Proxy modelling for multi-well simulations: enabling identification of major input variables and reduced computation time over Monte Carlo sampling
Thomas A. McCourt A B , Ryan Blackmore C , Iain Rodger D , Suzanne Hurter D , Bevan Thompson B , Mark Reilly E and Diane Donovan B FA Craigslea State High School, West Chermside, Qld 4032, Australia.
B School of Mathematics and Physics, The University of Queensland, Brisbane, Qld 4072, Australia.
C Cyber and Electronic Warfare Division/Electronic Warfare Operations/Radio Frequency Electronic Attack Group, Defence Science and Technology Group, Edinburgh, SA 5111, Australia.
D Centre for Coal Seam Gas, Faculty of Engineering, Architecture and Information Technology, University of Queensland, Brisbane, Qld 4072, Australia.
E School of Earth and Environmental Sciences, The University of Queensland, Brisbane, Qld 4072, Australia.
F Corresponding author. Email: dmd@maths.uq.edu.au
The APPEA Journal 59(1) 444-456 https://doi.org/10.1071/AJ18065
Submitted: 7 December 2018 Accepted: 8 January 2019 Published: 17 June 2019
Abstract
The petroleum industry uses high level dynamic simulations applied to geocellular models to guide forecasts of oil, gas and water production. Uncertainty in model choice and input variable selection is often addressed through large numbers of computationally slow Monte Carlo simulations designed around physics based models. Here, an alternate approach is proposed, which uses a relatively small amount of data and a reduced number of simulations of the high level physics model to train a fast (to evaluate) proxy or surrogate model based on a Polynomial Chaos Expansion. We give details of the theory and incorporated techniques, which significantly increase flexibility. Input variables (e.g. cell-by-cell variations in porosity and permeability) are sampled from unknown probability distributions and sensitivity analysis is based on low level proxy models. The theory is tested by developing proxy models to predict total gas production from a five-spot well configuration in the Hermitage area that taps into the Walloon Coal Measures of the Surat Basin in Queensland. Synthetic training data is simulated using commercial dynamic simulation software based on a high level physics model.
Keywords: dynamic modelling, polynomial chaos expansions, proxy models, static geological model.
Thomas McCourt is a mathematician whose research covers areas of applied analysis, modelling and simulation, metric geometry, topological graph theory, algebraic structures, graph decompositions and experimental designs. He holds a Doctorate in mathematics from the University of Queensland (UQ), a Graduate Diploma in Education from Queensland University of Technology, a Bachelor of Science (Honours) from UQ and a Post Graduate Certificate in Academic Practice from Plymouth University. Thomas is a mathematics teacher at Craigslea State High School, Brisbane, Australia. He has held the position of Lecturer in Pure Mathematics at Plymouth University, UK; a postdoctoral research position at the University of Bristol, UK; postdoctoral, teaching and research positions at UQ, Australia; and an instructor position at Auburn University, USA. Thomas has published in a range of highly respected international journals, and he has been invited to present at conferences, seminar series and workshops in Australia, America, Europe and Asia. |
Ryan Blackmore is currently working for the Cyber and Electronic Warfare Division in the Defence Science and Technology Group. His work has involved operating and maintaining radar simulators, participating in flight trails, trial preparation involving radio frequency system acceptance testing and conducting post-trial scientific analyses. Ryan graduated from UQ in July 2017 with a Bachelor of Engineering (Honours) where he majored in Electrical Engineering, specialising in radar and communication systems. Ryan graduated from UQ in December 2012 with a Bachelor of Science (Extended Major in Mathematics). |
Iain Rodger is a Petroleum Engineer currently working for the Centre for Coal Seam Gas (CCSG) at UQ. His work is focused on reservoir simulation; in particular, his research is related to unconventionals and Carbon Capture and Storage projects. Iain graduated from the University of Edinburgh with a BSc (Honours) in Chemistry with Environmental Chemistry, before completing an MSc in Petroleum Engineering at UQ. |
Suzanne Hurter joined UQ in 2015. Before that she held various roles in the oil and gas industry, working with Shell (Netherlands), Schlumberger (Netherlands and Australia), QGC (BG-Group) and Arrow Energy (Shell-PetroChina). She has also worked in academia at the Helmholtz Centre for Geosciences in Potsdam, the Leibnitz Institute of Applied Geophysics in Hannover (Germany) and at the University of Sao Paulo (Brazil). Suzanne's research interests and activities include hydrocarbon maturation and thermal evolution of sedimentary basins, carbon sequestration, coupled modelling of flow, heat in porous and fractured media and using numerical modelling to evaluate and improve onshore gas production. In 2017, the CCSG at UQ was awarded the Canadian Foundation CMG inaugural Industrial Research Chair in Onshore Gas Reservoir Modelling, and Suzanne was appointed to this chair. |
