Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime

Mutili Peter Mutisya

doi:doi:10.11648/j.ijamtp.20251104.11

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Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime

Mutili Peter Mutisya^*

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 4)

Received: 1 September 2025 Accepted: 16 September 2025 Published: 9 October 2025

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Abstract

Pressure distribution in horizontal wells mostly involves solving the diffusivity equation. Mathematical models take into account factors like permeability anisotropy, reservoir geometry and wellbore characteristics. The models use concepts like source and Green's functions to approximate pressure behavior particularly during various flow regimes like early radial, early linear, late pseudo radial and late linear. Mostly models consider the effects of boundaries like edge and bottom water drives and well completion strategies on pressure distribution. In this study reservoir characterization and description of the reservoir system form ways of actualizing prolonged clean oil production which is fundamental in all aspects of reservoir and petroleum engineering based on dimensionless pressure and dimensionless pressure derivative distribution in horizontal wells in oil reservoirs with water drive mechanisms. MATLAB programming was deployed in solving complex models and generating pressure and pressure derivative distribution plots. Since discrete data was used, cubic spline interpolation technique was used in generating smooth curves log – log plots. The results of the study show, at dimensionless breakthrough time (t_DBr), the dimensionless pressure derivative collapses to zero when dimensionless pressure exhibits a constant trend. In addition, t_DBrvaries as h_Dy_eD/L_Dand the dimensionless pressure varies directly as x2_eD and inversely as y_eD, P_D varies as x2_eD / y_eD. Further from this study, x_D = x_wD, y_eD = 2y_D = 2y_wD and z_eD = h_D = 2z_wD and prolonged L_D mar clean oil production. The information presented in this paper will assist the reservoir and petroleum engineers in designing correctly the horizontal well in the oil reservoir in order to enhance prolonged clean oil production.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 11, Issue 4)
DOI	10.11648/j.ijamtp.20251104.11
Page(s)	49-68
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Horizontal Well, Reservoir, Double Edge, Water Drive, Late Time Flow

References

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[2]	Adewole E. S. (August, 2012), “Interference Tests Analyses of Horizontal and Vertical Well Combinations in a Reservoir Subject to Double-Edged Water Drive”. Paper presented at the Nigeria Annual International Conference and Exhibition, Lagos, Nigeria. SPE-163033-MS. https://doi.org/10.2118/163033-MS
[3]	Henry Idudje and Steve Adewole (August, 2020), “A Method of Estimating Reservoir Pressure using Drawdown Test Data” Paper presented Virtually at the SPE Nigeria Annual International Conference and Exhibition. SPE-203700-MS. https://doi.org/10.2118/203700-MS
[4]	Malekzadeh D. and Tiab D. (October, 1991), “Interference Testing of Horizontal Wells.” Paper presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, U.S.A. https://doi.org/10.2118/22733-MS
[5]	Emumena E., Adewole E.S. and Ojah M.G. (July, 2023), “Dimensionless Pressures and Derivative Distribution of Two Interfering Horizontal and Vertical Wells in an Infinite Acting Reservoir.” Paper presented at the SPE Nigeria Annual International Conference and Exhibition, Lagos, Nigeria. SPE-217170-MS. https://doi.org/10.2118/217170-MS
[6]	Ojah M.G., Adewole E.S. and Emumena E. (July, 2023), “Pressures and Pressure Derivatives of Vertical and Horizontal Wells Located Within Intersecting Sealing Fault and Constant Pressure Boundary.” Paper presented at the SPE Nigeria Annual International Conference and Exhibition, Lagos, Nigeria. SPE-217103-MS. https://doi.org/10.2118/217103-MS
[7]	Edobhiye O. and Adewole E.S. (August, 2014), “Effects of Both Wellbore and Reservoir Properties on Dimensionless Pressure and Dimensionless Pressure Derivative Distribution of a Horizontal Well in a Reservoir Subject to Bottom Water, Gas Cap and Single Edge Water Drive Mechanisms.” Paper presented at the SPE Nigeria Annual International Conference and Exhibition, Lagos, Nigeria. SPE-172384-MS. https://doi.org/10.2118/172384-MS
[8]	Ogbue M.C. (June, 2025), “Comprehensive Pressure Distribution Model for Horizontal Well in Bottom-Water Reservoir.” Caritas Journal of Engineering Technology. Volume 4(01). www.caritasuniversityjournals.org
[9]	Ozkan Erdal and Raghavan Rajagopal (August, 1990), “Performance of Horizontal Wells Subject to Bottom water Drive.” SPE-18559-PA. Volume 5(03). https://doi.org/10.2118/18559-PA
[10]	Ogbamikhumi A.V. and Adewole E.S. (January, 2020), “Pressure Behaviour of a Horizontal Well Sandwiched Between Two Parallel Sealing Faults”, Nigerian Journal of Technology (NIJOTECH), Vol. 39(1), pp. 148-153. http://dx.doi.org/10.4314/njt.v39i1.16
[11]	Oloro J.O, Adewole E.S. and Olafuyi O.A. (December, 2014), “Pressure Distribution of Horizontal Wells in a Layered Reservoir with Simultaneous Gas Cap and Bottom Water Drives.” American Journal of Engineering Research. Vol. 3(12). pp. 41-53. www.ajer.org
[12]	Oloro J.O and Adewole S.E (April, 2020), “Performance and Behavior of a Horizontal Well in Reservoir Subject to Double-Edged Water Drive.” Nigerian Journal of Technology (NIJOTECH). Vol. 39(2). pp. 417 – 423. http://dx.doi.org/10.4314/njt.v39i2.11
[13]	Ogbue M.C. and Adewole E.S. (September, 2013), “Theoretical Investigation of Factors Affecting Water Breakthrough Time in a Horizontal Well Subject to Bottom Water Drive.” Advanced Materials Research. Trans Tech Publications, Switzerland. Volume 824(2013). pp. 394-400. https://doi.org/10.4028/www.scientific.net/AMR.824.394
[14]	Oloro J. and Adewole E.S. (January, 2015), “Factors that affect pressure distribution of horizontal wells in a layered reservoir with simultaneous gas cap and bottom water drive.” Journal of Petroleum and Gas Engineering. Vol. 6(1), pp. 1-9. http://www.academicjournals.org/JPGE
[15]	Joshi, S.D., (September, 1990), “Horizontal well technology”, Joshi Technologies International, Inc. Tulsa, Oklahoma, U.S.A.

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APA Style

Mutisya, M. P. (2025). Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime. International Journal of Applied Mathematics and Theoretical Physics, 11(4), 49-68. https://doi.org/10.11648/j.ijamtp.20251104.11

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ACS Style

Mutisya, M. P. Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime. Int. J. Appl. Math. Theor. Phys. 2025, 11(4), 49-68. doi: 10.11648/j.ijamtp.20251104.11

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AMA Style

Mutisya MP. Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime. Int J Appl Math Theor Phys. 2025;11(4):49-68. doi: 10.11648/j.ijamtp.20251104.11

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@article{10.11648/j.ijamtp.20251104.11,
  author = {Mutili Peter Mutisya},
  title = {Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {4},
  pages = {49-68},
  doi = {10.11648/j.ijamtp.20251104.11},
  url = {https://doi.org/10.11648/j.ijamtp.20251104.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251104.11},
  abstract = {Pressure distribution in horizontal wells mostly involves solving the diffusivity equation. Mathematical models take into account factors like permeability anisotropy, reservoir geometry and wellbore characteristics. The models use concepts like source and Green's functions to approximate pressure behavior particularly during various flow regimes like early radial, early linear, late pseudo radial and late linear. Mostly models consider the effects of boundaries like edge and bottom water drives and well completion strategies on pressure distribution. In this study reservoir characterization and description of the reservoir system form ways of actualizing prolonged clean oil production which is fundamental in all aspects of reservoir and petroleum engineering based on dimensionless pressure and dimensionless pressure derivative distribution in horizontal wells in oil reservoirs with water drive mechanisms. MATLAB programming was deployed in solving complex models and generating pressure and pressure derivative distribution plots. Since discrete data was used, cubic spline interpolation technique was used in generating smooth curves log – log plots. The results of the study show, at dimensionless breakthrough time (tDBr), the dimensionless pressure derivative collapses to zero when dimensionless pressure exhibits a constant trend. In addition, tDBr varies as hDyeD/LD and the dimensionless pressure varies directly as x2eD and inversely as yeD, PD varies as x2eD / yeD. Further from this study, xD = xwD, yeD = 2yD = 2ywD and zeD = hD = 2zwD and prolonged LD mar clean oil production. The information presented in this paper will assist the reservoir and petroleum engineers in designing correctly the horizontal well in the oil reservoir in order to enhance prolonged clean oil production.},
 year = {2025}
}

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TY  - JOUR
T1  - Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime
AU  - Mutili Peter Mutisya
Y1  - 2025/10/09
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251104.11
DO  - 10.11648/j.ijamtp.20251104.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 49
EP  - 68
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251104.11
AB  - Pressure distribution in horizontal wells mostly involves solving the diffusivity equation. Mathematical models take into account factors like permeability anisotropy, reservoir geometry and wellbore characteristics. The models use concepts like source and Green's functions to approximate pressure behavior particularly during various flow regimes like early radial, early linear, late pseudo radial and late linear. Mostly models consider the effects of boundaries like edge and bottom water drives and well completion strategies on pressure distribution. In this study reservoir characterization and description of the reservoir system form ways of actualizing prolonged clean oil production which is fundamental in all aspects of reservoir and petroleum engineering based on dimensionless pressure and dimensionless pressure derivative distribution in horizontal wells in oil reservoirs with water drive mechanisms. MATLAB programming was deployed in solving complex models and generating pressure and pressure derivative distribution plots. Since discrete data was used, cubic spline interpolation technique was used in generating smooth curves log – log plots. The results of the study show, at dimensionless breakthrough time (tDBr), the dimensionless pressure derivative collapses to zero when dimensionless pressure exhibits a constant trend. In addition, tDBr varies as hDyeD/LD and the dimensionless pressure varies directly as x2eD and inversely as yeD, PD varies as x2eD / yeD. Further from this study, xD = xwD, yeD = 2yD = 2ywD and zeD = hD = 2zwD and prolonged LD mar clean oil production. The information presented in this paper will assist the reservoir and petroleum engineers in designing correctly the horizontal well in the oil reservoir in order to enhance prolonged clean oil production.
VL  - 11
IS  - 4
ER  -

