### Mathematical models

Optimum well placement (determining optimum number, type and siting of w=
ells) is a crucial step in field development

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

### Model=
ing and algorithmic considerations:

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 m=
ost complex constraints come from the interaction between withdrawal rates =
and pressures at all the wells, that must be defined by the nonlinear gas f=
low equation. However this nonlinear constraint has a very good linearizati=
on substitute, called influence equations. In these equations, the pressure=
drop at well i is a linear function of withdrawal flow rates from all dril=
led wells. This is defined by influence function matrices. After this prope=
r linearization, the resulting problem is a mixed integer programming =
problem, which can be solved by well known techniques.

**Contributor:**

Dr Fabrizio Lacalandra, QuanTek

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