Advection versus Diffusion in the Invert Rev 00, ICN 00

ANL-EBS-MD-000063

August 2003

1. PURPOSE 
The purpose of this analysis report is to provide supporting analyses to several other reports 
regarding Engineered Barrier System (EBS) Flow and Transport as described in Technical Work 
Plan for: Engineered Barrier System Department Modeling and Testing FY03 Work Activities 
(BSC 2003a). These reports include revisions to Multiscale Thermohydrologic Model 
(BSC 2001a) and EBS Radionuclide Transport Abstraction (BSC 2003b). Multiscale 
Thermohydrologic Model (BSC 2001a) requires hydrological properties for a dual-porosity 
material in the invert comprised of crushed tuff. This report develops the retention and 
unsaturated flow properties for a dual-porosity media that includes an intragranular porosity for 
crushed tuff and an intergranular porosity between particles for crushed tuff: 
� Intergranular permeability of the crushed tuff 
� Intergranular saturated moisture content 
� Intergranular porosity 
� van Genuchten properties 
� Residual moisture content 
� Maximum saturation 
� Residual saturation. 
This analysis report also supports EBS Radionuclide Transport Abstraction (BSC 2003b) in 
determining whether advection or diffusion in each media is the dominant transport mechanism. 
The analysis of advection or unsaturated flow through the invert requires that the retention and 
flow properties of the invert be used in a flow analysis in conjunction with initial conditions and 
boundary conditions. For this purpose, the NUFT software code was used to analyze the dualcontinuum 
media. The resulting analysis indicates the degree to which the fine intragranular 
porosity and the coarse intergranular porosity retain and flow water, and thus it provides an 
estimate of advection through the invert. A NUFT analysis at ambient temperature uses a refined 
mesh to determine advection within the invert in cases where no dripping in the drift occurs other 
than what has been used in other previous analyses (BSC 2001a; BSC 2001b). 
Engineered Barrier System Features, Events, and Processes (CRWMS M&O 2001a) examines 
coupled processes and provides a list of the EBS features, events, and processes (FEPs) and 
supporting analyses. However, this analysis report examines the processes of advection, 
dispersion, and diffusion in the invert for the breakthrough of radionuclides. These processes 
require an assessment of the following properties based upon the treatment of the invert as a 
dual-porosity medium: 
� Intergranular porosity 
� Bulk volumetric moisture content 
� Dispersivity 
� Effective diffusion coefficient. 
The work contained in this analysis falls within the scope of work described by Technical Work 
(BSC 2003a). Primary tasks include performing sensitivity studies to quantify the importance of 
10 of 80 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
uncertainty in key parameters, such as the bulk permeability of the host rock and invert 
hydrological and diffusion properties. 
2. QUALITY ASSURANCE 
This analysis has been prepared in accordance with the Quality Assurance program. The 
direction for preparing the analysis was obtained from AP-SIII.9Q, Scientific Analyses. 
The work scope described in this report has been determined to be subject to Quality Assurance 
Requirements and Description (DOE 2003). The work scope of this report involves conducting 
investigations or analyses of the invert barrier that have been classified as Quality Level 2 by QList 
(YMP 2001). Furthermore, this report provides analysis of data indirectly supporting 
performance assessment activities for the Multiscale Thermohydrologic Model non-dripping 
case, and directly supporting the dripping case as presented in Section 6.10. Since the work 
scope provides supporting analyses to EBS Radionuclide Transport Abstraction (BSC 2003b), 
the activity provides analysis of the data used to assess the potential dispersion of radioactive 
materials from the licensed facility. 
Electronic data used as inputs in the preparation of this document were obtained from the Yucca 
Mountain Project (YMP) Technical Data Management System (TDMS) as appropriate, in 
accordance with controls specified in Section 8 of Technical Work Plan for: Engineered Barrier 
System Department Modeling and Testing FY03 Work Activities (BSC 2003a). 
3.1 DESCRIPTION OF SOFTWARE 
The software described in this section is used to develop the diagrams, graphs, and tables used in 
Chapter 6 and the Appendices. The computer software used was run on computers located at 
Bechtel SAIC Company, Las Vegas, Nevada. 
3. USE OF SOFTWARE 
11 of 80 
3.1.1 NUFT v3.0s 
NUFT v3.0s (NUFT V3.0s, STN: 10088-3.0s-01) is classified as a qualified software program 
per AP-SI.1Q, Software Management. It is used in this scientific analysis to calculate pore-water 
velocities and saturation in the invert. NUFT provides a numerical solution of a nonisothermal 
unsaturated-saturated flow and transport in porous media, with application to subsurface 
contaminant transport problems. 
NUFT v3.0s was obtained from Software Configuration Management and was run on a Sun 
Workstation computer. NUFT v 3.0s is appropriate for use in this analysis, and has been used 
only within the range of validation as identified in its user documentation (CRWMS M&O 
2000a). 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
3.1.2 XTOOL V10.1 
XTOOL V10.1 (XTOOL V10.1, STN: 10208-10.1-00) is classified as a qualified software 
program per AP-SI.1Q, Software Management. XTOOL v10.1 was obtained from Software 
Configuration Management for use on the SUN Ultra Sparc operating platform. This software 
has been used within the range of validation defined for the software, and is appropriate for use 
in this analysis. 
XTOOL V10.1 is a post-processor for NUFT. It is used to plot the time history of the liquid 
saturation (S) and moisture potential (�) computed by NUFT V3.0s, and provides an 
approximated liquid flow pattern showing the direction (but not the magnitude) of flow. 
3.2 DESCRIPTION OF THE USE OF OFF-THE-SHELF SOFTWARE 
3.2.1 Mathcad 2001 Professional 
Mathcad 2001 Professional was used in this analysis to calculate the pore water velocity through 
the invert and to generate the contaminant transport breakthrough curves. Mathcad 2001 
Professional was also used in Attachment VI in the derivation of the invert �packed bed� 
properties from the data by Brooks and Corey (1964). Mathcad 2001 Professional is an off-theshelf 
software program that performs calculations using standard functions, the results of which 
are not dependent on the software itself. The results are documented sufficiently that they can be 
reproduced and checked by hand calculation. Therefore, in accordance with Section 2.1.6 of APSI.
1Q, Software Management, Mathcad 2001 Professional is software that does not need to be 
qualified. 
3.2.2 Microsoft Excel 97 
Microsoft Excel 97 is classified as a commercial off-the-shelf program per AP-SI.1Q, Software 
Management. Microsoft Excel is designed as a spreadsheet program to assist in routine 
calculations. Microsoft Excel was used to perform van Genuchten retention relationship curve 
fitting (results in Attachments IV, V, VII, XIII, and XVI). The Solver is an add-in function in 
Microsoft Excel. The Solver can minimize a target cell that involves multiple cell variables that 
might be subject to multiple constraints. The Solver is used specifically to solve for several 
variables under the constraint for a target value. In this case, the constraint is the minimization 
of the least squares of the volumetric moisture content for curve fitting. 
4. INPUTS 
12 of 80 
4.1 DIRECT INPUTS 
This section presents the direct inputs to the analysis. Section 6.11 presents the uncertainty 
analysis for these parameters. 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
4.1.1 Retention Data for Volcanic Sand, Fine Sand, Glass Beads, and Touchet Silt Loam 
The input data for determining the retention and flow parameters for the invert include the 
permeability, the porosity, and the retention data for volcanic sand, fine sand, glass beads, and 
Touchet silt loam used to calculate retention characteristics from the measurements by Brooks 
and Corey (1964, Appendix 3, Table 1). Table 4-1 presents the permeability data (k) and the 
porosity data (�) of various unconsolidated materials. These inputs are used in Section 6.3 to 
calculate the hydrologic flow and retention properties of crushed tuff. Table 4-2 presents the 
retention data for the same materials. These properties are used in Section 6.4 of this analysis. 
Value 
(m2) 
1.80E-11 
2.50E-12 
6.30E-12 
6.00E-13 
Value 
Material a Porosity (-) 
�vs 
�fs 
�gb 
Volcanic Sand 
Fine Sand 
Glass Beads 
Touchet Silt Loam �ts 
NOTE: 
Intrinsic Permeability 
(m2) 
The data presented by Brooks and Corey (1964, Appendix 3, Table 1) is prominent data from 
reputable soil scientists whose work is widely recognized. The data, and an analysis of curve-fit 
parameters derived from the data, are corroborated by the analysis presented in Sections 6.5 and 
6.8. The data for various size particles (as subsequently presented), as well as the curve-fits to 
that data, are in agreement with the Campbell retention relation for the same-size particles. 
0.351 kvs 
0.377 kfs 
0.37 kgb 
0.485 kts 
Source: Brooks and Corey 1964, Appendix 3, Table 1 
a See Section 6.4 for a description of these materials. 
Table 4-1. Summary of Permeability and Porosity of Various Unconsolidated Materials 
August 2003 
4.1.2 Retention Data for Topopah Spring Tuff at the Repository Horizon 
Tuff matrix retention and flow hydrologic properties for the TSw35 and the TSw36 units of the 
Topopah Spring Formation are used in Section 6.3 of this analysis. Table 4-3 presents 
hydrologic properties (DTN: LB990861233129.001) that were used in the NUFT analysis runs 
reported in Section 6.0. A comparison of this data with data in DTN: LB0207REVUZPRP.002 
is provided in Table 4-4 and in Attachment IX. Note that this DTN is unqualified and is used for 
reference only. 
13 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Table 4-2. Retention Data for Various Materials 
Touchet Silt Loam Glass Beads Fine Sand Volcanic Sand 
Capillary 
Rise(cm) 
Water 
Capillary Capillary 
Rise(cm) Rise(cm) 
Water Saturation Hydrocarbon 
Capillary Capillary 
Rise(cm) Rise(cm) 
Water Saturation Hydrocarbon 
Capillary Capillary 
Rise(cm) Rise(cm) 
Water Saturation Hydrocarbon 
Capillary 
Rise(cm) 
Hydrocarbon Saturation 
0.99 12 24 0.99 12.8 0. 25.6 995 5.9 0. 11.8 998 32.8 65.6 
0.986 13.5 27 0.98 27.8 55.6 0.989 11.8 23.6 0.995 42.8 85.6 
0.98 14.5 29 0.962 30.8 61.6 0.985 17.8 35.6 0.992 52.8 105.6 
0.974 15.5 31 0.95 31.8 63.6 0.98 23.8 47.6 0.984 62.8 125.6 
0.948 16 32 0.926 34.8 69.6 0.971 26.9 53.8 0.978 67.8 135.6 
0.895 17 34 0.901 36.8 73.6 0.938 28.8 57.6 0.967 72.5 145 
0.875 17.2 34.4 0.855 39.8 79.6 0.912 29.3 58.6 0.946 77.8 155.6 
0.638 21 42 0.788 42.8 85.6 0.764 30.4 60.8 0.892 82.3 164.6 
0.479 24.8 49.6 0.716 45.8 91.6 0.681 31 62 0.821 87.7 175.4 
0.277 36.9 73.8 0.627 48.8 97.6 0.579 32.1 64.2 0.719 97.8 195.6 
0.188 67.7 135.4 0.503 52.8 105.6 0.465 32.7 65.4 0.641 107.6 215.2 
0.158 136.6 273.2 0.393 57.7 115.4 0.337 33.9 67.8 0.562 123 246 
14 of 80 
� � � 0.314 64.8 129.6 0.269 35.7 0. 71.4 492 142.6 285.2 
� � � 0.273 71.7 143.4 0.19 39 78 0.424 177 354 
� � � 0.262 74.4 148.8 0.13 43.8 0. 87.6 383 207.2 414.4 
� � � 0.217 92.1 184.2 0.099 53.5 107 � � � 
� � � 0.174 150.1 300.2 0.097 150.4 300.8 � � � 
Source: Brooks and Corey 1964, Appendix 3, Table 1 
August 2003 Advection versus Diffusion in the Invert 
Table 4-3. Tuff Matrix Hydrologic Properties for TSw35 and TSw36 
Parameter TSw36 TSw35 
Porosity of the rock matrix in an individual granule (��matrix) 0.112 0.131 
Full Saturation * (Ss) 1.0 1.0 
Residual Saturation (Sr) 0.18 0.12 
van Genuchten Air-Entry Parameter (��) 1/Pa 3.55E-6 6.44E-6 
van Genuchten Parameter (m) 0.380 0.236 
Intrinsic Permeability (k) m2 5.71E-18 3.04E-17 
DTN: LB990861233129.001 
Table 4-4. Summary Tuff Matrix Hydrologic Properties for TSw35 and TSw36 
TSw35 a TSw36 a Cell 
Reference d 
H21 
TSw36 
(Tptpln) 
2.3�~10-20 
TSw35 
(Tptpll) b 
5.71�~10-18 3.04�~10-17 3.70�~10-17 
Statigraphic Unit 
Parameter (��) 1/Pa 
van Genuchten Parameter (m) 
Intrinsic Permeability (k) m2 
NOTES: a DTN: LB990861233129.001. 
Cell 
Reference c 
O20 
Porosity of the rock matrix in 0.112 0.131 0.131 C20 0.103 C21 
an individual grain (��matrix) 
Residual Saturation (Sr) 0.18 0.12 0.12 V20 0.20 V21 
3.55�~10-6 6.44�~10-6 1.66�~10-5 e van Genuchten Air-Entry 2.84�~10-7 O21 
0.380 0.236 0.216 S20 0.442 S21 
H20 
b DTN: LB0207REVUZPRP.002. Values reported in BSC 2003c. Note that this DTN is unqualified and is 
used for reference only. The retention data from this DTN are obtained from the Microsoft Excel 
workbook Matrix_Props from worksheet Matrix Hydrologic Properties Row 20. 
c For the TSw35 (Tptpll) of DTN: LB0207REVUZPRP.002. 
d For the TSw36 (Tptpln) of DTN: LB0207REVUZPRP.002. 
e This value is obtained by inverting the value shown in the DTN as 6.01�~104 Pa. 
4.1.3 Retention Measurements of the Crushed Tuff Invert 
Measurements by the U.S. Geological Survey (USGS) (DTN: GS980808312242.015) are used 
to assess porosity and retention for crushed tuff. The measurements provide retention data for 
calculating a combined porosity of the intergranular porosity (��intergrain, i.e., between the crushed 
tuff particles) and an intragranular porosity (��matrix, i.e., within crushed tuff particles). This input 
is based upon measurements used in Section 6.3, 6.8 and Attachments XI and XVI. 
4.1.4 Porosity of Poorly Graded Sands in the Loose State 
Winterkorn and Fang report an intergranular porosity (��matrix) range of 0.40 to about 0.48 
(Winterkorn and Fang 1975, p. 257). These values are used in Section 6.3 to assess the range of 
intergranular porosity. 
4.1.5 Geometry of the Invert 
The emplacement drift configuration is shown on Repository Design Project, Repository/PA IED 
Emplacement Drift Configuration 1 of 2 (BSC 2003d). The configuration in this drawing is not 
August 2003 15 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
to scale, but references Repository Design, Emplacement Drift Steel Invert Plan and Details 
(BSC 2001c), which illustrates the invert geometry. The current design for the repository shows 
that the maximum depth of the invert is 0.806 m at the drift centerline. This value is used in 
Attachment XV to develop the path length for breakthrough analysis presented in Section 6.9. 
4.1.6 Percolation Rate at the Repository Horizon 
Percolation fluxes from the PTn to the TSw unit for mean infiltration-flux, the upper-bound 
infiltration flux and the lower-bound infiltration flux cases for the various climates are presented 
in DTN: LB0302PTNTSW9I.001, as discussed in Section 6 of UZ Flow Models and Submodels 
(BSC 2003e). This analysis uses the upper bound distribution for the glacial climate. This input 
is used for comparison to the NUFT analysis in Section 6.6. 
4.1.7 Thermal and Hydrologic Properties of Stratigraphic Units 
The thermal and hydrologic properties of stratigraphic units used to calculate the hydrologic 
properties of the tuff matrix (TSw36) were based on DTN: LB990861233129.001 and the 
calibrated one-dimensional property set (DTN: LB0207REVUZPRP.002) published in 
Calibrated Properties Model (BSC 2003c). The properties for the other stratigraphic units are 
presented in Tables 4-4 through 4-7. These inputs are used in Section 6.6, and Attachments IX 
and X. 
4.1.8 Thermal Properties of Crushed Tuff 
Additional measurements of geotechnical and thermal properties (DTN: GS000483351030.003) 
have been performed. These inputs are used in the statistical analysis of the data presented in 
Table XI-1 of Attachment XI. 
16 of 80 
4.1.9 Properties of Water 
The properties of water at ambient temperature are given by Incropera and DeWitt (1996). The 
water density (��) equals approximately 1000 kg/m3, and the absolute viscosity (��) equals 
8.935�~10-4 N.s/(m2). The surface tension of water equals 72 dynes/cm. These data are used in 
Sections 6.4, 6.5, and Attachments IV, V, VI, VII, and VIII. 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
Table 4-5. Summary of the Matrix Hydrologic Properties for the NUFT Analysis 
van 
Genuchten Residual Satiated 
Saturation Saturation 
van 
Porosity Genuchten � 
�m (-) 
M (�) 
mm (-) Slsm (-) �m (1/Pa) km (m2) 
Model Permeability 
Layer 
DTN: LB990861233129.002 
Slrm (-) 
tcw11 3.98E-15 0.253 4.27E-5 0.484 0.07 1.00 
tcw12 3.26E-19 0.082 2.18E-5 0.229 0.19 1.00 
tcw13 1.63E-16 0.203 2.17E-6 0.416 0.31 1.00 
ptn21 1.26E-13 0.387 1.84E-4 0.199 0.23 1.00 
ptn22 5.98E-12 0.439 2.42E-5 0.473 0.16 1.00 
ptn23 3.43E-13 0.254 4.06E-6 0.407 0.08 1.00 
ptn24 3.93E-13 0.411 5.27E-5 0.271 0.14 1.00 
ptn25 1.85E-13 0.499 2.95E-5 0.378 0.06 1.00 
ptn26 6.39E-13 0.492 3.54E-4 0.265 0.05 1.00 
tsw31 9.25E-17 0.053 7.79E-5 0.299 0.22 1.00 
tsw32 5.11E-16 0.157 4.90E-5 0.304 0.07 1.00 
tsw33 1.24E-17 0.154 1.97E-5 0.272 0.12 1.00 
tsw34 7.94E-19 0.110 3.32E-6 0.324 0.19 1.00 
tsw35 1.42E-17 0.131 7.64E-6 0.209 0.12 1.00 
tsw36 1.34E-18 0.112 3.37E-6 0.383 0.18 1.00 
tsw37 7.04E-19 0.094 2.70E-6 0.447 0.25 1.00 
ch1v 4.36E-14 0.273 4.23E-5 0.363 0.03 1.00 
ch2z 1.16E-17 0.331 1.13E-6 0.229 0.28 1.00 
ch4z 1.16E-17 0.331 1.13E-6 0.229 0.28 1.00 
August 2003 17 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Table 4-6. Summary of the Fracture Hydrologic Properties for the NUFT Analysis 
Active 
van Fracture van Residual Satiated 
Porosity Genuchten � Genuchten M Saturation Saturation Parameter Frequency 
�F (-) �F (1/Pa) (�) Mf (-) Slrf (-) Slsf (-) F (1/M) 
Fracture To Matrix 
Connection Area 
A (M 2 /M 3 ) 
Model 
Layer 
Permeability 
Kf (M 2 ) � (-) 
1.56 2.75E-12 2.8E-2 4.67E-3 0.636 0.01 1.00 0.31 0.92 
1.00E-10 2.0E-2 2.18E-3 0.633 0.01 1.00 0.31 1.91 
2.26E-12 1.5E-2 1.71E-3 0.631 0.01 1.00 0.31 2.79 
1.00E-11 1.1E-2 2.38E-3 0.611 0.01 1.00 0.08 0.67 
1.00E-11 1.2E-2 1.26E-3 0.665 0.01 1.00 0.08 0.46 
1.96E-13 2.5E-3 1.25E-3 0.627 0.01 1.00 0.08 0.57 
4.38E-13 1.2E-2 2.25E-3 0.631 0.01 1.00 0.08 0.46 
6.14E-13 6.2E-3 1.00E-3 0.637 0.01 1.00 0.08 0.52 
3.48E-13 3.6E-3 3.98E-4 0.367 0.01 1.00 0.08 0.97 
2.55E-11 5.5E-3 1.78E-4 0.577 0.01 1.00 0.09 2.17 
7.08E-12 9.5E-3 1.32E-3 0.631 0.01 1.00 0.38 1.12 
1.50E-12 6.6E-3 1.50E-3 0.631 0.01 1.00 0.38 0.81 
4.63E-13 1.0E-2 4.05E-4 0.579 0.01 1.00 0.38 4.32 
5.09E-12 1.1E-2 9.43E-4 0.627 0.01 1.00 0.38 3.16 
1.48E-12 1.5E-2 8.21E-4 0.623 0.01 1.00 0.38 4.02 
1.48E-12 1.5E-2 8.21E-4 0.623 0.01 1.00 0.38 4.02 
7.90E-13 6.9E-4 1.66E-3 0.656 0.01 1.00 0.10 0.10 
2.64E-14 4.3E-4 8.45E-4 0.628 0.01 1.00 0.10 0.14 
2.64E-14 4.3E-4 8.45E-4 0.628 0.01 1.00 0.10 0.14 
tcw11 
tcw12 
tcw13 
ptn21 
ptn22 
ptn23 
ptn24 
ptn25 
ptn26 
tsw31 
tsw32 
tsw33 
tsw34 
tsw35 
tsw36 
tsw37 
ch1v 
ch2z 
ch4z 
13.39 
3.77 
1.00 
1.41 
1.75 
0.34 
1.09 
3.56 
3.86 
3.21 
4.44 
August 2003 18 of 80 
13.54 
9.68 
12.31 
12.31 
0.30 
0.43 
0.43 
DTN: LB990861233129.002 Advection versus Diffusion in the Invert 
Table 4-7. Summary of the Thermal Properties for the NUFT Analysis 
Model Layer 
Rock Grain 
Rock Grain Specific 
Density Heat 
�G (Kg/M3) Cp (J/Kg K) 
2550 
2510 
2470 
2380 
2340 
2400 
2370 
2260 
2370 
2510 
2550 
2510 
2530 
2540 
2560 
2560 
2310 
2350 
2350 
823 
851 
857 
1040 
1080 
849 
1020 
1330 
1220 
834 
866 
882 
948 
900 
865 
865 
1060 
1150 
1150 
tcw11 
tcw12 
tcw13 
ptn21 
ptn22 
ptn23 
ptn24 
ptn25 
ptn26 
tsw31 
tsw32 
tsw33 
tsw34 
tsw35 
tsw36 
tsw37 
ch1v 
ch2z 
ch4z 
DTN: LB990861233129.002 
4.2 CRITERIA 
Section 1.2.4 of Technical Work Plan for: Engineered Barrier System Department Modeling 
and Testing FY03 Work Activities (BSC 2003a) has identified the scope for analyses such as this 
one: 
The EBS flow and transport models and analyses are used to quantify the postclosure 
release of radionuclides from the EBS. . . Advection and diffusion may 
transport radionuclides mobilized as dissolved or colloidal species (BSC 2003a, 
Section 1.2.4). 
The scope of this analysis is specifically defined in Section 1.2.4.1 of the technical work plan 
(BSC 2003a). 
In addition, the following acceptance criteria (AC), based on the requirements listed in Project 
Requirements Document (Canori and Leitner 2003) and Yucca Mountain Review Plan, Final 
Report (NRC 2003) apply: 
1. System Description and Demonstration of Multiple Barriers (NRC 2003, 
Section 4.2.1.1.3; Canori and Leitner 2003, PRD-002/T-014, PRD-002/T-016) 
Specific requirements involve identification of multiple barriers (natural and 
engineered), describing the capabilities of these barriers to isolate waste, and providing 
ANL-EBS-MD-000063 REV 00 
Dry Conductivity 
�Dry (W/M K) 
1.60 
1.24 
0.54 
0.50 
0.35 
0.44 
0.46 
0.35 
0.23 
0.37 
1.06 
0.79 
1.56 
1.20 
1.42 
1.42 
0.70 
0.61 
0.61 
19 of 80 
Wet Conductivity 
�Wet (W/M K) 
2.00 
1.81 
0.98 
1.07 
0.50 
0.97 
1.02 
0.82 
0.67 
1.00 
1.62 
1.68 
2.33 
2.02 
1.84 
1.84 
1.31 
1.20 
1.20 
Tortuosity 
� (-) 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 
August 2003 Advection versus Diffusion in the Invert 
technical bases for capabilities descriptions consistent with the post-closure 
performance objectives. To comply with these requirements, the following acceptance 
criteria are identified in Technical Work Plan for: Engineered Barrier System 
Department Modeling and Testing FY03 Activities (BSC 2003a). 
- AC1: Identification of Barriers is Adequate. 
- AC2: Description of the Capability of Identified Barriers is Acceptable. 
- AC3: Technical Basis for Barrier Capability is Adequately Presented. 
2. Scenario Analysis and Event Probability (NRC 2003, Section 4.2.1.2.1.3; Canori and 
Leitner 2003, PRD-002/T-015) 
Specific requirements include providing technical bases for inclusion or exclusion of 
specific FEPs. In order to meet these requirements, the following acceptance criteria 
are identified in the EBS Department Technical Work Plan for: Engineered Barrier 
System Department Modeling and Testing FY03 Activities (BSC 2003a). 
- AC1: The Identification of the Initial List of Features, Events, and Processes is 
Adequate. 
- AC2: Screening of the Initial List of FEPs is Appropriate. 
- AC3: Formation of Scenario Classes Using the Reduced Set of Events is 
Adequate. 
- AC4: Screening of Scenario Classes is Appropriate. 
3. Degradation of Engineered Barriers (NRC 2003, Section 4.2.1.3.1.3; Canori and 
Leitner 2003, PRD-002/T-015): 
Specific requirements include describing deterioration or degradation of engineered 
barriers and modeling degradation processes using data for performance assessment, 
including total system performance assessment (TSPA). Consideration of 
uncertainties and variabilities in model parameters and alternative conceptual models 
is also required. To fulfill these requirements, the following acceptance criteria are 
identified in the EBS Department Technical Work Plan for: Engineered Barrier 
System Department Modeling and Testing FY03 Activities (BSC 2003a). 
- AC1: System Description and Model Integration are Adequate. 
- AC2: Data are Sufficient for Model Justification. 
- AC3: Data Uncertainty is Characterized and Propagated Through the Model 
Abstraction. 
- AC4: Model Uncertainty is Characterized and Propagated Through the Model 
Abstraction. 
- AC5: Model Abstraction Output is Supported by Objective Comparisons. 
4. Quantity and Chemistry of Water Contacting Waste Packages and Waste Forms (NRC 
2003, Section 4.2.1.3.3.3; Canori and Leitner 2003, PRD-002/T-015): 
Specific requirements include quantifying the amount and chemistry of water 
contacting the waste package and the waste forms. To comply with these 
August 2003 20 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
requirements, the following acceptance criteria are identified in the EBS Department 
Technical Work Plan for: Engineered Barrier System Department Modeling and 
Testing FY03 Activities (BSC 2003a). 
AC5: Model Abstraction Output is Supported by Objective Comparisons. 
-
- 
AC1: System Description and Model Integration are Adequate. 
AC2: Data are Sufficient for Model Justification. 
- AC3: Data Uncertainty is Characterized and Propagated Through the Model 
Abstraction. 
- AC4: Model Uncertainty is Characterized and Propagated Through the Model 
Abstraction. 
- 
4.3 CODES AND STANDARDS 
No codes or standards were used in the preparation of this document. 
5. ASSUMPTIONS 
The following assumptions have been used in this analysis. 
5.1 DIRECTION OF ADVECTIVE TRANSPORT 
Assumption: It is assumed that advective transport in the invert occurs in the vertical direction at 
constant flux rates for purposes of breakthrough analysis. 
Rationale: The technical basis for this assumption is that the general flow in the vadose zone is 
in the vertical direction, and that any tendency for flow to occur locally in the invert�s horizontal 
direction would tend to decrease the breakthrough time in the vertical direction. Therefore, the 
assumption of one-dimensional flow in the vertical direction is a bounding, conservative 
assumption, and requires no further confirmation. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Sections 6.1 and 6.9.5. 
5.2 EFFECTS OF TRANSVERSE DISPERSION NEGLECTED 
Assumption: It is assumed that only longitudinal dispersion is important to consider in this 
analysis, and that the effects of transverse dispersion can be neglected. 
Rationale: The basis for using this assumption is the fact that analyzing breakthrough to neglect 
transverse dispersion is conservative, since transverse dispersion results in a lateral dispersion 
perpendicular to the direction of flow, and is slow compared to longitudinal dispersion. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Sections 6.1 and 6.9.5. 
21 of 80 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
5.3 TRAVEL TIME OF CONTAMINANTS THROUGH HOMOGENEOUS INVERT 
MATERIAL 
Assumption: The shortest travel time for breakthrough of a contaminant through a homogeneous 
material is one-dimensional flow along a straight line. 
Rationale: The technical basis for this assumption is that if flow were directed along a path other 
than a straight line, the travel time would be longer. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Section 6.1. 
5.4 RADIONUCLIDE CONCENTRATION RELEASED OVER TIME IS CONSTANT 
AND AT THE CENTERLINE OF THE DRIFT 
Assumption: The concentration of radionuclides released after waste package failure is assumed 
to be constant over time and at the centerline of the drift. 
Rationale: This is a bounding assumption; therefore, it is adequate for the purpose of performing 
sensitivity studies. Therefore, an analytical solution for comparing advection to diffusion can be 
used. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Section 6.1. 
5.5 THE EFFECTS OF RADIOACTIVE DECAY IS NEGLECTED 
Assumption: It is assumed that the effects of radioactive decay can be neglected. 
Rationale: The technical basis for this assumption is that the contaminant breakthrough will 
occur rapidly relative to the half-life of long-lived radionuclides. This is a conservative 
assumption for analyzing breakthrough times. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Section 6.1. 
5.6 MOLECULAR DIFFUSION OCCURS AT A CONSTANT TEMPERATURE 
Assumption: It is assumed that the breakthrough occurs at a common temperature in the 
sensitivity studies presented in this analysis. 
Rationale: Previous models and analyses have characterized the environment of the repository 
to be near ambient temperature in the invert at a time when a drip shield failure and the first 
waste package breach potentially occurs (~11,000 years). Multiscale Thermohydrologic Model 
(BSC 2001a, Figure 6-68) indicates the temperatures in the repository might be about 40�C at 
that time (~11,000 years). Note that the analysis considers waste package failure except for 
juvenile failures that would occur over an extended period of time where temperatures are 
August 2003 22 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
slightly elevated. The analysis does not consider juvenile waste package failure over shorter 
periods of time where repository temperatures would be higher. 
In addition, based on measurements by Mills (1973, pp. 687 and 688) and Fetter (1993, p. 44), 
values for molecular diffusion or binary diffusion coefficient are well known, and fall in the 
range of 1�10-5 to 2�10-5 cm2/sec. Section 6.2 provides a more detailed discussion on reported 
values for diffusion, and the technical basis for this assumption. The analysis uses a reasonable 
bounding value of 1.707�10-7 cm2/sec or 5.388�10-4 m2/yr for the molecular diffusion coefficient 
of water at 45�C. The calculations are presented in Attachment III. The measurements show a 
1.1�10-5 to 3.5�10-5 cm2/sec (0.11�10-9 � 0.35�10-9 m2/yr) for 1�C and 45�C range, respectively. 
Based on this information, the foregoing correction of the molecular diffusion coefficient of 
water for temperature is not expected to have an impact on the contaminant breakthrough time 
analysis. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Section 6.9 and Attachment III. 
5.7 SOLUTE VAPOR PHASE NEGLIGIBLITY 
Assumption: It is assumed that the vapor phase of the solute is negligible for advection or 
dispersion in the invert material. 
Rationale: The technical basis for this assumption is that while it is possible for contaminants to 
be transported by vapor diffusion, the critical radionuclides from the standpoint of individual 
release are not volatile, particularly at ambient temperatures. They may be soluble in water, 
however, and therefore would only be carried by the liquid phase. This is a bounding 
assumption. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Section 6.1 
23 of 80 
5.8 INVERT MATERIAL 
Assumption: For the purpose of this analysis, it is assumed that the crushed tuff is from any of 
the TSw2 thermal/mechanical units that comprise the repository horizon. 
Rationale: This assumption is based on the fact that crushed tuff will be taken from the surface 
of the muckpile that may be obtained from several lithostratigraphic units (e.g., the Tptpul, 
Tptpmn, Tptpll, and Tptpln units). The technical basis for this assumption is that the matrix 
retention and flow properties of the Tptpmn, Tptpll, and the Tptpln units are very similar, since 
the mineralogic composition and matrix porosity are similar. Therefore, the results of analysis 
for any of these units are similar. 
Confirmation Status: This assumption is used throughout the calculation and does not require 
further confirmation. 
Use in the Model: This assumption is used throughout the report. 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
5.9 THE CRUSHED TUFF PARTICLE SIZE DISTRIBUTION FOLLOWS A LOG 
NORMAL DISTRIBUTION 
Assumption: The particle sizes in the Campbell retention relation used to calculate the moisture 
retention relationship are assumed to follow a log normal distribution (Campbell 1985, pp. 9 
and 10). 
Rationale: The technical basis for this assumption is that the data for the soil texture diagram 
presented by Campbell (1985) follows a log normal distribution. 
Confirmation Status: This assumption does not require further confirmation. 
Use in the Model: This assumption is used in Section 6.5. 
August 2003 
6. SCIENTIFIC ANALYSIS DISCUSSION 
Section 6.1 develops the governing relationships for the scientific analysis discussion of the 
advection-dispersion at the centerline of the drift. The degree to which either diffusion or 
advection/dispersion dominates the flow system depends on the fundamental mass transport 
properties, and on the hydrological environment in the invert. In turn, the fundamental mass 
transport properties depend on other more fundamental geotechnical properties of the invert. 
Section 6.2 presents the effective dispersion-diffusion and solute properties of the invert, and 
develops these properties as a function of the porosity, and the degree of saturation in the invert. 
Advection through the invert occurs by unsaturated flow that depends on the fundamental 
retention and unsaturated hydraulic conductivity properties of the invert. 
Section 6.3 develops the retention and unsaturated hydraulic conductivity properties of the invert 
based upon a dual-porosity medium that consists of the intragranular porosity within the crushed 
tuff particles, and the intergranular porosity between the crushed tuff particles. The development 
of the fundamental retention and unsaturated hydraulic properties is assisted by the use of phase 
diagrams. 
Section 6.4 presents the non-dimensionalized van Genuchten retention relationship based upon 
the original work of Brooks and Corey (1964) for a given range of conditions. The scientific 
analysis uses an alternate approach for developing the retention properties of the crushed tuff 
medium. Section 6.5 presents this alternate approach analysis using the Campbell retention 
relationship for a similar range of conditions that corroborates the non-dimenionalized van 
Genuchten retention relationships. 
Sections 6.6 through 6.8 present the NUFT advection calculation using the non-dimensionalized 
van Genuchten retention relation and the Campbell retention relation. Table 6-9 present the 
comparison of invert conditions at steady state for the van Genuchten method. Table 6-10 
presents the same information for the Campbell method that provides corroborating or 
supporting technical information on the constitutive relations. 
Section 6.9 presents containment transport results, including calculations of bulk porosity, 
diffusion and dispersion coefficients, breakthrough analyses, and an analysis of retardation. 
24 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Sections 6.10 through 6.12 present the TSPA-LA uncertainties and limitations, as well as the 
YMRP acceptance criteria addressed in this analysis. 
  �� z . 
.. Dz . 
.. 
.. 
 . 
6.1 ADVECTION-DISPERSION-DIFFUSION AT THE CENTERLINE OF THE DRIFT 
The three-dimensional advection-dispersion-diffusion relation for transport or breakthrough of a 
solute is shown in the general Equation 6-1 (Fetter 1993, p. 53): 
where 
. �� . 
z 
. �� . �� C .. . �� 
. .  
�� x 
.. 
Dx . 
��
��
C 
x . . 
. + 
��
�� 
y 
. ... 
Dy . 
��
��
C 
y .
.
.. 
+ 
��
�� 
z 
.. 
. 
Dz . 
�� z 
.. 
... 
. .
  �� x 
( vx . C ) + 
��
�� 
y 
( vy . C ) + 
��
�� 
z 
( vz . C )
... 
= 
��
�� 
C 
t 
(Eq. 6-1) 
Dx = dispersion/diffusion coefficient in the x direction (m2/sec) 
Dy = dispersion/diffusion coefficient in the y direction (m2/sec) 
Dz = dispersion/diffusion coefficient in the z direction (m2/sec) 
C = solute concentration at location x, y, z and time t(mg/l) 
vx = pore-water velocity in the x direction (m/sec) 
vy = pore-water velocity in the y direction (m/sec) 
vz = pore-water velocity in the z direction (m/sec) 
t = time (sec) 
The dispersion coefficients (i.e., Dx, Dy, and Dz in the x, y, and z directions) include the process 
of both advection and hydrodynamic dispersion (which includes molecular diffusion and 
mechanical dispersion combined) (Fetter 1993, p. 51). This equation has been applied to 
homogeneous, anisotropic, saturated media. Jury et al. (1991, pp. 221 to 223) extend the 
application of the general equation to unsaturated media. 
Applying Assumptions 5.1 through 5.4, the radionuclides are released at the centerline of the 
drift and flow occurs in the vertical direction. Transverse-flux and transverse-dispersion are 
neglected (see Assumptions 5.1 and 5.2). Three-dimensional Equation 6-1 reduces to the onedimensional 
Equation 6-2a: 
. . Dz . 
�� C .. 
..
. . �� ( vz . C ). .
= 
��C (Eq. 6-2a) �� z �� t . 
.
. .
. . 
August 2003 ANL-EBS-MD-000063 REV 00 
..   .
�� z 
Expanding the expression for advection according to the chain rule, the following partial 
differential equation (Equation 6-2b) for mass transport is obtained: 
. . 
  . 
C 
�� z .. 
. .  
vz �� z .. 
= 
�� t 
(Eq. 6-2b) �� 2C .. . �� vz . . ��C . ��C 
�� z2 
Consider a uniform velocity (vz) in the media in which the velocity gradient is zero 
( �� v / �� z = 0 ). The above equation reduces to the one-dimensional advection-diffusiondispersion 
equation that can then be solved using a closed form analytical solution. This 
equation can be written with the effective Dispersion/Diffusion Coefficient (Dz) being 
independent of position (Jury et al. 1991, p. 223): 
25 of 80 Advection versus Diffusion in the Invert 
(Eq. 6-3) z . D . 
�� z 2 
l (Eq. 6-4) � �
+ exp. 
�� 2C .
�
. . v 
��C . ��C 
. . 
. �
. 
.. 
z ��z . . 
��t 
= 
Since the centerline of the drift represents a line of symmetry, the horizontal Darcy flux is zero. 
The vertical Darcy flux increases with depth into the invert and the host rock because, for an 
unsaturated flow in the vadose zone to occur in the absence of localized indrift seepage, the drift 
acts as a capillary barrier to flow. The flow pattern is then similar to the problem of fluid flow 
around an inclusion in which a stagnation point forms on the downstream side. Note that while 
the release of the radionuclides might advect, disperse or diffuse in the radial direction from the 
point of release, it is conservative to assume flow in the vertical direction. If the concentration is 
set to equal one at the top of the invert (Assumption 5.4), and the Darcy flux is constant for a 
steady state flow, then the one-dimensional advection-dispersion equation (Equation 6-3) can be 
solved using a closed form analytical solution. 
A solution to the above relation is presented for non-retarded transport in one dimension, with 
initial concentration (Co) moving at a continuous rate where the vapor phase transport is 
negligible (see Assumption 5.7) (Freeze and Cherry 1979, p. 391). 
. L . V . t . . L + V . t .. . V . L . 
. D . D . t . 
1 . 
2 
C
C0 
� . erfc 
. . 
. 
2 . D . t . � �. � � 
.. 
.. 
.. 
= . . erfc 
. . 
. 2 . . 
The pore water velocity (V) presented above equals the Darcy Flux (Jw) (the vertical Darcy flux 
rate) divided by the porosity (�� ) in the vertical direction of flow or the porewater velocity (V) for 
saturated flow. For unsaturated flow, the average linear velocity, or the porewater velocity, 
equals the Darcy Flux (Jw) divided by the volumetric moisture content (�� ) (Jury et al. 1991, 
p. 221). 
(Eq. 6-5) 
6.2 EFFECTIVE DISPERSION-DIFFUSION AND SOLUTE TRANSPORT 
PROPERTIES 
The flux/transport of a dissolved solute is governed by the processes of advection and 
hydrodynamic dispersion. Liquid advection is the bulk transport (also referred to as convection) 
of solutes moving with a flowing soil solution. The hydrodynamic dispersion process includes 
both molecular diffusion and mechanical dispersion (Fetter 1993, pp. 43 to 51). Molecular 
diffusion is characterized by transport due to a concentration gradient, and is expressed by Fick's 
First Law in which the mass flux of solute per unit area per unit time is the product of the 
diffusion coefficient and the concentration gradient. When liquid advection is characterized as 
��plug flow,�� the mass transport yields a sharp concentration front (Fetter 1993, p. 48). However, 
for systems where concentrations are changing with time, and in the absence of liquid advection, 
the governing equation (Equation 6-3) reduces to Fick��s Second Law which is the transient 
partial differential equation governing diffusion (Equation 6-5) (Fetter 1993, p. 44). 
