For the pressure diffusion with constant diffusion coefficients and linear 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 |

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 ]