Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Uncertainty Quantification
IF: 4.911 5-Year IF: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2020033602
pages 375-398

DATA-DRIVEN CALIBRATION OF P3D HYDRAULIC FRACTURING MODELS

Souleymane Zio
Mechanical Engineering, COPPE, Federal University of Rio de Janeiro, P.O. Box 68503, 21941972, Rio de Janeiro, Brazil; 1Institut du Génie des Systèmes Industriels et Textiles, École Polytechnique de Ouagadougou, 18 BP 234 Ouagadougou, Burkina Faso, West Africa
Fernando A. Rochinha
Mechanical Engineering Department, Federal University of Rio de Janeiro, RJ 21945-970, Brazil

ABSTRACT

Modeling the propagation of a hydraulic fracture is challenging due to the complex nonlinear equations and the presence of multiple scales behavior at the fracture tips. These complexities have motivated researchers to propose simplified models relying on constrained fracture geometry patterns. Amongst them, in the oil and gas industry domain, the Pseudo-3D (P3D), that computes fracture evolution in reservoirs confined by symmetric stress barriers, is frequently employed. The different assumptions made to obtain the P3D model lead to inaccurate predictions, like the overestimation of fracture height for important operating conditions. To correct such drawbacks, we propose a model discrepancy term in a multi-fidelity approach. The main difficulty associated is to construct a mapping between the inputs and outputs that faithfully represents the error between high and low fidelity models. In this paper, we investigated the efficiency modeling such discrepancy by using Gaussian processes and artificial neural networks. In this analysis, we employ three data sets with different input-output dimensions. The best model obtained is then used for carrying out an Uncertainty Quantification analysis aiming at identifying the impact of parametric uncertainty in fracture propagation computation.

REFERENCES

  1. Khristianovic, S.A. and Zheltov, Y.P., Formation of Vertical Fractures by Means of Highly Viscous Fluids, Proc. 4th World Petrol. Cong., 2:579-586, 1955.

  2. Perkins, T. and Kern, L., Widths of Hydraulic Fractures, J. Petrol. Tech. 13(9):937-949, 1961.

  3. Nordgren, R.P., Propagation of a Vertical Hydraulic Fracture, Soc. Petrol. Eng. J., 12(4):SPE-3009-PA, 1972.

  4. Settari, A., Development and Testing of a Pseudo-Three-Dimensional Model of Hydraulic Fracture Geometry (P3DH), Proc. of the 6th SPE Symposium on Reservoir Simulation of the Society of Petroleum Engineers, SPE-10505, pp. 185-214, 1986.

  5. Gladkov, I. and Linkov, A., Khristianovich-Geertsma-De Klerk Problem with Stress Contrast, Fluid Dynam., arXiv:1703.05686, 2017.

  6. Dontsov, E. and Peirce, A., An Enhanced Pseudo-3D Model for Hydraulic Fracturing Accounting for Viscous Height Growth, Non-Local Elasticity, and Lateral Toughness, Eng. Fracture Mech., 142:116-139, 2015.

  7. Zhang, X., Wu, B., Jeffrey, R.G., Connell, L.D., and Zhang, G., A Pseudo-3D Model for Hydraulic Fracture Growth in a Layered Rock, Int. J. Solids Struct., 115-116:208-223, 2017.

  8. Bochkarev, A., Budennyy, S., Nikitin, R., and Mitrushkin, D., Pseudo-3D Hydraulic Fracture Model with Complex Mechanism of Proppant Transport and Tip Screen Out, ECMOR XV-15th European Conf. on the Mathematics of Oil Recovery, pp. cp-494-00037, 2016.

  9. Dontsov, E. and Peirce, A., Comparison of Toughness Propagation Criteria for Blade-Like and Pseudo-3D Hydraulic Fractures, Eng. Fracture Mech., 160:238-247, 2016.

  10. Peirce, A. and Detourney, E., An Implicit Level Set Method for Modeling Hydraulically Driven Fractures, Comput. Meth. Appl. Mech. Eng., 197:2858-2885, 2008.

  11. Ling, Y., Mullins, J., and Mahadevan, S., Selection Model Discrepancy Priors in Bayesian Calibration, J. Comput. Phys., 276:665-680, 2014.

  12. Arendt, P.D., Apley, D.W., and Chen, W., Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability, J. Mech. Design, 134(10):100908, 2012.

  13. Arendt, P.D., Apley, D.W., Chen, W., Lamb, D., and Gorsich, D., Improving Identifiability in Model Calibration Using Multiple Responses, J. Mech. Design, 134(10):100909, 2012.

  14. Absi, G.N. and Mahadevan, S., Multi-Fidelity Approach to Dynamics Model Calibration, Mech. Syst. Signal Proc., 68-69:189-206, 2016.

  15. Sargsyan, K., Najm, H., and Ghanem, R., On the Statistical Calibration of Physical Models, Int. J. Chem. Kinetics, 47(4):246-76, 2015.

  16. Qian, Z. andWu, C.F.J., Bayesian Surrogate Models for Integrating Low-Accuracy and High-Accuracy Experiments, Technometrics, 50(2):192-204, 2008.

  17. Higdon, D., Kennedy, M., Cavendish, J.C., Cafeo, J.A., and Ryne, R.D., Combining Field Data and Computer Simulations for Calibration and Prediction, SIAM J. Sci. Comput., 26(2):448-466, 2004.

