PHD-Department of Mathematics

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Browsing PHD-Department of Mathematics by Subject "Horizontal Well"

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    Simulating Pressure Distribution of a Horizontal Well in an oil Reservoir Subject to Single Edged and Bottom Constant Pressure
    (Kenyatta University, 2022) Mutisya, Mutili Peter; Kennedy Otieno Awuor; Stephen Ezizanami Adewole; Daniel Okang’a Oyoo
    In this study the pressure distribution in an oil reservoir with a horizontal well is investigated. A horizontal well with single-edged and constant bottom pressure is outlined. A reservoir bounded with two constant pressure boundaries, like an edge and bottom water, requires that the production engineer should adhere diligently to a production schedule, developed by a reservoir engineer, for clean oil production to be possible. This means that arbitrary production practices through selection of production rates could lead to production of these external fluids. This can mar the economics of the project. Production schedules or plans show acceptable rates, well design and production time that can guarantee only clean oil production. In this study, pressure behaviour of a horizontal well drilled and completed in a reservoir subject to with simultaneous single-edged and bottom water drives is investigated in detail. All possible flow periods or patterns that can be exhibited by the well are determined. Fluid flow in oil reservoirs in real time is governed by a heterogenous diffusivity equation, which describes reservoir pressure as a function of reservoir, fluid and wellbore properties. To solve this unsteady state problem, Green’s functions were deployed to represent the boundaries of the reservoir selected for study. The Green’s functions selected are for flow from start of transient to late time, when all the external boundaries are felt. Newman product rule was used to derive a dimensionless pressure expression for the reservoir system oil flow. The source of pressure transient was production throughout. All the resulting integrals were performed numerically. MATLAB programming was used to plot the curves by applying spline functions interpolation. Influence of reservoir, fluid and wellbore properties on reservoir pressure was investigated in real time. To assist interpretation, dimensionless pressure derivatives were also computed. Near wellbore problems, like skin and wellbore storage, which affect well performance only at very early time, were not considered in the study. From the results, 𝑃𝐷 and 𝑃𝐷′ vary directly with ℎ𝐷 and inversely as 𝐿𝐷. The 𝑃𝐷′ gradually reduces to zero when 𝑃𝐷 begins to exhibit a constant trend. 𝑃𝐷′ vary inversely with ℎ𝐷 and 𝑦𝑒𝐷 at all flow times. The number of flow periods varies with reservoir size, well length and production time. The time at which the 𝑃𝐷′ starts to exhibit a downward trend is the external fluid breakthrough time. The breakthrough time is affected by well design. Longer wells exhibit delayed breakthrough time because of lower pressure drawdown associated with increased well length. If production rate is sustained for any particular well design, the well will completely water-out. Finally, infinite conductivity 𝑥𝐷=0.732 and uniform flux condition do not really affect 𝑃𝐷 and 𝑃𝐷′ at early time.
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    Well Test Analysis of a Horizontal Well in a Completely Bounded Reservoir
    (Kenyatta University, 2022) Kitungu, Nzomo Timothy; Kennedy Awuor; Stephen Ezizanami; Daniel Okang'a Oyoo
    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.