Arp's mathematical model of Decline Curve Analysis is based on the following equation:
(1) | q(t)=q_0 \cdot \left( 1+b \cdot D \cdot t \right)^{-1/b} |
where
q_0 = q(t=0) | Initial production rate of a well (or groups of wells) |
D=-\frac{1}{q}\frac{dq}{dt} | decline decrement (the higher the D the stronger is decline) |
0 \leq b \leq 1 | defines the type of decline (see below) |
The cumulative production is then:
(2) | Q(t)=\int_0^t q(t) dt |
Arp's model splits into three types based on the value of
b coefficient:
Exponential | Hyperbolic | Harmonic | ||||||
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b=0 |
0<b<1 | b=1 | ||||||
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The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves Q_{\rm max} \leq \infty while Harmonic decline is associated with production of infinite reserves Q_{\rm max} = \infty.
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 V_e with constant BHP: p_{wf}(t) = \rm const.
Harmonic and Hyperbolic declines are both empirical.
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis