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Contaminant Flow and Transport Simulation in Cracked Porous Media Using Locally Conservative Schemes

Published online by Cambridge University Press:  03 June 2015

Pu Song  and
Shuyu Sun
Pu Song*
Affiliation:
Department of Mathematical Sciences Clemson University, Clemson, SC 29634, USA
Shuyu Sun*
Affiliation:
Department of Mathematical Sciences Clemson University, Clemson, SC 29634, USA Computational Transport Phenomena Laboratory (CTPL), Division of Physical Sciences and Enginerring (PSE), King Abdullah University of Science and Technology (KAUST), 4700 King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Abrabia
*
Email: pus6@pitt.edu
Corresponding author. URL:http://web.kaust.edu.sa/faculty/ShuyuSun/, Email: shuyu.sun@kaust.edu.sa
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Abstract

The purpose of this paper is to analyze some features of contaminant flow passing through cracked porous medium, such as the influence of fracture network on the advection and diffusion of contaminant species, the impact of adsorption on the overall transport of contaminant wastes. In order to precisely describe the whole process, we firstly build the mathematical model to simulate this problem numerically. Taking into consideration of the characteristics of contaminant flow, we employ two partial differential equations to formulate the whole problem. One is flow equation; the other is reactive transport equation. The first equation is used to describe the total flow of contaminant wastes, which is based on Darcy law. The second one will characterize the adsorption, diffusion and convection behavior of contaminant species, which describes most features of contaminant flow we are interested in. After the construction of numerical model, we apply locally conservative and compatible algorithms to solve this mathematical model. Specifically, we apply Mixed Finite Element (MFE) method to the flow equation and Discontinuous Galerkin (DG) method for the transport equation. MFE has a good convergence rate and numerical accuracy for Darcy velocity. DG is more flexible and can be used to deal with irregular meshes, as well as little numerical diffusion. With these two numerical means, we investigate the sensitivity analysis of different features of contaminant flow in our model, such as diffusion, permeability and fracture density. In particular, we study Kd values which represent the distribution of contaminant wastes between the solid and liquid phases. We also make omparisons of two different schemes and discuss the advantages of both methods.

Keywords

65M1265M1565N3076D0776S05Mixed finite element methodsdiscontinuous Galerkin methodsreactive transport equationflow equation
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 4 , August 2012 , pp. 389 - 421
Copyright
Copyright © Global-Science Press 2012

