Multiperiod optimization model for oilfield production planning: bicriterion optimization and two-stage stochastic programming model

Abstract

In this work, we present different tools of mathematical modeling that can be used in oil and gas industry to help improve the decision-making for field development, production optimization and planning. Firstly, we formulate models to compare simultaneous multiperiod optimization and sequential single period optimization for the maximization of net present value and the maximization of total oil production over long term time horizons. This study helps to identify the importance of multiperiod optimization in oil and gas production planning. Further, we formulate a bicriterion optimization model to determine the ideal compromise solution between maximization of the two objective functions, the net present value (NPV) and the total oil production. To account for the importance of hedging against uncertainty in the oil production, we formulate a two-stage stochastic programming model to compute an improved expected value of NPV and total oil production for uncertainties in oil prices and productivity indices.

Appendix

1.1 Bicriterion optimization

The Pareto analysis of conflicting objective functions results in the formulation of a bicriterion optimization problem. The bicriterion optimization yields tradeoff solutions between two objective functions such as the net present value and the total oil production. To generate the Pareto cure the ϵ—constrained method (Hanes et al. 1975) is used. In this method one of the objective functions is optimized subject to a constraint ϵ on the other objective function. Further, after obtaining the Pareto curve one can determine the ideal compromise solution. In this case, the resulting Pareto front clearly shows that as the NPV value is increased the total oil production is decreased as shown in Fig. 9.

Fig. 9

Pareto curve of NPV vs Total oil production

The ideal solution would be the one in which we obtain the maximum NPV and maximum total oil production. This point is denoted as the utopia point (Freimer and Yu 1976; Yu 1973) as it has the best value for the objective, but it is infeasible. The ideal compromise solution corresponds to the point in the Pareto curve that has the shortest distance to the Utopia point.

1.2 Model

Let f1: Net present value, f2: total oil production. (Grossmann et al. 1982)

• The utopia point in Fig. 10 corresponds to [f1^U, f2^U], the maximum of both variables, where superscript U represent upper bound.

  Fig. 10

  

  Scaled Net present value (f1) vs Total oil produced (f2)

• An ideal compromise solution can be obtained by finding the point on the curve closest to utopia point i.e. minimizing the distance (δ_p) for a norm p, where:

  $$ \delta_{p} = \, \left[ {\left( {f1^{U} {-} \, f1} \right)^{p} + \, \left( {f2^{U} - f2} \right)^{p} } \right]^{1/p} \quad 1 \le \, p \le \infty $$

  (21)

• The variables f1 and f2 are scaled from zero to one.

• After scaling of the functions to f1′ and f2′. The utopia point for scaled variables is (1, 1).

• The Euclidean norm p = 2 is considered for minimizing the fractional deviations 1 − f1′ and 1 − f2′.

• To obtain the ideal compromise solution. For p = 2, solve,

  $$ \hbox{min} \, \left( {\left( {1 - \, f1^{\prime } } \right)^{2} + \left( {1 - \, f2^{\prime } } \right)^{2} } \right)^{1/2} , $$

  (22)

  s.t. Constraints.