# 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

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

(B.1)

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

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