Gas Hydrate as Contact Cement

The third modeling possibility is the case where gas hydrate forms as an intergranular cement. The basis for this model is the contact cement theory (CCT) of Dvorkin et al. (1994), which calculates the bulk and shear moduli of a dense, random packing of spherical elastic grains with small amounts of elastic cement at the grain contacts. This model is appropriate for granular sediments whose original porosity is approximately that of a sphere pack, namely 36 to 40%. The original CCT theory was only valid for small concentrations of cement or, equivalently, residual porosities greater than approximately 25%. Recently, Dvorkin et al. (1999) extended CCT to high cement concentrations (i.e., residual porosities < 25%) where even the entire pore space may be filled with cement. Together, these theories provide a method for calculating the elastic moduli of dry sediment comprised of sand grains cemented by gas hydrate. The remainder of the pore space is assumed filled with water and Gassmann's equation is used to calculate the bulk modulus of the saturated sediment. The inputs for CCT and its extension to high cement concentration are the porosity of the original (without gas hydrate) sediment; the amount of cement (gas hydrate); the elastic moduli of the original mineral phase; the elastic moduli of the cement (gas hydrate); and the coordination number of the grain pack.

The CCT model is mathematically realized for two specific distributions of gas hydrate (Figure 2): (1) Scheme 1, gas hydrate forms at the grain contacts and (2) Scheme 2, gas hydrate evenly coats the grains.

Figure 2. Hydrate as contact cement. Scheme 1 and 2.

The arrangement represented by Scheme 1 is much stiffer than that given by Scheme 2 for the same amount of cement. Also, the residual pore spaces have very different shapes. Scheme 1 produces roughly equidimensional residual porosity while Scheme 2 is star-shaped. The modeling at high cement concentrations is greatly simplified by assuming equidimensional residual porosity (Dvorkin et al., 1999). For this reason, only Scheme 1 is used to model high gas hydrate cement concentration.

To construct the general solution, we begin with the CCT model which describes the elastic moduli of a dense, random pack of identical elastic spheres with a small amount of gas hydrate elastic cement. The effective bulk and shear moduli of the dry cemented sphere pack are predicted to be (Dvorkin et al., 1994; Dvorkin and Nur, 1996):

where Kc and Gc are the bulk and shear moduli of the cement, respectively; (j)c is the (critical) porosity of the uncemented grain pack ((¡>c = 36 to 40%); and n (about 8.5) is the average number of contacts per grain.

Parameters Sn and ST are the solutions to integral equations. Dvorkin and Nur (1996) supply the following approximate solution:

S„ = An{hn)a2 + Bn{An)a + C„(A„), A„{A„) = -0.024153 • A„-U646, Bn( AJ = 0.20405 • A/'89008, C„( A J = 0.00024649 • A/1'9864; Sr = AT(AT, v)a2 + Br{AT, v)a + CT(Ar, v), Ar( AT, v) = -10-2 -(2.26v2 +2.07v + 2.3)-AT0079v2+0 1754v-U42, Bt{ At, v) = (0.0573 V2 +0.0937v + 0.202)-AT°0274l"+0(>529,'-()-8765,

CT(AT, v) = 10"4 -(9.654V2 +4.945V + 3.1)-A/'018671"+0-40IIV-1'8186; _ 2GC (1 - v)(l - vc) n nG 1 - 2 vc ' r tiG'

where G and V are the shear modulus and Poisson's ratio of the grain material, respectively; and Gc and Vc are the shear modulus and Poisson's ratio of the cement, respectively.

Parameter Ot depends on the cement distribution. For Scheme 1 (Figure 2) a has the form:

The details of calculating the elastic moduli of the pack at low residual porosities are given in Dvorkin et al. (1999).

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