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## Empirical relationships

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 vp, Vpf, and Vps are the p-wave sonic velocities of the saturated rock, the pore fluid, and the mineral material making up the rock respectively, and where . The interpretation of this expression is that the total transit time is the sum of the transit time of the elastic wave in the mineral and the transit time in the pore fluid. It is therefore often referred to as the time-averaged equation. Raymer [Raymer 1980] suggested improvements to Wyllie's empirical equation. For low porosities they proposed the relationship (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( ) = P1 + P2(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 Y0 and v0 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.

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