The sonic porosity is usually abbreviated SPHI or PHIS  on log panels and denoted as  \phi_s  in equations.

The key measurement is the p-wave velocity V_{p \ log} from sonic tool readings.

The key model parameter is rock matrix sonic velocity  V_{p \ m}  which is calibrated for each facies individually and can be can be assessed as vertical axis cut-off on  V_{p \ log} cross-plot against the core-data porosity  \phi_{\rm air}. 

The model also accounts for saturating rock fluids with p-wave velocity  V_{p \ f} value.

In overbalance drilling across permeable rocks the saturating fluid is usually mud filtrate. 

In underbalance drilling this the saturating fluid is identified from resistivity logs.  


WGG Equation (Wyllie)

 

The WGG sonic porosity  \phi_s equation is :

(1) \frac{1}{V_{p \ log}} = \frac{1-\phi_s \ C_p}{V_{p \ m}} + \frac{\phi_s \ C_p}{V_{p \ f}}

where   C_p is compaction factor, accounting for the shaliness specifics and calculated as:

(2) C_p = \frac{V_{shс}}{V_{sh}}

where 

V_{sh} – p-wave velocity for adjacent shales,

V_{shc} – p-wave velocity reference value for tight shales (usually 0.003 ft/μs).

 


GGG Equation (Gardner, Gardner, Gregory)


The GGG sonic porosity  \phi_s equation is :

(3) \frac{1}{V^{1/4}_{p \ log}} = \frac{(1-\phi_s)}{V^{1/4}_{p \ m}} + \frac{\phi_s}{V^{1/4}_{p \ f}}


The above equation is based on the Gardner correlation for sonic density:

(4) \rho_s = 171 \cdot V_{p \ m}^{1/4}

where  \rho_s is measured in  \rm \big[ \frac{m^3}{kg} \big] and  V_{p \ m} is measured in  \rm \big[ \frac{m}{\mu s} \big] 


and mass balance equation:


(5) \rho_s = (1-\phi_s)\rho_m + \phi_s \rho_f



RHG Equation (Raymer, Hunt, Gardner)


The RHG sonic porosity  \phi_s equation is :

(6) V_{p \ log} = (1-\phi_s)^2 V_{p \ m} + \phi_s V_{p \ f}

and only valid for  \phi_s < 0.37.