% Document Type: Latex
% Master File: ground.tex

\newpage
\appendix
\section{Hydraulic Conductivity} \label{append:hydr}
The Hydraulic Conductivity is the factor of proportionality in Darcy's flow equation.  It is a function of both the fluid and the porous medium through which the fluid is flowing. It can be expressed as:
\[K = \frac{kg}{v},
\]where
\begin{tabbing}
$K$ \hspace{.4in} \= = hydraulic conductivity [LT$^{-1}$],\\ 
$k$ \>= intrinsic permeability [L$^2$],\\
$g$ \>= gravitational constant [LT$^{-2}$], and\\ 
$v$ \>= kinematic viscosity of the fluid [L$^2$T$^{-1}$].\\ 
\end{tabbing}
The intrinsic permeability or what is commonly referred to now as the permeability is independent of the fluid and depends only on the porous medium.
The permeability property is:
\[k = cd^2,\]
where
\begin{tabbing}
$k$ \hspace{.4in} \= = intrinsic permeability [L$^2$],\\ 
$c$ \>= \= constant. (This constant is dimensionless and is controlled by\\
\> \> factors other than the diameter of the grains in the porous medium, \\
\> \> such as packing, distribution and shape.) \\ 
$d$ \>= mean grain diameter [L].\\ 
\end{tabbing}

$k$ has a high value for high porosity mediums (sands, gravels)
and a low value for low porosity mediums (silts, clay).

\newpage
%***********************************************************************
\section{Reynolds Number} \label{append:Rey}
Reynolds number is the indicator of flow mode
\begin{equation}
Re = \frac{inertial forces}{viscous forces} = \frac{\nu d}{v},
\label{eq:RN}
\end{equation}
where
\begin{tabbing}
$Re$ \hspace{.4in} \= = Reynolds number,\\ 
$\nu$ \>= Darcian flow velocity,\\
$d$ \>= characteristic length or in porous media, the average grain diameter, and\\ 
$v$ \>= kinematic viscosity.
\end{tabbing}

Reynolds number is the ``critical velocity'' that defines the transition between laminar and turbulent flow.  It is a dimensionless ratio that is the indicator of the flow mode. Equation~\ref{eq:RN} is the ratio action on the inertial forces  on the fluid per unit length of flow to the viscous force acting on the same fluid.
\subsection{Bernoulli Formula}
The Darcian flow velocity in Equation~\ref{eq:RN} relates the velocity of the flowing water with the hydraulic head
\[\nu = \sqrt{2gh},\]
where
\begin{tabbing}
$\nu$ \hspace{.4in} \= = flow velocity [LT$^{-1}$],\\ 
$h$ \>= velocity head [L], and\\
$g$ \>= gravitational constant [LT$^{-2}].$\\ 
\end{tabbing}
\newpage
%***********************************************************************
\section{Darcian Velocities} \label{append:Darcian}
Darcy's Law can be written in a differential form where the velocity of the flow in the $s$ direction is:
\begin{equation}
v_s = \frac{- \kappa \partial (\frac{\rho}{\gamma} + z)}{\partial s}.
\label{eq:Dar1}
\end{equation}
Differentiating the head:
\[h(x,y,t) = \frac{\rho}{\gamma} + z, \]
with respect to $s$, we obtain:
\begin{equation}
\frac{\partial h}{\partial s} = \frac{\partial(\frac{\rho}{\gamma} + z)}{\partial s}.
\label{eq:Dar2}
\end{equation}
Substituting Equation~\ref{eq:Dar2} into Equation~\ref{eq:Dar1}, we get:
\begin{equation}
v_s = - \kappa \frac{\partial h}{\partial s}, 
\end{equation}
where
\begin{tabbing}
$\kappa$ \hspace{.4in} \= = hydraulic conductivity [LT$^{-1}$],\\ 
$\frac{\rho}{\gamma}$ \>= hydrostatic pressure potential [L], and \\
$z$ \>= gravitational potential [L].\\ 
\end{tabbing}
The negative sign indicates that the flow moves in the direction of decreasing head. $v_s$ defines the flow rate in any direction through a porous medium is proportional to the negative rate of change of head in that direction.

