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Hydraulic fractures of the cusp shape in a Hele-Shaw cell

Published online by Cambridge University Press:  23 September 2024

Zhong Zheng Open the ORCID record for Zhong Zheng [Opens in a new window]
Zhong Zheng*
Affiliation:
State Key Laboratory of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China MOE Key Laboratory of Hydrodynamics, School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
*
†Email addresses for correspondence: zzheng@alumni.princeton.edu, zhongzheng@sjtu.edu.cn
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Abstract

We study the dynamics of hydraulic fracturing of an elastic solid in a Hele-Shaw cell. Compared with hydraulic fractures in an infinite elastic bulk, the viscous resistance comes mainly from the drag by the two parallel plates that forms the Hele-Shaw cell rather than by the fluid–solid interface. Such a feature leads to a different nonlinear differential–integral system that describes the coupled evolution of the fracture shape and pressure field. Our theory leads to hydraulic fractures of cusp shapes in the neighbourhood of the fracture tip, which is consistent with recent experimental observations. Accordingly, there exists no pressure singularity at the location of the fracture tip, which is also fundamentally different from our previous understandings of hydraulic fracturing of elastic solids.

JFM classification

Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films Waves/Free-surface Flows: Elastic waves

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Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 995 , 25 September 2024 , A17
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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