3.8 Expression of effective matrix viscosity
Equations 10, 11, and 13 (explicitly in the case of the latter two) all
include some dependence on the effective matrix viscosity, \(\xi^{{}^{\prime}}\).
The dependence of the effective matrix viscosity on melt fraction
remains elusive and depends on the selected deformation mechanism. Here,
we choose an expression for the effective matrix viscosity formulated on
the basis of experimental and theoretical studies investigating the
rheology of immersed granular media at intermediate melt fractions (ca.
0.4 – 0.6) (Bachmann & Huber, 2019, Guazzelli & Pouliquen, 2018).
Granular mechanics studies have offered insight into how expressions of
the effective matrix viscosity depend on porosity (equivalent to melt
fraction) when considering granular repacking and not crystal
deformation as the dominant process for compaction. For instance, Boyeret al. (2011) performed deformation experiments on immersed
granular media in annular shear cells with applied shear stress and a
controlled confining pressure. They measured the resulting strain rates
and porosity and observed that the shear and bulk viscosity of the
suspensions evolved as \(\left(\phi-\ \phi_{m}\right)^{-2},\) where \(\phi_{m}\) is the maximum packing fraction in terms of porosity. When
the imposed macroscopic shear rate is small, as is true in the case of
the experiments of Hoyos et al. (2022), the shear and bulk matrix
viscosities take the following functional forms with respect to\(\phi\):
\begin{equation}
\left(14\right)\ f_{\eta_{\text{repacking}}}=\ 1+\frac{5}{2}\left(1-\phi\right)\left[1-\frac{1-\phi}{1-\ \phi_{m}}\right]^{-1}+\mu_{\text{friction}}\left[\frac{1-\phi}{\phi-\ \phi_{m}\ }\right]^{2}\nonumber \\
\end{equation}
where \(\mu_{\text{friction}}\) is a friction coefficient, taken to be
ca. 0.3 (Boyer et al. , 2011), and
\begin{equation}
\left(15\right)\ f_{K_{\text{repacking}}}=\ \left[\frac{1-\phi}{\phi-\ \phi_{m}}\right]^{2}.\nonumber \\
\end{equation}
This leads to an effective matrix viscosity of:
\begin{equation}
\left(16\right)\ \xi_{\text{repacking}}=\xi_{\text{ref}}\left[{\frac{4}{3}f}_{\eta}+\ f_{K}\right],\nonumber \\
\end{equation}
where \(\xi_{\text{ref}}\) is a constant of proportionality for
effective matrix viscosity. The unknown parameters for a given granular
medium include \(\xi_{\text{ref}}\ \) and \(\phi_{m}\) for eq. (14),
(15), and (16).