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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}


Fracture width at well site

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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} \, t^{1/5


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_f = 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

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bodyt = Q(t)/q

injection time

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bodyq

injection rate

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bodyQ(t)

cumulative injection over time 

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bodyt

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bodyh_f

fracture height

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body--uriencoded--E' =\frac%7BE%7D%7B1-\nu%5e2%7D

plain stress

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bodyE

Young modulus

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body\nu

Poisson ratio

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body\mu

fluid viscosity

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