A Numerical IMPES Discontinuous Galerkin method for Immiscible Groundwater Contaminations Flow Using Lax-Wendroff scheme

Document Type : Complete scientific research article

Authors

1 Faculty of Engineering, Shohadaye Hoveizeh University of Technology, Susangerd

2 Faculty of Engineering, Khatam-Al Anbia University of Technology, Behbahan

3 Faculty of Agriculture, University of Shahrekord, Shahrekord0913

Abstract

Abstract
Background and Objectives:
The numerical modeling of the immiscible flows in the porous media is one of the issues which have always been considered by researchers due to their application in the monitoring of the groundwater pollutions, water and oil behavior in the petroleum reservoirs and hydrology sciences. In this study, we present a two-dimensional discontinuous Galerkin numerical model of immiscible flows in a porous media using the high order implicit pressure-explicit saturation (IMPES) strategy for governing equations. Here, the primary unknowns are wetting phase-pressure and saturation. In this hybrid numerical scheme, for the first time we developed the second-order Lax-Wendroff method to solve the water saturation equation which is considered as the main novelty of this paper.
Materials and Methods:
For the numerical modeling of immiscible groundwater pollutions, it has been utilized the local conservative discontinuous Galerkin scheme as the spatial discretization. The backward Euler and second-order Lax-Wendroff scheme are applied as temporal discretization for pressure and saturation equations respectively. Also, we stabilized the exchanging numerical flux and used projection of the velocity field in the H (div) vectorial interpolation space for improvement of results at the heterogeneities.at the end of each time step, non-physical oscillations omitted using modified Chaven-Jaffre slope limiter and the results are stabilized.
Results:
The second-order Lax-Wendroff scheme based on the Taylor expansion and the high order time derivatives is comparable with conventional IMPES strategy schemes such as multi stage Runge-kutta Method (RKDG) while has less computation cost than multi stage schemes. However, the time step size and the Courant number have some restrictions with respect to the explicit solving of the saturation equation.
Conclusion:
In order to validation of the model, the Buckley-Leverett benchmark problem is considered. The results of the developed model are compared with of other authors and a good agreement is observed between them. Also, model efficiency and ability have been evaluated with two test cases for high heterogeneous aquifers. Also employing various techniques improved the discontinuities resolution in highly heterogeneous media. Numerical models showed good non-oscillatory resolution of saturation around the less permeable subdomains and frontal interface between the wetting and nonwetting phases. In this study, the penalty parameter varies between 50 and 100. In SWIP version of DG method, the penalty parameter should be chosen greater than 50 while in OBB-DG method zero values could be allocated. The sensitivity analysis of the model has been considered for various effective parameters in modeling.

