Patching Hele-Shaw Cells to Investigate the Flow at Low Reynolds Number in Fracture Networks
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Patching Hele‑Shaw Cells to Investigate the Flow at Low Reynolds Number in Fracture Networks Pouria Aghajannezhad1 · Mathieu Sellier1 · Sid Becker1 Received: 8 May 2020 / Accepted: 2 November 2020 © Springer Nature B.V. 2020
Abstract This research has found a novel computationally efficient method of modelling flow at low Reynolds number through fracture networks. The numerical analysis was performed by connecting Hele-Shaw cells to investigate the effect of intersections on the pressure field and hydraulic resistance for given inlet and outlet pressure values. In this analysis, the impact of intersecting length, intersecting angle and fracture aperture on the fluid flow was studied. For this purpose, two models with different topologies were established. The HeleShaw simulation results for hydraulic resistance, pressure and velocity agreed well with results obtained by solving the full Navier–Stokes equations (NSE). The results indicated an approximately linear relationship between intersection length and hydraulic resistance. Specifically, an increase in the intersection length increases the flow rate and as a result, the pressure along the intersection length decreases. The error associated with employing the Hele-Shaw approximation in comparison with NSE is less than 2%. All investigations were performed in the Reynolds Number range of 1–10. Keywords Hele-Shaw approximation · Rock fractures · Intersection angle · Intersection length · Fracture aperture · Fluid flow List of Symbols 𝜌 Density (kg/m3 ) u Velocity vector (m/s) P Pressure (Pa) 𝜇 Viscosity ((Pa s) ) Q Flow rate m3 ∕s ( ) QH_S Evaluated flow rate by Hele-Shaw approximation( m3 ∕s ) QN_S Evaluated flow rate by Navier–Stokes equations m3 ∕s h Fracture aperture (m) w Fracture depth (m) L Fracture length (m) ΔP Global pressure difference (Pa) ) ( R Hydraulic resistance Pa s−1 m−3 * Mathieu Sellier [email protected] 1
Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
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i Intersection length (m) 𝜃 Intersection angle Uav Average flow velocity along a fracture ( ) A Cross-sectional area ( A = w × h) m2
1 Introduction From a geotechnical point of view, fractures play an essential role in groundwater fluid movement. Fractures allow the transport of geothermal energy, liquid fossil fuels and groundwater (National Research Council 1996). A better understanding of the flow in fracture networks will enable a better evaluation of geothermal resources. Even though several studies have been dedicated to investigating fluid flow in fractures, fracture networks with a complex topology still present a challenge and are as yet far from being fully understood (Lei et al. 2017; Yu et al. 2017). To deal with fracture intersections in numerical studies, most previous studies have focused on the Navier–Stokes equations (NSE) (Li et al. 2016; Liu et al. 2018b; Zimmerman et al. 2004). Zhang and Sun (2019) have used a Lattice Boltzmann scheme able to
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