https://github.com/cran/ReacTran
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Tip revision: 6a2a464424c6eaca2ee6747a2ff1fc6e15105936 authored by Karline Soetaert on 24 December 2014, 00:00 UTC
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ReacTran.rnw
\documentclass[article,nojss]{jss}
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\usepackage{graphicx}
\usepackage{amsmath}


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\newcommand{\rt}{\textbf{\textsf{ReacTran }}}
\newcommand{\ds}{\textbf{\textsf{deSolve }}}
\newcommand{\rs}{\textbf{\textsf{rootSolve }}}
\newcommand{\R}{\proglang{R }}
\title{
  \proglang{R}-package \rt: Reactive Transport Modelling in \R
}
\Plaintitle{Reactive transport modelling in R}

\Keywords{
  reactive-transport, diffusion, advection, reaction, porous media, rivers,
  estuary, water column, \proglang{R}
}

\Plainkeywords{
  reactive-transport, diffusion, advection, reaction, porous media, rivers,
  estuary, water column, R
}


\author{Karline Soetaert\\
NIOZ-Yerseke\\
The Netherlands
\And
Filip Meysman\\
NIOZ-Yerseke\\
The Netherlands
}

\Plainauthor{Karline Soetaert, and Filip Meysman}

\Abstract{
  \R package \rt \citep{ReacTran_paper} contains functions for creating reactive-
  transport models in \R.
}

%% The address of (at least) one author should be given
%% in the following format:
\Address{
  Karline Soetaert\\
  Centre for Estuarine and Marine Ecology (CEME)\\
  Royal Netherlands Institute of Sea Research (NIOZ)\\
  4401 NT Yerseke, Netherlands\\
  E-mail: \email{karline.soetaert@nioz.nl}\\
  URL: \url{http://www.nioz.nl}\\
  \\
  Filip Meysman\\
  Royal Netherlands Institute of Sea Research (NIOZ)\\
  4401 NT Yerseke, Netherlands\\
  E-mail: \email{filip.meysman@nioz.nl}\\
  URL: \url{http://www.nioz.nl}\\
}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R/Sweave specific LaTeX commands.
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%\VignetteIndexEntry{R-package ReacTran: reactive transport modelling in R}
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\begin{document}
\SweaveOpts{engine=R,eps=FALSE}
\SweaveOpts{keep.source=TRUE}

<<preliminaries,echo=FALSE,results=hide>>=
library("ReacTran")
options(prompt = "> ")
options(width=75)
@

\maketitle

\section{General reaction transport equation in one dimension}

  \subsection {The reaction-transport equation}
  The general 1-D reaction-transport equation in multi-phase environments and
  for shapes with variable geometry is:
  
  \[
    \frac{\partial \xi C}{\partial t} = -\frac{1}{A}\cdot \frac{\partial
       (A \cdot J)}{\partial x} + reac
  \]
  where
  \begin{itemize}
    \item t is time
    \item x is space
    \item C is concentration of a substance in its respective phase (units of
      e.g. $M L^{-3} liquid$ for sediment solutes).
      \footnote{here we use \emph{M} for mass, \emph{L} for
        length and \emph{t} for time
      }
    \item $\xi$ is the volume fraction (-), i.e. the fraction of a phase in
      the bulk volume (see figure).
      In many cases, only one phase is considered and $\xi$ = 1;
      For sediments, $\xi$ would be porosity (solutes), or
      1-porosity (solids).
    \item $A$ is the (total) surface area ($L^2$).
    \item J are fluxes, (units of $M L^{-2} t^{-1}$)
  \end {itemize}
  
  The Fluxes (J), which are estimated per unit of total surface, consist of a
  dispersive and an advective component:
  \[
    J = -\xi D  \cdot \frac{\partial C}{\partial x} + \xi u  \cdot C
  \]

  where
  \begin{itemize}
    \item D is the diffusion (or dispersion) coefficient, units of $L^2 t^{-1}$
    \item u is the advection velocity, units of $L t^{-1}$
  \end {itemize}

  \subsection{Boundary conditions in 1-D models}
  The boundaries (at the extremes of the model domain, e.g. at x=0)
  can be one of the following types:
  \begin{itemize}
    \item A concentration boundary, e.g. $C |_{x=0}=C_0$
    \item A diffusive + advective flux boundary $J_{x=0}=J_0$
    \item A boundary layer convective exchange flux boundary
      $J_{x=0} = a_{bl}\cdot(C_{bl}-C_0)$
  \end{itemize}

  \subsection{Numerical approximation}
  The reaction-transport formula consists of a partial differential equation
  (PDE).
  
  To solve it, the spatial gradients are approximated using so-called
  numerical differences (the method-of-lines, MOL, approach). This converts
  the partial differential equations into ordinary differential equations
  (ODE).
  
