US20190106978A1 - Method for producing an oil well - Google Patents

Abstract

This disclosure addresses the vibration problems that occur during drilling operations. Due to the rotational motion effected on the drill string while drilling, vibrations occur, and when these vibrations become excessive, the drill string may oscillate in a manner that could damage the pipes and damage other tools attached to the drill string. Machine learning is used to identify the vibration prone zones and provide recommendations to the driller to change the operating weight on bit (WOB) and rotation speed (RPM) to achieve drilling efficiency while reducing the possibility of damages downhole.

Description

TECHNICAL FIELD
• The present disclosure relates to the field drilling. In particular, the present disclosure relates to drilling parameters and their effect on drill string vibrations.
BACKGROUND
• To achieve improved drilling efficiency and better productivity of the driller, there is a need for real-time optimization of drilling parameters during drilling operations through each formation in order to optimize weight on bit and bit rotation speed to increase drilling rate as well as reduce the drilling cost. The driller only sees the surface data but there is usually a deviation in the downhole drilling parameters. The driller needs to make better decisions as he manipulates the drilling variables to improve drilling and deal with various issues that may arise during drilling operations.
• The drilling data collected during drilling include weight on bit (WOB), rotary speed (RPM), pump parameters (SPM), depth, inclination, azimuth and rate of penetration (ROP). These parameters have a significant impact on the entire optimization process of the WOB and RPM. The success of drilling optimization is closely related with the quality of the recorded drilling data. However, the driller has to make those important decisions in real time when drilling problems arise.
• Several methods have been used to optimize the drilling parameters. In 1975, Tansev explained how to improve drilling performance. His method involves the interaction of raw data, regression and an optimization technique in order to predict ROP and the life of the bit (Tansev 1975). Karlsson et al. in 1985, observed the use of a BHA design that included a navigation sub. They noticed that the tool allowed the driller to always know the direction of the well and make required trajectory changes while drilling (Karlsson et al. 1985). In 1997, Kamata et al. explained a drill-bit seismic technique which provides important subsurface structure information by using acoustic energy radiated during drilling operations. Sensors, placed at the top of drill string, were used to record the information. They achieved drilling optimization from the information gathered thereby improving safety records and saving cost (Kamata et al. 1997). Paes et al in 2005, focused on the use of sensors for pressure-while-drilling (PWD) and vibration sensors to reduce the drilling cost, non-productive time (NPT), and improve drilling effectiveness without adding more cost to the cost of the routine measurement while drilling (Paes et al 2005). Elshafei et al in 2015 determined the right combination of drilling parameters to reduce drilling time and minimize deviation from planned drilling path by inputting control commands on angular velocity and torque for a quad bit drilling system (Elshafei et al 2015). In 2017, Torres-Cabrera et al observed the difficulty in predicting BHA behaviour which leads to low ROP, unnecessary tripping, and occasionally lost pipe in hole. They addressed the issues through a series of drilling improvements based on real-time and post-well analyses (Torres-Cabrera et al 2017).
• Another method that can be applied to optimize drilling parameters is “machine learning.” Machine learning isn't new; it has been around at least since the 1970s, when the first related algorithms appeared. The general idea behind most machine learning is that a computer learns to perform a task by studying a training set of examples. The computer (or system of distributed or embedded computers and controllers) then performs the same task with data it hasn't encountered before (Louridas et al 2016). Machine learning has been applied to other aspects in the oil industry. Zhang et al in 1991, applied machine learning to rock mechanics and observed that all of the factors governing the rock mass behaviors could be considered as input variables to predict the varying rock behaviors. They made these observations without limiting the amount of input variables that could be used (Zhang et al 1991). Alvarado et al in 2002 used machine learning in their aim to adapt EOR/IOR (enhanced oil recovery/improved oil recovery) technologies to rejuvenate a large number of the mature fields in Venezuela. They used machine learning algorithms to draw rules for screening (Alvarado et al 2002). In 2016, Cao et al used machine learning algorithms to predict production for several wells using pressure and production data, geological maps, and constraints during operations. They used a well-known machine learning method—Artificial Neural Network (ANN). Without assuming a prearranged model, ANN learns from large volume of data points and can change based on the flexibility of the data available (Cao et al 2016). In 2017, Bangert proposed the use of machine learning in order to conduct smart condition monitoring. He realized that his proposed method was more successful than standard condition monitoring thus preventing false alarms and always alarming unhealthy states of plants or equipment (Bangert et al 2017).
• Frequent vibrations of the drill string may lead to poor drilling performance and non-productive time. The concerns arising from drilling vibration are: wasted energy input, low ROP, lengthy drilling time, spoilt bit, damage to the steerable motor leading to unintended trips, damaged Measurement-While-Drilling (MWD)/Logging-While-Drilling (LWD) tools causing lost data, increased fatigue in the drill string, higher caving due to borehole wall damage, discrepancy in data due to meddling with downhole tool telemetry during vibrations, increased cost of rig equipment repairs and increased downtime.
• Two kinds of vibration are of significant concern. First is Stick-Slip. In this case, the bit periodically stops rotating in a torque up moment then spins freely, this goes on through a non-uniform rotation of the drill string. During stick slip, the downhole RPM can be 3× to 15× the average surface RPM. The consequences of Stick-slip are bit damage, lower ROP, connection over-torque, back-off and drill string twist-offs. Stick slip occurrence also leads to wear on bit gauge and stabilizer as well as interruption in mud pulse telemetry.
• The second vibration type is drill string whirling. The bulk of drill string whirling happens in the BHA. During whirling, parts of the BHA face lateral displacements which generate bending stresses and lateral shocks when the BHA contacts the borehole wall (JPT Staff 1998). Having the drill string moving around the wellbore and not rotating about its centerline is the whirling phenomenon. Three types of whirling can occur; forward whirling is a scenario where the drill string is rotating around the wellbore in the same direction with its rotation around its own centerline; backward whirling is a situation where the drill string is rotating around the wellbore in a direction opposite the direction of its rotation around its own centerline. Chaotic whirling occurs where the bits moves in a zig-zag manner with no consistent direction. Whirling creates an over gauge hole reinforcing the tendency for the bit and BHA to whirl.
• The driller has to constantly manipulate available parameters to mitigate vibration problems. A driller's dilemma emerges when increasing the WOB induces stick-slip whereas increasing the RPM induces whirl. Keeping both WOB and RPM low reduces vibration levels but it negatively affects ROP. As a result, the drilling operation either suffers low ROP or experiences higher ROP but with severe vibrations (Wu et al 2010).
• Therefore, improvements in determining optimized parameters for drilling are desirable.
SUMMARY
