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Abstract Recent advances in multiscale methods have shown great promise in modeling multiphase flow in highly detailed heterogeneous domains. Existing multiscale methods, however, solve for the flow field (pressure and total-velocity) only. Once the fine-scale flow field is reconstructed, the saturation equations are solved on the fine scale. With the efficiency in dealing with the flow equations greatly improved by multiscale formulations, solving the saturation equations on the fine scale becomes the relatively more expensive part. In this paper, we describe an adaptive multiscale finite-volume (MSFV) formulation for the nonlinear transport (saturation) equations. A general algebraic multiscale formulation consistent with the operator based framework proposed by Zhou and Tchelepi (SPEJ 13:267โ173) is presented. Thus, the flow and transport equations are solved in a unified multiscale framework. Two types of multiscale operators, namely restriction and prolongation, are used to construct the multiscale saturation solution. The restriction operator is defined according to the local sum of the fine-scale transport equations in a coarse gridblock. Three adaptive prolongation operators are defined according to the local saturation history at a particular coarse block. The three operators have different computational complexity, and they are used adaptively in the course of a simulation run. When properly used, they yield excellent computational efficiency while preserving accuracy. This adaptive multiscale formulation has been tested using several challenging problems with strong heterogeneity, large buoyancy effects, and changes in the well operating conditions (e.g., switching injectors and producers during simulation). The results demonstrate that adaptive multiscale transport calculations are in excellent agreement with fine-scale reference solutions, but with a much lower computational cost.
Abstract We propose an upscaling method that is based on dynamic simulation of a given model in which the accuracy of the upscaled model is continuously monitored via indirect error-measures. If the indirect measures are bigger than a specified tolerance, the upscaled model is dynamically updated with approximate fine scale information that is reconstructed by a multi-scale finite volume method (Jenny et al., JCP 217; 627โ641, 2006). Upscaling of multi-phase flow entails a detailed flow information in the underlying fine scale. We apply adaptive prolongation and restriction operators for flow and transport equations in constructing an approximate fine scale solution. This new method eliminates inaccuracy associated with the traditional upscaling method which relies on prescribed inaccurate boundary conditions in computing upscaled variables. The new upscaling algorithm is validated for two-phase, incompressible flow in two dimensional porous media with heterogeneous permeabilities. It is demonstrated that the dynamically upscaled model achieves high numerical efficiency than the fine-scale models and also provides an excellent agreement with the reference solution computed from fine-scale simulation. Introduction The displacement process of multi-phase flow in porous media shows a strong dependency on process and boundary conditions. These process and boundary condition dependency, as a result, has hampered effort to construct a general coarse grid model that can be applied for multi-phase flow with various operational conditions. In addition, the conventional process in developing coarse-grid models lacks, in a general, a priori error estimate that will guide homogenization or upscaling process. Upscaling of single-phase and multiphase flow in porous media is reviewed by Farmer (2002), Christie (2001) and Barker and Thibeau (1997). Upscaling of multiphase flow in porous media is much more complex than that of single phase flow because it is difficult to delineate the effects of heterogeneous permeability distribution and multi-phase flow parameters and variables. To alleviate this difficulty, Efendiev and Durkofsky (2002, 2004) derived a generalized convection-diffusion equation to describe multi-phase flow, in place of the usual multi-phase extension of Darcy's equation with coarse grid (volume averaged) parameters and variables. Chen and Durkofsky (2006) combined the local-global upscaling and the generalized convection-diffusion equation to obtain upscaling of two-phase flow. This combined approach consistently provided reasonably accurate solutions for test cases.
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 10-10 grid cells. However, this level of resolution is far beyond the computational capability of state-of-the-art reservoir simulators (10 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, and solving 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.
Abstract Recently, multiscale methods have been developed for accurate and efficient numerical solution of large-scale heterogeneous reservoir problems. A scalable and extendible Operator Based Multiscale Method (OBMM) is described here. OBMM is cast as a general algebraic framework of the multiscale method. It is very natural and convenient to incorporate more physics in OBMM for multiscale computation. In OBMM, two multiscale operators are constructed: prolongation and restriction. The prolongation operator can be constructed by assembling basis functions, and the specific form of the restriction operator depends on the coarse-scale discretization formulation (e.g., finite-volume or finite-element). The coarse-scale pressure equation is obtained algebraically by applying the prolongation and restriction operators on the finescale flow equations. Solving the coarse-scale equation results in a high quality coarse-scale pressure. The fine scale 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 equation. As an application example, we study multiscale modeling of compressible flow. We show that the extension of modeling from incompressible to compressible flow is really straightforward for OBMM. No special treatment for compressibility is required. The efficiency of multiscale methods over standard fine-scale methods is retained by OBMM. The accuracy of OBMM is demonstrate by several challenging cases including highly compressible multiphase flow 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 O(107) - O(108) grid cells. However, this level of resolution is far beyond the computational capability of state-of-the-art reservoir simulators (O(106) grid cells). Moreover, some applications need to run many reservoir simulations (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. The multiscale method is very promising due to its 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 Wu4 proposed a multiscale finite-element method (MsFEM) that captures the fine-scale information by constructing special finite element basis functions within each element. However, the reconstructed fine-scale velocity is not conservative. Later, Chen and Hou proposed a conservative mixed finite-element multiscale method. Another multiscale mixed finite-element method has been presented by Arbogast1 and Arbogast and Bryant2. Numerical Green functions were used to resolve the fine-scale information, which are then coupled with coarse-scale operators to obtain the global solution. These methods considered incompressible flow in heterogeneous porous media where the flow equation is elliptic.