Experimental measurements have often shown that relatively simple empirical relationships can be used to describe the properties of sedimentary rocks. Measurements by Wyllie et al. [Wyllie 1956, Wyllie 1958, Wyllie 1963] revealed that a relatively simple monotonic relationship exists between the sonic velocity and porosity in fluid saturated sedimentary rocks with relatively uniform mineralogy. They approximated these relationships with the expression

(9) |

where *v _{p}*,

(10) |

A comparison of the predicted and numerically derived velocities for water saturated Fontainebleau sandstone is shown in Figure 8. The Raymer empirical equation along with the numerical data give a reasonable match to the measured data.

**Figure 8:** Comparison of the results of the simulations
(squares and dashed line)
for water-saturated sandstone to experimental data (circles) and the empirical
equations of Wyllie (Eqn. 9) and Raymer (Eqn.
10). The fit of the numerical data and the Raymer equation is satisfactory.
The Wyllie equation gives a poor fit.

Nur et al. [Nur 1991, Nur 1995] proposed that the moduli of rocks should trend
between the mineral grain modulus at low porosity to a value for a
mineral-pore suspension at some limiting high porosity. The idea is based on
the observation that for most porous materials there is a critical porosity
_{c}
which separates the mechanical behavior into two distinct domains.
For porosities lower than _{c}
the mineral grains are load bearing whereas
at porosities greater than the material falls apart. Theoretical
models may be modified by incorporating percolation behavior at any desired
_{c}
by simply redefining the endpoint porosity. The simplest model of this
type is based on a modified Voigt equation, where the original Voigt
upper bound for a property *P* is given by
*P*() =
*P*_{1} + *P*_{2}(1 - ).
This empirical model [Nur 1995] based on the
observation that the modulus of
porous rocks often trends linearly with porosity between the two values in the
load bearing domain is given by

The critical porosity for sandstones was found empirically in [Nur 1991] to be _{c} = 0.40. Recently [Roberts and Garboczi 2000] developed empirical equations for
the elastic properties of overlapping sphere packs under dry conditions:

where *Y*_{0} and *v*_{0} are the Young's modulus and the Poisson ratio of the spherical grains. A comparison of the above equations with the numerical and experimental data
is shown in Figure 9. The agreement is excellent over the full range of
.

**Figure 9:**
Comparison of the simulation results to (a) the critical porosity
model of [Nur 1995] and of [Roberts and Garboczi 2000]
for the dry case and (b) the critical porosity model of [Nur 1995]
for the water saturated case. Under both conditions the fit of the empirical
equations to experiment and numerical data is excellent and comparable to the
data obtained from images.