Commits

Richard Mills committed 3f5b381 Merge

Automerge.

Comments (0)

Files changed (2)

 *.stdout
 *.regression
 regression_tests/default/anisothermal/thc_1d.h5
+*.nav
+*.snm
+*.vrb
+shortcourse/exercises/CLM-CN/doc/CLM-CN.pdf

shortcourse/exercises/CLM-CN/doc/CLM-CN.tex

-\documentclass{beamer}
-
-\usepackage{comment}
-\usepackage{color}
-\usepackage{listings}
-\usepackage{verbatim}
-\usepackage{multicol}
-\usepackage{booktabs}
-\definecolor{green}{RGB}{0,128,0}
-
-\def\EQ#1\EN{\begin{equation*}#1\end{equation*}}
-\def\BA#1\EA{\begin{align*}#1\end{align*}}
-\def\BS#1\ES{\begin{split*}#1\end{split*}}
-\newcommand{\bc}{\begin{center}}
-\newcommand{\ec}{\end{center}}
-\newcommand{\eq}{\ =\ }
-\newcommand{\degc}{$^\circ$C}
-
-\def\p{\partial}
-\def\qbs{\boldsymbol{q}}
-\def\Dbs{\boldsymbol{D}}
-\def\A{\mathcal A}
-\def\gC{\mathcal C}
-\def\gD{\mathcal D}
-\def\gL{\mathcal L}
-\def\M{\mathcal M}
-\def\P{\mathcal P}
-\def\Q{\mathcal Q}
-\def\gR{\mathcal R}
-\def\gS{\mathcal S}
-\def\X{\mathcal X}
-\def\bnabla{\boldsymbol{\nabla}}
-\def\bnu{\boldsymbol{\nu}}
-\renewcommand{\a}{{\alpha}}
-%\renewcommand{\a}{{}}
-\newcommand{\s}{{\sigma}}
-\newcommand{\bq}{\boldsymbol{q}}
-\newcommand{\bz}{\boldsymbol{z}}
-\def\bPsi{\boldsymbol{\Psi}}
-
-\def\Li{\textit{L}}
-\def\Fb{\textbf{f}}
-\def\Jb{\textbf{J}}
-\def\cb{\textbf{c}}
-
-\def\Dt{\Delta t}
-\def\tpdt{{t + \Delta t}}
-\def\bpsi{\boldsymbol{\psi}}
-\def\dbpsi{\delta \boldsymbol{\psi}}
-\def\bc{\textbf{c}}
-\def\bx{\textbf{x}}
-\def\dbc{\delta \textbf{c}}
-\def\dbx{\delta \textbf{x}}
-\def\arrows{\rightleftharpoons}
-
-\newcommand{\bGamma}{\boldsymbol{\Gamma}}
-\newcommand{\bOmega}{\boldsymbol{\Omega}}
-%\newcommand{\bPsi}{\boldsymbol{\Psi}}
-%\newcommand{\bpsi}{\boldsymbol{\psi}}
-\newcommand{\bO}{\boldsymbol{O}}
-%\newcommand{\bnu}{\boldsymbol{\nu}}
-\newcommand{\bdS}{\boldsymbol{dS}}
-\newcommand{\bg}{\boldsymbol{g}}
-\newcommand{\bk}{\boldsymbol{k}}
-%\newcommand{\bq}{\boldsymbol{q}}
-\newcommand{\br}{\boldsymbol{r}}
-\newcommand{\bR}{\boldsymbol{R}}
-\newcommand{\bS}{\boldsymbol{S}}
-\newcommand{\bu}{\boldsymbol{u}}
-\newcommand{\bv}{\boldsymbol{v}}
-%\newcommand{\bz}{\boldsymbol{z}}
-\newcommand{\pressure}{P}
-
-\newcommand\gehcomment[1]{{{\color{orange} #1}}}
-\newcommand\add[1]{{{\color{blue} #1}}}
-\newcommand\remove[1]{\sout{{\color{red} #1}}}
-\newcommand\codecomment[1]{{{\color{green} #1}}}
-\newcommand\redcomment[1]{{{\color{red} #1}}}
-\newcommand\bluecomment[1]{{{\color{blue} #1}}}
-\newcommand\greencomment[1]{{{\color{green} #1}}}
-\newcommand\magentacomment[1]{{{\color{magenta} #1}}}
-
-\begin{comment}
-\tiny
-\scriptsize
-\footnotesize
-\small
-\normalsize
-\large
-\Large
-\LARGE
-\huge
-\Huge
-\end{comment}
