(240c) Dynamic Optimal Well Placement In Oil Reservoirs



Dynamic
Optimal Well Placement in Oil Reservoirs

M. S. Tavallali1, K. M. Teo2, I. A.
Karimi1*

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.




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