• In a first aspect, the present disclosure provides a method for producing an oil well. The method comprises: drilling into the Earth, the drilling being effected by a drill string, the drill string having a drill bit; obtaining real-time data from the drill string, the real-time data comprising, measured depth, drilling time, drill bit depth, weight on drill bit (WOB) data, revolution per minute (RPM) data, torque (TOR) data and rate of penetration (ROP) data; in accordance with the real-time data and in accordance with pre-determined rules, obtaining a drill string data classification scheme, which defines an optimum drilling parameter zone; performing a principal component analysis (PCA) of the real-time data, to obtain a set of principle components associated to the real-time data; selecting a subset of the set of principle components; in accordance with the subset of principles components, performing an inverse of the PCA, to obtain modified data; classifying the modified data in accordance with the drill string data classification scheme, to obtain classified modified data; comparing the classified modified data to the optimum drilling parameter zone, to obtain a comparison result; and adjusting at least one of the WOB and the RPM in accordance with the comparison result.
• Other aspects and features of the present disclosure will become apparent to those ordinarily skilled in the art upon review of the following description of specific embodiments in conjunction with the accompanying figures.
BRIEF DESCRIPTION OF THE DRAWINGS
• FIG. 1 shows prior art examples of machine learning methods.
• FIG. 2 shows an example of a prior art optimum Zone Chart.
• FIG. 3A shows a block diagram representation of an embodiment of a method in accordance with the present disclosure.
• FIG. 3B shows a flowchart of an embodiment of a method in accordance with the present disclosure.
• FIG. 3C shows an embodiment of a classification tree in accordance with an embodiment of the present disclosure.
• FIG. 4 shows an example of an operational process to determine the upper limit of RPM, in accordance with the present disclosure.
• FIG. 5 shows an example of how change in ROP and change in time versus time plot might to look like.
• FIG. 6 shows the ideal position the upper and lower limits of WOB and RPM in the optimum zone plot, in accordance with an embodiment of the present disclosure.
• FIG. 7 shows the plotting of principal components on data set on the X-Y coordinate system.
• FIG. 8 shows the effect of dimension reduction using Principal Components Analysis
• FIGS. 9A and 9B show that principal components are actually the eigenvectors of the covalent matrix of the original data in the X-Y coordinate system.
• FIG. 9C shows a plot of WOB vs. RPM, as determined for real-time data in an experiment in accordance with the present disclosure, also shown is an optimum zone as determined for the real-time data.
• FIG. 9D shows a plot of WOB vs. RPM, for the data of FIG. 9C, after PCA of that data.
• FIG. 10 shows how the safety factors affect the optimum zone to form the safe zone in the optimum zone chart, in accordance with an embodiment of the present disclosure.
• FIG. 11 shows a centroid in the safety zone of FIG. 10, in accordance with the present disclosure.
• FIG. 12 shows a plot of bit depth, measured depth versus time for the portion of a well under study.
• FIG. 13 shows the first 3.5 minutes of depth versus time plot in stand one (shallow depth).
• FIG. 14 shows the first 3.5 minutes of depth versus time plot in stand two (intermediate depth).
• FIG. 15 shows the first 3.5 minutes of depth versus time plot in stand three (deep depth).
• FIG. 16 shows the Torque versus WOB plot for Stand Two Update One which helps to obtain the corresponding constants.
• FIG. 17 shows the Depth of Cut versus WOB plot for Stand Two Update One which helps to obtain the corresponding constants.
• FIG. 18 shows a combined plot of change in ROP divided by Change in Time versus Time and also ROP and WOB versus Time in order to get the minimum WOB for stand two update one.
• FIG. 19 shows the optimum zone plot for stand two update one.
DETAILED DESCRIPTION
• The present disclosure enables a driller, drilling an oil well, to assess, during drilling, the appropriateness of the drilling parameters being used and to correct these during drilling. The drilling parameters are monitored/measured during drilling and the values of those measured parameters are used to define an optimum drilling zone in the WOB-RPM space. The optimum zone is displayed to the user in addition to WOB-RPM data points. The displayed WOB-RPM data points are obtained by subjecting the measured parameter values to a principal component analysis in order to obtain only the most significant WOB-RPM data points, which are the ones displayed. The principle component analysis essentially filters out less important data, which in turn provides the driller better insight into the drilling process and the best drilling parameters to use.
Abbreviations
• Abbreviations used throughout the present disclosure include:
• • ANN Artificial Neural Network
  • BHA Bottom Hole Assembly
  • LWD Logging While Drilling
  • MSE Mechanical Specific Energy
  • MWD Measurement-While-Drilling
  • NPT Non-productive Time
  • PCA Principal Component Analysis
  • PDC Polycrystalline Diamond Compact
  • PWD Pressure-While-Drilling
  • ROP Rate of Penetration
  • RPM Revolutions per Minute
  • WOB Weight on Bit
  • TOR Torque
  • DOC Depth of Cut
  • QRA Quantitative Risk Analysis
The Concept of Machine Learning
• Machine learning gives computers the ability to optimize performance criterion based on sample data or past knowledge. The goal of machine learning is to identify and reveal hidden patterns linked with the data being analyzed. The world today is circled with applications of machine learning. A perfect example is the use of Google™ search which learns to display the best results. Another example is the anti-spam software which filters email messages.
• As shown in FIG. 1, there are two major types of machine learning. First is supervised (predictive) learning where for a given input variables (x) and output variables (Y), one can use an algorithm to learn the mapping function from the input to the output: Y=f(x). The goal is to approximate the mapping function so well that when there is a new input data (x), accurate predictions can be made to obtain the output variables (Y) for that data.
• Unsupervised (descriptive) learning is the second major type of machine learning. Unsupervised learning is where for a given input data (x) there are no corresponding output variables. The concept behind unsupervised learning is identify the underlying pattern in the data in order to learn more about the data.
How Machine Learning is Utilized for Vibration Problems
• WOB and RPM causing whirling and stick slip can be predetermined if the total drilling conditions are known (Wu et al 2010). A boundary condition for stable drilling can be obtained in a plot with WOB on the Y axis and RPM on the X axis, as shown in FIG. 2. This means if the driller maintains the drilling parameters such as to keep the bit in the optimum zone, then drilling will be stable depending on the bit and mechanical properties of the rock.
• The boundaries of the optimum zone help determine the best combination of WOB and RPM for optimum ROP. The hard question to answer is if the stick slip and whirling zone is predicted accurately.
• In order to identify the optimum zone effectively, an exemplary embodiment of a method, in accordance with the present disclosure, is shown in FIG. 3A. This method is adopted to ensure that all the monitored/measured drilling parameters have an impact on the optimum zone. The method represented at FIG. 3A uses available real-time data 100 obtained from a drilling rig 102. The exemplary method performs a variable transformation and reduction (e.g., at steps 104, 106, 108, 110, 112, 114), and then utilizes machine learning algorithms to identify the optimum drilling parameter zone and display it to the driller.
• FIG. 3B shows a flowchart of an embodiment of a method in accordance with the present disclosure. The method of FIG. 3B has drilling—into the Earth—being carried out, at action 300. As the drilling is carried out, Measured Depth, Drilling Time, Bit Depth, WOB, ROP, RPM and TOR are obtained (e.g., measured or determined), in real-time, at action 302. All these can be referred to as surface parameters in that they can be obtained as the drilling progresses, in real-time, without requiring physical access to the bottom hole assembly. In addition to Measured Depth, Drilling Time, Bit Depth, WOB, ROP, RPM and TOR, any other parameter that can be measured in real-time is to be considered within the scope of the present disclosure. For example, MSE can also be measured. At action 304, the real-time data is processed, in accordance with pre-determined rules, in order to obtain a classification scheme for the real-time data. The classification scheme defines an optimum drilling parameter zone. As will be described further below, the pre-determined rules produce upper and lower limits for the WOB and for the RPM. These rules are based accepted practices in the art of drilling.
