Specific implementation of pressure spatial decomposition based on recognising the time pattern of flowrate variation pulse sequence in pressure response.
It constitutes the key interpretation procedure for PCT.
The decomposition algorithm is based on the minimisation of correlation functional X between generating well pressure variation \delta p_G(t) and pressure trend p_{R,{\rm tr}}(t) at receiving well after deducting the simulated pressure pulse-code response:
| (1) | X[p_{uGG}] = \left < | p_{G,{\rm tr}}(t) | , \, |\delta p_G(t)| \right > \rightarrow \min |
| (2) | X[p_{uGR}] = \left < | p_{R,{\rm tr}}(t) | , \, |\delta p_R(t)| \right > \rightarrow \min |
where
| (3) | p_G(t) = p_{G,{\rm tr}}(t) - \delta p_G(t), \quad \delta p_G(t) = \int_0^t p_{uGG}(t-\tau) \, dq_G(\tau) |
| (4) | p_R(t) = p_{R,{\rm tr}}(t) - \delta p_R(t), \quad \delta p_R(t) = \int_0^t p_{uGR}(t-\tau) \, dq_G(\tau) |
The generator's flowrate history q_G(t), generator's wellbore pressure history p_{G,{\rm tr}}(t) and receiver's wellbore pressure history p_{R,{\rm tr}}(t) are assumed to be known for the whole period of the test.
The result of decomposition is the set of the unit-rate transient responses, DTRs p_{uGG}(\tau) and CTRs p_{uGR}(\tau)), which characterise reservoir properties round generator and between generator and receiver.
The pressure trends at generator p_{G,{\rm tr}}(t) and receiver p_{R,{\rm tr}}(t) may have unknown origin but in order for decomposition to work they should have minimum correlation with generating well pressure variation \delta p_G(t). In terms of spectral analysis this means that trend's spectrum contains minimum overlap with spectrum of pressure variation \delta p_G(t) at generator.
The contamination components of the trends should either have low frequencies (created by monotonous activity of distant wells) or high frequencies (created by lift instability at generating/receiving well).
If for some reasons the trends contain components correlated with \delta p_G(t) (for example receiving well or distant wells have been synchronously varying the rates with generating well ) then decomposition is not unique and should not be considered.
See Also
Petroleum Industry / Upstream / Subsurface E&P Disciplines / Well testing / Pressure Pulse Test / Pressure Pulse-Code Test (PCT)