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 $\begin{array}{l}{\bf p} = \{ q^*_0, \, \tau_0, \, b \}\end{array}$
a training data set: $\begin{array}{l}\{ (t_k, q_k)\}_{k = 1..N} = \{ (t_0, q_0), (t_1, q_1), ..., (t_N, q_N) \}\end{array}$

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

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

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 (or a training dataset within):

Unconstrained matching
Constrained matching:
- Match the value of the initial rate $\begin{array}{l}q^*(t=0) = q^*_0 = q_0\end{array}$
- 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 $\begin{array}{l}q^*(t=t_N) = q_N\end{array}$ , $\begin{array}{l}Q^*(t=t_N) = Q_N\end{array}$

Unconstrained matching

All three model parameters $\begin{array}{l}\{ q^*_0, \, \tau_0, \, b \}\end{array}$ are being varied to achieve the best fit to the training dataset.

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)	$\begin{array}{l}\displaystyle q(t)=q^*_0 \exp \left( - t/\tau_0 \right)\end{array}$

(2)	$\begin{array}{l}\displaystyle q(t) = \frac{q^*_0}{ \left( 1+b \cdot t/\tau_0 \right)^{1/b} }\end{array}$

(3)	$\begin{array}{l}\displaystyle q(t)=\frac{q^*_0}{1+t/\tau_0}\end{array}$

The best-fit model may not match:

the initial production rate $\begin{array}{l}q^*(t=0) \neq q_0\end{array}$
the current production rate $\begin{array}{l}q^*(t=t_N) \neq q_N\end{array}$
the current cumulative production $\begin{array}{l}Q^*(t=t_N) \neq Q_N\end{array}$

Match the value of the initial rate $\begin{array}{l}q^*(t=0) = q_0\end{array}$

The value of the model rate at the initial time moment is set to training dataset: $\begin{array}{l}q^*(t=0) = q^*_0 = q_0\end{array}$ and the two other model properties $\begin{array}{l}\{ \tau_0, \, b \}\end{array}$ are being varied to achieve the best fit to the training dataset.

Exponential Production Decline

Hyperbolic Production Decline

Harmonic Production Decline

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

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

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

(4)	$\begin{array}{l}\displaystyle q(t)=q_0 \exp \left( -t/\tau_0 \right)\end{array}$

(5)	$\begin{array}{l}\displaystyle q(t) = \frac{q_0}{ \left( 1+b \cdot t/\tau_0 \right)^{1/b} }\end{array}$

(6)	$\begin{array}{l}\displaystyle q(t)=\frac{q_0}{1+t/\tau_0}\end{array}$

The best-fit model may not match:

the current production rate $\begin{array}{l}q^*(t=t_N) \neq q_N\end{array}$
the current cumulative production $\begin{array}{l}Q^*(t=t_N) \neq Q_N\end{array}$

Match the value of the current rate $\begin{array}{l}q^*(t=t_N) = q_N\end{array}$

The value of the model rate at the current time moment is set to training dataset: $\begin{array}{l}q^*(t=t_N) = q_N\end{array}$ and the two other model properties $\begin{array}{l}\{ \tau_0, \, b \}\end{array}$ are being varied to achieve the best fit to the training dataset.

Exponential Production Decline

Hyperbolic Production Decline

Harmonic Production Decline

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

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

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

(7)	$\begin{array}{l}\displaystyle q(t)=q_N \cdot \exp \big[ -(t-t_N)/\tau_0 \big]\end{array}$

(8)	$\begin{array}{l}\displaystyle q(t) = q_N \cdot \left[ \frac{1+b \cdot t_N/\tau_0 } { 1+b \cdot t/\tau_0 } \right]^{1/b}\end{array}$

(9)	$\begin{array}{l}\displaystyle q(t) = q_N \cdot \left[ \frac{1+t_N/\tau_0 } { 1+ t/\tau_0 } \right]\end{array}$

This ensures 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 $\begin{array}{l}[(t_{N+1},q_{N+1}), ...]\end{array}$ .

The best-fit model may not match:

the initial production rate $\begin{array}{l}q^*(t=0) \neq q_0\end{array}$
the current cumulative production $\begin{array}{l}Q^*(t=t_N) \neq Q_N\end{array}$

Match the value of the current cumulative $\begin{array}{l}Q^*(t=t_N) = Q_N\end{array}$

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

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

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

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

(10)	$\begin{array}{l}\displaystyle q(t)=\end{array}$

(11)	$\begin{array}{l}\displaystyle q(t) = \frac{q^*_0}{ \left( 1+b \cdot t/\tau_0 \right)^{1/b} }\end{array}$

(12)	$\begin{array}{l}\displaystyle q(t)=\frac{q^*_0}{1+t/\tau_0}\end{array}$

(13)	$\begin{array}{l}\displaystyle Q(t) - Q_N = [ q^*_N - q(t)] \, \tau_0\end{array}$

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

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

The best-fit model may not match:

the initial production rate $\begin{array}{l}q^*(t=0) \neq q_0\end{array}$
the current production rate $\begin{array}{l}q^*(t=t_N) \neq q_N\end{array}$

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

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

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

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

(16)	$\begin{array}{l}\displaystyle q(t)=q_N \cdot \exp \big[ -(t-t_N)/\tau_0 \big]\end{array}$

(17)	$\begin{array}{l}\displaystyle q(t) = q_N \cdot \left[ \frac{1+b \cdot t_N/\tau_0 } { 1+b \cdot t/\tau_0 } \right]^{1/b}\end{array}$

(18)	$\begin{array}{l}\displaystyle q(t) = q_N \cdot \left[ \frac{1+t_N/\tau_0 } { 1+ t/\tau_0 } \right]\end{array}$

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

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

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

The best-fit model may not match:

the initial production rate $\begin{array}{l}q^*(t=0) \neq q_0\end{array}$

Page tree

Unconstrained 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 $\begin{array}{l}q^(t=t_N) = q_N\end{array}$ , $\begin{array}{l}Q^(t=t_N) = Q_N\end{array}$

See Also

Page tree

DCA Arps Matching @model

Unconstrained matching

Match the value of the initial rate q^*(t=0) = q_0//<![CDATA[ \begin{array}{l}q^*(t=0) = q_0\end{array} //]]>

Match the value of the current rate q^*(t=t_N) = q_N//<![CDATA[ \begin{array}{l}q^*(t=t_N) = q_N\end{array} //]]>

Match the value of the current cumulative Q^*(t=t_N) = Q_N//<![CDATA[ \begin{array}{l}Q^*(t=t_N) = Q_N\end{array} //]]>

Match the value of the current rate and cumulative q^*(t=t_N) = q_N//<![CDATA[ \begin{array}{l}q^*(t=t_N) = q_N\end{array} //]]> , Q^*(t=t_N) = Q_N//<![CDATA[ \begin{array}{l}Q^*(t=t_N) = Q_N\end{array} //]]>

See Also

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 $\begin{array}{l}q^(t=t_N) = q_N\end{array}$ , $\begin{array}{l}Q^(t=t_N) = Q_N\end{array}$