Groundwater Contaminant Transport Calculation:
Step Injection

Chemical injected during a finite time period.  Compute concentration as a function of time and distance or back-calculate injected concentration.  Advection, dispersion, and retardation.

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Contaminant transport

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Topics on this page:   Introduction    Equations    Applications    Variable Definitions   Property Data    Error Messages    References

Introduction                                                Top of Page
This calculation simulates one-dimensional (x-direction) transport of a chemical in a confined groundwater aquifer.  It is also valid for transport in an unconfined aquifer if the head gradient (dh/dx) is nearly constant.  There are two common boundary conditions for chemical transport:   One is a step (i.e. continuous) injection of chemical - the chemical is added at x=0 from time t=0 to t=T.  The other common boundary condition is a pulse input where a mass of chemical is added instantaneously at x=0.  This web page uses the first boundary condition though a pulse input can be simulated by using a short injection time T.  The calculation solves for concentration at whatever time and distance is desired by the user.

The calculation includes advection, dispersion, and retardation.  Advection is chemical movement via groundwater flow due to the groundwater hydraulic (i.e. head) gradient.  Dispersion is the longitudinal (forward and backward) spreading of the contaminant.  If there were no dispersion, all of the contaminant would travel at the mean chemical velocity.  With dispersion, some chemical travels faster and some slower than the mean velocity; the chemical "spreads out."  Retardation causes the mean chemical velocity to be slower than the groundwater velocity.  If your chemical exhibits no dispersion, set both the dispersivity (a) and diffusion coefficient (D*) to zero.  If the chemical is not retarded, then uncheck the retardation check box or use the chemical drop-down menu to select "User enters Koc" and set  Koc =0.0.

Equations                                                         Top of Page

Governing Equation and Boundary Conditions
The governing equation for one-dimensional chemical transport in groundwater with advection, dispersion, and retardation is (Van Genuchten and Alves, 1982):

Governing equation for contaminant transport

Solution
The solution to the governing equation and boundary conditions shown above is (Van Genuchten and Alves, 1982):

Solution for contaminant transport

erfc( ) is called the "complementary error function."  Our calculation uses the most accurate numerical representation of erfc( ) given in Abramowitz and Stegun (1972, eqn 7.1.26).

Applications                                                             Top of Page
The following graphs were developed to demonstrate effects of dispersion for trichloroethylene (TCE) in a sandy aquifer as predicted by the calculation.  The following data were used:
Co = 10,000 mg/l, d = 1.6 g/cm3, dh/dx = -0.007 m/m, D* = 0,   foc = 0.1%,
K = 0.001 cm/s,  Koc = 100 cm3/g, n = 35%, ne = 25%.
    Click for variable definitions
Therefore, Kd = 0.1 cm3/g,  Rf  = 1.46,   Vw = 2.8x10-7 m/s,  and Vc = 1.92x10-7 m/s.

Two injection durations were used:  T=1,000,000 days in Figure 1 and T=100 days in Figure 2.  For Figure 1, T was selected large enough to simulate an infinite duration injection.  In both figures, the concentration front occurs at  x = Vc t = 16.6 m  when  a=0.   In Figure 2, the trailing edge of the concentration front occurs at  x = Vc( t-T) =14.9 m  when a=0.   a is dispersivity.

Figure 1.  TCE concentration profile at 1000 days for an injection of duration 1,000,000 days

Long duration step injection

Figure 2.  TCE concentration profile at 1000 days for an injection of duration 100 days.

