Page 254 - Intelligent Digital Oil And Gas Fields

204 Intelligent Digital Oil and Gas Fields

Alpak et al., 2015; Ramirez et al., 2017). In optimal control theory the
dynamic system, which usually corresponds to a nonlinear operator such
as finite-difference reservoir simulator, is introduced in the objective
function along with optimization constraints with a set of Lagrangian
multipliers, λ(n). In Lagrangian notation, the modified NPV objective
function (see Eq. 6.2) now becomes

N 1
X T
Q¼ QnðÞ + λ n +1Þ g nðÞ (6.4)
ð
n¼1
where n corresponds to the time-step of the simulation performed with
reservoir simulator g(n). The steepest descent (Brouwer et al., 2004)or
steepest ascent (Wang et al., 2009) method can be used to update the
estimates of the optimal control vector. For example, Sarma et al.
(2006) demonstrate that the approach is quite efficient and can render
a 70% increase in cumulative oil production in open-loop implemen-
tation and a 60% increase in closed-loop implementation. However,
a disadvantage of the adjoint approach is that it requires explicit
knowledge of the model equations as well as extensive programming
to implement the equations.
• Gradient-free algorithms, where solving the optimization problem is inde-
pendent of the model equations used and does not require implementa-
tion of adjoint equations. They can be beneficial when large-scale
distributed computing resources are scarce or not available and when
the optimization problem suggests using many starting points.
Lorentzen et al. (2006) develop a gradient-free approach using
ensemble-based statistics, more specifically ensemble Kalman filter
(EnKF) to optimize the NPV and total cumulative oil production. In
addition, Wang et al. (2002) use the partial enumeration method
(PEM), a discrete nongradient-based method, and Isebor et al. (2014)
use particle swarm optimization (PSO) with mesh-adaptive direct search
(PSO-MADS) for NPV optimization. Echeverria Ciaurri et al. (2011)
use derivative-free (i.e., noninvasive, blackbox) metaheuristic methods
(e.g., PSO) for optimization of oilfield operations, specifically the well
choke settings.
In the last decade, powered by the rapid evolution of high-performance dis-
tributed and parallel computing (HPC), the multiobjective optimization is
being used more often in the oil and gas industry. The multioptimization
problem generally consists of addressing two or more different, and usually