Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions

Abdou Njifenjou; Clement Obaker Nzonda Noussi; Moussa Sali

doi:doi:10.11648/j.ajam.20251303.13

Research Article |

| Peer-Reviewed

Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions

Abdou Njifenjou^*, Clement Obaker Nzonda Noussi, Moussa Sali

Published in American Journal of Applied Mathematics (Volume 13, Issue 3)

Received: 10 April 2025 Accepted: 1 June 2025 Published: 13 June 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

This work develops mathematical aspects of Conventional Finite Volume schemes for flow problems in porous media governed by discontinuous absolute permeability. Focusing on incompressible one-phase flow problems in heterogeneous porous media, a particular attention is put on the homogenized absolute permeability involved in the discrete Darcy velocity over the “interaction zone” between two adjacent control volumes. The first key-step of our presentation consists in putting in place a discrete-function-space frame-work endowed with inner products and their associated norms. Then after adequate mathematical tools are deployed as projection and interpolation operators with their fundamental properties. A discrete version of the Poincaré-Friedrichs inequality is also established and used to get equivalent discrete norms. Interpolation Operators are used to define cellwise-constant and linear-spline approximate solutions. A discrete variational formulation of the finite volume problem is stated and the Lax-Milgram theorem applies (upon projection operator continuity) to show the well posedness of the discrete variational problem. A first order convergence in L²-norm and in some discrete energy norm has been shown. Sufficient conditions to get higher order convergence rate in L²-norm and in H₀¹-norm have been stated for linear-spline solutions.

Published in	American Journal of Applied Mathematics (Volume 13, Issue 3)
DOI	10.11648/j.ajam.20251303.13
Page(s)	205-224
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

Conventional Finite Volumes, Incompressible Flows, Discrete Function Space Frame-Work, Cellwise-constant and Linear-spline Solutions, Rate Convergence

References

[1]	S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics (TAM), 2008.
[2]	H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer (2011).
[3]	Ph. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, SIAM (2013).
[4]	Ph. G. Ciarlet, Introduction à l’Analyse Numérique Matricielle et à l’Optimisation, Dunod (1998).
[5]	R. Eymard, Th. Gallouet and R. Herbin, Finite Volume Methods, HandBook of Numerical Analysis, Editors: Ph.G. Ciarlet and J. L. Lions, 2000.
[6]	R. F. Gabbasov, Generalized equations of the finite difference method in polar coordinates on problems with discontinuous solutions. Resistance of materials and theory of structures. Kiev: Budivelnik. 1984; 45: (55- 58).
[7]	R. F. Gabbasov and S. Moussa, Generalized Equations of Finite Diffrence Method and their Application for Calculation of Variable Stiffess Curved Plates. Ed. News of Higher Educational Institutions Construction. 2004.
[8]	T. Gallouet and D. Guérillot, An optimal method for averaging the absolute permeability, Proceedings of the Third International Reservoir Charaterization Technical Conference, Tulsa, Oklahoma, 3-5 November, 1991.
[9]	Th. Hontans, Homogénéisation Numérique de Parametres Pétrophysiques Pour des Maillages Déstructurés en Simulation de Réservoir, PhD thesis, University of “Pau et des Pays de l’Adour”, (France), 2000.
[10]	P. Lemonnier and B. Bourbiaux, Simulation of Naturally Fractured Reservoirs. State of the Art, Oil / Gas Science and Technology, ”Institut Francais du Pétrole” Review, March 2010.
[11]	C. M. Marle, Multiphase flow in porous media, Editor: Technip, 2000.
[12]	A. Njifenjou, Introduction to Finite Element Methods, https://www.researchgate.net/publication/354459495
[13]	A. Njifenjou, Eléments finis mixtes hybrides duaux et Homogénéisation des parametres pétrophysiques, PhD thesis, University Paris 6, 1993.
[14]	A. Njifenjou, Expression en termes d’énergie pour la perméabilité absolue effective, Revue de l’Institut Francais du Pétrole, vol. 49, No 4, pp 345-358 (1994).
[15]	A. Njifenjou, Discrete maximum principle honored by finite volume schemes for diffusion-convection- reaction problems: Proof with geometrical arguments, ResearchGate preprint, March 2025, https://doi.org/10.13140/RG.2.2.15483.43044
[16]	B. Noetinger, The effective permeability of heterogeneous porous media, Transport in Porous Media, Vol. 15, pp. 99-127, 1994.
[17]	M. Quintard and S. Whitaker, Two-Phase Flow in Heterogeneous Porous Media: The Method of Large-Scale Averaging, Transport in Porous Media, 1988.
[18]	P. A. Raviart and J. M. Thomas, Introduction a l’Analyse Numérique des Equations aux Dérivées Partielles, Dunod (2004).
[19]	Ph. Renard and R. Ababou, Equivalent Permeability Tensor of Heterogeneous Media: Upscaling Methods and Criteria (Review and Analyses), Geosciences, 2022, 12, 260.
[20]	S. Youssoufa, S. Moussa, A. Njifenjou, J. Nkongho Anyi, and A. C. Ngayihi, Application of generalized equations of finite difference method to computation of bent isotropic stretched and/or compressed plates of variable stiffess under elastic foundation, De Gruyter, Curved and Layer. Struct. 2022.
[21]	S. Youssoufa, Calcul Numérique des Plaques et Coques Isotropes et Homogenes a Epaisseur Variable par les Equations Généralisées de la Méthode des Différences Finies, PhD thesis, University of Douala (Cameroon), 2022.
[22]	Abdou Njifenjou, Abel Toudna Mansou, Moussa Sali. (2004). A New Second-order Maximum-principle- preserving Finite-volume Method for Flow Problems Involving Discontinuous Coefficients.American Journal of Applied Mathematics, 12(4), 91-110.

