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Normalised dimensionless difference between the sandface bottomhole pressure (BHP) 

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bodyp_{wf}(t)
and a model of a full-entry vertical well with homogeneous reservoir and non-damaged near-well reservoir zone 
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bodyp^*p_{wf}(t,r)
:

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S = \frac{p_{wf}(t)  - p(t, r_w)}{ \left[ r \cdot \frac{\partial p}{\partial r} \right]_{r=r_w} }

where

qt

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body

r_

total sandface rate

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body\sigma

formation transmissibility outside the damaged reservoir zone 

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bodyA_s

w

wellbore radius


By definition the skin-factor is a pressure adjustment at the well-reservoir contact and does not affect pressure distribution in reservoir away from wellbore

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bodyr > r_s
.

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p_{wf}(t) = \frac{q_t}{2 \pi \sigma} \, S +  p^*_{wf}p(t,r_w)

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where

LaTeX Math Inline
bodyq_t

total sandface rate

LaTeX Math Inline
body\sigma

formation transmissibility outside the damaged reservoir zone 

LaTeX Math Inline
body

...

A_s

LaTeX Math Inline
bodyp(t,r)

a model of a full-entry vertical well with homogeneous reservoir and non-damaged near-reservoir zone

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