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A new calculation model for the stress field of hydraulic fracture propagation at the formation interface

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Abstract

Reservoirs with different formation interfaces are becoming increasingly common in the exploitation of unconventional oil and gas reservoirs, such as shale oil and gas. Therefore, it is essential to calculate the stress field of fracture propagation to understand the mechanism of propagating hydraulic fractures near formation interfaces. This study adopts the superposition principle of same-type fractures and considers the model propagating hydraulic fractures at interfaces as the superposition of three issues: actuating the fluid pressure inside the fracture without far-field stress, analysis with far-field stress and without fractures, and calculating the self-balancing surface force effect on the surface of fractures without far-field stress. Then, the study establishes the stress field calculation model for propagating hydraulic fractures at the formation interfaces, which considers different rock properties on both sides of the formation interfaces, as well as the influence of the dip angle based on the complex variable function method and fracture mechanics theory. Finally, this paper proposes corresponding numerical methods. It is determined by analyzing different stress concentration which occurs around the fractures and stress singularity which occurs at the hydraulic fracture tips, analyzing the plastic zone at hydraulic fracture tips which propagates contacts with formation interfaces; the maximum compressive stress and normal stress change gradually from tensile stress to compressive stress as the distance between the interface and the hydraulic fracture tip increases, and the exhibiting stress reverses on the formation interfaces.

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Abbreviations

\( a \) :

Half fracture height (m)

\( D_{\text{TV}} \) :

Vertical depth (m)

\( E \) :

Young’s modulus (MPa)

\( f_{x} \) :

Concentrated force along x direction (MPa)

\( f_{y} \) :

Concentrated force along y direction (MPa)

\( F \) :

Stress function

\( g \) :

Acceleration of gravity (m/s2)

\( g_{v} \) :

Friction pressure drop gradient in the direction of fracture height (MPa/m)

\( g_{p} \) :

Fluid gravity gradient in the direction of fracture height (MPa/m)

\( H \) :

Depth of formation (m)

\( k_{\text{c}} \) :

Consistency coefficient of fracturing fluid (Pa sn)

\( K_{\text{H}} \) :

Structural coefficient in the direction of maximum principle stress

\( K_{\text{h}} \) :

Structural coefficient in the direction of minimum horizontal principle stress

\( L \) :

Fracture length (m)

\( n \) :

Flow index of fracturing fluid

\( O \) :

Offset value (MPa)

\( p_{\text{f}} \) :

Fluid pressure (MPa)

\( P_{\text{p}} \) :

Formation pore pressure (MPa)

\( Q \) :

Pump rate (m3/min)

\( R_{\text{e}} \) :

Real part of complex variable function

\( t \) :

Point on the formation interface

\( u \) :

x displacement (m)

\( v \) :

y displacement (m)

\( z \) :

Point in the complex coordinate system

\( \bar{z} \) :

The conjugate of complex number z

\( \beta \) :

Dip angle (°)

\( \theta \) :

Angle between hydraulic fracture and formation interface (°)

\( \varphi \) :

Argument of complex number z

\( \mu \) :

Elastic shear modulus (MPa)

\( \upsilon \) :

Poisson’s ratio

\( \rho \) :

Modulus of complex number

\( \rho_{\text{b}} \) :

Bulk density (kg/m3)

\( \rho_{\text{w}} \) :

Pore fluid density (kg/m3)

\( \sigma_{\text{h}} \) :

Minimum horizontal principal stress (MPa)

\( \sigma_{\text{H}} \) :

Maximum horizontal principal stress (MPa)

\( \sigma_{{1{\text{h}}}} \) :

Minimum horizontal principal stress in the lower formation (MPa)

\( \sigma_{{1{\text{z}}}} \) :

Vertical principal stress in the lower formation (MPa)

\( \sigma_{{2{\text{h}}}} \) :

Minimum horizontal principal stress in the upper formation (MPa)

\( \sigma_{{2{\text{z}}}} \) :

Vertical principal stress in the upper formation (MPa)

\( \sigma_{nn} \) :

Normal stress component along fracture plane of f x (MPa)

\( \sigma_{nt} \) :

Shear stress component along fracture plane of f y (MPa)

\( \sigma_{xx} \) :

Normal stress in x direction (MPa)

\( \sigma_{xy} \) :

Shear stress (MPa)

\( \sigma_{yy} \) :

Normal stress in y direction (MPa)

\( \sigma_{\text{z}} \) :

Vertical stress (MPa)

\( \varepsilon_{xx} \) :

Normal strain in x direction

\( \varepsilon_{yy} \) :

Normal strain in y direction

\( \gamma_{xy} \) :

Shear strain

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Acknowledgments

This article was prepared under the auspices of the State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation at Southwest Petroleum University and was supported by Open Fund (PLN1420) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), Key projects of Sichuan Provincial Department of Education (15ZA0045), and National Science and Technology Major Project of the Ministry of Science and Technology of China (2016ZX05006002-005).

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Lu, C., Guo, Jc. & Liu, Lm. A new calculation model for the stress field of hydraulic fracture propagation at the formation interface. Environ Earth Sci 75, 1178 (2016). https://doi.org/10.1007/s12665-016-5947-0

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