Petroleum field exploration and planning is a challenging area that involves complex nonlinear reservoir models as well as long term economic evaluations. Facilities required for exploration and production require very expensive investments that remain in operation typically for 10-30 years. Therefore, decisions related to the investment in these facilities greatly impact the profitability of the entire project. This in turn has attracted significant interest in the development of optimization models for exploration of petroleum and gas fields. Although the existence of oil or gas in a petroleum field is determined by seismic surveys and preliminary exploration test, the actual amount of oil or gas in reserves remains uncertain until the investment decisions are made. Therefore, in this industry a major challenge is how to make optimal investment decisions in the presence of uncertainty. Most of the available literature for exploration and planning of petroleum fields is based on deterministic models (Ierapetritou, Floudas, Vasantharajan, Cullick, 1999; Grothey, McKinnon, 2000; Van den Heever, Grossmann, 2000; Van den Heever, Grossmann, 2001; Barnes, Linke, Kokossis, 2000; Kosmidis, Perkins, Pistikopoulos, 2002; Lin, Floudas, 2003; Ortiz-Gomez, Rico-Ramirez, Hernandez-Castro, 2002), while there are only few papers that deal with uncertainty (Jornsten, 1992; Haugen, 1996; Meister, Clark, Shah, 1996; Jonsbraten, 1998; Lund, 2000; Aseeri, Gorman, Bagajewicz, 2004; Goel, Grossmann, 2004, 2006. In this paper, we consider a multistage stochastic programming approach for the design and planning of a petroleum field infrastructure over a planning horizon where there are uncertainties in the size of the reserves, and initial deliverabilities of reservoirs. Uncertainties are incorporated into the model in terms of different scenarios. In the conventional stochastic programming approach, there is only exogenous uncertainty where decisions do not affect the uncertainties, providing a fixed scenario tree. Different from the conventional stochastic programming problems, we have endogenous uncertainty where decisions affect the uncertainty, giving rise to variable scenario trees as described in Goel and Grossmann (2004). Compared to previous papers, the unique aspect of this presentation is that it considers nonlinear performance equations for the reservoirs, which leads to a stochastic MINLP problem.

Specifically, we consider a single petroleum field consisting of several reservoirs where each reservoir contains several possible well sites. The petroleum field infrastructure has only one well platform that is connected to the production platform. The location of production and well platforms are fixed while the location of wells is given. Also, it is assumed that the well platform is currently in operation with enough capacity. Some of the possible well sites have to be exploited for oil/gas over a planning horizon. In order to produce oil/gas from a field, some wells need to be drilled and connected to the well platform. Investment decisions for the project include the selection of the well sites to drill and the time to drill. Operation decisions include the oil/gas production rates from each field which are affected by the pressure differential between the reservoir and the well bore as well as the productivity index of the reservoir. It is also assumed that the oil flow rate profile from a well should be non-increasing. The main uncertainties are coming from the size of the reserves, initial deliverabilities of reservoirs, which are characterized through parameters of the nonlinear functions that predict reservoir properties. These uncertainties are implicitly taken into account during the reservoir simulations for generating possible scenarios. Given the above assumptions, the goal is to maximize the net present value of the project which is found by considering the revenues, investment and operation costs. In order to capture all the complex trade-offs, we present a nonlinear mixed-integer/disjunctive programming model that is composed of scenario linking and non-anticipativity constraints taking into account the variable scenario trees. In order to solve this difficult nonlinear stochastic problem we reformulate it as mixed-integer nonlinear program, which relies on convex envelopes for handling nonconvexities and that is solved through a duality-based branch and bound algorithm. In order to strengthen the bounds, we propose the derivation of cutting planes that are based on Lagrangean decomposition. We describe results on a variety of test problems to illustrate the capabilities of the proposed model.