Implication that pressure p(t, {\bf r}) at any point {\bf r} of a porous reservoir is a linear sum of pressure responses \delta p_k(t, {\bf r}) to individual rate variations q_k(t) in all wells connected to this reservoir:
| (1) | p(t, {\bf r}) = p_i + \sum_k \delta p_k(t, {\bf r}) = p_i + \sum_k \int_0^t p_{uk}(t-\tau, {\bf r}) \, dq_k(\tau) |
In case reservoir point {\bf r} defines location of m-well the superposition principle can be rewritten as:
| (2) | p_m(t) = p_{mi} + \sum_k \delta p_{mk}(t) = p_{mi} + \sum_k \int_0^t p_{umk}(t-\tau) \, dq_k(\tau) = p_{mi} + \int_0^t p_{umm}(t-\tau) \, dq_m(\tau) + \sum_{k \neq m} \int_0^t p_{umk}(t-\tau) \, dq_k(\tau) |
where
p_{mi} | initial formation pressure in m-well |
\delta p_{mk}(t) | specific component of m-well pressure variation caused by k-well flowrate history q_k(t) |
p_{umm}(\tau) | bottomhole pressure response in m-well to unit-rate production in the same well (DTR) |
p_{umk}(\tau) | bottomhole pressure in m-well to unit-rate production in k-well (CTR), k \neq m |