...
| Exponential Production Decline | Hyperbolic Production Decline | Harmonic Production Decline |
|---|
| | |
| LaTeX Math Block |
|---|
| q(t)=q_0 \exp \left( -D_0 \, t \right) |
| | LaTeX Math Block |
|---|
| q(t) = \frac{q_0}{ \left( 1+b \cdot D_0 \cdot t \right)^{1/b} } |
| | LaTeX Math Block |
|---|
| q(t)=\frac{q_0}{1+D_0 \, t} |
|
| LaTeX Math Block |
|---|
| Q(t)=\frac{q_0-q(t)}{D_0} |
| | LaTeX Math Block |
|---|
| Q(t)=\frac{q_0}{D_0 \, (1-b)} \, \left[ 1- \left( \frac{q(t)}{q_0} \right)^{1-b} \right]
|
| | LaTeX Math Block |
|---|
| 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) |
|
| LaTeX Math Block |
|---|
| Q_{\rm max}=\frac{q_0}{D_0} |
| | LaTeX Math Block |
|---|
| Q_{\rm max}=\frac{q_0}{D_0 \cdot (1-b)} |
| | LaTeX Math Block |
|---|
| Q_{\rm max}=\infty |
|
| LaTeX Math Block |
|---|
| D(t) = D_0 = \rm const |
| | LaTeX Math Block |
|---|
| anchor | D_hyper |
|---|
| alignment | left |
|---|
| D(t) =\frac{D_0}{1+ b \cdot D_0 \cdot t} |
| | LaTeX Math Block |
|---|
| anchor | D_harm |
|---|
| alignment | left |
|---|
| D(t) = \frac{D_0}{1+ D_0 \cdot t} |
|
| LaTeX Math Block |
|---|
| anchor | tau_exp |
|---|
| alignment | left |
|---|
| \tau(t) = \tau_0 = \rm const |
| | LaTeX Math Block |
|---|
| anchor | tau_hyper |
|---|
| alignment | left |
|---|
| \tau(t) = \tau_0 + b \cdot t |
| | LaTeX Math Block |
|---|
| anchor | tau_harm |
|---|
| alignment | left |
|---|
| \tau(t) = \tau_0 + t |
|
| LaTeX Math Block |
|---|
| anchor | RPR_exp |
|---|
| alignment | left |
|---|
| \mathrm{RPR}(t) = \tau(t) = \tau_0 = \rm const |
| | LaTeX Math Block |
|---|
| anchor | RPR_hyper |
|---|
| alignment | left |
|---|
| \mathrm{RPR}(t) = \tau_0 \, \left[
1 + \frac{b }{(1-b)} \cdot \frac{q_0}{q(t)}
\right]
|
| | LaTeX Math Block |
|---|
| anchor | RPR_harm |
|---|
| alignment | left |
|---|
| \mathrm{RPR}(t) = \infty |
|
To ensure the smooth transition from historical data
| LaTeX Math Inline |
|---|
| body | [(t_1,q_1)... (t_N, q_N)] |
|---|
|
to the production forecasts in future time moments | LaTeX Math Inline |
|---|
| body | --uriencoded--[(t_%7BN+1%7D,q_%7BN+1%7D), ...] |
|---|
|
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:...
...
...
| LaTeX Math Block |
|---|
|
q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N) \big] |
...
| LaTeX Math Block |
|---|
|
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} |
...
| LaTeX Math Block |
|---|
|
q(t) = q_N \cdot \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t } \right] |
...
| LaTeX Math Block |
|---|
|
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0 |
...
| LaTeX Math Block |
|---|
|
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] |
...
| LaTeX Math Block |
|---|
|
Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)} |
The Exponential and Hyperbolic decline are applicable for Boundary Dominated Flow with finite reserves
| LaTeX Math Inline |
|---|
| body | --uriencoded--Q_%7B\rm max%7D \leq \infty |
|---|
|
while
Harmonic decline is associated with production from the reservoir with infinite reserves
| LaTeX Math Inline |
|---|
| body | --uriencoded--Q_%7B\rm max%7D = \infty |
|---|
|
.
...
