Title 40 CFR Part
191
Subparts B and C
Compliance Recertification
Application
for the
Waste Isolation Pilot Plant
Appendix
PA2009
Performance Assessment
United States
Department of Energy
Waste Isolation Pilot Plant
Carlsbad Field
Office
Carlsbad, New Mexico
Appendix PA2009
Performance Assessment
PA2.0 Overview and Conceptual Structure of the PA
PA2.1 Overview of Performance Assessment
PA2.1.1 Changes in the CRA2009 PA
PA2.1.2 Conceptual Basis for PA
PA2.1.3 Undisturbed Repository Performance
PA2.1.4 Disturbed Repository Performance
PA2.1.4.1 Cuttings and Cavings
PA2.1.4.4 Mobilization of Actinides in Repository Brine
PA2.1.4.5 LongTerm Brine Flow up an Intrusion Borehole
PA2.1.4.6 Groundwater Flow in the Culebra
PA2.1.4.7 Actinide Transport in the Culebra
PA2.1.4.8 Intrusion Scenarios
PA2.1.5 Compliance Demonstration Method
PA2.2 Conceptual Structure of the PA
PA2.2.1 Regulatory Requirements
PA2.2.2 Probabilistic Characterization of Different Futures
PA2.2.3 Estimation of Releases
PA2.2.4 Probabilistic Characterization of Parameter Uncertainty
PA2.3.1 Identification and Screening of FEPs
PA2.3.2 Scenario Development and Selection
PA2.3.2.1 Undisturbed Repository Performance
PA2.3.2.2 Disturbed Repository Performance
PA2.3.2.2.1 Disturbed Repository M Scenario
PA2.3.2.2.2 Disturbed Repository E Scenario
PA2.3.2.2.5 The E1E2 Scenario
PA2.3.2.3 Disturbed Repository ME Scenario
PA2.3.2.4 Scenarios Retained for Consequence Analysis
PA2.3.3 Calculation of Scenario Consequences
PA3.0 Probabilistic Characterization of Futures
PA3.4 Penetration of Excavated/Nonexcavated Area
PA3.6 Penetration of Pressurized Brine
PA3.10 Scenarios and Scenario Probabilities
PA4.1 Results for Specific Futures
PA4.2 TwoPhase Flow: BRAGFLO
PA4.2.1 Mathematical Description
PA4.2.3 Creep Closure of Repository
PA4.2.4 Fracturing of MBs and DRZ
PA4.2.6 Capillary Action in the Waste
PA4.2.8 Option D Panel Closures
PA4.2.8.1 Panel Closure Concrete
PA4.2.8.2 Panel Closure Abutment with MBs
PA4.2.8.3 DRZ Above the Panel Closure
PA4.2.8.4 Empty Drift and Explosion Wall Materials
PA4.2.10 Castile Brine Reservoir
PA4.2.12 Gas and Brine Flow across Specified Boundaries
PA4.2.13 Additional Information
PA4.3 Radionuclide Transport in the Salado: NUTS
PA4.3.1 Mathematical Description
PA4.3.2 Calculation of Maximum Concentration ST( Br, Ox, El)
PA4.3.3 Radionuclides Transported
PA4.3.4 NUTS Tracer Calculations
PA4.3.5 NUTS Transport Calculations
PA4.3.7 Additional Information
PA4.4 Radionuclide Transport in the Salado: PANEL
PA4.4.1 Mathematical Description
PA4.4.4 Additional Information
PA4.5 Cuttings and Cavings to Surface: CUTTINGS_S
PA4.5.2.2 Turbulent Flow Model
PA4.5.3 Additional Information
PA4.6 Spallings to Surface: DRSPALL and CUTTINGS_S
PA4.6.1 Summary of Assumptions
PA4.6.2.1 Wellbore Flow Model
PA4.6.2.1.1 Wellbore Initial Conditions
PA4.6.2.1.2 Wellbore Boundary Conditions
PA4.6.2.2 Repository Flow Model
PA4.6.2.3 Wellbore to Repository Coupling
PA4.6.2.3.1 Flow Prior to Penetration
PA4.6.2.3.2 Flow After Penetration
PA4.6.2.3.3 Cavity Volume After Penetration
PA4.6.2.3.5 Waste Fluidization
PA4.6.3.1 Numerical Method—Wellbore
PA4.6.3.2 Numerical Method—Repository
PA4.6.3.3 Numerical Method—Wellbore to Repository Coupling
PA4.6.5 Additional Information
PA4.7 DBR to Surface: BRAGFLO
PA4.7.1 Overview of Conceptual Model
PA4.7.2 Linkage to TwoPhase Flow Calculation
PA4.7.3 Conceptual Representation for Flow Rate rDBR(t)
PA4.7.4 Determination of Productivity Index Jp
PA4.7.5 Determination of Waste Panel Pressure pw( t) and DBR
PA4.7.6 Boundary Value Pressure pwf
PA4.7.7 Boundary Value Pressure pwE1
PA4.7.7.1 Solution for Open Borehole
PA4.7.7.2 Solution for SandFilled Borehole
PA4.7.10 Additional Information
PA4.8 Groundwater Flow in the Culebra Dolomite
PA4.8.1 Mathematical Description
PA4.8.3 Computational Grids and Boundary Value Conditions
PA4.8.5 Additional Information
PA4.9 Radionuclide Transport in the Culebra Dolomite
PA4.9.1 Mathematical Description
PA4.9.1.1 Advective Transport in Fractures
PA4.9.1.2 Diffusive Transport in the Matrix
PA4.9.1.3Coupling Between Fracture and Matrix Equations
PA4.9.1.5 Cumulative Releases
PA4.9.2.1 Discretization of Fracture Domain
PA4.9.2.2 Discretization of Matrix Equation
PA4.9.2.3 FractureMatrix Coupling
PA4.9.2.4 Cumulative Releases
PA4.9.3 Additional Information
PA5.0 Probabilistic Characterization of Subjective Uncertainty
PA5.2 Variables Included for Subjective Uncertainty
PA5.4 Correlations and Dependencies
PA5.5 Separation of Aleatory and Epistemic Uncertainty
PA6.0 Computational Procedures
PA6.2 Sample Size for Incorporation of Subjective Uncertainty
PA6.3 Statistical Confidence on Mean CCDF
PA6.4 Generation of Latin Hypercube Samples
PA6.5 Generation of Individual Futures
PA6.7 Mechanistic Calculations
PA6.7.5 CUTTINGS_S Calculations
PA6.7.6 BRAGFLO Calculations for DBR Volumes
PA6.7.8 SECOTP2D Calculations
PA6.8.2.1 Construction of Cuttings and Cavings Releases
PA6.8.2.2 Construction of Spallings Releases
PA6.8.2.3 Construction of DBRs
PA6.8.3 Radionuclide Transport Through the Culebra
PA6.8.4 Determining Initial Conditions for Direct and Transport Releases
PA6.8.4.1 Determining Repository and Panel Conditions
PA6.8.4.2 Determining Distance from Previous Intrusions
PA6.9.3 Stepwise Regression Analysis
PA7.0 Results for the Undisturbed Repository
PA7.1.1 Pressure in the Repository
PA7.1.2 Brine Saturation in the Waste
PA7.1.3 Brine Flow Out of the Repository
PA7.2.1 Radionuclide Transport to the Culebra
PA7.2.2 Radionuclide Transport to the Land Withdrawal Boundary
PA8.0 Results for a Disturbed Repository
PA8.3.1 Pressure in the Repository
PA8.3.3 Brine Flow out of the Repository
PA8.4.1 Radionuclide Source Term
PA8.4.2 Transport through MBs and Shaft
PA8.4.3 Transport to the Culebra
PA8.4.3.1 Single Intrusion Scenarios
PA8.4.3.2 Multiple Intrusion Scenario
PA8.4.4 Transport Through the Culebra
PA8.4.4.1 Partial Mining Results
PA8.4.4.2 Full Mining Results
Figure PA1. Overall Mean CCDFs for Total Normalized Releases: CRA2009 PA and CRA2004 PABC
Figure PA2. Computational Models Used in PA
Figure PA3. Construction of the CCDF Specified in 40 CFR Part 191 Subpart B
Figure PA4. Distribution of CCDFs Resulting from Possible Values for the Sampled Parameters
Figure PA6. Example CCDF Distribution From CRA2009 PA (Replicate 1)
Figure PA7. Logic Diagram for Scenario Analysis
Figure PA8. Conceptual Release Pathways for the UP Scenario
Figure PA9. Conceptual Release Pathways for the Disturbed Repository M Scenario
Figure PA10. Conceptual Release Pathways for the Disturbed Repository Deep Drilling E2 Scenario
Figure PA11. Conceptual Release Pathways for the Disturbed Repository Deep Drilling E1 Scenario
Figure PA12. Conceptual Release Pathways for the Disturbed Repository Deep Drilling E1E2 Scenario
Figure PA13. CDF for Time Between Drilling Intrusions
Figure PA14. Discretized Locations for Drilling Intrusions
Figure PA15. Computational Grid Used in BRAGFLO for PA
Figure PA16. Definition of Element Depth in BRAGFLO Grid
Figure PA17. Identification of Individual Cells in BRAGFLO Grid
Figure PA19. Schematic Side View of Option D Panel Closure
Figure PA20. Representation of Option D Panel Closures in the BRAGFLO Grid
Figure PA21. Selecting Radionuclides for the Release Pathways Conceptualized by PA
Figure PA22. Detail of Rotary Drill String Adjacent to Drill Bit
Figure PA23. Schematic Diagram of the Flow Geometry Prior to Repository Penetration
Figure PA24. Schematic Diagram of the Flow Geometry After Repository Penetration
Figure PA25. Effective Wellbore Flow Geometry Before Bit Penetration
Figure PA26. Effective Wellbore Flow Geometry After Bit Penetration
Figure PA27. FiniteDifference Zoning for Wellbore
Figure PA28. DBR Grid Used in PA
Figure PA29. Assignment of Initial Conditions for DBR Calculation
Figure PA30. Borehole Representation Used for PoettmannCarpenter Correlation
Figure PA31. Areas of Potash Mining in the McNutt Potash Zone
Figure PA33. Boundary Conditions Used for Simulations of Brine Flow in the Culebra
Figure PA34. FiniteDifference Grid Showing Cell Index Numbering Convention Used by MODFLOW
Figure PA35. ParallelPlate, DualPorosity Conceptualization
Figure PA36. Schematic of FiniteVolume Staggered Mesh Showing Internal and Ghost Cells
Figure PA37. Illustration of Stretched Grid Used for Matrix Domain Discretization
Figure PA38. Correlation Between S_HALITE:PRMX_LOG and S_HALITE:COMP_RCK
Figure PA39. Correlation between CASTILLER:PRMX_LOG and CASTILLER:COMP_RCK
Figure PA40. Dependency between WAS_AREA:GRATMICI and WAS_AREA:GRATMICH
Figure PA41. Logic Diagram for Determining the Intrusion Type
Figure PA42. Processing of Input Data to Produce CCDFs
Figure PA43. Pressure in the Waste Panel Region, Replicate R1, Scenario S1, CRA2009 PA
Figure PA45. Brine Saturation in the Waste Panel Region, Replicate R1, Scenario S1, CRA2009 PA
Figure PA47. Brine Flow Away From the Repository, Replicate R1, Scenario S1, CRA2009 PA
Figure PA48. Brine Flow via All MBs Across the LWB, Replicate R1, Scenario S1, CRA2009 PA
Figure PA50. Pressure in the Waste Panel Region for Replicate R1, Scenario S2, CRA2009 PA
Figure PA51. Pressure in the Waste Panel Region for Replicate R1, Scenario S4, CRA2009 PA
Figure PA54. Brine Saturation in the Waste Panel Region for Replicate R1, Scenario S2, CRA2009 PA
Figure PA55. Brine Saturation in the Waste Panel Region for Replicate R1, Scenario S4, CRA2009 PA
Figure PA58. Total Cumulative Brine Outflow in Replicate R1, Scenario S2, CRA2009 PA
Figure PA59. Brine Flow via All MBs Across the LWB in Replicate R1, Scenario S2, CRA2009 PA
Figure PA60. Total Cumulative Brine Outflow in Replicate R1, Scenario S4, CRA2009 PA
Figure PA61. Brine Flow via All MBs Across the LWB in Replicate R1, Scenario S4, CRA2009 PA
Figure PA64. Total Mobilized Concentrations in Salado Brine, Replicate R1, CRA2009 PA
Figure PA65. Total Mobilized Concentrations in Castile Brine, Replicate R1, CRA2009 PA
Figure PA66. Cumulative Normalized Release to the Culebra, Scenario S2, CRA2009 PA
Figure PA67. Cumulative Normalized Release to the Culebra, Scenario S3, CRA2009 PA
Figure PA68. Cumulative Normalized Release to the Culebra, Scenario S4, CRA2009 PA
Figure PA69. Cumulative Normalized Release to the Culebra, Scenario S5, CRA2009 PA
Figure PA70. Cumulative Normalized Release to the Culebra, Replicate R1, Scenario S6, CRA2009 PA
Figure PA71. Scatterplot of Waste Particle Diameter Versus Spallings Volume, CRA2009 PA
Figure PA72. Scatterplot of Waste Permeability Versus Spallings Volume, CRA2009 PA
Figure PA74. Overall Mean CCDFs for Cuttings and Cavings Releases: CRA2009 PA and CRA2004 PABC
Figure PA75. Overall Mean CCDFs for Spallings Releases: CRA2009 PA and CRA 2004 PABC
Figure PA76. Overall Mean CCDFs for DBRs: CRA2009 PA and CRA2004 PABC
Figure PA78. The Preponderance and Distribution of Zeros Can Control the Regression
Figure PA79. Total Normalized Releases, Replicates R1, R2, and R3, CRA2009 PA
Figure PA80. Confidence Interval on Overall Mean CCDF for Total Normalized Releases, CRA2009 PA
Figure PA81. Mean CCDFs for Components of Total Normalized Releases, Replicate R1, CRA2009 PA
