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Appendix B

# Approximate Solution of the Diffusivity Crossflow Problem in n-Layer Reservoirs (Chapter 3)

Suppose a well produces from all layers in an n-layer reservoir at a common wellbore pressure and a constant total rate, q, for $t>0$ . Under the assumptions we made, the problem can be described as follows. The equations are

$\begin{array}{c}\hfill \frac{1}{{\eta }_{i}}\frac{\partial {p}_{i}}{\partial t}-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {p}_{i}}{\partial r}\right)+\frac{{\stackrel{˜}{k}}_{i-1}}{{k}_{i}{h}_{i}}\left({p}_{i}-{p}_{i-1}\right)+\frac{{\stackrel{˜}{k}}_{i}}{{k}_{i}{h}_{i}}\left({p}_{i}-{p}_{i+1}\right)=0,\hfill \\ \hfill i=1,2,\dots ,n;{\stackrel{˜}{k}}_{0}={\stackrel{˜}{k}}_{n}=0\hfill \end{array}$ (B.1)

where ${\eta }_{i}=\frac{{k}_{i}}{{\varphi }_{i}\mu {c}_{t}}$ and $\stackrel{˜}{k}$ is the semipermeability ...

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