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In case the phases have the same pressure the compressibility of multi-phase fluid can be expressed via compressibilities of single-phase fluids as: Compressibility of multiphase fluid in thermodynamic equilibrium at a given pressure 

LaTeX Math Inline
bodyp
 and temperature 
LaTeX Math Inline
bodyT
  is a linear sum of its single-phase components:

LaTeX Math Block
anchorcf1
alignmentleft
c_f(p, T) = c\sum_w {\,alpha} s_w\alpha +\cdot c_o \, s_o + c_g \, s_g\alpha(p,T)

where

LaTeX Math Inline
bodys_\alpha

LaTeX Math Inline
body\alpha
-phase volume share, subjected to
LaTeX Math Inline
body\sum_{\alpha} s_\alpha = 1

LaTeX Math Inline
bodyc_\alpha(p, T)

LaTeX Math Inline
body\alpha
-phase compressibility as function of pressure 
LaTeX Math Inline
bodyp
 and temperature 
LaTeX Math Inline
bodyT
 



Expand
titleDerivation


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borderColorwheat
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The total multiphase volume:

LaTeX Math Block
anchor1
alignmentleft
V
_f
 =
V_w
 \
, s_w + 
sum V_
o \, s_o + V_g \, s_g
where 
\alpha


where

LaTeX Math Inline
bodyV_\alpha
are volumes, occupied by individual phases.


The volume fraction of individual phase is defined as:

LaTeX Math Block
anchor1
alignmentleft
s_
w = \frac{V_w}{V_f}, \ s_o 
\alpha = \frac{V_
o}{V_f}, \ s_g = \frac{V_g
\alpha}{V
_f
}


This leads to:

LaTeX Math Block
anchor1
alignmentleft
c_f = \frac{1}{V
_f
} \, \frac{\partial V
_f
}{\partial p} = 
\frac{1}{V
_f
} \
frac{V_w}{p} + 
sum_\alpha \frac{
1}{
\partial V_
f} \frac{V_o
\alpha}{
p} + \frac{1}{V_f} \frac{V_g}{p}
LaTeX Math Block
anchor1
alignmentleft
c_f
\partial p} = 
 \
frac{V_w}{V_f}
sum_\alpha \frac
{1}
{V_
w
\alpha}
\frac
{V
_w
}
{p}
+ \frac{V_o}{V_f}
\, \frac{1}{V_
o
\alpha} \frac{\partial V_
o
\alpha}{\partial p}
+
=
 \
frac{V_g}{V_f} \frac{1}{V_g} \frac{V_g}{p}
sum_\alpha s_\alpha \, c_\alpha




In most popular practical case of a 3-phase fluid model this will be: 

1cf
, c_g
LaTeX Math Block
anchor
cf
alignmentleft
c_f =
\frac{V
s_w
}{V_f}
 \
frac{1}{V_w} \frac{V_w}{p}
, c_w +
\frac{V
s_o
}{V_f}
 \
frac{1}{V_o} \frac{V_o}{p}
, c_o +
\frac{V
s_g
}{V_f}
 \
frac{1}{V_g} \frac{V_g}{p}which leads to
LaTeX Math Block Reference
anchor

where 

LaTeX Math Inline
body\{ w, \, o, \, g \}
 mean water phase, oil phase and gas phase.


See also

...

Physics / Mechanics / Continuum mechanics / Fluid Mechanics / Fluid Statics / Fluid Compressibility / Fluid Compressibility @model