Figure PA82. Mean CCDFs for Components of Total Normalized Releases, Replicate R2, CRA2009 PA
Figure PA83. Mean CCDFs for Components of Total Normalized Releases, Replicate R3, CRA2009 PA
Figure PA84. Overall Mean CCDFs for Total Normalized Releases: CRA2009 PA and CRA2004 PABC
Table PA1. WIPP Project Changes and Cross References
Table PA3. Parameter Values Used in Representation of TwoPhase Flow
Table PA4. Models for Relative Permeability and Capillary Pressure in TwoPhase Flow
Table PA5. Initial Conditions in the Rustler
Table PA7. Permeabilities for Drilling Intrusions Through the Repository
Table PA8. Boundary Value Conditions for Pg and Pb
Table PA9. Auxiliary Dirichlet Conditions for Sg and Pb
Table PA10. Calculated Values for Dissolved Solubility (see Appendix SOTERM2009, Table SOTERM16)
Table PA13. Combination of Radionuclides for Transport
Table PA14. Initial and Boundary Conditions for Cbl( x, y, t) and Csl( x, y, t)
Table PA15. Uncertain Parameters in the DRSPALL Calculations
Table PA16. Initial Porosity in the DBR Calculation
Table PA17. Boundary Conditions for pb and Sg in DBR Calculations
Table PA18. Radionuclide Culebra Transport Diffusion Coefficients
Table PA19. Variables Representing Epistemic Uncertainty in the CRA2009 PA
Table PA20. Sampled Parameters Added Since the CRA2004 PA
Table PA21. Sampled Parameters Removed Since the CRA2004 PA
Table PA24. Algorithm to Generate a Single Future
Table PA25. BRAGFLO Scenarios in the CRA2009 PA
Table PA26. NUTS Release Calculations in the CRA2009 PA
Table PA27. CUTTINGS_S Release Calculations in the CRA2009 PA
Table PA28. MODFLOW Scenarios in the CRA2009 PA
Table PA29. SECOTP2D Scenarios in the CRA2004 PA
Table PA32. CRA2009 PA Cuttings and Cavings Area Statistics
Table PA33. CRA2009 PA Spallings Volume Statistics
% percent
AIC active institutional control
An actinide
BRAGFLO BRine And Gas FLOw computer code
C Celsius
CCA Compliance Certification Application
CCDF complementary cumulative distribution function
CDF cumulative distribution function
CFR Code of Federal Regulations
CHTRU contacthandled transuranic
Ci curies
CL Confidence Limit
CPR cellulosic, plastic, and rubber
CRA Compliance Recertification Application
DBR direct brine release
DDZ drilling damaged zone
DOE U.S. Department of Energy
DP disturbed repository performance
DRZ disturbed rock zone
E deep drilling scenario
EPA U.S. Environmental Protection Agency
ERDA U.S.Energy Research and Development Administration
FEP feature, event, and process
FMT FractureMatrix Transport
FVW fraction of excavated repository volume occupied by waste
gal gallon
GWB Generic Weep Brine
in inch
J Joule
K Kelvin
K_{d} distribution coefficient
kg kilogram
km kilometer
km^{2} square kilometers
L liter
LHS Latin hypercube sampling
LWB land withdrawal boundary
M mining scenario
m meter
m^{2} square meters
m^{3} cubic meters
MB marker bed
ME mining and drilling scenario
mi miles
mol mole
MPa megapascal
MTHM metric tons of heavy metal
MWd megawattdays
N Newton
Pa Pascal
PA performance assessment
PABC performance assessment baseline calculation
PAVT Performance Assessment Verification Test
PCC partial correlation coefficient
PCS panel closure system
PDE partial differential equation
PDF probability distribution function
PIC passive institutional control
PRCC partial rank correlation coefficient
RHTRU remotehandled transuranic
RKS RedlichKwongSoave
RoR Rest of Repository
s second
s^{2} seconds squared
SCF/d standard cubic feet per day
SMC Salado Mass Concrete
SNL Sandia National Laboratories
SRC standardized regression coefficient
T field transmissivity field
TRU transuranic
TVD Total Variation Diminishing
UP undisturbed repository performance
WIPP Waste Isolation Pilot Plant
yr year
Al aluminum
Am americium
C carbon
C_{6}H_{10}O_{5} generic formula for CPR
Ca calcium
CH_{4} methane
Cm curium
CO_{2} carbon dioxide
Cr chromium
Cs cesium
Fe iron
H_{2} hydrogen gas
H_{2}O water
H_{2}S hydrogen sulfide
I iodine
Mg magnesium
Mg(OH)_{2 } brucite
Mg_{5}(CO_{3})_{4}(OH)_{2}×4H_{2}O hydromagnesite (5424)
MgO magnesium oxide, or periclase
Mn manganese
Ni nickel
NO_{3}^{ } nitrate
Np neptunium
Pb lead
Pm promethium
Pu plutonium
Ra radium
Sn tin
SO_{4} sulfate
SO_{4}^{2} sulfate ion
Sr strontium
Tc technetium
Th thorium
U uranium
V vanadium
This appendix presents the mathematical models used to evaluate performance of the Waste Isolation Pilot Plant (WIPP) disposal system and the results of these models for the 2009 Compliance Recertification Application (CRA2009) Performance Assessment (PA). The term PA signifies an analysis that (1) identifies the processes and events that might affect the disposal system; (2) examines the effects of these processes and events on the performance of the disposal system; and (3) estimates the cumulative releases of radionuclides, considering the associated uncertainties, caused by all significant processes and events (40 CFR § 191.12 [U.S. Environmental Protection Agency 1993]). PA is designed to address three primary questions about the WIPP:
Q1: What processes and events that might affect the disposal system could take place at the WIPP site over the next 10,000 years?
Q2: How likely are the various processes and events that might affect the disposal system to take place at the WIPP site over the next 10,000 years?
Q3: What are the consequences of the various processes and events that might affect the disposal system that could take place at the WIPP site over the next 10,000 years?
In addition, accounting for uncertainty in the parameters of the PA models leads to a further question:
Q4: How much confidence should be placed in answers to the first three questions?
These questions give rise to a methodology for quantifying the probability distribution of possible radionuclide releases from the WIPP repository over the next 10,000 years and characterizing the uncertainty in that distribution due to imperfect knowledge about the parameters contained in the models used to predict releases. The containment requirements of 40 CFR § 191.13 require this probabilistic methodology.
This appendix is organized as follows: Section PA2.0 gives an overview and describes the overall conceptual structure of the CRA2009 PA. The WIPP PA is designed to address the requirements of section 191.13, and thus involves three basic entities: (1) a probabilistic characterization of different futures that could occur at the WIPP site over the next 10,000 years, (2) models for both the physical processes that take place at the WIPP site and the estimation of potential radionuclide releases that may be associated with these processes, and (3) a probabilistic characterization of the uncertainty in the models and parameters that underlies the WIPP PA. Section PA2.0 is supplemented by Appendix SCR2009, which documents the results of the screening process for features, events, and processes (FEPs) that are retained in the conceptual models of repository performance.
Section PA3.0 describes the probabilistic characterization of different futures. This characterization plays an important role in the construction of the complementary cumulative distribution function (CCDF) specified in section 191.13. Regulatory guidance and extensive review of the WIPP site identified exploratory drilling for natural resources and the mining of potash as the only significant disruptions at the WIPP site with the potential to affect radionuclide releases to the accessible environment. Section PA3.0 summarizes the stochastic variables that represent future drilling and mining events in the PA. The results of the PA for CRA2009, as documented in Section PA7.0, Section PA8.0, and Section PA9.0 of this appendix, confirm that direct releases from drilling intrusions are the major contributors to radionuclide releases to the accessible environment.
Section PA4.0 presents the mathematical models for both the physical processes that take place at the WIPP and the estimation of potential radionuclide releases. The mathematical models implement the conceptual models as prescribed in 40 CFR § 194.23 (2004), and permit the construction of the CCDF specified in section 191.13. Models presented in Section PA4.0 include twophase (i.e., gas and brine) flow in the vicinity of the repository; radionuclide transport in the Salado Formation (hereafter referred to as the Salado); releases to the surface at the time of a drilling intrusion due to cuttings, cavings, spallings, and direct brine releases (DBRs); brine flow in the Culebra Dolomite Member of the Rustler Formation (hereafter referred to as the Culebra); and radionuclide transport in the Culebra. Section PA4.0 is supplemented by Appendices MASS2009, TFIELD2009, and PORSURF2009. Appendix MASS2009 discusses the modeling assumptions used in the WIPP PA. Appendix TFIELD2009 discusses the generation of the transmissivity fields (T fields) used to model groundwater flow in the Culebra. Appendix PORSURF2009 presents results from modeling the effects of excavated region closure, waste consolidation, and gas generation in the repository.
Section PA5.0 discusses the probabilistic characterization of parameter uncertainty, and summarizes the uncertain variables incorporated into the CRA2009 PA, the distributions assigned to these variables, and the correlations between variables. Section PA5.0 is supplemented by Fox (2008) and Appendix SOTERM2009. Fox (2008) catalogs the full set of parameters used in the CRA2009 PA, previously referred to as the CCA Appendix PAR and the CRA2004, Appendix PA, Attachment PAR. Appendix SOTERM2009 describes the actinide (An) source term for the WIPP performance calculations, including the mobile concentrations of actinides that may be released from the repository in brine.
Section PA6.0 summarizes the computational procedures used in the CRA2009 PA, including sampling techniques (i.e., random and Latin hypercube sampling (LHS)); sample size, statistical confidence for mean CCDF, generation of sample, generation of individual futures, construction of CCDFs, calculations performed with the models discussed in Section PA4.0, construction of releases for each future, and the sensitivity analysis techniques in use.
Section PA7.0 presents the results of the PA for an undisturbed repository. Releases from the undisturbed repository are determined by radionuclide transport in brine flowing from the repository to the land withdrawal boundary (LWB) through the marker beds (MBs) or shafts. Releases in the undisturbed scenario are used to demonstrate compliance with the individual and groundwater protection requirements in 40 CFR Part 191 (40 CFR § 194.51 and 40 CFR § 194.52).
Section PA8.0 presents PA results for a disturbed repository. As will be discussed in Section PA2.3.1, the only future events and processes in the analysis of disturbed repository performance are those associated with mining and deep drilling. Release mechanisms include direct releases at the time of the intrusion via cuttings, cavings, spallings, and DBR, and longterm releases via radionuclide transport up abandoned boreholes to the Culebra and thence to the LWB.
