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

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

where

$\begin{array}{l}q_i = q(t=0)\end{array}$

Initial production rate of a well (or groups of wells)

(2)	$\begin{array}{l}\displaystyle D=-\frac{1}{q}\frac{dq}{dt}\end{array}$

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:

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

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

Exponential

Harmonic

Hyperbolic

Power Loss

b = 1

b = 0

0 < b < 1

$\begin{array}{l}\displaystyle D=D_{\infty} + \frac{t^{n-1}}{\tau^{n}}\end{array}$

(4)	$\begin{array}{l}\displaystyle q(t)=q_{i} \exp \big [ -D \, t \big ]\end{array}$

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

(6)	$\begin{array}{l}\displaystyle q(t)=\frac{q_{i}}{[1+b \, D \, t]^{\frac{1}{b}}}\end{array}$

(7)	$\begin{array}{l}\displaystyle q(t)=q_{i} \exp \big [ -D_{\infty}t- \bigg(\frac{t}{\tau} \bigg)^{n} \big]\end{array}$

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

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

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

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

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