-
-\begin{document}
-\title{CLM-CN Reaction\ldots in a Nutshell}
-\author{Glenn Hammond}
-\date{\today}
-
-%\frame{\titlepage}
-
-%-----------------------------------------------------------------------------
-\section{Description of CLM-CN Reaction}
-
-\subsection{Batch CLM-CN Reaction Conceptual Model}
-
-\frame{\frametitle{Schematic of CLM-CN Reaction Network}
-\includegraphics[width=\linewidth]{./CLM-CN_cycle}
-}
-
-%-----------------------------------------------------------------------------
-\subsection{Governing Equations}
-
-\frame{\frametitle{Reaction Expression}
-
-\Large
-
-\EQ\label{CN_rxn}
-\text{CN}_u \eq \left(1-f\right) \text{CN}_d + f \text{CO}_2 + n \text{N}_\text{mineral}
-\EN
-
-\EQ\label{n_calc}
-n \eq u - \left(1-f\right) d
-\EN
-
-\footnotesize
-\BA
-\text{CN}_u &\eq \text{upstream carbon pool } [\text{mol C}/m^3] \\
-\text{CN}_d &\eq \text{downstream carbon pool } [\text{mol C}/m^3] \\
-\text{CO}_2 &\eq \text{nitrogen concentration } [\text{mol CO}_2/m^3] \\
-\text{N}_\text{mineral} &\eq \text{nitrogen concentration } [\text{mol N}/m^3] \\
-u &\eq \text{\text{C}/\text{N} atomic weight ratio divided by upstream \text{C}/\text{N} ratio} \\
-d &\eq \text{\text{C}/\text{N} atomic weight ratio divided by downstream \text{C}/\text{N} ratio} \\
-f &\eq \text{respiration fraction} \\
-\EA
-}
-
-\frame{\frametitle{Kinetic Rate Expression}
-
-\Large
-
-\EQ\label{CN_kinetic_rxn}
-rate \eq f_T f_\theta f_\text{pi} k \text{CN}_u
-\EN
-
-%\bigskip
-%\normalsize
-%\footnotesize
-\scriptsize
-\BA
-f_T &\eq \exp\left[308.56 \left(\frac{1}{71.02}-\frac{1}{T - 227.13}\right)\right]\\
-f_\theta &\eq \frac{\log\left(\theta_\text{min}/\theta\right)}{\log\left(\theta_\text{min}/\theta_\text{max}\right)}\\
-f_\text{pi} &\eq \frac{\text{N}_\text{mineral}}{\text{N}_\text{mineral} + K_{\text{N}_\text{mineral}}} \\
-k &\eq \text{kinetic rate constant} [s^{-1}]\\
-T &\eq \text{temperature } [K] \\
-\theta &\eq \text{moisture content or saturation } [-] \\
-\text{CN}_u &\eq \text{upstream carbon pool } [\text{mol C}/m^3] \\
-\text{N}_\text{mineral} &\eq \text{nitrogen concentration } [\text{mol N}/m^3] \\
-K_\text{N} &\eq \text{nitrogen half saturation constant } [\text{mol N}/m^3]\\
-\EA
-
-}
-
-\frame{\frametitle{Kinetic Reaction Equations}
-
-
-\EQ
-\text{CN}_u \eq \left(1-f\right) \text{CN}_d + f \text{CO}_2 + n \text{N}_\text{mineral}
-\EN
-
-\EQ
-rate \eq f_T f_\theta f_\text{pi} k \text{CN}_u
-\EN
-
-\Large
-\BA
-\frac{\p}{\p t} \left(\text{CN}_u\right) &\eq -rate \\
-\frac{\p}{\p t} \left(\text{CN}_d\right) &\eq \left(1-f\right) rate \\
-\frac{\p}{\p t} \left(\text{CO}_2\right) &\eq f \cdot rate \\
-\frac{\p}{\p t} \left(\text{N}_\text{mineral}\right) &\eq n \cdot rate \\
