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Appendix F

# Case for Variable Bottom-Hole Rate (Chapter 8)

Using the dimensionless expressions from Eqs. (8.11) to (8.15), the problem described by Eqs. (8.4)–(8.10) can be written as

$\left\{\begin{array}{l}{\omega }_{1}\frac{\partial {p}_{D1}}{\partial {t}_{D}}-\frac{{w}_{1}}{{r}_{D}}\frac{\partial }{\partial {r}_{D}}\left({r}_{D}\frac{\partial {p}_{D1}}{\partial {r}_{D}}\right)+2{\stackrel{˜}{k}}_{D}\left({p}_{D1}-{p}_{D3}\right)=0\\ {\omega }_{3}\frac{\partial {p}_{D3}}{\partial {t}_{D}}+2{\stackrel{˜}{k}}_{D}\left(2{p}_{D3}-{p}_{D1}-{p}_{D2}\right)=0\\ {\omega }_{2}\frac{\partial {p}_{D2}}{\partial {t}_{D}}-\frac{{w}_{2}}{{r}_{D}}\frac{\partial }{\partial {r}_{D}}\left({r}_{D}\frac{\partial {p}_{D2}}{\partial {r}_{D}}\right)+2{\stackrel{˜}{k}}_{D}\left({p}_{D2}-{p}_{D3}\right)=0\end{array}\right\$

(F.1)

${p}_{Dj}\to 0\mathrm{when}{t}_{D}\to 0,j=1,2,3$

(F.2)

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

(F.3)

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

(F.4)

${{p}_{w}}_{Dj}={\left({p}_{Dj}-{s}_{j}{r}_{D}\frac{\partial {p}_{Dj}}{\partial {r}_{D}}\right)}_{{r}_{D}=1},j=1,2$

(F.5)

${q}_{D2}=f\left({p}_{wD1}\right)$

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