The general form of the Water-Oil Ratio (WOR) regerssion is:
| (1) | WOR = WOR_0 + \mbox{Regression}(\{q_k\}, \{Q_k\}), \quad k=[1..N] |
Power regression
| (2) | WOR = WOR_0 + \sum_{k=1..N} Q_{O,k} \cdot \big[ a_{O,k} \, Q_{O,k}^{gQ_{O,k}} + a_{W,k} \, Q_{W,k}^{gQ_{W,k}} + b_{O,k} \, q_{O,k}^{gq_{O,k}} + b_{W,k} \, q_{W,k}^{gq_{W,k}} \big] |
Pade regression
| (3) | WOR = WOR_0 + \frac {Q_{O} \cdot \sum_{k=1..N} \big[ a_{O,k} \, Q_{O,k} + a_{W,k} \, Q_{W,k} + b_{O,k} \, q_{O,k} + b_{W,k} \, q_{W,k} \big] }{1 + \sum_{k=1..N}\big[ c_{O,k} \, Q_{O,k} + c_{W,k} \, Q_{W,k} + d_{O,k} \, q_{O,k} + d_{W,k} \, q_{W,k} \big] } |
The general form of the watercut YW regression is:
| (4) | Y_W^{-1} = Y_{W0}^{-1} + \mbox{Regression}(\{q_k\}), \quad k=[1..N] |
where
(\{q_k\} = \{ q_1, \, q_2, \, ... q_N \} | sandface flowrates |
One can build various types of regression including the Artificial Neural Network or the closed-form regressions.
The simplest form of the linear closed-form regression is:
| (5) | Y_W^{-1} = Y_{W0}^{-1} + \sum_{k=1}^N w_k \cdot q_k, \quad k=[1..N] |
The simplest form of the non-linear closed-form regression is polynomial:
| (6) | Y_W^{-1} = Y_{W0}^{-1} + \sum_{k=1}^N w_k \cdot q_k ^ {n_k} |
The more general of the non-linear closed-form regression is rational fraction:
| (7) | Y_W^{-1} = Y_{W0}^{-1} + \frac{\sum_{k=1}^N w_k \cdot q_k ^ {n_k} }{1 + \sum_{k=1}^N z_k \cdot q_k ^ {m_k} }, \quad |z| = \sqrt{\sum_k z_k} >0 |
The other option is to perform neural network regression:
| (8) | Y_W^{-1} = Y_{W0}^{-1} + ANN(q_1, q_2 , ... q_N) |
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing (WT) / Flowrate Testing / Flowrate / Production Water cut (Yw)
[ WOR ] [ Watercut Diagnostics ][ Watercut Fractional Flow @model ]