...
Watercut WΣW History Matching @model predicts production adjustments coefficients
| LaTeX Math Inline |
|---|
| body | \alpha_W(t), \ \alpha_O(t) |
|---|
|
from
WΣW plot.The production adjustments coefficients
| LaTeX Math Inline |
|---|
| body | \alpha_W(t), \ \alpha_O(t) |
|---|
|
...
| LaTeX Math Block |
|---|
|
WOR_{\rm hist} = \frac{\alpha_W(t) \cdot q_W^\uparrow}{\alpha_O(t) \cdot q_O^\uparrow} |
...
| LaTeX Math Block |
|---|
|
ΣWOR_{\rm hist} = \frac{ \int\limits_0^t \alpha_W(t) \cdot q_W^\uparrow \, dt}{ \int\limits_0^t \alpha_O(t) \cdot q_O^\uparrow \, dt} |
by minimizing the following goal function:The production adjustments coefficients
| LaTeX Math Inline |
|---|
| body | \alpha_W(t), \ \alpha_O(t) |
|---|
|
can be found from minimisation of the goal function: | LaTeX Math Block |
|---|
| E[\alpha_W(t), \alpha_O(t)] = \sum_t \ \min D \big( P_{\rm mod}(t), P_{\rm hist}(t) \big) \rightarrow 0 |
|
where is the distance between each historical point and model curve on WΣW plot.:
| LaTeX Math Block |
|---|
| D \big( P_{\rm mod}}(t), P_{\rm hist}(t) \big) = \sqrt{ \big( WOR_{\rm mod} - WOR_{\rm hist} \big)^2 + \big( ΣWOR_{\rm mod} - ΣWOR_{\rm hist} \big)^2 } |
|
where
| LaTeX Math Block |
|---|
| WOR_{\rm hist} = \frac{\alpha_W(t) \cdot q_W^\uparrow}{\alpha_O(t) \cdot q_O^\uparrow} |
|
| LaTeX Math Block |
|---|
| ΣWOR_{\rm hist} = \frac{ \int\limits_0^t \alpha_W(t) \cdot q_W^\uparrow \, dt}{ \int\limits_0^t \alpha_O(t) \cdot q_O^\uparrow \, dt} |
|
and
| LaTeX Math Inline |
|---|
| body | --uriencoded--\min D \big( P_%7B\rm mod%7D%7D, P_%7B\rm hist%7D \big) |
|---|
|
means minimal distance between point | LaTeX Math Inline |
|---|
| body | --uriencoded--P_%7B\rm hist%7D = \big( ΣWOR_%7B\rm hist%7D , WOR_%7B\rm hist%7D \big) |
|---|
|
and the model curve | LaTeX Math Inline |
|---|
| body | --uriencoded--P_%7B\rm mod%7D = \big( ΣWOR_%7B\rm mod%7D , WOR_%7B\rm mod%7D \big) |
|---|
|
.
The results of the history matching are illustrated on Fig. 1 below.
...
