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Experimental and numerical investigation of groundwater head losses on and nearby short intersections between disc-shaped fractures
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Abstract: Discrete fracture models are used for investigating precise processes of groundwater flow in fractured rocks, while a disc-shaped parallel-plates model for a single fracture is more reasonable and efficient for computational treatments. The flow velocity has a large spatial differentiation which is more likely to produce non-linear flow and additional head losses on and nearby intersections in such shaped fractures, therefore it is necessary to understand and quantify them. In this study, both laboratory experiments and numerical simulations were performed to investigate the total head loss on and nearby the intersections as well as the local head loss exactly on the intersections, which were not usually paid sufficient attention or even ignored. The investigation results show that these two losses account for 29.17%-84.97% and 0-73.57% of the entire total head loss in a fracture, respectively. As a result, they should be necessarily considered for groundwater modeling in fractured rocks. Furthermore, both head losses become larger when aperture and flow rate increase and intersection length decreases. Particularly, the ratios of these two head losses to the entire total head loss in a fracture could be well statistically explained by power regression equations with variables of aperture, intersection length, and flow rates, both of which achieved high coefficients of determination. It could be feasible through this type of study to provide a way on how to adjust the groundwater head from those obtained by numerical simulations based on the traditional linear flow model. Finally, it is practicable and effective to implement the investigation approach combining laboratory experiments with numerical simulations for quantifying the head losses on and nearby the intersections between disc-shaped fractures.
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Figure 3. The schematic diagram of three losses (
$ {H}_{l} $ ,$ {H}_{n} $ and$ {H}_{i} $ ) defined in this paper
Figure 4. The relationships between the hydraulic gradient (
$ {\nabla \mathit{H}}_{\mathit{i}\mathit{n}-\mathit{o}\mathit{u}\mathit{t}\_\mathit{e}\mathit{x}\mathit{p}} $ and$ {\nabla \mathit{H}}_{\mathit{p}1-2\_\mathit{e}\mathit{x}\mathit{p}} $ ) and the flow rate (Q) in different experimental scenarios.
Figure 5. The percentage of
$ \Delta {\mathit{H}}_{\mathit{s}\_\mathit{e}\mathit{x}\mathit{p}} $ to$ \Delta {\mathit{H}}_{\mathit{i}\mathit{n}-\mathit{o}\mathit{u}\mathit{t}\_\mathit{e}\mathit{x}\mathit{p}} $ in different experimental scenarios
Figure 7. The percentage of
$ \Delta {H}_{i\_der} $ to$ \Delta {H}_{in-out\_exp} $ in different experimental scenariosTable 1. The parameters assigned in the corresponding numerical models
Parameters Units Values Boundary (flow in and flow out) Length (L) cm 10, 8, 6, 4 corresponding with each experiment Specified flux (Q) cm3/s corresponding with each experiment Aperture (b) cm 0.215, 0.125 corresponding with each experiment Reference fluid density ($ \rho ) $ kg/cm3 9.98E-4 Reference fluid viscosity (μ) kg/(cm·s) 9.53E-6 Acceleration due to gravity ($ g $) cm/s2 9.81E+2 
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Table 2. The experimental results of
