جایگذاری خوب بهینه تولید کننده و برنامه ریزی تولید در مخزن نفت

Most of the available literature on optimal well placement has employed numerical simulators in a black box manner linked to an external search engine. In this work, we formulate the contents of that box inside a mixed integer nonlinear programming model for optimal well placement. We provide a unified model that integrates the subsurface, wells, and surface levels of an upstream production project. It links the production plan with the aforementioned elements, and economics and market. This results in a complex spatiotemporal mixed integer nonlinear model, for whose solution we modify and augment an existing outer approximation algorithm. The model solution provides the optimal number of new producers, their locations, and optimal production plan over a given planning horizon. To our knowledge, this is the first contribution that uses mathematical programming in a real dynamic sense by honoring the constituent partial differential equations.

The continuous depletion of oil reserves and rise in global oil demand have created a challenge for the oil exploration and production (E&P) industry. In 2011, the global oil production and demand were 88.4 mb/d and 88.2 mb/d respectively (OPEC, 2011b). OPEC estimates the demand to be 109.7 mb/d in 2035 (OPEC, 2011b). To meet this demand, the oil companies are expanding (OPEC, 2011a) their drilling activities (see Fig. 1). However, drilling oil wells is highly expensive and uncertain, and involves potential environmental hazards and economic risks. For instance, a vertical onshore (offshore) well can cost MM$2–5 (MM$8.3) on an average and a horizontal one can cost MM$2.6–6.5 (MM$10.2). Even after such expense, there is no guarantee that a well will be productive. In 2010, 56 of 227 exploration wells and 5 of 726 development wells of Shell Company ( SHELL, 2010) turned out to be dry holes. BP's recent drilling blowout and resulting oil spill in the Gulf of Mexico keeps attracting news even now and BP has so far spent ( BP, 2012) more than B$8 in compensation. With such high financial and environmental stakes and significant uncertainty, there exist clear incentive and much recent interest to increase the overall economic efficiency and success rate of the hydrocarbon recovery processes by using systematic optimization approaches to obtain the best drilling and production scenarios. Full-size image (21 K) Fig. 1. Total upstream (exploration and development) expenditure of oil majors and number of active drilling rigs worldwide. Figure options In practice, the industry uses a variety of data, tools, and heuristics to select well locations. Typically, the entire task involves two stages. In the first stage, the engineering team defines a variety of development scenarios. In the second, it evaluates those scenarios via extensive simulations and develops various field production/injection profiles. Although intuitive and useful, such a sequential procedure is inherently empirical, ad hoc, and myopic, and has shortcomings. Much scope and benefits exist for the application of advanced optimization methods. A systematic model-based approach that simultaneously considers the drilling decisions along with the production/injection profiles over the planning horizon can yield significant returns in terms of economics, success, and recovery. An integrated strategy for optimal well placement would encompass at least five elements in one single optimization model: (a) subsurface physics, (b) well geometry and dynamics, (c) surface facilities, (d) production/injection profiles, and (e) market and economics. Formulating and solving such an optimization model is a tremendous challenge. First, the myriad of decisions such as potential well locations, types, functionalities (producer/injector) (Yeten, Durlofsky, & Aziz, 2003), trajectories and inclinations (Ayodele, 2004), drilling schedules (Beckner & Song, 1995), and flow distributions (Isebor, 2009, Shafiee et al., 2010 and Zerafat et al., 2009) make this a highly combinatorial optimization problem. Second, the physics of multi-phase flow in the reservoir is highly nonlinear and spatiotemporal, which makes the optimization problem large, complex, and nonconvex. Guaranteeing the best solution becomes a huge challenge. Last, the inevitable discretization of the governing continuity equations renders the problem non-differentiable in the spatial domain and limits the application of derivative-based optimization algorithms. The existing literature has studied three main approaches for optimal well placement: (a) mathematical programming (b) evolutionary and direct search (Afshari et al., 2011, Bouzarkouna et al., 2012, Güyagüler and Horne, 2000, Onwunalu and Durlofsky, 2009, Wang et al., 2012 and Yeten et al., 2003), (c) gradient-based search (Ebadat and Karimaghaee, 2013, Sarma and Chen, 2008, Vlemmix et al., 2009 and Zandvliet et al., 2008). Biegler and Grossmann (2004) and Grossmann and Biegler (2004) present an excellent overview of these