Section PA9.0 presents the set of CCDFs resulting from the CRA2009 PA. This material supplements 40 CFR § 194.34, which demonstrates compliance with the containment requirements of section 191.13. Section PA9.0 presents the most significant output variables from the PA models, accompanied by sensitivity analyses to determine which subjectively uncertain parameters are most influential in the uncertainty of PA results.
The overall structure of the CRA2009 PA does not differ from that presented in the first WIPP Compliance Certification Application (CCA) (U.S. Department of Energy 1996), the CRA2004 (U.S. Department of Energy 2004) or the CRA2004 Performance Assessment Baseline Calculation (PABC) (Leigh et al. 2005). This recertification application appendix follows the approach used by Helton et al. (1998) to document the mathematical models used in the CCA PA and the results of that analysis. Much of the content of this appendix derives from Helton et al. (1998); these authors’ contributions are gratefully acknowledged.
Because of the amount and complexity of the material presented in Appendix PA2009, an introductory summary is provided below, followed by detailed discussions of the topics in the remainder of this section, which is organized as follows:
Section PA2.1 – Overview of PA and the results
Section PA2.2 – The conceptual structure of the PA used to evaluate compliance with the containment requirements
Section PA2.3 – The overall methodology used to develop FEPs, the screening methodology applied to the FEPs, the results of the screening process, and the development of the scenarios considered in the systemlevel consequence analysis
The U.S. Department of Energy (DOE) continues to use the same PA methodology as in the CCA and CRA2004 because changes that have been made since the U.S. Environmental Protection Agency (EPA) certified WIPP do not impact PA methodology. A corresponding detailed presentation for the CRA2004 PA methodology is provided in the CRA2004, Chapter 6.0, Section 6.1, and a detailed presentation for the CRA2004 PABC implementation is provided in Leigh et al. (2005). A corresponding detailed presentation for the CCA PA methodology is provided in Helton et al. (1998, Section 2).
A demonstration of future repository performance was required by the disposal standards in Part 191. The EPA required a PA to demonstrate that potential cumulative releases of radionuclides to the accessible environment over a 10,000year period after disposal are less than specified limits based on the nature of the materials disposed (section 191.13). The PA is to determine the effects of all significant processes and events that may affect the disposal system, consider the associated uncertainties of the processes and events, and estimate the probable cumulative releases of radionuclides.
A PA was included in the CCA. This was the first demonstration of compliance by the DOE with the EPA’s disposal standards. The EPA required a verification PA based on the CCA PA that included revised parameters and distributions. This PA was termed the CCA Performance Assessment Verification Test (PAVT) (Trovoto 1997). The EPA based the original certification of the WIPP on the information in the CCA and the results of the CCA PAVT (U.S. Environmental Protection Agency 1998a). The CCA PAVT is documented in Sandia National Laboratories (SNL) 1997 and U.S. Department of Energy 1997.
The WIPP is required to be recertified every five years after first waste receipt (Public Law 02579). A revised PA was included in the first recertification application in 2004. This PA is termed the CRA2004 PA, and is documented in the CRA2004, Chapter 6.0. The EPA again required a verification PA using revised modeling assumptions and parameters (Cotsworth 2005). This PA was termed the CRA2004 PABC and was documented in Leigh et al. (2005).
As part of the fiveyear recertification cycle, a PA is included in the CRA2009. The CRA2009 PA is a culmination of the previous PAs and is not significantly different from the CRA2004 PABC, as the methodologies, conceptual models, and assumptions have not changed. Updates to parameters and improvements to computer codes are incorporated in the CRA2009.
A list of changes to PA since the CRA2004 and citations for where they are discussed is shown in Table PA1. In addition to the changes discussed in Table PA1, the terminology used to describe uncertainty has been updated to reflect the usage now prevalent in the risk assessment literature. Previously, uncertainty in model parameters was referred to as “subjective uncertainty,” and that due to stochastic processes was referred to as “stochastic uncertainty.” In the years since these terms were first employed, these concepts have matured, and the term “epistemic uncertainty” is now used to describe uncertainty from lack of knowledge, while the term “aleatory uncertainty” now describes uncertainty due to natural variability, e.g. uncertainty arising from future events whose occurrence can be defined in terms of probabilities. In this text, the terms epistemic uncertainty and aleatory uncertainty are used in place of the terms subjective uncertainty and stochastic uncertainty, respectively.
Table PA1. WIPP Project Changes and Cross References
WIPP Project Change 
Cross Reference 
CRA2004 to CRA2004 PABC Changes^{a} 

Inventory Information 

An Solubility Values 

An Solubility Uncertainty Ranges 
Garner and Leigh 2005 
Microbial Gas Generation Model 

Culebra T Fields Mining Modification 

Anhydrite Material Parameters 

SPALL Model Parameters 

CRA2004 PABC to CRA2009 Changes^{b} 

DBR Maximum Duration Parameter 

Conditional Relationship Between Humid and Inundated Cellulosic, Plastic, or Rubber (CPR) Degradation Rates 

BRAGFLO Code Improvements 

Capillary Pressure and Relative Permeability Model 
Nemer and Clayton 2008 
Drilling Rate 

Parameter Error Corrections · Emplaced CPR · Halite/Disturbed Rock Zone (DRZ) Parameter · Fraction of Repository Occupied by Waste · NUTS and DBR Calculation Input Files 
Nemer 2007a, Ismail 2007a, Dunagan 2007, Ismail 2007b, Clayton 2007 
^{a} See Leigh et al. 2005 for additional discussion of these changes and their implementation in the CRA2004 PABC. ^{b} See Clayton et al. 2008 for additional discussions of these changes and their implementation in the CRA2009 PA. 
From this assessment, the DOE has demonstrated that the WIPP continues to comply with the containment requirements of section 191.13. The containment requirements are stringent and state that the DOE must demonstrate a reasonable expectation that the probabilities of cumulative radionuclide releases from the disposal system during the 10,000 years following closure will fall below specified limits. The PA analyses supporting this determination must be quantitative and consider uncertainties caused by all significant processes and events that may affect the disposal system, including future inadvertent human intrusion into the repository. A quantitative PA is conducted using a series of loosely coupled computer models in which epistemic parameter uncertainties are addressed by a stratified Monte Carlo sampling procedure on selected input parameters, and uncertainties related to future intrusion events are addressed using simple random sampling.
As required by regulation, results of the PA are displayed as CCDFs showing the probability that cumulative radionuclide releases from the disposal system will exceed the values calculated for scenarios considered in the analysis. These CCDFs are calculated using reasonable and, in some cases, conservative conceptual models based on the scientific understanding of the disposal system’s behavior. Parameters used in these models are derived from experimental data, field observations, and relevant technical literature. Changes to the CCA and CRA2004 parameters and models since the original certification have been incorporated into the CRA2009 PA (Clayton 2008a). The overall mean CCDF continues to lie entirely below the specified limits, and the WIPP therefore continues to be in compliance with the containment requirements of 40 CFR Part 191 Subpart B (see Section PA2.1.6). Sensitivity analysis of results shows that the location of the mean CCDF is dominated by radionuclide releases that could occur on the surface during an inadvertent penetration of the repository by a future drilling operation (Section PA9.0). Releases of radionuclides to the accessible environment from transport in groundwater through the shaft seal systems and the subsurface geology are negligible, with or without human intrusion, and do not significantly contribute to the mean CCDF. No releases are predicted to occur at the ground surface in the absence of human intrusion. The natural and engineered barrier systems of the WIPP provide robust and effective containment of transuranic (TRU) waste, even if the repository is penetrated by multiple boreholes.
The foundations of PA are a thorough understanding of the disposal system and the possible future interactions of the repository, waste, and surrounding geology. The DOE’s confidence in the results of PA is based in part on the strength of the original research done during site characterization, experimental results used to develop and confirm parameters and models, and robustness of the facility design.
The progression of compliance applications document these aspects of PA leading up to the CRA2009 PA (i.e., the CCA, the CCA PAVT [Sandia National Laboratories 1997 and U.S. Department of Energy 1997], the CRA2004 PA, and the CRA2004 PABC [Leigh et al. 2005]).
The interactions of the repository and waste with the geologic system, and the response of the disposal system to possible future inadvertent human intrusion, are described in Section PA2.1.4.
An evaluation of undisturbed repository performance, which is defined to exclude human intrusion and unlikely disruptive natural events, is required by regulation (see 40 CFR § 191.15 and 40 CFR § 191.24). Evaluations of past and present natural geologic processes in the region indicate that none has the potential to breach the repository within 10,000 years (see the CCA, Appendix SCR, Section SCR.1). Disposal system behavior is dominated by the coupled processes of rock deformation surrounding the excavation, fluid flow, and waste degradation. Each of these processes can be described independently, but the extent to which they occur is affected by the others.
Rock deformation immediately around the repository begins as soon as excavation creates a disturbance in the stress field. Stress relief results in some degree of brittle fracturing and the formation of a DRZ, which surrounds excavations in all deep mines including the WIPP repository. For the WIPP, the DRZ is characterized by an increase in permeability and porosity, and it may ultimately extend a few meters (m) from the excavated region. Salt will also deform by creep processes resulting from deviatoric stress, causing the salt to move inward and fill voids. Salt creep will continue until deviatoric stress is dissipated and the system is once again at stress equilibrium (see the CRA2004, Chapter 6.0, Section 6.4.3.1).
The ability of salt to creep, thereby healing fractures and filling porosity, is one of its fundamental advantages as a medium for geologic disposal of radioactive waste, and one reason it was recommended by the National Academy of Sciences (see the CCA, Chapter 1.0, Section 1.3). Salt creep provides the mechanism for crushed salt compaction in the shaft seal system, yielding properties approaching those of intact salt within 200 years (see the CCA, Appendix SEAL, Appendix D, Section D5.2). Salt creep will also cause the DRZ surrounding the shaft to heal rapidly around the concrete components of the seal system. In the absence of elevated gas pressure in the repository, salt creep would also substantially compact the waste and heal the DRZ around the disposal region. Fluid pressures can become large enough through the combined effect of salt creep reducing pore volumes, and gas generation from waste degradation processes, to maintain significant porosity (greater than 20%) within the disposal room throughout the performance period (see also the CRA2004, Chapter 6.0, Section 6.4.3).
Characterization of the Salado indicates that fluid flow from the far field does not occur on time scales of interest in the absence of an artificially imposed hydraulic gradient (see the CRA2004, Chapter 2.0, Section 2.1.3.4 for a description of Salado investigations). This lack of fluid flow is the second fundamental reason for choosing salt as a medium for geologic disposal of radioactive waste. Lack of fluid flow is a result of the extremely low permeability of evaporite rocks that make up the Salado. Excavating the repository has disturbed the natural hydraulic gradient and rock properties, resulting in fluid flow. Small quantities of interstitial brine present in the Salado move toward regions of low hydraulic potential, and brine seeps are observed in the underground repository. The slow flow of brine from halite into more permeable anhydrite MBs, and then through the DRZ into the repository, is expected to continue as long as the hydraulic potential within the repository is below that of the far field. The repository environment will also include gas, so the fluid flow must be modeled as a twophase process. Initially, the gaseous phase will consist primarily of air trapped at the time of closure, although other gases may form from waste degradation. In the PA, the gaseous phase pressure will rise due to creep closure, gas generation, and brine inflow, creating the potential for flow from the excavated region (see also the CRA2004, Chapter 6.0, Section 6.4.3.2).
An understanding of waste degradation processes indicates that the gaseous phase in fluid flow and the repository’s pressure history will be far more important than if the initial air were the only gas present. Waste degradation can generate significant additional gas by two processes (see also the CRA2004, Chapter 6.0, Section 6.4.3.3 for historical perspective, and Leigh et al. 2005, Section 2.3 and Section 2.4 for changes):
1. The generation of hydrogen (H_{2}) gas by anoxic corrosion of steels, other iron (Fe)based alloys, and aluminum (Al) and Albased alloys
2. The generation of carbon dioxide (CO_{2}), hydrogen sulfide (H_{2}S), and methane (CH_{4}) by anaerobic microbial consumption of waste containing CPR materials
Coupling these gasgeneration reactions to fluidflow and saltcreep processes is complex. Gas generation will increase fluid pressure in the repository, thereby decreasing the hydraulic gradient and deviatoric stress between the far field and the excavated region and inhibiting the processes of brine inflow and salt creep. Anoxic corrosion will also consume brine as it breaks down water to oxidize steels and other Febased alloys and release H_{2}. Thus, corrosion has the potential to be a selflimiting process, in that as it consumes all water in contact with steels and other Febased alloys, it will cease. Microbial reactions also require water, either in brine or the gaseous phase. It is assumed that microbial reactions will result in neither the consumption nor production of water.
