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 given by:

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

and the ultimate recovery is defined as:

(3)	$\begin{array}{l}\displaystyle Q_{\rm max}=Q(t=\infty)=\int_0^\infty q(t) \, dt =\frac{q_0}{D \cdot (1-b)}\end{array}$

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

Exponential

Hyperbolic

Harmonic

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

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

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

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

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

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

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

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

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

(10)	$\begin{array}{l}\displaystyle Q_{\rm max}=\frac{q_0}{D}\end{array}$

(11)	$\begin{array}{l}\displaystyle Q_{\rm max}=\frac{q_0}{D \cdot (1-b)}\end{array}$

(12)	$\begin{array}{l}\displaystyle Q_{\rm max}=\infty\end{array}$

The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves $\begin{array}{l}Q_{\rm max} \leq \infty\end{array}$ while Harmonic decline is associated with production from the reservoir with infinite reserves $\begin{array}{l}Q_{\rm max} = \infty\end{array}$ .

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 into Exponential or Hyperbolic decline.

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

Harmonic and Hyperbolic declines are both 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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