Abstract
Optimal management of subsurface processes requires the characterization of the uncertainty in reservoir description and reservoir performance prediction. For fractured reservoirs, the location and orientation of fractures are crucial for predicting production characteristics. With the help of accurate and comprehensive knowledge of fracture distributions, early water/CO 2 breakthrough can be prevented and sweep efficiency can be improved. However, since the rock property fields are highly non-Gaussian in this case, it is a challenge to estimate fracture distributions by conventional history matching approaches. In this work, a method that combines vector-based level-set parameterization technique and ensemble Kalman filter (EnKF) for estimating fracture distributions is presented. Performing the necessary forward modeling is particularly challenging. In addition to the large number of forward models needed, each model is used for sampling of randomly located fractures. Conventional mesh generation for such systems would be time consuming if possible at all. For these reasons, we rely on a novel polyhedral mesh method using the mimetic finite difference (MFD) method. A discrete fracture model is adopted that maintains the full geometry of the fracture network. By using a cut-cell paradigm, a computational mesh for the matrix can be generated quickly and reliably. In this research, we apply this workflow on 2D two-phase fractured reservoirs. The combination of MFD approach, level-set parameterization, and EnKF provides an effective solution to address the challenges in the history matching problem of highly non-Gaussian fractured reservoirs.
Similar content being viewed by others
References
Agbalaka, C.C., Oliver, D.S.: Application of the EnKF and localization to automatic history matching of facies distribution and production data. Math Geosci. 40, 353–374 (2008). doi:10.1007/s11004-008-9155-7
Agbalaka, C.C., Oliver, D.S.: Joint updating of petrophysical properties and discrete facies variables from assimilating production data using the EnKF. SPE J 16(2), 318–330 (2011). doi:10.2118/118916-PA
Ahmed, R., Edwards, M.G., Lamine, S., Huisman, B.A.H., Pal, M.: Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model. J. Comput. Phys. 284, 462–489 (2015). doi:10.1016/j.jcp.2014.12.047
Alboin, C., Jaffré, J., Roberts, J.E., Serres, C.: Modeling fractures as interfaces for flow and transport in porous media Fluid flow and transport in porous media: mathematical and numerical treatment (South Hadley, MA, 2001), pp 13–24. Contemporary Mathematics of the American Mathematical Society, Providence, RI (2002)
Al-Hinai, O., Wheeler, M.F., Yotov, I.: A generalized mimetic finite difference method and two-point flux schemes over Voronoi diagrams. ESAIM: Mathematical Modelling and Numerical Analysis. doi:10.1051/m2an/2016033 (2016)
Al-Hinai, O., Singh, G., Pencheva, G., Almani, T., Wheeler, M.F.: Modeling multiphase flow with nonplanar fractures. Society of Petroleum Engineers. doi:10.2118/163605-MS (2013)
Al-Hinai, O., Dong, R., Srinivasan, S., Wheeler, M.F.: A new equi-dimensional fracture model using polyhedral cells for microseismic data sets. J. Petrol. Sci. Eng. 154, 49–59 (2017). doi:10.1016/j.petrol.2017.04.004
Antonietti, P.F., Formaggia, L., Scotti, A., Verani, M., Verzott, N.: Mimetic finite difference approximation of flows in fractured porous media. ESAIM: Math. Modell. Numer. Anal. 50, 809–832 (2016). doi:10.1051/m2an/2015087
Arbogast, T., Douglas, J.J., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21, 823–836 (1990). doi:10.1137/0521046
Benedetto, M.F., Berrone, S., Pieraccini, S., Scialò, S.: The virtual element method for discrete fracture network simulations. Comput. Methods Appl. Mech. Eng. 280, 135–156 (2014). doi:10.1016/j.cma.2014.07.016
Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16, 275–297 (2006). doi:10.1142/S0218202506001157
Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43, 1872–1896 (2005). doi:10.1137/040613950
Cardiff, M., Kitanidis, P.K.: Bayesian inversion for facies detection: an extensible level set framework. Water Resour. Res. 45, W10416 (2009). doi:10.1029/2008wr007675
Chang, H., Zhang, D., Lu, Z.: History matching of facies distribution with the EnKF and level set parameterization. J. Comput. Phys. 229, 8011–8030 (2010)
Chen, Y., Oliver, D.S.: Ensemble-based closed-loop optimization applied to Brugge field. SPE Reserv. Eval. Eng. 13, 56–71 (2010). doi:10.2118/118926-PA
Dai, C., Xue, L., Zhang, D., Guadagnini, A.: Data-worth analysis through probabilistic collocation-based Ensemble Kalman Filter. J. Hydrol. 540, 488–503 (2016). doi:10.1016/j.jhydrol.2016.06.037
Dorn, O., Villegas, R.: History matching of petroleum reservoirs using a level set technique. Inverse Probl. 24, 35015 (2008)
Dovera, L., Della Rossa, E.: Multimodal ensemble Kalman filtering using Gaussian mixture models. Comput. Geosci. 15, 307–323 (2011)
Evensen, G.: Data assimilation: the ensemble Kalman filter. Springer, New York (2007)
Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003)
Evensen, G.: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99, 10143–10162 (1994)