Bevan Thompson is an Honorary Associate Professor in UQ’s School of Mathematics and Physics. He has worked with industry partners Tarong Energy, Queensland Gas Corporation, Santos, APLNG, Arrow Energy and Origin Energy. He has attracted research funding from Tarong Energy and the Australian Research Council. Recent collaborations with Diane Donovan and members of the CCSG (UQ) have focused on an exploration of polynomial chaos proxy models for applications in the gas industry, particularly gas extraction. He is a Fellow of the Australian Mathematical Society and has over 130 publications. His research interests range from topological algebra, financial mathematics, nonlinear analysis, differential equations, approximation theory and Gaussian processes. |
Mark Reilly is an experienced Petroleum Geologist, having undertaken projects on the North West shelf, the Bowen/Surat Basin, the Georgina Basin and in Libya, amongst other destinations. He has extensive experience in field geology and geomorphology and is particularly familiar with Central Queensland, the Flinders Ranges and Lake Eyre. Mark holds a Bachelor of Applied Science in Geology from the Queensland University of Technology and a Bachelor of Science (Honours) from the National Centre for Petroleum Geology and Geophysics at the University of Adelaide. Mark leads industry field geology training courses and is a staff member at the CCSG at UQ. Mark has published in AAPG special publication, EABS journal and the APPEA journal, and he has presented at many local and international conferences. |
Diane Donovan is a Professor in the School of Mathematics and Physics at UQ. Professor Donovan has worked in discrete mathematics with applications for over 30 years. She is a Foundation Fellow of the Institute of Combinatorics and its Applications, a Fellow of the Australian Mathematical Society and a Life Member of the Combinatorial Mathematical Society of Australasia. She has worked with industry partners Tarong Energy, Queensland Gas Corporation, Santos, APLNG, Arrow Energy and Origin Energy and has been awarded over $2 million in research funding from industry partners, the Australian Research Council and the Australian Teaching and Learning Council. Recent collaborations include research with the CCSG (UQ) that focuses on an exploration of polynomial chaos proxy models for applications in the gas industry, particularly gas extraction. Other recently funded projects include a study of the theoretical aspects of discrete choice experiments, used in the design of marketing research. She has over 100 publications in discrete mathematics theory and its applications, including experimental design and cryptography. |
References
Alkhatib, A., and King, P. (2014). Robust quantification of parametric uncertainty for surfactant-polymer ooding. Computers & Geosciences 18, 77–101.| Robust quantification of parametric uncertainty for surfactant-polymer ooding.Crossref | GoogleScholarGoogle Scholar |
Aminian, K., and Ameri, S. (2009). Predicting production performance of CBM reservoirs. Journal of Natural Gas Science and Engineering 1, 25–30.
| Predicting production performance of CBM reservoirs.Crossref | GoogleScholarGoogle Scholar |
Babaei, M., Pan, I., and Alkhatib, A. (2015a). Robust optimization of subsurface flow using polynomial chaos and response surface surrogates. Computers & Geosciences 19, 979–998.
| Robust optimization of subsurface flow using polynomial chaos and response surface surrogates.Crossref | GoogleScholarGoogle Scholar |
Babaei, M., Pan, I., and Alkhatib, A. (2015b). Robust optimization of well location to enhance hysteretical trapping of CO2: Assessment of various uncertainty and quantification methods and utilization of mixed response surface surrogates. Water Resources Research 51, 9402–9424.
| Robust optimization of well location to enhance hysteretical trapping of CO2: Assessment of various uncertainty and quantification methods and utilization of mixed response surface surrogates.Crossref | GoogleScholarGoogle Scholar |
Bhavsar, A. B. (2005). Prediction of coalbed methane reservoir performance with type curves. PhD thesis, Morgantown WV West Virginia University.
Blatman, G., and Sudret, B. (2011). Adaptive sparse polynomial chaos expansion based on least angle regression. Journal of Computational Physics 230, 2345–2367.
| Adaptive sparse polynomial chaos expansion based on least angle regression.Crossref | GoogleScholarGoogle Scholar |
Collins, P. W., and Badessich, M. F. (2015). Addressing forecasting non-uniqueness and uncertainty in unconventional reservoir systems using experimental design. (Society of Petroleum Engineers.) https://doi.org/10.2118/175139-MS
Donovan, D., McCourt, T. A., Hurter, S., Thompson, B., and Blackmore, R. (2018). Uncertainty modelling with polynomial chaos expansions: Final Report. Centre for Coal Seam Gas Report, UQ.
Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004). Least angle regression. Annals of Statistics 32, 407–499.
| Least angle regression.Crossref | GoogleScholarGoogle Scholar |
Eldred, M., Webster, C. G., and Constantine, P. G. (2008). Evaluation of non-intrusive approaches for Wiener-Askey generalized polynomial chaos. In “49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Schaumburg, IL, 07−10 April 2008”. (American Institute of Aeronautics and Astronautics Inc, Reston VA)
Elsheikh, A. H., Hoteit, I., and Wheeler, M. F. (2014). Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates. Computer Methods in Applied Mechanics and Engineering 269, 515–537.
| Efficient Bayesian inference of subsurface flow models using nested sampling and sparse polynomial chaos surrogates.Crossref | GoogleScholarGoogle Scholar |
Fajraoui, N., Ramasomanana, F., Younes, A., Mara, T. A., Ackerer, P., and Guadagnini, A. (2011). Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory-scale porous media. Water Resources Research 47, 1–14.
| Use of global sensitivity analysis and polynomial chaos expansion for interpretation of nonreactive transport experiments in laboratory-scale porous media.Crossref | GoogleScholarGoogle Scholar |
Gray, I. (1987). Reservoir engineering in coal seams: part 1 − the physical process of gas storage and movement in coal seams. SPE Reservoir Engineering 2, 28–34.
| Reservoir engineering in coal seams: part 1 − the physical process of gas storage and movement in coal seams.Crossref | GoogleScholarGoogle Scholar |
Jansen, J., Bosgra, O. H., and Van den Hof, P. M. J. (2008). Model-based control of multiphase flow in subsurface oil reservoirs. Journal of Process Control 18, 846–855.
| Model-based control of multiphase flow in subsurface oil reservoirs.Crossref | GoogleScholarGoogle Scholar |
Keim, S. (2011). Optimization of coalbed methane completion strategies, selection criteria and production prediction: a case study in China’s Qinshui Basin. PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
Li, H., Sarma, P., and Zhang, D. (2011). A comparative study of the probabilistic-collocation and experimental design methods for petroleum-reservoir uncertainty quantification. SPE Journal 16, 429–439.
| A comparative study of the probabilistic-collocation and experimental design methods for petroleum-reservoir uncertainty quantification.Crossref | GoogleScholarGoogle Scholar |
McCourt, T. A., Hurter, S., Lawson, B., Zhou, F., Thompson, B., Tyson, S., and Donovan, D. (2017). Uncertainty quantification of coal seam gas production prediction using polynomial chaos. Journal of Petroleum Science Engineering 157, 1148–1159.
| Uncertainty quantification of coal seam gas production prediction using polynomial chaos.Crossref | GoogleScholarGoogle Scholar |
Montgomery, D. C. (2007). ‘Design and analysis of experiments. Seventh edition’. (John Wiley & Sons: New York).
Oladyshkin, S., and Nowak, W. (2012). Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion. Reliability Engineering & System Safety 106, 179–190.
| Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion.Crossref | GoogleScholarGoogle Scholar |
Oladyshkin, S., Class, H., Helmig, R., and Nowak, W. (2011). A concept for data-driven uncertainty quantification and its application to carbon dioxide storage in geological formations. Advances in Water Resources 34, 1508–1518.
| A concept for data-driven uncertainty quantification and its application to carbon dioxide storage in geological formations.Crossref | GoogleScholarGoogle Scholar |
Sarma, P., and Xie, J. (2011). Efficient and robust uncertainty quantification in reservoir simulation with polynomial chaos expansions and non-intrusive spectral projection. (Society of Petroleum Engineers.) https://doi.org/10.2118/141963-MS
Schlumberger (2014). Eclipse version 2014.2 technical description.
Scott, S. G. (2008). The geology, stratigraphy and coal seam gas characteristics of the Walloon subgroup- northeastern Surat Basin. PhD thesis, James Cook University, QLD.
Seidle, J. P. (2011). Fundamentals of coalbed methane reservoir engineering. PennWell. URL http://app.knovel.com/hotlink/toc/id:kpFCMRE001/fundamentals-coalbed
Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansion. Reliability Engineering & System Safety 93, 964–979.
| Global sensitivity analysis using polynomial chaos expansion.Crossref | GoogleScholarGoogle Scholar |
Tyson, S., Donovan, D., Thompson, B., Lynch, S., and Tas, M. (2015). Uncertainty modelling with polynomial chaos expansions: Stage 1 − Final Report. The University of Queensland, QLD.
Tyson, S., Donovan, D., Thompson, B., and Lawson, B. (2016). Uncertainty modelling with polynomial chaos expansions: Stage 2 –Report. The University of Queensland, QLD.
Yeten, B., Castellini, A., Guyaguler, B., and Chen, W. H. (2005). A comparison study on experimental design and response surface methodologies. (Society of Petroleum Engineers.) https://doi.org/10.2118/93347-MS
Zhou, F. (2014). A study on predicting coalbed methane production depending on reservoir properties. Geosystem Engineering 17, 89–94.
| A study on predicting coalbed methane production depending on reservoir properties.Crossref | GoogleScholarGoogle Scholar |
