Identification of gas hydrate based on velocity cross plot analysis

Abstract

Gas hydrate is regarded as a kind of important energy resource in the recent decades. The identification of gas hydrate is a hot issue for researchers all over the world in order to exploration and exploitation. Previous research shows some methods to identify gas hydrate, but these methods also have shortcomings in different actual conditions. It is still a difficult problem to distinguish gas hydrate accurately from marine sediments. Here we propose a cross plot analysis of velocity to identify gas hydrate. Firstly, we simulate different minerals bearing sediments by rock physics models to obtain different physical properties. After a series of experiments, the results show that there are obvious differences between gas hydrate bearing sediments and other minerals bearing sediments in P-wave and S-wave velocity. Therefore, the cross plot of velocity can be used to distinguish gas hydrate qualitatively. Finally, we verify the cross plot method using the data in the Ocean Drilling Program (ODP) Leg 204 expedition of the Hydrate Ridge along the Oregon continental margin, where gas hydrate is confirmed via logging. The hydrate identification results of actual velocity data by cross plot analysis are generally in keeping with the results of core samples. Moreover, using the cross plot method also can estimate gas hydrate saturation range, which is in accordance with the measurement of chloride (Cl⁻) concentration in most cases.

Appendices

Appendix A

Effective medium theory

The elastic velocities of gas hydrate bearing sediments are given by:

$${\text{V}}_{p} = \sqrt {\frac{{K_{sat} + 4G{}_{sat}/3}}{{\rho_{b} }}} ,$$

(A1)

$${\text{V}}_{S} = \sqrt {\frac{{G_{sat} }}{{\rho_{b} }}} ,$$

(A2)

The density of gas hydrate bearing sediments can be given by:$\rho_{{\text{b}}} = \left( {1 - \varphi } \right)\sum\nolimits_{i = 1}^{{\text{N}}} {f_{i} } \rho_{i} + \varphi \rho_{w} \left( {1 - S_{h} } \right) + \varphi \rho_{h} S_{h}$, where N represents the number of mineral components; $f_{i}$ and $\rho_{i}$ is the volume fraction and density of the ith mineral constituent, respectively; $\rho_{b}$, $\rho_{{\text{w}}}$ and $\rho_{h}$ denote the densities of gas hydrate bearing sediments, water and hydrate, respectively; ɸ is the porosity of sediments. S_h denotes the saturation of hydrate. The modulus of mineral components are given by (Hill 1952):

$${\text{X}} = \frac{1}{2}\left[ {\sum\limits_{i = 1}^{{\text{N}}} {f_{i} X_{i} + \left( {\sum\limits_{i = 1}^{N} {\frac{{f_{i} }}{{X_{i} }}} } \right)^{ - 1} } } \right],$$

(A3)

where X denotes the bulk modulus (K) or shear modulus (G). The bulk and shear modulus of brine-saturated sediments are given by (Gassmann 1951):

$${\text{K}}_{sat} = K\frac{{\varphi K_{dry} - (1 + \varphi )K_{f} K_{dry} /K + K_{f} }}{{(1 - \varphi )/K_{f} + \varphi K - K_{f} K_{dry} /K}},$$

(A4)

$${\text{G}}_{sat} = G_{dry} ,$$

(A5)

The bulk and shear modulus of dry frame are given by (Dvorkin and Nur 1996; Mindlin 1949):

$${\text{K}}_{dry} = \left[ {\frac{{(1 - \varphi )/(1 - \varphi_{c} )}}{{K_{hm} + 4G_{hm} /3}} + \frac{{(\varphi - \varphi_{c} )/\left( {1 - \varphi_{c} } \right)}}{{4G_{hm} /3}}} \right]^{ - 1} - \frac{{4G_{hm} }}{3},$$

