Simulating Solute Transport in Porous Media Using Model Reduction Techniques

Bruce A. Robinson, Zhiming Lu, Donatella Pasqualini

Civilian Nuclear Programs, Los Alamos National Laboratory, Los Alamos, USA.
Computational Earth Sciences Group, Los Alamos National Laboratory, Los Alamos, USA.
Energy and Infrastructure Analysis, Los Alamos National Laboratory, Los Alamos, USA.

DOI: 10.4236/am.2012.310170                

Abstract

In this study, we introduce a numerical method to reduce the solute transport equation into a reduced form that can replicate the behavior of the model described by the original equation. The basic idea is to collect an ensemble of data of state variables (say, solute concentration), called snapshots, by running the original model, and then use the proper orthogonal decomposition (POD) techniques (or the Karhunen-Loeve decomposition) to create a set of basis functions that span the snapshot collection. The snapshots can be reconstructed using these basis functions. The solute concentration at any time and location in the domain is expressed as a linear combination of these basis functions, and a Galerkin procedure is applied to the original model to obtain a set of ordinary differential equations for the coefficients in the linear representation. The accuracy and computational efficiency of the reduced model have been demonstrated using several one-dimensional and two-dimensional examples

Keywords

Model Reduction; Solute Transport; Proper Orthogonal Decomposition

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	G. Berkooz, P. Holmes and J. L. Lumley, “The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows,” Annual Review of Fluid Mechanics, Vol. 25, No. 1, 1993, pp. 539-575. doi:10.1146/annurev.fl.25.010193.002543
[2]	P. G. Cizmas, A. Palacios, T. O. O-Brien and M. Syamlal, “Proper-Orthogonal Decomposition of Spatio-Temporal Patterns in Fluidized Beds,” Chemical Engineering Science, Vol. 58, No. 19, 2003, pp. 4417-4427. doi:10.1016/S0009-2509(03)00323-3
[3]	H. M. Park and D. H. Cho, “The Use of the KarhunenLoeve Decomposition for the Modeling of Distributed Parameter Systems,” Chemical Engineering Science, Vol. 51, No. 1, 1996, pp. 81-98. doi:10.1016/0009-2509(95)00230-8
[4]	H. V. Ly and H. T. Tran, “Modeling and Control of Physical Processes Using Proper Orthogonal Decomposition,” Report CRSC-TR98-37, Center for Research in Scientific Computation, North Carolina State University, Raleigh, 1999.
[5]	A. J. Newman, “Model Reduction via the KarhunenLoeve Expansion Part II: Some Elementary Examples,” Tech. Report T.R. 96-33, Inst. Systems Research, 1996.
[6]	H. V. Ly and H. T. Tran, “Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor,” Report CRSC-TR98-13, Center for Research in Scientific Computation, North Carolina State University, Raleigh, 1998.
[7]	R. Markovinovic, E. L. Geurtsen, T. Heijin and J. D. Jansen, “Generation of Low-Order Reservoir Models Using POD, Empirical Gramians, and Subspace Identification,” 8th European Conference on the Mathematics of Oil Recovery, Freiberg, 3-6 September 2002.
[8]	T. Heijin, R. Markovinovic and J. D. Jansen, “Generation of Low-Order Reservoir Models Using System-Theoretical Concepts,” SPE Paper 79674, SPE Reservoir Symposium, Houston, 2003.
[9]	P. T. M. Vermeulen, A. W. Heemink and C. B. M. Te Stroet, “Reduced Models for Linear Groundwater Flow Models Using Empirical Orthogonal Functions,” Advances in Water Resources, Vol. 27, No. 1, 2004, pp. 57-69. doi:10.1016/j.advwatres.2003.09.008
[10]	A. J. Newman, “Model Reduction via the KarhunenLoeve Expansion Part I: An Exposition,” Tech. Report T.R. 96-32, Inst. Systems Research, 1996.
[11]	G. A. Zyvoloski, B. A. Robinson, Z. V. Dash and L. L. Trease, “Summary of the Models and Methods for the FEHM Application—A Finite-Element Heatand MassTransfer Code,” Los Alamos National Laboratory Report LA-13307-MS, Los Alamos, 1997.
[12]	W. L. Polzer, M. G. Rao, H. R. Fuentes and R. J. Beckman, “Thermodynamically Derived Relationships between the Modified Langmuir Isotherm and Experimental Parameters,” Environmental Science & Technology, Vol. 26, 1992, pp. 1780-1786. doi:10.1021/es00033a011
[13]	V. Batu, “Applied Flow and Solute Transport Modeling in Aquifers: Fundamental Principles and Analytical and Numerical Methods,” CRC Press, London, 2006.
[14]	C. V. Deutsch and A. G. Journel, “GSLIB: Geostatistical Software Library,” Oxford University Press, New York, 1992.