-\EA
-
-}
-
-
-\frame{\frametitle{Newton-Raphson Method}
-\LARGE
-\BA
-\Fb\left(\bx^{k+1,i}\right) &\eq \frac{\bx^{k+1,i}-\bx^k}{\Dt} - \sum_{irxn} \bnu_{irxn} \cdot rate_{irxn} \\
-\\
-\Jb \dbx &\eq -\Fb\left(\bx^{k+1,i}\right) \\
-\\
-\bx^{k+1,i+1} &\eq \bx^{k+1,i} + \dbx
-\EA
-\small
-\BA
-k &\eq \text{time step} \\
-i &\eq \text{iteration}
-\EA
-}
-
-%-----------------------------------------------------------------------------
-\subsection{PFLOTRAN Input Specification}
-
-\begin{frame}[fragile,allowframebreaks]\frametitle{CHEMISTRY}
-\small
-\begin{semiverbatim}
-CHEMISTRY
-  ...
-  IMMOBILE_SPECIES
-    N
-    C
-    SOM1
-    SOM2
-    SOM3
-    SOM4
-    LabileC
-    CelluloseC
-    LigninC
-    LabileN
-    CelluloseN
-    LigninN
-  /
-  ...
-\end{semiverbatim}
-\newpage
-\begin{semiverbatim}
-
-  ...
-  REACTION_SANDBOX
-    CLM-CN
-      POOLS   \bluecomment{! CN ratio}
-        SOM1   12.d0   \bluecomment{! -> u or d}
-        SOM2   12.d0
-        SOM3   10.d0
-        SOM4   10.d0
-        Labile
-        Cellulose
-        Lignin
-      /
-      ...
-\end{semiverbatim}
-\newpage
-\begin{semiverbatim}
-      ...
-      REACTION
-        UPSTREAM_POOL Labile         \bluecomment{! CN_u}
-        DOWNSTREAM_POOL SOM1         \bluecomment{! CN_d}
-        TURNOVER_TIME 20. h          \bluecomment{! -> k}
-        RESPIRATION_FRACTION 0.39d0  \bluecomment{! f}
-        N_INHIBITION 1.d-10          \bluecomment{! K_N}
-      /
-      REACTION
-        UPSTREAM_POOL SOM1
-        DOWNSTREAM_POOL SOM2
-        TURNOVER_TIME 14. d
-        RESPIRATION_FRACTION 0.28d0
-        N_INHIBITION 1.d-10
-      /
-      ...
-    / \bluecomment{! END CLM-CN}
-  / \bluecomment{! END REACTION_SANDBOX}
-/ \bluecomment{! END CHEMISTRY}
-\end{semiverbatim}
-\end{frame}
-
-\begin{frame}[fragile]\frametitle{CONSTRAINT}
-%\small
-\footnotesize
-\begin{semiverbatim}
-CONSTRAINT initial
-  CONCENTRATIONS   \bluecomment{! moles/L}
-    A(aq)  1.d-40  T
-  /
-  IMMOBILE       \bluecomment{! moles/m^3}
-    N     1.d-6
-    C     1.d-6
-    SOM1  1.d-10 \bluecomment{! moles C/m^3}
-    SOM2  1.d-10
-    SOM3  1.d-10
-    SOM4  1.d-10
-    LabileC     0.1852d-3
-    CelluloseC  0.4578d-3
-    LigninC     0.2662d-3
-    LabileN     0.00508954d-3
-    CelluloseN  0.01258096d-3
-    LigninN     0.00731553d-3
-  /
-END
-\end{semiverbatim}
-\end{frame}
-
-\end{document}
+\documentclass{beamer}
+
+\usepackage{comment}
+\usepackage{color}
+\usepackage{listings}
+\usepackage{verbatim}
+\usepackage{multicol}
+\usepackage{booktabs}
+\definecolor{green}{RGB}{0,128,0}
+
+\def\EQ#1\EN{\begin{equation*}#1\end{equation*}}
+\def\BA#1\EA{\begin{align*}#1\end{align*}}
+\def\BS#1\ES{\begin{split*}#1\end{split*}}
