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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



initial formation pressure at zero time t=0

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


p_{n, \, 0}

Initial formation pressure at zero time t=0 for the n-th well


Drawdown Transient Response in the n-th well to the unit-rate production


Cross-well Transient Response
in the n-th well to the unit-rate production in m-th well

\dot q_m(\tau) = \frac{dq_m}{d\tau}

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 ]

Convolution @math ]

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