Numerical studies of gas hydrate systems: sensitivity to porosity-permeability models

Abstract

The formation of sub-seafloor gas hydrates in marine environments can be described as a coupled transport and thermodynamic process inside a host sediment matrix undergoing structural evolution. The transport processes are driven by the sedimentary load and induced overpressure gradients, controlled by sediment permeability. In order to accurately model the resulting fluid flow profile, the decrease of sediment permeability during hydrate precipitation has to be taken into account, which affects both the transport of solutes and sediment compaction. In this paper, we investigate how total hydrate abundance is affected by regions of low permeability which deflect the flow field in their vicinity. For this purpose, a two-dimensional numerical hydrate system model was set up which permits to quantify this effect in scenarios where changes in water depth cause lateral variations of the thickness of the hydrate stability field, as well as of hydrate saturation and sediment permeability. The microscopic structure of gas hydrate crystals in the host sediment matrix defines the evolution of the permeability reduction during hydrate formation. Grain-coating precipitates have a stronger tendency to clog flow paths through pore throats than do pore-filling precipitates. Our results clearly show that these pore-scale processes affect the large-scale flow field and hydrate abundance. The sensitivity depends on the model geometry and, for a 5° slope of the seafloor, 4.1% relative difference is predicted for the hydrate saturation according to different porosity-permeability relationships.

Appendix

The hydraulic equation in an evolving hydrate system

The coupled pressure and transport model for an evolving hydrate system requires the formulation of the hydraulic equation with fluid sources added from sediment compaction and methane phase transitions. As this complete formulation is not derived elsewhere, we append the calculation of the hydraulic overpressure from the continuity equations for fluid transport and sediment transport, relating pressure to transport velocities through Darcy’s Law.

For incompressible fluids and sediment grains, mass conservation requires

$$ \frac{\partial }{{\partial t}}{S_{\text{w}}}\phi + \nabla \cdot \left( {{S_{\text{w}}}\phi {{\mathbf{v}}_{\text{f}}}} \right) = {q_{\text{f}}} $$

(3)

and

$$ \frac{\partial }{{\partial t}}\left( {1 - \phi } \right) + \nabla \cdot \left( {\left( {1 - \phi } \right){{\mathbf{v}}_{\text{s}}}} \right) = {q_{\text{s}}} $$

(4)

where hydrate dissociation can add a fluid source q _f, while we assume no change (q _s = 0) of solid material through diagenetic processes. S _w is the water fraction of the pore space, and v _f and v _s are the intrinsic fluid and sediment velocity respectively.

We now rearrange Eq. 4 into

$$ \nabla \cdot {{\mathbf{v}}_{\text{s}}} = \frac{1}{{1 - \phi }}\frac{{{\text{d}}\phi }}{{{\text{d}}t}} $$

and apply Darcy’s Law to relate the filtration rate ${{\mathbf{u}}_{\text{D}}} = {S_{\text{w}}}\phi \left( {{{\mathbf{v}}_{\text{f}}} - {{\mathbf{v}}_{\text{s}}}} \right)$ to the overpressure $p\prime = p - {p^{\text{h}}}$ at the permeability κ and viscosity ν,

$$ {S_{\text{w}}}\phi \left( {{{\mathbf{v}}_{\text{f}}} - {{\mathbf{v}}_{\text{s}}}} \right) = - \frac{\kappa }{\nu }\nabla p\prime $$

both of which inserted into Eq. 3 gives

$$ \phi \frac{{{\text{d}}{S_{\text{w}}}}}{{{\text{d}}t}} - \nabla \cdot \frac{\kappa }{\nu }\nabla p\prime + \frac{{{S_{\text{w}}}}}{{1 - \phi }}\frac{{{\text{d}}\phi }}{{{\text{d}}t}} = {q_{\text{f}}} $$

(5)

where we made use of the material derivative defined as

$$ \frac{{{\text{d}}}}{{{\text{d}}t}}\left( \cdot \right) \equiv \left( {\frac{\partial }{{\partial t}} + {{\mathbf{v}}_{\text{s}}} \cdot \nabla } \right)\left( \cdot \right) $$

