Given:
- a function q^*(t, {\bf p}) of real-value argument t \in \R and set of model parameters {\bf p} = \{ p_m\}_{m = 1..M} = \{p_1, p_2, ... p_M\}
- a training data set: \{ (t_k, q_k)\}_{k = 1..N} = \{ (t_0, a_0), (t_1, q_1), ..., (t_N, q_N) \}
the matching procedure assumes searching for thee specific set of model parameters {\bf p} to minimize the goal function:
F({\bf p}) = \sum_{k=1}^N \, \Psi \left( q^*(t_k) - q_k \right) \rightarrow \textrm{min} |
where \Psi(z) is the discrepancy distance function.
Most popular choices are \Psi(z) = z^2 and \Psi(z) = |z|.
There are few approaches to match the Arps decline to the historical data:
- Unconstrained matching
- 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
Unconstrained matching
All three model parameters \{ q^*_0, \, D_0, \, b \} are being varied to achieve the best fit to the training dataset.
Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline | ||||||
---|---|---|---|---|---|---|---|---|
b=0 |
0<b<1 | b=1 | ||||||
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The best-fit model may not match:
- the initial production rate q^*(t=0) \neq q_0
- the current production rate q^*(t=t_N) \neq q_N
- the current cumulative production Q^*(t=t_N) \neq Q_N
Match the value of the initial rate q^*(t=0) = q_0
The value of the model rate at the initial time moment is set to training dataset: q^*(t=0) = q_0 and the two other model properties \{ D_0, \, b \} are being varied to achieve the best fit to the training dataset.
Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline | ||||||
---|---|---|---|---|---|---|---|---|
b=0 |
0<b<1 | b=1 | ||||||
|
|
|
The best-fit model may not match:
- the current production rate q^*(t=t_N) \neq q_N
- the current cumulative production Q^*(t=t_N) \neq Q_N
Match the value of the current rate 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 | ||||||
---|---|---|---|---|---|---|---|---|
b=0 |
0<b<1 | b=1 | ||||||
|
|
| ||||||
|
|
|
The best-fit model may not match:
- the initial production rate q^*(t=0) \neq q_0
- the current cumulative production Q^*(t=t_N) \neq Q_N
Match the value of the current 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 | ||||||
---|---|---|---|---|---|---|---|---|
b=0 |
0<b<1 | b=1 | ||||||
|
|
| ||||||
|
|
|
The best-fit model may not match:
- the initial production rate q^*(t=0) \neq q_0
- the current production rate q^*(t=t_N) \neq q_N
Match the value of the current rate and cumulative 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 | ||||||
---|---|---|---|---|---|---|---|---|
b=0 |
0<b<1 | b=1 | ||||||
|
|
| ||||||
|
|
|
The best-fit model may not match:
- the initial production rate q^*(t=0) \neq q_0
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis / DCA Arps @model