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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:

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

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

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

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

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