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A specific type of mathematical model of Decline Curve Analysis, 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 given by:

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

and the ultimate recovery is defined as:

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

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

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- \left( \frac{q(t)}{q_0} \right)^{1-b}  \right]

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

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

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

Q_{\rm max}=\infty

The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves

Since all physical reserves are finite the true meaning of Harmonic decline is that up to date it did not reach the boundary of these reserves and at certain point in future it will transform intoa finite-reseves decline (possibly Exponential or Hyperbolic).

Exponential decline has a physical meaning of declining production from finite drainage volume 

Harmonic and Hyperbolic declines are both empirical.

The DCA Arps do not cover all types of production decline, but their application is quite broad and mathematics is quite simple which gained popularity.

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