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 p_m\%7D_%7Bm = 1..M%7D = \%7Bp_1, p_2, ... p_M\%7D

• a training data set: 

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

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

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

where 

Most popular choices are

There are few approaches to match the Arps decline to the historical data:

• Unconstrained matching (free)
• Constrained matching:
  • Match the value of the initial rate 

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

To ensure the smooth transition from historical data

q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big]

q(t) = q_N \cdot \left[ \frac{1+b \cdot D_0 \cdot t_N }
{ 1+b \cdot D_0 \cdot t  } \right]^{1/b}

q(t) =  q_N \cdot  \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t  } \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)}

See Also

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