Continuous gas lifting is the process of injecting compressed natural gas into the production tubing of an oil well, in order to increase the pressure of the reservoir and enhance its production [1]. The allocation of the limited amount of available gas among a set of neighboring oil wells constitutes an interesting optimization problem. Due to the piecewise linear nature of the gas injection vs. oil production curves, the problem can be formulated as an MILP, and solved to global optimality [2]. In this study, we exploit four different mathematically equivalent representations of the problem, utilizing different applicable formulations presented in the literature, such as the linear segmentation [3] and convex hull [4,5] formulations. All variants succeed in solving the case studies to global optimality, including large cases involving 200 wells. It is shown that the novel formulation of [6] outperforms the competition in terms of both number of iterations and CPU time.

Pooling problems correspond to maximizing the profit of operating a network of blending pools, while satisfying various flow rate and material composition constraints. These problems appear in a wide variety of fields, including oil refining and water treatment, and their efficient solution has been an active field of research for many years. The pooling problem is formulated as a bilinear NLP, and its relaxations are typically constructed with use of bilinear convex envelopes [7]. We present here piecewise linearization schemes that can be employed to refine the tightness of these standard relaxations. In particular, we compare the computational performance of various big-M [8], convex hull [9], and incremental cost [10] formulations on a variety of benchmark problems. Despite the fact that all formulations eventually result into the same root node lower bound, our study demonstrates that some of them are consistent in exhibiting superior computational performance, and should therefore be preferred in the construction of pooling problem relaxations.

[1] T. L. Mason, A. Bagirov, J. Klunder, and J. van Berkel. Development and application of the discrete gradient method for non-smooth non-convex production optimization. In International Oil Conference and Exhibition, Veracruz, Mexico, 2007. Society of Petroleum Engineers.

[2] V. D. Kosmidis, J. D. Perkins, and E. N. Pistikopoulos. Optimization of well oil rate allocations in petroleum fields. Ind. Eng. Chem. Res., 43(14):3513–3527, 2004.

[3] C. A. Floudas. Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications. Oxford University Press, New York, NY, 1995.

[4] H. D. Sherali. On mixed-integer zero-one representations for seperable lower-semicontinuous piecewise-linear functions. Operations Research Letters, 28(4):155–160, 2001

[5] E. Camponogara and P. Nakashima. Optimizing gas-lift production of oil wells: piecewise linear formulation and computational analysis. IIE Transactions, 38(2):173–182, 2006.

[6] A. B. Keha, I. R. de Farias Jr., and G. L. Nemhauser. Models for representing piecewise linear cost functions. Operations Research Letters, 32(1):44–48, 2004.

[7] F.A. Al-Khayyal and J.E. Falk. Jointly constrained biconvex programming. Mathematics of Operations Research, 8(2):273–286, 1983.

[8] C.A. Meyer and C.A. Floudas. Global optimization of a combinatorially complex generalized pooling problem. AIChE Journal, 52(3):1027–1037, 2006.

[9] M.L. Bergamini, P. Agguirre, I.E. Grossmann. Logic-based outer approximation for globally optimal synthesis of process networks. Computers & Chemical Engineering, 29(9):1914–1933, 2005.

[10] D.S. Wicaksono and I.A. Karimi. Piecewise MILP under- and overestimators for global optimization of bilinear programs. AIChE Journal, 54(4):991–1008, 2008.