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| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
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| q(t)=q_0 \exp \left( -D_0 \, t \right) |
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| q(t) = \frac{q_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} } |
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| q(t)=\frac{q_0}{1+D_0 \, t} |
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| Q(t)=\frac{q_0-q(t)}{D_0} |
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| Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]
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| 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) |
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| Q_{\rm max}=\frac{q_0}{D_0} |
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| Q_{\rm max}=\frac{q_0}{D_0 \cdot (1-b)} |
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| Q_{\rm max}=\infty |
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| D(t) = D_0 = \rm const |
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| anchor | D_hyper |
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| alignment | left |
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| D(t) =\frac{D_0}{1+ b \cdot D_0 \cdot t} |
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| anchor | D_harm |
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| alignment | left |
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| D(t) = \frac{D_0}{1+ D_0 \cdot t} |
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| anchor | tau_exp |
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| alignment | left |
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| \tau(t) = \tau_0 = \rm const |
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| anchor | tau_hyper |
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| alignment | left |
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| \tau(t) = \tau_0 + b \cdot t |
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| anchor | tau_harm |
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| alignment | left |
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| \tau(t) = \tau_0 + t |
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| anchor | RPR_exp |
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| alignment | left |
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| \mathrm{RPR}(t) = \tau(t) = \tau_0 = \rm const |
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| anchor | RPR_hyper |
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| alignment | left |
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| \mathrm{RPR}(t) = \tau_0 \, \left[
1 + \frac{b }{(1-b)} \cdot \frac{q_0}{q(t)}
\right]
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| anchor | RPR_harm |
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| alignment | left |
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| \mathrm{RPR}(t) = \infty |
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To ensure the smooth transition from historical data
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| body | [(t_1,q_1)... (t_N, q_N)] |
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to the production forecasts in future time moments | LaTeX Math Inline |
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| body | --uriencoded--[(t_%7BN+1%7D,q_%7BN+1%7D), ...] |
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one may wish to constrain the model by firm matching the production at the last historical moment which leads to the following form of Arp's model:...
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q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big] |
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q(t) = q_N \cdot \left[ \frac{1+b \cdot D_0 \cdot t_N }
{ 1+b \cdot D_0 \cdot t } \right]^{1/b} |
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q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t } \right] |
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Q(t) - Q_N = [ q_N - q(t)] \, \tau_0 |
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Q(t) - Q_N = \frac{q_N^b \, (\tau_0 + b \, t_N)}{1-b} \left[ q_N^{1-b} - q^{1-b}(t) \right] |
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Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)} |
The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves
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| body | --uriencoded--Q_%7B\rm max%7D \leq \infty |
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while
Harmonic decline is associated with production from the reservoir with infinite reserves
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| body | --uriencoded--Q_%7B\rm max%7D = \infty |
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.
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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
with constant
BHP:
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| body | p_{wf}(t) = \rm const |
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(a specific type of
Boundary Dominated Flow under
Pseudo Steady State (PSS) conditions).
Harmonic and Hyperbolic declines are both empiricalThere 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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