Abstract
Hydraulic fracturing processes are surrounded by uncertainty, as available data is typically scant. In this work, we present a sampling-based stochastic analysis of the hydraulic fracturing process by considering various system parameters to be random. Our analysis is based on the Perkins-Kern-Nordgren (PKN) model for hydraulic fracturing. This baseline model enables computation of high fidelity solutions, which avoids pollution of our stochastic results by inaccuracies in the deterministic solution procedure. In order to obtain the desired degree of accuracy of the computed solution, we supplement the employed time-dependent moving-mesh finite element method with two new enhancements: (i) global conservation of volume is enforced through a Lagrange multiplier; (ii) the weakly singular behavior of the solution at the fracture tip is resolved by supplementing the solution space with a tip enrichment function. This tip enrichment function enables the computation of the tip speed directly from its associated solution coefficient. A novel incremental-iterative solution procedure based on a backward-Euler time-integrator with sub-iterations is employed to solve the PKN model. Direct Monte-Carlo sampling is performed based on random variable and random field input parameters. The presented stochastic results quantify the dependence of the fracture evolution process—in particular the fracture length and fracture opening—on variations in the elastic properties and leak-off coefficient of the formation, and the height of the fracture.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adachi, J., Detournay, E.: Self-similar solution of a plane-strain fracture driven by a power-law fluid. Int. J. Numer. Anal. Methods Geomech. 26(6), 579–604 (2002)
Adachi, J., Siebrits, E., Peirce, A., Desroches, J.: Computer simulation of hydraulic fractures. Int. J. Rock Mech. Mining Sci. 44(5), 739–757 (2007)
Adachi, J.I., Peirce, A.P.: Asymptotic analysis of an elasticity equation for a finger-like hydraulic fracture. J. Elast. 90(1), 43–69 (2008)
Advani, S., Lee, J.: Finite element model simulations associated with hydraulic fracturing. Society of Petroleum Engineers Journal (1982)
Barker, E., Barker, W., Burr, W., Polk, W., Smid, M.: Recommendation for key management part 1: General (revision 3). NIST Spec. Publ. 800(57), 1–147 (2012)
Batchelor, G.K.: An Introduction to Fluid Dynamics. Cambridge University Press (2000)
Detournay, E.: Mechanics of hydraulic fractures. Annual Review of Fluid Mechanics (2016)
England, A.H., Green, A.E.: Some two-dimensional punch and crack problems in classical elasticity. Math. Proc. Camb. Philos. Soc. 59(2), 489–500 (1963)
Ganis, B., Mear, M.E., Sakhaee-Pour, A., Wheeler, M.F., Wick, T.: Modeling fluid injection in fractures with a reservoir simulator coupled to a boundary element method. Comput. Geosci. 18(5), 613–624 (2014)
Garagash, D., Detournay, E.: The tip region of a fluid-driven fracture in an elastic medium. Trans.-Amer. Soc. Mech. Eng. J. Appl. Mech. 67(1), 183–192 (2000)
Geertsma, J., De Klerk, F., et al.: A rapid method of predicting width and extent of hydraulically induced fractures. J. Petrol. Tech. 21(12), 1–571 (1969)
Gordeliy, E., Peirce, A.: Coupling schemes for modeling hydraulic fracture propagation using the xfem. Comput. Methods Appl. Mech. Eng. 253, 305–322 (2013)
Gutiérrez, M., Krenk, S.: Stochastic Finite Element Methods, chap. 20, pp 1–25. Wiley, New York (2017)
Hanson, M.E., Shaffer, R.J., Anderson, G.D., et al.: Effects of various parameters on hydraulic fracturing geometry. Soc. Pet. Eng. J. 21(04), 435–443 (1981)
Howard, G.C., Fast, C.R.: API-57-261, chap. Optimum Fluid Characteristics for Fracture Extension. American Petroleum Institute, New York (1957)
Kemp, L.: Study of Nordgren’s Equation of Hydraulic Fracturing. SPE Production Engineering (1990)
Kenney, J.F.: Mathematics of Statistics. D Van Nostrand Company Inc, Toronto (2013). Affiliated East-West Press Pvt-Ltd; New Delhi
Kiureghian, A.D., Liu, P.: Structural reliability under incomplete probability information. J. Eng. Mech. 112(1), 85–104 (1986)
Kovalyshen, Y., Detournay, E.: A reexamination of the classical PKN model of hydraulic fracture. Transp. Porous Media 81, 317–339 (2010)
Lecampion, B.: An extended finite element method for hydraulic fracture problems. Commun. Numer. Methods Eng. 25(2), 121–133 (2009)
Lowengrub, M.: A note on Griffith cracks. Proc. Edinb. Math. Soc. 15(2), 131–134 (1966)
Meakin, P., Li, G., Sander, L., Louis, E., Guinea, F.: A simple two-dimensional model for crack propagation. J. Phys. A Math. Gen. 22(9), 1393 (1989)
Melenk, J.M., Babuška, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139(1-4), 289–314 (1996)
Miehe, C., Mauthe, S.: Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media. Comput. Methods Appl. Mech. Eng. 304, 619–655 (2016)
Mikeliċ, A., Wheeler, M., Wick, T.: Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput. Geosci. 19(6), 1171–1195 (2015)
Munjiza, A., Owen, D., Bicanic, N.: A combined finite-discrete element method in transient dynamics of fracturing solids. Eng. Comput. 12(2), 145–174 (1995)