��C �� 2C 
�� t 
= Dd . 
�� x2 
26 of 80 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
In porous media, diffusion is slower than it is in water because solutes must follow longer 
pathways around mineral grains/particles in the invert, which introduces the coefficient related to 
the tortuosity (�) into the equation where, in porous media, the effective diffusion coefficient, 
D* must be used (Fetter 1993, p. 44): 
D* = �Dd (Eq. 6-6) 
Groundwater containing solute does not all travel at the same velocity due to the mixing that 
occurs along the flow path. This mixing is called mechanical dispersion, and is a result of solute 
dilution at the advancing edge of the flow. The mixing that occurs along the direction of the 
flow path is called longitudinal dispersion. Where the solute tends to spread and mix in the 
normal flow path direction, it is referred to as transverse dispersion (Fetter 1993, p. 49, Figure 
2.4). Factors causing longitudinal dispersion on the scale of individual pores are: 
1. As fluid moves through the pore space, fluid moves more rapidly in the pore space 
centers rather than along the edges. 
2. Some flow paths through a porous media are longer than other flow paths due to flow 
tortuosity. 
3. Some flow paths are larger due to grain size effects. 
These causal factors combined together result in dispersive flux. The three fluxes (liquid 
advection, soil-liquid diffusion, and hydrodynamic dispersion) can be expressed mathematically 
as follows (Jury et al. 1991, pp. 220 to 223): 
Soil-Liquid Diffusion 
where 
Liquid Advection (Bulk Flow or Convection) 
where 
Jlc = J . CL (Eq. 6-7) 
Jlc is the liquid advection flux (kg/s/m2) 
Jw is the vertical Darcy flux rate (m/s) 
CL is the solute concentration of the solute at location x = 0 (kg/m3 or mg/L) 
Jsl = . D 
Jsl is the soil liquid flux (kg/s/m2) 
Dsl is the solute diffusion coefficient of the solute in water (m2/sec) 
27 of 80 
Hydrodynamic Dispersion Flux 
ANL-EBS-MD-000063 REV 00 
w 
sl . dCL (Eq. 6-8) 
dz 
August 2003 Advection versus Diffusion in the Invert 
J = . D lh lh . dCL (Eq. 6-9) 
dz 
where 
Jlh is the hydrodynamic dispersion flux (kg/s/m2) 
Dlh is the hydrodynamic dispersion coefficient (m2/sec) 
The combined flux (Jl), which is the sum of liquid advection (Jk), soil-liquid diffusion (Dsl), and 
hydrodynamic dispersion flux (Jlh) (Equations 6.7 through 6.9) is expressed as: 
w w . Dlh . 
dCL (Eq. 6-10) Jl = J . CL + J k = J . CL . Dsl . 
dz 
dCL 
dz 
220 to 223) combine the last two terms in Equation 6-10 to express the combined flux (J 
It is important to understand the properties associated with soil-liquid diffusion (Dsl) and 
hydrodynamic dispersion (Dlh), and their dependence on other parameters. Jury et al. (1991, pp. 
l) as: 
e w Jl = J . CL . D . dCL (Eq. 6-11) 
dz 
in which the effective dispersion-diffusion coefficient (De) equals the sum of the hydrodynamic 
dispersion (Dlh) and soil-liquid diffusion coefficients (Dsl) (Jury et al. 1991, p. 222): 
(Eq. 6-12) De = Dlh + Dsl 
Further, the dispersion-diffusion coefficient (D) in m2/sec equals the effective dispersiondiffusion 
coefficient (De) divided by the volumetric moisture content (� ) (Jury et al. 1991, p. 
223). 
The hydrodynamic dispersion coefficient (Dlh) has frequently been observed to be proportional 
to the pore-water velocity V, also known as the average linear velocity, where � is the 
dispersivity in cm and V = Jw/� (Jury et al. 1991, p. 221): 
D lh � V 
(Eq. 6-13) 
. 
Dispersivity (� ) is defined as the degree of kinematic dispersion in a porous medium. Fetter 
provides a relationship between dispersivity and length that shows a conservative estimate of 
dispersivity to be 0.1 m (10 cm) for a path length (L) of one meter (Fetter 1993, p. 73, 
Figure 2.18). Jury et al. discuss a range from 0.5 to 2 cm for packed laboratory columns, and 
from 5 to 20 cm in field experiments (Jury et al. 1991, p. 222). Jury et al. state that the 
dispersivity can be considerably larger for regional groundwater flow. However, due to the scale 
of the invert, these values would not apply. Based upon information provided by Jury et al. 
(1991, p. 222) and Fetter (1993, p. 73), a reasonable bounding range of values for dispersivity 
would be 0.4 to 10 cm. Note that a value of 10 cm was used in the analysis in Attachment III to 
provide an upper bounding analysis for hydrodynamic dispersion. 
28 of 80 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
The soil-liquid diffusion coefficient (Dsl) is also a function of the binary diffusion coefficient 
(Dwl) in water, and a function of ��(��), the liquid tortuosity factor, which accounts for the 
increased path length and decreased cross-sectional area of solute diffusion in the invert. The 
soil-liquid diffusion coefficient (Dsl) is given by Equation 6-14 (Jury et al. 1991, p. 221) and 
applying Assumption 5.6. 
Dsl (��) = �� (��) Dwl (Eq. 6-14) 
The soil-liquid diffusion coefficient (Dsl) is applied to crushed tuff, assumed to be the 
composition of the invert in Assumption 5.8. The dependence of the soil-liquid diffusion 
coefficient on the saturation (S), the porosity (��), and the binary diffusion coefficient of water is 
represented mathematically in EBS Radionuclide Transport Abstraction (BSC 2003b) as: 
(S, ��) = D . ��1.863 . S1.863 (Eq. 6-15) D wl sl 
Substituting in the volumetric moisture content (��) for the saturation (S) and the porosity (��) 
values yields: 
(Eq. 6-16) Dsl ( ��) = Dwl . ��1 .863 
In the solution of the contaminant transport equation for porous media flow, de Marsily writes 
several basic contaminant transport equations that are based upon the flow through the pore 
space that entails the porosity (��) for saturated flow, and that includes the Darcy Flux (Jw) 
(de Marsily 1986, p. 267). The form of the soil-liquid diffusion coefficient as presented in the 
advection-dispersion-diffusion contaminant equations by de Marsily (1986) is similar to the 
relationships presented above. 
As noted in Section 5.6, variations in temperature do not significantly change the diffusion. The 
solute diffusion coefficient (Dsl) is calculated on the basis of Archie�fs Law in Section 4.1.1 of 
Invert Diffusion Properties Model (CRWMS M&O 2000b) that modifies the binary 
diffusion/molecular diffusion coefficient of water (Dwl) and on a diffusion versus temperature 
relationship presented in Attachment III. The self-diffusion coefficients of tritiated water in 
normal and heavy water were measured over a temperature range from 1��C to 45��C using the 
diaphragm-cell technique (Mills 1973, p. 685). These coefficients have been tabulated at various 
temperatures with the molecular mass for water taken into account. The measurements were 
compared with the molecular dynamics and Nuclear Magnetic Resonance (NMR) data. The 
measurements show temperature dependence from 1.1�~10-5 to 3.5�~10-5 cm2/sec (0.11�~10-9 to 
0.35�~10-9 m2/y) for 1��C and 45��C, respectively. Recent NMR studies provided values at 
different temperatures that were in reasonable agreement with measurements by Mills (1973, pp. 
687 to 688). Fetter (1993, p. 44) states that the values for molecular diffusion or a binary 
diffusion coefficient are well known, and fall in the range of 1�~10-5 to 2�~10-5 cm2/sec. Based 
upon Mills (1973, pp. 687 to 688), a reasonable bounding value of 2.30�~10-5 cm2/sec or 0.073 
m2/yr for the molecular diffusion coefficient of water at 25��C is used. Based on the relationship 
for correcting the molecular diffusion coefficient of water for temperature, the calculated value 
for diffusion at 45��C is 1.707�~10-7 cm2/sec (5.388�~10-4 m2/yr). The reduction in temperature 
August 2003 29 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
does not have a significant effect on breakthrough times, particularly when advection is the 
dominant mechanism of radionuclide transport through the invert (e.g., after drip shield failure). 
In addition, greater uncertainty exists with parameters of greater significance to radionuclide 
transport (i.e., volumetric moisture content, and Darcy flux) than the molecular diffusion 
coefficient for water. For example, the molecular diffusion coefficient for water at 45�C is only 
approximately three times greater than the value at 1�C (Mills 1973, pp. 687 to 688). 
tm 
6.3 RETENTION AND UNSATURATED HYDRAULIC CONDUCTIVITY 
PROPERTIES OF THE INVERT 
Inputs to the various models (Section 4.0) require an evaluation of the retention and hydraulic 
properties of the crushed tuff that comprise the invert. In Attachment XI, the retention and 
hydraulic properties were determined for a single-porosity material. For the calculations used in 
this analysis, the invert is characterized as a dual-porosity material in which the properties for 
each component of the material are determined. NUFT is run in this analysis using the new 
retention and hydraulic properties of the crushed tuff based on its characterization as a dualporosity 
invert material. This is done in order to demonstrate the concept of a fully-saturated 
matrix intragranular space and an unsaturated intergranular space in the invert. 
Prior analyses have estimated the intragranular porosity (�inatrix) of crushed tuff to be 0.112 
(Table 4-3) (DTN: LB990861233129.001). However, the total porosity includes both the 
intergranular porosity (�intergrain), related to the voids between the crushed tuff particles, as well as 
the intragranular porosity, related to the voids within the crushed tuff particles (�matrix). A total 
porosity of 0.55 has been calculated (Attachment XI) for sieved samples of crushed tuff ranging 
from 2 to 4.75 mm from TSw (DTN: GS980808312242.015) and as presented in detail in 
Attachment XI. 
The phase diagram for crushed tuff is shown in Figure 6-1. The diagram has been developed to 
illustrate and characterize a matrix made up of tuff solids, intragranular voids, and intergranular 
voids. This phase diagram and the relationship between void volume and porosity are used to 
calculate the intergranular porosity (�intergrain) based upon the reported values of 0.112 for 
intragranular porosity, and 0.55 for the total porosity. 
In this analysis, Vtm equals the volume of the voids in the intragranular void space. Vc is equal to 
the volume of the voids in the intergranular void space and the total volume of the solids (Vsm) is 
equal to 1 cm3, in accordance with standard soil mechanics conventions. Using this relationship, 
the void volume of the tuff matrix can be calculated. The porosity, �matrix, as defined in Equation 
6-17 and 6-18, is the volume of the voids divided by the total volume of solids. The volume of 
the voids within the tuff matrix (Vtm) can be determined by Equation 6-19. 
V = 0 . 112 (Eq. 6-17) �matrix = 
Vsm + Vtm 
tm 
Vtm (Eq. 6-18) = 0 .112 
1 + V
30 of 80 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
= 0.126 (Eq. 6-19) 0.112 Vtm = 
1 . 0.112 
When the total porosity (�) is set to 0.55, the volume of the coarse fraction is calculated as 
follows: 
c tm = 55 . 0 (Eq. 6-20) 
0 . 1 c tm 
= 55 . 0 
V + V 
+ V +V 
. 0 126 + 
0 . 1 + 0 .126 + Vc 
c V 
c = 1.10 cm3 (Eq. 6-21) 
Phase 
c Intergranular 
Voids - Coarse 
V = 1.23 m3 
v 
Vtm 
Intragranular Tuff 
Matrix Voids - Fin e 
V 
Having calculated the volume of the coarse pore space (Vc) to be 1.10 cm3, the calculated 
intergranular porosity (�intergrain) that pertains to this coarse void space is equal to 0.49. This 
value is relatively high, and indicates that the samples laboratory tested by the USGS using the 
unsaturated flow apparatus were high (DTN: GS980808312242.015). Winterkorn and Fang 
evaluated uniformly graded or poorly sorted sands and measured intergranular porosity (�matrix) 
ranges from 0.40 to about 0.48 when considering the maximum void ratio or porosity 
(Winterkorn and Fang 1975, p. 257). Because the intergranular porosity value of 0.49 exceeds 
the range 0.40 to 0.48, it can be concluded that the USGS values for measured bulk porosity 
(DTN: GS980808312242.015) are consistent with a poorly graded sand in a loose state. 
Considering that crushed tuff may settle over time, a median value of 0.45 is adopted for the 
intergranular porosity (�intergrain) for purposes of analysis. 
V = 1.10 m3 
= 0.13 m3 
V = 1.00 m3 
sm Solids 
Figure 6-1. Phase Diagram for Crushed Tuff 
An estimation of the water retention properties for the coarse intergranular porosity (�intergrain) 
requires developing a retention relationship. This is for the following reasons. First, the 
retention characteristics obtained in the laboratory from the Unsaturated Flow Apparatus (UFA) 
measurements in Attachment XI (DTN: GS980808312242.015) were not tested above a 
31 of 80 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
moisture potential (�) of approximately 0.04 bars (-40 cm). The measurements reported 
previously for the 2 mm to 4.75 mm had a minimum moisture potential of -100 cm. Since the 
higher negative pressures indicate that the intragranular pore space would retain water while the 
intergranular pore space would not (Attachment XI), these measurements reflect near saturation 
of the intragranular pore space but no retention of water in the intergranular pore space. 
6.4 NON-DIMENSIONALIZED VAN GENUCHTEN RETENTION RELATION 
Brooks and Corey retention data were obtained for such materials as volcanic sand, fine sand, 
and glass beads data (Brooks and Corey 1964). Figure 6.2 presents the capillary rise of these 
unconsolidated samples. Figure 6-3 presents the normalized capillary rise for various materials 
using the combined nondimensional van Genuchten retention relation measurements. The 
Brooks and Corey moisture potential measurements include a range of particle sizes. The 
measurements are combined by transforming the capillary pressures to nondimensional capillary 
pressures (Attachment IV). This is accomplished by using a nondimensional relationship 
dependent on particle size for moisture potential and for transforming volumetric moisture 
content to saturation for the respective materials (Leverett 1941, p. 159). (Volumetric moisture 
content is converted to saturation by dividing the volumetric moisture content by the porosity.) 
A description of these data sets follows: 
� Volcanic Sand�This material comes from a wind-blown deposit along Crab Creek in 
Washington State. It consists of dark-colored aggregates that can be broken down into 
finer particles by pressure. It is not known to what degree these aggregates are 
themselves permeable, but they undoubtedly have some permeability. This sand has a 
degree of structure and has both primary and secondary porosity. 
� Fine Sand�This sand was supplied by the Hanford Laboratories of General Electric 
Company at Richland, Washington, and apparently contains some volcanic minerals. 
This material contains a wide range of particle sizes, ranging down to silt size. Most of 
the particles are angular and not as rounded as most river bed sands. 
� Glass Beads�This material is an example of media having a very narrow range of pore 
sizes. In this respect, however, it is not much different from many clean river sands. 
� Touchet Silt Loam�This soil comes from the Columbia River basin and as also 
supplied by the Hanford Laboratory. It is extremely fine-textured in that it contains 
practically no coarse sand, but it is somewhat unusual in that it contains a smaller 
amount of clay than would be expected in such a fine-textured soil. It is, in fact, nearly 
pure silt mixed with some extremely fine sand. It contains enough clay, however, to 
create a structure with secondary porosity. 
August 2003 32 of 80 ANL-EBS-MD-000063 REV 00 Figure 6-2. Capillary Rise of Unconsolidated Samples 
Advection versus Diffusion in the Invert 
Source: Brooks and Corey 1964 
Dimensionless Capillary Rise 
1.4 
1.2 
1 
0.8 
0.6 
0.4 
0.2 
0 
0.3 0.2 0.1 0 
Source: Brooks and Corey 1964 
DTN: MO0307SPAVGHYD.000 
Figure 6-3. Normalized Capillary Rise for Various Materials 
The methodology for determining the relationship of unsaturated hydraulic conductivity (Kus) 
from the retention curve using the two-parameter nondimensional van Genuchten relationship 
ANL-EBS-MD-000063 REV 00 
Volcanic Sand 
Fine Sand 
Glass Beads 
Touchet Silt Loam 
Least Squares Fit 
1 0.9 0.8 0.7 0.6 0.5 0.4 
Saturation 
August 2003 33 of 80 Advection versus Diffusion in the Invert 
and the size of the crushed tuff is provided in Attachment IV. The intrinsic permeability (k) is 
determined from the Kozeny-Carman formula shown by Equation IV-4 (Bear 1972, p. 166) that 
relates intrinsic permeability (k) to the grain size or pore diameter (dm) and porosity (��). On the 
basis of the selected grain size, the saturated hydraulic conductivity (Ks) and the van Genuchten 
relationship for relative permeability, the relationship of unsaturated hydraulic conductivity (Kus) 
to moisture potential (��) can be determined. 
A qualitative assessment can be made over the range of moisture potentials (��) of interest (0.01 
to 0.1 bars) as to whether liquid flow or advection in the coarse fraction of the crushed tuff 
would occur for a range of particle diameters. The analysis is performed for grain size diameters 
of 0.317 mm, 3 mm, 10 mm, and 20 mm for the intergranular porosity (��intergrain), respectively, to 
cover a broad range of particle diameters. The equations used in these derivations are shown in 
Attachment VI. 
The van Genuchten parameters can be used to determine the moisture retention relationship for 
the tuff matrix and the intergranular pore space. Table 6-1 presents the tuff matrix hydrologic 
properties used for these determinations for TSw36 (DTN: LB990861233129.001). 
Additional analyses were performed for grain sizes of 3 mm, 10 mm, and 20 mm and these 
calculations are summarized in Tables 6-2 through 6-4, respectively, for each particle size 
evaluated. The detailed calculations for developing the parameters for the van Genuchten 
moisture retention relationships are presented in Tables IV-1 through IV-5 of Attachment IV. 
Figure 6-4 presents the retention relationships for the intergranular and intragranular pore space 
for comparison to the retention relationship for the tuff matrix. Figure 6-5 presents the 
unsaturated hydraulic conductivity relationship. 
An intrinsic permeability (k) of 1.68�~10-10 m2 corresponds to a grain size diameter of 0.317 mm 
and a saturated hydraulic conductivity (Ks) of 0.165 cm/sec (see footnote 5 to Table 6-1 for 
conversion factor). The relative permeability function scales the saturated conductivity (Ks) 
allowing the unsaturated hydraulic conductivity (Kus) function to be determined. The 
unsaturated hydraulic conductivity relationships are presented in Figure 6-5. 
The retention relationship (Equation IV-2) shows that for a fine intergranular porosity (��intergrain) 
associated with a particle size of 0.317 mm over the range of moisture potentials (��) of 0.01 to 
0.1 bars, water would be retained and would flow in the fine intergranular void space. If the 
unsaturated hydraulic conductivity (Kus) for the intergranular porosity (��intergrain) is higher than 
the tuff matrix over the range of moisture potential of interest (0.01 to 0.1 bars), it can be 
concluded that the water flowing in the intergranular pore space would be the dominant flow 
path. However, if Kus for the intergranular porosity is lower than the Kus of the tuff matrix over 
the range of moisture potential of interest (0.01 to 0.1 bars), it can be concluded that water would 
not be retained in the intergranular porosity (��intergrain), and the flow of water would not occur. 
August 2003 34 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table 6-1. Summary of Tuff Matrix Hydrologic Properties for TSw36 
(��matrix). 
2 
3 
Parameter Value 
Porosity of the rock matrix in an individual grain (��matrix) 0.112 
Saturation at saturation (Ss) 1.0 
Residual Saturation (Sr) 0.18 
Saturated Volumetric moisture content (��s)1 0.112 
Residual Volumetric moisture content (��r)2 0.02016 
van Genuchten air-entry parameter (��) 3.55�~10-6 (Pa)-1 
van Genuchten air-entry parameter (��) 0.355 bars-1 
van Genuchten parameter estimated from the water 0.380 
retention curve (m) 
van Genuchten n parameter (n)3 1.61 
Intrinsic permeability (k) 6�~10-18 m2 
Saturated hydraulic conductivity (Ks)4 6 �~10-9 cm/sec 
DTN: LB990861233129.001 
NOTES: 1 The saturated volumetric moisture content (��s) equals the porosity 
The residual volumetric moisture content (��r) equals the residual 
saturation (Sr) times the porosity (��) 
4 
The value of n is given by 1/(1-m) (Fetter 1993, p. 172). 
The value of the saturated hydraulic conductivity (Ks) is obtained by 
the equation that converts an saturated intrinsic permeability (k) to a 
saturated hydraulic conductivity (Ks) (Freeze and Cherry 1979, p. 27). 
K = s 
��. g . k 
�� 
The properties of water at ambient temperature are given by Incropera 
and DeWitt (1996, p. 846). The water density (��) equals 1000 kg/m3 
and the absolute viscosity (��) equals 8.935�~10-4 N.s/(m2). 
August 2003 35 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Comparison of Retention Models 
1000 
100 
Hydraulic Conductivity (cm/s) 
Moisture Potential (-bars) 
Tuff Matrix 
10 Van Genuchten 
(0.317 mm) 
1 Van Genuchten 
(3 mm) 0.1 
Van Genuchten 
(10 mm) 0.01 
0.001 Van Genuchten 
(20 mm) 
0.0001 
0.00 0.10 0.30 0.40 0.50 0.20 
1000 
100 
10 
1 
0.1 
0.01 
0.001 
0.0001 
1E-05 
1E-06 
1E-07 
1E-08 
1E-09 
1E-10 
1E-11 
1E-12 
1E-13 
1E-14 
1E-15 
1E-16 
1E-17 
1000 100 10 0.001 0.0001 
20 mm 10 mm Tuff Matrix k 
NOTE: See Table 6-4. 
Volumetric Water Content 
NOTE: See Tables 6-1 and 6-3. 
Figure 6-4. Retention Relationships Based Upon the Non-Dimensionalized van Genuchten Retention 
Relation 
1 0.1 0.01 
Tuff Fracture k 
Moisture Potential (bars) 
"3 mm" 0.317 mm 
Figure 6-5. Comparison of Conductivity Relationships Non-Dimensionalized van Genuchten Retention 
Relation 
August 2003 36 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table 6-2. Summary of Hydrologic Properties Based on the Non-Dimensionalized Moisture Potential 
Retention Relation 
20 mm 3 mm Parameter 
DTN: MO0307SPAVGSUM.000 
Particle Size (dm) 
10 mm 
Saturated Volumetric Moisture Content (��s)2 0.450 0.450 0.450 0.450 
0.050 0.050 0.050 Residual Volumetric Moisture Content (��r) 0.050 
van Genuchten Air-entry Parameter (bars-1) (��)1 65.9 624. 2080. 4160. 
van Genuchten Air-entry Parameter (cm-1) (��) 0.0647 0.612 2.04 4.08 
van Genuchten n Value (n) 8.013 8.013 8.013 8.013 
van Genuchten m Value (m)3 0.875 0.875 0.875 0.875 
Saturated Intrinsic Permeability(m2) (k)4 1.68E-10 1.51E-08 1.67E-07 6.69E-07 
Saturated Hydraulic Conductivity (cm/sec) (Ks)5 0.184 16.48 183.1 732.5 
NOTES: 1See text and Table IV-1 for the calculation of the van Genuchten Air-Entry Parameter (��). 
2The saturated volumetric moisture (��s) content equals the porosity (��). 
3The value of n is given by 1/(1-m) (Fetter 1993, p. 172). 
4The intrinsic permeability (k) is calculated from Equation IV-4. 
5The value of the saturated hydraulic conductivity (Ks) is obtained by the equation that converts an 
saturated intrinsic permeability (k) to a saturated hydraulic conductivity (Ks) (Freeze and Cherry 1979, p. 27). 
s 
0.317 mm 
K = 
��. g . k 
�� 
The properties of water at ambient temperature are given by Incropera and DeWitt (1996, p. 846). The water 
density (��) equals 997 kg/m3 and the absolute viscosity (��) equals 8.935�~10-4 N.s/(m2). Note that these 
values compare well with the values for sands and gravels from Freeze and Cherry (1979, p. 29). 
Table 6-3. Moisture Retention Calculations for the Non-Dimensionalized Moisture Potential Retention 
Relation 
Volumetric Moisture Content (��) 
Particle Size (dm) Moisture Potential(��) 
(bars) 0.317 mm 3 mm 10 mm 20 mm 
0.0001 0.450 0.450 0.450 0.450 
0.0002 0.450 0.450 0.450 0.384 
0.001 0.450 0.450 0.238 0.052 
0.001 0.450 0.442 0.052 0.050 
0.002 0.450 0.124 0.050 0.050 
0.005 0.450 0.050 0.050 0.050 
0.010 0.438 0.050 0.050 0.050 
0.020 0.103 0.050 0.050 0.050 
0.050 0.050 0.050 0.050 0.050 
0.100 0.050 0.050 0.050 0.050 
0.200 0.050 0.050 0.050 0.050 
0.500 0.050 0.050 0.050 0.050 
DTN: MO0307SPAVGHYD.000 
NOTE: The detailed calculations for developing the moisture 
retention relationships are presented in Tables IV-1 
through IV-5 of Attachment IV. 
August 2003 37 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table 6-4. Unsaturated Hydraulic Conductivity (Kus) Calculations for the Non-Dimensionalized Moisture 
Potential (��) Retention Relation 
Unsaturated Hydraulic Conductivity (Kus) (cm/sec)1 
Particle Size (dm) 
10 mm 3 mm 20 mm 
0.0001 6.23E-09 1.84E-01 1.65E+01 1.83E+02 7.29E+02 
0.0002 6.21E-09 1.84E-01 1.65E+01 1.82E+02 3.97E+02 
0.001 6.15E-09 1.84E-01 1.52E+01 8.60E-05 4.56E-10 
0.0015 6.13E-09 1.84E-01 4.54E+00 3.15E-08 1.66E-13 
0.002 6.10E-09 1.84E-01 1.16E-01 1.14E-10 6.01E-16 
0.005 5.99E-09 1.84E-01 2.81E-09 1.93E-18 1.01E-23 
0.01 5.86E-09 1.63E-01 3.71E-15 2.54E-24 1.36E-29 
0.02 5.66E-09 5.02E-04 4.89E-21 3.41E-30 2.02E-34 
0.05 5.24E-09 1.08E-11 8.27E-29 1.22E-35 5.45E-36 
0.1 4.74E-09 1.42E-17 1.57E-34 1.36E-36 5.92E-37 
0.2 4.02E-09 1.86E-23 7.34E-37 1.48E-37 6.30E-38 
NOTES: 1The values for the saturated hydraulic conductivity (Ks) (cm/sec) as a function 
of the moisture potential (��) are obtained by calculating the intrinsic 
permeability (k) for a given set of van Genuchten parameters from Equation 
IV-4 and then applying the conversion from intrinsic permeability (k) to 
saturated hydraulic conductivity (Ks) (Freeze and Cherry 1979, p. 27) 
Moisture 
Potential (��) 
(bars) 
Attachment XI. 
Welded Tuff Matrix 0.317 mm 
DTN: MO0307SPAVGSUM.000 
K = s 
��. g . k 
�� 
The properties of water at ambient temperature are given by Incropera and 
DeWitt (1996, p. 846). The water density (��) equals 1000 kg/m3 and the 
absolute viscosity (��) equals 8.935�~10-4 N.s/(m2). The values for the 
unsaturated hydraulic conductivity are then determined by scaling the saturated 
hydraulic conductivity relationship by the relative permeability relationship 
presented in Equation IV-11 for a given moisture potential. 
Subsequent analyses were developed based upon a grain sizes of 3 mm, 10 mm, and 20 mm 
particles that correspond approximately to the average grain size of the material used in 
The value of the van Genuchten air-entry parameter (��) for the 3 mm diameter grain size in 
terms of bars-1 is calculated to be 624 (bars)-1 (see Attachment IV, Equation IV-10). The 
intrinsic permeability (k) corresponding to a grain size of 3 mm is equal to 1.51�~10-8 m2. This 
corresponds to a saturated hydraulic conductivity (Ks) value of 14.76 cm/s (see footnote 5 to 
Table 6-1 for conversion). The hydraulic conductivity relationship suggests that, over the range 
of moisture potential (��) of interest, the coarse intergranular porosity (��intergrain) would not retain 
water, and matrix flow would be dominant. 
August 2003 38 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
s 
g 
g 
6.5 CAMPBELL RETENTION RELATION 
In the following discussion, a moisture retention relation proposed by Campbell (1985, 
pp. 45 to 47) is used to develop the moisture potential relation. The prediction is based upon 
developing parameters for soil texture. Campbell presents a texture diagram for clays, silts, and 
sands that is based upon the assumption that the particle size distribution in soil is approximately 
a log-normal distribution (Assumption 5.9) characterized by the geometric mean and a geometric 
standard deviation (Campbell 1985, pp. 9 and 10). For sands that have diameters greater than 
0.8 mm, the approximate geometric standard deviation is approximately one. 
Two components of the soil-water potential depend on volumetric moisture content; these 
include the matrix and the osmotic potentials. Campbell (1985) terms the relationship between 
moisture potential (� ) and volumetric water or moisture content (� ) as the soil moisture 
characteristic or moisture release curve. For the moisture potential � <� e, Campbell states that 
the relationship is determined by the function (Campbell 1985, p. 43): 
�= � . ( � / � ) . b (Eq. 6-22) e 
The air-entry moisture potential (� e) is the water potential at which the largest water filled pore 
in the soil will drain. As the mean pore diameter becomes smaller the air-entry moisture 
potential decreases (becomes more negative). Note that the �b� parameter increases as the 
standard deviation (� g) of the pore size increases. The following approximate relationships can 
be used to develop a moisture retention relationship based on the assumption that particle sizes 
follow a log-normal distribution (Assumption 5.9) (Campbell 1985, p. 45): 
� = .0.5 . d . 1/2 (Eq. 6-23) es 
b = . 2 . � + 0.2 . � (Eq. 6-24) es 
According to Campbell, the geometric mean diameter (� g) can be calculated for any combination 
of sand, silt, and clay particle sizes (Campbell 1985, p. 8). The log normal distribution can be 
represented by a geometric mean particle diameter (dg), and a geometric standard deviation (� g). 
The �b� parameter (i.e., the slope) increases with the geometric standard deviation of the pore 
size. The geometric standard deviation depends on the soil texture. For sand particles, the 
geometric standard deviation can be estimated from a soil texture diagram as equal to 1 
(Campbell 1985, p. 10). Further, Campbell provides an empirical correction for the effects of 
bulk density (Campbell 1985, p. 46): 
�e = �es . ( �b / 3 . 1 ) . 0 67b (Eq. 6-25) 
The results for the Campbell retention relation for particle diameters of 0.317 mm, 3 mm, 
10 mm, and 20 mm are presented in Tables 6-5 through 6-8, based upon the detailed calculations 
in Attachment V. Table 6-5 presents a summary of the calculations for the parameters for the 
Campbell retention relation based upon the relations presented in Equations 6-22 through 6-25. 
Table 6-6 presents a summary of the van Genuchten curve fit parameters (� , n) based upon the 
van Genuchten curve fit to the Campbell retention relation. Table 6-7 presents the moisture 
39 of 80 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
3 mm 10 mm 
(��s) 4.50�~10-1 4.50�~10-1 4.50�~10-1 
-8.88�~10-1 -2.89�~10-1 -1.58�~10-1 
2.78 7.77�~10-1 5.16�~10-1 
-8.88�~10-3 -2.89�~10-3 -1.58�~10-3 
-9.06 -2.94 -1.61 
(��e) (Bars) 1.006�~10-2 2.99�~10-3 1.62�~10-3 
mass density (�� = 1000 kg/m3), and the acceleration due to gravity. 
0.317 mm 3 mm 
retention calculations for the Campbell retention relation. Table 6-8 presents the unsaturated 
hydraulic conductivity calculations for the Campbell retention relation. The coarse fraction will 
have low hydraulic conductivity over the range of the absolute values of moisture potentials from 
0.01 to 0.1 bars. 
Table 6-5. Summary of Parameter Determinations for the Campbell Retention Relation 
Particle Size (dg) 
0.317 mm 
(��es) 
(J/kg)1 
(��es) 
(Bars)3 
��es 
(cm)4 
Parameter 
Saturated Volumetric Moisture 
Content 
Air-entry water potential or the 
potential at which the largest 
water filled pores just drain 
Standard Deviation (��g) 
Slope of the ln(��) versus ln(��) 
retention curve (b)2 
Air-entry water potential or the 
potential at which the largest 
water filled pores just drain 
Corrected Air-entry water 
potential or the potential at 
which the largest water filled 
pores just drain 
20 mm 
4.50�~10-1 
-1.12�~10-1 
(-) 5 1 1 1 
4.24�~10-1 
-1.12�~10-3 
-1.14 
1.141.006�~10-3 
NOTES: 1 Air-entry water potential or the potential at which the largest water filled pores just drain 
(J/kg) is calculated from Equation 6-23. 
2 The b value is calculated from Equation 6-24 with ��g equal to 5 for a fine grained material 
(0.317 mm), and 1 for the coarser materials (Campbell 1985, Figure 2.1). 
3 The conversion from (J/kg) is performed by multiplying by the density (�� = 1000 kg/m3), and 
then expressing the pressure in bars. 
4 The conversion from bars to cm is performed by dividing the pressure by the product of the 
Table 6-6. Summary of van Genuchten Curve Fit Parameters Based Upon the Campbell Retention 
Relation1 
Parameter 20 mm2 
Particle Size (dg) 
10 mm 
Saturated Volumetric Moisture Content (��s) 0.450 0.450 0.450 0.450 
Residual Volumetric Moisture Content (��r) 0.020 0.010 0.010 0.010 
van Genuchten Air-entry Parameter (��) (bars-1) 47.64 230.84 476.91 561.61 
van Genuchten Air-entry Parameter (��) (cm-1) 0.06 0.24 0.48 0.56 
van Genuchten (n) Value 1.53 3.04 4.03 11.11 
van Genuchten (m) Value 0.35 0.67 0.752 0.91 
Saturated Intrinsic Permeability (k)(m2) 1.68E-10 1.51E-08 1.67E-07 6.69E-07 
Saturated Hydraulic Conductivity (Ks)(cm/sec) 0.184 16.48 183.1 732.5 
NOTES: 1 See Attachment V for the details of the van Genuchten curve fit to the Campbell retention 
relation. 
2 Note that the hydraulic conductivity under saturated conditions may have a Reynolds 
Number that exceeds the range of validity for Darcy�fs Law. 
August 2003 40 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table 6-7. Moisture Retention Calculations for the Campbell Retention Relation 1 
20 mm 
Volumetric Moisture Content2 
Particle Size (dg) 
10 mm 
Moisture 
Potential 
(�) bars 0.317 mm 3 mm 
0.076 0.010 0.010 0.010 
0.059 0.010 0.010 0.010 
0.044 0.010 0.010 0.010 
0.037 0.010 0.010 0.010 
0.032 0.010 0.010 0.010 
0.027 0.010 0.010 0.010 
0.025 0.010 0.010 0.010 
0.023 0.010 0.010 0.010 
0.022 0.010 0.010 0.010 
0.021 0.010 0.010 0.010 
0.0001 0.450 0.450 0.450 0.450 
0.0002 0.450 0.450 0.450 0.450 
0.0005 0.450 0.450 0.449 0.450 
0.001 0.449 0.447 0.434 0.449 
0.002 0.446 0.424 0.290 0.119 
0.005 0.435 0.245 0.041 0.010 
0.01 0.410 0.086 0.014 0.010 
0.02 0.363 0.029 0.010 0.010 
0.05 0.271 0.013 0.010 0.010 
0.1 0.203 0.011 0.010 0.010 
0.2 0.149 0.010 0.010 0.010 
0.5 0.100 0.010 0.010 0.010 
1 
2 
5 
10 
20 
50 
100 
200 
500 
1000 
NOTES: 1 See Attachment V for the detailed calculations. 
2 Note that Tables V-3, V-5, V-7, and V-9 present 
the calculations for the retention data presented 
above based upon the curve fit to the Campbell 
relationship and determination of van 
Genuchten parameters. 
August 2003 41 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
1000 
100 
10 
1 
0.1 
0.0001 
0.000 0.100 
Water Potential (-Bars) 
i Hydraulic Conduct vity (cm/sec) 
0.01 
0.001 
Volumetric Moisture Content 
Figure 6-6. Retention Relationships Based Upon a Normalized Moisture Potential for the Campbell 
Retention Relation (Attachment V) 
1.00E+03 
1.00E+02 
1.00E+01 
1.00E+00 
1.00E-01 
1.00E-02 
1.00E-03 
1.00E-04 
1.00E-05 
1.00E-06 
1.00E-07 
1.00E-08 
1.00E-09 
1.00E-10 
1.00E-11 
1.00E-12 
0.0001 
NOTE: The hydraulic conductivity for 20 mm under saturated conditions may have a Reynolds Number that 
exceeds the range of validity for Darcy�s Law. 
Figure 6-7. Comparison of Conductivity Relationships Based Upon the Campbell Retention Relation 
(Attachment V) 
0.200 
ANL-EBS-MD-000063 REV 00 
Van Genuchten (0.317 mm) 
Van Genuchten (3 mm) 
Van Genuchten (10 mm) 
Van Genuchten (20 mm) 
0.500 0.400 0.300 
Tuff Matrix 
0.317 mm 
3 mm 
10 mm 
20 mm 
1 0.1 0.01 0.001 
Moisture Potential (-bars) 
August 2003 42 of 80 Advection versus Diffusion in the Invert 
Table 6-8. Unsaturated Hydraulic Conductivity (Kus) Calculations for the Campbell Retention Relation 
3 mm 20 mm2 0.317 mm 
Moisture 
Potential (��) 
(bars) 
0.0001 1.65E-01 1.48E+01 1.64E+02 6.53E+02 
0.0002 1.65E-01 1.48E+01 1.63E+02 3.56E+02 
0.001 1.65E-01 1.48E+01 1.64E+01 3.08E-04 
0.001 1.65E-01 1.36E+01 7.70E-05 4.09E-10 
0.002 1.65E-01 1.04E-01 1.02E-10 5.39E-16 
0.005 1.65E-01 2.52E-09 1.72E-18 9.09E-24 
0.010 1.46E-01 3.32E-15 2.27E-24 1.22E-29 
0.020 4.50E-04 4.38E-21 3.04E-30 1.81E-34 
0.050 9.65E-12 7.40E-29 1.10E-35 4.88E-36 
0.100 1.27E-17 1.40E-34 1.22E-36 5.30E-37 
0.200 1.68E-23 6.58E-37 1.60E-37 6.72E-38 
0.500 2.84E-31 3.27E-38 7.68E-39 2.70E-39 
1.000 7.15E-37 3.48E-39 7.93E-40 3.72E-40 
2.000 3.67E-39 3.64E-40 9.29E-41 3.27E-41 
5.000 2.44E-40 1.99E-41 3.74E-42 1.89E-42 
10.000 2.65E-41 2.02E-42 4.22E-43 1.49E-43 
20.000 2.82E-42 2.02E-43 4.64E-44 1.47E-44 
50.000 1.59E-43 1.03E-44 2.07E-45 8.82E-46 
100.000 1.86E-44 1.12E-45 2.01E-46 7.76E-47 
200.000 1.63E-45 9.82E-47 1.94E-47 7.46E-48 
500.000 7.48E-47 4.36E-48 7.80E-49 3.55E-49 
1000.000 9.28E-48 4.21E-49 8.87E-50 3.38E-50 
1 NOTE: The values for the saturated hydraulic conductivity (Ks) 
(cm/sec) as a function of the moisture potential (��) are 
obtained by calculating the intrinsic permeability (k) for a 
given set of van Genuchten parameters from Equation IV-4 
and then applying the conversion from intrinsic permeability 
(k) to saturated hydraulic conductivity (Ks) (Freeze and 
Cherry 1979, p. 27): 
Unsaturated Hydraulic Conductivity (Kus) 
(cm/sec)1 
Particle Size (dg) 
10 mm 
K = s 
��. g . k 
�� 
The properties of water at ambient temperature are given by 
Incropera and DeWitt (1996, p. 846). The water density (��) 
equals 1000 kg/m3 and the absolute viscosity (��) equals 
8.935�~10-4 N.s/(m2). The values for the unsaturated 
hydraulic conductivity are then determined by scaling the 
saturated hydraulic conductivity relationship by the relative 
permeability relationship presented in Equation IV-11 for a 
given moisture potential. 
August 2003 43 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
6.6 NUFT ADVECTION CALCULATION 
In In-Drift Thermal-Hydrological-Chemical Model (BSC 2001b), a single-continuum 
approximation was used to represent hydrologic properties and performance of the invert ballast 
material. 