• As will be understood by the skilled worker, the measured depth is the length of the path of the drill string, including the bends. The bit depth is the same as the measured depth during drilling. When drilling stops, the bit depth will be less when pulled up from the bottom of the well being drilled.
• At action 305, a principal component analysis (PCA) of the real-time data is performed to obtain a set of principle components associated to the real-time data. Subsequently, at action 307, a subset of the principal components is selected. For example, only the principal components that account for 99% (or any other suitable percentage) of the data points can be selected to be part of the subset. At action 309, using only the subset of principal components, an inverse PCA is performed to obtain a modified data, which no longer includes the original real-time data related to the principal components that were not identified as important (for example, the principal components that accounted for the remaining 1% of the data points).
• At action 311, the modified data is classified in accordance with the classification scheme obtained at 304, to obtain classified modified data, which is then compared, at action 313, to the optimum drilling parameter zone. This results in a comparison result on which an adjustment of the WOB and/or the RPM can be effected, at action 315. Visualization of the data points in the optimum zone chart will show the driller which zones have most of the data points. Regardless of whether there are data points in the optimum zone or not, the upper and the lower limits of RPM and WOB are the boundaries within which the driller can run the operations with.
• Subsequently, after waiting for a pre-determined amount of time at action 317 (for example, 3.5 minutes or any other suitable time duration), the method loops back to action 304 where the classification scheme is defined (re-defined) in accordance with real-time data acquired since the definition of the previous classification scheme. As will be understood by the skilled worker, this re-defines the optimum drilling parameter zone. In addition to looping back to action 304, the method also loops back to action 305 where a PCA is performed on in accordance with real-time data acquired since the previous PCA.
• As will be understood by the skilled worker, the aforementioned comparison can be automated through any conventional means. The automated process can include the step of identifying data points that have values comprised within the optimum zone, compare those points to the current WOB and RPM settings, and automatically adjust those settings so that they correspond to one of the data points identified as being within the optimum zone.
• In other embodiments, as will be detailed further below, a safe zone within the optimum zone can be determined by quantitative risk analysis (QRA) and the comparison action can entail comparing post-PCA data comprised within the safe zone with the current settings of WOB and RPM, and automatically adjust those settings so that they correspond to one of the data points identified as being within the optimum zone.
• In further embodiments, and as will be detailed further below, a centroid of the post-PCA data points that are within the safe zone, or within the optimum zone, can be calculated by, for example, a clustering operation, and the current settings of the WOB and RPM can be compared to the WOB and RPM values of the centroid. The drilling WOB and RPM settings can automatically be set to the WOB and RPM values of the centroid if they differ from those values.
• In instances where the process is not automated, the driller in charge of the drilling operation can be provided with a display showing a plot of the WOP versus RPM post-PCA data and the optimum zone (for an example of such a plot, see FIG. 9D further below) and, based on the displayed data, the driller can set the WOB and the RPM to any suitable value found in the optimum zone. Similarly, the driller can be provided with a display showing a plot of the WOP versus RPM post-PCA data and the safe zone and, based on the displayed data, the driller can set the WOB and the RPM to any suitable value found in the safe some. Further, the driller can be provided with a display showing the aforementioned centroid and, based on the WOB and RPM values of the centroid, the driller can set the drilling parameters to those values.
Classification Scheme
• The following relates to action 302 in FIG. 3B.
• Classification is a kind of arrangement where like data are classed together and separated from unlike data; the main reasons behind classification is to (a) put knowledge in shape and storage, (b) do structural analysis of the data being stored; and (c) figure out the relationship existing among different parts of the structure found (Mirkin 1996).
• A decision tree classification is used, as an example in the present disclosure. Decision trees are based on algorithms which split data into branches. Unlike a tree where the root is at the bottom, a decision tree has its root node at the apex of the tree (Ville et al 2013). The basis for building the decision tree is echoed in this root node: the name of the field of data and the arrangement of the values that are contained in that field.
• There are 3 types of nodes in a decision tree:
• • Decision nodes;
  • Chance nodes;
  • Leaf or terminal or end nodes (Bloomsbury Publishing 2013).
• In each internal node of the tree reflects certain characteristics of the system, and each leaf node represents a class label. There are 3 steps to contrasting the decision tree:
• • Step 1: At the root of the tree, place the most defining feature of the dataset
  • Step 2: The training set is then split into subsets with values corresponding to their respective attributes.
  • Step 3: Redo step 1 and step 2 on each subset till there are terminal nodes in all the branches of the tree.
• In the generic classification tree in FIG. 3C, there are four key values: the upper limit of WOB, the lower limit of WOB, the upper limit of RPM and the lower limit of RPM. These values represent the boundaries for stick slip, forward whirling, backward whirling and low ROP zones respectively. These values change for each stand on a 3.5 minutes basis.
Obtaining the Upper and Lower Limits of RPM and WOB Upper Limit of RPM
• Conventionally, the upper limit of RPM is calculated by first determining the mean RPM value and then increasing that value by 10% three times. See FIG. 4.
• Increasing the average RPM by 10% three times means
• 
  RPM_upper=(1.1)³(Mean RPM)=1.331(Mean RPM)
• After several iterations with field data, the need to further reduce this value arose, hence a new formula for the upper limit of RPM.
• 
  RPM_upper=1.331*mean(RPM)−((0.95*mean(RPM))/3))
Lower Limit of RPM
• The lower limit of RPM (RPM lower) can be obtained by first finding the minimum depth of cut, which can be obtained based on equation below, which was derived from the mechanical specific energy (MSE) equation introduced by Teale (Teale 1965).
• 
  B ₂*WOB⁴+2B ₁ B ₂*WOB³+(B ₁ ²+2B ₂ B ₀−2πA ₁ B ₂)*WOB²+(2B ₁ B ₀−4πA ₀ B ₂)*WOB+B ₀ ²+2A ₁ B ₀−2πA ₀ B ₁)=0
• Four values of WOB would be gotten from this quartic equation, only the positive value has physical meaning. The positive value of WOB can be plugged into the known equation for depth of cut to obtain the optimum depth of cut. The constants in the equation above can be calculated from their source equations below (Hamrick 2011).
• 
  Depth of Cut=DOC=g(WOB)=B ₂*WOB² +B ₁*WOB+B ₀
• 
  Torque=f(WOB)=A ₀ +A ₁*WOB
• By plotting a chart of incoming torque, depth of cut and WOB data, the constants A and B can be calculated. The minimum depth of cut would then be 50% of the optimum depth of cut. Just by unit conversion using ROP, the minimum RPM can be calculated.
• ${\left(\mathrm{DOC}\right)}_{\min }=\frac{{\left(\mathrm{DOC}\right)}_{\mathrm{opt}}}{2}$ ${\left(\mathrm{RPM}\right)}_{\min }=\frac{{\left(\mathrm{ROP}\right)}_{\mathrm{avg}}}{{\left(\mathrm{DOC}\right)}_{\min }}$
Upper Limit WOB
• The upper limit of WOB is determined based on stick slip index. It is expected that the optimum zone chart would be updated every 3.5 minutes or 210 seconds. The stick slip index would be calculated every 20 seconds. This makes 10 test of stick slip index within each update of the optimum zone.
• $\mathrm{Stick}\mathrm{Slip}\mathrm{Index}=\frac{\left({\mathrm{Torque}}_{\max }-{\mathrm{Torque}}_{\min }\right)}{{\mathrm{Torque}}_{\mathrm{avg}}}\%$
• Based on that calculation, the severity of the stick slip calculation can be estimated which is shown in the table 3 below:

