commit ca0f8b4895f832ea82a675f340ba72be6889a4e2
parent af4236d06fe96489729f665e9e70c8568027130a
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 24 Jun 2019 11:40:28 +0200
Clean up methods text
Diffstat:
1 file changed, 10 insertions(+), 17 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -59,7 +59,7 @@ We generalize these previous models to a water-saturated granular flow, by compa
\subsection{Granular flow}%
\label{sub:granular_flow}
In our model, the sediment deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit for a cohesionless material.
-We expand a steady-state continuum model for granular flow by \citet{Henann2013} with a coupling to pore water, including transient dynamics during shear zone formation and vanishment.
+We expand a steady-state continuum model for granular flow by \citet{Henann2013} with a coupling to pore water.
We assume that the elasticity is negligible and set the total shear rate $\dot{\gamma}$ to consist of a plastic contribution $\dot{\gamma}^\text{p}$:
\begin{equation}
\dot{\gamma} \approx \dot{\gamma}^\text{p} = g(\mu_\text{c}, \sigma_\text{n}') \mu,
@@ -88,38 +88,33 @@ where
\end{equation}
-Unlike \citet{Pailha2009} we do not implicitly prescribe the viscous drag during dilation and equation, as we instead solve for the fluid pressure which is described in the following section.
+Unlike \citet{Pailha2009} we do not implicitly prescribe the viscous drag during dilation and equation, and instead solve for the fluid pressure.
\subsection{Fluid-pressure evolution}%
\label{sub:fluid_pressure_evolution}
-
-The transient evolution of pore-fluid pressure ($p_\text{f}$) is governed by Darcian pressure diffusion and local forcing by volumetric changes to the pore volume \citep{Goren2010, Goren2011, Damsgaard2017}:
+The transient evolution of pore-fluid pressure ($p_\text{f}$) is governed by Darcian pressure diffusion \citep[e.g.]{Goren2010, Goren2011, Damsgaard2017}:
\begin{equation}
\frac{\partial p_\text{f}}{\partial t} = \frac{1}{\phi\mu_\text{f}\beta_\text{f}} \nabla \cdot (k \nabla p_\text{f})
\label{eq:p_f}
\end{equation}
where $\mu_\text{f}$ denotes dynamic fluid viscosity [Pa s], $\beta_\text{f}$ is adiabatic fluid compressibility [Pa$^{-1}$], and $k$ is intrinsic permeability [m$^2$].
The sediment is assumed to be in the critical state throughout the domain, as in the original formulation by \citet{Henann2013}.
-For that reason there is no need to correct for porosity advection as in \citet{Damsgaard2015}.
-However, we include transient dynamics due to the interplay of sediment and water during shear zone formation and vanishment, as these deviations in water pressure can contribute hardening and weakening, respectively \citep[e.g.][]{Iverson1998, Moore2002, Pailha2008, Damsgaard2015, Damsgaard2016}.
\subsection{Numerical solution procedure}%
\label{sub:numerical_solution_procedure}
-
-In sum of the above, the material parameters consist of nonlocal amplitude $A$ [-], representative grain diameter $d$ [m], static yield coefficient $\mu_\text{s}$ [-], and the rate dependence beyond yield $b$ [-].
-Hydraulic properties are given by the intrinsic permeability prefactor $k$ [??], the fluid viscosity $\mu_\text{f}$, and the adiabatic fluid compressibility $\beta_\text{f}$.
-During deformation, the porosity evolves towards the critical-state value ($\phi_\text{c}$).
+The material parameters consist of nonlocal amplitude $A$ [-], representative grain diameter $d$ [m], static yield coefficient $\mu_\text{s}$ [-], and the rate dependence beyond yield $b$ [-].
+Hydraulic properties are given by the permeability $k$ [m$^2$], the fluid viscosity $\mu_\text{f}$ [Pa s], and the adiabatic fluid compressibility $\beta_\text{f}$ [Pa$^{-1}$].
These parameters do not change over the course of a simulation, and are kept constant everywhere in the domain where the material is of identical origin.
-While the above formulation is applicable to any spatial dimensionality, we apply it in a 1D framework in the following study.
-Shear deformation is restricted to occur in horizontal (x) shear zones with dilative and contractive movements along the vertical axis ($z$).
+The above formulation is applicable to any spatial dimensionality, for the purposes of this study we apply it in a 1D spatial reference system.
+Shear deformation is restricted to occur in horizontal (x) shear zones.
The axis $z$ is pointed upwards with a domain length of $L_z$.
-We assign spatial depth coordinates $z_i$ and fluidity $g_i$ to a regular grid with ghost nodes and cell spacing $\Delta z$.
-The normal stress is assumed to increase with depth due to lithostatic pressure from the overburden ($\sigma_\text{n}(z) = \int^{z'=L_z}_{z'=z} \rho_\text{s} \phi G dz' + \sigma_\text{n}$), where G is the magnitude of gravitational acceleration and $\sigma_\text{n}$ is the normal stress applied on the top of the domain.
+We assign depth coordinates $z_i$ and fluidity $g_i$ to a regular grid with ghost nodes and cell spacing $\Delta z$.
+The normal stress is assumed to increase with depth due to lithostatic pressure from the overburden ($\sigma_\text{n}(z) = \int^{z'=L_z}_{z'=z} \rho_\text{s} \phi G dz' + \sigma_\text{n,t}$), where G is the magnitude of gravitational acceleration and $\sigma_\text{n,t}$ is the normal stress applied on the top of the domain.
The fluidity field $g$ is solved for a set of mechanical forcings ($\mu$, $\sigma_\text{n}'$, boundary conditions for $g$), and material parameters ($A$, $b$, $d$).
We rearrange Eq.~\ref{eq:g} and split the Laplace operator ($\nabla^2$) into a 1D central finite difference 3-point stencil.
-We then apply an iterative scheme to relax the following equation at each grid node $i$:
+An iterative scheme is applied to relax the following equation at each grid node $i$:
\begin{equation}
g_i = {\left(1 + \alpha_i\right)}^{-1}
\left(\alpha_i g_\text{local}(\sigma_{\text{n},i}', \mu_i)
@@ -142,5 +137,3 @@ where
\printbibliography{}
\end{document}
-
-% vim: tw=1000
