Dynamic Optimal Well Placement In Oil Reservoirs

Dynamic Optimal Well Placement in Oil Reservoirs

M. S. Tavallali¹, K. M. Teo², I. A. Karimi^1*

¹Department of Chemical & Biomolecular Engineering, 4 Engineering Drive 4

² Department of Industrial & Systems Engineering, 1 Engineering Drive 2
National University of Singapore, Singapore 117576

Abstract

OPEC predicts the oil industry to increase its production by almost 23% of the current level to be able to fulfill the world oil demand in the next 20 years. However, the hydrocarbon resources are limited, and hence it is highly crucial to exploit the new, marginal and existing mature fields in an optimal way. In this situation, optimal well placement can play an important role with long-term financial impact. However, determining optimal well locations is inherently a complex nonlinear and combinatorial problem due to the spatial-temporal continuity equations and their non-differentiability with respect to well location, nonlinearities of the multiphase flow, existence of different uncertainties and a myriad of related decisions such as the number of potential well positions, their types, functionalities (producer / injector), trajectories, inclinations, drilling schedules, etc.

          In this work, we address this challenging problem using a combination of mathematical programming and local search for locating optimum drilling sites. While mathematical programming is a potential tool for analyzing such problems, very few contributions have used this approach for well placement (Rosenwald and Green 1974; Ierapetritou, Floudas et al. 1999; Cullick, Vasantharajan et al. 2004). Even these efforts have used only static features of the system. Therefore, to use mathematical programming and bridge the gap between the static and dynamic information, one must address two key issues. First, one must capture the dynamics of the multiphase flow and temporal behavior over the production horizon. Second, one needs a specialized algorithm that exploits the model structure to solve such a large and complex nonlinear, nonconvex, discrete optimization problem. These goals motivated the present study.

          For model development, we assume that all deterministic geological and PVT information are supplied and try to maximize the profit of oil production over a planning horizon. The model involves three sets of constraints (physical, logical and operational) related to the structure and dynamics of a reservoir over time. These constraints result in a multiperiod, dynamic, mixed integer nonlinear (MINLP) model with discretized partial differential equations. For model solution, we have developed an outer approximation algorithm based on the work of (Viswanathan and Grossmann 1990), which exploits the model structure. While the master problem identifies new promising well locations, the primal problem assesses their quality. Since the primal problem is also large, PDE-constrained, and highly nonlinear, we further decompose it into a series of smaller subproblems over the planning horizon to improve solution speed.

          This algorithm while effective in identifying regions of optimal well locations, terminated prematurely without spotting actual locations rigorously. Therefore, we made two further modifications. For the model, we considered convective flows amongst neighboring cells and the concept of upstream flow to relax some of the structural constraints. For the algorithm, we added a local neighborhood search after the termination. We probe the neighborhood for possible improvement and then continue the search using the KKT points of a neighbor with higher objective value (if any). We also simplified our linearization scheme to improve solution time and avoid eliminating potential solutions.

          We use two examples to successfully demonstrate the impact of the above modifications. The examples involve up to three producer wells to be placed in an oil reservoir with two existing producer and injector wells. We consider a planning horizon of 1470 days. The results confirm that the decomposition of the primal problem and simplification in linearization are essential to solve this large scale PDE constraint MINLP model. However, since it is a nonlinear nonconvex problem, guaranteeing a global optimal solution is difficult. However, our hybrid algorithm certainly improves the solution and likelihood of reaching a global optimum. For example, for the case of mono well placement, we compared our results with complete enumeration, using a commercial reservoir simulator (ECLISPE). Our algorithm reached the global optimum successfully.

 

Acknowledgement: We would like to thank National University of Singapore and SINGA program for the financial support of this research, and also Schlumberger Company for granting the academic license of ECLIPSE. We are also thankful to Mr. David Baxendale from RPS Energy Limited for his valuable industrial insights.

References

Cullick, A. S., S. Vasantharajan, et al. (2004). Determining Optimal Well Locations From A 3D Reservoir Model. EUROPEAN PATENT APPLICATION.

Ierapetritou, M. G., C. A. Floudas, et al. (1999). "Optimal location of vertical wells: Decomposition approach." AIChE Journal 45(4): 844-859.

Rosenwald, G. W. and D. W. Green (1974). "Method For Determining The Optimal Location Of Wells In A Reservoir Using Mixed-Integer Programming." Soc Pet Eng AIME J 14(1): 44-54.

Viswanathan, J. and I. E. Grossmann (1990). "A combined penalty function and outer-approximation method for MINLP optimization." Computers and Chemical Engineering 14(7): 769-782.

 

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* Corresponding author: Tel.: +65 6516-6359, Fax: +65 6779-1936, Email – cheiak@nus.edu.sg