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p(t,x) = p_i - \frac{q_t}{\sigma \, d} \bigg[ \sqrt{\frac{4 \chi t}{\pi}} \exp \bigg( -\frac{x^2}{4 \chi t} \bigg) - x \, \bigg[ 1- {\rm erf} \bigg(\frac{x}{\sqrt{4 \, \chi \, t}} \bigg) \bigg]  \bigg]



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p_{wf}(t) = p(t,x=0)= p_i - \frac{q_t}{\sigma \, d} \,  \sqrt{\frac{4 \chi t}{\pi}}




Scope of Applicability

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Pressure Testing

Pressure Drop


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\delta p = p_i - p_{wf}(t) \sim t^{1/2}



Log derivative


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t \frac{d (\delta p)}{dt}  \sim t^{1/2}


Fig. 2. PTA Diagnostic plot for LFS

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