(6)

The change in porosity is related to the sediment compressibility C by

$$ \frac{{{\text{d}}\phi }}{{{\text{d}}t}} = \frac{{\partial \phi }}{{\partial \sigma }}\frac{{{\text{d}}\sigma }}{{{\text{d}}t}} = - C\frac{\text{d}}{{{\text{d}}t}}\left( {{p^{\text{l}}} - \left( {{p^{\text{h}}} + p\prime} \right)} \right) $$

(7)

where σ is the effective stress, and p ^h and p ¹ are the hydrostatic and lithostatic pressure respectively.

If we assume that only vertical stress and solid material transport occurs $\left( {{{\mathbf{v}}_{\text{s}}} = {v_{{\text{s,z}}}}{{\hat e}_{\text{z}}}} \right)$, then the compressibility is calculated as

$$ \begin{gathered} C = - \frac{{\partial \phi }}{{\partial \sigma }} = - \frac{{\partial \phi }}{{\partial z}}\frac{{\partial z}}{{\partial \sigma }} = - \frac{{\partial \phi }}{{\partial z}}{\left( {\frac{\partial }{{\partial z}}\left( {{p^{\text{l}}} - \left( {{p^{\text{h}}} + p\prime} \right)} \right)} \right)^{ - 1}} \\ = - \frac{{\partial \phi }}{{\partial z}}{\left( {g({\rho_{\text{s}}} - {\rho_{\text{f}}})(1 - \phi ) - \frac{{\partial p\prime}}{{\partial z}}} \right)^{ - 1}} \\ \end{gathered} $$

(8)

and the lithostatic potential p ¹−p ^h grows according to

$$ \begin{gathered} \frac{\text{d}}{{{\text{d}}t}}\left( {{p^{\text{l}}} - {p^{\text{h}}}} \right) = \left( {\frac{\partial }{{\partial t}} + {{\mathbf{v}}_{\text{s}}} \cdot \nabla } \right)\left( {{p^{\text{l}}} - {p^{\text{h}}}} \right) \\ = {v_{{\text{s,z}}}}g\left( {{\rho_{\text{s}}} - {\rho_{\text{f}}}} \right)\left( {1 - \phi } \right) \\ \end{gathered} $$

(9)

with the gravitational acceleration g, and sediment and fluid density ρ _s and ρ _f.

Hydrate dissociation (formation)—most prevalent at the base (top) of the hydrate zone—represents a fluid source (sink) equivalent to the released (fixed) amount of fresh water,

$$ {\rho_{\text{f}}}{q_{\text{f}}} = - \left( {1 - {r_{\text{g}}}} \right){\rho_{\text{h}}}\phi \frac{{{\text{d}}{S_{\text{h}}}}}{{{\text{d}}t}} $$

(10)

controlled by the expansion factor r _g (Xu and Germanovich 2006).

Finally, we substitute S _w=1−S _h−S _g, as the water, hydrate fraction S _h and gas fraction S _g occupy the total pore space, and the basic equation for the hydraulic overpressure p′ follows when Eqs. 7, 9 and 10 are substituted into Eq. 5:

$$ \left( {\left( {1 - {r_{\text{g}}}} \right)\frac{{{\rho_{\text{h}}}}}{{{\rho_{\text{f}}}}} - 1} \right)\phi \frac{\text{d}}{{{\text{d}}t}}{S_{\text{h}}} - \phi \frac{\text{d}}{{{\text{d}}t}}{S_{\text{g}}} - \nabla \cdot \frac{\kappa }{\nu }\nabla p\prime = C\frac{{1 - {S_{\text{h}}} - {S_{\text{g}}}}}{{1 - \phi }}\left( {{\nu_{{\text{s,z}}}}g\left( {{\rho_{\text{s}}} - {\rho_{\text{f}}}} \right)\left( {1 - \phi } \right) - \frac{{{\text{d}}p\prime}}{{{\text{d}}t}}} \right) $$

(11)