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principal subspace (PS) or signal subspace, while the subspace covered by MC is called minor
subspace (MC) or noise subspace. The PCs and MCs result from converging matrix differential
equations derived from the symmetric matrix’s principal and minor component analyzers (Kong
et al. 2017). Since the PC is the direction correlated with the eigenvector affiliated with the largest
eigenvalue of the autocorrelation matrix of the data vectors and the MC is the direction correlated
with the smallest eigenvalue of the autocorrelation matrix of the data vectors, batch eigenvalue
decomposition (ED) of the sample correlation matrix or singular value decomposition (SVD) of the
data matrix can be used for PCA and MCA.
The concept of PCA can be described with math equations. The goal of PCA is to reduce the
dimensionality of the matrix X(cid:15)RN∗D to matrix ZN∗M by projecting X into lower dimension space
M where D>>M.
X = [x x x ...x ]T X(cid:15)RN∗D (3.7)
1 2 3 N
Z = [z z z ...z ]T Z(cid:15)RN∗M (3.8)
1 2 3 N
By using transformation matrix U, the dimensionality of matrix X should be reduced from D to
M.
Z = U ∗X U(cid:15)RD∗M where; (3.9)
1
The covariance matrix of S = S = ZT ∗Z S (cid:15)RM∗M (3.10)
Z Z
N
Optimization should be done to maximize the covariance matrix S, and an upper boundary condi-
tion should be added to avoid an infinite number of results.
max S (3.11)
u Z
1
max ZT ∗Z where;Z = UX (3.12)
u
N
1
max (XU)T(XU) (3.13)
u
N
1
max UTXTXU where;XTX = S (3.14)
u x
N
1
max UTS U (3.15)
u x
N
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N
1 (cid:88)
Cov(X) = XXT (3.35)
N
n=1
TheequationaboveshowsthatXT Xisthescaledversionofthecovariancematrixofdesignmatrix
X. The matrix V contains the eigenvectors of the design matrix, and the matrix D contains the
eigenvalues of the design matrix. By applying the same process as eigenvalue decomposition, the
information represented by each eigenvalue can be calculated.
3.3.2 The Results of PCA
The fundamental concept of principal component analysis is explained with mathematical ex-
pressionsintheprevioussection. Asmentionedbefore,themainobjectiveofPCAistomeasurethe
variance within the data set and provide insight into how the variance contribution changes with
each principal component. The measurement of variance contribution by each principal component
is called explained variance analysis. Also, PCA can be used to indicate the variance contribution
of each variable by simply measuring the number of data points transferred to each principal com-
ponent. This process is called Feature Importance Analysis, and it can be used to indicate the
influence of each variable on principal components. In this study, both explained variance analysis
and feature importance analysis will be applied to the complete data set to indicate explained
variance by each PC and how these PCs are influenced by each variable in the data set. Implemen-
tation of PCA helped to gain insight into how the variance within the data set changes with each
variable, and the insight eventually led to changes in this study. The PCA will be applied only to
the training data set as the objective is to study the variance contribution of each variable. These
variables are MSE, ROP, Torque, N-FPI, WOB, RPM, and Depth.
3.3.2.1 Explained Variance Analysis
Explained variance can be defined as the variance retained by each eigenvalue and eigenvector
described by the covariance matrix of principal components. . These retained values by each
principal component is calculated by using scikit-library (Pedregosa et al. 2011). The explained
variance of each component is indicated in Figure 3.5. As can be observed in Figure 3.5, PC1
reflects the highest variance within the dataset as the eigenvalues and eigenvectors are sorted out
after applying eigenvalue decomposition. The observation from this graph is the uniform decrease
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3.3.2.2 Feature Importance Analysis
A feature matrix indicates the importance of each feature that is reflected by the magnitude of
corresponding values in eigenvectors. The numbers are shown in the feature matrix range from -1
to1, indicatingtheinfluenceoffeaturesoneachprincipalcomponent. Thehighernumberindicates
a more significant influence. The feature matrix of the training data set is presented in Figure 3.7.
Thecoefficientvaluesreflecttheinfluenceofeachvariableonprincipalcomponents. Thecalculated
coefficient value can be negative or positive since these values indicate a distance, aka magnitude,
from orthogonal lines set for each principal component. The observation from the feature matrix
is a significant influence of variables on PC7. Commonly, the degree of influence decreases with
each principal component added, but similar to the anomaly observed from explained variance
percentages, the influence of variables on principal components is not decreasing. Especially, the
influence of MSE and Torque are high on PC7, which indicates that valuable information from the
original data set will be lost if less than seven PCs are used to train the algorithm.
The explained variance and feature importance analysis indicated that a possible implementa-
tionofPCAtoreducedimensionalitywillresultinthelossofvaluableinformation. Thisobservation
is simple yet essential as it impacted this study and led to changes. A detailed discussion on the
possible loss of information is given in Chapter 4.
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CHAPTER 4
MODELING AND TESTING
Once negative values are removed and the data set re-indexed successfully, the drilling param-
eters are fed into the Random Forest regression algorithm as a training data set. The objective is
to develop a regression model that estimates unconfined compressive strength from drilling param-
eters. This chapter covers machine learning methods and includes model performance evaluation
methods while building a regression model. Also, the final architecture of the Random Forest
regression algorithm is explained.
4.1 Machine Learning Methods
Machinelearningisstatedasthescienceofprogrammingcomputerssotheycanlearnfromdata
(Geron 2019). Machine learning can be a valuable asset for solving extremely complicated prob-
lems that can not be explained using traditional physics-based models. Due to drilling operations’
complex and dynamic nature, machine learning methods to predict geomechanical and operational
parameters perform better than traditional models that commonly consider several assumptions.
The machine learning methods can be grouped into two main categories, supervised learning and
unsupervised learning. The main distinction between supervised and unsupervised learning meth-
ods is the data set used to train models. Supervised learning methods use labeled data sets for
classification and regression problems, whereas unsupervised learning methods use unlabeled data
sets for clustering, association, and dimensionality reduction problems.
This study uses a supervised learning method called Random Forest regression algorithm to
estimate UCS instantaneously from drilling parameters. The algorithm built includes two main
sections. The first section consists of a main Random Forest regression algorithm and hyper-
parameter tuning tools for RF model to ensure the best fit while avoiding overfitting. These
hyper-parameter tuning tools are called Randomized-Search and Grid-Search, which have a built-
in K-fold cross-validation function. The K-fold cross-validation function is essential while tuning
hyper-parameters to ensure the best bias-variance trade-off. The second section includes common
model performance evaluation techniques such as root-mean-squared-error, and mean-absolute-
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error. In this chapter, the most common regression models and performance evaluation techniques
are evaluated, the algorithm’s final architecture is provided, and the potential implementation of
the model for drilling operations is discussed.
4.1.1 Regression Models
Regression problems can be defined as problems that require a quantitative approach to solve.
A quantitative relation between one or more dependent or independent features can be measured
using regression models. Regression models can be used to solve the simple or complex relations
betweenoneormultiplefeatures. Theregressionmodelcomplexitywillincreasewiththeincreasing
complexity of relations between the features and target values. This relation can be as simple as
linear,whichcanbesolvedwithafirst-degreepolynomialequation,orincrediblycomplex,requiring
more sophisticated methods such as decision trees. More information regarding these sophisticated
methods is given as four different regression models are evaluated in this section.
4.1.1.1 Linear Regression
Linearregressionisthemostcommonandsimplestregressionmethodthatcanbedefinedasthe
simplest supervised machine learning algorithm for regression problems. The lack of complexity of
linear regression can be an advantage as long as the relation between dependent and independent
features can be explained with a first-degree polynomial equation. Linear regression is still robust
and practical approach to building simple quantitative models. The main linear regression types
can be divided into simple and multiple linear regression. Simple linear regression is a powerful
regression method when a parameter is estimated using only one predictor or feature, aka a single
independent variable. Even though simple and multiple linear regression methods follow similar
techniques to estimate features, a multiple linear regression provides the complexity needed to
estimate more than one feature. The multiple linear regression model is required to estimate UCS
from drilling parameters as more than one feature is present in the data set. A mathematical
description of the multiple regression method is given below in Equation 4.1.
Y = β +(β X )+(β X )+β (X ∗X )+β (X ∗X ∗X )+...+(cid:15) (4.1)
0 1 1 2 2 3 1 2 4 1 2 3
where:
Y: Output Parameter or Predictor
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X : Input Parameters or Features
i
β : Regression Coefficients
i
(cid:15): Error
Here m can be defined as average impact of X or X *X on Y.
i i i i+1
The goal of implementing a multiple regression model is to estimate the regression coefficients
(β )foreachinputparameterorfeaturewhilesatisfyingtherequirementofhavingtheminimumsum
i
of square residuals. This simple approach performs well while building a robust multiple regression
model, butitisessentialtomeasurethefeatureimportancetorecognizearelationbetweenfeatures
and predictors. By recognizing the relation between features and predictor, features with a small
impact on predictor can be eliminated. The most common method to estimate feature importance
is the null hypothesis. The null hypothesis is essentially replacing each regression coefficient with
integer zero to identify the importance of each feature. Mathematically, the null hypothesis for a
model with n features can be described as (James et al. 2013)
H = β = β = β = ... = β = 0 OR H = β (cid:54)= 0 (4.2)
o 0 1 2 n a i
The hypothesis is tested by performing the F-statistic. The F-statistic is defined as (James
et al. 2013)
(TSS −RSS)/p
F = where;
RSS/(n−p−1)
TSS = (cid:88) (y −y)2 and RSS = (cid:88) (y −y )2 (4.3)
i i (cid:98)i
In equation 4.3, y refers to the predicted Y on the ith value of X, y refers to the mean of y
(cid:98)i i
values,y referstotheactualith valueofy,n,andpreferstostatisticalfactorsdescribingF-statistics
i
distribution. TSS is the measurement of total variance in response to Y, and RSS is the amount of
variation left unexplained after applying regression (James et al. 2013). Here, p value indicates the
importance of the feature. If p-value is low after implementing the null hypothesis for a feature, it
will indicate a significant impact on output, whereas the feature will have a small or no indication
if p-value is high. A process called backward elimination can be applied to implement a null
hypothesis to each feature. The process of backward elimination is described below in Figure 4.1
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Figure 4.1: The backward elimination flowchart
Even though the multiple linear regression method is robust and can be used for complex
regression problems, the multiple linear regression approach is not complex enough to cover the
drilling data patterns due to the dynamic and complex nature of drilling.
4.1.1.2 Polynomial Regression
Polynomialregressionisdefinedasadaptingalinearregressionmodeltofitanon-linearrelation
using power features. The polynomial regression can be described as:
y = β +β ∗x +β ∗(x )2+β ∗(x )3+(cid:15) (4.4)
i 0 1 i 2 i 3 i i
Polynomial regression can be a robust method to fit a non-linear model. The process to fit a
non-linear model is the same as a linear regression model, but it should be noted that polynomial
regression models tend to overfit, which is essentially memorizing patterns in training data set and
replicating the same results. The overfitting causes the model to be non-generic and results in a
model that can only work with the same training data set. This can be avoided by implementing
cross-validation methods. The cross-validation methods will be explained later in this chapter.
Overfitting can be observed clearly for 300th degree polynomial regression case on Figure 4.2. A
linear regression, 2nd degree polynomial regression, and 300th degree polynomial regression models
fitthesametrainingdatasettoobservepossibleoverfitting. For300th degreepolynomialregression
case, a model recognizes even insignificant patterns in the data set, which results in the model that
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will perform poorly if another data set is used to estimate target values.
Figure 4.2: Linear Regression, 2nd, 300th degree Polynomial Regression models on the same train-
ing dataset (Geron 2019) ©2019 Kiwisoft S.A.S. Published by O’Reilly Media, Inc. Used with
permission
4.1.1.3 Support Vector Regression
Support Vector Machines (SVM) is a type of supervised ML algorithm that sorts the data
into two categories based on classification differences, and it uses these two categories to identify
difference within the dataset. Even though SVM commonly used for classification problems such
as image classification, recognition of handwritten characters, it can be a robust and efficient
algorithm for regression problems. Support Vector Regression (SVR) is an extension SVM that is
used to bring solution to regression problems. Similar to SVM, SVR creates (n-1) number of linear
or non-linear hyper-planes based on predefined function or kernel with bounds (cid:15) distance away
from these hyper-planes. SVR estimates target values based on the data points laying outside of
margins set at a distance (cid:15) from hyper-plane, and feeding the model with more data points to the
margin placed within the boundaries don’t necessarily impact the estimations (Geron 2019). The
mathematical approach to SVR formulation can be best described with geometrical perspective
by approximating continuous-valued function for one-dimensional data (Awad and Khanna 2015).
The example model of one-dimensional linear SVR is given in Figure 4.3. As it can be observed in
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Figure 4.3, possible support vector are determined by data points outside of margin determined by
(cid:15) distance from hyper-plane.
y = f(x) =< w,x > +b =(cid:80) M(w x +b) where; y,b ∈ R and x,w ∈ RM (4.5)
j=1 j j
Figure 4.3: One-dimensional linear SVR example (Awad and Khanna 2015)
4.1.1.4 Random Forest Regression
Random forest is a tree-based model that can be a useful method to build efficient regression
models. Decisiontreeandrandomforestmodelsfollowthesamesplittingrulesbasedoninformation
gained by splitting the training data set. Detailed information about decision tree methods and
random forest are provided in Section 2.3 and 2.3.1, respectively. Decision tree models can be a
robust solution to complex regression problems. The regression tree models split the training data
set into branches (leafs) until the limit data set allows. While training decision tree regression
models, data will be split into small batches until reaching the terminal node. The predicted value
is estimated based on the mean of the terminal node that the input vector activates (Joshi 2021).
Figure 4.4 clearly shows how the data split among leaves. The estimation process of ROP based
on WOB and RPM can be observed in the decision tree below. In Figure 4.4, the relation between
ROP, WOB, and RPM as data sets is split into small batches by applying various split rules (Joshi
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2021). By using the decision tree below, ROP = 160 ft/hour can be predicted for WOB = 5 klb
and RPM = 150.
Figure 4.4: (Left) The relation between WOB,ROP, and RPM based on split rules, (Right) The
decision tree to predict ROP. Modified from Joshi 2021.
Eventhoughdecisiontreeregressionmodelsarepowerfultoolstobuildregressionmodelsaround
a vast amount of data, there are several issues regarding their low generalization and high variance.
In other words, small changes in the data set can drastically impact predictions. A group of trees
can be built to reduce this impact, and collective decisions made on each tree can be combined
to provide a final prediction. This method is called bagging or bootstrapping, and it is explained
in detail in Section 2.3. Bagging or bootstrapping is using the same batch of data set to train
different predictors to build a group of weak predictors instead of a single strong predictor. The
bagging can significantly reduce the variance in predictions as each prediction delivered from a
group of predictors is taken into account and averaged to estimate the final output. Random forest
implements bagging as part of its function and splits each predictor’s data set based on randomly
sampled input features.
4.2 Possible Problems and Solutions
Thissection presentspossibleproblemsregarding theimplementationofmachinelearningmod-
elsforregressionproblems. Generally, complexregressionproblemscanbesolvedbyusingmachine
learning models, but these models require hyper-parameter tuning to prevent common problems
such as low accuracy, underfitting, overfitting due to bias, and variance errors. The difference
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between predicted and actual output value is called error, and any supervised machine learning
algorithm is compromised to noise, bias, and variance error. Even though the noise error, aka
irreducible error, can not be eliminated, bias and variance error can be minimized to a certain
degree.
4.2.1 Bias and Variance
Bias is defined as a generalization error caused by incorrect assumptions, such as modeling 4th
polynomial regression as linear regression (Geron 2019). Also, bias can be defined as the difference
betweenpredictedandactualvalues. Themodeldoesnotlearnenoughtomakeaccuratepredictions
if it has a high bias error, and eventually, it will cause an underfitting of the training data.
Contrary to bias, Variance error is caused by learning from training data very well. Variance
error is defined as excess sensitivity of the model to small patterns in the training data set (Geron
2019). The high variance error is generally caused by giving a high degree of freedom to a model
since a model with the freedom to choose a degree of fit has a tendency to fit a higher polynomial
regression. The high variance error eventually causes overfitting of the training data.
4.2.2 Underfitting and Overfitting
As mentioned in the previous section, underfitting and overfitting are the results of high bias
and high variance.
Overfitting is a phenomenon of well-performing model on the same training data set, but the
modelcan’tgeneralizeorperformspoorlyondifferentdatasets(Geron2019). Areal-lifeexampleof
overfitting can be generalizing an attitude of a bad taxi driver and assuming that every taxi driver
will act in the same manner (Geron 2019). Overfitting is caused by high variance error, and it can
be described as the significant impact of insignificant variations or patterns in the training data
set on the overall learning of the model. An example of overfitting can be observed in Figure 4.5.
The model assumed that the training data is more complex due to small variations in the data
set, which led to building a high-degree polynomial regression model instead of a simple linear
regression model. Some ways to avoid overfitting are re-processing the training data to fix data
errors and remove outliers, using a bigger training data set, simplifying the model by dropping the
features that have small or no impact on the learning process (Geron 2019).
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Figure 4.5: An Example of Overfitting the training data (Geron 2019)©2019 Kiwisoft S.A.S.
Published by O’Reilly Media, Inc. Used with permission
As mentioned, underfitting is the opposite of overfitting, and it is caused by high bias error.
