manus_continuum_granular1

manuscript files for first continuum-till paper
git clone git://src.adamsgaard.dk/manus_continuum_granular1
Log | Files | Refs

commit 49c88bd8a879487f69798fd57c98e291ea11e7f8
parent 9af4c78d1c79e35a777d652691aad0825cc34c4e
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date:   Wed,  9 Oct 2019 14:43:05 +0200

Work on results section

Diffstat:
Mcontinuum-granular-manuscript1.tex | 81+++++++++++++++++++++++++++++++++++++++++++++++--------------------------------
1 file changed, 48 insertions(+), 33 deletions(-)

diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex @@ -289,6 +289,21 @@ The simulated velocities are for the most part far greater than any glacial sett Parameter values and their references are listed in Table~\ref{tab:params}. For the first experiment with variable water pressure, we apply a water-pressure forcing amplitude of 50 kPa that modulates effective stress at the top around 100 kPa (Fig.~\ref{fig:stick_slip}). +\begin{figure*}[htbp] + \begin{center} + \includegraphics{experimental-setup.pdf} + \caption{\label{fig:experimental-setup}% + Experimental setup for 1d shear experiments. + The upper boundary conditions for the granular solver are fixed shear stress $\tau(z=L_z)$ (stress controlled), or fixed shear velocity $v_x(z=L_z)$ (rate controlled). + The upper normal stress ($\sigma_\text{n}$) is constant, but effective normal stress ($\sigma'_\text{n}$) varies if $p_\text{f}$ changes. + The granular phase lower boundary condition is no slip. + For the fluid solver, the top is fixed fluid pressure $p_\text{f}(z=L_z)$, which can be constant or vary in time. + The lower fluid boundary condition is a constant hydrostatic gradient (von Neumann, $dp_\text{f}/dz(z=0) = \rho_\text{f}G$). + } + \end{center} +\end{figure*} + + \begin{table*}[htbp] {\scriptsize \begin{tabular}{lllllll} @@ -326,34 +341,6 @@ For the first experiment with variable water pressure, we apply a water-pressure \section{Results}% \label{sec:results} -% Calibration of A against prior experiments - -% Unfortunately, there isn't a laboratory experiment in the literature where the -% effects of normal stress are analysed for changes in strain distribution in -% the till. So we will have to do with my discrete-element simulations. - -% In the DEM, low normal stresses produce shallow deformation and higher normal -% stresses deepen deformation. - -% There are field observations from glaciers indicating similar trends, but the -% physical setting is less well controlled. - -% By plugging in the corresponding stresses and material properties to the -% non-local continuum model, we can almost exactly replicate the DEM result. - -% The simulations for this figure took about two months to compute with a -% powerful graphics processing unit, whereas this was done in a fraction of a -% second. - -% Stress dependence of sediment advection is very interesting because it is -% relevant for non-planar ice-bed interfaces and theories of landform -% instability (drumlins, ribbed moraines, etc) - - - - - - \begin{figure*}[htbp] \begin{center} \includegraphics[width=15cm]{experiments/fig1.pdf} @@ -362,7 +349,6 @@ For the first experiment with variable water pressure, we apply a water-pressure Rate dependence in till friction from laboratory experiments \citep[after][]{Iverson2010}. \textbf{b:} Influence of rate-dependence factor $b$ in Eq.~\ref{eq:g_local} on post-failure friction in continuum model. - The friction value can be shifted up and down by adjusting $\mu_\text{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}. Here, $\mu_\text{s} = 0.5$ and $\sigma_\text{n}' = 100$ kPa. \textbf{c:} Mohr-Coulomb analysis of till samples in laboratory experiments \citep[after][]{Iverson2010}. @@ -372,6 +358,14 @@ For the first experiment with variable water pressure, we apply a water-pressure \end{center} \end{figure*} +We first compare the modeled mechanical behavior to various tills tested in laboratory settings (Fig.~\ref{fig:rate_dependence}), after \citet{Iverson2010}. +Over five orders of magnitude some tills show slight rate weakening, and others are slightly rate strengthening (Fig.~\ref{fig:rate_dependence}a). +Shear strain rates up to 5.000 a$^{-1}$ are realistic for natural glacier systems \citep{Cuffey2010}. +Our model is effectively rate-independent over most of the range, but higher $b$ values provide larger frictional resistance at extreme shear-strain rates (Fig.~\ref{fig:rate_dependence}b). +The modeled friction value can be shifted up and down by adjusting $\mu_\text{s}$ in Eqs.