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

## 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

- Adachi J, Siebrits E, Peirce A, Desroches J (2007) Computer simulation of hydraulic fractures. Int J Rock Mech Min Sci 44(5):739–757CrossRefGoogle Scholar
- Ausas RF, Jai M, Buscaglia GC (2009) A mass-conserving algorithm for dynamical lubrication problems with cavitation. ASME J Tribol 131(3):031702–031702-7CrossRefGoogle Scholar
- Bayada G, Chambat M (1983) Analysis of a free boundary problem in partial lubrication. Q Appl Math 40(4):369–375CrossRefGoogle Scholar
- Bunger AP, Detournay E, Jeffrey RG (2005) Crack tip behavior in near-surface fluid-driven fracture experiments. C R Mec 333(4):299–304CrossRefGoogle Scholar
- Buscaglia GC, Talibi MEA, Jai M (2015a) Mass-conserving cavitation model for dynamical lubrication problems. Part I: mathematical analysis. Math Comput Simul 118:130–145CrossRefGoogle Scholar
- Buscaglia GC, Talibi MEA, Jai M (2015b) Mass-conserving cavitation model for dynamical lubrication problems. Part II: numerical analysis. Math Comput Simul 118:146–162CrossRefGoogle Scholar
- Chiaramonte MM, Shen Y, Keer LM, Lew AJ (2015) Computing stress intensity factors for curvilinear cracks. Int J Numer Methods Eng 104(4):260–296CrossRefGoogle Scholar
- 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
- Garagash DI (2006) Propagation of a plane-strain hydraulic fracture with a fluid lag: early-time solution. Int J Solids Struct 43:5811–5835CrossRefGoogle Scholar
- Garagash D, Detournay E (2000) The tip region of a fluid-driven fracture in an elastic medium. J Appl Mech 67(1):183–192CrossRefGoogle Scholar
- Geertsma J, de Klerk F (1969) A rapid method of predicting width and extent of hydraulically induced fractures. J Petrol Technol 21(12):1571–1581CrossRefGoogle Scholar
- 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):602–629CrossRefGoogle Scholar
- Gordeliy E, Peirce A (2013) Coupling schemes for modeling hydraulic fracture propagation using the XFEM. Comput Methods Appl Mech Eng 253:305–322CrossRefGoogle Scholar
- 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):993–1015CrossRefGoogle Scholar
- Jakobsson B, Floberg L (1957) The finite journal bearing considering vaporization. Transactions of Chalmers University Technology, Goteborg, Sweden, vol 190Google Scholar
- 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
- 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
- Lecampion B, Detournay E (2007) An implicit algorithm for the propagation of a hydraulic fracture with a fluid lag. Comput Methods Appl Mech Eng 196:4863–4880CrossRefGoogle Scholar
- Medlin WL, Massé L (1984) Laboratory experiments in fracture propagation. Soc Petrol Eng J 24(3):256–268CrossRefGoogle Scholar
- Nitzschke S, Woschke E, Schmicker D, Strackeljan J (2016) Regularised cavitation algorithm for use in transient rotordynamic analysis. Int J Mech Sci 113:175–183CrossRefGoogle Scholar
- Olsson KO (1965) Cavitation in dynamically loaded bearings. Transactions of Chalmers University Technology, Goteborg, Sweden, vol 308Google Scholar
- 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):632–670CrossRefGoogle Scholar
- Shen Y (2014) A variational inequality formulation to incorporate the fluid lag in fluid-driven fracture propagation. Comput Methods Appl Mech Eng 272:17–33CrossRefGoogle Scholar
- Shen Y, Wu C, Wan Y (2017) Universal meshes for a branched crack. Finite Elem Anal Des 129:53–62CrossRefGoogle Scholar
- Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New YorkGoogle Scholar
- 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
- 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):1317–1340CrossRefGoogle Scholar