text:
\section{Introduction}
\label{sec:introduction}

Carbon capture and storage in deep saline aquifers has been proposed
as a potential means to limit carbon emissions into the atmosphere,
while enabling the continued supply of energy from fossil fuels. Much
research has been undertaken to explore the processes which control
the storage of CO$_2$ over very long periods, and in particular the
integrity of a geological storage facility in terms of the possible
migration of CO$_2$ back to the surface \citep{hesse:2007, boait:2012,
  verdon:2013, pruess:2003}. Owing to the buoyancy of CO$_2$ relative
to water at depths of 1--2~km, CO$_2$ tends to migrate along permeable
sedimentary layers and ultimately ponds in structural highs, for
example an anticline, which represents the upper part of a fold or
other deformation in the geological strata (see
Figure~\ref{fig:cartoon}).  Such structural traps offer a possible
storage site providing there is a competent seal rock above the
anticline (e.g.,~\citealt{IPCC_2005}). However, CO$_2$ is soluble in
groundwater, which may accommodate concentrations of a few wt\% CO$_2$
in solution.  This in turn leads to an increase in density of the
water. With the dependency of water density on CO$_{2}$ concentration,
convectively-driven dissolution may develop. Water below the trapped
CO$_2$ plume becomes increasingly concentrated in CO$_2$ until it
becomes convectively unstable and sinks into the underlying permeable
rock, to be replaced by less dense, unsaturated water
(c.f.~\citealt{riaz:2006, pau:2009, hewitt:2014, lindeberg:1997}).
Eventually, the water below the CO$_2$ plume becomes fully saturated
and the continued dissolution requires a more distal supply of
undersaturated groundwater.
\begin{figure}
  \begin{subfigure}[b]{\textwidth}
    \center\includegraphics[width=0.75\textwidth]{images/diagram}
    \caption{Cartoon of the geological problem with region of interest
      indicated by dashed box. The large red arrows represent the
      direction of the background hydrological flow, while the smaller
      black curved arrows below the CO$_2$, which is trapped at the
      top of the anticline, represent the convective mixing of the
      CO$_2$ saturated and undersaturated water. The smaller blue
      arrow inside the dashed box represents the buoyancy driven flow
      of dense CO$_2$ saturated water flowing upstream into the
      background hydrological flow.}
    \label{fig:cartoon}
  \end{subfigure}
  %
  \begin{subfigure}[b]{\textwidth}
    \vspace{1.5em}
    \centering
    \begin{tikzpicture}[scale=16]
      \draw[very thick] (0,0) -- (0, .1)  node [midway, anchor=east]
    	     {$\Gamma_{c}$};
   	\draw[very thick] (0.5,0) -- (0.5, .1)  node [midway, anchor=west]
    	{$\Gamma_{B}$};
    	\draw[very thick] (0,0) -- (0.5, 0)  node [midway, anchor=north]
    	{$\Gamma_{\text{cap}}$};
    	\draw[very thick] (0,.1) -- (0.5, .1)  node [midway, anchor=south]
    	{$\Gamma_{\text{cap}}$};
     	\draw[solid, <->] (0.54,0) -- (0.54, .1)  node
      	[midway,anchor=west] {$H$};
      	\draw[solid, <->] (0,-0.033) -- (0.5, -0.033)  node
      	[below,midway,anchor=north] {$L$};

     	\draw[solid, ->] (-0.031,-0.05) -- (-0.031, 0.002)  node
     	[above,anchor=south]{$y$};
     	\draw[solid, ->] (-0.031,-0.05) -- (0.01, -0.05)
     	node [above,anchor=west]{$x$};

     	\draw[ultra thick, <-] (0.45,0.015) -- (0.51, 0.015) ;
     	\draw[ultra thick, <-] (0.45,0.035) -- (0.51, 0.035) ;
     	\draw[ultra thick, <-] (0.45,0.065) -- (0.51, 0.065) ;
     	\draw[ultra thick, <-] (0.45,0.085) -- (0.51, 0.085) ;
     	\draw[ultra thick, ->] (-0.02,0.015) -- (0.05, 0.015) ;
     	\draw[ultra thick, ->] (0.05,0.085) -- (-0.02, 0.085) ;
     	\node[text width=.1cm] at (0.44,0.05){$u_{B}, c_{0}$};
     	\node[text width=.1cm] at (0.06,0.015){$c_{D}$};
     	\node[text width=.1cm] at (0.25,0.05){$c_{0}$};
    \end{tikzpicture}
    \caption{Model problem for analysis and simulation.}
    \label{fig:boundaries}
  \end{subfigure}
  \caption{Problem of interest.}
  \label{fig:problem}
\end{figure}

\citet{szulczewski_carbon_2013} examined the longer-time dissolution
by examining the convective exchange flow which can develop in a
horizontal aquifer. They established that following the initial
dissolution and near-saturation of the groundwater directly below the
plume of CO$_2$, the lateral convective exchange flow leads to the
slow horizontal spreading of a zone of CO$_2$ enriched groundwater
associated with the continued dissolution. Eventually, the dynamics of
this zone may become controlled by a balance between: (a)~the
buoyancy-driven shear, as the dense groundwater spreads along the base
of the aquifer; and (b) the vertical diffusion of CO$_2$ from this
outward spreading dense fluid to the return flow of under-saturated
groundwater higher in the aquifer. By itself, such buoyancy-driven
shear dispersion leads to a progressively waning rate of dissolution,
and, owing to the relatively low solubility of CO$_2$ in the ground
water, the prediction that the plume of CO$_2$ may be trapped in the
anticline for a very long time \citep{szulczewski_carbon_2013}.

However, at long times the slow background hydrological flows which
transport fluid laterally through aquifers will become important in
controlling the flux of unsaturated water from far upstream,
especially as other transport processes wane.  It is the purpose of
this paper to explore the long term influence of a background
hydrological flow on the process.  In this context, \citet{woods:2012}
established some non-linear bounds on the flux of groundwater that may
reach an anticline along a weakly tilted aquifer resulting from the
interaction of a background hydrological flow with a convective
exchange flow for intermediate times, during which the cross-aquifer
diffusive transport of CO$_2$ is small. In the present work, we
account for the effects of such diffusion and this leads to a more
complex problem involving the interaction of the background advection,
the buoyancy-driven flow and the diffusive transport of CO$_2$
upstream of the anticline.  We note that in our modelling we assume
the background hydrological flows are constant in time and that there
are no mineralogical reactions of the CO$_2$ with the formation; these
are simplifications but provide a reference with which the effects of
mineral precipitation or changes in the background forcing over time
could be compared.

We develop a series of idealised, analytical solutions for the
governing equations and then test these solutions using a full
numerical simulation of the two-dimensional governing equations.  In
modelling the flow in porous rock, we assume that the dynamics are
governed by Darcy's Law, which relates to slow viscous flow.  This is
an appropriate model in the present content of slow hydrological flows
and the slow buoyancy driven flow of CO$_2$ saturated water
(e.g.~\citealt{woods:2015, bear:1972}). We thereby establish that when
the buoyancy-driven flow of the dense CO$_2$ saturated water is large
compared to the background flow speed, which is typical, three
different regimes may become established: (i)~the weak diffusion limit
in which there is a nearly static intrusion of CO$_2$ saturated water
upstream; (ii)~an intermediate regime in which there is a balance of
buoyancy-driven flow and vertical diffusion with the oncoming flow;
and (iii)~a strong diffusion limit, in which there is a balance
between the upstream diffusion of CO$_2$ and the downstream advective
transport of unsaturated water.  In each case, the dissolution rate is
proportional to the groundwater flow, even though the controls on the
extent of the CO$_2$ enrichment of the groundwater upstream of the
anticline may be very different.

\section{Model system}
\label{sec:full_model}

In our analysis, we consider a two-dimensional flow geometry, shown in
Figure~\ref{fig:problem}. This corresponds to an anticline produced by
a fold in the geological strata that extends for a relatively long
distance in the direction normal to the page compared to the width of
the fold.  We consider a background hydrological flow that supplies
fluid from the right-hand side, and an aquifer that is of uniform
thickness and horizontal.  We assume that directly below the plume of
CO$_2$ which is trapped in the anticline, the water is fully saturated
in CO$_2$ as a result of the vertical convective dissolution, and that
this is carried downstream (to the left in
Figure~\ref{fig:cartoon}). The primary purpose of this paper is to
examine how the concentration of CO$_2$ varies in the upstream
direction, shown in Figure~\ref{fig:boundaries}.  In our numerical
model, we choose the location of the upstream boundary of the flow
domain to be upstream of the region containing elevated concentrations
of CO$_2$, so that we can
impose a simple uniform flow of fluid with uniform background CO$_2$
concentration.

The full model involves Darcy flow with a buoyancy term that is
dependent on the dissolved CO$_{2}$ concentration~$c \in [c_{0},
c_{D}]$, where $c_{0} \ge 0$ is the initial CO$_2$ mass fraction of
the groundwater and $c_{D}$ is the mass fraction of the CO$_2$
saturated groundwater below the trapped plume of CO$_{2}$ (see
Figure~\ref{fig:boundaries}). We assume a fluid density $\rho$ given
by
\begin{equation}
  \rho = \rho_{0} +  \beta (c - c_{0}) \rho_{0},
\end{equation}
where the constant $\rho_{0} \ge 0$ is the initial water density and
$\beta \ge 0$ is the expansion coefficient of dissolved CO$_2$ in
groundwater.  We work with scaled concentration $c^{\star} \in [0,1]$,
given by
\begin{equation}
  c = (c_{D} - c_{0}) c^{\star} + c_{0}.
 \end{equation}

To formulate the governing equations in non-dimensional form, we
denote dimensionless variables by the superscript~`$\star$' and we
introduce:
\begin{align}
  \boldsymbol{u} & = \frac{\kappa\beta(c_{D}-c_{0})\rho_{0}g}
  {\mu}\boldsymbol{u}^{\star},
  \\
  p & = \beta(c_{D}-c_{0}) \rho_{0} g H p^{\star},
  \\
  t & = \frac{H \mu }{\kappa\beta(c_{D}-c_{0})\rho_{0}g}t^{\star},
  \\
  \boldsymbol{x} &=  H\boldsymbol{x}^{\star},
\end{align}
where $\boldsymbol{u}$ is the Darcy velocity, the constants $\kappa
\ge 0$ and $\mu > 0$ are the permeability and viscosity, respectively,
$g$ is the gravitational acceleration, $p$ is the pressure field, $H$
is the characteristic height of the domain, $t$ is time and
$\boldsymbol{x}$ is spatial position.

