GLOSSARY

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LaTeX Math Inline
body\Sigma_k

well-reservoir contact of the 

LaTeX Math Inline
bodyk
-th well

LaTeX Math Inline
body--uriencoded--d %7B\bf \Sigma%7D

normal vector of differential area on the well-reservoir contact, pointing inside wellbore

LaTeX Math Inline
bodym_k(t)

mass rate at 

LaTeX Math Inline
bodyk
-th well 
LaTeX Math Inline
bodym_k(t) = \rho(p) \cdot q_k(t)

LaTeX Math Inline
bodyq_k(t)

sandface flowrate at 

LaTeX Math Inline
bodyk
-th well 

LaTeX Math Inline
body\rho(p)

fluid density as function of reservoir fluid pressure 

LaTeX Math Inline
bodyp


Then use the following equality:

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LaTeX Math Block
anchorS8TNB
alignmentleft
\rho \, \phi \, c_t  \cdot \frac{\partial p}{\partial t} + \nabla \, ( \rho \, {\bf u}) = 0\rho \, \sum_k q_k(t) \cdot \delta({\bf r}-{\bf r}_k)
LaTeX Math Block
anchorqk
alignmentleft
\int_{\Sigma_k} \, {\bf u} \,  d {\bf A} = q_k(t)

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