Page 252 - Intelligent Digital Oil And Gas Fields

202 Intelligent Digital Oil and Gas Fields

6.2.1 Single- vs. Multiobjective Optimization
A mathematicaloptimizationproblem essentially combinesthree components:
• objective or cost function,
• optimization constraints, and
• control variables.
The objective of optimization is to determine a feasible combination of opti-
mization (control) variables within the boundaries of defined constraints that
maximizes or minimizes the objective or cost function of choice.
Based on the nature of the search for an optimal value of the objective
function, the optimization problems can be classified as a single- or a mul-
tiobjective optimization problem. An example of a single-objective optimi-
zation is finding an extrema, a minimum, or maximum, of a nonlinear
convex problem, such as a quadratic function. A common oil and gas opti-
mization problem is (dynamic) model calibration or history matching,
which seeks a least-square fit of reservoir simulation response to the observed
or measured data. The misfit objective function Q is represented as (Ferraro
and Verga, 2009):

n
X 2
Q ¼ R (6.1)
i
i¼1
with R i ¼w i (X m X o ) i defined as a residual, where X m , X o , and w i corre-
spond to the model data (reservoir simulation response), observed (mea-
sured) data (e.g., pressures, fluid rates, gas-oil ratio) and the weighting
factor, respectively.
The optimization problem can be approached as a single-objective
optimization in which an aggregate of all the quantities to be matched are
grouped into a single, joint objective function, or as a multiobjective
optimization approach, which usually considers two or more different
objectives, addressed separately during the optimization process. Mathe-
matically, the single-objective optimization is defined as (Hutahaean
et al., 2015)

minimizef xðÞ
l
subjectto h x k h u (6.2)
k k
x ¼ x 1 , x 2 , …, x k , …, x N g
f
where x¼{x 1 ,x 2 ,…,x k ,…,x N } is the vector of the N variables in the
l u
parameterization and h k and h k , respectively, correspond to the lower and