  This means that the model domain is divided into
  a number of grid cells, and for each grid cell i, we write:
  
  \[
    \frac{d \xi_i C_i}{d t} = -\frac{1}{A_i}\cdot \frac{\Delta_i
       (A \cdot J)}{\Delta x_i} + reac_i
  \]
  where $\Delta_i$ denotes that the flux gradient is to be taken over
  box i, and $\Delta x_i$ is the thickness of box i:
  \[
  \Delta_i (A \cdot J) =A_{i,i+1} \cdot J_{i,i+1}-A_{i-1,i} \cdot J_{i-1,i}
  \]
  where \code{i,i+1} denotes the interface between box i and i+1.
  
  The fluxes at the box interfaces are discretized as:
  \[
    J_{i-1,i} = -\xi_{i-1,i} D_{i-1,i}  \cdot \frac{C_{i}-C_{i-1}}{\Delta x_{i-1,i}} +
                 \xi_{i-1,i}  u_{i-1,i} \cdot (\vartheta_{i-1,i} \cdot C_{i-1} +
                 (1-\vartheta_{i-1,i}) \cdot C_{i})
  \]
  with  $\Delta x_{i-1,i}$ the distance between the centre of grid cells i-1
  and i, and $\vartheta$ the upstream weighing coefficients for the
  advective term.

  \clearpage
 \section{one-dimensional finite difference grids and properties in ReacTran}

  \subsection{Generating a spatial discretization grid }

  The 1-D spatial discretization grid can best be generated with \rt function
  \code{setup.grid.1D}.

\setkeys{Gin}{width=0.5\textwidth}
\begin{figure}
\begin{center}
\includegraphics{reactran-phase}
\end{center}
\caption{An example of multiple phases in \rt, adapted from figure 3.9
 from Soetaert and Herman, 2009}
\label{fig:fig1}
\end{figure}


\setkeys{Gin}{width=0.5\textwidth}
\begin{figure}
\begin{center}
\includegraphics{reactran-fig1}
\end{center}
\caption{Nomenclature for the spatial discretization grid in \code{grid.1D}}
\label{fig:fig1}
\end{figure}


  Function \code{setup.grid.1D} creates a grid, which can comprise several
  zones:
  \begin{verbatim}
  setup.grid.1D <- function(x.up = 0,	x.down = NULL, L = NULL, N = NULL,
     dx.1 = NULL, p.dx.1 = rep(1,length(L)), max.dx.1 = L,
     dx.N = NULL, p.dx.N = rep(1,length(L)), max.dx.N = L)
  \end{verbatim}
  with the following arguments:
  \begin{itemize}
    \item \code{x.up}. The position of the upstream boundary.
    \item \code{x.down}. The positions of the downstream boundaries in
      each zone.
    \item \code{L, N}, the thickness and the number of grid cells in each zone.
    \item \code{dx.1, p.dx.1, max.dx.1}, the size of the first grid cell,
      the factor of increase near the upstream boundary,
      and maximal grid cell size in the upstream half of each zone.
    \item \code{dx.N, p.dx.N, max.dx.N}, the size of the last grid cell,
      the factor of increase near the downstream boundary,
      and maximal grid cell size in the downstream half of each zone.
  \end{itemize}
  It returns an element of class \code{grid.1D} that contains the following
  elements (units L), see figure 2:
  \begin{itemize}
    \item \code{x.up, x.down}. The position of the upstream and downstream
      boundary.
    \item \code{x.int}, the position of the grid cell interfaces,
      where the fluxes are specified, a vector of length N+1.
    \item \code{x.mid}, the position of the grid cell centres, where the
      concentrations are specified, a vector of length N.
    \item \code{dx}, the thickness of boxes , i.e. the distance between the
      grid cell interfaces, a vector of length N. This is equivalent to
       $\Delta x_i$
    \item \code{dx.aux}, the distance between the points where the
      concentrations are specified, a vector of length N+1. This is equivalent
      to  $\Delta x_{i-1,i}$ .
  \end{itemize}

  For example, to subdivide an estuary, 100 km long into 50 boxes, with the
  first box of size 1 km, we write \footnote{note that by embracing the
  statement within brackets, we execute it AND print the result to the screen}:
<<>>=
 (grid <- setup.grid.1D(L = 100, dx.1 = 1, N = 50))
@
which can be plotted as follows:
<<label=Grid,include=FALSE>>=
 plot(grid)
@
\setkeys{Gin}{width=0.6\textwidth}
\begin{figure}
\begin{center}
<<label=grid,fig=TRUE,echo=FALSE>>=
<<Grid>>
@
\end{center}
\caption{exponential grid size  - see text for \R-code}
\label{fig:grid}
\end{figure}

  \subsection{Other grid properties}
  In the 1-D discretization formula, some properties (e.g. concentrations)
  are defined at the centre of boxes ($C_i$), while other properties
  (transport coefficients) are prescribed at the box interfaces
  ($\vartheta$, $\xi$, D).

