1d verification examples - geo-slope...
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1D Verification Examples
1 Introduction
Software verification involves comparing the numerical solution with an analytical solution. The
objective of this example is to compare the results from CTRAN/W analyses to well-established closed-
form solutions for four different transport scenarios. The transport scenarios include:
Case 1: no adsorption and no decay
Case 2: adsorption included
Case 3: decay included
Case 4: adsorption and decay included.
Equations for the closed-form solutions are solved using a Microsoft Excel spreadsheet. The solutions
require the use of a complimentary error function (erfc). This function is available in MS Excel 2007 as
part of the Analysis ToolPak Add-in, which can be installed under the Excel Options | Add-Ins dialogue
box .
2 Feature Highlights
GeoStudio feature highlights include:
1. Comparing CTRAN/W to closed-form solutions for the advection-dispersion equation;
2. Including adsorption and decay processes; and,
3. Comparison of backward difference and central difference iteration schemes.
3 Closed-Form Solutions
Closed-form analytical solutions to the advection-dispersion equation are available in the literature for
one-dimensional problems involving steady-state seepage flow. The advection-dispersion equation is
written as:
2
2L x d
C C C SnD nv n n C n S
x x t t
[1]
where
C = concentration of solute in liquid phase,
t = time,
DL = longitudinal hydrodynamic dispersion,
d = dry density,
n = porosity or volumetric water content (),
S = amount of solute sorbed per unit weight of soil, and
= coefficient of decay.
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The first and second terms on the left side of Equation 1 represent the dispersive and advective transport
of the solute, respectively. All of the terms on the right side account for changes in mass or concentration
that may occur with time due to mass flux (term 1), sorption processes (term 2), and radioactive decay of
mass in solution (term 3) and mass sorbed onto the solid (term 4).
3.1 Case 1: No Adsorption and No Decay
Ogata (1970) provided an analytical solution to the advection-dispersion equation for a homogeneous,
isotropic, and saturated porous geological medium. The concentration at any lateral distance for a given
elapsed time can be determined:
exp2 2 2
oC x vt vx x vtC erfc erfc
DDt Dt
[2]
where
C = concentration,
Co = specified concentration in the source boundary,
D = hydrodynamic dispersion coefficient,
v = average linear velocity,
t = elapsed time,
x = distance from the source boundary, and
erfc = complementary error function.
The solution assumes that the concentration at a distance of x = 0 is maintained at Co for all time (i.e.
C(0,t) = Co), the initial concentration everywhere in the flow domain is zero (i.e. C(x,0) = 0), and the flow
domain is infinitely long with a concentration of zero at the far boundary (i.e. C(,t) = 0). It should be
noted that the complimentary error function is related to the error function (erf) by the following
Erfc(x) = 1 – erf(x)
and that
Erf(0) = 0; erf() = 1; erf(–x) = –erf(x)
3.2 Case 2: Adsorption Included
Adsorption is the physical process by which a solute adheres to a solid surface. The relationship between
the mass of solute sorbed onto the solid (S) and the concentration of the solute (C) can take a variety of
linear and nonlinear forms (Figure 1). The slope of the linear sorption isotherm is often referred to as the
distribution coefficient (Kd). The amount of solute sorbed per dry unit weight of solid is given by
dS K C [3]
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Figure 1 - Linear and nonlinear sorption isotherms
If the equation for the linear sorption isotherm is substituted into the advection-dispersion equation [1]
and ignoring the decay terms, the governing partial differential equation is
2
2
d
L x d
K CC C CnD nv n
x x t t
[4]
Ignoring dispersive transport and dividing by the porosity (n) yields the following
1 dx d
C Cv K
x n t
[5]
where
1 ddK
n
is referred to as the retardation factor (R).
For this specific case, the sorption process reduces the velocity by a factor of R (i.e., vx/R). Accordingly,
the arrival time of the C/Co = 0.5 front for a transport problem involving advection-only is reduced by the
factor R.
The solution to Equation 4 for the same boundary conditions discussed above is given by (Bear, 1972)
exp
2 2 2
o
v vx t x tC vxR RC erfc erfc
DDt DtR R
[6]
S (g
/g)
Concentration (g/m3)
Linear Isotherm
Non-Linear Isotherm
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It should be noted that the governing partial differential equation is formulated in CTRAN using the more
general form shown in Equation 1. Accordingly, the formulation can accommodate unsaturated soil and
non-linear adsorption functions. The retardation factor is therefore given by
d
SR
C
[7]
where
= volumetric water content, and
S
C
= slope of the sorption isotherm.
3.3 Case 3: Decay Included
Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation.
This process will reduce the concentration of radionuclides in both the dissolved and adsorbed phases.
