Page 272 - Intelligent Digital Oil And Gas Fields

222 Intelligent Digital Oil and Gas Fields

multi-Gaussian notation (see Eq. 6.12) using the iterative Gauss-Newton
method (Le et al. 2015):

n 1
i
m i + 1 ¼ m i β i m m pr + C M G T G i C M G + C D
T
i
i
(6.13)
o
i i

gm d obs G i m m pr
where i is the iteration index, β i is the iteration step-size in the search direc-
T T -1
tion, and G i is the sensitivity matrix at m i . The term C M G [GC M G +C D ]
isusuallyreferredtoastheKalmangain.Thecovariancetermsintheensemble
update of Eq. (6.13) usually satisfy approximations C MD C M G and
e
T
C DD GC M G , where C MD corresponds to the cross-variance between
e
e
the vector of model parameters m and the vector of predicted data d, while
C DD is the auto-covariance matrix of the predicted data.
e
While EnKF is generally considered as a robust, efficient, and
easy-to-implement tool for sequential data assimilation and uncertainty
quantification, the main disadvantages of the method are the Gaussian
approximation applied in the model update scheme and the suboptimal
performance in terms of assimilation convergence when the relations
between the (reservoir) model parameters and the data predicted by
the forward estimator (e.g., reservoir simulator) are highly nonlinear.
These issues may lead to a well-known problem of ensemble collapse,
which is particularly evident for small ensemble sizes (Jafarpour and
McLaughlin, 2009). Variants of ensemble design and update have been
developed to alleviate these issues through more efficient handling of
model constraints with Kernel-based EnKF (Sarma and Chen, 2011)as
well as introduction of subspace EnKF and ES with Kernel PCA param-
eterization (Sarma and Chen, 2013). The ES also fits the category of
ensemble-based data-assimilation methods but, in comparison to EnKF,
which updates both model parameters and the states of the system, the
ES is an alternative data assimilation method that computes the update
of global reservoir model parameters in real time, by assimilating all data
simultaneously. The ES was originally proposed by van Leeuwen and
Evensen (1996) in an application with an ocean circulation model.
Skjervheim et al. (2011) and Chen and Oliver, 2013 describe recent
applications of ES for AHM of reservoir simulation models.
Emerick and Reynolds (2012, 2013) further propose the ES with the
multiple data assimilation (ES-MDA) algorithm, which repeats the ES