×

Adaptive POD model reduction for solute transport in heterogeneous porous media. (English) Zbl 1405.76057

Summary: We study the applicability of a model order reduction technique to the solution of transport of passive scalars in homogeneous and heterogeneous porous media. Transport dynamics are modeled through the advection-dispersion equation (ADE) and we employ Proper Orthogonal Decomposition (POD) as a strategy to reduce the computational burden associated with the numerical solution of the ADE. Our application of POD relies on solving the governing ADE for selected times, termed snapshots. The latter are then employed to achieve the desired model order reduction. We introduce a new technique, termed Snapshot Splitting Technique (SST), which allows enriching the dimension of the POD subspace and damping the temporal increase of the modeling error. Coupling SST with a modeling strategy based on alternating over diverse time scales the solution of the full numerical transport model to its reduced counterpart allows extending the benefit of POD over a prolonged temporal window so that the salient features of the process can be captured at a reduced computational cost. The selection of the time scales across which the solution of the full and reduced model are alternated is linked to the Péclet number (Pe), representing the interplay between advective and dispersive processes taking place in the system. Thus, the method is adaptive in space and time across the heterogenous structure of the domain through the combined use of POD and SST and by way of alternating the solution of the full and reduced models. We find that the width of the time scale within which the POD-based reduced model solution provides accurate results tends to increase with decreasing Pe. This suggests that the effects of local-scale dispersive processes facilitate the POD method to capture the salient features of the system dynamics embedded in the selected snapshots. Since the dimension of the reduced model is much lower than that of the full numerical model, the methodology we propose enables one to accurately simulate transport at a markedly reduced computational cost.

MSC:

76S05 Flows in porous media; filtration; seepage
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Software:

SGeMS
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Ballio, F., Guadagnini, A.: Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour. Res. 40(4) (2004)
[2] Baroli, D., Cova, C.M., Perotto, S., Sala, L., Veneziani, A.: Hi-POD solution of parametrized fluid dynamics problems: preliminary results. MS&A series. In press (2017) · Zbl 1457.76096
[3] de Barros, F.P.J., Ezzedine, S., Rubin, Y.: Impact of hydrogeological data on measures of uncertainty, site characterization and environmental performance metrics. Adv. Water Resour. 36, 51-63 (2012) · doi:10.1016/j.advwatres.2011.05.004
[4] de Barros, F.P.J., Fiori, A.: First-order based cumulative distribution function for solute concentration in heterogeneous aquifers: Theoretical analysis and implications for human health risk assessment. Water Resour. Res. 50(5), 4018-4037 (2014) · doi:10.1002/2013WR015024
[5] de Barros, F.P.J., Nowak, W.: On the link between contaminant source release conditions and plume prediction uncertainty. J. Contam. Hydrol. 116(1), 24-34 (2010) · doi:10.1016/j.jconhyd.2010.05.004
[6] de Barros, F.P.J., Rubin, Y.: A risk-driven approach for subsurface site characterization. Water Resour. Res. 44(1) (2008)
[7] Bear, J.: Dynamics of fluids in porous media. Courier Corporation (2013) · Zbl 1191.76001
[8] Bergmann, M., Bruneau, C.H., Iollo, A.: Enablers for robust POD models. J. Comput. Phys. 228(2), 516-538 (2009) · Zbl 1409.76099 · doi:10.1016/j.jcp.2008.09.024
[9] Cao, Y., Zhu, J., Luo, Z., Navon, I.: Reduced-order modeling of the upper tropical pacific ocean model using proper orthogonal decomposition. Comput. Math. Appl. 52(8-9), 1373-1386 (2006) · Zbl 1161.86002 · doi:10.1016/j.camwa.2006.11.012
[10] Cardoso, M., Durlofsky, L., Sarma, P.: Development and application of reduced-order modeling procedures for subsurface flow simulation. Int. J. Numer. Methods Eng. 77(9), 1322-1350 (2009) · Zbl 1156.76420 · doi:10.1002/nme.2453
[11] Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808-817 (2000)
[12] Crommelin, D., Majda, A.: Strategies for model reduction: comparing different optimal bases. J. Atmosph. Sci. 61(17) (2004)
[13] Dagan, G., Neuman, S.P.: Subsurface flow and transport: a stochastic approach. Cambridge University Press (2005)
[14] Dentz, M., Le Borgne, T., Englert, A., Bijeljic, B.: Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120, 1-17 (2011) · doi:10.1016/j.jconhyd.2010.05.002
[15] van Doren, J., Markovinovic, R., Jansen, J.D.: Reduced-order optimal control of waterflooding using POD. In: 9th European Conference on the Mathematics of Oil Recovery (2004) · Zbl 1161.86304
[16] van Doren, J.F., Markovinović, R., Jansen, J.D.: Reduced-order optimal control of water flooding using proper orthogonal decomposition. Comput. Geosci. 10(1), 137-158 (2006) · Zbl 1161.86304 · doi:10.1007/s10596-005-9014-2
[17] Edery, Y., Guadagnini, A., Scher, H., Berkowitz, B.: Origins of anomalous transport in heterogeneous media: structural and dynamic controls. Water Resour. Res. 50(2), 1490-1505 (2014) · doi:10.1002/2013WR015111
[18] Efendiev, Y., Gildin, E., Yang, Y.: Online adaptive local-global model reduction for flows in heterogeneous porous media. Computation 4(2), 22 (2016) · doi:10.3390/computation4020022
[19] Esfandiar, B., Porta, G., Perotto, S., Guadagnini, A.: Impact of space-time mesh adaptation on solute transport modeling in porous media. Water Resour. Res. 51(2), 1315-1332 (2015) · doi:10.1002/2014WR016569
[20] Fetter, C.W., Fetter, C. Jr.: Contaminant Hydrogeology, vol. 500. Prentice Hall,New Jersey (1999)
[21] Ghommem, M., Presho, M., Calo, V.M., Efendiev, Y.: Mode decomposition methods for flows in high-contrast porous media. Global-local approach. J. Comput. Phys. 253, 226-238 (2013) · Zbl 1349.76209 · doi:10.1016/j.jcp.2013.06.033
[22] Henri, C.V., Fernàndez-Garcia, D., Barros, F.P.: Probabilistic human health risk assessment of degradation-related chemical mixtures in heterogeneous aquifers: risk statistics, hot spots, and preferential channels. Water Resour. Res. 51(6), 4086-4108 (2015) · doi:10.1002/2014WR016717
[23] Jolliffe, I.: Principal Component Analysis. Wiley Online Library (2005) · Zbl 1011.62064