methods, while our detailed literature survey (Tavallali, Karimi, Teo, Ayatollahi, & Baxendale, 2013) and Nasrabadi, Morales, and Zhu (2012) specifically discuss their applications to well placement. While mathematical programming has been the first reported approach (Rosenwald & Green, 1974), the other two approaches (call them search methods) have received much more attention. The two methods usually search for better well locations (Güyagüler et al., 2002 and Wang et al., 2007), and then use commercial reservoir simulators in a black box manner to evaluate the performance of these locations. Thus, in a sense, they parallel the conventional industrial approach. The simulator acts as a mere function evaluator that numerically solves the system of governing equations for a given set of heuristic control policies. The optimizer then uses the black box to determine a feasible production/injection plan. To obtain a near optimal solution, the optimizer must evaluate many such plans and simulate many scenarios. This can easily become computationally expensive for the evolutionary and direct search methods due to the dynamic nature of the reservoir and time-dependent decisions regarding entire production/injection profiles. While the gradient-based methods (Jansen, 2011) used in the work of Forouzanfar, Li, and Reynolds (2010), Li and Jafarpour (2012), Wang et al. (2007), and Zhang, Li, Reynolds, Yao, and Zhang (2010), have the potential to fare better, they all assume constant well/field production rates over the planning horizon. This makes it difficult for them to handle dynamic events (e.g. water breakthrough) and infeasible pre-fixed production profiles. By opening the black box of reservoir simulation and embedding its governing spatiotemporal equations inside an optimization model, one could provide quicker and better guidance to the optimization engine, and reduce the computational burden. To this end, the powerful and versatile technique of mathematical programming offers significant potential and promise. It has been successfully used in a variety of industries and applications such as energy systems (Khalilpour and Karimi, 2011 and Khalilpour and Karimi, 2012), petroleum refining and blending (Li and Karimi, 2011, Li et al., 2010 and Méndez et al., 2006), pharmaceutical enterprises (Susarla & Karimi, 2011), chemical process design and integration (Faruque Hasan et al., 2010, Floudas and Ciric, 1989, Huang et al., 2012, Razib et al., 2012 and Yee and Grossmann, 1990), chemical logistics (Bansal et al., 2005 and Bansal et al., 2007), and others. It offers several advantages over other methods. First, it is a constrained optimization approach that can naturally handle the geometrical and logical constraints such as minimum well-to-well spacing and well segment connection criteria. The other two methods are unconstrained optimization approaches that are unable to do well in the presence of discrete constraints. They produce many infeasible configurations, which deteriorates their performance (Ciaurri, Mukerji, & Durlofsky, 2011), and require external intervention to recover from infeasibility (Emerick et al., 2009). Second, the mathematical programming approach is flexible and versatile. It can embed the reservoir physics inside its model and benefit from its mathematical structure. This allows one to include the production/injection profiles along with the location decisions in the model and improve computational speed. For a more thorough discussion of the advantages of mathematical programming, please refer (Tavallali et al., 2013). Most previous mathematical programming work on well placement and production planning has focused on the surface issues and related problems such as numbers, types, capacities, locations, and allocations of wells and platforms (Barnes et al., 2007, Carvalho and Pinto, 2006, Devine and Lesso, 1972, Garcia-Diaz et al., 1996, Grimmett and Startzman, 1988, Kosmidis et al., 2002, Lin and Floudas, 2003, Van Den Heever et al., 2000 and Van Den Heever et al., 2001). Those that have included the subsurface issues have usually empirically approximated the reservoir response to various production scenarios. These approximations being largely linear have resulted in mixed integer linear programs (MILPs). Rosenwald and Green (1974) developed an MILP model by using influence function and superposition to approximate the flow dynamics. While the former is an approximation derived from several reservoir simulations, the latter relates pressure drop at each well to production rate (see Murray & Edgar, 1978). Using the same approach, Haugland, Hallefjord, and Asheim (1988) studied well placement and scheduling, platform capacity, and production planning. Later, Iyer, Grossmann, Vasantharajan, and Cullick (1998) used piecewise linear approximations of reservoir pressure and gas-oil-ratio (GOR) versus cumulative oil production to describe the subsurface dynamics. They also included the well and surface elements in their MILP model. Although comprehensive, their model uses