• TABLE 1
  Vibration Severity Levels Based on Downhole
  Measurements (Al Dushaishi et al 2015)

  Stick-Slip

  Lateral Acc

  Lateral RMS Acc

  Severity

  (g's)	Severity Level	(g's)	Severity Level	(—)	Level
  0-15	Normal	0-2.5	Normal	0-0.5	Low
  15-35	Moderate			0.5-1	Moderate
  35+	Severe	2.5+	Severe	1+	Severe

• The upper limit of WOB can then be derived based on the following rules:
• • when one test has stick slip index greater than 0.5, make the upper limit of WOB equal to the minimum WOB of the test
  • when two or more tests have stick slip index greater than 0.5, make the upper limit of WOB equal to the least minimum WOB of all the tests with stick slip index greater than 0.5
  • when all the tests have stick slip index less than 0.5, make upper limit of WOB equal the maximum WOB of all the tests
Lower Limit WOB
• The lower limit of WOB can be based on the hardness of the formation being drilled. This is the WOB which corresponds to the time when the slope of the ROP versus time plot becomes constant. This is shown in FIG. 5.
Rules of the Classification Tree to Obtain the Optimum Zone
• The optimum zone, and the lower and upper limits for RPM and WOB are shown at FIG. 6. In this figure:
• • Zone 1 is the Stick Slip Zone
  • Zone 4 is the Low ROP Zone
  • Zone 5 is the Forward Whirling Zone
  • Zone 3 is the Backward Whirling Zone
  • Zone 2 is the Optimum Zone
  • WOB upper limit is based on stick slip index calculations
  • WOB lower limit is based on formation hardness (ROP change)
  • RPM lower Limit is based on minimum depth of cut calculations
  • RPM upper limit is still based on reversal of conventional operational processes leading to vibrations
• With this knowledge, a decision tree can be formed based on the fact that any data point above the stick slip line is in the stick slip zone and would most likely be experiencing stick slip, any data point behind the low ROP line is in the low ROP zone and would be experiencing less efficient drilling, any data point ahead of the backward whirling line would be in the backward whirling zone and would be experiencing backward whirling and finally any data point below the forward whirling line would be in the forward whirling zone and most likely be experiencing forward whirling. FIG. 3C, discussed above, is based on FIG. 6.
• At every 3.5 minutes or 3 feet interval (or any other suitable time interval or distance), the optimum zone cab updated by calculating, based on real-time data obtained at action 302, FIG. 3B, new lower and upper limits for WOB and RPM. All the data points will belong to one of the zones.
• As will be understood by the skilled worker, the real-time data could be classified and represented in the same plot as the optimum zone. However, representing all acquired data in in the same plot as the optimum zone would result in a very dense plot and provide little or no insight to the driller, when the real-time data is acquired at any reasonable rate (e.g., 100 data points per second). As such, the present disclosure uses a dimensionality reduction technique to obtain a modified data set that has considerably less data point.
• After dimensionality reduction, the driller can see how much of the data points are in stick slip or whirling. Based on the arrangement, the driller can either select the readings of the data points in the optimum zone or ask the system to generate a range of data points that are in the optimum zone. However, if there is a significant change in drilling parameters, the optimum zone will shift its location and new safe ranges would have to be generated. This will be discussed further below in relation to FIGS. 9C and 9D.
Principal Component Analysis (PCA)
• In an example provided in the present disclosure, PCA is used to form a lean data set that best represents the drilling process. A summary of PCA is provided below.
• PCA can be used for searching out veiled patterns in high dimension data (i.e., where the number of features exceed the number of observation). In this research, PCA is used for reducing the dimension of the input data without losing important information in the original data (Lindsay 2002). Three steps govern the PCA process.
• The first step is to determine the covariance of the matrix. Covariance is the measure how two different variables relate with each other during changes in values. The formula for covariance is an adjustment of the variance formula which only analysis the dataset in one variable.
• $\mathrm{Variance}={\mathrm{<figure-callout\; id="2"\; label="\sigma "\; filenames="US20190106978A1-20190411-D00008.png,US20190106978A1-20190411-D00009.png"\; state="\{\{state\}\}">\sigma </figure-callout>}}^2=\frac{\sum {\left(x-\mu \right)}^2}{N}$
• For the variable x, μ is the mean and N is quantity of data points in variable x. This formula is then modified the give the formula for covariance between two variables. Consider two variables x and y
• $\mathrm{Covariance}=\mathrm{cov}\left(x,y\right)=\frac{\sum _{i=1}^n\left(x_i-\stackrel{\_}{x}\right)\left(y_i-\stackrel{\_}{y}\right)}{n-1}$
• If multiple variables are involved, the covariance matrix will be symmetrical; meaning the transpose of the matrix will be the same as the original matrix. Assuming there are four variables, w, x, y and z. The covariance matrix will be as follows:
• $C=\left(\begin{array}{cccc}\mathrm{cov}\left(w,w\right)& \mathrm{cov}\left(w,x\right)& \mathrm{cov}\left(w,y\right)& \mathrm{cov}\left(w,z\right)\\ \mathrm{cov}\left(x,w\right)& \mathrm{cov}\left(x,x\right)& \mathrm{cov}\left(x,y\right)& \mathrm{cov}\left(x,z\right)\\ \mathrm{cov}\left(y,w\right)& \mathrm{cov}\left(y,x\right)& \mathrm{cov}\left(y,y\right)& \mathrm{cov}\left(y,z\right)\\ \mathrm{cov}\left(z,w\right)& \mathrm{cov}\left(z,x\right)& \mathrm{cov}\left(z,y\right)& \mathrm{cov}\left(z,z\right)\end{array}\right)$
• Note that the diagonal are the variances of each variable.
• Next would be to estimate the eigenvalues and eigenvectors of the covariance matrix. Let A be an n×n matrix. The number λ is an eigenvalue of A if there exist a non-zero vector v, such that Av=λv The eigen values of A are the roots of the characteristic polynomial
• $p\left(\lambda \right)=\det \left(A-\lambda I\right);\mathrm{where}I\mathrm{is}\mathrm{the}\mathrm{identity}\mathrm{matrix}.I=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\mathrm{or}I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
• For each eigenvalue λ, the corresponding eigenvectors are
• $v=\left[\begin{array}{c}{v}_1\\ v_2\\ :\\ .\\ v_n\end{array}\right]$
• obtained by solving the linear system (A−λI)v=0
