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Dynamics of viscous backflow from a model fracture network

Published online by Cambridge University Press:  12 December 2017

Asaf Dana Open the ORCID record for Asaf Dana [Opens in a new window] ,
Zhong Zheng Open the ORCID record for Zhong Zheng [Opens in a new window] ,
Gunnar G. Peng Open the ORCID record for Gunnar G. Peng [Opens in a new window] ,
Howard A. Stone ,
Herbert E. Huppert  and
Guy Z. Ramon Open the ORCID record for Guy Z. Ramon [Opens in a new window]
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
Zhong Zheng
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
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 School of Mathematics and Statistics, University of New South Wales, Kensington NSW 2052, Australia Faculty of Science, University of Bristol, Bristol BS2 6BB, 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
*
†Email address for correspondence: [email protected]
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Abstract

Hydraulic fracturing for production of oil and gas from shale formations releases fluid waste, by-products that must be managed carefully to avoid significant harm to human health and the environment. These fluids are presumed to result from a variety of fracture relaxation processes, and are commonly referred to as ‘flowback’ and ‘produced water’, depending primarily on the time scale of their appearance. Here, a model is presented for investigating the dynamics of backflows caused by the elastic relaxation of a pre-strained medium, namely a single fracture and two model fracture network systems: a single bifurcated channel and its generalization for $n$ bifurcated fracture generations. Early- and late-time asymptotic solutions are obtained for the model problems and agree well with numerical solutions. In the late-time period, the fracture apertures and backflow rates exhibit a time dependence of $t^{-1/3}$ and $t^{-4/3}$, respectively. In addition, the pressure distributions collapse to universal curves when scaled by the maximum pressure in the system, which we calculate as a function of $n$. The pressure gradient along the network is steepest near the outlet while the bulk of the network serves as a ‘reservoir’. Fracture networks with larger $n$ are less efficient at evicting fluids, manifested through a longer time required for a given fractional reduction of the initial volume. The developed framework may be useful for informing engineering design and environmental regulations.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 828 - 849
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Copyright
© 2017 Cambridge University Press 

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