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Fracture half-length

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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

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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

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\bar w_f =  \frac{\pi}{5} \, w_{f0} 


Net pressure at the wellbore

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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}

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