Given:
- a function
of real-value argument 
and set of model parameters 
- a training data set:
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 
is the discrepancy distance function.
Most popular choices are 
and 
.
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
- Match the value of the current rate
- Match the value of the current cumulative
- Match the value of the current rate and cumulative ,
To ensure the smooth transition from historical data 
to the production forecasts in future time moments 
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:
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
|---|
| | |
q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big] |
| 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} |
| q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t } \right] |
|
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0 |
| 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] |
| Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)} |
|
See Also
Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis / DCA Arps @model