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

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

where

$\begin{array}{l}q_i = q(t=0)\end{array}$	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 stringer 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 four types based on the value of $\begin{array}{l}b\end{array}$ coefficient:

Exponential

Harmonic

Hyperbolic

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

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

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

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

(4)	$\begin{array}{l}\displaystyle q(t)=\frac{q_{i}}{1+D \, t}\end{array}$

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

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

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

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

Exponential decline has a clear physical meaning of pseudo-steady state production with finite drainage volume.

References

Arps, J.J.: “ Analysis of Decline Curves,” Trans. AIME,160, 228-247, 1945.

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