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} = \{ (q_0, t_0), (q_1, t_1), ..., (t_N, q_N) \}
the matching procedure assumes minimizing the goal function:
F({\bf p}) = \sum_{k=1}^N \, \Psi(q^*(t_k) - q_k) \rightarrow \textrm(min) |
where \Psi(x) is the discrepancy distance function.
Most popular choices are \Psi(x) = x^2 and \Psi(x) = |x|.
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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b=0 |
0<b<1 | b=1 | ||||||
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See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis / DCA Arps @model