Estimating the probability of rare events is an extremely challenging task. For example, estimating the probability of leakage of CO2 from a saline aquifer using a direct Monte-Carlo approach would in general require a number of simulations proportional to the inverse of the rare event probability. Since it is likely that any action will require a probability of failure of less than $10^{−6}$, requiring $10^7$ to $10^8$ simulations, it is understandable that few such studies have been published.

In this paper, we propose a means of simulating such rare events using a multilevel splitting algorithm called subset simulation (SS)[ ]. SS is an iterative algorithm that introduces intermediate events which are easier to sample from, and then iteratively samples within each constrained region in the probability space until the rare event threshold is reached.

We show results for a standard benchmark for CO2 leakage through a leaky well [ ]. In this test case, CO2 is injected into a deep aquifer, then spreads within the aquifer and, upon reaching an abandoned well, it rises to a shallower aquifer. We show that subset simulation is an effective algorithm for estimating the probability of rare events with significant computational advantages over a direct Monte-Carlo approach. The SS algorithm relies on two parameters that have to be adjusted at the start of the simulation; we show the effect of these parameters on the quality of estimated rare event probabilities and propose guidelines on how to select these parameters.

References: Siu-Kui Au and James L. Beck. Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 16(4):263 – 277, 2001. ISSN 0266-8920. doi: http://dx.doi.org/10.1016/S0266-8920(01)00019-4. URL http://www.sciencedirect.com/science/article/pii/S0266892001000194.

Holger Class, Anozie Ebigbo, Rainer Helmig, Helge K. Dahle, Jan M. Nordbotten, Michael A. Celia, Pascal Audigane, Melanie Darcis, Jonathan Ennis-King, Yaqing Fan, Bernd Flemisch, Sarah E. Gasda, Min Jin, Stefanie Krug, Diane Labregere, Ali Naderi Beni, Rajesh J. Pawar, Adil Sbai, Sunil G. Thomas, Laurent Trenty, and Ling li Wei. A benchmark study on problems related to CO2 storage in geologic formations. Comput. Geosci., 13(4):409-434, 2009. ISSN 1420-0597. doi: 10.1007/s10596-009-9146-x. URL http://dx.doi.org/10.1007/s10596-009-9146-x.