The invert ballast material is comprised of crushed tuff. Crushed tuff contains two types of 
porosity, an intragranular matrix porosity (�matrix) component (within the grains) and an 
intergranular porosity (�intergrain) (between the grains). Each of these porosity components has 
distinct intrinsic hydrologic properties. The single-continuum approach, generally referred to as 
Equivalent Continuum Model (ECM), does not explicitly represent the hydrologic properties for 
each porosity component, but instead represents hydrologic behavior of the crushed tuff with a 
single set of average properties (Attachment XI). 
An alternate analysis approach is to assess invert hydrologic performance with a dual 
permeability model (DKM) approach wherein each porosity component is represented explicitly. 
This approach allows the intragranular and intergranular porosities to behave in a manner 
consistent with their respective intrinsic hydrologic properties (e.g., capillary suction potentials). 
In general, when the grain size is small, the intragranular and intergranular component behave 
like a porous matrix medium and thus the single-continuum approach may be justified. 
However, as the grain size increases, the intergranular and intragranular components would tend 
to behave more independently of each other and accordingly, the DKM approach is a more 
appropriate method to use in modeling. 
r 
A series of NUFT simulations have been performed to corroborate the hydrologic performance 
of the invert using the van Genuchten and Campbell retention relations for different grain sizes 
as examined by this analysis. The input parameters to the DKM model of NUFT are porosity 
(�), van Genuchten air-entry parameter (�) and �m� values, residual and maximum saturation (S 
and Ss), and intrinsic permeability (k) for the intergranular and intergranular components, as 
derived in Section 6.3. 
The percolation flux at the PTn/TSw contact for the analysis was selected as 35 mm per year or 
70 mm per year. Percolation fluxes for the upper bound glacial climate that represents most 
extreme case for nine cases analyzed are presented in DTN: LB0302PTNTSW9I.001. The nine 
cases are for the three climate conditions, and the lower, mean, and upper bound distributions for 
each climate. The mean value for the upper bound glacial is 35.63 mm per year. A value of 35 
mm per year was selected for analysis. The distribution function approximately follows an 
exponential distribution in which the mean equals the standard deviation (Hahn and Shapiro 
1967, pp. 122 to 124). To provide a more extreme condition that corresponds approximately to 
the mean, plus one standard deviation (70 mm per year) was applied. Note that, as discussed 
subsequently in Section 6.8, in the coarse pore space, the results were not sensitive to the 
percolation flux. 
The matrix properties for the crushed tuff grains are the same as those of the host rock 
surrounding the drift (TSw36). Simulations are conducted for different intergranular pore space 
properties based on a uniform grain diameter size of 0.317, 3, 10, or 20 mm. The 0.317-mm size 
tends to wick more liquid into the invert than the larger grain size and thus is conservative for 
radionuclide transport. 
August 2003 44 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
The hydraulic response of the intergranular pore space is represented as a relationship between 
moisture potential and moisture content. This relationship is based on the van Genuchten 
retention relation (Fetter 1993, p. 172) discussed previously in Section 6.4, and the Campbell 
retention relation (Campbell 1985, p. 45) discussed previously in Section 6.5. Figures 6-4 and 66 
present the van Genuchten and Campbell curve fit to moisture retention data for the different 
grain sizes. The corresponding van Genuchten and Campbell relationships (Fetter 1993, p. 182) 
between unsaturated hydraulic conductivity (Kus) and moisture potential are presented in Figures 
6-5 and 6-7. Although the van Genuchten and the Campbell curves show the same saturated 
hydraulic conductivity (Ks) for a given grain size, the van Genuchten relationship, in general, 
provides a lower unsaturated hydraulic conductivity, for the same moisture potential for the 
invert material. 
6.7 NUFT SIMULATIONS 
The key changes for the current calculations are the use of a dual-continuum approach and better 
spatial resolution for EBS components. Attachment I presents a list of the NUFT input file 
names. These and other minor modifications are listed below: 
� Backfill is not present in the space between the drip shield and the drift wall. 
� Heating from the waste package is ignored since the purpose of this analysis is to predict 
a steady-state liquid saturation (S) and flux distribution in the invert. 
� An approximate-round shape for the waste package and a letter-box shape for the drip 
shield are included in the model. 
� The air gap below the drip shield extends through the space between the waste package 
and the invert. That is, the waste package is not in contact with the invert. 
� A Dual-Permeability Model (DKM), as discussed in Water Distribution and Removal 
Model (CRWMS M&O 2001b, Section 6.2.1), has been used rather than a singlecontinuum 
or equivalent continuum model. DKM is used to describe the permeability 
for the geologic media surrounding the drift and for the invert ballast material (DTN: 
LB990861233129.001). It is appropriate to model the invert with the dual-permeability 
approach because the crushed tuff has an intragranular matrix component and an 
intergranular pore space component. The dual-permeability approach is consistent with 
the anticipated response of the tuff grains, where the matrix component will behave like 
the host rock and the large, intergranular pore spaces will act as a capillary barrier to 
incoming flow from the host rock. 
� The depth of the invert of 0.806 m is used to determine the path length, as discussed in 
Attachment XV. Note that, as discussed in Attachment XV, the path length of 0.5 was 
selected to account for possible variations in the location of contaminants entering the 
invert from the waste package, and to account for settlement with time. 
� The simulation grid is finer than that used in In-Drift Thermal-Hydrological-Chemical 
Model (BSC 2001b) to provide more spatial resolution in the invert and to provide better 
definition of the shapes of the waste package, drip shield, invert, and the air gap between 
waste package and invert. 
ANL-EBS-MD-000063 REV 00 August 2003 45 of 80 Advection versus Diffusion in the Invert 
The thermal and hydrologic properties of stratigraphic units of the Natural Barrier System (NBS) 
from the ground surface to the water table were developed in Calibrated Properties Model 
(BSC 2003c) as presented in DTN: LB990861233129.002. These properties (Tables 4-5, 4-6, 
and 4-7) are the same as those used in In-Drift Thermal-Hydrological-Chemical Model (BSC 
2001b). Attachment VIII presents the hydrologic properties of other EBS components used in 
the NUFT runs. For the specific location of the repository used in the In-Drift Thermal- 
Hydrological-Chemical Model, the drift would be surrounded by the Tptpln (TSw36) formation 
even though the majority of the repository elsewhere will be in the Tptpll (TSw35) formation. 
Similarly in these calculations, it is assumed that the TSw36 unit surrounds the drift. The 
sensitivity of replacing TSw36 by the TSw35 units around the drift on the fluxes in the invert is 
analyzed in Section 6.8. 
The calculations were performed with an ambient geothermal gradient (no heating from waste 
package) because the purpose of the analyses is to predict the steady-state liquid saturation (S) 
and flux distribution in the invert at late times, after waste package failure, when the thermal 
pulse has passed through the repository system. 
6.8 NUFT RESULTS 
The detailed NUFT results are presented in Attachment X. Attachment II presents a list of the 
NUFT output files. A model simulation grid is presented on Figure X-1. All discussions here 
will pertain to steady state conditions. Figures X-2a and X-2b show the general flow directions 
in the rock fractures and matrix, respectively, around the drift. They both show essentially 
identical pattern. The flow directions as displayed by XTOOL, a post-processor of NUFT, are 
only approximate. The size of the vector does not reflect the magnitude of flow and the vectors 
may extend slightly to a neighboring cell that may not have any flow at all. This description 
applies to Figures X-3 through X-18 that show liquid flux directions in the intergranular and 
intragranular components of the invert for different grain sizes using the van Genuchten and 
Campbell models. The blank space on these figures represents areas with zero flux. The exact 
flow direction and magnitude of flux in a cell, as computed by NUFT, can be displayed on a 
monitor by positioning the cursor in that cell while running XTOOL. 
6.8.1 NUFT Analyses with the Nondimensionalized van Genuchten and Campbell 
Relationships 
Tables 6-9 and 6-10 summarize the computational results at steady state for a grain size varying 
from 0.317-mm diameter to 20-mm diameter for the two retention relations. The tables compare 
conditions in the invert in the intergranular, and intragranular components. The data in Tables 
6-9 and 6-10 and Figures X-2 through X-17 demonstrate that: 
� The saturation of the rock matrix in individual grains (Smatrix) of the grain matrix is 
approximately equal to the saturation of the host rock. This is a reasonable response 
because the tight pores in the tuff grains have high capillarity that draws water from the 
fractures of the host rock into the grains. 
� The vertical pore-water velocity (V) in the matrix component is low and is controlled by 
the saturated hydraulic conductivity (Ks), the pressure gradient in the matrix, and also by 
the pressure gradient across the intergranular and intragranular components. 
August 2003 46 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
� The Campbell retention relation predicts that for a grain size of 0.317-mm in diameter, 
the intergranular component behaves similarly to the intragranular matrix component, 
and the moisture potential equilibrates across the two components. As the grain size 
increases to 3-mm diameter or larger, the intergranular component breaks away and 
behaves as a capillary barrier, as shown by a sharp difference in pressure potential across 
the two components. This pressure gradient steepens as the grain size increases from 
3-mm to 20-mm diameter, inducing a slightly higher flux in the matrix component. This 
matrix flux tends to stabilize between the 10- and 20-mm grain size models. 
� The van Genuchten retention relation predicts that for all grain sizes of the crushed tuff 
0.317-mm diameter or larger, the intergranular component acts as a capillary barrier, 
whereas the intragranular component behaves as a matrix medium, as expected in a 
DKM model. 
� For those cells in the invert showing zero intergranular fluxes (represented by blank 
space in the figures), the saturation (Sintergrain) of the intergranular pore spaces is zero, 
confirming that the pore space acts like a capillary barrier to flow. 
� Except for the 0.317-mm Campbell retention relation, the intergranular components of 
all other models behave as a capillary barrier to flow from the rock to the invert. 
6.8.2 NUFT Sensitivity Analyses 
Sensitivity analyses were conducted to evaluate the impact of: 1) different grain/void contact 
areas, 2) host-rock properties, 3) infiltration rates, 4) invert thickness, and 5) grid spacing in the 
invert on fluxes in the invert for the van Genuchten retention relation. These analyses are 
discussed in Attachment X and presented in Figures X-2 through X-19. The analyses show that: 
1. An increase of the specific grain/void contact area from 90 m-1 to 8,517 m-1 would 
increase the matrix fluxes by only a factor of 1.2 to 1.4. The intergranular fluxes are 
the same (pr2D35rmntinvrs in Attachment I). 
2. A change of the host-rock properties from tsw36 (Tptpln) to tsw35 (Tptpll) and the 
corresponding invert matrix properties would increase the matrix fluxes by a factor of 
1.3 to 1.5. Intergranular fluxes are essentially the same (pr2D35rmntinv35 in 
Attachment I). 
3. An increase of infiltration rate at the ground surface from 35- to 70-mm per year 
(Section 3.14) would increase the intergranular fluxes by a factor of 3 to 5 along the 
outer edge of the invert. Matrix fluxes remain practically unchanged (pr2D70rmni in 
Attachment I). 
4. An increase of invert thickness from 0.5 to 0.6 meters would not change the fluxes 
(pr2D35rmntinv2 in Attachment I). 
5. Refinement of grid spacing in the invert would not significantly change the fluxes in 
the invert. This indicates that the mesh is fine enough to produce reasonable results 
(pr2D35rmnt2 in Attachment I). 
ANL-EBS-MD-000063 REV 00 August 2003 47 of 80 Advection versus Diffusion in the Invert 
The NUFT_Input.txt file (Attachment I) contains information from the output of the NUFT 
calculation. The first line states the number of grid points minus one (n-1) at the centerline of 
the NUFT model that are input. The program then loops from 0 to (n-1) to read in each line the 
elevation (m), depth(m), pore-water velocity (V) (m/sec), the degree of saturation (S), and the 
Darcy flux (Jw) (m/sec) for each grid point. Note that the Darcy flux for the invert material 
equals the pore-water velocity times the degree of saturation (S) times the porosity (�). 
6.8.3 Corroboration of the NUFT Analyses 
The following presents a discussion of the retention measurements made on crushed tuff 
presented in Attachment XI based upon the Unsaturated Flow Apparatus (UFA) as described in 
� 
The results of the analysis as shown in Figure XI-1 show that over a broad range of moisture 
potential that would be expected in the repository that the measured volumetric moisture content 
was approximately equal to the matrix porosity. The UFA measurements indicate that the fine 
pore space as represented by the intragranular porosity was nearly saturated while the 
intergranular porosity was free of water. The UFA measurements provide a corroboration of the 
NUFT analysis results. 
Appendix C of Engineered Barrier System Performance Requirements Systems Study Report 
(CRWMS M&O 1996). The measurements are from DTN: GS980808312242.015. 
The UFA mainly consists of an ultracentrifuge in which a soil sample is subject to centrifugal 
force. The volumetric moisture content (�) as a function of the moisture potential (�) as 
discussed subsequently below can be determined by allowing the sample to drain until the 
moisture potential equals the centrifugal force per unit area divided by the unit weight in a state 
of equilibrium. The volumetric moisture content (�) is determined gravimetrically using the bulk 
density of the sample. The UFA represents an efficient method for testing fine-grained soils at 
higher moisture potential (�). 
As discussed Attachment XI, the van Genuchten curve-fitting parameters were determined by 
fitting the curve to the retention data for crushed tuff (Section 4.1.5). Attachment XVI presents 
the Microsoft Excel 97 spreadsheet that uses the Equation Solver to calculate the curve-fitting 
parameters. The results from the curve-fitting process presented in Attachment XVI (p. XVI-22) 
are: 
r = 0.05 
�i = 0.12 (1/cm) 
ni = 2.75 
August 2003 
6.9 CONTAMINANT TRANSPORT RESULTS 
The results of the NUFT calculations, as summarized in Tables 6-9 and 6-10, show in general 
that the coarse intergranular pore space is drained and free of water while the intragranular pore 
space is near saturation. If consideration is given to the advection-dispersion-diffusion relation 
presented in Equation 6-4, with calculation of the diffusion coefficient by the relationship 
presented in Equation 6-11 to flow, bounding calculations can be performed to estimate the 
range of breakthrough times near the centerline of the model. The NUFT results suggest that 
48 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
diffusion would be the dominant mass transport mechanism in the intergranular pore space that is 
drained of water, and near the top surface of the invert. As depth into the invert increases, the 
NUFT results suggest that advection becomes more dominant. 
The diffusion relation (Equation 6-12) is used to estimate the invert liquid diffusion property. 
Note that the experiments performed, as presented in Invert Diffusion Properties Model 
(CRWMS M&O 2000b, Figure 3), did not distinguish between intergranular and intragranular 
porosities in the calculation of the diffusion coefficient. The following presents a calculation of 
the bulk porosity (��), and the bulk volumetric moisture content (��) based upon the summary 
material parameters. 
. 1 1 �� intergrain 
1 �� intergrain 
V 
�� matrix 
. 
6.9.1 Calculation of Bulk Porosity 
Using the following definitions for the total volume, Vtotal, of grains and pore space and for the 
total volume of the grains, VGrain,Total, the total volume and the grain total volume can be 
expressed as, respectively: 
(Attachment XIII) 
. 1 �� intergrain 
�� Bulk 
�� Bulk 
ANL-EBS-MD-000063 REV 00 
The detailed calculations for the dripping case and the nondripping case are presented in 
Attachment XIII. The bulk moisture content for the dripping case can be expressed as 
. 
1 �� matrix 1 �� intergrain 
For the nondripping case, the coarse pore space has a zero percent saturation as determined from 
the NUFT analysis, and the fine pore space may be partially saturated at a fairly high saturation. 
The bulk volumetric moisture content can be expressed as (Attachment XIII): 
�� matrix 
1 �� matrix 
V = Total . V , c 
1 
1 �� matrix 1 �� intergrain �� intergrain �� r 
. 
�� matrix 
1 �� matrix 1 �� intergrain 
. 
. 
�� matrix 
49 of 80 
�� Vs + Vtm 
VTotal �� V s + Vtm + V c , 
Grain , Total 
.
S matrix 
1 
1 
1 
(Eq. 6-26) 
�� �� r �� matrix 
1 �� matrix (Eq. 6-27) 
�� matrix 
1 �� matrix 
(Eq. 6-28) 
�� matrix 
1 �� matrix 
August 2003 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Table 6-9. Comparison of Invert Conditions at Steady State for the van Genuchten Method of Formulating Retention Data 
Component 
Parameters Intergranular Component Intragranular (Matrix) Component 
Saturation 
at Center 
of Invert 
Moisture 
Potential (�) at 
Center (Pa) 
Moisture 
Potential (�) 
at Center (Pa) 
Vertical Pore 
Velocity (V) Along 
Centerline (m/s) 
Saturation at 
Center of 
Invert (Smatrix) 
Vertical Pore 
Velocity (V) Along 
Centerline (m/s) 
Grain 
Diameter 
(dm) (mm) 
Porosity 
(�matrix) 
Porosity 
(�intergrain) (Sintergrain) 
3377 0.0 0.450 0.317 0.0 0.999 0.112 9368 7.6E-11 to 1.8E-10 
357 0.0 0.450 3 0.0 0.998 0.112 1.41E4 5.9E-11 to 1.3E-10 
111 0.0 0.450 10 0.0 0.997 0.112 1.52E4 5.8E-11 to 1.3E-10 
56 0.0 0.450 20 0.0 0.997 0.112 1.54E4 5.9E-11 to 1.3E-10 
Table 6-10. Comparison of Invert Conditions at Steady State for the Campbell Method of Formulating Retention Data 
Component 
Parameters Intergranular Component Intragranular (Matrix) Component 
50 of 80 
Saturation 
at Center 
of Invert 
Moisture 
Potential (�) at 
Center (Pa) 
Moisture 
Potential (�) 
at Center (Pa) 
Vertical Pore 
Velocity (V) Along 
Centerline (m/s) 
Saturation at 
Center of 
Invert (Smatrix) 
Vertical Pore 
Velocity (V) Along 
Centerline (m/s) 
Grain 
Diameter 
(dm) (mm) 
Porosity 
(�matrix) 
Porosity 
(�intergrain) (Sintergrain) 
5.5E-10 to 1.6E-9 7370 0.487 0.450 0.317 0.112 0.999 7370 4.4E-14 to 2.1E-13 
2981 0.0 0.450 3 0.0 0.998 0.112 1.16E4 3.7E-11 to 8.3E-11 
830 0.0 0.450 10 0.0 0.997 0.112 1.50E4 5.7E-11 to 1.3E-10 
267 0.0 0.450 20 0.0 0.997 0.112 1.54E4 5.8E-11 to 1.3E-10 
August 2003 Advection versus Diffusion in the Invert 
Table 6-11. Summary of Material Parameters 
Parameter Symbol Value Source 
Porosity of the rock matrix in 
an individual grain 
Porosity of the large pore 
spaces between grains 
Saturation of the rock matrix in 
an individual grain 
Saturation of the large pore 
spaces between grains 
The calculated volumetric moisture contents for the dripping and nondripping cases are 
presented in Attachment XIII as 0.073 and 0.058, respectively, for purposes of illustration are 
presented in Attachment XIII. 
��matrix 
��intergrain 
smatrix 
0.112 DTN: LB990861233129.001 
0.45 See Section 6.3 
0.999 Table 6-9 
0.0 Table 6-9 sintergrain 
August 2003 
6.9.2 Calculation of the Diffusion and Dispersion Coefficient 
The calculated diffusion coefficient is based on a freewater diffusion coefficient of 2.3�~10-5 
cm2/sec (Assumption 5.6) and near saturation of the intergranular pore space from Attachment 
III. Equation III.7 is 1.7�~10-7 cm2/sec for the dripping case, as presented in the breakthrough 
analysis in Attachment III. 
The hydrodynamic dispersion coefficient for each of the breakthrough analyses is calculated by 
taking the pore-water velocities from Tables 6-9 and 6-10 of this calculation, and multiplying by 
the dispersivity of 10 cm or 0.1 m, as presented in Attachment III. The effective dispersion 
coefficient is then the sum of the diffusion coefficient, and the hydrodynamic dispersion for each 
case analyzed, as presented in Attachment III. 
Attachment XII presents a range of diffusion coefficients for the dripping case for a broad range 
of waste package flow rates. The results of the analysis shows that diffusion rates are on the 
order of 10-8 to 10-7 cm2/sec. 
51 of 80 
6.9.3 Breakthrough Analysis 
The closed form analytical solution for the one-dimensional advection-dispersion equation 
(Equation 6-4) is used to develop breakthrough curves for a range of conditions. Figure 6-8 
presents the breakthrough analysis for 0.317 mm grain diameter with the retention properties 
given by the van Genuchten retention relation. Attachment III presents the detailed calculations. 
The analysis is subject to the assumptions that the direction of the advective flux is in the vertical 
direction (Assumption 5.1); the effects of lateral dispersion are neglected (Assumption 5.2); and 
the invert is homogeneous (Assumption 5.3). The radionuclides are assumed to be released at 
the centerline of the drift (Assumption 5.4), and that the effects of radioactive decay are 
neglected (Assumption 5.5). Solute vapor phase is assumed to be negligible (Assumption 5.7). 
The range of pore water velocities for the intergranular pore space was from 7.6�~10-11 m/s to 
1.8�~10-10 m/s for the low and high advection cases, respectively (Table 6-9). For comparison, 
the breakthrough curve for diffusion is presented. The breakthrough analysis for the case of 
variable pore-water velocity would fall between these extremes for high and low advection. The 
ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
results of the analysis show that breakthrough would occur within a time frame of 400 to 1,000 
years while a diffusion dominated process in the intergranular pore space would occur within a 
time frame of 1,000 to 10,000 years. 
The closed form analytical solution for the one-dimensional advection-dispersion equation 
(Equation 6-4) is used to develop breakthrough curves for grain diameters from 3 to 20 mm with 
the retention properties given by the van Genuchten and Campbell retention relations 
(Section 6.3). Figures 6-9 through 6-10 present breakthrough curves for high and low advection 
for the nondimensionalized van Genuchten and Campbell relations, respectively. The range of 
pore water velocities for the intergranular pore space was from 5.9�10-11 m/s to 1.3�10-10 m/s for 
the low and high advection cases, respectively (Table 6-9). The breakthrough analysis for the 
case of variable pore-water velocity would fall between these extremes for high and low 
advection. The results of the analysis show that breakthrough would occur again within the same 
time frame of 600 to 2,000 years while a diffusion dominated process such as would occur in the 
intergranular pore space would be of the order of a 1,000 to 10,000 years. Since advection 
through the intergranular pore space of the tuff matrix is not affected by pore size, the range for 
breakthrough for grain sizes between 3 to 20 mm is very similar. 
The closed form solution for (Equation 6-4) is used to develop breakthrough curves for grain 
diameters from 3 to 20 mm with the retention properties given by the Campbell retention 
relation. The range of pore water velocities for the intergranular pore space was from 3.7�10-11 
m/s to 1.3�10-10 m/s for the low and high advection cases, respectively (Table 6-10). For this 
range of grain sizes, the results of the breakthrough analysis for the intergranular pore space are 
nearly identical to the range given by the van Genuchten retention relation. 
1 
0.75 
0.5 
0.25 
1 
0 
0.1 
Diffusion Dominated 
Low Advection 
High Advection 
Figure 6-8. Breakthrough Curves for the 0.317 mm van Genuchten Retention Relation (Attachment III) 
ANL-EBS-MD-000063 REV 00 August 2003 52 of 80 
C(t)/C0 
Van Genuchten Retention 0.317 mm 
10 1 .105 1 .104 1 .103 100 
Time(yr) Advection versus Diffusion in the Invert 
R = 1+ 
�b Kd (Eq. 6-29) 
1000 . 3 3 m m = 2385 . kg (Eq. 6-30) 3 m 
6.9.4 Retardation 
This section provides a sensitivity analysis for radionuclide transport in solute or radionuclides 
adsorbed onto colloidal particles in the invert. This section applies the solution to the onedimensional 
solute transport equation (Equation 6-4) for a range of retardation. 
The retardation factor for unsaturated flow is obtained from Jury et al. (1991, p. 227): 
�b = 
Sheppard and Thibault (1990, Tables A-1 through A-4) present data on the mean and standard 
deviation of the natural logarithm of the partition coefficient for individual elements for sands, 
loams, clays, and organic soils that would have application to the invert. This information 
includes the number of measurements. Sheppard and Thibault (1990, p. 472) also cite previous 
work in which the partition coefficients are log normally distributed. Finally, Sheppard and 
Thibault (1990, p. 472) calculated the geometric mean and the geometric standard deviation for 
each element by soil texture for the mineral soils and also for organic soils. All distributions 
developed were log-uniform because the partition coefficient (Kd) data is often observed to be 
logarithmically distributed, because a log-distribution maintains uniformity of sampling in each 
decade, and because little is known about the distribution of partition coefficient (Kd) values 
throughout a given range. 
To take into account a broad range of values, the partition coefficients, an analysis was 
performed using the closed form solution presented in Equation 6-4 for partition coefficient (Kd) 
values of 1, 10, 100, and 1,000 ml/gram. The calculated values for the retardation coefficients 
are presented in Table 6-12 using Equation 6-29. 
Equation 6-4 is used to calculate breakthrough curves taking the pore-water velocity, and the 
diffusion coefficient as calculated above, and dividing by the retardation factor as calculated in 
Table 6-12. The results of these calculations are presented in Figures 6-11 through 6-14. The 
results of the calculations show that for the case of low sorption (Kd = 1.0 ml/gm), the retardation 
factor is low, and breakthrough is delayed to 1,000 to 5,000 years. For the case of intermediate 
and high sorption (10 - 100 ml/gm), the delay in breakthrough time is more substantial. In the 
case of the intermediate sorption, breakthrough would be delayed from 1,000 to 10,000 years. In 
the case of high sorption (1,000 ml/gm), breakthrough is delayed by at least one million years. 
ANL-EBS-MD-000063 REV 00 
� 
The bulk density used above is taken as the saturated bulk density. The following calculation is 
used to estimate the saturated bulk density. The volume of the voids (Vtm) is calculated in 
Equation 6-19 as 0.126 m3. The grain density is given as 2560 kg/m3 for TSW36 
(DTN: LB990861233129.001). The saturated bulk density is the weight of the water (Ww) and 
the weight of the solids (Ws) divided by the volume of the solids (Vs = 1 m3) and the volume of 
the water (VWtm): 
kg . 126 . m3 + 2560 . kg . 0 
3 1. m3 + 0 .126 . m 
53 of 80 August 2003 Advection versus Diffusion in the Invert 
Table 6-12. Retardation Calculations 
Figure No. 
Saturation 
(S) (�b) 
Volumetric 
Moisture 
Content (�) 
Partition Coefficient 
(Kd) 
ml/gm m3/kg (-) 
C(t)/C0 
1 
0.75 
0.5 
0.25 
10 1 
0 
0.1 
Bulk Density Retardation 
Factor R 
(-) (kg/m3) (-) 
1 0.001 0.997 0.11 2385 22.7 6-11 
10 0.01 0.997 0.11 2385 217.8 6-12 
100 0.1 0.997 0.11 2385 2169.2 6-13 
1000 1 0.997 0.11 2385 21682.8 6-14 
Van Genuchten Retention 3-20. mm 
1 .105 1 .104 1 .103 100 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection 
Figure 6-9. Breakthrough Curves for High and Low Advection (3-20 mm van Genuchten Retention 
Relation) 
August 2003 54 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
1 
0.75 
0.5 
0.25 
1 
0 
0.1 
Diffusion Dominated 
Low Advection 
High Advection 
Figure 6-10. Breakthrough Curves for High and Low Advection (3-20 mm Campbell Retention Relation) 
1 
0.75 
0.5 
0.25 
1 
0 
0.1 
Diffusion Dominated 
Low Advection 
High Advection 
C(t)/C0 
Kd = 1 ml/gm 
10 1 .103 100 
Time(yr) 
1 .105 1 .104 
Figure 6-11. Breakthrough Curves for the Case of Low Partition Coefficient (Kd = 1.0 ml/gm) 
ANL-EBS-MD-000063 REV 00 55 of 80 August 2003 
C(t)/C0 
Campbell Retention 3-20. mm 
10 1 .103 100 
Time(yr) 
1 .105 1 .104 Advection versus Diffusion in the Invert 
1 
0.75 
0.5 
0.25 
1 
0 
0.1 
Diffusion Dominated 
Low Advection 
High Advection 
Figure 6-12. Breakthrough Curves for the Case of Low Partition Coefficient (Kd = 10.0 ml/gm) 
1 
0.75 
0.5 
0.25 
1 
0 
0.1 
Diffusion Dominated 
Low Advection 
High Advection 
C(t)/C0 
Kd = 100 ml/gm 
10 100 1 .103 1 .104 1 .105 1 .106 1 .107 
Time(yr) 
Figure 6-13. Breakthrough Curves for the Case of High Partition Coefficient (Kd = 100 ml/gm) 
ANL-EBS-MD-000063 REV 00 August 2003 56 of 80 
C(t)/C0 
Kd = 10 ml/gm 
10 1 .105 1 .106 1 .103 1 .104 100
Time(yr) Advection versus Diffusion in the Invert 
Kd = 1000 ml/gm 
C(t)/C0 
1 
0.5 
10 100 1 . 103 1 . 104 1 . 105 1 . 106 1 . 107 1 . 108 1 
0 
0.1 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection 
Figure 6-14. Breakthrough Curves for the Case of High Partition Coefficient (Kd = 1000 ml/gm) 
6.10 TOTAL SYSTEM PERFORMANCE ASSESSMENT LICENSE APPLICATION 
ABSTRACTION FOR THE DRIPPING CASE 
The following analysis presents a method for assessment of the breakthrough for radionuclides in 
the invert for the dripping case and the nondripping case. As discussed in Section 6.9.3, and as 
summarized in Tables 6-9 and 6-10, the results of the analysis show that if the grain sizes were 
equal or greater than 3 mm that the intergranular porosity is free of water, while the intragranular 
porosity is near saturation. If consideration is given to the moisture potential for the cases in 
which the grain sizes are 3 mm or greater, then an estimate of the vertical porewater velocity is 
obtained from the relationship presented by Jury et al. (1991, p. 127): 
J = . K ( � ) (Eq. 6-31) w 
Considering the retention relationship presented by Fetter (1993, p. 172), this relationship can be 
restated as: 
J = . K ( � ) (Eq. 6-32) w 
As stated by Jury et al. (1991), for prolonged downward flow, the invert water content adjusts to 
that value necessary to drain the invert under gravity at the imposed rate. This provides a means 
of estimating the fluxes in the coarse and fine pore space, and on the basis of saturation, the 
moisture content, and diffusion rates from the basic constitutive relations: 
� For the case of zero seepage within the drift, the results of the two-dimensional NUFT 
calculations in which saturations are determined in the coarse and fine pore space are 
used. The NUFT calculations provide the advective fluxes directly, and the pore 
August 2003 57 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
velocities can be determined from the advective fluxes and the volumetric moisture 
content. 
� The estimate of the diffusion coefficient for the invert requires an assessment of the bulk 
volumetric moisture content. Attachment XIII presents several derived relationships for 
assessing the bulk volumetric moisture content. From Attachment XIII, the bulk 
volumetric moisture content is given by for the case of zero seepage into the drift: 
� matrix S matrix 1 � matrix � Bulk 
. 
� matrix 1 � matrix . 1 1 � intergrain 1 � matrix (Eq. 6-33) 
. 1 863 (Eq. 6-34) 
. 
1 � matrix 1 � intergrain 
� The diffusion relationship for the bulk saturation is given by (Section 6.2): 
wl . �bulk Dsl 
6-12). 
p. 223). 
(Eq. 6-35) w A WASTE _ PACKAGE _ PLAN 
( �) = D 
A solution is presented by Freeze and Cherry (1979, Equation 6-4) for non-retarded transport in 
one dimension with initial concentration (Co) moving at a continuous rate where the vapor phase 
transport is negligible (Freeze and Cherry 1979, p. 391). This premise is supported by 
Assumptions 5.1, 5.2, 5.3, 5.4, 5.5, and 5.7 and is appropriate for use in this scientific analysis. 
The effective dispersion-diffusion coefficient (De) equals the sum of the hydrodynamic 
dispersion (Dlh) and soil-liquid diffusion coefficients (Dsl) (Jury et al. 1991, p. 222) (Equation 
Further, the dispersion-diffusion coefficient (D) in m2/sec, equals the effective dispersiondiffusion 
coefficient (De) divided by the volumetric moisture content (�) (Jury et al. 1991, 
This hydrodynamic dispersion coefficient (Dlh) has frequently been observed to be proportional 
to the pore-water velocity V, also known as the average linear velocity, where � is the 
dispersivity in cm and V = Jw/�matrix (Jury et al. 1991, p. 221) (Equation 6-13). 
Note that for the dripping case, the unsaturated flow analysis takes into account any potential 
flow focusing to the emplacement drift that tends to increase saturation in the intergranular 
porosity. On the other hand the fine pore structure would result from wicking of water to the 
crushed tuff matrix. Further, the drip shield tends to divert flow around the waste package and 
would reduce the advective flux. For flow that occurs through the drip shield, and waste 
package, the flux to the invert for the dripping case can be approximated as: 
Q WASTE _ PACKAGE J = 
58 of 80 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment XIV presents the analysis over a range of waste package flow rates to illustrate 
calculations for the dripping case for the four different particle sizes. The attachment first inputs 
the unsaturated hydraulic conductivity relationships from Attachment IV. The following 
equation is solved through the use of an interpolation table: 
J . K ( �r ) = 0 (Eq. 6-36) w 
Or substituting in the unsaturated hydraulic conductivity as determined from the van Genuchten 
relationship (Fetter 1993, p. 182): 
2 .. 
(Eq. 6-37) ) n . � . 
.. 
= 0 w J . K . 
s 
. . 1 + 
n 
.. 
. 
.. 
1
+ ( �r .. 
. . 
1 .
. . 
1 . . 
. 
. 
. 
. 
... 
.. 
. 1
. ( �\ �r ).. . 
n . 1.� 
. 
. �\ 
. 
.. 
1 
.
. 
2 . .
2n .
� .
.. 
. . . 
. 
. 
. 
. . . 
..1 + ( �\ �r )n . . 
.. 
.. 
. 
. 
The lookup tables use logarithmic interpolation to solve for the moisture potential in the coarse 
intergranular pore space. Solving for �r : 
�r = K .1 (J ) (Eq. 6-38) w 
The moisture potential can then be used to determine the volumetric moisture content in the 
coarse pore space assuming the fine pore space is saturated, and then used with the 
dispersivity/diffusion relationships presented above to calculate the diffusion, and the dispersion 
coefficients. The breakthrough analysis then proceeds as presented above. 
As a practical matter, the retention relationships show that there is a ��wet�� range of moisture 
potential near saturation, and a very broad ��dry�� range of moisture potential in which the 
intergranular porosity is completely devoid of water. It is much less likely that the invert would 
retain moisture at intermediate saturations. If dripping occurs to saturate the coarse pore space of 
the invert, both advection and diffusion would be high, and breakthrough would occur rapidly. 
On the other hand, if the intergranular porosity remains unsaturated then breakthrough occurs 
slowly as suggested by the analysis presented in Attachment XIV. 
6.11 DISCUSSION ON UNCERTAINTIES AND LIMITATIONS 
The analyses presented in this report develop the hydrological and thermal properties for crushed 
tuff in the invert for a dual-porosity media comprised of an intergranular porosity between 
crushed tuff particles, and an intragranular porosity within crushed tuff. The following 
discussion presents the breakthrough performance measure, the evolution of the results with 
time, and the range of results. It discusses uncertainties as they apply to the intergranular and 
intragranular porosities for percolation rate, and the range of crushed tuff particle sizes as they 
affect the intergranular porosity. Further, the discussion considers diffusion, and retardation. 
The relative importance of these uncertainties is ranked in terms of their importance. 
59 of 80 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
The fundamental performance measure in this analysis is the time for breakthrough, as presented 
in Section 6.9.5. The radionuclides are released at a constant rate as a step function beginning at 
time zero, and breakthrough depending on the mechanisms of advection, dispersion, and 
diffusion. The breakthrough time performance measure provides an indication as to when a 
concentration front of radionuclides released at a constant rate at the top of the invert would 
travel from the top of the invert to the bottom of the invert through the active pore space, and be 
released into the unsaturated zone. The breakthrough analysis also presents the evolution of 
radionuclide breakthrough with time since a high advection, dispersion, and diffusion results in a 
more rapid breakthrough of radionuclides while a low advection, dispersion, and diffusion results 
in a slow breakthrough of radionuclides. Note that concentration fronts that are time varying can 
be analyzed by convolution or superposition methods to assess a time varying concentration 
front. 
The NUFT sensitivity analysis presented in Section 6.8 considered the effect of percolation on 
advection in the invert. Percolation fluxes for the upper bound glacial climate that represents 
most extreme case were discussed in Section 6.6. The range discussed in that section was 
selected to cover a broad range of percolation rates and as discussed previously, the unsaturated 
flow the matrix was found to be insensitive to the percolation rate. The NUFT results show that 
the intragranular pore space within crushed tuff particles in the invert is near saturation with 
advection is bounded by the matrix saturated hydraulic conductivity (10-8 cm/sec) (see Figure 
6-5), and independent of the percolation rate. 
Uncertainties as they apply to the coarse intergranular porosity can be divided into uncertainty of 
the constitutive relations to be applied, and parameter uncertainty. Note that while the van 
Genuchten retention constitutive relationship was adopted for analysis, other constitutive 
relationships such as the Brook-Corey retention relationship would yield very similar results. 
Further, the analysis using the empirical relationships in Section 6.4 developed by Campbell 
corroborates the calculations using the non-dimensionalized van Genuchten retention relation. 
Both sets reflect the fact that for coarse pore sizes (greater than 3 mm), it is expected that water 
at the given moisture potential at the repository horizon would not be retained in the coarse 
interstices. The analysis shows that this would be true for even coarser pore spaces than those 
considered in this report. 
The analyses were carried out with different uniform particle-size diameters ranging from 
0.3 mm to 20 mm. These correspond to a broad range of particle sizes (Winterkorn and Fang 
1975, p. 287, Figure 7.42) for sand and gravel. The results show that for larger particle sizes, 
water would not be retained in the coarse pore space while the intragranular pore space is near 
saturation. The dual-porosity invert material would restrict flow from the surrounding formation 
to the invert due to the combined capillarity of both the intragranular and the intergranular 
porosities of the crushed tuff. The flux rates through the intragranular porosity of the crushed 
tuff would be bounded by the matrix saturated hydraulic conductivity (10-8 cm/sec) and would be 
independent of crushed tuff particle size diameters. 
The results of the analysis presented in this report have used the active fracture model for the 
welded tuff repository host horizon that are representative of the rock media surrounding the 
crushed tuff invert. The invert was analyzed with intragranular matrix retention and intrinsic 
matrix permeability equal to the properties of the host horizon, and with intergranular properties 
August 2003 60 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
selected on the basis of the data by Brooks and Corey (1964) that is named the 
nondimensionalized van Genuchten properties in this report. The Campbell properties were used 
as an alternate model, and the NUFT results (Section 6.8) showed very similar behavior to the 
nondimensionalized van Genuchten properties in which the crushed tuff intragranular matrix 
showed high saturation, and the intergranular porosity was free of water. 
The results of this analysis are also consistent with the results presented in Figure 6.3.4 of Drift- 
Scale Radionuclide Transport (BSC 2003f) that shows similar behavior in the matrix host rock 
for similar hydrologic properties sets. In one analysis reported in Figure 6.3.4 (BSC 2003f), the 
far field flux rate was selected as 10 mm per year which is lower than the far field flux of 35 mm 
per year used in the NUFT analysis. The percolation rate of 10 mm per year according to Figure 
6.3.4 (BSC 2003f) would result in a lower saturation than the higher saturation presented in this 
analysis. Figure 6.3.4 (BSC 2003f) shows that the saturation is approximately 0.87. Substituting 
into the van Genuchten relation for unsaturated hydraulic conductivity (Jury et al. 1991, p. 109): 
where 
and Cherry 1979, p. 27): 
The calculated value for the saturated hydraulic conductivity is approximately 10-8 cm per 
second (2 mm per year). Substituting in the following properties from Table 4-4: 
. 
... 
�� = 0 . 0157 
= 0 .112 
m = 0.236 
Now substituting in the relationship of hydraulic conductivity to intrinsic permeability that 
converts a saturated intrinsic permeability (k) to a saturated hydraulic conductivity (Ks) (Freeze 
. 1 . 
... 
... 
= 
��s 
K = s 
2 k S 
. 1 . 
. . 
... 
61 of 80 
�� 
r 
s 
ANL-EBS-MD-000063 REV 00 
1 
M 2 . 1 (1 K(S ) = K s . S 
S 
. �� g 
�� 
. S M ) 
1 
1 (1 . . S M 
(Eq. 6-40) 
. k (Eq. 6-41) 
... 
)M K (S ) = 
. �� g 
�� 
�� . �� 
��
r 
. r 
... 
2 
(Eq. 6-39) 
. . 
... 
2 
(Eq. 6-42a) 
.... 
August 2003 Advection versus Diffusion in the Invert 
Solving for the volumetric moisture content: 
1 �� = S 
) r ��+ 
( r ��. 
M 
. 
1 
M 2 ) 
. 
.  
099 �� = 0 . 
Solving for the hydraulic conductivity: 
K ( S ) = K s . S 
. 1 . 
..  
..  
