Fracture half-length

X_f =  \frac{Q}{2 \, w_f \, h_f}

Average Fracture width

w_f = \frac{2 \pi \, p_{\rm net} \, h_f}{5 \, E'}

Net pressure at the wellbore

p_{\rm net} =  \left(  \frac{20}{\pi^2} \cdot \frac{E'^4 \, q \, Q \, \mu}{h_f^6} \right)^{1/5}

where

	fracture fluid injection time
	fracture fluid injection rate
	cumulative fracture fluid injection over time
	fracture height
	plane stress
	Young modulus
	Poisson's ratio
	fracture fluid viscosity

Fracture half-length

X_f = 0.524 \, \left( \frac{q^3 E'}{\mu \, h_f^4} \right)^{1/5} \, t^{4/5} = 0.524 \, \left( \frac{E'}{\mu \, h_f^4}  \frac{Q^4}{q}\right)^{1/5} = 0.8 \cdot \frac{E' \, Q}{p_{\rm net} \, h_f^2} = \frac{Q}{w_f \, h_f}

Fracture width at wellbore

w_{f0} = 3.04 \, \left( \frac{q^2 \mu}{E' \, h_f} \right)^{1/5} \, t^{1/5}= 3.04 \, \left( \frac{q \, Q \mu}{E' \, h_f} \right)^{1/5} = \frac{2 \, p_{\rm net} \, h_f}{E'}

Average Fracture width

w_f =  \frac{\pi}{5} \, w_{f0} = \frac{2 \pi \, p_{\rm net} \, h_f}{5 \, E'}

Net pressure at the wellbore

p_{\rm net} = 1.524 \, \left( \frac{E'^4 \, q^2 \, \mu}{h_f^6} \right)^{1/5} \, t^{1/5} = 1.524 \, \left( \frac{E'^4 \, q \, Q \, \mu}{h_f^6} \right)^{1/5}

where

	injection time
	injection rate
	cumulative injection over time
	fracture height
	plain stress
	Young modulus
	Poisson ratio
	fluid viscosity

Reference

Perkins, Kern and Nordgren

https://dnicolasespinoza.github.io/node56.html

See Also

Reference