(A6)

$${\text{G}}_{dry} = \left[ {\frac{{(1 - \varphi )/(1 - \varphi_{c} )}}{{G_{hm} + Z}} + \frac{{(\varphi - \varphi_{c} )/(1 - \varphi_{c} )}}{Z}} \right]^{ - 1} - Z,$$

(A7)

$${\text{Z}} = \frac{{G_{hm} }}{6}\left[ {\frac{{9K_{hm} + 8G_{hm} }}{{K_{hm} + 2G_{hm} }}} \right],$$

(A8)

$${\text{K}}_{hm} = \left[ {\frac{{m^{2} \left( {1 - \varphi_{c} } \right)^{2} G^{2} P}}{{18\pi^{2} \left( {1 - \nu } \right)^{2} }}} \right]^{\frac{1}{3}} ,$$

(A9)

$${\text{G}}_{hm} = \frac{5 - 4\nu }{{5\left( {2 - \nu } \right)}}\left[ {\frac{{3m^{2} \left( {1 - \varphi_{c} } \right)^{2} G^{2} P}}{{2\pi^{2} \left( {1 - \nu } \right)^{2} }}} \right]^{\frac{1}{3}} ,$$

(A10)

where ɸ_c represents critical porosity (ɸ_c = 0.35 ~ 0.4, generally) (we set 0.4) (Nur et al. 1998), and $\varphi \ge \varphi_{c}$ (the marine sediment porosity is above the critical porosity in our study); m is the contacts’ number per grain (m = 8 ~ 10, generally) (we set 8.5) (Murphy 1982); the effective pressure (P) is given by: $P = \left( {\rho_{b} - \rho_{w} } \right)gh$; g denotes gravity acceleration; h represents the depth below seafloor; $\nu = 0.5\left( {K - 2/3G} \right)/\left( {K + 1/3G} \right)$ is the Poisson’s ratio. The gas hydrate is regarded as a part of the rock frame in our study (Lee and Collett 2013), the porosity reduces to $\varphi_{f} = \varphi \left( {1 - S_{h} } \right)$, the frame volume fraction of gas hydrate ($f_{h}$) is $f_{h} = S_{h} \varphi /\left( {1 - \varphi_{f} } \right)$.

Appendix B

Double porosity model

There are two equations to describe the double porosity model: the governing equations and fluid transport equations (Pride and Berryman 2003a, 2003b). The governing equations describe the propagation of waves in a dual porous medium. And the fluid transport equations describe the attenuation effect of seismic waves. This model is composed of two lithology with different degree of consolidation: consolidation part (solid phase 1) and unconsolidation part (solid phase 2). The P-wave and S-wave velocity are given by:

$$V_{p} = 1/{\text{Re}} \left\{ {S_{B} } \right\},$$

(B1)

$$V_{s} = \sqrt {G/\rho } ,$$

(B2)

where S_B denotes the Biot theory’s slowness, which is given by:

$$S_{B}^{2} = b + \sqrt {b^{2} - \frac{{\rho \tilde{\rho } - \rho_{f}^{2} }}{{M_{B} H_{B} - C_{B} }}} ,$$

(B3)

$$b = \frac{{\rho M_{B} + \tilde{\rho }H_{B} - 2\rho_{f} C_{B} }}{{2(M_{B} H_{B} - C_{B}^{2} )}},$$

(B4)

where the M_B, H_B and C_B are the poroelastic modulus of Biot theory (Biot 2004):

$$M_{B} = \frac{{B^{2} }}{{1 - K_{D} /K_{U} }}K_{U} ,$$

(B5)

$$H_{B} = K_{U} + 4G/3,$$

(B6)

$$C_{B} = BK_{U} ,$$

(B7)

$$\tilde{\rho } = - \eta /\left[ {i\omega k\left( \omega \right)} \right],$$

(B8)

where B is the effective coefficient of Skempton, K_D is the effective bulk modulus of drained rock frame, K_U is the effective bulk modulus.