The total volume of gas generated by corrosion and microbial consumption may be sufficient to result in repository pressures that approach lithostatic. Sustained pressures above lithostatic are not physically reasonable within the disposal system, because the more brittle anhydrite layers are expected to fracture if sufficient gas is present. The conceptual model implemented in the PA causes anhydrite MB permeability and porosity to increase rapidly as pore pressure approaches and exceeds lithostatic. This conceptual model for pressuredependent fracturing approximates the hydraulic effect of pressureinduced fracturing and allows gas and brine to move more freely within the MBs at higher pressures (see the CRA2004, Chapter 6.0, Section 6.4.5.2).
Overall, the behavior of the undisturbed disposal system will result in extremely effective isolation of the radioactive waste. Concrete, clay, and asphalt components of the shaft seal system will provide an immediate and effective barrier to fluid flow through the shafts, isolating the repository until salt creep has consolidated the compacted crushed salt components and permanently sealed the shafts. Around the shafts, the DRZ in halite layers will heal rapidly because the presence of the solid material within the shafts will provide rigid resistance to creep. The DRZ around the shaft, therefore, will not provide a continuous pathway for fluid flow (see the CRA2004, Chapter 6.0, Section 6.4.4). Similarly, the panel closure will rigidly resist creep, leading to a buildup of compressive stress which in turn will cause a rapid elimination of the DRZ locally. In PA, it is conservatively assumed that the DRZ does not heal around either the disposal region or the operations and experimental regions, and pathways for fluid flow may exist indefinitely to the overlying and underlying anhydrite layers (e.g., MB 139 and Anhydrites A and B). Some quantity of brine will be present in the repository under most conditions and may contain actinides mobilized as both dissolved and colloidal species. Gas generation by corrosion and microbial degradation is expected to occur, and will result in elevated pressures within the repository. These pressures will not significantly exceed lithostatic because the more brittle anhydrite layers will fracture and provide a pathway for gas to leave the repository. Fracturing due to high gas pressures may enhance gas and brine migration from the repository, but gas transport will not contribute to the release of actinides from the disposal system. Brine flowing out of the waste disposal region through anhydrite layers may transport actinides as dissolved and colloidal species. However, the quantity of actinides that may reach the accessible environment boundary through the interbeds during undisturbed repository performance is insignificant and has no effect on the compliance determination. No migration of radionuclides is expected to occur vertically through the Salado (see Section PA7.0, and Ismail and Garner 2008).
The WIPP PA is required by the performance standards to consider scenarios that include intrusions into the repository by inadvertent and intermittent drilling for resources. The probability of these intrusions is based on a future drilling rate. This rate was calculated using the method outlined in Section 33, which analyzes the past record of drilling events in the Delaware Basin. Active institutional controls (AICs) are assumed to prevent intrusion during the first 100 years after closure (40 CFR § 194.41). Passive institutional controls (PICs) were assumed in the CCA to effectively reduce the drilling rate by two orders of magnitude for the 600year period following 100 years of active control. However, in certifying the WIPP, the EPA denied credit for the effectiveness of passive controls for 600 years (U.S. Environmental Protection Agency 1998a). Although the CRA2009 PA does not include a reduced drilling intrusion rate to account for PICs, future PAs may do so. Future drilling practices are assumed to be the same as current practice, also consistent with regulatory criteria. These practices include the type and rate of drilling, emplacement of casing in boreholes, and the procedures implemented when boreholes are plugged and abandoned (see 40 CFR § 194.33).
Human intrusion by drilling may cause releases from the disposal system through five mechanisms:
1. Cuttings, which include material intersected by the rotary drilling bit
2. Cavings, which include material eroded from the borehole wall during drilling
3. Spallings, which include solid material carried into the borehole during rapid depressurization of the waste disposal region
4. DBRs, which include contaminated brine that may flow to the surface during drilling
5. Longterm brine releases, which include the contaminated brine that may flow through a borehole after it is abandoned
The first four mechanisms immediately follow an intrusion event and are collectively referred to as direct releases. The accessible environment boundary for these releases is the ground surface. The fifth mechanism, An transport by longterm groundwater flow, begins when concrete plugs are assumed to degrade in an abandoned borehole and may continue throughout the regulatory period. The accessible environment boundary for these releases is the lateral subsurface limit of the controlled area (CRA2004, Chapter 6.0, Section 6.0.2.3).
Repository conditions prior to intrusion will be the same as those for undisturbed repository performance, and all processes active in undisturbed repository performance will continue to occur following intrusion. Because an intrusion provides a pathway for radionuclides to reach the ground surface and enter the geological units above the Salado, additional processes will occur that don’t in the undisturbed repository performance. These processes include the mobilization of radionuclides as dissolved and colloidal species in repository brine and groundwater flow, and subsequent An transport in the overlying units. Flow and transport in the Culebra are of particular interest because it is the most transmissive unit above the repository. Thus, the Culebra is a potential pathway for lateral migration of contaminated brine in the event of a drilling intrusion accompanied by significant flow up the intrusion borehole (see the CRA2004, Chapter 6.0, Section 6.4.6.2).
In a rotary drilling operation, the volume of material brought to the surface as cuttings is calculated as the cylinder defined by the thickness of the unit and the diameter of the drill bit. The quantity of radionuclides released as cuttings is therefore a function of the activity of the intersected waste and the diameter of the intruding drill bit. The DOE uses a constant value of 0.31115 m (12.25 inches [in]), consistent with bits currently used at the WIPP depth in the Delaware Basin (see the CRA2004, Chapter 6.0, Section 6.4.12.5). The intersected waste activity may vary depending on the type of waste intersected. The DOE considers random penetrations into remotehandled (RH) transuranic (TRU) (RHTRU) waste and each of the 690 different waste streams (Leigh, Trone, and Fox 2005, Section 4.4) identified for contacthandled (CH) transuranic (TRU) (CHTRU) waste (569 and 693 waste streams were used in the CCA and the CRA2004, respectively; see the CRA2004, Chapter 6.0, Section 6.0.2.3.1).
The volume of particulate material eroded from the borehole wall by the drilling fluids and brought to the surface as cavings may be affected by the drill bit diameter, effective shear resistance of the intruded material, speed of the drill bit, viscosity of the drilling fluid and rate at which it is circulated in the borehole, and other properties related to the drilling process. The most important of these parameters, after drill bit diameter, is the effective shear resistance of the intruded material (Leigh et al. 2005, Section 7.2). In the absence of data describing the reasonable and realistic future properties of degraded waste and magnesium oxide (MgO) backfill, the DOE used conservative parameter values based on the properties of finegrained sediment. Other properties are assigned fixed values consistent with current practice. The quantity of radionuclides released as cavings depends on the volume of eroded material and its activity, which is treated in the same manner as the activity of cuttings (see also Section PA4.5 and Section PA6.8.2.1).
Unlike releases from cuttings and cavings, which occur with every modeled borehole intrusion, spalling releases will occur only if pressure in the wastedisposal region exceeds the hydrostatic pressure in the borehole. At lower pressures, below about 8 megapascals (MPa), fluid in the wastedisposal region will not flow toward the borehole. At higher pressures, gas flow toward the borehole may be sufficiently rapid to cause additional solid material to enter the borehole. If spalling occurs, the volume of spalled material will be affected by the physical properties of the waste, such as its tensile strength and particle diameter. The DOE based the parameter values used in the PA on reasonable and conservative assumptions. Since the CCA, a revised conceptual model for the spallings phenomena has been developed (see the CRA2004, Appendix PA, Section PA4.6 and Appendix PA, Attachment MASS, Section MASS16.1.3). Model development, execution, and sensitivity studies necessitated implementing parameter values pertaining to waste characteristics, drilling practices, and physics of the process. The parameter range for particle size was derived by expert elicitation (U.S. Department of Energy 1997).
The quantity of radionuclides released as spalled material depends on the volume of spalled waste and its activity. Because spalling may occur at a greater distance from the borehole than cuttings and cavings, spalled waste is assumed to have the volumeaveraged activity of CHTRU waste, rather than the sampled activities of individual waste streams. The low permeability of the region surrounding the RHTRU waste means it is isolated from the spallings process and does not contribute to the volume or activity of spalled material (see also Section PA4.6 and Section PA6.8.2.2 for more description of the spallings model).
Radionuclides may be released to the accessible environment if repository brine enters the borehole during drilling and flows to the ground surface. The quantity of radionuclides released by direct brine flow depends on the volume of brine reaching the ground surface and the concentration of radionuclides contained in the brine. As with spallings, DBRs will not occur if repository pressure is below the hydrostatic pressure in the borehole. At higher repository pressures, mobile brine present in the repository will flow toward the borehole. If the volume of brine flowing from the repository into the borehole is small, it will not affect the drilling operation, and flow may continue until the driller reaches the base of the evaporite section and installs casing in the borehole (see also Section PA4.7 and Section PA6.8.2.3).
Actinides may be mobilized in repository brine in two principal ways:
1. As dissolved species
2. As colloidal species
The solubilities of actinides depend on their oxidation states, with the more reduced forms (for example, III and IV oxidation states) being less soluble than the oxidized forms (V and VI). Conditions within the repository will be strongly reducing because of large quantities of metallic Fe in the steel containers and the waste, and—in the case of plutonium (Pu)—only the lowersolubility oxidation states (Pu(III) and Pu(IV)) will persist. Microbial activity will also help create reducing conditions. Solubilities also vary with pH. The DOE is therefore emplacing MgO in the wastedisposal region to ensure conditions that reduce uncertainty and establish low An solubilities. MgO consumes CO_{2} and buffers pH, lowering An solubilities in WIPP brines (see Appendix SOTERM2009 and Appendix MgO2009). Solubilities in the PA are based on the chemistry of brines that might be present in the wastedisposal region, reactions of these brines with the MgO engineered barrier, and strongly reducing conditions produced by anoxic corrosion of steels and other Febased alloys.
The waste contains organic ligands that could increase An solubilities by forming complexes with dissolved An species. However, these organic ligands also form complexes with other dissolved metals, such as magnesium (Mg), calcium (Ca), Fe, lead (Pb), vanadium (V), chromium (Cr), manganese (Mn), and nickel (Ni), that will be present in repository brines due to corrosion of steels and other Febased alloys. The CRA2009 PA speciation and solubility calculations include the effect of organic ligands but not the beneficial effect of competition with Fe, Pb, V, Cr, Mn, and Ni. (Appendix SOTERM2009, Section SOTERM2.3.6 and Section SOTERM4.6, and Brush and Xiong 2005).
Colloidal transport of actinides has been examined, and four types of colloids have been determined to represent the possible behavior at the WIPP. These include microbial colloids, humic substances, An intrinsic colloids, and mineral fragments. Concentrations of An mobilized as these colloidal forms are included in the estimates of total An concentrations used in PA (Appendix SOTERM2009, Section SOTERM3.8.1 and Section SOTERM4.7, and Garner and Leigh 2005).
Longterm releases to the ground surface or groundwater in the Rustler Formation (hereafter referred to as the Rustler) or overlying units may occur after the borehole has been plugged and abandoned. In keeping with regulatory criteria, borehole plugs are assumed to have properties consistent with current practice in the basin. Thus, boreholes are assumed to have concrete plugs emplaced at various locations. Initially, concrete plugs effectively limit fluid flow in the borehole. However, under most circumstances, these plugs cannot be expected to remain fully effective indefinitely. For the purposes of PA, discontinuous borehole plugs above the repository are assumed to degrade 200 years after emplacement. From then on, the borehole is assumed to fill with a siltysandlike material containing degraded concrete, corrosion products from degraded casing, and material that sloughs into the hole from the walls. Of six possible plugged borehole configurations in the Delaware Basin, three are considered either likely or adequately representative of other possible configurations; one configuration (a twoplug configuration) is explicitly modeled in the flow and transport model (see Section PA3.7 and Appendix MASS2009, Section MASS16.3).
