Page 262 - Intelligent Digital Oil And Gas Fields

212 Intelligent Digital Oil and Gas Fields

The many model and asset parameters of IAMod and IAM workflows are
uncertain. This uncertainty introduces a new level of complexity in the opti-
mization process, because optimization functions and/or constraints are no
longer deterministic functions but probabilistic/stochastic distributions.
However, if statistical principles are considered, the optimization frame-
works as discussed above can be modified to incorporate stochastic func-
tions. In reservoir characterization, an example of uncertainty
quantification may represent a set (ensemble) of model realizations, each
of them honoring a set of historic data. In retrospect, a stochastic production
optimization problem may represent maximization of expected NPV over
all available realizations. Note that the statistical nature of such a problem
will render the mean (expected) NPV value with associated confidence
intervals; however, the optimization will require many reservoir flow sim-
ulations and may be prohibitively time consuming. Echeverria Ciaurri et al.
(2012) propose the approach of retrospective optimization (RO), which
replaces a stochastic optimization problem by a sequence of optimization
problems where constraining statistics are approximated with gradually
increasing levels of quality.
An alternative approach is the use of stochastic programming (SP)
(Nemirovski et al., 2009) where the optimization problem with objective
function F 0 (x) formulates as follows:

minimize F 0 xðÞ ¼ Ef 0 x, ωð Þ
(6.6)
subject to F i xðÞ ¼ Ef i x, ωð Þ 0, i ¼ 1,…,m

where E represents the expected value operator on objective and constrain
functions f i (x,ω), which depend on x and ω, optimization and random vari-
ables, respectively. The value of ω is not known, but its distribution is and
the goal is to select x so that constraints are satisfied on average or with high
probability and the objective is minimized on average or with high proba-
bility. The stochastic constraint E f(x)<0 is classified as a standard quadratic
inequality.
It is interesting to note that neither RO nor SP are markedly represented
in the area of oil and gas production optimization problems. However, the
E&P industry has been rapidly adopting a complementary ensemble-based
approach to assisted history matching (AHM) with uncertainty, using for
example Bayesian inversion techniques such as ensemble Kalman filter
(EnKF) (Evensen, 1994), ensemble smoother with multiple data assimilation
(ES-MDA) (Emerick and Reynolds, 2013), or sequential, Markov-chain