We denote our domain of interest by $\Omega \subset \mathbb{R}^{2}$,
with boundary $\Gamma = \partial\Omega$ and outward unit normal vector
to the boundary~$\boldsymbol{n}$.  The boundary is partitioned as
depicted in Figure~\ref{fig:boundaries}.  We formulate a
time-dependent model, with time interval of interest denoted by~$I =
[0, t_N)$. We are interested in steady solutions, hence $t_{N}$ will
be chosen to be suitably large in numerical simulations.  In terms
of non-dimensional quantities, the continuity equation, the Darcy equation
and the boundary conditions read:
\begin{alignat}{2}
  \label{eq:contmassnd}
  \nabla^{\star} \cdot \boldsymbol{u}^{\star} &= 0
  &\quad \textrm{on} \ \Omega \times I,
  \\
  \label{eq:darcynd}
  \boldsymbol{u}^{\star} &= - \nabla^{\star} p^{\star} + c^{\star}
  \boldsymbol{e}_{k}
  &\quad \textrm{on} \ \Omega \times I,
  \\
  \label{eq:pdbcnd}
  p^{\star} &= p^{\star}_{D}
  &\quad \textrm{on} \ \Gamma_{c} \times I,
  \\
  \label{eq:pnbcnd2}
  \boldsymbol{u}^{\star} \cdot \boldsymbol{n} &= 0
  &\quad \textrm{on} \ \Gamma_{\text{cap}} \times I,
  \\
  \label{eq:pnbcnd}
  \boldsymbol{u}^{\star} \cdot \boldsymbol{n} &= - u_{B}^{\star}
  &\quad \textrm{on} \ \Gamma_{B} \times I,
\end{alignat}
where $\boldsymbol{e}_{k}$ is the unit vector in the direction in
which gravity acts, $p^{\star}_{D}$ is a prescribed pressure and
$u_{B}^{\star} \geq 0$ is the prescribed fluid velocity across the
inflow boundary.  The condition in~\eqref{eq:pnbcnd} gives a
background flow from right-to-left in Figure~\ref{fig:boundaries}.

The concentration of dissolved CO$_2$ is modelled by:
\begin{alignat}{2}
  \label{eq:advdiffnd}
  \frac{\partial c^{\star}}{\partial t^{\star}}
  + \nabla^{\star} c^{\star} \cdot \boldsymbol{u}^{\star}
  - \frac{1}{\Ray} \nabla^{\star} \cdot \nabla^{\star} c^{\star}
     &= 0  &\quad \textrm{on} \ \Omega\times I,
  \\
  \label{eq:cbc1And}
  c^{\star} \boldsymbol{u}^{\star} \cdot \boldsymbol{n}
  &= \boldsymbol{u^{\star}} \cdot \boldsymbol{n}
  &\quad \textrm{on} \ \Gamma_{c, {\rm in}} \times I,
  \\
  \label{eq:dbc1and}
  \frac{1}{\Ray}\nabla^{\star} c^{\star}\cdot\boldsymbol{n} &= 0
  &\quad \textrm{on} \ \Gamma_{c} \times I,
  \\
  \label{eq:cdbc1bnda}
  (-\frac{1}{\Ray}\nabla^{\star} c
   + c^{\star}\boldsymbol{u}^{\star}) \cdot \boldsymbol{n} &= 0
  &\quad\textrm{on} \ \del{\Gamma_{\text{cap}} \cup \Gamma_{B}} \times I,
  \\
  c^{\star}(x,0) &= 0 &\quad \textrm{on} \ \Omega,
\end{alignat}
where $\Gamma_{c, {\textrm{in}}}$ is the portion of $\Gamma_{c}$ on which
$\boldsymbol{u}^{\star} \cdot \boldsymbol{n} < 0$, the constant
$\Ray$ is a Rayleigh number,
\begin{equation}
  \Ray = \frac{\kappa\beta(c_{D}-c_{0})\rho_{0} gH}{\mu D},
\end{equation}
and $D \ge 0$ is the pore-scale diffusivity.  The boundary condition
in~\eqref{eq:cbc1And} ensures that the advective flux of CO$_2$ at
the CO$_2$ trap boundary ($x=0$) has dimensionless concentration
unity on the inflow parts of the
boundary, while it is not prescribed on the outflow parts of the
boundary.  In steady state,
equations~\eqref{eq:advdiffnd}--\eqref{eq:cdbc1bnda} require that:
\begin{equation}
  \int_{\Gamma_c} c^{\star} \boldsymbol{u}^{\star} \cdot
  \boldsymbol{n} \dif s = 0
\end{equation}
We work from this point onward with the non-dimensional equations,
hence we drop the~`${\star}$' superscript in the following.

\section{Physical discussion}
\label{sec:defeq}

The above non-dimensionalisation identifies two controlling
parameters: $u_{B}$ represents the ratio of the background flow speed
to the buoyancy-driven flow speed, and $\Ray$ represents the
buoyancy-driven flow speed compared to the effective speed associated
with vertical diffusive transport across the flow domain.  These two
parameters may be used to delineate the different flow regimes which
may develop. We explore this below.

\subsection{Gravity intrusion model}
\label{sec:gravity_intrusion}
In the case of weak diffusion, we expect that a nearly static
intrusion of the dense CO$_2$ saturated fluid extends upstream into
the aquifer, and that this is balanced by the pressure gradient of the
oncoming hydrostatic flow. There will be a thin diffusive boundary
layer between the intrusion of CO$_2$ saturated water and the oncoming
flow of groundwater. If the intrusion extends far into the aquifer ($X
\gg 1$), then the continuity equation suggests that the background
flow will be largely parallel to the boundary of the domain.  To model
this regime, we assume a sharp interface in the concentration field at
a height $h(x)$  above the lower boundary of the aquifer, where
$0 \le h(x) \le 1$. This interface delineates the CO$_2$ saturated
intrusion and the overlying
groundwater.  Assuming the pressure in the intrusion is approximately
hydrostatic (c.f.~\citet{woods:2015, huppert:1995}), then in
equilibrium the buoyancy driven pressure gradient in the $x$-direction
along the intrusion matches the pressure gradient associated with the
background flow above the intrusion, which has speed $ u_{B}/(1 -
h)$. This leads to the balance
\begin{equation}
  -(1 - h) \od{h}{x} = u_{B}.
\end{equation}
The shape of the intrusion is therefore given by
\begin{equation}
  h(x) = 1 - \sqrt{2u_{B}x},
  \label{eq:intrusion_height}
\end{equation}
and it follows that the extent of the intrusion into the aquifer is
$X_{\rm int}= 1/2u_{B}$.  This implies that if $u_{B}$ is small, the
intrusion extends far upstream into the aquifer, relative to the
vertical extent of the aquifer, and the assumption
that the flow is one-dimensional is valid.

The dimensionless time-of-travel of the oncoming flow past this
intrusion is given by $\tau_{\rm int} = \int_{X_{\rm int}}^{0}
(1/u) \dif x = 1/(3 u^{2}_{B})$.  For the interface to remain sharp,
the time should be small relative
to the diffusion time, $\tau_{\rm diff} = \Ray$. This requires that
$\Ray \gg 1/(3u_{B}^{2})$ which may be expressed in the form:
\begin{equation}
  u_{B}^{2} \Ray \gg \frac{1}{3}.
  \label{eq:intrusion}
\end{equation}
For simplicity, we will henceforth use the condition~$u_{B}^{2} \Ray
\gg 1$.

When $u_{B}$ is large, we expect any intrusion will become progressively
smaller, with $X_{\rm int} \le 1/u_{B}$ (for example
Figure~\ref{fig:flowplots2-intrusion}, Section~\ref{sec:stong_bg_flow})
and so we now expect the time-scale $1/u^{2}_{B}$ to be an upper bound on
the advection time, $\sim X_{\rm int}/u_{B}$. Since $X_{\rm int}<1$ for
large $u_{B}$, we compare this with the diffusion time along, rather
than across, the aquifer. This diffusion time scales as $X^2_{\rm int}
\Ray$ and suggests that the line $\Ray =1$ provides an upper bound on
the transition from the diffusion to the advection regime. We return to
this case in Section~\ref{sec:stong_bg_flow}.

\subsection{Buoyancy-driven shear dispersion model}

In the case $u_{B}^{2} \Ray \ll 1$, diffusion in the vertical
direction will be relatively fast, hence the vertical gradient in
concentration across the aquifer will be small. However, there may be
a significant gradient in the along-aquifer direction.  This can lead
to different regimes in which diffusion is important and we now
establish conditions which determine whether a buoyancy-driven shear
flow develops or a simple advection--diffusion balance controls the
transport.  We explore these two limits by starting from the full
equations and allowing for variations in the concentration of CO$_2$
in the fluid associated with the diffusive flux. We follow largely the
analysis of \citet{szulczewski_carbon_2013} and \citet{woods:2015} to
formulate a one-dimensional asymptotic model for the long-time
evolution of the vertically averaged concentration field, but now in
the presence of a background flow.

We decompose the CO$_2$ concentration of the groundwater in the form
$c(x, y, t) = \bar{c}(x, t) + \hat{c}(x, y, t)$, where $\bar{c}$
is the average concentration across the depth of the aquifer:
\begin{equation}
  \bar{c} =  \int^{1}_{0} c \dif y.
\end{equation}
Under the assumptions that the concentration fluctuations $\hat{c}$
are small, as expected in the limit $u_{B}^{2} \Ray \ll 1$, and that
the horizontal scale of the flow is much larger than the thickness of
the aquifer, as expected in the case $u_{B} \ll 1$, the
non-hydrostatic vertical pressure gradient is relatively small and the
flow is approximately parallel to the boundaries of the flow domain.
Therefore the pressure may be approximated by:
\begin{equation}
  p = p_0 - y\Bar{c},
  \label{eq:pressure}
\end{equation}
where $p_{0} = p_{0}(x, t)$ is the pressure at the base of the aquifer.

