We discuss the multi-block algorithm for an implicit black-oil model as implemented in a multi-phase simulator framework IPARS. The multi-block algorithm consists of decomposition of the simulation domain into multiple non-overlapping subdomains or blocks according to the geometric, geological and physical/chemical properties, distribution of wells, etc. Each block can have its own grid system, and the grids of the neighboring blocks can be non-matching on the interface, which allows, for example, for local grid refinement, or discrete fault or fracture modeling. Adjacent blocks are coupled across the interface by a set of conditions imposing continuity of the primary variables and component mass fluxes, which is realized with the use of special interface mortar variables. The resulting system is solved by an interface Newton procedure. Regularization techniques and preconditioners are proposed to improve the performance of the solver. The multi-block technique is effective and scalable, as shown by our numerical experiments. In addition, we present how the multi-block black-oil model has been used in the coupling of different physical models.
The main thrust of this paper is to investigate accurate and efficient numerical techniques for simulation of flow and transport phenomena in porous media, which are of major importance in the environmental and petroleum industries. We propose to emphasize a novel numerical methodology, which is called the multi-block algorithm. This algorithm is based on decomposing the simulation domain into multiple subdomains (blocks) according to their geological, geometric and physical / chemical properties. One then applies the most efficient grid, numerical scheme and physical model in each subdomain so that the computational cost is reduced and accuracy is preserved.
Multi-block (also known as macro-hybrid) formulations1-7,23 provide numerical models consistent with the physical and engineering description of the underlying equations. That is, the equations hold with their usual meaning on the subdomains and physically meaningful conditions are imposed on interfaces between the subdomains. In particular it is possible to couple different discretizations on non-matching multi-block grids as well as to couple different physical models in different parts of the simulation domain. These two features make the multiblock approach one of great computational interests.
For instance, in many applications the geometry and physical properties of the domain or the behavior of the solution may require the use of different grids in different parts of the domain, which possibly do not match on the interface. For example, the geology of the subsurface may involve modeling of faults, pinchouts and other internal boundaries, where the discontinuities of coefficients (e.g., mobilities) reduce the accuracy of traditional single-block algorithm near discontinuities. By splitting the domain into multiple subdomains along the boundaries of discontinuities, solutions in each subdomain may have smooth properties and local convergence rates are regained. Furthermore, locally refined grids may be needed for accurate approximation of local phenomena such as high gradients around wells.
More generally, multi-block decomposition can be induced by differences in the physical processes or mathematical models or by differences in the numerical discretization models applied to different parts of the simulation domain8,9,10. The overall computational cost can be reduced by selecting the most appropriate model in a given part of the reservoir. For example, only a single-phase or a two-phase model is needed for the aquifer part of the reservoir, while a black-oil model or a compositional model is necessary if the gas phase is present in a subdomain.