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Topics on groundwater page: Introduction Equations
Definitions Property Data Error Messages References
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This groundwater calculation simulates one, two, or three-dimensional transport of a chemical in a
confined groundwater aquifer. It is also valid for transport in an unconfined groundwater
aquifer if the head gradient (dh/dx) is nearly constant. The groundwater calculation
simulates instantaneous injection of a chemical having a mass M. The groundwater
calculation solves for concentration at whatever time and distances are desired by the
user. The groundwater calculation also can back-calculate mass or distances.
The groundwater 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 three-dimensional dispersion, the
chemical additionally disperses upward (and downward) and laterally, respectively,
relative to the groundwater plume centerline. The maximum concentration occurs at x=Vt,
y=0, and z=0. If you solve for x, y, or z, the groundwater calculation
provides two solutions since the plume spreads forward and backward (in the x-direction),
up and down (in the y-direction), and in and out (in the z-direction).
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Groundwater Governing Equations
The governing equations for one, two, and three - dimensional chemical transport in
groundwater with advection and dispersion are (Bear, 1972):
Groundwater Boundary Conditions and Solutions
Initially the groundwater aquifer has C=0 everywhere. Then at t=0 and x=0,
a chemical slug of mass, M, is injected instantaneously in the groundwater. If M is
injected uniformly across a groundwater aquifer's width and height, then there is dispersion only in
the x-direction (one-dimensional dispersion). If M is injected
uniformly across the groundwater aquifer's width, then dispersion occurs in the x and y
directions (two-dimensional dispersion). If M is injected at a point in the groundwater, then
dispersion occurs in all three dimensions x, y, and z.
The following solutions have been coded in our groundwater calculation (Bear, 1972):
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The following graph was developed to demonstrate effects of three-dimensional dispersion
in a sandy groundwater aquifer as predicted by the calculation. The following data were used:
(Click for variable definitions)
M = 100 kg, t = 1000 days, y = z = 0, ay = az
= ax /10.
K = 0.001 cm/s, n = 35%, ne = 25%, dh/dx = -0.007 m/m,
D* = 1.0x10-9 m2/s.
Therefore, V = 2.8x10-7 m/s, and Vt = 24.192 m.
Figure 1. Concentration profile at 1000 days for an injection of
Groundwater Variable Definitions
Units: [L]=Length, [M]=Mass, [T]=Time
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The variables used on this groundwater web page are:
ax , ay , az = 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 groundwater test. Large scale
groundwater tests have higher a than small lab column tests. An approximate value for ax
is 0.1 times the scale of your system (Fetter, 1993). If you are simulating
groundwater contaminant transport in a 1 m long laboratory column, then ax~0.1
m. However, if you are simulating transport in a large groundwater aquifer greater than 1 km in
extent, then use ax~100 m. ay and az
are approximately ax /10 (Javandel et al., 1984, p. 12).
C = Chemical concentration [M/L3].
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 = (h2-h1)/(x2-x1).
Dx , Dy , Dz = Dispersion
coefficients in x, y, and z directions [L2/T]. The equation D=a V +
D*/ne shown above 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).
H = Groundwater aquifer height [L]. User enters if one-dimensional
K = Hydraulic conductivity of aquifer [L/T].
M = Chemical mass injected [M].
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.
t = Time [T]. Time at which C is to be computed.
V = Pore water velocity in x-direction [L/T].
Also known as groundwater velocity.
W = Aquifer width [L]. User enters if one or
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.
Groundwater Property Data
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The following are tables of groundwater 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 groundwater calculation 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
Error Messages given by groundwater calculation
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"Cannot have 1D and Solve for y or z ." No
computations. For one-dimensional dispersion, the chemical plume is uniformly
distributed across the groundwater 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 two-dimensional dispersion, the chemical plume is uniformly
distributed across the groundwater aquifer's width. z is not a variable, so it cannot
"H, W must be > 0." No computations.
This error message will only appear in one-dimensional dispersion. Groundwater 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 two-dimensional dispersion. Groundwater 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
"Certain inputs must be ≥ 0." No
computations. C, D*, dh/dx, K, M, n, ne , and t must
all be ≥ 0 if entered. x, y, and z can be
positive, negative, or zero.
"n and ne must be ≤ 100%." No
computations. Total porosity and effective porosity cannot exceed 100%.
"ne must be ≤ n." No
computations. Effective porosity cannot exceed total porosity.
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Bear, J. 1972. Dynamics of Fluids in Porous Media. American Elsevier
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.
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,
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Software, Ltd. All rights reserved.
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LMNO Engineering home page (more calculations)
1-D Step injection with advection, dispersion, retardation
Compute gradient from well head measurements
Drawdown for unsteady groundwater flow to pumping well in confined aquifer (Theis calculator)
Transmissivity T=Kb (and K table)