Underfitting can be defined as the model’s lack of complexity due to incorrect assumptions (Geron
2019). Commonly, linear regression models tend to underfit the training data. An example of
underfitting can be observed in Figure 4.2. The linear regression shown in the red line in Figure 4.2
indicates that the simple regression model does not cover the variations in the training data enough
to make accurate predictions. Some of the ways to avoid underfitting are choosing a more complex
androbustmodelwithmorefeatures,reducingthemodel’slimitationsbytuninghyper-parameters,
and using better features to train the model (Geron 2019).
4.2.3 Bias and Variance Trade-off
As explained in the previous sections above, bias and variance introduce opposite types of
error to a regression model. The high variance causes overfitting, whereas the high bias causes
underfitting. A trade-off is required to develop the best approach to limit both bias and variance
as an increase in variance generates a more complex model, leading to high bias. The relation
between bias and variance is visualized and provided in Figure 4.6. There are several methods to
achieve optimum model complexity. These methods are cross-validation, bagging, removing outlier
data, and hyper-parameter tuning. While building Random Forest regression model for this study,
bagging, cross-validation, and hyper-parameter tuning has been utilized to optimize the model. As
mentioned in Section 4.1.1.4, bagging is a built-in function of the Random Forest regression model,
and it reduces variance error, which is a common problem of decision-tree regression models. Also,
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accuracy of predictions. While applying K-fold Cross-validation, the training data set is sub-
sampled to have k mutually exclusive data batches, and the model is tested with one of the k
data batches on each iteration, whereas remaining (k-1) batches are used to fit a model. K-fold
Cross-validationimplementsaniterativeapproachtotesttheaccuracyofpredictions, soeverydata
batch is used as both train and test subset. The implementation of K-fold cross-validation is an
example presented in Figure 4.7 to clarify the concept of cross-validation.
Figure 4.7: K-fold Cross Validation Example
Both Randomized-Search and Grid-Search are optimization tools developed as a built-in func-
tion in scikit-learn library (Pedregosa et al. 2011). Both methods use an iterative approach to find
best-fitting parameters as they continuously fit the model based on a preset list of parameters. The
advantage of using these methods is implementing an iterative approach to try out combinations
of the listed parameters to achieve optimum accuracy in predictions. The fundamental difference
between Randomized-Search and Grid-Search is how the combinations of parameters are selected.
While using Randomized-Search, the parameters are selected randomly from a range of numbers
to fit that individual instance, whereas these parameters are selected from a list of numbers preset
by the author while using Grid-Search. These methods are powerful tools to improve the model’s
generalization as K-fold Cross-Validation is used for each fit. The implementation of these methods
is similar and is presented in Figure 4.8 as a flowchart.
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Figure 4.8: RandomizedSearchCV-GridSearchCV Steps of Implementation
4.4 Evaluating Performance of the Models
The performance parameters of a regression model can be calculated by using several methods.
The process of validating a model is simply estimating the performance of the trained model based
on the error margin between actual and predicted values. There are several statistical approaches
to measure the accuracy of a model. For regression models, the most common methods to evaluate
the accuracy are Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE) (Geron
2019). This section evaluated three main methods, including RMSE and MAE (Swalin 2018).
Root Mean Squared Error (RMSE): Root mean squared error is the most common method
to estimate the accuracy of a regression model. For a regression model, the equation below is used
to calculate RMSE. In Equation 4.6, y refers to the actual value at ith observation, y refers to
i l
the predicted value at ith observation, where n refers to the number of predicted values for each
observation. The values of RMSE will be small, if the predicted values are close to the actual
values.
(cid:118)
(cid:117) n
(cid:117)1 (cid:88)
RMSE = (cid:116) (y i−y (cid:98)l)2 (4.6)
n
i=1
Mean Absolute Error (MAE): Mean absolute error is fairly similar to RMSE, and it can be
calculated using Equation 4.7. The fundamental difference between RMSE and MAE is that MAE
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treats errors equally, whereas RMSE penalizes significant errors by square.
n
1 (cid:88)
MAE = |y −y | (4.7)
i (cid:98)l
n
i=1
R2 and Adjusted R2: R2 can be defined as estimating variations within the predictions based on
features fed into the model. The range of R2 is between 0 and 1. The higher R2 value represents
accurate predictions and less variance within the prediction compared to actual variance within the
training dataset. The R2 can be calculated by using Equation 4.8. On the other hand, adjusted
R2 considers the number of features fed to the model by considering how useful the feature is. If
a feature with a significant impact is fed to the model, adjusted R2 will increase, whereas it will
decrease if a feature with no impact is fed to the model (Swalin 2018).
(cid:80)n (y −y )2
R2 = 1− i=1 i (cid:98)l (4.8)
(cid:80)n (y −y)2
i=1 i
The last step of evaluating these models was choosing the most efficient method. R2 and Adjusted
R2 were powerful methods to estimate the accuracy, but Ford (2015) argues against using this
method based on the initial arguments proposed by Shalizi (2015). Ford (2015) argues that R2
can estimate the accuracy of the model’s accuracy as close to 1 even though the fitted model is
completely wrong if the variance in the training data set is high.
Considering a high variance and noise in the drilling data, R2 is considered the least preferred
method to measure the model’s accuracy. RMSE and MAE were suitable options to measure the
accuracyofUCSestimations. Hence,bothRMSEandMAEareusedastheperformanceevaluation
method in this study.
4.5 Final Architecture of the Model
As mentioned before, the final architecture of the regression algorithm trained to estimate UCS
from drilling parameters is explained in this section. The process of training the model is described
anprovidedinFigure4.9. TheoverallobjectiveofthealgorithmistotrainarobustRandomForest
regression model while optimizing the accuracy of estimation without overfitting the drilling data.
The training process is completed in three main steps, and at each step, a different function is
used. These steps prepare the training data set, optimize the model, and evaluate the final model’s
performance.
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Figure 4.9: The Final Architecture of prediction model.
• Preparation of the training dataset: The overall goal of this step is to implement Prin-
cipal Component Analysis to prepare the training data set, gain insights into the correlation
between features, and estimate explained variance in the data set by each principal compo-
nent. Also, the data set is split into three subsets. The first set was testing, aka blind test
data set. The blind test data set was 10% of the original data set, and it was never shown
to the model until the final model was fit. The main reason to split the testing data set was
to ensure that the model never learned from this sub-set of original data. Hence, the testing
data set can be used to check if the final model is overfitted. The remaining data set is split
into two subsets, training and validation data. The training and validation subsets consisted
of 80% and 20% of the remaining data set.
• Optimization of the Model: In this step, the best parameters are searched by using
Randomized-Search and Grid-Search. These methods fit the model numerous times continu-
ously by using a preset list of parameters. Each fit uses the K-fold cross-validation method to
avoid overfitting. At the end of each iterative method, a combination of the best parameters
is chosen. The model is fit one more time based on these best parameters, and the accuracy
of predictions is evaluated before initiating the next step.
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• Training and Evaluating accuracy of the model: This is the final step of the model.
After implementing Randomized-Search and Grid-Search, the best parameters for the final
model are selected. Based on these parameters, the final model is fit, and the accuracy of
predictions is evaluated. Also, the testing subset split from the original data set is used to
evaluate the performance of the final model. This testing subset is called a blind testing data
set, and it is an essential step that helps identify overfitted training data.
4.6 Discussion on Challenges and Changes
The path of this study changed with time due to the particular problem of high accuracy in
predictions of Random Forest regression model trained by the principal components of drilling
parameters. The accuracy of UCS predictions was as high as 97% on the blind testing data set,
which indicated that the model is overfitted. These results were highly unexpected as bagging and
bootstrapping were used while training each decision tree of Random Forest regression algorithm
to reduce variance error. In addition to this, K-fold cross-validation was used on each model fit
while tuning hyper-parameters to reduce both bias and variance error. Even though the initial
exploratory data analysis on drilling data indicated a lack of variability in UCS as only seven
different UCS values were present, however, this issue was never assumed to cause an overfitting
problem while building a regression model. After realizing the overfitting problem, the model was
fitted numerous times with a different list of parameters fed into hyper-parameter tuning tools
Random-Search and Grid-Search, but the model was performing poorly, and the generalization of
the model was low. To achieve the study’s overall objective, the following hypotheses are tested by
conducting comprehensive research to indicate the root cause of the overfitting problem.
• A possible linear correlation between MSE and UCS can lead to perfect predictions.
• A loss of valuable information due to the implementation of PCA can cause overfitting.
• A lack of variability on UCS can reduce the variability of predictions, which can cause over-
fitting.
Pearson correlation coefficient between MSE and UCS is calculated to test the first hypothe-
sis. Pearson correlation coefficient is defined as a measurement of linear correlation between two
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variables. The coefficient value ranges from -1 to 1, and perfect positive correlation is indicated
by 1, while perfect negative correlation is indicated by -1. Pearson coefficient between MSE and
UCS is calculated as -0.189, which indicates no correlation. The first hypothesis is tested and elim-
inated. The second hypothesis was tested by going through the implementation of PCA on each
data subset. These data subsets are data files in feather format. Initially, the data set collected by
Joshi (2021) was provided in four different feather files. These data files were initially divided as it
was computationally expensive to load the complete data set. The initial feature importance and
explained variance analyses are conducted on these four subsets separately. The explained variance
retained by the first four principal components was over the threshold value of 85% for all subsets,
as observed in Figure 4.10. This value was set as a target explained variance retained for this study
to use principal components of drilling parameters as a training data set.
Figure 4.10: The Explained Variance Retained by PCs for Datasets 1 to 4.
After analyzing retained explained variance percentages, feature importance analysis is con-
ducted to indicate the influence of variables on each principal component. The feature matrix for
each subset is visualized to conduct a feature importance analysis. As a result of observations from
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Figure 4.11 and Figure 4.12, the feature importance analysis indicated that there was small or no
influence of variables on principal components 5, 6, and 7. The initial observations from explained
variance and feature importance analyses indicated that using principal components of drilling pa-
rameters did not lead to information loss. With these results, theoretically, a similar result should
have been observed on the complete data set. However, the observations in the results of feature
importance and explained variance were significantly different. The observations from Figure 3.5
and Figure 3.7 indicated that explained variance ratio by four principal components was lower
than the threshold value of 85% set for this study. Also, feature importance analysis indicated a
significant influence of variables on Principal Component 7. These results indicated an information
loss introduced by implementing PCA, leading to an overfitting problem. As a result, PCA has
been taken out of the algorithm to observe changes in the accuracy of predictions. Unfortunately,
the removal of PCA from the algorithm completely did not solve the problem of high accuracy, as
the accuracy of fitted models was approximately 98%. With this result, the second hypothesis is
tested and eliminated. However, PCA is not added back to the algorithm as information loss on
removing Principal Component 7 is observed in the analyses.
Figure 4.11: The Feature Matrix of Datasets 1 - 2.
Testing the last hypothesis was not as straightforward as other hypotheses since variability
should be introduced to target outputs, and collecting a new data set were impossible. In addition,
therawdatasetwasnotaccessiblesincethealgorithmbuiltbyJoshi(2021)wasworkingasawhole
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Figure 4.12: The Feature Matrix of Datasets 3 - 4.
system that takes collected drilling data as an input and predicts Torque values after classifying
the raw data as drilling and non-drilling data. Also, high accuracy in UCS predictions is observed
by Joshi (2021), as it is stated that the UCS regression model predicted UCS values for analog and
cryogenic samples with less than a 2.5% error margin. However, the issue of overfitting UCS did
not introduce a significant generalization problem as the purpose of the final algorithm built by
Joshi (2021) was to indicate drilling dysfunctions while drilling, which is essentially a classification
problem. Thefindingfromapreviousstudyconductedbythisdatasetshowsthatanotherapproach
should be developed to indicate if the regression algorithm built for this study is working or not.
Potential performance changes in the Multi-Output Random Forest algorithm are studied with
the idea of increasing variability by introducing another target value to the model. The study
conducted by Linusson (2013) on Multi-output Random Forests indicates that the performance of
multi-outputrandomforestmodelsshouldbesimilartoorthesameassingleoutputmodels. Then,
oneofthevariablesshouldbemovedtotargettoincreasevariabilitywithintheoutputvalues. MSE
is selected as a variable to move to the target values. As a result of moving MSE to target values,
approximately a 5-10% decrease in average accuracy was achieved on every fitted model. With
this, the third hypothesis is tested, and the performance improvement indicated that the lack of
variability in UCS data was leading to perfect predictions and causing the model to overfit the
training data.
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4.7 Potential Implementation of the Model for Field Applications
Throughout this study, the improvements that can be introduced by implementing data-driven
solutions to the field applications are mentioned and discussed based on previous studies conducted
with similar intent. This section discusses the potential implementation of the regression model
built step by step. As mentioned in previous sections, the regression model built aims to estimate
UCS from the drilling parameters instantaneously and avoid potential wellbore stability problems
and drilling accidents by providing changes in UCS while drilling. The model should be re-trained
with necessary data collected from the field where it will be used as a UCS prediction tool since
data-driven solutions are as unique as the data fed into the model. The potential development of
the UCS prediction tool by using this model for field applications is studied, and necessary steps
are provided in Figure 4.13. The initial implementation step will be using available drilling and
Figure 4.13: Potential Implementation on Field Application Workflow
geomechanicaldatafrompreviouswellsdrilledtotrainthemodelandtestthemodel’sperformance.
Then, the model can be tested based on different scenarios to indicate the model’s reaction to real-
life drilling operations (i.e., drilling break, high dogleg severity). After running these tests, a demo
product can be produced using the model if satisfactory test results are obtained. The demo
product can be tested in a field with development potential by training the model after collecting
necessary drilling and geomechanical from 1st exploration well. The model’s ability to predict UCS
while drilling can be tested for every appraisal well drilled and the additional data collected can be
fed to the model to improve its ability to indicate previous problems faced while drilling. A final
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CHAPTER 5
RESULTS AND TECHNICAL EVALUATION
The analysis was conducted on data set, and the findings are previously mentioned in Chapter
4. The final architecture of the algorithm is given, and possible problems that might occur while
training RF regression algorithm are explained in detail in Chapter 4. The final and third step
of the algorithm is described in this chapter. In addition, the overall results of models fitted by
using different approaches are provided. The changes in the algorithm as a consequence of findings
comprehensivelydiscussedinthepreviouschapterarebrieflysummarized. Aproblemofoverfitting
indicated due to a lack of variability within UCS data for each model is evaluated by visualizing
predicted and actual UCS values. Finally, the final model fit by using UCS and MSE as target
values, and the results are visualized and evaluated in this section.
5.1 The development phase of the algorithm
As mentioned before, the RF algorithm built to predict UCS is fit numerous times, and hyper-
parameter tuning tools are implemented at each fit. Initially, the model was fitted numerous times
with only 1% of the complete data set to gain more knowledge about RF regression model and
achieve the overall objective of building a robust algorithm. These subsets are sampled randomly
based on the distribution within the data set. A five K-fold cross validation method is applied to
reduce variance error at each fit. These initial tests on the algorithm helped to eliminate minor
errors in the code.
5.1.1 Initial Results
While conducting this study, the original plan was to use principal components of drilling
parameterstotraintheRFregressionmodel. Asmentionedearlier,85%explainedvarianceretained
principal components set as a threshold value. Initially, only 5% of data sampled from each file
are fed into the PCA algorithm to transform into principal components. The results acquired by
implementing PCA before feeding the training data into the algorithm were promising, and using
only fraction of the complete data set helped avoid minor errors. In addition to hyper-parameter
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tuning, the percentage of validation data split and implementation of K-fold cross-validation are
tested in these initial trials. The performance of each fit is evaluated with MAE, MSE, RMSE,
and MAPE. The overall accuracy is calculated from MAPE. The results are given in Table 5.1.
After running through the performance evaluation algorithm, it is clearly indicated that K-fold
cross-validation reduced variance error by preventing overfitting.
Table 5.1: Initial models fitted with sub datasets after implementing PCA
Number K-Fold Validation MAE MSE RMSE MAPE Overall
of PCs CV Dataset Size Accuracy
3 No CV 20% 0.17 1.15 1.07 0.23 99.77%
2 No CV 20% 3.61 103.31 10.16 4.29 95.71%
2 No CV 30% 3.61 103.46 10.17 4.28 95.72%
3 No CV 30% 0.17 1.2 1.09 0.25 99.75%
5 No CV 30% 0.046 0.018 0.13 0.03 99.97%
3 No CV 30% 0.18 1.24 1.11 0.25 99.75%
5 3 K-Fold 20% 20.33 1042 32.38 28.34 71.66%
5 4 K-Fold 30% 16.43 682.9 26.13 21.9 78.10%
Later, PCA was implemented on each data file provided, andexploratory analysis onthese data
setswaspromisingsinceexplainedvarianceratioretainedbyfourprincipalcomponentswerehigher
than 85%, and the influence of variables on principal components was low, as discussed in detail
Chapter 4. After implementing PCA, seven drilling parameters are replaced with four principal
components,asfourPCsexplainedmostofthedatavariancepresentinthecompletedataset. After
these steps, PCs are fed to the algorithm as a training dataset. The hyper-parameters are searched
using Randomized - Search and Grid-Search while fitting the model numerous times to indicate the
best possible fit. The final parameters used while fitting the last mode is given in Table 5.2. The
followings refer to the terms indicated in Table 5.2.
• Numberofestimators: Numberoftreescreatedbyusingtrainingdata,andthefinalprediction
is made by the contribution of each of these estimators.
• Maximum features: Decision criteria for maximum amount of features considered while split-
ting node.
