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Normalised dimensionless difference between the sandface bottomhole pressure (BHP) and sandface reservoir pressure 

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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 
LaTeX Math Inline
bodyp^*_{wf}(t)
:

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S = \frac{2 \pi \sigma}{q_t} \cdot \left[ p_{wf}(t)  - p({\bf r}, t) |_{{bf r} \nin A_s} \right]

where

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bodyq_t

total sandface rate

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body\sigma

formation transmissibility outside the damaged reservoir zone 

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bodyA_s

LaTeX Math Inline
bodyA_s

damaged reservoir zone


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
.


The total skin is usually decomposed into a sum of two components:


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S_T = S_G + S_M

where

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bodyS_G

Geometrical skin, related to deviation of the well-reservoir contact from the simplest model

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bodyS_M

Mechanical skin, related to pressure drop caused by the near-reservoir zone formation damage


Based on definition the wellbore pressure dynamics 

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bodyp_{wf}(t)
of the well with skin-factor can be writen as:

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

where 

LaTeX Math Inline
bodyp^*_{wf}(t)
 is a model of a full-entry vertical well with homogeneous reservoir and non-damaged near-reservoir zone.