Practical Desurvey

created: 2019-06-27 modified: 2020-12-15

Introduction

Desurvey is the practice of obtaining the geometry of a drillhole through its deviation survey data.

Here we present two methods: tangent and spherical arc.

Definitions

Considering a right-handed coordinate system, with $x$ pointing east, $y$ north and $z$ up,

$x_i$ — the spatial location vector of each survey point, being $x_0$ given as the collar of the drillhole.
$v_i$ — the plunge vector of the drillhole at the survey points. It is usually calculated from the measured trend $\theta$ and plunge $\phi$, with positive plunge downwards: $$ v_i = \begin{bmatrix} \sin(\theta)\cos(\phi)\\ \cos(\theta)\cos(\phi)\\ -\sin(\phi)\\ \end{bmatrix} $$
$a_i$ — the position along the drillhole on where each survey was made.
$s_i$ — the length of the segment between each $i$ and $i+1$ survey points, that is, $$s_i = a_{i+1} - a_i$$

Tangent

The simplest method is to consider the drillhole a sequence of straight segments with the same down-hole orientation as the last survey point. For that, we simply accumulate $s_i v_i$ from $x_0$ to obtain each $x_{i+1}$ in sequence: $$ x_{i+1} = x_0 + \sum_{j=0}^{i} s_j v_j $$

To obtain the coordinates of an arbitrary point $p$ at position $d$ along the drillhole find $i$ such that $a_{i+1} > d$, then $$ p = x_i + (d - a_i) v_i \label{tangent}\tag{1} $$

Spherical arc

The spherical arc method, also known as minimum curvature, treats each segment of the drillhole as a great circle arc with length $s_i$ that encompass the angle¹ $\alpha_i$ between each survey point $i$ and its successor $i+1$. Being $\alpha$ in radians, the radius $r_i$ of the sphere can be calculated as $$ r_i = {s_i \over \alpha_i} $$

To obtain the center $c_i$ of the sphere, we must offset $x_i$ a distance $r_i$ perpendicular to $v_i$ on the plane spanning $v_i$ and $v_{i+1}$. The normal $n_i$ to this plane is obtained through $$ n_i = {{v_i \times v_{i+1}} \over {|v_i \times v_{i+1}|}} $$

from this, the vector $q_i$ perpendicular to $v_i$ is simply $$ q_i = n_i \times v_i $$

being the center then $$ c_i = x_i + r n_i \times v_i. $$

Using $v_i$ and $q_i$ as an orthogonal pair spanning the plane, the position of the next survey point $x_{i+1}$ is built as $$ x_{i+1} = c_i - r q_i cos (\alpha_i) + r v_i sin (\alpha_i) \label{sphere_survey}\tag{2} $$

Similarly, an arbitrary point $p$ at position $d$ along the drillhole is found using $i$ such that $a_{i+1} > d$. Its angular position $\theta$ along the great circle is $$ \theta = {{d - a_i} \over r_i} $$

This angle can be plugged into equation $\ref{sphere_survey}$ instead of $\alpha$ so that we get $$ p = c_i - r q_i cos (\theta) + r v_i sin (\theta) \label{sphere}\tag{3} $$

Interactive Example

Collar

Survey

Lith

Options

^{1. Always check if the angle between two consecutive survey points is
not zero or near zero; if so, use the tangent method for the segment
instead of spherical arc, as the numerical precision might blow up and
give you bad results.

On the subject of angles, you can either use the arccosine of the dot
product or the arctangent of the magnitude of the cross product over
the dot product. The second method is much more precise for small
angles and will allow you to use a closer tolerance before considering
that the angle between two survey points is zero. It's costlier in
terms of processing, though.
↩}