A Machine Learning Approach to Estimate Geo-mechanical
Parameters from Core Samples: A Comparative Approach
JWNGSAR BRAHMA
Department of Mathematics, School of Technology,
Pandit Deendayal Energy University,
Rysan, Gandhinagar-382421,
INDIA
Abstract: - Geo-mechanical parameters and Thomsen parameters play very important roles to design a stable
wellbore in a challenging environment. The main objective of this paper is to estimate Thomsen's parameters
(ε, γ, δ) and geo-mechanical properties from the core samples by Machine Learning and a comparative analysis
with the conventional mathematical approach; to place emphasis on the use of Machine Learning and Artificial
Intelligence in the Oil & Gas industry and to highlight its future potential to help in the digital transformation of
the industry. Two different Machine Learning models, the Ordinary Least Square method and the Random
Forest method, were used to predict the aforementioned geo-mechanical properties from the wave velocity and
confining pressure data. In this study, it has been observed that the approaches employed in the estimate of geo-
mechanical properties are rapid and reliable (about 93.5 percent accuracy) and may be applied in geo-
mechanical modeling for wellbore stability analysis for safe and cost-effective well plan and design on a large
scale. The analysis in this work indicates that Young’s modulus and Poisson’s ratio are heavily influenced by
the anisotropy parameters. Finally, a comparison is made with mathematical approaches. The machine learning
and artificial intelligence approaches shown here are excellently matched with mathematical approaches.
The geo-mechanical parameters and Thomsen parameters and be computed with reasonable accuracy with the
help of our proposed ML algorithms. Our proposed ML model can predict the geo-mechanical parameters and
Thomsen parameters from the velocity profile directly without complex mathematical computation. The
mathematical model would have required us to first determine the stiffness constants for the prediction of that
parameters.
Additionally, we may conclude that a machine learning model needs to be trained with more modeling data
to predict the right values with a smaller error margin. The number of data points required to train a model has
a significant impact on the model's overall accuracy. Therefore, additional modeling data is needed to learn
about and comprehend the intricacies, patterns, and interactions between provided input and output variables.
Key-Words: - Machine Learning, Artificial Intelligence, Core Samples, Thomsen Parameter, Geo-mechanical.
Received: February 2, 2023. Revised: May 25, 2023. Accepted: June 17, 2023. Published: July 18, 2023.
1 Introduction
The anisotropy in a rock is defined as its properties
that vary with the direction of observation. This
anisotropy may present everywhere within the
subsurface. It is the different geological origins of
each rock that creates different sedimentary
features in them, [1]. These properties are often
measured parallelly or perpendicularly to the
sedimentary features (bedding planes), with the
earth’s anisotropic response to changes investigated
with the help of seismic, sonic, or ultrasonic
surveys, [2].
Fundamental geo-mechanical properties
include stress, strain, Young's modulus, Poisson's
ratio, and compressive strength. Geo-mechanical
evaluation is required in petroleum engineering for
rock failure prediction, determination of in-situ
stress, wellbore stability analysis, hydraulic
fracturing design, and anisotropy measurement.
Geo-mechanical rock properties are a subset of
petrophysical parameters that can be calculated
directly in rock mechanics labs or field
experiments, [3]. However, since they are more or
less highly correlated with other petrophysical
parameters (e.g., elastic wave velocities), an
“indirect" derivation from geophysical
measurements is being researched and applied.
Three parameters characterize anisotropy, in
addition to the normal VP, VS, and ρ. Thomsen's
parameters include δ (delta), ϵ (epsilon), and γ
(gamma), [4]. The short offset effect, δ or delta,
captures the relationship between the velocity
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required to flatten gathers (the NMO velocity) and
the zero-offset average velocity recorded by check
shots. It is simple to calculate, but it may be
challenging to comprehend physically. ϵ or epsilon,
the long offset impact is "the fractional difference
between vertical and horizontal P velocities, i.e., it
is the parameter commonly referred to as 'the'
anisotropy of a rock." Horizontal velocity, on the
other hand, is difficult to measure, [4]. The shear
wave effect, often known as γ or gamma, contrasts
a horizontal shear wave with horizontal polarisation
to a vertical shear wave.
Shales comprise about 70% of sedimentary
basins. However, due to the friable nature of shales,
there are very few laboratory measurements of
velocity anisotropy, [5], [6], [7], [8]. More recently,
the influence of pore fluid on the elastic properties
of shale has been investigate by Hornby in 1998.
He measured compressional and shear wave
velocities up to 80MPa on two fluid-saturated shale
samples under drained conditions, [9]. One sample
was Jurassic outcrop shale that was recovered from
undersea and stored in its natural fluid, and the
other is Kimmeridge clay taken from a North Sea
borehole. Measurements were made on cores
parallel, perpendicular, and at 450 to bedding.
Values of anisotropy were up to 26% for
compressional and 48% for shear wave velocity
and were found to decrease with increasing
pressure, [10]. The effect of reduced porosity was
therefore concluded to be more influential on
anisotropy than an increased alignment of minerals
at higher pressure. The elastic constants, velocities,
and anisotropies in shales can be obtained from
traditionally measured on multiple adjacent core
plugs with different orientations, [10]. To derive
the five independent constants for transversely
isotropic (TI) rock, Wang measured three plugs
separately (one parallel, one perpendicular, and one
to the symmetric axis). The advantage of this
three-plug method is redundancy for the calculation
of the five independent elastic constants since each
core plug measurement yields three velocities.
Unlike shale, clean sandstone is intrinsically
isotropic. Sandstones are rarely clean; they often
contain minerals other than quartz, such as clay
minerals which can affect their reservoir qualities
as well as their elastic properties. The presence of
clay minerals and clastic sheet silicates strongly
influences the physical and chemical properties of
both sandstones and shales, [11]. Clay can be
located between the grain contacts as structural clay
in the pore space as dispersed clay or as
laminations, [12]. The distribution of the clay will
depend on the conditions at deposition on
compaction, bioturbation, and diagenesis. While
most reservoirs are composed of relatively isotropic
sandstones or carbonates, their properties may be
modified by stress. Non-uniform compressive
stress will have a major effect on randomly
distributed microcracks in a reservoir. When the
rock is unstressed all of the cracks may be open,
however, compressional stresses will close cracks
oriented perpendicular to the direction of maximum
compressive stress, while cracks parallel to the
stress direction will remain open. Elastic waves
passing through the stressed rock will travel faster
across the closed cracks (parallel to maximum
stress) than across the open ones.