[24] Kowalski, M.E., Jin, J.M.: Model-order reduction of nonlinear models of electromagnetic phased-array hyperthermia. IEEE Trans. Biomed. Eng. 50(11), 1243-1254 (2003) · doi:10.1109/TBME.2003.818468
[25] Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90(1), 117-148 (2001) · Zbl 1005.65112 · doi:10.1007/s002110100282
[26] Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40(2), 492-515 (2002) · Zbl 1075.65118 · doi:10.1137/S0036142900382612
[27] Le Borgne, T., Dentz, M., Carrera, J.: Lagrangian statistical model for transport in highly heterogeneous velocity fields. Phys. Rev. Lett. 101(9), 090,601 (2008) · doi:10.1103/PhysRevLett.101.090601
[28] Li, H., Luo, Z., Chen, J.: Numerical simulation based on POD for two-dimensional solute transport problems. Appl. Math. Model. 35(5), 2489-2498 (2011) · Zbl 1217.74082 · doi:10.1016/j.apm.2010.11.064
[29] Li, X., Hu, B.X.: Proper orthogonal decomposition reduced model for mass transport in heterogenous media. Stoch. Env. Res. Risk A. 27(5), 1181-1191 (2013) · doi:10.1007/s00477-012-0653-2
[30] Lumley, J.L.: The structure of inhomogeneous turbulent flows. Atmosph. Turb. Radio Wave Propag. 166-178 (1967)
[31] Luo, Z., Li, H., Zhou, Y., Xie, Z.: A reduced finite element formulation based on POD method for two-dimensional solute transport problems. J. Math. Anal. Appl. 385(1), 371-383 (2012) · Zbl 1243.65123 · doi:10.1016/j.jmaa.2011.06.051
[32] Ly, H.V., Tran, H.T.: Proper orthogonal decomposition for flow calculations and optimal control in a horizontal cvd reactor. Tech. rep., DTIC Document (1998) · Zbl 1146.76631
[33] Mojgani, R., Balajewicz, M.: Lagrangian basis method for dimensionality reduction of convection dominated nonlinear flows. arXiv:1701.04343 (2017)
[34] Moslehi, M., Rajagopal, R., de Barros, F.P.J.: Optimal allocation of computational resources in hydrogeological models under uncertainty. Adv. Water Resour. 83, 299-309 (2015) · doi:10.1016/j.advwatres.2015.06.014
[35] Pasetto, D., Guadagnini, A., Putti, M.: POD-based monte carlo approach for the solution of regional scale groundwater flow driven by randomly distributed recharge. Adv. Water Resour. 34(11), 1450-1463 (2011) · doi:10.1016/j.advwatres.2011.07.003
[36] Pasetto, D., Putti, M., Yeh, W.W.G.: A reduced-order model for groundwater flow equation with random hydraulic conductivity: application to monte carlo methods. Water Resour. Res. 49(6), 3215-3228 (2013) · doi:10.1002/wrcr.20136
[37] Perotto, S.: A survey of hierarchical model (hi-mod) reduction methods for elliptic problems. In: Numerical Simulations of Coupled Problems in Engineering, pp. 217-241. Springer (2014)
[38] Porta, G., Bijeljic, B., Blunt, M., Guadagnini, A.: Continuum-scale characterization of solute transport based on pore-scale velocity distributions. Geophys. Res. Lett. (2015)
[39] Porta, G., Thovert, J. F., Riva, M., Guadagnini, A., Adler, P.: Microscale simulation and numerical upscaling of a reactive flow in a plane channel. Phys. Rev. E 86(3), 036,102 (2012) · doi:10.1103/PhysRevE.86.036102
[40] Rapún, M.L., Vega, J.M.: Reduced order models based on local POD plus galerkin projection. J. Comput. Phys. 229(8), 3046-3063 (2010) · Zbl 1187.65111 · doi:10.1016/j.jcp.2009.12.029
[41] Remy, N., Boucher, A., Wu, J.: Applied Geostatistics with SGeMS: A User’s Guide. Cambridge University Press (2009)
[42] Rubin, Y.: Applied Stochastic Hydrology. Oxford University Press, New York (2003)
[43] Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley (2011) · Zbl 1219.76002
[44] Siade, A.J., Putti, M., Yeh, W.W.G.: Snapshot selection for groundwater model reduction using proper orthogonal decomposition. Water Resour. Res. 46(8) (2010)
[45] Siena, M., Guadagnini, A., Riva, M., Bijeljic, B., Nunes, J.P., Blunt, M.: Statistical scaling of pore-scale lagrangian velocities in natural porous media. Phys. Rev. E 90(2), 023,013 (2014) · doi:10.1103/PhysRevE.90.023013
[46] Tartakovsky, D.M.: Assessment and management of risk in subsurface hydrology: a review and perspective. Adv. Water Resour. 51, 247-260 (2013) · doi:10.1016/j.advwatres.2012.04.007
[47] Zhang, D.: Stochastic methods for flow in porous media: coping with uncertainties. Academic Press (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.