Abstract
In this paper, a recently developed parameterization procedure based on principal component analysis (PCA), which is referred to as optimization-based PCA (O-PCA), is generalized for use with a wide range of geological systems. In O-PCA, the mapping between the geological model in the full-order space and the low-dimensional subspace is framed as an optimization problem. The O-PCA optimization involves the use of regularization and bound constraints, which act to extend substantially the ability of PCA to model complex (non-Gaussian) systems. The basis matrix required by O-PCA is formed using a set of prior realizations generated by a geostatistical modeling package. We show that, by varying the form of the O-PCA regularization terms, different types of geological scenarios can be represented. Specific systems considered include binary-facies, three-facies and bimodal channelized models, and bimodal deltaic fan models. The O-PCA parameterization can be applied to generate random realizations, though our focus here is on its use for data assimilation. For this application, O-PCA is combined with the randomized maximum likelihood (RML) method to provide a subspace RML procedure that can be applied to non-Gaussian models. This approach provides multiple history-matched models, which enables an estimate of prediction uncertainty. A gradient procedure based on adjoints is used for the minimization required by the subspace RML method. The gradient of the O-PCA mapping is determined analytically or semi-analytically, depending on the form of the regularization terms. Results for two-dimensional oil-water systems, for several different geological scenarios, demonstrate that the use of O-PCA and RML enables the generation of posterior reservoir models that honor hard data, retain the large-scale connectivity features of the geological system, match historical production data, and provide an estimate of prediction uncertainty. MATLAB code for the O-PCA procedure, along with examples for three-facies and bimodal models, is included as online Supplementary Material.
Similar content being viewed by others
References
Aanonsen, S.I., Naevdal, G., Oliver, D.S., Reynolds, A.C., Vallès, B.: The Ensemble Kalman Filter in reservoir engineering—a review. SPE J. 14(3), 393–412 (2009)
Awotunde, A.A., Horne, R.N.: Reservoir description with integrated multiwell data using two-dimensional wavelets. Math. Geosci. 45(2), 225–252 (2013)
Brouwer, D.R., Jansen, J.D.: Dynamic optimization of waterflooding with smart wells using optimal control theory. SPE J. 9(4), 391–402 (2004)
Caers, J.: Comparing the gradual deformation with the probability perturbation method for solving inverse problems. Math. Geol. 39(1), 27–52 (2007)
Castro, S.A.: A Probabilistic Approach to Jointly Integrate 3D/4D Seismic, Production Data and Geological Information for Building Reservoir Models. Ph.D. thesis, Department of Energy Resources Engineering, Stanford University (2007)
Chang, H., Zhang, D., Lu, Z.: History matching of facies distribution with the EnKF and level set parameterization. J. Comput. Phys. 229(19), 8011–8030 (2010)
Dorn, O., Villegas, R.: History matching of petroleum reservoirs using a level set technique. Inverse Prob 24(3), 035,015 (2008). doi:10.1088/0266--5611/24/3/035,015
Gao, G., Zafari, M., Reynolds, A.C.: Quantifying uncertainty for the PUNQ-S3 problem in a Bayesian setting with RML and EnKF. SPE J. 11(4), 506–515 (2006)
Gavalas, G.R., Shah, P.C., Seinfeld, J.H.: Reservoir history matching by Bayesian estimation. SPE J. 16(6), 337–350 (1976)
Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM Rev 47(1), 99–131 (2005)
Gill, P.E., Murray, W., Wright, M.H.: Practical optimization, 1st edn. Academic Press, New York (1981)
Hu, L.Y.: Gradual deformation and iterative calibration of Gaussian-related stochastic models. Math. Geol. 32(1), 87–108 (2000)
Hu, L.Y., Blanc, G., Noetinger, B.: Gradual deformation and iterative calibration of sequential stochastic simulations. Math. Geol. 33(4), 475–489 (2001)
Jafarpour, B., Goyal, V., McLaughlin, D.B., Freeman, W.T.: Compressed history matching: Exploiting transform-domain sparsity for regularization of nonlinear dynamic data integration problems. Math. Geosci. 42(1), 1–27 (2010)
Jafarpour, B., McLaughlin, D.B.: Efficient permeability parameterization with the Discrete Cosine Transform. Paper SPE 106453 presented at the SPE Reservoir Simulation Symposium, Houston, Texas, USA (2007)
Khaninezhad, M.M., Jafarpour, B.: Bayesian history matching and uncertainty quantification under sparse priors: a randomized maximum likelihood approach. Paper SPE 163656 presented at SPE Reservoir Simulation Symposium, Woodlands, Texas, USA (2013)
Khaninezhad, M.M., Jafarpour, B.: Sparse randomized maximum likelihood (SpRML) for subsurface flow model calibration and uncertainty quantification. Adv. Water Resour. 69, 23–37 (2014)
Khaninezhad, M.M., Jafarpour, B., Li, L.: History matching with learned sparse dictionaries. Paper SPE 133654 presented at the SPE Annual Technical Conference and Exhibition, Florence, Italy (2010)
