Well Test Analysis for Multilayered Reservoirs with Formation Crossflow

Appendix G

Case for Constant Wellhead Rate (Chapter 8)

Suppose the well flows at a constant wellhead rate q_r in the testing period. The wellbore storage of the second layer is negligible. The first layer has wellbore storage coefficient C_D. In this case, condition Eq. (F.8) is replaced by

$C_D\frac{d{p}_{wD1}}{d{t}_D}-{\displaystyle \sum _{j=1}^2{w}_j{\left(r_D\frac{\partial p_{Dj}}{\partial r_D}\right)}_{r_D=1}=1}$

  (G.1)

Its Laplace transform is

$C_D{\sigma }_D{\overline{p}}_{wD1}-{\displaystyle \sum _{j=1}^2{w}_j{\left(r_D\frac{d{\overline{p}}_{Dj}}{d{r}_D}\right)}_{r_D=1}=\frac{1}{{\sigma }_D}}$

  (G.2)

or

${\displaystyle \sum _{j=1}^2{w}_j{\left(r_D\frac{d{\overline{p}}_{Dj}}{d{r}_D}\right)}_{r_D=1}=-\frac{1}{{\sigma }_D}}\left(1-C_D{\sigma }_D^2{\overline{p}}_{wD1}\right)$