Abstract
Prior to the estimation process of channelized reservoirs, in the context of any Assisted History Matching method, the parameterization of facies field is a necessary task. The parameterization of the facies field consists of defining a numerical field (parameter field) on the reservoir domain so that, using a projection function, we are able to recover the facies field from the values of parameter field. One of the most important issues encountered is the loss of the multipoint geostatistical properties in the updates (channel continuity). In this study, we start from an initial (global) parameterization of the channelized field and infer from it a low-dimensional parameterization obtained after a high-order singular value decomposition of a tensor built with the parameter fields. We decompose the parameter field as a linear combination of some basis functions with coefficients. The decomposition is followed by a truncation so that we keep the relevant information from the channel continuity perspective, but with a small number of coefficients. The coefficients will represent the low-dimensional parameterization and are further introduced in the estimation process of facies field, using the Ensemble Smoother with Multiple Data Assimilations (ES-MDA). For a fair assessment of the parameterization, we perform a comparison of the results with those obtained by applying the traditional truncated singular value decomposition and the global parameterization. In addition, we compare the parameterization with a low-dimensional parameterization defined with the PCA decomposition. The comparison is done from the perspective of multipoint geostatistical characteristics of the updates and predictions. We show that the new parameterization is able to better keep the multipoint geostatistical structure in the updates than the other parameterizations, while the prediction capabilities are the same.
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References
Afra, S., Gildin, E.: Tensor based geology preserving reservoir parameterization with higher order singular value decomposition (HOSVD). Comput. Geosci. 94, 110–120 (2016)
Bergqvist, G., Larsson, E.: Higher-order singular value decomposition: theory and an application. IEEE Signal Process. Mag. 27(3), 151–154 (2010)
Caers, J., Zhang, T.: Multiple-point geostatistics: a quantitative vehicle for integrating geologic analogs into multiple reservoir models (2004)
De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21(4), 1253–1278 (2000)
Deutsch, C.V., Wang, L.: Hierarchical object-based stochastic modeling of fluvial reservoirs. Math. Geol. 28(7), 857–880 (1996)
Emerick, A.A.: Investigation on principal component analysis parameterizations for history matching channelized facies models with ensemble-based data assimilation. Math. Geosci. 49(1), 85–120 (2017). https://doi.org/10.1007/s11004-016-9659-5
Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013). https://doi.org/10.1016/j.cageo.2012.03.011
Evensen, G.: The ensemble kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53(4), 343–367 (2003)
Golmohammadia, A., Khaninezhad, M.R., Jafarpour, B.: Pattern-based calibration of complex subsurface flow models against dynamic response data. Adv. Water Resour. https://doi.org/10.1016/j.advwatres.2018.04.007 (2018)
Hanea, R., Ek, T., Sebacher, B.: Consistent joint updates of facies and petrophysical heterogeneities using an ensemble based assisted history matching. In: Petroleum Geostatistics, 2015. EAGE. https://doi.org/10.3997/2214-4609.201413598 (2015)
Insuasty, E., Van den Hof, P., Weiland, S., Jansen, J.: Low-dimensional Tensor Representations for the Estimation of Petrophysical Reservoir Parameters. In: SPE Reservoir Simulation Conference, SPE-182707-MS. SPE Society of Petroleum Engineers. https://doi.org/10.2118/182707-MS (2017)
Jafarpour, B.: Wavelet reconstruction of geologic facies from nonlinear dynamic flow measurements. IEEE Trans. Geosci. Remote Sens. 49(5), 1520–1535 (2011)
Jafarpour, B., Khodabakhshi, M.: A probability conditioning method (PCM) for nonlinear flow data integration into multipoint statistical facies simulation. Math. Geosci. 43(2), 133–164 (2011)
Jafarpour, B., McLaughlin, D.B.: History matching with an ensemble Kalman filter and discrete cosine parameterization. Comput. Geosci. 12(2), 227–244 (2008)
Keogh, K., Martinius, A.W., Osland, R.: The development of fluvial stochastic modelling in the Norwegian oil industry: a historical review, subsurface implementation and future directions. Sediment. Geol. 202(1–2), 249–268 (2007)
Khaninezhad, M., Golmohammadi, A., Jafarpour, B.: Discrete regularization for calibration of geologic facies against dynamic flow data water resources research. https://doi.org/10.1002/2017WR022284 (2018)
Khaninezhad, M.M., Jafarpour, B., Li, L.: Sparse geologic dictionaries for subsurface flow model calibration Part i. inversion formulation. Adv. Water Resour. 39, 106–121 (2012)
Rajwade, A., Rangarajan, A., Banerjee, A.: Image denoising using the higher order singular value decomposition. IEEE Trans. Pattern Anal. Mach. Intell. 35(4), 849–862 (2012)
