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Ground Water Contaminant One-Dimensional Model

by Daniel Gallagher

 

 


Click on image to see sample screen.

 The goal of this program is to graph the analytical solution to the advective-dispersive equation for ground water flow.  The model assumes a homogeneous, 1-dimensional aquifer.  Either the concentration profile with distance, or the concentration profile with time can be graphed.  Inputs include distance or time, average linear velocity (not Darcy velocity), dispersion coefficient, retardation factor (for sorption), decay rate (for 1st order decay), and duration of the release.  All the parameters are set using sliders.  Simply move the slider bar up and down to change the parameter value.  The graph is automatically redrawn as you move the slider.  This approach allows for the sensitivity of the predicted concentrations to be easily assessed.  Boxes at the top and bottom of the sliders can be used to set the range for each parameter.

Only one of the time and distance sliders is available depending on the x-axis.  For example, if you are plotting concentration with distance, the distance slider is automatically turned off. 

If the retardation factor is set greater than 1, or the decay rate is greater than 0, two predicted curves are shown to illustrate the impact that these mechanisms have on the concentration profile.  The curve in red is for the contaminant with sorption and decay.  The curve in blue is for a nonsorbing, nonreacting contaminant. 

The model assumes a contaminant release over a fixed duration.  To model a continuous release, set the duration to be greater than the time on the graph.  Although the duration can be set quite low, the model is not designed to handle an instantaneous spill.

 

The one-dimensional advection-dispersion equation with constant velocity and dispersion coefficient is

     

Assuming:

  • 1-dimensional
  • homogenous saturated media
  • steady state flow

    •  

    where

       =  average linear velocity = v/n

        where v = Q/A

      C = liquid phase concentration

      D = dispersion coefficient

      rb = bulk phase  density

      n = porosity

      k = 1st order decay rate

       

       

       

The solid phase concentration S needs to be converted to an aqueous concentration

     

If adsorption/desorption is fast and reversible, an isotherm can apply, S/C represents the isotherm or partitioning.

If a linear isotherm is used

     

Substituting into the mass transport equation yields

     

    Rearranging terms yields

       

      Or

       

      Where

         

        and

         

        and

         

The retardation factor is

 

 

Sometimes, the advective-dispersive equation for contaminant transport in ground water can be solved analytically. Such a solution generally requires extreme simplifications, but the results can still be used for approximate solutions. They are also very useful to illustrate the impact that the different parameters have on results. Numerical models (finite difference or finite element) are more commonly used for real world problems.

Below are some analytical solutions for a conservative (k=0), nonsorbing (R=1) contaminant.

1) fixed concentration boundary condition (step input) (type 1 in Fetter)

    Boundary conditions and initial conditions are:

    • C (x, 0) = 0 for x > 0
    • C (0,t) = C0 for t > 0 (the step  input)
    • C („ , t) = 0 for t > 0

    The solution is:

       

 

      if Dx, x, or t is very large, second term Ž 0

 

          ERF and ERFC: error function and complimentary  error function often appear in solutions to PDE  describing dispersion and chemical reactions.

            • The definition of erf(x) is

              •  

              where y is dummy integration variable

              erf (0) = 0 erf („ ) = 1 erf (-x) = - erf (x)

            • The complimentary error function is

              • erfc(x) = 1 - erf(x)

 

2) fixed gradient boundary condition (type 2 in Fetter)

    • C (x, 0) = 0 for x > 0
    •   for t>0
    • C („ , t) = 0 for t > 0

    The solution is:

       

 

3) variable flux boundary condition (type 3 in Fetter)

  • C (x, 0) = 0 for x > 0

      •  

       

    The solution is:

       

INSTRUCTIONS

  1. Web Based Version

The model will run within your browser if you are  using MS Internet Explorer 4 or higher. This version uses  several ActiveX controls and is not compatible with  Netscape or other browsers. The first time you run the  model, the download times may be rather long. After that,  the model should download very quickly, since only  changes are checked. You will need to set your browser's security setting  to low the first time you run the model. I urge you to  reset it back to medium afterwards.

NOTE:  ActiveX Document technology is somewhat temperamental at this point and may not be able to run on your particular machine.  If a 'File Download' dialog box appears, press 'Cancel' and press the button below again.  If a 'Security Warning' dialog box appears, press 'Yes'.  You may need to try this sequence a couple times.  If still unsuccessful, download the Setup version, by pressing the 'Download Version' button below.

  1. Download Version

A downloadable version is available for Windows 9x or  Windows NT.

Comments:
     Daniel Gallagher
     Department of Civil and Environmental Engineering
     Virginia Tech
     Blacksburg, VA 24061-0246
     email: dang@vt.edu

Copyright © 1998 Daniel Gallagher
Last Modified: 10-1-1999

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