$ {\Delta \mathit{H}}_{\mathit{i}\mathit{n}-\mathit{o}\mathit{u}\mathit{t}\_\mathit{e}\mathit{x}\mathit{p}} $ and$ \Delta {\mathit{H}}_{\mathit{p}1-2\_\mathit{e}\mathit{x}\mathit{p}} $ along with the percentage of$ \Delta {\mathit{H}}_{\mathit{s}\_\mathit{e}\mathit{x}\mathit{p}} $ to$ \Delta {\mathit{H}}_{\mathit{i}\mathit{n}-\mathit{o}\mathit{u}\mathit{t}\_\mathit{e}\mathit{x}\mathit{p}} $ b=0.152 cm b=0.215 cm Q
(cm3/s)$ {\Delta H}_{in-out\_exp} $
(cm)$ \Delta {H}_{p1-2\_exp} $
(cm)$ \dfrac{\Delta {H}_{s\_exp}}{\Delta {H}_{in-out\_exp}} $
(-)Q
(cm3/s)$ {\Delta H}_{in-out\_exp} $
(cm)$ \Delta {H}_{p1-2\_exp} $
(cm)$ \dfrac{\Delta {H}_{s\_exp}}{\Delta {H}_{in-out\_exp}} $
(-)L=10 cm 14.41 0.48 0.34 29.17% 35.34 0.45 0.29 35.56% 26.53 0.95 0.65 31.58% 43.55 0.60 0.35 41.67% 35.54 1.35 0.88 34.81% 54.91 0.88 0.48 45.45% 42.88 1.72 1.10 36.05% 69.03 1.22 0.66 45.90% 70.97 3.38 1.89 44.08% 75.28 1.48 0.71 52.03% L=8 cm 14.53 0.47 0.35 25.53% 35.28 0.49 0.30 38.78% 26.54 0.97 0.66 31.96% 43.57 0.68 0.38 44.12% 35.68 1.36 0.89 34.56% 54.70 0.97 0.48 50.52% 43.51 1.77 1.08 38.98% 68.48 1.36 0.66 51.47% 71.83 3.40 1.87 45.00% 75.06 1.58 0.67 57.59% L=6 cm 14.61 0.54 0.35 35.19% 35.58 0.55 0.28 49.09% 26.30 1.04 0.65 37.50% 43.10 0.71 0.36 49.30% 35.68 1.49 0.88 40.94% 55.17 1.08 0.47 56.48% 43.43 1.89 1.07 43.39% 68.55 1.48 0.57 61.49% 71.66 3.60 1.83 49.17% 75.78 1.75 0.62 64.57% L=4 cm 14.61 0.59 0.36 38.98% 35.25 0.97 0.29 70.10% 26.51 1.22 0.66 45.90% 42.94 1.31 0.31 76.34% 35.57 1.81 0.86 52.49% 55.76 2.00 0.41 79.50% 42.94 2.37 1.05 55.70% 69.71 2.92 0.53 81.85% 71.07 4.86 1.74 64.20% 74.82 3.46 0.52 84.97%
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Table 3. The derivate results of
$ {\Delta \mathit{H}}_{\mathit{i}\mathit{n}-\mathit{o}\mathit{u}\mathit{t}\_\mathit{d}\mathit{e}\mathit{r}} $ and the ratios of$ \Delta {\mathit{H}}_{\mathit{i}\_\mathit{d}\mathit{e}\mathit{r}} $ to$ \Delta {\mathit{H}}_{\mathit{i}\mathit{n}-\mathit{o}\mathit{u}\mathit{t}\_\mathit{e}\mathit{x}\mathit{p}} $ b=0.152 cm b=0.215 cm Q
(cm3/s)$ \dfrac{\Delta {H}_{in-out\_num}}{\Delta {H}_{p1-2\_num}} $
(-)$ \Delta {H}_{in-out\_der} $
(cm)$ \dfrac{\Delta {H}_{i\_der}}{\Delta {H}_{in-out\_exp}} $
(-)Q
(cm3/s)$ \dfrac{\Delta {H}_{in-out\_num}}{\Delta {H}_{p1-2\_num}} $
(-)$ \Delta {H}_{in-out\_der} $
(cm)$ \dfrac{\Delta {H}_{i\_der}}{\Delta {H}_{in-out\_exp}} $
(-)L=10 cm 14.41 1.46 0.49 0.00% 35.34 1.46 0.42 6.23% 26.53 1.46 0.95 0.44% 43.55 1.46 0.52 15.12% 35.54 1.46 1.28 5.15% 54.91 1.46 0.69 20.63% 42.88 1.46 1.60 6.94% 69.03 1.46 0.97 21.28% 70.97 1.46 2.74 18.65% 75.28 1.46 1.04 30.20% L=8 cm 14.53 1.52 0.54 0.00% 35.28 1.52 0.47 6.96% 26.54 1.52 1.00 0.00% 43.57 1.52 0.58 15.08% 35.68 1.52 1.36 0.55% 54.70 1.52 0.72 24.81% 43.51 1.52 1.65 7.27% 68.48 1.52 1.01 26.25% 71.83 1.52 2.84 16.42% 75.06 1.52 1.03 35.56% L=6 cm 14.61 1.61 0.57 0.00% 35.58 1.61 0.46 17.90% 26.30 1.61 1.05 0.00% 43.10 1.61 0.57 18.23% 35.68 1.61 1.41 4.76% 55.17 1.61 0.76 29.82% 43.43 1.61 1.73 8.70% 68.55 1.61 0.92 37.90% 71.66 1.61 2.96 18.02% 75.78 1.61 1.01 42.87% L=4 cm 14.61 1.76 0.63 0.00% 35.25 1.76 0.52 47.42% 26.51 1.76 1.16 4.85% 42.94 1.76 0.54 58.38% 35.57 1.76 1.51 16.44% 55.76 1.76 0.73 63.95% 42.94 1.76 1.85 22.09% 69.71 1.76 0.94 68.08% 71.07 1.76 3.06 37.05% 74.82 1.76 0.91 73.57%
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Major loss Minor loss $ \Delta {H}_{l} $ $ \Delta {H}_{n} $ $ \Delta {H}_{i} $ $ \Delta {H}_{in-out\_exp} $ √ √ √ $ \Delta {H}_{p1-2\_exp} $ √ √ $ \Delta {H}_{s\_exp} $ √ √ √ $ \Delta {H}_{in-out\_der} $ √ √ $ \Delta {H}_{i\_der} $ √ $ \Delta {H}_{in-out\_num} $ √ $ \Delta {H}_{p1-2\_num} $ √
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