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• Maximum depth: Maximum number of levels created while building each decision tree (esti-
mator).
• Minimum samples split: Minimum number of data points placed in a node before it splits.
• Minimum sample leaf: Minimum number of data points to be kept in each leaf node.
• Bootstrap: Method to sample data points.
Afterfittingthemodel,MAPEisusedtoindicatemodelperformance,and98%accuracyinUCS
predictions was observed. Extremely accurate predictions were a clear indication of overfitting.
Table 5.2: Hyper-parameters of the model fitted with four PCs
Hyper - Parameter Selected Parameter
Number of estimators 35
Maximum features Square root
Maximum depth 40
Minimum sample split 9
Minimum sample leaf 2
Bootstrap True
The UCS predictions are visualized to observe possible problems introduced by the lack of
variability in UCS data. Initially, the results from a model trained with one of the subsets are
visualized to observe how the lack of variability within UCS data affects the final visuals. As
expected, raw predictions collected from the model could not be visualized properly since the
predictions were too noisy. The moving average method is used to smooth the results. The moving
average can be defined as a statistical method for smoothing noisy predictions by continuously
taking the average values of the range defined. The advantage of moving average is keeping the
impact of each variable taken into consideration while continuously calculating the mean of a given
range of numbers. For example, the moving average of integers from 1 through 5 in the range of 2
would be 1.5, 2.5, 3.5, and 4.5. A closer match between actual and predicted values was observed
after using the moving average. The actual and predicted UCS values from the model trained
with a subset is plotted after applying a moving average with a range of 20, 50, and 100, given in
Figure 5.1.
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Figure 5.1: Comparison between predicted and actual UCS with 20, 50, 100 moving averages.
The significant noise in predictions is observed from the model trained with 4 PCs of complete
data set also indicated a presence within the predictions, and a higher range of moving average
should be used to observe a proper correlation between prediction and testing data. Then, a
10,000 moving average is applied to 5+ million predictions, and the observations on results were
more promising to indicate almost a perfect match between actual and predicted UCS values, as
observed in Figure 5.2. These results show a high variance error and indicate overfitting. As
mentioned in Chapter 4, PCA has been removed from the final algorithm.
5.1.2 Models Fitted without PCA
To test the 2nd hypothesis discussed in detail in 4, PCA is removed from the main algorithm.
Later, PCA is used to conduct explained variance and feature importance analysis. After removing
PCA, the number of variables fed to the algorithm is changed to seven; the overall objective was
to predict a single target value, UCS. The variables fed to the algorithm are Depth, RPM, WOB,
FPI, Torque, ROP, and MSE. Different cases are considered by varying training data set size and
number of iterations.
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Figure 5.2: Predicted and actual UCS data with 10000 moving average - Predicted Values (Blue),
Actual Values(Black).
Initially, RMSE was planned to evaluate the model’s performance, but MAPE has replaced it
since RMSE values were incredibly low, and it was impossible to indicate any difference between
models. MAPE values indicated a performance change between fitted models in the third decimal
point. The motivation for removing PCA was to study a possible information loss by using only
four principal components.
ThesecondhypothesiswastestedbyremovingPCAtoindicateifinitialinformationlosscaused
the overfitting observed in previous results. Similar to previous graphs describing predicted and
tested UCS versus depth created for the model fitted without PCA. Two different cases are eval-
uated. For the 1st case, only 50% of the complete data set is fitted through 280 iterations with
four K-fold cross-validation. For the 2nd case, the entire data set is fitted with five K-fold cross-
validation and 250 iterations. Unfortunately, the overall results did not change as UCS accuracy
in predictions was approximately 98%. This outcome proved that the information loss through the
implementation of PCA was not causing the overfitting problem. For 2nd case, the comparison
between UCS values from testing data and predicted UCS values is completed by observing the
results in Figure 5.3.
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Figure 5.3: Comparison between prediction and test UCS data with 10000 moving average.
5.1.3 Multi-output Random Forest Model
To test the 3rd hypothesis, one of the variables moved to target values. In this hypothesis, the
theory of how introducing a variation to target values would change the prediction accuracy. From
variables, MSEismovedtotargetvalues. Thenthemodelisfittedwithsixvariables(Depth, RPM,
WOB, PFI, Torque, ROP) and two target values (UCS, MSE). To the hypothesis, the model fit
four times based on different cases. A similar performance was expected as a multi-output random
forest should perform similar to a single output (Linusson 2013). In these cases, the number of
iterations and percentage of data used to fit the model is varied. These cases are given below in
Table 5.3. Based on these findings, it was decided to fit the model with only 100 iterations as the
number of iterations was not affecting prediction accuracy significantly. Hyper-parameter tuning
tools are used to search for the best model by using K-Fold cross-validation 100 times. The final
parameters decided to use while fitting the final model are presented in Table 5.4.
After fitting the model for each case, model prediction accuracy is calculated using MAPE.
The prediction of accuracy was not sensitive to the number of iterations, but it increased with the
increasingnumberofdatapointsusedtotrainthemodel. Thisfindingindicatedalackofvariability
within the data set even though two target values were used to increase variability. The prediction
accuracy of each case evaluated with train, validation, and test sets are given in Table 5.5.
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The predicted UCS values are compared with actual values in Figure 5.4. This comparison was
completed after applying a 10,000 moving average to actual and predicted UCS as a combination
of lack of variability within UCS and 5+ million data points made it impossible to visualize these
values on a scatter plot. The perfect match between predicted and actual UCS was expected as
the average prediction accuracy of MSE and UCS is 94%. Since moving average had to be applied
to see a clear match between actual predicted UCS, prediction accuracy is taken as an indication
of model performance. 10,000 moving average on UCS and MSE predictions from the final model
is applied. The comparison of predicted and actual MSE is plotted and presented in Figure 5.5.
Also, the actual versus predicted values are visualized using a scatter plot after applying a 10,000
moving average and presented in Figure 5.6.
Figure 5.4: Target UCS (Black) vs Predicted UCS(Blue).
Figure 5.5: Actual and Predicted MSE with the Final Model.
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CHAPTER 6
SUMMARY AND CONCLUSIONS
The objective of this work was to develop a complex machine learning algorithm by using
Random Forest Regression method to process high-frequency drilling data and train a model that
can estimate UCS instantaneously from drilling parameters. The importance of gaining knowledge
about subsurface conditions and geomechanical parameters is one of the key criteria for efficient
drilling operations. Current methods of estimating geomechanical parameters are time-consuming
and expensive. These constraints motivate the oil and gas industry to seek much more efficient
methods to indicate subsurface conditions and geomechanical parameters. A decrease in the cost
of computational power and robust implementation of data-driven solutions in other industries
encourages the oil and gas industry to search for more efficient solutions to solve common problems
through data collection. By predicting UCS while drilling, the model will allow to show changes in
UCS in different zones and give a key input to indicate potential wellbore stability problems.
Initially, this study was proposed to be conducted with a different data set and core samples
collected by using a coring rig located at the Edgar mine. The coring rig is equipped with after-
market sensors to collect RPM, ROP, WOB, and Torque data. The UCS and Young’s Modulus
of formation drilled through with the coring rig was planned to be measured by conducting a
laboratory experiment at the Colorado School of Mines campus using the MTS Rock Mechanics
System. Eventhoughtheuniquedrillingdatawascollected, UCSdatacouldnotbecollectedasthe
project milestones did not match this study. A similar data set, collected Joshi (2021), is utilized
for this study as it was purposed to build a similar stack of machine learning and deep learning
algorithms. The data set collected by Joshi (2021) was re-purposed to achieve the main objective
of building a regression algorithm to predict UCS from drilling parameters. The main objectives
of this thesis are:
• Develop a machine-learning model that can be trained to estimate unconfined compressive
strength from drilling parameters instantaneously.
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• Provide changes in Unconfined Compressive Strength based on drilling parameters instanta-
neously to avoid possible wellbore failures and drilling accidents.
• Utilize principal component analysis to analyze feature importance using available drilling
data.
• StudytheimplementationofRandomForestregressionalgorithmtobuildarobustregression
model to estimate certain geomechanical parameters (i.e., UCS, and Mechanical Specific
Energy).
The initial path of this study was to implement PCA to the complete data set and use only four
principal components of drilling parameters instead of seven variables (Depth, RPM, WOB, PFI,
Torque,ROP,MSE)totrainRandomForestregressionmodeltopredictUCS.Theearlyexploratory
analysisconductedonthedatasetindicatedalackofvariabilityinUCSdataasonlysevendifferent
values were present even though the complete data set had 55+ million UCS observations. This
lack of variability is assumed to be a unique part of the data set similar to predicted Torque values.
However, as the study developed further by training the model with complete data set, a high
variance error is indicated as the accuracy of UCS predictions was 98%. The high variance error
was not resolved as the model was fitted numerous times with a different list of hyper-parameters.
To bring a solution to this problem, a root cause of high variance error is tested through three
different hypotheses. These hypotheses are as followings:
• A possible linear correlation between MSE and UCS can lead to perfect predictions.
• A loss of valuable information due to the implementation of PCA can cause overfitting.
• A lack of variability on UCS can reduce the variability of predictions, which can cause over-
fitting.
These hypotheses are tested to indicate the root cause of high variance error. As mentioned
in Chapter 4, 1st and 2nd hypotheses are tested by changing the algorithm by removing PCA and
measuring the correlation between MSE and UCS. Both 1st and 2nd hypotheses are found wrong.
The 3rd hypothesis is tested by moving MSE from variables to target values as MSE would cover
the variation within the data set most. The accuracy of predictions decreased by 5-10%, and the
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accuracy reduction changed with respect to the amount of data used to train the model. These
observations indicated that the model was robust, but the lack of variability within UCS was
causing a high variance error as Multi-Output Random Forest performs similar to single output
RF (Linusson 2013).
The final model is fitted with six variables (Depth, RPM, WOB, PFI, Torque, ROP) and two
target values (UCS, MSE). The following conclusions are found from this study;
• Implementation of PCA on this data set will cause a valuable information loss.
• A small explained variance contribution by a variable doesn’t reflect its importance in regres-
sionmodelsasthevariablemightcarryanimportantpieceofinformationthatwillcontribute
to predictions of the model.
• A lack of variability within target values while training a regression model will cause high
variance error, aka overfitting, as the number of data points fed to algorithm increase.
• This data set does not reflect the real performance of the algorithm as a high variance error
is observed if the model is fitted with a single target value, even though most common tools
are used to avoid overfitting.
• The model is robust enough to indicate variation change in target values as UCS predictions
accuracy decreased when another target value is added to target values to introduce more
variation.
• The model can be trained to estimate UCS from drilling parameters accurately if the data
set with higher variations among variables is used.
ThemainobjectivesofthisthesisarefulfilledasthemachinelearningmodelbuiltbyusingRan-
dom Forest regression algorithm were successfully estimate UCS and MSE from drilling parameters
instantaneously. Also,PCAisutilizedtoindicatethefeatureimportanceandtheretainedexplained
variance by each principal component to observe the possible loss of valuable information. As a
result of the implementation of PCA, it is decided to use six drilling parameters directly instead of
training the model with only four principal components.
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CHAPTER 7
FUTURE WORK
The work completed in this study can be developed further to implement the model as an
efficient data-driven solution for field applications. The suggested approaches and methods can be
a part of the further development of this study as follow:
• The model built for this study can be trained with field data collected at the drilling site,
and a comparison between UCS predictions from the model and UCS measurements from
laboratory experiments can be made.
• Different variables affecting UCS can be introduced to the model as a part of the training
data set. These variables can be porosity and elemental spectroscopy (Negara et al. 2017).
• The empirical equations derived to estimate UCS can be added to the model as a lower
boundary condition (Chang et al. 2006), if the empirical equation is derived for the same
formation where the data set is collected.
• The core samples retrieved from Edgar Mine by using Apache Coring Rig can be tested to
measure UCS and Young’s Modulus, and these measurements can be integrated into the
drilling data collected. The integrated drilling parameters and UCS can be used to train the
model.
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ABSTRACT
Drilling technology has advanced rapidly over the past few decades. The petroleum
industry is motivated to automate directional drilling operations, making them safer, more cost-
effective, and efficient. The main input parameters fed to directional drilling tools (e.g., rotary
steerable systems) are inclination, azimuth, and distance. Inclination and azimuth are measured
downhole; however, the distance drilled is computed using a pipe tally system installed on the
surface. Having continuous downhole measurements of distance, inclination, and azimuth allow
the directional drilling tool to determine the drilling bit's location. Moreover, if most calculations
were made downhole, then the bit could follow a preprogrammed trajectory and drill
autonomously.
Many researchers have proposed different methods for measuring the distance drilled in
the bottomhole as a solution for directional drilling automation. However, the suggested
approaches have experienced challenges during field applications. For this reason, this
dissertation focuses on solving one of the main limiting factors, i.e., real-time downhole
measurements of the distance drilled.
This work proposes a method that senses the distance drilled at the bottom of the well
based on automated image analysis. The dissertation involves the development of a tool
prototype with multiple identical imaging sensors spaced at known distances. The sensors
acquire images of the formation at synchronized times, with the imaging sensors capturing an
image of the same formation location at different times. Based on this concept, an algorithm was
developed to identify the "fingerprints" of the images captured and register similar images
according to those fingerprints. Under wellbore conditions, each fingerprint reveals unique
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marks left by the bit, natural geological features of the rocks drilled, and the topology of the
borehole wall, all of which contribute to the image registration process. Therefore, when two
images captured by two sensors match, the timestamps of each image and the distance between
the sensors can be used to calculate the average rate of penetration in that time interval.
Subsequently, integrating the rate of penetration results in an estimate of the interval distance
drilled. Adding all intervals drilled is then equivalent to the total distance of the well, and
combining this information with azimuth and inclination can provide an estimate of the depth
drilled. This dissertation aims to demonstrate that an image-matching process is an attractive
candidate for downhole measurements that compute the distance drilled.
In this work, an image-matching algorithm is developed with the sole intention of
estimating the distance drilled. Tests were designed to analyze the effect of different variables on
the accuracy of the distance calculation. Experiments were devised to investigate how sampling
criteria, the distance between the sensors, velocity, matches missed, and noise affect the method's
efficacy. The method was tested in homogeneous and heterogeneous scenarios (i.e., images with
distinct and repetitive features) to ensure that the algorithm can match images when exposed to
analog scenarios that model wellbore conditions. The results show that the image-matching
algorithm can accurately match images even when wellbore features are considerably repetitive.
Additionally, having an optimized velocity-dependent combination of sampling criterion and
distance between the sensors can minimize errors, especially those associated with missed
matches, and therefore, increase the accuracy of the distance calculation.
The main contribution of this Ph.D. dissertation is that it demonstrates that having
identical sensors capture images showing topological features can be a robust method to measure
the distance traveled. Additionally, this work provides initial research on how to optimize
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ACKNOWLEDGMENTS
I thank Allah (God) for giving me the strength and power to complete this dissertation
and granting me the opportunity to be a part of Colorado School of Mines and learn from
outstanding professors.
I also want to take this opportunity to thank my advisor, Dr. Luis Zerpa, for all of his
unconditional support and guidance. I wouldn't have been able to complete this dissertation
without him.
I also owe a great amount of gratitude to my co-advisor, Dr. Jorge Sampaio. I want to
thank him for his guidance, unconditional support, and giving me the chance to be a part of a
great project. He has helped me in every step of this dissertation, and it has been an honor
working by his side.
I would also like to express my sincere appreciation to my committee members, Dr.
Alfred Eustes, Dr. Jeffrey Shragge, and Dr. Manika Prasad. It is a privilege to be working with
them, and I want to thank them for all the help and guidance they gave me.
A very special thanks goes to Dr. Heba Mahgoub, my mum, and Dr. Khaled Mansour,
my dad, for all their emotional support and unconditional love. They have always been by my
side and have been the source of my inspiration and motivation. They have raised me to be the
man I am today, and without them, I would have never achieved this goal. In addition to my
parents, I would also like to thank my brother, Mohamed Mansour, for always encouraging me to
work hard and supporting me. He has been a blessing in my life and has always been there for
me whenever I needed him.
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CHAPTER 1
INTRODUCTION
This chapter introduces the problem statement for this doctoral research dissertation. It
explains the different kinds of depth measurements and discusses their importance during drilling
operations. Additionally, it introduces and describes the differences between surface and current
bottomhole depth measurement techniques along with their associated errors and limitations.
1.1 Definition and Importance of Well Depth
In drilling operations, the well(cid:182)s depth acts as a standard reference for most data observed
and measured on the well. For example, formation evaluation, a technique where a measuring
tool is deployed down the borehole to measure physical rock properties of subsurface geologic
formations, is highly dependent on well depth for data interpretation. Other data such as drilling
parameters, cuttings description, deviation surveys, and cores also rely on well depth. However,
well depth is a distinct measurement that needs to be made during drilling operations (Storey
2013).
Well depth is defined as a parameter that characterizes the position of an object within
the wellbore concerning a reference point at the surface. Five attributes are required to specify
the depth of a point along the wellbore (Storey 2013):
1. Wellbore name from which depth will be measured.
2. Point of reference, which represents the depth measurement starting point.
3. Depth unit (e.g., in feet).
4. The measurement process used (e.g., driller(cid:182)s or logger(cid:182)s depth).
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5. Type of path measured (e.g., Measured Depth (MD), also known as Along the Hole
Depth (AHD), or True Vertical Depth (TVD), as seen in Figure 1-1).
