commit a000b38002b3aaa8a2024d4e3d264770be16b2cc
parent 45b047c1cf80423a60305390c0a1d4d1a35ff46d
Author: Anders Damsgaard <anders@adamsgaard.dk>
Date: Tue, 2 Jul 2019 13:22:10 +0200
Update figures, add another form of skin depth equations
Diffstat:
4 files changed, 7 insertions(+), 4 deletions(-)
diff --git a/continuum-granular-manuscript1.tex b/continuum-granular-manuscript1.tex
@@ -222,13 +222,16 @@ For the first experiment with variable water pressure, we apply a water-pressure
The water pressure variations vary with the same periodocity as the forcing, but with exponential decay in amplitude and increasing lag at depth.
The skin depth is defined as the distance where the fluctuation amplitude decreases to $1/e$ of its surface value.
-As long as fluid and diffusion properties are constant,
- an analytical solution to skin depth in our system follows the form \citep[after Eq.~4.90 in][]{Turcotte2002},
+As long as fluid and diffusion properties are constant, an analytical solution to skin depth $d_\text{s}$ in our system follows the form \citep[after Eq.~4.90 in][]{Turcotte2002},
\begin{equation}
- d_\text{s} = \left( \frac{k}{\phi\mu_\text{f}\beta_\text{f}\pi f} \right)^{1/2}
+ d_\text{s}
+ = \sqrt{ \frac{D P}{\pi} }
+ = \sqrt{ \frac{k}{\phi\mu_\text{f}\beta_\text{f}\pi f} },
\label{eq:skin_depth}
\end{equation}
-The above relation implies that the amplitude in water-pressure forcing does not influence the maximum depth of slip.
+where $D$ is the hydraulic diffusivity and $P$ is the period of the oscillations.
+The remaining terms were previously defined.
+The relation implies that the amplitude in water-pressure forcing does not influence the maximum depth of slip.
Figure~\ref{fig:skin_depth} shows the skin depth for water under a range of permeabilities and forcing frequencies.
The stick-slip experiments (Fig.~\ref{fig:stick_slip} to~\ref{fig:stick_slip_depth_normalized}) correspond to a skin depth of 2.2 meter.
Practically all of the shear strain through a perturbation cycle occurs within the upper half to the skin depth ($\sim$1 m in Fig.~\ref{fig:stick_slip_depth}).
diff --git a/experiments/fig2.pdf b/experiments/fig2.pdf
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diff --git a/experiments/fig3.pdf b/experiments/fig3.pdf
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diff --git a/experiments/fig4.pdf b/experiments/fig4.pdf
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