Short duration step injection

Variable Definitions                                                  Top of Page
The variables used on this web page are:
a = dispersivity [L].  Varies from 0.1 to 100 m.   Field and laboratory tests have indicated that a varies with the scale of the test.  Large scale tests have higher a than small lab column tests.  An approximate value for a is 0.1 times the scale of your system (Fetter, 1993).   If you are simulating contaminant transport in a 1 m long laboratory column, then a~0.1 m.  However, if you are simulating transport in a large aquifer greater than 1 km in extent, then use a~100 m.
C = Chemical concentration [M/L3].
Co = Injected concentration at x=0 [M/L3].
d = Dry bulk density of the aquifer [M/L3].
dh/dx = Hydraulic (or head) gradient [L/L].  Since dh/dx is negative, we ask you to enter -dh/dx so that you can enter a positive number for convenience.  You determine dh/dx from two head measurements using the equation, dh/dx = (h2-h1)/(x2-x1).
D = Dispersion coefficient [L2/T].  The equation D=a Vc + D*/ne is adapted from Ingebritsen and Sanford (1998).
D* = Molecular diffusion coefficient [L2/T].  Varies somewhat for different chemicals but a typical value to use is 1.0x10-9 m2/s (Fetter, 1993).
foc = Organic carbon fraction in soil [%].  (Mass organic carbon per mass soil) x 100%.
K = Hydraulic conductivity of aquifer [L/T].
Kd = Distribution coefficient [L3/M].  Represents chemical partitioning between groundwater and soil.
Koc = Organic carbon partition coefficient [L3/M].   Represents chemical partitioning between organic carbon and water in soil.  Good discussion in Lyman et al. (1982).
n = Total porosity of soil [%].  (Void volume/total volume) x 100%.
ne = Effective porosity [%].  Porosity through which flow can occur.  A thin film of water bound to soil particles by capillary forces does not move through the aquifer.  ne is always <= n.
Pe = Peclet number.  Pe=(Vc x ) / D.  It is a commonly used dimensionless parameter indicating the relative impact of inertial effects to dispersive effects.
Rf  = Retardation factor.  Rf =1 if there is no retardation which implies that Vc=VxRf =1 would occur for a conservative tracer; that is, a tracer that does not sorb to the aquifer soil.
t = Time [T].  Time at which C is to be computed.
T = Duration of injection [T].  Co is injected from t=0 to t=T.
Vc = Mean chemical velocity [L/T].
Vw = Pore water velocity [L/T].  Also known as groundwater velocity.
x = Distance [L].  Distance at which to compute C.

Property Data                                                                Top of Page
The following are tables of hydraulic conductivity, total porosity, effective porosity, bulk density, and organic carbon partition coefficient.   Parameter values have been compiled from a variety of sources such as Coduto (1994), Fetter (1993), Freeze and Cherry (1979), Hillel (1982), and Sanders (1998).   The values used in the calculation above are typical numbers within the ranges given below.

Table of Soil Properties

Soil Type Hydraulic
Conductivity
K (cm/s)
Total
Porosity
n (%)
Effective
Porosity

ne (%)
Bulk
Density

d (g/cm3)
Clayey 10-9 - 10-6 40-60 0-5 1.2-1.8
Silty 10-7 - 10-3 35-50 3-20 1.1-1.8
Sandy 10-5 - 10-1 20-50 10-35 1.3-1.9
Gravelly 10-1 - 102 25-40 12-30 1.6-2.1


Table of Organic Carbon Partition Coefficient, Koc

Chemical Koc (cm3/g) Chemical Koc (cm3/g)
Benzene 20-400 Pyrene 2000-200,000
Ethyl Benzene 90-1500 Tetrachloroethylene 100-3000
Dichloroethane 10-250 Trichloroethylene 20-500
Naphthalene 100-2500    

 

Error Messages given by calculation                            Top of Page
"Certain inputs must be > 0."  No computations.   Co , C, d, -dh/dx, K, n, ne, and T must all be  > 0 if entered.

"Certain inputs must be >= 0."  No computations.  a, D*,  foc , Koct, and x must all be >= 0 if entered.

"n, ne , and  foc must be <= 100%."  No computations.   Total porosity, effective porosity, and soil organic carbon cannot exceed 100%.

"ne must be <= n."  No computations.   Effective porosity cannot exceed total porosity.

"Co cannot be determined" or "Co=infinity."  Co not computed.  Certain input combinations result in computing erfc(infinity), and erfc(infinity)=0.0.  Therefore, the equation Co=C/0.0   results, in which Co cannot be evaluated or is evaluated as infinity.


References  
                                                        Top of Page
Abramowitz, M. and I. A. Stegun.  1972.  Handbook of Mathematical Functions.   Dover Publications, Inc.

Coduto, D. P.  1994.  Foundation Design Principles and Practices.  Prentice Hall, Inc.

Fetter, C. W.  1993.  Contaminant Hydrogeology.  Macmillan Pub. Co.

Freeze, R. A. and J. A. Cherry.  1979.  Groundwater.  Prentice Hall, Inc.

Hillel, D. H.  1982.  Introduction to Soil Physics.  Academic Press, Inc.

Ingebritsen, S. E. and W. E. Sanford.  1998.  Groundwater in Geologic Processes.  Cambridge University Press.

Lyman, W. J.  Adsorption coefficient for soils and sediments.  In Handbook of Chemical Property Estimation Methods.  Lyman, W. J., W. F. Reehl, and D. H. Rosenblatt, eds.  McGraw-Hill Book Co.  1982.  pp. 4-1 thru 4-33.

Sanders, L. L.  1998.  A Manual of Field Hydrogeology.  Prentice Hall, Inc..

Van Genuchten, M. Th. and W. J. Alves.  1982.  Analytical Solutions of the One-Dimensional Convective-Dispersive Solute Transport Equation.  United States Department of Agriculture, Agricultural Research Service, Technical Bulletin 1661.

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