The type of path referred to will depend on the type of calculations needed. For example,
the total MD refers to the total distance drilled by the bit and is, therefore, the total length of the
hole, while TVD refers to the vertical depth of the hole. MD is used for directional drilling
surveys and volumetric calculations, while TVD can be used for pressure and mud weight
calculations. It is important to note that the method in this dissertation is estimating MD and not
TVD.
Figure 1-1 Different paths are measured, where “Q(cid:180) is a point on a vertical well, and “P(cid:180) is a
point on a directional well. The depth of P and Q is the same, but the length is different due to
the difference in inclination. AHD refers to the length, and TVD refers to the true vertical depth
(Storey 2013).
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1.2 Problem Statement
The oil and gas industry has been motivated to automate directional drilling operations,
and many researchers have investigated ways to make such automation possible. As of 2021,
directional drilling tools, such as Rotary Steerable Systems (RSS), depend on MD
measurements, inclination, and azimuth. The RSS collects the target formation's coordinates sent
from the surface through the Measurement While Drilling (MWD) tool. Finally, it adjusts the
azimuth, tool-face, and inclination based on its sensors until the target formation is reached
(Yiyong et al. 2009). The limitation encountered with this directional drilling system is that for it
to be fully automated, it requires a downhole MD measurement method that is relatively accurate
to the measurements made on the surface, which is currently not the case (Starkey 2018; Stoner
2018).
If a directional drilling tool could measure inclination, azimuth, and the MD downhole, it
could determine the location of the drilling bit. The engineer could then pick the best possible
trajectory that minimizes torque and drag (Sampaio and Mansour 2019). Programming this pre-
defined trajectory alongside the data gathered downhole would allow the autonomous drilling
tool to know where it is and where it needs to go, thus achieving directional drilling automation.
1.3 Depth Measurement Techniques
There are different approaches and methods that engineers have used to measure the MD
of the well. This section discusses different measurement techniques and inventions used today
during drilling operations.
1.3.1 Surface Depth Measurements
Two conventional surface-related MD measurement techniques are used in the field
today, and these are the driller(cid:182)s and the logger(cid:182)s depth. During drilling operations, the driller(cid:182)s
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depth is computed by summing up individual lengths of the drill bit, drill collar, and drill pipe
joints below the kelly bushing or the rig floor (Chia et al. 2006). Most of the lengths are
measured on the surface using a steel measuring tape before lowering the pipe in the hole. This
measurement technique implies that the pipe(cid:182)s length is measured before getting attached to the
drill string and placed downhole (Reistle and Sikes 1938).
The way these measurements are made is a problem because they do not account for pipe
length changes that occur downhole due to forces acting on the pipe, which can alter its length.
Therefore, the pipe length measured on the surface is different from the actual pipe length in the
bottomhole (Chia et al. 2006).
The two most significant factors that contribute to measurement errors are thermal
expansion and mechanical stretch. Temperature changes contribute to thermal expansion, while
well profile, frictional forces, and pipe weight contribute to the mechanical stretch. Axial
pressure effects, ballooning, buckling, and drill pipe twists can also contribute to the pipe's
mechanical deformation. Complex algorithms may identify these errors, and a torque and drag
model can then adjust for the mechanical stretch and thermal expansion (Chia et al. 2006).
The automated version of the driller's depth is referred to as the pipe tally measurement
system. A typical system includes a drawwork's encoder or a geolograph that records depth
continuously on a log by tracking the block's movement (Figure 1-2). An installed software
monitors the hookload sensor and analyzes the bit's in and out of slips motion. The movement of
the bit can only be updated when the pipe is in "out of slips" motion. (Chia et al. 2006).
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Figure 1-2 Pipe tally measurement system consisting of a drawworks encoder that continuously
records depth on a log and a hook load sensor that analyzes in and out of slips motion. (Chia et
al. 2006).
Lai et al. (2016) analyzed the errors associated with the pipe tally measurement system.
They studied 29 onshore wells with an average MD of 16,800 ft located in Canada and the U.S.
The objective of their study was to determine the frequency of depth corrections and the absolute
error magnitude resulting from the pipe tally. The results show that the frequency of error
corrections ranged between 40 to 143 corrections. Additionally, the pipe tally's absolute average
measurement error for all 29 wells was 1965 ft, where 779 ft was due to hookload threshold
error, and 1186 ft attributed to human errors. Hookload threshold errors occurred due to failure
in detecting in-slip motion and incorrect tracking. Human errors occurred due to incorrect
settings in the pipe tally's system, typing mistakes, and unfamiliarity with the tally's software
user interface.
The second type of MD surface-related measurements is the logger(cid:182)s depth, and it is
made using a wireline system that consists of an electric logging cable, guide rollers, and a
measuring wheel. The electric cable acts as a measuring tape and is attached to a calibrated
measuring wheel with a known circumference. As the electric cable moves along the hole,
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friction occurs between the cable and the wheel, causing it to rotate. Each measuring wheel
revolution is counted and correlated to MD (Epstein 2005). This type of measurement is accurate
if the wellbore is vertical. However, as the wellbore shape increases in complexity, the wireline
measurement system starts accumulating errors (Chia et al. 2006). Additionally, errors may
result from different suspended loads, incorrect counting of measuring wheel revolutions, and
cable stretch resulting from acceleration forces (Reistle and Sikes 1938; Bolt 2017)
In summary, there are two different ways that MD can be measured on the surface. The
standard measurement method for drilling operations is the driller's depth that uses the automated
pipe tally measurement system. However, as shown by Lai et al. (2016), the system has errors
and requires constant corrections. Thus, the advantage of developing a bottomhole MD
measurement system is not to enhance the system's accuracy but to automate directional drilling
and facilitate a continuous decision-making process made by a downhole computer instead of a
discrete process made by an engineer on the surface. An ongoing decision-making process will
lead to a smooth directionally drilled trajectory, decreasing torque, drag, and drilling costs.
Nevertheless, while the bottomhole measurement system will have errors, it should not
continuously disrupt drilling operations due to low accuracies. The bottomhole measurement
system should work effectively under normal drilling conditions.
1.3.2 Bottomhole Depth Measurements
Due to the growing interest in automated directional drilling systems, researchers have
aimed to decrease dependence on surface measurements by finding new ways to obtain
downhole real-time MD measurements.
Patton (1994) invented a new system for controlling the direction of the bit and
measuring MD at the bottomhole. The system consisted of a standard bottomhole assembly
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accompanied by a magnetic marker system. This system generated pulses via magnetic markers
spaced out at a known distance and magnetized the formation. The system reads the markers and
computes the distance drilled. This invention faced challenges because the formation, most often,
could not be magnetized. According to Jain (2015), field applications showed that failure to
magnetize the formation led to significant errors associated with the MD measurements.
Hassan and Kurkoski (2007) proposed acquiring formation characteristics through
logging tools to measure the MD at the bottom. The method consisted of similar sensors placed
at different depths, gathering specific petrophysical information about the rock, such as porosity
and neutron density. Since two or more sensors will pass the same location in the formation, and
the distance between the sensors is known, a downhole computer can correlate the similar data
gathered.
According to Jain (2015), this geological correlation method faced a strong challenge.
Horizontal wellbores usually extend within the same formation and are characterized by the
same lithology, meaning that rock characteristics will not significantly change due to
homogeneity. Field results show that using a sensor such as a gamma-ray in a homogeneous
section caused MD errors to be approximately 95%.
Estes et al. (2012) proposed measuring MD in the bottomhole using an inertial navigation
system involving an accelerator installed on the drillstring that transmits the recorded data to a
programmed memory module. Then, a downhole processor installed between the accelerometer
and the memory module gathers data from both objects to calculate the acceleration of the
drillstring. Finally, the computed acceleration data is integrated to calculate the total MD of the
well. According to Jain (2015), this method goes through an integration drift process and needs a
secondary system that constantly corrects these errors.
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Jain (2015) proposed an MD measurement method similar to the geological correlation
method proposed by Hassan and Kurkoski (2007) that uses the Normalized Cross-Correlation
(NCC) algorithm or the Hybrid Matching (HM) algorithm. The NCC algorithm can identify a
time lag between two identical signals. This algorithm generates all possible matches and
calculates a correlation coefficient for each match, with the highest value considered the greatest
possible match. The HM algorithm uses brute-force to search for potential matches that depend
on tolerance criteria and fit. If several outputs fit the input criteria, then the best match is defined
using the minimum square distance method (Jain 2015).
The simulation results using both NCC and HM algorithms showed that even if sensor
drift caused an offset in data, the correlation algorithms were not affected. Additionally, when
data input counts are higher, depth measurements become more accurate. However, MD
measurement errors occurred due to noise and ROP averaging. Additionally, based on the
gamma-ray tests performed, errors were still high in homogeneous sections due to a lack of
change in rock properties (Jain 2015).
Sugiura (2015) proposed another geological correlation technique for estimating MD.
Firstly, two sensors are installed to measure the same borehole parameter and generate two time-
based logs. Secondly, a downhole computer processor measures the time shift by comparing both
logs. A correlation factor is then generated by time-shifting the logs using a simplified version of
Pearson(cid:182)s correlation method, measuring the degree of the linear relationship between two data
points. Finally, the drilling speed is then calculated and integrated to calculate the distance
drilled.
Researchers have proposed many techniques to measure the MD at the bottom of the
well; however, every solution proposed was accompanied by challenges that made the approach
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difficult for routine field implementation. Geological correlation methods typically fail in
homogeneous formations due to a lack of change in rock properties. Since homogeneous
formations are common (especially in horizontal drilling), it is appropriate to have a
measurement method that works during such conditions, which is the focus of this dissertation.
The method proposed here is similar to the geological correlation methods but focuses on the
topology of the borehole wall rather than the petrophysical rock properties mentioned above.
1.4 Dissertation Structure
This dissertation is divided into six chapters: Chapter 1 explains the importance of depth
measurements in drilling operations, the different methods used to measure depth, and the
limitations of existing methods. Chapter 2 explains the proposed distance measurement
methodology and the goal and objectives of this dissertation. Chapter 3 consists of the
background and literature review covering important topics such as the rate of penetration,
drilling automation, and logging while drilling. Chapter 4 explains the operating principle of the
proposed methodology and the design of the image-matching algorithm and experiments.
Chapter 5 presents and discusses the results obtained from all the experiments performed in this
dissertation. Chapter 6 discusses the potential of the proposed methodology and its limitations.
Chapter 7 concludes the dissertation and provides recommendations for future work.
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CHAPTER 2
RESEARCH PROPOSAL AND OBJECTIVES
This chapter discusses the proposed downhole distance measurement method, explains
the method(cid:182)s theory, and demonstrates the objectives of this dissertation.
2.1 Proposed Bottomhole Measurement Technique
As discussed in the previous chapter, current bottomhole MD measurement techniques
still face challenges. These complications create a barrier to fully automating a directional
drilling system. At present, a reliable bottomhole distance measurement technique does not exist.
Therefore, this work proposes a new approach to sense the distance drilled at the bottom of the
hole.
The proposed method includes identical sensors at known positions that synchronously
capture images of the wellbore. The images captured by the sensors show unique marks left by
the bit, the topology of the borehole wall, and the natural geological features of the rocks drilled.
Therefore, each image has an identified “fingerprint.(cid:180) An image-matching algorithm will then
correlate the fingerprints and obtain the times each sensor records the same formation image.
The algorithm uses time and distance information to estimate the local average rate of
penetration. The algorithm then integrates the rate of penetration to give the interval distance
drilled, which is then summed to estimate the MD.
The methodology is based on the theory that comparing images that show the topology of
the wellbore is more accurate than comparing data from rock characteristics. This is because the
images will contain important features such as fractures, marks left by the bit, and the borehole
geometry, which are constantly changing and are independent of the rock(cid:182)s characteristics. This
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independence allows the methodology to work in homogeneous sections where petrophysical
data collected from the Logging While Drilling (LWD) tool rarely change. As a result, two
crucial questions need to be answered:
1. Which image-matching algorithms are capable of efficiently and accurately matching
wellbore images?
2. Which type of sensors or methods should be used to capture wellbore feature images?
2.2 Image and Pattern Matching
Pattern matching is a method used to find a specific pattern within an image. The process
uses a concept called “masking,(cid:180) where a predefined mask consisting of an image pattern is
placed over all possible pixels positioned in the picture. A factor-matching score is then
generated for the masking pattern and the image, allowing comparisons with a preset matching
score. If the score is higher than a predefined value, the pattern matches the image (Walia and
Suneja 2010). According to Balletti and Guerra (2009), there are three different patterns and
image-matching methods. These methods are feature-based, area-based, and symbolic. Each
method has its way of generating a matching process for the images (Balletti and Guerra 2009).
This research focuses on pattern matching techniques that use feature-based methods which
extract specific geometrical features at various scales. The comparison depends on the size and
shape of these features (Balletti and Guerra 2009).
2.2.1 Feature and Key-Points Identification Techniques
Various programs, simulators, and open-source codes for image feature identification
exist today. Lowe (2004) developed a model for distinctive image-matching and named it Scale
Invariant Feature Transform (SIFT). This innovative method identifies key features from an
existing image that are not affected by image rotation or scaling and act like a “fingerprint(cid:180) for a
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(cid:3105)(cid:3118)𝐺(cid:4666)𝑥,𝑦,(cid:3097)(cid:4667)
where 𝐿 (cid:4666)𝑥,𝑦,𝜎(cid:4667) (cid:3404) ∗𝐼(cid:4666)𝑥,𝑦(cid:4667). This method speeds up the image-matching technique
𝑥𝑥 (cid:3105)𝑥(cid:3118)
and replaces the use of difference-of-Gaussian functions (Bay et al. 2008).
Calonder et al. (2010) developed a feature point detection and image-matching model
called Binary Robust Independent Elementary Features (BRIEF). The authors aimed to build an
image-matching technique that works faster than SIFT and SURF. The computational speed was
increased by comparing the intensity of point pairs using short descriptors. However, one of the
main challenges this technique faces is that it is affected by rotation and may not be used in
many applications.
Rublee et al. (2011) developed a feature-point detection and image-matching model
called Oriented FAST and Rotated BRIEF (ORB). For this model, the authors combine an
accurate feature point detection method, FAST, proposed by Rosten and Drummond (2006), with
an updated version of BRIEF that is not affected by image rotation. ORB works very similar to
SIFT and SURF but uses a different mathematical approach. ORB has been known to be fast and
accurate for many applications.
Karami et al. (2017) measured the effect of an image(cid:182)s deformation on three different
image and pattern matching techniques. These effects included scale, rotation, noise,
illumination, and translation. The methods compared were SIFT, SURF, and ORB. It was
determined that ORB was the fastest algorithm when compared to the other techniques.
However, SIFT provided the highest matching rates and accuracy when compared to the other
methods.
2.2.2 Image and Pattern Matching Techniques
Key-points identification algorithms allow a unique “fingerprint(cid:180) for each image taken.
However, the algorithm itself does not provide any matching methods between similar
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“fingerprints.(cid:180) Therefore, pattern-matching algorithms are needed to compare the images and
evaluate them based on their similarities.
The brute-force technique is one of the simplest pattern-matching methods used today.
This technique matches the key-points in the images using the Euclidean distance, 𝑑(cid:4666)𝑃 ,𝑃 (cid:4667),
1 (cid:2870)
which is calculated as:
𝑑(cid:4666)𝑃 ,𝑃 (cid:4667) (cid:3404) (cid:3493)(cid:4666)𝑥 (cid:3398)𝑥 (cid:4667)(cid:2870) (cid:3397)(cid:4666)𝑦 (cid:3398)𝑦 (cid:4667)(cid:2870) (2.9)
1 (cid:2870) 1 (cid:2870) 1 (cid:2870)
where 𝑃 and 𝑃 are key-points on the first and second image, respectively. This technique
1 (cid:2870)
requires many trials because each key-point from an image will be compared with all the other
key-points from another image. A match is produced based on the closest points found
(Mordvintsev and Abid 2013).
Another commonly used pattern-matching algorithm is the Fast Library for Approximate
Nearest Neighbors (FLANN). This algorithm works very similarly to a brute-force technique.
However, FLANN is more efficient and allows large datasets to be matched much faster than
brute-force because it uses parallel linear brute-force searching. Therefore, FLANN decreases
the computational time for large datasets while still maintaining the accuracy of the brute-force
technique (Mordvintsev and Abid 2013; Muja and Lowe 2009).
Other pattern-matching methods that have been used in many applications are the Boyer-
Moore technique and the Karp-Rabin algorithm. The Boyer-Moore technique is a pattern-
matching method that only works when string matching elements have a finite fixed size. This
matching method uses information based on previous unsuccessful matches and decides what to
do next with such information. The decision is based on precomputed timetables, where the
method tries to match key-points by going back in time (Korber 2006). Alternatively, the Karp-
Rabin algorithm generates hash-codes for each key-point. The algorithm will then process a
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matching technique that compares similar hash-codes together from key-points located on
different images. However, once the matches are obtained, they should be checked to ensure that
there are no collisions and that the hash-codes matched correctly (Korber 2006).
All of the algorithms mentioned above provide matching techniques between images.
However, the main challenge is knowing how good a match is. Therefore, researchers specified
different methods to identify the accuracy of the matching process and the ability to find
successful matches. Out of all the research made, the Good-Ratio Test (GRT) proposed by Lowe
(2004) and the Random Sample Consensus (RANSAC) algorithm proposed by Fischler and
Bolles (1981) are sufficient techniques for the method proposed.