The calculated value for matrix percolation flux is approximately 0.1 mm per year that is in 
approximate agreement with the matrix percolation flux of 0.15 mm per year reported in 
Figure 6.3-4 of Drift-Scale Radionuclide Transport (BSC 2003f). It should be noted that small 
variations in matrix saturation can result in large variations in flux since the unsaturated matrix 
hydraulic conductivity is a strong nonlinear function of volumetric moisture content or 
saturation. 
The results reported in Figure 6.3-4 of Drift-Scale Radionuclide Transport (BSC 2003f) indicate 
that the open drift behaves like an impermeable inclusion with flow being diverted around the 
inclusion. The analysis results (Section 6.7 and Attachment IX) show a similar behavior in flow 
being diverted around the open drift. 
The crushed tuff matrix flow results can be interpreted with the aid of a closed form analytical 
solution for flow around an inclusion (CRWMS M&O 2001b, Attachment XII). The report 
developed a closed form solution for flow. Figure 6-15 shows the results of a comparison of the 
fluxes in the invert to the closed form solution for far field matrix flow around an inclusion. This 
far field flow is approximately 1.3 mm per year consistent with NUFT analysis. The figure 
shows that the Darcy fluxes in the invert at a percolation rate of 35 mm per year are near 
saturation, and are responding to the impermeable inclusion as represented by the emplacement 
drift. 
This discussion shows that with the upper percolation fluxes of 35 mm per year to 70 mm per 
year used in this analysis that flow within the invert would be of the order of several mm per 
year. If the lower percolation flux of 10 mm per year resulting in a matrix flux of 0.15 mm per 
year is considered as in Drift-Scale Radionuclide Transport (BSC 2003f), the fluxes in the invert 
would be of the order of tenths of a mm per year, and the system would be more diffusion 
dominated. This shows that the principal uncertainties affecting whether advection versus 
diffusion in the invert occurs or not includes the percolation flux, the intrinsic permeability of the 
matrix at the host horizon, and the retention characteristics of the crushed tuff matrix. 
The results of the NUFT analysis presented in this report show that for a percolation flux of 
35 mm per year that flow dominantly occurs in the fractures in the surrounding rock with the 
coarse intergranular porosity acting as a capillary barrier to flow at the drift invert boundary. 
. ( s �� 
. 
... 
. . 
... 
��s 
. 1 (1. S 
62 of 80 ANL-EBS-MD-000063 REV 00 
.. 
2 
r ��. ) 
= 0 . 055 . (Eq. 6-42b) mm 
yr 
August 2003 Advection versus Diffusion in the Invert 
Fracture flow is affected by gravity to a more significant degree than for matrix flow. For 
fracture flow within the rock, a dry shadow forms below the invert with a lower degree of 
fracture flow. To the side of the drift fracture flow is dominated by gravity, and there is a 
tendency for �drip lobes� to occur as was the case in Drift-Scale Radionuclide Transport (BSC 
2003f). Further, the flow through the fractures that comprises a large percentage of the 
percolation flux would be diverted around the crushed tuff invert as the coarse pore space forms 
a capillary barrier to flow within the intergranular pore space of the invert. The results also show 
that matrix flow enters the micopores or intragranular porosity of the crushed tuff from the sides. 
This flow exits the drifts along the base of the drift into the rock matrix. 
However, it should be noted that a significant source of uncertainty in these calculations is the 
interface zone between the crushed tuff and the surrounding host formation. The actual 
boundary that would exist between the two media might provide resistance to flow, and reduce 
the potential advection occurring through the invert such that the invert is actually a part of the 
rigid inclusion as is modeled in Drift-Scale Radionuclide Transport (BSC 2003f). This would 
result in a diffusion-dominated process in the invert. At present, there exist a number of 
unquantified uncertainties affecting the interface zone of the invert. Among these include the 
degree of consolidation of the crushed tuff, stresses acting across the grain to rock contacts from 
a transfer of loads from the waste package support system; interface retention of zones that are 
subject to local crushing, and stress corrosion cracking of the crushed tuff particles under stress. 
Near saturation, the crushed tuff to rock interface might represent a preferential pathway for flow 
while at elevated temperature during drying, it might represent a capillary barrier to flow. This 
analysis has conservatively assumed the interface offers no resistance to flow in the absence of 
definitive data to the contrary. 
Since advection through the invert is relatively independent of percolation rate, and crushed tuff 
particle size for larger crushed tuff particle sizes, the moisture potential in the invert is relatively 
constant. The analyses use the hydrologic properties of welded tuff rock for the intergranular 
porosity. Calibrated Properties Model (BSC 2003c, Section 6.4.2) presents a discussion on the 
quantification of parameter uncertainty that applies to the intragranular porosity and permeability 
for crushed tuff in this report. The uncertainties in advection then reflect uncertainties in the 
matrix retention properties of the crushed tuff that would account for variations in pore size, and 
the distribution of pore sizes within the intragranular porosity. 
The depth of the invert represents the path length as discussed in Attachment XV. Note that as 
discussed in Attachment XV, the path length of 0.5 was selected to account for possible 
variations in the location of contaminants entering the invert from the waste package, and to 
account for settlement with time. 
Hydrodynamic dispersion would reflect uncertainties in advection (Section 6.8) and uncertainty 
in the dispersivity (Equation 6-13). The hydrodynamic dispersion affects the degree of spreading 
of the concentration breakthrough curve. The analysis showed that advection dispersion was 
dominant over diffusion in the intragranular pore space in the invert since the concentration 
breakthrough times occurred more rapidly (100 years to 1,000 years) while the breakthrough for 
diffusion was from 1,000 to 100,000 years (see Section 6.9.5). 
August 2003 63 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
1.4 
1.2 
1 
0.8 
0.6 
0.4 
0.2 
0 
0 1 
Matrix Percolation Flux 1.3 mm per yr 
Matrix Percolation 0.15 mm per yr 
NUFT Results 
Figure 6-15. Comparison of a Closed Form Solution for Matrix Flow at Saturation with the Results of the 
NUFT Analysis 
Since diffusion depends on the moisture content (Equation 6-16) in both the intragranular and 
intergranular porosity which has been found to be relatively insensitive to the effects of 
percolation rate, and crushed tuff pore size, diffusion is relatively constant with diffusion 
coefficients of the order of 10-7 cm2/sec (see Attachment XII, Figure XII-1). As discussed in 
EBS Radionuclide Transport Abstraction (BSC 2003b, Section 6.3.4.1.1), the uncertainty 
approximates a normal distribution for the residuals in the log-log space. The log- normal 
distribution is truncated to plus or minus three standard deviations. The log-normal distribution 
has a mean value of 0.033, and a standard deviation of 0.218. A comparison of the results in 
Figure XII-1 over a broad range of percolation rates suggests that the principal source for 
uncertainty for diffusion is with the statistical curve fit, as presented in EBS Radionuclide 
Transport Abstraction (BSC 2003b, Section 6.3.4.1.1). 
The analysis in this report presents a sensitivity analysis of the effects of retardation 
(Section 6.9.6). The retardation calculations considered a broad range of partition coefficients 
(Table 6-12). The results show that breakthrough is strongly dependent on retardation, and that 
concentration breakthrough could be retarded for very long periods of time. 
ANL-EBS-MD-000063 REV 00 64 of 80 August 2003 
Darcy Flux (mm/yr) 
Percolation Rate 35 mm per yr 
4 3 2 5 
Depth in Floor (m) 
9 8 7 6
Advection versus Diffusion in the Invert 
The analysis presented in this model was performed for long periods of time and analyzes 
radionuclide migration after waste package failure due to corrosion. The results of the analysis 
presented in this report are weakly coupled to temperature after extended periods of time. 
Juvenile failures have not been considered in the analysis. Such failures might occur at earlier 
periods of time when temperatures are more elevated, and potential dryout zone exists. Under 
these circumstances the potential flux through the invert are strongly coupled to elevated 
temperature. For temperatures above boiling, it would be expected that a dryout zone in the 
invert would form, and moisture potentials would be lower. This would mean that advection 
within the invert would be smaller. Also, diffusion would be smaller because of the lower 
moisture content. 
In summary, the uncertainties in terms of the their rank importance on breakthrough time 
through the invert are: 
� Uncertainties in retardation for individual radionuclides is the most significant source of 
uncertainty in these calculations. High partition coefficients (Kds) have the potential of 
delaying the release of radionuclides for very long periods of time. 
� Uncertainties in the matrix retention properties and saturated hydraulic conductivity is 
also a significant source of uncertainty. Variations in the unsaturated hydraulic 
conductivity as it is affected by retention in the fine intragranular pore space, and the 
saturated hydraulic conductivity has a significant effect on breakthrough times. 
� Since the analysis shows that advection/dispersion dominates over diffusion in the invert 
in the intragranular porosity, a source of uncertainty of intermediate importance is the 
uncertainty in hydrodynamic dispersion. 
� Since advection/dispersion is dominant over diffusion, uncertainties in diffusion, and the 
temperature dependence of diffusion is of low importance. 
� Advection/dispersion occurs through the fine intergranular pore space and, therefore, the 
effects of particle size are of low importance provided that the particle sizes are coarse 
enough the intergranular porosity remains dry over the range of moisture potentials. 
The properties and their uncertainties are adequately analyzed in this report because the van 
Genuchten flow properties are well documented. The retention relationship for the coarse pore 
space as determined from analysis of the data by Brooks and Corey (1964) corroborated by the 
Campbell relationship. Further, the results from the NUFT analysis is corroborated by the UFA 
flow measurements presented in Attachment XI. The analysis shows that advection/dispersion 
within the invert is the dominant transport mechanism with diffusion being of secondary 
importance. The analysis results show that the diffusion coefficient is relatively constant. 
The finest pore space considered in the analysis was 0.3 mm. During the preclosure and 
postclosure periods, it is anticipated that fugitive dust from suspended particulates would settle at 
the top surface of the invert, and potentially migrate into the invert. It is unknown at the current 
time as to how much dust would settle, and migrate into the invert. While the formation of dust 
August 2003 65 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
would be expected to locally increase the retention of water within the coarse pore space, it is not 
expected to alter the retention characteristics of the intragranular porosity, and the predominance 
of flow within the fine pore space of the crushed tuff particles. 
6.12 DISCUSSION OF YMRP ACCEPTANCE CRITERIA 
The acceptance criteria from Yucca Mountain Review Plan, Final Report (NRC 2003) for the 
advection versus diffusion through the invert are summarized in Section 4.2. These criteria 
apply to system description and multiple barriers (Section 4.2.1); scenario analysis and event 
probability (Section 4.2.2); degradation of engineered barriers (Section 4.2.3); and quantity and 
chemistry of water contacting waste packages and waste forms (Section 4.2.4). Table 6-13 
presents which criteria are addressed by this report; and which sections of the report present the 
detailed discussions addressing these criteria. A summary of how this report addresses the 
criteria is as follows: 
6.12.1 System Description and Demonstration of Multiple Barriers 
� The report describes the geometry, the relevant hydrologic material properties, and the 
advection-dispersion-diffusion transport phenomena in detail adequate for barrier 
identification (Sections 6.1 through 6.4, Section 6.7, and Attachment XV). 
� The selection of engineered materials for the invert is made to make the performance of 
the barrier acceptable (Section 6.8). 
� The technical basis for the relevant hydrologic properties is adequate and corroborated. 
The NUFT analysis presented is corroborated by experiments on samples of crushed tuff 
(Section 6.8.3). 
6.12.2 Scenario Analysis and Event Probability 
� Section 6.1 describes the physical processes of advection, dispersion, and diffusion for 
radionuclide transport through the invert. The descriptions of the transport phenomena 
are adequate. 
6.12.3 Degradation of Engineered Barriers 
� Model integration is adequate. 
- Section 6.4 provides the van Genuchten properties and the intrinsic permeability 
for intergranular porosity for the invert that are used in the Multiscale 
Thermohydrologic Model. When these properties for the dual-permeability media 
are input to NUFT for the nondripping case, along with the properties for other 
components within the EBS, and other initial and boundary conditions required 
for the thermal hydrological model, the analysis will produce the flows within 
both intergranular and intragranular porosities. 
- Section 6.10 provides the information for evaluating breakthrough for the 
dripping case. In this case, flux rates can be calculated using the method outlined 
66 of 80 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
in Section 6.10 for the coarse intergranular porosity under the assumption that the 
intragranular porosity is saturated. Flows can be calculated for both components. 
� Data are sufficient for the models supported. The models for the retention properties are 
well established, and accepted for use in vadose zone hydrology. The Campbell 
relationships provide corroboration of the nondimensionalized van Genuchten 
relationship. The NUFT analysis predicts that the coarse intergranular porosity is free of 
water over the range of expected moisture potentials in the repository. 
� Data uncertainty is characterized and propagated through the model abstraction as 
described in Section 6.11. The report is a supporting analysis to EBS Radionuclide 
Transport Abstraction (BSC 2003b) and Multiscale Thermohydrologic Model (BSC 
2001a). 
� Analysis output is supported by objective comparisons. The measured retention on 
crushed tuff as reported in Attachment XI corroborates the NUFT analysis. 
6.12.4 Quantity and Chemistry of Water Contacting Waste Packages and Waste Forms 
� System description and model integration are adequate. Section 6.4 provides the van 
Genuchten properties for the invert used in the Multiscale Thermohydrologic Model. 
The quantity of water is calculated in that model. Section 6.10 presents a method for 
calculating water fluxes through the invert for the dripping case. 
� The data for the calculation of the quantity of water are sufficient. The report is a 
supporting analysis to EBS Radionuclide Transport Abstraction (BSC 2003b) and 
Multiscale Thermohydrologic Model (BSC 2001a), where the quantities of water are 
calculated in detail. 
� Data uncertainty is characterized and propagated through the model abstraction in 
Section 6.11. The report is a supporting analysis to EBS Radionuclide Transport 
Abstraction (BSC 2003b) and Multiscale Thermohydrologic Model (BSC 2001a). 
Note that since this report is a supporting report to other models, Table 6-13 directs the reader to 
those model reports where the model criteria are addressed in detail. Further, the reader is 
directed to Engineered Barrier System Features, Events, and Processes (CRWMS M&O 2001a), 
which addresses scenario analysis and event probability in more detail. 
August 2003 67 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Table 6-13. Report Sections that Address the YMRP Acceptance Criteria 
Major YMRP 
Categories Criteria No. 
Section Addressing 
Criteria Remarks Criteria 
System Description and Demonstration of Multiple Barriers (NRC 2003, Section 4.2.1.1.3; Canori and Leitner 2003, PRD-002/T-014, PRD-002/T- 1 
016): 
Specific requirements involve identification of multiple barriers (natural and engineered), describing the capabilities of these barriers to isolate waste, and 
providing technical bases for capabilities descriptions consistent with the post-closure performance objectives. To comply with these requirements, the following 
acceptance criteria are identified in Technical Work Plan for: Engineered Barrier System Department Modeling and Testing FY03 Activities (BSC 2003a). 
Identification of Barriers is Adequate. 6.1,6.2,6.3, 1.0, 6.4,6.8 The report is a supporting analysis to EBS Radionuclide AC1: 
Transport Abstraction (BSC 2003b) and Multiscale 
Thermohydrologic Model (BSC 2001a). 
6.1, 6.9 AC2: The breakthrough analysis as presented in Section 6.9 
describes the invert barrier's capability. 
Description of the Capability of Identified 
Barriers is Acceptable. 
6 AC3: The technical basis for the hydrological properties for a 
dual-porosity material in the invert comprised of crushed 
tuff is presented in the supporting analysis. 
Technical Basis for Barrier Capability is 
Adequately Presented. 
2 Scenario Analysis and Event Probability (NRC 2003, Section 4.2.1.2.1.3; Canori and Leitner 2003, PRD-002/T-015): 
68 of 80 
Specific requirements include providing technical bases for inclusion or exclusion of specific FEPs. In order to meet these requirements, the following 
acceptance criteria are identified in Technical Work Plan for: Engineered Barrier System Department Modeling and Testing FY03 Activities (BSC 2003a). 
6.1 T AC1: Identificat he ion of the Initial List of Features, Section 6.1 describes the processes of advection, 
dispersion, and diffusion for radionuclide transport through 
the invert. 
Events, and Processes is Adequate. 
NA AC2: The screening of FEPs for EBS is presented in Engineered Screening of the Initial List of FEPs is 
Appropriate. Barrier System Features, Events, and Processes (CRWMS 
M&O 2001a). 
NA AC3: The formation of FEPs for EBS is presented in Engineered Formation of Scenario Classes Using the 
Reduced Set of Events is Adequate. Barrier System Features, Events, and Processes (CRWMS 
M&O 2001a). 
AC4: The screening of scenario classes for EBS is presented in Screening of Scenario Classes is Appropriate. NA 
Engineered Barrier System Features, Events, and 
Processes (CRWMS M&O 2001a). 
August 2003 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Table 6-13. Report Sections that Address the YMRP Acceptance Criteria (Continued) 
Major YMRP 
Categories Criteria No. 
Section Addressing 
Criteria Remarks Criteria 
3 Degradation of Engineered Barriers (NRC 2003, Section 4.2.1.3.1.3; Canori and Leitner 2003, PRD-002/T-015): 
Specific requirements include describing deterioration or degradation of engineered barriers and modeling degradation processes using data for performance 
assessment, including total system performance assessment (TSPA). Consideration of uncertainties and variabilities in model parameters and alternative 
conceptual models are also required. To fulfill these requirements, the following acceptance criteria are identified in Technical Work Plan for: Engineered Barrier 
System Department Modeling and Testing FY03 Activities (BSC 2003a). 
1.0,6.4,6.10 Section 6.4 provides the van Genuchten properties for the AC1: System Description and Model Integration are 
invert used in the Multiscale Thermohydrologic Model. 
Section 6.10 provides the information for evaluating 
Adequate. 
breakthrough for the dripping case. 
6.2-6.5, 6.8 and 
Attachment XI 
Data are Sufficient for Model Justification. AC2: The Campbell relationship provide corroboration of the 
nondimensionalized van Genuchten relationship. The 
NUFT analysis predict that the coarse intergranular 
porosity is free of water over the range of expected 
moisture potentials in the repository. The measured 
retention on crushed tuff as reported in Attachment XI 
corroborates the NUFT analysis. 
69 of 80 
Data Uncertainty is Characterized and 
Propagated Through the Model Abstraction. 
6.11 AC3: The report is a supporting analysis to EBS Radionuclide 
Transport Abstraction (BSC 2003b). 
Model Uncertainty is Characterized and 
Propagated Through the Model Abstraction. 
6.11 AC4: The report is a supporting analysis to EBS Radionuclide 
Transport Abstraction (BSC 2003b). 
NA AC5: Model Output is S Abstraction upported by The report is a supporting analysis to EBS Radionuclide 
Transport Abstraction (BSC 2003b). Objective Comparisons. 
August 2003 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Table 6-13. Report Sections that Address the YMRP Acceptance Criteria (Continued) 
Major YMRP 
Categories Criteria No. 
Section Addressing 
Criteria Remarks Criteria 
Quantity and Chemistry of Water Contacting Waste Packages and Waste Forms (NRC 2003, Section 4.2.1.3.3.3; Canori and Leitner 2003, PRD- 4 
002/T-015): 
Specific requirements include quantifying the amount and chemistry of water contacting the waste package and the waste forms. To comply with these 
requirements, the following acceptance criteria are identified in Technical Work Plan for: Engineered Barrier System Department Modeling and Testing FY03 
Activities (BSC 2003a). 
AC1: System Description and Model Integration are 1.0,6.4,6.10 Section 6.4 provides the van Genuchten properties for the 
invert used in the Multiscale Thermohydrologic Model. Adequate. 
Section 6.10 provides the information for evaluating 
breakthrough for the dripping case. 
The report is a supporting analysis to EBS Radionuclide 6.4,6.5,6.10 Data are Sufficient for Model Justification. AC2: 
Transport Abstraction (BSC 2003b) and Multiscale 
Thermohydrologic Model (BSC 2001a). 
6.11 AC3: Section 6.11 describes data uncertainties, and their Data Uncertainty is Characterized and 
Propagated Through the Model Abstraction. propagation through the model abstraction. 
NA AC4: The report is a supporting analysis to EBS Radionuclide Model Uncertainty is Characterized and 
Propagated Through the Model Abstraction. Transport Abstraction (BSC 2003b) and Multiscale 
August 2003 70 of 80 
Thermohydrologic Model (BSC 2001a). 
The report is a supporting analysis to EBS Radionuclide NA AC5: Model Output is S Abstraction upported by 
Transport Abstraction (BSC 2003b) and Multiscale Objective Comparisons. 
Thermohydrologic Model (BSC 2001a). Advection versus Diffusion in the Invert 
7. CONCLUSIONS 
Technical Work Plan for: Engineered Barrier System Department Modeling and Testing FY03 
Work Activities (BSC 2003a) provides the basis for conducting an Advection versus Diffusion 
Analysis to support the development of several reports for EBS Flow and Transport Model 
Development, Analyses and Testing. These include Multiscale Thermohydrologic Model report 
and the EBS Radionuclide Abstraction Analysis report. 
The Advection versus Diffusion Analysis presented in this analysis report is a supporting analysis 
to the Multiscale Thermohydrologic Analysis for the selection of hydrological properties for a 
dual-porosity material in the invert. The dual-porosity media for the invert required the 
development of retention and unsaturated hydraulic conductivity properties for the invert 
(Section 6.3). The dual-porosity media is comprised of the intragranular porosity of the crushed 
tuff that has retention properties equivalent to the tuff matrix that is characteristic of the site, and 
an intergranular porosity in which retention properties were developed. 
Section 6.4 developed the retention and unsaturated flow properties for a dual-porosity media 
using the Non-Dimensionalized van Genuchten Retention Relation. The nondimensionalized 
van Genuchten Moisture Potential Retention relation was developed from the retention data for 
volcanic sand, fine sand, and glass beads. The basis for the nondimensional retention 
relationship for the first retention relation is a least squares curve fit for a series of retention 
measurements made on these materials that gives equal weighting to the several materials. The 
van Genuchten air-entry parameter �� was determined from a scaling relationship while the �n� 
parameter is held constant. The saturated hydraulic conductivity (Ks) is determined from the 
Kozeny-Karmen formula (Bear 1972, p. 166) that relates saturated hydraulic conductivity (Ks) to 
grain size or pore diameter (dm) and porosity (�). The properties developed for the Multiscale 
Thermohydrologic Analysis include (Table 6-2): 
� Intergranular permeability of the crushed tuff 
� Intergranular saturated moisture content 
� Intergranular porosity 
� van Genuchten properties 
� Residual moisture content 
� Maximum saturation 
� Residual saturation. 
The Campbell retention relations provide an alternate method for the assessment of the 
hydrological properties for the intergranular porosity. This retention relation was developed 
from empirical relations presented by Campbell (1985, p. 43) based on the assumption that 
particle sizes in soils follow a log-normal distribution (Assumption 5.9). The analysis using the 
Campbell relation corroborates the nondimensionalized van Genuchten Moisture Potential 
relation. Tables 6-5 and 6-6 provide a summary of the Campbell hydrological properties. 
In performing analysis of potential flow through the invert, the Multiscale Thermohydrologic 
Model would provide similar results using properties from the nondimensionalized van 
Genuchten relation based upon the data by Brooks and Corey (1964) as presented in Section 6.4 
71 of 80 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
and the Campbell relation (Campbell 1985, p. 43) as presented in Section 6.5. The 
nondimensionalized van Genuchten relation based upon data by Brooks and Corey (1964) is 
selected for analysis for the reason that the hydrologic properties can be derived from 
fundamental data. Output DTNs (see Table 7-1) are included for the hydrological properties. 
This analysis report also supports EBS Radionuclide Transport Abstraction (BSC 2003b) in 
determining whether advection or diffusion in each media is the dominant transport mechanism. 
NUFT analyses were performed to compare the contaminant breakthrough in the invert for 
advection versus dispersion-diffusion to assess the significance of advection to EBS radionuclide 
transport. NUFT simulations for flow through the invert used dual-porosity hydrological 
properties (Section 6.4) to estimate saturations and advection in the invert (Sections 6.6 through 
6.8). A NUFT analysis at ambient temperature using a refined mesh in the invert (Sections 6.4 
and 6.5) determined advection within the invert under the case of no seepage in the drift. 
The NUFT results utilized a refined mesh and showed that the saturation of the grain matrix 
equals the matrix saturation. This response is reasonable because of the high capillarity that 
draws water from the fractures of the host rock into grains (Section 6.7). The NUFT results 
show that the vertical pore-water velocity in the matrix component is low and is controlled by the 
saturated hydraulic conductivity (Ks), the pressure gradient in the matrix, and the pressure 
gradient across the intergranular and intragranular components. 
NUFT analyses were also performed using the Campbell hydrological properties. The Campbell 
retention relation predicted that for a grain size of 0.317 mm in diameter, the intergranular 
component behaves similarly to the intragranular matrix component, and the moisture potential 
equilibrates across the two components. As the grain size increases to 3-mm diameter or larger, 
the intergranular component breaks away and behaves as a capillary barrier, as shown by a sharp 
difference in pressure potential across the two components. This pressure gradient steepens as 
the grain size increases from 3-mm to 20-mm diameter inducing a slightly higher flux in the 
matrix component. This matrix flux tends to stabilize between 10- and 20-mm grain size 
models. The van Genuchten retention relation predicts that for all grain sizes of the crushed tuff 
0.317-mm diameter or larger, the intergranular component acts as a capillary barrier whereas the 
intragranular component behaves as a matrix medium, as expected in a DKM model. 
The one-dimensional contaminant transport equation (Section 6.1, Equation 6-4) was used to 
evaluate contaminant transport over the range of pore-water velocities and saturations from the 
NUFT calculation (Section 6.9.5). The closed form analytical solution for the one-dimensional 
advection-dispersion equation (Equation 6-4) was used to perform a sensitivity analysis of 
breakthrough times for a range of conditions in the invert. The sensitivity analysis is subject to 
the assumptions that the direction of the advective flux is in the vertical direction 
(Assumption 5.1); the effects of lateral dispersion are neglected (Assumption 5.2); and the invert 
is homogeneous (Assumption 5.3). The radionuclides are assumed to be released at the 
centerline of the drift (Assumption 5.4), and that the effects of radioactive decay are neglected 
(Assumption 5.5). Solute vapor phase is assumed to be negligible (Assumption 5.7). The range 
of pore water velocities for the intergranular pore space was from 7.6�10-11 m/s to 1.8�10-10 m/s 
for the low- and high-advection cases, respectively (Table 6-9). For comparison, the 
breakthrough curve for diffusion is presented. The breakthrough analysis for the case of variable 
pore-water velocity would fall between these extremes for high and low advection. The results 
August 2003 72 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
of the analysis show that breakthrough times would occur within a time frame of 400 to 1,000 
years, while a diffusion-dominated process such as would occur in the intergranular pore space 
would occur within a time frame of 1,000 to 10,000 years (Figure 6-9). 
Additional sensitivity studies using the one-dimensional advection-diffusion-dispersion equation 
were used to estimate the effects of retardation. Sheppard and Thibault (1990, Tables A-1 
through A-4) present data on the mean and standard deviation of the natural logarithm of the 
partition coefficient for individual elements for sands, loams, clays, and organic soils. To take 
into account a broad range of values, the partition coefficients the sensitivity study for 
retardation used a range of partition coefficients from 1 to 1,000 ml/gm (Table 6-12). The results 
of the calculations show that for the case of low sorption (Kd = 1.0 ml/gm), the retardation factor 
is low, and the breakthrough time is delayed to 1,000 to 5,000 years. For the case of 
intermediate and high sorption (10 - 100 ml/gm), the delay in breakthrough time is more 
substantial. In the case of the intermediate sorption, breakthrough would be delayed from 1,000 
to 10,000 years. In the case of high sorption (1,000 ml/gm), breakthrough is delayed by at least 
one million years. 
The NUFT analysis presented in Sections 6.7 and 6.8 apply to the nondripping case in which 
flow is diverted around the open drift. Section 6.10 presents a method of analysis discussed in 
detail in Attachment XIV for the dripping case for estimating the flux and diffusion coefficient 
through invert. Table 7-1 presents the recommendations for the properties for the dripping and 
nondripping cases. Note that the properties for the nondripping case are implemented in the 
Multiscale Thermohydrologic Model that calculates flux rates through the invert. For the 
dripping case for flow through the matrix, the results of Multiscale Thermohydrologic Model can 
be used. For that portion of the flow carried by the large pore space, the moisture potential can 
be estimated from the flux from an interpolation table presented in Attachment XIV based upon 
the van Genuchten unsaturated hydraulic conductivity relation. This moisture potential can used 
to find the moisture content of the large pore space that then can be used to calculate the 
diffusive release. Noting that the results are not sensitive to the grain-size distribution, it is 
recommended that a particle size of 3 mm be adopted for TSPA analysis since the results of the 
analysis. 
As noted in Section 6.11, uncertainties exist in the constitutive relations applied, and the 
calculated parameters. While the van Genuchten retention constitutive relationship was adopted 
for analysis, other constitutive relationships such as the Brooks-Corey relationship would yield 
similar results. Both the van Genuchten retention constitutive relationship and the Campbell 
retention relationship show that for pore sizes greater than 3 mm, it is expected that water at the 
given moisture potential at the repository horizon would not be retained in the coarse pore space. 
The results of this analysis are also consistent with the results presented in Figure 6.3.4 of Drift- 
Scale Radionuclide Transport (BSC 2003f) that shows similar behavior in the matrix host rock 
for similar hydrologic properties sets. The calculated value for matrix percolation flux is 
approximately 0.1 mm per year that is in approximate agreement with the matrix percolation flux 
of 0.15 mm per year reported in Figure 6.3-4 of Drift-Scale Radionuclide Transport (BSC 
2003f). It should be noted that small variations in matrix saturation can result in large variations 
in flux since the unsaturated matrix hydraulic conductivity is a strong nonlinear function of 
volumetric moisture content or saturation. 
August 2003 73 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
The crushed tuff matrix flow results can be interpreted with the aid of a closed form analytical 
solution for flow around an inclusion (CRWMS M&O 2001b, Attachment XII). 
The report developed a closed form solution for flow. Figure 6-15 shows the results of a 
comparison of the fluxes in the invert to the closed form solution for far field matrix flow around 
an inclusion. This far field flow is approximately 1.3 mm per year consistent with NUFT 
analysis. The figure shows that the Darcy fluxes in the invert at a percolation rate of 35 mm per 
year are near saturation, and are responding to the impermeable inclusion as represented by the 
emplacement drift. The closed form solution presented in Attachment XVI represents an 
alternate method for evaluating the advection in the invert. 
A significant source of uncertainty is the interface zone flow between the crushed tuff and the 
surrounding host rock. As discussed in Section 6.11, the actual boundary may provide more 
resistance to flow than what has been analyzed. This analysis has conservatively assumed the 
interface offers no resistance to flow in the absence of definitive data to the contrary. 
The finest pore space considered in the analysis is 0.3 mm. During the preclosure and 
postclosure period, it is anticipated that fugitive dust from suspended particulates would settle at 
the top surface of the invert, and potentially migrate into the invert. It would be expected that 
such fugitive dust would locally increase the retention of water within the coarse pore space. 
However, fugitive dust would not be expected to alter the retention characteristics of the 
intragranular porosity, and the predominance of flow within the fine pore space of the crushed 
tuff particles. 
This analysis has three output DTNs as summarized in Table 7-2. The first two DTNs support 
the Multiscale Thermohydrologic Model in providing van Genuchten curve fit parameters, and 
the intrinsic permeability. The last DTN supports TSPA for the dripping case within the invert. 
Table 7-1. Recommended Values for TSPA Based Upon a Grain Size of 3mm 
74 of 80 
Parameter 
Porosity of the Rock Matrix in an Individual Grain for Crushed Tuff 
(�matrix) for the Dripping and Nondripping Cases 
Porosity of Large Pore Spaces Between Grains (�intergraqin) for the 
Dripping and Nondripping Cases 
Residual Volumetric Moisture Content (�r) of Large Pore Spaces 
for the Dripping Case 
Saturated Volumetric Moisture Content (�s) of Large Pore Spaces 
for the Dripping Case 
van Genuchten Air-Entry Parameter of Large Pore Spaces for the 
Dripping and Nondripping Cases 
van Genuchten n Parameter of Large Pore Spaces for the 
Dripping and Nondripping Cases 
Saturated Intrinsic Permeability of the Large Pore Spaces for the 
Dripping and Nondripping Cases 
ANL-EBS-MD-000063 REV 00 
Recommended Value for TSPA Units 
0.131 (-) 
0.45 (-) 
0.05 (-) 
0.45 (-) 
624 (bars)-1 
8.013 (-) 
1.51E-08 (m2) 
August 2003 Advection versus Diffusion in the Invert 
Table 7-2. Output DTNs 
Sources of 
Uncertainty Output Name DTN 
MO0307SPAVGHYD.000 See Section 6.11 See Attachment IV 
MO0307SPAVGSUM.000 See Section 6.11 Table 6-2 
MO0307SPAFTPDC.000 
Output Description 
Unsaturated Hydraulic 
Conductivity Data 
van Genuchten Retention and 
Hydrologic 
Parameters 
Genuchten Parameters 
van Genuchten Summary van 
Summary 
Parameters 
TSPA Flow and Lookup Table, and 
Transport 
Abstraction 
Technical Approach for 
Assessing Advection 
and Diffusion for the 
Dripping Case 
Characteristic 
Values 
See Section 6.11 Attachment XIV 
August 2003 
8. INPUTS AND REFERENCES 
75 of 80 
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NUREG-1804, Rev. 2. Washington, D.C.: U.S. Nuclear Regulatory Commission, Office of 
Nuclear Material Safety and Safeguards. TIC: 254568. 
August 2003 77 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Saxena, N.S.; Chohan, M.A.; and Gustafsson, S.E. 1986. �Effective Thermal Conductivity of 
Loose Granular Materials.� Journal of Physics D: Applied Physics, 19, (9), 1625-1630. 
London, England: The Institute of Physics. TIC: 249120. 
Sheppard, M.I. and Thibault, D.H. 1990. �Default Soil Solid/Liquid Partition Coefficients, Kds, 
for Four Major Soil Types: A Compendium.� Health Physics, 59, (4), 471-482. New York, 
New York: Pergamon Press. TIC: 249329. 
van Genuchten, M.T. 1980. �A Closed-Form Equation for Predicting the Hydraulic 
Conductivity of Unsaturated Soils.� Soil Science Society of America Journal, 44, (5), 892-898. 
Madison, Wisconsin: Soil Science Society of America. TIC: 217327. 
Weast, R.C. and Astle, M.J., eds. 1981. CRC Handbook of Chemistry and Physics. 62nd Edition. 
Boca Raton, Florida: CRC Press. TIC: 240722. 
Willhite, G.P.; Kunii, D.; and Smith, J.M. 1962. �Heat Transfer in Beds of Fine Particles (Heat 
Transfer Perpendicular to Flow).� American Institute of Chemical Engineers Journal, 8, (3), 340345. 
New York, New York: American Institute of Chemical Engineers. TIC: 249121. 
Winterkorn, H.F. and Fang H-Y., eds. 1975. Foundation Engineering Handbook. Pages 37, 72, 
78, 79, 117, 156, 256, 257, 266, and 267. New York, New York: Van Nostrand Reinhold. 
TIC: 241820. 
YMP (Yucca Mountain Site Characterization Project) 2001. Q-List. YMP/90-55Q, Rev. 7. 
Las Vegas, Nevada: Yucca Mountain Site Characterization Office. ACC: 
MOL.20010409.0366. 
8.2 CODES, STANDARDS, REGULATIONS, AND PROCEDURES 
AP-SI.1Q, Rev. 5, ICN 1. Software Management. Washington, D.C.: U.S. Department of 
Energy, Office of Civilian Radioactive Waste Management. ACC: DOC.20030708.0001. 
AP-SIII.9Q, Rev. 1, ICN 0. Scientific Analyses. Washington, D.C.: U.S. Department of Energy, 
Office of Civilian Radioactive Waste Management. ACC: DOC.20030708.0002. 
August 2003 
8.3 SOURCE DATA, LISTED BY DATA TRACKING NUMBER 
GS000483351030.003. Thermal Properties Measured 12/01/99 to 12/02/99 Using the 
Thermolink Soil Multimeter and Thermal Properties Sensor on Selected Potential Candidate 
Backfill Materials Used in the Engineered Barrier System. Submittal date: 11/09/2000. 
GS980808312242.015. Water Retention and Unsaturated Hydraulic Conductivity Measurements 
for Various Size Fractions of Crushed, Sieved, Welded Tuff Samples Measured Using a 
Centrifuge. Submittal date: 08/21/1998. 
LB0207REVUZPRP.002. Matrix Properties for UZ Model Layers Developed from Field and 
Laboratory Data. Submittal date: 07/15/2002. 
78 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
LB0302PTNTSW9I.001. PTN/TSW Interface Percolation Flux Maps for 9 Infiltration 
Scenarios. Submittal date: 02/28/2003. 
LB990861233129.001. Drift Scale Calibrated 1-D Property Set, FY99. Submittal date: 
08/06/1999. 
LB990861233129.002. Drift Scale Calibrated 1-D Property Set, FY99. Submittal date: 
08/06/1999. 
8.4 OUTPUT DATA, LISTED BY DATA TRACKING NUMBER 
MO0307SPAFTPDC.000. Total System Performance Assessment (TSPA) Flow and Transport 
Parameters for the Dripping Case. Submittal date: 07/25/2003. 
MO0307SPAVGHYD.000. van Genuchten Hydrologic Parameters. Submittal 
date: 07/25/2003. 
MO0307SPAVGSUM.000. van Genuchten Hydrologic Parameters. Submittal 
date: 07/26/2003. 
8.5 SOFTWARE CODES 
Software Code: NUFT. V3.0s. Sun, Sun O.S. 5.8. 10088-3.0s-01. 
Software Routine: XTOOL V10.1. V10.1. Sun Ultra10. 10208-10.1-00. 
Attachment I 
Attachment II 
Attachment III 
Attachment V 
Attachment VI 
Attachment VII 
Attachment IX 
ANL-EBS-MD-000063 REV 00 
List of NUFT Input File Names 
List of NUFT Output File Names 
EBS Radionuclide Transport Analysis 
Attachment IV Analysis of Nondimensionalized van Genuchten Moisture Potential 
Retention Relation 
Analysis of the Campbell Retention Relation 
Derivation of Invert �Packed Bed� Properties from the Brooks and Corey 
Data 
Verification of the Mathcad Calculations Using Microsoft Excel 
Attachment VIII Hydrologic Properties of Engineered Barrier System Components Used for 
NUFT Runs 
Comparison of Tuff Matrix Hydrologic Properties 
9. LIST OF ATTACHMENTS 
79 of 80 August 2003 Advection versus Diffusion in the Invert 
Attachment X Liquid Flux Patterns in Matrix Component of Invert and Rock Matrix and in 
the Intergranular Component of Invert and Rock Fractures for the Campbell 
and van Genuchten Grain Size Retention Relations 
Hydrologic and Thermal Properties of the Invert Attachment XI 
Calculation of the Diffusion Coefficient for the Dripping Case Attachment XII 
Attachment XIII Derivation of the Formulas for the Calculation of the Bulk Volumetric 
Moisture Content 
Attachment XIV TSPA Calculation of the Diffusion Coefficient for the Dripping Case 
Calculation of the Thickness of the Invert for Analysis Attachment XV 
Attachment XVI Moisture Retention for Crushed Tuff 
Attachment XVII Estimate of the Boundary Between Advection and Diffusion 
Attachment XVIII Alternate Pore Water Velocity Calculation 
Attachment XIX CD-ROMs With Input and Output Files 
August 2003 80 of 80 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT I 
LIST OF NUFT INPUT FILE NAMES 
I-1 of I-4 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 I-2 of I-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment I - List of NUFT Input Files Names 
These files are found in Directory �Input� of the CD-ROM. 
File Name (size in kilobytes), date) 
genmsh4rm 13 kb, Nov. 8, 2001 
genmsh4rm2 (13kb, Nov. 8, 2001) 
genmsh4rm35 (13 kb, Nov. 8, 2001) 
genmsh5rmni (13 kb, Nov. 8, 2001) 
genmsh5rm2 (13kb, Nov. 8, 2001) 
genmsh5rmni35 (14kb, Nov. 8, 2001) 
genmsh5rmnt (14kb, Nov. 8, 2001) 
genmsh5rmnt2 (14 kb, Nov. 8, 2001) 
dkm-afc-NBS-WDR4 (64 kb, Nov. 8, 2001) 
dkm-afc-EBS_Rev10-WDR4 (9 kb, Nov. 8, 2001) 
dkm-afc-NBS-WDR (65 kb, Nov. 8, 2001) 
dkm-afc-EBS_Rev10-WDR (13 kb, Nov. 8,2001) 
dkm-afc-EBS_Rev10-WDRinv135 (13 kb, Nov. 8, 
2001) 
dkm-afc-EBS_Rev10-WDRinvrs (12 kb, Sept. 20, 
2001) 
ir2D35rm (2kb, Nov. 8, 2001) 
ir2D35rm2 (2kb, Nov. 8, 2001) 
ir2D70rm (2kb, Nov. 8, 2001) 
ir2D35rm35 (2 kb, Nov. 8, 2001) 
pr2D35rmntinvr (4 kb, Sept. 19, 2001 - using 
genmsh5rmnt and dkm-afc-EBS_Rev10-WDRinv 
pr2D35rmntinv (4 kb, Nov. 8, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10-WDRinv1 the invert. 
pr2D35rmntinvr3 (4 kb, Oct. 26, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10- 
WDRinvr3 
pr2D35rmntinvr2 (4 kb, Oct. 26, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10- 
WDRinvr2 
area. 
production run. 
in production runs. 