Fisher, M.K., Wright, C.A., Davidson, B.M., Goodwin, A.K., Fielder, E.O., Buckler, W.S., Steinsberger, N.P.: Integrating fracture mapping technologies to optimize stimulations in the Barnett Shale. Society of Petroleum Engineers. doi:10.2118/77441-MS (2002)
Gu, Y., Oliver, D.S.: History matching of the PUNQ-S3 reservoir model using the ensemble Kalman filter. SPE J. 10, 217–224 (2005). doi:10.2118/89942-PA
Hægland, H., Assteerawatt, A., Dahle, H.K., Eigestad, G.T., Helmig, R.: Comparison of cell- and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture–matrix system. Adv. Water Resour. 32, 1740–1755 (2009). doi:10.1016/j.advwatres.2009.09.006
Hoteit, H., Firoozabadi, A.: An efficient numerical model for incompressible two-phase flow in fractured media. Adv. Water Resour. 31, 891–905 (2008). doi:10.1016/j.advwatres.2008.02.004
Iglesias, M.A., Lin, K., Stuart, A.M.: Well-posed Bayesian geometric inverse problems arising in subsurface flow. Inverse Probl. 30, 114001 (2014). doi:10.1088/0266-5611/30/11/114001
Jafarpour, B.: Wavelet reconstruction of geologic facies from nonlinear dynamic flow measurements. IEEE T. Geosci. Remote IEEE T. Geosci. Remote 49, 1520–1535 (2011)
Jafarpour, B., McLaughlin, D.: History matching with an ensemble Kalman filter and discrete cosine parameterization, SPE 108761. Presented at the SPE Annual Technical Conference and Exhibition (2007)
Karimi-Fard, M., Durlofsky, L.J., Aziz, K.: An efficient discrete-fracture model applicable for general-purpose reservoir simulators. SPE J. 9, 227–236 (2004). doi:10.2118/88812-PA
Khodabakhshi, M., Jafarpour, B.: Multipoint statistical characterization of geologic facies from dynamic data and uncertain training images. Society of Petroleum Engineers. doi:10.2118/146935-MS (2011)
Lie, K.-A., Krogstad, S., Ligaarden, I.S., Natvig, J.R., Nilsen, H.M., Skaflestad, B.: Open-source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci. 16, 297–322 (2012). doi:10.1007/s10596-011-9244-4
Liu, N., Oliver, D.S.: Critical evaluation of the ensemble Kalman filter on history matching of geologic facies. SPE Reserv. Eval. Eng. 8, 470–477 (2005)
Lorentzen, R.J., Flornes, K.M., Nævdal, G.: History matching channelized reservoirs using the ensemble Kalman filter. SPE J. 17, 137–151 (2012)
Lu, L., Zhang, D.: Assisted history matching for fractured reservoirs by use of Hough-transform-based parameterization. SPE J. 20, 0942–0961 (2015). doi:10.2118/176024-PA
Moreno, D.L., Aanonsen: Stochastic facies modeling using the level set method. Presented at the Petroleum Geostatistics, pp. 10–14 (2007)
Naevdal, G., Johnsen, L.M., Aanonsen, S.I., Vefring, E.H.: Reservoir monitoring and continuous model updating using ensemble Kalman filter. SPE J. 10, 66–74 (2005)
Nejadi, S., Leung, J.Y.W., Trivedi, J.J.: Integration of production data for estimation of natural fracture properties in tight gas reservoirs using ensemble Kalman filter. Society of Petroleum Engineers. doi:10.2118/162783-MS (2012)
Ochs, S., Hinkelmann, R., Neunhäuserer, N., Suess, M., Helmig, R., Gebauer, S.: Adaptive methods for the equidimensional modelling of flow and transport processes in fractured aquifers. Presented at the 5th international conference on hydro-science and engineering, Warsaw, Poland (2002)
Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)
Osher, S.J., Santosa, F.: Level set methods for optimization problems involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171, 272–288 (2001)
Ping, J., Zhang, D.: History matching of channelized reservoirs with vector-based level-set parameterization. SPE J. 19, 514–529 (2014). doi:10.2118/169898-PA
Ping, J., Zhang, D.: History matching of fracture distributions by ensemble Kalman filter combined with vector based level set parameterization. J. Petrol. Sci. Eng. 108, 288–303 (2013). doi:10.1016/j.petrol.2013.04.018
Ping, J., Al-Hinai, O., Srinivasan, S., Wheeler, M.F., Min, B.: History matching for fractured reservoirs using mimetic finite differences and ensemble Kalman filter. AGU Fall Meeting, San Francisco, California, pp. 12–16 (2016)
Sarma, P., Chen, W.: Generalization of the ensemble Kalman filter using kernels for nongaussian random fields (2009)
Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. Soc. Petrol. Eng. J. 3, 245–255 (1963). doi:10.2118/426-PA
Zhang, Y., Oliver, D.: Evaluation and error analysis: Kalman gain regularization versus covariance regularization. Comput. Geosci. 15, 489–508 (2011). doi:10.1007/s10596-010-9218-y
Zhao, Y., Reynolds, A.C., Li, G.: Generating facies maps by assimilating production data and seismic data with the ensemble Kalman filter, SPE 113990. Presented at the SPE/DOE Symposium on Improved Oil Recovery (2008)
Acknowledgments
This research is partially funded by DOE grant DE-FE0023314. The authors acknowledge the financial support from the King Abdullah University of Science and Technology Academic Excellence Alliance.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ping, J., Al-Hinai, O. & Wheeler, M.F. Data assimilation method for fractured reservoirs using mimetic finite differences and ensemble Kalman filter. Comput Geosci 21, 781–794 (2017). https://doi.org/10.1007/s10596-017-9659-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-017-9659-7