If sufficient brine is available in the repository, and if pressure in the repository is higher than in the overlying units, brine may flow up the borehole following plug degradation. In principle, this brine could flow into any permeable unit or to the ground surface if repository pressure were high enough. For modeling purposes, brine is allowed to flow only into the higherpermeability units and to the surface. Lowerpermeability anhydrite and mudstone layers in the Rustler are treated as if they were impermeable to simplify the analysis while maximizing the amount of flow into units where it could potentially contribute to disposal system releases. Model results indicate that essentially all flow occurs into the Culebra, which has been recognized since the early stages of site characterization as the most transmissive unit above the repository and the most likely pathway for subsurface transport (see also the CRA2004, Chapter 2.0, Section 2.2.1.4.1.2).
Site characterization activities in the units above the Salado have focused on the Culebra. These activities have shown that the direction of groundwater flow in the Culebra varies somewhat regionally, but in the area that overlies the repository, flow is southward. These characterization and modeling activities conducted in the units above the Salado confirm that the Culebra is the most transmissive unit above the Salado. The Culebra is the unit into which actinides are likely to be introduced from longterm flow up an abandoned borehole. Regional variation in the Culebra’s groundwater flow direction is influenced by the transmissivity observed, as well as the lateral (facies) changes in the lithology of the Culebra in the groundwater basin where the WIPP is located. Site characterization activities have provided no evidence of karst groundwater systems in the controlled area, although groundwater flow in the Culebra is affected by the presence of fractures, fracture fillings, and vuggy pore features (see Appendix HYDRO2009 and the CRA2004, Chapter 2.0, Section 2.1.3.5). Other laboratory and field activities have focused on the behavior of dissolved and colloidal actinides in the Culebra.
Basinscale regional modeling of threedimensional groundwater flow in the units above the Salado demonstrates that it is appropriate, for the purposes of estimating radionuclide transport, to conceptualize the Culebra as a twodimensional confined aquifer (see the CRA2004, Chapter 2.0, Section 2.2.1.1). Uncertainty in the flow field is incorporated by using 100 different geostatistically based T fields, each of which is consistent with available head and transmissivity data (Appendix PA2009, Appendix TFIELD2009).
Groundwater flow in the Culebra is modeled as a steadystate process, but two mechanisms considered in the PA could affect flow in the future. Potash mining in the McNutt Potash Zone (hereafter referred to as the McNutt) of the Salado, which occurs now in the Delaware Basin outside the controlled area and may continue in the future, could affect flow in the Culebra if subsidence over mined areas causes fracturing or other changes in rock properties (see the CRA2004, Chapter 6.0, Section 6.3.2.3). Climatic changes during the next 10,000 years may also affect groundwater flow by altering recharge to the Culebra (see the CRA2004, Chapter 6.0, Section 6.4.9 and the CCA, Appendix CLI).
Consistent with regulatory criteria of 40 CFR § 194.32, mining outside the controlled area is assumed to occur in the near future, and mining within the controlled area is assumed to occur with a probability of 1 in 100 per century (adjusted for the effectiveness of AICs during the first 100 years after closure). Consistent with regulatory guidance, the effects of mine subsidence are incorporated in PA by increasing the transmissivity of the Culebra over the areas identified as mineable by a factor sampled from a uniform distribution between 1 and 1000 (U.S. Environmental Protection Agency 1996a, p. 5229). T fields used in PA are therefore adjusted and steadystate flow fields calculated accordingly; once for mining that occurs only outside the controlled area, and once for mining that occurs both inside and outside the controlled area (Appendix TFIELD2009, Section 9.0). Mining outside the controlled area is considered in both undisturbed and disturbed repository performance.
The extent to which the climate will change during the next 10,000 years and how such a change will affect groundwater flow in the Culebra are uncertain. Regional threedimensional modeling of groundwater flow in the units above the Salado indicates that flow velocities in the Culebra may increase by a factor of 1 to 2.25 for reasonably possible future climates (see the CCA, Appendix CLI). This uncertainty is incorporated in PA by scaling the calculated steadystate specific discharge within the Culebra by a sampled parameter within this range.
Field tests have shown that the Culebra is best characterized as a doubleporosity medium for estimating contaminant transport in groundwater (see the CRA2004, Chapter 2.0, Section 2.2.1.4.1.2 and Appendix HYDRO2009). Groundwater flow and advective transport of dissolved or colloidal species and particles occurs primarily in a small fraction of the rock’s total porosity and corresponds to the porosity of open and interconnected fractures and vugs. Diffusion and slower advective flow occur in the remainder of the porosity, which is associated with the lowpermeability dolomite matrix. Transported species, including actinides (if present), will diffuse into this porosity.
Diffusion from the advective porosity into the dolomite matrix will retard An transport through two mechanisms. Physical retardation occurs simply because actinides that diffuse into the matrix are no longer transported with the flowing groundwater. Transport is interrupted until they diffuse back into the advective porosity. In situ tracer tests have demonstrated this phenomenon. Chemical retardation also occurs within the matrix as actinides are sorbed onto dolomite grains. The relationship between sorbed and liquid concentrations is assumed to be linear and reversible. The distribution coefficients (K_{d}s) that characterize the extent to which actinides will sorb on dolomite were based on experimental data (see the CRA2004, Chapter 6.0, Section 6.4.6.2).
Human intrusion scenarios evaluated in the PA include both single intrusion events and combinations of multiple boreholes. Two different types of boreholes are considered: those that penetrate a pressurized brine reservoir in the underlying Castile Formation (hereafter referred to as the Castile), and those that do not.
The presence of a brine reservoir under the repository is speculative, but on the basis of current information cannot be ruled out. A pressurized brine reservoir was encountered at the WIPP12 borehole within the controlled area to the north of the disposal region, and other pressurized brine reservoirs associated with regions of deformation in the Castile have been encountered elsewhere in the Delaware Basin (see the CRA2004, Chapter 2.0, Section 2.2.1.2.2). Based on a geostatistical analysis of the geophysical data of brine encounters in the region, the DOE estimates that there is a 0.08 probability that a random borehole penetrating waste in the WIPP will also penetrate an underlying brine reservoir (see the CCA, Appendix MASS, Attachment 186). Upon their review of the CCA, the EPA determined that the DOE should treat this probability as uncertain, ranging from 0.01 to 0.60 in the CCA PAVT. The EPA also required the DOE to modify the assumptions concerning Castile properties to increase the brine reservoir volumes (U.S. Environmental Protection Agency 1998b; Technical Support Document for 194.23: Parameter Justification Report, Section 5). The EPA determined that changing the rock compressibility and porosity of the Castile effectively modified the sampled brine reservoir volume to include the possibility of larger brine reservoir volumes like those encountered by the WIPP12 borehole.
The primary consequence of penetrating a pressurized reservoir is to provide an additional source of brine beyond that which might flow into the repository from the Salado. Direct releases at the ground surface resulting from the first repository intrusion would be unaffected by additional Castile brine, even if it flowed to the surface, because brine moving straight up a borehole will not significantly mix with waste. However, the presence of Castile brine could significantly increase radionuclide releases in two ways. First, the volume of contaminated brine that could flow to the surface may be greater for a second or subsequent intrusion into a repository that has already been connected by a previous borehole to a Castile reservoir. Second, the volume of contaminated brine that may flow up an abandoned borehole after plug degradation may be greater for combinations of two or more boreholes that intrude the same panel if one of the boreholes penetrates a pressurized reservoir. Both processes are modeled in PA.
The DOE’s approach to demonstrating continued compliance is the PA, which is based on the criteria indicated in section 194.34. The PA process comprehensively considers the FEPs relevant to disposal system performance (see Appendix SCR2009). Those FEPs shown by screening analyses to potentially affect performance are included in quantitative calculations using a system of loosely coupled computer models to describe the interaction of the repository with the natural system, both with and without human intrusion. Uncertainty in parameter values is incorporated in the analysis by a Monte Carlo approach, in which multiple simulations (or realizations) are completed using sampled values for the imprecisely known input parameters (see the CRA2004, Chapter 6.0, Section 6.1.5). Distribution functions characterize the state of knowledge for these parameters, and each realization of the modeling system uses a different set of sampled input values. A sample size of 300 results in 300 different values of each parameter. Thus, there are 300 different sets (vectors) of input parameter values. These 300 vectors were divided among 3 replicates. Quality assurance activities demonstrate that the parameters, software, and analysis used in PA were the result of a rigorous process conducted under controlled conditions (40 CFR § 194.22).
Of the FEPs considered, exploratory drilling for natural resources was identified as the only disruption with sufficient likelihood and consequence of impacting releases from the repository. For each vector of parameters values, 10,000 possible futures (realizations) are simulated, where a single future is defined as a series of intrusion events that occur randomly in space and time (Section PA2.2). Each of these futures is assumed to have an equal probability of occurring; hence a probability of 0.0001. Cumulative radionuclide releases from the disposal system are calculated for each future, and CCDFs are constructed by sorting the releases from smallest to largest and then summing the probabilities across the future. Mean CCDFs were then computed for the three replicates of sampled parameters (Section PA2.2).
This section summarizes the results of the CRA2009 PA and demonstrates that the WIPP continues to comply with the quantitative containment requirements in 40 CFR § 191.13(a). The CRA2009 PA is different than the original certification PA in the CCA and the PA in the CRA2004 PABC because it includes additional information, changes, and new data required by 40 CFR § 194.15 recertification application requirements. Table PA1 details the changes and new information included in this PA.
The results of the CRA2009 PA demonstrate that the repository continues to comply with the disposal standards. The key metric for regulatory compliance is the mean CCDF. Figure PA1, which compares the overall mean CCDF for the CRA2009 PA to the overall mean CCDF for the CRA2004 PABC, demonstrates two key points. First, the overall mean CCDF lies entirely below the limits specified in section 191.13(a). Thus, the WIPP is in compliance with the containment requirements of Part 191. Second, for any probability, the expected releases in the CRA2009 PA are only slightly higher than those in the CRA2004 PABC, primarily because of the increased drilling rate.
Figure PA1. Overall Mean CCDFs for Total Normalized Releases: CRA2009 PA and CRA2004 PABC
Detailed results of the CRA2009 PA are contained in Section PA9.0, which describes sensitivity analyses conducted as the final step in the Monte Carlo analysis. These sensitivity analyses indicate the relative importance of each sampled parameter in terms of its contribution to uncertainty in estimating disposal system performance. Analyses also examine the sensitivity of intermediate performance measures to the sampled parameters. Examples of such intermediate performance measures include the quantity of radionuclides released to the accessible environment by any one mechanism (for example, cuttings or DBRs), and other model results that describe conditions of interest, such as disposal region pressure.
Section PA9.0 presents CCDF distributions for each replication of the analysis, mean CCDFs, and an overall mean CCDF with the 95% confidence interval estimated from the 3 independent mean distributions. All 300 individual CCDFs, as well as the overall mean CCDF determined from the 3 replicates, lie entirely below and to the left of the limits specified in section 191.13(a) (see Figure PA79). Thus, the WIPP continues to comply with the containment requirements of Part 191. Comparing the results of the 3 replicates indicates that the sample size of 100 in each replicate is sufficient to generate a stable distribution of outcomes (see Figure PA80). Within the region of regulatory interest (that is, at probabilities greater than 10^{ }^{3}/10^{4} year [yr]), the mean CCDFs from each replicate are essentially indistinguishable from the overall mean.
As discussed in Section PA9.1, Section PA9.2, and Section PA9.3, examining the normalized releases from cuttings and cavings, spallings, and DBRs provides insight into the relative importance of each release mode’s contribution to the mean CCDF’s location and the compliance determination. Releases from cuttings and cavings dominate the mean CCDF at high probabilities, while DBRs dominate the mean CCDF at low probabilities. Spallings are less important and have very little effect on the location of the mean. Subsurface releases from groundwater transport are less than 10^{ }^{6} EPA units and make no contribution to the mean CCDF’s location.
Uncertainties characterized in the natural system and the interaction of waste with the disposal system environment show little variation between the location of the mean CCDFs of the three replicates, providing additional confidence in the compliance determination. The natural and engineered barrier systems of the WIPP provide robust and effective containment of TRU waste even if the repository is penetrated by multiple borehole intrusions.
This section outlines the conceptual structure of the WIPP PA. First, a discussion of the regulatory requirements is given. The requirements of section 191.13 and section 194.34 (summarized in Section PA2.2.1) lead to the identification of three main PA components:
1. A probabilistic characterization of the likelihood for different futures to occur at the WIPP site over the next 10,000 years
2. A procedure for estimating the radionuclide releases to the accessible environment associated with each possible future that could occur at the WIPP site over the next 10,000 years
3. A probabilistic characterization of the uncertainty in the parameters used to estimate potential releases
The probabilistic methods employed in WIPP PA give rise to the CCDF specified in section 191.13(a) and the distributions specified by 40 CFR § 194.34(b).
