Given:

• a function  of real-value argument  and set of model parameters 
• a training data set: 

the matching procedure assumes searching for thee specific set of model parameters  to minimize the goal function:

G({\bf p}) = \sum_{k=1}^N \, \Psi \left( q^*(t_k) - q_k \right) \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 (or a training dataset within):

• Unconstrained matching 
• 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 ,

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  are being varied to achieve the best fit to the training dataset.

q(t)=q^*_0 \exp \left( -  t/\tau_0 \right)

q(t) = \frac{q^*_0}{ \left( 1+b \cdot  t/\tau_0 \right)^{1/b} }

q(t)=\frac{q^*_0}{1+t/\tau_0} 

The best-fit model may not match: 

• the initial production rate 
• the current production rate 
• the current cumulative production 

Match the value of the initial rate 

The value of the model rate at the initial time moment is set to training dataset:  and the two other model properties  are being varied to achieve the best fit to the training dataset.

q(t)=q_0 \exp \left( -t/\tau_0 \right)

q(t) = \frac{q_0}{ \left( 1+b \cdot t/\tau_0 \right)^{1/b} }

q(t)=\frac{q_0}{1+t/\tau_0} 

The best-fit model may not match: 

• the current production rate 
• the current cumulative production 

Match the value of the current rate 

The value of the model rate at the current time moment is set to training dataset:  and the two other model properties  are being varied to achieve the best fit to the training dataset.

q(t)=q_N \cdot \exp \big[ -(t-t_N)/\tau_0 \big]

q(t) = q_N \cdot \left[ \frac{1+b \cdot t_N/\tau_0 }
{ 1+b \cdot t/\tau_0  } \right]^{1/b}

q(t) =  q_N \cdot  \left[ \frac{1+t_N/\tau_0 }
{ 1+ t/\tau_0  } \right]

This ensures the smooth transition from historical data  to the production forecasts in future time moments .

The best-fit model may not match: 

• the initial production rate 
• the current cumulative production 

Match the value of the current cumulative 

The value of the model rate at the initial time moment is set to achieve the match between the values of current cumulative from model prediction and training dataset  :

q(t) = \frac{Q_N/\tau_0}{1-\exp(-t_N/\tau_0)}  \cdot \exp(-t/\tau_0) 

q(t) = \frac{(1-b) \cdot Q_N/\tau_0}{ 1 - \left( 1 + b \, t_N/\tau_0 \right) ^{-b/(1-b)}} \cdot \frac{1}{\left(1 + b\, t/\tau_0 \right)^{1/b}}

q(t)=\frac{Q_N/\tau_0}{\ln (1+ t_N/\tau_0)} \cdot \frac{1}{1+t/\tau_0}

The best-fit model may not match: 

• the initial production rate 
• the current production rate 

Match the value of the current rate and cumulative , 

The value of the model rate at the current time moment and decline pace are set to match both current rate  and current cumulative .

This makes Exponential Production Decline and Harmonic Production Decline are fully set while Hyperbolic Production Decline has opportunity to vary one model parameter to achieve the best fit to the training dataset.

q(t) = \frac{Q_N/\tau_0}{1-\exp(-t_N/\tau_0)}  \cdot \exp(-t/\tau_0) 

q(t) = \frac{(1-b) \cdot Q_N/\tau_0}{ 1 - \left( 1 + b \, t_N/\tau_0 \right) ^{-b/(1-b)}} \cdot \frac{1}{\left(1 + b\, t/\tau_0 \right)^{1/b}}

q(t)=\frac{Q_N/\tau_0}{\ln (1+ t_N/\tau_0)} \cdot \frac{1}{1+t/\tau_0}

The best-fit model may not match: 

• the initial production rate 

See Also

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