Neyman, J.: Outline of a theory of statistical estimation based on the classical theory of probability. Philos. Trans. R. Soc. London. Series A Math. Phys. Sci. 236(767), 333–380 (1937)
Ng, A.K., Small, J.C.: A case study of hydraulic fracturing using finite element methods. Can. Geotech. J. 36(5), 861–875 (1999)
Nordgren, R.: Propagation of vertical hydraulic fractures. Society of Petroleum Engineers Journal (1972)
Peirce, A., Siebrits, E.: An Eulerian Finite Volume Method for Hydraulic Fracture Problems. Finite Volumes for Complex Applications IV, ISTE. London, pp. 655–664 (2005)
Perkins, T.K., Kern, L.R.: Widths of hydraulic fractures. Journal of Petroleum Technology (1961)
Remij, E.W., Remmers, J.J.C., Huyghe, J.M., Smeulders, D.M.J.: The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials. Comput. Methods Appl. Mech. Eng. 286, 293–312 (2015)
Remij, E.W., Remmers, J.J.C., Huyghe, J.M., Smeulders, D.M.J.: On the numerical simulation of crack interaction in hydraulic fracturing. Comput. Geosci. 22(1), 423–437 (2018)
Settari, A., Cleary, M.P., et al.: Development and testing of a pseudo-three-dimensional model of hydraulic fracture geometry. SPE Prod. Eng. 1(06), 449–466 (1986)
Simonson, E., Abou-Sayed, A., Clifton, R., et al.: Containment of massive hydraulic fractures. Soc. Pet. Eng. J. 18(01), 27–32 (1978)
Sneddon, I.N.: The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc. R. Soc. London A: Math. Phys. Eng. Sci. 187(1009), 229–260 (1946)
Sneddon, I.N.: A note on the problem of the penny-shaped crack. Math. Proc. Camb. Philos. Soc. 61(2), 609–611 (1965)
Spanos, P.D., Ghanem, R.: Stochastic finite element expansion for random media. J. Eng. Mech. 115(5), 1035–1053 (1989)
Valko, P., Economides, M.: Hydraulic Fracture Mechanics. Wiley (1995)
Wangen, M.: Finite element modeling of hydraulic fracturing on a reservoir scale in 2d. J. Pet. Sci. Eng. 77 (3), 274–285 (2011)
Warpinski, N., Abou-Sayed, I., Moschovidis, Z., Parker, C.: Hydraulic fracture model comparison study: complete results. Tech. rep., Sandia National Labs., Albuquerque, NM (United States); Gas Research Inst., Chicago, IL (United States (1993)
Wheeler, M.F., Wick, T., Wollner, W.: An augmented-Lagrangian method for the phase-field approach for pressurized fractures. Comput. Methods Appl. Mech. Eng. 271, 69–85 (2014)
Wilson, Z.A., Landis, C.M.: Phase-field modeling of hydraulic fracture. J. Mech. Phys. Solids 96, 264–290 (2016)
Youn, D., Griffiths, D.: Stochastic analysis of hydraulic fracture propagation using the extended finite element method and random field theory. In: Integrating Innovations of Rock Mechanics: Proceedings of the 8th South American Congress on Rock Mechanics, 15–18 November 2015, p 189. IOS Press, Buenos Aires (2015)
Zhang, G., Liu, H., Zhang, J., Wu, H., Wang, X.: Three-dimensional finite element simulation and parametric study for horizontal well hydraulic fracture. J. Pet. Sci. Eng. 72(3), 310–317 (2010)
Zhao, H., Li, Z., Zhu, C., Ru, Z.: Reliability analysis models for hydraulic fracturing. J. Pet. Sci. Eng. 162, 150–157 (2018)
Zheltov, A.K., et al.: 3. Formation of vertical fractures by means of highly viscous liquid. In: 4th World Petroleum Congress. World Petroleum Congress (1955)
Zimmerman, R., In-Wook, Y.: Fluid Flow in Rock Fractures: From the Navier-Stokes Equations to the Cubic Law, pp. 213–224. American Geophysical Union (AGU) (2013)
Acknowledgements
We acknowledge the support from the European Commission EACEA Agency, Framework Partnership Agreement Erasmus Mundus Action 1b, as a part of the EM Joint Doctorate Simulation in Engineering and Entrepreneurship Development (SEED). The work of S. Zlotnik and P. Díez was funded by the project DPI2017-85139-C2-2-R of the Spanish Ministry and by grant 2017-SGR-1278 from the Generalitat de Catalunya.
Funding
The work of S. Zlotnik and P. Díez was funded by the project DPI2017-85139-C2-2-R of the Spanish Ministry and by grant 2017-SGR-1278 from the Generalitat de Catalunya.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Benchmark results
Appendix: Benchmark results
In Section 5.1, we have considered the deterministic benchmark result based on the case study by Warpinski et al. [41]. The parameters for the considered simulation can be found in Table 1. Figure 5 in Section 5.1 shows the results for this benchmark case as obtained using the finite element technique developed in this manuscript, as well as the results for the simulators included in Ref. [41]. Note that the results of these simulators have been reported with intervals of 1200 s. For completeness, in Table 4, we report the results obtained by the method proposed herein with a mesh size of Δx = 1 m and a time step size of Δt = 1 s. Note that the presented results have been rounded off to 4 decimals.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Garikapati, H., Verhoosel, C.V., van Brummelen, E.H. et al. Sampling-based stochastic analysis of the PKN model for hydraulic fracturing. Comput Geosci 23, 81–105 (2019). https://doi.org/10.1007/s10596-018-9784-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-018-9784-y