Application of Finite Element Method in One-Dimensional Instability Phenomenon Arising in Inclined Porous Media

Authors

  • Tejaskumar Sharma Department of Mathematics, N V Patel College of Pure and Applied Sciences, The Charutar Vidya Mandal University, Vallabh Vidyanagar-388120, India https://orcid.org/0000-0003-3185-4706
  • Shreekant Pathak Department of Applied Science and Humanities, Madhuben & Bhanubhai Patel Institute of Technology (MBIT), Anand, Vallabh Vidyanagar, Gujarat, India, 388 120
  • Gargi Trivedi Department of Applied Mathematics, Faculty of Technology and Engineering, The Maharaja Sayajirao University of Baroda, Vadodara, India https://orcid.org/0000-0001-6597-1420
  • Shital Patel Department of Chemistry, N V Patel College of Pure and Applied Sciences, The Charutar Vidya Mandal University, Vallabh Vidyanagar-388120, India https://orcid.org/0000-0001-6402-1482

Keywords:

Finite Element Method, Viscous Fingering, Inclined Porous Media,, Secondary oil recovery, Non linear PDE, Saturation Dynamics, Homogeneous Porous Matrix

Abstract

Viscous fingering, an instability phenomenon during secondary oil recovery, occurs
when water is injected into an oil-saturated porous medium inclined at θ = 10◦. This
study employs the Finite Element Method (FEM) to solve the non-linear partial differen-
tial equation (PDE) governing water saturation dynamics in a one-dimensional homoge-
neous porous medium, incorporating viscosity differences, capillary pressure (ε = 1 Pa),
and gravitational effects. Using a Galerkin approach with linear basis functions and im-
plicit time-stepping, the FEM framework discretizes the PDE, satisfying exponential initial
and boundary conditions. Numerical solutions, implemented in MATLAB, are validated
against semi-analytical results (5). Saturation increases exponentially with distance and
time, with inclination reducing saturation due to gravity. Tables present saturation values
for inclined (θ = 10◦) and non-inclined (θ = 0◦) cases across 10 values of X and T , and graphs illustrate profiles. A comparative study quantifies inclination effects. This work
provides a robust, scalable framework for optimizing oil recovery.

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Published

June 30, 2026

Issue

Vol. 17 No. 2 (2026)

Section

Research Article

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How to Cite

Sharma, T., Pathak, S., Trivedi, G., & Patel, S. (2026). Application of Finite Element Method in One-Dimensional Instability Phenomenon Arising in Inclined Porous Media. Communications in Mathematics and Applications, 17(2). https://doi.org/10.26713/cma.v17i2.3359