For the pressure diffusion with constant diffusion coefficients and homogeneous boundary conditions the pressure response p(t) in one well to a complex flowrate history q(t) in the same well honours the convolution equation:
| (1) | p(t) = p_0 + \int_0^t p_u(t-\tau) \, dq(\tau) = p_0 + \int_0^t p_u(t-\tau) \, \dot q(\tau) \, d\tau |
where
p_0 | initial formation pressure at zero time t=0 |
p_u(\tau) | Drawdown Transient Response |
\dot q(\tau) = \frac{dq}{d\tau} | a pace of sandface flow rate variation per unit time |
In case a well is interfering with the offset wells the pressure in a given well n may respond to the offset wells m \neq n and the multi-well form of convolution is going to be:
| (2) | p_n(t) = p_{n, 0} + \sum_{m=1}^N \int_0^t p_{u,nm}(t-\tau) \, dq_m(\tau) = p_{n, 0} + \sum_{m=1}^N \int_0^t p_{u,nm}(t-\tau) \, \dot q_m(\tau) \, d\tau = p_{n, 0} + \int_0^t p_{u,nn}(t-\tau) \, \dot q_n(\tau) \, d\tau + p_{n, 0} + \sum_{m \neq n}^N \int_0^t p_{u,nm}(t-\tau) \, \dot q_m(\tau) \, d\tau |
where
p_{n, \, 0} | Initial formation pressure at zero time t=0 for the n-th well |
p_{u,nm}(\tau) | Drawdown Transient Response in the n-th well to the unit-rate production |
| |
A speed of n-th well total sandface flow rate variation |
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing
[ Well & Reservoir Surveillance ] [ Pressure Diffusion ] [ Pressure drawdown ]