The effects of anisotropy in seismic data will
reveal a lot about the Earth's processes and
mineralogy. Seismic anisotropy has garnered a lot
of attention from academia and business in the
recent two decades. Many seismic processing and
inversion procedures now use anisotropic models,
which provide significant improvements in seismic
imaging efficiency and resolution. The employment
of an anisotropic velocity model in conjunction
with seismic imaging has greatly reduced the
uncertainty surrounding internal and bounding-fault
locations, reducing the likelihood of making an
investment choice simply based on seismic
interpretation.
In addition, the discovery of a link between
anisotropy parameters, fracture orientation, and
density has resulted in the development of realistic
reservoir characterization methodologies. If
fractures are considered throughout the drilling
choice process, the drainage area of each producing
well can be greatly expanded, due to the acquisition
of such parameters as fracture spatial distribution
and density. The drilling cost of exploration and
production (E&P) projects will be greatly
decreased because there will be fewer wells due to
the expanded drainage area.
One of the most critical aspects of preparing a
strategy for hydraulic fracking is understanding the
geo-mechanical rock properties. Reduce
operational risk and optimize production while
spending as little money as possible, especially in
ultra-tight complicated formations like shale, where
operational risk is significant owing to formation
uncertainty. As a result, machine learning and
artificial intelligence (AI) are becoming important
in the oil business, as they generate accurate
information by merging log and core data. This
method is essential for forecasting the geo-
mechanical parameters of shale, the most
heterogeneous rock with the least desired
wettability for hydrocarbon flow. As a result,
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machine learning and artificial intelligence (AI) can
aggregate and correlate data more accurately than a
human can. Manual integration and correlation are
less reliable, effective, and, most crucially, less
expensive and time-consuming. Many researchers
have focused on combining core and log data to
quantify geo-mechanical parameters and output
sweet spots in the Eagle Ford and Barnett
formations using machine learning and artificial
intelligence, [13].
Wave velocities, Poisson's ratio, Young's,
shear, and bulk modulus are all significant rock
mechanical properties in the geo-mechanical study
of petroleum reservoirs. Due to the unavailability
of data or the steep costs for testing, direct
measurement of these parameters is typically
impossible, particularly in older wells. Hence, to
predict these parameters from available data,
indirect methods are frequently used. Empirical
equations are the most basic and widely used tool.
These correlations, on the other hand, are very
susceptible to various types of fluids or lithology
and are frequently unrelated to local geology. In
recent years, intelligent systems have been applied
in a variety of fields of science and technology, and
they have consistently proven to be useful in
prediction and optimization challenges, [14].
Challenges that the industry currently faces
include complex mathematical modeling that is
extremely difficult and time-consuming. Apart
from that, several complex calculations are
performed to calculate the stiffness constant using
the Voigt matrix, which can have dimensions up to
81 x 81. The time needed to solve the matrix of
such large dimensions is extraordinarily complex
and takes several months for the solution to be
reached. These challenges can be easily overcome
using AI/ML to calculate geo-mechanical and
anisotropic parameters, giving reasonably high
accuracy.
In this research work, data from lab studies of
four types of cores – Dry Sandstone, Shale, Sandy
Shale, and Saturated Sandstone – was utilized. The
goal of this study is to determine how elastic
anisotropy affects Young's modulus and Poisson's
ratio. VP, VSh, and VSV wave velocity data were
measured as a function of confining pressure from
1 MPa to 40.3 MPa, obtained through the ultrasonic
transmission method. The study aims to predict
here stress, strain, Young's modulus, Poisson's
ratio, and Thomsen's parameters using a data-
driven approach with the help of Machine Learning
algorithms. Pandas (W. McKinney and Pandas) and
Numpy (T. Oliphant) library of the phyton are used
in this computation. During the prediction of these
Geomechanical parameters we used VP, VSh, and
VSV wave velocity and confining pressures as an
input and we predicted Thomsen's parameters,
Young's modulus, and Poisson's ratio as an output.
2 Methodology
2.1 Mathematical Analysis
Core data (velocity and density) of four different
types of sedimentary rock is taken to calculate
Thomsen's anisotropic parameters and geo-
mechanical properties concerning different
confining pressure. Cores of different rock types
were used for the study of dry sandstone, shale,
sandy shale, and saturated sandstone from a
particular basin.
Transverse isotropy is commonly seen in
sedimentary rocks. Each layer has similar qualities
in-plane but distinct properties as it progresses in
thickness. Each layer's plane is the isotropy plane,
and the vertical axis is the symmetry axis. The
horizontal velocity differs from the vertical velocity
in vertical transverse isotropy, [15].
For the measurement of transverse isotropy, the
standard three-plug method is used. Using the
standard three-plug method, each rock sample is
cut in three different orientations: parallel,
perpendicular, and typically 45˚ to the vertical
symmetry axis, [15]. The details of measurement
methods are shown in Figure 1 and Figure 2
(Appendix). Phase velocity measurement for each
core sample by two orthogonal and one
compressional shear wave is performed using
dynamic or static ultrasonic laboratory
measurements. So, each core sample's total three
velocities are determined; therefore, each rock
sample's total of nine velocities is determined.
In vertical transverse isotropy (VTI), the
following equation describes the relationship
between strain (ekl) and stress (σij):
(1)
where Cijkl represents the Voigt matrix, [16]. If the
vertical axis is noted by Z, then the other two
principal axes (X and Y) are parallel to the
transversely isotropic plane.
In linear elasticity, Hooke's law states that the
stress and strain are related, i.e.
(2)
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Here stress is in N/mm2 and strain and elastic
constants are dimensionless quantities.
Here is stress, is strain and C is elastic constant
and are defined as below:
;
and
The equation (2) becomes as below:
(3)
For Vertical isotropic material, this Voigt matrix is
reduced to only five non-zero independent elastic
constants, which are C11, C33, C44, C66, and C13.