We decompose the velocity of the fluid $\boldsymbol{u} = (u, v)$ into
a sum of the average across the depth of the aquifer
$\Bar{\boldsymbol{u}} = (\Bar{u}, \Bar{v})$ and the fluctuation
$\Hat{\boldsymbol{u}} = (\Hat{u}, \Hat{v})$, where:
\begin{equation}
  \Bar{\boldsymbol{u}} = \int^{1}_{0} \boldsymbol{u} \dif y.
\end{equation}
Using the approximation for the pressure~\eqref{eq:pressure}, Darcy's
law implies that
\begin{equation}
  \label{eq:uplusp}
  u = -\frac{\partial p_{0}}{\partial x}
  + y\frac{\partial{\bar{c}}}{\partial x},
\end{equation}
and so
\begin{equation}
  \label{eq:uflucprof}
  \hat{u}
  = \frac{\partial \bar{c}}{\partial x}\del{y - \frac{1}{2}}.
\end{equation}

Taking the vertical average of the transport
equation~\eqref{eq:advdiffnd}, and combining with the continuity
equation~\eqref{eq:contmassnd}, it may be shown that
\begin{equation}
  \label{eq:transportbuoyancy}
  \frac{\partial \bar{c}}{\partial t}
  + \bar{u}\frac{\partial \bar{c}}{\partial x}
  + \overline{\hat{u}\frac{\partial\hat{c}}{\partial x}}
  + \overline{\frac{\partial\hat{u}}{\partial x}\hat{c}}
  = \frac{1}{\Ray}\frac{\partial^{2}\bar{c}}{\partial x^{2}}.
\end{equation}
Subtracting~\eqref{eq:transportbuoyancy} from the transport equation,
we obtain an equation governing the evolution of the concentration
fluctuation:
\begin{equation}
  \frac{\partial \hat{c}}{\partial t}
  + \Hat{u}\frac{\partial{\bar{c}}}{\partial{x}}
  + \Bar{u}\frac{\partial \Hat{c}}{\partial x}
  + \Hat{u}\frac{\partial \Hat{c}}{\partial x}
  + \Hat{v}\frac{\partial \Hat{c}}{\partial y}
  = \frac{1}{\Ray}\del{ \frac{\partial^{2}\hat{c}}{\partial x^2}
    + \frac{\partial^{2}\hat{c}}{\partial y^2}}
  + \overline{\hat{u}\frac{\partial \hat{c}}{\partial x}}
  +\overline{\frac{\partial\hat{u}}{\partial x}\hat{c}}.
  \label{eq:fluctuationsinc}
\end{equation}

After long time periods, we expect the dominant balance in
equation~\eqref{eq:fluctuationsinc} to be between the distortion of
the mean concentration due to the shear flow and the cross layer
diffusion~\citep{taylor:1953}.  This gives rise to the following
dominant balance, which can be shown \emph{a posteriori}:
\begin{equation}
  \label{eq:balance}
  \frac{1}{\Ray}\frac{\partial^{2}\hat{c}}{\partial y^{2}}
  = \hat{u}\frac{\partial \bar{c}}{\partial x}.
\end{equation}
Inserting~\eqref{eq:uflucprof} into~\eqref{eq:balance} and integrating
leads to the expression
\begin{equation}
  \hat{c} = \Ray \del{\frac{\partial \bar{c}}{\partial x}}^{2}
  \del{\frac{y^3}{6} - \frac{y^2}{4} + \frac{1}{24}}.
\end{equation}
Combining this expression with the expression for $\Hat{u}$
in~\eqref{eq:uflucprof}, the depth-averaged transport
equation~\eqref{eq:transportbuoyancy} becomes:
\begin{equation}
  \label{eq:transportbuoyancyfull}
  \frac{\partial \bar{c}}{\partial t}
  - u_{B}\frac{\partial\bar{c}}{\partial x}
  = \frac{1}{\Ray}\frac{\partial^{2}\bar{c}}{\partial x^{2}}
  + \frac{\Ray}{120}\frac{\partial}{\partial x}
  \del{\frac{\partial \bar{c}}{\partial x}}^{3}.
\end{equation}

At long times,~\eqref{eq:transportbuoyancyfull} admits steady
solutions in which $\bar{c} \rightarrow 0$ as $x \rightarrow
\infty$. In the limit that $u^2_B \Ray \ll 1$, the vertical gradient
of concentration is small, and so in this limit it follows from the
boundary condition in~\eqref{eq:cbc1And} that these solutions also
require $\bar{c} \approx 1$ at $x = 0$.  To help interpret these
solutions, it is convenient to re-scale the horizontal coordinate
according to
\begin{align}
  x = \del{\frac{\Ray}{120 u_{B}}}^{\frac{1}{3}} \Tilde{x},
\end{align}
leading to the relation
\begin{equation}
  \label{eq:simplificationalpha}
  -\frac{\partial \bar{c}}{\partial \Tilde{x}}
  = \alpha\frac{\partial^{2}\bar{c}}{\partial \Tilde{x}^{2}}
  + \frac{\partial}{\partial \Tilde{x}}
  \del{\frac{\partial\bar{c}}{\partial \Tilde{x}}}^{3},
\end{equation}
where
\begin{align}
  \label{eq:alpha}
  \alpha = \del{\frac{120}{\Ray^{4}u_{B}^{2}}}^{\frac{1}{3}}.
\end{align}
We see that for large $\alpha$ diffusion dominates ($-\Bar{c} = \alpha
\Bar{c}^{\prime} $), and for small $\alpha$ dispersion is dominant
($-\Bar{c} = \Bar{c}^{\prime 3} $).
We have not found an analytic solution
to~\eqref{eq:simplificationalpha}, but in the two limits $\alpha \gg
1$ and $\alpha \ll 1$ there are useful analytical approximations.

In the limit $\alpha \ll 1 $, the buoyancy-driven dispersion balances
the advection. The solution to~\eqref{eq:simplificationalpha} when
$\alpha = 0$ is:

\begin{equation}
  \label{eq:cdispersion}
  \bar{c}
  = \del{1 -
  \frac{2}{3}\del{\frac{120u_{B}}{\Ray}}^{\frac{1}{3}}x}^{\frac{3}{2}}.
\end{equation}
Substitution of this solution into equation~\eqref{eq:fluctuationsinc}
and comparison of terms identifies that the dominant balance is indeed
given by~\eqref{eq:balance}, in the limit $u_{B}^{2} \Ray \ll 1$
and~$u_{B} < 1$. The flow extends a large distance upstream compared
to the thickness of the aquifer and the cross-flow diffusion is fast
compared to the time for the background flow to pass through the
region in which there is an elevated CO$_2$ concentration. The
solution~\eqref{eq:cdispersion} suggests that the region of enhanced
concentration advances upstream a non-dimensional distance
\begin{equation}
  \label{eq:xdis}
  X_{\textrm{dis}}
  = \frac{3}{2}\del{\frac{\Ray}{120u_{B}}}^{\frac{1}{3}}.
\end{equation}

In the limit $\alpha \gg 1$ the steady-state is dominated by a balance
of advection and diffusion, and the solution may be approximated by
\begin{equation}
  \label{eq:cdiffusion}
  \bar{c} =  e^{-\Ray u_{B} x},
\end{equation}
with a characteristic length scale of
\begin{equation}
  \label{eq:xdif}
  X_{\textrm{diff}} = - \frac{\ln c_{\text{diff}}}{\Ray  u_{B}},
\end{equation}
where $c_{\text{diff}}$ is the concentration at which we consider it
to be negligible.

Equating $X_{\text{diff}}$ and $X_{\text{dis}}$, we find that
\begin{equation}
  \alpha_{e} = \frac{3}{2 \ln c_{\text{diff}}},
\end{equation}
and this provides an indication of the transition between diffusive
and dispersive mechanisms. For $c_{\text{diff}} = 0.01$, we
find~$\alpha_{e} = 0.326$.  Figure~\ref{fig:xs_ra} shows
$X_{\text{dis}}$ and $X_{\text{diff}}$ as function of $\Ray$ for two
different values of the background flow, using~$c_{\text{diff}} =
0.01$.  For larger Rayleigh numbers, dispersion controls the distance
that the CO$_{2}$ front extends upstream.

\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{images/x_ra}
  \caption{The distance the CO$_{2}$ front extends upstream in the
    dispersion limit and in the diffusion limit as a function of
    $\Ray$ for two different values of background flow~$u_{B}$.
    Distances $X$ correspond to multiples of the aquifer height~$H$.}
  \label{fig:xs_ra}
\end{figure}

\subsection{Regime differentiation}

Combining the analysis of the gravity intrusion with the model of the
buoyancy-driven shear dispersion, we infer that for $u_{B} < O(1)$ three
regimes may arise, as shown in Figure~\ref{fig:regimecomparison}.
Gravity intrusion occurs when $u_{B}^{2} \Ray \gg 1$. When $u_{B}^{2}
\Ray \ll 1$, either a diffusion or dispersion dominated flow results,
depending on the parameter $\alpha$ (see equation~\eqref{eq:alpha}).

\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{images/regimes}
  \caption{Illustration of the different regimes. The dashed line
    denotes $\alpha = 1$ (see equation~\eqref{eq:alpha}) which
 	delineates the  dispersive and diffusive regimes for $u_B < O(1)$.
    The solid line shows $\Ray = 1/u_{B}^{2}$ (see
    equation~\eqref{eq:intrusion}) and delineates the boundary between
    the intrusion regime and (i) the dispersive regime for $u_B<O(1)$
    and (ii) the diffusive  regime for $u_{B}>O(1)$. The dotted line
    $\Ray= 4.93$ denotes an upper bound on the transition between the
    intrusive and diffusive regimes in the case $u_{B}>1$, when the
    intrusion is relatively short, and the along-aquifer diffusive
    transport dominates the cross-aquifer diffusion; we have chosen
    $\Ray = 4.93$, so that this line intersects the point where the
    dashed and solid lines converge.}
  \label{fig:regimecomparison}
\end{figure}

As $u_{B}$ approaches unity, the above analysis shows that the
transition between the dispersion and diffusion regimes and between
the dispersion and intrusion regimes converge. In the case $u_{B}
\geq O(1)$, the flow becomes more restricted in lateral extent upstream
of the anticline, and our analysis of time-scales for $u_{B}>1$ given at
the end of Section~\ref{sec:gravity_intrusion}, suggests that with
$u_{B}>O(1)$, an upper  bound for the case in which the along-aquifer
diffusion dominates the intrusion regime is  $ \Ray = 0(1)$.  In the
diffusion dominated regime, from the boundary
conditions~\eqref{eq:cbc1And} and~\eqref{eq:dbc1and}, we expect
that at $x=0$, $\bar{c} < 1$ and we explore this further in
Section~\ref{sec:stong_bg_flow} below. In Figure~\ref{fig:regimecomparison},
we illustrate this upper bound with a dotted line, which for convenience we
show as $\Ray = 4.93$ so that it intersects the point at which the solid
and dashed lines converge.