File Description 
Grid spacing and material designation for initialization run 
(no EBS components). 
Same as genmsh4rm except with refined mesh in the invert 
Same as genmsh4rm except using TSw35 properties 
around the drift. 
Same as genmsh4rm but EBS components included for 
Same as genmsh5rmni except with refined mesh in the 
invert area. 
Same as genmsh5rmni except using tsw35 properties 
around the drift and with rounded outer invert boundary. 
Same as genmsh5rmni except with rounded outer invert 
boundary. 
Same as genmsh5rmnt except with invert thickness 
increased from 0.5 to 0.6 meters. 
Natural rock properties for initialization runs. 
EBS component properties specified but not used in 
initialization runs. 
Natural Rock properties for production runs. 
ECM invert properties and other EBS properties specified 
TSw35 matrix properties for m-invert and 3mm-grain-size 
(van Genuchten) properties for f-invert (sensitivity analysis). 
DKM properties of invert for the .3 mm van Genuchten 
retention relation with contact area = 90 m2/m3 . 
Initialization run using 35 mm infiltration rate at ground 
surface. 
Initialization run for 35 mm infiltration @ ground surface 
with refined mesh in invert. 
Initialization run for 70 mm infiltration @ ground surface. 
Invert properties based on TSw35 rather than TSw36 
(TSw35 surrounding drift). 
Production run for van Genuchten 0.317 mm grain diameter 
for the invert. 
Production run for van Genuchten 3 mm grain diameter for 
Production run for van Genuchten 10 mm grain diameter 
for the invert. 
Production run for van Genuchten 20 mm grain diameter 
for the invert. 
ANL-EBS-MD-000063 REV 00 August 2003 I-3 of I-4 Advection versus Diffusion in the Invert 
pr2D70rmntinv (4 kb, Nov. 8, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10-WDRinv1 relation) except with 70 mm /yr Infiltration. 
File Description 
Production run for Campbell 0.317 mm grain diameter for 
the invert. 
Production run for Campbell 3 mm grain diameter for the 
invert. 
Production run for Campbell 10 mm grain diameter for the 
invert. 
Production run for Campbell 20 mm grain diameter for the 
invert. 
Sensitivity run on change of specific contact area to 90 
m2/m3 for the .3-mm van Genuchten retention relation. 
Production run with 35 mm/yr of infiltration rate, using ECM 
properties of invert. 
Same as pr2D35rmntinv (3mm van Genuchten retention 
Sensitivity run similar to pr2D35rmni but using infiltration of 
70mm/yr. 
Sensitivity run using refined mesh in center of invert. 
Sensitivity run using thicker invert (DKM Model). 
Sensitivity run using TSw35 properties for the m-invert and 
around the drift (3 mm retention relation). 
File Name (size in kilobytes), date) 
pr2D35rmntinvc (4 kb, Oct. 2, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10-WDRinc 
pr2D35rmntinc (4 kb, Nov. 8, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10-WDRinvc 
pr2D35rmntinc3 (4 kb, Nov. 8, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10- 
WDRinvc3 
pr2D35rmntinc2 (4 kb, Nov. 8, 2001) - using 
genmsh5rmnt and dkm-afc-EBS_Rev10- 
WDRinvc2 
pr2D35rmntinvrs (4 kb, Sept. 20, 2001) - using 
genmsh5rmnt dkm-afc-EBS_Rev10-WDRinvrs 
pr2D35rmni (4 kb, Aug. 1, 2001) - using 
genmsh5rmni and dkm-afc-EBS_Rev10-WDR 
pr2D70rmni (4 kb, Aug. 3, 2001) - using 
genmsh5rmni and dkm-afc-EBS_Rev10-WDR 
pr2D35rmnt2 (4 kb, Nov. 8, 2001) - using 
genmsh5rm2 dkm-afc-EBS_Rev10-WDR 
pr2D35rmntinv2 (4 kb, Nov. 8, 2001) - using 
genmsh5rmnt2 dkm-afc-EBS_Rev10-WDRinv1 
pr2D35rmntinv35 (4 kb, Nov. 8, 2001) - using 
genmsh5rmni35 dkm-afc-EBS_Rev10- 
WDRinv135 
ANL-EBS-MD-000063 REV 00 August 2003 I-4 of I-4 Advection versus Diffusion in the Invert 
ATTACHMENT II 
LIST OF NUFT OUTPUT FILE NAMES 
II-1 of II-4 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 II-2 of II-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment II - List of NUFT Output File Names 
�Output� of the CD-ROM. 
Note: All .ext files may be graphically displayed by XTOOL to show saturations, moisture 
potentials, and liquid fluxes for different times of simulation. These files are found in Directory 
Table II-1. List of NUFT Files 
pr2D35rmntinv. 
pr2D35rmntinvr3. 
File Description 
Output from ir2D35rm used as restart file for all production 
runs with 35 mm/yr infiltration rate. 
Output from ir2D35rm2 used as restart file for sensitivity 
run with a finer mesh. 
Output from ir2D70rm used as restart file for sensitivity run 
with 70 mm/yr infiltration rate. 
Output from ir2D35rm35 used as restart file for sensitivity 
run pr2D35rmntinv35. 
Output for intergranular component for run pr2D35rmntinvr. 
Output for intra-granular (matrix) component for run 
pr2D35rmntinvr. 
Output for intergranular component for run pr2D35rmntinv. 
Output for intra-granular (matrix) component for run 
Output for intergranular component for run 
pr2D35rmntinvr3 
Output for intra-granular (matrix) component for run 
Output for intergranular component for run 
pr2D35rmntinvr2. 
Output for intra-granular (matrix) component for run 
pr2D35rmntinvr2. 
Output for intergranular component for run pr2D35rmntinvc. 
Output for intra-granular (matrix) component for run 
pr2D35rmntinvc. 
Output for intergranular component for run pr2D35rmntinc. 
Output for intra-granular (matrix) component for run 
pr2D35rmntinc. 
Output for intergranular component for run pr2D35rmntinc3. 
Output for intra-granular (matrix) component for run 
pr2D35rmntinc3. 
Output for intergranular component for run pr2D35rmntinc2 
Output for intra-granular (matrix) component for run 
pr2D35rmntinc2. 
File Name (size in megabytes, date) 
ir2D35rm.res (4 mb, Nov. 9, 2001) 
ir2D35rm2.re1 (4 mb, Nov. 9, 2001) 
ir2D70rm.res (4 mb, Nov. 9, 2001) 
ir2D35rm35.re1 (4 mb, Nov. 9, 2001) 
pr2D35rmntinvr.f.ext (54 mb, Sept. 24, 2001) 
pr2D35rmntinvr.m.ext (54 mb, Sept. 24, 2001) 
pr2D35rmntinv.f.ext (54 mb, Nov. 9, 2001) 
pr2D35rmntinv.m.ext (54 mb, Nov. 9, 2001) 
pr2D35rmntinvr3.f.ext (54 mb, Nov. 1, 2001) 
pr2D35rmntinvr3.m.ext (54 mb, Nov. 1, 2001) 
pr2D35rmntinvr2.f.ext (54 mb, Nov. 1, 2001) 
pr2D35rmntinvr2.m.ext (54 mb, Nov. 1, 2001) 
pr2D35rmntinvc.f.ext (54 mb, Oct. 26, 2001) 
pr2D35rmntinvc.m.ext (54 mb, Oct. 26, 2001) 
pr2D35rmntinc.f.ext (54 mb, Nov. 9, 2001) 
pr2D35rmntinc.m.ext (54 mb, Nov. 9, 2001) 
pr2D35rmntinc3.f.ext (54 mb, Nov. 9, 2001) 
pr2D35rmntinc3.m.ext (54 mb, Nov. 9, 2001) 
pr2D35rmntinc2.f.ext (54 mb, Nov. 9, 2001) 
pr2D35rmntinc2.m.ext (54 mb, Nov. 9, 2001) 
August 2003 II-3 of II-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table II-2. List of Microsoft Excel Files in Directory �Excel� of CD-ROM 
potential retention relation. 
Attachment IV REV01Ahm.xls Analysis of Nondimensionalized van Genuchten moisture 
(62 kb 5/23/2003) 
Attachment V Rev01B.xls (133 Analysis of the Campbell retention relation. 
kb 8/14/2003) 
Verification of the Mathcad Calculation Using Microsoft 
Excel. 
Retention Property.xls (48 kb Retention Properties for Welded Tuff (Attachment IX). 
8/11/2003) 
Calculation of the Diffusion Coefficient For The Dripping 
Case (Attachment XII). 
Upper Bound Glacial Climate Statistical analysis of the Upper Bound Glacial Climate 
Percolation.xls (598 kb Percolation Flux. 
8/09/2003 
Thermal properties and statistical analysis of crushed tuff. 
Invert Velocity Profiles Rev � Develops a plot of the Peclet Number in the Invert. 
1.xls (31 kb 8/09/2003) 
Moisture Retention for Crushed Tuff 
Table II-3. List of Mathcad Files in Directory �Mathcad� of CD-ROM 
Description File Name 
Attachment VII.xls (105 kb 
5/28/2003) 
Total System Performance 
Abstraction.xls (21 kb 
7/16/2003) 
Thermolink Properties of 
Crushed Tuff.xls(38 kb 
8/09/2003) 
Attachment XVI.xls (31 kb 
8/16/03) 
6/28/2003) 
Breakthrough Analysis Rev 01.mcd EBS Radionuclide Analysis (Attachment III). 
(48 kb 8/14/2003) 
Invert Properties mcad7.mcd (145 Derivation of Invert "Packed Bed" Properties from Brooks 
kb 7/24/2003) 
Description 
and Corey Data (Attachment VI). 
Attachment XIV TSPA Calculation of the Diffusion 
coefficient for the Dripping Case. 
Attachment XIII Derivation of the Formulas for Calculation 
of the Bulk Volumetric Moisture Content. 
Attachment XVIII Alternate Pore Water Velocity 
Calculation. 
Calculation of the Peclet Number for Diffusion. 
File Name 
Total System Performance 
Assessment Abstraction Dripping 
Ca.mcd (59 kb 8/15/2003) 
Calculation of the Bulk Volumetric 
Moisture Content.mcd (44 kb 
Attachment XVIII Alternate Pore 
Water Velocity Calculation Rev 
01.mcd (47 kb 08/15/2003) 
Calculation of the Peclet Number 
for Diffusion Rev 01.mcd (10 kb 
08/16/2003) 
ANL-EBS-MD-000063 REV 00 August 2003 II-4 of II-4 Advection versus Diffusion in the Invert 
ATTACHMENT III 
EBS RADIONUCLIDE TRANSPORT ANALYSIS 
III-1 of III-10 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
D 
(Eq. III-1) 
(Eq. III-2) 
Attachment III - EBS Radionuclide Transport Analysis 
This attachment presents the results of the EBS Radionuclide Transport Analysis. The model 
used for this calculation is documented in Attachment IV of the EBS Radionuclide Transport 
Model (CRWMS M&O, 2000 Attachment IV)[DIRS 150793]. The documentation and 
verification of the software routine is presented in this attachment. The attachment is 
incorporated by the Mathcad reference statement below. 
Calculate breakthrough curves using the closed form analytical solution for contaminant 
transport (Equation 6 4) for the van Genuchten retention relation for the retention properties. 
Use the NUFT steady state results that considers the van Genuchten retention model and the 
Campbell model for the coarse intergranular pore space. The pore water velocities for the van 
Genuchten Model are presented in Table 6-9. Compare the results to the diffusion dominated 
process. Note that the saturation is nearly 1.0. Using the relationship developed for invert 
diffusion (Equation 6 16), calculate the soil-liquid diffusion constant using the free water 
diffusion coefficient (Section 5.6). The volumetric moisture content is calculated in Attachment 
XIII for the nondripping case as 0.058. 
Note in the following calculation in Mathcad that := means an assignment of a constant to a 
variable, and = means to output the constant in the default SI units. Note that the constants can 
be output in alternate units. 
Calculate the change in diffusivity DT due to a change in temperature from 25��C to 45��C. The 
diffusivity DT is proportional to the absolute temperature and inversely proportional to the 
viscosity ��; i.e., DT is proportional T/��T (Cussler 1997, p. 114). It follows that if the diffusivity 
0 is known at some temperature T0, the diffusivity at temperature T can be found by: 
T 
T0 DT 
D ��T 0 
��0 
0. The temperature dependence of viscosity is given by 
where DT is the diffusion coefficient (m2/s) at temperature T(K), D0 is the free water diffusion 
coefficient (m2/s) at T0(K), ��T is the viscosity of water (Pa s) at temperature T(K) and ��0 is the 
viscosity of water (Pa s) at temperature T 
Weast and Astle (1981, p. F-42): 
  . . 1.3272.(293.15 T) .0.001053.(T 293.15)2 . . 
T.168.15 
T 
. 
10 
  . . 1.3272.( 293.15 T0.).0.001053.( T0.293.15)2
. . 
T0.168.15 
10 
III-2 of III-10 
�� 
��0 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
and the diffusion coefficient at temperature T is presented in EBS Radionuclide Transport 
Abstraction (BSC 2003b). 
1.3272.( . 293.15 T0. ) .0.001053 .( T0.293.15) 2 
1.3272.( 293.15 T) 
T0.168.15 
10 
10 
1.3272 .( 293.15 T .0.001053.( T0.293.15)2 .1.3272 ( 293.15 .T) .0.001053 
. 
T0 := 25 + 273.15 
T := 45 + 273.15 
sec 
The diffusion coefficient is approximately fifty percent higher. 
As discussed in Section 6.2, the dependence of the soil-liquid diffusion coefficient on the 
saturation (Smatrix), the porosity (��matrix), and the binary diffusion coefficient of water is 
represented mathematically in EBS Radionuclide Transport Abstraction (BSC 2003b, 
Equation 6-15) as: 
From Attachment XIII, Section XIII.2 for the nondripping case, the bulk volumetric moisture 
content is calculated as 0.058. Calculate the solute diffusion coefficient at an elevated 
temperature of 45��C using the calculation presented above and the binary coefficient as given in 
Section 6.2 at ambient temperature. The diffusivity is increased by a factor of 1.494 as 
calculated above. 
. 
D D0 .��matrix 
1.863 
.Smatrix 
1.863 
. 5 cm 
2.30 . 10 
10 
10 
0) . 
T0.168.15 
2 
. 7 cm 
sec 
2 
. 4 m
yr 
Consider the case of the van Genuchten Model with 0.317 mm. From Table 6-9, the pore water 
velocities are given as: 
D := 
.. 
. DT 
D0 
L := 
ANL-EBS-MD-000063 REV 00 
.  
Assign the temperatures for the calculation. The measured diffusion is at ambient temperature 
that equals 25��C. 
D = sl 1.707�~ 
D = sl 5.388�~ 
V . 11 m 
7.6 . 10 . 
sec 
. 0.001053.( T 293.15) 
(Eq. III-3) 
2 .. 
. 
(Eq. III-4) 
(Eq. III-5) 
. 
.(T.293.15) 2..   
. 
T . 168.15 
T . 168.15 
.. 
= 1.494 (Eq. III-6) 
(Eq. III-7) 
2
. 
1.863 .1..(0.058) . 1.494 sl . 
III-3 of III-10 August 2003 Advection versus Diffusion in the Invert 
. 3 m 
10 2.398 �� VL yr 
VL 2.398 
yr 
mm 
. 10 m 
1.8 10 . VH := 
�� 10 5.68 VH 
VH 5.68 
yr 
mm 
Calculate the effective diffusion coefficient using a dispersivity of 0.1 m (Section 6.2) based 
upon Equations 6-12 and 6-13. 
:= 0.1.m 
= 10cm 
�f
�f 
+ := DL Dsl �f VL . 
2 
10. 7 cm (Eq. III-8) DL = 2.467�� 
sec 
H Dsl + �f. VH 
2 
sec 
(Eq. III-9) Co := 1.0 
L . 
C 
...
1 1 
.� 
.� 
.. 
. 
sec 
. 3 m
yr 
. 7 cm 
10 
C V ( ,t , L, R , D) := . + . . . . . (Eq. III-10) erfc 
Co 
2 
. V L 
1 D 2 2 
o 
erfc 
2 .. 
exp .�
. . t t 
.L + V 
R 
.t 
D 
R 
V 
R 
.t 
D 
R 
.
. 
. 
.. 
. 
.. 
. 
.. 
.. 
. 
.. 
.. 
V = pore water velocity (m/yr) 
t = time (yr) 
�f = dispersivity (m) 
R = retardation factor 
D = effective dispersion/diffusion coefficient (m2/yr) 
C = solute concentration at location x, y, z and time t (mg/l) 
C0 = solute concentration of the solute at location x = 0 (mg/l) 
L = length in the vertical direction (m) 
=
=
=
= 
D := 
DH = 3.507�� 
In the following analysis, the user defined function is defined by Equation 6-4 and Jury et al. 
(1991, p. 227, Equation 7.25) as: 
where 
ANL-EBS-MD-000063 REV 00 
.. 
t ( logt ) := 1 . yr 0logt 
logt := . 1 , . 0.9 .. 6 (Eq. III-11) 
III-4 of III-10 August 2003 Advection versus Diffusion in the Invert 
1 
. m 
C . ( . 0.0 . , t logt) , 0.5 m , 1, Dsl 
. 
� . yr 
C V . ( L, t ( logt) , 0.50 m , 1, DL) 0.5 
C V . ( H , t ( logt) , 0.608 m , 1, DH) 
0 
0.1 
Figure III-1. Breakthrough Curves for the 0.317 mm van Genuchten Retention 
Consider the case of the van Genuchten retention relation with 3-20 mm particles sizes (see 
Table 6-9). 
VL . := 5.9 10 . 
sec 
. 3 m 
VL = 1.862�� 10 
L = 1.862 
yr 
V 
mm 
VH . := 1.3 10 . 
sec 
. 3 m 
VH = 4.102�� 10 
H = 4.102 
yr 
V 
mm 
DL := Dsl + �f.VL 
. 7 cm 
DL = 2.297�� 10 
DH := Dsl + �f.VH 
III-5 of III-10 ANL-EBS-MD-000063 REV 00 
. 11 m
yr 
. 10 m
yr 
2 
sec 
C(t)/C0 
Van Genuchten Retention 0.317 mm 
10 1 1 .103 1 .104 1 .105 100 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection 
August 2003 Advection versus Diffusion in the Invert 
. 7 cm 
DH = 3.007�� 10 
1 
. m 
C . ( . 0.0 . , t logt) , 0.5 m , 1, Dsl 
. 
� . yr 
C V . ( L , t ( logt) , 0.50 m , 1, DL) 0.5 
C V . ( H, t ( logt) , 0.608 m , 1, DH) 
0 
0.1 
Figure III-2. Breakthrough Curves for the 3-20 mm van Genuchten Retention Relation 
Consider the case of the 3 mm-20 mm Campbell retention relation (see Table 6-10). 
VL . := 3.7 10 . 
sec 
. 3 m 
VL = 1.168�� 10 
L = 1.168 
yr 
V 
mm 
VH . := 1.3 10 . 
sec 
. 3 m 
VH = 4.102�� 10 
H = 4.102 
yr 
V 
mm 
DL := Dsl + �f.VL 
. 7 cm 
DL = 2.077�� 10 
. 7 cm 
DH = 3.007�� 10 
III-6 of III-10 ANL-EBS-MD-000063 REV 00 
. 11 m
yr 
. 10 m
yr 
2 
sec 
2 
sec 
August 2003 
C(t)/C0 
2 
sec 
Van Genuchten Retention 3-20. mm 
10 1 .103 1 .104 1 .105 1 100 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection Advection versus Diffusion in the Invert 
DH := Dsl + �f.VH 
1 
. m 
C . ( . 0.0 . , t logt) , 0.5 m , 1, Dsl 
.
� . yr 
C V . ( L , t ( logt) , 0.50 m , 1, DL) 0.5 
C V . ( H, t ( logt) , 0.608 m , 1, DH) 
0 
0.1 
ml = 1 10 . 3 m
kg 
3 
�� 
Figure III-3. Breakthrough Curves for the 3-20 mm Campbell Retention Relation 
Define the units of ml for purposes of analysis. 
ml := 
0.01. 
gm 
ml = 1 10 . 5 m
kg 
3 
1. 
gm 
1000. 
R := 1 + 
ANL-EBS-MD-000063 REV 00 
liter 
1000 
3 
gm 
ml = 1 
m
kg 
Use Equation 6-9 to calculate the retardation factor for a low sorption coefficient of 1 ml/gm and 
10 ml/gm. See Table 6-4 for properties. 
2385. 
kg .1. 
ml 
3 gm m
0.11 
R = 22.7 
III-7 of III-10 
�� 
C(t)/C0 
Campbell Retention 3-20. mm 
10 1 100 1 .103 1 .104 1 .105 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection 
August 2003 Advection versus Diffusion in the Invert 
1 
. m 
C 0.0 . . , t logt) , 0.5 m , R, Dsl ( . 
� . yr 
0.5 
. 
C V . ( L, t ( logt) , 0.50 m , R, DL) 
C V . ( H, t ( logt) , 0.608 m , R, DH) 
0 
0.1 
.
� 
Figure III-4. Breakthrough Curves for the Case of Low Sorption Coefficient (Kd = 1.0 ml/gm) 
R := 1 + 
1 
. m 
C . ( . 0.0 . , t logt) , 0.5 m , R, Dsl . yr 
C V . ( L, t ( logt) , 0.50 m , R, DL) 0.5 
C V . ( H, t ( logt) , 0.608 m , R, DH) 
0 
0.1 
C(t)/C0 
2385. 
kg .10. 
ml 
3 gm m
0.11 
R = 217.8 
Kd = 10 ml/gm 
10 1 100 1 .103 1 .104 1 .105 1 .106 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection 
Figure III-5. Breakthrough Curves for the Case of Low Sorption Coefficient (Kd = 10.0 ml/gm) 
III-8 of III-10 ANL-EBS-MD-000063 REV 00 August 2003 
C(t)/C0 
Kd = 1 ml/gm 
10 1 100 1 .103 1 .104 1 .105 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection Advection versus Diffusion in the Invert 
Use Equation 6-9 to calculate the retardation factor for an intermediate sorption coefficient of 
100 ml/gm. See Table 6-4 for properties. 
. m 
C . ( . 0.0 . , t logt) , 0.5 m , R, Dsl . yr 
C V . ( L, t ( logt) , 0.50 m , R, DL) 
C V . ( H, t ( logt) , 0.608 m , R, DH) 
Figure III-6. Breakthrough Curves for the Case of Intermediate Sorption Coefficient (Kd = 100.0 ml/gm) 
Use Equation 6-9 to calculate the retardation factor for an intermediate sorption coefficient of 
1000 ml/gm. See Table 6-4 for properties. 
R = 21682.8 
ANL-EBS-MD-000063 REV 00 
2385. 
kg .1000. 
ml 
3 gm m 
R := 1 + 
0.11 
August 2003 III-9 of III-10 
C(t)/C0 
logt , := 3 3.1 .. 8 
2385. 
kg .100. 
ml 
3 gm 
R := 1 + 
m
0.11 
R = 2169.2 
Kd = 100 ml/gm 
1 
. 
� 
0.5 
10 100 1 .103 1 .104 1 .105 1 .106 1 .107 1 
0 
0.1 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection Advection versus Diffusion in the Invert 
Kd = 1000 ml/gm 
1 
. m 
C . ( . 0.0 . , t logt) , 0.5 m , R, Dsl 
. 
� . yr 
C V . ( L, t ( logt) , 0.50 m , R, DL) 0.5 
C V . ( H, t ( logt) , 0.608 m , R, DH) 
0 
0.1 1 10 100 1 .103 1 .104 1 .105 1 .106 1 .107 1 .108 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection 
High Advection 
Figure III-7. Breakthrough Curves for the Case of High Sorption Coefficient (Kd = 1000.0 ml/gm) 
III-10 of III-10 ANL-EBS-MD-000063 REV 00 August 2003 
C(t)/C0 Advection versus Diffusion in the Invert 
ATTACHMENT IV 
ANALYSIS OF NONDIMENSIONALIZED VAN GENUCHTEN MOISTURE 
POTENTIAL RETENTION RELATION 
IV-1 of IV-14 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
Attachment IV - Analysis of Nondimensionalized van Genuchten Moisture 
Potential Retention Relation 
This attachment presents the calculations and plots of the retention curves for the 
nondimensionalized van Genuchten moisture potential retention relation. The summary of the 
calculations of the air entry parameters is presented in Table IV-1. The van Genuchten (n) value 
is 8.013 as discussed below. The individual retention curves with curve fits are presented in 
Tables IV-2 through IV-5 and are illustrated in Figures IV-1 through IV-4. 
Table IV-1. van Genuchten Air Entry Parameter (�) Calculations 
Particle Diameter(dm) 
3 mm Parameter 
van Genuchten air-entry parameter(�) (cm)-1 
ANL-EBS-MD-000063 REV 00 
10 mm 20 mm 
0.06471 0.6123 2.04 4.08 
van Genuchten air entry parameter(bars)-1 65.92 6244 2080 4160 
August 2003 
0.317 mm 
NOTES: 1See the calculation presented in Equation IV-6. 
2See the calculation presented in Equation IV-7. 
3See the calculation presented in Equation IV-10. 
4See the calculation presented in Equation IV-11. 
IV-2 of IV-14 Advection versus Diffusion in the Invert 
Table IV-2a. van Genuchten Curve Parameters for 0.317 mm Crushed Tuff 
Alpha (�) 
Parameter Value Units 
0.45 (no units) Moisture Content at Saturation (�s) 
0.05 (no units) Residual Moisture Content (�r) 
65.91 bars-1 
n 8.01 (no units) 
m 0.88 (no units) 
NOTES: The alpha parameter is also equal to 0.07 cm-1. 
Table IV-2b. Retention Analysis for 0.317 mm Crushed Tuff 
Moisture Potential 
y(bars) 
Predicted 
Moisture Content 
(�) 
0.0001 0.450 
0.0002 0.450 
0.0005 0.450 
0.001 0.450 
0.002 0.450 
0.005 0.450 
0.01 0.438 
0.02 0.103 
0.05 0.050 
0.1 0.050 
0.2 0.050 
0.5 0.050 
1 0.050 
2 0.050 
5 0.050 
10 0.050 
20 0.050 
50 0.050 
100 0.050 
200 0.050 
500 0.050 
1000 0.050 
NOTE: Predicted moisture content 
obtained from Equation IV-1. 
August 2003 IV-3 of IV-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table IV-3a. van Genuchten Curve Parameters for 3.0 mm Crushed Tuff 
Alpha (�) 
Parameter Value Units 
Moisture Content at Saturation (�s) 
Residual Moisture Content (�r) 
bars-1 
0.45 (no units) 
0.05 (no units) 
624.0 
0 
n 8.01 (no units) 
m 0.88 (no units) 
NOTES: The alpha parameter is also equal to 0.62 cm-1. 
Table IV-3b. Retention Analysis for 3.0 mm Crushed Tuff 
Moisture Potential 
y(bars) 
Predicted 
Moisture Content 
(�) 
0.0001 0.450 
0.0002 0.450 
0.0005 0.450 
0.001 0.442 
0.002 0.124 
0.005 0.050 
0.01 0.050 
0.02 0.050 
0.05 0.050 
0.1 0.050 
0.2 0.050 
0.5 0.050 
1 0.050 
2 0.050 
5 0.050 
10 0.050 
20 0.050 
50 0.050 
100 0.050 
200 0.050 
500 0.050 
1000 0.050 
NOTE: Predicted moisture content 
obtained from Equation IV-1. 
August 2003 IV-4 of IV-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table IV-4a. van Genuchten Curve Parameters for 10.0 mm Crushed Tuff 
Alpha (�) 
Parameter Value Units 
0.45 (no units) Moisture Content at Saturation (�s) 
0.05 (no units) Residual Moisture Content (�r) 
2079.32 bars-1 
n 8.01 (no units) 
m 0.88 (no units) 
NOTES: The alpha parameter is also equal to 2.05 cm-1. 
Table IV-4b. Retention Analysis for 10.0 mm Crushed Tuff 
Moisture Potential 
y(bars) 
Predicted 
Moisture Content 
(�) 
0.0001 0.450 
0.0002 0.450 
0.0005 0.238 
0.001 0.052 
0.002 0.050 
0.005 0.050 
0.01 0.050 
0.02 0.050 
0.05 0.050 
0.1 0.050 
0.2 0.050 
0.5 0.050 
1 0.050 
2 0.050 
5 0.050 
10 0.050 
20 0.050 
50 0.050 
100 0.050 
200 0.050 
500 0.050 
1000 0.050 
NOTE: Predicted moisture content 
obtained from Equation IV-1. 
August 2003 IV-5 of IV-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table IV-5a. van Genuchten Curve Parameters for 20.0 mm Crushed Tuff 
Alpha (�) 
Parameter Value Units 
0.45 (no units) Moisture Content at Saturation (�s) 
0.05 (no units) Residual Moisture Content (�r) 
4158.80 bars-1 
n 8.01 (no units) 
m 0.88 (no units) 
NOTES: The alpha parameter is also equal to 4.09 cm-1. 
Table IV-5b. Retention Analysis for 20.0 mm Crushed Tuff 
Moisture Potential 
y(bars) 
Predicted 
Moisture Content 
(�) 
0.0001 0.450 
0.0002 0.384 
0.0005 0.052 
0.001 0.050 
0.002 0.050 
0.005 0.050 
0.01 0.050 
0.02 0.050 
0.05 0.050 
0.1 0.050 
0.2 0.050 
0.5 0.050 
1 0.050 
2 0.050 
5 0.050 
10 0.050 
20 0.050 
50 0.050 
100 0.050 
200 0.050 
500 0.050 
1000 0.050 
NOTE: Predicted Moisture content 
obtained from Equation IV-1. 
August 2003 IV-6 of IV-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
1000 
100 
10 
1 
Water Potential (-bars) 
0.1 
0.01 
0.001 
0.0001 
0.00 
1000 
100 
10 
1 
0.1 
0.01 
0.001 
Figure IV-1. Nondimensionalized Model for Mean Particle Diameter of 0.317 mm 
Volum etric Water Content 
0.40 0.20 
0.0001 
0.00 
Figure IV-2. Nondimensionalized Model for Mean Particle Diameter of 3.0 mm 
Volum etric Water Content 
ANL-EBS-MD-000063 REV 00 
Water Potential (-bars) 
0.60 0.40 0.20 
0.60 
August 2003 IV-7 of IV-14 Advection versus Diffusion in the Invert 
1000 
100 
10 
1 
0.1 
0.01 
0.001 
0.0001 
0.00 0.10 0.60 0.70 0.20 0.30 0.40 0.50 
1000 
100 
10 
1 
0.1 
0.01 
0.001 
Volum etric Wate r Content 
Figure IV-3. Nondimensionalized Model for Mean Particle Diameter of 10.0 mm 
0.50 0.60 0.30 0.40 0.10 0.20 
0.0001 
0.00 0.70 
Volum etric Wate r Content 
Figure IV-4. Nondimensionalized Model for Mean Particle Diameter of 20 mm 
August 2003 ANL-EBS-MD-000063 REV 00 IV-8 of IV-14 
Water Potential (-bars) Water Potential (-bars) Advection versus Diffusion in the Invert 
The following discussion presents the methodology and the analysis for determining the 
relationship of unsaturated hydraulic conductivity (Kus) from the retention curve using the twoparameter 
nondimensional van Genuchten relationship and the size of the crushed tuff. The 
basis for the nondimensional retention relationship is a least squares curve fit for a series of 
retention measurements made on the materials presented above. Equal weighting is given to 
these materials (Attachment VII). The nondimensional van Genuchten air-entry parameter (� ) is 
determined from a scaling relationship while the nondimensional (n) parameter is constant. The 
intrinsic permeability (k) is determined from the Kozeny-Carman formula shown by Equation 
IV-4 (Bear 1972, p. 166) that relates intrinsic permeability (k) to the grain size or pore diameter 
(dm) and porosity (�). On the basis of the selected grain size, the saturated hydraulic 
conductivity (Ks) and the van Genuchten relationship for relative permeability, the relationship 
of unsaturated hydraulic conductivity (Kus) relationship to moisture potential (� ) can be 
determined. It should be noted that the hydraulic conductivity under saturated conditions may 
have a Reynolds Number that exceeds the range of validity for Darcy�s Law. Some other power 
law relating flow rate to hydraulic conductivity might apply under saturated conditions. 
A qualitative assessment can be made over the range of moisture potentials (� ) of interest (0.01 
to 0.1 bars) as to whether liquid flow or advection in the coarse fraction of the crushed tuff 
would occur for a range of particle diameters. The analysis presented below is performed for 
grain size diameters of 0.317 mm, 3 mm, 10 mm, and 20 mm for the intergranular porosity 
(� intergrain) respectively to cover a broad range of particle diameters. 
Equation IV-1 is used to define the moisture potential (� ) (capillary pressure divided by weight 
density) versus moisture content (� ) relation (Fetter 1993, p. 172): 
�) = � + 
( �s 
. �r ) �( �, n, s r r 
IV-1) [ 1+ ( ��)n ]m (Eq. 
where 
� , � , 
� = Volumetric moisture/water content 
� = van Genuchten air-entry parameter (Pa-1) or (fcm-1) or (bars-1) 
n = van Genuchten nondimensional �n� parameter 
� s = Saturated volumetric moisture content 
� r = Residual volumetric moisture content 
� = Nondimensional moisture/matric potential 
m = van Genuchten nondimensional �m� parameter 
Fetter designates the (m) parameter for the van Genuchten relationship as: m = 1-1/n 
(Fetter 1993, p. 172). This value of �m� can be substituted into Equation IV-1 to derive the 
volumetric moisture content (� ) as shown in Equation IV-2. Figure IV-5 reflects this 
relationship. The parameter �n� is a van Genuchten nondimensional parameter estimated from 
the soil-water retention curve (van Genuchten 1980, p. 895). 
IV-9 of IV-14 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
r s � (Eq. IV-2) �( , n, s r r [ 1+ ( ��)n ]( 1. 
1
n 
) 
where 
( � . � ) 
� 
� = moisture potential 
� = mass density of water ((kg/m3) 
g = acceleration due to gravity 
k = intrinsic permeability 
� w 
= interfacial tension between the pore water and mineral surface (dyne/cm) 
� = porosity 
The least squares regression analysis of the data is presented in Attachment VI using the above 
relationship. Attachment VII presents independent verification of the Mathcad calculations 
using Microsoft Excel. The van Genuchten nondimensional parameter (n) is calculated, where 
the van Genuchten dimensionless air entry parameter, � ND = 2.455 such that n = 8.01 
(Attachments VI and VII). The original Brooks and Corey data have been plotted with a curve 
fit using Equation IV-3 (Brooks and Corey 1964). For specific sized grains, the Kozeny-Carman 
equation expresses the relationship of the intrinsic permeability (k) to the porosity (� ) as 
expressed in Equation 6-25 (Bear 1972, p. 166). 
) = � + � , � , � 
Equation IV-3 is used to define the nondimensional moisture potential parameter (� ND) for 
analysis (Leverett 1941, p. 159). Figure IV-5 shows this relationship. 
�ND . = �. 
�. g k (Eq. IV-3) 
W � 
IV-10 of IV-14 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
600 
500 
400 
300 
200 
100 
0 
0.2 0.1 0 
Figure IV-5. Capillary Rise of Unconsolidated Samples 
dm = mean particle diameter 
� = porosity 
This equation can be rearranged to obtain: 
ANL-EBS-MD-000063 REV 00 
d 2 
m 
2 Intrinsic permeability = k = 
180 
. (1 . �) 
(Eq. IV-4) 
k = 
dm . 
1 
� w 
Capillary Rise (cm) 
where 
The Leverett Equation, as presented in Equation IV-3, expresses the relationship of the 
nondimensional moisture potential to the dimensional moisture potential. Since the air-entry 
parameter � is inversely proportional to the moisture potential (see Equation IV-2), the 
relationship of the nondimensional air-entry parameter as determined from the regression 
analysis for Equation IV-2 is given by: 
� 
ND 
Volcanic Sand 
Fine Sand 
Predicted Volcanic Sand 
Predicted Fine Sand 
Glass Beads 
Predicted Glass Beads 
Touchet Silt Loam 
Predicted Touchet Silt Loam 
Capillary Rise of Unconsolidated Samples 
0.9 0.6 0.4 0.8 0.7 0.5 1 0.3 
Saturation 
� 
180 ( 1 . �) 
�3 
(Eq. IV-5) 
k (Eq. IV-6) . 
� 
�. g = 1 . 
� � 
IV-11 of IV-14 August 2003 Advection versus Diffusion in the Invert 
Solving for the air-entry parameter �� in terms of the nondimensional air-entry parameter: 
. ND 
material is given in Equation IV-7: 
��. g 
�� 
�� �� = 
g k . 2 455 
Substituting in the value of 0.317 mm as a particle size with an intergranular porosity (�� intergrain) 
of 0.45, the calculated value for the van Genuchten air-entry parameter (�� ) in cm-1 for this size 
w 
. 2 455 = 0. 0647 . cm .1 (Eq. IV-9) 
�� w 
Substituting Equation IV-5 into Equation IV-7 to express the air-entry parameter �� in terms of 
the particle diameter (dm); the fluid density (�� ); the surface tension (�� w 
); the porosity (�� ), the 
gravitational constant (g); and the nondimensionless air entry parameter ( �� ND) as determined 
from the curve fitting analysis in Attachments VI and VII: 
��. g 
�� 
2 .455 = 
( 1 2 
K = us 
= 
This analysis uses the following referenced properties for water: mass density, �� = 1000 kg/m3, 
�� g, geometric standard deviation, an interfacial tension between the pore water and mineral 
surface, �� w = 72 dynes/cm at ambient temperature, and gravity, g = 9.81 m/sec2 (Incropera and 
DeWitt 1996, p. 846). The air-entry parameter (��) in term s of (bars)-1: 
The wetting-phase relative permeability as a function of moisture potential (�� ) for van 
Genuchten curve fit is restated from Fetter (1993, p. 182). The unsaturated hydraulic 
conductivity (Kus) (wetting-phase relative permeability times saturated hydraulic conductivity) as 
a function of moisture potential (�� ) is derived using Equation IV-11 (Fetter 1993, Equation 4.17) 
��w 
��= 
1000 . 
K( ��, n, ��,K 
��. 
�� 
ANL-EBS-MD-000063 REV 00 
�� ND 
w 
m . 
d 
180 w 
k 
1 
. 
1 
.�. 
( . 
.. 
. 
�� 
��) 
��. g 0 .0317 . cm 
180
. cm 
81 . 9 . . 
1 . �� �� . 
. 
.... 
�� =
(1. 
. 
0 .06466 
kg 
3 m 
s ) = K s 
45 . 0 
. 45 . 0 
( �� �� n ) 
+ 
.. 
. 
. 
= 9 . 65 (bars) 1 (Eq. IV-10) 
) 
�� �� n 
. 
..... 
. 
. 1 
m 
sec 
. 
1 + 
).. n . 
.. 
. 
...... 
IV-12 of IV-14 
�� 
(Eq. IV-7) 
m (Eq. IV-8) 
�� 
. 
��. g 
�� (1 
n 
. 
2 1 . 
.... 
��) 
(Eq. IV-11) 
.���. 
. ) 
. 
. 
..... 
( 
d
180 
. 
+ 
...... 
. 
. 1 
..... 
. 
.. 
1 1 
2 2n ... 
.�. 
August 2003 Advection versus Diffusion in the Invert 
1000 
100 
10 
1 
0.1 
0.01 
0.001 
0.0001 
1E-05 
1E-06 
1E-07 
1E-08 
1E-09 
1E-10 
1E-11 
1E-12 
1E-13 
1E-14 
1E-15 
1E-16 
1E-17 
Hydraulic Conductivity (cm/s) 
0.0001 
A second analysis was developed based upon a grain sizes of 3 mm. The van Genuchten airentry 
parameter (��) for this grain size from Equations IV-3 and IV-5 is given as: 
Figure IV-6. Unsaturated Hydraulic Conductivity Relationships 
�� 
��. g k . 2 .455 = 
��. g 0 .3. cm . 
w w 
3 
IV-13 of IV-14 
. . 2 455 = . 0 612 . cm .1 (Eq. IV-12) . 0 45 
180 (1 . . 0 45) 
624 . bars .1 (Eq. IV-13) 
sec 
Moisture Potential (bars) 
"3 mm" 0.317 mm 
The intrinsic permeability (k) corresponding to a grain size of 3 mm using Equation IV-4 as 
1.51�~10-8 m2: 
�� = 
The value of the van Genuchten air-entry parameter (��) for the 3 mm diameter grain size in 
terms of bars-1 is calculated using Equation IV-9 to be 624 (bars)-1: 
0.612 . cm.1 
1000 . kg . 9.81. m
2 m 
��
180 ( 1 . ��)2 
Comparison of Conductivity Relationships 
0.01 
Tuff Fracture k 
10 1 0.1 
= 
k = 
dm
2 
. 
