Operator-Based Multiscale Method for Compressible Flow

Zhou, Hui (Stanford University) | Tchelepi, Hamdi A.

OnePetro 

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.