\section{Comparison of analytical and numerical models}
\label{sec:examples}

To support the asymptotic analysis, the full problem in
Section~\ref{sec:full_model} has been solved on a domain of length $L
= 100$ and height~$H = 1$.  We use a mixed finite element method for
the Darcy flow, and an upwinded discontinuous Galerkin method for the
transport equation. Problems are advanced in time until a steady-state
is reached.  A detailed description of the numerical method and the
complete computer code used to produce all examples is provided in the
supporting material~\citep{unwin:supporting}.  The computer code is
built on the FEniCS libraries~\citep{fenics:book}.

\subsection{Weak background flows ($u_{B} < 1 $)}

To illustrate the form of the velocity and concentration fields for
the three different regimes when $u_{B} < 1$, we show in
Figure~\ref{fig:flowplots} the computed concentration field for three
different values of $\Ray$ when~$u_{B} = 0.1$. The values of $\Ray$
correspond to points in the gravity intrusion, dispersion dominated,
and diffusion dominated regimes.
\begin{figure}
  \begin{subfigure}[b]{\textwidth}
    \center
    \vspace{1.0em}
    \includegraphics[width=0.6\textwidth]{images/large_c}
    \caption{Gravity intrusion: $\Ray = 3000 $}
    \label{fig:grav_int}
  \end{subfigure}
  %
  \begin{subfigure}[c]{\textwidth}
    \center
    \vspace{1.0em}
    \includegraphics[width=0.6\textwidth]{images/medium_c}
    \caption{Dispersive regime: $\Ray = 50$}
  \end{subfigure}
  %
  \begin{subfigure}[c]{\textwidth}
    \center
    \vspace{1.0em}
    \includegraphics[width=0.6\textwidth]{images/small_c}
    \caption{Diffusive regime: $\Ray = 1.0$}
  \end{subfigure}
  %
  \begin{subfigure}[b]{\textwidth}
    \center
    \includegraphics[width=0.3\textwidth]{images/conc_scale}
  \end{subfigure}
  \caption{Concentration contours for $u_{B} = 0.1$ and different
    values of~$\Ray$.  The colours show the concentration locally as
    defined by the scale in the figure and the contour lines are shown
    at equal intervals of 0.1 from 0.1 to 0.9. The $x$-axis is longer
    in (c) than (a) or (b) to display the full diffusive regime.}
  \label{fig:flowplots}
\end{figure}

\subsubsection{Concentration profiles}
\label{sec:concprofile}

Figure~\ref{fig:conc_profiles} shows the vertically averaged
concentration as a function of position along the aquifer for the
one-dimensional model as given by ~\eqref{eq:simplificationalpha} (dashed lines)
and the  two-dimensional full numerical calculations (solid lines) for $u_{B} = 8 \times
10^{-4}$ and values of $\Ray$ ranging from $100$ to~$10000$.
\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{images/conc_profile}
  \caption{Computed and analytical variation of $\Bar{c}$ with $x$ for
    different values of~$\Ray$ when $u_{B} = 8 \times 10^{-4}$.  The
    solid lines represents the two-dimensional numerical solution and
    the dashed lines represent the one-dimensional solution.}
    \label{fig:conc_profiles}
\end{figure}
The one-dimensional model (equation~\eqref{eq:simplificationalpha})
was solved using the FEniCS libraries and the full code is included in
the supporting material~\citep{unwin:supporting}.  There is very good
agreement between the two models when $\Ray \le 1000$ and the flow is
in either the dispersion or diffusion dominated regime. The limiting
dispersive~\eqref{eq:cdispersion} and diffusive~\eqref{eq:cdiffusion}
cases (heavy black lines as shown in legend) are also shown in the figure, and coincide with
the two-dimensional numerical simulation when in the appropriate limit
regimes.

When $\Ray = 10000$, the problem is entering the gravity intrusion
regime. In Figure~\ref{fig:conc_profiles} the two-dimensional
numerical solution for this case no longer matches the one-dimensional
dispersion dominated model in~\eqref{eq:simplificationalpha} since
there are significant fluctuations in concentration across the height
of the aquifer.  In this regime   the solution has a
narrow vertical region of adjustment in the concentration field from
the CO$_2$ saturated fluid at the base of the aquifer to the
unsaturated oncoming groundwater at the top of the aquifer,
reminiscent of the intrusion model given in
Section~\ref{sec:gravity_intrusion}.

Indeed, in
Figure~\ref{fig:intrusion} we compare the numerical solution for the
concentration in the case $\Ray=3000$ and $u_B=0.1$ with the prediction of the interface height $h$
as predicted by equation~\eqref{eq:intrusion_height} for the gravity
intrusion model. That model
treats the adjustment of the concentration from the CO$_2$ saturated
fluid to the oncoming groundwater flow, as   a sharp
interface.  There is a reasonable match for $x < 3.5$.  However, the
diffusive boundary layer which is present in the full numerical solution leads
to a weak recirculation that is not included in the intrusion model.
\begin{figure}
  \center
  \includegraphics[width=0.7\textwidth]{images/intrusion}
  \caption{Concentration as a function of position  for the gravity intrusion case with
    $u_{B} = 0.1$ and $\Ray = 3000$  and the prediction of the  intrusion model
    (equation~\ref{eq:intrusion_height}) for the interface position
    overlaid (dotted line).}
  \label{fig:intrusion}
\end{figure}

\subsubsection{Velocity fluctuation profiles}

In the buoyancy driven dispersion regime, which arises for  small $\alpha$ and when $u_{B}^{2} \Ray \ll 1$,
equation~\eqref{eq:uflucprof} predicts that the velocity variation
from the mean flow $\hat{u}$ will vary linearly with depth.
If~$\hat{u}$ is divided by the depth-averaged concentration gradient,
the profiles are predicted to pass through $-0.5$ at $y = 0$ and $0.5$
at~$y = 1$.  Figure~\ref{fig:scaled_velocity_profile} shows the scaled
velocity profiles computed from the two-dimensional model for $\Ray =
1000$ and $u_{B} = 8 \times 10^{-4}$ ($\alpha = 0.0527$).  For $x
\lesssim (3/4)X_{dis}$ the velocity profiles are linear, whereas close
to the stall point ($x/X_{dis} = 1$) the profile begins to deviate
from the simplified theory.
\begin{figure}
  \centering
  \begin{subfigure}[b]{0.7\textwidth}
    \centering \includegraphics[width=\textwidth]{images/v_profile_dis}
    \caption{Velocity fluctuations $\hat{u}$ scaled by concentration
      gradient for $\Ray = 1000$ and $u_{B} = 8 \times 10^{-4}$
      ($\alpha = 0.0527$).}
    \label{fig:scaled_velocity_profile}
  \end{subfigure}
  \\
  \begin{subfigure}[b]{0.7\textwidth}
    \centering
    \includegraphics[width=\textwidth]{images/v_comparison_dispersion}
    \caption{Velocity fluctuations $\hat{u}$ for different values of
      $\alpha$ at different values of~$x$.}
    \label{fig:v_alpha}
  \end{subfigure}
  \caption{Scaled velocity profiles at different points along the
    domain for different values of~$\alpha$ with $u_B=8 \times 10^{-4}$.}
  \label{fig:v_profiles}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.7\textwidth]{images/streamlines}
    \caption{Typical streamlines of the flow for $u_{B} = 0.01$ and
    $\Ray = 1, 10$ and $100$.}
    \label{fig:streamfunctions}
\end{figure}

When in the regime where diffusion is important ($u_{B}^{2} \Ray \ll
1$), as $\alpha$ increases and the flow transitions from dispersive to
diffusive, the one-dimensional analytical model becomes less
applicable for~$\hat{u}$.  This can be seen in
Figure~\ref{fig:v_alpha}, where the scaled horizontal velocity
fluctuations are shown at different distances into the domain for
different values of~$\alpha$.  For $0.0572 \leq \alpha \leq 0.1442$,
the scaled horizontal velocity fluctuations at different distances
into the domain all lie on a straight line between $-0.5$ and~$0.5$.
As $\alpha$ increases, the numerically computed profiles deviate from
the model, as the lateral diffusive transport becomes comparable to
and then progressively more significant than the shear dispersion, but in this diffusion limit
 these perturbation velocities are very small compared to the background
hydrological flow.
When $\Ray = 10000$ and $u_{B} = 8 \times 10^{-4}$ ($\alpha = 0.0027$)
the gravity intrusion region is being approached and the vertical concentration
fluctuations across the domain are no longer small and so the velocity
fluctuations are no longer governed by
equation~\eqref{eq:uflucprof}.

In Figure~\ref{fig:streamfunctions}, we present a picture of the
typical streamlines for the flow when $u_B = 0.01$ and $\Ray = 1, 10$
and $100$. The streamlines are computed, approximately, by solving
equation~\eqref{eq:simplificationalpha} numerically and inserting the
result into~\eqref{eq:uflucprof}, and then adding on the background
hydrological flow.  The case $\Ray = 1$ corresponds to
the diffusion limit, the case $\Ray = 100$ corresponds to the
dispersion regime, and the case $\Ray = 10$ is an intermediate
case. The figure shows that in the diffusion limit the oncoming
groundwater flow is dominant and the flow remains nearly uniform, but as $\Ray$ increases, some
fluid begins to flow upstream leading to a small circulation in the
lower part of the aquifer. As $\Ray$ increases further, and the flow
is controlled by the buoyancy driven dispersion, a strong
recirculation develops upstream of the anticline, leading to the
diversion of the oncoming groundwater flow towards the top of the
aquifer.