(4)
2
-
4
From the equation (3), Voigt matrix C has five non-
zero independent elastic constants: C11, C33, C44,
C66, and C13. The sixth elastic constant is C12 = C11
- 2C66, where,
C11 = in-plane compressional modulus,
C33 = out-of-plane compressional modulus,
C44 = out-of-plane shear modulus,
C66 = the in-plane shear modulus,
C13 = important constant that controls the
shape of the wave surfaces.
It is essential to establish a relationship
between five non-zero independent elastic
constants with geo-mechanical and Thomsen
parameters. We need to use a compliance matrix
(inverse of the elastic stiffness matrix) to establish
this relationship. The elastic constant can be
estimated from phase velocity data.
For a hexagonal material (VTI material), two
dynamic Poisson's ratios can be obtained using five
elastic stiffness Cij, one vertical Young's modulus
(E3), and one horizontal Young's modulus (E1),
[17]:
These dynamic Poison’s ratios ϑij, are indirect
measures of the ratio of the lateral to axial strains
when the uniaxial stress is applied in the same
direction of axial strain.
Using five elastic stiffness Cij, Thomsen’s
parameters can be estimated by the following
mathematical equations suggested by Thomsen
(1986):
(7)
(8)
(9)
2.2 Machine Learning Approaches
2.2.1 Data Preparation
For machine learning applications, once the
measurements data from the core samples are
collected there is a complete procedure that must be
carried out before training the machine learning
model.
1. Data Selection
The purpose of this stage is to choose a subset of all
available data. In this step, pandas and the numpy
library of Python are used. In this process, the most
considerable data has been selected which is best
suitable for our problem statement.
(3)
(4)
(5)
(6)
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2. Data Pre-Processing
Three common data pre-processing steps
include:
Formatting: It's possible that the data that
is chosen isn't in the right format to work
with. As a result, the data has been
converted to an Excel spreadsheet.
Cleaning: The removal or correction of
missing data is known as data cleaning.
Some data instances may be incomplete,
and they'll need to be eliminated.
Additionally, some of the attributes may
contain sensitive information, and these
attributes may need to be anonymized or
eliminated from the data. The outlier
removal method has been used to eliminate
outliers from the data using the is null ()
function for the null value of the data set.
Sampling: There may be a lot more
selected data than is required. More data
can lead to substantially longer algorithm
execution times as well as increased
computing and memory needs. As a result,
before evaluating the entire dataset, a
smaller subset sample of the selected data
works considerably faster for exploring and
developing ideas.
3. Data Transformation
Scaling, attribute decompositions, and attribute
aggregations are three popular data
transformations. This process is called feature
engineering.
Scaling: The pre-processed data could
have a mix of scales for different quantities
like dollars, kilograms, and sales volume.
Many machine learning algorithms prefer
data attributes with the same scale, such as
0 to 1 for the smallest and highest value for
a specific feature. Consider any feature
scaling that may be required.
Decomposition: There may be features
that indicate a complicated notion that,
when broken down into basic bits, are more
valuable to a machine learning method. A
date, for example, may comprise day and
time components that could be separated
further. Perhaps only the time of day has
any bearing on the problem at hand.
Aggregation: There may be features that
can be combined into a single feature to
make the problem more meaningful. For
example, each time a client logged into a
system, there may be data instances for the
number of logins, allowing the extra
instances to be discarded. Consider the
several types of feature aggregations that
could be used.
4. Outlier Detection and Removal
Outliers are extraordinary results that differ
significantly from the rest of the data. Outliers
in a normal distribution. Outlier mining, outlier
modeling, novelty identification, and anomaly
detection are all terms used in data mining and
machine learning to describe the process of
discovering outliers.
Here are some outlier detection methods:
Extreme Value Analysis: Calculate the
statistical tails of the data's underlying
distribution univariate data.
Probabilistic and Statistical Models:
Determine unlikely events using a
probabilistic data model. Gaussian mixture
models were optimized via expectation
maximization.
Linear Models: Methods for projecting
data into lower dimensions based on linear
correlations. Principal component analysis
(PCA) and data with substantial residual
errors may be considered outliers.
Proximity-based Models: Cluster, density,
or closest neighbor analysis is used to
isolate data instances from the rest of the
data.
Information Theoretic Models: Outliers
are data occurrences that add to the
complexity of a dataset (minimum code
length).
High-Dimensional Outlier Detection:
Distance-based metrics in higher
dimensions are broken down using
methods that explore subspaces for
outliers. (Curse of dimensionality).
2.2.2 Data Splitting
Taking a dataset and separating it into two
subgroups is the technique. The training dataset is
the first subset, which is used to fit the model. The
second subset is not used to train the model;
instead, the dataset's input element is given to the
model, which then makes predictions and compares
them to the predicted values. The test dataset is the
name given to the second dataset.
Train Dataset: Used to train by feeding to
the machine learning model.
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Test Dataset: Used to evaluate the metrics
of an already trained machine learning
model and to check for accuracy.
The size of the train and test sets, which is usually
stated as a percentage between 0 and 1 for either
the train or test datasets, is the procedure's key
configuration parameter. Here in the training set,
we consider a size of 80% of the total samples 1000
and test set with the remaining 20%.
Here, using the train_test_split() syntax of Python,
the data has been split into the train data set and test
dataset.
2.2.3 Dummy Data
The amount of available original data is not
sufficient to train the machine learning model, so
instead, we created dummy data (1000 samples)
with the help of Python functions using the original
data. The Python function has some basic input
arguments which use the minimum, maximum,
mode value, and the quantity of the data points to
be generated. Suppose we want 50 dummy data
points of the available original data; we need to
input the fundamental above-listed argument
values. The function generates values that can be
utilized better to understand the complexities of our
machine learning model.
2.2.4 Data Visualization
In applied statistics and machine learning, data
visualization is a must-have talent. It can be useful
for identifying patterns, faulty data, outliers, and
other things when studying and getting to know a
dataset. Data visualizations can express and
demonstrate crucial links in plots and charts that
are more visceral to yourself and stakeholders than
measurements of association or importance with
just a little subject knowledge. The matplotlib() and
seaborn() libraries of Python are used for the data
visualization.
