Summary
Kalman filter-based methods have been widely applied for assimilating new
measurements to continuously update the estimate of state variables, such as
reservoir properties and responses. The standard Kalman filtering scheme
requires computing and storing the covariance matrix of state variables, which
is computationally expensive for large-scale problems with millions of
gridblocks. In the ensemble Kalman filter (EnKF), this problem is alleviated
with sampling from a limited number of realizations and computing the required
subset of the covariance matrix at each update. However, the goodness of the
(ensemble) covariance approximated from the limited ensemble depends on the
number of realizations used and the representativity of a given ensemble. In
this study, we propose an efficient, dimension-reduced Kalman filtering scheme
based on Karhunen-Loeve (KL) and other orthogonal polynomial decompositions of
the state variables. We consider flow in heterogeneous reservoirs with
spatially variable permeability. The reservoir responses such as pressure are
measured at some locations at various time intervals. The aim is to dynamically
characterize the reservoir properties and to predict the reservoir performance
and its uncertainty at future times. In our scheme, the covariance of the
reservoir properties is approximated by a small set of eigenvalues and
eigenfunctions using the KL decomposition and the reconstruction of the
covariance from the KL decomposition can be done whenever needed. In each
update, the forecast step is solved using the KL-based moment method, giving a
set of functions from which the mean and covariance of the state variables can
be constructed, when needed. The statistics of both the reservoir properties
and the reservoir responses are then updated with the available measurements at
this time using the auto- and cross-covariances obtained from the forecast
step. The new approach is illustrated on a heterogeneous reservoir with dynamic
measurements and the results are compared with those from the EnKF method, in
terms of accuracy and efficiency.
Introduction
Owing to the high cost associated with direct measurements of reservoir
properties, for instance permeability and porosity, the number of direct
observations is always limited. However, the reservoir exhibits a high degree
of spatial variability at all length scales resulted from its intrinsically
complicated nature. This combination of significant spatial heterogeneity with
a relatively small number of direct observations leads to uncertainty in
characterizing reservoir properties, which in turn results in uncertainty in
estimating or predicting the corresponding reservoir responses.
© 2007. Society of Petroleum Engineers
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History
- Original manuscript received:
14 July 2006
- Meeting paper published:
9 October 2005
- Revised manuscript received:
14 October 2006
- Manuscript approved:
25 October 2006
- Version of record:
20 March 2007
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