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q(t)=q_0 \cdot \left( 1+b \cdot D_0 \cdot t \right)^{-1/b}

where

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bodyq_0 = q(t=0)

Initial production rate of a well (or groups of wells)

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bodyD=-\frac{1}{q}\frac{dq}{dt}_0

model parameter characterizing the decline rate decline decrement (the higher the

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bodyD_0
the stronger is decline)

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body0 \leq b \leq 1

defines the type of decline (see below)

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Q_{\rm max}=Q(t=\infty)=\int_0^\infty q(t) \, dt =\frac{q_0}{D_0 \cdot (1-b)}


Arp's model splits into three types based on the value of 

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bodyb
 coefficient:

Exponential Production DeclineHyperbolic Production DeclineHarmonic Production Decline

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bodyb=0

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body0<b<1

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bodyb=1


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q(t)=q_0 \exp \left( -D_0 \, t \right)



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q(t)=q_0 \cdot \left( 1+b \cdot D_0 \cdot t \right)^{-1/b}



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q(t)=\frac{q_0}{1+D_0 \, t} 



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Q(t)=\frac{q_0-q(t)}{D_0}



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Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b}  \right]



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Q(t)=\frac{q_0}{D_0} \, \ln \left[ \frac{q_0}{q(t)} \right]



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Q_{\rm max}=\frac{q_0}{D_0}



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Q_{\rm max}=\frac{q_0}{D_0 \cdot (1-b)}



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Q_{\rm max}=\infty


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