Summary
Accurate modeling of flow in oil/gas reservoirs requires a detailed
description of reservoir properties such as permeability and porosity. However,
such reservoirs are inherently heterogeneous and exhibit a high degree of
spatial variability in medium properties. Significant spatial heterogeneity and
a limited number of measurements lead to uncertainty in characterization of
reservoir properties and thus to uncertainty in predicting flow in the
reservoirs. As a result, the equations that govern flow in such reservoirs are
treated as stochastic partial differential equations. The current industrial
practice is to tackle the problem of uncertainty quantification by Monte Carlo
simulations (MCS). This entails generating a large number of equally likely
random realizations of the reservoir fields with parameter statistics derived
from sampling, solving deterministic flow equations for each realization, and
post-processing the results over all realizations to obtain sample moments of
the solution. This approach has the advantages of applying to a broad range of
both linear and nonlinear flow problems, but it has a number of potential
drawbacks. To properly resolve high-frequency space-time fluctuations in random
parameters, it is necessary to employ fine numerical grids in space-time.
Therefore, the computational effort for each realization is usually large,
especially for large-scale reservoirs. As a result, a detailed assessment of
the uncertainty associated with flow-performance predictions is rarely
performed.
In this work, we develop an accurate yet efficient approach for solving flow
problems in heterogeneous reservoirs. We do so by obtaining higher-order
solutions of the prediction and the associated uncertainty of reservoir flow
quantities using the moment-equation approach based on Karhunen-Loéve
decomposition (KLME). The KLME approach is developed on the basis of the
Karhunen-Loéve (KL) decomposition, polynomial expansion, and perturbation
methods. We conduct MCS and compare these results against different orders of
approximations from the KLME method. The 3D computational examples demonstrate
that this KLME method is computationally more efficient than both Monte Carlo
simulations and the conventional moment-equation method. The KLME approach
allows us to evaluate higher-order terms that are needed for highly
heterogeneous reservoirs. In addition, like the Monte Carlo method, the KLME
approach can be implemented with existing simulators in a straightforward
manner, and they are inherently parallel. The efficiency of the KLME method
makes it possible to simulate fluid flow in large-scale heterogeneous
reservoirs.
Introduction
Owing to the heterogeneity of geological formations and the incomplete
knowledge of medium properties, the medium properties may be treated as random
functions, and the equations describing flow and transport in these formations
become stochastic. Stochastic approaches to flow and transport in heterogeneous
porous media have been extensively studied in the last 2 decades, and many
stochastic models have been developed (Dagan 1989; Gelhar 1993; Zhang 2002).
Two commonly used approaches for solving stochastic equations are MCS and the
moment-equation method. A major disadvantage of the Monte Carlo method, among
others, is the requirement for large computational efforts. An alternative to
MCS is an approach based on moment equations, the essence of which is to derive
a system of deterministic partial differen- tial equations governing the
statistical moments [usually the first two moments (i.e., mean and
covariance)], and then solve them analytically or numerically.
© 2006. Society of Petroleum Engineers
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History
- Original manuscript received:
6 December 2004
- Revised manuscript received:
27 December 2005
- Manuscript approved:
3 January 2006
- Version of record:
20 June 2006
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