\newpage
%************************************************************************
\section{Transformation into Polar Coordinates} \label{append:polar}
To transform Equation~\ref{eq:gov}:
\[ \frac{\partial ^2h}{\partial x^2} +\frac{\partial ^2h}{\partial y^2}= \frac{S_s}{\kappa} \frac{\partial h}{\partial t} = \frac{S}{T} \frac{\partial h}{\partial t} \]
into polar coordinates, use the facts that :
\begin{enumerate}
\item Polar coordinates $P(r, \theta)$ have the identity: \[r = \sqrt{x^2 + y^2}.\]
\item The chain rule for \[ \frac{\partial s}{\partial r} = \frac{\partial s}{\partial r} \frac{\partial r}{\partial x}.\]
\end{enumerate}

For the $x$ direction:
\[ \frac{\partial r}{\partial x} = \frac{\partial x}{\partial \sqrt{x^2 + y^2}} = \frac{x}{r}\]
\[ \frac{\partial^2 r}{\partial x^2} = \frac{r - \frac{\partial r}{\partial x} x}{r^2} = \frac{r - \frac{x}{r} x}{r^2} = \frac{r^2 - x^2}{r^3}.\]

Likewise in the $y$ direction:
\[ \frac{\partial r}{\partial y} = \frac{\partial y}{\partial \sqrt{x^2 + y^2}} = \frac{y}{r}\]
\[ \frac{\partial^2 r}{\partial y^2} = \frac{r - \frac{\partial r}{\partial y} x}{r^2} = \frac{r - \frac{y}{r} x}{r^2} = \frac{r^2 - y^2}{r^3}.\]

Thus the Chain Rule gives us:
\[ \frac{\partial h}{\partial x} = \frac{\partial s}{\partial r}\frac{\partial r}{\partial x} = \frac{\partial s}{\partial r} \frac{x}{r}\]

\[ \frac{\partial^2 h}{\partial x^2} = \frac{\partial^2 s}{\partial r^2} \left( \frac{\partial r}{\partial x} \right)^2 + \frac{\partial s}{\partial r} \frac{\partial^2 r}{\partial x^2}\]

\[ = \frac{\partial^2 s}{\partial r^2} \left( \frac{x}{r} \right)^2 + \frac{\partial s}{\partial r}  \left(\frac{r^2 - x^2}{r^3}\right). \]

Similiarily:
\[ \frac{\partial^2 h}{\partial y^2} \frac{\partial^2 s}{\partial r^2}  \left( \frac{y}{r} \right)^2 + \frac{\partial s}{\partial r}  \left(\frac{r^2 - y^2}{r^3}\right) \]
Thus:
\[ \frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} = \frac{\partial^2 s}{\partial r^2}  \frac{x^2 + y^2}{r^2} + \frac{\partial s}{\partial r} \frac{ 2r^2 - (x^2 + y^2)}{r^3}\]
\[ = \frac{\partial^2 s}{\partial r^2} + \frac{\partial s}{\partial r} \left(\frac{r^2}{r^3}\right) \]
\[ = \frac{\partial^2s}{\partial r^2} + \frac{1}{r} \frac{\partial s}{\partial r}.\]
This is the right hand side of Equation~\ref{eq:gov}.
\newpage
%************************************************************************
\section{Plot} 
\label{append:plot}
This is a simple program written in Matlab that plots the cone of depression as a function of the drawdown versus the values of $r = .5$ to $r = 200$ meters at the times: $t = 1 $ hour; $t = 1 $ day and $t = 30 $ days.\vspace{.25in}
{\bf Note: You must use the units indicated.}\\
See Appendix~\ref{append:conv} to convert to the correct  units.\vspace{.2in}\\
\underline{Code} \vspace{.1in}\\
\setlength{\baselineskip}{12pt}
\begin{verbatim}
global Q S T t r

Q = input('Enter the Discharge Rate Q (cubic meters per day) = ');