2.2.5 Machine Learning Model Selection
Once the Excel file with all the features (confining
pressure and shear waves in different directions)
variables and output (Thomsen's parameters)
variables have been imported, then comes the data
pre-processing, which is a very crucial step that
helps us understand the data and any flaws that are
associated with it which may affect the overall
accuracy of the machine learning model.
To make predictions, visualization is an
important step that gives us a graphic
representation of our data. When we have very
numerous data, by just looking at the numbers, we
cannot interpret anything out of it unless we
visualize the data in a graphical representation. For
instance, if we have some density-neutron log data,
just looking at numbers would not help the
geologist drive important decisions. When we put
these data into a graphical visual context, we can
have a better understanding of the logs and their
features (for example, the crossovers). So,
visualization helps to know the trends of the data,
the patterns, and the outliers.
Accuracy is a crucial parameter for the
selection of the machine learning model. However,
our focus of the study is to predict the Thompson
parameter values with given shear wave values in a
different direction, and it falls under the linear
regression problem. In the regression
problem/model, there are specific evaluation
metrics. Let's have a look at some critical
evaluation metrics.
R-squared value (R2): R2 is a statistical
measure of the level of the correlation between
the observed outcome and the predicted value
given by the model. So, if a model achieves an
R2 score of 1, then it can be understood that
both variables are perfectly correlated to each
other, which implies no variance. In another
way, (total variance explained by the model)/
(total variance) the value of this equation
signifies the quality of the correlation between
the variables. The closer the value of R2 to 1,
the better the model is considered to fit.
Adjusted R-squared: It's a modified version
of R-squared that adjusts for predictors in a
regression model that are no longer significant.
It demonstrates whether or not increasing the
number of independent variables improves the
model. The Adjusted R-squared value is always
less than the R-squared value.
Root Mean Squared Error (RMSE): Another
popular method is where the regression
prediction errors are calculated. It's essentially
the average of the squared errors or the
difference between the dataset's observed value
and the model's predicted value. The square
root of the mean squared error (MSE), which is
the average squared difference between the
observed actual outcome values and the values
predicted by the model, is the RMSE in
mathematics. The better the fit, the lower the
RMSE number.
Mean Absolute error (MAE): It's similar to
RMSE, except that MAE gauges the model's
prediction error. It is the average absolute
difference between observed and projected
results in mathematics. MAE is somewhat
unaffected by outliers in the sample.
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R-squared tells us how the independent
variables explain much variance in the dependent
variable. In a general way, if we add more amount
of observations and more independent variables,
the value of the R-squared increases, but when the
R-squared value does not increase any further with
adding more independent variables, we need to
understand that the added variables are
uncorrelated with the dependent variable. In most
complex predictions, it is recommended to use
Adjusted R-squared in place of R-squared for
model evaluation as it gives some penalty for an
extra variable if the previous variable does not
explain the dependent variable more correctly.
However, in our case, we have a less complex
situation, so it is better to use the R-squared method
for model evaluation and selection in our study.
2.2.6 Algorithms Used in Machine Learning
Ordinary Least Squares (OLS)
For estimating the unknown parameters in a linear
regression model, ordinary least squares (OLS) is a
sort of linear least-squares method. By minimizing
the sum of the squares of the differences between
the observed dependent variable (values of the
variable being observed) in the given dataset and
those predicted by the linear function of the
independent variable, OLS chooses the parameters
of a linear function of a set of explanatory
variables.
Geometrically, the total of the squared
distances between each data point in the set and the
corresponding point on the regression surface,
measured parallel to the axis of the dependent
variable—the lower the differences, the better the
model fits the data. The resulting estimator,
especially in a basic linear regression with a single
regressor on the right side of the regression
equation, can be stated by a simple formula.
The OLS coefficient estimates for the simple linear
regression are as follows:
(10)
(11)
where the “hats” above the coefficients indicate
that it concerns the coefficient estimates, and the
“bars” above the x and y variables mean that they
are the sample averages, which are computed as
follows:
(12)
(13)
Random Forest
The bagging approach is used to train Random
Forests. Bagging, also known as Bootstrap
Aggregating, entails picking subsets of the training
data at random, fitting a model to these smaller data
sets, and then aggregating the results. Given that
we are sampling with replacement, this strategy
permits numerous instances to be utilized again for
the training step. Sampling subsets of the training
set, fitting a Decision Tree to each, and aggregating
the results is what tree bagging is all about. The
systematic diagrammatic representation of the
Random Forest is shown in Figure 4 (Appendix).
By applying the bagging approach to the feature
space, the Random Forest method offers more
randomness and diversity. Instead of searching
greedily for the best predictors to construct
branches, it randomly samples elements of the
predictor space, increasing variety and lowering
variance while maintaining or increasing bias. This
strategy is also known as "feature bagging," and it
leads to a more robust model.
Random Forests is an algorithm where each
new data point goes through the same process, but
in the ensemble, it visits all the different trees,
which were grown using random samples of both
training data and features. The functions for
aggregation will differ according to the task at
hand. It uses the mode or most frequent class
predicted by the individual trees (also known as a
majority vote) for classified problems, but in
Regression tasks, the average prediction of each
tree is used.
Building and Training Random Forest Models
with Scikit-Learn
Assuming any two child nodes, Scikit – Learn
calculates the importance of a node using Gini
Importance for each decision tree.
(16)
Where
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To calculate the importance of each feature on a
decision tree, the following equation can be used.
(17)
At the random forest level, the final feature
importance by the following equation:
(18)
Where,
Where,
It can be normalized to a value between 0 and 1 by
the following formula:
(19)
Building and Training Random Forest Models
with Spark
Spark determines the relevance of a feature by
adding the gain multiplied by the number of
samples that pass through the node for each
decision tree as below:
(20)
Where
The normalization for each tree can be estimated by
the following equation:
(21)
Where
Finally, the feature importance values for each tree
can be calculated using the following equation:
(22)
Where
.
The accuracy of all the models is going to estimate
with the parameter R2. The graphical representation
of R2 and how it can be calculated is given in detail
in Figure 3 (Appendix).
