International Journal of Fracture

, Volume 211, Issue 1–2, pp 183–202

# An Elrod–Adams-model-based method to account for the fluid lag in hydraulic fracturing in 2D and 3D

• Mostafa Mollaali
• Yongxing Shen
Original Paper

## Abstract

An efficient method to model the fluid lag in hydraulic fracturing has been developed based on the Elrod–Adams model. The main feature of this method is the absence of the need to explicitly track the free end of the fracturing fluid, but rather, the fluid front is obtained by solving the pressure field (zero for the lag) and an auxiliary field for the entire fracture. An important advantage of this method is that no change of formulation, and hence no contact detection, is needed when the fluid reaches the fracture tip. Moreover, the method works for both the injection phase and the liquid withdrawal phase. Based on the latter case studies can be developed to investigate the quantity of the remaining fluid after the fracturing process in order to assess the environmental impact of fracturing. The method applies to both 2D and 3D problems.

## Keywords

Hydraulic fracturing Elrod–Adams model Fluid lag Lubrication equation

## Notes

### Acknowledgements

The authors would like to acknowledge the helpful discussion with Gustavo Buscaglia from University of São Paulo. This work is supported by the National Natural Science Foundation of China with Grant No. 11402146. YS also acknowledges the financial support by the Young 1000 Talent Program of China.

## References

1. Adachi J, Siebrits E, Peirce A, Desroches J (2007) Computer simulation of hydraulic fractures. Int J Rock Mech Min Sci 44(5):739–757
2. Ausas RF, Jai M, Buscaglia GC (2009) A mass-conserving algorithm for dynamical lubrication problems with cavitation. ASME J Tribol 131(3):031702–031702-7
3. Bayada G, Chambat M (1983) Analysis of a free boundary problem in partial lubrication. Q Appl Math 40(4):369–375
4. Bunger AP, Detournay E, Jeffrey RG (2005) Crack tip behavior in near-surface fluid-driven fracture experiments. C R Mec 333(4):299–304
5. Buscaglia GC, Talibi MEA, Jai M (2015a) Mass-conserving cavitation model for dynamical lubrication problems. Part I: mathematical analysis. Math Comput Simul 118:130–145
6. Buscaglia GC, Talibi MEA, Jai M (2015b) Mass-conserving cavitation model for dynamical lubrication problems. Part II: numerical analysis. Math Comput Simul 118:146–162
7. Chiaramonte MM, Shen Y, Keer LM, Lew AJ (2015) Computing stress intensity factors for curvilinear cracks. Int J Numer Methods Eng 104(4):260–296
8. Elrod HG, Adams ML (1975) A computer program for cavitation and starvation problems. In: Dowson D, Godet M, Taylor CM (eds) Cavitation and related phenomena in lubrication: proceedings of the first Leeds-Lyon symposium on tribology, pp 37–42Google Scholar
9. Garagash DI (2006) Propagation of a plane-strain hydraulic fracture with a fluid lag: early-time solution. Int J Solids Struct 43:5811–5835
10. Garagash D, Detournay E (2000) The tip region of a fluid-driven fracture in an elastic medium. J Appl Mech 67(1):183–192
11. Geertsma J, de Klerk F (1969) A rapid method of predicting width and extent of hydraulically induced fractures. J Petrol Technol 21(12):1571–1581
12. Gordeliy E, Detournay E (2011) A fixed grid algorithm for simulating the propagation of a shallow hydraulic fracture with a fluid lag. Int J Numer Anal Methods Geomech 35(5):602–629
13. Gordeliy E, Peirce A (2013) Coupling schemes for modeling hydraulic fracture propagation using the XFEM. Comput Methods Appl Mech Eng 253:305–322
14. Hunsweck MJ, Shen Y, Lew AJ (2013) A finite element approach to the simulation of hydraulic fractures with lag. Int J Numer Anal Methods Geomech 37(9):993–1015
15. Jakobsson B, Floberg L (1957) The finite journal bearing considering vaporization. Transactions of Chalmers University Technology, Goteborg, Sweden, vol 190Google Scholar
16. Johnson E, Cleary MP (1991) Implications of recent laboratory experimental results for hydraulic fractures. In: Low permeability reservoirs symposium, pp 413–428. Society of Petroleum EngineersGoogle Scholar
17. Khristianovic SA, Zheltov YP (1955) Formation of vertical fractures by means of highly viscous liquid. In: Fourth world petroleum congress proceedings, pp 579–586Google Scholar
18. Lecampion B, Detournay E (2007) An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag. Comput Methods Appl Mech Eng 196:4863–4880
19. Medlin WL, Massé L (1984) Laboratory experiments in fracture propagation. Soc Petrol Eng J 24(3):256–268
20. Nitzschke S, Woschke E, Schmicker D, Strackeljan J (2016) Regularised cavitation algorithm for use in transient rotordynamic analysis. Int J Mech Sci 113:175–183
21. Olsson KO (1965) Cavitation in dynamically loaded bearings. Transactions of Chalmers University Technology, Goteborg, Sweden, vol 308Google Scholar
22. Rangarajan R, Chiaramonte MM, Hunsweck MJ, Shen Y, Lew AJ (2015) Simulating curvilinear crack propagation in two dimensions with universal meshes. Int J Numer Methods Eng 102(3–4):632–670
23. Shen Y (2014) A variational inequality formulation to incorporate the fluid lag in fluid-driven fracture propagation. Comput Methods Appl Mech Eng 272:17–33
24. Shen Y, Wu C, Wan Y (2017) Universal meshes for a branched crack. Finite Elem Anal Des 129:53–62
25. Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New YorkGoogle Scholar
26. Van Dam DB, De Pater CJ, Romijn R (1998) Analysis of hydraulic fracture closure in laboratory experiments. In: SPE/ISRM rock mechanics in petroleum engineering, pp 365–374. Society of Petroleum EngineersGoogle Scholar
27. Zhang X, Jeffrey R, Detournay E (2005) Propagation of a hydraulic fracture parallel to a free surface. Int J Numer Anal Methods Geomech 29(13):1317–1340