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

(1)	$\begin{array}{l}\displaystyle q(t)=q_0 \cdot \left( 1+b \cdot D \cdot t \right)^{-1/b}\end{array}$

where

$\begin{array}{l}q_0 = q(t=0)\end{array}$	Initial production rate of a well (or groups of wells)
$\begin{array}{l}D=-\frac{1}{q}\frac{dq}{dt}\end{array}$	decline decrement (the higher the $\begin{array}{l}D\end{array}$ the stronger is decline)
$\begin{array}{l}0 \leq b \leq 1\end{array}$	defines the type of decline (see below)

The cumulative production is then:

(2)	$\begin{array}{l}\displaystyle Q(t)=\int_0^t q(t) dt\end{array}$

Arp's model splits into three types based on the value of $\begin{array}{l}b\end{array}$ coefficient:

Exponential

Hyperbolic

Harmonic

$\begin{array}{l}b=0\end{array}$

$\begin{array}{l}0< b< 1\end{array}$

$\begin{array}{l}b=1\end{array}$

(3)	$\begin{array}{l}\displaystyle q(t)=q_0 \exp \left( -D \, t \right)\end{array}$

(4)	$\begin{array}{l}\displaystyle q(t)=q_0 \cdot \left( 1+b \cdot D \cdot t \right)^{-1/b}\end{array}$

(5)	$\begin{array}{l}\displaystyle q(t)=\frac{q_0}{1+D \, t}\end{array}$

(6)	$\begin{array}{l}\displaystyle Q(t)=\frac{q_0-q(t)}{D}\end{array}$

(7)	$\begin{array}{l}\displaystyle Q(t)=\frac{q_0}{D \, (1-b)} \, \left[ 1-{\frac{q(t)}{q_0}}^{1-b} \right]\end{array}$

(8)	$\begin{array}{l}\displaystyle Q(t)=\frac{q_0}{D} \, \ln \left[ \frac{q_0}{q(t)} \right]\end{array}$

(9)	$\begin{array}{l}\displaystyle Q_max=\frac{q_0}{D}\end{array}$

(10)	$\begin{array}{l}\displaystyle Q_max=\frac{q_0}{D \, (1-b)}\end{array}$

(11)	$\begin{array}{l}\displaystyle Q_max=\infty\end{array}$

Arps decline only work for Boundary Dominated Flow.

Exponential decline has a physical meaning of declining production from finite drainage volume $\begin{array}{l}V_e\end{array}$ with constant BHP: $\begin{array}{l}p_{wf}(t) = \rm const\end{array}$ .

Harmonic and Hyperbolic declines are empirical.

References

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

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