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In general , the matching procedure assumes minimizing the goal function:
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F = \sum_{k=1}^N \, (q^*(t_k) - q_k)^2 \rightarrow \textrm(min) |
There are few approaches to match the Arps decline to the historical data:
- Unconstrained matching (free)
- Constrained matching:
- Match the value of the initial rate
q^*(t=0) = q_0
- Match the value of the current rate
q^*(t=t_N) = q_N
- Match the value of the current cumulative
Q^*(t=t_N) = Q_N
- Match the value of the current rate and cumulative
q^*(t=t_N) = q_N,
Q^*(t=t_N) = Q_N
To ensure the smooth transition from historical data
[(t_1,q_1)... (t_N, q_N)] to the production forecasts in future time moments
[(t_{N+1},q_{N+1}), ...] one may wish to constrain the model by firm matching the production at the last historical moment
(t_N, q_N) which leads to the following form of Arp's model:
Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
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| | |
(1) |
q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big] |
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(2) |
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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(3) |
q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t } \right] |
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(4) |
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0 |
|
(5) |
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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(6) |
Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)} |
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See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis / DCA Arps @model