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Author Information

Mutili Peter Mutisya

Department of Mathematics and Actuarial Science, Kenyatta University, Nairobi, Kenya

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http://orcid.org/0009-0006-4412-8155

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APA Style

Mutisya, M. P. (2025). Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime. International Journal of Applied Mathematics and Theoretical Physics, 11(4), 49-68. https://doi.org/10.11648/j.ijamtp.20251104.11

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ACS Style

Mutisya, M. P. Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime. Int. J. Appl. Math. Theor. Phys. 2025, 11(4), 49-68. doi: 10.11648/j.ijamtp.20251104.11

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AMA Style

Mutisya MP. Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime. Int J Appl Math Theor Phys. 2025;11(4):49-68. doi: 10.11648/j.ijamtp.20251104.11

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@article{10.11648/j.ijamtp.20251104.11,
  author = {Mutili Peter Mutisya},
  title = {Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {11},
  number = {4},
  pages = {49-68},
  doi = {10.11648/j.ijamtp.20251104.11},
  url = {https://doi.org/10.11648/j.ijamtp.20251104.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20251104.11},
  abstract = {Pressure distribution in horizontal wells mostly involves solving the diffusivity equation. Mathematical models take into account factors like permeability anisotropy, reservoir geometry and wellbore characteristics. The models use concepts like source and Green's functions to approximate pressure behavior particularly during various flow regimes like early radial, early linear, late pseudo radial and late linear. Mostly models consider the effects of boundaries like edge and bottom water drives and well completion strategies on pressure distribution. In this study reservoir characterization and description of the reservoir system form ways of actualizing prolonged clean oil production which is fundamental in all aspects of reservoir and petroleum engineering based on dimensionless pressure and dimensionless pressure derivative distribution in horizontal wells in oil reservoirs with water drive mechanisms. MATLAB programming was deployed in solving complex models and generating pressure and pressure derivative distribution plots. Since discrete data was used, cubic spline interpolation technique was used in generating smooth curves log – log plots. The results of the study show, at dimensionless breakthrough time (tDBr), the dimensionless pressure derivative collapses to zero when dimensionless pressure exhibits a constant trend. In addition, tDBr varies as hDyeD/LD and the dimensionless pressure varies directly as x2eD and inversely as yeD, PD varies as x2eD / yeD. Further from this study, xD = xwD, yeD = 2yD = 2ywD and zeD = hD = 2zwD and prolonged LD mar clean oil production. The information presented in this paper will assist the reservoir and petroleum engineers in designing correctly the horizontal well in the oil reservoir in order to enhance prolonged clean oil production.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Performance of a Horizontal Well in an Anisotropic Oil Reservoir Subject to Simultaneous Double Edge Water Drive Mechanisms at Late Time Flow Regime
AU  - Mutili Peter Mutisya
Y1  - 2025/10/09
PY  - 2025
N1  - https://doi.org/10.11648/j.ijamtp.20251104.11
DO  - 10.11648/j.ijamtp.20251104.11
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 49
EP  - 68
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20251104.11
AB  - Pressure distribution in horizontal wells mostly involves solving the diffusivity equation. Mathematical models take into account factors like permeability anisotropy, reservoir geometry and wellbore characteristics. The models use concepts like source and Green's functions to approximate pressure behavior particularly during various flow regimes like early radial, early linear, late pseudo radial and late linear. Mostly models consider the effects of boundaries like edge and bottom water drives and well completion strategies on pressure distribution. In this study reservoir characterization and description of the reservoir system form ways of actualizing prolonged clean oil production which is fundamental in all aspects of reservoir and petroleum engineering based on dimensionless pressure and dimensionless pressure derivative distribution in horizontal wells in oil reservoirs with water drive mechanisms. MATLAB programming was deployed in solving complex models and generating pressure and pressure derivative distribution plots. Since discrete data was used, cubic spline interpolation technique was used in generating smooth curves log – log plots. The results of the study show, at dimensionless breakthrough time (tDBr), the dimensionless pressure derivative collapses to zero when dimensionless pressure exhibits a constant trend. In addition, tDBr varies as hDyeD/LD and the dimensionless pressure varies directly as x2eD and inversely as yeD, PD varies as x2eD / yeD. Further from this study, xD = xwD, yeD = 2yD = 2ywD and zeD = hD = 2zwD and prolonged LD mar clean oil production. The information presented in this paper will assist the reservoir and petroleum engineers in designing correctly the horizontal well in the oil reservoir in order to enhance prolonged clean oil production.
VL  - 11
IS  - 4
ER  -

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