Kitanidis, P.: Quasi-linear geostatistical theory for inversing. Water Resour. Res. 31(10), 2411–2419 (1995)
Liu, L., Oliver, D.S.: Experimental assessment of gradual deformation method. Math. Geol. 36(1), 65–77 (2004)
Liu, N., Oliver, D.S.: Evaluation of Monte Carlo methods for assessing uncertainty. SPE J. 8(2), 188–195 (2003)
Liu, N., Oliver, D.S.: Automatic history matching of geologic facies. SPE J. 9(4), 429–436 (2004)
Liu, N., Oliver, D.S.: Ensemble Kalman filter for automatic history matching of geologic facies. J. Pet. Sci. Eng. 47(3–4), 147–161 (2005)
Lu, P., Horne, R.: A multiresolution approach to reservoir parameter estimation using wavelet analysis. Paper SPE 62985 presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, USA (2000)
Ma, X., Zabaras, N.: Kernel principal component analysis for stochastic input model generation. J. Comput. Phys. 230(19), 7311–7331 (2011)
Mannseth, T.: Relation between level set and truncated pluri-Gaussian methodologies for facies representation. Math. Geosci. 46(6), 711–731 (2014)
Moskowitz, M.A., Paliogiannis, F.: Functions of several real variables. World Scientific (2011)
Nocedal, J., Wright, S.J.: Numerical optimization, 2nd edn. Springer, Berlin Heidelberg (2006)
Oliver, D.S.: Multiple realizations of permeability field from well test data. SPE J. 1(2), 145–154 (1996)
Oliver, D.S., Chen, Y.: Recent progress in history matching: a review. Comput. Geosci. 15(1), 185–221 (2011)
Oliver, D.S., He, N., Reynolds, A.C.: Conditioning permeability fields to pressure data. Paper presented at the 5th European Conference on the Mathematics of Oil Recovery, Leoben, Austria (1996)
Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press, Cambridge (2008)
Ping, J., Zhang, D.: History matching of fracture distributions by ensemble Kalman filter combined with vector based level set parameterization. J. Pet. Sci. Eng. 108, 288–303 (2013)
Ping, J., Zhang, D.: History matching of channelized reservoirs with vector-based level-set parameterization. SPE J. 19(3), 514–529 (2014)
Remy, N., Boucher, A., Wu, J.: Applied Geostatistics with SGeMS: A User’s Guide. Cambridge University Press, Cambridge (2009)
Reynolds, A.C., He, N., Chu, L., Oliver, D.S.: Reparameterization techniques for generating reservoir description conditioned to variograms and well-test pressure data. SPE J. 1(4), 413–426 (1996)
Reynolds, A.C., He, N., Oliver, D.S.: Reducing uncertainty in geostatistical description with well testing pressure data. In: Reservoir Characterization – Recent Advances, pp. 149–162. American Association of Petroleum Geologists (1999)
Sahni, I., Horne, R.: Multiresolution wavelet analysis for improved reservoir description. SPE Reserv. Eval. Eng. 8(1), 53–69 (2005)
Sarma, P., Durlofsky, L.J., Aziz, K.: Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics. Math. Geosci. 40(1), 3–32 (2008)
Sarma, P., Durlofsky, L.J., Aziz, K., Chen, W.H.: Efficient real-time reservoir management using adjoint-based optimal control and model updating. Comput. Geosci. 10(1), 3–36 (2006)
Sarma, P., Durlofsky, L.J., Aziz, K., Chen, W.H.: A new approach to automatic history matching using kernel PCA. Paper SPE 106176 presented at the SPE Reservoir Simulation Symposium, Houston, Texas, USA (2007)
Shirangi, M.G.: History matching production data and uncertainty assessment with an efficient TSVD parameterization algorithm. J. Pet. Sci. Eng. 113, 54–71 (2014)
Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geosci. 34(1), 1–21 (2002)
Tavakoli, R., Reynolds, A.C.: Monte Carlo simulation of permeability fields and reservoir performance predictions with SVD parameterization in RML compared with EnKF. Comput. Geosci. 15(1), 99–116 (2011)
Vo, H.X., Durlofsky, L.J.: A new differentiable parameterization based on principal component analysis for the low-dimensional representation of complex geological models. Math. Geosci. 46(7), 775–813 (2014)
Zafari, M., Reynolds, A.C.: Assessing the uncertainty in reservoir description and performance predictions with the ensemble Kalman filter. SPE J. 12(3), 382–391 (2007)
Zhao, H., Li, G., Reynolds, A.C., Yao, J.: Large-scale history matching with quadratic interpolation models. Comput. Geosci. 17(1), 117–138 (2013)
Zhou, H., Gómez-Hernández, J.J., Franssen, H.H., Li, L.: An approach to handling non-Gaussianity of parameters and state variables in ensemble Kalman filtering. Adv. Water Resour. 34(7), 844–864 (2011)
Zhou, Y.: Parallel General-Purpose Reservoir Simulation with Coupled Reservoir Models and Multi-Segment Wells. Ph.D. thesis, Department of Energy Resources Engineering. Stanford University, Stanford (2012)
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Vo, H.X., Durlofsky, L.J. Data assimilation and uncertainty assessment for complex geological models using a new PCA-based parameterization. Comput Geosci 19, 747–767 (2015). https://doi.org/10.1007/s10596-015-9483-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-015-9483-x
Keywords
- Non-Gaussian fields
- Geological parameterization
- Model order reduction
- Reservoir simulation
- History matching
- Data assimilation
- Uncertainty assessment
- Regularization
- Soft-thresholding
- Histogram transform