Watercut WΣW History Matching @model predicts production adjustments coefficients $\begin{array}{l}\alpha_W(t), \ \alpha_O(t)\end{array}$ from WΣW plot by minimizing the following goal function:

(1)	$\begin{array}{l}\displaystyle E[\alpha_W(t), \alpha_O(t)] = \sum_t \ \min D \big( P_{\rm mod}(t), P_{\rm hist}(t) \big) \rightarrow 0\end{array}$

where $\begin{array}{l}D\end{array}$ is the distance between each historical point and model curve on WΣW plot:

(2)	$\begin{array}{l}\displaystyle D \big( P_{\rm mod}}(t), P_{\rm hist}(t) \big) = \sqrt{ \big( WOR_{\rm mod} - WOR_{\rm hist} \big)^2 + \big( ΣWOR_{\rm mod} - ΣWOR_{\rm hist} \big)^2 }\end{array}$

where

(3)	$\begin{array}{l}\displaystyle WOR_{\rm hist} = \frac{\alpha_W(t) \cdot q_W^\uparrow}{\alpha_O(t) \cdot q_O^\uparrow}\end{array}$

(4)	$\begin{array}{l}\displaystyle ΣWOR_{\rm hist} = \frac{ \int\limits_0^t \alpha_W(t) \cdot q_W^\uparrow \, dt}{ \int\limits_0^t \alpha_O(t) \cdot q_O^\uparrow \, dt}\end{array}$

and $\begin{array}{l}\min D \big( P_{\rm mod}}, P_{\rm hist} \big)\end{array}$ means minimal distance between point $\begin{array}{l}P_{\rm hist} = \big( ΣWOR_{\rm hist} , WOR_{\rm hist} \big)\end{array}$ and the model curve $\begin{array}{l}P_{\rm mod} = \big( ΣWOR_{\rm mod} , WOR_{\rm mod} \big)\end{array}$ .

The results of the history matching are illustrated on Fig. 1 below.


Fig. 1.1 – WΣW plot before history matching	Fig. 1.2 – WΣW plot after history matching


Fig. 1.2 – The production adjustments coefficients from Watercut WΣW History Matching @model

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