  These properties can be generated conform a previously defined grid
  with \rt function \code{setup.prop.1D}. Its syntax is:
  \begin{verbatim}
  setup.prop.1D(func = NULL, value = NULL, xy = NULL,
    interpolate = "spline", grid, ...)
  \end{verbatim}
  They can be specified as a function, or as a (constant) value, or as
  an (x,y) data series, which is interpolated using a spline or linearly.
  
  The function returns a list of class \code{prop.1D} that contains:
  \begin{itemize}
    \item \code{int}, the property value at the grid cell interfaces,
      a vector of length N+1.
    \item \code{mid}, the property value at the middle of grid cells,
      a vector of length N.
  \end{itemize}

  A number of commonly used property functions are implemented in \rt:
  \begin{itemize}
    \item \code{p.exp} for an exponentially decreasing transition
    \item \code{p.lin} for a linearly decreasing transition
    \item \code{p.sig} for a sigmoidally decreasing transition
  \end{itemize}

  Below, we demonstrate the use of \code{setup.prop.1D} to create the
  properties for use in a sediment biogeochemical model.

  After defining a grid, we first use utility function \code{p.exp} to
  calculate a bioturbation profile, assuming there is a constant bioturbation
  in an upper layer (2 cm), declining exponentially below this layer.:
<<>>=
grid <- setup.grid.1D(L = 10, N = 100)
Db   <- setup.prop.1D(func = p.exp, grid = grid, 
                      y.0 = 5, y.inf = 0, x.L = 2)
@

  We then define a function (\code{exp.inc}) that specifies an
  exponentially increasing profile to calculate the volume fraction
  of solid substances in the sediment. These are represented by 1-porosity,
  where the porosity is an exponentially decreasing function, which can
  be calculated using \code{p.exp}.

<<>>=
exp.inc <- function(x, y.0 = 1, y.inf = 0.5, x.L = 0, x.att = 1)
       return(1 - p.exp(x, y.0, y.inf, x.L, x.att))

VFsolid <- setup.prop.1D(func = exp.inc, grid = grid, y.0 = 0.9, y.inf = 0.7)
@

  A \code{plot} method is defined for class \code{prop.1D}; to invoke it
  one has to pass both the property as the grid on which it is based;
  \code{xyswap=TRUE} swaps the x- and y-axis, which is useful
  for plotting vertical profiles.
<<label=prop,include=FALSE>>=
par(mfrow = c(1, 2))
plot(VFsolid, grid, xyswap = TRUE, type = "l", main = "1-porosity")
plot(Db, grid, xyswap = TRUE, type = "l", main = "Db")
par(mfrow = c(1, 1))
@

\setkeys{Gin}{width=0.6\textwidth}
\begin{figure}
\begin{center}
<<label=prop,fig=TRUE,echo=FALSE>>=
<<prop>>
@
\end{center}
\caption{Two exponentially declining properties  - see text for \R-code}
\label{fig:prop}
\end{figure}
\clearpage
 \section{one-dimensional reactive-transport modelling in ReacTran}
 Implementing a reactive-transport model in \rt proceeds in several steps.
 
 \begin{itemize}
   \item Setting up a finite difference grid and defining properties attached
     to this grid (see previous section)
   \item Specifying a model function that describes the rate of change of
     substances due to transport and reaction.
     The transport of properties is done with \rt function \code{tran.1D}
   \item Solving the model. Depending on whether a transient or a
     steady-state solution is desired, solving the model makes use of
     functions \code{ode.1D} or \code{steady.1D} from packages \pkg{deSolve}
     and \pkg{rootSolve}.
 \end{itemize}
 Below we first explain how to use \rt function \code{tran.1D}, after which it
 is used in a model, which is consequently solved.

  \subsection{R-function tran.1D}
  The default input for the \code{tran.1D()} function in \proglang{R} is:
  \begin{verbatim}
    tran.1D(C, C.up = C[1], C.down = C[length(C)],
      flux.up = NULL, flux.down = NULL, 
      a.bl.up = NULL, a.bl.down = NULL, 
      D = 0, v = 0, AFDW = 1, VF = 1, A = 1, dx,
      full.check = FALSE, full.output = FALSE)
  \end{verbatim}
  with the following arguments:
  \begin{itemize}
    \item \code{C, C.up, C.down}. The concentrations, per unit of phase
      volume, in the centre of each grid cell, a vector of length N
      (\code{C}) and at the upstream or downstream boundary, one value
      (\code{C.up, C.down}).
    \item  \code{flux.up, flux.down}.  The fluxes, per unit of total surface,
      at the upstream and downstream boundaries, ($M L^{-2} t^{-1}$).
    \item \code{a.bl.up, a.bl.down}, the convective
      transfer coefficients.
    \item \code{D}, the diffusion (dispersion) coefficients, either
      one value, or a vector (\code{D}) or packed as a \code{grid},
      ($L^2 t^{-1}$).
    \item \code{v}, the advective velocity ($L t^{-1}$).
    \item \code{AFDW} the weights used in the finite difference
      approximation for advection (-).
    \item \code{VF}, the volume fractions (-).
    \item \code{A} the surface areas ($L^2$)
    \item \code{dx}, the distances between cell interfaces (L), the
      discretization grid. Must be specified.
    \item \code{full.check}, when \code{TRUE}, the consistency of the
      input is checked.
    \item \code{full.output}, when \code{TRUE} full output is returned.
  \end{itemize}
  