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References

[1]Aizinger, V., Dawson, C. N., Cockburn, B. and Castillo, P., The local discontinuous Galerkin method for contaminant transport, Adv. Water. Res., 24 (2001), pp. 73–87.CrossRefGoogle Scholar
[2]Arbogast, T., Brayant, S., Dawson, C. N., Saaf, F. and Wang, C., Compuational methods for multiphase flow and reactive transport problems arising in subsurface contaminant remediation, J. Comput. Appl. Math., 74 (1996), pp. 19–32.Google Scholar
[3]Arbogast, T. and Wheeler, M. F., A characteristics-mixed finite element method for advec-tion dominated transport problems, SIAM J. Numer. Anal., 32 (1995), pp. 404–424.CrossRefGoogle Scholar
[4]Arbogast, T., Wheeler, M. F. and Zhang, N. Y., A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal., 33 (1996), pp. 1669–1687.CrossRefGoogle Scholar
[5]Aziz, A. K. and Liu, J. L., A Galerkin method for the forward-backward heat equation, Math. Comput., 56 (1991), pp. 35–44.Google Scholar
[6]Aziz, A. K. and Monk, P., Continuous finite element in space and time for the heat equation, Math. Comput., 52 (1989), pp. 255–274.Google Scholar
[7]Cowsar, L. C., Dupont, T. F. and Wheeler, M. F., A priori estimates for mixed finite element methods for the wave equation, Comput. Methods. Appl. Mech. Eng., 82 (1990), pp. 205–222.Google Scholar
[8]Chen, Z., Finite Element Methods and Their Applications, Springer-Verlag, Heidelberg, ISBN-3-540-24078-0.Google Scholar
[9]Dawson, C. N., Godunov-mixed methods for advection-diffusion equations in one space dime-sion, SIAM J. Numer. Anal., 28 (1991), pp. 1282–1309.CrossRefGoogle Scholar
[10]Dawson, C. N., Analysis of an upwind-mixed finite element method for nonlinear contaminant transport equations, SIAM J. Numer. Anal., 35(5) (1998), pp. 1709–1724.CrossRefGoogle Scholar
[11]Dawson, C. N., Sun, S. and Wheeler, M. F., Compatible algorithms for coupled flow and transport, Comput. Methods. Appl. Mech. Eng., 193 (2004), pp. 2565–2580.Google Scholar
[12]Dong, C., Sun, S. and Taylor, G. A., Numerical modeling of contaminant transport in fractured porous media using mixed finite element and finite volume methods, J. Porous. Media., 14(3) (2011), pp. 219–242.Google Scholar
[13]Duguid, J. O. and Lee, P. C. Y., Flow in fractured porous media, Water. Res. Research., 13(3) (1977), PP. 558–566, doi:10.1029/WR013i003p00558.Google Scholar
[14]French, D. A. and Peterson, T. E., A continuous space-time finite element method for the wave equation, Math. Comput., 65 (1996), pp. 491–506.Google Scholar
[15]Gallo, C. and Manzini, G., 2D Numerical modeling of bioremediation in heterogeneous saturated soils, Trans. Porous. Media., 31 (1998), pp. 67–88.Google Scholar
[16]Gallo, C. and Manzini, G., Mixed finite element/volume approach for solving biodegradation transport in groundwater, Int. J. Number. Meth. Fluids., 26 (1998), pp. 533–556.Google Scholar
[17]Glowinski, R., Kinton, W. A. and Wheeler, M. F., A mixed finite element formulation for boundary controllability of the wave equation, Int. J. Numer. Methods. Eng., 27 (1989), pp. 623–635.CrossRefGoogle Scholar
[18]Jiang, L. and Mishev, I. D., Mixed multiscale finite volume methods for elliptic problems in two-phase flow simulations, Commun. Comput. Phys., 11 (2012), pp. 19–47.CrossRefGoogle Scholar
[19]Kang, Q.-J., Lichtner, P. C. and Janecky, D. R., Lattice Boltzmann Method for Reacting Flows in Porous Media, Adv. Appl. Math. Mech., 2 (2010), pp. 545–563.Google Scholar
[20]Kim, J. and Deo, M., Finite element, discrete fracture model for multiphase flow in porous media, AIChE J., 46(6) (2000), pp. 1120–1130.Google Scholar
[21]Karimifard, M. and Firoozabadi, A., Numerical simulation of water injection in 2-d fractured media using discrete-fracture model, SPE Reservoir. Eval. Eng., 4 (2003), pp. 117–126.Google Scholar
[22]Layton, W. J., Schieweck, F. and Yotov, I., Coupling fluid flow with porous media flow, SIAM J. Numer. Anal., 40(6) (2003), pp. 2195–2218.CrossRefGoogle Scholar
[23]Nakayama, S., Takagi, I. and Higashi, K., A semi-analytical solution for advection-dispersion migration of radionuclides through two-layered geologic media, Mere. Fac. Eng., 48 (1986), pp. 227–239.Google Scholar
[24]Sahimi, M., Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches, 1995.Google Scholar
[25]Siegel, P., Mose, R., Ackerer, P. and Jaffre, J., Solution of the advection-diffusion equation using a combination of discontinuous and mixed finite element, Int. J. Number. Meth. Fluids., 24 (1997), pp. 595–613.Google Scholar
[26]Sun, S., Discontinuous Galerkin Methods for Reactive Transport in Porous Media, Ph.D. dissertation, The University of Texas at Austin, 2003.Google Scholar
[27]Sun, S., Riviere, B. and Wheeler, M. F., A combined mixed finite element and discontinuous Galerkin method for miscible displacement problems in porous media, in: Proceedings of International Symposium on Computational and Applied PDEs held at Zhangjiajie National Park of China, pp. 321–348, 2002.Google Scholar
[28]Sun, S. and Wheeler, M. F., Discontinuous Galerkin methods for coupled flow and reactive transport problems, Appl. Numer. Math., 52(2-3) (2005), pp. 273–298.CrossRefGoogle Scholar
[29]Sun, S. and Wheeler, M. F., A posteriori error estimation and dynamic adaptivity for symmetric discontinuous Galerkin approximations of reactive transport problems, Comput. Methods. Appl. Mech. Eng., 195 (2006), pp. 632–652.CrossRefGoogle Scholar
[30]Sun, S. and Wheeler, M. F., Projections of velocity data for the compatibility with transport, Comput. Methods. Appl. Mech. Eng., 195 (2006), pp. 653–673.CrossRefGoogle Scholar
[31]Sun, S. and Wheeler, M. F., A dynamic, adaptive, locally conservative and nonconforming solution strategy for transport phenomena in chemical engineering, Chem. Eng. Commun., 193(12) (2006), pp. 1527–1545.Google Scholar
[32]Sarkar, S. and Toksöz, M. N., Fluid Flow Simulation in Fractured Reservoirs, Report, Annual Consortium Meeting, 2002.Google Scholar
[33]Yu, M. and Dougherty, D. E., FCT Model of contaminant Transport on Unstructured Meshes, Volume 1, pp. 199–206, Computational Mechanicas Publications, 1998.Google Scholar