Modeling fault activation due to fluid production: Bayesian update by seismic data

  • Claudia ZoccaratoEmail author
  • Massimiliano Ferronato
  • Andrea Franceschini
  • Carlo Janna
  • Pietro Teatini
Original Paper


The reactivation of faults and the generation of fractures can be caused by stress changes due to injection and/or production of fluids into and/or from the subsurface. The simulation of these processes, which could be associated with (micro-)seismicity, is affected by a high uncertainty. The aim of this work is at developing a mathematical framework to quantify and possibly reduce the prior modeling uncertainties by assimilation of seismic data. The mechanics of fault (re-)activation is simulated by a Finite Element (FE) numerical model where the discontinuous displacements between the fault surfaces are suitably considered using appropriate Interface Elements (IEs). The study is carried out by using a stochastic approach, with a global sensitivity analysis (gSA) based on Sobol’ indices initially performed to estimate the influence of the input parameters on the model solution. Then, a Markov Chain Monte Carlo (MCMC) sampling technique based on the generalized Polynomial Chaos expansion (gPC) surrogate solution is used to update the prior information conditioned on seismic observations. The methodology is tested on a 3D synthetic test case. The uncertain input is the natural stress regime and the Mohr-Coulomb parameters characterizing the fault activation criterion. A good reduction of the prior uncertainty is obtained, showing that the assimilation of seismic data can have a promising potential for improving the subsurface characterization.


Fault reactivation Uncertainty quantification Polynomial chaos expansion Bayesian updating FE-IE model 

List of Symbols


Cartesian coordinates of a point in the three-dimensional space


time/loading step variable


state vector of the system


forcing term of the system


vector of the model parameters

\(\mathcal {G}\)

forward problem operator


vector of the quantities of interest

\(\mathcal {M}\)

map from y to q

\(\mathcal {S}\)

solution operator of the forward problem


random vector of the model parameters


random vector of the quantities of interest


random vector of the model output


joint probability density function of a given random vector


random vector of the model parameters defined over a unit hypercube




first order Sobol’ index

S1,…, s

higher order Sobol’ index


random vector of the noisy measurements


random vector of the true observable


random vector of the observational error

\(\mathcal {H}\)

map from the model state to the true observable


conditioned probability density function


shear stress limit


fault cohesion


fault friction angle


normal effective stress acting on the fault


k-th univariate basis function

\(\tilde {\boldsymbol {Y}}\)

gPC approximation of Y


vector of coefficients of the gPC expansion


multivariate gPC basis functions

α, β

multi-index vectors

\(\tilde {\boldsymbol {c}}_{\boldsymbol {\alpha }}\)

approximation of the gPC expansion coefficients


integration weights

σ1, σ2, σ3

principal stresses


ratio of the principal stress σ1 and σ2 to σ3


effective normal stress vector


shear stress vector


displacement field vector


activated area of the fault


sliding of the fault


hydraulic conductivity


pore pressure


slippage of the fault elements


seismic moment


shear modulus of the rock formation


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The research has been carried out in the framework of the UNESCO IGCP 641 project “Mechanisms, Monitoring and Modeling Earth Fissure generation and Fault activation due to subsurface Fluid exploitation (M3EF3)”.

Funding information

This work has been partially funded by the University of Padova project “Data Assimilation algorithms for reservoir geomechanics and induced seismicity.”