  Note that several properties (\code{dx, D, v, AFDW, VF, A}) can be passed
  in different ways:
  \begin{itemize}
    \item a single number, in which case they are assumed constant
    \item a vector of length N+1, i.e. defined on the grid interfaces
    \item a list of type \code{grid.1D} or of type \code{prop.1D}, as
      created by \code{setup.grid} (spatial discretization) or by
      \code{setup.prop.1D} (see previous section).
  \end{itemize}

  Function \code{reac.1D} returns the rate of change of \code{C} due to
  transport, and the fluxes up-and downstream; if \code{full.output}
  is \code{TRUE} then also the advective and dispersive fluxes at all
  layer interfaces are returned.
  
  For example:
<<>>=
tran.1D(C = 1:20, D = 0, flux.up = 1, v = 1, dx = 1)
tran.1D(C = 1:20, D = 0, flux.up = 1, v = 1, dx = 1, full.output = TRUE)
@
  \subsection{A 1-D reaction transport model}
  Function \code{tran.1D} estimates the \emph{rate of change} of substances
  as a function of \emph{transport} processes.

  If, in addition to transport, reaction terms are added, a 1-D
  \emph{reaction-transport model} is obtained.

  In line with the way ordinary differential equations are solved in \R,
  a reaction-transport model is specified in an \R-function that computes
  the derivatives in the ODE at a certain time \code{t}.
  
  This function will be called by the solution methods as:
  \code{func(t,y,parms,...)} where  \code{t} is the current time point,
  \code{y} are the current values of the state variables in the ODE system
  and \code{parms} are model parameters.
  
  It should return a list whose first element is a vector containing the
  derivatives of y. (for more details, see packages \ds, \rs)
  
  For instance, the function representing the following model

  \[
    \frac{\partial C}{\partial t} = - v \cdot \frac{\partial C}{\partial x} - k C
  \]
  with boundary condition:
  \[
    v \cdot C|_{x=0} = F_0
  \]

  can be implemented in \R as:

<<>>=
parms <- c(F0 = 1, v = 1, k = 0.1, dx = 1)
@
<<>>=
advModel <- function(t, C, parms) {
    
   with (as.list(parms), {

     Tran <- tran.1D(C = C, D = 0, flux.up = F0, v = v, dx = dx)
     Consumption <-  k*C
     dC   <- Tran$dC - Consumption
        
     return (list(dC = dC, Consumption= Consumption,
                  flux.up = Tran$flux.up, flux.down = Tran$flux.down))
      })

    }
@

  Note the use of \code{with (as.list(parms),...} which allows to access by
  name the previously defined model parameters (\code{parms}) within the function.

  \subsection{Solving a 1-D reaction transport model}
  In \R, 1-D models consisting of ordinary differential equations can
  be solved in two ways.
  \begin{itemize}
    \item by estimating the \emph{steady-state condition}, using function
      \code{steady.1D} from \R-package rootSolve \citep{rootSolve}.
    \item by running the model \emph{dynamically}, using functions
      \code{ode.1D}  from \R-package deSolve \citep{deSolve}.
  \end{itemize}

  To solve the above model to steady-state, we invoke \code{steady.1D}:
<<>>=
out <- steady.1D(func = advModel, y = runif(25), parms = parms,
                 nspec = 1, positive = TRUE)
@
  Function \code{steady.1D} estimates the steady-state iteratively and
  requires an initial guess of the state variables. For most cases, the
  exact values are not important; the initial guess of the state variables
  in the above example consists simply of 25 uniformly distributed random
  numbers ([0,1]);

  we specify that the model comprises only one species (\code{nspec} and
  that we are interested only in a solution consisting of positive numbers
  (\code{positive=TRUE}), as negative concentrations do not exist (in the
  real environment; they may exist mathematically).
  
  The outcome of this model is a list called \code{out}, which contains
  the steady-state condition of the state-variables (item \code{y}), the
  consumption rate and the fluxes at the upper and lower boundary as
  defined in function \code{advModel}. In addition,
  attribute \code{precis} gives a measure of how far the system is from
  steady-tate at each iteration of the steady-state solver; only two
  iterations were needed.

