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

$\{\begin{cases} ω_{1} \frac{\partial p_{D 1}}{\partial t_{D}} - \frac{w_{1}}{r_{D}} \frac{\partial}{\partial r_{D}} (r_{D} \frac{\partial p_{D 1}}{\partial r_{D}}) + 2 {\tilde{k}}_{D} (p_{D 1} - p_{D 3}) = 0 \\ ω_{3} \frac{\partial p_{D 3}}{\partial t_{D}} + 2 {\tilde{k}}_{D} (2 p_{D 3} - p_{D 1} - p_{D 2}) = 0 \\ ω_{2} \frac{\partial p_{D 2}}{\partial t_{D}} - \frac{w_{2}}{r_{D}} \frac{\partial}{\partial r_{D}} (r_{D} \frac{\partial p_{D 2}}{\partial r_{D}}) + 2 {\tilde{k}}_{D} (p_{D 2} - p_{D 3}) = 0 \end{cases}$

si1_e (F.1)

$p_{D j} \to 0 when t_{D} \to 0, j = 1, 2, 3$

(F.2)

$p_{D j} \to 0 when r_{D} \to \infty, j = 1, 2$

(F.3)

$w_{j} {(r_{D} \frac{\partial p_{D j}}{\partial r_{D}})}_{r_{D} = 1} = - q_{D j}, j = 1, 2$

(F.4)

${p_{w}}_{D j} = {(p_{D j} - s_{j} r_{D} \frac{\partial p_{D j}}{\partial r_{D}})}_{r_{D} = 1}, j = 1, 2$

(F.5)

$q_{D 2} = f (p_{w D 1})$

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Well Test Analysis for Multilayered Reservoirs with Formation Crossflow by Hedong Sun, Chengtai Gao

Case for Variable Bottom-Hole Rate (Chapter 8)

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