Given:

• a function 

  LaTeX Math Inline
  body --uriencoded--q%5e*(t, %7B\bf p%7D)

   of real-value argument 

  LaTeX Math Inline
  body t \in \R

   and set of model parameters 

  LaTeX Math Inline
  body --uriencoded--%7B\bf p%7D = \%7B q%5e*_0, \, \tau_0, \, b \%7D

• a training data set: 

  LaTeX Math Inline
  body --uriencoded--\%7B (t_k, q_k)\%7D_%7Bk = 1..N%7D = \%7B (t_0, q_0), (t_1, q_1), ..., (t_N, q_N) \%7D

the matching procedure assumes searching for thee specific set of model parameters 

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

where 

Most popular choices are

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 

    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

Anchor
Unconstrained Unconstrained

Unconstrained matching

All three model parameters 

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 

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

• the current production rate 

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

• the current cumulative production 

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

Anchor
Initial Initial

Match the value of the initial rate 

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

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

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

• the current cumulative production 

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

Anchor
Current_rate Current_rate

Match the value of the current rate 

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

The value of the model rate at the current time moment is set to 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

The best-fit model may not match: 

• the initial production rate 

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

• the current cumulative production 

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

Anchor
Current_cumulative Current_cumulative

Match the value of the current cumulative 

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

The value of the model rate at the initial time moment

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}

Q(t) - Q_N = [ q^*_N - q(t)] \, \tau_0

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]

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

The best-fit model may not match: 

• the initial production rate 

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

• the current production rate 

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

Anchor
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

To ensure the smooth transition from historical data

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]

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

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]

Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)}

The best-fit model may not match: 

• the initial production rate 

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

See Also

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