# Wyllie time-average equation

The Wyllie time-average equation is an equation that relates sonic velocities with the porosity of a rock. It was proposed by Wyllie, Gregory and Gardner in 1956.[1]. The equation basically holds that the total travel time recorded on the log is the sum of the time the sonic wave spends travelling the solid part of the rock, called the rock matrix and the time spent travelling through the fluids in the pores.

The equation is as follows:

$\frac{1}{v}=\frac{\phi}{v_f}+\frac{1-\phi}{v_m}$

and can be rewritten in terms of interval travel-times as

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

where, $\phi$ is the total porosity, $v$ is the wave's measured phase velocity, $v_m$ is the velocity of the rock matrix, and $v_f$ is the velocity of the fluid in the pores, $\Delta t$ is the measured interval travel-time, $\Delta t_m$ is the interval travel-time of the rock matrix, $\Delta t_f$ is the interval travel-time of the saturating fluid.

## References

1. Wyllie, M. R. J., Gregory, A. R., and Gardner, L. W., 1956, Elastic wave velocities in heterogeneous and porous media: Geophysics, 21, no. 1, 41-70