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Figure 2.1 Common methods for estimating the shear wave velocity.

Each of these methods has its own unique advantages that make it suitable to estimate the shear wave velocity. The dependency of wave propagation velocity to lithology, porosity, temperature, pressure and type of pore fluid, has introduced a variety of investigation methods in geophysics and rock physics.

      Laboratory measurement of shear wave velocity in a core is known as a standard procedure and the obtained velocities are comparable with that of the other methods. There are other field methods to estimate shear wave velocity such as Dipole Shear Sonic Imager and sonic logs. Although these methods are common, they must be performed in a large number of wells to obtain the velocity distribution in the entire field, and core extraction or running of sonic tools in a large field is very expensive [4]. Another common method to estimate the shear wave velocity is based on theoretical evaluation and modeling. In the past few decades, several empirical formulas have been introduced for estimating the shear wave velocity in rocks with different lithology, based on physical parameters of rock, especially the P-wave velocity and porosity. Many scientists, including Pickett [5], Milholand [6], Domenico [7], Thomsen [8], Han [9], Krief [10], Castagna [11] and Greenberg [12] have done very useful research in this area and various relationships have been developed and presented. These relations are valid for the saturated rocks with brine.

      Gassman formulas have been introduced to extend these experimental formulas to other fluids contents. In 1986, Han offered empirical regression formulas for elastic waves in laboratory condition which would estimate the speed based on porosity and clay content. In 1989, Eberhart added the pressure parameter to Han’s equation for shale sand rocks. Years before, Tosaya and Nur [13] and Castagna et al. [11] presented empirical formulas for shale sand rocks based on velocity, porosity and clay parameters.

      Other methods are artificial intelligence techniques such as neural networks and fuzzy logic [14–16]. Although their estimates are associated with less error, these methods also present a specific model for each different field, and the results from one field cannot be applied to other fields.

There are two methods to calculate the elastic coefficients. The first one is the static method (destructive) and the other one is the dynamic (non-destructive) method. In the static method, elastic parameters of rock are calculated from laboratory measurements such as the uniaxial compressive strength of cores. But in a dynamic method, assuming that rocks are elastic, the coefficients are determined from propagating shear and compressional waves in rock and measuring waves travel time, without damaging the rock [17].

      Currently, there is no unique empirical formula or comprehensive theory that could be utilized to determine the elastic wave velocity and elastic coefficients at different environmental conditions, type of fluid and rock. In this paper, experimental data (elastic wave propagation velocity) has been gathered using a sample saturated with water and supercritical carbon dioxide at different reservoir pressure. Then while the common fluid (brine) was replaced by critical CO₂, the elastic wave velocity values were calculated using Gassman formulas and Greenberg - Castagna empirical equations and the results were compared with direct laboratory observations.

2.2 Methodology

      2.1.2 Estimating the Shear Wave Velocity

      A major part of the seismic signal analysis in regards to rock physics models relates shear wave velocity to mineralogy and porosity. Rock physics analysis based on logs and cores and the relation of these to the geological model, leads to the establishment of a relationship between velocity and porosity. Formulation of the relation between rock velocity and rock

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