~\ref{eq:g_local} and~\ref{eq:cooperativity}. +Tills are Mohr-Coulomb materials, where the shear stress linearly depends on effective normal stress (Fig.~\ref{fig:rate_dependence}c). +Our modified NGF model can simulate any combination of effective friction (or friction angle $\varphi = \tan^{-1}(\mu_s)$) and cohesion (Fig.~\ref{fig:rate_dependence}d). + \begin{figure*}[htbp] \begin{center} \includegraphics[width=0.48\textwidth]{experiments/damsgaard2013-fig8.pdf}\\ @@ -382,19 +376,32 @@ For the first experiment with variable water pressure, we apply a water-pressure \end{center} \end{figure*} +The NGF model contains parameter $A$ for adjusting the degree of material non-locality (Eq.~\ref{eq:cooperativity}). +Unfortunately, no laboratory experiment exists in the literature where the effects of normal stress are analysed for changes in strain distribution in the till. +Instead, we compare the modeled strain distribution with discrete-element results from \citet{Damsgaard2013}. +By inserting relevant material parameters for grain size, friction, stress, and shear velocity (Table~\ref{tab:params}), we almost exactly replicate the strain distribution with the NGF model (Fig.~\ref{fig:strain_distribution}). +Sediment advection is pressure dependent, with low effective normal stresses producing shallow deformation, and high effective normal stresses deepening the material mobilization. +The DEM results took more than two months of computational time, whereas the continuum model is completed in a fraction of a second, albeit without detail of individual particle kinematics. + \begin{figure}[htbp] \begin{center} \includegraphics[width=15cm]{experiments/fig3.pdf} - \caption{\label{fig:till_simulation}% + \caption{\label{fig:parameter_test}% Analysis of parameter influence on steady-state strain distribution and bulk friction during shear. - All experiments are at a shear rate of 300 m a$^{-1}$ and a normal stress of $\sigma_\text{n}'$ = 100 kPa. Parameter values marked with an asterisk (*) are used outside of the individual parameter sensitivity tests. } \end{center} \end{figure} -Figure~\ref{fig:stick_slip} shows the water-pressure forcing and observed shear dynamics over a simulation time of seven days. -The shear velocities during the first cycle ($t<1$ d) is slightly different from later cycles ($t>1$ d) since the model is initialized with a hydrostatic water-pressure distribution. +Figure~\ref{fig:parameter_test} is a systematic analysis of parameter influence under a constant shear rate. +All experiments are at a shear rate of 300 m a$^{-1}$ and a normal stress of $\sigma_\text{n}'$ = 100 kPa. +The grain size $d$ has a major influence on the strain distribution, where finer materials show deeper deformation. +The material is slightly weaker with larger grain sizes. +The shear zone is more narrow with higher material static friction coefficients ($\mu_\text{s}$), as the material is less willing to fail. +Our implementation of cohesion does not influence strain. +Static friction and cohesion both linearly scale the bulk friction, as expected with Mohr-Coulomb materials (see also Fig.~\ref{fig:rate_dependence}). +The non-local amplitude $A$ slightly changes the curvature of the shear strain profile, but does not affect the overall friction. +There is a significant strengthening when the bed thickness $L_z$ begins to constrict the shear zone thickness. \begin{figure}[htbp] \begin{center} @@ -409,6 +416,10 @@ The shear velocities during the first cycle ($t<1$ d) is slightly different from \end{center} \end{figure} +Next we vary the top water pressure and observe the shear dynamics over a simulation time of seven days Figure~\ref{fig:stick_slip}. +We perform tests under both stress- and rate-controlled configurations. +The shear velocities during the first cycle ($t<1$ d) is slightly different from later cycles ($t>1$ d) since the model is initialized with a hydrostatic water-pressure distribution. + \begin{figure}[htbp] \begin{center} \includegraphics[width=0.49\textwidth]{experiments/fig5.pdf} @@ -476,6 +487,10 @@ Practically all of the shear strain through a perturbation cycle occurs above th \citet{Tulaczyk1999} and \citet{Tulaczyk2000} demonstrated that diffusion of pore-pressure variations into the bed can distribute strain away form the ice-bed interface. +% Stress dependence of sediment advection is very interesting because it is +% relevant for non-planar ice-bed interfaces and theories of landform +% instability (drumlins, ribbed moraines, etc) + \section{Conclusion}% \label{sec:conclusion}