Appendix E

Suppose a homogeneous two-layer reservoir is penetrated by a well. Layer 1 and Layer 2 produce independently with constant dimensionless rates q_{D1} and q_{D2}, respectively. The problem is expressed as follows:

$\frac{1}{{\eta}_{D1}}\frac{\partial {p}_{D1}}{\partial {t}_{D}}-\frac{1}{{r}_{D}}\frac{\partial}{\partial {r}_{D}}\left({r}_{D}\frac{\partial {p}_{D1}}{\partial {r}_{D}}\right)+\frac{{\tilde{k}}_{D1}}{{w}_{1}}\left({p}_{D1}-{p}_{D2}\right)=0$

and

$\frac{1}{{\eta}_{D2}}\frac{\partial {p}_{D2}}{\partial {t}_{D}}-\frac{1}{{r}_{D}}\frac{\partial}{\partial {r}_{D}}\left({r}_{D}\frac{\partial {p}_{D2}}{\partial {r}_{D}}\right)+\frac{{\tilde{k}}_{D1}}{{w}_{2}}\left({p}_{D2}-{p}_{D1}\right)=0$

(E.1)

${w}_{i}{\left({r}_{D}\frac{\partial {p}_{Di}}{\partial {r}_{D}}\right)}_{{r}_{D}=1}=-{q}_{Di},i=1,2$

(E.2)

${p}_{Di}\to 0,i=1,2,\mathrm{when}{r}_{D}\to \infty $

(E.3)

and

${p}_{Di}\to 0,i=1,2,\mathrm{when}{t}_{D}\to 0$

(E.4)

Using

${p}_{D}=$

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