Essentially non-oscillatory schemes applied to Buckley-Leverett equation with diffusive term

  • Raphael de Oliveira Garcia Federal University of São Paulo
  • Graciele Paraguaia Silveira Federal University of São Carlos
Keywords: Numerical methods, Immiscible two-phase fluid, Petroleum flow


The purpose of this work was to investigate the flow of two-phase fluids via the Buckley-Leverett equation, corresponding to three types of scenarios applied in oil extraction, including a diffusive term. For this, a weighted essentially non-oscillatory scheme, a Runge-Kutta method and a central finite difference were computationally implemented. In addition, a numerical study related to the precision order and stability was performed. The use of these methods made it possible to obtain numerical solutions without oscillations and without excessive numerical dissipation, sufficient to assist in the understanding of the mixing profiles of saturated water and petroleum fluids, inside pipelines filled with porous material, in addition to allowing the investigation of the impact of adding the diffusive term in the original equation.


Download data is not yet available.

Author Biography

Graciele Paraguaia Silveira, Federal University of São Carlos

Mathematics, Physics and Chemisty Department


S. E. Buckley, and M. C. Leverett, “Mechanism of fluid displacement in sands,” Trans. AIME, vol. 146, pp. 187–196, 1942.

F. L. Fayers, and J. W. Sheldon, “The effect of capillary pressure and gravity on two-phase fluid flow in porous medium,” Trans. AIME, vol. 216, pp. 147-155, 1959.

W. Proskurowski, “A note on solving the Buckley-Leverett equation in the presence of gravity,” J. Comput. Phys., vol. 41, pp. 136-141, 1981.

D. A. DiCarlo, “Experimental measurements of saturation overshoot on infiltration,” Water Resour. Res., vol. 40, pp. 4215-1-4215.9, 2004.

Y. Wang, and C-Y. Kao, “Central schemes for the modified Buckley–Leverett equation,” J. Comput. Sci., vol. 4, pp. 12-23, 2013.

E. Abreu, and J. Vieira, “Computational numerical solutions of the pseudo-parabolic Buckley-Leverett equation with dynamic capillary pressure,” Math. Comput. Simulation, vol. 137, issue C, pp. 29-48, 2017.

R. O. Garcia, and G. P. Silveira, “Essentially non-oscillatory schemes applied to Buckley-Leverett equation,” in XXV ENMC - Encontro Nacional de Modelagem Computacional, XIII ECTM - Encontro de Ciência e Tecnologia de Materiais, 9º MCSul - Conferência Sul em Modelagem Computacional, and IX SEMENGO - Seminário e Workshop em Engenharia Oceânica, 2022 [Online]. Available: [Acessed: Feb. 2, 2023].

R. J. Leveque, Finite Volume Methods for Hyperbolic Problems. New York: Cambridge University Press, 2002.

C. J. van Duijin, L. A. Peletier, and I. S. Pop, “A new class of entropy solutions of the Buckley-Leverett equation,” , SIAM J. Math. Anal., vol. 39, no. 2, pp. 507-536, 2007.

A. Harten, and S. Osher, “Uniformly high-order accurate non-oscillatory schemes I,” SIAM J. Numer. Anal., vol. 24, pp. 279-309, 1987.

A. Harten, S. Osher, B. Engquist, and S. Chakravarthy, “Uniformly high-order accurate non-oscillatory schemes III,” J. Comput. Phys., vol. 71, pp. 231-303, 1987.

C-W. Shu, and S. Osher, “Efficient implementation of essentially non-oscillatory shock capturing schemes,” J. Comput. Phys., vol. 77, pp: 439-471, 1988.

G-S. Jiang, and C-W Shu, “Efficient Implementation of Weighted ENO Schemes,” J. Comput. Phys., vol. 126, pp: 202-228, 1996.

G. A. Gerolymos, D. Sénéchal, and I. Vallet, “Very-high-order WENO schemes,” J. Comput. Phys., vol. 228, pp. 8481-8524, 2009.

G. P. Silveira, and L. C. Barros, “Numerical methods integrated with fuzzy logic and stochastic method for solving PDEs: An application to dengue,” Fuzzy Sets Syst., vol. 225, pp. 39-57, 2013.

J. W. Thomas, Numerical Partial Differential Equations: Finite Difference Methods. New York: Springer, 1995.

M. Motamed, C. B. Macdonald, and S. J. Ruuth, “On the linear stability of the fifth-order WENO discretization,” J. Sci. Comput., vol. 47, pp. 127-149, 2011.

How to Cite
R. Garcia and G. Silveira, “Essentially non-oscillatory schemes applied to Buckley-Leverett equation with diffusive term”, LAJC, vol. 11, no. 1, pp. 42-55, Jan. 2024.
Research Articles for the Regular Issue