Remy, N.: S-gems: the Stanford geostatistical modeling software: a tool for new algorithms development. Geostatistics Banff 2004, 865–871 (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., Chen, W.H., et al.: Generalization of the Ensemble Kalman Filter using kernels for nongaussian random fields. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers (2009)
Sebacher, B., Stordal, A., Hanea, R.: Bridging multi point statistics and truncated gaussian fields for improved estimation of channelized reservoirs with ensemble methods. Comput. Geosci. 19(2), 341–369 (2015). https://doi.org/10.1007/s10596-014-9466-3
Sebacher, B., Stordal, A., Hanea, R.: Complex geology estimation using the iterative adaptive Gaussian mixture filter. Comput. Geosci. 20(1), 133–148 (2016). https://doi.org/10.1007/s10596-015-9553-0
Strebelle, S.: Conditional simulation of complex geological structures using multiple-point statistics. Math. Geol. 34(1), 1–21 (2002)
Tahmasebi, P., Sahimi, M., Shirangi, M.: Rapid learning-based and geologically consistent history matching. Transp. Porous Media 122(2), 279–304 (2018)
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). https://doi.org/10.1007/s10596-010-9200-8
Tene, M.: Ensemble-Based History Matching for Channelized Petroleum Reservoirs. MSc Thesis Applied Mathematics, Delft University of Technology (2013)
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 (Oct 2014). https://doi.org/10.1007/s11004-014-9541-2
Vo, H.X., Durlofsky, L.J.: Regularized kernel PCA for the efficient parameterization of complex geological models. J. Comput. Phys. 322, 859–881 (2016)
Zhang, Y., Oliver, D., Chen, Y., Skaug, H.: Data assimilation by use of the iterative ensemble smoother for 2D facies models. SPE J. 20(1), 169–185 (2015)
Zhao, Y., Forouzanfar, F., Reynold, A.C.: History matching of multi-facies channelized reservoirs using ES-MDA with common basis DCT. Comput. Geosci. 1–22. November 2016. https://doi.org/10.1007/s10596-016-9604-1 (2016)
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Appendix: Truncation analysis of the HOSVD of tensor
Appendix: Truncation analysis of the HOSVD of tensor
The reason for truncation of the decomposition from Eq. 5 is to obtain an approximation for the tensor but, with a small loss of information. This approximation generates an approximation for each layer (third mode) of the tensor and consequently, an approximation of the facies fields. Because of the nature of the HOSVD procedure, the relevant information is contained in the superior layers of the core tensor σ, so for a good tensor approximation are necessary only small values for nx and ny. However, in our geostatistical context, we have to be very careful when choosing these parameters because we have to take into account the geological meaning of the new facies fields generated with the parameter fields obtained after truncation (\(\overline {\theta _{r}}\)). Consequently, in order to set values for nx and ny, a sensitivity analysis is necessary and we verify the behavior of the transformed facies fields for all possible values of nx and ny. In Fig. 27 is shown the mean difference in percents between the original facies fields and the facies fields obtained after the truncation of the parameter \(\overline {\theta _{r}}\) (\(r\in \overline {1,120})\), for all values of parameters \(n_{x}, n_{y}\in \overline {1,35}\). The red curve in the middle of the picture represents the pairs (nx,ny) for which the average, from the ensemble of 120 members, is the interval [2.95%, 3.05%]. The values nx = 30,ny = 15 are those values that minimize the product nx ⋅ ny and yield the minimum coefficients in the truncation. From this picture one can observe that the tensor truncation is only a bit more sensitive in the Ox direction than in the Oy direction (the matrix g associated with the picture has the property g(i, j) ≤ g(j, i) for j ≤ i). In Fig. 28 are shown the differences in percents between the original facies fields and transformed facies fields (for nx = 30 and ny = 15) for all ensemble members, and we can see here that the bounds are [2.5%, 3.6%], so we have a small variance. In Fig. 29 is shown the third ensemble member in four situations: the original field (a) (simulated from the training image) and three facies fields obtained after truncation of the parameter field \(\overline {\theta _{3}}\) for the cases nx = 30,ny = 15 (b), nx = 15,ny = 30 (c), and nx = 15,ny = 15 (c). The main difference between them consists of the smaller width of the channel for the transformed facies fields compared with the original (the lower width is obtained for small values of nx and ny). The conclusion is that with the truncated HOVSD we are able to control better the channel continuity than with the truncated SVD where the channel continuity is broken (see Fig. 6).
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Sebacher, B., Hanea, R. Channelized reservoir estimation using a low-dimensional parameterization based on high-order singular value decomposition. Comput Geosci 24, 509–531 (2020). https://doi.org/10.1007/s10596-019-09856-1
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DOI: https://doi.org/10.1007/s10596-019-09856-1