Operator-Based Multiscale Method for Compressible Flow

Summary Multiscale methods have been developed for accurate and efficient numerical solution of flow problems in large-scale heterogeneous reservoirs. A scalable and extendible Operator-Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework. It is natural and convenient to incorporate more physics in OBMM for multiscale computation. In OBMM, two operators are constructed: prolongation and restriction. The prolongation operator is constructed by assembling the multiscale basis functions. The specific form of the restriction operator depends on the coarse-scale discretization formulation (e.g., finitevolume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators to the fine-scale flow equations. Solving the coarse-scale equation results in a high-quality coarse-scale pressure. The finescale pressure can be reconstructed by applying the prolongation operator to the coarse-scale pressure. A conservative fine-scale velocity field is then reconstructed to solve the transport (saturation) equation. We describe the OBMM approach for multiscale modeling of compressible multiphase flow. We show that extension from incompressible to compressible flows is straightforward. No special treatment for compressibility is required. The efficiency of multiscale formulations over standard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrated using several numerical examples including a challenging depletion problem in a strongly heterogeneous permeability field (SPE 10). Introduction The accuracy of simulating subsurface flow relies strongly on the detailed geologic description of the porous formation. Formation properties such as porosity and permeability typically vary over many scales. As a result, it is not unusual for a detailed geologic description to require 107-108 grid cells. However, this level of resolution is far beyond the computational capability of state-of-the-art reservoir simulators (106 grid cells). Moreover, in many applications, large numbers of reservoir simulations are performed (e.g., history matching, sensitivity analysis and stochastic simulation). Thus, it is necessary to have an efficient and accurate computational method to study these highly detailed models. Multiscale formulations are very promising due to their ability to resolve fine-scale information accurately without direct solution of the global fine-scale equations. Recently, there has been increasing interest in multiscale methods. Hou and Wu (1997) proposed a multiscale finite-element method (MsFEM) that captures the fine-scale information by constructing special basis functions within each element. However, the reconstructed fine-scale velocity is not conservative. Later, Chen and Hou (2003) proposed a conservative mixed finite-element multiscale method. Another multiscale mixed finite element method was presented by Arbogast (2002) and Arbogast and Bryant (2002). Numerical Green functions were used to resolve the fine-scale information, which are then coupled with coarse-scale operators to obtain the global solution. Aarnes (2004) proposed a modified mixed finite-element method, which constructs special basis functions sensitive to the nature of the elliptic problem. Chen et al. (2003) developed a local-global upscaling method by extracting local boundary conditions from a global solution, and then constructing coarse-scale system from local solutions. All these methods considered incompressible flow in heterogeneous porous media where the pressure equation is elliptic. A multiscale finite-volume method (MsFVM) was proposed by Jenny et al. (2003, 2004, 2006) for heterogeneous elliptic problems. They employed two sets of basis functions--dual and primal. The dual basis functions are identical to those of Hou and Wu (1997), while the primal basis functions are obtained by solving local elliptic problems with Neumann boundary conditions calculated from the dual basis functions. Existing multiscale methods (Aarnes 2004; Arbogast 2002; Chen and Hou 2003; Hou and Wu 1997; Jenny et al. 2003) deal with the incompressible flow problem only. However, compressibility will be significant if a gas phase is present. Gas has a large compressibility, which is a strong function of pressure. Therefore, there can be significant spatial compressibility variations in the reservoir, and this is a challenge for multiscale modeling. Very recently, Lunati and Jenny (2006) considered compressible multiphase flow in the framework of MsFVM. They proposed three models to account for the effects of compressibility. Using those models, compressibility effects were represented in the coarse-scale equations and the reconstructed fine-scale fluxes according to the magnitude of compressibility. Motivated to construct a flexible algebraic multiscale framework that can deal with compressible multiphase flow in highly detailed heterogeneous models, we developed an operator-based multiscale method (OBMM). The OBMM algorithm is composed of four steps:constructing the prolongation and restriction operators,assembling and solving the coarse-scale pressure equations,reconstructing the fine-scale pressure and velocity fields, andsolving the fine-scale transport equations. OBMM is a general algebraic multiscale framework for compressible multiphase flow. This algebraic framework can also be extended naturally from structured to unstructured grid. Moreover, the OBMM approach may be used to employ multiscale solution strategies in existing simulators with a relatively small investment.