+\newcommand{\bc}{\begin{center}}
+\newcommand{\ec}{\end{center}}
+\newcommand{\eq}{\ =\ }
+\newcommand{\degc}{$^\circ$C}
+
+\def\p{\partial}
+\def\qbs{\boldsymbol{q}}
+\def\Dbs{\boldsymbol{D}}
+\def\A{\mathcal A}
+\def\gC{\mathcal C}
+\def\gD{\mathcal D}
+\def\gL{\mathcal L}
+\def\M{\mathcal M}
+\def\P{\mathcal P}
+\def\Q{\mathcal Q}
+\def\gR{\mathcal R}
+\def\gS{\mathcal S}
+\def\X{\mathcal X}
+\def\bnabla{\boldsymbol{\nabla}}
+\def\bnu{\boldsymbol{\nu}}
+\renewcommand{\a}{{\alpha}}
+%\renewcommand{\a}{{}}
+\newcommand{\s}{{\sigma}}
+\newcommand{\bq}{\boldsymbol{q}}
+\newcommand{\bz}{\boldsymbol{z}}
+\def\bPsi{\boldsymbol{\Psi}}
+
+\def\Li{\textit{L}}
+\def\Fb{\textbf{f}}
+\def\Jb{\textbf{J}}
+\def\cb{\textbf{c}}
+
+\def\Dt{\Delta t}
+\def\tpdt{{t + \Delta t}}
+\def\bpsi{\boldsymbol{\psi}}
+\def\dbpsi{\delta \boldsymbol{\psi}}
+\def\bc{\textbf{c}}
+\def\bx{\textbf{x}}
+\def\dbc{\delta \textbf{c}}
+\def\dbx{\delta \textbf{x}}
+\def\arrows{\rightleftharpoons}
+
+\newcommand{\bGamma}{\boldsymbol{\Gamma}}
+\newcommand{\bOmega}{\boldsymbol{\Omega}}
+%\newcommand{\bPsi}{\boldsymbol{\Psi}}
+%\newcommand{\bpsi}{\boldsymbol{\psi}}
+\newcommand{\bO}{\boldsymbol{O}}
+%\newcommand{\bnu}{\boldsymbol{\nu}}
+\newcommand{\bdS}{\boldsymbol{dS}}
+\newcommand{\bg}{\boldsymbol{g}}
+\newcommand{\bk}{\boldsymbol{k}}
+%\newcommand{\bq}{\boldsymbol{q}}
+\newcommand{\br}{\boldsymbol{r}}
+\newcommand{\bR}{\boldsymbol{R}}
+\newcommand{\bS}{\boldsymbol{S}}
+\newcommand{\bu}{\boldsymbol{u}}
+\newcommand{\bv}{\boldsymbol{v}}
+%\newcommand{\bz}{\boldsymbol{z}}
+\newcommand{\pressure}{P}
+
+\newcommand\gehcomment[1]{{{\color{orange} #1}}}
+\newcommand\add[1]{{{\color{blue} #1}}}
+\newcommand\remove[1]{\sout{{\color{red} #1}}}
+\newcommand\codecomment[1]{{{\color{green} #1}}}
+\newcommand\redcomment[1]{{{\color{red} #1}}}
+\newcommand\bluecomment[1]{{{\color{blue} #1}}}
+\newcommand\greencomment[1]{{{\color{green} #1}}}
+\newcommand\magentacomment[1]{{{\color{magenta} #1}}}
+
+\begin{comment}
+\tiny
+\scriptsize
+\footnotesize
+\small
+\normalsize
+\large
+\Large
+\LARGE
+\huge
+\Huge
+\end{comment}
+
+\begin{document}
+\title{CLM-CN Reaction\ldots in a Nutshell}
+\author{Glenn Hammond}
+\date{\today}
+
+%\frame{\titlepage}
+
+%-----------------------------------------------------------------------------
+\section{Description of CLM-CN Reaction}
+
+\subsection{Batch CLM-CN Reaction Conceptual Model}
+
+\frame{\frametitle{Schematic of CLM-CN Reaction Network}
+\includegraphics[width=\linewidth]{./CLM-CN_cycle}
+}
+
+%-----------------------------------------------------------------------------
+\subsection{Governing Equations}
+
+\frame{\frametitle{Reaction Expression}
+
+\Large
+
