Abstract
Unintentional variations of fluid pressure within a hydraulic fracture will disturb the surrounding stress state, affect the stability of fracture propagation and complicate the fracture intersections during the process of hydraulic fracturing. However, little attention has been paid to the effect of nonuniform fluid pressure inside the hydraulic fracture. This paper presents a semianalytical solution for a Griffith crack nonuniformly pressurized by internal fluid in an impermeable elastic plane. The fluid pressure is hypothetically designated as a general polynomial with respect to the location of the fluid, such that the effect of an arbitrary form of fluid pressure distribution can be explored by polynomial fitting. The semianalytical solution is capable of being degenerated into constant pressure forms which confirms that the solution is an extension of classic constant pressure. In addition, the critical propagation conditions and stress distributions (e.g., σxx) under constant and nonuniform pressures are compared and discussed. The comparison results indicate that the effect of nonuniform fluid pressure accumulated by the number of polynomial terms increases the magnitude of stress (or displacement) but does not change its distribution. Subsequently, the semianalytical solution is validated by comparing the fracture intersection predicted by the semianalytical solution with the laboratory experimental observations and published predictions under constant fluid pressure. Good agreement with the experiments and sufficient advantages over previous predictions are observed. Finally, a sensitivity analysis of existing parameters in the semianalytical solution, including crack length, initial pressure, number of terms and number of subintervals, is conducted to evaluate their influence on surrounding stresses and critical propagation conditions, which further demonstrates the applicability and reliability of the presented semianalytical solution. The new solution enriches hydraulic fracturing theory by considering the nonuniform fluid pressure effect and provides important reference for fracture network design during hydraulic fracturing.
Similar content being viewed by others
Abbreviations
- a :
-
Crack length
- a C :
-
Critical fracture length at the beginning of fracture expansion under constant fluid pressure
- a N :
-
Critical fracture length at the beginning of fracture expansion under nonuniform fluid pressure
- b k :
-
Dimensionless coefficient variable of the assumed distribution of internal pressure
- E :
-
Young’s modulus
- \(E^{\prime}\) :
-
Plane-strain elastic modulus
- K I :
-
Stress intensity factor (SIF)
- K IC :
-
Mode I fracture toughness of rock (critical stress intensity factor)
- \(K_{\text{I}}^{\text{C}}\) :
-
Stress intensity factor under the action of constant fluid pressure
- \(K_{\text{I}}^{\text{N}}\) :
-
Stress intensity factor under the action of nonuniform fluid pressure
- m :
-
Number of subintervals in the composite Simpson’s rule
- n :
-
Number of terms of the polynomial that is assumed as the distribution of internal pressure
- P(x):
-
Assumed net fluid pressure inside the crack
- P 0 :
-
Constant term of the assumed distribution of internal pressure (initial fluid pressure excluding the effect of far-field stress)
- P hf :
-
Critical fluid pressure required to drive the crack tip to move ahead
- r c :
-
Critical radius of the plastic zone ahead of fracture tip
- u x :
-
Displacement in the direction of x-axis
- u y :
-
Displacement in the direction of y-axis
- v :
-
Poisson’s ratio
- μ :
-
Friction coefficient
- β :
-
Intersection angle (≤ 90°) between a hydraulic fracture and a natural fracture
- κ :
-
Dimensionless form of the critical intensity factor
- σ H :
-
Maximum horizontal in situ stress
- σ h :
-
Minimum horizontal in situ stress
- σ xx :
-
Normal stress in the direction of x-axis induced by internal pressure
- σ xy :
-
Tangential stress induced by internal pressure
- σ yy :
-
Normal stress in the direction of y-axis induced by internal pressure
References
Adachi JI (2001) Fluid-driven fracture in permeable rock. PhD thesis, Minneapolis University of Minnesota
Barati R, Liang JT (2014) A review of fracturing fluid systems used for hydraulic fracturing of oil and gas wells. J Appl Polym Sci 131(6):40735. https://doi.org/10.1002/app.40735
Beugelsdijk LJL, Pater C, Sato K (2000) Experimental hydraulic fracture propagation in a multi-fractured medium. In: SPE Asia Pacific conference on investigated modelling for asset management. Society of Petroleum Engineers. https://doi.org/10.2118/59419-ms