The methodology employed in PA derives from the EPA’s standard for the geologic disposal of radioactive waste, Environmental Radiation Protection Standards for the Management and Disposal of Spent Nuclear Fuel, HighLevel and Transuranic Radioactive Wastes (Part 191) (U.S. Environmental Protection Agency 1993), which is divided into three subparts. 40 CFR Part 191 Subpart A applies to a disposal facility prior to decommissioning and establishes standards for the annual radiation doses to members of the public from waste management and storage operations. Part 191 Subpart B applies after decommissioning and sets probabilistic limits on cumulative releases of radionuclides to the accessible environment for 10,000 years (section 191.13) and assurance requirements to provide confidence that section 191.13 will be met (40 CFR § 191.14). Part 191 Subpart B also sets limits on radiation doses to members of the public in the accessible environment for 10,000 years of undisturbed repository performance (section 191.15). 40 CFR Part 191 Subpart C limits radioactive contamination of groundwater for 10,000 years after disposal (section 191.24). In this recertification application, the DOE must demonstrate a reasonable expectation that the WIPP will continue to comply with the requirements of Part 191 Subparts B and C.
The following is the central requirement in Part 191 Subpart B, and the primary determinant of the PA methodology (U.S. Environmental Protection Agency 1985, p. 38086).
§ 191.13 Containment Requirements:
(a) Disposal systems for spent nuclear fuel or highlevel or transuranic radioactive wastes shall be designed to provide a reasonable expectation, based upon performance assessments, that cumulative releases of radionuclides to the accessible environment for 10,000 years after disposal from all significant processes and events that may affect the disposal system shall:
(1) Have a likelihood of less than one chance in 10 of exceeding the quantities calculated according to Table 1 (Appendix A); and
(2) Have a likelihood of less than one chance in 1,000 of exceeding ten times the quantities calculated according to Table 1 (Appendix A).
(b) Performance assessments need not provide complete assurance that the requirements of 191.13(a) will be met. Because of the long time period involved and the nature of the events and processes of interest, there will inevitably be substantial uncertainties in projecting disposal system performance. Proof of the future performance of a disposal system is not to be had in the ordinary sense of the word in situations that deal with much shorter time frames. Instead, what is required is a reasonable expectation, on the basis of the record before the implementing agency, that compliance with 191.13(a) will be achieved.
Section 191.13(a) refers to “quantities calculated according to Table 1 (Appendix A),” which means a normalized radionuclide release to the accessible environment based on the type of waste being disposed, the initial waste inventory, and the size of release that may occur (U.S. Environmental Protection Agency 1985, Appendix A). Table 1 of Appendix A specifies allowable releases (i.e., release limits) for individual radionuclides and is reproduced as Table PA2. The WIPP is a repository for TRU waste, which is defined as “waste containing more than 100 nanocuries of alphaemitting TRU isotopes, with halflives greater than twenty years, per gram of waste” (U.S. Environmental Protection Agency 1985, p. 38084). The normalized release R for TRU waste is defined by
where Q_{i} is the cumulative release of radionuclide i to the accessible environment during the 10,000year period following closure of the repository (curies [Ci]), L_{i} is the release limit for radionuclide i given in Table PA2 (Ci), and C is the amount of TRU waste emplaced in the repository (Ci). In the CRA2009 PA, C = 2.32 ´ 10^{6} Ci (Leigh and Trone 2005, Section 3). Further, “accessible environment” means (1) the atmosphere, (2) land surfaces, (3) surface waters, (4) oceans, and (5) all of the lithosphere beyond the controlled area. “Controlled area” means (1) a surface location, to be identified by PICs, that encompasses no more than 100 square kilometers (km^{2}) and extends horizontally no more than 5 kilometers (km) in any direction from the outer boundary of the original radioactive waste’s location in a disposal system, and (2) the subsurface underlying such a location (section 191.12).
PAs are the basis for addressing the containment requirements. To help clarify the intent of Part 191, the EPA promulgated 40 CFR Part 194 (2004), Criteria for the Certification and
Table PA2. Release Limits for the Containment Requirements (U.S. Environmental Protection Agency 1985, Appendix A, Table 1)
Radionuclide 
Release Limit L_{i} per 1000 MTHM^{a} or Other Unit of Waste^{b} 
Americium241 or 243 
100 
Carbon14 
100 
Cesium135 or 137 
1,000 
Iodine129 
100 
Neptunium237 
100 
Pu238, 239, 240, or 242 
100 
Radium226 
100 
Strontium90 
1,000 
Technetium99 
10,000 
Thorium (Th) 230 or 232 
10 
Tin126 
1,000 
Uranium (U) 233, 234, 235, 236, or 238 
100 
Any other alphaemitting radionuclide with a halflife greater than 20 years 
100 
Any other radionuclide with a halflife greater than 20 years that does not emit alpha particles 
1,000 
^{a} Metric tons of heavy metal (MTHM) exposed to a burnup between 25,000 megawattdays (MWd) per metric ton of heavy metal (MWd/MTHM) and 40,000 MWd/MTHM. ^{b} An amount of TRU wastes containing one million Ci of alphaemitting TRU radionuclides with halflives greater than 20 years. 
Recertification of the Waste Isolation Pilot Plant’s Compliance with the Part 191 Disposal Regulations. There, an elaboration on the intent of section 191.13 is prescribed.
§ 194.34 Results of performance assessments.
(a) The results of performance assessments shall be assembled into “complementary, cumulative distributions functions” (CCDFs) that represent the probability of exceeding various levels of cumulative release caused by all significant processes and events.
(b) Probability distributions for uncertain disposal system parameter values used in performance assessments shall be developed and documented in any compliance application.
(c) Computational techniques, which draw random samples from across the entire range of the probability distributions developed pursuant to paragraph (b) of this section, shall be used in generating CCDFs and shall be documented in any compliance application.
(d) The number of CCDFs generated shall be large enough such that, at cumulative releases of 1 and 10, the maximum CCDF generated exceeds the 99th percentile of the population of CCDFs with at least a 0.95 probability.
(e) Any compliance application shall display the full range of CCDFs generated.
(f) Any compliance application shall provide information which demonstrates that there is at least a 95% level of statistical confidence that the mean of the population of CCDFs meets the containment requirements of § 191.13 of this chapter.
The DOE’s methodology for PA uses information about the disposal system and waste to evaluate performance over the 10,000year regulatory time period. To accomplish this task, the FEPs with potential to affect the future of the WIPP are first defined (Section PA2.3.1). Next, scenarios that describe potential future conditions in the WIPP are formed from logical groupings of retained FEPs (Section PA2.3.2). The scenario development process results in a probabilistic characterization for the likelihood of different futures that could occur at the WIPP (Section PA2.2.2). Using the retained FEPs, models are developed to estimate the radionuclide releases from the repository (Section PA2.2.3). Finally, uncertainty in model parameters is characterized probabilistically (Section PA2.2.4).
As discussed in Section PA2.3.1, the CCA PA scenario development process for the WIPP identified exploratory drilling for natural resources as the only disruption with sufficient likelihood and consequence of impacting releases from the repository (see the CCA, Appendix SCR). In addition, Part 194 specifies that the occurrence of mining within the LWB must be included in the PA. This has not changed for the CRA2009 PA. As a result, the projection of releases over the 10,000 years following closure of the WIPP is driven by the nature and timing of intrusion events.
The collection of all possible futures x_{st} forms the basis for the probability space (S_{st}, S_{st}, p_{st}) characterizing aleatory uncertainty, where S_{st} = {x_{st} : x_{st} is a possible future of the WIPP}, S_{st} is a suitably restricted collection of sets of futures, called “scenarios” (Section PA3.10), and p_{st} is a probability measure for the elements of S_{st} . A possible future, x_{st,i}, is thus characterized by the collection of intrusion events that occur in that future:
where
n is the number of drilling intrusions
t_{j} is the time (year) of the j^{th} intrusion
l_{j} designates the location of the j^{th} intrusion
e_{j} designates the penetration of an excavated or nonexcavated area by the j^{th} intrusion
b_{j} designates whether or not the j^{th} intrusion penetrates pressurized brine in the Castile Formation
p_{j} designates the plugging procedure used with the j^{th} intrusion (i.e., continuous plug, two discrete plugs, three discrete plugs)
a_{j} designates the type of waste penetrated by the j^{th} intrusion (i.e., no waste, CHTRU waste, RHTRU waste, and, for CHTRU waste, the waste streams encountered)
t_{min} is the time at which potash mining occurs within the LWB
The subscript st indicates that aleatory (i.e., stochastic) uncertainty is being considered; the letters st are retained for historical context. The subscript i indicates that the future x_{st} is one of many sample elements from S_{st}.
The probabilistic characterization of n, t_{j}, l_{j},and e_{j} is based on the assumption that drilling intrusions will occur randomly in time and space at a constant average rate (i.e., follow a Poisson process); the probabilistic characterization of b_{j} derives from assessed properties of brine pockets; the probabilistic characterization of a_{j} derives from the volumes of waste emplaced in the WIPP in relation to the volume of the repository; and the probabilistic characterization of p_{j} derives from current drilling practices in the sedimentary basin (i.e., the Delaware Basin) in which the WIPP is located. A vector notation is used for a_{j} because it is possible for a given drilling intrusion to miss the waste or to penetrate different waste types (CHTRU and RHTRU) as well as to encounter different waste streams in the CHTRU waste. Further, the probabilistic characterization for t_{min} follows from the criteria in Part 194 that the occurrence of potash mining within the LWB should be assumed to occur randomly in time (i.e., follow a Poisson process with a rate constant of l_{m} = 10^{ }^{4} yr^{ }^{1}), with all commercially viable potash reserves within the LWB extracted at time t_{min} . In practice, the probability measure p_{st} is defined by specifying probability distributions for each component of x_{st}, as discussed further in Section PA3.0.
Based on the retained FEPs (Section PA2.3.1), release mechanisms include direct transport of material to the surface at the time of a drilling intrusion (i.e., cuttings, spallings, and brine flow) and release subsequent to a drilling intrusion due to brine flow up a borehole with a degraded plug (i.e., groundwater transport). The quantities of releases are determined by the state of the repository through time, which is determined by the type, timing, and sequence of prior intrusion events. For example, pressure in the repository is an important determinant of spallings, and the amount of pressure depends on whether the drilling events that have occurred penetrated brine pockets and how long prior to the current drilling event the repository was inundated.
Computational models for estimating releases were developed using the retained FEPs; these models are summarized in Figure PA2. These computational models implement the conceptual models representing the repository system as described in section 194.23 and the mathematical models for physical processes presented in Section PA4.0. Most of the computational models involve the numerical solution of partial differential equations (PDEs) used to represent processes such as material deformation, fluid flow, and radionuclide transport.
The collection of computation models can be represented abstractly as a function f (x_{st}v_{su}), which quantifies the release that could result from the occurrence of a specific future x_{st} and a specific set of values for model parameters v_{su}. Because the future of the WIPP is unknown, the values of f (x_{st}v_{su}) are uncertain. Thus, the probability space (S_{st}, S_{st}, p_{st}), together with the function f (x_{st}v_{su}), give rise to the CCDF specified in section 191.13(a), as illustrated in Figure PA3. The CCDF represents the probability that a release from the repository greater than R will be observed, where R is a point on the abscissa (xaxis) of the graph (Figure PA3).
Formally, the CCDF depicted in Figure PA3 results from an integration over the probability space (S_{st}, S_{st}, p_{st}):
where d_{R}( f (x_{st}v_{su})) = 1 if f (x_{st}v_{su}) > R, d_{R}( f (x_{st}v_{su})) = 0 if f (x_{st}v_{su}) £ R, and d_{st}(x_{st}v_{su}) is the probability density function associated with the probability space (S_{st}, S_{st}, p_{st}). In practice, the integral in Equation (PA.3) is evaluated by a Monte Carlo technique, where a random sample x_{st,i}, i = 1, nR, is generated from S_{st} consistent with the probability distribution p_{st}. Using this random sample, Equation (PA.3) is numerically evaluated as
The models in Figure PA2 are too complex to permit a closedform evaluation of the integral in Equation (PA.4) that defines the CCDF specified in Part 191. In WIPP PA, these probability distribution functions (PDFs) are constructed using Monte Carlo simulation to sample the entire possible set of release outcomes. As long as the sampling is conducted properly and a sufficient number of samples is collected, the PDF of the sample should successfully approximate the PDF of the sample “universe” of all possible releases.
