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The Friction Factor in the Forchheimer Equation for Rock Fractures


The friction factor is an important dimensionless parameter for fluid flow through rock fractures that relates pressure head loss to average flow velocity; it can be affected by both fracture geometry and flow regime. In this study, a theoretical formula form of the friction factor containing both viscous and inertial terms is formulated by incorporating the Forchheimer equation, and a new friction factor model is proposed based on a recent phenomenological relation for the Forchheimer coefficient. The viscous term in the proposed formula is inversely proportional to Reynolds number and represents the limiting case in Darcy flow regime when the inertial effects diminish, whereas the inertial term is a power function of the relative roughness and represents a limiting case in fully turbulent flow regime when the fracture roughness plays a dominant role. The proposed model is compared with existing friction factor models for fractures through parametric sensitivity analyses and using experimental data on granite fractures, showing that the proposed model has not only clearer physical significance, but also better predictive performance. By accepting proper percentages of nonlinear pressure drop to quantify the onset of Forchheimer flow and fully turbulent flow, a Moody-type diagram with explicitly defined flow regimes is created for rock fractures of varying roughness, indicating that rougher fractures have a large friction factor and are more prone to the Forchheimer flow and fully turbulent flow. These findings may prove useful in better understanding of the flow behaviors in rock fractures and improving the numerical modeling of non-Darcy flow in fractured aquifers.

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f :

Friction factor


Reynolds number

ρ :

Fluid density

Q :

Volumetric flow rate

μ :

Fluid viscosity

w :

Fracture width

ξ :

Mean peak asperity height

e h :

Hydraulic aperture

P :

Pressure gradient (drop)

k :

Intrinsic permeability of fracture

A h :

Cross-sectional area

β :

Non-Darcy coefficient or inertial resistance

A, B :

Forchheimer coefficients describing pressure losses due to viscous and inertial dissipation mechanisms, respectively

a D, b D, a, b :

Dimensionless coefficients

D h :

Hydraulic diameter

L :

Fracture length

v :

Average velocity

σ 3 :

Confining stress

T 0, T a :

Intrinsic and apparent transmissivity of fracture

β c :

Nonlinear deviation factor

α :

Non-Darcy effect factor


Normalized objective function

γ :

Slope of the regression line in the plot of theoretical versus experimental values of friction factor


Root mean square error between the theoretical and experimental values of friction factor

X :

Overall mean of the experimental values of friction factor


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The authors Gratefully thank the anonymous reviewers for their kind efforts and constructive comments in improving this study. Financial supports from the National Natural Science Foundation of China (No. 51579188), the National Basic Research Program of China (No. 2011CB013503), and the China Postdoctoral Science Foundation (No. 2015M580672) are Gratefully acknowledged.

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Correspondence to Yi-Feng Chen.

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Zhou, JQ., Hu, SH., Chen, YF. et al. The Friction Factor in the Forchheimer Equation for Rock Fractures. Rock Mech Rock Eng 49, 3055–3068 (2016).

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