Given:

• a discrete flowrate history  (see Fig. 1)
  
  
• vertical well and homogeneous reservoir with no boundaries
  
  
• duration of the last drawdown interval  reaching radial flow regime

the superposition time is defined  as:

\ln t_s = \ln (t-t_{N-1}) - \sum_{i=1}^{N-1} \frac{q_i - q_{i-1}}{q_{N-1}}  \ln (t-t_{i-1}) = \ln \Delta t - \sum_{i=1}^{N-1} \frac{q_i - q_{i-1}}{q_{N-1}}  \ln \left( \Delta t + \sum_{i=1}^{N-i} \Delta t_i \right)

The pressure response at time moment  will be given by formula:

p_{wf}(t_s) = p_i + \frac{q_{N-1} B}{4 \pi \sigma} \ \ln t_s

which has the same format as for a single rate drawdown test with flowrate , and duration  and can be interpreted using Type-Curve Matching.

Multi-rate build-up test

Given:

• a discrete flowrate history  followed by shut-in period   (see Fig. 2)
  
  
• vertical well and homogeneous reservoir with no boundaries
  
  
• duration of the last shut-in  reaching radial flow regime

the superposition time is defined  as:

\ln t_s = \ln (t-t_{N-1}) - \sum_{i=1}^{N-1} \frac{q_i - q_{i-1}}{q_{N-1}}  \ln (t-t_{i-1}) = \ln \Delta t - \sum_{i=1}^{N-1} \frac{q_i - q_{i-1}}{q_{N-1}}  \ln \left( \Delta t + \sum_{i=1}^{N-i} \Delta t_i \right)

The pressure response at time moment  will be given by formula:

p_{wf}(t_s) = p_i + \frac{q_{N-1} B}{4 \pi \sigma} \ \ln t_s

which has the same format as for a single rate drawdown test with flowrate , and duration  and can be interpreted using Type-Curve Matching.