Machine-learning-based modeling of coarse-scale error, with application to uncertainty quantification
- 243 Downloads
The use of upscaled models is attractive in many-query applications that require a large number of simulation runs, such as uncertainty quantification and optimization. Highly coarsened models often display error in output quantities of interest, e.g., phase production and injection rates, so the direct use of these results for quantitative evaluations and decision making may not be appropriate. In this work, we introduce a machine-learning-based post-processing framework for modeling the error in coarse-model results in the context of uncertainty quantification. Coarse-scale models are constructed using an accurate global single-phase transmissibility upscaling procedure. The framework entails the use of high-dimensional regression (random forest in this work) to model error based on a number of error indicators or features. Many of these features are derived from approximations of the subgrid effects neglected in the coarse-scale saturation equation. These features are identified through volume averaging, and they are generated by solving a fine-scale saturation equation with a constant-in-time velocity field. Our approach eliminates the need for the user to hand-design a small number of informative (relevant) features. The training step requires the simulation of some number of fine and coarse models (in this work we perform either 10 or 30 training simulations), followed by construction of a regression model for each well. Classification is also applied for production wells. The methodology then provides a correction at each time step, and for each well, in the phase production and injection rates. Results are presented for two- and three-dimensional oil–water systems. The corrected coarse-scale solutions show significantly better accuracy than the uncorrected solutions, both in terms of realization-by-realization predictions for oil and water production rates, and for statistical quantities important for uncertainty quantification, such as P10, P50, and P90 predictions.
KeywordsUpscaling Machine learning Uncertainty quantification Reservoir simulation Classification Error modeling Random forest Surrogate model
Unable to display preview. Download preview PDF.
We thank the sponsors of the Stanford Smart Fields Consortium for financial support. We are grateful to Hai X. Vo for providing the channelized geological models used in this study, and to Olav Møyner for his support with MRST. We also thank Stanford’s Center for Computational Earth & Environmental Science for providing computational resources.
- 3.Bakay, A., Demyanov, V., Arnold, D.: Uncertainty quantification in fractured reservoirs based on outcrop modelling from northeast Brazil. In: 7th EAGE international conference and exhibition (2016)Google Scholar
- 4.Bardy, G., Biver, P.: Sorting reservoir models according to flow criteria: A methodology, using fast marching methods and multi-dimensional scaling. In: Mathematics of Planet Earth: Proceedings of the 15th Annual Conference of the International Association for Math. Geosci., pp. 643–646. Springer. https://doi.org/10.1007/978-3-642-32408-6_140 (2014)
- 6.Breiman, L., Cutler, A., Liaw, A., Wiener, M.: Package random forest version 4.6–12. https://cran.r-project.org/web/packages/randomForest/randomForest.pdf (2015)
- 14.Durlofsky, L.J.: Upscaling and gridding of fine scale geological models for flow simulation. In: 8th International Forum on Reservoir Simulation (2005)Google Scholar
- 15.Durlofsky, L.J., Chen, Y.: Uncertainty quantification for subsurface flow problems using coarse-scale models. In: Numerical Analysis of Multiscale Problems, pp. 163–202. Springer (2012)Google Scholar
- 16.Efendiev, Y., Datta-Gupta, A., Ma, X., Mallick, B.: Efficient sampling techniques for uncertainty quantification in history matching using nonlinear error models and ensemble level upscaling techniques. Water Resources Research 45(11) (2009)Google Scholar
- 21.Guyon, I., Elisseeff, A.: An introduction to variable and feature selection. J. Mach. Learn. Res. 3, 1157–1182 (2003)Google Scholar
- 22.Hastie, T., Tibshirani, R., Friedman, J., Franklin, J.: The elements of statistical learning: Data mining, Inference, and prediction, vol. 27. Springer-Verlag, New York (2005)Google Scholar
- 26.Khodabakhshi, M., Jafarpour, B., King, M.J.: Field applications of a multi-scale multi-physics history matching approach. In: SPE Reservoir Simulation Symposium, SPE 173239-MS (2015)Google Scholar
- 28.Krogstad, S., Lie, K.A., Møyner, O., Nilsen, H.M., Raynaud, X., Skaflestad, B.: MRST-AD–An open-source framework for rapid prototyping and evaluation of reservoir simulation problems. In: SPE Reservoir Simulation Symposium, SPE 173317-MS (2015)Google Scholar
- 33.Lødøen, O.P.: Bayesian calibration of reservoir models using a coarse-scale reservoir simulator in the prior specification. In: EAGE Conference on Petroleum Geostatistics (2007)Google Scholar
- 35.Lødøen, O.P., Omre, H., Durlofsky, L.J., Chen, Y.: Assessment of uncertainty in reservoir production forecasts using upscaled flow models. In: Geostatistics Banff, pp. 713–722. Springer (2005)Google Scholar
- 38.Ng, L.W.T., Eldred, M.: Multifidelity uncertainty quantification using non-intrusive polynomial chaos and stochastic collocation. In: 14th AIAA Non-Deterministic Approaches Conference, vol. 43 (2012)Google Scholar
- 42.Salehi, A., Voskov, D., Tchelepi, H.: Thermodynamically consistent transport coefficients for upscaling of compositional processes. In: SPE Reservoir Simulation Symposium, SPE 163576-MS (2013)Google Scholar
- 48.Shook, G.M., Mitchell, K.M.: A robust measure of heterogeneity for ranking earth models: The F-PHI curve and dynamic Lorenz coefficient. In: SPE Annual Technical Conference and Exhibition, SPE 124625-MS (2009)Google Scholar
- 50.Suzuki, S., Caers, J.K.: History matching with an uncertain geological scenario. In: SPE Annual Technical Conference and Exhibition, SPE 102154-MS (2006)Google Scholar
- 51.Trehan, S.: Surrogate modeling for subsurface flow: A new reduced-order model and error estimation procedures. Ph.D. thesis, Stanford University (2016)Google Scholar