Keywords

References

1.Amaziane, B., and Jurak, M. 2008. A new
formulation of immiscible compressible
two-phase flow in porous media, Comptes
Rendus Mécanique, 336: 7. 600-605.
2.Amaziane, B., Pankratov, L., and
Piatnitski, A. 2017. An improved
homogenization result for immiscible
compressible two-phase flow in porous
media, NHM, 12: 1. 147-171.
3.Arbogast, T., Juntunen, M., Pool, J.,
and Wheeler, M.F. 2013. A discontinuous
Galerkin method for two-phase flow in
a porous medium enforcing H
(div) velocityand continuous capillary
pressure, Computational Geosciences,
17: 6. 1055-1078.
4.Bastian, P., and Riviere, B. 2004.
Discontinuous Galerkin methods for
two-phase flow in porous media,
Technical Reports of the IWR (SFB 359)
of the Universität Heidelberg.
5.Brooks, R., and Corey, T. 1964.
Hydraulic Properties of Porous Media.
Colorado State University.
6.Buckley, S.E., and Leverett, M.
1942. Mechanism of fluid displacement
in sands, Transactions of the AIME,
146: 1. 107-116.
7.Bürger, R., Kenettinkara, S.K., and Zorío,
D. 2017. Approximate Lax-Wendroff
discontinuous Galerkin methods for
hyperbolic conservation laws, Computers
and Mathematics with Applications,
74: 6. 1288-1310.
8.Burri, A. 2004. Implementation of a
multiphase flow simulator using a fully
upwind galerkin method within the CSP
multiphysics toolkit, Unpublished Diploma
Thesis, Eidgenössische Technische
Hochschule Zürich, Switzerland.
9.Chavent, G., and Jaffré, J. 1986.
Mathematical models and finite elements
for reservoir simulation: single phase,
multiphase and multicomponent flows
through porous media. Elsevier.
10.Di Pietro, D.A., and Ern, A. 2011.
Mathematical aspects of discontinuous
Galerkin methods. Springer69.
11.Donea, J. 1991. Generalized Galerkin
methods for convection dominated
transport phenomena, Applied
Mechanics Reviews, 44: 5. 205-214.
12.Donea, J. 1984. A Taylor-Galerkin
method for convective transport
problems, Inter. J. Num. Method. Engin.
20: 1. 101-119.
13.Donea, J., Giuliani, S., Laval, H., and
Quartapelle, L. 1984. Time-accurate
solution of advection-diffusion problems
by finite elements, Computer Methods
in Applied Mechanics and Engineering,
45: 1-3. 123-145.
14.Donea, J., Quartapelle, L., and Selmin,
V. 1987. An analysis of time
discretization in the finite element
solution of hyperbolic problems, J. Com.
Physic. 70: 2. 463-499.
15.Ern, A., Stephansen, A.F., and Zunino,
P. 2008. A discontinuous Galerkin
method with weighted averages for
advection–diffusion equations with
locally small and anisotropic diffusivity,
IMA J. Num. Anal. 29: 2. 235-256.
16.Eslinger, O.J. 2005. Discontinuous
galerkin finite element methods applied
to two-phase, air-water flow problems,
Ph.D Thesis, University of Texas at
Austin.
17.Geiger Boschung, S. 2004. Numerical
simulations of the hydrodynamics and
thermodynamics of NaCl-H₂O fluids,
Ph.D Thesis, ETH Zurich.
18.Gottlieb, S. 2005. On high order strong
stability preserving Runge-Kutta and
multi step time discretizations, J. Sci.
Com. 25: 1. 105-128.
19.Gottlieb, S., Ketcheson, D.I., and Shu,
C.W. 2009. High order strong stability
preserving time discretizations, J. Sci.
Com. 38: 3. 251-289.
20.Gottlieb, S., Shu, C.W., and Tadmor,
E. 2001. Strong stability-preserving
high-order time discretization methods,
SIAM review, 43: 1. 89-112.
21.Hadad, A., Bensabat, J., and Rubin,
H. 1996. Simulation of immiscible
multiphase flow in porous media: a
focus on the capillary fringe of
oil-contaminated aquifers, Transport in
porous media, 22: 3. 245-269.
22.Hoteit, H., Ackerer, P., Mosé, R., Erhel,
J., and Philippe, B. 2004. New two‐
dimensional slope limiters for
discontinuous Galerkin methods on
arbitrary meshes, Inter. J. Num. Method.
Engin. 61: 14. 2566-2593.
23.Jamei, M., Raeisi Isa Abadi, A., and
Ahmadianfar, I. 2019. A Lax–WendroffIMPES scheme for a two-phase flow
in porous media using interior penalty
discontinuous Galerkin method,
Numerical Heat Transfer, Part B:
Fundamentals, 75: 5. 325-346.
24.Jamei, M., and Ghafouri, H. 2016. An
efficient discontinuous Galerkin method
for two-phase flow modeling by
conservative velocity projection, Inter. J.
Num. Method. Heat & Fluid Flow,
26: 1. 63-84.
25.Jamei, M., and Ghafouri, H. 2016. A
novel discontinuous Galerkin model for
two-phase flow in porous media using
an improved IMPES method, Inter.
J. Num. Method. Heat Fluid Flow.
26: 1. 284-306.
26.Jamei, M., and Ghafouri, H.R. 2016. A
discontinuous Galerkin method for twophase flow in porous media using
modified MLP slope limiter, Modares
Mechanical Engineering, 15: 12. 326-336.
27.Kirby, R.C. 2004. Algorithm 839:
FIAT, a new paradigm for computing
finite element basis functions, ACM
Transactions on Mathematical Software
(TOMS), 30: 4. 502-516.
28.Klieber, W., and Riviere, B. 2006.
Adaptive simulations of two-phase flow
by discontinuous Galerkin methods,
Computer methods in applied mechanics
and engineering, 196: 1. 404-419.
29.Kou, J., and Sun, S. 2010. A
new treatment of capillarity to improve
the stability of IMPES two-phase flow
formulation, Computers & Fluids,
39: 10. 1923-1931.
30.Kubatko, E.J., Dawson, C., and Westerink,
J.J. 2008. Time step restrictions for
Runge-Kutta discontinuous Galerkin
methods on triangular grids, J. Com.
Physic. 227: 23. 9697-9710.
31.Osborne, M., and Sykes, J. 1986.
Numerical modeling of immiscible
organic transport at the Hyde Park
landfill, Water Resources Research,
22: 1. 25-33.
32.Pruess, K. 1991. TOUGH2-A generalpurpose numerical simulator for
multiphase fluid and heat flow.
33.Raeisi Isaabadi, A., Ghafouri, H.R., and
Rostamy, D. 2017. A new numerical
method based on discontinuous galerkin
for simulation of seawater intrusion into
coastal aquifers, Gorgan University of
Agricultural Sciences and Natural
Resources, 24: 4. 23-41.
34.Riaz, A., and Tchelepi, H.A. 2006.
Numerical simulation of immiscible
two-phase flow in porous media,
Physics of Fluids, 18: 1. 014104.
35.Rivière, B. 2008. Discontinuous
Galerkin methods for solving elliptic
and parabolic equations: theory and
implementation. Society for Industrial
and Applied Mathematics.
36.Roig, B. 2007. One-step Taylor–
Galerkin methods for convection–
diffusion problems, J. Com. Appl. Math.
204: 1. 95-101.
37.Shu, C.W. 1988. Total-variationdiminishing time discretizations, SIAM
J. Sci. Stat. Com. 9: 6. 1073-1084.
38.Toulorge, T., and Desmet, W. 2011.
CFL conditions for Runge–Kutta
discontinuous Galerkin methods on
triangular grids, J. Com. Physic.
230: 12. 4657-4678.
39.Van Genuchten, M.T., and Nielsen, D.
1985. On describing and predicting the
hydraulic properties of unsaturated soils,
Ann. Geophys. 3: 5. 615-628.
Journal of Water and Soil Conservation
Volume 26, Issue 2
May and June 2019
Pages 1-27

Files

Share

How to cite

Statistics