Effective viscoelastic representation of gas-hydrate bearing sediments from finite-element harmonic experiments

Abstract

We present a novel numerical upscaling technique for modeling the wave response of gas-hydrate bearing sediments composed of a rock frame, gas-hydrate and water, where the hydrate consists of ice-like lattice of water molecules with methane trapped inside. These sediments are highly heterogeneous at mesoscopic scales, much smaller than the wavelength but much larger than the pore size, inducing substantial seismic wave attenuation and dispersion due to mode conversions. The proposed numerical upscaling procedure simulates the wave-induced fluid-flow loss mechanism by computing an average effective viscoelastic medium having the same behavior of the original sediment. The method determines the complex stiffness coefficients associated with the viscoelastic medium by solving numerically boundary value problems formulated in the space-frequency domain, representing compressibility and shear experiments. The procedure is applied to composite media with regions of different amounts of hydrate with patchy or periodic-layer distributions, which define an anisotropic effective viscoelastic medium, respectively. The examples demonstrate that variations in hydrate content induce strong attenuation and dispersion effects on seismic waves due to the mesoscopic loss mechanism.

Appendices

Appendix A

The coefficients in the constitutive relations Eqs. 5–7 are computed as follows. Let K_s1,m, K_s3,m, μ_s1,m and μ_s3,m denote the bulk and shear modulus of the two solid (dry) frames, respectively. We assume that K_s1,m is known and K_s3,m, μ_s1,m and μ_s3,m can be determined using a percolation-type model. See formulas A8-A9 of Appendix A in [17] for details. Then with K_s1, μ_s1, K_s3 and μ_s3 denoting the bulk and shear moduli of the grains in the two solid phases, respectively, and K_f the bulk modulus of the fluid phase, the coefficients μ₁, μ₃ and μ₁₃ are:

$$ \begin{array}{@{}rcl@{}} \mu_{1} &=& [(1-g_{1})\phi_{1}]^{2} \mu_{av} + \mu_{s1,m},\\ \mu_{3} &=& [(1-g_{3})\phi_{3}]^{2} \mu_{av} + \mu_{s3,m},\\ \mu_{13} &=& (1-g_{1})(1-g_{3})\phi_{1}\phi_{3} \mu_{av},\\ g_{1} &=& \frac{\mu_{s1,m}}{\phi_{1} \mu_{s1}},\quad\quad g_{3} = \frac{\mu_{s3,m}}{\phi_{3} \mu_{s3}},\\ \mu_{av}&=& \big[\frac{(1-g_{1})\phi_{1}}{\mu_{s1}} + \frac{\phi}{2 \omega \eta} +\frac{(1-g_{3})\phi_{3}}{\mu_{s3}}\big]^{-1}. \end{array} $$

(34)

The symbol ω in the definition of μ_av above denotes the angular frequency, taken to be 2 π in the examples. The remaining elastic coefficients in Eqs. 5–7 are given by the following expressions [17]

$$ \begin{array}{@{}rcl@{}} K_{G1} &=& K_{1} + ({S_{1}})^{2} K_{2} + 2 {S_{1}} C_{12},\\ K_{G3} &=& K_{3} + ({S_{3}})^{2} K_{2} + 2 {S_{3}} C_{23},\\ B_{1} &=& \frac{{S_{1}} K_{2} + C_{12}}{\phi},\\ B_{2} &=& \frac{{S_{3}} K_{2} + C_{23}}{\phi},\\ B_{3} &=& C_{13} +{S_{3}} C_{12} + {S_{1}} C_{23} + {S_{3}} {S_{1}} K_{2}, \end{array} $$

