The plot of water production rate (along y-axis) against the oil production rate (along x-axis).

It can be used for express Watercut Diagnostics of thief water production.

The mathematical model of the thief water production from aquifer is based on the following equation:

q_W = a \, \cdot q_O + b

a = J^{-1}_{1O} \cdot ( J_{1W} + J_{2W})

b = J_{2W} \cdot (p^*_2 - p^*_1)

where

	water production rate	oil production rate
formation pressure in petroleum reservoir	water productivity index of petroleum reservoir	oil productivity index of petroleum reservoir
formation pressure in aquifer	water productivity index of aquifer

For the case of aquifer pressure is higher than that of petroleum reservoir:

For the case of aquifer pressure is lower than that of petroleum reservoir:

In practical applications, the equation is often considered through the weighted average values:

\langle q_W \rangle = a \, \cdot \langle q_O \rangle  + \, b

where

are weighted average of and

There are different ways to calculate weighted average of the dynamic variable, for example:

\langle A \rangle_t \ = \frac{1}{t} \int_o^t A(t) \, dt

\langle A \rangle_q \ = \frac{1}{Q(t)} \int_o^t A(t) \, q(t) \, dt

See Also