Well Test Analysis for Multilayered Reservoirs with Formation Crossflow

Appendix D

Approximate Solution for a Well Producing From All Layers at a Constant Rate (Chapter 7)☆

Suppose a well penetrates an n-layer reservoir completely and produces at a constant total rate q from $t=0$. Under the assumptions stated above, the problem can be expressed as follows:

$\frac{\partial p_i}{\partial t}-\frac{{\eta }_i}{r}\frac{\partial }{\partial r}\left(r\frac{\partial p_i}{\partial r}\right)+\frac{{\tilde{k}}_{i-1}}{b_i}\left(p_i-p_{i-1}\right)+\frac{{\tilde{k}}_i}{b_i}\left(p_i-p_{i+1}\right)=0,{\tilde{k}}_0={\tilde{k}}_n=0,i=1,2,\dots ,n$

${\displaystyle \sum _{i=1}^n\frac{2\pi k_i{h}_i}{\mu }}{\left(r\frac{\partial p_i}{\partial r}\right)}_{r=r_w}=B{\displaystyle \sum _{i=1}^n{q}_i}=\mathit{Bq}$

$p_{wf}=p_1\left(r_w,t\right)-\frac{B{q}_1\mu }{2\pi k_1{h}_1}{s}_1=\cdots =p_n\left(r_w,t\right)-\frac{B{q}_n\mu }{2\pi k_n{h}_n}{s}_n$

$p_i\to p_0,i=$