For the pressure diffusion with constant diffusion coefficients and homogeneous boundary conditions the pressure response in one well to a complex flowrate history in the same well honours the convolution equation:

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

initial formation pressure at zero time

Drawdown Transient Response

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 may respond to the offset wells and the multi-well form of convolution is going to be:

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

	Initial formation pressure at zero time for the -th well
	Drawdown Transient Response in the -th well to the unit-rate production
	Cross-well Transient Response in the -th well to the unit-rate production in -th well
	A speed of -th well total sandface flow rate variation

See also

See Also