• The principal components are the eigenvectors. The principal components are ranked according to their corresponding eigenvalues. If the characteristic polynomial of A has 4 as its highest power then there would be 4 eigenvalues. The highest eigenvalue would produce the first principal component; the second highest eigenvalue would produce the second principal component (eigenvector).
• In FIG. 7, the data is first plotted on X and Y coordinates. The principal direction is where the highest variance lies. In this case, the U direction is the principal direction with the highest importance. The V direction must be orthogonal to the U direction. It is expected that when X and Y coordinates are transformed into U and V coordinates, the covariance between X and Y variables becomes zero. U and V variables are called principal components (Gillies et al). In reality, they are the eigenvectors of the covariance matrix of the original dataset. The level of importance is based on the eigenvalues; the eigenvector with the highest eigenvalue is the most significant and is termed the first principal component. The eigenvector orthogonal to the first principal component with the next highest eigenvalue is the second principal component and so on (Gillies et al). The reduction aspect is done after the original dataset has been transformed to principal components. Before inverse PCA is done to get the original variables, some dimensions are zeroed out which have low eigenvalues. The resulting original dataset is leaner and very distinct on what values are to be used as shown in FIG. 8.
• Let's assume that the drilling parameters inputted into PCA are WOB, RPM, TOR, ROP or any other drilling parameter desired to have an impact on the optimum zone, for example, MSE. If we represent their values by x₁, x₂, . . . , x_k:
• From k original variables: x₁, x₂, . . . , x_k: PCA aims to produce k new variables: y₁, y₂, . . . , y_k: where
• $\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="y"\; filenames="US20190106978A1-20190411-D00008.png,US20190106978A1-20190411-D00010.png"\; state="\{\{state\}\}">y</figure-callout>}}_1=a_{11}x_1+a_{12}x_2+...+a_{1k}{\mathrm{<figure-callout\; id="2"\; label="x\; k\; y"\; filenames="US20190106978A1-20190411-D00008.png,US20190106978A1-20190411-D00009.png"\; state="\{\{state\}\}">x</figure-callout>}}_k& \\ y_{}{}_2=a_{21}x_1+a_{22}x_2+...+a_{2k}x_k\\ \cdots \\ y_k=a_{k1}x_1+a_{k2}x_2+...+a_{\mathrm{kk}}x_k\end{array}$
• yk's are uncorrelated (orthogonal)
  y₁ explains as much as possible of original variance in data set
  y₂ explains as much as possible of remaining variance
  {a₁₁, a₁₂, . . . , a_1k} is 1st Eigenvector, λ₁
  {a₂₁, a₂₂, . . . , a_2k} is 2nd Eigenvector, λ₂
• FIGS. 9A and 9B simply refreshes the understanding of how principal components relate to each other in PCA. λ₁ & λ₂ are the eigenvectors of the correlation/covariance matrix and λ₁ & λ₂ are the coefficients of the principal components. If y₁ and y₂ explains 99% of original data, {a₃₁, a₃₂, . . . , a_3k} up to {a_k1, a_k2, . . . , a_kk} are equated to zero. Therefore
• $\begin{array}{c}{y}_1=a_{11}x_1+a_{12}x_2+...+a_{1k}x_k\\ y_2=a_{21}x_1+a_{22}x_2+...+a_{2k}x_k\\ y_3=a_{31}x_1+a_{32}x_2+...+a_{3k}x_k\\ y_4=a_{41}x_1+a_{42}x_2+...+a_{4k}x_k\\ y_5=a_{51}x_1+a_{52}x_2+...+a_{5k}x_k\\ \cdots \\ y_k=a_{k1}x_1+a_{k2}x_2+...+a_{kk}x_k\end{array}$ $\mathrm{becomes}$ $\begin{array}{c}{y}_1=a_{11}x_1+a_{12}x_2+...+a_{1k}x_k\\ y_2=a_{21}x_1+a_{22}x_2+...+a_{2k}x_k\\ y_3=\left(0\right)x_1+\left(0\right)x_2+...+\left(0\right){\mathrm{<figure-callout\; id="4"\; label="x\; k\; y"\; filenames="US20190106978A1-20190411-D00008.png,US20190106978A1-20190411-D00009.png"\; state="\{\{state\}\}">x</figure-callout>}}_k& \\ y_{}{}_4=\left(0\right)x_1+\left(0\right)x_2+...+\left(0\right){\mathrm{<figure-callout\; id="5"\; label="x\; k\; y"\; filenames="US20190106978A1-20190411-D00006.png,US20190106978A1-20190411-D00008.png"\; state="\{\{state\}\}">x</figure-callout>}}_k& \\ y_{}{}_5=\left(0\right)x_1+\left(0\right)x_2+...+\left(0\right)x_k\\ \cdots \\ y_k=\left(0\right)x_1+\left(0\right)x_2+...+\left(0\right)x_k\end{array}$
• Based on the new values of y₃ . . . y_k, inverse PCA is performed to produce new set of x₁, x₂, . . . , x_k. At this point, the reduction has already happened.
• FIG. 9C shows real-time, WOB vs. RPM data points and the optimum zone (rectangle) determined in accordance with the real-time data. FIG. 9D shows, on an expanded scale, the PCA data calculated based on the real-time data of FIG. 9C, and the optimum zone. These FIGS. 9C and 9D) are the result of a field test conducted on a well in the continental Unites States. In FIG. 9C, there are data points in every zone even though more dominant in the stick slip and forward whirling zones. After PCA, FIG. 9D, there is a clear definition of where the data points lie. Most of the points are in the stick slip zone while the forward whirling zone has more data points than the optimum zone.
• Safe Zone within the Optimum Zone
• The concept of the safe zone is to account for the risk of having data points lie in the optimum zone when they should actually outside the optimum zone, in vibration prone zone. The following process takes note of this risk.
• For the stick slip zone, a safety factor is obtained and is subtracted from the upper limit of the WOB, while for the forward whirling zone, the corresponding safety factor is added to the lower limit of WOB. For the backward whirling zone, the corresponding the safety factor is subtracted from the upper limit of RPM. The safety factor can be obtained through quantitative risk analysis.
Quantitative Risk Analysis (QRA)
• QRA has been used widely in the construction industries and has also been used in casing design and well planning by the oil and gas industries. The QRA approach considers the uncertainty of each input variable and provides comprehensive statistical properties of WOB, RPM, ROP, MSE, TOR and other drilling parameters. The parameters needed for quantitatively calculating the risks are discussed generally below.
• A mean value, m, is the expected value or the weighted average of a number N of data points x.
• $m=\frac{\sum x}{N}$
• Standard deviation, s, is a measure of dispersion or variability. Standard deviation measures the closeness of each random variable to the mean value (Liang 2002). It is given as
• $s=\sqrt{\frac{\sum {\left(x_i-m\right)}^2}{N}}$
• Coefficient of Variance (COV) evaluates the distribution of the standard deviation over the mean value (Liang 2002) The data is more uncertain as the COV goes higher.
• $\mathrm{COV}=\frac{s}{m}$
