Numerical Study for Two-Phase Flow with Gravity in Homogeneous and Piecewise-Homogeneous Porous Media

ARAUJO, I. L. N.; RODRÍGUEZ-BERMÚDEZ, P.; RODRÍGUEZ-NÚÑEZ, Y.

doi:10.5540/tema.2020.021.01.0021

Acessibilidade / Reportar erro

Brasil

Español English

sumário « anterior atual seguinte »

Sumário

Articles • TEMA (São Carlos) 21 (1) • Jan-Apr 2020 • https://doi.org/10.5540/tema.2020.021.01.0021 copy

Numerical Study for Two-Phase Flow with Gravity in Homogeneous and Piecewise-Homogeneous Porous Media

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT

In this work we study two-phase flow with gravity either in 1-rock homogeneous media or 2-rocks composed media. These phenomena can be modeled by a non-linear scalar conservation law with continuous flux function or discontinuous flux function, respectively. Our study is essentially from a numerical point of view, we apply the new Lagrangian-Eulerian (LEH) finite difference method developed by Abreu and Pérez ¹1. E. Abreu, V. Matos, J. Perez & P. Rodriguez-Bermudez. A class of Lagrangian-Eulerian shock-capturing schemes for first-order hyperbolic problems with forcing terms. sent, (2019).^{), (}²2. E.C. Abreu, W. Lambert, J. Perez & A. Santo. A new finite volume approach for transport models and related applications with balancing source terms. Mathematics and Computers in Simulation, 137 (2017), 2-28.^{), (}³3. E.C. Abreu & J.A. Perez. A Lagrangian-Eulerian algorithm scheme for hyperbolic conservation laws and balance laws. HYP2014XV International Conference on Hyperbolic Problems, (2014). Acesso em 31 de agosto de 2018.^{), (}⁴4. E.C. Abreu & J.A. Perez. A new locally Conservative Lagrangian Eulerian method for hyperbolic and Balance laws. VIII Pan-American Workshop Applied and Computational Mathematics, (2014).^{), (}⁵5. E.C. Abreu & J.A. Perez. A fast, robust, and simple Lagrangian-Eulerian solver for balance laws and applications. Computers & Mathematics with Applications, 77(9) (2019), 2310-2336.^{), (}²⁵25. J.A. Perez. “Lagrangian-Eulerian approximation methods for balance laws and hyperbolic conservation laws”. Ph.d. thesis, UNICAMP (2015). and the classical Lax-Friedrichs method to obtain numerical entropic solutions. Comparisons between numerical and analytical solutions show the efficiency of the methods even for discontinuous flux function. Our main contribution, is the comparison and error analysis between the new LEH and the classical Lax-Friedrichs (LF) methods, in order to show the good performance of the LEH scheme for models with discontinuous flux functions.

Keywords:
conservation laws; finite differences; Lagrangian-Eulerian approach; two-phase flow; heterogeneous porous medium

Cells	$L E H {‖ u - U ‖}_{l_{h}^{1}}$	$L E H {‖ u - U ‖}_{l_{h}^{2}}$	$L F {‖ u - U ‖}_{l_{h}^{1}}$	$L F {‖ u - U ‖}_{l_{h}^{2}}$
128	1.57 × 10⁻²	5.04 × 10⁻²	2.52 × 10⁻²	6.79 × 10⁻²
256	9.80 × 10⁻³	4.00 × 10⁻²	1.60 × 10⁻²	5.38 × 10⁻²
512	6.10 × 10⁻³	3.23 × 10⁻²	1.03 × 10⁻²	4.32 × 10⁻²
1024	3.50 × 10⁻³	2.37 × 10⁻²	6.20 × 10⁻³	3.27 × 10⁻²
2048	2.20 × 10⁻³	1.97 × 10⁻²	3.90 × 10⁻³	2.65 × 10⁻²
4096	1.30 × 10⁻³	1.54 × 10⁻²	2.30 × 10⁻³	2.07 × 10⁻²

Cells	$L F {‖ u - U ‖}_{l_{h}^{1}}$	$L F {‖ u - U ‖}_{l_{h}^{2}}$	$L E H {‖ u - U ‖}_{l_{h}^{1}}$	$L E H {‖ u - U ‖}_{l_{h}^{2}}$
256	1.48 × 10⁻²	5.59 × 10⁻²	1.00 × 10⁻²	4.76 × 10⁻²
512	8.90 × 10⁻³	4.28 × 10⁻²	5.80 × 10⁻³	3.55 × 10⁻²
1024	5.30 × 10⁻³	3.27 × 10⁻²	3.30 × 10⁻³	2.66 × 10⁻²
2048	3.00 × 10⁻³	2.41 × 10⁻²	1.80 × 10⁻³	1.91 × 10⁻²

Sociedade Brasileira de Matemática Aplicada e Computacional Rua Maestro João Seppe, nº. 900, 16º. andar - Sala 163 , 13561-120 São Carlos - SP, Tel. / Fax: (55 16) 3412-9752 - São Carlos - SP - Brazil
E-mail: sbmac@sbmac.org.br

Acompanhe os números deste periódico no seu leitor de RSS

[1] *Corresponding author: I. L. N. Araujo -- E-mail: isamara-landim@hotmail.com