3 Results and Discussion
Dry sandstone, saturated sandstone, Shale, and
Sandy shale are four types of rock samples that
accurately describe elastic and anisotropic behavior
in the sedimentary column. Because each of these
rock types has a varied sensitivity to confining
pressure, the anisotropies are influenced differently
as the confining pressure changes.
The anisotropy parameters are plotted. Each
meaning of the Thomsen anisotropy parameters is
given as follows: The first parameter is γ; from the
original equation, it is evident that γ only depends
on the S-wave velocity component, whether fast or
slow S-wave velocity. Hence, the parameter γ
describes the anisotropy condition of the S-wave
velocity. The anisotropy of the S-wave grows as the
value of the anisotropy parameter γ increases. The
second parameter is ε; ε describes P-wave velocity.
The anisotropy of the S-wave grows as the value of
the anisotropy parameter ε increases. Moreover, the
third parameter is δ; δ describes both P-wave and
S-wave. An increase in the anisotropy parameter δ
indicates an increase in both P-wave and S-wave
anisotropy.
Poisson's ratio (horizontal and vertical) and
Young's modulus (horizontal and vertical) are
important rock mechanical parameters in the geo-
mechanical study of petroleum reservoirs. Due to
the high expense of testing or a lack of relevant
data, direct measurement of these parameters is
frequently not possible, especially in older wells.
As a result, machine learning models are frequently
employed to forecast these parameters based on
available data. Empirical equations are the most
basic and widely used strategy. These correlations,
on the other hand, are particularly sensitive to
different types of fluids or lithology and are
frequently irrelevant to local geology. In recent
years, AI and Machine learning has been used in
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different sciences and technologies and have often
been demonstrated to help predict and optimize
problems. Herein, a set of geo-mechanical
parameters for different types of rocks has been
predicted using Machine learning models (Ordinary
Least Square method and Random Forest method).
For this purpose, the mechanical properties of
rocks, belonging to different lithologies, were
predicted from wave velocities measured in the
experimental studies on the core. The results
depicted that the used methodologies were swift
and reliable (93.5% accuracy) in the estimation of
geo-mechanical properties and can be used in the
geo-mechanical modeling of petroleum reservoirs
on the industrial scale.
3.1 The Effect of Confining Pressure on
Thomsen Anisotropic Parameters
Dry Sandstone
With increasing confining pressure, the Thomsen
anisotropic parameters show different trends. The
P-wave anisotropy parameter ɛ increases linearly
with increasing confining pressure and the S-wave
anisotropy parameter γ, which shows a similar
trend and increases in confining pressure.
Furthermore, the third parameter is δ which
describes that both P-wave and S-wave remain
constant at zero and do not change with an increase
in confining pressure (Figure 5, Appendix).
Sandy Shale
With increasing values of confining pressure, the P-
wave anisotropy parameter ɛ increases gradually,
showing high values, while the S-wave anisotropy
parameter γ also shows a similar trend, but with
negative values, as it approaches zero. The third
anisotropy parameter δ, which describes both P-
wave and S-wave, remains constant at zero and
does not change (Figure 6, Appendix).
Shale
With increasing values of confining pressure, the P-
wave anisotropy parameter ɛ increases gradually,
showing high values, while the S-wave anisotropy
parameter γ also shows a similar trend but with
slightly lower values. The third anisotropy
parameter δ, which describes both P-wave and S-
wave, remains constant at zero and does not change
(Figure 7, Appendix).
Saturated Sandstone
With increasing values of confining pressure, the P-
wave anisotropy parameter ɛ shows a relatively
sharper decline before becoming more gradual. In
contrast, the S-wave anisotropy parameter γ moves
in the opposite direction, remaining constant
initially before showing an upward trend. The third
anisotropy parameter δ, which describes both P-
wave and S-wave, remains constant at zero and
does not change (Figure 8, Appendix).
3.2 Estimation of Geo-mechanical
Parameters using Predictive Modelling
Techniques and Comparison with
Mathematical Model
Two different Machine Learning models, Ordinary
Least Square (OLS) and Random Forest (RF), were
used to predict horizontal & vertical Young's
Modulus (E1& E3) and horizontal & vertical
Poisson's Ratio (V12& V31) for the below samples.
The predicted values were then compared with the
values from the mathematical model, and
calculated using empirical equations, to determine
the accuracy.
Dry Sandstone
While calculating E1 and E3 for a given set of
confining pressures through the OLS method and
Random Forest method, the OLS method gives an
accuracy of approximately 98% and 94%,
respectively, compensating the number of
predictions only three out of six input values.
Comparing it to the Random Forest method gives
an accuracy of 92% and 98%, respectively, to
predict the outcome for all six input values (Table
1, Appendix).
While calculating V12 and V31 for a given set of
confining pressures through the OLS method and
Random Forest method, the OLS method gives an
accuracy of approximately 98% and 75%,
respectively, compensating the number of
predictions only three out of six input values.
Comparing it to the Random Forest method gives
an accuracy of 89% and 74%, respectively, to
predict the outcome for all six input values in the
first case (Table 2, Appendix).
For this particular sample, the RF method was
vastly superior to the OLS method, and the
obtained values were pretty accurate compared to
those of the mathematical model.
Shale
While calculating E1 and E3 for a given set of
confining pressures through the OLS method,
approximately 99% was obtained for both cases.
The predicted values were found to be almost
identical to those calculated using the mathematical
model. RF method was not considered for the given
Shale sample because the OLS method was deemed
accurate enough (Table 3, Appendix).
While calculating V12 and V31 for a set of
confining pressures through the OLS method, it
gives an accuracy of approximately 98% and 97%,
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respectively, which is almost identical to those
calculated using the mathematical model (Table 4,
Appendix).
RF method was not considered for the given
Shale sample because the OLS method was deemed
accurate enough.
Sandy Shale
While calculating E1 and E3 for a given set of
confining pressures through the OLS method and
Random Forest method, the OLS method gives an
accuracy of 98% for both cases, but with the slight
drawback of accurately predicting outcomes, only
three of the five input values. This irregularity is
due to the complexity of calculating the constant
elastic C13, which becomes a complex number at
the associated pressures. On the other hand, the RF
method faces no such complexities and can predict
97% and 99% accuracy for E1 and E3, respectively
(Table 5, Appendix).