$$\frac{1}{{K_{D} }} = a_{11} - \frac{{a_{13}^{2} }}{{a_{33} - \gamma /i\omega }},$$

(B9)

$$B = \frac{{ - a_{12} (a_{33} - \gamma /i\omega ) + a_{13} (a_{23} + \gamma /i\omega )}}{{(a_{22} - \gamma /i\omega )(a_{33} - \gamma /i\omega ) - (a_{23} + \gamma /i\omega )^{2} }},$$

(B10)

$$\frac{1}{{K_{U} }} = \frac{1}{{K_{D} }} + B\left( {a_{12} - \frac{{a_{13} (a_{23} + \gamma /i\omega )}}{{a_{33} - \gamma /i\omega }}} \right),$$

(B11)

where $a_{ij}$ is real and corresponds to high frequency responses, and does not change with changes in internal fluid pressure (Pride and Berryman 2003a).

$$a_{11} = 1/K,$$

(B12)

$$a_{22} = \frac{{v_{1} \alpha_{1} }}{{K_{1}^{d} }}\left( {\frac{1}{{B_{1} }} - \frac{{\alpha_{1} \left( {1 - Q_{1} } \right)}}{{1 - K_{1}^{d} /K_{2}^{d} }}} \right),$$

(B13)

$$a_{33} = \frac{{v_{2} \alpha_{2} }}{{K_{2}^{d} }}\left( {\frac{1}{{B_{2} }} - \frac{{\alpha_{2} \left( {1 - Q_{2} } \right)}}{{1 - K_{2}^{d} /K_{1}^{d} }}} \right),$$

(B14)

$$a_{12} = - v_{1} Q_{1} \alpha_{1} /K_{1}^{d} ,$$

(B15)

$$a_{13} = - v_{2} Q_{2} \alpha_{2} /K_{2}^{d} ,$$

(B16)

$$a_{23} = - \frac{{\alpha_{1} \alpha_{2} K_{1}^{d} /K_{2}^{d} }}{{(1 - K_{1}^{d} /K_{2}^{d} )}}\left( {\frac{1}{K} - \frac{{v_{1} }}{{K_{1}^{d} }} - \frac{{v_{2} }}{{K_{2}^{d} }}} \right),$$

(B17)

$$\alpha_{i} = (1 - K_{i}^{d} /K_{i}^{u} )/B_{i} ,$$

(B18)

$$K_{i}^{u} = K_{i}^{d} /(1 - B_{i} (1 - K_{i}^{d} /K_{i}^{s} )),$$

(B19)

where $\alpha_{i}$ is the Biot (1957) parameter of solid phase i, $K_{i}^{u}$ is the non-draining bulk modulus, Q_i is auxiliary constant (the solid phases i = 1,2), which can be given by:

$$v_{1} Q_{1} = \frac{{1 - K_{2}^{d} /K}}{{1 - K_{2}^{d} /K_{1}^{d} }},$$

(B20)

$$v_{2} Q_{2} = \frac{{1 - K_{1}^{d} /K}}{{1 - K_{1}^{d} /K_{2}^{d} }},$$

(B21)

where v₁ and v₂ are the volume fractions of the solid phases. In our study, v₂ = S_h, v₁ = 1−S_h, S_h is the hydrate saturation. The K and G are respectively the total frequency bulk and shear modulus, given by Hashin and Shtrikman (1963).

$$\frac{1}{{K + 4G_{i} /3}} = \frac{{v_{1} }}{{K_{1}^{d} + 4G_{i} /3}} + \frac{{v_{2} }}{{K_{2}^{d} + 4G_{i} /3}},$$

(B22)

$$\frac{1}{{G + \varsigma_{i} }} = \frac{{v_{1} }}{{G_{1} + \varsigma_{i} }} + \frac{{v_{2} }}{{G_{2} + \varsigma_{i} }},$$

(B23)

$$\varsigma {}_{i} = \frac{{G_{i} (9K_{i}^{d} + 8G_{i} )}}{{6(K_{i}^{d} + 2G_{i} )}},$$

(B24)

Naturally, we define solid phase 2 to be less consolidated than solid phase 1, that is to say, K₂^d < K₁^d and G₂ < G₁.