+\EQ\label{CN_rxn}
+\text{CN}_u \eq \left(1-f\right) \text{CN}_d + f \text{CO}_2 + n \text{N}_\text{mineral}
+\EN
+
+\EQ\label{n_calc}
+n \eq u - \left(1-f\right) d
+\EN
+
+\footnotesize
+\BA
+\text{CN}_u &\eq \text{upstream carbon pool } [\text{mol C}/m^3] \\
+\text{CN}_d &\eq \text{downstream carbon pool } [\text{mol C}/m^3] \\
+\text{CO}_2 &\eq \text{nitrogen concentration } [\text{mol CO}_2/m^3] \\
+\text{N}_\text{mineral} &\eq \text{nitrogen concentration } [\text{mol N}/m^3] \\
+u &\eq \text{\text{C}/\text{N} atomic weight ratio divided by upstream \text{C}/\text{N} ratio} \\
+d &\eq \text{\text{C}/\text{N} atomic weight ratio divided by downstream \text{C}/\text{N} ratio} \\
+f &\eq \text{respiration fraction} \\
+\EA
+}
+
+\frame{\frametitle{Kinetic Rate Expression}
+
+\Large
+
+\EQ\label{CN_kinetic_rxn}
+rate \eq f_T f_\theta f_\text{pi} k \text{CN}_u
+\EN
+
+%\bigskip
+%\normalsize
+%\footnotesize
+\scriptsize
+\BA
+f_T &\eq \exp\left[308.56 \left(\frac{1}{71.02}-\frac{1}{T - 227.13}\right)\right]\\
+f_\theta &\eq \frac{\log\left(\theta_\text{min}/\theta\right)}{\log\left(\theta_\text{min}/\theta_\text{max}\right)}\\
+f_\text{pi} &\eq \frac{\text{N}_\text{mineral}}{\text{N}_\text{mineral} + K_{\text{N}_\text{mineral}}} \\
+k &\eq \text{kinetic rate constant} [s^{-1}]\\
+T &\eq \text{temperature } [K] \\
+\theta &\eq \text{moisture content or saturation } [-] \\
+\text{CN}_u &\eq \text{upstream carbon pool } [\text{mol C}/m^3] \\
+\text{N}_\text{mineral} &\eq \text{nitrogen concentration } [\text{mol N}/m^3] \\
+K_\text{N} &\eq \text{nitrogen half saturation constant } [\text{mol N}/m^3]\\
+\EA
+
+}
+
+\frame{\frametitle{Kinetic Reaction Equations}
+
+
+\EQ
+\text{CN}_u \eq \left(1-f\right) \text{CN}_d + f \text{CO}_2 + n \text{N}_\text{mineral}
+\EN
+
+\EQ
+rate \eq f_T f_\theta f_\text{pi} k \text{CN}_u
+\EN
+
+\Large
+\BA
+\frac{\p}{\p t} \left(\text{CN}_u\right) &\eq -rate \\
+\frac{\p}{\p t} \left(\text{CN}_d\right) &\eq \left(1-f\right) rate \\
+\frac{\p}{\p t} \left(\text{CO}_2\right) &\eq f \cdot rate \\
+\frac{\p}{\p t} \left(\text{N}_\text{mineral}\right) &\eq n \cdot rate \\
+\EA
+
+}
+
+
+\frame{\frametitle{Newton-Raphson Method}
+\LARGE
+\BA
+\Fb\left(\bx^{k+1,i}\right) &\eq \frac{\bx^{k+1,i}-\bx^k}{\Dt} - \sum_{irxn} \bnu_{irxn} \cdot rate_{irxn} \\
+\\
+\Jb \dbx &\eq -\Fb\left(\bx^{k+1,i}\right) \\
+\\
+\bx^{k+1,i+1} &\eq \bx^{k+1,i} + \dbx
+\EA
+\small
+\BA
+k &\eq \text{time step} \\
+i &\eq \text{iteration}
+\EA
+}
+
+%-----------------------------------------------------------------------------
+\subsection{PFLOTRAN Input Specification}
+
+\begin{frame}[fragile,allowframebreaks]\frametitle{CHEMISTRY}
+\small
+\begin{semiverbatim}
+CHEMISTRY
+  ...
