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Units:
cm=centimeter, ft=foot, g=gram, gal=U.S. gallon, gpd=U.S. gallon per day, 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
Application Variable
Definitions Property Data Error Messages References
Introduction
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This calculation simulates one, two, or threedimensional 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. The calculation
simulates instantaneous injection of a chemical having a mass M. The
calculation solves for concentration at whatever time and distances are desired by the
user. It also can backcalculate mass or distances.
The calculation includes advection and dispersion. Advection is
chemical movement via groundwater flow due to the groundwater hydraulic (i.e. head)
gradient. Dispersion causes spreading of the contaminant. If there were no
dispersion, all of the contaminant would travel at the groundwater velocity. With
dispersion, some chemical travels faster and some slower than the mean velocity; the
chemical "spreads out." In two and threedimensional dispersion, the
chemical additionally disperses upward (and downward) and laterally, respectively,
relative to the plume centerline. The maximum concentration occurs at x=Vt,
y=0, and z=0. If you solve for x, y, or z, the calculation
provides two solutions since the plume spreads forward and backward (in the xdirection),
up and down (in the ydirection), and in and out (in the zdirection).
Equations
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Governing Equations
The governing equations for one, two, and three  dimensional chemical transport in
groundwater with advection and dispersion are (Bear, 1972):
Boundary Conditions and Solutions
Initially the aquifer has C=0 everywhere. Then at t=0 and x=0,
a chemical slug of mass, M, is injected instantaneously. If M is
injected uniformly across an aquifer's width and height, then there is dispersion only in
the xdirection (onedimensional dispersion). If M is injected
uniformly across the aquifer's width, then dispersion occurs in the x and y
directions (twodimensional dispersion). If M is injected at a point, then
dispersion occurs in all three dimensions x, y, and z.
The following solutions have been coded in our calculation (Bear, 1972):
where:
Application
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The following graph was developed to demonstrate effects of threedimensional dispersion
in a sandy aquifer as predicted by the calculation. The following data were used:
(Click for variable definitions)
M = 100 kg, t = 1000 days, y = z = 0, a_{y} = a_{z}
= a_{x} /10.
K = 0.001 cm/s, n = 35%, n_{e} = 25%, dh/dx = 0.007 m/m,
D* = 1.0x10^{9} m^{2}/s.
Therefore, V = 2.8x10^{7} m/s, and Vt = 24.192 m.
Figure 1. Concentration profile at 1000 days for an injection of
100 kg
Variable Definitions
Units: [L]=Length, [M]=Mass, [T]=Time
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The variables used on this web page are:
a_{x }, a_{y} , a_{z} = Dispersivities
in x, y, and z directions [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_{x}
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_{x}~0.1
m. However, if you are simulating transport in a large aquifer greater than 1 km in
extent, then use a_{x}~100 m. a_{y} and a_{z}
are approximately a_{x} /10 (Javandel et al., 1984, p. 12).
C = Chemical concentration [M/L^{3}].
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 = (h_{2}h_{1})/(x_{2}x_{1}).
D_{x }, D_{y} , D_{z} = Dispersion
coefficients in x, y, and z directions [L^{2}/T]. The equation D=a V +
D*/n_{e} shown above is adapted from Ingebritsen and Sanford (1998).
D* = Molecular diffusion coefficient [L^{2}/T].
Varies somewhat for different chemicals but a typical value to use is 1.0x10^{9}
m^{2}/s (Fetter, 1993).
H = Aquifer height [L]. User enters if onedimensional
dispersion.
K = Hydraulic conductivity of aquifer [L/T].
M = Chemical mass injected [M].
n = Total porosity of soil [%]. (Void volume/total volume)
x 100%.
n_{e} = 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. n_{e} is always ≤ n.
t = Time [T]. Time at which C is to be computed.
V = Pore water velocity in xdirection [L/T].
Also known as groundwater velocity.
W = Aquifer width [L]. User enters if one or
twodimensional dispersion.
x, y, z = Distances [L]. Distances at which to compute C.
x is the direction of groundwater flow. y is the vertical
distance from the centerline of the plume. z is lateral distance (distance
into "the computer monitor") from the plume centerline.
Property Data
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The following are tables of hydraulic conductivity, total porosity, and
effective porosity. Parameter values have been compiled from a variety of sources
such as Freeze and Cherry (1979), Hillel (1982), and Sanders (1998). The values
used in the calculation are typical numbers within the ranges given below.
Table of Soil Properties
Soil Type 
Hydraulic
Conductivity
K (cm/s) 
Total
Porosity
n (%) 
Effective
Porosity
n_{e} (%) 
Clayey 
10^{9}  10^{6} 
4060 
05 
Silty 
10^{7}  10^{3} 
3550 
320 
Sandy 
10^{5}  10^{1} 
2050 
1035 
Gravelly 
10^{1}  10^{2} 
2540 
1230 
Error Messages given by calculation
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"Cannot have 1D and Solve for y or z ." No
computations. For onedimensional dispersion, the chemical plume is uniformly
distributed across the aquifer's height and width. y and z are not
variables, so they cannot be computed.
"Cannot have 2D and Solve for z ." No
computations. For twodimensional dispersion, the chemical plume is uniformly
distributed across the aquifer's width. z is not a variable, so it cannot
be computed.
"H, W must be > 0." No computations.
This error message will only appear in onedimensional dispersion. Aquifer
height and width must be entered, and they must be positive.
"W must be > 0." No computations.
This error message will only appear in twodimensional dispersion. Aquifer width
must be entered, and it must be positive.
"Infeasible input." No computations.
This error message will only appear if x, y, or z is being computed. This message
will be shown if the ratio of mass to concentration entered is too low to be physically
achievable.
"Certain inputs must be ≥ 0." No
computations. C, D*, dh/dx, K, M, n, n_{e} , and t must
all be ≥ 0 if entered. x, y, and z can be
positive, negative, or zero.
"n and n_{e} must be ≤ 100%." No
computations. Total porosity and effective porosity cannot exceed 100%.
"n_{e} must be ≤ n." No
computations. Effective porosity cannot exceed total porosity.
References
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Bear, J. 1972. Dynamics of Fluids in Porous Media. American Elsevier
Pub. Co.
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.
Javandel, I, C. Doughty, and C. F. Tsang. 1984. Groundwater Transport:
Handbook of Mathematical Models. American Geophysical Union.
Sanders, L. L. 1998. A Manual of Field Hydrogeology. Prentice Hall,
Inc.
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Software, Ltd. All rights reserved.
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fax: (740) 5921890
LMNO@LMNOeng.com http://www.LMNOeng.com

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