Algebraic multiscale finite-volume methods for reservoir simulation [electronic resource]
Keywords: Energy Resources Engineering

One of the major challenges in reservoir simulation is posed by the existence of multiple scales in reservoirs and the resulting high resolution geophysical models. It is usually too expensive to compute directly on the finest geo-cellular scale. On the other hand, the accuracy of simulating subsurface flow relies strongly on the detailed geophysical properties of natural heterogeneous formations. Multiscale methods have been shown to be very promising to bridge the gap between the geological and flow-simulation scales. However, there are a few limitations in existing multiscale methods, e.g., the extension to physical mechanisms (such as compressibility, gravity and capillary pressure), and difficulties for cases with channelized permeability or high anisotropy. Moreover, the multiscale method has been applied only to the flow problem for efficient solutions of the pressure and velocity fields, while the transport problem is solved on the fine scale. In this work, we develop an algebraic multiscale framework for coupled flow and transport problems in heterogeneous porous media. An operator-based multiscale method (OBMM) is proposed to solve general multiphase flow problems. The key ingredients of the method are two algebraic multiscale operators, prolongation and restriction, with which the multiscale solution can be constructed algebraically. It is straightforward to extend OBMM to general flow problems that involve more physical mechanisms, such as compressibility, gravity and capillary pressure. The efficiency and accuracy of OBMM are demonstrated by a wide range of problems. An adaptive multiscale formulation for the saturation equations is developed within the algebraic multiscale framework, which is the first multiscale treatment of transport problems. Our multiscale formulation employs a conservative restriction operator and three adaptive prolongation operators. For the time interval of interest, the physical domain is divided into three distinct regions according to the coarse-scale saturation solution. Then, different prolongation operators are defined and used adaptively in different regions to construct the fine-scale saturation field. The multiscale computations of coupled flow and transport further improve the computational efficiency over the original multiscale finite-volume method, which is already significantly more efficient than fine-scale methods. An efficient two-stage algebraic multiscale (TAMS) method is also developed, which overcomes the limitations of the multiscale finite-volume method for channelized permeability fields and highly anisotropic problems. The TAMS method consists of two stages, one global and one local. In the global stage, a multiscale solution is obtained purely algebraically from the fine-scale matrix. The prolongation operator is obtained algebraically using the wirebasket ordered reduced system of the original fine-scale coefficient matrix. In the second stage, a local solution is constructed from a simple block preconditioner, such as Block ILU(0) (BILU), or an Additive Schwarz (AS) method. The TAMS method is purely algebraic and only needs the fine-scale coefficient matrix and the wirebasket ordering information of the multiscale grid. Thus, the TAMS method can be applied as a preconditioner for solving the large-scale linear systems associated with the flow problem. The TAMS method converges rapidly even for problems with channelized permeability fields and high anisotropy ratios. TAMS also preserve the favorable property of local mass conservation of the multiscale finite-volume method. Therefore, the TAMS method can be applied as either an efficient linear solver or a fast approximation approach with very good accuracy.