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Appendix E

# Approximate Solution and Its Extension When Each Layer of a Two-Layer Reservoir Produces Independently (Chapter 7)☆

Suppose a homogeneous two-layer reservoir is penetrated by a well. Layer 1 and Layer 2 produce independently with constant dimensionless rates qD1 and qD2, respectively. The problem is expressed as follows:

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

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