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

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

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

...

To ensure the smooth transition from historical data

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[ -D_0 \cdot (t-t_N)/\tau_0 \big]

q(t) = q_N \cdot \left[ \frac{1+b \cdot Dt_0 \cdot t_NN/\tau_0 }
{ 1+b \cdot Dt/\tau_0 \cdot t   } \right]^{1/b}

q(t) =  q_N \cdot  \left[ \frac{1+Dt_0 \cdot t_NN/\tau_0 }
{ 1+ Dt/\tau_0 \cdot t   } \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

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

q(t) = [ q_N \frac{(1-b) \cdot Q_N/\tau_0}{ 1 - q(t)] \, \tau_0\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)} - Q_N\cdot \frac{1}{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

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

...

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

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

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

See Also

...

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

...