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: (1) constructing the prolongation and restriction
operators, (2) assembling and solving the coarse-scale pressure equations, (3)
reconstructing the fine-scale pressure and velocity fields, and (4) 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.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
13 December 2006
- Meeting paper published:
26 February 2007
- Revised manuscript received:
15 November 2007
- Manuscript approved:
30 November 2007
- Version of record:
25 June 2008
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