\subsection{Strong background flows ($u_{B} \gtrsim 1 $)}
\label{sec:stong_bg_flow}

We now look at the case with a stronger  background flow  ($u_{B} \gtrsim1$). Three examples of full
numerical solutions are shown in Figure~\ref{fig:flowplots2}
corresponding to $\Ray = 1000$, $2$ and $0.5$ for~$u_{B}=1.0$.
\begin{figure}
  \begin{subfigure}[b]{\textwidth}
    \center
    \includegraphics[width=0.6\textwidth]{images/largec}
    \caption{Gravity intrusion regime: $\Ray = 1000$}
    \label{fig:flowplots2-intrusion}
  \end{subfigure}
  %
  \begin{subfigure}[c]{\textwidth}
    \center
    \vspace{1.0em}
    \includegraphics[width=0.6\textwidth]{images/mediumc}
    \caption{Transitional regime: $\Ray = 2 $}
    \label{fig:flowplots2-transition}
  \end{subfigure}
  %
  \begin{subfigure}[c]{\textwidth}
    \center
    \vspace{1.0em}
    \includegraphics[width=0.6\textwidth]{images/smallc}
    \caption{Diffusive regime: $\Ray = 0.5$}
    \label{fig:flowplots2-diffusive}
  \end{subfigure}
  %
  \begin{subfigure}[b]{\textwidth}
    \center
    \vspace{1.0em}
    \includegraphics[width=0.3\textwidth]{images/conc_scale_2}
  \end{subfigure}
  \caption{Concentration fields for different values of~$\Ray$ for
    $u_{B} = 1.0$. The white contour in (b) and (c) is the~$c = 0.01$
    contour. The presented domains have been truncated at different
    lengths to best show each regime.  Note that the maximum
    concentration for the colour bar is set equal to 0.3, which is
    greater than the highest concentration for panels (b) and~(c). }
  \label{fig:flowplots2}
\end{figure}
In contrast to the case $u_{B} < O(1)$, only two distinct regimes
develop (figure 3) .  Now, the  flow transitions from the gravity intrusion regime to the
diffusion dominated adjustment of the concentration since, with large $u_B$,  the upstream extent of the buoyancy driven flow
is insufficient for the buoyancy driven dispersion to develop before the along aquifer diffusion becomes significant.  With large
$\Ray$, the solution is similar to the gravity-driven intrusion
solution (Figure~\ref{fig:flowplots2-intrusion}), while for smaller
values of $\Ray$ the solution evolves towards the along aquifer
diffusion solution (Figure~\ref{fig:flowplots2-diffusive}).

We have calculated the vertically
averaged mean concentration at $x = 0$ from the numerical solutions
for the cases when $\Ray = 10, 1, 0.1$ and $0.01$, as shown in
Figure~\ref{fig:cbarx0}.  As $u_{B}$ increases, $\bar{c}$ decreases at
$x = 0$ since there is a progressively stronger flow from the upstream
region which suppresses the upstream buoyancy driven flow of dense,
CO$_2$ saturated fluid from below the anticline
\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{images/cbar_x0}
  \caption{Computed vertically averaged concentration for various
    values of $\Ray$ and $u_{B}$ at~$x = 0$.}
  \label{fig:cbarx0}
\end{figure}

In the diffusive regime, which we expect to apply for small  $\Ray   $,
 the vertically averaged concentration may be approximated
by the diffusive solution of equation~\eqref{eq:simplificationalpha},
given by:
\begin{equation}
  \bar{c}(x) = \bar{c}(0) e^{-\Ray u_{B} x}.
  \label{eq:cbar_c0}
\end{equation}
Figure~\ref{fig:conc_profile_diffusion} shows the vertically-averaged
concentration profiles for the three values of $\Ray$ and $u_{B} =
1.0$. Using the numerically determined value for the mean
concentration at $x=0$, we have compared the vertically averaged
concentration with the diffusion solution given by
equation~\eqref{eq:cbar_c0}. When $\Ray = 0.1$, the system is in the
diffusion regime and the numerical solution for $\Bar{c}$ matches the
diffusion profile. When $Ra \geq 1$, the simulations move towards the
intrusion regime and the numerical solutions evolve away from the
approximate analytical solution.
\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{images/conc_profile_diffusion}
  \caption{Computed and analytical variation of $\Bar{c}$ with $x$ for
    different values of~$\Ray$ when~$u_{B} = 1.0$.  The solid line
    represents the two-dimensional numerical solution and the dashed
    line represents the diffusion limit if using the numerically
    computed $\Bar{c}$ as the boundary condition at~$x = 0$.}
  \label{fig:conc_profile_diffusion}
\end{figure}

\section{Conclusions}

We have explored both analytically and numerically the long-term
dissolution of a plume of CO$_2$ trapped in an anticline and driven by
a steady background flow of CO$_2$ unsaturated water from upstream. We
have focused on the role of diffusion and buoyancy-driven flow in
regulating the distribution of CO$_2$ in the aquifer fluid upstream of
the anticline. In the case $u_{B} < 1$, where $u_{B}$ is the
dimensionless background hydrological flow, the buoyancy-driven speed
of the dense CO$_2$ saturated water exceeds the oncoming flow speed
and the CO$_2$ extends a significant distance upstream of the
anticline ($x \gg H$).  In this case, we have established that three
different regimes may develop. With a small diffusive flux across the
aquifer ($u_{B}^{2} \Ray \gg 1$, where $\Ray$ is the Rayleigh number
associated with the dense CO$_2$ laden fluid), a static intrusion of
dense CO$_2$ saturated fluid develops and extends a distance
$1/2u_{B}$ upstream of the anticline. This is balanced by the pressure
gradient in the oncoming flow.  With larger diffusive fluxes across
the aquifer ($u_{B}^{2} \Ray \ll 1$) we have established that a
buoyancy-driven shear dispersion flow regime may develop and a
convective recirculation develops just upstream of the aquifer,
regulated by (i)~the supply of unsaturated aquifer fluid from
upstream; (ii)~the buoyancy-driven flow associated with the dense
CO$_2$ saturated fluid from downstream; and (iii)~the vertical
diffusion of CO$_2$ across the aquifer.  However, if the diffusive
transport is too rapid ($\alpha \ll 1$) then a simple
advection-diffusion balance regulates the distribution of the CO$_2$
in solution in the water upstream of the anticline.  In the case
$u_{B} > 1$, the CO$_2$ extends a much smaller distance upstream from
the anticline, and in this case, either only the advection-diffusion
balance or the intrusion regimes develop.

In the context of CO$_2$ sequestration in deep saline aquifers, this
analysis is important as it demonstrates the strong effect that a
background hydrological flow has on the long-term dissolution of
CO$_2$ in a structural trap.  We now show that under some typical
conditions the dynamics may indeed be controlled by a balance between
the buoyancy-driven shear dispersion and the background hydrological
flow.  This leads to new estimates of the maximum upstream migration
of CO$_2$ rich groundwater.  The solubility of CO$_2$ in groundwater
is only a few wt\%, so if we consider a plume of CO$_2$ of order 10~m
deep, trapped in a structural anticline and connected to a laterally
extensive aquifer of order 20--30~m deep, then vertical convective
dissolution alone will only lead to dissolution of order 0.2--0.6~m.
Continued dissolution will require the lateral supply of
undersaturated water from the aquifer and this may be achieved through
a combination of buoyancy-driven lateral dispersion of the dense
CO$_2$ saturated water from below the CO$_2$ plume and supply of water
resulting from a background hydrological flow. For typical conditions,
with permeability of order $0.1$--$0.01$ Darcy, a density difference
of order a few percent between the undersaturated and saturated water,
and an aquifer diffusivity of order $10^{-9}$--$10^{-10}$~m$^{2}$/s,
the Rayleigh number will be of order~$10^{3}$--$10^{4}$. With a
hydrological flow speed of order $10^{-8}$--$10^{-9}$~m/s, the
dimensionless velocity $u_{B}$ will be of order~$0.01$--$1.0$.  From
Figure~\ref{fig:regimecomparison} we see that it is the transport
associated with the shear dispersion that balances the steady
background flow.  We estimate that the length scale of the dispersive
transport (equation~\eqref{eq:xdis}) will be of order $100$--$400$~m.
Once the steady flow regime is established, the continued dissolution
will occur at a rate proportional to the supply of undersaturated
water in the hydrological flow, as given in non-dimensional form by
$u_{B}(c_{D} - c_{0})$. For an anticline whose extent in the direction
of the flow is of order $1000$~m, and with a CO$_2$ plume with initial
depth of order $10$~m, then in order to dissolve, this will require a
net flow of groundwater of order $10^{6}$~m$^{2}$, which will require
a time of order $10^{12}$--$10^{13}$~s corresponding to
$10^{5}$--$10^{6}$ years.

\subsubsection*{Acknowledgements}

HJTU was funded by an EPSRC Doctoral Training Partnership scheme
(grant EP/J500380/1). Data relating to this publication is available
in~\citet{unwin:supporting}.

\bibliographystyle{abbrvnat}

redpajama_set_name: RedPajamaArXiv


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All local authorities have a statutory responsibility to identify children who are missing from education (CME), and those pupils at risk of missing education. This applies to children of compulsory school age who aren't on a school roll, and who aren't receiving any other suitable form of education in place of being at school (for example, being educated at home, privately, or in alternative provision).
In Middlesbrough, the Access to Education Service is responsible for children who are missing from education. The team establishes and monitors a proactive system for seeking information regarding CME, recording the details, and tracking/monitoring the progress of such pupils as they are re-engaged in education.
The CME Officer monitors information made available to the council by parents, schools, and other agencies regarding children missing from education, e.g. excluded pupils, children whose families are seeking a school place, and pupils not on school roll for other reasons.
Guidance on children missing from education or at risk of missing education is available for schools, academies, and other agencies.