Cerasi P, Ladva HK, Bradbury AJ et al (2001) Measurement of the mechanical properties of filtercakes. In: Proceedings of the SPE European Formation damage conference, The Hague
Chuprakov D, Melchaeva O, Prioul R (2014) Injection-sensitive mechanics of hydraulic fracture interaction with discontinuities. Rock Mech Rock Eng 47(5):1625–1640. https://doi.org/10.1007/s00603-014-0596-7
Dahi Taleghani A, Klimenko D (2015) An analytical solution for microannulus cracks developed around the wellbore. J Energy Resour Technol 10(1115/1):4030627
Desroches J, Detournay E, Lenoach B et al (1994) The crack tip region in hydraulic fracturing. Proc R Soc 447(1929):39–48. https://doi.org/10.1098/rspa.1994.0127
Detournay E (2004) Propagation regimes of fluid-driven fractures in impermeable rocks. Int J Geomech 4(1):35–45. https://doi.org/10.1061/(asce)1532-3641(2004)4:1(35)
Detournay E, Peirce AP, Bunger AP (2007) Viscosity-dominated hydraulic fractures. In: Proceedings 1st Canada-U.S. rock mechanics symposium, Vancouver. American Rock Mechanics Association, pp 1649–1656. https://doi.org/10.1201/NOE0415444019-c207
Economides MJ, Boney C (2000) Reservoir stimulation in petroleum production. In: Economides MJ, Nolte KG (eds) Reservoir stimulation, 3rd edn. Wiley, Chichester
Garagash D, Detournay E (2000) The tip region of a fluid-driven fracture in an elastic medium. J Appl Mech 67:183–192. https://doi.org/10.1115/1.321162
Garagash DI, Detournay E (2002) Viscosity-dominated regime of a fluid-driven fracture in an elastic medium. In: Symposium on analytical and computational fracture mechanics of non-homogeneous materials, IUTAM pp 25–29. http://doi.org/10.1007/978-94-017-0081-8
Garagash DI, Sarvaramini E (2012) Equilibrium of a pressurized plastic fluid in a wellbore crack. Int J Solids Struct 49(1):197–212. https://doi.org/10.1016/j.ijsolstr.2011.09.022
Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series and products. Academic, Boston
Griffith AA (1921) The phenomena of rupture and flow in solids. Philos Trans R Soc Lond 221(582–593):163–198. https://doi.org/10.1098/rsta.1921.0006
Haimson B, Fairhurst C (1967) Initiation and extension of hydraulic fractures in rocks. Soc Petrol Eng J 7(03):310–318. https://doi.org/10.2118/1710-pa
Hanson MT, Keer LM (1992) An analytical life prediction model for the crack propagation occurring in contact fatigue failure. Tribol Trans 35:451–461. https://doi.org/10.1080/10402009208982143
Hubbert M, Willis D (1957) Mechanics of hydraulic fracturing. Trans Soc Petrol Eng AIME 210:153–168
Khristianovic SA, Zheltov YP (1955) Formation of vertical fractures by means of highly viscous liquid. In: Proceedings of the fourth world petroleum congress, Rome, pp 579–586
Lai J, Ioannides E, Wang J (2009) Fluid-crack interaction in lubricated rolling-sliding contact. In: Proceedings of the STLE/ASME international joint tribology conference 2008, pp 437–439. https://doi.org/10.1115/IJTC2008-71254
Liu JZ, Wu XR (1997) Analytical expressions for crack opening displacements of edge cracked specimens under a segment of uniform crack face pressure. Eng Fract Mech 58:107–119
Mu W, Li L, Yang T et al (2019) Numerical investigation on grouting mechanism with slurry-rock coupling and shear displacement in single rough fracture. Bull Eng Geol Environ 78(8):6159–6177
Nilson RH, Proffer WJ (1984) Engineering formulas for fractures emanating from cylindrical and spherical holes. J Appl Mech Trans ASME 51(4):929–933. https://doi.org/10.1115/1.3167748
Renshaw CE, Pollard DD (1995) An experimentally verified criterion for propagation across unbounded frictional interfaces in brittle, linear elastic-materials. Int J Rock Mech Min Sci Geo-Mech Abstr 32(3):237–249
Rice JR (1968) Mathematical analysis in the mechanics of fracture. In: Liebowitz H (ed) Fracture, an advanced treatise, vol II. Academic, New York, pp 191–311
Sneddon IN, Elliot HA (1946) The opening of a Griffith crack under internal pressure. Q Appl Math 4(3):262–267. https://doi.org/10.1090/qam/17161
Sneddon IN, Lowengrub M (1969) Crack problems in the classical theory of elasticity. Wiley, New York
Teufel LW, Clark JA (1984) Hydraulic fracture propagation in layered rock—experimental studies of fracture containment. Soc Petrol Eng J 24:19–32. https://doi.org/10.2118/9878-Pa
Titchmarsh EC (1937) Introduction to the theory of Fourier intergals. Oxford University Press, Oxford
Wang C, Pan L, Yu Z et al (2019) Analysis of the pressure-pulse propagation in rock: a new approach to simultaneously determine permeability, porosity, and adsorption capacity. Rock Mech Rock Eng. https://doi.org/10.1007/s00603-019-01874-w
Warpinski NR, Teufel LW (1987) Influence of geologic discontinuities on hydraulic fracture propagation. J Petrol Technol 39:209–220. https://doi.org/10.2118/13224-Pa
