#
Backflow Dynamics of Newtonian Fluids in an Elastic Fracture with Slip Walls^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory

**u**the velocity vector, and $\tau $ the deviatoric stress tensor. Considering an incompressible, steady, and laminar flow propagating in the x-direction (no movement in the z-direction), the momentum equation in the Cartesian system of Figure 1 becomes

#### 2.1. Linear Navier Slip Law

_{s}and shear stress is linear:

#### 2.2. Governing Equations

#### 2.2.1. Dimensionless Governing Equations

#### 2.2.2. Nonlinear Ordinary Differential Equation (ODE)

## 3. Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Montgomery, C.; Smith, M. Hydraulic fracturing: History of an enduring technology. J. Pet. Technol.
**2010**, 62, 26–32. [Google Scholar] [CrossRef] - Britt, L. Fracture stimulation fundamentals. J. Nat. Gas. Sci. Eng.
**2012**, 8, 34–51. [Google Scholar] [CrossRef] - Osiptov, A. Fluid mechanics of hydraulic fracturing: A review. J. Pet. Sci. Eng.
**2017**, 156, 513–535. [Google Scholar] [CrossRef] - Lai, C.-Y.; Zheng, Z.; Dressaire, E.; Ramon, G.; Huppert, H.E.; Stone, H. Elastic relaxation of fluid-driven cracks and the resulting backflow. Phys. Rev. Lett.
**2016**, 15, 24–36. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dana, A.; Zheng, Z.; Peng, G.G.; Stone, H.A.; Huppert, H.E.; Ramon, G.Z. Dynamics of viscous backflow from a model fracture network. J. Fluid Mech.
**2018**, 836, 828–849. [Google Scholar] [CrossRef] [Green Version] - McLennan, J.; Walton, I.; Moore, J.; Brinton, D.; Lund, J. Proppant backflow: Mechanical and flow considerations. Geothermics
**2015**, 57, 224–237. [Google Scholar] [CrossRef] [Green Version] - Garagash, D.I. Transient solution for a plane-strain fracture driven by a shear thinning, power-law fluid. Int. J. Numer. Anal. Meth. Geomech.
**2006**, 30, 1439–1475. [Google Scholar] [CrossRef] - Chiapponi, L.; Ciriello, V.; Longo, S.; Di Federico, V. Non-Newtonian backflow in an elastic fracture. Water Resour. Res.
**2019**, 55, 10144–10158. [Google Scholar] [CrossRef] - Ciriello, V.; Lenci, A.; Longo, S.; Di Federico, V. Relaxation-induced flow in a smooth fracture for Ellis rheology. Adv. Water Resour.
**2021**, 152, 103914. [Google Scholar] [CrossRef] - Lenci, A.; Chiapponi, L.; Longo, S.; Di Federico, V. Experimental investigation on backflow of power-law fluids in planar fractures. Phys. Fluids
**2021**, 33, 083111. [Google Scholar] [CrossRef] - Zeighami, F.; Lenci, A.; Di Federico, V. Drainage of power-law fluids from fractured or porous finite domains. J. Non-Newton. Fluid Mech.
**2022**, 305, 104832. [Google Scholar] [CrossRef] - Neto, C.; Evans, D.R.; Bonaccurso, E.; Butt, H.J.; Craig, V.S. Boundary slip in Newtonian liquids: A review of experimental studies. Rep. Prog. Phys.
**2005**, 68, 2859. [Google Scholar] [CrossRef] - Lauga, E.; Brenner, M.P.; Stone, H.A. Handbook of Experimental Fluid Dynamics. In Microfluidics: The No-Slip Boundary Condition; Springer: New York, NY, USA, 2005. [Google Scholar]
- Lee, H.-B.; Yeo, I.W.; Lee, K.-K. Water flow and slip on NAPL-wetted surfaces of a parallel-walled fracture. Geophys. Res. Lett.
**2007**, 34, L19401. [Google Scholar] [CrossRef] - Zheng, L.; Wang, L.; Wang, T.; Singh, K.; Wang, Z.-L.; Chen, X. Can homogeneous slip boundary condition affect effective dispersion in single fractures with Poiseuille flow? J. Hydrol.
**2020**, 581, 124385. [Google Scholar] [CrossRef] - Landau, L.D.; Lifshitz, E.M.; Kosevich, A.M.; Pitaevskii, L.P. Theory of Elasticity; Elsevier: Amsterdam, The Netherlands, 1986; Volume 7. [Google Scholar]
- Kerr, A.D. Elastic and viscoelastic foundation models. Trans. ASME J. Appl. Mech.
**1964**, 31, 491–498. [Google Scholar] [CrossRef]

**Figure 1.**The schematics of a two-dimensional fracture with a finite length L and an initial aperture $h\left(t=0\right)={h}_{0}$

**Figure 2.**Wing symmetric fractures with the spacing l. The blue arrows in the figure demonstrate the fluid flow.

**Figure 3.**Numerical solutions of the governing ODE of Equation (25) for the different values of P

_{e}. The analytical result of the case P

_{e}= 0 obtained from Equation (A3) is highlighted by the red dots.

**Figure 4.**Time evolution of the dimensionless pressure distribution of Equation (24) inside the fracture when the pressure gradient is null (P

_{e}= 0).

**Figure 5.**Numerical solution of the ODE for the different N

_{s}when the external pressure is null (P

_{e}= 0).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zeighami, F.; Lenci, A.; Longo, S.; Di Federico, V.
Backflow Dynamics of Newtonian Fluids in an Elastic Fracture with Slip Walls. *Environ. Sci. Proc.* **2022**, *21*, 45.
https://doi.org/10.3390/environsciproc2022021045

**AMA Style**

Zeighami F, Lenci A, Longo S, Di Federico V.
Backflow Dynamics of Newtonian Fluids in an Elastic Fracture with Slip Walls. *Environmental Sciences Proceedings*. 2022; 21(1):45.
https://doi.org/10.3390/environsciproc2022021045

**Chicago/Turabian Style**

Zeighami, Farhad, Alessandro Lenci, Sandro Longo, and Vittorio Di Federico.
2022. "Backflow Dynamics of Newtonian Fluids in an Elastic Fracture with Slip Walls" *Environmental Sciences Proceedings* 21, no. 1: 45.
https://doi.org/10.3390/environsciproc2022021045