��3 
= 1 .51.10 .8 . m2 (Eq. IV-14) 
ANL-EBS-MD-000063 REV 00 
�� 
0.001 
Tuff Matrix k 
1000 100 
10 mm 20 mm 
August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 IV-14 of IV-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT V 
ANALYSIS OF THE CAMPBELL RETENTION RELATION 
V-1 of V-14 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
diagram as: 
gm 
cm 3 
Attachment V - Analysis of the Campbell Retention Relation 
This attachment presents calculations to develop the retention curve based upon the Campbell 
retention relation for the intergranular porosity. Based upon the retention data, the van 
Genuchten curve fit parameters (� , n) are estimated using Microsoft Excel Solver as outlined in 
Water Diversion Model (CRWMS M&O 2000c, Section 3 and Attachment IV). The Solver is an 
add-in function in Microsoft Excel. The Solver can minimize a target cell that involves multiple 
cell variables that might be subject to multiple constraints. The Solver is used specifically to 
solve for the van Genuchten curve fit parameters (� , n) based upon a minimization of the least 
squares of the volumetric moisture content for curve fitting to the Campbell retention relation. 
Table V-1 presents a summary of hydrologic properties based upon the Campbell retention 
relation. Tables V-2 through V-9 present the detailed calculations. 
The following discussion presents a sample calculation for the purpose of calculating the 
retention relationship using the Campbell method. The first step is to calculate the air entry 
pressure for a particle diameter of 0.317 mm. Apply Equation 6-23 to the calculation of the airentry 
moisture potential. 
= . 0 . 5 . 0 . d . 1 / 2 = .0 .5. ( 317).0 .5 = . 0 .888. J / kg g es 
g es 
v = 45 . 0 V
VT 
Vv = 45 . 0 
0 . 1 + Vv 
V . ( 1 . 45 . 0 ) = 45 . 0 V 
. 0 
V = 
The specific gravity of solids equals 2.52 gm/cm3. Therefore, the dry mass density is calculated 
as: 
�d = 
1 + 818 
= 1 .39 52 . 2 
V-2 of V-14 
� 
Calculate the value of b as reported in Table 6-5 from Equation 6-24: 
b = .2 . � + 0 .2 . � = . 2 . .0 .888 + 0 .2 . (5) = 2 .78 
For purposes of calculating the retention relationship, calculate the air-entry moisture potential 
for the crushed tuff using Equation 6-25. The dry unit weight is calculated from the phase 
Using the soils mechanics convention of setting the volume of solids equal to one: 
Solving for the volume of the voids: 
818 V 0 .45/( 1. 0 .45) = 0 . 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
Applying this correction to the air-entry moisture potential (Equation 6-25): 
. ( 39 . 1 / 3 . 1 ) 67 . 0 b = . 1 .006 �r = . 0 .888 e = .0 .01006Bars J 
kg 
. �p= . . 
Calculate the volumetric moisture content by solving Equation 6-22 for volumetric moisture 
content for a value of moisture potential of 0.1 bars. 
. . 1 . 
776 
. . 1 . 
� . .
. . 
2 . � . 
. �p= 0.197 
. �r .. . 
b � . 
�c= ..�r e � . .
. 
The results of this calculation are in agreement with the spreadsheet calculation presented in 
Table V-2. 
Table V-1. Summary of Hydrologic Properties Based Upon the Campbell Retention Relation 
Parameter 
Saturated Volumetric Moisture Content (�c s 
)1 
Residual Volumetric Moisture Content (�c r)2 
van Genuchten Air-entry Parameter (�\ ) (bars-1)3 
van Genuchten Air-entry Parameter (�\ ) (cm-1) 
van Genuchten n Value4 
van Genuchten m Value 
Saturated Intrinsic Permeability (k)(m2)6 
Saturated Hydraulic Conductivity (Ks)(cm/sec)7 
6 
NOTES: 1 See Section 6.3 for the selection of the porosity (�p ). The saturated volumetric moisture content (�c s 
) 
equals the porosity. 
2 The residual volumetric moisture content (�cr ) is selected as 0.02 based upon a plot of the empirical 
retention curve. 
3 The van Genuchten air-entry parameter(�\ ) is determined from a regression analysis using the Microsoft 
Excel Solver as explained in the text. 
4 The van Genuchten n value is determined from a regression analysis using the Microsoft Excel Solver 
as explained in the text. The value of n is given by 1/(1-m) (Fetter 1993, p. 172). 
5 Note that m = 1-1/n (Fetter 1993, p. 172). 
7 
The intrinsic permeability (k) is calculated from Equation IV-4. 
The value of the saturated hydraulic conduct ivity (Ks) is obtained by the equation that converts an 
saturated intrinsic permeability (k) to a saturated hydraulic conductivity (Ks) (Freeze and Cherry 1979, 
20 mm 
0.450 0.450 0.450 0.450 
0.020 0.010 0.010 0.010 
47.64 230.84 476.91 561.61 
0.06 0.24 0.48 0.56 
1.53 3.04 4.03 11.11 
0.35 0.67 0.752 0.91 
1.68E-10 1.51E-08 1.67E-07 6.69E-07 
0.184 16.48 183.1 732.5 
p. 27): 
Particle Size (dg) 
0.317 mm 
K 
The properties of water at ambient temperature are given by Incropera and DeWitt (1996, p. 846). The 
water density (�l ) equals 1000 kg/m3 and the absolute viscosity (�g ) equals 8.935�� 10-4 N. s/(m2). 
g . �l 
�g 
ANL-EBS-MD-000063 REV 00 
1 . 0 
. 0 .01006 
3 mm 10 mm 
k . s = 
August 2003 V-3 of V-14 Advection versus Diffusion in the Invert 
Table V-2a. Campbell Model Parameters1 for 0.317 mm Crushed Tuff 
2 
3 
4 
Parameter Value 
3.17E-01 
4.50E-01 
Grain Diameter (dg) (mm) 
Saturated Volumetric Water Content (�S) 
Air-entry water potential or the potential at which the largest waterfilled 
pores just drain �es (J/kg)2 -8.88E-01 
Geometric Standard Deviation3 5.00E+00 
Slope of the ln(�) versus ln(�) retention curve (b)4 2.78E+00 
�e (bars) -1.01E-02 
�e (cm) -1.03E+01 
NOTES: 1 See text and Table 6-5 for a description of the parameters. 
This value is calculated from Equation 6.23. 
This value is estimated from Campbell (1985, p. 10, Figure 2.1). 
This value is calculated from Equation 6.24. 
Table V-2b. Campbell Model Moisture Data for 0.317 mm Crushed Tuff 
Moisture 
Potential (bars) 
Moisture 
Content 
0.0001 0.450 
0.0002 0.450 
0.0005 0.450 
0.001 0.450 
0.002 0.450 
0.005 0.450 
0.01 0.450 
0.02 0.351 
0.05 0.253 
0.1 0.197 
0.2 0.153 
0.5 0.110 
1 0.086 
2 0.067 
5 0.048 
10 0.037 
20 0.029 
50 0.021 
100 0.016 
200 0.013 
500 0.009 
1000 0.007 
August 2003 V-4 of V-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table V-3a. van Genuchten Curve Fit Parameters for 0.317 mm Crushed Tuff 
Alpha (�) 
n 
m 
Parameter Value Units 
Moisture Content at Saturation (�s) 0.45 
Residual Moisture Content (�r) 
(no units) 
(no units) 
bars-1 
(no units) 
(no units) 
0.02 
47.64 
1.53 
0.35 
NOTES: The alpha parameter is also equal to 0.06 cm-1. 
The sum of the residuals is 3.22E-03 
Table V-3a. van Genuchten Retention Analysis Results for 0.317 mm Crushed Tuff 
Predicted 
Moisture 
Content2 
Volumetric 
Moisture 
Content1 
Moisture 
Potential 
(bars) 
1000 
Residuals 
Squared 
0.450 0.0001 0.450 1.77E-09 
0.450 0.0002 0.450 1.47E-08 
0.450 0.0005 0.450 2.41E-07 
0.450 0.001 0.449 1.99E-06 
0.450 0.002 0.446 1.61E-05 
0.450 0.005 0.435 2.39E-04 
0.450 0.01 0.410 1.56E-03 
0.351 0.02 0.363 1.29E-04 
0.253 0.05 0.271 3.29E-04 
0.197 0.1 0.203 3.85E-05 
0.153 0.2 0.149 1.51E-05 
0.110 0.5 0.100 9.53E-05 
0.086 1 0.076 9.93E-05 
0.067 2 0.059 6.56E-05 
0.048 5 0.044 1.73E-05 
0.037 10 0.037 7.52E-07 
0.029 20 0.032 5.41E-06 
0.021 50 0.027 3.74E-05 
0.016 100 0.025 7.36E-05 
0.013 200 0.023 1.14E-04 
0.009 500 0.022 1.68E-04 
0.007 0.021 2.05E-04 
NOTES: 1 The volumetric moisture content is 
determined from Equation 6-22 to the 
Campbell data. 
2 The predicted moisture content is 
determined from the minimization of the 
least squares using the Microsoft Excel 
Solver as explained in the text. 
August 2003 V-5 of V-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table V-4. Campbell Model Parameters1 for 3.0 mm Crushed Tuff 
2 
3 
4 
Parameter Value 
3 
4.50E-01 
Grain Diameter (dg) (mm) 
Saturated Volumetric Water Content (�S) 
Air-entry water potential or the potential at which the largest water-filled 
pores just drain �es (J/kg)2 
Geometric Standard Deviation3 
Slope of the ln(�) versus ln(�) retention curve (b)4 
�e (bars) 
�e (cm) 
-2.89E-01 
1.00E+00 
7.77E-01 
-2.99E-03 
-3.05E+00 
NOTES: 1 See text and Table 6-5 for a description of the parameters. 
This value is calculated from Equation 6-23. 
This value is estimated from Campbell (1985, p. 10, Figure 2.1). 
This value is calculated from Equation 6-24. 
Table V-4. Campbell Model Moisture Data for 3.0 mm Crushed Tuff 
(bars) 
Moisture 
Potential Moisture 
Content 
0.0001 0.450 
0.0002 0.450 
0.0005 0.450 
0.001 0.450 
0.002 0.450 
0.005 0.232 
0.01 0.095 
0.02 0.039 
0.05 0.012 
0.1 0.015 
0.2 0.012 
0.5 0.011 
1 0.010 
2 0.010 
5 0.010 
10 0.010 
20 0.010 
50 0.010 
100 0.010 
200 0.010 
500 0.010 
1000 0.010 
August 2003 V-6 of V-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table V-5a. van Genuchten Curve Fit Parameters for 3.0 mm Crushed Tuff 
Moisture Content at Saturation (�s) 0.45 (no units) 
Residual Moisture Content (�r) 0.01 (no units) 
Alpha (�) 230.84 bars-1 
n 3.04 (no units) 
m 0.67 (no units) 
NOTES: The alpha parameter is also equal to 0.24 cm-1. 
The sum of the residuals is 1.06E-03. 
Table V-5a. van Genuchten Retention Analysis Results for 3.0 mm Crushed Tuff 
Parameter Value Units 
Volumetric 
Moisture 
Content1 
0.450 
0.450 
0.450 
0.450 
0.450 
0.232 
0.095 
0.039 
0.012 
0.015 
0.012 
0.011 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
NOTES: 1 The volumetric moisture content is 
determined from Equation 6-22 to the 
Campbell data. 
2 The predicted moisture content is 
determined from the minimization of the 
least squares using the EX Microsoft Excel 
Solver as explained in the text. 
Residuals 
Squared 
9.48E-12 
6.45E-10 
1.70E-07 
1.14E-05 
6.78E-04 
1.61E-04 
9.06E-05 
9.69E-05 
9.21E-07 
1.77E-05 
3.40E-06 
3.53E-07 
6.15E-08 
1.06E-08 
1.02E-09 
1.72E-10 
2.90E-11 
2.75E-12 
4.62E-13 
7.78E-14 
7.37E-15 
1.24E-15 
ANL-EBS-MD-000063 REV 00 
Predicted 
Moisture 
Content2 
0.450 
0.450 
0.450 
0.447 
0.424 
0.245 
0.086 
0.029 
0.013 
0.011 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
Moisture 
Potential 
(bars) 
0.0001 
0.0002 
0.0005 
0.001 
0.002 
0.005 
0.01 
0.02 
0.05 
0.1 
0.2 
0.5 
1 
2 
5 
10 
20 
50 
100 
200 
500 
1000 
V-7 of V-14 August 2003 Advection versus Diffusion in the Invert 
Table V-6a. Campbell Model Parameters for 10 mm Crushed Tuff 1 
2 
Parameter Value 
10 
0.45 
Grain Diameter (dg) (mm) 
Saturated Volumetric Water Content (�S) 
Air-entry water potential or the potential at which the largest 
water filled pores just drain �es (J/kg)2 -0.158113883 
Geometric Standard Deviation3 1 
Slope of the ln(�) versus ln(�) retention curve (b)4 
�es (Bars) 
�es (cm) 
0.516227766 
-0.001618173 
-1.650787548 
NOTES: 1 See text and Table 6-5 for a description of the parameters. 
This value is calculated from Equation 6.23. 
3 This value is estimated from Campbell (1985, p. 10, Figure 2.1) 
4 This value is calculated from Equation 6.24. 
Table V-6b. Campbell Moisture Content Data for 10 mm Crushed Tuff 
Moisture 
Potential 
(bars) Moisture Content 
0.0001 0.45 
0.0002 0.450 
0.0005 0.450 
0.001 0.450 
0.002 0.299 
0.005 0.051 
0.01 0.013 
0.02 0.013 
0.05 0.011 
0.1 0.010 
0.2 0.010 
0.5 0.010 
1 0.010 
2 0.010 
5 0.010 
10 0.010 
20 0.010 
50 0.010 
100 0.010 
200 0.010 
500 0.010 
1000 0.010 
August 2003 V-8 of V-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table V-7a. van Genuchten Curve Fit Parameters for 10 mm Crushed Tuff 
Moisture Content at Saturation (�s) 0.45 (no units) 
Residual Moisture Content (�r) 0.01 (no units) 
Alpha (�) 476.91 bars-1 
n 4.03 (no units) 
m 0.75 (no units) 
NOTES: The alpha parameter is also equal to 0.48 cm-1. 
The sum of the residuals is 3.42E-04. 
Table V-7a. van Genuchten Curve Fit Parameters for 10 mm Crushed Tuff 
Parameter Value Units 
Volumetric 
Moisture 
Content1 
0.450 
0.450 
0.450 
0.450 
0.285 
0.048 
0.013 
0.013 
0.011 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
2 
NOTES: 1 The volumetric moisture content is 
determined from Equation 6-22 to the 
Campbell data. 
The predicted moisture content is 
determined from the minimization of the 
least squares using the Microsoft Excel 
Solver as explained in the text. 
Residuals 
Squared 
2.51E-12 
6.65E-10 
1.06E-06 
2.58E-04 
1.87E-05 
5.38E-05 
1.59E-06 
7.95E-06 
2.80E-07 
2.03E-08 
1.42E-09 
4.14E-11 
2.84E-12 
1.94E-13 
5.57E-15 
3.80E-16 
2.59E-17 
7.45E-19 
5.08E-20 
3.47E-21 
9.95E-23 
6.79E-24 
ANL-EBS-MD-000063 REV 00 
Predicted 
Moisture 
Content2 
0.450 
0.450 
0.449 
0.434 
0.290 
0.041 
0.014 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
Moisture 
Potential 
(bars) 
0.0001 
0.0002 
0.0005 
0.001 
0.002 
0.005 
0.01 
0.02 
0.05 
0.1 
0.2 
0.5 
1 
2 
5 
10 
20 
50 
100 
200 
500 
1000 
V-9 of V-14 August 2003 Advection versus Diffusion in the Invert 
Table V-8a. Campbell Model Parameters for 20 mm Crushed Tuff 1 
Grain Diameter (dg) (mm) 20 
Saturated Volumetric Water Content (�S) 0.45 
Air-entry water potential or the potential at which the largest 
water filled pores just drain �es (J/kg)2 -0.111803399 
Geometric Standard Deviation3 1 
Slope of the ln(�) versus ln(�) retention curve (b)4 
�e (Bars) 
�e (cm) 
NOTES: 1 See text and Table 6-5 for a description of the parameters. 
2 This value is calculated from Equation 6.23. 
3 This value is estimated from Campbell (1985, p. 10, Figure 2.1) 
4 This value is calculated from Equation 6.24. 
Table V-8b. Campbell Moisture Content Data for 10 mm Crushed Tuff 
Moisture 
Potential 
(bars) Moisture Content 
0.0001 
0.0002 
0.0005 
0.001 
0.002 
0.005 
0.01 
0.02 
0.05 
0.1 
0.2 
0.5 
1 
2 
5 
10 
20 
50 
100 
200 
500 
1000 
0.4500000 
0.4500000 
0.4500000 
0.4500000 
0.1192454 
0.0137099 
0.0126693 
0.0105197 
0.0100598 
0.0100116 
0.0100023 
0.0100003 
0.0100001 
0.0100000 
0.0100000 
0.0100000 
0.0100000 
0.0100000 
0.0100000 
0.0100000 
0.0100000 
0.0100000 
V-10 of V-14 ANL-EBS-MD-000063 REV 00 
0.423606798 
-0.001139478 
-1.162444233 
August 2003 Advection versus Diffusion in the Invert 
Table V-9a. van Genuchten Curve Fit Parameters for 20 mm Crushed Tuff 
Moisture Content at Saturation (�s) 0.45 (no units) 
Residual Moisture Content (�r) 0.01 (no units) 
Alpha (�) 561.61 bars-1 
n 11.11 (no units) 
m 0.91 (no units) 
NOTES: The alpha parameter is also equal to 0.56 cm-1. 
The sum of the residuals is 1.86E-05. 
Table V-9a. van Genuchten Curve Fit Parameters for 20 mm Crushed Tuff 
Parameter Value Units 
Volumetric 
Moisture 
Content1 
0.450 
0.450 
0.450 
0.450 
0.119 
0.014 
0.013 
0.011 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
2 
NOTES: 1 The volumetric moisture content is 
determined from Equation 6-22 to the 
Campbell data. 
The predicted moisture content is 
determined from the minimization of the 
least squares using the Microsoft Excel 
Solver as explained in the text. 
Residuals 
Squared 
2.55E-29 
1.25E-22 
8.78E-14 
4.30E-07 
3.53E-10 
1.08E-05 
5.10E-06 
1.20E-08 
1.23E-07 
1.59E-07 
1.66E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
1.68E-07 
ANL-EBS-MD-000063 REV 00 
Predicted 
Moisture 
Content2 
0.450 
0.450 
0.450 
0.449 
0.119 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
0.010 
Moisture 
Potential 
(bars) 
0.0001 
0.0002 
0.0005 
0.001 
0.002 
0.005 
0.01 
0.02 
0.05 
0.1 
0.2 
0.5 
1 
2 
5 
10 
20 
50 
100 
200 
500 
1000 
V-11 of V-14 August 2003 Advection versus Diffusion in the Invert 
1000 
100 
10 
1 
0.1 
0.01 
0.001 
0.0001 
0.00 0.60 0.40 0.20 
1000 
100 
10 
1 
0.1 
0.01 
0.001 
Volumetric Water Content 
Figure V-1. Comparison of Data to Curve Fit Campbell Model (0.317 mm) 
0.40 0.20 
0.0001 
0.00 0.60 
Volumetric Water Content 
Figure V-2. Comparison of Data to Curve Fit Campbell Model (3 mm) 
August 2003 V-12 of V-14 ANL-EBS-MD-000063 REV 00 
Water Potential (-bars) 
Water Potential (-bars) Advection versus Diffusion in the Invert 
1000 
100 
10 
1000 
Water Potential (-bars) 
1 
0.1 
0.01 
0.001 
0.0001 
0.00 
100 
10 
1 
0.1 
0.01 
0.001 
0.0001 
0.00 
Figure V-3. Comparison of Data to Curve Fit Campbell Model (10 mm) 
Figure V-4. Comparison of Data to Curve Fit Campbell Model (20 mm) 
Volum e tric Wate r Conte nt 
Volumetric Water Content 
0.40 0.20 
ANL-EBS-MD-000063 REV 00 
Water Potential (-bars) 
0.60 0.40 0.20 
0.60 
August 2003 V-13 of V-14 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 V-14 of V-14 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT VI 
DERIVATION OF INVERT "PACKED BED" PROPERTIES FROM BROOKS AND 
August 2003 ANL-EBS-MD-000063 REV 00 
COREY DATA 
VI-1 of VI-16 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 VI-2 of VI-16 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment VI - Derivation of Invert �Packed Bed� Properties from 
Brooks and Corey Data 
This attachment uses the nondimensional form for the capillary pressure curve presented by 
Leverett (1941, p. 159) with the capillary pressure data generated by Brooks and Corey (1964) to 
provide a capillary pressure curve for the invert. The nondimensionalized data is then used to fit 
with a van Genutchen relationship (Equation 6-22) for use in developing retention curves for the 
coarse intergranular porosity of the invert. 
The invert will consist of crushed tuff, probably supplied from the drift itself, crushed to a size of 
fine gravel to coarse �sand.� The analysis using NUFT presented in Section 6.6 requires the 
permeability, capillary pressure, and relative permeability of the interparticle void space. 
Because the invert particle diameter is not fixed, a means of computing these properties as a 
function of particle diameter is desired. 
VI.1 DEVELOPMENT OF PERMEABILITY AND RETENTION RELATIONSHIPS 
FOR THE INTERGRANULAR POROSITY 
The permeability is supplied by the Kozeny-Carman equation (Bear 1972, p. 166). This equation 
relates the permeability to the mean particle diameter and the porosity. The mean particle 
diameter used in this analysis is based upon the specific surface area of the particle size 
distribution. 
The dependence of the capillary pressure curve on particle diameter and porosity is analyzed in 
the same manner as Leverett (1941, p. 159). Leverett showed that capillary pressure data for 
various sands could be correlated using a nondimensional group that included the mean particle 
diameter and the porosity. This approach is applied to the data published by Brooks and Cory 
(1964) to produce a nondimensional capillary pressure/saturation curve (Brooks and Cory 1964). 
This nondimensional data is then fitted with the functional form posed by van Genuchten (1980, 
Equation 21). 
The relative permeability is based upon the work of Mualem, as reported by van Genuchten 
(1980). Maulem developed a semi-empirical relationship between the capillary pressure curve 
and the relative permeability curve. Maulem's correlation coefficient is combined with the fit of 
the nondimensional capillary pressure to produce a liquid relative permeability curve appropriate 
for the invert interparticle porosity. The relative permeability of the gas phase is set to unity in 
anticipation of the very low interparticle saturation anticipated in the invert. 
Specifically, the Brooks and Corey (1964) database (Attachment VI) for unconsolidated 
particulate media is most applicable to the proposed invert configuration. Both are composed of 
particles that are not consolidated together. Angularity of the particulate will be reflected in the 
data for sands. This, of course, presumes that the invert �rock� is �roughly� spherical, as 
opposed to a plate-like shape. 
The assumed bed porosity should provide everything needed. 
August 2003 VI-3 of VI-16 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Input the properties for the four materials from Attachment VI (Brooks and Corey 1964, 
Appendix III and Table 1): 
Note that only the first few values are printed out in the Mathcad format. 
Input the intrinsic permeability and porosity: 
18 . 
2.5 
(Eq. VI-1) .( 
10. 6 .m) 2 
. 
. 
. 
. 
. 
.. 
. 
. 
. 11.3 
:= . 6.3 
. 
0.6 
. kVS . 
FS k
kFM 
. GB k 
kTSL 
������� 
������ 
30 . 
. 
.. 
k . FFH . 
According to Brooks and Corey (1964, p. 9, Equation 17), the capillary rise of water was about 
twice that of the hydrocarbon used in the measurements. van Genuchten (1980) used the same 
factor in his analysis. 
. 
.. 
VI-4 of VI-16 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
VS.1. . . VS.1. 
.2 . 
. . FS 1 FS 1 2 . 
. . 
. . FM 1 . FM 1 2 
. . 
:= 
GB 1 . . 
. GB 1 2 
. . 
. . TSL 1 2 . TSL 1 . . 
������� 
FFH 1 . 
. . 2 . 
������� 
FFH 1 . . 
. 
.......... 
w �� := 72 
dyne 
cm 
. 
�� 
�� := 998 . 
kg 
m3 
w 
��HC := 
2 
:= for i �� 1.. 9999 
........... 
To completely specify the problem, we need the surface tension. Because Brooks and Corey 
(1964) do not specify surface tension or temperature, use an ambient temperature to represent 
laboratory conditions. Input the surface tension of 72 dynes/cm. The water density is set to 998 
kg/m3 (Incropera and DeWitt 1996, p. 846): 
fplot 
i 
si �� 
10000 
VI.2 DATA FITTING 
In the following analysis, curve fits are developed for the four materials separately for 
comparison to the measured retention data. A function representing the van Genuchten function 
is defined. Error functions are defined that for a set of van Genuchten parameters (��, n, Sr) 
provide a summation of the residuals squared between the predicted value for capillary pressure, 
and the measured capillary pressure. After defining the error function, the Mathcad Minerr 
function is used to calculate the set of van Genuchten parameters (��, n, Sr) that minimizes the 
sum of the residuals squared. Define a vector of points for plotting purposes: 
ANL-EBS-MD-000063 REV 00 
s 
VI-5 of VI-16 
(Eq. VI-2) 
(Eq. VI-3) 
August 2003 Advection versus Diffusion in the Invert 
The van Genuchten�s fitting function (van Genuchten 1980, Equation 21) is: 
� s � r 
r 
( � � . )n m 
1 
1 
m 1 . 
n 
Equation VI-5 can be rewritten as follows to solve for the exponent: 
1 n 
n .1 
� � 
� � 
van Genuchten includes the residual saturation in the fit. The van Genuchten m parameter is 
defined in terms of n (van Genuchten 1980, Equation 22): 
r 1
n 
m n 1 . 
1 
1 n . 
Substituting in the value of m into Equation VI-4: 
� s � r 
1 
( � � . )n 
Define the values for saturation: 
� s 
� s 
S s 1 
By definition, the value of Ss at saturation is one: 
� S � s 
. 
. 
r S r � s 
m 
S s 
� 
VI-6 of VI-16 ANL-EBS-MD-000063 REV 00 
(Eq. VI-4) 
(Eq. VI-5) 
(Eq. VI-6) 
(Eq. VI-7) 
(Eq. VI-8) 
August 2003 Advection versus Diffusion in the Invert 
Substitute these definitions into Equation VI-5: 
. S � s S r � s 
Restating the equation: 
. 
Factor out �s from both sides of the equation: 
S Sr 
S Sr 
1 Sr 
1 ( � � . )n 
1 ( � � . )n 
1 ( � � . )n 
1 ( � � . )n 
1 ( � � . )n 
Solve for the value of moisture potential as a function of saturation (S): 
S S n 1 
r 
1 Sr 
. 
1 
n 1 
( � � . )n 
Solving for the moisture potential in terms of the satuation, and defining a function for the 
moisture potential: 
VI-7 of VI-16 ANL-EBS-MD-000063 REV 00 
1 S r � s 
1 
1 
n 
1 
1 
n 
1 
1 
n 
1 
n 1 Sr 
S Sr 
n 1 Sr 
S Sr 
n 
. 
1 ( � � . )n 
1 1 S r 
1 
S S n 1 
r 
1 Sr 
n 
1 
(Eq. VI-9) 
(Eq. VI-10) 
(Eq. VI-11) 
(Eq. VI-12) 
(Eq. VI-13) 
(Eq. VI-14) 
August 2003 Advection versus Diffusion in the Invert 
G VS , Sr , �� . VS   
. 0.1 . . 
��   
.S S . . 
. .
1 S . r . 
.. 
:= 
1 .
. 
G S ( ,S , , n) r �� 
i 2 , 
. �� . := . 0.02 . 
6 . 
.. 
. n . . 
Given 
Define a function in terms of the van Genuchten parameters that represents the sum of the 
residuals squared between the measured capillary pressure and the predicted capillary pressure 
Use the Minerr function to obtain the lease squares fit to the volcanic sand data: 
error (Sr �� , , n) 0 (Eq. VI-18) 
.. 
:= ( Minerr Sr �� , , n) 
. 
.. 
i = 1 
Define an initial estimate of the parameters: 
VS 
VS 
. VS 
0.156 . 
rows VS ( ) 
�� .( ( 
. Sr 
�� 
n
. 
. Sr . . 
.
.
. 
. SrVS .. 
. 
.. 
4.413 . 
.
.
. 
VI.2.1 Volcanic Sand 
for the volcanic sand: 
error (Sr , �� , n) := 
ANL-EBS-MD-000063 REV 00 
. �� VS = . 0.021 . 
. 
n . VS 
VI-8 of VI-16 
1 
n 
1 n . 
r 
(Eq. VI-15) 
. n 
. 
. 
. 1 
, n) )2. i 1 . , 
(Eq. VI-16) 
(Eq. VI-17) 
(Eq. VI-19) 
(Eq. VI-20) 
August 2003 Advection versus Diffusion in the Invert 
Output the sum of the residuals squared: 
VI-21) error ( SrVS , �� VS , nVS) = 536.779 (Eq. 
Define a function for plotting purposes: 
(Eq. VI-22) 
VI.2.2 Fine Sand 
Define a function in terms of the van Genuchten parameters that represents the sum of the 
residuals squared between the measured capillary pressure and the predicted capillary pressure 
for the fine sand: 
( G FSi 2 , , Sr , ��, n) . FSi 1 , )2. 
Splot . VS := SrVS + ( 1 SrVS).f plot 
i = 1 
Define an initial estimate of the parameters: 
. Sr . . 0.01 . ..
�� � := . 0.1 � 
. . . n . . 2 . 
Use the Minerr function to obtain the lease squares fit to the fine sand data: 
Given 
error ( Sr �� , , n) 0 
FS . 
rows FS ( ) 
�� . .
( 
. 
. Sr 
. 
. 
(Eq. VI-23) 
(Eq. VI-24) 
(Eq. VI-25) 
. 
. nFS . 
. ( �� FS � := Minerr Sr �� , , n) 
FS . . 0.170 . . 
.. 
Sr 
.
. �� FS � = . 0.010 � 
. nFS . . 5.664 . 
(Eq. VI-26) 
error ( Sr �� , , n) := 
ANL-EBS-MD-000063 REV 00 
. 
VI-9 of VI-16 August 2003 Advection versus Diffusion in the Invert 
Output the sum of the residuals squared: 
error ( SrFS , �� FS , nFS) = 675.283 (Eq. 
Define a function for plotting purposes: 
VI.2.3 Glass Beads 
Define a function in terms of the van Genuchten parameters that represents the sum of the 
residuals squared between the measured capillary pressure and the predicted capillary pressure 
for the glass beads: 
Splot . FS := SrFS + ( 1 S rFS).f plot 
i = 1 
Define an initial estimate of the parameters: 
. Sr . . 0.01 . ..
�� � := . 0.03 � 
. . . n . . 7 . 
Use the Minerr function to obtain the lease squares fit to the glass beads data: 
Given 
rows ( GB) 
�� . .
( 
. Sr 
. 
error ( Sr �� , , n) 0 
GB . 
. 
. 
. nGB . 
. ( �� GB � := Minerr Sr �� , , n) 
GB . .. 
Sr 
error ( Sr �� , , n) := 
ANL-EBS-MD-000063 REV 00 
.. 
nGB . . 8.122966 . 
. �� GB � = . 0.016780 � 
VI-10 of VI-16 
( G GBi 2 , , Sr , ��, n) . GBi 1 , )2. 
. 0.096991 . . 
. 
VI-27) 
(Eq. VI-28) 
(Eq. VI-29) . 
(Eq. VI-30) 
(Eq. VI-31) 
(Eq. VI-32) 
August 2003 Advection versus Diffusion in the Invert 
Output the sum of the residuals squared: 
error ( SrGB , �� GB , nGB) = 2.472 �� 103 
Define a function for plotting purposes: 
Splot . GB := SrGB + ( 1 SrGB).f plot 
i = 1 
Define an initial estimate of the parameters: 
. Sr . . 0.001 . ..
�� � := . 0.01 � 
. . . n . . 3 . 
Use the Minerr function to obtain the lease squares fit to the Touchet silt loam data: 
Given 
(Eq. VI-37) 
rows ( TSL) 
�� . .
( 
. Sr 
. 
error ( Sr �� , , n) 0 
TSL . 
. 
.. 
Sr 
VI.2.4 Touchet Silt Loam 
Define a function in terms of the van Genuchten parameters that represents the sum of the 
residuals squared between the measured capillary pressure and the predicted capillary pressure 
for the Touchet silt loam: 
error ( Sr �� , , n) := 
ANL-EBS-MD-000063 REV 00 
. 
. nTSL . 
. ( �� TSL � := Minerr Sr �� , , n) 
.. 
nTSL . . 5.808 . 
. 
. �� TSL � = . 4.775 �� 10. 3 � 
TSL . . 0.360 . 
VI-11 of VI-16 
( G TSLi 2 , , Sr , ��, n) . TSLi 1 , )2. 
. 
(Eq. VI-33) 
(Eq. VI-34) 
(Eq. VI-35) . 
(Eq. VI-36) 
(Eq. VI-38) 
(Eq. VI-39) 
August 2003 Advection versus Diffusion in the Invert 
Output the sum of the residuals squared: 
error ( SrTSL , �� TSL , nTSL) = 1.872 �~ 103 
Define a function for plotting purposes: 
Splot . TSL := SrTSL + (1 S rTSL).fplot 
The data sets have now been fitted individually. The residual saturation (Sr) for each set is 
inferred from the data. The results of the analysis are presented in Figure VI-1. 
1 
0.5 
0 
0 
VI.2.5 Nondimensional Capillary Pressure Correlation Using Leverett's Nondimensional 
Group 
The next analysis applies Leverett's nondimensional group (Leverett 1941, p. 159) to the four 
data sets. It collapses the data reasonably well, with the volcanic sand providing the major 
deviation from the group. It compares favorably with Leverett's drainage curve for clean 
unconsolidated sands (Leverett 1941, Figure 4). 
N. D. Pc of Unconsolidated Samples 
1.5 
0.3 0.2 0.1 
Capillary Rise (cm) 
0.9 0.8 0.7 0.6 0.5 0.4 1 
August 2003 
Volcanic Sand 
Fine Sand 
Glass Beads 
Touchet Silt Loam 
. 
Figure VI-1. Capillary Rise of Unconsolidated Samples 
ANL-EBS-MD-000063 REV 00 VI-12 of VI-16 
(Eq. VI-40) 
(Eq. VI-41) 
Saturation Advection versus Diffusion in the Invert 
Composite Set 
Define the function using the := notation for assigning a function: 
G2 ( S ) 
G2 
Define the nondimensional parameter for analysis (Leverett 1941, p. 159). 
Define a function in terms of the dimensionless capillary pressure and the nondimensional van 
Genuchten parameters that represents the sum of the residuals squared between the measured 
dimensionless capillary pressure and the predicted dimensionless capillary pressure for all data 
sets: 
��w 
�� 
Pc 
error(�� , n) := 
... 
1 
eff �� , , n := . eff S 
VS 
. 
�� 
.  
. 
1 = i 
1 S . r 
, 
FS �� + 
.. 
1 = i 
G2 �� , , n . 
. 
G2 
, �� , n 
.. 
. 
, FSi 1 . 
�� , , n . 
. .. 
. 
. . GB .cm.�� .g k 
. 
TSL , i 1 TSLi 2 . SrTSL 
TSL 
. 
1 S . r 
, 
GB 1 S . r 
, 
1 S . r 
..= i 
rows ( TSL) 
�� + 
. 
... 
.  
.  
rows ( GB) 
�� + 
.. 
.. 
. GBi 2 . SrGB 
G2 .  
.  
.  
.  
. . . 
. 
G2 .. 
. 
. 
.
. , . rows ( ) VS . . VSi 2 SrVS 
rows ( ) FS . . FSi 2 SrFS 
i ( ) := 1 .. rows VS 
Define an initial estimate of the parameters: 
. �� 
n 
... 
1 = i 
1 
ANL-EBS-MD-000063 REV 00 VI-13 of VI-16 
.. 
. 
1 
n n 
1 n . 
. 
. 1 
k 
. 
�� 
, VSi 1
�� 
.. 
. 
. �� 
i 1 , 
�� 
. 
.cm.�� .g 
. 
w 
.cm.�� .g 
. 
w 
w 
.. 
.. 
. .. 
�� , , n . 
�� 
.cm.�� .g 
. 
w . 
:= 
. 4 . 
. 
2.3 . . 
(Eq. VI-42) 
(Eq. VI-43) 
2 
kVS 
(Eq. VI-44) 
VS �� 
k
�� 
... 
2 
�� 
.
.. 
. 
2
. ... 
. 
.
. 
.. 
.. .
. 
FS 
FS 
GB 
GB 
. . 
. . 
. 
.. 
.. 
.
.
.. 
.
. 
.. 
kTSL 
�� TSL 
. ... 
. 
.
. 
2 
. 
.. 
.
. 
(Eq. VI-45) 
August 2003 Advection versus Diffusion in the Invert 
Use the Minerr function to obtain the lease squares fit to the volcanic sand data: 
Given 
error(�\ , n) 0 (Eq. VI-46) 
. 
(Eq. VI-47) := Minerr(�\ , n) 
. 
. �\ . comp 
. n . comp � 
comp . 2.455 . 
= (Eq. VI-48) 
8.013 � 
..
n . comp � 
. �\ .. 
Output the sum of the residuals squared: 
j ( ) := 1 .. rows FS 
VI-49) 
k ( := 1 .. rows GB) 
( error �\ comp , ncomp ) = 4.054 (Eq. 
Define a function for plotting purposes: 
Splotcomp := fplot 
(Eq. VI-50) l ( := 1 .. rows TSL) 
The results of the analysis are presented in Figure VI-2. 
VI-14 of VI-16 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
0.3 0.2 0.1 
0 
0 
Saturation 
Volcanic Sand 
Fine Sand 
Glass Beads 
Touchet Silt Loam 
. 
Figure VI-2. Nondimensional Capillary Pressure for Unconsolidated Samples 
August 2003 VI-15 of VI-16 ANL-EBS-MD-000063 REV 00 
Capillary Rise (cm) 
N. D. Pc of Unconsolidated Samples 
1.5 
1 
0.5 
1 0.9 0.8 0.7 0.6 0.5 0.4 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 VI-16 of VI-16 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT VII 
VERIFICATION OF THE MATHCAD CALCULATIONS USING MICROSOFT EXCEL 
August 2003 VII-1 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 VII-2 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment VII - Verification of the Mathcad Calculations Using Microsoft Excel 
This attachment presents an independent verification of the Mathcad 2001 Professional error 
functions presented in Attachment VI that are used for curve fitting to volcanic sand, fine sand, 
glass beads, and Touchet silt loam individually, and then collectively by transforming the 
capillary rise data to a nondimensional capillary rise. Microsoft Excel is used to perform curve 
fitting to the van Genuchten retention relationship. The Solver is an add-in function in Microsoft 
Excel. The Solver can minimize a target cell that involves multiple cell variables that might be 
subject to multiple constraints. The Solver is used specifically to solve for several variables 
under the constraint for a target value. In this case, the minimization of the least squares of the 
capillary rise is the target value for curve fitting. Tables VII-1 through VII-4 present individual 
curve fits to the data for volcanic sand, fine sand, glass beads, and Touchet silt loam from the 
measurements by Brooks and Corey (1964, Appendix 3, Table 1), as presented in Attachment 
VI. The results of the Microsoft Excel analysis for these materials are in agreement with the 
Mathcad 2001 Professional results presented in Attachment VI. 
The nondimensional curve fit presented in Attachment VI requires identification of the porosity 
and permeability of each of these materials (Attachment VI, Figure VI-2), and the properties of 
water. Table VII-5 presents these properties (Incropera and DeWitt 1996, p. 846). The van 
Genuchten curve fit parameters are presented in Table VII-6. Table VII-7 presents the 
nondimensionalized data used to determine the van Genuchten parameters. 
August 2003 VII-3 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table VII-1a. van Genuchten Curve Fit Parameters for Volcanic Sand 
Alpha (�) 
Parameter Value Units 
Saturation at Complete Saturation (no units) 1.000 
Residual Saturation (Sr) (no units) 0.156 
0.021 bars-1 
n 4.413 (no units) 
m 0.773 (no units) 
NOTES: The sum of the residuals is 5.37E+02. 
