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LaTeX Math Block
anchorqGamma
alignmentleft
{\rm F}_{\Gamma}(p, {\bf u}) = 0

where

LaTeX Math Inline
body\Gamma

reservoir boundary

LaTeX Math Inline
bodyp

reservoir pressure

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

fluid velocity

LaTeX Math Inline
body--uriencoded--%7B\rm F%7D_%7B\Gamma%7D(p, %7B\bf u%7D)

some function


The popular form of the Reservoir boundary flow condition @model is:

LaTeX Math Block
anchorNeyman
alignmentleft
{\rm F}_{\Gamma}(p, {\bf u}) = \big[ a \cdot (p({\bf r}) - p_0) + \epsilon \cdot {\bf n} \cdot M \,  (\nabla p - \rho \, {\bf g})  \big]_{{\bf r} \in \Gamma} = 0

where

LaTeX Math Inline
body--uriencoded--p(t, %7B\bf r%7D)

reservoir pressure

LaTeX Math Inline
bodyt

time

LaTeX Math Inline
body--uriencoded--\rho(%7B\bf r%7D,p)

fluid density 

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

position vector

LaTeX Math Inline
bodyM = k / \mu

LaTeX Math Inline
body\nabla

gradient operator

LaTeX Math Inline
bodyk

formation permeability to a given fluid

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

gravity vector

LaTeX Math Inline
body\mu

dynamic viscosity of a given  fluid

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

fluid velocity 

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

external normal to the reservoir boundary

LaTeX Math Inline
body\Gamma

LaTeX Math Inline
body--uriencoded--\epsilon \in \%7B 0,1 \%7D

a binary value


The two extreme cases of 

LaTeX Math Block Reference
anchorNeyman
 are:

Constant PressureNo flow

LaTeX Math Inline
body--uriencoded--p(%7B\bf r%7D)_%7B%7B\bf r%7D \in \Gamma%7D = p_0 = \rm const

LaTeX Math Inline
body--uriencoded--%7B\bf n%7D \cdot %7B\bf u%7D \, \Big%7C_%7B%7B\bf%7D in \Gamma%7D = %7B\bf n%7D \cdot M \, (\nabla p - \rho \, %7B\bf g%7D) \, \Big%7C_%7B%7B\bf%7D in \Gamma%7D = 0

See Also


Petroleum Industry / Upstream / Subsurface E&P Disciplines / Petroleum Geology / Reservoir boundary

Infinite reservoir boundary  ] [ Reservoir flow boundary ] [ Multiwell Retrospective Testing (MRT) ]