PHD-Department of Statistics and Actuarial Sciencehttp://ir-library.ku.ac.ke/handle/123456789/185402022-12-01T15:31:30Z2022-12-01T15:31:30ZWell Test Analysis of a Horizontal Well in a Completely Bounded ReservoirKitungu, Nzomo Timothyhttp://ir-library.ku.ac.ke/handle/123456789/239382022-08-18T00:11:35Z2022-01-01T00:00:00ZWell Test Analysis of a Horizontal Well in a Completely Bounded Reservoir
Kitungu, Nzomo Timothy
Well test analysis as an important part of reservoir engineering has gone through
tremendous improvement through the years since the discovery of oil. This is in terms
of the tools used, the technology involved, and the mathematical modeling involved.
Since most oil reservoirs are underground and in some cases thousands of feet from
the surface, it’s impossible to physically observe them and see how they behave or
determine their character. Mathematical models play an important role in reservoir
system characterization by predicting the well and reservoir behaviour and properties.
Over the years horizontal wells have proved that they are more productive compared
to vertical wells. In this dissertation, possible mathematical models that can be applied
in well test analysis for horizontal wells in a completely bounded oil reservoir are
developed. In developing the models, source and Green’s functions are used. Using
these functions, dimensionless pressure and dimensionless pressure derivative
distributions in real time are derived. Mathematical analyses of the models developed
and how they can be applied to characterize a completely bounded oil reservoir
penetrated with a horizontal well are presented. All possible flow periods and the
effects of reservoir, fluid properties and well design on horizontal well performance
are investigated and presented. The effects of reservoir anisotropy on well
performance are also investigated. The results of this study show that assuming
isotropic cases might reduce the accuracy and reliability of the results obtained and
thus recommend consideration of anisotropy in computations. This should be in all
the three directions. Further, the results of this study show that well design, directional
permeability and reservoir geometry will affect the horizontal well performance
differently at early flow time. This applies when the infinite-acting flow is considered
as compared to the pseudosteady state flow at late time. It is also noted that the
number of flow periods can be many; four full and at least three transitional flow
periods from inception of early transient, when the well is infinite-acting, to late
transient, when all the external reservoir boundaries are felt. The results also suggest
that oil wells that are in the same reservoir and closely spaced will experience
pressure communication (interference) faster and vice-versa. Thus, well pressure
interference affects well performance. The results obtained in this study can be used
for complete reservoir system characterization and to investigate the best well design
for optimum oil recovery in a bounded well reservoir penetrated with a horizontal
well.
A Thesis Submitted in Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics in the School of Pure and Applied Sciences of Kenyatta University, April, 2022
2022-01-01T00:00:00ZComparison of fuzzy and crisp classification trees using gini index, chi-square statistic and the gain ratioMuchai, Eunice Wambuihttp://ir-library.ku.ac.ke/handle/123456789/185422021-06-28T06:28:05Z2017-07-01T00:00:00ZComparison of fuzzy and crisp classification trees using gini index, chi-square statistic and the gain ratio
Muchai, Eunice Wambui
ABSTRACT
Discriminant (classification) analysis is a classification problem where a new individual is allocated
into one of known populations or classes based on the measured characteristics of the individual.
Different models are used in allocating the new individual into one of the populations (classes). Some
of the models depend on the underlying distribution of the populations, hese are known as parametric
models. If the model does not depend on any underlying distribution it is known as a distribution free
or non parametric model. In this work a distribution free model known as classification tree is used.
A classification tree is a presentation of edges and nodes. It is a model that is used to assign an
individual to one of many classes or populations. At each node a test is applied on a value of one of
the attributes (variables) of the individual. The individual moves to the next node (child node) along
an edge depending on the result of the test. The attribute, on which the test is applied, is known as the
splitting attribute and the value the splitting value. Tests are carried out at each node until it is not
possible to carry out more tests. The final nodes are known as terminal or leaf nodes. Classification
is done at the terminal nodes by assigning all the individuals on that node to a class. If the splitting
value is a fuzzy value, then the tree is known as a fuzzy classification tree otherwise the tree is known
as a crisp classification tree. When there are only two possible answers to the test at each node, the
resulting tree is known as a binary tree. Classification trees have been used to model many situations.
These include speech recognition, data mining and market surveys among others. In this study the
performance of crisp and fuzzy classification trees was compared. The performance was based on
probabilities of correct allocation and probabilities of misclassification. Simulated data and real
data were used. Data was simulated using R and the real data was obtained from machine learning
repository. Gini Index, Chi-Square Statistic and Gain Ratio impurity measures were applied to both
the simulated data and real data. The performance of Gini Index, Chi-Square Statistic and Gain
Ratio impurity measures was also compared. Finally the performance of the trees using varied
sample sizes was compared. It was found that for the simulated data, fuzzy classification tree
performed better than the crisp classification tree when all the three impurity measures were applied.
It was found that the Gini Index and Chi-Square Statistic impurity measures were appropriate as
impurity measures for the data used in the study and gave similar results. However the Gain Ratio
impurity measure did not perform as well as the other two impurity measures. It was also found that
there was no significant difference in the probabilities of misclassification irrespective of different
sample sizes in the populations.
A thesis submitted in fulfilment of the requirements for the award of degree of doctor of philosophy (statistics) in the school of pure and applied sciences of Kenyatta University. July 2017
2017-07-01T00:00:00Z