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\title[] 
{A Porous Medium Model}
 
\author{Eric Stachura \\{\small eric.stachura@kennesaw.edu}}
\institute{Department of Mathematics \\ Kennesaw State University \\ Marietta, GA 30060 USA}
 
%\author[] 
%{Author}
 
\date{\today}




\begin{document}

\maketitle


\begin{frame} \frametitle{Problem}
We will derive a model for the flow of water under gravity through a homogeneous, isotropic, porous medium. 
\end{frame}

\begin{frame} {Definitions}
\begin{enumerate}
\item \underline{Homogeneous}: The material looks the same at every location. It has uniform properties throughout; no clumps of other material. 
\vfill
\item \underline{Isotropic}: The material looks the same in all directions. That is, the material properties do not depend on the direction.\\
\end{enumerate}

\end{frame}

\begin{frame} {Porous Medium}
 Intuitively, such a medium has ``pores", or voids, which are typically filled with a fluid. Examples include rocks, soil, biological tissues, etc. You can imagine something as in the following figure:

\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.2]{pore_scale_flow_structure}
\caption{A porous medium.}
\end{center}
\end{figure}

\end{frame}

\begin{frame} {The Model}
Let us define:
\begin{itemize}
\item $u(x,t)$ to be the volume of water per unit volume of the porous media; \\
\item $q(x,t)$ to be the volume of water flowing across a unit area per unit time\\ \ps
\end{itemize}

If the density of water is assumed to be constant, which we will assume, then the \emph{moisture content} $u(x,t)$ and the \emph{seepage velocity} $q(x,t)$ of the water are governed by the continuity equation
\begin{align} \label{continuityequation}
\dfrac{\partial u}{\partial t}+\dfrac{\partial q}{\partial x} = 0
\end{align}


\end{frame}

\begin{frame} {The Model}
as well as Darcy's Law
\begin{align} \label{DarcyLaw}
q(x,t)=-K(u) \dfrac{\partial G}{\partial x}
\end{align}
where $G$ is some potential function and $K(u)$ is called the \emph{hydraulic conductivity}. \ps %For more details, see \cite{porousmedia}. 

\vfill

If chemical and thermal effects are ignored, then for unsaturated flow, $G$ can be expressed as a sum of a gravitational potential and a potential due to capillary suction:
$$G=H(u) +x$$
where $H(u)$ is called a \emph{hydrostatic potential}. 
\end{frame}

\begin{frame} {The Model}
Combining all of this we obtain the equation
\begin{align} \label{eq1}
\dfrac{\partial u}{\partial t} =\dfrac{\partial}{\partial x} \left( K(u) \dfrac{\partial H}{\partial x} \right) + \dfrac{\partial}{\partial x} K(u)
\end{align} \\

Empirical expressions are known for $D(u) \coloneqq K(u) \dfrac{dH}{du}$, and in particular, we have $D(u)=D_0 u^{m-1}$ and $K(u)=K_0u^n$, for $D_0, K_0, m, n$ positive constants. \ps Substituting these values into (\ref{eq1}) and rescaling, we obtain the main model:
\begin{align} \label{mainPDEmodel}
\dfrac{\partial u}{\partial t} = \dfrac{\partial u^n}{\partial x} + \dfrac{\partial^2 u^m}{\partial x^2}, \quad n, m \geq 1, \; x \in \mathbb{R}, \; t>0
\end{align}

\end{frame}

\begin{frame}{Comments on the Model Derivation}
\begin{itemize}
\item Darcy's Law can be derived from basic principles for porous media, but the derivation is quite involved. If interested, see\\
S. Whitaker, \textit{Flow in Porous Media I: A theoretical derivation of Darcy's Law}. Transport in Porous Media, 1 (1986), 3-25. \ps
\item The continuity equation is derived by considering a unit volume of the medium and equating the rate of increase of the fluid mass within that volume to the net mass flux into the volume. 
\end{itemize}


\end{frame}


\begin{frame} {Movie}
Let's watch a simulation of this type of flow!

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\end{frame}



\begin{frame} {Your task}
Your task is to analyze the model (\ref{mainPDEmodel}) by developing analytical solutions. \ps

\vfill

You will consider the case when $n=3, m=2$. A typical approach to solving such problems is by seeking \emph{traveling wave solutions}; that is, seeking solutions of the form
$$u(x,t) = u(x-ct)$$
which represents a traveling wave with speed $c$. 
\end{frame}


 \end{document}