The GRT is a test that filters all "bad matches" out to ensure the accuracy of the pattern-
matching algorithm. Instead of only comparing the distance between the key-points and their
closest neighbors, the test also examines the gap between the nearest and second nearest
neighbors. Therefore, the second-closest match provides an estimate of "bad matches" occurring
within the pattern-matching algorithm. Generally, if the distance ratio is set to 0.8, approximately
90% of bad matches will be eliminated (Lowe 2004).
Figure 2-1 shows an example of two photographs of the same book yet from a different
camera perspective. The SIFT algorithm was applied to these images to identify the key-points
of both pictures. Next, the brute-force matching algorithm was combined with the GRT to match
both images together. The result illustrated in Figure 2-2 shows the similarity between both
images. The lines connecting both photos are the "good matches" defined by the GRT test.
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The same photographs presented in Figure 2-1 are used to show an example using
RANSAC. For this example, the SIFT algorithm was applied to identify key-points in these
images. Afterward, the brute-force matching method and the RANSAC algorithm are used to
match both photos together. Figure 2-3 shows the result where the green lines represent "good
matches" between the image, and the square indicates the region where the two photos are
similar.
Figure 2-3 RANSAC pattern-matching algorithm. The orange square is generated by the
RANSAC algorithm and represents the exact area where both images are similar.
It can be seen from the examples mentioned above that GRT and RANSAC algorithms
can provide reliable matches. Both algorithms can accurately identify similarities between two
images. Based on preliminary testing, it was determined that SIFT, FLANN, and GRT are the
most suitable algorithms for the work proposed in this dissertation.
2.3 Ultrasonic Imaging
My research determined that ultrasonic imaging is the most suitable type of imaging for
the proposed method. Ultrasonic imaging is highly reactive to fractures and bit marks and, when
implemented properly, can show the whole surface topology (Yao 2018). It is important to note
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that other sensors can be used, especially those that clearly show topological features.
Nevertheless, based on the current technology available in the oil and gas industry, ultrasonic
sensors are preferred over other sensors because of their strong capability of identifying
topological features.
Ultrasonic imaging is used in many engineering and medical applications. The imaging
process uses a piezoelectric crystal that is placed in a transducer. As the electric current passes
through the crystal, the current transforms to mechanical pressure waves, i.e., sound waves. The
thickness of the crystals controls the frequency of the waves released, and they are usually higher
than 20 kHz (Wang 2020; Boston Piezo Optics 2020; The Green Age 2020).
Once the waves are generated, they travel through a medium, and the crystal switches
from the “sending(cid:180) phase to the “listening(cid:180) phase. The waves travel through the medium until
they get reflected. The reflected wave travels back to the crystal, and the whole process is
repeated. The time it takes for the waves to come back depends on the velocity of the sound
waves in the medium through which they travel and the distance to the reflector. An image is
then created using the time it took for the waves to be reflected and the relationship between the
amplitude of the waves sent and received. It is also possible to have more than one crystal to do
this process (Wang 2020)
An ultrasonic Imaging While Drilling (IWD) tool works using the same concept
mentioned above. The transducer sends sound waves through the drilling fluid, and then they are
reflected from the borehole wall and sent back to the transducer or another receiver. All data
acquired are assigned a toolface position and azimuthal orientation. When the data are received,
environmental corrections for sound velocity and denoising are made before generating the final
image. The environmental modifications are made on parameters that affect sound velocity.
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These parameters include mud weight, mud type, temperature, pressure, and cuttings (Hassan et
al. 2004; Steinseik et al. 2010).
Hassan et al. (2004) created a device that measures ultrasound velocity in drilling fluids.
Such a device could be applied to our proposed methodology because it prevents environmental
corrections made on the surface. The device consists of an acoustic transmitter that generates
ultrasound waves in the drilling fluid and two acoustic receivers that detect the ultrasound waves
over two paths. The ultrasound velocity is calculated using the time that the wave arrived and the
distance.
Ultrasonic imaging tools are mainly used to detect fractures and understand rock
properties. However, the proposed method does not focus on petrophysical properties but rather
on the topology of the borehole wall. Yao (2018) designed a MATLAB code that reconstructs a
wrench surface from data gathered from a focused acoustic transmitter (Figure 2-4). The wrench
was submerged in water, and the objective of the experiment was to reconstruct the word
“DUTY(cid:180) written on the wrench. The reconstruction is a function of the reflection coefficient, 𝑅 ,
(cid:3027)
which is calculated as:
𝑅 (cid:3404)
(cid:3053)(cid:3117)−(cid:3053)(cid:3118)
(2.10)
(cid:3027)
(cid:3053)(cid:3117)+(cid:3053)(cid:3118)
Where 𝑧 is the acoustic impedance (i.e., opposition to wave propagation) of the surface and 𝑧
(cid:2870) 1
is the acoustic impedance of water. The focused acoustic transmitter can scan a small area one at
a time, allowing many reflection coefficients to be calculated over the whole surface area of the
wrench. As a result, the authors were able to reconstruct the word accurately.
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Figure 2-4 Reconstruction of the word “DUTY(cid:180) embedded on a wrench (right) using a focused
acoustic transmitter (left) (Yao 2018).
2.4 Research Goal and Objectives
This research aims to assist in the automation of directional drilling by estimating the
distance drilled in the bottomhole and minimizing dependence on surface information. The
method depends on capturing images of the formation while drilling and comparing them until a
successful match is reached. Therefore, the main goal of this dissertation is to demonstrate that
an image-matching algorithm can be used to calculate the distance traveled/drilled. The
objectives of this dissertation are as follows:
1. Explain the concept and theory of the bottomhole MD measurement method.
2. Develop a computer algorithm for automatic image processing that identifies key-
points (i.e., image fingerprints), compares time-lapse images, identifies matches, and
calculates the distance drilled. The algorithm should have the following features:
a. Flexible and accurately differentiates between good and bad matches.
b. Designed to work in real-time applications.
c. Work in areas where rock features may be homogeneous.
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CHAPTER 3
BACKGROUND
This chapter explains the theory behind this research. The method proposed in this work
includes an image-matching algorithm that estimates the rate of penetration to calculate the
distance drilled. Therefore, this chapter explains the meaning of the rate of penetration and the
factors that affect it. Additionally, this chapter discusses drilling automation and rotatory
steerable systems. The proposed method aims to automate directional drilling by adding a
necessary input parameter (i.e., distance drilled) to the directional drilling tools. Finally, this
chapter discusses imaging while drilling methods because the proposed method depends on
capturing images of the formation.
3.1 Rate of Penetration
The proposed method calculates the rate of penetration (ROP) in real-time while drilling
and uses a mathematical integration procedure to get the interval distance drilled. Therefore, this
section discusses ROP and the factors that affect it.
3.1.1 Definition and Importance of Rate of Penetration
A drilling bit is usually used to break the rock. Simultaneously, drilling fluid is circulated
within the hole to cool down and lubricate the bit and carry the cuttings to the surface. To
minimize costs, the well needs to be drilled in a fast and efficient manner. For the drilling
engineer to know how fast a well is drilled, ROP needs to be calculated. ROP is defined as the
advancement of the drilling bit per unit time while drilling a formation and is calculated as:
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Distance Drilled
ROP = (3.1)
Time
ROP calculations can be divided into two types: (1) instantaneous ROP measured while
the drilling operation is still occurring and (2) average ROP measured over the total distance
interval after the formation has been drilled (Mensa-Wilmot et al. 2010).
ROP is considered to be one of the main factors that influence drilling efficiency and
operational costs. Most performance quantifying metrics such as Mechanical Specific Energy
(MSE), Cost per Foot (CPF), and feet per day are strongly affected by ROP. An increase in ROP
generally leads to an increase in drilling efficiency and a decrease in costs (Mensa-Wilmot et al.
2010).
Researchers have identified the parameters that affect ROP as well as examined ways to
remediate and optimize ROP to enhance drilling efficiency. The main motivation for undertaking
ROP research is to decrease drilling costs. For example, in BP operations, onshore drilling time
accounted for up to 30% of the well(cid:182)s cost (Fear 1999). The costs associated with offshore
drilling are even higher (Hemphill et al. 2001; Mansour 2017).
Even though serious financial costs may occur due to drilling time, ROP is still a
complex problem that is not well understood. Therefore, to minimize costs, drilling engineers
need to find the best method to quickly drill a well while maintaining wellbore integrity.
3.1.2 Factors Affecting Rate of Penetration
There are five aspects affecting ROP: formation characteristics, hydraulic factors, rig
efficiency, mud properties, and mechanical factors. Additionally, these aspects are categorized
into either controllable or uncontrollable factors. Drilling engineers can change controllable
factors, while uncontrollable factors are challenging to alter due to geological or economic
reasons. It is difficult to analyze an individual aspect by itself because the factors affect each
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other as well as ROP (Elkatatny et al. 2017). Many factors such as lithology types, bit design,
drilling fluids, drillstring rotation speed (RPM), and weight on bit (WOB) can affect ROP
(Mensa-Wilmot et al. 2010).
Brantly and Clayton (1939) studied various factors affecting ROP while drilling: RPM,
circulating fluid volume and pump rate, WOB, and equipment and personnel. The authors used
the data from tour reports, bit records, questionnaires, and formation records in many different
fields over seven U.S. states. The authors conclude that all factors studied affected ROP (Table
3-1).
Table 3-1 ROP study results (Brantly and Clayton 1939).
Factors Studied Effect
When the rotating speed increases, ROP increases.
However, it will not increase ROP by much when the
Drillstring rotation speed
rotating speed has reached a critical limit. To enhance ROP,
the engineer will have to alter the other factors.
The higher the fluid volume and pump rate, the higher the
Fluid volume and pump rate ROP. However, there is also a critical limit for fluid volume
and pump rate.
WOB will only increase ROP if the bit is penetrating a hard
formation. For soft formations, rotating speeds and fluid
Weight on bit volumes are more effective. For brittle formations, there is a
certain limit that WOB can be increased to alter ROP
positively. Above that limit will cause the ROP to decrease.
The better the design of the drilling bit, drilling pipes, and
Drilling Equipment
drilling fluid, the higher the ROP.
Hellums (1952) studied the effect of pump horsepower (PH) on ROP. The authors
analyzed data obtained from drilling records in eight different southern U.S. states. The results
showed that ROP is almost proportional to PH; in other words, when PH is increased, ROP
increases.
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Bielstrein and Cannon (1950) investigated the effect of bit design and hydraulic factors
on ROP. All tests were made in the field via various wells. The authors chose formations that
were classified as either soft or medium hard and nonabrasive. The authors studied the effect of
nozzle fluid velocity, nozzle efficiency and power factor, and fluid distribution around the bit on
ROP (Table 3-2).
Table 3-2 ROP Study results (Bielstrein and Cannon 1950).
Factors Studied Effect
When nozzle fluid velocity increases, ROP increases.
Nozzle fluid velocity
The effect of nozzle efficiency and power factor on
Nozzle efficiency and power factor ROP varies and does not show a constant
relationship.
The more optimized the fluid distribution is around
the bit, the higher the ROP due to hole cleaning.
Fluid distribution around the bit Fluid forced through the teeth of the bit will cause
better distribution, ensure circulation of the cuttings,
and therefore, clean the hole.
Tables 3-1 and 3-2 show that many variables affect ROP. This implies that ROP can vary
highly from well to well. Therefore, the proposed method should work effectively for a wide
range of velocities.
3.2 Directional Drilling Automation
Drilling automation is a complex problem that has attracted many researchers. Over the
years, various studies have examined ways to automate drilling operations. Many factors need to
be analyzed to have a fully automated drilling system. Drilling automation can be defined and
classified through three different terms (Annaiyappa 2013):
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1. Machine Automation: this process is achieved when mechanical power replaces
human labor. Equipment from various vendors should be integrated to achieve the
same outcome that humans can do.
2. Drilling Process Automation: this process is achieved when equipment from the rig
and service companies can work together without manual instructions to the driller.
3. Drilling Advisory Automation: this process is achieved when drilling parameters(cid:182)
computations are integrated with the rig control system instead of being handed to the
driller manually.
In the earliest times of drilling operations, very simple equipment was required to start
the drilling job. A big part of the drilling procedure depended on heavy human labor. Human
safety was a significant concern because of the unguarded equipment and hazardous operations
occurring during drilling. Safety, as well as the intention of decreasing labor costs, motivated
automating drilling operations (Eustes 2007).
The main objective of drilling automation is to increase rig safety, maintain operations
during harsh weather conditions, reduce the overall costs, reduce people on the rig and control
the rig remotely, improve drilling efficiency and ROP, and reduce environmental footprint
(Eustes 2007).
A typical onshore drilling rig has seven different systems working together. These are the
rig power system, the hoisting system, the circulating system, the rotary system, the well-control
system, the trajectory control system, and the well-monitoring system (Bourgoyne et al. 1986).
The rig power system provides power to the rig through internal combustion diesel engines. The
hoisting system is mainly used to lower or raise casing strings, drillstrings, or other subsurface
equipment. The circulating system ensures the removal of the formation(cid:182)s cuttings by circulating
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fluid around the wellbore. The rotary system applies torque to the drilling bit and allows rotation
for cutting. The well-control system maintains uncontrolled formation fluids within the wellbore
and prevents unwanted fluid flow to the surface. The trajectory control system (e.g., a rotary
steerable system) allows drilling of a specific well path (e.g., a deviated well). Finally, the well-
monitoring system allows the driller to continually monitor information such as depth, rotary
speed, rate of penetration, and directional drilling data (Bourgoyne et al. 1986; Schaaf et al.
2000a).
A fully automatic drilling system can only be achieved when all these systems are
working together automatically in an efficient and unified manner. The focus of this dissertation
is the trajectory control system and its progress towards directional drilling automation.
Therefore, only rotary steerable systems are discussed.
3.2.1 Rotary Steerable Systems (RSS)
RSS is a type of trajectory control system used to automate directional drilling
operations. RSS was initially developed for Extended Reach Drilling (ERD) to prevent problems
caused by drag (Schaaf et al. 2000a). Today, RSS is used for performance drilling, geosteering,
and improved hole cleaning (Schaaf et al. 2000b; May et al. 2015). RSS is classified under two
categories: (1) point-the-bit systems and (2) push-the-bit systems (Schaaf et al. 2000a).
Point-the-bit systems have a rotating shaft within a non-rotating stabilizer that can be
deflected at an angle away from the wellbore axis. This type of system works by placing a bend
along the trajectory. The system is divided into three components: a power generation section,
electronics and sensor section, and steerability enhancement section (Figure 3-1) (Schaaf et al.
2000a; Schaaf et al. 2000b).
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Figure 3-1 A schematic showing the different components of point-the-bit rotary steerable
system (Schaaf et al. 2000a).
The power generation section provides power for the tool to work. The steering section
uses the weight and torque on the bit. Additionally, it allows the system to move at an angle that
offsets the tool(cid:182)s axis. The electronics and sensor section allows control of the system(cid:182)s
direction. Sensors, such as magnetometers and gyroscopes, analyze the coordinates of the current
trajectory and compare it with the target(cid:182)s coordinates. A signal is then sent to the steering
section via the electronics to allow the drilling trajectory to reach the target(cid:182)s coordinates. In
summary, the system allows continuous data feedback and control of the trajectory until the
desired coordinates are reached. (Schaaf et al. 2000a; Schaaf et al. 2000b)
Push-the-bit systems include steering pads that perform a side-force on the formation,
which facilitates directional drilling. These steering pads can be positioned on either a rotating or
a non-rotating housing (Schaaf et al. 2000a; Bryant 2019).
Push-the-bit systems are usually divided into four sections: head, power, control, and
MWD system. The RSS head contains annular ports, oil fill ports, and hydraulic steering pads.
The annular ports prevent excessive pressure buildup in case the bit gets plugged. A centralizer
channels the mud to the hydraulic oil fill ports. The oil is pressurized and forces pistons to
activate a specific number of pads (Bryant 2019).
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Secondly, the RSS power contains batteries or a turbine that provides electrical power for
the system to work. Thirdly, the control system contains sensors that analyze the coordinates of
the drilled trajectory and control the number and location of pads activated based on the
predefined trajectory. Finally, the MWD system provides two-way communication between the
surface and the bottomhole to allow the RSS to drill across the preset drilling path (Bryant 2019).
This research intends to provide the RSS with a bottomhole MD measurement to prevent
the necessity of communication between the surface and the bottomhole. The distance calculated
by the proposed method should be sent to the RSS, where the drilled and predefined trajectories
are compared. Then, the RSS should be able to continue drilling the preset path without waiting
for and depending on new surface distance measurements.
3.3 Logging While Drilling
The proposed method includes an IWD tool used to estimate the distance drilled at the
bottom of the well and provide real-time ROP data. Since the IWD tool is a logging tool, logging
while drilling techniques are discussed in this section. This section focuses on how information
from the LWD tool is collected, processed, and sent to the surface. Finally, it discusses how
imaging while drilling tools operate.
LWD is a method of gathering data while the well is being drilled. A data-gathering tool,
such as a resistivity tool, is attached above the drilling bit in the bottomhole assembly (Figure 3-
2). This design makes the tool follow the drilling bit (Allen et al. 1989).
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Figure 3-2 A schematic showing LWD and MWD tools combined. Here, the type of LWD tool is
a Compensated Dual Resistivity (CDR) tool that measures the resistivity of the formation (ODP
Legacy 2000).
The design of the LWD tool allows it to survive the harsh conditions occurring downhole
while drilling. It can withstand problems such as a bent or a vibrating bottomhole assembly and
still transmit accurate data (Allen et al. 1989). The LWD is almost always accompanied by
another tool and method called MWD that provides information regarding the azimuth and
inclination of the well.