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  1. Ashraf, M., S., O. and Nowak, W.
    [2013] Geological storage of CO2: global sensitivity analysis and risk assessment using arbitrary polynomial chaos expansion. International Journal of Greenhouse Gas Control, 19, 704–719, doi:10.1016/j.ijggc.2013.03.023.
    https://doi.org/10.1016/j.ijggc.2013.03.023 [Google Scholar]
  2. Au, S. and Beck, J.
    [2003] Subset simulation and its application to seismic risk based on dynamic analysis. J. Eng. Mech., 129(8), 901–917, doi:10.1061/(ASCE)0733‑9399(2003)129:8(901).
    https://doi.org/10.1061/(ASCE)0733-9399(2003)129:8(901) [Google Scholar]
  3. Au, S.K. and Beck, J.L.
    [2001] Estimation of small failure probabilities in high dimensions by subset simulation. Probabilistic Engineering Mechanics, 16(4), 263 – 277, ISSN 0266-8920, doi:10.1016/S0266‑8920(01)00019‑4.
    https://doi.org/http://dx.doi.org/10.1016/S0266-8920(01)00019-4 [Google Scholar]
  4. Class, H. et al.
    [2009] A benchmark study on problems related to CO2 storage in geologic formations. Comput. Geosci., 13(4), 409–434, ISSN 1420-0597, doi:10.1007/s10596‑009‑9146‑x.
    https://doi.org/10.1007/s10596-009-9146-x [Google Scholar]
  5. Ebigbo, A., Class, H. and Helmig, R.
    [2007] CO2 leakage through an abandoned well: problem-oriented benchmarks. Comput. Geosci., 11(2), 103-115, ISSN 1420-0597, doi:10.1007/s10596‑006‑9033‑7.
    https://doi.org/10.1007/s10596-006-9033-7 [Google Scholar]
  6. Elsheikh, A.H., Hoteit, I. and Wheeler, M.F.
    [2014] Efficient bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates. Computer Methods in Applied Mechanics and Engineering, 269(0), 515 – 537, ISSN 0045-7825, doi:10.1016/j.cma.2013.11.001.
    https://doi.org/10.1016/j.cma.2013.11.001 [Google Scholar]
  7. Elsheikh, A.H., Jackson, M.D. and Laforce, T.C.
    [2012] Bayesian reservoir history matching considering model and parameter uncertainties. Math. Geosci., 44(5), 515–543, doi:10.1007/s11004‑012‑9397‑2.
    https://doi.org/10.1007/s11004-012-9397-2 [Google Scholar]
  8. Elsheikh, A.H., Wheeler, M.F. and Hoteit, I.
    [2013a] An iterative stochastic ensemble method for parameter estimation of subsurface flow models. J. Comput. Phys., 242, 696 – 714, ISSN 0021-9991, doi:10.1016/j.jcp.2013.01.047.
    https://doi.org/10.1016/j.jcp. 2013.01.047 [Google Scholar]
  9. [2013b] Nested sampling algorithm for subsurface flow model selection, uncertainty quantification and nonlinear calibration. Water Resour. Res., 49(12), 8383 – 8399, ISSN 1944-7973, doi:10.1002/2012WR013406.
    https://doi.org/10.1002/2012WR013406 [Google Scholar]
  10. Flemisch, B. et al.
    [2011] Dumux: {DUNE} for multi-{phase, component, scale, physics, …} flow and transport in porous media. Advances in Water Resources, 34(9), 1102 – 1112, ISSN 0309-1708, doi:10.1016/j.advwatres.2011.03.007, new Computational Methods and Software Tools.
    https://doi.org/http://dx.doi.org/10.1016/j.advwatres.2011.03.007 [Google Scholar]
  11. Hansson, A. and Bryngelsson, M.
    [2009] Expert opinions on carbon dioxide capture and storage – A framing of uncertainties and possibilities. Energy Policy, 37, 2282–.
    [Google Scholar]
  12. IPCC
    [2005] Special report on carbon dioxide capture and storage, Technical report, Intergovernmental Panel on Climate Change (IPCC), prepared by Working Group III. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.
    [Google Scholar]
  13. Kopp, A., Class, H. and Helmig, H.
    [2009] Investigations on CO2 storage capacity in saline aquifers - part 1: Dimensional analysis of flow processes and reservoir characteristics. Int. J. of Greenhouse Gas Control, 3, 276–.
    [Google Scholar]
  14. Li, H. and Zhang, D.
    [2007] Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods. Water Resour. Res., 43, 48–.
    [Google Scholar]
  15. Oladyshkin, S., Class, H., Helmig, R. and Nowak, W.
    [2011a] A concept for data-driven uncertainty quantification and its application to carbon dioxide storage in geological formations. Advances in Water Resources, 34, 1508–1518, doi:10.1016/j.advwatres.2011.08.005.
    https://doi.org/10.1016/j.advwatres.2011.08.005 [Google Scholar]
  16. [2011b] An integrative approach to robust design and probabilistic risk assessment for CO2 storage in geological formations. Computational Geosciences, 15(3), 565-577, doi:10.1007/s10596‑011‑9224‑8.
    https://doi.org/10.1007/s10596-011-9224-8 [Google Scholar]
  17. Oladyshkin, S., Class, H. and Nowak, W.
    [2013a] Bayesian updating via Bootstrap filtering combined with data-driven polynomial chaos expansions: methodology and application to history matching for carbon dioxide storage in geological formations. Computational Geosciences, 17(4), 671–687, doi:10.1007/s10596‑013‑9350‑6.
    https://doi.org/10.1007/s10596-013-9350-6 [Google Scholar]
  18. Oladyshkin, S. and Nowak, W.
    [2012] Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliability Engineering and System Safety, 106, 179–190, doi:10.1016/j.ress.2012.05.002.
    https://doi.org/10.1016/j.ress.2012.05.002 [Google Scholar]
  19. Oladyshkin, S., Schröder, P., Class, H. and Nowak, W.
    [2013b] Chaos expansion based bootstrap filter to calibrate co2 injection models. Energy Procedia, 40, 407–.
    [Google Scholar]
  20. Pappenberger, F. and Beven, K.J.
    [2006] Ignorance is bliss: Or seven reasons not to use uncertainty analysis. Water Resources Research, 42(5), 1–8.
    [Google Scholar]
  21. Robert, C.P. and Casella, G.
    [2004] Monte carlo methods. New York: Springer.
    [Google Scholar]
  22. Skilling, J.
    [2006] Nested sampling for general Bayesian computation. Bayesian Anal., 1(4), 833–860, doi:10.1214/06‑BA127.
    https://doi.org/10.1214/06-BA127 [Google Scholar]
  23. Tavakoli, R., Yoon, H., Delshad, M., ElSheikh, A.H., Wheeler, M.F. and Arnold, B.W.
    [2013] Comparison of ensemble filtering algorithms and null-space monte carlo for parameter estimation and uncertainty quantification using co2 sequestration data. Water Resources Research, 49(12), 8108–8127, ISSN 1944–7973, doi:10.1002/2013WR013959.
    https://doi.org/10.1002/2013WR013959 [Google Scholar]
  24. Walter, L., Binning, P., Oladyshkin, S., Flemisch, B. and Class, H.
    [2012] Brine migration resulting from CO2 injection into saline aquifers - an approach to risk estimation including various levels of uncertainty. International Journal of Greenhouse Gas Control, 9(495–506), doi:10.1016/j.ijggc.2012.05.004.
    https://doi.org/10.1016/j.ijggc.2012.05.004 [Google Scholar]

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