The total compressibility of porous rock and pore-saturating fluid is going to be:

$\begin{array}{l}\displaystyle c_t = c_r + c_f\end{array}$

where

Derivation

The pore-saturating fluid volume is related to total rock volume $\begin{array}{l}V_r\end{array}$ and reservoir porosity $\begin{array}{l}\phi\end{array}$ as:

(1)	$\begin{array}{l}\displaystyle V_f = V_r \phi\end{array}$

The total reservoir compressibility is going to be:

(2)

$\begin{array}{l}\displaystyle c_t = -\frac{1}{V_r} \, \frac{\partial V_r}{\partial p} = - \frac{\phi}{V_f} \, \frac{\partial}{\partial p} \left( \frac{V_f}{\phi} \right) = - \frac{\phi}{V_f} \, \left( V_f \, \frac{\partial}{\partial p} \frac{1}{\phi} + \frac{1}{\phi} \frac{\partial V_f}{\partial p} \right) = \frac{1}{\phi} \, \frac{\partial \phi}{\partial p} - \frac{1}{V_f} \, \frac{\partial V_f}{\partial p} = c_r + c_f\end{array}$

Some applications (like multi-phase pressure diffusion) account for the impact of cross-phase fluid exchange on the total compressibility which require some corrections to equation

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:

(3)	$\begin{array}{l}\displaystyle c_t(s,P) = c_r + c_w s_w + c_o s_o + c_g s_g + s_o [ R_{sp} + (c_r + c_o) R_{sn} ] + s_g [ R_{vp} + R_{vn}(c_r + c_g) ]\end{array}$

where

$\begin{array}{l}R_{sn}, \; R_{vn}\end{array}$	normalized cross-phase exchange ratios as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$
$\begin{array}{l}R_{sp}, \; R_{vp}\end{array}$	normalized cross-phase exchange derivatives as functions of reservoir pressure $\begin{array}{l}p\end{array}$ and temperature $\begin{array}{l}T\end{array}$

See Non-linear multi-phase pressure diffusion @model for derivation of (3).

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Total compressibility @model

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