GATE-2018 (24) :Wyllie – Time average equation

24. Given that the velocity of P-waves in a sandstone matrix is 5600 m/s and that in oil is 1200 m/s, the velocity of P-wave propagation in oil saturated sandstone with 30% porosity is ______________ m/s, (Use Wyllie time average equation.)

Solution:

Velocity of the matrix (V_m) = 5600 m/s

Velocity of the oil (V_oil) = 1200 m/s

Porosity = 30%

From Wyllie time average equation 

$\frac{1}{V}= \frac{\phi}{V_f}+\frac{1-\phi}{V_m}$--(1)

Substitute above values in equation (1)

$\frac{1}{V}= \frac{0.3}{1200}+\frac{1-0.3}{5600}$


$\frac{1}{V}= \frac{0.3}{1200}+\frac{0.7}{5600}$


$\frac{1}{V}=0.00025+0.000125$

$\frac{1}{V}= 0.000375$

V=2666m/s

Wyllie – Time average equation:-

It relates the sonic velocities with the porosity of the rock

$\frac{1}{V}= \frac{\phi}{V_f}+\frac{1-\phi}{V_m}$

V -Velocity

V_f -Velocity of the fluid

Φ – Porosity

V_m – Velocity of the matrix

It can be written in terms of interval travel-times

$\triangle t=\phi \triangle t_f +(1-\phi)\triangle t_m$

Reference: http://www.subsurfwiki.org/wiki/Wyllie_time-average_equation