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Arp's mathematical model of Decline Curve Analysis is based on the following equation: 

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

where

The cumulative production is then:

Q(t)=\int_0^t q(t) dt

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

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

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

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

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

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

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

Q_max=\frac{q_0}{D}

Q_max=\frac{q_0}{D \, (1-b)}

Q_max=\infty

Arps decline only work for Boundary Dominated Flow.

Exponential decline has a physical meaning of declining production from finite drainage volume  with constant BHP: .

Harmonic and Hyperbolic declines are empirical.

See Also

Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis

References

Arps, J. J. (1945, December 1). Analysis of Decline Curves. Society of Petroleum Engineers. doi:10.2118/945228-G