  1. 1.
    Aagaard, B.T., Knepley, M.G., Williams, C.A.: A domain decomposition approach to implementing fault slip in finite-element models of quasi-static and dynamic crustal deformation. J. Geophys. Res.: Solid Earth 118(6), 3059–3079 (2013)CrossRefGoogle Scholar
  2. 2.
    Baisch, S., Vörös, R., Rothert, E., Stang, H., Jung, R., Schellschmidt, R.: A numerical model for fluid injection induced seismicity at Soultz-sous-Forêts. Int. J. Rock Mech. Mining Sci. 47(3), 405–413 (2010). CrossRefGoogle Scholar
  3. 3.
    Baù, D., Ferronato, M., Gambolati, G., Teatini, P.: Basin-scale compressibility of the Northern Adriatic by the radioactive marker technique. Geotechnique 52(8), 605–616 (2002)CrossRefGoogle Scholar
  4. 4.
    Beer, G.: An isoparametric joint/interface element for finite element analysis. Int. J. Numer. Methods Eng. 21(4), 585–600 (1985)CrossRefGoogle Scholar
  5. 5.
    Cappa, F., Rutqvist, J.: Modeling of coupled deformation and permeability evolution during fault reactivation induced by deep underground injection of CO2. Int. J. Greenhouse Gas Control 5(2), 336–346 (2011)CrossRefGoogle Scholar
  6. 6.
    Castelletto, N., Teatini, P., Gambolati, G., Bossie-Codreanu, D., Vincké, O., Daniel, J.M., Battistelli, A., Marcolini, M., Donda, F., Volpi, V.: Multiphysics modeling of CO2 sequestration in a faulted saline formation in italy. Adv. Water Resour. 62, 570–587 (2013). CrossRefGoogle Scholar
  7. 7.
    Castiñeira, D., Jha, B., Juanes, R.: Uncertainty quantification and inverse modeling of fault poromechanics and induced seismicity: application to a synthetic carbon capture and storage (ccs) problem. ARMA-2016-151, American Rock Mechanics Association, 50th U.S Rock Mechanics/Geomechanics Symposium, 26-29, June, Houston (2016)Google Scholar
  8. 8.
    Cesca, S., Grigoli, F., Heimann, S., González, A.́, Buforn, E., Maghsoudi, S., Blanch, E., Dahm, T.: The 2013 September-October seismic sequence offshore Spain: a case of seismicity triggered by gas injection? Geophys. J. Int. 198(2), 941–953 (2014). CrossRefGoogle Scholar
  9. 9.
    Cescotto, S., Charlier, R.: Frictional contact finite elements based on mixed variational principles. Int. J. Numer. Methods Eng. 36(10), 1681–1701 (1993)CrossRefGoogle Scholar
  10. 10.
    Chang, H., Chen, Y., Zhang, D.: Data assimilation of coupled fluid flow and geomechanics using the ensemble kalman filter. Soc. Petroleum Eng., 15(2). (2010)CrossRefGoogle Scholar
  11. 11.
    Constantine, P., Eldred, M., Phipps, E.: Sparse pseudospectral approximation method. Comput. Methods Appl. Mech. Eng. 229–232, 1–12 (2012). CrossRefGoogle Scholar
  12. 12.
    Crestaux, T., Le Maître, O., Martinez, J.M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Safety 94(7), 1161–1172 (2009). CrossRefGoogle Scholar
  13. 13.
    Fajraoui, N., Ramasomanana, F., Younes, A., Mara, T.A., Ackerer, P., Guadagnini, A.: Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory-scale porous media. Water Resour. Res. 47(2), W02521 (2011). CrossRefGoogle Scholar
  14. 14.
    Ferronato, M., Gambolati, G., Janna, C., Teatini, P.: Numerical modelling of regional faults in land subsidence prediction above gas/oil reservoirs. Int. J. Numer. Anal. Methods Geomech. 32(6), 633–657 (2008)CrossRefGoogle Scholar
  15. 15.
    Fokker, P.A., Wassing, B.B., van Leijen, F.J., Hanssen, R.F., Nieuwland, D.A.: Application of an ensemble smoother with multiple data assimilation to the Bergermeer gas field, using PS-InSAR. Geomech. Energy Environ. 5, 16–28 (2016). CrossRefGoogle Scholar
  16. 16.
    Formaggia, L., Guadagnini, A., Imperiali, I., Lever, V., Porta, G., Riva, M., Scotti, A., Tamellini, L.: Global sensitivity analysis through polynomial chaos expansion of a basin-scale geochemical compaction model. Comput. Geosci. 17(1), 25–42 (2013). CrossRefGoogle Scholar
  17. 17.
    Franceschini, A., Castelletto, N., Ferronato, M.: Block preconditioning for fault/fracture mechanics saddle-point problems. Comput. Methods Appl. Mech. Eng. 344, 376–401 (2019). CrossRefGoogle Scholar
  18. 18.
    Franceschini, A., Ferronato, M., Janna, C., Teatini, P.: A novel Lagrangian approach for the stable numerical simulation of fault and fracture mechanics. J. Comput. Phys. 314, 503–521 (2016). CrossRefGoogle Scholar
  19. 19.