Weng X (2015) Modeling of complex hydraulic fractures in naturally fractured formation. J Unconv Oil Gas Resour 9:114–135. https://doi.org/10.1016/j.juogr.2014.07.001
Xu J, Zhai C, Qin L (2017) Mechanism and application of pulse hydraulic fracturing in improving drainage of coalbed methane. J Nat Gas Sci Eng 40:79–90. https://doi.org/10.1016/j.jngse.2017.02.012
Yang H, Liu J, Liu B (2018) Investigation on the cracking character of jointed rock mass beneath TBM disc cutter. Rock Mech Rock Eng 51(4):1263–1277. https://doi.org/10.1007/s00603-017-1395-8
Yao Y, Wang W, Keer LM (2017) An energy based analytical method to predict the influence of natural fractures on hydraulic fracture propagation. Eng Fract Mech. https://doi.org/10.1016/j.engfracmech.2017.11.02
Yew CH, Weng X (2014) Mechanics of hydraulic fracturing. Gulf Professional Publishing, Houston
Zhao Y, Wang CL, Zhang YF, Liu Q (2019a) Experimental study of adsorption effects on shale permeability. Nat Resour Res 28:1575–1586. https://doi.org/10.1007/s11053-019-09476-7
Zhao Yu, Yongfa Zhang, Pengfei He (2019b) A composite criterion to predict subsequent intersection behavior between a hydraulic fracture and a natural fracture. Eng Fract Mech 209:61–78. https://doi.org/10.1016/j.engfracmech.2019.01.015
Zhao Y, He PF, Zhang YF, Wang L (2019c) A new criterion for a toughness-dominated hydraulic fracture crossing a natural frictional interface. Rock Mech Rock Eng 52:2617–2629. https://doi.org/10.1007/s00603-018-1683-y
Zhao Y, Bi J, Zhou X, Wang C (2019d) Effect of HTHP (high temperature and high pressure) of water on micro-characteristic and splitting tensile strength of gritstone. Front Earth Sci. https://doi.org/10.3389/feart.2019.0030
Zhou X, Yang H (2007) Micromechanical modeling of dynamic compressive responses of mesoscopic heterogeneous brittle rock. Theor Appl Fract Mech 48(1):1–20
Zhou J, Chen M, Jin Y, Zhang GQ (2008) Analysis of fracture propagation behavior and fracture geometry using tri-axial fracturing system in naturally fractured reservoirs. Int J Rock Mech Min Sci Geo-Mech Abstr 45:1143–1152
Acknowledgements
This work was financially supported by the Research Fund for Talents of Guizhou University (Grant No. 201901), the Special Research Funds of Guizhou University (Grant No. 201903), the Project supported by graduate research and innovation foundation of Chongqing, China (Grant No. CYB19015) and the National Natural Science Foundation of China (Nos. 51374257, 50804060).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: ξ-Integrals Function
A pair of general form functions composed of Bessel, trigonometric, exponential and power functions (ξ-integrals) can be expressed as:
where c, d and v are integers ranging from zero to infinity.
Gradshteyn and Ryzhik (2007) also provided the following solution of ξ-integrals when c and d both equal v + 1:
A recursion formula of the Bessel function is given by:
Making v in Eq. (43) equal to w + 1 (w is an integer greater than or equal to 0) and substituting this equation into the ξ-integrals, we obtain:
Using integration by parts, Eq. (44) can be expressed as:
After simplification, Eq. (45) becomes
In Eq. (46), two binary linear equations of \(\int\nolimits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{{w{ + }1}} (u\xi )\cos (\xi x){\text{d}}\xi }\) (X) and \(\int\nolimits_{0}^{\infty } {\xi^{w + 1} {\text{e}}^{ - \xi y} J_{{w{ + }1}} (u\xi )\sin (\xi x){\text{d}}\xi }\) (Y) can be solved using M1 and M2 derived by Eq. (42). Notably, these deductions, held as variables of c, d, v and w, are all integers ranging from zero to infinity. Thus, it is appropriate to replace variable w with v in X and Y for the sake of consistent expression with Eq. (45). Thus, the unknown functions X and Y are separately expressed as:
In the case of c = d=v + 2, using the recurrence relation of Eq. (43), we can obtain:
Then, this relation reduces to:
Substituting Eq. (49) into ξ-integrals, we can obtain:
Based on integration by parts, Eq. (50) can be further simplified to:
When v in Eq. (42) equals v + 1, it is easy to derive:
Substituting Eqs. (47) and (52) into Eq. (51), ξ-integrals in the case of c = d=v + 2 are solved:
Appendix 2: Closed Form of F(ξ)
The function F(ξ) in Eq. (24) can be expressed as:
where 2F1 is a hypergeometric function.
Rights and permissions
About this article
Cite this article
Zhang, Y., Zhao, Y., Yang, H. et al. A Semianalytical Solution for a Griffith Crack Nonuniformly Pressurized By Internal Fluid. Rock Mech Rock Eng 53, 2439–2460 (2020). https://doi.org/10.1007/s00603-020-02052-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00603-020-02052-z