Well Test Analysis for Multilayered Reservoirs with Formation Crossflow

Appendix E

Approximate Solution and Its Extension When Each Layer of a Two-Layer Reservoir Produces Independently (Chapter 7)☆

Suppose a homogeneous two-layer reservoir is penetrated by a well. Layer 1 and Layer 2 produce independently with constant dimensionless rates q_D1 and q_D2, respectively. The problem is expressed as follows:

$\frac{1}{{\eta }_{D1}}\frac{\partial p_{D1}}{\partial t_D}-\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial p_{D1}}{\partial r_D}\right)+\frac{{\tilde{k}}_{D1}}{w_1}\left(p_{D1}-p_{D2}\right)=0$

and

$\frac{1}{{\eta }_{D2}}\frac{\partial p_{D2}}{\partial t_D}-\frac{1}{r_D}\frac{\partial }{\partial r_D}\left(r_D\frac{\partial p_{D2}}{\partial r_D}\right)+\frac{{\tilde{k}}_{D1}}{w_2}\left(p_{D2}-p_{D1}\right)=0$

  (E.1)

$w_i{\left(r_D\frac{\partial p_{Di}}{\partial r_D}\right)}_{r_D=1}=-q_{Di},i=1,2$

  (E.2)

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

  (E.3)

and

$p_{Di}\to 0,i=1,2,\mathrm{when}{t}_D\to 0$

  (E.4)

Using

$p_D=$