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 Hele-Shaw 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.
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Abbreviations
- \(\rho\) :
-
Density \(({\text{kg/m}}^{3} )\)
- \(\varvec{u}\) :
-
Velocity vector \(\left( {\text{m/s}} \right)\)
- \(P\) :
-
Pressure \(\left( {\text{Pa}} \right)\)
- \(\mu\) :
-
Viscosity \(\left( {{\text{Pa}}\,{\text{s}}} \right)\)
- \(Q\) :
-
Flow rate \(\left( {{\text{m}}^{3} / {\text{s}}} \right)\)
- \(Q_{{{\text{H}}\_{\text{S}}}}\) :
-
Evaluated flow rate by Hele-Shaw approximation \(\left( {{\text{m}}^{3} / {\text{s}}} \right)\)
- \(Q_{{{\text{N}}\_{\text{S}}}}\) :
-
Evaluated flow rate by Navier–Stokes equations \(\left( {{\text{m}}^{3} / {\text{s}}} \right)\)
- \(h\) :
-
Fracture aperture \(\left( {\text{m}} \right)\)
- \(w\) :
-
Fracture depth \(\left( {\text{m}} \right)\)
- \(L\) :
-
Fracture length \(\left( {\text{m}} \right)\)
- \(\Delta P\) :
-
Global pressure difference \(\left( {\text{Pa}} \right)\)
- \(R\) :
-
Hydraulic resistance \(\left( {{\text{Pa}}\,{\text{s}}^{ - 1} \,{\text{m}}^{ - 3} } \right)\)
- \(i\) :
-
Intersection length \(\left( {\text{m}} \right)\)
- \(\theta\) :
-
Intersection angle
- \(U_{\text{av}}\) :
-
Average flow velocity along a fracture
- \(A\) :
-
Cross-sectional area (\(A = w \times h)\) \(\left( {{\text{m}}^{2} } \right)\)
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We thank Tan Sri Dr Ngau Boon Keat for providing the grant for this research.
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Aghajannezhad, P., Sellier, M. & Becker, S. Patching Hele-Shaw Cells to Investigate the Flow at Low Reynolds Number in Fracture Networks. Transp Porous Med 136, 147–163 (2021). https://doi.org/10.1007/s11242-020-01505-x
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DOI: https://doi.org/10.1007/s11242-020-01505-x