S = input('Enter the Storativity S (dimensionless) = ');


T = input('Enter the Transmissivity T (square meters per day) = ');

r=[.5:2:200];
r1=100;

%At time t = 1 hour
t=1;
draw1=draw(r,Q,S,T,t);
time1=draw(r1,Q,S,T,t)+.5;

%At time t = 1 day
t=24;
draw2=draw(r,Q,S,T,t);
time2=draw(r1,Q,S,T,t)+.5;

%At time t = 30 days
t=720;
draw3=draw(r,Q,S,T,t);
time3=draw(r1,Q,S,T,t)+.5;

%Draw the plot for t = 1 hour; t = 1 day; and t = 30 days
plot(r,-draw1,'y-',r,-draw2,'m--',r,-draw3,'g-.');
title('Cone of Depression');
xlabel('Radial Distance (in meters)');
ylabel('Drawdown');
text(100,-time1,'Time = 1 Hour')
text(100,-time2,'Time = 1 Day')
text(100,-time3,'Time = 30 Days')
\end{verbatim}
\newpage
\begin{verbatim}

function d=draw(r,Q,S,T,t)
%
%  function d = draw(r, Q, S, T, t)
%
%  This is C. V. Theis's drawdown function
%
%  It is a function of the radial distance, r, and the time, t, since the 
%  pumping began. r may be a vector, but t must be a scalar.
%
%  The parameters are:
%     discharge rate, Q, [L^3/T]
%     storativity, S, [dimensionless]
%     transmissivity, T [L^2/T].

u = (S/(4*T*t))*r.*r;
d=(Q/(4*pi*T)).*well(u);


function w=well(u)
%
%  function w= well(u)
%
%  Theis's well function
%

Econst = 0.577215664901532860606512;

n = 10;
term = 1;
for k = n:-1:2  
   term =  1 + ((1-k)/k^2)*u.*term;
end
w = -Econst - log(u) + u.*term;
\end{verbatim}
\setlength{\baselineskip}{24pt}
\newpage
\subsection{Example of a Plot}
Figure~\ref{exam} is a specific example of the plot of the cone of depression where
\begin{tabbing}
$Q$ \hspace{.4in} \= = 1600 $\frac{\mbox{m}^3}{\mbox{day}}$\\
$S$ \>= .0004\\
$T$ \>= 208.8 $\frac{\mbox{m}^2}{\mbox{day}}$
\end{tabbing}
\begin{figure}[hbt]
\centerline{\psfig{figure=exam.ps,height=6.5in}}
\caption{Specific Example of a Cone of Depression}
\label{exam}
\end{figure}

