Given:

a function $\begin{array}{l}q^*(t, {\bf p})\end{array}$ of real-value argument $\begin{array}{l}t \in \R\end{array}$ and set of model parameters $\begin{array}{l}{\bf p} = \{ p_m\}_{m = 1..M} = \{p_1, p_2, ... p_M\}\end{array}$
a training data set: $\begin{array}{l}\{ (t_k, q_k)\}_{k = 1..N} = \{ (t_0, a_0), (t_1, q_1), ..., (t_N, q_N) \}\end{array}$

the matching procedure assumes searching for thee specific set of model parameters $\begin{array}{l}{\bf p}\end{array}$ to minimize the goal function:

$\begin{array}{l}\displaystyle F({\bf p}) = \sum_{k=1}^N \, \Psi(q^*(t_k) - q_k) \rightarrow \textrm(min)\end{array}$

where $\begin{array}{l}\Psi(z)\end{array}$ is the discrepancy distance function.

Most popular choices are $\begin{array}{l}\Psi(z) = z^2\end{array}$ and $\begin{array}{l}\Psi(z) = |z|\end{array}$ .

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 $\begin{array}{l}q^*(t=0) = q_0\end{array}$
- Match the value of the current rate $\begin{array}{l}q^*(t=t_N) = q_N\end{array}$
- Match the value of the current cumulative $\begin{array}{l}Q^*(t=t_N) = Q_N\end{array}$
- Match the value of the current rate and cumulative $\begin{array}{l}q^*(t=t_N) = q_N\end{array}$ , $\begin{array}{l}Q^*(t=t_N) = Q_N\end{array}$

To ensure the smooth transition from historical data $\begin{array}{l}[(t_1,q_1)... (t_N, q_N)]\end{array}$ to the production forecasts in future time moments $\begin{array}{l}[(t_{N+1},q_{N+1}), ...]\end{array}$ one may wish to constrain the model by firm matching the production at the last historical moment $\begin{array}{l}(t_N, q_N)\end{array}$ which leads to the following form of Arp's model:

Exponential Production Decline

Hyperbolic Production Decline

Harmonic Production Decline

$\begin{array}{l}b=0\end{array}$

$\begin{array}{l}0< b< 1\end{array}$

$\begin{array}{l}b=1\end{array}$

(1)	$\begin{array}{l}\displaystyle q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big]\end{array}$

(2)	$\begin{array}{l}\displaystyle q(t) = q_N \cdot \left[ \frac{1+b \cdot D_0 \cdot t_N } { 1+b \cdot D_0 \cdot t } \right]^{1/b}\end{array}$

(3)	$\begin{array}{l}\displaystyle q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N } { 1+ D_0 \cdot t } \right]\end{array}$

(4)	$\begin{array}{l}\displaystyle Q(t) - Q_N = [ q_N - q(t)] \, \tau_0\end{array}$

(5)	$\begin{array}{l}\displaystyle Q(t) - Q_N = \frac{q_N^b \, (\tau_0 + b \, t_N)}{1-b} \left[ q_N^{1-b} - q^{1-b}(t) \right]\end{array}$

(6)	$\begin{array}{l}\displaystyle Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)}\end{array}$

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DCA Arps Matching @model

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