  18. Kennedy, M.C. and O'Hagan, A., Bayesian Calibration of Computer Models, J. R. Stat. Soc., 63(3):425-464, 2001.

  19. Templeton, J.A. and Blaylock, M., Calibration and Forward Uncertainty Propagation for Large-Eddy Simulations of Engineering Flows, Sandia National Laboratories Tech. Rep. SAND2015-7938, 2015.

  20. Oliver, T.A., Terejanu, G., Simmons, C.S., and Moser, R.D., Validating Predictions of Unobserved Quantities, Comput. Methods Appl. Mech. Eng., 283(1):1310-1335, 2015.

  21. Sargsyan, K., Huan, X., and Najm, H.N., Embedded Model Error Representation for Bayesian Model Calibration, Int. J. Uncertainty Quantif., 9(4):365-394, 2019.

  22. Brynjarsdottir, J. and O'Hagan, A., Learning about Physical Parameters: The Importance of Model Discrepancy, Inverse Problems, 30(11):114007, 2014.

  23. Chan, S. and Elsheikh, A.H., A Machine Learning Approach for Efficient Uncertainty Quantification Using Multiscale Methods, Comput. Sci. Mach. Lear., arXiv:1711.04315, 2017.

  24. He, K., Zhang, X., Ren, S., and Sun, J., Deep Residual Learning for Image Recognition, 2016 IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), pp. 770-778, 2015.

  25. Hesthaven, J.S. and Ubbiali, S., Non-Intrusive Reduced Order Modeling of Nonlinear Problems Using Neural Networks, J. Comput. Phys., 363:55-78, 2018.

  26. Tripathy, R.K. and Bilionis, I., Deep UQ: Learning Deep Neural Network Surrogate Models for High Dimensional Uncertainty Quantification, J. Comput. Phys., 375:565-588, 2018.

  27. Neal, K., Hu, Z., Mahadevan, S., and Zumberge, J., Discrepancy Prediction in Dynamical System Models under Untested Input Histories, J. Comput. Nonlin. Dyn., 14(2):021009, 2019.

  28. Subramanian, A. and Mahadevan, S., Bayesian Estimation of Discrepancy in Dynamics Model Prediction, Mech. Syst. Signal Proc., 123:351-368, 2019.

  29. Neal, R.M., Bayesian Learning for Neural Networks, PhD, University of Toronto, 2014.

  30. Zio, S. and Rochinha, F.A., A Stochastic Collocation Approach for Uncertainty Quantification in Hydraulic Fracture Numerical Simulation, Int. J. Uncertainty Quantif., 2(2):145-160, 2012.

  31. Bonfiglio, L., Perdikaris, P., Brizzolara, S., and Karniadakis, G., Multi-Fidelity Optimization of Super-Cavitating Hydrofoils, Comput. Methods Appl. Mech. Eng., 332:63-85, 2018.

  32. Lee, T., Bilionis, I., and Tepole, A.B., Propagation of Uncertainty in the Mechanical and Biological Response of Growing Tissues Using Multi-Fidelity Gaussian Process Regression, Comput. Methods Appl. Mech. Eng., 359:112724, 2020.

  33. Bilionis, I., Zabaras, N., Konomi, B.A., and Lin, G., Multi-Output Separable Gaussian Process: Towards an Efficient, Fully Bayesian Paradigm for Uncertainty Quantification, J. Comput. Phys., 241:212-239, 2013.

  34. Ebrahimi, S., Elhoseiny, M., Darrell, T., and Rohrbach, M., Uncertainty-Guided Continual Learning with Bayesian Neural Networks, Comput. Sci. Mach. Learning, arXiv:1906.02425, 2019.

  35. Rochinha, F.A. and Peirce, A., Monitoring Hydraulic Fractures: State Estimation Using an Extended Kalman Filter, Inv. Problems, 2:025009, 2010.

  36. Peirce, A. and Rochinha, F., An Integrated Extended Kalman Filter-Implicit Level Set Algorithm for Monitoring Planar Hydraulic Fractures, Inv. Problems, 1:015009, 2011.


Articles with similar content:

GRADIENT-BASED STOCHASTIC OPTIMIZATION METHODS IN BAYESIAN EXPERIMENTAL DESIGN
International Journal for Uncertainty Quantification, Vol.4, 2014, issue 6
Youssef Marzouk, Xun Huan
A STOCHASTIC COLLOCATION APPROACH FOR UNCERTAINTY QUANTIFICATION IN HYDRAULIC FRACTURE NUMERICAL SIMULATION
International Journal for Uncertainty Quantification, Vol.2, 2012, issue 2
Souleymane Zio , Fernando A. Rochinha
INTEGRATED DATA ANALYSIS AND MODELING OF A HIGHLY HETEROGENEOUS CARBONATE RESERVOIR
Special Topics & Reviews in Porous Media: An International Journal, Vol.7, 2016, issue 1
Shahab Hejri, Ali Esfandyari Bayat, Mohammad-Reza Rasaei, Nader Ghadami
OPTIMAL SENSOR PLACEMENT FOR THE ESTIMATION OF TURBULENCE MODEL PARAMETERS IN CFD
International Journal for Uncertainty Quantification, Vol.5, 2015, issue 6
Costas Papadimitriou, Dimitrios I. Papadimitriou
USING PARALLEL MARKOV CHAIN MONTE CARLO TO QUANTIFY UNCERTAINTIES IN GEOTHERMAL RESERVOIR CALIBRATION
International Journal for Uncertainty Quantification, Vol.9, 2019, issue 3
M. J. O'Sullivan, G. K. Nicholls, C. Fox, Tiangang Cui