-
VP = SQR[(k + 4µ/3)/
]
The effect of gas hydrate on velocities of unconsolidated sediments is modeled using the BGTL (Lee, 2002, 2003). The effect of free gas on velocities of unconsolidated sediments is modeled by the BGT with parameters derived from water-saturated sediments using the BGTL as demonstrated by Lee (2004). The difference between the BGT and BGTL is in the calculation of shear modulus of water-saturated sediments. This section summarizes the essence of the BGTL.
Elastic velocities (i.e., compressional velocity [VP] and shear velocity [VS]) of sediments can be computed from the elastic moduli using the following formulas:
]
and
), (1)
where
= density of the formation.
The formation density is given by
= (1 – 
)ma + fl, (2)
where
= porosity,
ma = matrix density, and
fl = pore fluid density.
Under the low-frequency assumption, the bulk modulus of fluid-saturated sediment (k) is given by the following formula in terms of the Biot coefficient, 
(Biot, 1941, 1956; Gassmann, 1951):
) + 2M, (3)
where
– )/kma] + (/kfl), (4)
and
The term 2M represents the interaction of fluid filling the porosity with the solid skeleton of the formation, and the Biot coefficient measures the ratio of volume change of the fluid to the volume change of the formation (Krief et al., 1990).
If it is assumed that the velocity ratio, VP/VS, is constant irrespective of porosity and is equal to the velocity ratio of matrix, the shear modulus of dry rock (µd) is given by (Krief et al., 1990):
), (5)
where µma is the shear modulus of the matrix. Conventionally, it is assumed that fluid in the pore space does not change the shear modulus of sediments. Therefore, the shear modulus of fluid-saturated sediment (µ) is the same as the dry shear modulus, (i.e., µ = µd), which is consistent with the BGT.
Lee (2004) derived the following shear modulus of water-saturated sediment assuming VP/VS is a function of porosity:
)G2(1 – )2n + µma2MG2(1 – )2n]/
)2n]/3}, (6)
where
As opposed to equation 5, the shear modulus predicted from the BGTL depends on the bulk modulus of fluid through M. Even though equation 5 is not the BGT (but is consistent with the BGT), Lee (2002) designated the use of equations 3 and 5 as the BGT to differentiate from the BGTL, which uses equations 3 and 6 to calculate elastic velocities.
For soft formations or unconsolidated sediments, the following Biot coefficient is used for the BGTL (Lee, 2002):
= [–68.7421/(1 + e( + 0.40635)/0.09425)] + 0.98469. (7)
As indicated in Lee (2004), the appropriate Biot coefficient for velocities of partially gas saturated sediments (bgt) is given by the following formula:
bgt = 1 – (µbgtl/µma), (8)
and in equations 3 through 5, bgt is substituted for .
Two kinds of gas saturation in the pore space are considered: uniform and patch. Brie et al. (1995) suggested the following empirical mixing law based on the in situ downhole data:
where
When e = 1, Brie et al.'s formula is the same as the isostrain (Voigt) average, proposed by Domenico (1977). As e increases, the patch saturation approaches the characteristics of uniform saturation and approaches nearly a uniform saturation at e = 40. Brie et al. (1995) showed that most downhole data they analyzed fit the mixing law with calibration constants between e = 2 and e = 5. However, other e values are appropriate depending on the frequency of measurement and the degree of consolidation.
In the BGTL formulation, two parameters are introduced to match the predicted velocity ratio with the measured velocity ratio or velocity. As shown in Lee (2003), these parameters are somewhat constrained by the nature of sediments, such as differential pressure, the degree of consolidation, and clay volume content. Lee (2003) suggested the following formula for n, based on laboratory data compiled by Prasad (2002):
where
Usually, for unconsolidated sediments m varies from 1 to 2 and for consolidated sediments m varies from 4 to 5.
The scale G is introduced to compensate for the effect of clay or gas hydrate on the matrix material. Based on the data given by Han et al. (1986) and well logging data at the Mallik 5L-38 well, the following formula for G has been proposed (Lee, 2003; Lee and Collett, in press):
where
For a clean sandstone without gas hydrate, G = 1.
In summary, in the framework of BGTL, the parameter n is the only free parameter to choose, and it can be estimated by fitting the measured velocities to calculated velocities. G is an empirically derived parameter that depends on the clay volume content and gas hydrate saturations. The difference between the BGT and the BGTL is the method of calculating the shear modulus of the water-saturated sediments.