Table VII-1b. Retention Analysis for Volcanic Sand 
Predicted 
Capillary 2 
Rise 
Residuals 
Squared 
1.89E+01 2.60E+01 
2.04E+01 4.33E+01 
2.22E+01 4.65E+01 
2.36E+01 5.50E+01 
2.78E+01 1.74E+01 
3.32E+01 5.95E-01 
3.48E+01 1.73E-01 
4.92E+01 5.24E+01 
5.95E+01 9.90E+01 
8.41E+01 1.06E+02 
1.26E+02 9.03E+01 
2.73E+02 5.61E-02 
Saturation 
2 
Capillary Rise 1 
(cm) 
0.99 24 
0.986 27 
0.98 29 
0.974 31 
0.948 32 
0.895 34 
0.875 34.4 
0.638 42 
0.479 49.6 
0.277 73.8 
0.188 135.4 
0.158 273.2 
NOTES: 1The capillary pressure is determined from 
Equation 6-22. 
The capillary rise is determined from the 
minimization of the least squares using the 
Microsoft Excel Solver as explained in the text. 
August 2003 VII-4 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table VII-2a. van Genuchten Curve Fit Parameters for Fine Sand 
Alpha (�) 
Parameter Value Units 
Saturation at Complete Saturation (no units) 1.000 
Residual Saturation (Sr) (no units) 0.170 
0.010 bars-1 
n 5.664 (no units) 
m 0.823 (no units) 
NOTES: The sum of the residuals is 6.75E+02. 
Table VII-2b. Retention Analysis for Fine Sand 
Predicted 
Capillary 
Rise 2 
Residuals 
Squared 
4.53E+01 3.90E+02 
5.14E+01 1.79E+01 
5.78E+01 1.46E+01 
6.08E+01 7.66E+00 
6.56E+01 1.61E+01 
6.95E+01 1.67E+01 
7.53E+01 1.84E+01 
8.22E+01 1.17E+01 
8.87E+01 8.50E+00 
9.64E+01 1.41E+00 
1.08E+02 6.12E+00 
1.21E+02 3.66E+01 
1.36E+02 3.79E+01 
1.47E+02 1.31E+01 
1.51E+02 4.48E+00 
1.76E+02 7.44E+01 
3.00E+02 4.31E-02 
Capillary Rise 1 
(cm) 
0.99 25.6 
0.98 55.6 
0.962 61.6 
0.95 63.6 
0.926 69.6 
0.901 73.6 
0.855 79.6 
0.788 85.6 
0.716 91.6 
0.627 97.6 
0.503 105.6 
0.393 115.4 
0.314 129.6 
0.273 143.4 
0.262 148.8 
0.217 184.2 
0.174 300.2 
NOTES: 1 The capillary pressure is determined from 
Equation 6-22. 
The capillary rise is determined from the 
minimization of the least squares using the 
Microsoft Excel Solver as explained in the text. 
Saturation 
2 
August 2003 VII-5 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table VII-3a. van Genuchten Curve Fit Parameters for Glass Beads 
Alpha (�) 
Parameter Value Units 
Saturation at Complete Saturation (no units) 1.000 
Residual Saturation (Sr) (no units) 0.096 
0.016 bars-1 
n 10.271 (no units) 
m 0.903 (no units) 
NOTES: The sum of the residuals is 1.33E+03. 
Table VII-3b. Retention Analysis for Glass Beads 
Predicted 
Capillary 
Rise 2 
Residuals 
Squared 
3.71E+01 6.40E+02 
4.01E+01 2.72E+02 
4.13E+01 3.28E+01 
4.25E+01 2.58E+01 
4.41E+01 9.34E+01 
4.77E+01 9.77E+01 
4.95E+01 8.23E+01 
5.57E+01 2.64E+01 
5.81E+01 1.52E+01 
6.09E+01 1.09E+01 
6.41E+01 1.69E+00 
6.84E+01 4.04E-01 
7.15E+01 1.96E-02 
7.71E+01 8.57E-01 
8.65E+01 1.21E+00 
1.13E+02 3.21E+01 
1.27E+02 3.03E+04 
Capillary Rise 1 
(cm) 
0.995 11.8 
0.989 23.6 
0.985 35.6 
0.98 47.6 
0.971 53.8 
0.938 57.6 
0.912 58.6 
0.764 60.8 
0.681 62 
0.579 64.2 
0.465 65.4 
0.337 67.8 
0.269 71.4 
0.19 78 
0.13 87.6 
0.099 107 
0.097 300.8 
NOTES: 1 The capillary pressure is determined from 
Equation 6-22. 
The capillary rise is determined from the 
minimization of the least squares using the 
Microsoft Excel Solver as explained in the text. 
Saturation 
2 
August 2003 VII-6 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
TableVII-4a. van Genuchten Curve Fit Parameters for Touchet Silt Loam 
Parameter Value 
Saturation at Complete Saturation 1.000 
Residual Saturation(Sr) 0.360 
Alpha (�) (1/cm) 4.775E-03 
n 5.808 
m 0.828 
TableVII-4a. van Genuchten Retention Analysis Results 
Residuals 
Squared 
Predicted 
Capillary Rise 2 Saturation 
2 
Capillary Rise 1 
(cm) 
0.998 65.6 8.02E+01 2.13E+02 
0.995 85.6 9.40E+01 7.02E+01 
0.992 105.6 1.02E+02 1.30E+01 
0.984 125.6 1.15E+02 1.08E+02 
0.978 135.6 1.22E+02 1.87E+02 
0.967 145 1.31E+02 1.91E+02 
0.946 155.6 1.44E+02 1.41E+02 
0.892 164.6 1.65E+02 1.49E-01 
0.821 175.4 1.85E+02 9.21E+01 
0.719 195.6 2.10E+02 2.03E+02 
0.641 215.2 2.30E+02 2.07E+02 
0.562 246 2.54E+02 5.65E+01 
0.492 285.2 2.83E+02 4.63E+00 
0.424 354 3.35E+02 3.67E+02 
0.383 414.4 4.19E+02 1.83E+01 
NOTES: 1 The capillary pressure is determined from 
Equation 6-22. 
The capillary rise is determined from the 
minimization of the least squares using the 
Microsoft Excel Solver as explained in the text. 
Table VII-5. Properties Used in the Analysis 
Unit 
72 dynes/cm 
Parameter Symbol Value 
998 kg/m3 Water Density 
Surface Tension 
� 
Acceleration g 9.802 m/sec2 
�w 
Table VII-6. van Genuchten Curve Fit Parameters 
Parameter Symbol Value 
Saturation at Complete Saturation 1.000 
van Genuchten Alpha (�) (1/cm) � 2.459E+00 
n n 8.013 
m m 0.875 
August 2003 VII-7 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Table VII-7. Nondimensional Retention Analysis 
Capillary 
Effective Rise(cm) 
Material1 Saturation Saturation2 Hydrocarbon 
1.20E+01 2.40E+01 2.34E-01 Verification 
9.90E-01 9.88E-01 1.20E+01 
9.86E-01 9.83E-01 1.35E+01 
9.80E-01 9.76E-01 1.45E+01 
9.74E-01 9.69E-01 1.55E+01 
9.48E-01 9.38E-01 1.60E+01 
Fine Sand 
8.75E-01 8.52E-01 1.72E+01 
6.38E-01 5.71E-01 2.10E+01 
4.79E-01 3.83E-01 2.48E+01 
2.77E-01 1.43E-01 3.69E+01 
1.88E-01 3.79E-02 6.77E+01 
1.58E-01 2.37E-03 1.37E+02 
9.90E-01 9.88E-01 1.28E+01 
9.80E-01 9.76E-01 2.78E+01 
9.62E-01 9.54E-01 3.08E+01 
9.50E-01 9.40E-01 3.18E+01 
9.26E-01 9.11E-01 3.48E+01 
9.01E-01 8.81E-01 3.68E+01 
8.55E-01 8.25E-01 3.98E+01 
7.88E-01 7.45E-01 4.28E+01 
7.16E-01 6.58E-01 4.58E+01 
6.27E-01 5.51E-01 4.88E+01 
5.03E-01 4.01E-01 5.28E+01 
3.93E-01 2.69E-01 5.77E+01 
3.14E-01 1.73E-01 6.48E+01 
2.73E-01 1.24E-01 7.17E+01 
2.62E-01 1.11E-01 7.44E+01 
2.17E-01 5.66E-02 9.21E+01 
1.74E-01 4.82E-03 1.50E+02 
Volcanic Sand 
8.95E-01 8.76E-01 1.70E+01 3.40E+01 3.31E-01 3.25E-01 3.89E-05 
3.44E+01 3.35E-01 3.33E-01 3.13E-06 
4.20E+01 4.09E-01 4.01E-01 5.48E-05 
4.96E+01 4.83E-01 4.43E-01 1.54E-03 
7.38E+01 7.18E-01 5.29E-01 3.58E-02 
1.35E+02 1.32E+00 6.47E-01 4.50E-01 
2.73E+02 2.66E+00 9.63E-01 2.87E+00 
2.56E+01 8.96E-02 2.39E-01 2.22E-02 
5.56E+01 1.95E-01 2.61E-01 4.37E-03 
6.16E+01 2.16E-01 2.83E-01 4.58E-03 
6.36E+01 2.23E-01 2.94E-01 5.06E-03 
6.96E+01 2.44E-01 3.10E-01 4.38E-03 
7.36E+01 2.58E-01 3.23E-01 4.24E-03 
7.96E+01 2.78E-01 3.41E-01 3.95E-03 
8.56E+01 2.99E-01 3.63E-01 4.02E-03 
9.16E+01 3.20E-01 3.83E-01 3.87E-03 
9.76E+01 3.41E-01 4.06E-01 4.11E-03 
1.06E+02 3.69E-01 4.39E-01 4.82E-03 
1.15E+02 4.04E-01 4.75E-01 5.13E-03 
1.30E+02 4.53E-01 5.13E-01 3.52E-03 
1.43E+02 5.02E-01 5.41E-01 1.55E-03 
1.49E+02 5.21E-01 5.51E-01 9.09E-04 
1.84E+02 6.44E-01 6.10E-01 1.21E-03 
3.00E+02 1.05E+00 8.70E-01 3.25E-02 
VII-8 of VII-10 ANL-EBS-MD-000063 REV 00 
Capillary Nondimensional 
Capillary 
Rise 
Rise(cm) 
Water3 Capillary Rise 
Predicted 
Nondimensional Residuals 
Squared 
2.40E+01 2.34E-01 2.38E-01 2.14E-05 
2.70E+01 2.63E-01 2.49E-01 2.01E-04 
2.90E+01 2.82E-01 2.60E-01 4.88E-04 
3.10E+01 3.02E-01 2.69E-01 1.06E-03 
3.20E+01 3.11E-01 2.95E-01 2.82E-04 
August 2003 Advection versus Diffusion in the Invert 
Capillary 
Effective Rise(cm) 
Material1 Saturation Saturation2 Hydrocarbon 
Glass Beads 
Touchet 
Silt Loam 
Sum of Residuals 
NOTES: 1 See Attachment V for the data. 
2 The effective saturation is defined as the ratio (S-Sr)(1-Sr) 
3 According to Brooks and Corey (1964, Equation 17, p. 9), the capillary rise of water was about twice 
that of the hydrocarbon used in the measurements. 
4 Equation 4.22 is applied the the capillary rise (cm) to convert to a nondimensional capillary rise. 
9.95E-01 9.94E-01 5.90E+00 
9.89E-01 9.88E-01 1.18E+01 
9.85E-01 9.83E-01 1.78E+01 
9.80E-01 9.78E-01 2.38E+01 
9.71E-01 9.68E-01 2.69E+01 
9.38E-01 9.31E-01 2.88E+01 
9.12E-01 9.03E-01 2.93E+01 
7.64E-01 7.39E-01 3.04E+01 
6.81E-01 6.47E-01 3.10E+01 
5.79E-01 5.34E-01 3.21E+01 
4.65E-01 4.08E-01 3.27E+01 
3.37E-01 2.66E-01 3.39E+01 
2.69E-01 1.90E-01 3.57E+01 
1.90E-01 1.03E-01 3.90E+01 
1.30E-01 3.66E-02 4.38E+01 
9.90E-02 2.22E-03 5.35E+01 
9.70E-02 9.97E-06 1.50E+02 
9.98E-01 9.97E-01 3.28E+01 
9.95E-01 9.92E-01 4.28E+01 
9.92E-01 9.88E-01 5.28E+01 
9.84E-01 9.75E-01 6.28E+01 
9.78E-01 9.66E-01 6.78E+01 
9.67E-01 9.48E-01 7.26E+01 
9.46E-01 9.16E-01 7.78E+01 
8.92E-01 8.31E-01 8.23E+01 
8.21E-01 7.20E-01 8.77E+01 
7.19E-01 5.61E-01 9.78E+01 
6.41E-01 4.39E-01 1.08E+02 
5.62E-01 3.16E-01 1.23E+02 
4.92E-01 2.06E-01 1.43E+02 
4.24E-01 1.00E-01 1.77E+02 
3.83E-01 3.59E-02 2.07E+02 
ANL-EBS-MD-000063 REV 00 
Table VII-7. Nondimensional Retention Analysis (Continued) 
VII-9 of VII-10 
Capillary Nondimensional 
Capillary 
Rise 
Rise(cm) 
Water3 Capillary Rise 
Predicted 
Nondimensional Residuals 
Squared 
1.18E+01 6.62E-02 2.16E-01 2.26E-02 
2.36E+01 1.32E-01 2.39E-01 1.14E-02 
3.56E+01 2.00E-01 2.49E-01 2.40E-03 
4.76E+01 2.67E-01 2.58E-01 8.16E-05 
5.38E+01 3.02E-01 2.70E-01 9.72E-04 
5.76E+01 3.23E-01 2.99E-01 5.78E-04 
5.86E+01 3.29E-01 3.14E-01 2.25E-04 
6.08E+01 3.41E-01 3.64E-01 5.49E-04 
6.20E+01 3.48E-01 3.85E-01 1.41E-03 
6.42E+01 3.60E-01 4.09E-01 2.42E-03 
6.54E+01 3.67E-01 4.37E-01 5.00E-03 
6.78E+01 3.80E-01 4.76E-01 9.26E-03 
7.14E+01 4.00E-01 5.05E-01 1.09E-02 
7.80E+01 4.37E-01 5.57E-01 1.43E-02 
8.76E+01 4.91E-01 6.50E-01 2.53E-02 
1.07E+02 6.00E-01 9.72E-01 1.38E-01 
3.01E+02 1.69E+00 2.10E+00 1.72E-01 
6.56E+01 9.91E-02 2.01E-01 1.05E-02 
8.56E+01 1.29E-01 2.26E-01 9.33E-03 
1.06E+02 1.60E-01 2.40E-01 6.43E-03 
1.26E+02 1.90E-01 2.62E-01 5.19E-03 
1.36E+02 2.05E-01 2.73E-01 4.61E-03 
1.45E+02 2.19E-01 2.88E-01 4.66E-03 
1.56E+02 2.35E-01 3.07E-01 5.22E-03 
1.65E+02 2.49E-01 3.40E-01 8.24E-03 
1.75E+02 2.65E-01 3.69E-01 1.07E-02 
1.96E+02 2.96E-01 4.03E-01 1.16E-02 
2.15E+02 3.25E-01 4.30E-01 1.10E-02 
2.46E+02 3.72E-01 4.61E-01 7.99E-03 
2.85E+02 4.31E-01 4.98E-01 4.51E-03 
3.54E+02 5.35E-01 5.60E-01 6.07E-04 
4.14E+02 6.26E-01 6.52E-01 6.51E-04 
3.99E+00 
August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 VII-10 of VII-10 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT VIII 
HYDROLOGIC PROPERTIES OF ENGINEERED BARRIER SYSTEM COMPONENTS 
USED FOR NUFT RUNS 
VIII-1 of VIII-4 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 VIII-2 of VIII-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment VIII - Hydrologic Properties of Engineered Barrier System 
(�) 
0.0 
Drip Shield 0.0 0.0 
Air 0.990 1e-08 1.0 
Table VIII-2. Thermal Properties of Engineered Barrier System Components 
Thermal Conductivity 
(W/(m.K)) 
Components Used for NUFT Runs 
Tables VIII-1 and VIII-2 present the hydrologic and thermal properties of the waste package, the 
drip shield, and the drift air required to run the NUFT model. Since this analysis evaluates 
advection versus diffusion in the invert at ambient temperature, in the absence of drift seepage, 
these inputs serve only as placeholders necessary to initiate the NUFT simulations and do not 
impact the non-thermal analysis presented in this document. 
Table VIII-1. Hydrologic Properties of Engineered Barrier System Components 
Porosity 
Component 
Waste Package 
Permeability 
(m2) Tortuosity Factor 
0.0 0.0 
0.0 
Specific Heat Capacity 
(J/(kg.K) 
VIII-3 of VIII-4 
Component 
Waste Package1 
Drip Shield2 
Air3 
Source: BSC 2001d, Table 5-10, 5-11, 5-14, and 5-15 
NOTES: 1 Values are for waste package at 46�C made from Alloy 22. 
2 Values are for drip shield at 100�C made from titanium Grade 7. 
3 Values are from Incropera and DeWitt 1996. 
ANL-EBS-MD-000063 REV 00 
Mass Density 
(kg/m3) 
10.1 414.0 8690 
20.708 540.82 4512.00 
0.0263 1.007 1.61 
August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 VIII-4 of VIII-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT IX 
COMPARISON OF TUFF MATRIX HYDROLOGIC PROPERTIES 
August 2003 IX-1 of IX-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 IX-2 of IX-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
The information in this attachment comes from DTN: LB990861233129.001 and 
DTN: LB0207REVUZPRP.002. 
Attachment IX - Comparison of Tuff Matrix Hydrologic Properties 
The tuff matrix hydrologic properties for TSw35 and TSw36 inputs used in this analysis 
(DTN: LB990861233129.001) are compared to more recent data 
(DTN: LB0207REVUZPRP.002). Although there are some differences in moisture potential at 
low and intermediate saturations, the moisture potential at high saturations of interest for this 
analysis, are equivalent. 
Table IX-1. Comparison of Tuff Matrix Hydrologic Properties for TSw35 and TSw36. 
(��matrix) 
Porosity of the rock matrix in an individual grain 
Full Saturation * (Ss) 
Residual Saturation (Sr) 
3Saturated Volumetric Moisture Content (��s) 
Residual Volumetric Moisture Content (��r) 
van Genuchten Air-Entry Parameter (��) 1/Pa 
van Genuchten Parameter (m) 
5van Genuchten Parameter (n) 
Intrinsic Permeability (k) m2 
NOTES: 1 DTN: LB990861233129.001. 
Parameter TSw36 TSw351 TSw352 
2 
3 
4 
5 
1 TSw36 
(Tptpln) 
0.112 0.131 0.131 0.103 
1.0 1.0 1.0 1.0 
0.18 0.12 0.12 0.20 
0.112 0.131 0.131 0.103 
0.02016 0.0157 0.0157 0.0206 
3.55E-6 6.44E-6 1.66E-05 2.84�~10-7 
0.380 0.236 0.216 0.442 
1.612 1.309E-9 1.280 1.750 
5.71E-18 3.04E-17 3.7E-17 2.3�~10-20 
DTN: LB0207REVUZPRP.002. Values reported in Calibrated Properties Model (BSC 2003c). 
��s is a calculated value. See Section 6. 
��r is a calculated value. See Section 6. 
The value of n is given by 1/(1-m) (Fetter 1993, p. 172). 
ANL-EBS-MD-000063 REV 00 August 2003 IX-3 of IX-4 Advection versus Diffusion in the Invert 
Figure IX-1. Retention Properties for Welded Tuff 
DTN: LB990861233129.001 
LB0207REVUZPRP.002 
August 2003 IX-4 of IX-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT X 
LIQUID FLUX PATTERNS IN MATRIX COMPONENT OF INVERT AND ROCK 
MATRIX AND IN THE INTERGRANULAR COMPONENT OF INVERT AND ROCK 
FRACTURES FOR THE CAMPBELL AND VAN GENUCHTEN GRAIN SIZE 
RETENTION RELATIONS 
August 2003 X-1 of X-22 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 X-2 of X-22 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
The purpose of the NUFT analyses presented in this attachment is to provide a detailed 
calculation of the occurrence of water and advection in the unsaturated media of the invert. The 
NUFT calculation provides an analysis of advection and the retention of water in both the 
intergranular and intragranular pore space of the invert. These calculations show that for the 
properties adopted for analysis for the moisture potential at the repository horizon that the fine 
pore space within the crushed tuff grains retains water while the coarse pore space between the 
particles of crushed tuff is free of water. Figures X-1 through X-18 corroborate the concept of 
one-dimensional transport and shows where water flows in the invert. The colors show different 
zones of flow. Note that the arrows show the general direction of flow but not magnitude. Also 
note that flow does not occur in the open void space of the drift, and that this is an artifact of the 
XTOOL plotting routine. 
Ground 
Surface 
Approximate 
Location of 
Drift 
Model Line of 
Symmetry 
Water Table 
Figure X-1. NUFT Simulation Grid 
August 2003 X-3 of X-22 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Figure X-2a. Liquid Flux Pattern in Rock Fractures Around the Drift 
ANL-EBS-MD-000063 REV 00 
Drift wall 
Approximate 
Boundary of 
Drip Shield 
Approximate 
Boundary of 
Waste Package 
Invert 
Boundary 
August 2003 X-4 of X-22 Advection versus Diffusion in the Invert 
Figure X-2b. Liquid Flux Pattern in Rock Matrix Around the Drift 
ANL-EBS-MD-000063 REV 00 
Drift wall 
Approximate 
Boundary of 
Drip Shield 
Approximate 
Boundary of 
Waste Package 
Invert 
Boundary 
August 2003 X-5 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-3. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the van Genuchten 0.317-mm Grain Size Retention 
Relation 
X-6 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-4. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the van Genuchten 3-mm Grain Size Retention 
Relation 
X-7 of X-22 Advection versus Diffusion in the Invert 
Invert 
Boundary 
Line of 
Symmetry 
ANL-EBS-MD-000063 REV 00 August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-5. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the van Genuchten 0.317-mm Grain Size 
X-8 of X-22 Advection versus Diffusion in the Invert 
Invert 
Boundary 
Line of 
Symmetry 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-6. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the van Genuchten 3-mm Grain Size Retention Relation 
ANL-EBS-MD-000063 REV 00 August 2003 X-9 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-7. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the van Genuchten 10-mm 
Grain Size Retention Relation 
X-10 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-8. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the van Genuchten 10-mm Grain Size Retention Relation 
X-11 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-9. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the van Genuchten 20-mm Grain Size Retention 
Relation 
X-12 of X-22 Advection versus Diffusion in the Invert 
Invert Boundary 
Line of 
Symmetry 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-10. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the van Genuchten 20-mm Grain Size Retention Relation 
August 2003 X-13 of X-22 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Invert 
Boundary 
Line of 
Symmetry 
ANL-EBS-MD-000063 REV 00 August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-11. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the Campbell 0.317-mm Grain 
Size Retention Relation 
X-14 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-12. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the Campbell 0.317-mm Grain Size Retention Relation 
X-15 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-13. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the Campbell 3-mm Grain Size Retention Relation 
X-16 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-14. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the Campbell 3-mm Grain Size Retention Relation 
X-17 of X-22 Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 X-18 of X-22 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-15. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the Campbell 10-mm Grain Size Retention Relation Advection versus Diffusion in the Invert 
ANL-EBS-MD-000063 REV 00 
Invert 
Boundary 
Line of 
Symmetry 
August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-16. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the Campbell 10-mm Grain Size Retention Relation 
X-19 of X-22 Advection versus Diffusion in the Invert 
Invert 
Boundary 
Line of 
Symmetr 
ANL-EBS-MD-000063 REV 00 August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-17. Liquid Flux Pattern in Intergranular Component of Invert and Rock Fractures for the Campbell 20-mm Grain Size Retention Relation 
X-20 of X-22 Advection versus Diffusion in the Invert 
Invert 
Boundary 
Line of 
Symmetry 
ANL-EBS-MD-000063 REV 00 August 2003 
NOTE: Arrows show general flow directions, but do not represent magnitude of flow. 
Figure X-18. Liquid Flux Pattern in Matrix Component of Invert and Rock Matrix for the Campbell 20-mm Grain Size Retention Relation 
X-21 of X-22 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 X-22 of X-22 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT XI 
HYDROLOGIC AND THERMAL PROPERTIES OF THE INVERT 
August 2003 XI-1 of XI-12 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XI-2 of XI-12 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment XI - Hydrologic and Thermal Properties of the Invert 
Crushed tuff is selected for the invert to provide geochemical compatibility with the surrounding 
host rock. The basis for the selection of the crushed is that the material provides diffusionbarrier 
performance when transport from the waste package to the rock floor is diffusion 
dominated. This could occur if a waste package is breached but the protecting drip shield is 
intact, so that the invert ballast material immediately below the drip shield is unsaturated and 
protected from advective flow from other engineered barrier components. 
Crushed welded tuff sieved between 2.0 and 4.75 mm (Section 6.3) has been selected for pilot 
testing and the properties are described below for this material. The final design may require a 
different size distribution or material type, or both. 
XI.1 BULK DENSITY AND POROSITY 
The invert material is crushed tuff from the Tptpll lithostratigraphic unit that is part of the TSw2 
thermal/mechanical unit. The repository host horizon is located mainly in the TSw2 unit. The 
invert material hydrological properties are presently unavailable for the Tptpll formation. It is 
valid to substitute the Tptpmn properties in place of Tptpll values because they are both part of 
the TSw2 thermal/mechanical unit. The matrix porosities of these materials are similar , and this 
would mean the crushed materials would have similar retention characteristics. 
The USGS measured the bulk density, water retention, and unsaturated hydraulic conductivity. 
These properties were measured in conjunction with the UFA measurements as described 
subsequently. The hydrologic and geotechnical properties for the crushed tuff are taken from 
DTN: GS980808312242.015. These data sets are illustrated in Figures XI-1 and XI-2. The 
curve fits to the retention data shown in these figures are presented in Attachment XVI. 
XI-3 of XI-12 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
1 .10 6 
1 .10 5 
1 .10 4 
1 .10 3 
100 
10 
0.2 
1 
0 
Volumetric Moisture Content 
Curve fit 
Data 2-4.75 mm 
NOTE: See Attachment XVI for the curve fit. 
Figure XI-1. Moisture Retention Relationship for the Invert 
1 
0.1 
0.01 
1 10 3 
1 10 4 
1 10 5 
1 10 6 
1 10 7 
1 10 8 
1 10 9 
1 10 10 
1 10 11 
0 
Data 
Curve Fit 
/s) Hydraulic Conductivity (cm 
0.6 0.4 0.2 
Volumetric Moisture Content 
0.8 
Figure XI-2. Unsaturated Hydraulic Conductivity versus Volumetric Moisture Content for the Invert 
XI-4 of XI-12 ANL-EBS-MD-000063 REV 00 August 2003 
Moisture Potential (cm) 
0.6 0.4 0.8
DTN: GS980808312242.015 
DTN: GS980808312242.015 Advection versus Diffusion in the Invert 
For materials sieved between 2.00 and 4.75 mm, used for hydraulic conductivity measurements, 
the measured dry bulk density was 1.15 g/cm3, as presented in Water Distribution and Removal 
Model (CRWMS M&O 2001b, Attachment XIV). The grain density is 2.53 gm/cm3 as presented 
in Water Distribution and Removal Model (CRWMS M&O 2001b, Attachment XIV). Calculate 
the porosity using the soil phase convention of setting the volume of the solids (Vs) equal to 1.0 
cm3, developing a formula for the bulk density, and then calculating the volume of the voids. 
The dry bulk density (�) is defined as in the Water Distribution and Removal Model (CRWMS 
M&O 2001b, Attachment XIV): 
� = GsVs/Vt (Eq. XI-1) 
where 
� = Dry bulk mass density (g/cm3) 
Gs = Specific gravity of solids 
Vs = Solids volume (cm3) 
Vt = Total volume (cm3) 
Substituting in for the total volume which is equal to the volume of the solids and volume of the 
voids (Vt = Vs + Vv): 
� = GsVs/(Vs+Vv) (Eq. XI-2) 
where 
Vv = Void volume (cm3) 
Substituting in the values for Gs, �, and Vs: 
1.15 cm3 = 2.53 gm/cm3 (1.0 cm3) / (1.0 cm3 +Vv) (Eq. XI-3) 
Solve for Vv: 
Vv = (2.53/1.15-1.0) cm3 (Eq. XI-4) 
Vv = 1.20 cm3 (Eq. XI-5) 
Solving for the porosity (�): 
(Eq. XI-6) � = 1.20/(1.0 +1.20) = 0.55 
XI.2 MOISTURE RETENTION 
Moisture retention measurements were performed on the crushed tuff using the UFA 
measurements (CRWMS M&O 1996, Appendix C). 
The UFA consists of an ultracentrifuge with a constant ultra low flow pump that provides fluid to 
the sample through a rotating seal assembly and microdispersal system. The volumetric moisture 
content (�) as a function of the moisture potential (�) can be determined by allowing the sample 
to drain until the moisture potential equals the centrifugal force per unit area divided by the unit 
ANL-EBS-MD-000063 REV 00 August 2003 XI-5 of XI-12 Advection versus Diffusion in the Invert 
weight in a state of equilibrium. The sample is then weighed to determine the volumetric 
moisture content (� ). 
The moisture retention data obtained from the two methods can be plotted and a curve fitting 
performed for the retention model based upon the van Genuchten two-parameter model n and m 
(m=1-1/n) (Fetter 1993, p. 172). Define the moisture potential (capillary pressure divided by 
weight density) versus moisture content relation: 
r r s �= [ 1+ ( �. �n ]. m 
. ( � . � ) + � (Eq. XI-7) 
where 
n = van Genuchten curve-fitting parameter 
m = van Genuchten curve-fitting parameter 
� = van Genuchten or exponential curve-fitting parameters (cm-1) 
� = Volumetric moisture con 
� r = Residual volumetric moisture content 
� s = Saturated volumetric moisture content and 
� = Moisture potential (cm of water) 
Substituting the value of (m) into Equation XI-7 for the two-parameter model, gives: 
�= [ 1 + ( �. �n )]. 1 . 
n 
1 
. ( � . � ) + � (Eq. XI-8) r r s 
The estimated curve-fitting parameters are from Attachment XIV of Water Distribution and 
Removal Model (CRWMS M&O 2001b): 
� r = 0.05 
� i = 0.12 (1/cm) 
ni = 2.75 
From the definition of van Genuchten m (m = 1-1/n) given above: 
= 64 . 0 1 . 1
75 . 2 
The residual saturation equals the residual moisture content divided by the porosity 
(0.05/0.55) = 0.091. The satiated saturation is by definition. 
Note that the measurements were performed near the residual moisture saturation. To establish 
the curve at higher moisture contents, the volumetric moisture content at saturation was 
estimated from the porosity, as presented in Water Distribution and Removal Model (CRWMS 
M&O 2001b, Attachment XIV). The volumetric moisture content � s equals the porosity of 0.63 
which corresponds to the loose state. It should be noted that while the UFA testing was 
August 2003 XI-6 of XI-12 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
performed on the crushed tuff in a loose state (�� = 0.63) than what would be anticipated in the 
repository (�� = 0.55) allowing for consolidation over time, the moisture retention scaled to the 
saturation level would not be significantly different. 
XI.3 INTRINSIC PERMEABILITY 
The saturated hydraulic conductivity (Ks) of the invert is estimated from the curve fitting 
analysis presented in Water Distribution and Removal Model (CRWMS M&O 2001b, 
Attachment XV) using the combined UFA unsaturated hydraulic conductivity (Ku) to moisture 
potential (�� ) and retention measurements. The calculated value from Water Distribution and 
Removal Model (CRWMS M&O 2001b, Attachment XV) is 0.60 cm/sec. This value 
corresponds to an approximate intrinsic permeability conversion value of 6.0�� 10-6 cm2 or 
6.0�� 10-10 m2 (Freeze and Cherry 1979, p. 29). 
2 
1 . . 
1. 1 
� .
. 1
1 .. n . 
r � . 
1. 
n . . . 
(Eq. XI-9) 
. 
�� 
. 
. 2 . .. � . . 1 
. �� . . 
��r . �
. . . �� . ��r . �
. . 
1 . .
. s 
. . .. 
.. 
The relative permeability function scales the saturated conductivity (Ks) to allow the unsaturated 
hydraulic conductivity function to be determined as shown in Figure XI-2. 
The wetting-phase relative permeability as a function of moisture potential for this model is 
restated from Fetter (1993 , p. 182, Equation 4.17). The unsaturated hydraulic conductivity 
(wetting-phase relative permeability times saturated hydraulic conductivity) as a function of 
moisture potential is from Water Distribution and Removal Model (CRWMS M&O 2001b, 
Attachment XIV). The relative permeability function scales the saturated conductivity (Ks) to 
allow the unsaturated hydraulic conductivity function to be determined for crushed tuff as shown 
in Figure XI-3. 
XI.4 RELATIVE PERMEABILITY 
The UFA test apparatus described above can be used to determine the relationship between the 
unsaturated hydraulic conductivity (Ku) and volumetric moisture content through a direct 
application of Darcy��s Law (CRWMS M&O 1996, Appendix C). By measuring the flow rates to 
0.001 ml/hr and measuring the effluent collected from the sample in a volumetrically calibrated 
chamber that determines volumetric moisture content (�� ), the unsaturated hydraulic conductivity 
can be determined from the ratio of the flow rate to the centrifugal force per unit volume 
(CRWMS M&O 1996, p. C-2). 
The relationship of the unsaturated hydraulic conductivity with volumetric moisture content is 
given by Jury et al. (1991, p. 109): 
r K . s 
�� . �� 
. ��s 
.... 
where 
K ( ��) = 
ANL-EBS-MD-000063 REV 00 
. 
Ks is the saturated hydraulic conductivity (cm/sec) 
XI-7 of XI-12 August 2003 Advection versus Diffusion in the Invert 
XI.5 THERMAL PROPERTIES 
The rock grain specific heat for crushed tuff is estimated to be 948 J/(kg.K). The specific heat 
for the crushed tuff with a porosity of 0.55 and a bulk density of 1.15 g/cm3 equals the specific 
heat of the grains since specific heat capacity depends on mass which is independent of volume. 
The volumetric heat (Cp) equals the specific heat (Cs) 948 J/(kg.K) times the bulk density (�) 
1.15 g/cm3. The thermal emissivity of the invert is taken equal to the emissivity for quartz on a 
rough surface 0.93 (Holman 1997, p. 649). 
Additional measurements (DTN: GS000483351030.003) of geotechnical and thermal properties 
have been performed to characterize the thermal properties of crushed tuff. Also, it includes 
thermal-property measurements of oven dry samples of crushed tuff using the Thermolink Probe. 
This device uses a dual-probe, short-duration, heat pulse technique to simultaneously measure 
the volumetric specific heat and thermal diffusivities of granular materials. The measurements 
were performed for a �fine� crushed tuff, and �4-10� crushed tuff. The average properties are 
summarized in Table XI-1 for oven dry conditions at ambient temperature. 
Table XI-1. Summary of Thermolink Results for Crushed Tuff 
Thermal 
Diffusivity 
(mm2/s) 
Volumetric 
Specific Heat 
(J/cm3/K) Material 
4-10 Crushed Tuff 0.930 � 0.074 0.16 � 0.01 
Fine Crushed 
Tuff Group 1 
Temperature 
(�C) 
0.175 � 0.014 17.3 � 1.0 
0.153 � 0.005 24.0 � 2.4 
0.159 � 0.007 19.2 � 0.17 
Thermal 
Conductivity 
(W/m/K) 
0.919 � 0.061 0.14 � 0.01 
0.972 � 0.036 0.16 � 0.01 
XI-8 of XI-12 
Fine Crushed 
Tuff Group 2 
DTN: GS000483351030.003 
Crane et al. (1977) compared a number of models to the results of experimental studies. These 
included the model developed by Willhite et al. (1962) and the Dietz model. The Dietz model is 
a Fourier model for thermal conductivity of a packed bed. The Dietz model considered a special 
case of the packed bed-hexagonal array of touching spheres. However, it was found that the 
resulting expression for the effective bed conductivity was only a weak function of bed 
geometry, allowing the expression to be applied to a variety of packings. 
These models were evaluated by comparing the predicted values for thermal conductivity on a 
separate and independent set of data developed by Saxena et al. (1986). Saxena et al. performed 
thermal conductivity measurements on porous materials. Measurements of effective thermal 
conductivity of these materials were made using three different experimental methods via the 
thermal probe method. The thermal probe method reported by Saxena et al. consisted of a line 
heat source method in which a steel hypodermic needle of length 10 cm and outer diameter 0.125 
cm is used as the source and sensor for temperature (Saxena et al. 1986). 
The measured data are regressed against the predicted data as presented in Water Distribution 
and Removal Model (CRWMS M&O 2001b, Attachment XIV) and illustrated in Figure XI-4. 
The plot shows the ratio of the thermal conductivity to the continuous or gas phase thermal 
conductivity for measured data and predicted values for the Dietz model, as presented in Water 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
Distribution and Removal Model (CRWMS M&O 2001b, Attachment XIV). The Dietz model 
was found to produce a better result for this data. 
The results of the analysis on the data by Saxena et al. (1986) over a range of porosities are 
presented in Figure XI-4. Also, Figure XI-4 shows a data point for crushed tuff using the grain 
thermal conductivity for TSw34 as discussed below. 
The Dietz model (Equation XI-11) can be applied to measured data for crushed tuff (TSw34) 
performed by the YMP. The value for the solids phase thermal conductivity for welded tuff is 
given in Water Distribution and Removal Model (CRWMS M&O 2001b, Table 4-5) as 1.56 
W/(m.K). Considering the air thermal conductivity is given by Chapman (1974, p. 593) is 0.029 
W/(m.K) at 60��C, the calculated value for thermal conductivity of crushed tuff is predicted to be 
0.15 W/(m.K) which compares reasonably well with the measured values presented in Table XI1. 
The Dietz model can be used to predict the thermal conductivity under saturated conditions by 
substituting the value of thermal conductivity for water into Equation XI-11. Considering the 
water thermal conductivity given by Chapman (1974, p. 586) at 60��C, the value is 0.65 W/(m.K) 
as determined in Water Distribution and Removal Model (CRWMS M&O 2001b, 
Attachment XIV). Substituting in this value yields a value for thermal conductivity under 
saturated conditions of 1.03 W/(m.K) as determined in Water Distribution and Removal Model 
(CRWMS M&O 2001b, Attachment XIV). 
The volumetric heat capacity under saturated conditions may be estimated by simple volumetric 
averaging. According to Jury et al. (1991, p. 179): 
N 
C = X . C + X . C + w w a a c s j �� Xs j 
. C (Eq. XI-11) 
j=1 
where 
Cc = Average volumetric heat capacity 
Xa = Void fraction of air 
Xw = Void fraction of water 
Xsj = Void fraction of the jth solids component 
Ca = Heat capacity per unit volume of the air 
Cw =Heat capacity per unit volume of the water 
Csj = Heat capacity per unit volume of the jth solids component 
Note that NUFT will calculate the volume averaged specific heat capacity based upon the 
volume fractions and their respective volumetric heats for the solids, water, and air. The 
following calculation is provided for reference only, and illustrates how specific heat, and 
thermal diffusivity would change when the degree of saturation is increased from zero to one. 
August 2003 XI-9 of XI-12 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
1 
0.1 
0.01 
Figure XI-4. Comparison of Experimental Results with Calculated Thermal Conductivity for the Dietz 
Model 
ANL-EBS-MD-000063 REV 00 XI-10 of XI-12 August 2003 
Unsaturated Hydraulic Conductivity(cm/s) 
1 .10 3 
1 .10 4 
1 .10 5 
1 .10 6 
1 .10 7 
1 .10 8 
1 .10 9 
1 .10 10 
1 .10 11 
1 .10 12 
1 
Source: CRWMS M&O 2001b, Attachment XIV 
Figure XI-3. Unsaturated Hydraulic Conductivity versus Moisture Potential for the Invert 
Measured Thermal Conductivity Ratio (-) 
12 
10 
8 
6 
4 
2 
4 2 
Predicted Thermal Conductivity Ratio (-) 
Saxena et al. Data 
Regression Analysis 
4-10 Crushed Tuff 
8 6 12 10 
10 100 
Moisture Potential (cm) 
1 .103
Advection versus Diffusion in the Invert 
Calculate the volumetric heat capacity for air. From Chapman (1974, p. 593), the properties of 
air at 60�� C (140�� F) are given as: 
a = 0 .2409 . BTU 
lb . R 
lb 
Cp
a �� = 0 .06614. 
ft3 
Converting to the SI system of units: 
. a = 1009 . J 
kg . K 
�� = 1 .059 . kg 
a m 3 
C = Cp . ��a a a 
J 
m 3 . K 
30 . 9 s a a a 
Cp 
Calculate the volumetric heat capacity for air: 
C = 1069 .0. a 
Calculate the volumetric capacity of the tuff from Equation X-11 under dry conditions by 
considering 4-10 crushed tuff that has a volumetric heat capacity of 0.930 J/(cm3 K) for TSw34 
(CRWMS M&O 2001b, Table XI-3). The air void fraction is determined from Figure 6-1 from 
the ratio of 1.23/(1+1.23) = 0.55. Converting the value to SI units, the equation to be solved is: 
5 . 10 = C . X + (1. X ) . C (Eq. XI-12 
Solving for Cs, the value of 2.07�~ 106 J/(m3 K) is obtained which is approximately twice the 
value for the porous crushed tuff (9.30�~ 105 J/(m3 K). As noted by Jury et al. (1991, p. 180), the 
volumetric heat capacity of air is small. 