+  IMMOBILE_SPECIES
+    N
+    C
+    SOM1
+    SOM2
+    SOM3
+    SOM4
+    LabileC
+    CelluloseC
+    LigninC
+    LabileN
+    CelluloseN
+    LigninN
+  /
+  ...
+\end{semiverbatim}
+\newpage
+\begin{semiverbatim}
+
+  ...
+  REACTION_SANDBOX
+    CLM-CN
+      POOLS   \bluecomment{! CN ratio}
+        SOM1   12.d0   \bluecomment{! -> u or d}
+        SOM2   12.d0
+        SOM3   10.d0
+        SOM4   10.d0
+        Labile
+        Cellulose
+        Lignin
+      /
+      ...
+\end{semiverbatim}
+\newpage
+\begin{semiverbatim}
+      ...
+      REACTION
+        UPSTREAM_POOL Labile         \bluecomment{! CN_u}
+        DOWNSTREAM_POOL SOM1         \bluecomment{! CN_d}
+        TURNOVER_TIME 20. h          \bluecomment{! -> k}
+        RESPIRATION_FRACTION 0.39d0  \bluecomment{! f}
+        N_INHIBITION 1.d-10          \bluecomment{! K_N}
+      /
+      REACTION
+        UPSTREAM_POOL SOM1
+        DOWNSTREAM_POOL SOM2
+        TURNOVER_TIME 14. d
+        RESPIRATION_FRACTION 0.28d0
+        N_INHIBITION 1.d-10
+      /
+      ...
+    / \bluecomment{! END CLM-CN}
+  / \bluecomment{! END REACTION_SANDBOX}
+/ \bluecomment{! END CHEMISTRY}
+\end{semiverbatim}
+\end{frame}
+
+\begin{frame}[fragile]\frametitle{CONSTRAINT}
+%\small
+\footnotesize
+\begin{semiverbatim}
+CONSTRAINT initial
+  CONCENTRATIONS   \bluecomment{! moles/L}
+    A(aq)  1.d-40  T
+  /
+  IMMOBILE       \bluecomment{! moles/m^3}
+    N     1.d-6
+    C     1.d-6
+    SOM1  1.d-10 \bluecomment{! moles C/m^3}
+    SOM2  1.d-10
+    SOM3  1.d-10
+    SOM4  1.d-10
+    LabileC     0.1852d-3
+    CelluloseC  0.4578d-3
+    LigninC     0.2662d-3
+    LabileN     0.00508954d-3
+    CelluloseN  0.01258096d-3
+    LigninN     0.00731553d-3
+  /
+END
+\end{semiverbatim}
+\end{frame}
+
+%-----------------------------------------------------------------------------
+\subsection{PFLOTRAN Numerical Implementation}
+\frame{\frametitle{Kinetic Reaction Equations}
+\BA
+\text{Lit1C} + \frac{1}{\text{CN}_{\text{Lit1C}}} {\text {Lit1N}} &\eq \left(1-f_1\right) \text{SOM1} + f_1 \text{CO}_2 + n_1 \text{N}_\text{mineral} \\
+\text{Lit2C} + \frac{1}{\text{CN}_{\text{Lit2C}}} {\text {Lit2N}} &\eq \left(1-f_2\right) \text{SOM2} + f_2 \text{CO}_2 + n_2 \text{N}_\text{mineral} \\