Cite This Article

Plain Text BibTeX RIS

APA Style

Njifenjou, A., Noussi, C. O. N., Sali, M. (2025). Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions. American Journal of Applied Mathematics, 13(3), 205-224. https://doi.org/10.11648/j.ajam.20251303.13

Copy | Download

ACS Style

Njifenjou, A.; Noussi, C. O. N.; Sali, M. Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions. Am. J. Appl. Math. 2025, 13(3), 205-224. doi: 10.11648/j.ajam.20251303.13

Copy | Download

AMA Style

Njifenjou A, Noussi CON, Sali M. Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions. Am J Appl Math. 2025;13(3):205-224. doi: 10.11648/j.ajam.20251303.13

Copy | Download

@article{10.11648/j.ajam.20251303.13,
  author = {Abdou Njifenjou and Clement Obaker Nzonda Noussi and Moussa Sali},
  title = {Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {3},
  pages = {205-224},
  doi = {10.11648/j.ajam.20251303.13},
  url = {https://doi.org/10.11648/j.ajam.20251303.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251303.13},
  abstract = {This work develops mathematical aspects of Conventional Finite Volume schemes for flow problems in porous media governed by discontinuous absolute permeability. Focusing on incompressible one-phase flow problems in heterogeneous porous media, a particular attention is put on the homogenized absolute permeability involved in the discrete Darcy velocity over the “interaction zone” between two adjacent control volumes. The first key-step of our presentation consists in putting in place a discrete-function-space frame-work endowed with inner products and their associated norms. Then after adequate mathematical tools are deployed as projection and interpolation operators with their fundamental properties. A discrete version of the Poincaré-Friedrichs inequality is also established and used to get equivalent discrete norms. Interpolation Operators are used to define cellwise-constant and linear-spline approximate solutions. A discrete variational formulation of the finite volume problem is stated and the Lax-Milgram theorem applies (upon projection operator continuity) to show the well posedness of the discrete variational problem. A first order convergence in L2-norm and in some discrete energy norm has been shown. Sufficient conditions to get higher order convergence rate in L2-norm and in H01-norm have been stated for linear-spline solutions.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions
AU  - Abdou Njifenjou
AU  - Clement Obaker Nzonda Noussi
AU  - Moussa Sali
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251303.13
DO  - 10.11648/j.ajam.20251303.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 205
EP  - 224
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251303.13
AB  - This work develops mathematical aspects of Conventional Finite Volume schemes for flow problems in porous media governed by discontinuous absolute permeability. Focusing on incompressible one-phase flow problems in heterogeneous porous media, a particular attention is put on the homogenized absolute permeability involved in the discrete Darcy velocity over the “interaction zone” between two adjacent control volumes. The first key-step of our presentation consists in putting in place a discrete-function-space frame-work endowed with inner products and their associated norms. Then after adequate mathematical tools are deployed as projection and interpolation operators with their fundamental properties. A discrete version of the Poincaré-Friedrichs inequality is also established and used to get equivalent discrete norms. Interpolation Operators are used to define cellwise-constant and linear-spline approximate solutions. A discrete variational formulation of the finite volume problem is stated and the Lax-Milgram theorem applies (upon projection operator continuity) to show the well posedness of the discrete variational problem. A first order convergence in L2-norm and in some discrete energy norm has been shown. Sufficient conditions to get higher order convergence rate in L2-norm and in H01-norm have been stated for linear-spline solutions.
VL  - 13
IS  - 3
ER  -

