Well Test Analysis for Multilayered Reservoirs with Formation Crossflow

Appendix H

Solution of the Problem (Chapter 9)

Using the dimensionless expressions in Eqs. (9.28)–(9.38) and the following approximate expressions at interfaces z=0 and z=h₃,

${\left.\frac{\partial p_1^{\prime }}{\partial z}\right|}_{z=h_3}=\frac{{\left.p_1-p_1^{\mathrm{\prime }}\right|}_{z=h_3}}{h_1/2}\mathrm{and}{\left.\frac{\partial p_2^{\prime }}{\partial z}\right|}_{z=0}=\frac{{\left.p_2^1\right|}_{z=0}-p_2}{h_2/2}$

  (H.1)

Eqs. (9.8)–(9.21) becomes

${\omega }_1\frac{\partial p_{D1}}{\partial t_D}-w_1{\mathrm{\Delta }}_D{p}_{D1}+2k_{vD1}\left(p_{D1}-{\left.p_{D1}^{\prime }\right|}_{z_D=1}\right)=0$

  (H.2)

${\omega }_2\frac{\partial p_{D2}}{\partial t_D}-w_2{\mathrm{\Delta }}_D{p}_{D2}+2k_{vD2}\left(p_{D2}-{\left.p_{D2}^{\prime }\right|}_{z_D=0}\right)=0$

  (H.3)

$p_{Dj}\to 0\mathrm{when}{t}_D\to 0,j=1,2$

  (H.4)

$p_{Dj}\to 0\mathrm{when}{r}_D\to \infty ,j=1,2$

  (H.5)

$w_j{\left(r_D\frac{\partial p_{Dj}}{\partial r_D}\right)}_{r_D=1}=-q_{Dj},j=1,2$

  (H.6) ...