Given:

• a function  $//<![CDATA[ \begin{array}{l}q^*(t, {\bf p})\end{array} //]]>$ of real-value argument  $//<![CDATA[ \begin{array}{l}t \in \R\end{array} //]]>$ and set of model parameters  $//<![CDATA[ \begin{array}{l}{\bf p} = \{ q^*_0, \, \tau_0, \, b \}\end{array} //]]>$
• a training data set:  $//<![CDATA[ \begin{array}{l}\{ (t_k, q_k)\}_{k = 1..N} = \{ (t_0, q_0), (t_1, q_1), ..., (t_N, q_N) \}\end{array} //]]>$

the matching procedure assumes searching for thee specific set of model parameters  $//<![CDATA[ \begin{array}{l}{\bf p}\end{array} //]]>$ to minimize the goal function:

where  $//<![CDATA[ \begin{array}{l}\Psi(z)\end{array} //]]>$ is the discrepancy distance function.

Most popular choices are $//<![CDATA[ \begin{array}{l}\Psi(z) = z^2\end{array} //]]>$ and $//<![CDATA[ \begin{array}{l}\Psi(z) = |z|\end{array} //]]>$.

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  $//<![CDATA[ \begin{array}{l}q^*(t=0) = q^*_0 = q_0\end{array} //]]>$
  • Match the value of the current rate $//<![CDATA[ \begin{array}{l}q^*(t=t_N) = q_N\end{array} //]]>$
  • Match the value of the current cumulative $//<![CDATA[ \begin{array}{l}Q^*(t=t_N) = Q_N\end{array} //]]>$
  • Match the value of the current rate and cumulative $//<![CDATA[ \begin{array}{l}q^*(t=t_N) = q_N\end{array} //]]>$, $//<![CDATA[ \begin{array}{l}Q^*(t=t_N) = Q_N\end{array} //]]>$

The constrained matching is used to one may wish to ensure the smooth transition from the training dataset to future model predictions.

Unconstrained matching

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

The best-fit model may not match: 

• the initial production rate  $//<![CDATA[ \begin{array}{l}q^*(t=0) \neq q_0\end{array} //]]>$
• the current production rate  $//<![CDATA[ \begin{array}{l}q^*(t=t_N) \neq q_N\end{array} //]]>$
• the current cumulative production  $//<![CDATA[ \begin{array}{l}Q^*(t=t_N) \neq Q_N\end{array} //]]>$

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

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

The best-fit model may not match: 

• the current production rate  $//<![CDATA[ \begin{array}{l}q^*(t=t_N) \neq q_N\end{array} //]]>$
• the current cumulative production  $//<![CDATA[ \begin{array}{l}Q^*(t=t_N) \neq Q_N\end{array} //]]>$

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

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

This ensures the smooth transition from historical data $//<![CDATA[ \begin{array}{l}[(t_1,q_1)... (t_N, q_N)]\end{array} //]]>$ to the production forecasts in future time moments $//<![CDATA[ \begin{array}{l}[(t_{N+1},q_{N+1}), ...]\end{array} //]]>$.

The best-fit model may not match: 

• the initial production rate  $//<![CDATA[ \begin{array}{l}q^*(t=0) \neq q_0\end{array} //]]>$
• the current cumulative production  $//<![CDATA[ \begin{array}{l}Q^*(t=t_N) \neq Q_N\end{array} //]]>$

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

The value of the model rate at the initial time moment $//<![CDATA[ \begin{array}{l}q^*(t=0) = q^*_0\end{array} //]]>$ is set to achieve the match between the values of current cumulative from model prediction and training dataset   $//<![CDATA[ \begin{array}{l}Q^*(t=t_N) = Q_N\end{array} //]]>$:

The best-fit model may not match: 

• the initial production rate  $//<![CDATA[ \begin{array}{l}q^*(t=0) \neq q_0\end{array} //]]>$
• the current production rate  $//<![CDATA[ \begin{array}{l}q^*(t=t_N) \neq q_N\end{array} //]]>$

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

The value of the model rate at the current time moment and decline pace are set to match both current rate  $//<![CDATA[ \begin{array}{l}q^*(t=t_N) = q_N\end{array} //]]>$ and current cumulative  $//<![CDATA[ \begin{array}{l}Q^*(t=t_N) = Q_N\end{array} //]]>$.

This makes Exponential Production Decline and Harmonic Production Decline are fully set while Hyperbolic Production Decline has opportunity to vary one model parameter $//<![CDATA[ \begin{array}{l}\{ b \}\end{array} //]]>$ to achieve the best fit to the training dataset.

The best-fit model may not match: 

• the initial production rate  $//<![CDATA[ \begin{array}{l}q^*(t=0) \neq q_0\end{array} //]]>$

See Also

Petroleum Industry / Upstream /  Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis / DCA Arps @model