While calculating V12 and V31 for a given set of
confining pressures, the OLS method gives an
accuracy of 68% and 78%, respectively, while the
Random Forest method gives an accuracy of 99%
and 74%, respectively. In V31 however, both the
ML models were able to predict outcomes for only
two of the five input values. This complication is
due to the issue of data redundancy (Table 6,
Appendix).
For this particular sample, the RF method was
vastly superior to the OLS method, and the
obtained values were reasonably accurate compared
to those of the mathematical model.
Saturated Sandstone
While calculating E1 and E3 for a given set of
confining pressures, the OLS method accuracy is
approximately 68% and 66%, respectively.
Comparing it to the Random Forest method gives
an accuracy of 92% and 94%, respectively (Table
7, Appendix).
While calculating V12 and V31 for a given set of
confining pressures, the OLS method gives an
accuracy of approximately 59% and 57%,
respectively. Comparing it to the Random Forest
method gives an accuracy of 88% and 86%,
respectively (Table 8, Appendix).
For this particular sample, the RF method was
found to be vastly superior to the OLS method, and
the obtained values were found to be reasonably
accurate when compared to those of the
mathematical model.
3.3 Relationship between Geo-mechanical
and Thomsen Parameters
The application of a correlation matrix interprets
the relationship between the Geo-mechanical
parameters and Thomsen parameters. A correlation
number close to zero shows no linear link between
the variables, whereas an absolute value of 1 in the
correlation table indicates a complete positive
linear relationship between the variables. The sign
of correlation, either positive or negative, shows the
direction of the relationship. If the variables are
likely to decrease or increase together, then the
coefficient is positive. Similarly, if one variable
increases concerning a decrease in the other
variable, then there is a negative correlation and
coefficient if negative.
Dry Sandstone
For a dry Sandstone sample, ɛ and γ are positively
correlated to E1, E3, and V12 to a great degree, but
they show a high negative correlation with V31. On
the contrary, δ shows the opposite trend whereby it
exhibits a high negative correlation with E1, E3, and
V12 whereas V31 is positively correlated to δ. The
detailed parameters are listed in Table 9,
(Appendix).
Shale
For a Shale sample, ɛ is positively correlated to E1,
E3, and V12 to a great degree, but it shows a high
negative correlation with V31. In the case of γ, it
shows a high positive correlation with E3 and V12,
but it shows a high negative correlation with E1 and
V31. On the contrary, δ shows the opposite trend
whereby it displays a high negative correlation with
E3 and V12 whereas E1 and V31 are positively
correlated to δ. The detailed parameters are listed in
Table 10, (Appendix).
Sandy Shale
For a Sandy shale sample, ɛ and γ are positively
correlated with V31 to a great degree, but it shows a
moderately negative correlation with V12.
Moreover, it does not hold any relation with E1 and
E3. On the contrary, δ shows a high correlation with
all the geo-mechanical parameters. The detailed
parameters are listed in Table 11, (Appendix).
Saturated Sandstone
For a Saturated sandstone sample, ɛ is positively
correlated with E1 and V12 to a great degree, but it
shows a high negative correlation with E3 and V31.
Whereas in the case of γ, there is a stark difference
as it shows a high positive correlation with E3 and
V31, but it exhibits a high negative correlation with
V12. Moreover, γ has no relation with E1. Similarly,
δ exhibits a moderate and high negative correlation
with E1 and V12, respectively, whereas E3 and V31
are positively correlated to δ. The detailed
parameters are listed in the Table 12, (Appendix).
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4 Conclusion
Currently, the conventional approach of selectively
adopting many technologies and applying
digitalization may not be the best way forward.
Instead, the industry would gain more if it pursued
a transformative agenda with digitalization as the
foundation itself. A digital transformation at this
stage can revolutionize not only the industry but
also benefit society. A centered digital strategy and
a culture of creativity and technology adoption
would be required for such a transition.
Through this study, we may conclude that elastic
anisotropy parameters are the primary determinants
of hydrocarbon reservoir characterization
parameters estimation. To estimate more precise
hydrocarbon reservoir characterization parameters,
vertical P-wave and S-wave velocities, as well as
the three anisotropy values, are required. Surface
seismic data of good quality and high resolution
can be used to estimate the anisotropy parameters ε,
γ, and δ. We must rely on downhole data, wireline
measurements for sonic profiling, and other seismic
profiling methods to determine the remaining
parameters. The lab studies on core samples would
only aid in the development of the initial model by
providing empirical connections between some of
the parameters.
After applying the ML algorithms, the anisotropy
parameters and the geo-mechanical properties
could be estimated with reasonable accuracy. Using
the mathematical model would have required us to
find out the stiffness constants first, which has been
eliminated using ML algorithms, which facilitate
the direct estimation of geo-mechanical properties
through velocity profile inputs.
Moreover, we can also conclude that for a machine
learning model to predict correct values with less
error margin; the model needs to be trained with
more modeling data. The amount of data points we
need for training a model has a substantial effect on
the overall accuracy of the models. So, to be able to
learn and understand the complexities, patterns, and
relationships between given input and output
variables, requires more modeling data.
We can understand the effect of fewer modeling
data points on the overall accuracy of the OLS
method, where the OLS (Ordinary Least Square)
model fails miserably in predicting some data
points of the original data. It means the model does
not entirely understand the relationships between
the variables on fewer data.
References:
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[2] Barton, N., Quadros, E., Anisotropy is
Everywhere, to See, to Measure, and to
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[3] Schön, J. H., Handbook of Petroleum
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[5] Kaarsberg, E. A., Introductory studies of
natural and artificial argillaceous aggregates
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[6] Podio, A.L., Gregory, A. R., and Gry, K.E,
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[7] Jones, L. E. A., and Wang, H. F., Ultrasonic
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pp. 288–297.
[8] Lo, T. W., Coyner, K. B., and M. N.,
Experimental determination of elastic
anisotropy of Berea Sandstone Chicopee
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Vol. 51, 1986, pp. 164-171.