<<>>=
out
@
  It can be plotted using \code{rootSolve}s \code{plot} method:
<<label=st1,include=FALSE>>=
plot (out, xlab = "x", ylab = "Conc", main = "advection")
@

\setkeys{Gin}{width=0.4\textwidth}
\begin{figure}
\begin{center}
<<label=figst1,fig=TRUE,echo=FALSE>>=
<<st1>>
@
\end{center}
\caption{Solution of the uni-component reactive-transport model in 1-D
  - see text for \R-code}
\label{fig:st1}
\end{figure}

  It is good modelling practice to test mass conservation of a model,
  i.e. as a check that mass is not created or destroyed by numerical
  means (programming or modelling error).

  For this simple example, it
  is obvious that mass is conserved, but we will test it to exemplify
  the procedures.

  If mass is conserved and the system is at steady-state, then the net
  input by transport in the system should equal the total net consumption.
  For our model, the net input to the system is given by
  \code{flux.up - flux.down}, while
  the total consumption is simply the sum of the consumption in each box.
<<>>=
with (out, print(sum(Consumption)-(flux.up-flux.down)))
@
  The fact that the quantity is not completely zero is due to the small
  deviation from steady-state (as is clear from attribute \code{precis}).
  
\subsection{example: 1-D transport in a porous spherical body}
  This, somewhat more complex example models oxygen consumption in a
  spherical aggregate. The example is more complex because it assumes
  that both the surface area (\code{A}) and the volume fraction (\code{VF})
  (here the "porosity") vary along the spatial axis.
  
  We start by formulating the model function (\code{Aggregate.Model}).

<<>>=

Aggregate.Model <- function(time, O2, pars) {

  tran <- tran.1D(C = O2, C.down = C.ow.O2,
                  D = D.grid, A = A.grid,
                  VF = por.grid, dx = grid )

  reac <- - R.O2*(O2/(Ks+O2))
  return(list(dCdt = tran$dC + reac, reac = reac,
              flux.up = tran$flux.up, flux.down = tran$flux.down))
}
@

  next the parameters are defined:
<<>>=
C.ow.O2 <- 0.25     # concentration O2 water [micromol cm-3]
por     <- 0.8      # porosity
D       <- 400      # diffusion coefficient O2 [cm2 yr-1]
v       <- 0        # advective velocity [cm yr-1]
R.O2    <- 1000000  # O2 consumption rate [micromol cm-3 yr-1]
Ks      <- 0.005    # O2 saturation constant [micromol cm-3]
@
  and the spatial extent of the model discretized (grid definition).
<<>>=
R <- 0.025           # radius of the agggregate [cm]
N <- 100             # number of grid layers

grid <- setup.grid.1D(x.up = 0, L = R, N = N)
@
  We assume that porosity (the volume fraction) and the diffusion
  coefficient are constant. Both properties are defined as a grid list.
<<>>=
por.grid <- setup.prop.1D(value = por, grid = grid)
D.grid <- setup.prop.1D(value = D, grid = grid)
@

  In this model, each "grid cell" is equivalent to a thin spherical region;
  the further away from the origin, the larger the surface area of this layer.
  
  To take into account this expanding surface, we define a grid with the
  surfaces of these spherical layers.  Function \code{sphere.surf}
  calculates the surface of a sphere.

<<>>=
sphere.surf <- function (x)   4*pi*x^2

A.grid  <- setup.prop.1D(func = sphere.surf, grid = grid)
@

The model is then solved to steady-state
<<>>=
O2.agg <- steady.1D (y = runif(N), func = Aggregate.Model, nspec = 1,
                     positive = TRUE, atol = 1e-10)
@

and the output plotted
<<label=agg,include=FALSE>>=
plot(O2.agg, grid = grid$x.mid, xlab = "distance from centre, cm", 
     ylab = "mmol/m3", 
     main = "Diffusion-reaction of O2 in a spherical aggregate")
@

\setkeys{Gin}{width=0.4\textwidth}
\begin{figure}
\begin{center}
<<label=figagg,fig=TRUE,echo=FALSE>>=
<<agg>>
@
\end{center}
\caption{Solution of the aggregate model - see text for \R-code}
\label{fig:st1}
\end{figure}
@
  Note that in this model, we have imposed a zero-flux boundary at the
  centre of the sphere (or the "upstream" boundary) (zero-gradient is the default
  boundary condition). This makes sense as we assume that the sphere is
  symmetrical. However, oxygen is dffusing into the aggregate, at the
  "downstream" boundary.
  
  The influx, per unit of surface area is in \code{O2.agg$flux.down}.
<<>>=
  O2.agg$flux.up
  O2.agg$flux.down
@

  Calculating the mass conservation for this model is slightly more complex than
  in previous example, as we need to take volumetric averages of the rates,
  AND correct for the porosity effect.
  