Copy | Download

Author Information

Abdou Njifenjou

Department of Mathematics, National Advanced School of Engineering, University of Yaounde 1, Yaounde, Cameroon

Contact Email
Clement Obaker Nzonda Noussi

Department of Mathematics, Faculty of Science, University of Douala, Douala, Cameroon

Contact Email
Moussa Sali

Department of Civil Engineering and Architecture, National Advanced School of Engineering, University of Maroua, Maroua, Cameroon

Contact Email

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Njifenjou, A., Noussi, C. O. N., Sali, M. (2025). Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions. American Journal of Applied Mathematics, 13(3), 205-224. https://doi.org/10.11648/j.ajam.20251303.13

Copy | Download

ACS Style

Njifenjou, A.; Noussi, C. O. N.; Sali, M. Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions. Am. J. Appl. Math. 2025, 13(3), 205-224. doi: 10.11648/j.ajam.20251303.13

Copy | Download

AMA Style

Njifenjou A, Noussi CON, Sali M. Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions. Am J Appl Math. 2025;13(3):205-224. doi: 10.11648/j.ajam.20251303.13

Copy | Download

@article{10.11648/j.ajam.20251303.13,
  author = {Abdou Njifenjou and Clement Obaker Nzonda Noussi and Moussa Sali},
  title = {Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions},
  journal = {American Journal of Applied Mathematics},
  volume = {13},
  number = {3},
  pages = {205-224},
  doi = {10.11648/j.ajam.20251303.13},
  url = {https://doi.org/10.11648/j.ajam.20251303.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20251303.13},
  abstract = {This work develops mathematical aspects of Conventional Finite Volume schemes for flow problems in porous media governed by discontinuous absolute permeability. Focusing on incompressible one-phase flow problems in heterogeneous porous media, a particular attention is put on the homogenized absolute permeability involved in the discrete Darcy velocity over the “interaction zone” between two adjacent control volumes. The first key-step of our presentation consists in putting in place a discrete-function-space frame-work endowed with inner products and their associated norms. Then after adequate mathematical tools are deployed as projection and interpolation operators with their fundamental properties. A discrete version of the Poincaré-Friedrichs inequality is also established and used to get equivalent discrete norms. Interpolation Operators are used to define cellwise-constant and linear-spline approximate solutions. A discrete variational formulation of the finite volume problem is stated and the Lax-Milgram theorem applies (upon projection operator continuity) to show the well posedness of the discrete variational problem. A first order convergence in L2-norm and in some discrete energy norm has been shown. Sufficient conditions to get higher order convergence rate in L2-norm and in H01-norm have been stated for linear-spline solutions.},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Conventional Finite Volumes for Flow Problems in Porous Media Involving Discontinuous Permeability: Convergence Rate Analysis of Cellwise-constant and Linear-spline Solutions
AU  - Abdou Njifenjou
AU  - Clement Obaker Nzonda Noussi
AU  - Moussa Sali
Y1  - 2025/06/13
PY  - 2025
N1  - https://doi.org/10.11648/j.ajam.20251303.13
DO  - 10.11648/j.ajam.20251303.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 205
EP  - 224
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20251303.13
AB  - This work develops mathematical aspects of Conventional Finite Volume schemes for flow problems in porous media governed by discontinuous absolute permeability. Focusing on incompressible one-phase flow problems in heterogeneous porous media, a particular attention is put on the homogenized absolute permeability involved in the discrete Darcy velocity over the “interaction zone” between two adjacent control volumes. The first key-step of our presentation consists in putting in place a discrete-function-space frame-work endowed with inner products and their associated norms. Then after adequate mathematical tools are deployed as projection and interpolation operators with their fundamental properties. A discrete version of the Poincaré-Friedrichs inequality is also established and used to get equivalent discrete norms. Interpolation Operators are used to define cellwise-constant and linear-spline approximate solutions. A discrete variational formulation of the finite volume problem is stated and the Lax-Milgram theorem applies (upon projection operator continuity) to show the well posedness of the discrete variational problem. A first order convergence in L2-norm and in some discrete energy norm has been shown. Sufficient conditions to get higher order convergence rate in L2-norm and in H01-norm have been stated for linear-spline solutions.
VL  - 13
IS  - 3
ER  -

Copy | Download