Any information gathered from the tool is sampled with respect to time, for example, 0.1
Hz (Fagin 1994). The sampling rate can only be set on the surface and cannot be changed unless
the tool returns to the surface. Dividing the sampling rate by the ROP results in the number of
samples generated for each interval distance drilled. Since ROP values change throughout
drilling, the sampling rate must be altered to account for these changes. Therefore, when ROP
increases, samples per foot must increase. Typically, the sampling criteria can range from
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capturing one image every 10.0 s to capturing one image every 120.0 s, depending on the ROP
and the desired amount of data (Allen et al. 1989).
There are two ways that the engineer can get the information captured by the LWD: (1)
mud-pulse telemetry (MPT) system and (2) memory chip. If data are wanted in real-time, then
the MPT system is used to send them to the surface. In contrast, nonessential data are stored in
the LWD's memory chip and can be read when the tool is brought to the surface. Figure 3-3
shows an example of LWD data sent to the surface for real-time analysis (Allen et al. 1989;
Hutin 2017).
Figure 3-3 Example of logging data where the plots show from left to right: (1) gamma ray, (2)
resistivity, (3) travel time, (4) bulk density, and (5) total organic carbon (Mahmoud et al. 2017).
The MPT system allows for two-way data transmission between the surface and the
bottomhole. Firstly, the drilling fluid is forced through a valve, where the flow is restricted, and
pressure pulses are generated. These pressure pulses represent the data measured in the
bottomhole, which are then sent to the surface and read by a pressure transducer that decodes the
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frequency (𝜔), cut-off frequency (𝜔 ), number of poles (n), and the maximum passband
𝑐
gain (𝜀). After removing the high frequencies representing the noise, the signal can then be
reconstructed in its denoised form. There are also other types of filters similar to the Butterworth
filter that can denoise the signal.
3.3.2 Imaging While Drilling
As mentioned before, the proposed method relies on taking images of the formation, and
therefore, an IWD tool is needed. IWD is a process of measuring real-time images of the
wellbore while it is being drilled. These images can be taken from different formation evaluation
measurements and can be in the form of ultrasonic, resistivity, density, electromagnetic, or
gamma-ray images.
As of 2021, various service companies have provided different measurement tools and
methods for imaging the wellbore. The main reason for performing IWD is to have an image that
reveals the formation and its geological features. Once this image has been generated, various
engineering analysis techniques can be performed to understand the characteristics of the
formation that is being drilled.
Schlumberger researcher Montaron (1996) was one of the first researchers to develop a
real-time wellbore imaging tool and method using MWD telemetry. Following this breakthrough
invention, other service companies started manufacturing IWD tools. The IWD tool (Figure 3-5)
works by measuring real-time formation characteristics, such as gamma-ray measurements, using
logging sensors. These measurements are made around the whole wellbore (i.e., 360 degrees)
and are then compressed using a downhole processor and transmitted to the surface via MWD
telemetry (Montaron 1996).
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Figure 3-5 A schematic showing the different parts of an IWD tool. The azimuthal gamma ray is
capturing information for the full circumference (360 degrees) of the wellbore (Columbia
University 2020; Schlumberger 2000).
Once the compressed data have reached the surface, each data point is combined with its
toolface position and azimuthal orientation. Each datum is assigned a translational distance value
where the data points are correlated to their depth position. Finally, the captured data are then
decompressed and converted to pixels. The pixel arrangement generates a high-resolution
resistivity image of the formation (Montaron 1996; Fulda et al. 2007).
IWD tools have many different field applications. Bazara et al. (2013a) used the IWD
tool to identify fractures causing excessive lost circulation. The tool provided engineers with
ultra-high-resolution images of the borehole. Based on these images, the engineers could
characterize the geometry, morphology, and conductivity of each fracture, and calculate the
structural dip. The analysis of the fractures assisted them in finding solutions to remediate lost
circulation.
Bazara et al. (2013b) used the IWD tool for geological steering, often known as
geosteering, which is the art of drilling the well based on well logging data instead of three-
dimensional targets in space. Since the IWD tool provides a method formation evaluation and
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structural dip calculation, the engineers could geosteer the well more accurately than when using
standard LWD tools (Bazara et al. 2013b)
For this research(cid:182)s purpose, the IWD tool is a source of measuring the distance drilled.
Therefore, instead of correlating the images to depth like other typical applications, the images
will be correlated to time. Additionally, the type of sensors used should be ultrasonic to show the
topology of the borehole wall. A good example of an ultrasonic IWD is the TerraSphere tool, a
registered trademark of Schlumberger. The tool has four focused pulse-echo transducers and
generates high-resolution images of the wellbore(cid:182)s topological features.
3.4 Summary
This chapter discussed ROP, drilling automation, RSS, and IWD. The proposed method
is designed to capture images using an IWD, calculate the ROP, and then integrate it to estimate
the distance drilled. Combining this distance-calculation process with an RSS that measures the
inclination and azimuth, the bit(cid:182)s location can be found. Continuously locating the bit at the
bottom of the hole facilitates directional drilling automation.
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CHAPTER 4
RESEARCH PLAN
This chapter discusses the operating principle of the proposed method, the design of the
image-matching algorithm, and the different experimental procedures performed.
4.1 Operating Principle
The method proposed in this dissertation calculates the distance drilled in the bottomhole
using identical imaging sensors. The concept proposed will be explained using an example of a
theoretical imaging tool (Figure 4-1). The sensors on this tool are imaging sensors that capture
formation images and are placed within a known distance of each other (e.g., D1). Additionally,
while the figure shows a 2D drawing of the tool, a 3D illustration would show the locations of
four different sensors located around the tool to capture 360° images. In addition to the sensors,
the tool contains a Programmable Memory Chip (PMC) with a key-point identification
algorithm, such as SIFT, to locate key-points for all images taken. The PMC also contains a
pattern-matching algorithm, such as FLANN, which compares images taken from one sensor
with images taken from the other sensor until a good match is reached.
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where 𝐶 is the depth of the casing shoe, D1 is the distance between Sensor 1 and Sensor 2, and
𝐷
𝐿 is the total length of the drilling bit, drill collar, and PMC.
𝑏
Additionally, because the timestamps of each image are known, the PMC can identify the
difference between the times that both sensors took an image of Location “A(cid:180) and use the
distance between the sensors to calculate the average ROP for that time interval using the
following equation:
𝑅𝑂𝑃 (cid:3404)
T(cid:3177)(cid:3118)−𝑇(cid:3294)(cid:3117)
(4.2)
𝐷1
where T is the time that Sensor 2 captured an image of Location “A(cid:180) and T is the time that
s(cid:2870) s1
Sensor 1 took an image of Location “A(cid:180).
The method described above (Method 1) is straightforward, and the process can be
repeated many times until the drilling operation is terminated. However, there are two main
challenges associated with this method:
1. The total MD consists of a summation of the discrete calculations representing the
distance drilled. Each discrete calculation is equal to the distance between the sensors,
and therefore, drilled distances that are shorter than the distance between the sensors
cannot be computed.
2. If Sensor 2 does not capture the image needed to match the image from Sensor 1, a
“match miss(cid:180) is introduced into the system. This miss will jeopardize the whole MD
measurement because the tool does not know its location. A match miss can also be
introduced in the system from a noisy environment that alters one of the image(cid:182)s
features, and the algorithm cannot recognize the pair as the optimum match.
Based on the challenges mentioned above, the algorithm should use a different method
that calculates the distance drilled to prevent disruptions. Therefore, both Sensors 1 and 2 will
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capture images at a specific rate, allowing them to constantly find optimum image pairs and
calculate many rates of penetration (Method 2). Subsequently, the algorithm will compute the
following equation,
MD (cid:3404) 𝐶 (cid:3397) 𝐿 (cid:3397)∫𝑅𝑂𝑃𝑑𝑡 (4.3)
𝐷 𝑏
and constantly integrate the ROP, continuously providing an estimate of the distance drilled. The
accuracy of Method 2 depends on the sampling rate and sensor location but is not as affected by
match misses compared to Method 1.
In some cases, more than two sensors may be needed, especially during high drilling
rates, to ensure image-matching. This is because one sensor may miss capturing an essential
image due to the high speed of drilling. Therefore, adding multiple sensors can assist in
mitigating missed matches because it increases the likelihood of at least one sensor capturing an
image similar to the first sensor(cid:182)s image
The proposed method will decrease the RSS dependence on the measured depth
calculated from the surface. The RSS will frequently obtain the distance drilled, process all
required data to determine the current coordinates, and take the appropriate actions to guide the
trajectory to the target with minimum dependence on surface measurements. This method is not
designed to entirely cut off communications with the surface. Information will always be sent to
the surface, and if unexpected events occur, such as the tool being incapable of finding a good
match, then surface corrections will be requested by the tool.
To summarize, the sensors are continuously imaging, registering, analyzing (obtaining
the “fingerprints(cid:180)), and comparing the fingerprints and times as a moving window. This
procedure allows a continuous measurement of the ROP. Continuously integrating the ROP
produces an estimation of the total distance drilled.
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4.2 Experimental Design and Procedure
This section explains and describes the experimental design and procedures made to
evaluate the concept of the method proposed in this dissertation.
4.2.1 General Design of Experiments
An IWD tool with multiple similar sensors is not currently designed or readily available.
Therefore, the theoretical tool illustrated in Figure 4-1 is redesigned using cost-effective
laboratory equipment that tests this concept. For this simple design, optical cameras will be used
to capture images of natural or artificial “sceneries(cid:180) representing the borehole wall instead of
ultrasonic imaging sensors capturing images of a drilled well. It was mentioned previously in
Section 2.3 that the type of imaging sensor required should be able to identify the borehole wall
topology. Based on this, it was proposed to use an ultrasonic IWD because ultrasonic images
could give information about fractures and rock geometry, among others. Similarly, when
captured in clear fluids, optical images are highly accurate and show the geometric features
required for the image-matching process. It is understood that the use of optical cameras during
drilling operations is challenging because most drilling fluids are not transparent to light.
Nevertheless, using optical cameras to prove the concept is justified because both ultrasonic and
optical images provide similar features required for the key-points identification (Figures 4-4 and
4-5).
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Figure 4-9 Scenery 4: three actual ultrasonic image well logs (CT-GW 22 (left), US-53-21-703
(middle), and G-2996 (right)) combined (USG 2021).
4.2.2 Image-Matching Algorithm
The image-matching algorithm finds the fingerprint of each image taken from the
different sensors and uses the equations described in Chapter 2 to find a match. The algorithm
processes the data until a match occurs. Then, it extracts the images' timestamps and uses the
distance between the sensors to calculate the velocities. Finally, it integrates the velocities to
compute the distance traveled. The algorithm is designed to work in real-time or non-real-time,
depending on the type of experiments made. The steps of the algorithm are:
Step 1: All the experiments use optical cameras. In some experiments, videos are
recorded, and in others, the algorithm captures images at a defined sampling criterion (e.g., one
image every one second).
Step 2: If images are captured and not videos, then this step is skipped. The videos taken
by the cameras have a constant frame rate of 30 frames per second. This step in the algorithm
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applies a preset sampling criterion to extract the images from the videos and assign a timestamp
to each image. For example, if the videos taken by Sensors 1 and 2 are of 60.0 s duration, then
the videos can be subsampled at a rate of one image every 2.0 s. Therefore, a total of 30 images
is taken by each camera, where Image 1 from both sensors will have a timestamp of 2.0 s.
Step 3: The SIFT algorithm combined with FLANN and a GRT of 0.45 are used to match
the images from the different sensors. The algorithm is based on the theory that the match rate
will increase to the optimum match and then decrease. Therefore, one image from Sensor 1 is
matched with images from Sensor 2 until the match rate starts declining. The code waits and
matches a predefined number of images after the decrease to ensure that this is the maximum
point. The algorithm then moves on to the next image pair, and the process keeps repeating until
the experiment/drilling operation is terminated.
SIFT and a GRT value of 0.45 were chosen based on preliminary testing that used
multiple sceneries and images. The SIFT method provided the highest accuracy in terms of
match ratings and had considerably low computational times. The GRT of 0.45 ensures that only
similar images match, while everything else gives a match rate very close to 0%. The following
equation is used to calculate the GRT match rate:
𝐺𝑅𝑇 𝐺𝑜𝑜𝑑 𝑀𝑎𝑡𝑐ℎ𝑒𝑠
𝐺𝑅𝑇 𝑀𝑎𝑡𝑐ℎ 𝑅𝑎𝑡𝑒 (cid:4666)%(cid:4667) (cid:3404) (cid:3400)100 (4.4)
(cid:3286)(cid:3291)(cid:3294)(cid:3117)(cid:3126) (cid:3286)(cid:3291)(cid:3294)(cid:3118)
(cid:3118)
where 𝐺𝑅𝑇 𝐺𝑜𝑜𝑑 𝑀𝑎𝑡𝑐ℎ𝑒𝑠 is the good matches calculated by the Good Ratio Test, 𝑘𝑝𝑠 is the
1
number of key-points computed using the image from the first sensor, and 𝑘𝑝𝑠 is the number of
(cid:2870)
key-points computed using the image from the second sensor.
It is important to note that this equation only holds if images have key-points. If the SIFT
algorithm cannot find key-points for at least one of the images, then both images are disregarded,
and the algorithm analyzes the next pair of images.
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After the interpolation, the algorithm finds the timestamp that gave the highest match
rate. In this case, the interpolation shows that the match happened at 23.56 s and not 24.0 s. This
interpolation estimates the time the match occurred since the sampling criterion may have missed
it.
Step 5: The algorithm checks for any times that the cameras were stationary. For
example, suppose that two images from Sensor 1 at times 8.0 s and 6.0 s matched with the same
image from Sensor 2 at time 10.0 s. This means that from time 6.0 to 8.0 s, Sensors 1 and 2 were
stationary and not moving because they were capturing the same images for this period. In this
case, Sensor 1 at time 6.0 s matches with Sensor 2 at 10.0 s, and the match that occurred at time
8.0 is disregarded because the object was stationary.
Another way to test for the stationary position is to match Sensor 2 images together and
ensure that the match rate changes. If the match rate is constant, then the object is stationary.
Only the technique mentioned in the example is used for this research, not the Sensor 2 image-
matching method.
Step 6: The following equation is used by the algorithm to calculate the velocity:
Distance (cid:2886)etween (cid:2903)ensors
v(cid:4666)t(cid:4667) (cid:3404) (4.5)
𝑇𝑠(cid:3118)−𝑇𝑠(cid:3117)
For example, suppose the algorithm found the matched timestamps presented in Table 4-1.
(cid:2873) cm cm
Therefore, the velocity of the row highlighted in blue is v(cid:4666)t(cid:4667) (cid:3404) (cid:3404) 0.9 . Here, the
(cid:2870)3.(cid:2873)(cid:2874) s−1(cid:2876).00 s s
distance between the sensors is assumed to be 5 cm.
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Table 4-1 An example of matched data gathered from an experiment.
Timestamp of Sensor 1 (s) Timestamp of Sensor 2 (s)
0.00 2.28
4.00 9.62
6.00 12.91
14.00 20.24
18.00 23.56
22.00 28.12
28.00 35.49
40.00 43.12
Step 7: The algorithm uses the following equation to calculate the distance traveled
(trapezoidal integration):
D (cid:3404) D (cid:3397) 1 (cid:4670)𝑣 (cid:3397)𝑣 (cid:4671)(cid:3427)𝑡𝑖 (cid:3398)𝑡𝑖−1(cid:3431) (4.6)
i i−1 𝑖 𝑖−1 𝑠(cid:2870) 𝑠(cid:2870)
(cid:2870)
where D is the current distance traveled, D is the previous distance calculated, 𝑣 is the
i i−1 𝑖
velocity calculated at the current match, 𝑣 is the velocity calculated at the previous match, 𝑡𝑖
𝑖−1 𝑠(cid:2870)
is the timestamp of the image taken by Sensor 2 for the current match, and 𝑡𝑖−1 is the timestamp
𝑠(cid:2870)
of the image taken by Sensor 2 for the previous match. Finally, the algorithm assumes a linear
interpolation to connect the distance data points together.
4.2.3 Experiment 1 - Image-Matching Code Accuracy and Key Variables
This section explains the procedures performed to evaluate the effect of different
variables on the image-matching algorithm and the distance calculation. The main objective of
this experiment is to test the effectiveness of the algorithm. This experiment aims to determine if
the algorithm can accurately differentiate between the images captured from similar sensors and
correctly find the optimal match for each image. Additionally, it seeks to identify the factors and
variables that affect the distance calculation. Those factors can be sampling criteria, the distance
between sensors, noise, and match misses. This experiment is not meant to generate accurate
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Step 4: Hold the frame above the scenery and ensure that the camera lens on the first
cellphone (i.e., Sensor 1) is vertically aligned above the start marker (Figure 4-15). At this point,
start recording a video using both cameras at the same time.
Figure 4-15 Performing Experiment 1 by holding the frame above the started marker and moving
towards the end marker.
Step 5: Once the cameras are recording, start moving towards the second marker. Identify
the times that the lens of the second cellphone (i.e., Sensor 2) is vertically aligned above the start
and end markers. This step requires a second person constantly monitoring the position of the
camera relative to the marked points. Keep moving until the end marker and then stop recording
both videos simultaneously.
The movement in this experiment is not controlled or automated and is done randomly,
i.e., a person holding the cameras attached to the gate and walking. The purpose is to see which
variables cause the accuracy to change and not to increase the accuracy of the distance
calculation. Therefore, random movement is not a problem for this experiment.
