The oil well placement problem, a crucial problem in reservoir engineeri= ng, consists in determining the optimum number, type, and location of oil w= ells so as to optimize the hydrocarbon production and the drilling costs. I= n industry, the decision to drill a well or not and its location is taken b= y reservoir engineers trusting their professional expertise. These decision= s strongly relate with the understanding of the impact of different influen= cing engineering and geological parameters. However, such influence is very= complex (non linear) and changing over time, thus a deep understanding of = such phenomena requires more than human experience. Satisfying solutions co= uld be provided by practitioners, but optimization methods can lead to impr= oved configurations.

From a mathematical modelling viewpoint, the number of injector and prod= ucer wells, the number of branches could be represented by integer variable= s, completed by continuous variables as wells and branches location in the = reservoir, the length of the branches, etc. The functions to optimize and t= he constraints are generally computed from the outputs of a reservoir fluid= flow simulator, costly in computational time: the outputs to optimize are = the quantities of produced oil and water, and the quantities of injected (t= o facilitate the production). As we do not have access an analytic formula = of the objective function, we have a Black-Box optimization problem. Hence = we have no knowledges of the continuity, differentiability or convexity of = the objective function. Localization constraints and number of wells constr= aints could also be included to problem.

Thus, the oil well placement problem can be modeled as a Black-Box MINLP= problem, a very challenging problem both from a theoretical and a computat= ional viewpoint. Note also that, as no convexity assumption holds, in optim= al well placement one should perform some kind of global search in order to= avoid being trapped in local minima.

A few more details on the model:

The two most widely used objective function are:

i. maximize the quantity of produced oil;

ii. evaluate the revenue of a well configuration with Net Present Value = (NPV) function. This function combines oil revenue, water management (water= injection and separation), and drilling costs.

Automatic well placement optimization is an iterative procedure tha= t can be divided into following procedures:

- Using engineering judgment, guess initial well(s) location
- Use an optimization engine based on user-defined decision variable= s to suggest possible improved well location(s).
- Apply a reservoir response model to report to the optimization eng= ine the performance of the proposed well locations.
- Include the effect of uncertainty in reservoir properties, economi= c factors, etc, which can be an optional step.
- Calculate the objective function (e.g. Net Present Value or NPV).=
- Repeat steps 2 to 5 until stopping criteria (set by user) are met=
.

The approaches to problems 1 to 5, may differ in the optimization a= lgorithm, reservoir response modeling technique, and available decisio= n variables and constraints. We now turn our focues to the modelling aspect= of the problem

Concentrating our attention to the optimization problem after the i= nitial guess (problem n. 2), the well placement problem is translated into = optimization of an objective function (NPV or cumulative hydrocarbon produc= tion). Applying MILP models one can for instance, solves this problem throu= gh finding locations of a given number of wells (out of total possible well= locations) that minimizes the difference between the production and schedu= led demand. Hence, the drilling decision can only be made at particular loc= ations i which have to be identified beforehand (guess problem n. 1).

The most complex constraints come from the interaction between withdrawa= l rates and pressures at all the wells, that must be defined by the nonline= ar gas flow equation. However this nonlinear constraint has a very good lin= earization substitute, called influence equations. In these equations, the = pressure drop at well i is a linear function of withdrawal flow rates from = all drilled wells. This is defined by influence function matrices. After th= is proper linearization, the resulting problem is a mixed integer programmi= ng problem, which can be solved by well known techniques.

Constraints are generally physical ones, ensuring the practical realizab= ility of the solution and the correct behavior of the simulator. A useful c= onstraint is also the water cut constraint that consists in applying some r= eactive control on each producer to avoid producing much water which impact= s negatively on the NPV. Such reactive control shuts off producers when the= water cut, i.e., the ratio between the water rate produced and the sum of = water and oil rates produced, is higher than a given threshold. It is also = possible to add constraints during the production, e.g., produce a mi= nimal quantity of oil for instance.

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Dr. Claudia D'Ambrosio, CNRS LIX, Ecole Polytechnique