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:
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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
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:
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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
The pressure convolution principle has some limitations and may not be adequate for some practical cases.
For example, changing reservoir conditions, high compressibility – everything which breaks linearity of diffusion equations.
There are some workarounds on these cases but the best practice is to check the validity of pressure convolution (and therefore the applicability of MDCV) on the simple synthetic 2-well Dynamic Flow Model (DFM) with the typical for the given case reservoir-fluid-production conditions.
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well Testing / Pressure Testing
[ Well & Reservoir Surveillance ] [ Pressure Diffusion ] [ Pressure drawdown ] [ MDCV ]
[ Convolution @math ]
