An effective method for shear-wave velocity prediction in sandstones

Abstract

Shear-wave velocity is essential in various seismic exploration applications, including seismic modeling, amplitude variation with offset analysis, multicomponent seismic data interpretation and other exploration applications. This study presents a simple but effective method for S-wave velocity prediction from P-wave velocity based on granular media model. In the proposed method, the soft-sand, intermediate stiff-sand and stiff-sand models are unified with the expression of the soft-sand model with an effective coordination number. The shear modulus of dry rocks can be related to bulk modulus through the coordination number in the unified model. Elastic-wave velocities of water-saturated rocks at low frequencies can be predicted from the moduli of dry rock using Gassmann’s equation. Thus, the coordination number can be inverted from P-wave velocity of saturated rocks by combining the unified granular media model and Gassmann’s equation. Eventually, S-wave velocity of saturated rock is computed with the inverted effective coordination number. The numerical results indicate that the predicted S-wave velocities agree well with the measured velocities for the laboratory data and well logging data. The proposed method is applicable to sandstones with lithification of large ranges because the unified granular media model accounts for the media between Hashin–Shtrikman lower bound and upper bound. Moreover, the new method is quite suitable for the prediction of S-wave velocity in sandstones deposited in deep-water environment, particularly for turbidite sediment.

Appendix

A. Gassmann equation

The Gassmann (1951) equation has been widely used for calculating the effect of fluid substitution on elastic properties using the frame properties. It assumes a homogeneous mineral modulus and statistical isotropy of the pore space but is free of assumptions about the pore geometry. To be most noteworthy, the Gassmann theory is based on low frequency assumption such that the induced pore pressures are equilibrated throughout the pore space (Mavko et al. 2009). The bulk modulus of a fluid-saturated porous medium is calculated using the bulk moduli of the solid grains, the frame, and the pore fluid through the following equation:

$$\frac{{K^{*} }}{{K_{s} - K^{*} }} = \frac{{K_{d} }}{{K_{s} - K_{d} }} + \frac{{K_{f} }}{{\varphi \left( {K_{s} - K_{f} } \right)}},$$

(16)

where $K^{*}$ is the bulk modulus of a rock saturated with a fluid of bulk modulus $K_{f}$, $K_{d}$ is the frame bulk modulus, $K_{s}$ is the grain bulk modulus, and $\varphi$ is porosity. The shear modulus $G^{*}$ of the rock is not affected by pore fluid, thus

$$G^{*} = G_{d} ,$$

(17)

where $G_{d}$ is the frame shear modulus of the rock. The density $\rho^{*}$ of the saturated rock is computed by the follow equation,

$$\rho^{*} = \rho_{d} + \varphi \rho_{f} ,$$

(18)

where $\rho_{d}$ is the density of dry rock, and $\rho_{f}$ is the pore fluid’s density. To be noted, $\rho_{d} = \left( {1 - \varphi } \right)\rho_{s}$, where $\rho_{s}$ is the grain density (Wang 2012).

When the pore fluid consists of various phases, the bulk modulus $K_{f}$ of a fluid mixture can be calculated using Wood’s equation (Wood 1941):

$$\frac{1}{{K_{f} }} = \frac{{S_{w} }}{{K_{w} }} + \frac{{S_{o} }}{{K_{o} }} + \frac{{S_{g} }}{{K_{g} }},$$

(19)

where $K_{w}$, $K_{o}$, and $K_{g}$ are the bulk modulus of water, oil and gas, respectively; $S_{w}$, $S_{o}$, and $S_{g}$ are the water, oil and gas saturations respectively. Equation 19 indicates that the pore fluid is uniformly distributed in the pores of the rocks.