  The volume in each spherical layer is simply equal to the surface at the
  centre of the layer (\code{A.grid$mid}) times the grid size (\code{grid$dx}).
  
<<>>=
 Volume <- A.grid$mid * grid$dx
(Consump <- - sum(O2.agg$reac * Volume * por.grid$mid))
@
  The total flux into the aggregate equals the flux per unit surface times
  the surface area at the downstream boundary:
<<>>=
 (Fluxin <- - O2.agg$flux.down * A.grid$int[N+1])
@

  The flux equals the total consumption, up to a certain numerical precision,
  hence mass is conserved.
<<>>=
  Consump - Fluxin
@
\clearpage
\section{volumetric advective-diffusive transport in an aquatic system}

  The volumetric reaction-transport equation in 1-D is best derived in
  two steps: first we rewrite the discretisation over a numerical grid:
  \[
    \frac{d C_i}{d t} = -\frac{1}{A}\cdot \frac{\Delta_i (A \cdot J)}
    {\Delta x_i}+ reac_i
  \]
  as
  \[
    \frac{d C_i}{d t} = - \frac{\Delta_i (A \cdot J)}{\Delta V_i}+ reac_i
  \]

  where we defined cell-volume as the product of mid-surface and cell thickness
  \[
  V_i=A_i \Delta x_i
  \]
  and then redefine the mass fluxes:
  \[
   A \cdot J = -D  \cdot A \frac{\Delta C}{\Delta x} + A  \cdot u  \cdot C
  \]
  as :
  \[
    A \cdot J = - E \Delta C + Q  \cdot C
  \]

  where
  $ Q=A \cdot u $ and $ E= \frac{D \cdot A}{\Delta x} $
  are the  flow rate  and the bulk dispersion coefficient
  respectively, both in units of $L^3 t^{-1}$.


  Volumetric transport implies the use of total flows (mass per unit of time)
  rather than fluxes (mass per unit of area per unit of time) as is done
  in \code{tran.1D}.

  \code{tran.volume.1D} implements this in \rt.

  The \code{tran.volume.1D} routine is particularly suited for modelling
  channels (like rivers, estuaries) where the cross-sectional area changes,
  but where this area change is not explicitly modelled (as a function of time).

  \subsection{R-function tran.volume.1D}
  The default input for the \code{tran.volume1D()} function in \proglang{R} is:
  \begin{verbatim}
    tran.volume.1D(C, C.up = C[1], C.down = C[length(C)],
      C.lat = 0, F.up = NULL, F.down = NULL, F.lat = NULL,
      Disp = NULL, flow = 0, flow.lat = NULL, AFDW = 1, V = NULL,
      full.check = FALSE, full.output = FALSE)
  \end{verbatim}
  with the following arguments:
  \begin{itemize}
    \item \code{C, C.up, C.down}. The concentrations in the centre of each
      grid cell, a vector of length N (\code{C}) and at the upstream or
      downstream boundary, one value (\code{C.up, C.down}).
    \item  \code{F.up, F.down}.  Total input at the upstream and
      downstream boundaries, ($M t^{-1}$).
    \item  \code{F.lat}.  Total input laterally, defined at the grid cells,
      ($M t^{-1}$).
    \item \code{Disp}, the bulk dispersion coefficients, ($L^3 t^{-1}$).
    \item \code{Q}, the water flow rate, ($L^3 t^{-1}$).
    \item \code{AFDW} the weights used in the finite difference
      approximation for flow (-).
    \item \code{V}, the volume of each grid cell ($L^3$).
    \item \code{full.check}, when \code{TRUE}, the consistency of the
      input is checked.
    \item \code{full.output}, when \code{TRUE} full output is returned.
  \end{itemize}

  \subsection{An estuarine model}
  Consider the following example that models organic carbon decay in an
  estuary.

  Two scenarios are simulated: the baseline includes only input
  of organic matter at the upstream boundary. The second scenario simulates the
  input of an important side river halfway the estuary.

  The model is formulated in function \code{river.model}

<<>>=
river.model <- function (t = 0, OC, pars = NULL)  {

  tran <- tran.volume.1D(C = OC, F.up = F.OC, F.lat = F.lat, 
                         Disp = Disp, flow = flow, V = Volume,
                         full.output = TRUE)
  reac <- - k*OC

  return(list(dCdt = tran$dC + reac,
            F.up = tran$F.up, F.down = tran$F.down,
            F.lat = tran$F.lat))
}
@