• To calculate the risk of data points in the optimum zone fall into the vibration prone zones, there is a need to first determine the means and standard deviations of the stick slip zone (M_SS and S_SS), the backward whirling zone (M_BW and S_BW), the forward whirling zone (M_FW and S_FW) and the optimum zone (M_OP and S_OP).
• • For normally distributed stick slip and optimum zone data, the margin between the two probability density functions (PDFs) has a mean margin of
• 
  M _SO =M _SS −M _OP
• And standard deviation margin of
• 
  S _SO=√{square root over ((S _SS)²+(S _OP)²)}
• The risk of having optimum zone data points in stick slip
• $\mathrm{zone}=R_{\mathrm{SO}}=\left(\frac{M_{\mathrm{SO}}}{S_{\mathrm{SO}}}\right);$
• • In order to give the driller some more space to change parameters, 20% of the risk can be allowed
• Therefore,
• $R_{\mathrm{SO}}=80\%\left(\frac{M_{\mathrm{SO}}}{S_{\mathrm{SO}}}\right);$
• this is the safety factor for the stick slip zone.
• • For normally distributed optimum zone and forward whirling data, the margin between the two probability density functions (PDEs) has a mean margin of
• 
  M _OF =M _OP −M _FW
• And standard deviation margin of
• 
  S _OF=√{square root over ((S _OP)²+(S _FW)²)}
• The risk of having forward whirling zone data points in optimum
• $\mathrm{zone}=R_{\mathrm{OF}}=\left(\frac{M_{\mathrm{OF}}}{S_{\mathrm{OF}}}\right);$
• • In order to give the driller some more space to change parameters, can take 20% of the risk can be allowed
• Therefore,
• $R_{\mathrm{OF}}=80\%\left(\frac{M_{\mathrm{OF}}}{S_{\mathrm{OF}}}\right);$
• this is the safety factor for the forward whirling zone.
• • For normally distributed backward whirling and optimum zone data, the margin between the two probability density functions (PDEs) has a mean margin of
• 
  M _BO =M _BW −M _OP
• And standard deviation margin of
• 
  S _BO=√{square root over ((S _BW)²+(S _OP)²)}
• The risk of having optimum zone data points in backward whirling
• $\mathrm{zone}=R_{\mathrm{BO}}=\left(\frac{M_{\mathrm{BO}}}{S_{\mathrm{BO}}}\right);$
• • In order to give the driller some more space to change parameters, 20% of the risk can be allowed
• Therefore,
• $R_{\mathrm{BO}}=80\%\left(\frac{M_{\mathrm{BO}}}{S_{\mathrm{BO}}}\right);$
• this is the safety factor for the backward whirling zone.
• FIG. 10 shows a safety zone (safe zone) within the optimum zone of FIG. 6. The safety factor is calculated based on the real-time data, not on data obtained post PCA.
Clustering and Centroid of Optimum Zone
• Clustering is a process forming groups whose objects are somewhat similar. A cluster is grouping of objects which are alike and different from objects in other clusters. K-means clustering is a known type of clustering used, as an example, in the present disclosure. Widely used in data mining, K-means algorithm is a type of clustering analysis based on partitioning. The centre of each cluster represents the cluster as the algorithm ensures convergence towards stable centroids of clusters. The centroid is the centre or mean point, of the cluster. K is the number of clusters. After initialization, there are 3 steps in the K-means process.
• Initialization: set seed points (randomly)
• • Step 1: Each object (compressed data point) is placed in a cluster of the nearest seed point (centroid) measured with a specific distance metric (Euclidean distance)
  • Step 2: Estimate new centroid for each cluster in the current partitioning
  • Step 3: Repeat Step 1; continue iterating until there are no more changes in membership in each cluster.
• A centroid obtained from Kmeans Clustering (or any other suitable method) can be used to obtain the recommended WOB and RPM values of the safe zone which the driller can operate with when there are vibration issues. The centroid of the safe zone is shown in FIG. 11. The centroid in FIG. 11 is obtained by clustering the data points in the optimum zone. If the optimum zone has no data points, the centroid would be based on the polygon formed by the upper and lower limits of WOB and RPM. Referring now to FIG. 9D above, the centroid there was determined by clustering the post-PCA data points in the optimum zone.
Example
• In the following example, the data is drawn from a well in Western Canada. The results presented here are the outcome of each step in the machine learning process. The first set of results relate to PCA done on all the field data fed to the system. The principal components and their respective percentage of significance are derived. The principal components that make up at least 99% of the data were chosen while the other principal components are zeroed out before an inverse PCA is performed to obtain the leaner original data. Based on the decision tree classification, each data point is then classified into one of the five zones in the WOB and RPM plot. The quantitative risk analysis results are shown and then applied to the optimum zone chart to produce the safe zone plot.
• This analysis was done on the first 3.5 minutes of three stands of drill string (that is the first 3 updates of three stands). For this well, a depth of 3.5 feet is drilled in 3.5 minutes. For this post analysis, the entire data for the region for the selected stand would be analysed for vibration issues and classified into the five zones. The stand chosen is one with no obvious issues. The visible signs of problems with the data from a stand are inequalities between the bit depth and the measured depth. It is the bit depth that is very important; it tells that the drill string is moving into the formation and not just rotating at a spot. Any stand that has a constant depth for a while is an indication of stoppage in drilling or pause in drilling forward. FIG. 12 shows the plot of bit depth, measured depth versus time for the portion of the well being studied.
Results
• FIGS. 13 to 15 show the first 3.5 minutes of the three stands. Each 3.5 minutes of each stand is called the first update of that stand. Usually each stand would have an average of 5 updates. Results from Stand 2 Update 1 are the focus of this example.
RPM_upper Calculations
• The upper limit of RPM was calculated in accordance with the details provided further above.
• • For stand one, RPM_upper=58.4993 rpm
  • For stand two, RPM_upper=59.8457 rpm
  • For stand three, RPM_upper=30.4300 rpm
RPM_min (Rev/Min) Calculations
• In order to find the constants for the depth of cut and torques equations, graphs of torque versus WOB and depth of cut versus WOB were plotted and the constants were obtained for the first update from stand two.
• The value for the constants in the Torque equation are shown in the table 2 below are obtained from FIG. 16, the Torque versus WOB plot.

