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A New Criterion for a Toughness-Dominated Hydraulic Fracture Crossing a Natural Frictional Interface

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Abstract

Hydraulic fracturing is a powerful technology, especially in stimulating fluid production from reservoirs. However, the problem of the intersection between hydraulic fractures and natural fractures is inevitable in engineering practice due to naturally fractured formations. This paper presents a new criterion for a toughness-dominated hydraulic fracture crossing a natural frictional interface through coupling the fluid flow and elastic deformation of the hydraulic fracture prior to intersecting with the natural frictional interface. The critical condition for the hydraulic fracture crossing the natural frictional interface is that the total superimposed stress does not satisfy the failure condition of the Mohr–Coulomb criterion. Simultaneously, the new criterion considers nonorthogonal intersection angles and six independent parameters relating to fluid flow (hydraulic fracture half-length, approaching distance and injection rate), rock mechanic properties (rock fracture toughness and Young’s modulus) and in situ stress. The prediction outcomes show good agreement with laboratory experiments as well as sufficient advantages compared with the analytical criteria of Blanton, extended Renshaw-Pollard and Llanos. Parameter sensitivity analysis is conducted using the control variable method. The parametric analysis results reveal that the influence sphere of different parameters is limited to a certain extent by the variations in the intersection angle except for Young’s modulus and the injection rate, which show slight effects on the intersection behaviors.

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Abbreviations

HF:

Hydraulic fracture

NF:

Natural fracture with frictional interface

K I :

Stress intensify factor

K IC :

Mode I rock fracture toughness (critical stress intensity factor)

β :

Intersection angle between HF and NF

P f :

Internal fluid pressure within HF

σ h :

Minimum horizontal in situ stress

σ H :

Maximum horizontal in situ stress

∆σ :

Stress difference between σH and σh

t :

Injection time

l(t):

Half-length of HF

ξ :

Ratio of x and l(t) (ranging from 0 to 1)

γ(ρ):

Dimensionless half-length of HF

L(t):

Length scale, which has the dimension of length

w(x, t):

Crack opening of HF

Ω(ξ, t):

Dimensionless crack opening of HF

P net(x, t):

Net fluid pressure inside HF

Π(ξ, t):

Dimensionless net fluid pressure inside HF

ε(t):

Small dimensionless parameter that guarantees the variation of Ω and Π from zero to infinity

ρ(t):

Dimensionless evolution parameter

\(E'\) :

Plane-strain elastic modulus

v :

Poisson’s ratio

Q 0 :

Injection rate

\({\sigma _{xx}}\) :

Normal stress in the direction of x-axis induced by net pressure

\({\sigma _{yy}}\) :

Normal stress in the direction of y-axis induced by net pressure

\({\tau _{xy}}\) :

Shear stress induced by net pressure

\({\sigma _{\beta x}}\) :

Total normal stress applied parallel to NF interface (along the direction of βx-axis)

\({\sigma _{\beta y}}\) :

Total normal stress applied perpendicularly on NF interface (along the direction of βy-axis)

\({\tau _{\beta xy}}\) :

Total shear stress applied on NF interface

\({\sigma _{\gamma ,\beta x}}\) :

Normal stress component generated by in situ stress field along the direction of βx-axis on NF interface

\({\sigma _{\gamma ,\beta y}}\) :

Normal stress component generated by in situ stress field along the direction of βy-axis on NF interface

\({\tau _{\gamma ,\beta }}\) :

Shear stress component generated by in situ stress field on NF interface

\({\sigma _{{P_{{\text{net}}}},\beta x}}\) :

Normal stress components along the direction of βx-axis generated by induced stress field on NF interface

\({\sigma _{{P_{{\text{net}}}},\beta y}}\) :

Normal stress components along the direction of βy-axis generated by induced stress field on NF interface

\({\tau _{{P_{{\text{net}}}},\beta }}\) :

Shear stress components generated by induced stress field on NF interface

µ :

Friction coefficient of the NF

c :

Cohesion of the NF

T 0 :

Rock tensile strength

r c :

Critical radius of nonlinear region where the stresses on the natural interface are maximized

∆l :

Hydraulic fracture approaching distance

∆:

Dimensionless stress difference

F :

Slip function

\(\overline {F}\) :

Dimensionless slip function

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (nos. 51374257, 50804060).

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Authors and Affiliations

  1. School of Civil Engineering, Guizhou University, Guiyang, 550025, China

    Yu Zhao

  2. Key Laboratory of New Technology for Construction of Cites in Mountain Area (Chongqing University), Ministry of Education, Chongqing, 400045, China

    Yu Zhao, Pengfei He, Yongfa Zhang & Chaolin Wang

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Zhao, Y., He, P., Zhang, Y. et al. A New Criterion for a Toughness-Dominated Hydraulic Fracture Crossing a Natural Frictional Interface. Rock Mech Rock Eng 52, 2617–2629 (2019). https://doi.org/10.1007/s00603-018-1683-y

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