commit 3ee98695a7355a63057ffefbc0065d1a9dfdec6a
parent 66f09cf042d4bd6f25a18b1877f58cae9f9ace1d
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Mon, 24 Jun 2019 11:27:36 +0200
Add csquotes, begin to weed out text
Diffstat:
1 file changed, 10 insertions(+), 70 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -12,6 +12,7 @@
\usepackage[charter]{mathdesign} % Math font
\usepackage[scale=0.9]{sourcecodepro} % Monospaced fontenc
\usepackage[lf]{FiraSans} % Sans-serif font
+\usepackage{csquotes}
%% Graphics
\usepackage{graphicx}
@@ -28,7 +29,7 @@ maxcitenames=2, backend=bibtex8]{biblatex}
\bibliography{BIBnew.bib}
%%% TITLE
-\title{A new continuum model for subglacial till based on granular rheology}
+\title{Deep slip in subglacial tills from water-pressure memory}
\author{Anders Damsgaard}
\date{Latest revision: \today}
@@ -48,36 +49,18 @@ We generalize these previous models to a water-saturated granular flow, by compa
\end{abstract}
+\section{Introduction}%
+\label{sec:introduction}
-\section{Governing equations}%
-\label{sec:governing_equations}
-\subsection{Mass conservation}%
-\label{subsec:mass_conservation}
-The model consists of a granular sediment phase with pores fully saturated by a fluid phase.
-The porosity is denoted $\phi$ [-], and the sediment material density is $\rho_\text{s}$ [kg/m$^3$].
-The equation for sediment phase mass conservation over time ($t$) is defined as:
-\begin{equation}
- \frac{\partial[(1-\phi)\rho_\text{s}]}{\partial t} +
- \nabla \cdot [ (1-\phi)\rho_\text{s} \boldsymbol{v}_\text{s}]
- = 0,
-\label{eq:mass_cons_solid}
-\end{equation}
-where $\boldsymbol{v}_\text{s}^{}$ denotes sediment flow velocity.
-The fluid phase continuity equation similarly contains fluid density ($\rho_\text{f}$) and flow velocity ($\boldsymbol{v}_\text{f}^{}$):
-\begin{equation}
- \frac{\partial[\phi\rho_\text{f}]}{\partial t} +
- \nabla \cdot [\phi\rho_\text{f} \boldsymbol{v}_\text{f}]
- = 0
-\label{eq:mass_cons_fluid}
-\end{equation}
+\section{Methods}%
+\label{sec:methods}
\subsection{Granular flow}%
\label{sub:granular_flow}
-The granular material deforms as a highly nonlinear Bingham material with yield beyond the Mohr-Coulomb failure limit for a cohesionless material.
+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.
-The total shear rate $\dot{\gamma}$ consists of elastic ($\dot{\gamma}^\text{e}$) and plastic ($\dot{\gamma}^\text{p}$) contributions.
-We assume that the elastic component is negligible:
+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,
\label{eq:shear-strain-rate}
@@ -104,63 +87,20 @@ where
\label{eq:cooperativity}
\end{equation}
-\subsubsection{Transient dynamics}%
-\label{ssub:transient_dynamics}
-
-The porosity in loose granular materials tends towards a critical state porosity ($\phi_\text{c}$) during early shear strain.
-If the porosity initially is lower than the critical-state value, the material dilates to accommodate relative movement of grains in the shear zone.
-Similarly, the porosity decreases during shear-driven compaction if the material is initially underconsolidated.
-We use the empirical relationship from \citet{Pouliquen2006} to describe critical state porosity $\phi_\text{c}$ under the assumption that the grains are rigid:
-\begin{equation}
- \phi_\text{c}(I) = \phi_\text{min} + (\phi_\text{max} - \phi_\text{min})I,
- \label{eq:phi_c}
-\end{equation}
-where $\phi_\text{min}$ and $\phi_\text{max}$ are the smallest observable porosity and the observed or inferred porosity at $I=1$, respectively.
-$I$ is the inertia number \citep{GDR-MiDi2004}:
-\begin{equation}
- I = \frac{\dot{\gamma}d}{\sqrt{\sigma_\text{n}' / \rho_\text{s}}}
- \label{eq:I}
-\end{equation}
-We describe the volumetric and shear stress dynamics before the critical state in a set of transient equations \citep[e.g.][]{Roux1998, Pailha2009}:
-\begin{equation}
- \frac{1}{\phi} \frac{\mathrm{d}\phi}{\mathrm{d}t} = \frac{\partial v_{\text{s},z}}{\partial z} = \tan \psi \dot{\gamma},
- \label{eq:dilation}
-\end{equation}
-\begin{equation}
- \tau = \tan \psi \sigma_\text{n}' + \tau_\text{c},
- \label{eq:shear_stress}
-\end{equation}
-\begin{equation}
- \tan \psi = K\left(\phi_\text{c}(I) - \phi\right).
- \label{eq:dilatancy_angle}
-\end{equation}
-The dilatancy angle $\psi$ is a state variable that reflects the materials tendency to compact or dilate from its present state during shear.
-Its value is positive for compacted materials with current porosity larger than their critical state porosity, and vice versa.
-The empirical constant $K$ links porosity deviations and the dilatancy angle.
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.
-The permeability of the pore network can for simple materials be scaled through empirical relations \citep[e.g.][]{Hazen1911, Kozeny1927, Carman1937, Krumbein1943, Harleman1963, Schwartz2003}.
-We apply the common Kozeny-Carman relationship \citep{Goren2011, Damsgaard2015}:
-
-\begin{equation}
- k(\phi) = \frac{d^2}{180} \frac{\phi^3}{{(1-\phi)}^2}
- \label{eq:k}
-\end{equation}
-https://www.overleaf.com/6999857524nqxtxjvksmvr
\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}:
\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})
- - \frac{1}{\beta_\text{f}(1-\phi)} \left(
- \frac{\partial \phi}{\partial t} + \boldsymbol{v}_\text{s} \cdot \nabla \phi \right),
\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 porosity change forcing ($\partial\phi/\partial t$) is corrected for porosity advection ($\boldsymbol{v}_\text{s} \cdot \nabla \phi$).
-If the sediment is assumed to be in the critical state throughout the domain, as in the original formulation by \citet{Henann2013}, the rightmost term in Eq.~\ref{eq:p_f} can be discarded as porosity is constant.
+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}%