[9] Hornby, B.E., Experimental laboratory
determination of the dynamic elastic
properties of wet, drained shales , J Geophys
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29,964.
[10] Wang, Z., Seismic anisotropy in sedimentary
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1415–1422.
[11] Bjørlykke, K., Clay mineral diagenesis in
sedimentary basins - a key to the prediction of
rock properties. Examples from the North Sea
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34.
[12] Sam, M.S. and Andrea, A., The effect of clay
distribution on the Elastic properties of
sandstones. Geophysical Prospecting, Vol.
49, 2001, pp. 128-150.
[13] Syed, F. I., Abdulla, A., Dahaghi, A. K.,
Neghabhan S., Application of ML & AI to
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model petrophysical and geomechanical
properties of shale reservoirs - A systematic
literature review, Petroleum, Vol. 8, 2020, pp.
158-166.
[14] Rajabi, M., & Tingay, M., Applications of
Intelligent Systems in Petroleum
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[15] Wang, Z., Seismic Anisotropy in Sedimentary
Rocks Part 1: A Single-Plug Laboratory
Method. Geophysics, Vol. 67, No.5, 2002, pp.
1415-1422.
[16] Mavko, G., Mukerji, T., & Dvorkin, J., Tools
for Seismic Analysis in Porous Media, Rock
Physics Handbook, Cambridge University
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[17] Pena, F. R., Elastic Properties of Sedimentary
Anisotropic. Dissertation, Massachusetts
Institute of Technology , 1998, pp. 19-26.
[18] Jain, D., ML, R-Squared in Regression
Analysis, www.geeksforgeeks (2023 ) , URL:
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[19] Reinstein, I. Random Forests, Explained,
www.kdnuggets.com (2017),
URL:https://www.kdnuggets.com/2017/10/ra
ndom-forests-explained.html, 2017.
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Appendix
Fig. 1: Experimental procedure for data acquisition from core same, [15].
Fig. 2: Velocity measured at different angles for core samples, [15].
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Fig. 3: Graphical representation of R2 value, [18].
Fig. 4: Diagrammatical representation of the Random Forest method, [19].
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Fig. 5: Thomsen's parameters for dry sandstone as a function of confining pressure.
Fig. 6: Thomsen's parameters for sandy shale as a function of confining pressure.
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Fig. 7: Thomsen's parameters for shale as a function of confining pressure.
Fig. 8: Thomsen's parameters for saturated sandstone as a function of confining pressure.
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Table 1. Comparison between the mathematical and predicted model for estimating Young's modulus in a dry sandstone sample.
Dry Sandstone
Confining pressure
(MPa)
E1 (Horizontal Young's Modulus (MPa))
E3 (Vertical Young's Modulus (MPa))
Mathematical
Model
Predicted Model
Mathematical
Model
Predicted Model
OLS
Method
Accuracy
RF
Method
Accuracy
OLS
Method
Accuracy
RF
Method
Accuracy
2.3
3.708012529
3.910
95%
3.79
98%
1.410995664
1.735
81%
1.4260
99%
5.1
3.210830011
3.216
100%
4.431
72%
1.402640235
1.4
100%
1.4240
99%
10.2
3.632629248
3.668
99%
3.955
92%
1.490805682
1.494
100%
1.5137
98%
20
5.465012748
5.67
96%
1.797465001
1.7470
97%
30
6.11027495
5.95
97%
1.838503204
1.7900
97%
30.2
6.384496232
6.12
96%
1.853176489
1.8140
98%
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
98%
92%
94%
98%
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Table 2. Comparison between the mathematical and predicted model for estimating Poisson’s ratio in a dry sandstone sample.
Dry Sandstone
Confining pressure
(MPa)
V31 (Vertical Poisson's ratio)
V12 (Horizontal Poisson's ratio)
Mathematical
Model
Predicted Model
Mathematical
Model
Predicted Model
OLS
Method
Accuracy
RF
Method
Accuracy
OLS
Method
Accuracy
RF
Method
Accuracy
2.3
-0.016860547
0.0049
29%
-
0.0340
50%
0.169340116
0.1780
95%
0.162
96%
5.1
-0.135471039
-
0.1390
97%
-
0.1220
90%
0.100161802
0.1000
100%
0.118
85%
10.2
-0.159511085
-
0.1550
97%
-
0.1300
81%
0.080528974
0.0810
99%
0.103
78%
20
0
0.176851374
0.189
94%
30
0
0.233715971
0.192
82%
30.2
0
0.198308787
0.195
98%
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
75%
74%
98%
89%
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Table 3. Comparison between the mathematical and predicted model for estimating Young’s modulus in a shale sample
Shale
Confining pressure (MPa)
E1 (Horizontal Young's Modulus (MPa))
E3 (Vertical Young's Modulus (MPa))
Mathematical Model
Predicted Model
Mathematical Model
Predicted Model
OLS Method
Accuracy
OLS Method
Accuracy
3.1
9.733428376
9.710
100%
6.003302606
6.010
100%
5.2
11.0755007
11.160
99%
6.158448828
6.316
98%
10.3
12.22255047
11.870
97%
6.341019392
6.390
99%
20.1
12.83280877
12.960
99%
6.503649957
6.411
99%
30.6
13.44198011
13.200
98%
6.491447506
6.594
98%
40.3
13.22874197
13.220
100%
6.620959109
6.624
100%
Avg. Accuracy
Avg. Accuracy
99%
99%
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Table 4. Comparison between the mathematical and predicted model for estimating Poisson’s ratio in a shale sample
Shale
Confining pressure (MPa)
V31 (Vertical Poisson's ratio)
V12 (Horizontal Poisson's ratio)
Mathematical Model
Predicted Model
Mathematical Model
Predicted Model
OLS
Method
Accuracy
OLS
Method
Accuracy
3.1
0.129480576
0.129
100%
0.427600388
0.428
100%
5.2
0.104723742
0.112
94%
0.449112072
0.476
94%
10.3
0.073195914
0.068
93%
0.430831071
0.429
100%
20.1
0.051288404
0.050
97%
0.445622109
0.431
97%
30.6
0.054635724
0.054
99%
0.462438597
0.471
98%
40.3
0.058807873
0.058
99%
0.459837903
0.460
100%
Avg. Accuracy
Avg. Accuracy
97%
98%
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Table 5. Comparison between the mathematical and predicted model for estimating Young’s modulus in a sandy shale sample.
