Appendix D

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:

$\frac{\partial {p}_{i}}{\partial t}-\frac{{\eta}_{i}}{r}\frac{\partial}{\partial r}\left(r\frac{\partial {p}_{i}}{\partial r}\right)+\frac{{\tilde{k}}_{i-1}}{{b}_{i}}\left({p}_{i}-{p}_{i-1}\right)+\frac{{\tilde{k}}_{i}}{{b}_{i}}\left({p}_{i}-{p}_{i+1}\right)=0,{\tilde{k}}_{0}={\tilde{k}}_{n}=0,i=1,2,\dots ,n$

$\sum _{i=1}^{n}\frac{2\pi {k}_{i}{h}_{i}}{\mu}}{\left(r\frac{\partial {p}_{i}}{\partial r}\right)}_{r={r}_{w}}=B{\displaystyle \sum _{i=1}^{n}{q}_{i}}=\mathit{Bq$

${p}_{wf}={p}_{1}\left({r}_{w},t\right)-\frac{B{q}_{1}\mu}{2\pi {k}_{1}{h}_{1}}{s}_{1}=\cdots ={p}_{n}\left({r}_{w},t\right)-\frac{B{q}_{n}\mu}{2\pi {k}_{n}{h}_{n}}{s}_{n}$

${p}_{i}\to {p}_{0},i=$

Start Free Trial

No credit card required