The Federation of the National Association of Home Builders (NAHB) exists to represent the home building industry by serving its members and affiliated state and local builders associations. The NAHB member, who are involved in all aspects of a diversified building industry, create jobs for millions of people and contribute significantly to the economic activity of their community, the nation and the world.
The Florida Home Builders Association is a professional trade organization comprised of local associations and affiliated with the National Association of Home Builders. It serves its members by striving for a successful building environment throughout the state.
The programs and services provided to FHBA's member enhance their professionalism and encourage their involvement in community leadership to improve the quality of life for all Florida citizens.
The association is dedicated to an active governmental affairs program which provides sage and affordable housing and the opportunity for home ownership.
CDBIA is a non-profit organization consisting of a group of individuals and firms who care about their community and believe that home ownership should be within reach of every American. They encourage research within the building industry and give their informed and vigorous support to all sound legislative proposals affecting the industry.
Association members come from all types of business-builders, large and small, architects, engineers, subcontractors, financial institutions, public utilities, manufacturers, suppliers, retail and wholesale firms and marketing professionals. These firms and individuals are proud of the part they play in building the communities where they work and live.
The FGBC's mission is to "improve the built environment" by helping all Floridians who are seeking to become more responsible stewards of the environment.
The following are industry designations that members of our SandStar Remodeling Team have earned through their dedication to ongoing education.
The CAPS designation program teaches the technical, business management and customer service skills essential to competing in the aging-in-place market.
CGB is an exclusive professional designation designed to emphasize business and project management skills.
Certified Graduate Remodeler is an exclusive professional designation designed to emphasize business management skills as the key to a professional remodeling operation.
The National Association of Home Builders Certified Green ProfessionalTM Designation teaches builders, remodelers and other industry professionals techniques for incorporating green building principles into homes without driving up the cost of construction.
The CSP designation was designed for specialists in new home sales to enhance their professional image, increase their marketability in the home building industry and sell more homes!
CMP is the mid-level designation for IRM students who have completed the marketing intensive IRM courses I though IV.
The MIRM designation is the top-level achievement for professionals in new home marketing.
CKD® – Certified Kitchen Designers have a minimum of 7 years experience designing residential kitchen spaces. They are highly skilled in design, space planning and product selection and have extensive knowledge of building codes, flooring materials, appliances, and mechanical systems. They write specifications and draw plans that are easily interpreted by plumbers, electricians, and installers. A CKD must meet specific educational requirements and pass a comprehensive academic and practical exam.
GMB is the designation for experienced contractors. Before beginning the GMB designation process, you must have the CGB designation, the CGR designation, or the CGA designation with five years of building experience, or ten years of building experience and have completed three CGB/CGR/CGA courses. GMB courses are more advanced with in-depth instruction geared for experienced building professionals.
SandStar has been very fortunate to have been the recipient of many remodeling and building awards over the years, but we feel that the best award is our many happy home owners who love the work we do for them.


Hi Ladies, I was wondering how many of y'all are going to send out birth announcements when your lilone arrives? If you are, where are you getting the announcements from etc? Thanks!
I saw some really cute ones on tinyprints.com - i am going to send some out i think.
i will be sending some out. you can upload your pictures to costco and go in and have them printed too.


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Q: Connect angular module and service dependencies using jest testing framework I'm tryin to configure jest testing framework for my simple application
And getting issues like that:
[$injector:modulerr] Failed to instantiate module dib due to:
Error: [$injector:modulerr] Failed to instantiate module angulartics due to:

And so on.
Can't understand how to connect all main module dependencies and service being tested dependencies with jest at one place.
Should i list all submodules in package.json, or in each test file?

redpajama_set_name: RedPajamaStackExchange


If you are, then I'd like to welcome you to the world of online marketing.
Believe it or not, people just like you and me, earn a substantial and comfortable living via the internet every day. In my experience, younger people find it easier to embrace this concept, because they have grown up in the digital age, however I have also met quite a few of their parents who are enjoying their golden years to the fullest, because they've harnessed the power of the internet to fund their lifestyle.
It's all right if you're skeptical, I certainly was in the beginning, and I was one of the first diamond dealers to sell diamonds online, way back in 1996. My web site Nice Ice is recognized throughout the diamond industry as one of the ones which blazed the trail for the internet diamond industry that we all benefit from today, and I'll tell you, those early days were a lot like the Wild West, every day was a new adventure, we never knew what to expect.
Right about this point in the conversation, a lot of my clients will begin to tell me all about how they want to learn how to earn money online; how they've heard of people earning stacks of cash online, but how those stories are probably urban legends; and how if they're true, those people must have gotten in on the ground floor of the opportunity, at just the right time, and how they could never do it; after all, what do they know about online marketing?
Create your own products and services, which provide a solution or solve a problem.
Promote and sell other people's products and services.
Advertise other people's products and services.
That's it, the only three ways to earn money online. Now people have a lot more ways of describing the processes of advertising and selling products and services online, but if we boil all those down, at the end of the day, we can place each of them into one of these three categories.
Obviously one of the most rewarding and profitable ways to earn money online is to create your own products and services, and market them through various forms of social media, such as blog posts, Facebook, Google +, Twitter, Instagram and Pinterest.
So while you're working on creating your own products and services, you can earn some money, and develop an audience by writing reviews for things which you have purchased and found to be beneficial… be very selective about this, don't promote everything that you think will sell, that's not how to develop an audience which will trust your opinion, limit your promotions to products and services which you have truly found to be great!
For instance, I talk a lot about Natalie Sisson's book, The Suitcase Entrepreneur, because Natalie has been instrumental in helping me to take my online business to the next level, by showing me that it is possible to run my online business from anywhere in the world where there is a stable internet connection, by doing so herself and blogging about it every step of the way!
Natalie's experience inspired me to sell my house, close my brick and mortar location, and embrace the challenge of operating an online business without being tied down to a physical location, the last year has been an amazing journey, and it's only going to get more interesting, as I prepare to continue the journey by driving through Mexico, down to Costa Rica, or wherever else my heart desires… maybe I'll see you there.
This happens to be Day 10 of Natalie's 30 Day Blog Challenge, during which she is freely sharing her advice about how to build a successful online business that will enable you to be location independent. Be sure to click here to visit The Suitcase Entrepreneur web site if you want to learn how to create freedom in business and adventure in life!
Now obviously this is a working example of how to market products and services created by other people, when you click on any of the links above, and purchase a book, program, or other kind of product, or service, I receive a small commission for making the introduction, this is basically a form of advertising, but it is based on trust and not just a desire to make money.
I'm comfortable promoting Natalie's business coaching programs because we have a working relationship built upon trust, and because I have personally benefited from following her advice. You can call this "affiliate marketing" if you'd like, but I prefer to think of it as "trust marketing" which is a phrase coined by internet marketing guru, Nick Reese.
And finally, you can use the space on your web site or blog, as advertising space for other companies, who will pay you small commissions for any click through traffic generated from your web site. One of the most popular services for this is Google Adwords, but note that I find this to be the least of my income streams.
On this, Day 10 of Natalie Sisson's 30 Day Blog Challenge, she asks "What are the key ways in which you want to make money online in your chosen business. ~ OR ~ Which are the key online revenue streams you want to focus on in your existing business?
And as you can see, I turned those questions into a blog post designed to teach you about the three ways to earn money online, by providing you with working examples of how I do so every day.
If you want to see a working example of how I use "trust marketing" to convert my Click & Mortar Business, into an educational blog, which provides consumers with in-depth diamond grading tutorials and diamond buying advice, while providing me with a stable income, visit my diamond web site Nice Ice, and read a few of the articles. You will see how I am able to use my years of experience as a diamond buyer for the mutual benefit of my customers, myself, and my affiliate partners. And the best part is that it doesn't cost my customers a dime, all fees are paid by the vendors as advertising expense.
And of course, I am in the process of creating my own coaching programs, which are designed to teach people strategies for improving their relationships and lives, and which will be sold online. So I incorporate all three methods of earning money online, and therein lies the clue to a successful online business.
Now I have to go "pin" the pictures in this article to my Pinterest Account, in hopes that people will like it well enough to re-pin it on one of their pin boards, so that more more traffic will be generated for this blog post, and so goes the process of marketing yourself online.
Oh that reminds me… if you're thinking about using Pinterest as a platform for your affiliate marketing endeavors, and I certainly recommend that you do, be sure to use a Link Cloaking Software because otherwise the software used on the back end of Pinterest will strip your link of your affiliate ID and replace it with their own, so that they make the money for the sale and not you! Don't bother with the more well known services such as Tiny URL and more well known URL shorteners, the programmers at Pinterest have already taken those into account.
I hope you learned a little something from the examples I've provided in this post about the three ways to earn money online, and look forward to your hearing your online success story!