The coefficient of decay () in equation [1] is given by:
1/2
ln 2
t [8]
where
1/2t = half-life of the radionuclide.
Bear (1972, 1979) provided the following analytical solution subject to the same boundary conditions
discussed above
2 24 4
exp exp exp2 2 2 2
ox t v D x t v DC vx
C x erfc x erfcD Dt Dt
[9]
where
2
2
xv
D D
[10]
3.4 Case 4: Adsorption and Decay Included
If radioactive decay and adsorption are included, the analytical solution becomes (Bear, 1972, 1979)
2 24 4
exp exp exp2 2 2 2
o
v D v Dx t x tC vx R R R RC x erfc x erfc
D Dt DtR R
where
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2
2
v R
D D
4 Boundary Conditions and Material Properties
Figure 2 presents the geometry and mesh of the model domain. The model is comprised of a one-
dimensional column that is 3 m in length and 0.2 m in height. The mesh consists of 80 elements and 162
nodes. Although the dimensions are arbitrary, the column length was chosen to ensure that the ‘far-field’
boundary condition has no effect on the results. This condition is in keeping with the analytical
solutions, which assume that the C = 0 boundary is located at infinite distance.
Figure 2 Model geometry and mesh
A screen capture of the KeyIn Analyses dialogue box is presented in Figure 3. A steady-state seepage
analysis forms the ‘parent’ analysis for each transport model. In the seepage analysis, a unit flux of 1×10-
4 m/sec was applied to the left boundary and the right boundary was assigned a hydraulic head of 1 m.
The soil was assigned a hydraulic conductivity and saturated volumetric water content of 1×10-5
m/sec
and 0.5, respectively. Accordingly, the average linear velocity (v) in the flow domain is 2×10-4
m/sec (i.e.
v = q/n).
Figure 3 Model structure for the 1D verification example
Steady-State Seepage
1D Advection-Dispersion
Distance (m)
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
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For each transport model, the left and right boundaries were set to constant concentrations of 1.0 and 0.0
g/m3, respectively. Table 1 presents the material properties used in the analyses. The coefficient of
diffusion (D) and dispersivity were set to zero and 0.1 m for all cases, yielding a hydrodynamic dispersion
(v) of 2×10-5
m2/sec. A distribution coefficient Kd of 1×10
-6 m
3/g was used for the cases with
adsorption, producing a retardation factor R equal to 4.0. Note that the activation concentration is set to 0
g/m3 for each material under the KeyIn Materials dialogue box. Each model was run for an elapsed time
of 6000 seconds using 60 time steps.
Table 1 Material properties for transport analyses
Case Disp. (m) Kd (m3/g) Dry Density (g/m
3) Decay Half-life (sec)
Case 1: No Adsorp./No Decay 0.1 – – –
Case 2: Adsorp. Included 0.1 1×10-6
1.5×106 –
Case 3: Decay Included 0.1 – – 6931.5
Case 4: Adsorp. & Decay 0.1 1×10-6
1.5×106 6931.5
5 Results and Discussion
5.1 Case 1: No Adsorption and No Decay
Figure 4 presents results for the Case 1 analyses (no adsorption or decay) at elapsed times of 2000, 4000,
and 6000 seconds. CTRAN/W was solved using the backward difference time integration scheme. The
CTRAN/W results compare very well to the analytical solution; however, the analytical solution is
slightly steeper than CTRAN/W. In other words, the CTRAN/W solution is slightly more spread out or
‘smeared’ compared to the closed-form solution. This phenomenon is due to numerical dispersion, which
is inherent in the finite element solution of the transport equation.
A better match can be achieved when CTRAN/W is solved using the central difference time integration
scheme (Figure 5). In general, the central difference technique provides a better solution than using
backward difference for most transport problems. However, the central difference technique is
susceptible to numerical oscillation, which can cause the computed concentrations to be larger or smaller
than the specified maximum or minimum concentrations. Figure 6 shows a more extreme case of both
numerical dispersion and oscillation. Figure 1Numerical dispersion and oscillation can only be
minimized, not eliminated. Techniques for minimizing numerical dispersion and oscillation are presented
in the CTRAN/W documentation.
