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LaTeX Math Block
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p(t,r) = p_i + \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right)



Approximations

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Middle Time Response and Late

-time response

Time Response

LaTeX Math Inline
body\displaystyle t \gg \frac{r^2}{4\chi}



LaTeX Math Block
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p(t,r) \sim p_i + \frac{q_t}{4 \pi \sigma} \left[  
\gamma + \ln \left(\frac{r^2}{4 \chi t} \right) \right] 

= p_i - \frac{q_t}{4 \pi \sigma} \ln \left(\frac{2.24585 \, t}{r^2} \right)



Diagnostic Plots

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Pressure Drop


LaTeX Math Block
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\delta p = p_i - p_{wf}(t) = \frac{q_t}{4 \pi \sigma} {\rm Ei} \left(-\frac{r^2}{4 \chi t} \right)    \sim  \ln t + {\rm const}


Log derivative


LaTeX Math Block
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p' = t \frac{d (\delta p)}{dt} = \frac{q_t}{2\pi\sigma} \exp \left( - \frac{r^2}{4\chi t} \right)  \sim \rm const



Fig. 1. PTA Diagnostic Plot for LSS pressure response for the 0.1 md reservoir in a close line source vicinity (0.1 m), which is about a typical wellbore size.

One can easily see that with wellbore effects neglected even for a very low permeability reservoir the IARF regime is getting formed very early at 0.01 hr (36 s).

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