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The BEST Episodes of Kimi ni Todoke: From Me To You
Sawako Kuronuma, called Sadako by her classmates for her resemblance to the character from The Ring, has always been feared and misunderstood because of her appearance. There are rumors that Sawako can see ghosts and curse people. However, despite her appearances, she is a sweet and timid girl who longs to be able to make friends with everyone and be like everyone else. When her idol, popular boy Kazehaya, begins talking with her, everything changes. She finds herself in a new world, trying to make friends and talking to different people and she can't thank Kazehaya enough for giving her these opportunities. Slowly, but surely, a sweet love is blossoming between the two as they will overtake any circumstance or clear obstacles in their way.
#1 - From Now On
The class finishes the costumes and Sawako performs her solo performance. Kazahaya ensures that Sawako knows his feelings.
Watch on Hulu Buy on iTunes
#2 - Birthday
Sawako and Kazehaya walk to the shrine silently. They finally look at each other after Kazehaya saves her before she falls down the stairs at the shrine. They are about to continue their awkward silence until Chizuru, Ayane, and Ryuu (who are also stealthily at the shrine) call Kazehaya to apologize for not coming along. The two go to get hot drinks at a stand and Kazehaya admits he normally doesn't like the drink, but this time he does because Sawako does. She's excited over getting to see his first time at something, but he lists several of his first times that relate to her (like wearing the stomach band). They exchange phone emails and Kazehaya realizes Sawako's birthday is that day. She tells him about her childhood as they walk, up until the present, then realizes she had always been alone until she met him.
Watch on Hulu Watch on Amazon Buy on iTunes
#3 - Two People
It's the last day of class before winter break. Sawako worries that she won't see her friends until classes start again. She reluctantly says goodbye to Kazehaya then goes out to eat with Ayane and Chizuru. They find out her birthday is on New Year's Eve and tell her about their plans to go to the shrine that day with Ryuu, inviting her as well. She finds out she can go, so her friends convince her to call Kazehaya and invite him as well. He agrees to come. On New Year's Eve, Chizuru and Ayane come to Sawako's house. Their present to her is to make her look really cute for Kazehaya. They style her hair and put on a little makeup and head out for the shrine. Before they arrive, Chizuru and Ayane leave, telling Sawako Ryuu actually isn't coming either. She continues alone, meeting Kazehaya. He is astonished by her extra-cuteness and they continue to the shrine. She reflects on her happiness to spend her birthday with him.
#4 - Christmas
Season 1 - Episode 22 - Aired Mar 9, 2010
Chizuru and Ryuu are back on good terms and everyone is eagerly anticipating Christmas. Their class is going to hold a party and Kazehaya asks Sawako to come. She really wants to, but thinks of her parents. Traditionally, her family spends all day celebrating, so she would feel badly about leaving them. She decides not to go to the party, but makes gifts for Chizuru and Ayane. She also makes a hat for Kazehaya, though she knows she won't have the nerve to give it to him. On Christmas Eve, she spends the day with her parents. Her dad is especially enthusiastic and ends up taking the hat Sawako made, thinking it was for him. She doesn't have the heart to correct him. Later that night, she gets a call from her friends at the party asking her to come. She becomes very distraught, so her mom takes the call instead. Her dad calls her over and gives her his present for her- a cellphone. He lets her go to the party and realizes she is growing up. Sawako runs there, but everyone has left. Only Kazehaya has been waiting for her out in the snow.
#5 - New Year
Sawako and Kazehaya continue talking while they walk. She begins to say she'll try harder this year so people will like her, but he interrupts and says they already do. They visit the shrine and go to pick out fortunes. Sawako is distressed by getting a "Bad Luck" slip, so Kazehaya gives her his- "Very Good Luck". She reads the part pertaining to love and realizes it could be talking about Kazehaya. They continue home, taking the path she walks to school. Kazehaya stops at one corner, and reminisces that he met Sawako for the first time at that spot. She's surprised he remembers, but has also been thinking the same thing. They continue home, each internally realizing their love for each other, but neither acting on it. They part, leaving the impression they will be together this year.
Watch on Hulu Watch on Amazon
#6 - First Snow
School continues and Chizuru looks refreshed but still ignores Ryuu. While walking home with Sawako and Ayane, they discuss what Chizuru would say to Tooru, Ryuu's older brother if he was to return. Then Tooru actually shows up and tells the girls that he's going to steal Chizuru. The two of them walk together, as Tooru talks about his memories with Chizuru when they were younger. At the end of their walk Chizuru confesses her feelings for Tooru, only have to Tooru says that he likes her too, but as a little sister. They part, and when Tooru returns home Ryuu greets him and gives him bread from the other day the Chizuru left. Ryuu then goes to find Chizuru at docks where she was the last time they played with Tooru before he moved out. There Chizuru realizes that it was Ryuu that called Tooru to come and visit her and asks Ryuu to comfort her, and they embrace.
#7 - The Person I Like
Season 2 - Episode 5 - Aired Feb 9, 2011
With the first semester midterms over, everyone's recognizing Sawako for her review session, praising her. Sawako's class decides on having "Sawako's Black Magic Cafe" as their exhibition, using Sawako's herbs to make tea, and having an additional "Ask Sawako" advice booth. Sawako and Kazehaya are still awkward and silent towards each other, which Ayane notices and worries Chizuru. Next day, Ayane goes over to Chizuru's house to help her get ready for Toru's Wedding. The wedding ends without a hitch and Chi solidifies her sibling-like relationship with Toru. Afterwards, Chizuru asks Ryu to follow-up with Kazehaya, feeling guilty about what she said to him. Ryu initially refuses but concedes, saying he would only listen to his problems. Later, Kento approaches Sawako, who's tending to her garden. Back in the cafeteria, Ryu, still keeping in mind what Chizuru said, asked Kazehaya, bluntly, if he had confessed already or not. As Kazehaya's explaining his confusion and worry, Pin arrives, teases him about not having confessed yet and mocks him, telling him to confess to Sawako already. Kazehaya, determined, runs out, searching for Sawako (Pin told him where she was). Out in the courtyard, Kento and Sawako are discussing how Sawako's finally fitting into class (which she hadn't noticed, and shocks her). She receives a mental shock when Kento says that Kazehaya was probably looking out for her because she was an outcast, and since now she's not, he probably wouldn't talk to her anymore, and that he is in love with someone (both not realizing it's Sawako herself). Sawako, distraught, walks away and starts to cry. Kento tries to cheer her up, and ends up asking her out in the flow of things, only to be drowned out by Sawako's thoughts and the arrival of Kazehaya.
#8 - Reaching You
#9 - Present
Season 1 - Episode 20 - Aired Feb 23, 2010
Ryuu's brother leaves, he tells Ryuu to tell Chizuru to show up next time. Later Chizuru heads over to give Ryuu her present, where Ryuu states that she was late and he hopes that Chizuru will finally get over his brother, she gets angry and throws his present at him. Sawako and Ayane are grief stricken with worry about Chizuru since they can't do anything about the situation. Kazehaya shows up and explains to them that its good that Chizuru has them, and that she has them to worry and care for her. In class Chizuru ignores Ryuu, they eventually run into each other where Chizuru asks what he meant last night when he said that she was late. Eventually he shows that he wanted Chizuru to get rejected in order for her to finally get over his brother, Chizuru is upset that he thought the only reason she came over the other night was just to see his brother and leaves. Ayane and Sawako find her, where she eventually breaks down and tells them about how angry she is at Ryuu, not knowing that he is listening and then leaves, Chizuru continues saying what pissed her off the most was how she treated Ryuu. The three girls then go out and enjoy the rest of the day.
#10 - Favour and Imposition
Season 2 - Episode 6 - Aired Feb 16, 2011
Kazehaya walks out on Sawako and Kento, with Kento grasping Sawako's arms and Sawako crying. Kazehaya asks what Kento is doing, and why Sawako is crying. Three of Kazehaya's friends walk over, to find Kazehaya angry, Sawako crying, and Kento just being there. When the three of them fool around and say that Kazehaya and Kento are both going after Sawako, Kazehaya finally cracks and confesses his love to Sawako. Sawako takes the confession as a misunderstanding and tells Kazehaya that if he says it that way, people will get the wrong idea. Kazehaya finally walks away, and Sawako follows. Kazehaya and Sawako skip class, and when Kazehaya asks her if she likes him, she says yes, but Kazehaya takes it in the way as just being friends and walks off, leaving Sawako alone in the alley. Pin finally finds Sawako, and after Sawako tells Pin that she has been rejected by Kazehaya, Pin tries to cheer her up. Ayane and Chizuru are also looking for Sawako and they have a feeling that Kento was a major part of it. They ask Kento where she is, and Kento says that she's outside by the benches. Kento tries to follow, but is interrupted by Kurumi, who makes Kento tell her what happened. Pin is still outside comforting Sawako, but is interrupted by Ayane and Chizuru. They ask what happened to Sawako, and before Ayane finished explaining something, Chizuru breaks out in tears and tells Sawako that she's being wrong, and Kazehaya is only angry because of the actions that Sawako took. Chizuru finally takes off, Ayane following her. The episode ends with Sawako asking herself if she had really done her best with Kazehaya up until this point.
#11 - Just Give Up on Her
The class is finally starting to prepare for the school festival, and everyone is wondering where Sawako is, in which Chizuru and Ayane say that she had somewhere else to be. Ryu finds Kazehaya outside as soon as he is done with baseball practice, and asks Kazehaya how his confession went. Kazehaya tells Ryu he was rejected, and that the rejection hurt a lot more than he thought it would, though he is unaware that Kurumi overheard the conversation, and she runs off to find Sawako. After confronting Sawako, Kurumi snaps at Sawako and tells her that she is hurting Kazehaya, and that since Kurumi actually told Kazehaya how she felt for him, that they were not on the same level. That night, Kazehaya and Pin run into each other, and Pin starts yelling at Kazehaya for rejecting Sawako. Kazehaya explains that he was rejected, and Pin tells Kazehaya to just give up on Sawako, and walks off, leaving Kazehaya to think for himself. The next day people are talking about the "battlefield" between Kazehaya, Sawako, and Kento, and after Sawako hears people talking about it, she says it was all a misunderstanding and that the meaning of "like" is different, not realizing Kazehaya is standing right there. Ayane and Chizuru take Kento away, as he had said very loudly that Kazehaya liked Sawako in front of everyone. Ayane gets mad at Kento, and starts yelling at him, when Kurumi interrupts. Kento asks Kurumi what she did after she learned about what happened between Kazehaya and Sawako, and Ayane and Chizuru start questioning her. They find out that Kurumi stopped Sawako from giving Kazehaya the Valentine's Day Chocolate and told her off. Kento also finds out that Kurumi likes Kazehaya, and asks her if it would make it easier for her if he went out with Sawako. Kurumi slaps Kento across the face and tells him not to get ahead of himself. Kurumi also says that she didn't think she would lose to such a weak girl, and runs off. After Kento finally gets everything straight through his hea
#12 - Friends
Sawako is cornered in the girls bathroom by the some girls from class A until Yoshida and Yano intervene. Kazehaya makes a public statement on how he hates ridiculous rumors and realises he can't be alone with Sawako now she's close to Yano and Yoshida.
#13 - Holiday
Season 1 - Episode 17 - Aired Feb 2, 2010
Ayane and Chizuru visit Sawako's house and, despite appearing disreputable at first, impress her parents by being "such nice girls". Meanwhile, Kazehaya and Ryuu are together when Kazehaya gets a call from Pin. Pin explains that he met Sawako the previous night, and now thinks he is possessed because he doesn't feel good. The truth is that Sawako noticed he was catching a cold, and gave him advice on how to prevent it, although he misunderstood. Pin insists that Kazehaya bring Sawako to him right away. Kazehaya, Ryuu, Sawako, Ayane and Chizuru all end up at Pin's filthy apartment and realize he only has a cold. They spend the rest of the day cleaning for him. They leave at night and Kazehaya ends up walking Sawako home. Thanks to a prank played by Pin, the two of them almost kissed under the stars.
#14 - Confession
Despite being nervous and embarrassed, Sawako goes to the classroom to confront Kazehaya. Before Sawako has a chance to tell Kazehaya how she feels, Kazehaya apologizes to her for making her cry and angry, and that he would listen to her no matter what, because his feelings wouldn't change. He then says that Chizuru and Ayane are probably outside, but Sawako interrupts and says that she came to talk to Kazehaya, not to Chizuru or Ayane. Sawako tells Kazehaya about people apologizing to her for everything she did, and that she was thankful that Kazehaya smiled and said "thank you". She thanks him for his smiles, kindness, and telling her his hidden feelings. Kazehaya is about to say something when Sawako interrupts with "I like you", and keeps on repeating it. Kazehaya pulls Sawako into the classroom by the hand and shuts the door. Meanwhile, Joe is looking outside for Kazehaya and Pin is looking for Sawako. When Chizuru and Ayane tell Pin she went home, Pin heads towards the classroom to look for Kazehaya. However, Kazehaya and Sawako are hugging to the side of the classroom door, unaware of anything else. They stare right at each other, Kazehaya realizing they both have the same feelings, and Sawako still being confused. After a minute Joe walks in to the classroom and Kazehaya and Sawako pull away from each other. Sawako runs off, and Kazehaya becomes embarrassed and upset. Later that night, Ayane and Chizuru ask how the talk went, and Sawako says she wasn't able to say 1/10 of her feelings to Kazehaya. Pin walks in and pulls Sawako outside, where Sawako breaks down and tells Pin she can't look at Kazehaya without secret feelings. Pin replies that everyone has a heart, so Kazehaya probably has the same feelings. He runs off, telling her not to worry about it and for her to do well in the solo performance. He's heading home when he realizes he said something "super wonderful" and runs off to find Kazehaya. Kazehaya is sending a text to someone when Pin barges in te
#15 - Romantic Feelings
Season 1 - Episode 12 - Aired Dec 22, 2009
Kurumi tells Sawako to talk to Ryuu and maybe she'll realize something. Sawako doesn't really understand that Kurumi wants her to separate from Kazehaya, but she says she will still talk to Ryuu. During the softball game Sawako can't keep her eyes off Kazehaya, which makes Kurumi angry as she thinks she is the only one always watching Kazehaya, so she leaves. Kurumi then starts her plan by planting a note asking Ryuu to meet Sawako at the sports equipment room, where Kazehaya has to go later. Sawako, is told by Kurumi that Ryuu is there, and goes to thank him. When they meet, Sawako asks Ryuu about the feeling of love, which leads him to confess that he likes Chizuru. In the meantime, Kazehaya heads to the room carrying some equipment. Kurumi goes up to him and tells Kazehaya that Sawako has fallen in love with Ryuu, urging him to support their relationship. They see Sawako and Ryuu talking happily together.
#16 - Chizuru's Love
Sawako is happy about her growing relationships with her friends and Kazehaya. Ayane, however, complains about her clingy boyfriend, while Chizuru is in a good mood because she anticipates something at New Year's, which is more than a month away. Chizuru and Sawako go out to pick a present for Ryuu's birthday, and Sawako is sure by the affection Chizuru shows for him that Chizuru loves Ryuu. However, she finds out that Chizuru actually loves Ryuu's older brother. The next time everyone is in school together, it is found out that Ayane dumped her boyfriend. Amid the talk about broken hearts, Ryuu leaves the room and is encouraged by Kazehaya. Later, at Chizuru's house, Ryuu comes into her room unexpectedly. He appears to be about to confess to Chizuru when she notices that she left his present lying out in plain sight. She therefore forces Ryuu to leave, and he ends up going home in the rain.
#17 - Rival
Season 1 - Episode 15 - Aired Jan 19, 2010
Kurumi reflects on how she met and fell in love with Kazehaya in middle school. In the meantime, rumors about how she was rejected by Pin spread around the school. She realizes that she hates being misunderstood by Kazehaya, so ends up confessing to him. He apologizes, saying he has someone else he loves. She plays it off as "his loss" and says that he has bad taste in women. Meanwhile, Sawako confesses her love for Kazehaya to Yano and Chizu. Later, Sawako and Kurumi meet and Sawako asks if they can be friends. Kurumi responds by saying she isn't Sawako's friend, but her rival. Later, during the after-festival party, Sawako wonders when she fell in love with Kazehaya and ends up falling asleep on his shoulder.
#18 - Love
Season 1 - Episode 13 - Aired Jan 5, 2010
After seeing Sawako and Ryuu together, Kazehaya rushes up to Sawako and pulls her away, leaving Kurumi behind. Kazehaya then confirms with Sawako that she doesn't like Ryuu, while Sawako has just realized that she really likes Kazehaya. Kazehaya asks her what she thinks about people going out as girlfriend and boyfriend, which shocks her so much that she cannot answer properly. Later, Ayane finds out that Kurumi was the one who spread the rumours. Ryuu then shows them the note written by Kurumi, which confirms that she set Sawako up. Ayane and Chizuru go to find more evidence against Kurumi, who is in the meantime waiting for Kazehaya to return. Pin arrives at the equipment room and comes to the conclusion that Kurumi has a crush on him, and rejects her just as Kazehaya arrives. Ayane and Chizuru confront Kurumi, who will not admit to wrongdoing, but when Sawako arrives, Ayane tells her outright that Kurumi was the one who spread the rumors.
#19 - Switching Seats
The class has a seat changing day, but everyone dreads drawing the seat next to Sawako. Until Kazahaya picks the seat himself and Chizuru and Yano take the seats in front of her. Kazahaya adopts a puppy that Sawako has encountered in the street.
#20 - Prologue
Season 1 - Episode 1 - Aired Oct 7, 2009
Being an outsider in her class, Sawako meets the popular Kazehaya, who she has admired since the first day of school. He encourages her to try open up and make new friends. Sawako then volunteers to play as the ghost during a class event, the test of courage. That night, Kazehaya comes to keep her company. Sawako is also becomses a little closer to two classmates, Yoshida Chizuru and Yano Ayane.
#21 - Determination
Season 1 - Episode 5 - Aired Nov 3, 2009
After Sawako hears the rumors she avoids Kazehaya, Yano and Yoshida. Kazehaya is upset about this and confronts Sawako, while Yoshida and Yano make her realise that they are already friends. Sawako stands up against some girls badmouthing Yoshida and Yano.
#22 - Special?
Sawako admits that she cannot help Kurumi confess to Kazehaya. Kurumi yells at Sawako, telling her she did nothing to get Kazehaya's attention while "cute and popular girls" like herself need to work a lot harder. Sawako begins considering whether she only likes Kazehaya because he is nice to her, and whether she would like any guy who did the same. The Sports Day competitions begin. The girls in Sawako's class win their first soccer match, and Sawako convinces Kurumi to watch Kazehaya play in the boys' match together. As they talk, Sawako mentions that she has always thought of Kazehaya as special. Kurumi tells her that this is just because she doesn't talk to many guys, so she ought to try talking to others. Just then, Ryuu (Sanada) saves Sawako from a rogue pitch. Kurumi suggests that Sawako should talk to Ryuu more.
#23 - Saturday Night
Sawako is invited to eat ramen with Yano and Yoshida at the ramen shop of Ryu's father. After they eat, they head up to Sanada's room. They later invite Kazehaya, but the mood is ruined when Pin shows up. Yano makes sure Sawako and Kazehaya walk home together under the stars.
#24 - Rumour
Yano and Yoshida become the talk of the school as everyone starts spreading rumors about them. An interrupted conversation leads them to believe that Sawako doesn't think they are friends. Sawako believes she's ruining Kazahaya's reputation by hanging out with him.
#25 - After School
Sawako meets Kazehaya after school to see Maru-chan, the puppy he has adopted. The day after Kazehaya has her helping the class with their homework. Yano and Yoshida overhear some false rumours about themselves and are told that it is Sawako who's spreading them.
Re:ZERO -Starting Life in Another World-