    Franceschini, A., Teatini, P., Janna, C., Ferronato, M., Gambolati, G., Ye, S.J., Carreón-Freyre, D.: Modelling ground rupture due to groundwater withdrawal: Applications to test cases in China and Mexico. Proc. Int. Assoc. Hydrol. Sci. 372, 63 (2015)Google Scholar
  20. 20.
    Gan, Q., Elsworth, D.: Thermal drawdown and late-stage seismic-slip fault reactivation in enhanced geothermal reservoirs. J. Geophys. Res.: Solid Earth 119(12), 8936–8949 (2014). CrossRefGoogle Scholar
  21. 21.
    Gan, W., Frohlich, C.: Gas injection may have triggered earthquakes in the Cogdell oil field, Texas. Proc. Natl. Acad. Sci. 110(47), 18786–18791 (2013). CrossRefGoogle Scholar
  22. 22.
    Gaucher, E., Shoenball, M., Heidbach, O., Zang, A., Fokker, A.P., van Wees, J.D., Kohl, T.: Induced seismicity in geothermal reservoirs: A review of forecasting approaches. Renew. Sustain. Energy Rev. 52, 1473–1490 (2015). CrossRefGoogle Scholar
  23. 23.
    Hȧring, M.O., Schanz, U., Ladner, F., Dyer, B.C.: Characterisation of the Basel 1 enhanced geothermal system. Geothermics 37(5), 469–495 (2008). CrossRefGoogle Scholar
  24. 24.
    Hastings, W.K.: Monte Carlo sampling methods using Markov Chains and their applications. Biometrika 57, 97–109 (1970)CrossRefGoogle Scholar
  25. 25.
    Horton, S.: Disposal of hydrofracking waste fluid by injection into subsurface aquifers triggers earthquake swarm in central Arkansas with potential for damaging earthquake. Seismol. Res. Lett. 83(2), 250–260 (2012). CrossRefGoogle Scholar
  26. 26.
    Jha, B., Juanes, R.: Coupled multiphase flow and poromechanics: A computational model of pore pressure effects on fault slip and earthquake triggering. Water Resour. Res. 50, 3776–3808 (2014)CrossRefGoogle Scholar
  27. 27.
    Kanamori, H.L., Anderson, D.: Theoretical basis of some empirically relations in seismology. Bull. Seismol. Soc. Am. 65(5), 1073–1095 (1975)Google Scholar
  28. 28.
    King, G.C.P., Stein, R.S., Lin, J.: Static stress changes and the triggering of earthquakes. Bull. Seismol. Soc. Am. 84(3), 935–953 (1994)Google Scholar
  29. 29.
    Kraaijpoel, D.A., Nieuwland, D.A., Dost, B.: Microseismic monitoring and subseismic fault detection in an underground gas storage. In:. 4th Passive Seismic Workshop, pp. 80–82 (2013)Google Scholar
  30. 30.
    Labuz, J.F., Zang, A.: Mohr-Coulomb failure criterion. Rock Mech. Rock. Eng. 45, 975–979 (2012). CrossRefGoogle Scholar
  31. 31.
    Lei, X., Ma, S., Chen, W., Pang, C., Zeng, J., Jiang, B.: A detailed view of the injection-induced seismicity in a natural gas reservoir in Zigong, southwestern Sichuan Basin, China. J. Geophys. Res.: Solid Earth 118(8), 4296–4311 (2013)CrossRefGoogle Scholar
  32. 32.
    Li, H., Zhang, D.: Probabilistic collocation method for flow in porous media: Comparisons with other stochastic methods. Water Resour. Res. 43(9), 1–13 (2007). CrossRefGoogle Scholar
  33. 33.
    Li, W., Lu, Z., Zhang, D.: Stochastic analysis of unsaturated flow with probabilistic collocation method. Water Resour. Res. 45(8), 1–13 (2009). CrossRefGoogle Scholar
  34. 34.
    Li, Y.J., Kokkinaki, A., Darve, E.F., Kitanidis, P.K.: Smoothing-based compressed state kalman filter for joint state-parameter estimation: Applications in reservoir characterization and CO2 storage monitoring. Water Resour. Res. 53, 7190–7207 (2017). CrossRefGoogle Scholar
  35. 35.
    Liao, Q., Zhang, D.: Probabilistic collocation method for strongly nonlinear problems: 3. Transform by time. Water Resour. Res. 52, 2366–2375 (2016). CrossRefGoogle Scholar
  36. 36.
    Mazzoldi, A., Rinaldi, A.P., Borgia, A., Rutqvist, J.: Induced seismicity within geological carbon sequestration projects: Maximum earthquake magnitude and leakage potential from undetected faults. Int. J. Greenhouse Gas Control 10, 434–442 (2012). CrossRefGoogle Scholar
  37. 37.
    Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953). CrossRefGoogle Scholar
  38. 38.
    Nicol, A., Carne, R., Gerstenberger, M., Christophersen, A.: Induced seismicity and its implications for CO2 storage risk. Energy Procedia 4, 3699–3706 (2011). CrossRefGoogle Scholar
  39. 39.
    Ochoa-Gonzalez, G., Carreon-Freyre, D., Franceschini, A., Cerca, M., Teatini, P.: Overexploitation of groundwater resources in the faulted basin of Queretaro, Mexico: A 3D deformation and stress analysis. Eng. Geol. 245, 192–206 (2018). CrossRefGoogle Scholar
