like porosity was initiated by Gassman [18] and revised later on by Mavko and Mukerji [19] and Mavko et al. [20]. Other studies on this subject include Wyllie et al. [21], Raymer et al. [22], Castagna et al. [11], Han [23], Raiga-Clemenceau and colleagues [24], Eberhart [25], and the critical porosity model of Wang and Nur [26].
Greenberg - Castagna model is utilized in this study to estimate the shear wave velocity of a rock sample. Greenberg and Castagna (1993) presented an empirical formula for multi-mineral rocks saturated in brine:
(2.1)
where L is the number of lithology in the formation, Xi is the percentage of the volume of lithology, aij is regression coefficient, Ni is the degree of polynomial regression for the targeted lithology, Vp and Vs are compression and shear wave velocities (Km/s).
This formula estimates shear wave velocity using compressional wave velocity in pure unit minerals, saturated in water. Regression coefficients of the formula for four different lithologies were presented by Greenberg and Castagna [12].
To estimate the shear wave velocity of a brine saturated rock using Greenberg - Castagna, formula, one needs to find a way to replace the existing fluid with brine as a common fluid. This work is done by utilizing Gassman relations. In fact, by brine replacement, a similar condition is assumed for the whole environment. Then the compressional wave velocity is obtained for brine saturated situation using the following formulas. Finally, the shear wave velocity is obtained from the estimated compressional wave velocity.
Replacing different types of pore fluids with brine, and keeping the rest of the physical properties of the rock (e.g. porosity) intact, the compressional wave modulus of the rock will also be changed [27]. Compressional wave modulus is expressed as a linear combination of bulk modulus and shear modulus:
(2.2)
The usual process is initiated by replacing the primary fluid with a fluid with similar sets of velocities and rock densities, compared with the primary fluid. These velocities are usually obtained from logs, but sometimes they may also be the results of theoretical models. In this study, the velocity of the wave that has passed through the primary fluid (in our case, supercritical dioxide is injected into the water) is obtained through laboratory measurements. But the removal of the existing fluid effects and replacing it by the common fluid (brine) has been achieved through the following steps (Dvorkin, 2003):
In the first stage the effective bulk modulus of pore fluid composition, (
(2.3)
where, Sgas, Soil, Sbr, indicate gas, oil and brine saturation and Kgas, Koil, Kbr, correspond to the apparent modulus of gas, oil, and brine. In the next step, bulk modulus of rock (Klog), is calculated by equation (2.4):
where, ρb is the bulk density and Vp is the compressional wave velocity of the rock. At the next step, bulk modulus of dry rock (Kdry) is calculated using the equation (2.5) and the mineral bulk modulus:
where, ϕ is porosity and Kmineral is the apparent modulus in the mineral phase (Thomsen, 1986). The bulk modulus of rock saturated with brine (Kcommon) is determined by:
(2.6)
where, Kcf is the bulk modulus. The compressional wave modulus of the rock saturated with brine (Mcommon) is calculated using the following formula:
(2.7)
The compressional wave velocity after removal of the primary fluid and replacing it with brine is obtained by:
(2.8)
In this case, when the shear wave data is not available, compressional wave modulus (Mlog) is calculated from charts (logs) using the following relation:
(2.9)
The compressional wave modulus of the dry rock (Mdry) is also calculated using compressional wave modulus of the rock’s minerals:
(2.10)
(2.11)
where, ϕ is porosity, μmineral is shear modulus and Kmineral is the apparent modulus in the mineral phase. The changes in the elastic modules of different minerals as a whole have been estimated [28]. Finally, the compressional wave modulus of the brine saturated rock (Mcommon) is calculated as follows:
(2.12)
Greenberg - Castagna formula is defined for rocks completely saturated with brine. In this paper, the apparent modulus (Kcf) of 2/25 is assumed for brine saturated cases.
It is worth mentioning that the fluid changes have no effect on shear wave modulus, which is the same before and after complete saturation with brine.
2.2.2 Estimating Geomechanical Parameters
Here to determine the bulk modulus, shear modulus, and Young’s modulus of rocks, we assumed that they are elastic, homogeneous and isotropic and used the following formula:
(2.13)
(2.14)
(2.15)
2.3 Laboratory Set Up and Measurements
The data used in this study were collected by a core flooding system at the laboratory of petroleum engineering department, Curtin University of Technology, Australia. The tested sandstone sample has the volume of 2.350 (gr/cm3), 3.79 cm in diameter, 7.98 cm in length and the porosity of 36/43 deg. After drying and preliminary preparation of this sample, it was placed in a polymer sleeve to be protected from the fluid axial pressure applied. The sample was placed inside the core holder, and piezoelectric (transducers) were placed on the caps to send and receive the acoustic wave. Figure 2.2 shows schematically, how the