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| p(t,r) = p_i - \frac{{\rm w \,} q_t }{V_e \, \phi \, c_t} \, t + \frac{{\rm w \,} q_t }{4\pi \sigma} \left[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \right]
, \quad r_{wf} < r \leq r_e,
\quad {\rm w }= 1 - \frac{r_w^2}{r_e^2} |
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| p_{wf}(t) = p_e(t) - \frac{{\rm w\, }q_t}{2 \pi \sigma} \, \left[ \ln \frac{r_e}{r_w} - 0.5 + S \right] |
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| p_e(t) = p_i - \frac{q_t}{V_e \phi c_t}t |
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Approximations
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| title | Derivation
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| p(t,r) \approx p_i - \frac{q_t}{V_e \, \phi \, c_t} \, t + \frac{q_t}{4\pi \sigma} \bigg[ 2 \ln \frac{r}{r_e} - \frac{r^2}{r_e^2} + 1 \bigg]
, \quad r_{wf} < r \leq r_e |
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| p_{wf}(t) \approx p_e(t) - \frac{q_t}{2 \pi \sigma} \, \left[ \ln \frac{r_e}{r_w} - 0.5 + S \right] |
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| p_e(t) = p_i - \frac{q_t}{V_e \phi c_t}t |
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Applications
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Equation
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shows how the
basic diffusion model parameters impact the relation between
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| body | \Delta p = p_i - p_{wf} |
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and
total sandface flowrate and plays important methodological role as they are used in many algorithms and express-methods of
Pressure Testing. It also called Dupuis
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