Consider now the properties of water. From Chapman (1974, p. 586) at a temperature of 60�� C 
(140�� F): 
Cpw = 0 .998. BTU 
lb . R 
lb 
w �� = 39 . 61 . 
ft3 
Converting to SI units: 
August 2003 XI-11 of XI-12 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Cpw = 4178 . J 
kg . K 
983 .4 . kg 
w m 3 
6 (Eq. XI-13) 10 . s w w 
J 
m 3 . K 
. 
The volumetric heat capacity is increased by an approximate factor of three when compared to 
the volumetric heat capacity under dry conditions, as presented in Table XI-3. 
From Water Distribution and Removal Model (CRWMS M&O 2001b), the value for the 
saturated thermal conductivity(ksat) is 1.03 W/(m.K). The thermal diffusivity under saturated 
conditions is estimated from the thermal diffusivity relationship (Jury et al. 1991, p. 178): 
The calculated thermal diffusivity under saturated conditions is 0.32 mm2/s. The thermal 
diffusivity for the wet case increased by a factor of two when compared to the values for the dry 
case in Table XI-3. 
�� = 
The calculated volumetric heat capacity for water (Cw) is 4.11�~106 J/(m3 K). Substituting in 
Equation XI-12, the volumetric heat capacity under saturated conditions is given by: 
X . C + (1. X ) . C = 3 .19 w 
�� = ksat/ C 
XI-12 of XI-12 August 2003 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT XII 
CALCULATION OF THE DIFFUSION COEFFICIENT FOR THE DRIPPING CASE 
August 2003 XII-1 of XII-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XII-2 of XII-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment XII - Calculation of the Diffusion Coefficient for the Dripping Case 
Attachment XII contains summary calculations for the estimate of the diffusion coefficient for 
the dripping case for a waste package flow rate of 1 m3 per year. The attachment contains the 
�step by step� method that can be used to calculate the diffusion coefficient for the range of 
particle sizes of from 0.3 to 20 mm. 
A sensitivity study was developed to evaluate the diffusion coefficient over a broad range of 
percolation rates. Table XII-1 summarizes the diffusion analysis developed on the basis of the 
approach presented in Attachment XII. The analysis shows that over a broad range of 
percolation rates that the intergranular water content remains �low� with the coarse pore space 
remaining dry. When the diffusion relationship presented in Equation 6-16 is used to calculate 
the diffusion coefficient, the diffusion coefficient is calculated to be on the order of 10-8 to 10-7 
cm2/sec in the invert. 
A summary of the diffusion coefficients for the dripping case is presented in Figure XII-1. 
Table XII-1. Summary of Diffusion Coefficients for the Dripping Case 
10 
385 
1 
38.5 
0.01 
0.385 
0.1 
3.85 Flux Rate (Jw) 
Waste Package Flow Rate (m3/yr) 
mm/yr 
100 1000 10000 
3850 38500 385000 
Particle Sizes 
ANL-EBS-MD-000063 REV 00 
0.3 9.21E-08 9.45E-08 9.99E-08 1.13E-07 1.45E-07 2.29E-07 6.84E-07 
3 9.07E-08 9.12E-08 9.22E-08 9.47E-08 1.01E-07 1.14E-07 1.49E-07 
10 9.05E-08 9.07E-08 9.11E-08 9.21E-08 9.43E-08 1.01E-07 1.59E-07 
20 9.05E-08 9.06E-08 9.10E-08 9.20E-08 9.44E-08 1.01E-07 1.18E-07 
XII-3 of XII-4 August 2003 Advection versus Diffusion in the Invert 
1.00E-06 
1.00E-07 
1.00E-08 
0.1 
Figure XII-1. Summary of Diffusion Coefficients for the Dripping Case Over a Range of Percolation Rates 
ANL-EBS-MD-000063 REV 00 
Diffusion Coefficient (cm2/sec) 
3 Waste Package Flow Rate 0.1 m /yr 
3 Waste Package Flow Rate 1.0 m /yr 
0.3 mm 
3 mm 
10 mm 
20 mm 
10000 1000 100 10 1 
Flux Rate (Jw)(mm/yr) 
August 2003 XII-4 of XII-4 Advection versus Diffusion in the Invert 
ATTACHMENT XIII 
DERIVATION OF THE FORMULAS FOR THE CALCULATION OF THE BULK 
VOLUMETRIC MOISTURE CONTENT 
XIII-1 of XIII-6 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XIII-2 of XIII-6 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Moisture Content 
This attachment presents a derivation of the formulas to be applied for the bulk volumetric 
moisture content that are used to calculate the diffusion coefficient in the invert for the dripping 
case, and the nondripping case. The first derivation considers the dripping case. 
Attachment XIII - Derivation of the Formulas for the Calculation of the Bulk Volumetric 
August 2003 
XIII.1 DERIVATION FOR THE DRIPPING CASE 
The following parameters are considered for the dripping case (Table 4-3). 
0.051 
0.45 
� intergrain 
� intergrain 
� .051 
� matrix 0.131 
� r 0.05 
� s � intergrain 
Consider the phase diagram for the matrix, and solve for the volume of the voids within the 
crushed tuff particles. Consider the soil mechanics convention of expressing the volume of the 
solids equal to one. 
The matrix porosity as would be determined on core samples is defined as the volume of the 
matrix voids divided by the total volume (Jury et al. 1991, p. 29): Solve for the volume of the 
matrix. 
Vmatrix � matrix 1 Vmatrix 
� matrix Vmatrix 1 � matrix 
Vmatrix = 0.151 
XIII-3 of XIII-6 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Now calculate the volume between the grains. From the fundamental definition of porosity for 
the coarse pore space: 
Vintergrain � intergrain 1 V matrix Vintergrain 
1 V matrix . � intergrain Vintergrain 1 � intergrain 
Vintergrain = 0.942 
Now consider the definition of the saturation for the coarse pore space. The saturation is defined 
in terms of the volumetric moisture content of the coarse pore space as (Fetter 1993, p. 182): 
S e 
Vw Vmatrix 
� � r 1 � matrix � matrix . . 
1 � matrix 1 � intergrain � intergrain � r 1 � matrix 
� � r 
� s � r 
S e = 0.25 % 
This represents the saturation of the intergranular pore space. Now calculate the bulk density on 
the basis of the phase diagram assuming the fine intragranular pore space is saturated with water. 
Vw S e V intergrain 
� Bulk 
� Bulk 
1 � matrix � matrix 1 � intergrain 
. 1 . 
1 � matrix 1 � intergrain 1 � matrix 
. 
V = 2.354 10 3 
w 
Vmatrix = 0.151 
Vintergrain = 0.942 
The bulk volumetric moisture content is defined as: 
1 V intergrain Vmatrix 
� Bulk = 0.073 
� intergrain 
. 1 
� Bulk = 0.073 
XIII.2 DERIVATION FOR THE NONDRIPPING CASE 
Now consider the nondripping case. In this case the coarse pore space has a zero saturation as 
determined, and the fine pore space may by partially saturated. Therefore, Se = zero, and 
�intergrain = �r. 
XIII-4 of XIII-6 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
For purposes of calculation, assume that due to drying, the fine pore space in the invert is at 80 
percent saturation. 
0.80 S matrix 
Calculate the bulk volumetric moisture content. 
0 SmatrixVmatrix � Bulk 
� Bulk = 0.058 
� matrix � rmatrix . 
� matrix 
� matrix � rmatrix 1 � matrix � Bulk 
1 � matrix � matrix 
1 � matrix 
S matrix 
. 
1 � matrix 1 � intergrain 
� matrix 
1 � matrix � Bulk 
. 
� matrix 1 � matrix 1 
. 
1 V intergrain Vmatrix 
Develop a formula for the calculation of the bulk water content. 
1 � intergrain 
. 1 
1 � intergrain 
. 
1 � matrix 1 � intergrain 1 � matrix 
� Bulk = 0.058 
. 
ANL-EBS-MD-000063 REV 00 August 2003 XIII-5 of XIII-6 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XIII-6 of XIII-6 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT XIV 
TSPA CALCULATION OF THE DIFFUSION COEFFICIENT 
ANL-EBS-MD-000063 REV 00 
FOR THE DRIPPING CASE 
XIV-1 of XIV-8 August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XIV-2 of XIV-8 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment XIV - TSPA Calculation of the Diffusion Coefficient for the Dripping Case 
Develop a Mathcad 2001 TSPA-LA Abstraction to be used to assess the diffusion properties for 
the dripping case. 
Hydraulic Conductivity (cm/sec) 
Crushed Tuff 
dm = 10.0 mm dm =0.317 mm dm = 3.0 mm 
Reverse the order for the table lookup: 
Table_Lookup reverse( Table_Lookup ) 
XIV.1 INPUT THE TABLE LOOKUP FOR THE UNSATURATED HYDRAULIC 
CONDUCTIVITY AS A FUNCTION OF MOISTURE POTENTIAL 
The table lookup includes the four particle sizes. The lookup is contained on Worksheet 
Conductivity of Attachment IV. 
Table XIV-1. Table LookUp 
Tuff Matrix 
Moisture 
Potential 
(bars) 
dm = 20.0 mm 
0.00001 6.24655E-09 0.184 16.47999988 183.093696 729.1073774 
0.0002 6.23415E-09 0.184 16.47998446 182.2521233 397.4660442 
0.001 6.17408E-09 0.183999998 15.16743525 8.5956E-05 4.56172E-10 
0.0015 6.14714E-09 0.183999966 4.536805636 3.14417E-08 1.6582E-13 
0.002 6.12358E-09 0.183999745 0.115992667 1.14088E-10 6.0155E-16 
0.005 6.01436E-09 0.183835865 2.81183E-09 1.92525E-18 1.01515E-23 
0.01 5.88112E-09 0.162813422 3.70878E-15 2.53888E-24 1.35981E-29 
0.02 5.68014E-09 0.000502335 4.89062E-21 3.39937E-30 2.02283E-34 
0.05 5.25441E-09 1.07748E-11 8.2659E-29 1.2231E-35 5.44857E-36 
0.1 4.75272E-09 1.42106E-17 1.56708E-34 1.36209E-36 5.91883E-37 
0.2 4.03656E-09 1.87389E-23 7.34321E-37 1.79037E-37 7.49958E-38 
Note that the first column is in bars, and the remaining columns are in cm/sec. 
August 2003 
XIV.2 LOG-LOG INTERPOLATION FUNCTIONS 
Define the Log-Log Interpolation functions. Use the Mathcad interpolation functions for this 
analysis. Transform to log space for interpolation. 
i ( := 1.. rows Table_Lookup) 
Log_Table_Lookup i j , := log ( Table_Lookupi j , ) 
j ( := 1.. cols Table_Lookup) 
XIV-3 of XIV-8 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Log_Table_Lookup = 
.
.
.
.
.
. 
.
.
.
.
.
.
. 
.
.
.
.
.
.
.
. 
1.000 8.323 16.847 33.805 35.866 
1.301 8.279 10.968 28.083 34.913 
1.699 8.246 3.299 20.311 29.469 
2.000 8.231 0.788 14.431 23.595 
2.301 8.221 0.736 8.551 17.716 
2.699 8.213 0.735 0.936 9.943 
2.824 8.211 0.735 0.657 7.502 
3.000 8.209 0.735 1.181 4.066 . 
.
.
.
.
.
.
.
.
. 
.
.
.
.
.
.
.
.
. 3.699 8.205 0.735 
. . .0.699 .8.394 .22.727 .36.134 .36.747 37.125. 
36.228 
35.264 
33.694 
28.867 
22.993 
15.221 
12.780 
9.341 
1.217 2.261 2.599 
1.217 2.263 2.863 � 
........... 
. . 
.
.
.
.
.
.
.
.
. 
( 
0.000 . 
linterp Log_Table_Lookup 
.3. 
Log_Table_Lookup 1 , x ) 
8.204 .0.735 
, 
linterp ( Log_Table_Lookup 
.4. 
Log_Table_Lookup 
.1. 
, x ) 
linterp ( Log_Table_Lookup 
.5. 
, Log_Table_Lookup 
.1. 
, x) 
.............. 
Inter_Function_03 (x) := 
Inter_Function_3 (x) := 
Inter_Function_10 (x) := 
Inter_Function_20 (x) := 
, 
linterp ( Log_Table_Lookup 
.6. 
, Log_Table_Lookup 
.1. 
, x) 
XIV.3 WASTE PACKAGE FLOW RATE AND CROSS SECTIONAL AREA 
Input the waste package flow rate and the waste package cross sectional area. 
Q_Waste_Package := 0.1. 
3 m
yr 
The waste package area is taken from Seepage Model for PA Including Drift Collapse (CRWMS 
M&O 2000d, Section 6.3.1) 
2 A_Waste_Package_Plan := 28.05 . m 
XIV.4 FLUX RATE CALCULATION 
Calculate the flux rate through the invert. 
Q_Waste_Package 
Jw := 
A_Waste_Package_Plan 
. 8 cm 
10 1.130 �� Jw sec 
=
= Jw 3.565 
yr 
mm 
XIV.5 SOLUTION FOR MOISTURE POTENTIAL 
Solve for the moisture potential for each case. Note that a power function is defined to cover a 
broad range of values. 
XIV-4 of XIV-8 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
. . . sec 
Inter_Function_03 log Jw . 
� � cm �r 1 := 10 
. 
... 
.. 
1 = 0.034851878638221 
. . sec 
Inter_Function_3 log Jw . 
� � cm 
�r 
�r 2 := 10 
. 
... 
.. 
2 = 0.004649532359782 
. . sec 
Inter_Function_10 log Jw . 
� � cm 
�r 
3 := 10 �r 
3 = 0.001580705181856 
. 
... 
.. 
. . sec 
Inter_Function_20 log Jw . 
� � cm 
�r 
�r 4 := 10 
�r 
... 
.. 
4 = 0.000828715173741 
XIV.6 SOLUTION OF VOLUMETRIC MOISTURE CONTENT 
Solve for the volumetric moisture content. We first input the constants for the van Genuchten 
relationship (see Table 6-2). 
i := 1.. 4 
�\i := 
65.91 
624. 
2080. 
4160. 
n := 8.013 i 
r := 0.05 �c 
i 
s := 0.45 
ni 
XIV-5 of XIV-8 
�c 
i 
Calculate the m value for the three parameter van Genuchten relationship: 
mv := 1 . 1 
i 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
( 
i �� i) . + 
.1 ( �� 
. 0.051 . 
. 0.050 
. 0.050 
0.050 
�� 
. 
.. 
. . 
��matrix 
��
intergrain 
1 . 
matrix 
.. 
1 . �� 1 . �� . 
.. 
. = . 
. 
XIV.7 SOLVE FOR THE VOLUMETRIC MOISTURE CONTENT FROM THE VAN 
GENUCHTEN RETENTION RELATIONSHIP FROM FETTER (1993, P.172) 
s i 
�� 
i + �� i := �� ri mvi 
�� r ) . 
ni .. 
. 
. 
= �� 
��matrix 
��matrix 
August 2003 
XIV.8 SOLUTION OF EFFECTIVE WATER CONTENT 
Calculate the effective water content based upon the fine pore structure being saturated with 
water, and the coarse pore structure have the water content shown above. Assign the matrix 
porosity ��m to 0.131 (Table 4-3 for TSw35 tuff) and assume that the fine pore structure is 
saturated with water. From Attachment XIII, the following formula is developed for the dripping 
case. 
matrix := 0.131 ��
��intergrain := 0.45 
i �� r . ��matrix 1 i 
.. 
+ + . intergrain 1 . 
..
1 �� �� �� intergrain r 
�� . 
intergrain 
. 
matrix . i := Bulki 1 . 
+ 1 
. 
+ . 
. 
. 
. 1
matrix 
��intergrain .. 1 + 
1 �� 
. 
.. 
. ��
. matrix . 
�� 
�� 
Bulki 
. 
�� 
��
0.073 
0.072 
0.072 
. 0.072 
.
.. 
XIV-6 of XIV-8 
. 
ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
. 7 
.. 
. 
. 
2.568�~ 10 . 
XIV.9 SOLUTION OF SOIL-LIQUID DIFFUSION COEFFICIENT 
Calculate the soil-liquid diffusion coefficient from Equation 6-16, and accounting for an 
increased temperature to 45��C as calculated in Attachment III, Equations III-6 and III-7: 
T0 := 25 + 273.15 
T := 45 + 273.15 
1.3272 ( 293.15 T0) . .0.001053.( T0.293.15)2 . 
. ( . 1.3272 293.15 T ).0.001053 
T0.168.15 
10 
2 cm .1.494.( )1.863 
. 
2.30 10 . i Dsli 
:= . 
sec 
2 
:= Dsl Bulk 
�� 
) 1.863 
i i 
. 7 
. 5 
-5 3.44 10 . . 
sec 
.( cm 
10 2.648�~ 
2.578�~ 
. 7 . 
10 
sl 
�� 
2 cm 
sec 
D = . 
....
. 2.566�~ 
. 7 10 . 
.(T.293.15) 2.. 
. 
T . 168.15 
XIV-7 of XIV-8 
. 
ANL-EBS-MD-000063 REV 00 
= 1.494 
August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XIV-8 of XIV-8 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT XV 
CALCULATION OF THE THICKNESS OF THE INVERT FOR ANALYSIS 
August 2003 XV-1 of XV-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XV-2 of XV-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment XV - Calculation of the Thickness of the Invert for Analysis 
The analysis in this report (Section 6.10) provides a breakthrough analysis for radionuclides 
through invert. The following analysis takes information on the geometry of the invert as 
determined from the current repository design to calculate an average thickness of the invert. A 
second analysis is then performed to estimate the maximum settlement that would occur. Note 
that the analysis is conservative because the invert foundation will be engineered to place the 
crushed tuff near its critical void ratio, and this would reduce the amount of settlement with time. 
Nevertheless, the thickness of the invert during the postclosure period is estimated. The 
emplacement drift configuration is shown on Repository Design Project, Repository/PA IED 
Emplacement Drift Configuration 1 of 2 (BSC 2003d) and references Repository Design, 
Emplacement Drift Steel Invert Plan and Details (BSC 2001c, Figure 5). The dimension shown 
from center to center of the rail system on the typical invert elevation is 2.95 m. The thickness 
of the invert is shown as 0.806 m. Calculate the thickness of the invert considering that the 
emplacement drift radius is 2.75 m. Figure XV-1 shows the coordinate system and dimensions 
for the calculations. The distance from the centerline of the drift to the top of the invert is 1.94 m. 
1.94 m 
Source: BSC 2001c; BSC 2003d 
where 
X x coordinate, 
Y y coordinate, and 
R radius (2.75 m) 
ANL-EBS-MD-000063 REV 00 
Y 
X 
R = 2.75 m 
3.9 m 
August 2003 
The equation of a circle is given by: 
The invert thickness is given by the expression: 
Figure XV-1. Invert Geometry 
X 2 + Y 2 = R 2 (Eq. XV-1a) 
R 2 - x 2 = 94 . 1 
XV-3 of XV-4 Advection versus Diffusion in the Invert 
Using the equation solve for the point when y = 0: 
R2 - x2 - 1 .94 = 0 
The value for X0 equals 1.949 m. Use integral calculus to solve for the average depth of the 
invert. From integral calculus 
R 
R2 x2 dx x . R2 x2 R2 
.asin x 
2 2 
1.94 d x 1.94 .x 
Evaluating the definite integral for the area. 
0 x0 x0 0 2 R2 
= 1.088 1.94 x . R2 02 R2 
.asin .asin R2 x 0 0 R R 2 2 2 
. 
2 
The average height is given as by the area divided by the value for x0. 
0 
. 
x0 0 x0 2 R2 
R2 02 R2 
.asin .asin R2 x 1.94 x0 0 R R 2 2 2 = 0.558 
. 
The calculated value for the thickness of the invert is 0.558 m. Consider the maximum potential 
settlement that might occur with time. This report considers materials that have a range of 
particles from 0.3 to 20 mm. According to Foundation Engineering Handbook (Winterkorn and 
Fang 1975, p. 84), the materials would classify as a poorly graded sand (SP). Considering these 
materials and the potential for some settlement with time, the invert thickness used in these 
calculations is 0.5 m. 
. 
2 
x0 
XV-4 of XV-4 
. 
ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
ATTACHMENT XVI 
MOISTURE RETENTION FOR CRUSHED TUFF 
XVI-1 of XVI-4 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XVI-2 of XVI-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment XVI - Moisture Retention for Crushed Tuff 
The analysis method is presented in Water Diversion Model (CRWMS M&O 2000c, Section 3 
and Attachment V). 
Table XVI-1. van Genuchten Curve Fit Parameter Results for the Invert 
Parameter Value Unit 
Residual Moisture Content (�r) 
Moisture Content at Saturation (�s) 0.63 (no units) 
0.05 (no units) 
117.00 bars-1 
5.25E-04 (no units) 
�i 
ni 2.75 (no units) 
m 0.64 (no units) 
Sum of Residuals 
NOTE: The � parameter is calculated as 0.12 cm-1. 
Note that the parameters are calculated using the Microsoft Excel Equation Solver based upon 
the sum of the residuals as given above from Table XVI-2. 
Table IV-2. Retention Analysis Results for the Invert 
Residuals 
Moisture 
Potential 
(bars) 
Volumetric 
Moisture 
Content 
Predicted 
Moisture 
Content 
0.068 0.121 0.057 1.29E-04 
0.059 0.174 0.054 2.52E-05 
0.058 0.309 0.052 3.49E-05 
0.057 0.483 0.051 3.03E-05 
0.056 0.696 0.051 2.24E-05 
0.055 1.090 0.051 1.51E-05 
0.053 1.930 0.051 3.83E-06 
0.052 3.020 0.051 9.60E-07 
0.050 4.350 0.051 1.02E-06 
0.045 17.400 0.051 3.60E-05 
0.060 0.121 0.057 1.14E-05 
0.060 0.174 0.054 3.63E-05 
0.059 0.309 0.052 4.77E-05 
0.058 0.483 0.051 4.23E-05 
0.058 0.696 0.051 4.54E-05 
0.056 1.090 0.051 2.38E-05 
0.054 1.930 0.051 8.74E-06 
0.054 3.020 0.051 8.88E-06 
0.052 4.350 0.051 9.79E-07 
0.047 17.400 0.051 1.60E-05 
NOTE: Volumetric moisture content and moisture 
potential are obtained from Attachment XI and 
DTN: GS980808312242.015 for crushed tuff. 
Equation IV-1 is used for calculating the predicted moisture content. 
Residuals are calculated as the square of the difference between the actual volumetric moisture 
content and the predicted moisture content. 
August 2003 XVI-3 of XVI-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XVI-4 of XVI-4 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT XVII 
ESTIMATE OF THE BOUNDARY BETWEEN ADVECTION AND DIFFUSION 
August 2003 XVII-1 of XVII-6 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
Attachment XVII - Estimate of the Boundary Between Advection and Diffusion 
This attachment presents an analysis to estimate the boundary between a diffusion dominated 
mass transfer and an advection dominated mass transfer in the invert. Section 6.2 developed the 
relations describing advection and hydrodynamic dynamic dispersion restated as: 
w 
sl . dCL (Eq. XVII-2) 
dz 
sl . CL (Eq. XVII-3) 
L 
J . L (Eq. XVII-4) 
D w 
sl 
August 2003 
Liquid Advection (bulk flow or convection) 
Jlc = J . CL (Eq. XVII-1) 
where 
Jlc is the liquid advection flux (m/s) 
Jw is the vertical Darcy flux rate (m/s) 
CL is the solute concentration of the solute at location x = 0 (mg/L) 
J = . D sl 
J �� . D sl 
Pec = 
XVII-2 of XVII-6 
Soil-Liquid Diffusion 
where 
Jsl = soil liquid flux (m/sec) 
Jw = vertical darcy flux (m/sec) 
Z = vertical coordinate (m) 
Note that the second equation can be written as an approximation across the invert. 
where 
L = length in the vertical direction 
Noting that the ratio of advective transport to diffusive transport: 
where 
Pec = Peclet Number 
The dimensionless Peclet Number can then be defined as the ratio of the advective transport to 
the diffusive transport (Fetter 1993, p. 54): 
ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
v . L (Eq. XVII-5) 
sl 
. 
. m 
C 0.0 . ( . , t logt) , 0.500 m , 1 , Dsl
.
� 0.5 . yr 
Pec = xD 
It should be noted that the Peclet Number as calculated in this manner is a relative index of the 
ratio of advective to diffusive transport, and that the transition from a diffusion dominated 
system to an advective dominated system depends on the characteristic length used in the 
analysis. Fetter (1993, p. 55) defines the average diameter of the particles in cm. However, in 
this analysis the vertical length of the invert is selected as a more meaningful index. 
In the following analysis, the critical Peclet Number is determined as the value of pore velocity 
that would result in the fifty percent breakthrough occurring in approximately one half the time. 
The analysis estimates the critical Peclet Number for diffusion to occur. In this calculation, the 
contaminant transport equation for breakthrough is used to perform a diffusion analysis. The 50 
percent breakthrough is defined. The demarcation between diffusion and advection occurs for a 
pore velocity that is one-half of this value. The velocity is selected to result in a breakthrough at 
the specified time of 250 years. 
Consider diffusion alone. From Attachment III: 
. 7 cm 
D = 1.707�� 10 sl 
The breakthrough for diffusion alone is presented in Figure XVII-1. The fifty percent 
breakthrough time is about 500 years. 
Van Genuchten Retention 3-20. mm 
1 
1 
0 
0.1 
Diffusion Dominated 
Figure XVII-1. Breakthrough for a Diffusion Dominated System 
Calculate the velocity that results in a fifty percent breakthrough of 250 years. From Equations 
6-12 and 6-13 for a pore water velocity of 0.85 mm per year: 
XVII-3 of XVII-6 ANL-EBS-MD-000063 REV 00 
C(t)/C0 
2 
sec 
10 1 .103 1 .104 1 .105 100 
t ( logt) 
yr 
Time(yr) 
August 2003 Advection versus Diffusion in the Invert 
L := 0.85. 
yr 
V 
mm 
DL := �f.VL + Dsl 
1 
. 
. yr 
C . ( . 0.0 . , t logt) , 0.5 m , 1, Dsl� 
. m 
0.5 C V ( ( L . , t logt) , 0.5 m , 1, DL) 
0 
0.1 
Figure XVII-2. Breakthrough Analysis for Advection-Diffusion with Fifty Percent Breakthrough Occurring 
After 250 Years 
The results of the calculation are presented in Figure XVII-2. Calculate the critical Peclet 
Number based upon this velocity and the maximum depth of the invert using Equation XVII-4. 
D 
VL.0.5.m 
= 0.8 
sl 
The critical Peclet Numbers based upon this criterion is about 0.8. Use this Peclet Number in the 
analysis. 
An analysis was performed to calculate Peclet Numbers from the NUFT analysis for the case of 
35 mm per year with the nondimensional van Genuchten parameters, and the Campbell 
parameters for particle sizes of 3 mm and 10 mm, respectively. Figure XVII-3 presents the 
profile of the Darcy Flux within the invert close the open drift. The Darcy Flux is calculated 
from the NUFT pore water velocity results by multiplying by the cross-sectional flow area within 
the pore space. The cross-sectional area approximately equals the saturation (S) times the 
porosity (�c). It is seen that the percolation fluxes expressed in mm per year are less than the 
saturated percolation flux of about 3 mm per year. 
Figure XVII-4 presents the Peclet Number for these analyses for comparision of the value in 
which diffusion controls as presented above. The analysis shows that the approximate thickness 
of the diffusion controlled invert is approximately 0.1 m or about twenty percent of the 0.5 m 
thick invert. 
XVII-4 of XVII-6 ANL-EBS-MD-000063 REV 00 August 2003 
C(t)/C0 
Van Genuchten Retention 0.317 mm 
10 1 1 .103 1 .104 1 .105 100 
t ( logt) 
yr 
Time(yr) 
Diffusion Dominated 
Low Advection Advection versus Diffusion in the Invert 
Matrix Flux (mm y / r) Peclet Number 
0.8 
0.7 
0.6 
0.5 
0.4 
0.3 
0.2 
0.1 
0 
0.1 0.05 0 
5.00 
4.50 
4.00 
3.50 
3.00 
2.50 
2.00 Advection 
Dominated 
1.50 
1.00 
0.50 
0.1 0.05 
0.00 
0 
Profile of Matrix Advection in the Invert 
0.15 
Figure XVII-3. Profile of Matrix Advection in the Invert 
Depth(m) 
Profile of Peclet Number in the Invert 
Diffusion 
Dominated 
0.15 
Figure XVII-4. Profile of the Peclet Number in the Invert 
Critical Peclet Number = 0.8 
0.4 0.35 0.3 0.25 0.2 
Depth(m) 
0.4 0.35 0.3 0.25 0.2 
ANL-EBS-MD-000063 REV 00 
3 mm van Genuchten 
10 mm van Genuchten 
3 mm Campbell 
10 mm Campbell 
0.5 0.45 
van Genuchten 3 mm 
van Genuchten 10 mm 
Campbell 3 mm 
Campbell 10 mm 
0.5 0.45 
August 2003 XVII-5 of XVII-6 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XVII-6 of XVII-6 ANL-EBS-MD-000063 REV 00 Advection versus Diffusion in the Invert 
ATTACHMENT XVIII 
ALTERNATE POREWATER VELOCITY CALCULATION 
XVIII-1 of XVIII-8 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
From Jury et al. (1991, p. 109): 
. �c �c r 
S 
Solving for �c: 
�c := .S + 
..( �c s . �c r) 
1
m 
2 
m 
.
� 
�c s . �c r 
1 
r ( �c s . �c r ) .�c 
�c = 0.116 
1 . 
2 . 
K S) ( 
�s . . 
.. 
1 . 1 . S 
mm 
yr 
.. 
Solving for the unsaturated hydraulic conductivity and substituting into the van Genuchten 
relation for unsaturated hydraulic conductivity (Jury et al. 1991, p. 109): 
K S := ( ) Ks 
S . 
pi 
.
.. 
= 0.056 
The calculated value for matrix percolation flux is approximately 0.1 mm per year. This value is 
in approximate agreement with the matrix percolation flux of 0.15 mm per year reported in 
Figure 6.3-4 of Drift-Scale Radionuclide Transport (BSC 2003f). It should be noted that small 
variations in matrix saturation might result in large variations in flux, because the unsaturated 
matrix hydraulic conductivity is a strong nonlinear function of volumetric moisture content or 
saturation. In addition, differences in temperature affect unsaturated hydraulic conductivity 
between the two analyses. 
The results reported in Figure 6.3-4 of Drift-Scale Radionuclide Transport (BSC 2003f) indicate 
that the open drift behaves like an impermeable inclusion with flow being diverted around the 
inclusion. The analysis results (Section 6.7 and Attachment IX) show a similar behavior in flow 
being diverted around the open drift. 
The crushed tuff matrix flow results can be interpreted with the aid of a closed form analytical 
solution for flow around an inclusion (CRWMS M&O 2001b, Attachment XII). The report 
developed a closed form solution for flow. From Attachment XII, Equations XII-9 and XII-10: 
. 2 ko 
ki + k 
r . a 
... 
o 
.r . cos ( ) �c 
XVIII-3 of XVIII-8 ANL-EBS-MD-000063 REV 00 
(Eq. XVIII-1) 
(Eq. XVIII-2) 
.. 
(Eq. XVIII-3) .. 
(Eq. XVIII-4) 
August 2003 Advection versus Diffusion in the Invert 
( ki . ko ) a 2 
po 
i ko 2 
where 
. 
x � 
k + 
r . a 
�c atn. y . 
u 
d u 
dx 
.. 
+
y
x 
1 
1 
.y 
2 x 
2 
. 
. y 
1 + 
d 
.
. y
1 
2 
�s . . 1 . 
pi = capillary pressure within an inclusion 
po = capillary pressure outside an inclusion 
r = radius from the center of the inclusion 
�s = Darcy flux in the direction of flow 
ki = unsaturated intrinsic permeability within the inclusion 
ko = unsaturated intrinsic permeability outside the inclusion 
a = radius of the inclusion 
Consider the partial derivatives (from Beyer 1987, p. 205): 
r 
r 
dx 
( x 2 + y 2 
2
) 
r. 
.. 
cos ( ) �c . 
r . 
x 2 y 2 (Eq. XVIII-6) 
atn u ( ) 
1 u + d
d 
x 
u 
2 x2 
. 
( x + y 2 ) 2 
. x 
.
� 
�c 
d 
x 
�c 
d
d 
dx
�c 
r 
... 
x � 
. y 
1 
x 2 + y 
. 
.. 
XVIII-4 of XVIII-8 
.. 
. . 
d 
dx
�c 
Consider the derivative with respect to y: 
ANL-EBS-MD-000063 REV 00 
(Eq. XVIII-5) 
(Eq. XVIII-7) 
(Eq. XVIII-8) 
(Eq. XVIII-9) 
(Eq. XVIII-10) 
(Eq. XVIII-11) 
August 2003 Advection versus Diffusion in the Invert 
1 d �c 
dy 
. d y 
. y . 
2 dy x 
1 + . 
2 
. 
x
1 d �c 
dy 
. x � 
1 
1 + . 
. y d r 
dy 
. y . 
. x � 
1 
( x 2 + y 2). 2 � 
. 1 . . 
For the case inside the inclusion, the derivative is trivial: 
2 k . o 
. 1 
ki + ko 
d pi 
dx 
For the case outside the inclusion: 
po 
�s . . 
. ki + ko 2 . . 
po 
�s . .. . 
1 . 
( ki . ko) 
. 
a 2 
. . 
. r. cos ( ) �c 
r . 
r . a 
ki + ko r . . 
Use the chain rule: 
r . a 
�s . .. . 
r . 
( ki . ko) 
. 
a 2 
. . 
. 1. cos ( ) �c 
ki + ko r . 
. . 
( ki . ko) a2 
.. x 
. 
1 + ( ki + ko) . d po 
dx 
x + y 
. 
. 
o 
. 
�s . .. 1 + . ( k . 
. 
. . �s.. r . 
x . 
. �s.. 1 . �s . .
.. 
1 + 
( ki . ko) 
. 
a2 
..
. 
�s . . 
d p 
dx 
o 
d p 
dx . ( k 
. r2 . . 
. 
2 2 
. cos ( ) �c . �s.. .
r . 
( ki . ko) 
. 
a 2 
.
. 
. .sin ( ) �c . 
.y 
Substitute the definitions for sin and cos: 
x ( ki . ko) a2 . x . . 
( ki . ko) a2 
i + ko r . r ( k i + ko) r2 . . 
2 2 r 
x + y 
x . . k 
. 
x + y 
( ki . ko) a2 . 
i + ko 
.
.
. 
. y . 
.y 
.. . y 
r2 .. 
( x2 + y2) 
x2 + y2) 
2 
. 
i + ko) r2 . . 
2 2 2 2 x + y . 
XVIII-5 of XVIII-8 
. 
ANL-EBS-MD-000063 REV 00 
(Eq. XVIII-12) 
(Eq. XVIII-13) 
(Eq. XVIII-14) 
(Eq. XVIII-15) 
(Eq. XVIII-16) 
(Eq. XVIII-17) 
2 ( x 2 + y ) 
(Eq. XVIII-18) 
(Eq. XVIII-19) 
August 2003 Advection versus Diffusion in the Invert 
This agrees with Equation XII-56 of Water Distribution and Removal Model (CRWMS M&O 
2001b): 
d p 
d x 
a . �� . o ko 
d p 
d y i 2 r 
Take the derivative with respect to y: 
.  
y 
y 
o 
d p 
d y (k + k ) 
. 
.. 
1 + 
( ki . ko ) 2 . 
. (k + ) 
o i 2 r 
Substitute the definitions for sin and cos: 
a . 
. 
2 r 
x + y . 
  . 
1 + .. 
x . 
. y . 
Calculate the approximate area of inclusion. From Attachment XV: 
. sin ( ) �� 
  
2 x 
�� a . 
has solution(s) 
.  
.  
. 
.. 
1 + 
( ki . ko ) 2 . 
. 
Area_Invert . := 2 1.088 
Area_Invert = 2.176 
2 Area_Drift := ��. 2.75 
Area_Drift = 23.758 
Area_Inclusion := Area_Drift . Area_Invert 
Area_Inclusion = 21.582 
2 Area_Inclusion 
1 
1 ( ��.Area_Inclusion ) 2 
yr 
a := . 
�� 
a = 2.621 
The calculated radius is 2.62 m. Set the farfield flow in the matrix equal to 1.31 mm per year 
consistent with the NUFT analysis. Set the external hydraulic conductivity equal to one, and the 
gradient in the farfield equal to one for a deep water table. See Jury et al. 1993 p. 127), [DIRS 
102010]. The internal hydraulic conductivity is set to zero. 
a := 2.621 
mm 
1.31. ko := 
mm 
k := 0 . i 
�� . 
yr 
�� := .1. 0 
ANL-EBS-MD-000063 REV 00 
( ki . ko ) a2 . .. 
. . ��. . . (Eq. XVIII-20) 1 + o 
2 y 
2 
( ki . ko) a2 . 2 x 
2 ko ( ki + k ) o 2 i 2 r r r r 
1 
. 
.. 
r 
1 .
. 
k +
. . . . 
. 
k + 2 
. 
x
1 . ... 
. 
. 
. 
. 
. y . 
. x . 
. 
.. 
1 + . 
1 . 
.. 
.  
( ki . ko) 2 . a 
i ko 
a 
.cos ( ) �� . �� 
2 x + y 
x 
..
(Eq. XVIII-21) 
(Eq. XVIII-22) . 
�� . . 
2 
2 
. 
.. 
  
i ko 
r . 
r 
... 
1 . . 
.. 
r . 
( ki . ko ) 2 .
. 
.y �� . 
k + 
. 
r . 
.. 
.. 
.. 
. 
.. 
  
XVIII-6 of XVIII-8 August 2003 Advection versus Diffusion in the Invert 
k 
v := x(x y , ) ko. 
.. 
.�� . 
.. 
1 + 
( ko ) i . 
i ko ( 
.  
i . ko ) 
. vy(x y , ) := .ko 
k +
2 a 
2 2 x + y 
) .  
. . 
.. 
1 + 
( 
�� 
k 
(ki + ko ) . 
.. 
. 
.  
. 
. 
. 
.  
. 
. 
... 
( 
. 0.05 0.075516768 
0.225 0.332607053 
. 0.15 0.222774466 
Array_Invert := . 0.275 0.415758816 
. 0.35 0.491351328 
0.45 0.680332608. 
Perform an analysis out to about four times the diameter. 
Input the porewater velocity profile from NUFT analysis Attachment II and XVII. The data are 
from Figure XVII-2: 
. 
.. 
.  
XVIII-7 of XVIII-8 
. 
Consider a column of elements 0.15 meters from the centerline. 
ANL-EBS-MD-000063 REV 00 
2 2 a 2 a ki . ko y .. . . . . 
2 x 
2 2 2 2 2 y 2 x + y 2 x + y 2 x y + ) + 
..
(Eq. XVIII-23) 
) �� . . 
2 1 y 
) 
( ko i x ) 
. 
.. 
. 
.. 
.  
2 x + y . . �� 
.. 
) 
.
. 
2 x + y 
.. 
.. 
) ( 
. 
+ y 2 
1 . 
( 
k + 
( ki . ko 
k + 
.. 
. 
x 
1 2 
( 
y . 
2 2 x + y 
( 
2 a 
2 
) 
i ko 
1 . 
x 
(
. 
.. 
.. 
2 x ) 
. 
. . 
. . 
.. 
.. 
... 
. 
  . 
1 + .. 
x . 
. y . 
(Eq. XVIII-24) 
August 2003 Advection versus Diffusion in the Invert 
1.4 
1.2 
1 
0.8 
0.6 
0.4 
0.2 
0 
0 1 
Matrix Percolation Flux 1.3 mm per yr 
Matrix Percolation 0.15 mm per yr 
NUFT Results 
Figure XVIII-1. Comparison of a Closed Form Solution for Matrix Flow at Saturation with the Results of 
the NUFT Analysis 
Figure XVIII-1 shows the results of a comparison of the fluxes in the invert to the closed form 
solution for far field matrix flow around an inclusion. This far field flow is approximately 1.3 
mm per year consistent with NUFT analysis. The figure shows that the Darcy fluxes in the 
invert at a percolation rate of 35 mm per year are near saturation, and are responding to the 
impermeable inclusion as represented by the emplacement drift. 
ANL-EBS-MD-000063 REV 00 August 2003 XVIII-8 of XVIII-8 
Darcy Flux (mm/yr) 
Percolation Rate 35 mm per yr 
9 7 8 6 4 3 2 5 
Depth in Floor Advection versus Diffusion in the Invert 
ATTACHMENT XIX 
CD-ROMS WITH INPUT AND OUTPUT FILES 
XIX-1 of XIX-2 ANL-EBS-MD-000063 REV 00 August 2003 Advection versus Diffusion in the Invert 
INTENTIONALLY LEFT BLANK 
August 2003 XIX-2 of XIX-2 ANL-EBS-MD-000063 REV 00