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Given:
- a function
q^*(t, {\bf p}) of real-value argument
t \in \R and set of model parameters
{\bf p} = \{ p_m\}_{m = 1..M} = \{p_1, p_2, ... p_M\}
- a training data set:
\{ (t_k, q_k)\}_{k = 1..N} = \{ (t_0, a_0), (t_1, q_1), ..., (t_N, q_N) \}
the matching procedure assumes searching for thee specific set of model parameters
{\bf p} to minimize the goal function:
| |
F({\bf p}) = \sum_{k=1}^N \, \Psi \left( q^*(t_k) - q_k \right) \rightarrow \textrm{min} |
where
\Psi(z) is the discrepancy distance function.
Most popular choices are
\Psi(z) = z^2 and
\Psi(z) = |z|.
There are few approaches to match the Arps decline to the historical data:
Unconstrained matching
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
|---|
| | |
| (1) |
q(t)=q_0 \exp \left( -D_0 \, t \right) |
|
| (2) |
q(t) = \frac{q_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} } |
|
| (3) |
q(t)=\frac{q_0}{1+D_0 \, t} |
|
| (4) |
Q(t)=\frac{q_0-q(t)}{D_0} |
|
| (5) |
Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]
|
|
| (6) |
Q(t)=\frac{q_0}{D_0} \, \ln \left[ \frac{q_0}{q(t)} \right] = \frac{q_0}{D_0} \ln q_0 + \frac{q_0}{D_0} \cdot \ln q(t) |
|
Match the value of the initial rate
q^*(t=0) = q_0
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
|---|
| | |
| (7) |
q(t)=q_0 \exp \left( -D_0 \, t \right) |
|
| (8) |
q(t) = \frac{q_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} } |
|
| (9) |
q(t)=\frac{q_0}{1+D_0 \, t} |
|
| (10) |
Q(t)=\frac{q_0-q(t)}{D_0} |
|
| (11) |
Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]
|
|
| (12) |
Q(t)=\frac{q_0}{D_0} \, \ln \left[ \frac{q_0}{q(t)} \right] = \frac{q_0}{D_0} \ln q_0 + \frac{q_0}{D_0} \cdot \ln q(t) |
|
Match the value of the current rate
q^*(t=t_N) = q_N
To ensure the smooth transition from historical data
[(t_1,q_1)... (t_N, q_N)] to the production forecasts in future time moments
[(t_{N+1},q_{N+1}), ...] one may wish to constrain the model by firm matching the production at the last historical moment
(t_N, q_N) which leads to the following form of Arp's model:
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
|---|
| | |
| (13) |
q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big] |
|
| (14) |
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} |
|
| (15) |
q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t } \right] |
|
| (16) |
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0 |
|
| (17) |
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] |
|
| (18) |
Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)} |
|
Match the value of the current cumulative
Q^*(t=t_N) = Q_N
To ensure the smooth transition from historical data
[(t_1,q_1)... (t_N, q_N)] to the production forecasts in future time moments
[(t_{N+1},q_{N+1}), ...] one may wish to constrain the model by firm matching the production at the last historical moment
(t_N, q_N) which leads to the following form of Arp's model:
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
|---|
| | |
| (19) |
q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big] |
|
| (20) |
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} |
|
| (21) |
q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t } \right] |
|
| (22) |
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0 |
|
| (23) |
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] |
|
| (24) |
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