• TABLE 2
  Constants Obtained from the Torque Equation

  	Constants from
  	Torque =
  	f(WOB) = A0 +
  	A1 * WOB

  	Data Source	A0	A1
  	Stand Two Update One	3.7345	0.5447

• The value for the constants in the Depth of Cut equation are shown in the table 3 below are from FIG. 17, Depth of Cut versus WOB plot.

• TABLE 3
  Constants Obtined from the Depth of Cut Equation

  	Constants from
  	Depth of Cut = DOC =
  	g(WOB) = B2 * WOB2 +
  	B1 * WOB + B0

  	Data Source	B2	B1	B0
  	Stand Two Update One	0.0002	0.0049	0.0015

• The constants from the Torque and Depth of Cut equations are now substituted to find the WOB_opt, DOC_opt which will then be combined with the ROP_avg to find RPM_min. Four solutions will always be gotten from the WOB_opt equation, only the positive value has a physical meaning and only that value would be used in the DOC_opt equation.
• 
  B ₂*WOB_opt ⁴+2B ₁ B ₂*WOB_opt ³+(B ₁ ²+2B ₂ B ₀−2πA ₁ B ₂)*WOB_opt ²+(2B ₁ B ₀−4πA ₀ B ₂)*WOB_opt+(B ₀ ²+2A ₁ B ₀−2πA ₀ B ₁)=0
• 
  Depth of Cut=DOC_opt =g(WOB)=B ₂*WOB_opt ² +B ₁*WOB_opt +B ₀

• TABLE 4
  Calculations Breakdown for Obtaining Minimum RPM

  	Parameters	Stand Two Update One
  	B2 * WOBopt 4	0.0002WOBopt 4
  	2B1B2 *	0.00000196WOBopt 3
  	WOBopt 3
  	(B1 2 + 2B2B0 −	−0.000659969WOBopt 2
  	2πA1B2) *
  	WOBopt 2
  	(2B1B0 −	−0.0093723392WOBopt
  	4πA0B2) *
  	WOBopt
  	(B0 2 + 2A1B0 −	−0.1133548802
  	2πA0B1)
  	WOBopt	5.513049145597762
  	solution 1
  	WOBopt	−0.4938393454168616 +
  	solution 2	i * 4.735475187016882
  	WOBopt	−0.4938393454168616 −
  	solution 3	i * 4.735475187016882
  	WOBopt	−4.535170454764039
  	solution 4
  	Relevant	5.513049145597762
  	WOBopt
  	DOCopt	0.0345926827
  	DOCmin	0.0172963414
  	ROPavg	47.0425
  	RPMmin	2719.794834762
  	(rev/hr)
  	RPMmin	45.3299139127
  	(rev/min)

WOB_upper Calculations
• The stick slip index is used to find the upper limit of WOB. For stand two update one, there are ten test conducted and the results are as follows

