Page History

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• Unconstrained matching 
• Constrained matching:
  • Match the value of the initial rate 

    LaTeX Math Inline
    body --uriencoded--q%5e*(t=0) = q%5e*_0 = q_0

  • Match the value of the current rate

    LaTeX Math Inline
    body --uriencoded--q%5e*(t=t_N) = q_N

  • Match the value of the current cumulative

    LaTeX Math Inline
    body --uriencoded--Q%5e*(t=t_N) = Q_N

  • Match the value of the current rate and cumulative

    LaTeX Math Inline
    body --uriencoded--q%5e*(t=t_N) = q_N

    ,

    LaTeX Math Inline
    body --uriencoded--Q%5e*(t=t_N) = Q_N

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

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Unconstrained Unconstrained

Unconstrained matching

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The value of the model rate at the initial time moment is set to training dataset: 

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Current_cumulative Current_cumulative

Match the value of the current cumulative 

LaTeX Math Inline
body --uriencoded--Q%5e*(t=t_N) = Q_N

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To ensure the smooth transition from historical data

q(t) =  (\frac{Q_N/\tau_0) \cdot }{1-\exp(-t_N/\tau_0)}  \cdot \left[ 1 - \exp( -t_n/\tau_0) \right]^{-1}

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

q(t)= \cdot \frac{q^*_01}{1+t/\tau_0} 

left

Q(t)1 - Q_N = [ q^*_N - q(t)] + b\, t/\tau_0 \right)^{1/b}}

Qq(t) - Q_N = \frac{q^*_N^b \, (Q_N/\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)\ln (1+ t_N/\tau_0)} \cdot \frac{1}{1+t/\tau_0}

The best-fit model may not match: 

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Current_rate_cumulative Current_rate_cumulative

Match the value of the current rate and cumulative 

LaTeX Math Inline
body --uriencoded--q%5e*(t=t_N) = q_N

, 

LaTeX Math Inline
body --uriencoded--Q%5e*(t=t_N) = Q_N

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The value of the model rate at the current time moment and decline pace are set to match both current rate To ensure the smooth transition from historical data

This makes Exponential Production Decline and Harmonic Production Decline are fully set while Hyperbolic Production Decline has opportunity to vary one model parameter

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

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

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

Q(t) - Q_N = [ q_N - q(t)] \, \tau_0

Q(t) - Q_N = \frac{qQ_N^b \, (\N/\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)\ln (1+ t_N/\tau_0)} \cdot \frac{1}{1+t/\tau_0}

The best-fit model may not match: 

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