\newpage
%************************************************************************
\section{Conversion Table for Units} \label{append:conv}
\begin{table}[hb]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Unit & Multiply & By & To Obtain\\ \hline\hline
Length & ft & 3.048 $\times 10^{-1}$ & m\\ \hline
& ft & 3.048 $\times 10$ & cm\\ \hline
& ft & 3.048 $\times 10^{-4}$ & km\\ \hline
& mile & 1.609 $\times 10^{3}$ & m\\ \hline
& mile & 1.609 & km\\ \hline \hline
Area & ft$^2$ & 9.290 $\times 10^{-2}$ & m$^2$\\ \hline
& mi$^2$ & 2.590 & km$^2$\\ \hline
& acre & 4.047 $\times 10^3$ & m$^2$\\ \hline
& acre & 4.047 $\times 10^{-3}$ & km$^2$\\ \hline \hline
Volume & ft$^3$ & 2.832 $\times 10^{-2}$ & m$^3$\\ \hline
& U.S. gal & 3.785 $\times 10^{-3}$ & m$^3$\\ \hline
& U.K. gal & 4.546 $\times 10^{-3}$ & m$^3$\\ \hline
& ft$^3$ & 2.832 $\times 10$ & liter\\ \hline 
& U.S. gal & 3.785 & liter\\ \hline
& U.K. gal & 4.546 & liter\\ \hline \hline
Velocity & ft/s &  3.048 $\times 10^{-1}$ & m/s\\ \hline
& ft/s &  3.048 $\times 10$ & cm/s\\ \hline
& mi/h &  4.470 $\times 10^{-1}$ & m/s\\ \hline
& mi/h & 1.609& km/h\\ \hline \hline
Discharge & ft$^3$/s &  2.832$\times 10^{-2}$ & m$^3$/s\\ \hline
& ft$^3$/s &  2.832$\times 10$ & liter/s\\ \hline
& U.S. gal/min & 6.309 $\times 10^{-5}$ & m$^3$/s\\ \hline
& U.K. gal/min & 7.576 $\times 10^{-5}$ & m$^3$/s\\ \hline
& U.S. gal/min & 6.309 $\times 10^{-2}$ & liter/s\\ \hline
& U.K. gal/min & 7.576 $\times 10^{-2}$ & liter/s\\ \hline \hline
Hydraulic Conductivity & ft/s & 3.048 $\times 10^{-1}$ & m/s\\ \hline
& U.S. gal/day/ft$^2$ & 4.720 $\times 10^{-7}$ &m/s\\ \hline \hline
Transmissivity & ft$^2$/s & 9.290 $\times 10^{-2}$ & m$^2$/s\\ \hline
& U.S. gal/day/ft & 1.438 $\times 10^{-7}$ & m$^2$/s \\\hline
\end{tabular}
\end{center}
\caption{\label{conv} \em Conversion Table}
\end{table}

\newpage
%************************************************************************
\section{Glossary}
\begin{itemize}

\item   \underline{Active Porosity} - percentage of water that drains by gravity or "specific yield."

\item \underline{Anistrophic} - a formation is anistropic at a point if the hydraulic conductivity K varies with the direction of the measurement.~\cite{cherry}

\item  \underline{Aquifer} - formation or group of saturated geologic formations capable of storing and yielding freshwater in usuable quantities, formed from rocks or soils or both.
  
\item   \underline{Aquifer Conditions} - 
    \begin{itemize}
        \item   \underline{Unconfined}  
            \begin{itemize}
               \item receive recharge directly from the overlying surface
               \item shallow water level
               \item water at atmospheric pressure at water table
               \item other names- water table, ordinary or gravitational
            \end{itemize}
        \item   \underline{Confined}
            \begin{itemize}
               \item bounded below and above by aquitards
               \item does not receive significant amounts of percolation from the overlying surface
               \item under a pressure that includes the sum of the weight of the atmosphere and the overburden
               \item unconfined at their exposed edges
               \item other names - artesian or potentiometric aquifers
            \end{itemize}
    \end{itemize}
\item \underline{Aquitard} - semi-confining beds that allow a very slow flow of water through them. The total volume of water might be large though if the aquitard itself is large.
\newpage
\item \underline{Capillary Fringe} - lower boundary of the vadose zone where the soil pores are completely filled with water. \\
\underline{Note:} Coarse grained mediums with large pore spaces and a low ratio of surface area to volume have little or no capillary fringe, while fine particle mediums with small pore spaces and  a large surface area to volume ratio may have capillary fringes of 50 feet or more.
 
\item \underline{Cone of Depression} - potentiometric or gravity free pressure surface that extends radially outward from the well and is a function of storativity, transmissivity, the pumping rate of the well and time. 

\item \underline{Confining Beds} - beds that do not allow the easy penetration of water across them.

\item \underline{Continuity Equation} - is given as the mass of fluid flowing into a given volume is equal to the mass of fluid flowing out of the volume.

\item \underline{Density} $\rho$ - of a fluid is defined as its mass per unit volume.

\item \underline{Drawdown} - value of the groundwater potential below an arbitrary datum.

\item \underline{Effective Porosity} - a portion of the total porosity or interconnected void space that is available for groundwater flow.

\item \underline{Evapotranspiration} - the processes of evaporation and transpiration.

\item \underline{Groundwater} - subsurface water that occurs beneath the water table in soils and geological formations that are fully saturated.
 