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Figure 4 Case 1 (no adsorption or decay) results: CTRAN/W solved using backward difference time integration scheme
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
0 0.5 1 1.5 2 2.5 3
C/C
o
Distance (m)
Case 1: Backward Difference
2000 seconds
4000 seconds
6000 seconds
CTRAN 2000 seconds
CTRAN 4000 seconds
CTRAN 6000 seconds
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Figure 5 Case 1 results: CTRAN/W solved using central difference time integration scheme
0.000
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
0 0.5 1 1.5 2 2.5 3
C/C
o
Distance (m)
Case 1: Central Difference
2000 seconds
4000 seconds
6000 seconds
CTRAN 2000 seconds
CTRAN 4000 seconds
CTRAN 6000 seconds
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Figure 6 Example of numerical dispersion and oscillation
5.2 Case 2: Adsorption Included
Figure 7 presents results for the Case 2 analyses, in which adsorption (R = 4.0) is included. CTRAN/W
was solved using the central difference time integration scheme. Note that the scale of the x-axis ranges
from 0 to 1 m. Again, CTRAN/W is in close agreement with the analytical solution. The adsorption
process slows the arrival front as anticipated. For example, the position of C/Co = 0.5 front at a time of
4000 seconds is about 0.27 m, compared to 0.90 m without adsorption (i.e., approximately one-quarter;
v*t/R). The small discrepancy is due to the inclusion of dispersivity in this example.
0.0
0.2
0.4
0.6
0.8
1.0
0 0.5 1 1.5 2 2.5 3
C/C
o
Distance (m)
6000 seconds
Backward Difference
Central Difference
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Figure 7 Case 2 results (adsorption included)
5.3 Case 3: Decay Included
Figure 8 presents the results for Case 3 (decay included) at an elapsed time of 4000 seconds. The results
from Case 1 are also included for comparison. CTRAN/W compares very well with closed-form
analytical solution. The largest effect of decay occurs near the source boundary where the concentration
is the highest. As the concentration approaches zero, the decay component has less effect as there is little
mass to decay.
The affects of decay are more apparent when a slug of contaminant is introduced into the system. This
was modeled by creating a region 0.1 m in length on the left side of the model domain. The region was
assigned the same material used for Case 3, but with an ‘activation concentration’ of 1.0 g/m3.
Accordingly, the initial mass in the system is calculated as 0.1 m × 0.2 m × 1 m × 0.5 × 1 g/m3 = 0.01 g
(i.e. length×height×unit width×porosity×Concentration). The column length was extended to 6 m and the
model was run for a time of 18,000 seconds. Figure 9 presents the concentration verses distance profiles
for three elapsed times. The amount of mass in the model domain (i.e. area under the curve) decreases
with time due to decay. This can be checked by hand-calculation via the relationship M=Moe-t
, where
Mo is the initial mass and is the coefficient of decay. For example, the calculated mass remaining in the
system after an elapsed time of 6900 seconds should be about 0.005 g using a = 1×10-4
s-1
. CTRAN/W
reports the same value for the ‘Total System Mass’ under ‘View Mass Accumulation’.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.2 0.4 0.6 0.8 1
C/C
o
Distance (m)
Case 2: Central Difference
2000 seconds
4000 seconds
6000 seconds
CTRAN 2000 sec
CTRAN 4000 sec
CTRAN 6000 sec
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Figure 8 Case 3 results (decay included)
Figure 9 Effects decay on the transport of a contaminant slug
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.5 1 1.5 2 2.5 3
C/C
o
Distance (m)
Case 3: Central Difference
Case 1: 4000 Seconds
4000 seconds
CTRAN 4000 sec
0
0.05
0.1
0.15
0.2
0.25
0 1 2 3 4 5 6
C/C
o
Distance (m)
900 sec
6900 sec
13800 sec
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5.4 Case 4: Adsorption and Decay Included
A comparison between CTRAN/W and the analytical solution for Case 4 is presented in Figure 10, along
with the results from Case 1. There is excellent agreement between the CTRAN/W solution and the
analytical solution. In fact, the two solutions are almost identical for all three elapsed times.
Furthermore, a comparison between Case 1 and 4 demonstrates that CTRAN/W correctly captures the
effect of both adsorption and decay. Mass is lost due to radioactive decay and the contaminant front is
slowed due to adsorption.
Figure 10 Case 4 results (adsorption and decay) compared to Case 1 (no adsorption/no decay)
6 Summary and Conclusions
In this example, results from CTRAN/W are compared to the closed-form solution of the advection-
dispersion equation for four different transport scenarios. The results demonstrate that CTRAN/W is
capable of modeling geochemical processes such as adsorption and decay. In general, the results from
CTRAN/W provide a better-match to the analytical solution using the central difference time integration
scheme.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 0.5 1 1.5 2 2.5 3
C/C
o
Distance (m)
Case 4: Central Difference
2000 seconds
4000 seconds
6000 seconds
CTRAN 2000 sec
CTRAN 4000 sec
CTRAN 6000 sec
Case 1: 2000 sec
Case 2: 4000 sec
Case 3: 6000 sec