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A specific type of mathematical model of Decline Curve Analysis, based on the following equation for Production rate

q(t) = q_0 \cdot \left( 1+b \cdot D_0 \cdot t \right)^{-1/b}

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Q(t)=\int_0^t q(t) \, dt

The Production Decrement is given by value can be a given an explicit form:

D(t) = - \frac{dq}{dQ} = - \frac{d \, \ln q(t)}{dt} = \frac{D_0}{1+ b \cdot D_0 \cdot t}

The Production DecrementThe Recovery Pace is the inverse value to Production Decrement:

\tau(t) = \frac{1}{D(t)} = - \frac{dQ}{dq} = - \left[ \frac{d \ln q}{dt} \right]^{-1}  \quad \rightarrow  \quad  \tau(t) = \tau_0 + b\cdot t = \frac{1}{D_0} + b\cdot t

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RPR(t) = \frac{Q_{max}-Q(t)}{q(t)}

Arp's model splits  is usually split into three types based on the value of 

q(t)=q_0 \exp \left( -D_0 \, t \right)

q(t) = \frac{q_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} }

q(t)=\frac{q_0}{1+D_0 \, t} 

Q(t)=\frac{q_0-q(t)}{D_0}

Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b}  \right]

Q(t)=\frac{q_0}{D_0} \, \ln \left[ \frac{q_0}{q(t)} \right] =  \frac{q_0}{D_0} \ln q_0 + \frac{q_0}{D_0}  \cdot \ln q(t)

Q_{\rm max}=\frac{q_0}{D_0}

Q_{\rm max}=\frac{q_0}{D_0 \cdot (1-b)}

Q_{\rm max}=\infty

D(t) = D_0 = \rm const

D(t) =\frac{D_0}{1+ b \cdot D_0 \cdot t}

D(t) = \frac{D_0}{1+ D_0 \cdot t}

\tau(t) = \tau_0 = \rm const

\tau(t) = \tau_0 + b \cdot t

\tau(t) = \tau_0 + t

\mathrm{RPR}(t) = \tau(t) = \tau_0 = \rm const

\mathrm{RPR}(t) = \tau_0 +\, \frac{\tau_0 \, q_0 \, left[ 
1 + \frac{b }{(1-b)} \,cdot \frac{1q_0}{q(t)}
\right]

\mathrm{RPR}(t) = \rm{not applicable due to infinite volume}infty

The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves

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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 a certain point in future it will transform transit into a finite-reserves decline (possibly Exponential or Hyperbolic).

The Harmonic decline is also observed at the mature stage of waterflood projects.

Both Harmonic and Hyperbolic declines are empirical while Exponential decline has a physical meaning.
Exponential Production Decline has a physical meaning of declining production from finite drainage volume 

There are few approaches to match the Arps decline to the historical data which are covered in DCA Arps Matching @model.

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.

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[ Exponential Production Decline ][ Hyperbolic Production Decline ][ Harmonic Production Decline ][ Production Decrement ] 

[ DCA Arps Matching @model ]

References

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

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