1d verification examples - geo-slope...

GEO-SLOPE International Ltd, Calgary, Alberta, Canada www.geo-slope.com

CTRAN Example File: 1D Verification Examples (pdf)(gsz) of 12

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.



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]



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



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



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



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.



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



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



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



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)


2000 seconds

4000 seconds

6000 seconds

CTRAN 2000 sec

CTRAN 4000 sec

CTRAN 6000 sec



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 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



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)


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

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