  40. 40.
    Orlic, B., Wassing, B., Geel, C.: Field scale geomechanical modeling for prediction of fault stability during underground gas storage operations in a depleted gas field in the netherlands. ARMA-2013-300, American Rock Mechanics AssociationSource 47th U.S Rock Mechanics/Geomechanics Symposium, 23-26 June, San Francisco, California (2013)Google Scholar
  41. 41.
    Phillips, W., Rutledge, J., House, L., Fehler, M.: Induced microearthquake patterns in hydrocarbon and geothermal reservoirs: Six case studies. Pure Appl. Geophys. 159(1), 345–369 (2002). CrossRefGoogle Scholar
  42. 42.
    Priolo, E., Romanelli, M., Plasencia Linares, M.P., Garbin, M., Peruzza, L., Romano, M.A., Marotta, P., Bernardi, P., Moratto, L., Zuliani, D., Fabris, P.: Seismic monitoring of an underground natural gas storage facility: The collalto seismic network. Seismol. Res. Lett. 86(1), 109–123 (2015). CrossRefGoogle Scholar
  43. 43.
    Rutqvist, J., Birkholzer, J., Tsang, C.: Coupled reservoir-geomechanical analysis of the potential for tensile and shear failure associated with 2 injection in multilayered reservoir-caprock systems. Int. J. Rock Mech. Min. Sci 45, 132–143 (2008)CrossRefGoogle Scholar
  44. 44.
    Saltelli, A., Marco, R., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S.: Global Sensitivity Analysis. The Primer. Wiley, Chichester (2007)CrossRefGoogle Scholar
  45. 45.
    Sobol’, I.: Sensitivity estimates for nonlinear mathematical models. Matematicheskoe Modelirovanie 2, 112–118 (1990). (1990 in Russian, translated in English in Sobol’ (1993))Google Scholar
  46. 46.
    Sudret, B.: Meta-models for structural reliability and uncertainty quantification. In: Proc. 5th Asian-Pacific Symp. Struct. Reliab. (APSSRA’2012) Singapore, pp. 53–76. (2012)
  47. 47.
    Teatini, P., Baù, D., Gambolati, G.: Water-gas dynamics and coastal subsidence over chioggia mare field, northern adriatic sea. Hydrogeol. J. 8(5), 462–479 (2000)CrossRefGoogle Scholar
  48. 48.
    Verdon, J.P., Kendall, J.M., White, D.J., Angus, D.A.: Linking microseismic event observations with geomechanical models to minimise the risks of storing CO2 in geological formations. Earth Planet. Sci. Lett. 305(1-2), 143–152 (2011). CrossRefGoogle Scholar
  49. 49.
    White, J.A., Foxall, W.: Assessing induced seismicity risk at CO2 storage projects: Recent progress and remaining challenges. Inte. J. Greenhouse Gas Control 49, 413–424 (2016). CrossRefGoogle Scholar
  50. 50.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897–936 (1938). CrossRefGoogle Scholar
  51. 51.
    Wriggers, P.: Contact Mechanics. Springer, Berlin (2005)Google Scholar
  52. 52.
    Xiu, D.: Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2(2), 293–309 (2007)Google Scholar
  53. 53.
    Xiu, D.: Numerical Methods for Stochastic Computations. A Spectral Method Approach. Princeton University Press, Princeton (2010)CrossRefGoogle Scholar
  54. 54.
    Xiu, D., Karniadakis, G.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)CrossRefGoogle Scholar
  55. 55.
    Ye, S., Franceschini, A., Zhang, Y., Janna, C., Gong, X., Yu, J., Teatini, P.: A novel approach to model earth fissure caused by extensive aquifer exploitation and its application to the Wuxi case, China. Water Resour. Res. 54(3), 2249–2269 (2018). CrossRefGoogle Scholar
  56. 56.
    Zander, E.K.: Sglib - a matlab/octave toolbox for stochastic galerkin and stochastic collocation methods,
  57. 57.
    Zang, A., Oye, V., Jousset, P., Deichmann, N., Gritto, R., McGarr, A., Majer, E., Bruhn, D.: Analysis of induced seismicity in geothermal reservoirs - An overview. Geothermics 52, 6–21 (2014). CrossRefGoogle Scholar
  58. 58.
    Zoback, M.D., Gorelick, S.M.: Earthquake triggering and large-scale geologic storage of carbon dioxide. Proc. Natl. Acad. Sci. 109(26), 10164–10168 (2012). CrossRefGoogle Scholar
  59. 59.
    Zoccarato, C., Bau̇, D., Ferronato, M., Gambolati, G., Alzraiee, A., Teatini, P.: Data assimilation of surface displacements to improve geomechanical parameters of gas storage reservoirs. J. Geophys. Res.: Solid Earth 121(3), 1441–1461 (2016). CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Civil, Environmental and Architectural EngineeringUniversity of PadovaPadovaItaly

Personalised recommendations