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Step 6: Upload the videos recorded to the image-matching code for analysis and distance
calculation.
4.2.3.2 Variables Tested
The variables tested in this experiment are:
1) Sampling Criteria: this represents the time interval for which the algorithm captures
one image. The sampling criteria tested for the granite rock test was capturing one image every
2.0 s, 4.0 s, 6.0 s, and 8.0 s. The sampling criteria tested for the white-painted wall test were
similar but with an additional criterion of 10.0 s.
2) Missing Matches: the algorithm forces images not to match based on preset commands
that control a miss rate. For example, suppose the algorithm found an optimal match between
two images. If the miss rate is set to missing every other match, the algorithm is forced not to
match the second pair of images. The different miss rates tested were missing one match every
other match, two matches, five matches, and ten matches.
3) Distance Between Sensors: the distance that the sensors are placed away from each
other is fixed. However, this test aims to know what happens if the sensors are placed further
away or closer to each other. The distances between the sensors tested for the granite rock test
were 8.3 cm, 11.0 cm, and 13.3 cm. The distances between the sensors tested for the white-
painted wall were 11.0 cm, 15.0 cm, and 22.0 cm.
4) Random Noise: this test evaluates the effect that random noise has on the accuracy of
the distance calculation. It aims to find if noise will prevent the algorithm from finding accurate
matches. Additionally, it aims to see if an image denoiser can retrieve the image's key-points and
maintain the accuracy of the distance calculation. Therefore, this test contains three parts:
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4a) A normal image-matching process where no noise is added, and this is the control
experiment.
4b) Salt and pepper noise is randomly added at a 0.55 ratio to all images from all sensors
and compared to the results obtained from the control experiment (Figure 4-16). This is a type of
noise added in an image that presents itself as white or black pixels, hence the name.
Figure 4-16 Example of salt and pepper noise added to the experiment.
4c) The noisy images are denoised and compared to the control experiment.
Figure 4-17 shows a normal, noisy, and denoised image. The denoising process is made
by applying a 2D median filter to restore the image's features and then a bilateral filter. The
average time needed to match two images captured from the granite rock is 1.30 s without a
denoising algorithm and 0.75 s with a denoising algorithm. The average time needed to match
two images captured from the white-painted wall is 0.80 s and 1.20 s with and without a
denoising algorithm, respectively. The technical specifications of the laptop used to run the
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sensors. The distance between the sensors is only controllable before the design phase; however,
it cannot be changed once the sensors are fixed. Contrarily, the algorithm can continuously adjust
the sampling criterion based on preset commands.
Because the distances between the sensors are fixed, it is crucial to ensure that they will
be compatible with different velocities. For example, suppose the sensors are very close to each
other, and the tool is traveling at a relatively high velocity. In that case, there is a high chance
that one of the sensors will miss the image that gives a good match unless an optimal sampling is
chosen. Conversely, if the sensors are separated by long distance and the velocity is very low, it
will take a considerable amount of computer power to process and match the optimal images
unless the optimal sampling is chosen again. Additionally, a combination of low velocity and a
long DBS may cause errors because false positives can occur due to the increased number of
images used in the image-matching process, or misses can occur due to changes in wellbore
conditions. As a result, the distance between the sensors and the sampling needs to be optimized
to minimize errors.
To design an optimized system, at least three sensors are needed to provide three
different distances between the sensors. One of the distances will be short, one will be medium,
and one will be long. Additionally, each distance between the sensors will require a specific
sampling criterion or a range of optimum sampling criteria that mitigates misses and minimizes
errors. The combination of optimum sampling criteria and optimum distance between the sensors
will change when the velocity changes. Therefore, this experiment aims to find the optimum
combination of sampling criterion and distance between the sensors for various velocities.
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4.2.4.1 Experimental Procedure
Experiment 2 uses Scenery 3 mentioned in Section 4.2.1. The procedure for this experiment is:
Step 1: Build a motorized dolly cart with an adjustable speed controller (Figure 4-18).
The dolly imitates the theoretical IWD tool illustrated previously in Figure 4-1. Two different
motors with different rotations per minute (6 RPM and 20 RPM) were used interchangeably in
this experiment to allow a wide range of velocities to be tested. The speed controller enables the
velocity to be adjusted accordingly.
Figure 4-18 A motorized dolly cart with an adjustable speed controller.
Step 2: Mount Polyvinyl Chloride (PVC) pipes to the ground using a mounting putty. The
dolly is designed to travel on PVC pipes to ensure a smooth straight trajectory so that the cart can
run reliably with minimum interruptions when set to constant speeds.
Step 3: Attach the three cellphone cameras at predetermined locations to account for the
various distances between the sensors and place the dolly correctly on the pipes (Figure 4-19).
The distances between the sensors tested in this experiment are 15.0 cm, 30.0 cm, and 45.0 cm.
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Step 5: Subsample the videos using different sampling criteria because the experiment is
not performed in real-time. The sampling criteria tested in this experiment are 2.0 s, 5.0 s, 12.0 s,
20.0 s, and 30.0 s.
Step 6: Compare the average velocity calculated by the algorithm over the total 5-minute
run with the average velocity of the cart measured using the stopwatch.
Step 7: Calculate the average velocity and the percentage error for each combination of
sampling criterion and distance between the sensors. For example, suppose that the average
velocity of the cart is 63 ft/hour. Then, suppose that for the first and second runs, the algorithm
calculated many velocities using various timestamps from matched images (Table 4-2). Those
velocities (for each run) are then integrated with respect to time to calculate the total distance
traveled over the whole run. Therefore, dividing the total distance traveled by the total time
results in the average velocity of the whole run. Since two runs were made, the average velocity
calculated by the algorithm when the cart is moving at 63 ft/hour will be the mean of both
average velocities calculated from the two runs.
Table 4-2 Velocities calculated for two runs.
Velocity - first run (ft/hour) Velocity- second run (ft/hour)
58.20 59.38
58.39 59.45
58.62 59.74
58.32 60.70
58.43 60.92
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4.2.5 Experiment 3 - Real-time Distance Calculation on Actual Well Logs
Experiment 2 provided an optimum combination of sampling criterion and distance
between the sensors for each average velocity. For this experiment, the image-matching
algorithm is modified by including the optimal combinations from Experiment 2. Afterward,
various random velocities are tested during the same run to see if the algorithm will choose the
correct combination and correctly estimate the distance. This experiment was performed in real-
time on Scenery 4 described in Section 4.2.1. Additionally, three tests were made in this
experiment:
1. Test 1 is made at a constant velocity different from the velocities tested in Experiment
2. Figure 4-20 shows the constant velocity of the cart at an average velocity of 89
ft/hour. Note that the constant velocity test is not always maintained due to rough
velocity changes occurring at the pipe connections. This is a noteworthy limitation of
the experimental design and is demonstrated at time 280.0 s in the figure.
2. Test 2 is made at random velocities by constantly adjusting the speed controller.
Figure 4-21 shows the random velocity of the cart.
3. Test 3 is made at random velocities but is different from the second test. Figure 4-22
shows the random velocity of the cart.
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for low velocities. For example, suppose sampling at one image every 2.0 s and separating the
sensors by 15.0 cm provides the lowest errors for velocities below 60 ft/hour. In that case, the
algorithm will initially only match the images using this combination and calculate the velocity.
Therefore, even though the webcams capture one image every one second, the algorithm will
only match images at a two-second interval using the sensors separated by 15.0 cm. The
algorithm will keep all the other captured images stored but will not attempt to match them.
Once the velocity is calculated, the algorithm compares the velocity with the results
obtained from Experiment 2 and sees if the initial guess of the optimum combination of sampling
criterion and distance between the sensors is correct. If the initial guess is correct, then the
algorithm calculates the distance using Step 7 in Section 4.2.2. Conversely, if the initial guess of
the optimum combination is incorrect, the algorithm repeats the process using the correct
combination and the images stored. It then calculates the distance using Step 7 in Section 4.2.2.
This whole process is repeated until the cart comes to a complete stop.
To summarize, Experiment 1 helps in understanding the key variables that affect the
image-matching code. Once the key variables are known, Experiment 2 analyzes the controllable
variables to minimize errors and computational time. Finally, Experiment 3 tests the model
developed in Experiment 2 in real-time on real ultrasonic imaging logs to prove that the distance
can be calculated using an image-matching process and that misses can be minimized when
variables are optimized.
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CHAPTER 5
RESULTS
This chapter presents the results obtained from the three experiments described in
Chapter 4, and demonstrates the effects of different variables such as velocity, the distance
between the sensors (DBS), and sampling criteria on the accuracy of the distance calculations.
Additionally, it presents the error analysis performed on the relationship between sampling,
DBS, and velocity and shows the final model developed to optimize the image-matching
algorithm and minimize errors. Finally, the method's efficacy is demonstrated here by presenting
the results obtained from the real-time tests performed on real well logs obtained by acoustic
televiewers.
5.1 Experiment 1 – Image-Matching Code Accuracy and Key Variables
This section discusses the results obtained from Experiment 1. This experiment evaluates
the effects of noise, sampling, the distance between the sensors, and multiple sceneries on the
accuracy of the image-matching algorithm and distance calculation.
The results presented from Experiment 1 are demonstrated in terms of percentage errors
on a box and whisker plot. The symbol "X" represents the average data point, and "O"
represents internal points and outliers. The median is represented as a line and, for this
experiment, is always equal to the average because only two data points are used for each box.
Finally, when reading the graphs, a negative percent error means that the distance is
undercalculated, and a positive percent error means that the distance is overcalculated. The
following equation shows the percentage error calculation:
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Percentage Error (cid:3404)
𝐷(cid:3278)−𝐷(cid:3264)
(cid:3400)100 (5.1)
𝐷(cid:3264)
where 𝐷 is the distance calculated by the algorithm; and 𝐷 is the actual distance traveled and
𝑐 𝑂
checked by a measuring tape. Figure A-4 shows the general labels related to box and whisker
plots.
5.1.1 The Effect of Sampling and the Distance Between the Sensors
As presented in Section 4.2.3.2, the distance between the sensors was varied to observe
its effect on the distance calculation. The experiment was performed on two sceneries: the
granite rock and the white-painted wall.
5.1.1.1 Granite Rock
The objectives of this test are to see: (1) if the image-matching algorithm can correctly
match images captured from Scenery 1 (i.e., granite rock described in Section 4.2.1), and (2)
how varying the sampling criterion and the distances between the sensors affect the accuracy of
the distance calculation. The test was repeated once to give a rough estimate of the average
percentage error and the range of percentage errors for each combination of DBS and sampling
criterion. It is understood that two data points do not represent the true average percentage error.
However, this test only aims to understand how the DBS and sampling affect the algorithm.
Performing statistical analyses or acquiring accurate distance calculations are not the objectives
of this experiment.
To complete the first objective, all the images matched by the algorithm in the first run
were visually inspected and verified that they indeed matched. This inspection was made for all
sampling criteria and all the distances between the sensors. As a result, it was demonstrated that
the algorithm matched the granite images correctly.
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Table 5-1 shows the average percentage error from the two runs for each DBS and
sampling criterion combination. The table shows that the average error significantly varies when
the sampling criterion or the DBS changes. For example, for a DBS of 8.3 cm and a sampling
criterion of 6.0 s, the distance is overcalculated by 2.3%. However, if the DBS is kept the same
but the sampling changes to 8.0 s, the distance is undercalculated, and the error increases to an
absolute value of 11.9%.
Table 5-1 Granite rock DBS test - average percentage error for each combination of sampling
criterion and distance between the sensors.
Sampling Criteria 8.3 cm 13.3 cm 15.0 cm
2.0 s -4.4% -2.7% 12.5%
4.0 s -5.0% -5.7% 7.1%
6.0 s 2.3% -5.3% 6.2%
8.0 s -11.9% 2.9% -4.2%
Table 5-2 shows the standard deviation of the percentage errors calculated from the two
runs for each combination of DBS and sampling criterion. The table demonstrates that the range
of errors can also significantly vary when the sampling criterion or the DBS changes. A low
standard deviation suggests that the errors have a narrow range and are controlled. Conversely, a
high standard deviation indicates a wide range of errors that vary unexpectedly.
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Table 5-2 Granite rock DBS test – standard deviation of percentage error for each combination
of sampling criterion and distance between the sensors.
Sampling Criteria 8.3 cm 13.3 cm 15.0 cm
2.0 s ± 3.7% ± 8.7% ± 10.4%
4.0 s ± 3.2% ± 7.4% ± 7.6%
6.0 s ± 2.4% ± 5.4% ± 9.1%
8.0 s ± 2.3% ± 0.5% ± 4.9%
To visualize the results and qualitatively analyze the range of percentage errors, Figures
5-1 to 5-4 show the effect of the DBS on the distance calculation when sampled at one image
every 2.0 s, 4.0 s, 6.0 s, and 8.0 s, respectively.
When sampling at one image every 2.0 s, Figure 5-1 shows that the average percentage
error increases and shifts from negative to positive as the DBS increases. It can also be seen that
the range of errors increases when the DBS increases. Here, a DBS of 13.3 cm has the lowest
average percentage error when calculating the distance; however, the range of errors is wide.
Therefore, a DBS of 13.3 cm may look good, but it is a bad choice for this specific sampling
criterion.
Sample Rate = 1 Image/2 seconds Sampled at 1Image every 2 seconds Sample Rate = 1 Image/2 seconds
20
44
15 42
10 40 Distance between
Distance between sensors = 8.3 cm
38 5 sensors = 8.3 cm Distance between
Distance between 36 sensors = 13.3 cm
0
sensors = 13.3 cm Distance between
34
-5 Distance between sensors = 15 cm
32
sensors = 15 cm Tape Measurement =
-10
30 33.5 cm
-15 28
-20 26
Figure 5-1 Granite rock DBS test - the effect of the distance between the sensors on the accuracy
of the distance calculation when sampled every 2.0 s.
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When sampling at one image every 4.0 s, Figure 5-2 shows that the range of errors
increases when the DBS increases. When comparing the results to Figure 5-1, the average
percentage error for a DBS of 8.3 cm and a DBS of 13.3 cm increases and becomes more
negative. Additionally, when the sampling changes from 2.0 s to 4.0 s, the average percentage
error decreases for a DBS of 15.0 cm and becomes less positive. For this sampling criterion, a
DBS of 8.3 cm provides the lowest range of errors.
Sample Rate = 1 Image/4 seconds Sampled at 1Image every 4 seconds Sample Rate = 1 Image/4 seconds
20 40
15 38
Distance between 10
Distance between 36 sensors = 8.3 cm
5 sensors = 8.3 cm Distance between
34
Distance between sensors = 13.3 cm
0
sensors = 13.3 cm Distance between
32
-5 Distance between sensors = 15 cm
sensors = 15 cm 30 Tape Measurement =
-10
33.5 cm
-15 28
-20 26
Figure 5-2 Granite rock DBS test - the effect of the distance between the sensors on the accuracy
of the distance calculation when sampled every 4.0 s.
When sampling at one image every 6.0 s, Figure 5-3 shows a very different trend than
Figures 5-1 and 5-2. Firstly, a DBS of 8.3 cm or 15.0 cm provides a positive average percentage
error, while a DBS of 13.3 cm provides a negative average percentage error. Secondly, the
average percentage error for a DBS of 8.3 cm shifts to a positive value when the sampling
changes from 4.0 s to 6.0 s. Thirdly, the average percentage error for a DBS of 15.0 cm
decreases when the sampling changes from 4.0 s to 6.0 s. Finally, the average percentage error
for a DBS of 13.3 cm stays approximately the same when the sampling changes from 4.0 s to 6.0
s. For this sampling criterion, a DBS of 8.3 cm provides the most accurate distance calculation.
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Sample Rate = 1 Image/6 seconds Sampled at 1Image every 6 seconds Sample Rate = 1 Image/6 seconds
20 40
15 38
Distance between 10
Distance between 36 sensors = 8.3 cm
5 sensors = 8.3 cm Distance between
34
Distance between sensors = 13.3 cm
0
sensors = 13.3 cm Distance between
32
-5 Distance between sensors = 15 cm
sensors = 15 cm 30 Tape Measurement =
-10
33.5 cm
-15 28
-20 26
Figure 5-3 Granite rock DBS test - the effect of the distance between the sensors on the accuracy
of the distance calculation when sampled at 1 image every 6.0 s.
When sampling at one image every 8.0 s and comparing Figure 5-4 with Figure 5-3, the
average percentage error became negative instead of positive for both a DBS of 8.3 cm and 15.0
cm. Additionally, for a DBS of 13.3 cm, as the sampling changes from 6.0 s to 8.0 s, the average
percentage error becomes positive instead of negative. For this sampling criterion, a DBS of 13.3
cm provides the most accurate distance calculation.
Sample Rate = 1 Image/8 seconds Sampled at 1Image every 8 seconds Sample Rate = 1 Image/8 seconds
20 40
15 38
Distance between 10
Distance between 36 sensors = 8.3 cm
5 sensors = 8.3 cm Distance between
34
Distance between sensors = 13.3 cm
0
sensors = 13.3 cm Distance between
32
-5 Distance between sensors = 15 cm
sensors = 15 cm 30 Tape Measurement =
-10
33.5 cm
-15 28
-20 26
Figure 5-4 Granite rock DBS test - the effect of the distance between the sensors on the accuracy
of the distance calculation when sampled at 1 image every 8.0 s.