Sandy Shale
Confining pressure
(MPa)
E1 (Horizontal Young's Modulus (MPa))
E3 (Vertical Young's Modulus (MPa))
Mathematical
Model
Predicted Model
Mathematical
Model
Predicted Model
OLS
Method
Accuracy
RF
Method
Accuracy
OLS
Method
Accuracy
RF
Method
Accuracy
1.3
5.566921833
5.563
100%
5.66
98%
1.32
1.31
99%
1.322
100%
3.3
5.746127921
5.405
94%
1.33
1.3
98%
5.2
5.906146999
5.583
95%
1.33
1.311
98%
10.2
5.525406037
5.536
100%
5.446
99%
1.33
1.33
100%
1.327
100%
20.2
7.043371391
7.437
95%
6.837
97%
1.36
1.43
95%
1.36
100%
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
98%
97%
98%
99%
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Table 6. Comparison between the mathematical and predicted model for estimating Poisson’s ratio in a sandy shale sample
Sandy Shale
Confining pressure
(MPa)
V31 (Vertical Poisson's ratio)
V12 (Horizontal Poisson's ratio)
Mathematical
Model
Predicted Model
Mathematical
Model
Predicted Model
OLS
Method
Accuracy
RF
Method
Accuracy
OLS
Method
Accuracy
RF
Method
Accuracy
1.3
0
0.53
0.531
100%
0.53
99%
3.3
0
0.52
-
2.040
25%
0.51
98%
5.2
0
0.51
2.818
18%
0.50
99%
10.2
-0.036633466
-
0.04
92%
-
0.04
83%
0.49
0.490
99%
0.49
99%
20.2
0.032063781
0.05
64%
0.02
65%
0.49
0.500
97%
0.49
99%
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
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Table 7. Comparison between the mathematical and predicted model for estimating Young’s modulus in a saturated sandstone sample
Saturated Sandstone
Confining pressure
(MPa)
E1 (Horizontal Young's Modulus (MPa))
E3 (Vertical Young's Modulus (MPa))
Mathematical
Model
Predicted Model
Mathematical
Model
Predicted Model
OLS
Method
Accuracy
RF
Method
Accuracy
OLS
Method
Accuracy
RF
Method
Accuracy
6.7
2.60
2.600
100%
2.562
98%
1.97
1.9660
100%
2.0270
97%
11.3
2.35
5.032
47%
2.525
93%
2.54
5.4240
47%
2.1120
83%
15.6
1.89
0.540
29%
2.502
75%
1.92
0.4087
21%
2.2071
87%
20.1
2.32
1.059
46%
2.506
92%
2.27
0.9625
42%
2.2687
100%
30.3
2.36
2.737
86%
2.523
93%
2.28
2.6780
85%
2.3032
99%
40.2
2.54
2.537
100%
2.5327
100%
2.33
2.3340
100%
2.3033
99%
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
Avg.
Accuracy
68%
92%
66%
94%
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DOI: 10.37394/232011.2023.18.11
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Table 8. Comparison between the mathematical and predicted model for estimating Poisson’s ratio in a saturated sandstone sample
Saturated Sandstone
Confining pressure
(MPa)
V31 (Vertical Poisson's ratio)
V12 (Horizontal Poisson's ratio)
Mathematical
Model
Predicted Model
Mathematical
Model
Predicted Model
OLS
Method
Accura
cy
RF
Method
Accura
cy
OLS
Method
Accura
cy
RF
Method
Accura
cy
6.7
0.20
0.1950
100%
0.2131
92%
0.27
0.2650
100%
0.2534
96%
11.3
0.16
0.6700
23%
0.2365
66%
0.32
0.7870
41%
0.2390
74%
15.6
0.35
0.0460
13%
0.2641
74%
0.18
-
0.0580
32%
0.2163
83%
20.1
0.30
0.0750
25%
0.2823
94%
0.21
0.0126
6%
0.1960
93%
30.3
0.32
0.3833
82%
0.2955
94%
0.16
0.2230
73%
0.1780
91%
40.2
0.31
0.3000
98%
0.2967
97%
0.16
0.1620
100%
0.1740
93%
Avg.
Accurac
y
Avg.
Accura
cy
Avg.
Accura
cy
Avg.
Accura
cy
57%
86%
59%
88%
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DOI: 10.37394/232011.2023.18.11
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Table 9. Correlation between Geo-mechanical and Thomsen parameters for a dry sandstone sample
Dry Sandstone
E1
E3
V12
V31
ε
0.9997
0.9987
0.9994
-0.9999
γ
0.9987
0.9972
0.9985
-0.9998
δ
-0.9998
-0.9997
-0.9988
0.9989
Table 10. Correlation between Geo-mechanical and Thomsen parameters for a shale sample
Shale
E1
E3
V12
V31
ε
0.9997
0.9987
0.9994
-0.9999
γ
-0.9997
0.9972
0.9985
-0.9998
δ
0.9996
-0.9997
-0.9988
0.9989
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Table 11. Correlation between Geo-mechanical and Thomsen parameters for a sandy shale sample
Sandy Shale
E1
E3
V12
V31
ε
-0.0844
0.0111
-0.5469
0.9844
γ
-0.1241
-0.0308
-0.5821
0.985
δ
0.985
0.8983
0.8018
0.9704
Table 12. Correlation between geo-mechanical and Thomsen parameters for a saturated sandstone sample
Saturated Sandstone
E1
E3
V12
V31
ε
0.8515
-0.9906
0.9307
-0.985
γ
-0.1388
0.7323
-0.8713
0.7551
δ
-0.5919
0.9665
-0.997
0.9764
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Contribution of Individual Authors to the
Creation of a Scientific Article:
Dr. Jwngsar Brahma is responsible for overall
supervision, writing the original draft, the writing
review, editing Normal analysis, Investigation,
Validation, writing of the original draft and the
writing review.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself:
No funding for this research work.
Conflict of Interest
The author has no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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n_US
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