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| Unconstrained |
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| Unconstrained |
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Unconstrained matching
All three model parameters | LaTeX Math Inline |
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| body | --uriencoded--\%7B q%5e*_0, \, D_0, \, b \%7D |
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are being varied to achieve the best fit to the training dataset.
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
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| LaTeX Math Block |
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| q(t)=qq^*_0 \exp \left( -D_0 \, t \right) |
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| q(t) = \frac{qq^*_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} } |
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| q(t)=\frac{qq^*_0}{1+D_0 \, t} |
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| LaTeX Math Block |
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| Q(t)=\frac{qq^*_0-q(t)}{D_0} |
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| Q(t)=\frac{q^*_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]
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| Q(t)=\frac{qq^*_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) |
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Match the value of the initial rate | LaTeX Math Inline |
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| body | --uriencoded--q%5e*(t=0) = q_0 |
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The value of the model rate at the initial time moment is set to training dataset: | LaTeX Math Inline |
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| body | --uriencoded--q%5e*(t=0) = q_0 |
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and the two other model properties | LaTeX Math Inline |
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| body | --uriencoded--\%7B D_0, \, b \%7D |
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are being varied to achieve the best fit to the training dataset.
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
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| LaTeX Math Block |
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| q(t)=q_0 \exp \left( -D_0 \, t \right) |
| | LaTeX Math Block |
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| q(t) = \frac{q_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} } |
| | LaTeX Math Block |
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| q(t)=\frac{q_0}{1+D_0 \, t} |
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| LaTeX Math Block |
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| Q(t)=\frac{q_0-q(t)}{D_0} |
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| Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]
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| 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) |
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To ensure the smooth transition from historical data
| LaTeX Math Inline |
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| body | [(t_1,q_1)... (t_N, q_N)] |
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to the production forecasts in future time moments
| LaTeX Math Inline |
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| body | --uriencoded--[(t_%7BN+1%7D,q_%7BN+1%7D), ...] |
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one may wish to constrain the model by firm matching the production at the last historical moment
which leads to the following form of
Arp's model:
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
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| LaTeX Math Block |
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| q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big] |
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| 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} |
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| q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t } \right] |
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| Q(t) - Q_N = [ q_N - q(t)] \, \tau_0 |
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| 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] |
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| Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)} |
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See Also
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Petroleum Industry / Upstream / Production / Subsurface Production / Field Study & Modelling / Production Analysis / Decline Curve Analysis / DCA Arps @model
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