(35)

$$ \begin{array}{@{}rcl@{}} K_{av}&=&\left[(1 - c_{1}) \frac{\phi_{1}}{K_{s1}} +\frac{\phi}{K_{f}} +(1 - c_{3})\frac{\phi_{3}}{K_{s3}}\right]^{-1}, \end{array} $$

where

$$ \begin{array}{@{}rcl@{}} &&K_{1} = \left[(1 - c_{1})\phi_{1}\right]^{2} K_{av} + K_{s1,m}, \\ &&K_{2} = {\phi_{w}^{2}} K_{av}\\ &&K_{3} = \left[(1 - c_{3})\phi_{3}\right]^{2} K_{av} + K_{s3,m},\\ &&C_{12}= (1 - c_{1}) \phi_{1} \phi K_{av}, \\ &&C_{13}= (1 - c_{1}) (1 - c_{3})\phi_{1} \phi_{3} K_{av},\\ &&C_{23}= (1 - c_{3}) \phi \phi_{3} K_{av},\\ &&c_{1} = \frac{K_{s1,m}}{\phi_{1} K_{s1}}, \quad c_{3} = \frac{K_{s3,m}}{\phi_{3} K_{s3}}. \end{array} $$

(36)

Appendix B

Here we describe a procedure to determine the dissipation factors of an associated classic Biot model that in the low-frequency range is equivalent to a composite material. The following notation is used to define the associated classic Biot model [2].

The solid and fluid particle displacements are denoted as ${\widehat {{\mathbf u}}}^{(s)}, {\widehat {{\mathbf u}}}^{(f)}$, while ${\widehat {\boldsymbol \sigma }}$ and ${\widehat p}_{f}$ denote the total stress and fluid pressure.

The constitutive relations and the diffusion equation of the classic Biot model are

$$ \begin{array}{@{}rcl@{}} &&{\widehat \sigma}_{ij} = \big[K_{G} {\widehat \theta}^{(s)} -B {\widehat \theta}^{(f)}\big] \delta_{ij} + 2 \mu {\widehat d}_{ij}^{(s)}, \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} &&{\widehat p}_{f} = -B {\widehat \theta}^{(s)} - {\widehat K_{av}} {\widehat \theta}^{(f)}, \end{array} $$

(38)

$$ \begin{array}{@{}rcl@{}} && \nabla\cdot {\widehat {\boldsymbol \sigma}} = 0, \end{array} $$

(39)

$$ \begin{array}{@{}rcl@{}} && i \o \frac{\eta}{{\widehat \kappa}} {\widehat {{\mathbf u}}}^{(f)} + \nabla p_{f} = 0 , \end{array} $$

(40)

where

$$ {\widehat d}_{ij}^{(s)} = \epsilon_{ij}({\widehat {{\mathbf u}}}^{s)}) - \frac13 {\widehat \theta}^{(s)} \delta_{ij}, \quad {\widehat \theta}^{(m)} = \nabla\cdot {\widehat {{\mathbf u}}}^{(m)}, m=s,f. $$

In Eqs. 37–40, K_G and μ are the bulk and shear moduli of the saturated material, while B and ${\widehat K_{av}}$ are elastic coupling coefficients. Furthermore, ${\widehat \kappa }$ denotes the rock permeability and η the fluid viscosity.

Let ${K^{a}_{s}}, {K^{a}_{m}}, K_{f}$ denote the bulk moduli of the solid grains, dry matrix and fluid, respectively. The coefficient μ is the shear modulus of the dry matrix and the other coefficients in Eqs. 37–40 can be determined from the relations

$$ \begin{array}{@{}rcl@{}} &&K_{G} = {K^{a}_{m}} + \alpha^{2}\widehat K_{av}, \quad \alpha = 1 - \frac{{K^{a}_{m}}}{{K^{a}_{s}}}, \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} &&\widehat K_{av} = \left[\frac{\alpha - \phi_{w}}{{K^{a}_{s}}} + \frac{\phi_{w}}{K_{f}}\right]^{-1},\quad B = \alpha \widehat K_{av}.\label {defcoef2} \end{array} $$

(42)

Assume that u⁽¹⁾ = u⁽³⁾ ≡u^(s) (the low-frequency assumption is used here) and define the total stress tensor as

$$ \begin{array}{@{}rcl@{}} \tau_{ij} = \tau_{ij}^{(1)} + \tau_{ij}^{(3)}. \end{array} $$

(43)

Then adding Eqs. 5 and 6, we obtain

$$ \begin{array}{@{}rcl@{}} &&\tau_{ij} = \big[\left( K_{G1} + K_{G3} + 2 ~ B_{3}\right) \theta^{(s)} \end{array} $$