+\text{Lit3C} + \frac{1}{\text{CN}_{\text{Lit3C}}} {\text {Lit3N}} &\eq \left(1-f_3\right) \text{SOM3} + f_3 \text{CO}_2 + n_3 \text{N}_\text{mineral} \\
+\text{SOM1} &\eq \left(1-f_4\right) \text{SOM2} + f_4 \text{CO}_2 + n_4 \text{N}_\text{mineral} \\
+\text{SOM2} &\eq \left(1-f_5\right) \text{SOM3} + f_5 \text{CO}_2 + n_5 \text{N}_\text{mineral} \\
+\text{SOM3} &\eq \left(1-f_6\right) \text{SOM4} + f_6 \text{CO}_2 + n_6 \text{N}_\text{mineral} \\
+\text{SOM4} &\eq f_7 \text{CO}_2 + n_7 \text{N}_\text{mineral} \\
+\EA
+}
+
+\frame{\frametitle{Kinetic Reaction Equations}
+
+\BA
+\text{CN}_{\text{Lit1C}} & \eq  \frac{\text{Lit1C}}{\text{Lit1N}} \\
+\text{CN}_{\text{Lit2C}} & \eq \frac{\text{Lit2C}}{\text{Lit2N}} \\
+\text{CN}_{\text{Lit3C}} & \eq \frac{\text{Lit3C}}{\text{Lit3N}} \\
+\EA
+
+{\begin{center}
+\begin{tabular}{ c || c }
+  $\text{CN}_{\text{SOM1}} \eq 12$ & $\text{CN}_{\text{SOM2}} \eq 12$  \\
+  \hline
+  $\text{CN}_{\text{SOM3}} \eq 10$ & $\text{CN}_{\text{SOM4}} \eq 10$  \\
+  \hline
+  $f_1 \eq 0.39 $ & $f_2 \eq 0.55 $ \\
+  \hline
+  $f_3 \eq 0.29 $ & $f_4 \eq 0.28 $ \\
+  \hline
+  $f_5 \eq 0.46 $ & $f_5 \eq 0.55 $ \\
+  \hline
+  $f_7 \eq 1.0 $ & \\
+\end{tabular}
+\end{center}
+}
+}
+
+\frame{\frametitle{Kinetic Reaction Equations}
+\BA
+n_1 &\eq \frac{1}{\text{CN}_{\text{Lit1C}}} - \left(1-f_1\right) \frac{1}{\text{CN}_{\text{SOM1}}} \\
+n_2 &\eq \frac{1}{\text{CN}_{\text{Lit2C}}} - \left(1-f_2\right) \frac{1}{\text{CN}_{\text{SOM2}}} \\
+n_3 &\eq \frac{1}{\text{CN}_{\text{Lit3C}}} - \left(1-f_3\right) \frac{1}{\text{CN}_{\text{SOM3}}} \\
+n_4 &\eq \frac{1}{\text{CN}_{\text{SOM1}}} - \left(1-f_4\right) \frac{1}{\text{CN}_{\text{SOM2}}} \\
+n_5 &\eq \frac{1}{\text{CN}_{\text{SOM2}}} - \left(1-f_5\right) \frac{1}{\text{CN}_{\text{SOM3}}} \\
+n_6 &\eq \frac{1}{\text{CN}_{\text{SOM3}}} - \left(1-f_6\right) \frac{1}{\text{CN}_{\text{SOM4}}} \\
+n_7 &\eq \frac{1}{\text{CN}_{\text{SOM4}}} 
+\EA
+}
+
+\frame{\frametitle{Kinetic Reaction Equations}
+\BA
+R_1 \eq f_T f_\theta f_\text{pi} k \text{Lit1C} \\
+R_2 \eq f_T f_\theta f_\text{pi} k \text{Lit2C} \\
+R_3 \eq f_T f_\theta f_\text{pi} k \text{Lit3C} \\
+R_4 \eq f_T f_\theta f_\text{pi} k \text{SOM1} \\
+R_5 \eq f_T f_\theta f_\text{pi} k \text{SOM2} \\
+R_6 \eq f_T f_\theta f_\text{pi} k \text{SOM3} \\
+R_7 \eq f_T f_\theta f_\text{pi} k \text{SOM4} 
