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titleSolution



LaTeX Math Block
anchorpwf
alignmentleft
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}




LaTeX Math Block
anchorpwf
alignmentleft
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]



LaTeX Math Block
anchorpwf
alignmentleft
p_e(t) = p_i - \frac{q_t}{V_e \phi c_t}t




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titleDerivation



Approximations

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Derivation
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titleSolution
title



LaTeX Math Block
anchorpwf
alignmentleft
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





LaTeX Math Block
anchorpwf
alignmentleft
p_{wf}(t) \approx p_e(t) - \frac{q_t}{2 \pi \sigma} \, \left[  \ln \frac{r_e}{r_w}  - 0.5 + S \right]



LaTeX Math Block
anchorpwf
alignmentleft
p_e(t) = p_i - \frac{q_t}{V_e \phi c_t}t
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Applications

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Equation  

LaTeX Math Block Reference
anchorpwf
 shows how the basic diffusion model parameters impact the relation between drawdown 
LaTeX Math Inline
body\Delta p = p_i - p_{wf}
 and total sandface flowrate 
LaTeX Math Inline
bodyq_t
 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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