(44)

$$ \begin{array}{@{}rcl@{}} &&\qquad -\left( B_{1} + B_{2} \right) \theta^{(f)}] \delta_{ij} + 2 \left( \mu_{1} + \mu_{3} + \mu_{13}\right) d_{ij}^{(s)},\\ &&p_{f} = -(B_{1} + B_{2}) \theta^{(s)} - K_{av} \theta^{(f)}. \end{array} $$

(45)

Now from Eqs. 37, 38 and Eqs. 44, 45, we can identify the elastic coefficients of the associated classic Biot model as follows

$$ \begin{array}{@{}rcl@{}} &&K_{G} = K_{G1} + K_{G3} + 2 ~ B_{3}, \end{array} $$

(46)

$$ \begin{array}{@{}rcl@{}} &&B = B_{1} + B_{2}, \end{array} $$

(47)

$$ \begin{array}{@{}rcl@{}} &&\mu = \mu_{1} + \mu_{3} +\mu_{13}, \end{array} $$

(48)

$$ \begin{array}{@{}rcl@{}} &&{\widehat K_{av}} = K_{av}, \end{array} $$

(49)

with the coefficients in the right-hand side of Eqs. 46– 49 computed by the relations given in Appendix A.

Next, note that we can determine $\alpha , {K^{a}_{m}}$ and ${K^{a}_{s}}$ in Eqs. 41 and 42 as follows:

$$ \begin{array}{@{}rcl@{}} \alpha = \frac{B}{\widehat K_{av}}, \quad {K^{a}_{m}} = K_{G} - \alpha^{2} \widehat K_{av}\quad {K^{a}_{s}} = \frac{{K^{a}_{m}}}{1 - \alpha}, \end{array} $$

(50)

where B, $\widehat K_{av}$ and K_G are given in terms of the coefficients of the GH-bearing sediments by Eqs. 46, 47 and 49, respectively.

The procedure to determine the coefficients $K_{G}, B, {\widehat K_{av}}$ and μ in Eqs. 46–49 can be shown to give identical results to those given in Carcione et al. [7]

Furthermore, from equations (B5) and (B8) in Santos et al. [17]

$$ \begin{array}{@{}rcl@{}} &&\phi_{w}\left( S_{3} \phi_{w} f_{22} - f_{12} \right) = b_{23}, \end{array} $$

(51)

$$ \begin{array}{@{}rcl@{}} &&\phi_{w}\left( f_{12}+ S_{1} \phi f_{22} \right) = b_{12}, \end{array} $$

(52)

where

$$ \begin{array}{@{}rcl@{}} b_{12} = {\phi_{w}^{2}} \frac{\eta}{\kappa_{1}},\qquad b_{23} = {\phi_{w}^{2}} \frac{\eta}{\kappa_{3}}, \end{array} $$

(53)

with κ₁ and κ₃ denoting the permeabilities of two solid phases Thus, add Eqs. 51 and 52 to get

$$ \begin{array}{@{}rcl@{}} f_{22} = \eta\left( \frac{1}{\kappa_{1}} + \frac{1}{\kappa_{3}}\right). \end{array} $$

(54)

Next, from the low-frequency assumption, the f₁₂-terms in Eq. 3 cancel and this equation reduces to

$$ \begin{array}{@{}rcl@{}} i \o f_{22} {{\mathbf u}}^{(2)} + \nabla p_{f}=0. \end{array} $$

(55)

Thus Eqs. 40, 54 and 55 allow to identify the effective permeability ${\widehat \kappa }$ of the associated classic Biot model by the relation

$$ \begin{array}{@{}rcl@{}} \frac{1}{{\widehat \kappa}} = \left( \frac{1}{\kappa_{1}} + \frac{1}{\kappa_{3}}\right). \end{array} $$

(56)

Equations 48 and 50 allow to determine the coefficients in Eqs. 41–42 in the constitutive relations of the the associated classic Biot medium to the GH-bearing sediment.

Remark

Equations 46–49 and 56 may also be used in the case of shaley sandstones as presented in Santos et al. [17].