. However, parallelizing implicit methods on unstructured meshes, such as are required to accurately represent the fine geological details of the heterogeneous media considered, is notoriously difficult. It has also been suggested that time level parallelism could be a useful avenue to provide an extra degree of parallelism. Project DEDALES will show that space-time DD methods can provide this extra level, and can usefully be combined with parallel linear solvers at the subdomain level. To improve two-phase flow simulation in such context, we propose a two level approach with the outer level using physically based domain decomposition, parallelized with a distributed memory approach, while the inner level for the sub domain solver will exploit both MPI and thread level parallelism. This approach matches well with the hierarchical nature both of physics and of the hardware architecture. The resulting hybrid parallel solver will be integrated into an Open Source flow simulator, and validated against several large scale examples. This proposal will - Study how space time domain decomposition methods with local time stepping, which have been successfully used in saturated flow situations, can be extended to the two phase flow case; - Implement a hybrid linear solver by combining an outer iterative method (using MPI) with an inner direct solver on top of a runtime system like StarPU, and study how this solver can be adapted to deal with the specific issues arising from hydrogeological models, such as the very high degree of medium heterogeneity; - Implement the previous ideas to show that they provide a natural and efficient way to exploit two- or three-level parallelism, and validate the implementation on several challenging examples.