\item \underline{Hydraulic Conductivity} - constant of proportionality in Darcy' Law which is a function of the medium through which the fluid flows as well as the properties of the fluid.
\newpage 
\item \underline{Hydraulic Diffusivity(D)} - defined as 
\[ \frac{T}{S} = \frac{Transmissivity}{Storativity}\]
or
\[ \frac{\em K}{\em S_s} = \frac{\em Hydraulic Conductivity}{ \em Specific Storage}\] 

\item \underline{Hydraulic Gradient} - change in hydraulic head with respect to the change in the length of the path.
 
\item \underline{Hydraulic Head} - potential water levels.

\item \underline{Hydrologic Cycle} - the continuous circulation of water between the oceans, the atmosphere and the land.

\item \underline{Infiltration Capacity} - Maximum rate at which a soil can absorb water. The capacity is a limiting curve that describes the maximum possible rates of infiltration as a function of time.

\item \underline{Isotropic} - a formation is isotropic at a point if the hydraulic conductivity $K$ does not vary with the direction of the measurement.

\item \underline{Laminar flow} - on a macroscopic view, a very smooth or orderly flow

\item \underline{Leakage} - vertical seepage across the boundaries which is determined by calculation of Darcy's Law.

\item \underline{Manometers} - a tube or pipe used to determine the elevation of a water level. In the field this device is called a piezometer.

\item \underline{Phreatic Surface} - the water table surface or the level at which water stands. Also the upper boundary of an unconfined aquifer.

\item \underline{Phreatic Zone} - zone of saturation.

\item \underline{Piezometer} - a tube or pipe used to determine the elevation of a water level. In the laboratory this device is called a manometer.

\item \underline{Piezometric Surface} -  the surface created in a contour plot of the different elevations of water levels sampled in the investigation of a well site.

\item \underline{Porosity} - is simply the volume of the voids between the grains in the medium.
    
    \begin{itemize}
      \item \underline{Active porosity} - the specific yeild or the percentage of water that drains by the force of gravity.
      
      \item \underline{Effective Porosity} - is the portion of ithe total interconnected pore space through which water flows.  Not all interconnected pore space allows the flow of water.
    \end{itemize}
    \begin{itemize}
    \item \underline{3 Major types}
        \begin{enumerate}
        \item -  spaces between the grains which is \underline{primary porosity} or the void spaces formed as the medium is deposited
        \item - fractures or \underline{secondary porosity} void spaces created through structural deformation and erosion
        \item - tubes through which the water flows
        \end{enumerate}
     \end{itemize}

\item \underline{Pores} - the void spaces between the grains of a porous medium.

\item \underline{Porous Medium} - solid material with interconnecting void spaces.

\item \underline{Storativity} - addition or release of water to the storage space due to the increase or decrease of hydraulic head. Storativity is a function of the thickness of the saturated aquifer and the Hydraulic Conductivity.
\newpage
\item \underline{Streamline} - path followed by a sequence of water particles.
    \begin{itemize}
    \item \underline{Laminar flow} - on a macroscopic view, a very smooth or orderly flow.
    \item \underline{Turbulent flow} - irregular or rotational flow paths such as those found in rivers and streams.
    \end{itemize} 
  
\item \underline{Surface Retention} - process where electrical forces bind water molecules to grain surfaces in the vadose zone and the capillary fringe where pressures are less than atmospheric.

\item \underline{Transmissivity} - a function of the hydraulic conductivity and the thickness of the aquifer and descibes how easily the aquifer moves groundwater through its pore spaces.
    
\item \underline{Turbulent flow} - irregular or rotational flow paths such as those found in rivers and streams.

\item \underline{Vadose Zone} - first layer under the ground level and above the water table where the soil pores are only partially filled with water and the fluid pressure is less than atmospheric.
 
\item \underline{Viscosity} - property that allow fluids to resist relative motion and shear deformation during flow.\cite{cherry}
 
\item \underline{Watertable} - phreatic surface - upper boundary of an unconfined aquifer.

\end{itemize}