+\EA
+}
+
+
+\frame{\frametitle{Mass Conservation Equations}
+The CLM-CN reaction network results in 12 basis species for which the mass conservation equation are given by: 
+\BA
+\frac{\p}{\p t} \left(\text{N}_\text{mineral}\right) &\eq n_1R_1 + n_2R_2 + n_3R_3 +n_4R_4 +n_5R_5 +n_6R_6 + n_7R_7 \\
+\frac{\p}{\p t} \left(\text{CO}_2\right) &\eq  f_1R_1 + f_2R_2 + f_3R_3 + f_4R_4 + f_5R_5 + f_6R_6 + f_7R_7 \\
+\frac{\p}{\p t} \left(\text{SOM1}\right) &\eq  (1-f_1)R_1 - R_4\\
+\frac{\p}{\p t} \left(\text{SOM2}\right) &\eq  (1-f_2)R_2 + (1-f_4)R_4 - R_5\\
+\frac{\p}{\p t} \left(\text{SOM3}\right) &\eq  (1-f_3)R_3 + (1-f_5)R_5 - R_6\\
+\frac{\p}{\p t} \left(\text{SOM4}\right) &\eq  (1-f_6)R_6 - R_7
+\EA
+}
+
+\frame{\frametitle{Mass Conservation Equations}
+\BA
+\frac{\p}{\p t} \left(\text{Lit1C}\right) &\eq  -R_1 \\
+\frac{\p}{\p t} \left(\text{Lit2C}\right) &\eq  -R_2 \\
+\frac{\p}{\p t} \left(\text{Lit3C}\right) &\eq  -R_3 \\
+\frac{\p}{\p t} \left(\text{Lit1N}\right) &\eq   \frac{-1}{\text{CN}_{\text{Lit1C}}}R_1\\
+\frac{\p}{\p t} \left(\text{Lit3N}\right) &\eq  \frac{-1}{\text{CN}_{\text{Lit2C}}}R_2 \\
+\frac{\p}{\p t} \left(\text{Lit3N}\right) &\eq  \frac{-1}{\text{CN}_{\text{Lit3C}}}R_3 
+\EA
+}
+
+\frame{\frametitle{Numerical Implementation in PFLOTRAN}
+Applying finite-volume spatial discretization on mass conservation equation for Lit1C:
+\EQ
+\int \frac{\p}{\p t} \left(\text{Lit1C}\right) dV \eq  \int -R_1 dV
+\EN
+
+\EQ
+\frac{\p}{\p t} \left(\text{Lit1C }\Delta V\right)  \eq   -R_1 \Delta V
+\EN
+
+Implicit time discretization:
+\EQ
+\frac{\Delta V}{\Delta t} \left(\text{Lit1C}^{t+1} - \text{Lit1C}^t\right)  \eq   -R_1^{t+1} \Delta V
+\EN
+
+Residual equation:
+\EQ
+\mathcal{R}_7 \eq  \frac{\Delta V}{\Delta t} \left(\text{Lit1C}^{t+1} - \text{Lit1C}^t\right)  + R_1^{t+1} \Delta V
+\EN
+Note: The residual equation for Lit1C is $\mathcal{R}_7$ since Lit1C is the 7-th basis specie
+}
+
+\frame{\frametitle{Numerical Implementation in PFLOTRAN}
+Jacobian for Lit1C:
+\BA
+\mathcal{J}_{7,7} & \eq \frac{\p \mathcal{R}_7 }{\p \left(\text{Lit1C}^{t+1} \right)} \\
+\mathcal{J}_{7,1} & \eq \frac{\p \mathcal{R}_7 }{\p \left(\text{N}_\text{mineral}^{t+1} \right)} \\
+\mathcal{J}_{7,i} & \eq 0 \text{\hspace{2cm}i $\neq 1, 7$ }
+\EA
+}
+
+\end{document}