• TABLE 5
  Results of Stick Slip Index Calculations

  	Test	Stick Slip Index

  	1	0.3467
  	2	0.1934
  	3	0.1506
  	4	0.1559
  	5	0.1236
  	6	0.1232
  	7	0.0936
  	8	0.7406
  	9	0.2577
  	10	0.2684

• Based on the rules mentioned further above, test 8 shows potentials for stick slip since the index is above 0.5. Therefore, the upper limit of WOB would be the minimum WOB in test 8. The minimum WOB in test 8 is 2.2 kDaN. Therefore WOB_upper=2.2 kDaN.
WOB_min Calculations
• WOB lower (WOB min) is achieved by taking the slope of ROP versus time every 5 seconds for the entire update leading to 43 runs of slope calculations. The change in ROP versus time plot is fairly constant after the point chosen as where constant change begins. Ideally, the change in ROP versus time should remain constant but in reality, the change keeps dropping. So the point chosen would be the highest change in ROP before a consistent drop in change in ROP. The closest highest peak after this peak can be referred to as the Founder Point (that topic is not the focal point of this disclosure). From FIG. 18, the WOB_min=1.8 kDaN
The Optimum Zone Chart
• A combination of the upper and lower limits for WOB and RPM form the box that makeup the optimum zone plot, FIG. 19. The lack of data points in the optimum zone for this particular update (stand two update one) is the reason why all the safe factors are zero for this case. In this Figure, the dotted lines are the data points. The RPM is constant based on feed data. The start and end is an indication of when ROP starts occurring so the reader can see what is happening in relation with the optimum zone till the ROP comes to the last data point at the end.
• In the preceding description, for purposes of explanation, numerous details are set forth in order to provide a thorough understanding of the embodiments. However, it will be apparent to one skilled in the art that these specific details are not required. In other instances, well-known electrical structures and circuits are shown in block diagram form in order not to obscure the understanding. For example, specific details are not provided as to whether the embodiments described herein are implemented as a software routine, hardware circuit, firmware, or a combination thereof.
• Embodiments of the disclosure can be represented as a computer program product stored in a machine-readable medium (also referred to as a computer-readable medium, a processor-readable medium, or a computer usable medium having a computer-readable program code embodied therein). The machine-readable medium can be any suitable tangible, non-transitory medium, including magnetic, optical, or electrical storage medium including a diskette, compact disk read only memory (CD-ROM), memory device (volatile or non-volatile), or similar storage mechanism. The machine-readable medium can contain various sets of instructions, code sequences, configuration information, or other data, which, when executed, cause a processor to perform steps in a method according to an embodiment of the disclosure. Those of ordinary skill in the art will appreciate that other instructions and operations necessary to implement the described implementations can also be stored on the machine-readable medium. The instructions stored on the machine-readable medium can be executed by a processor or other suitable processing device, and can interface with circuitry to perform the described tasks.
• The above-described embodiments are intended to be examples only. Alterations, modifications and variations can be effected to the particular embodiments by those of skill in the art. The scope of the claims should not be limited by the particular embodiments set forth herein, but should be construed in a manner consistent with the specification as a whole.
• As detailed above, the present disclosure enables a driller to assess, during drilling, the appropriateness of the drilling parameters being used and to correct these during drilling. The drilling parameters are monitored/measured during drilling and the values of those measured parameters are used to define an optimum drilling zone in the WOB-RPM space. The optimum zone is displayed to the user in addition to WOB-RPM data points. The displayed WOB-RPM data points are obtained by subjecting the measured parameter values to a principal component analysis in order to obtain only the most significant WOB-RPM data points, which are the ones displayed. The principle component analysis essentially filters out less important data, which in turn provides the driller better insight into the drilling process and the best drilling parameters to use. In some embodiments, the method described can be automated.
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Claims (12)

1. A method for producing an oil well, the method comprising:

a. drilling into the Earth, the drilling being effected by a drill string, the drill string having a drill bit;

b. obtaining real-time data from the drill string, the real-time data comprising, measured depth, drilling time, drill bit depth, weight on drill bit (WOB) data, revolution per minute (RPM) data, torque (TOR) data and rate of penetration (ROP) data;

c. in accordance with the real-time data and in accordance with pre-determined rules, obtaining a drill string data classification scheme, which defines an optimum drilling parameter zone;

d. performing a principal component analysis (PCA) of the real-time data, to obtain a set of principle components associated to the real-time data;

e. selecting a subset of the set of principle components;

f. in accordance with the subset of principles components, performing an inverse of the PCA, to obtain modified data;

g. classifying the modified data in accordance with the drill string data classification scheme, to obtain classified modified data;

h. comparing the classified modified data to the optimum drilling parameter zone, to obtain a comparison result; and

i. adjusting at least one of the WOB and the RPM in accordance with the comparison result.

2. The method of claim 1 further comprising:

displaying the classified modified data and the optimum drilling parameter zone.

3. The method of claim 1 further comprising:

performing a quantitative risk analysis (QRA) of the real-time data to in accordance with the real-time data, to obtain QRA results; and

reducing a size of the optimum drilling parameter zone in accordance with the QRA results, to obtain a safe drilling parameter zone,

wherein comparing the modified data to the optimum drilling parameter zone consists in comparing the modified data to the safe drilling parameter zone.

4. The method of claim 3 further comprising:

determining a centroid of the safe drilling parameter zone, wherein comparing the modified data to the optimum drilling parameter zone consists in comparing the modified data to WOB and RPM values of the centroid.

5. The method of claim 1, wherein the pre-determined rules include rules for determining a lower WOB limit, an upper WOB limit, a lower RPM limit and an upper RPM limit.

6. The method of claim 5, wherein the rule for determining the upper RPM limit includes:

in accordance with the real-time data:

calculating a mean RPM; and

increasing the average RPM by 10% three, three times.

7. The method of claim 6, wherein the rule for determining the upper RPM limit further includes:

reducing the value obtained by increasing the average RPM by 10% three by 0.95*mean(RPM))/3.

8. The method of claim 4, wherein the rule for determining the lower RPM is based on a determination of a mechanical specific energy.

9. The method of claim 4, wherein the rule for determining the lower WOB is based on a hardness of a formation being drilled.

10. The method of claim 4, wherein the rule for determining the upper WOB is based on a determined stick slip index.

11. The method of claim 1, wherein comparing and adjusting are automated actions.

12. The method of claim 1, further comprising:

periodically repeating the actions b through i, as the drilling progresses.

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