A specific type of mathematical model of Decline Curve Analysis, based on the following equation:

(1)	$\begin{array}{l}\displaystyle q(t)=q_0 \cdot \left( 1+b \cdot D_0 \cdot t \right)^{-1/b}\end{array}$

where

$\begin{array}{l}q_0 = q(t=0)\end{array}$	Initial production rate of a well (or groups of wells)
$\begin{array}{l}D_0 > 0\end{array}$	initial Production decline rate (the higher the $\begin{array}{l}D_0\end{array}$ the stronger is decline)
$\begin{array}{l}0 \leq b \leq 1\end{array}$	defines the type of decline (see below)

It can be applied to any fluid production: water, oil or gas.

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

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

Exponential Production Decline

Hyperbolic Production Decline

Harmonic Production Decline

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

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

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

(4)	$\begin{array}{l}\displaystyle q(t)=q_0 \exp \left( -D_0 \, t \right)\end{array}$

(5)	$\begin{array}{l}\displaystyle q(t)=q_0 \cdot \left( 1+b \cdot D_0 \cdot t \right)^{-1/b}\end{array}$

(6)	$\begin{array}{l}\displaystyle q(t)=\frac{q_0}{1+D_0 \, t}\end{array}$

(7)	$\begin{array}{l}\displaystyle Q(t)=\frac{q_0-q(t)}{D_0}\end{array}$

(8)	$\begin{array}{l}\displaystyle Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]\end{array}$

(9)	$\begin{array}{l}\displaystyle Q(t)=\frac{q_0}{D_0} \, \ln \left[ \frac{q_0}{q(t)} \right]\end{array}$

(10)	$\begin{array}{l}\displaystyle Q_{\rm max}=\frac{q_0}{D_0}\end{array}$

(11)	$\begin{array}{l}\displaystyle Q_{\rm max}=\frac{q_0}{D_0 \cdot (1-b)}\end{array}$

(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 $\begin{array}{l}Q_{\rm max} = \infty\end{array}$ . In other words the Harmonic decline is very slow.

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 a finite-reserves decline (possibly Exponential or Hyperbolic).

Exponential Production Decline has a physical meaning of declining production from finite drainage volume $\begin{array}{l}V_e\end{array}$ with constant BHP: $\begin{array}{l}p_{wf}(t) = \rm const\end{array}$ (a specific type of Boundary Dominated Flow under Pseudo Steady State (PSS) conditions).

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 as quick estimation of production perspectives.

References

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

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