  The estuary is 100 km long (\code{lengthEstuary}; it is
  subdivided in 500 grid cells (\code{nbox}).
<<>>=
nbox          <- 500                # number of grid cells
lengthEstuary <- 100000             # length of estuary [m]
BoxLength     <- lengthEstuary/nbox # [m]
@
  The estuarine cross-sectional area widens sigmoidally towards
  the estuarine mouth (\code{CrossArea}); based on this area and
  the lenght of a box, the volume of each box is easily estimated.
<<>>=
Distance      <- seq(BoxLength/2, by = BoxLength, len = nbox) # [m]

CrossArea <- 4000 + 72000 * Distance^5 /(Distance^5+50000^5)

Volume  <- CrossArea*BoxLength
@

  The dispersion coefficient (\code{Disp}) and the upstream flow rate
  (\code{flow}) are parameters.
<<>>=
Disp    <- 1000   # m3/s, bulk dispersion coefficient
flow    <- 180    # m3/s, mean river flow
@
  The organic carbon input on upstream boundary (\code{F.OC}),
  the lateral input of carbon (\code{F.lat.0}) and the decay rate of
  organic carbon (\code{k}) are declared next:
<<>>=
F.OC    <- 180               # input organic carbon [mol s-1]
F.lat.0 <- F.OC              # lateral input organic carbon [mol s-1]

k       <- 10/(365*24*3600)  # decay constant organic carbon [s-1]
@
  In the first scenario, the lateral flux of material is zero.
<<>>=
F.lat <- rep(0, length.out = nbox)
@
  The model is solved using \rs function \code{steady.1D} which finds the
  steady-state solution, given an initial guess, \code{y}
  (here simply 500 random numbers).
<<>>=
sol  <- steady.1D(y = runif(nbox), fun = river.model, nspec = 1,
                  atol = 1e-15, rtol = 1e-15, positive = TRUE)
@
  In the second scenario, there is lateral input of organic carbon:
<<>>=
F.lat <- F.lat.0*dnorm(x = Distance/lengthEstuary,
                       mean = Distance[nbox/2]/lengthEstuary,
                       sd = 1/20, log = FALSE) /nbox
sol2 <- steady.1D(y = runif(nbox), fun = river.model, nspec = 1, 
                  atol = 1e-15, rtol = 1e-15, positive = TRUE)
@
We set the tolerances for the steady-state calculation (\code{atol} and
\code {rtol}) to a very small value so that the mass budget is very tight
(see below).

The summary for the outputs shows the ranges, means and standard deviations
of all quantities:
<<>>=
summary(sol)
summary(sol2)
@
Finally the output is plotted, using rootSolve's \code{plot} method.

<<label=est,include=FALSE>>=
plot(sol, sol2, grid = Distance/1000, lwd = 2,
        main = "Organic carbon decay in an estuary",xlab = "distance [km]",
        ylab = "OC Concentration [mM]")
legend ("topright", col = 1:2, lwd = 2, c("baseline", "with lateral input"))
@
\setkeys{Gin}{width=0.4\textwidth}
\begin{figure}
\begin{center}
<<label=est,fig=TRUE,echo=FALSE>>=
<<est>>
@
\end{center}
\caption{Solution of the estuarine model - see text for \R-code}
\label{fig:st1}
\end{figure}

and a budget estimated
(total input = consumption)
<<>>=
sum(sol$F.up) + sum(sol$F.lat) - sum(sol$F.down) - sum(sol$y*k*Volume)
@
\clearpage
  \section{Transport in two dimensions}
  
  The function that performs transport in two
  dimensions is similar to its 1-D equivalent.
  
  Here is its default input
\begin{verbatim}
tran.2D ( C, C.x.up = C[1,], C.x.down = C[nrow(C),],
  C.y.up = C[,1], C.y.down = C[,ncol(C)],
  flux.x.up = NULL, flux.x.down = NULL, 
  flux.y.up = NULL, flux.y.down = NULL,
  a.bl.x.up = NULL, a.bl.x.down = NULL, 
  a.bl.y.up = NULL, a.bl.y.down = NULL,  
  D.grid = NULL, D.x = NULL, D.y = D.x,
  v.grid = NULL, v.x = 0, v.y = 0,
  AFDW.grid = NULL, AFDW.x = 1, AFDW.y = AFDW.x,
  VF.grid = NULL, VF.x = 1, VF.y = VF.x,
  A.grid = NULL, A.x = 1, A.y = 1,
  grid = NULL, dx = NULL, dy = NULL,
  full.check = FALSE, full.output = FALSE)

}
\end{verbatim}
  where
  \begin{itemize}
    \item \code{C, C.x.up, C.x.down, C.y.up, C.y.down} are the concentration in
      the 2-D grid (\code{C}) and the boundary concentrations
    \item \code{flux.x.up, flux.x.down, ...} are the fluxes
      prescribed at the boundaries,
    \item \code{a.bl.up, ...} are the exchange coefficients for
      transfer across the various boundary layers
    \item \code{D.x, ...} are the diffusion coefficients
    \item \code{v.x, ...} are the advective velocities
    \item \code{AFDW.x, ...} are the weights used in the numerical
      approximation of the advective component
    \item \code{VF.x, ...} are the volume fractions of the various phases
    \item \code{A.x, ...} are the surface areas
    \item \code{dx, dy, grid} define the discretisation grid
    \item \code{full.check, full.output} whether full output needs writing
      or the input needs checking
  \end{itemize}
  By making clever use of the surface areas, it is possible to describe
  reactive transport processes in a 2-D channel with variable surface area,
  e.g. for widening estuaries that are stratified.
  
