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Groundwater Units:
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
Groundwater topics on this page: Introduction Equations
Applications Variable
Definitions Property Data Error Messages References
Groundwater Introduction
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This groundwater calculation simulates onedimensional (xdirection) transport of a chemical
in a confined groundwater aquifer. It is also valid for transport in an unconfined
aquifer if the groundwater 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 groundwater calculator uses the first boundary condition
though a pulse input can be simulated by using a short injection time T.
The calculator solves for concentration at whatever time and distance is desired by the
user.
The groundwater 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 in groundwater. 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 dropdown menu to select "User enters K_{oc}"
and set K_{oc} =0.0.
Groundwater Equations
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Groundwater Governing Equation and Boundary Conditions
The governing equation for onedimensional chemical transport in groundwater with
advection, dispersion, and retardation is (Van Genuchten and Alves, 1982):
Groundwater Solution
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
groundwater calculation uses the most accurate numerical representation of erfc( ) given in Abramowitz
and Stegun (1972, eqn 7.1.26).
Groundwater Applications
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The following graphs were developed to demonstrate effects of dispersion for
trichloroethylene (TCE) in a sandy groundwater aquifer as predicted by the groundwater calculation. The
following data were used:
C_{o} = 10,000 mg/l, d = 1.6 g/cm^{3}, dh/dx = 0.007 m/m, D* =
0, f_{oc} = 0.1%,
K = 0.001 cm/s, K_{oc} = 100 cm^{3}/g, n = 35%, n_{e} = 25%.
Click for groundwater variable definitions
Therefore, K_{d} = 0.1 cm^{3}/g, R_{f} = 1.46,
V_{w} = 2.8x10^{7} m/s, and V_{c} = 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 = V_{c} t = 16.6 m when a=0.
In Figure 2, the trailing edge of the concentration front occurs at x = V_{c}(
tT) =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.
Groundwater Variable Definitions Units: [L]=Length, [M]=Mass, [T]=Time
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The variables used in the groundwater calculator 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 groundwater tests have higher a than small lab column tests. An
approximate value for a is 0.1 times the scale of your groundwater system (Fetter, 1993).
If you are simulating groundwater contaminant transport in a 1 m long laboratory column, then a~0.1
m. However, if you are simulating groundwater transport in a large aquifer greater than 1 km in
extent, then use a~100 m.
C = Chemical concentration [M/L^{3}].
C_{o} = Injected concentration at x=0 [M/L^{3}].
d = Dry bulk density of the groundwater aquifer [M/L^{3}].
dh/dx = Groundwater 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 = Dispersion coefficient [L^{2}/T]. The equation
D=a V_{c} + D*/n_{e} 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).
f_{oc} = Organic carbon fraction in soil [%].
(Mass organic carbon per mass soil) x 100%.
K = Hydraulic conductivity of aquifer [L/T].
K_{d} = Distribution coefficient [L^{3}/M].
Represents chemical partitioning between groundwater and soil.
K_{oc} = Organic carbon partition coefficient [L^{3}/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%.
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 groundwater aquifer. n_{e} is always ≤ n.
Pe = Peclet number. Pe=(V_{c} x ) / D.
It is a commonly used dimensionless parameter indicating the relative impact of inertial
effects to dispersive effects.
R_{f} = Retardation factor. R_{f}
=1 if there is no retardation which implies that V_{c}=V_{w}.
R_{f} =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]. C_{o} is
injected from t=0 to t=T.
V_{c} = Mean chemical velocity [L/T].
V_{w} = Pore water velocity [L/T]. Also known as
groundwater velocity.
x = Distance [L]. Distance at which to compute C.
Groundwater Property Data
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The following are tables of groundwater hydraulic conductivity, total porosity,
effective porosity, bulk density, and organic carbon partition coefficient.
Groundwater 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 groundwater 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
n_{e} (%) 
Bulk
Density
d (g/cm^{3}) 
Clayey 
10^{9}  10^{6} 
4060 
05 
1.21.8 
Silty 
10^{7}  10^{3} 
3550 
320 
1.11.8 
Sandy 
10^{5}  10^{1} 
2050 
1035 
1.31.9 
Gravelly 
10^{1}  10^{2} 
2540 
1230 
1.62.1 
Table of Organic Carbon Partition Coefficient, K_{oc}
Chemical 
K_{oc} (cm^{3}/g) 
Chemical 
K_{oc} (cm^{3}/g) 
Benzene 
20400 
Pyrene 
2000200,000 
Ethyl Benzene 
901500 
Tetrachloroethylene 
1003000 
Dichloroethane 
10250 
Trichloroethylene 
20500 
Naphthalene 
1002500 


Error Messages given by groundwater calculation
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"Certain inputs must be > 0." No
groundwater computations. C_{o} , C, d, dh/dx, K, n, n_{e}, and T
must all be > 0 if entered.
"Certain inputs must be ≥ 0." No
groundwater computations. a, D*, f_{oc} , K_{oc}, t,
and x must all be ≥ 0 if entered.
"n, n_{e} , and f_{oc} must be ≤ 100%."
No groundwater computations. Total porosity, effective porosity, and soil organic carbon cannot
exceed 100%.
"n_{e} must be ≤ n." No
groundwater computations. Effective porosity cannot exceed total porosity.
"C_{o} cannot be determined" or
"C_{o}=∞." C_{o} not
computed. Certain input combinations result in computing erfc(∞), and
erfc(∞)=0.0. Therefore, C_{o} cannot be evaluated or is evaluated as infinity.
"erfc(x,t)=0. Cannot compute C_{o}",
"efrc(x,tT)=0. Cannot compute C_{o}",
"A(x,t)=0. Cannot compute C_{o}", or
"A(x,t)A(x,tT)=0. Cannot compute C_{o}."
C_{o} not computed since division by zero occurs.
Groundwater References
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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,
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. McGrawHill Book Co. 1982. pp. 41 thru 433.
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
OneDimensional ConvectiveDispersive Solute Transport Equation. United States
Department of Agriculture, Agricultural Research Service, Technical Bulletin 1661.
© 20032023 LMNO Engineering, Research, and
Software, Ltd. All rights reserved.
Please contact us for consulting or questions about groundwater contaminant transport with step injection.
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 7072614
LMNO@LMNOeng.com https://www.LMNOeng.com

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