Ensemble filter methods with perturbed observations applied to nonlinear problems |
| |
| Authors: | Yanfen Zhang Ning Liu Dean S Oliver |
| |
| Institution: | 1.Mewbourne School of Petroleum & Geological Engineering,University of Oklahoma,Norman,USA;2.Chevron Energy Technology Co.,Houston,USA |
| |
| Abstract: | The ensemble Kalman filter (EnKF) has been shown repeatedly to be an effective method for data assimilation in large-scale
problems, including those in petroleum engineering. Data assimilation for multiphase flow in porous media is particularly
difficult, however, because the relationships between model variables (e.g., permeability and porosity) and observations (e.g.,
water cut and gas–oil ratio) are highly nonlinear. Because of the linear approximation in the update step and the use of a
limited number of realizations in an ensemble, the EnKF has a tendency to systematically underestimate the variance of the
model variables. Various approaches have been suggested to reduce the magnitude of this problem, including the application
of ensemble filter methods that do not require perturbations to the observed data. On the other hand, iterative least-squares
data assimilation methods with perturbations of the observations have been shown to be fairly robust to nonlinearity in the
data relationship. In this paper, we present EnKF with perturbed observations as a square root filter in an enlarged state
space. By imposing second-order-exact sampling of the observation errors and independence constraints to eliminate the cross-covariance
with predicted observation perturbations, we show that it is possible in linear problems to obtain results from EnKF with
observation perturbations that are equivalent to ensemble square-root filter results. Results from a standard EnKF, EnKF with
second-order-exact sampling of measurement errors that satisfy independence constraints (EnKF (SIC)), and an ensemble square-root
filter (ETKF) are compared on various test problems with varying degrees of nonlinearity and dimensions. The first test problem
is a simple one-variable quadratic model in which the nonlinearity of the observation operator is varied over a wide range
by adjusting the magnitude of the coefficient of the quadratic term. The second problem has increased observation and model
dimensions to test the EnKF (SIC) algorithm. The third test problem is a two-dimensional, two-phase reservoir flow problem
in which permeability and porosity of every grid cell (5,000 model parameters) are unknown. The EnKF (SIC) and the mean-preserving
ETKF (SRF) give similar results when applied to linear problems, and both are better than the standard EnKF. Although the
ensemble methods are expected to handle the forecast step well in nonlinear problems, the estimates of the mean and the variance
from the analysis step for all variants of ensemble filters are also surprisingly good, with little difference between ensemble
methods when applied to nonlinear problems. |
| |
| Keywords: | |
| 本文献已被 SpringerLink 等数据库收录！ |
|
