## 4 Experiments and Results
### Feature Construction  
Some real-time data channels are shown in Figure 8. Taking, for instance, the stuck pipe scenario into account, it can be presumed a priori that the block position and hook load play some role. If the pipe is stuck, the block cannot be moved with the drill string connected. Attempting to move the block up will drastically increase the hook load without giving the block a chance to move--or, to be more precise, the block speed is nearly zero. So, the block speed, which is the first derivative with regards to time of the block position, might be a feature that simplifies the problem of stuck pipe detection. The same applies to drill string rotation and torque. In terms of real-world tasks, stuck pipe detection is an actual but rather simple problem. A more complex task is stuck pipe prediction, and, as a consequence, stuck pipe prevention through using some precautionary counter-actions. The assumption that hook load and torque contain information about a possibly emerging stuck pipe is still valid. How that information is provided is actually unknown, and therefore a challenge whose importance should not be underestimated. Figure 9 sketches a typical borehole drilled nearly vertical at its beginning which then changes direction to nearly horizontal. The main components of the hook load \(F_{Hock}\) are the acceleration force \(F_{A}\), the component of the total weight of the whole drill string (\(F_{T}\)) aligned with the drilling direction \(F_{W}\), (according to the principles of mechanics the component that is vertical to the drilling direction \(F_{N}\) is to be ignored), the friction forces \(F_{F}\) (that according to the principles of mechanics could depend on \(F_{N}\)), and some other non-quantifiable forces denoted as \(\varepsilon\). In case of creating a deterministic model, the mass influx of the drill string, the borehole trajectory, especially the inclination and friction factors, amongst others, need to be known. Since it appears unpromising to identify and estimate all input factors with a reasonable certainty and accuracy to predict and thus prevent a stuck pipe (as well as other crises), a heuristic approach incorporating deterministic know-how seems to be the most feasible solution. To incorporate as much deterministic know-how as possible, a systematic approach to generate features based on specific laws of physics appears to be appropriate. Figure 9: Forces influencing the hook load. For feature creation, the physical rules of kinematics and dynamics were applied to the drill string in a first approach to create some heuristics for the features created via the available data channels. For instance, the simple rule for the acceleration force \(F_{A}\) based on the mass of \(m\) and the acceleration of \(a\),  
\[F_{A}=m\cdot a, \tag{6}\]  
leads to the resulting heuristic that \(F_{A}\) can be expressed as  
\[F_{A}=c_{1}\left(d_{h}^{\ \ast}a+d_{s}a\right)+c_{2}a. \tag{7}\]  
In Equation (7), \(d_{h}\) denotes the length of the drill string above a rotary table and thus out of the borehole, \(d_{s}\) is the length of the drill string in the borehole, and a is the acceleration applied to the total drill string. The constants \(c_{1}\) and \(c_{2}\) need to be evaluated. If a deterministic model builder is used, they are probably automatically estimated using a heuristic model builder and thus can be ignored if the same features are derived. In fact, the features from Equation (7) are \(d_{h}*a\), \(d_{s}*a\) and the acceleration \(a\) of the drill string itself. Combining all the kinematic and dynamic rules heuristically and normalizing the results by the length of the drill string, a set of exactly 100 feature rules was obtained. Those rules were applied to the 10 base channels as shown in Table 4 as well as to the extended base channels shown in Table 5, resulting in a total of 1600 features.