redpajama_set_name: RedPajamaCommonCrawl


As a vegetarian, I know that finding good veggie options while eating out can be challenging. Often the only available veggie items are the usual suspects such as a Portobello mushroom "burger", which I personally detest, or boring and flavorless options such as an iceberg lettuce salad or a veggie wrap made-up of mostly lettuce and without any sauce or spices – do vegetarians not deserve flavor?
Fortunately for us non-meat eaters, many downtown St. Pete restaurants have delicious and flavorful veggie items; you just have to know where to go. Here's a list of my top five favorite veggie dishes.
1. Veggie Burger from Tryst (240 Beach Drive). This house-made black bean burger is the best veggie burger that I have ever had (which is saying a lot considering that I've had a lot of veggie patties). The burger is crumbly in a good, home-made, and whole ingredients way and filled with spicy, but not hot flavors. And the accompanying French fries are so good; thin and not greasy. Just add a St. Germaine cocktail or two and the night is golden.
2. Vegetarian Platos from Bodega (1120 Central Ave). This delicious vegetarian plate includes mojo grilled tofu, beans, rice, and plantains. The tofu is spiced in all the right ways with a green sauce and is extremely flavorful. The portion is large (some might say "yuge!") and could be enough for two meals; however this dish is so good that I doubt you'll need to worry about finding a place for the leftovers in the fridge.
3. Tofu Noodle Soup from La V (441 Central Ave). This tasty veggie pho is made with a meat free broth (finding veggie pho with a veggie broth is a rarity), fresh and tasty vegetables, fried tofu, and rice noodles. The menu at La V is extremely veggie friendly with a section devoted to veggie items called "Dearest Vegetarian." My other favorites are the fresh tofu rolls, the tofu vermicelli bowl, and the tofu rice dish, and I can attest that all of these items are delicious.
4. Mixed Berries Salad from Cassis (170 Beach Drive NE). The berries included in this salad seem to vary with the season (usually includes a mix of blue berries, raspberries, and strawberries), but it always includes spinach, candied almonds, apple, and blue cheese. The dressing is delicious and light. I prefer the full portion because the salad is so good, but the half size is plenty for a lunch or a small meal. Especially with the bread that comes along with it.
5. Custom Vegetarian Sandwich from Lonni's Sandwiches (425 Central Ave). This extremely tasty veggie sandwich includes cheese (choice of swiss, cheddar, or cream cheese), cucumbers, sprouts, onion, lettuce, tomato, mayo, and black olives. It also comes with your choice of Italian or honey French dressing. The bread is fresh, thick, and delicious. I'm partial to this sandwich with cheddar cheese and Italian dressing. This sandwich is so good that I crave it. Lonni's also has a great Very Vegetarian Salad that I love as well.
Top Five Veggie Dishes in Downtown St Petersburg, November 2017 as written by a 24 year vegetarian.
Christina has lived in St. Petersburg for six and a half years and is originally from the Pacific Northwest. She is an applied anthropologist. She has been a vegetarian for 24 years. Christina is married and is the mother of a sweet three year old son who loves meat, especially hot dogs.


So everything is engulfed in flames.. Good thing is that new things keep popping up after the flames have been extinguished.
More houses = larger population?
And of course, more advanced technologies and more crafting recipes.
Good news is that I have already updated my crafting and resources page - check it out!
Apparently our ladies can now give birth to twins and even triplets!
Special thanks to Blythe Lim for sharing!
Ok now to actually playing the game!


text:
Q: Unable to connect nbd.name openwrt carambola2 I have always the same problem, when I try to build the openwrt image for my carambola 2 board:
Checking out files from the git repository...
Cloning into 'ubus-2015-01-22'...
fatal: unable to connect to nbd.name:
nbd.name[0: 46.4.11.11]: errno=Connection refused

A ping on this ip adress does work, also the git web is up. But it seems that there is a problem with the git protocol.
Is there somewhere a mirror of this repository?

git://nbd.name/luci2/ubus.git

A: The old source is abandon now, you can get the package from
 http://git.openwrt.org/project/ubus.git