For consolidation part (solid phase 1):

$$K_{1}^{d} = K_{1}^{s} \frac{{1 - \varphi_{1} }}{{1 + c\varphi_{1} }},$$

(B25)

$$G_{1} = G_{1}^{s} \frac{{1 - \varphi_{1} }}{{1 + 3c\varphi_{1} /2}},$$

(B26)

where c is the degree of consolidation between the particles (c = 10). $K_{1}^{s}$ and $G_{1}^{s}$ are the bulk and shear modulus of rock frame.

Normally, double porosity model applies for conditions where the physical properties of the two solid phases are quite different. When solid phase 2 is more permeable than solid phase 1, Pride and Berryman (2003b) proposed the following equation:

$$\gamma_{m} = - \frac{{k_{1} K_{1}^{d} }}{{\eta L^{2} }}\left( {\frac{{a_{12} + B_{0} \left( {a_{22} + a_{33} } \right)}}{{R_{1} - B_{0} /B_{1} }}} \right),$$

(B27)

$$B_{0} = - \frac{{a_{12} + a_{13} }}{{a_{22} + 2a_{23} + a_{33} }},$$

(B28)

$$R{}_{1} = Q_{1} + \frac{{\alpha_{1} (1 - Q_{1} )B_{0} }}{{1 - K_{1}^{d} /K_{2}^{d} }} - \frac{{v_{2} }}{{v_{1} }}\frac{{\alpha_{2} (1 - Q_{2} )B_{0} }}{{1 - K_{2}^{d} /K_{1}^{d} }},$$

(B29)

where L is the average distance between the solid phases (L² = a²/15), a is the spherical envelop radius. The transfer frequency (w_m) can be given by:

$$\omega_{m} = \frac{{\eta B_{1} K_{1}^{d} }}{{k_{1} \alpha_{1} }}(\gamma_{m} \frac{V}{S})^{2} \left( {1 + \sqrt {\frac{{k_{1} B_{2} K_{2}^{d} \alpha_{1} }}{{k_{2} B_{1} K_{1}^{d} \alpha_{2} }}} } \right)^{2} ,$$

(B30)

$$\gamma = \gamma_{m} \sqrt {1 - i\omega /\omega_{m} } ,$$

(B31)

$$\omega = 2\pi f,$$

(B32)

where V/S is volume surface area ratio.

For unconsolidation part (solid phase 2):

$$K_{2}^{d} = \frac{1}{6}\left[ {\frac{{4\left( {1 - \varphi_{0} } \right)^{2} n_{0}^{2} P_{0} }}{{\pi^{4} C_{s}^{2} }}} \right]^{1/3} \frac{{\left( {P_{e} /P_{0} } \right)^{1/2} }}{{\left\{ {1 + \left[ {16P_{e} /\left( {9P_{0} } \right)} \right]^{4} } \right\}^{1/24} }},$$

(B33)

$$G_{2} = 3K_{2}^{d} /5,$$

(B34)

$$P_{e} = (1 - \varphi_{2} )(\rho_{2}^{s} - \rho_{f} )gh,$$

(B35)

where P_e is effective overburden pressure, g is the gravity constant (g = 9.81 m/s²), n₀ = 9, P₀ = 10Mpa, h is the overburden thickness. $\varphi_{0}$ is the primary porosity, Cs is the coupling parameter, given by:

$$C_{s} = \frac{1}{4\pi }\left(\frac{1}{{G_{2}^{s} }} + \frac{1}{{K_{2}^{s} + G_{2}^{s} /3}}\right),$$

(B36)

where $K_{2}^{s}$ and $G_{2}^{s}$ are the bulk and shear modulus of rock frame.