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cm=centimeter, ft=foot, g=gram, gal=U.S. gallon, hr=hour, kg=kilogram, km=kilometer, l=liter, lb=pound, m=meter, mg=milligram, min=minute, mm=millimeter, ppm=part per million (by mass), ppb=part per billion (by mass), s=second, yr=year, ug=microgram
Topics on this page: Introduction Equations
Definitions Property Data Error Messages References
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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
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.
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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):
The solution to the governing equation and boundary conditions shown above is (Van
Genuchten and Alves, 1982):
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).
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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
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
Figure 2. TCE concentration profile at 1000 days for an injection
of duration 100 days.
Variable Definitions Units: [L]=Length, [M]=Mass, [T]=Time
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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
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)
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=Vw.
Rf =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
x = Distance [L]. Distance at which to compute C.
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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
||10-9 - 10-6
||10-7 - 10-3
||10-5 - 10-1
||10-1 - 102
Table of Organic Carbon Partition Coefficient, Koc
Error Messages given by calculation
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"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 , Koc, t,
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
"ne must be ≤ n." No
computations. Effective porosity cannot exceed total porosity.
"Co cannot be determined" or
"Co=∞." Co not
computed. Certain input combinations result in computing erfc(∞), and
erfc(∞)=0.0. Therefore, Co cannot be evaluated or is evaluated as infinity.
"erfc(x,t)=0. Cannot compute Co",
"efrc(x,t-T)=0. Cannot compute Co",
"A(x,t)=0. Cannot compute Co", or
"A(x,t)-A(x,t-T)=0. Cannot compute Co."
Co not computed since division by zero occurs.
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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,
Hillel, D. H. 1982. Introduction to Soil Physics. Academic Press,
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,
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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