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Backflow from a model fracture network: an asymptotic investigation

Published online by Cambridge University Press:  14 February 2019

Asaf Dana
Affiliation:
The Nancy and Stephen Grand Technion Energy Program, and Department of Civil and Environmental Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
Gunnar G. Peng
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Howard A. Stone
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Herbert E. Huppert
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Guy Z. Ramon*
Affiliation:
The Nancy and Stephen Grand Technion Energy Program, and Department of Civil and Environmental Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
*Corresponding
Email address for correspondence: ramong@technion.ac.il

Abstract

We develop a model for predicting the flow resulting from the relaxation of pre-strained, fluid-filled, elastic network structures. This model may be useful for understanding relaxation processes in various systems, e.g. deformable microfluidic systems or by-products from hydraulic fracturing operations. The analysis is aimed at elucidating features that may provide insight on the rate of fluid drainage from fracturing operations. The model structure is a bifurcating network made of fractures with uniform length and elastic modulus, which allows for general self-similar branching and variation in fracture length and rigidity between fractures along the flow path. A late-time $t^{-1/3}$ power law is attained and the physical behaviour can be classified into four distinct regimes that describe the late-time dynamics based on the location of the bulk of the fluid volume (which shifts away from the outlet as branching is increased) and pressure drop (which shifts away from the outlet as rigidity is increased upstream) along the network. We develop asymptotic solutions for each of the regimes, predicting the late-time flux and evolution of the pressure distribution. The effects of the various parameters on the outlet flux and the network’s drainage efficiency are investigated and show that added branching and a decrease in rigidity upstream tend to increase drainage time.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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Footnotes

These two authors contributed equally.

References

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