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Summary

In this paper we discuss the multiblock algorithm for an implicit black-oil model as implemented in the multiphase simulator framework of IPARS (Integrated Parallel Accurate Reservoir Simulator). The multiblock algorithm decomposes the simulation domain into multiple nonoverlapping subdomains, or blocks, according to the geometric, geological, and physical/chemical properties, and well distribution. Each block can have its own grid system, and the grids of the neighboring blocks can be nonmatching on the interface, which allows for local grid refinement, or discrete fault or fracture modeling. Adjacent blocks are coupled across the interface by a set of conditions imposing a continuity of both primary variables and component mass fluxes that is realized through 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 multiblock technique is effective and scalable, as shown by our numerical experiments. In addition, we present how the multiblock black-oil model has been used in the coupling of different physical models.

Introduction

The main thrust of this paper is to investigate accurate and efficient numerical techniques for the 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 called the multiblock algorithm. This algorithm decomposes 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.

Multiblock (also known as macro-hybrid) formulations ^1-8 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 both to couple different discretizations on nonmatching multiblock grids and to couple different physical models in different parts of the simulation domain. These two features make the multiblock approach one of great computational interest.

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 that might not possibly match on the interface. For example, the geology of the subsurface may involve the modeling of faults, pinchouts, and other internal boundaries. In such cases, 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 the accurate approximation of local phenomena such as high gradients around wells.

More generally, multiblock decomposition can be induced by differences in the physical processes and mathematical models or by differences in the numerical discretization models applied to different parts of the simulation domain. ^9-11 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- or two-phase model is needed for the aquifer part of the reservoir, whereas a black-oil or compositional model is necessary if the gas phase is present in a subdomain.

In this paper, we discuss the formulation and implementation of a multiblock algorithm for an implicit black-oil model. This work represents a nontrivial extension of the multiblock algorithm for a two-phase oil-water model¹ as, in particular, it needs to address numerical regularization issues arising at phase transitions. Next, we briefly describe how the multiblock black-oil model is used in the multiphysics coupling with the two-phase oil-water model. We also address the issues that arise during implementation in the IPARS framework. In particular, we discuss the parallelism between the multimodel problem with the MPI multicommunicator and model-based load balancing strategies. In the end, we present numerical experiments that demonstrate the scalability of our approach.

Multiblock Black-Oil Model

The 3D reservoir domain Ω is divided into a series of n_bl nonoverlapping subdomains (blocks)Ω_k, k=1, ..., n _bl, owing to geological faults,¹² geometry irregularities, variations of rock properties, and physical/chemical properties of flow, well types, their distribution, etc. Each block has a smooth rectangular grid. The grids are constructed locally and may be nonmatching on the interfaces between neighboring blocks. Fig. 1 illustrates a typical geometry for a 2D domain decomposition. Note that the interfaces between blocks are filled with “mortars.” These are elements of a finite element space called mortar space, which is constructed on the 2D interface.

Black-Oil Subdomain Formulation.

The black-oil model is a three-phase (water, oil, and gas) model describing the flow in a petroleum reservoir.^13,14 It is assumed that no mass transfer occurs between the water phase and the other two phases. In the hydrocarbon (oil/gas) system, only two components are considered. The oil component (stock-tank oil) is the residual liquid at atmospheric pressure left after differential vaporization, leaving the gas component as the remaining fluid.