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5.1.1.2 White-painted Wall
This test has the same objectives as the granite rock test shown above. All the images
matched by the algorithm in the first run were visually inspected and verified that they indeed
matched. This inspection was made for all sampling criteria and all the distances between the
sensors. As a result, it was demonstrated that the algorithm correctly matched the wall images.
Table 5-3 shows the average percentage error calculated from the two runs for each DBS
and sampling criterion combination. The results show the same behavior demonstrated by the
granite rock test. The average error significantly varies when the sampling criterion or the DBS
changes. For example, a DBS of 15.0 cm and a sampling criterion of 6.0 s caused the distance to
be undercalculated by 1.6%. However, if the DBS is kept the same but the sampling changes to
8.0 s, the distance is overcalculated, and the error increases to an absolute value of 11%.
Table 5-3 White-painted wall DBS test - average percentage error for each combination of
sampling criterion and distance between the sensors.
Sampling Criteria 11.0 cm 15.0 cm 22.0 cm
2.0 s 1.3% 3.0% -4.5%
4.0 s 7.4% 0.8% -3.8%
6.0 s 6.0% -1.6% -6.7%
8.0 s 1.9% 11.0% -15.5%
10.0 s -2.4% 0.5% 0.2%
Table 5-4 shows the standard deviation of the percentage errors calculated from the two
runs for each DBS and sampling criterion combination. The results show that the range of errors
varies when the sampling criterion or the DBS changes. This observation is similar to the granite
rock test; however, the range of errors is lower than the granite rock test. Comparing Tables 5-2
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and 5-4 and focusing only on the 15.0 cm DBS shows that the range of errors has indeed
decreased. A reason for this behavior could be the differences in the distances tested where the
granite rock scenery tests 33.5 cm, while white-painted wall scenery tests 91.5 cm.
Table 5-4 White-painted wall DBS test – standard deviation of percentage error for each
combination of sampling criterion and distance between the sensors.
Sampling Criteria 11.0 cm 15.0 cm 22.0 cm
2.0 s ± 2.0% ± 0.9% ± 5.1%
4.0 s ± 2.9% ± 2.1% ± 4.4%
6.0 s ± 1.9% ± 2.5% ± 2.9%
8.0 s ± 7.2% ± 3.2% ± 0.9%
10.0 s ± 5.2% ± 0.4% ± 1.1%
To visualize the results and qualitatively analyze the range of percentage errors, Figures
5-5 to 5-9 show the effect of the DBS on the distance calculation when sampled at one image
every 2.0 s, 4.0 s, 6.0 s, 8.0 s, and 10.0 s, respectively.
When sampling at one image every 2.0 s, Figure 5-5 shows that the average percentage
error is positive and increases as the DBS increases from 11.0 cm to 15.0 cm. Additionally, the
average percentage error becomes negative and increases when the DBS is 22.0 cm. For this
sampling criterion, a DBS of 11.0 cm has the highest accuracy when calculating the distance, and
a DBS of 22.0 cm has the widest range of errors and least accuracy.
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Sample Rate = 1 Image/2 secondSsampled at 1Image every 2 secondSsample Rate = 1 Image/2 seconds
20 105
15 Distance between
100
sensors = 11 cm
10
Distance between
95 Distance between 5 sensors = 11 cm
sensors = 15 cm
Distance between
0 90
sensors = 15 cm
Distance between
-5 Distance between
85 sensors = 22 cm
sensors = 22 cm
-10 Tape
80 Measurement =
-15
91.5 cm
-20 75
Figure 5-5 White-painted wall DBS test - the effect of the distance between the sensors on the
accuracy of the distance calculation when sampled at 1 image every 2.0 s.
When sampling at one image every 4.0 s, Figure 5-6 shows a downward trend where the
average percentage error shifts from positive to negative as the DBS increases. Comparing those
results with Figure 5-5 shows that changing the sampling from 2.0 s to 4.0 s causes an increase in
the average percentage error when the DBS is 11.0 cm. Additionally, the average percentage
error for a DBS of 15.0 cm and a DBS of 22.0 cm decreases. Finally, the sign of the average
percentage error for each DBS stays the same and does not change. A DBS of 15.0 cm has the
highest accuracy and narrowest range of errors for this sampling criterion.
Sample Rate = 1 Image/4 secondSsampled at 1Image every 4 secondSsample Rate = 1 Image/4 seconds
20 105
15 Distance between
100
sensors = 11 cm
10
5 D si est na sn oc re s =b e 1t 1w ce men 95 Distance between
sensors = 15 cm
Distance between
0 90 sensors = 15 cm
Distance between
-5 Distance between 85 sensors = 22 cm
sensors = 22 cm
-10 Tape
80 Measurement =
-15
91.5 cm
-20 75
Figure 5-6 White-painted wall DBS test - the effect of the distance between the sensors on the
accuracy of the distance calculation when sampled at 1 image every 4.0 s.
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When sampling at one image every 6.0 s, Figure 5-7 shows a similar trend to Figure 5-6.
When the DBS increases, the average percentage error shifts from a positive to a negative value.
Comparing the results from Figure 5-6 with Figure 5-7 shows that changing the sampling from
4.0 s to 6.0 s causes the average percentage error and the range of errors for a DBS of 11.0 cm to
decrease. Additionally, the average percentage error for a DBS of 15.0 cm shifted from positive
to negative and had approximately the same absolute value. Finally, the average percentage error
for a DBS of 22 cm increases in value but maintains the same negative sign.
Sample Rate = 1 Image/6 secondSsampled at 1Image every 6 secondSsample Rate = 1 Image/6 seconds
20 105
15 Distance between
100
sensors = 11 cm
10
Distance between
95
5 sensors = 11 cm Distance between
sensors = 15 cm
Distance between
0 90
sensors = 15 cm
Distance between
-5 Distance between
85 sensors = 22 cm
sensors = 22 cm
-10 Tape
80 Measurement =
-15
91.5 cm
-20 75
Figure 5-7 White-painted wall DBS test - the effect of the distance between the sensors on the
accuracy of the distance calculation when sampled at 1 image every 6.0 s.
When sampling at one image every 8.0 s, Figure 5-8 shows that the results significantly
change compared to the previous sampling criterion. Firstly, for a DBS of 11.0 cm, the total
range of percentage errors in Figure 5-7 are all positive values, while in Figure 5-8, the range of
errors covers both positive and negative values. Secondly, for a DBS of 15.0 cm, the total range
of percentage errors in Figure 5-7 are all negative values, while the range of percentage errors in
Figure 5-8 are all positive values. Finally, for a DBS of 15.0 cm, the range of percentage errors
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are all negative values in both Figures 5-7 and 5-8. Additionally, changing the sampling from 6.0
s to 8.0 s causes the distance calculated by a DBS of 15.0 cm to be less accurate.
Sample Rate = 1 Image/8 secondSsampled at 1Image every 8 secondSsample Rate = 1 Image/8 seconds
20 105
15 100 Distance between
sensors = 11 cm
10
95 Distance between
5 sensors = 11 cm Distance between
90 sensors = 15 cm
Distance between
0
sensors = 15 cm
85 Distance between
-5 Distance between
sensors = 22 cm
sensors = 22 cm
80
-10 Tape
-15 75 Measurement =
91.5 cm
-20 70
Figure 5-8 White-painted wall DBS test - the effect of the distance between the sensors on the
accuracy of the distance calculation when sampled at 1 image every 8.0 s.
When sampling at one image every 10.0 s, Figure 5-9 shows that a DBS of 15.0 cm and
22.0 cm provides an accurate calculation of the distance traveled. However, when the DBS is
11.0 cm, the errors increase, and the distance calculation is less accurate.
Sample Rate = 1 Image/10 secondSasmpled at 1Image every 10 seconSdasmple Rate = 1 Image/10 seconds
20 105
15 100 Distance between
sensors = 11 cm
10
95 Distance between
Distance between 5 sensors = 11 cm
90 sensors = 15 cm
Distance between
0
sensors = 15 cm
85 Distance between
-5 Distance between
sensors = 22 cm
sensors = 22 cm
80
-10 Tape
-15 75 Measurement =
91.5 cm
-20 70
Figure 5-9 White-painted wall DBS test - the effect of the distance between the sensors on the
accuracy of the distance calculation when sampled at 1 image every 10.0 s.
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5.1.2 The Effect of Noise and the Accuracy of Denoising
This section discusses the effect of random speckle noise on the accuracy of the image-
matching code. Here, the sampling is varied, but the DBS is kept constant and is equal to 13.3
cm. The tests were made on two sceneries: the granite rock and a white-painted wall.
All the figures presented in this section have the same label. The blue “Original(cid:180) label
defines a test made without any added noise or de-noising. The orange “Noisy(cid:180) label represents a
test where salt and pepper noise is added to the original images at a ratio of 0.55. Then, the
algorithm used the noisy images to calculate the distance. The gray “Denoised(cid:180) label represents a
test where the noisy images are denoised, and then the algorithm uses the denoised images to
calculate the distance. Finally, the black “Measurement by tape(cid:180) label represents the actual
length of the scenery.
Each of the original, noisy, and denoised tests was repeated once to give a rough estimate
of the average percentage error and the range of percentage errors. It is also understood that two
data points do not represent the true average percentage error. However, this test only aims to
understand how noise affects the algorithm. Performing statistical analyses or acquiring accurate
distance calculations are not the objectives of this experiment.
5.1.2.1 Granite Rock
This section discusses the results associated with the tests made on the granite rock.
Table 5-5 presents the average percentage error and the standard deviation calculated from the
two runs for each test. The results show that the denoising algorithm raises the accuracy of the
distance calculation when compared to the noisy calculations. Additionally, the calculation
performed on the denoised images had similar average percentage errors compared to the
original test and, in most cases, better. Interestingly, the denoised algorithm removed the noise
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When sampling at one image every 4.0 s, Figure 5-11 shows that the error range in the
noisy system increases compared to sampling every 2.0 s. Again, the error range of the results
obtained from the denoising test falls within the range of the original test. Finally, the noisy
system provides the least accurate calculations.
Sample Rate = 1 Image/4 seconds Sampledat 1 image every 4 seconds Sample Rate = 1 Image/4 seconds
20 38.00
15 36.00
34.00 10 Original
32.00 5
Original 30.00 Noisy
0 Noisy
Denoised 28.00 Denoised
-5
26.00
Measurement by
-10
24.00 tape = 33.5 cm
-15 22.00
-20 20.00
Figure 5-11 Noise test - the effect of noise and denoising for the granite rock experiment when
sampled at 1 image every 4.0 s.
When sampling at one image every 6.0 s, Figure 5-12 demonstrate that the error range for
the original and denoised tests shifts from a negative value to a positive value compared to
sampling every 4.0 s. The denoised system had an average percentage error of 0% and a very
narrow range of errors for this sampling criterion. Again, the noisy system provides the least
accurate calculations.
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Sample Rate = 1 Image/6 seconds Sampledat 1 image every 6 seconds Sample Rate = 1 Image/6 seconds
20 38.00
15 36.00
34.00 10 Original
32.00 5
Original 30.00 Noisy
0 Noisy
Denoised 28.00 Denoised
-5
26.00
Measurement by
-10
24.00 tape = 33.5 cm
-15 22.00
-20 20.00
Figure 5-12 Noise test - the effect of noise and denoising for the granite rock experiment when
sampled at 1 image every 6.0 s.
When sampling at one image every 8.0 s, Figure 5-13 shows that the error range for the
original and denoised tests shifted from a positive value to a negative value compared to
sampling every 6.0 s. Additionally, the range of errors generated from all tests is very similar.
However, the absolute value of the average percentage error in the noisy test is still higher than
the values of the original and denoised tests.
Sample Rate = 1 Image/8 seconds Sampledat 1 image every 8 seconds Sample Rate = 1 Image/6 seconds
20 38.00
15 36.00
34.00 10 Original
32.00 5
Original 30.00 Noisy
0 Noisy
Denoised 28.00 Denoised
-5
26.00
Measurement by
-10
24.00 tape = 33.5 cm
-15 22.00
-20 20.00
Figure 5-13 Noise test - the effect of noise and denoising for the granite rock experiment when
sampled at 1 image every 8.0 s.
5.1.2.2 White-painted Wall
This section discusses the results associated with the tests made on the white-painted
wall. Table 5-6 presents the average percentage error and the standard deviation calculated from
85
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detaluclaC
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Colorado School of Mines
|
the two runs for each test. The results show that the average percentage errors of the noisy test
are very different from the average percentage errors of the original test. Additionally, the
denoising algorithm successfully removed the noise from the image and obtained average
percentage errors similar to the original test.
Table 5-6 White-painted wall - noise test - average percentage error and standard deviation for
noisy, denoised, and original tests.
Sample Criteria Noisy (%) Denoised (%) Original (%)
2.0 s 12.2 ± 13.5 2.1 ± 2.7 1.3 ± 2.2
4.0 s 2.3 ± 0.3 9.3 ± 6.1 7.4 ± 3.1
6.0 s -3.7 ± 3.4 2.6 ± 3.8 6.0 ± 2
8.0 s -0.8 ± 11.4 2.8 ± 8.2 1.9 ± 7.9
10.0 s -2.2 ± 6.5 -2.1 ± 6.2 -2.4 ± 5.7
To visualize the results and qualitatively analyze the range of percentage errors, Figures
5-14 to 5-18 show the effect of noise and denoising on the distance calculations when sampled at
one image every 2.0 s, 4.0 s, 6.0 s, 8.0 s, and 10.0 s, respectively.
When sampling at one image every 2.0 s, Figure 5-14 shows that the error range of the
results obtained from the original test and the denoising test are very similar. Those results were
also observed in the granite rock experiment for the same sampling criterion.
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Sample Rate = 1 Image/2 seconds Sampledat 1 image every 2 seconds Sample Rate = 1 Image/2 seconds
20 120
15 115
10 110 Original
5 105
Original Noisy
0 Noisy 100
Denoised Denoised
-5 95
Measurement by
-10 90
tape = 91.5
-15 85
-20 80
Figure 5-14 Noise test - the effect of noise and denoising for the white-painted wall experiment
when sampled at 1 image every 2.0 s.
When changing the sampling criterion from 2.0 s to 4.0 s, the average percentage error of
the denoised and original tests increases (Figure 5-15). Furthermore, the average percentage error
of the denoised and original tests is very similar. However, the range of the denoised tests is
wider, which means that denoising at this specific sampling criterion can generate results that
deviate from a non-noisy environment. Finally, the results generated from the noisy system are
more accurate than the original and denoised systems. Occasionally, such behavior should be
expected because noise causes images that are not similar to match. As a result of this incorrect
matching process, the accuracy of the distance calculated can fluctuate unexpectedly and
generate random results that may be very accurate or inaccurate.
87
)%(
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Colorado School of Mines
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Sample Rate = 1 Image/4 seconds Sampledat 1 image every 4 seconds Sample Rate = 1 Image/4 seconds
20 120
15 115
10 110 Original
5 105
Original Noisy
0 Noisy 100
Denoised Denoised
-5 95
Measurement by
-10 90
tape = 91.5
-15 85
-20 80
Figure 5-15 Noise test - the effect of noise and denoising for the white-painted wall experiment
when sampled at 1 image every 4.0 s.
When changing the sampling criterion from 4.0 s to 6.0 s, the average percentage error of
the original test remained relatively the same but decreased for the denoised test (Figure 5-16).
Also, the average percentage error of the denoised and original tests are very close to each other.
However, the range of the denoised tests is wider, and the average percentage error is lower.
Finally, the original and denoised tests generated positive errors, while the noisy tests generated
negative errors. This behavior again indicates that noise randomly affects the image-matching
process.
Sample Rate = 1 Image/6 seconds Sampledat 1 image every 6 seconds Sample Rate = 1 Image/6 seconds
20 120
15 115
10 110 Original
5 105
Original Noisy
0 Noisy 100
Denoised Denoised
-5 95
Measurement by
-10 90
tape = 91.5
-15 85
-20 80
Figure 5-16 Noise test - the effect of noise and denoising for the white-painted wall experiment
when sampled at 1 image every 6.0 s.
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Colorado School of Mines
|
When changing the sampling criterion from 6.0 s to 8.0 s, the range of errors increased
for all tests (Figure 5-17). Additionally, the average percentage error for the original and
denoised tests are similar to each other and are positive, while the average percentage error for
the denoised test is negative. Finally, the range of errors is very similar for the original and
denoised tests but wider for the noisy test.
Sample Rate = 1 Image/8 seconds Sampledat 1 image every 8 seconds Sample Rate = 1 Image/6 seconds
20 120
15 115
10 110 Original
5 105
Original Noisy
0 Noisy 100
Denoised Denoised
-5 95
Measurement by
-10 90
tape = 91.5
-15 85
-20 80
Figure 5-17 Noise test - the effect of noise and denoising for the white-painted wall experiment
when sampled at 1 image every 8.0 s.
When changing the sampling criterion from 8.0 s to 10.0 s, Figure 5-18 shows that the
range of errors decreased for all the tests but is more dominated by negative values. Interestingly,
the average percentage error for all the tests is similar.
Sample Rate = 1 Image/8 seconds Sampledat 1 image every 10 seconds Sample Rate = 1 Image/6 seconds
20 120
15 115
10 110 Original
5 105
Original Noisy
0 Noisy 100
Denoised Denoised
-5 95
Measurement by
-10 90
tape = 91.5
-15 85
-20 80
Figure 5-18 Noise test - the effect of noise and denoising for the white-painted wall experiment
when sampled at 1 image every 10.0 s.
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