Figure PA2. Computational Models Used in PA
Figure PA3. Construction of the CCDF Specified in 40 CFR Part 191 Subpart B
In PA, the number of samples nR used to construct a CCDF is 10,000. However, the models in Figure PA2 are also too computationally intensive to permit their evaluation for each of these 10,000 futures. Due to this constraint, the models in Figure PA2 are evaluated for a relatively small number of specific scenarios, and the results of these evaluations are used to construct CCDFs. The representative scenarios are labeled E0, E1, E2, and E1E2, and are defined in Section PA3.10; the procedure for constructing a CCDF from these scenarios is described in Section PA6.0.
If the parameters used in the processlevel models of Figure PA2 were precisely known and if the models could accurately predict the future behavior of the repository, the evaluation of repository performance alone would be sufficient to answer the first three questions related to repository performance. However, the models do not perfectly represent the dynamics of the system and their parameters are not precisely known. Therefore, it is necessary to estimate the confidence one has in the CCDFs being constructed. The confidence in the CCDFs is established using Monte Carlo methods to evaluate how the uncertainty in the model parameters impacts the CCDFs or releases. The probabilistic characterization of the uncertainty in the model parameters is the outcome of the data development effort for the WIPP (summarized in Section 8.0 in Fox 2008).
Formally, uncertainty in the parameters that underlie the WIPP PA can be characterized by a second probability space (S_{su}, S_{su}, p_{su}), where the sample space S_{su} is defined by
S_{su} = {v_{su}: v_{su} is a sampled vector of parameter values} (PA.5)
The subscript su indicates that epistemic (i.e., subjective) uncertainty is being considered; the letters su are retained for historical context. An element v_{su} Î S_{su} is a vector v_{su} = v_{su,1}, v_{su,2}, …, v_{su,N}) of length N, where each element v_{su,k} is an uncertain parameter used in the models to estimate releases. In practice, the probability measure p_{su} is defined by specifying probability distributions for each element of v_{su}, discussed further in Section PA5.0.
If the actual value for v_{su} were known, the CCDF resulting from evaluation of Equation (PA.4) could be determined with certainty and compared with the criteria specified in Part 191. However, given the complexity of the WIPP site, the 10,000year period under consideration, and the state of knowledge about the natural and engineered system, values for v_{su} are not known with certainty. Rather, the uncertainty in v_{su} is characterized probabilistically, as described above, leading to a distribution of CCDFs (Figure PA4) with each CCDF resulting from one of many vectors of values of v_{su}. The uncertainty associated with the parameters is termed epistemic uncertainty, and has been referred to in WIPP PA documentation as subjective uncertainty.
WIPP PA uses a Monte Carlo procedure for evaluating the effects of epistemic uncertainty on releases. The procedure involves sampling the distributions assigned to the uncertain parameters and generating a CCDF of releases based on the results of the processlevel models generated using those parameters values. By repeating this process many times, a distribution of the
Figure PA4. Distribution of CCDFs Resulting from Possible Values for the Sampled Parameters
CCDFs can be constructed. The requirements of section 191.13 are evaluated, in part, using the mean probability of release. The overall mean probability curve is created by averaging across the CCDFs for releases, i.e., averaging the CCDFs across vertical slices (Figure PA4) (a formal definition is provided in Helton et al. 1998). In addition, confidence limits on the mean are computed using standard tstatistics (Figure PA5). The proximity of these curves to the boundary line in Figure PA3 indicates the confidence with which Part 191 will be met. Confidence is also established by examining the distribution of the CCDFs in relation to the release limits (Figure PA6).
WIPP PA uses a stratified sampling design called LHS (McKay, Beckman, and Conover 1979) to generate a sample v_{su}, i = 1, …, nLHS, from S_{su} consistent with the probability distribution p_{su}. LHS is an efficient scheme for sampling the range of a distribution using a relatively small sample. Based on order statistics, the sample size of nLHS = 300 replicates would provide coverage of 99% of the CCDF distribution with a confidence of 95%.
In Part 194, the EPA decided that the statistical portion of the determination of compliance with Part 191 will be based on the sample mean. The LHS sample sizes should be demonstrated operationally to improve (reduce the size of) the confidence interval for the estimated mean. The underlying principle is to show convergence of the mean (U.S. Environmental Protection Agency 1996b, p. 841).
Figure PA5. Example Mean Probability of Release and the Confidence Limits on the Mean from CRA2009 PA
Figure PA6. Example CCDF Distribution From CRA2009 PA (Replicate 1)
The DOE has chosen to demonstrate repeatability of the mean and to address the associated criteria of Part 194 using an operational approach of multiple replication, as proposed by Iman (1982). The complete set of PA calculations was repeated three times with all aspects of the analysis identical except for the random seed used to initiate the LHS procedure. Thus, PA results are available for 3 replicates, each based on an independent set of 100 LHS vectors drawn from identical distributions for imprecisely known parameters and propagated through an identical modeling system. This technique of multiple replication allows the adequacy of the sample size chosen in the Monte Carlo analysis to be evaluated and provides a suitable measure of confidence in the mean CCDF estimation used to demonstrate compliance with section 191.13(a).
This section addresses scenarios formed from FEPs that were retained for PA calculations, and introduces the specification of scenarios for consequence analysis.
The EPA has provided criteria concerning the scope of PAs in 40 CFR § 194.32. In particular, criteria relating to the identification of potential processes and events that may affect disposal system performance are provided in 40 CFR § 194.32(e), which states
Any compliance application(s) shall include information which:
(1) Identifies all potential processes, events or sequences and combinations of processes and events that may occur during the regulatory time frame and may affect the disposal system;
(2) Identifies the processes, events or sequences and combinations of processes and events included in performance assessments; and
(3) Documents why any processes, events or sequences and combinations of processes and events identified pursuant to paragraph (e)(1) of this section were not included in performance assessment results provided in any compliance application.
Section 32 of this application fulfills these criteria by documenting the DOE’s identification, screening, and screening results of all potential processes and events consistent with the criteria specified in section 194.32(e).
The first two steps in scenario development involve identifying and screening FEPs that are potentially relevant to the performance of the disposal system. The CRA2004, Chapter 6.0, Section 6.2 discusses the development of a comprehensive initial FEPs set used in the CCA, the methodology and criteria used for screening, the method used to reassess the CCA FEPs for the CRA2004, and a summary of the FEPs retained for scenario development. Changes to FEPs since the CRA2004 are outlined in Section 32 and Appendix SCR2009 of this application.
The original FEPs generation and screening were documented in the CCA, and the resulting FEPs list became the FEPs compliance baseline. The baseline contained 237 FEPs and was documented the CCA, Appendix SCR. The EPA compliance review of FEPs was documented in EPA’s Technical Support Document 194.32: Scope of PA (U.S. Environmental Protection Agency 1998c). The EPA numbered each FEP with a different scheme than the DOE used for the CCA. The DOE has since adopted the EPA’s numbering scheme.
Logic diagrams illustrate the formation of scenarios for consequence analysis from combinations of events that remain after FEP screening (Cranwell et al. 1990) (Figure PA7). Each scenario shown in Figure PA7 is defined by a combination of occurrence and nonoccurrence for all potentially disruptive events. Disruptive events are defined as those that create new pathways or significantly alter existing pathways for fluid flow and, potentially, radionuclide transport within the disposal system. Each of these scenarios also contains a set of features and nondisruptive events and processes that remain after FEP screening. As shown in Figure PA7, undisturbed repository performance (UP) and disturbed repository performance (DP) scenarios are considered in consequence modeling for the WIPP PA. The UP scenario is used for compliance assessments (40 CFR § 194.54 and 40 CFR § 194.55). Important aspects of UP and DP scenarios are summarized in this section.
The UP scenario is defined in 40 CFR § 191.12 to mean “the predicted behavior of a disposal system, including consideration of the uncertainties in predicted behavior, if the disposal system is not disrupted by human intrusion or the occurrence of unlikely natural events.” For compliance assessments with respect to the Individual and Groundwater Protection Requirements (section 191.15; Appendix IGP2009), it is only necessary to consider the UP scenario. The UP scenario is also considered with DP scenario for PA with respect to the containment requirements (section 191.13).
No potentially disruptive natural events and processes are likely to occur during the regulatory time frame. Therefore, all naturally occurring events and processes retained for scenario construction are nondisruptive and are considered part of the UP scenario. Mining outside the LWB is assumed at the end of AIC for all scenarios. The M scenario involves future mining within the controlled area. The disturbed repository E scenario involves at least one deep drilling event that intersects the waste disposal region. The M scenario and the E scenario may both occur in the future. The DOE calls a future in which both of these events occur the ME scenario. More detailed descriptions are found in Section PA2.3.2.2.
The only natural features and waste and repositoryinduced FEPs retained after screening that are excluded in the UP scenario, but included in the DP scenario, are those directly associated with the potential effects of future deep drilling within the controlled area. Among the most significant FEPs that will affect the UP scenario within the disposal system are excavationinduced fracturing, gas generation, salt creep, and MgO in the disposal rooms.
· The repository excavation and consequent changes in the rock stress field surrounding the excavated opening will create a DRZ immediately adjacent to excavated openings. The DRZ will exhibit mechanical and hydrological properties different than those of the intact rock.
Figure PA7. Logic Diagram for Scenario Analysis
· Organic material in the waste may degrade because of microbial activity, and brine will corrode metals in the waste and waste containers, with concomitant generation of gases. Gas generation may result in pressures sufficient to both maintain or develop fractures and change the fluid flow pattern around the waste disposal region.
· At the repository depth, salt creep will tend to heal fractures and reduce the permeability of the DRZ and the crushed salt component of the longterm shaft seals to near that of the host rock salt.
· The MgO engineered barrier emplaced in the disposal rooms will react with CO_{2} and maintain mildly alkaline conditions. Metal corrosion in the waste and waste containers will maintain reducing conditions. These effects will maintain low radionuclide solubility.
Radionuclides can become mobile as a result of waste dissolution and colloid generation following brine flow into the disposal rooms. Colloids may be generated from the waste (humics, mineral fragments, and An intrinsic colloids) or from other sources (humics, mineral fragments, and microbes).
Conceptually, there are several pathways for radionuclide transport within the undisturbed disposal system that may result in releases to the accessible environment (Figure PA8). Contaminated brine may migrate away from the wastedisposal panels if pressure within the panels is elevated by gas generated from corrosion or microbial consumption. Radionuclide transport may occur laterally, through the anhydrite interbeds toward the subsurface boundary of the accessible environment in the Salado, or through access drifts or anhydrite interbeds to the base of the shafts. In the latter case, if the pressure gradient between the panels and overlying strata is sufficient, contaminated brine may migrate up the shafts. As a result, radionuclides may be transported directly to the ground surface, or laterally away from the shafts through permeable strata such as the Culebra, toward the subsurface boundary of the accessible environment. These conceptual pathways are shown in Figure PA8.
Figure PA8. Conceptual Release Pathways for the UP Scenario
The modeling system described in Section PA4.0 includes potential radionuclide transport along other pathways, such as migration through Salado halite. However, the natural properties of the undisturbed system make radionuclide transport to the accessible environment via these other pathways unlikely.
Assessments for compliance with section 191.13 need to consider the potential effects of future disruptive natural and humaninitiated events and processes on the performance of the disposal system. No potentially disruptive natural events and processes are considered sufficiently likely to require inclusion in analyses of either the UP or DP scenario. The only future humaninitiated events and processes retained after FEP screening are those associated with mining and deep drilling (but not the subsequent use of a borehole) within the controlled area or LWB when institutional controls cannot be assumed to eliminate the possibility of such activities (Section PA3.2 and the CRA2004, Chapter 6.0, Section 6.4.12.1). In total, 21 disturbed repository FEPs associated with future mining and deep drilling have been identified. These FEPs were assigned a screening designator of the DP scenario.
For evaluating the consequences of disturbed repository performance, the DOE has defined the mining scenario (M), the deep drilling scenario (E), and a mining and drilling scenario (ME). These scenarios are described in the following sections.
