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Equivalent Permeability of Fractured Media Incorporating Tortuosity and Nonlinear Flow

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Abstract

Permeability of fractured media plays an important role in many hydrogeologic applications. In this study, we propose an approach to quantify the equivalent permeability of fractal fractured media along with the fracture length direction, and both the laminar flow and nonlinear flow are considered according to the fracture features and critical Reynolds number. The analytical expression for equivalent permeability of fractured media is derived, which depends on fractal fracture properties and the applied conditions. The new approach is applicable for any Newtonian fluid. The effects of fractal dimension for aperture and length, maximum fracture aperture, ratio of minimum to maximum aperture, critical Reynolds number and pressure head loss across the fractures are explored and discussed. The fractal dimension for fracture aperture is positively correlated with the porosity of porous media, and a clear power-law correlation between the equivalent permeability and porosity can be observed in the results. The maximum aperture and the ratio of minimum to maximum aperture control the size of fracture apertures, and the increase in both parameters leads to high permeability. The larger fractal dimension for length causes more tortuous fractures, which reduce the equivalent permeability. A higher critical Reynolds number leads to a larger equivalent permeability. A larger pressure head loss results in a smaller equivalent permeability.

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Abbreviations

A :

Cross-sectional area (m2)

B :

Dimensionless coefficient in flowrate formula defined in Eq. (13)

C :

Dimensionless coefficient in flowrate formula defined in Eq. (17)

D l :

Fractal dimension for fracture length

D w :

Fractal dimension for fracture aperture

E :

Dimensionless coefficient in flowrate formula defined in Eq. (18)

F :

Dimensionless coefficient in flowrate formula defined in Eq. (19)

g :

Gravitational acceleration (m/s2)

H :

Height of representative volume element (m)

K :

Equivalent permeability of fractured media (m2)

K + :

Dimensionless equivalent permeability of fractured media

l :

Fracture length (m)

l avg :

Average fracture length of fractured media (m)

L :

Length of representative volume element (m)

N f :

Total number of fractures in media

q l :

Laminar flowrate in a fracture (m3/s)

q u :

Nonlinear flowrate in a fracture (m3/s)

Q :

Flowrate of fractured media (m3/s)

Q l :

Laminar flowrate in fractures (m3/s)

Q u :

Nonlinear flowrate in fractures (m3/s)

R ec :

Critical Reynolds number

u :

Flow velocity in a fracture (m/s)

u c :

Critical velocity of fractured media (m/s)

w :

Fracture aperture (m)

w avg :

Average fracture aperture of fractured media (m)

w c :

Critical fracture aperture (m)

w max :

Maximum fracture aperture (m)

w min :

Minimum fracture aperture (m)

W :

Width of representative volume element (m)

Δh :

Fluid pressure head loss across a fracture along the flow direction (m)

Δp :

Pressure drop across a fracture along the flow direction (Pa)

P :

Pressure gradient (Pa/m)

ε :

Dimensionless constant for nonlinear flow effect

μ :

Dynamic viscosity of the fluid (Pa s)

ν :

Kinematic viscosity of the fluid (m2/s)

ρ :

Density of the fluid (kg/m3)

ϕ :

Porosity of fractured media

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Acknowledgements

This work was partly supported by the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20150922) and was also partly supported by the Wyoming Center for Environmental Hydrology and Geophysics (WyCEHG) EPSCoR RII Track-1 Project and funded by the US National Science Foundation EPS-1208909.

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Correspondence to Peng Deng.

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Deng, P., Zhu, J. Equivalent Permeability of Fractured Media Incorporating Tortuosity and Nonlinear Flow. Transp Porous Med 132, 741–760 (2020). https://doi.org/10.1007/s11242-020-01410-3

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