Consequences of crystal shape and fabric on anisotropic permeability in magmatic mush

Abstract

Crystals that form an interconnected porous network can become preferentially oriented both prior to and during compaction of magmatic mush. This introduces anisotropy in the melt pore-space that can reduce permeability in the direction of compaction and in turn decrease melt flux and compaction rate. Using a number of grain-scale numerical models, the consequences of end-member magmatic fabrics on the directional dependence of permeability are tested over a range in melt fraction from 22 to 77%. As the crystal aspect ratio (i.e. ratio of long to short axis length) increases from 2 to 10, isotropic permeability decreases by a factor of 2 and 5 for randomly oriented prolate and oblate-shaped crystals, respectively, at a melt fraction of 22%. With a flattening fabric, permeability is reduced in the compaction direction no more than approximately a factor of 2 relative to the isotropic permeability at the same melt fraction and crystal shape for both oblate and triaxial prisms. However, permeability is enhanced in directions orthogonal to the compaction direction. For example, permeability is enhanced up to a factor of 11 relative to the isotropic permeability at a melt fraction of 22% for oblate prisms with a ratio of the long to short axis length of 10. Anisotropy in permeability increases as the melt fraction decreases and the crystal aspect ratio increases. Ratios of the principal permeabilities are sufficiently large based on the realistic crystal shapes tested here to warrant including anisotropic permeability into macroscale melt segregation models including those for compaction.

Appendix

The 3D lattice–Boltzmann (LB) model is based on the Bhatnager–Gross–Krook (BGK) approximation as developed from the literature (Chen et al. 1992; Zou and He 1997; Maier et al. 1998). The porous medium domain, in this case the simulated microstructures, are discretized into either solid or fluid nodes on a Cartesian grid. Groups of particles, with a mass much larger than the molecules comprising the fluid, travel between fluid nodes with fixed velocity in 15 possible directions given by a cubic lattice. The dimensionless displacement vectors, ${\user2{e}}_i,$ defining particle transport between fluid nodes are column vectors in the following matrix,

$$ {\user2{E}}=\left[ \begin{array}{lllllllllllllll} 0&1&-1&0&0&0&0&1&-1&1&-1&1&-1&1&-1\\ 0&0&0&1&-1&0&0&1&-1&1&-1&-1&1&-1&1\\ 0&0&0&0&0&1&-1&1&-1&-1&1&1&-1&-1&1\\ \end{array} \right] $$

(3)

where ${\user2{e}}_1$ marks the particle at rest (Zou and He 1997). The number of particles that move in direction, i, from a given node, ${\user2{x}},$ at time, t, is called the particle distribution function, $f_i({\user2{x}},t).$ During a time step, particles travel according to $f_i({\user2{x}},t)$ and collide at adjacent nodes where they are scattered according to simple algebraic equations that obey mass and momentum conservation. This is summarized by the LB equation,

$$ f_i({\user2{x}}+{\user2{e}}_i,t+1)-f_i ({\user2{x}},t)={\frac{1}{\tau}} \left[f_i^{(\rm eq)}({\user2{x}},t)-f_i({\user2{x}},t)\right] $$

(4)

where τ is the relaxation time that controls the rate of convergence and f ^(eq) _i is the equilibrium distribution function that describes particles colliding and scattering. Continuum scale dimensionless variables, such as density, ρ′ = ∑ _i f _i , momentum, $\rho^{\prime}{\user2{u}}^{\prime}=\sum_if_i {\user2{e}}_i,$ and pressure, p′ = ρ′/3 can be obtained from the particle distribution function (Zou and He 1997; Maier et al. 1998). A Chapman–Enskog expansion of Eq. 4 in time and space recovers the dimensionless Navier–Stokes equation $\partial{\user2{u}}^{\prime}/\partial{t}^{\prime}+{\user2{u}}^ {\prime}\cdot\nabla{\user2{u}}^{\prime}=-(1/\rho^{\prime}) \nabla{p}^{\prime}+\nu^{\prime}\nabla^2{\user2{u}}^{\prime},$ where the dimensionless kinematic viscosity is ν′ = (2τ − 1)/6. This expansion determines the coefficients of the equilibrium distribution, $f_i^{(\rm eq)}=c_i\rho\left(1+3{\user2{e}}_i\cdot{{\user2{u}}^{\prime}}+ {\frac{9}{2}}({\user2{e}}_i\cdot{\user2{u}}^{\prime})^2- {\frac{3}{2}}({\user2{u}}^{\prime}\cdot{\user2{u}}^{\prime})\right)$ where c ₀ = 4/9, c _i  = 1/9 for i equal 1 to 6, and c _i  = 1/36 for i equal 7 to 14.

The advantage of using an LB method over many other numerical methods in solving porous fluid flow at the grain scale is that the no-slip boundary condition at solid-fluid interfaces can be easily incorporated through a particle bounce-back scheme. The scheme is implemented by assigning the no-slip interface to be half-way between the fluid and solid node. Each f _i is assigned the value of the f _i of its opposite direction with relaxation on the bounceback nodes (Zou and He 1997). Porous fluid flow is induced by a horizontal pressure gradient that is implemented by prescribing constant densities ρ ^′₁ and ρ ^′₂ to the inlet and outlet respectively (Maier et al. 1996). For all LB simulations, the remaining four boundaries of the domain are periodic, a value of τ = 0.8 is used (Pan et al. 2001), and the pressure difference, Δρ′ = ρ ^′₁  − ρ ^′₂ , is set to 0.01 which is sufficiently low to ensure low Reynolds number flow (i.e. Re≪1). The flow is considered steady when the following convergence criteria is achieved, $\left|\sum_{{\user2{x}}}\left|u^\prime({\user2{x}},t)\right| /\sum_{{\user2{x}}}\left|u^\prime({\user2{x}},t+50)\right| -1\right|\leq10^{-5}.$ The LB model is validated against an analytical solution for flow in a duct (Pan et al. 2001).

Dimensional properties of the flow and porous medium can be determined using the steady dimensionless variables. Permeability, obtained from Darcy’s equation (Eq. 1), is κ = 8ν′L ²ρ′u ^′ _d /(3mΔρ′), where L is the dimensional length of the porous medium, m is the number of lattice nodes along the flow direction, and u ^′ _d is the dimensionless phase-average Darcy velocity in the direction parallel to the imposed pressure gradient. The Reynolds number, $Re=u_d^\prime\bar{D}m/(\nu^\prime{L}),$ is determined based on an estimated average crystal length, $\bar{D},$ for the length scale. Given a value for the kinematic viscosity, ν, a characteristic dimensional velocity is u _o  = νm/ν′L. Using u _o and ρ _o  = (ρ₁ + ρ₂)/2, other dimensional properties are time, t = Lt′/mu _o , velocity, ${\user2{u}}=u_o{\user2{u}}^{\prime},$ pressure, p = ρ _o u ² _o p′, and density, ρ = ρ _o ρ′.