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Well Trajectory Calculations

Permanent records of hole locations, and the accuracy of these records, can have significant impact on future drilling and completion operations, as well as on other technical, economic and legal issues relating to a field.

To describe the well trajectory, we must be able to determine the coordinates x_i, y_i and z_i from the measured angles θ_i and α_i at every station i along the well path. There are several calculation methods available, including the following:

Tangential method
Balanced tangential method
Minimum curvature method
Radius of curvature method
Angle-averaging method

The tangential method is the least accurate of all methods, and should not be used. The most accurate are the radius of curvature and minimum curvature methods. In this section, we summarize the equations representing the minimum curvature, radius of curvature and angle-averaging methods.

Minimum Curvature Method

The minimum curvature method uses the angles measured at two consecutive stations, i−1 and i, to describe a smooth, circular curve that represents the wellbore path. It uses a dog-leg severity ratio factor, R_ei, for each corresponding section of the curve.

The coordinates xi and y_i represent, respectively, the departures in the west-east and north-south directions, while z_i is the vertical departure. These coordinates are expressed as follows:

West-east departure:

$x_{i}= \dfrac{\Delta d_{i}}{2}\cdot \left ( \sin \theta_{i-1}\cdot \sin \alpha_{i-1} +\sin\theta_{i}\cdot \sin \alpha_{i} \right )\cdot R_{ei}$ (1)

North-south departure:

$y_{i}= \dfrac{\Delta d_{i}}{2}\cdot \left ( \sin \theta_{i-1}\cdot \cos \alpha_{i-1} +\sin\theta_{i}\cdot \cos \alpha_{i} \right )\cdot R_{ei}$ (2)

Vertical departure:

$z_{i} = \dfrac{\dfrac{\left ( \Delta d_{i} \right )}{2}}{\left ( \cos\theta_{i-1} +\cos\theta_{i} \right )\cdot R_{ei}}$ (3)

where

$R_{ei}= \dfrac{2}{e_{i}}\cdot \tan \dfrac{e_{i}}{2}$ (4)

ei = angle change of drill string between i and i−1, given by

$e_{i}= \cos^{-1}\left [ \sin\theta_{i}\cdot \sin\theta _{i-1}\cdot \cos\left ( \alpha_{i}-\alpha_{i-1} \right )+\cos\theta _{i}\cos\cdot \theta _{i-1} \right ]$ (5)

We can determine the coordinates at the nth point along the well path by algebraically summing over the total number (n) of survey points:

$x_{n}= \sum\limits_{i= 1}^{n}x_{i}$ (6)

$y_{n}= \sum\limits_{i= 1}^{n}y_{i}$ (7)

$z_{n}= \sum\limits_{i= 1}^{n}z_{i}+\left ( \textrm{Kick - off depth }\right )$ (8)

Radius of Curvature Method

Like the minimum curvature method, radius of curvature assumes that the wellbore is a smooth curve represented by circular or spherical segments. From survey data at two consecutive stations, i−1 and i, we can determine the coordinates x_i, y_i and z_i from the following equations:

$x_{i} = \dfrac{180\cdot \Delta d_{i}}{\pi \cdot \left ( \theta_{i}-\theta _{i-1}\right )\cdot \left ( \alpha_{i}-\alpha _{i-1}\right )}\cdot \left (\cos \theta_{i-1}-\cos\theta_{i} \right )\cdot \left ( \cos\alpha_{i-1}-\cos \alpha_{i} \right )$ (9)

$y_{i} = \dfrac{180\cdot \Delta d_{i}}{\pi \cdot \left ( \theta_{i}-\theta _{i-1}\right )\cdot \left ( \alpha_{i}-\alpha _{i-1}\right )}\cdot \left (\cos \theta_{i-1}-\cos\theta_{i} \right )\cdot \left ( \sin \alpha_{i}-\sin \alpha_{i-1} \right )$ (10)

$z_{i}= \dfrac{180 \cdot \Delta d_{i}}{\pi \cdot \left ( \theta _{i}-\theta_{i-1} \right )} \cdot \left ( \sin \theta _{i}-\sin \theta _{i-1} \right )$ (11)

where θ_i≠θ_i−1 and α_i≠α_i−1

Angle Averaging Method

The angle averaging method takes the mean of two measured sets of inclination and azimuth values [(θ_i, α_i) and (θ_i−1, α_i−1)], and assumes that the wellbore follows a tangential path. The coordinates at station i are given by

$x_{i}= \Delta d_{i}\cdot \sin \dfrac{\theta_{i}+\theta_{i-1}}{2}\cdot \cos \dfrac{\alpha_{i}+\alpha_{i-1}}{2}$ (12)

$y_{i}= \Delta d_{i}\cdot \sin \dfrac{\theta_{i}+\theta_{i-1}}{2}\cdot \sin \dfrac{\alpha_{i}+\alpha_{i-1}}{2}$ (13)

$z_{i}= \Delta d_{i}\cdot \cos \dfrac{\theta_{i}+\theta_{i+1}}{2}$ (14)

We may use Equations 6, 7 and 8 above to calculate the total departures.

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Directional Drilling Tools and Techniques

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