several simplifying assumptions such as linear pressure drop vs. flow relation for pipes, constant productivity index for each well throughout the planning horizon, non-interacting and independent wells, uniform fluid pressure and composition throughout the reservoir. These assumptions can affect production estimates significantly. On the other hand, their work also addresses several important issues: well selection in reservoirs belonging to multiple fields, well drilling and platform installation scheduling considering the drilling rig availability, and finally platform sizing and production planning. Among the surface-directed studies, Van Den Heever and Grossmann (2000) extended the MILP model of Iyer et al. (1998) by fitting an exponential function to describe reservoir pressure vs. cumulative oil flow, and quadratic functions to describe cumulative gas production and GOR vs. cumulative oil flow. In contrast to these works that have used dynamic approximations, several others have used static approximations. Dogru (1987) employed productivity index and oil-in-place data to formulate the offshore well platform and drilling location-allocation problem. Vasantharajan and Cullick (1997) used connected hydrocarbon pore volume in a specified drainage area, inversely weighted by tortuosity, to define another static metric. Ierapetritou, Floudas, Vasantharajan, and Cullick (1999) used a similar approach. Their MILP model allows multiple geo-objects and layers with perforated wells spanning multiple layers. Later, Cullick, Vasantharajan, and Dobin (2003) extended this approach and included deviated wells based on a sequential heuristic. Most of the above optimization models are large; hence several solution approaches have also been used or developed in the literature. These include heuristic or decomposition procedures (Cullick et al., 2003, Ierapetritou et al., 1999, Iyer et al., 1998 and Van Den Heever and Grossmann, 2000) and pre-processing steps (e.g. reservoir data, Ierapetritou et al., 1999 and Vasantharajan and Cullick, 1997). In spite of its potential and versatility, the application of mathematical programming in practice has been limited due to several reasons. One is the complexity in their model formulations and executions. Another is the lack of their awareness in the industry. Furthermore, their various approximations of the nonlinear multiphase flow dynamics have been largely problem-specific and far less accurate than rigorous numerical simulations. Significant advances in computing hardware and the solvers and tools for mathematical programming enable us to go beyond approximating the subsurface multi-phase flow. This is one of the main objectives of this work. In addition, we relate the subsurface flow to the flow inside the well tubing and consider the surface and economic constraints to obtain a very detailed and comprehensive model for the upstream drilling and production activities. In this article, we consider the deterministic problem of optimally locating the drilling sites for new/infill producers and deciding the optimal production and injection plans for all active wells. We first state and define the scope of our well placement and production planning problem. Then, we describe our modeling approach and devise a solution algorithm, as the commercial solvers fail to solve the formulated problem. We then present two case studies to demonstrate the effectiveness of our proposed approach, and conclude with a concise discussion.

We presented an integrated and practically useful model for oil well placement and production planning in a petroleum reservoir, and proposed an effective solution algorithm for the same. The major contribution of our work is that we considered subsurface flow dynamics much more rigorously than any other previous study. Most studies thus far on mathematical programming neglected or grossly approximated this dynamics. In fact, none of them, to our knowledge, considered optimal production plan using such a detailed spatiotemporal model. Thus, this is the first contribution to integrate most of the critical elements of upstream production and spatiotemporal subsurface dynamics in a multiperiod mathematical programming approach. Furthermore, in contrast to most previous work, our approach does not require pre-fixing wells and locations or production/injection rate patterns. We also successfully tailored and modified the OA/ER/AP algorithm to improve its success in solving this large and complex problem and improving its performance. Our modifications of the primal NLP and master MILP subproblems along with a 2-step local search before termination were critical in ensuring progress and good solutions for the two illustrative examples. While much further work is needed to address the size and complexity of this important problem, we have taken the first step in rigorously applying the powerful and versatile technique of mathematical programming and addressing some of the challenges associated with the industry-scale well placement problem.