  It is also possible to mimic certain 3-D applications, e.g. by assuming
  cylindrical symmetries.

  \subsection{example of transport in 2 dimensions}
  We model the dynamics of oxygen, on a 2-D grid, and subjected to diffusion
  in the two directions, to advection in y-direction, from left to right,
  first-order consumption and a source in a central spot (e.g. an
  animal ventilating its burrow).

  At the upper boundary, the concentration is 300 mM, the other boundaries
  are zero-flux boundaries.

  We start by defining the parameters and the grid
<<>>=
n     <- 100           # number of grid cells
dy    <- dx <- 100/n   # grid size

Dy    <- Dx <- 5   # diffusion coeff, X- and Y-direction
r     <- -0.02     # production/consumption rate
Bc    <- 300       # boundary concentration
irr   <- 20        # irrigation rate
vx    <- 1         # advection
@
  As initial concentrations we assume that oxygen is 0 everywhere.

<<>>=
y  <- matrix(nrow = n, ncol = n, 0)
@
  In the model function, the state variables are passed as one vector
  (\code{y}). They are recast as a matrix first (\code{CONC}).
<<>>=
Diff2D <- function (t, y, parms, N) {

  CONC <- matrix(nrow = N, ncol = N, y)
# Transport
  Tran    <-tran.2D(CONC, D.x = Dx, D.y = Dy, C.y.down = Bc,
                    dx = dx, dy = dy, v.x = vx)

# transport + reaction
  dCONC   <- Tran$dC + r*CONC

# Bioirrigation in a central spot
  mid <- N/2
  dCONC[mid, mid] <- dCONC[mid, mid]  + irr*(Bc - CONC[mid, mid])

  return (list(dCONC))
}
@
  The model is solved in two ways.
  \begin{itemize}
    \item \code{steady.2D} solves the steady-state condition
    \item \code{ode.2D} runs the model dynamically
  \end{itemize}
  The model consists of 100*100 = 10000 equations.
  We print the time it takes to obtain these solutions (in seconds).
  
  Note that for both these methods, we need to pass the dimension of the
  problem (e.g. the number of boxes in x and y-direction), and the work-
  space required (\code{lrw}).
  
<<>>=
print(system.time(
 std  <- steady.2D(func = Diff2D, y = as.vector(y), time = 0, N = n,
                   parms = NULL, lrw = 1000000, dimens = c(n, n),
                   nout = 0, positive = TRUE)
))
@
  We run the model for 200 time units, producing output every 5 units.
<<>>=
times <- seq(0, 100, 5)
print(system.time(
  out2 <- ode.2D(func = Diff2D, y = as.vector(y), times = times, N = n,
                 parms = NULL, lrw = 10000000, dimens = c(n, n))
))
@
We use method \code{select} to get the values of O2 after 20 days, and plot the 
concentrations using \code{filled.contour}
<<label=twod5,include=FALSE>>=
mat <- matrix(nrow = n, ncol = n, subset(out2, time ==  20))
filled.contour(mat, zlim = c(0, Bc), color = femmecol,
               main = "after 20 time units")
@
The same can be done using \code{deSolve}s \code{image} function:
\begin{verbatim}
image(out2, subset = time == 20, method = "filled.contour", main = "")
\end{verbatim}
\setkeys{Gin}{width=0.4\textwidth}
\begin{figure}
\begin{center}
<<label=twod,fig=TRUE,echo=FALSE>>=
<<twod5>>
@
\end{center}
\caption{Solution of the 2-dimensional diffusion and
irrigation model, after 20 time units - see text for \R-code}
\label{fig:twod}
\end{figure}
It is simple to plot the steady-state solution:
<<label=twodst,include=FALSE>>=
image (std, main = "steady-state", legend = TRUE)
@
\setkeys{Gin}{width=0.4\textwidth}
\begin{figure}
\begin{center}
<<label=twodst,fig=TRUE,echo=FALSE>>=
<<twodst>>
@
\end{center}
\caption{Steady-state solution of the 2-dimensional diffusion and
irrigation model - see text for \R-code}
\label{fig:twodst}
\end{figure}

  \section{finally}
This vignette was made with Sweave \citep{Leisch02}.


\clearpage
\bibliography{bibs}

\end{document}
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