Computational Geosciences

, Volume 22, Issue 1, pp 297–308 | Cite as

Adaptive POD model reduction for solute transport in heterogeneous porous media

  • Calogero B. Rizzo
  • Felipe P. J. de Barros
  • Simona Perotto
  • Luca Oldani
  • Alberto Guadagnini
Original Paper


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 (P e), 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 P e. 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.


Proper orthogonal decomposition Model reduction Heterogeneous porous media Flow and transport Computational efficiency 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Funding from the European Unions Horizon 2020 Research and Innovation programme (Project “Furthering the knowledge Base for Reducing the Environmental Footprint of Shale Gas Development” FRACRISK - Grant Agreement No. 640979) and from MIUR (Italian ministry of Education, University and Research, Water JPI, WaterWorks 2014, project: WE-NEED- Water NEEDs, availability, quality and sustainability) is acknowledged. The first author acknowledges the financial support from USC Provost’s Ph.D. Fellowship. The third author would like to thank the partial support of INdAM-GNCS 2017 project on “Advanced Numerical Methods Combined with Computational Reduction Techniques for Parametrized PDEs and Applications”.


  1. 1.
    Ballio, F., Guadagnini, A.: Convergence assessment of numerical Monte Carlo simulations in groundwater hydrology. Water Resour. Res. 40(4) (2004)Google Scholar
  2. 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)Google Scholar
  3. 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)CrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 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)CrossRefGoogle Scholar
  6. 6.
    de Barros, F.P.J., Rubin, Y.: A risk-driven approach for subsurface site characterization. Water Resour. Res. 44(1) (2008)Google Scholar
  7. 7.
    Bear, J.: Dynamics of fluids in porous media. Courier Corporation (2013)Google Scholar
  8. 8.
    Bergmann, M., Bruneau, C.H., Iollo, A.: Enablers for robust POD models. J. Comput. Phys. 228(2), 516–538 (2009)CrossRefGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 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)CrossRefGoogle Scholar
  11. 11.
    Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808–817 (2000)Google Scholar
  12. 12.
    Crommelin, D., Majda, A.: Strategies for model reduction: comparing different optimal bases. J. Atmosph. Sci. 61(17) (2004)Google Scholar
  13. 13.
    Dagan, G., Neuman, S.P.: Subsurface flow and transport: a stochastic approach. Cambridge University Press (2005)Google Scholar
  14. 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)CrossRefGoogle Scholar
  15. 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)Google Scholar
  16. 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)CrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 18.
    Efendiev, Y., Gildin, E., Yang, Y.: Online adaptive local-global model reduction for flows in heterogeneous porous media. Computation 4(2), 22 (2016)CrossRefGoogle Scholar
  19. 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)CrossRefGoogle Scholar
  20. 20.
    Fetter, C.W., Fetter, C. Jr.: Contaminant Hydrogeology, vol. 500. Prentice Hall,New Jersey (1999)Google Scholar
  21. 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)CrossRefGoogle Scholar
  22. 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)CrossRefGoogle Scholar
  23. 23.
    Jolliffe, I.: Principal Component Analysis. Wiley Online Library (2005)Google Scholar
  24. 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)CrossRefGoogle Scholar
  25. 25.
    Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90(1), 117–148 (2001)CrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 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)CrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 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)CrossRefGoogle Scholar
  30. 30.
    Lumley, J.L.: The structure of inhomogeneous turbulent flows. Atmosph. Turb. Radio Wave Propag. 166–178 (1967)Google Scholar
  31. 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)CrossRefGoogle Scholar
  32. 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)Google Scholar
  33. 33.
    Mojgani, R., Balajewicz, M.: Lagrangian basis method for dimensionality reduction of convection dominated nonlinear flows. arXiv:1701.04343 (2017)
  34. 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)CrossRefGoogle Scholar
  35. 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)CrossRefGoogle Scholar
  36. 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)CrossRefGoogle Scholar
  37. 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)Google Scholar
  38. 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)Google Scholar
  39. 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)CrossRefGoogle Scholar
  40. 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)CrossRefGoogle Scholar
  41. 41.
    Remy, N., Boucher, A., Wu, J.: Applied Geostatistics with SGeMS: A User’s Guide. Cambridge University Press (2009)Google Scholar
  42. 42.
    Rubin, Y.: Applied Stochastic Hydrology. Oxford University Press, New York (2003)Google Scholar
  43. 43.
    Sahimi, M.: Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches. Wiley (2011)Google Scholar
  44. 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)Google Scholar
  45. 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)CrossRefGoogle Scholar
  46. 46.
    Tartakovsky, D.M.: Assessment and management of risk in subsurface hydrology: a review and perspective. Adv. Water Resour. 51, 247–260 (2013)CrossRefGoogle Scholar
  47. 47.
    Zhang, D.: Stochastic methods for flow in porous media: coping with uncertainties. Academic Press (2001)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Sonny Astani Department of Civil and Environmental EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly
  3. 3.Dipartimento di Ingegneria Civile e AmbientalePolitecnico di MilanoMilanoItaly

Personalised recommendations