The M scenario involves future mining within the controlled area. Consistent with the criteria stated by the EPA in 40 CFR § 194.32(b) for PA calculations, the effects of potential future mining within the controlled area are limited to changes in hydraulic conductivity of the Culebra that result from subsidence (as described in Section PA3.9).
Radionuclide transport may be affected in the M scenario if a head gradient between the wastedisposal panels and the Culebra causes brine contaminated with radionuclides to move from the wastedisposal panels to the base of the shafts and up to the Culebra. The changes in the Culebra T field may affect the rate and direction of radionuclide transport within the Culebra. Features of the M scenario are illustrated in Figure PA9.
The three disturbed repository FEPs labeled M in the CRA2004, Chapter 6.0, Table 69 are related to the occurrence and effects of future mining. The modeling system used for the M scenario is similar to that developed for the UP scenario, but with a modified Culebra T field in the controlled area to account for the mining effects.
The disturbed repository E scenario involves at least one deep drilling event that intersects the waste disposal region. The EPA provides criteria for analyzing the consequences of future drilling events in PA in 40 CFR § 194.33(c).
Performance assessments shall document that in analyzing the consequences of drilling events, the Department assumed that:
(1) Future drilling practices and technology will remain consistent with practices in the Delaware Basin at the time a compliance application is prepared. Such future drilling practices shall include, but shall not be limited to: the types and amounts of drilling fluids; borehole depths, diameters, and seals; and the fraction of such boreholes that are sealed by humans; and
Figure PA9. Conceptual Release Pathways for the Disturbed Repository M Scenario
(2) Natural processes will degrade or otherwise affect the capability of boreholes to transmit fluids over the regulatory time frame.
Consistent with these criteria, there are several pathways for radionuclides to reach the accessible environment in the E scenario. Before any deep drilling intersects the waste, potential release pathways are identical to those in the undisturbed repository scenario.
If a borehole intersects the waste in the disposal rooms, releases to the accessible environment may occur as material entrained in the circulating drilling fluid is brought to the surface. Particulate waste brought to the surface may include cuttings, cavings, and spallings. Cuttings are the materials cut by the drill bit as it passes through waste. Cavings are the materials eroded by the drilling fluid in the annulus around the drill bit. Spallings are the materials forced into the circulating drilling fluid if there is sufficient pressure in the waste disposal panels. During drilling, contaminated brine may flow up the borehole and reach the surface, depending on fluid pressure within the waste disposal panels.
When abandoned, the borehole is assumed to be plugged in a manner consistent with current practices in the Delaware Basin as prescribed in 40 CFR § 194.33(c)(1). An abandoned intrusion borehole with degraded casing and/or plugs may provide a pathway for fluid flow and contaminant transport from the intersected waste panel to the ground surface if the fluid pressure within the panel is sufficiently greater than hydrostatic. Additionally, if brine flows through the borehole to overlying units, such as the Culebra, it may carry dissolved and colloidal actinides that can be transported laterally to the accessible environment by natural groundwater flow in the overlying units.
Alternatively, the units intersected by an intrusion borehole may provide sources for brine flow to a waste panel during or after drilling. For example, in the northern Delaware Basin, the Castile, which underlies the Salado, contains isolated volumes of brine at fluid pressures greater than hydrostatic (as discussed in the CRA2004, Chapter 2.0, Section 2.2.1.2.2). The WIPP12 penetration of one of these reservoirs provided data on one brine reservoir within the controlled area. The location and properties of brine reservoirs cannot be reliably predicted; thus, the possibility of a deep borehole penetrating both a waste panel and a brine reservoir is accounted for in consequence analysis of the WIPP, as discussed in the CRA2004, Chapter 6.0, Section 6.4.8. Such a borehole could provide a connection for brine flow from the Castile to the waste panel, thus increasing fluid pressure and brine volume in the waste panel.
A borehole that is drilled through a disposal room pillar, but does not intersect waste, could also penetrate the brine reservoir underlying the waste disposal region. Such an event would, to some extent, depressurize the brine reservoir, and thus would affect the consequences of any subsequent reservoir intersections. The PA does not take credit for possible brine reservoir depressurization.
The DOE has distinguished two types of deep drilling events by whether or not the borehole intersects a Castile brine reservoir. A borehole that intersects a waste disposal panel and penetrates a Castile brine reservoir is designated an E1 event. A borehole that intersects a waste panel but does not penetrate a Castile brine reservoir is designated an E2 event. The consequences of deep drilling intrusions depend not only on the type of a drilling event, but on whether the repository was penetrated by an earlier E2 event or flooded due to an earlier E1 event. The PA also does not take credit for depressurization of brine reservoirs from multiple drilling intrusions. These scenarios are described in order of increasing complexity in the following sections.
The E2 scenario is the simplest scenario for inadvertent human intrusion into a waste disposal panel. In this scenario, a panel is penetrated by a drill bit; cuttings, cavings, spallings, and brine flow releases may occur; and brine flow may occur in the borehole after it is plugged and abandoned. Sources for brine that may contribute to longterm flow up the abandoned borehole are the Salado or, under certain conditions, the units above the Salado. An E2 scenario may involve more than one E2 drilling event, although the flow and transport model configuration developed for the E2 scenario evaluates the consequences of futures that have only one E2 event. Features of the E2 scenario are illustrated in Figure PA10.
Figure PA10. Conceptual Release Pathways for the Disturbed Repository Deep Drilling E2 Scenario
Any scenario with exactly one inadvertent penetration of a waste panel that also penetrates a Castile brine reservoir is called E1. Features of this scenario are illustrated in Figure PA11.
Sources of brine in the E1 scenario are the brine reservoir, the Salado, and, under certain conditions, the units above the Salado. However, the brine reservoir is conceptually the dominant source of brine in this scenario. The flow and transport model configuration developed for the E1 scenario evaluates the consequences of futures that have only one E1 event.
Figure PA11. Conceptual Release Pathways for the Disturbed Repository Deep Drilling E1 Scenario
The E1E2 scenario is defined as all futures with multiple penetrations of a waste panel of which at least one intrusion is an E1. One example of this scenario, with a single E1 event and a single E2 event penetrating the same panel, is illustrated in Figure PA12. However, the E1E2 scenario can include many possible combinations of intrusion times, locations, and types of event (E1 or E2). The sources of brine in this scenario are those listed for the E1 scenario, and multiple E1 sources may be present. The E1E2 scenario has a potential flow path not present in the E1 or E2 scenarios: flow from an E1 borehole through the waste to another borehole. This flow path has the potential to (1) bring large quantities of brine in direct contact with waste and (2) provide a
Figure PA12. Conceptual Release Pathways for the Disturbed Repository Deep Drilling E1E2 Scenario
less restrictive path for this brine to flow to the units above the Salado (via multiple boreholes) compared to either the individual E1 or E2 scenarios. It is both the presence of brine reservoirs and the potential for flow through the waste to other boreholes that make this scenario different from combinations of E2 boreholes in terms of potential consequences. Estimates from flow and transport modeling are used to determine the extent of flow between boreholes and whether modeled combinations of E1 and E2 boreholes at specific locations in the repository should be treated as E1E2 scenarios or as independent E1 and E2 scenarios in the consequence analysis.
The M scenario and the E scenario may both occur in the future. The DOE calls a future in which both of these events occur the ME scenario. The occurrence of both mining and deep drilling do not create processes beyond those already described separately for the M and E scenarios. For example, the occurrence of mining does not influence any of the interactions between deep boreholes and the repository or brine reservoirs, nor does the occurrence of drilling impact the effects of mining on Culebra hydrogeology.
The scenarios described in Section PA2.3.2.1, Section PA2.3.2.2, and Section PA2.3.2.3 have been retained for consequence analysis to determine compliance with the containment requirements in section 191.13. The modeling systems used to evaluate the consequences of these undisturbed and disturbed scenarios are discussed in Section PA2.3.3.
Calculating scenario consequences requires quantitative modeling. This section discusses the conceptual and computational models and some parameter values used to estimate the consequence of the scenarios described in Section PA2.3.2. Additional discussion of conceptual models and modeling assumptions is provided in Section PA4.0. Additional descriptions of sampled parameter values are included in Fox (2008).
A single modeling system was used to represent the disposal system and calculate the CCDFs. The modeling system, however, can be conveniently described in terms of various submodels, with each describing a part of the overall system. The models used in the WIPP PA, as in other complex analyses, exist at four different levels.
1. Conceptual models are a set of qualitative assumptions that describe a system or subsystem for a given purpose. At a minimum, these assumptions concern the geometry and dimensionality of the system, initial and boundary conditions, time dependence, and the nature of the relevant physical and chemical processes. The assumptions should be consistent with one another and with existing information within the context of the given purpose.
2. Mathematical models represent the processes at the site. The conceptual models provide the context within which these mathematical models must operate, and define the processes they must characterize. The mathematical models are predictive in the sense that, once provided with the known or assumed properties of the system and possible perturbations to the system, they predict the response of the system. The processes represented by these mathematical models include fluid flow, mechanical deformation, radionuclide transport in groundwater, and removal of waste through intruding boreholes.
3. Numerical models are developed to approximate mathematical model solutions because most mathematical models do not have closedform solutions.
4. The complexity of the system requires computer codes to solve the numerical models. The implementation of the numerical model in the computer code with specific initial and boundary conditions and parameter values is generally referred to as the computational model.
Data are descriptors of the physical system being considered, normally obtained by experiment or observation. Parameters are values necessary in mathematical, numerical, or computational models. The distinction between data and parameters can be subtle. Parameters are distinct from data, however, for three reasons: (1) Data may be evaluated, statistically or otherwise, to generate model parameters to account for uncertainty in data. (2) Some parameters have no relation to the physical system, such as the parameters in a numerical model to determine when an iterative solution scheme has converged. (3) Many model parameters are applied at a different scale than one directly observed or measured in the physical system. The distinction between data and parameter values is described further in Fox (2008) and Tierney (1990), where distribution derivations for specific parameters are given. The interpretation and scaling of experimental and field data are discussed in Fox (2008) for individual and sampled parameters, as appropriate.
The PA for the WIPP identifies uncertainty in parameters and uncertainty in future events as distinctly different entities and requires sampling to be conducted in two dimensions. One dimension focuses on characterizing the uncertainty in terms of the probability that various possible futures will occur at the WIPP site over the next 10,000 years. The other dimension characterizes the uncertainty due to lack of knowledge about the precise values of model parameters appropriate for the WIPP repository. Each dimension of the analysis is characterized by a probability space. Monte Carlo methods are used with the WIPP PA modeling system to sample each of the two probability spaces.
Characterizing the probability distribution for the first dimension of the PA depends on identifying the kinds of events that could impact releases from the repository over the next 10,000 years. Screening analyses of possible future events concluded that the only significant events with the potential to affect radionuclide releases to the accessible environment are drilling and mining within the LWB (Appendix SCR2009, Section SCR5.0). Consequently, modeling the future states of the repository focuses on representing the occurrences and effects of these two events. CCDFGF uses stochastic processes to simulate intrusion events by drilling and the occurrence of mining for natural resources. CCDFGF assembles the results from the deterministic models and selects the most appropriate scenario data provided by these models to use as the simulation of a 10,000year future progresses. Ten thousand potential futures are simulated and used to create distributions of potential releases, and then compiled into a single CCDF of potential releases.
WIPP PA is required not only to estimate the likelihood of future releases, but to establish confidence in those estimates. Confidence is established using the second dimension of the analysis, which is based on the evaluation of uncertainty in the values of some of the parameters of the deterministic models. This uncertainty is assumed to represent a lack of knowledge about the true values of the parameters, and is labeled epistemic uncertainty. Epistemic uncertainty can be viewed as the representation of potential systematic errors in the results. The impact of epistemic uncertainty on the results is determined by generating 300 sets of parameter values using a stratified random sampling design, LHS, and then running the deterministic models and CCDFGF with each set of sampled parameters. Thus, 300 CCDFs are generated by CCDFGF. One set of parameters is often referred to as a vector. The 300 simulations are organized as 3 replicates of 100 vectors each. Because the uncertainty assigned to the parameters represents a lack of knowledge, this epistemic uncertainty could theoretically be reduced by collecting data to improve knowledge about the parameters. Epistemic uncertainty is represented in the projections of potential releases from the repository by the variability among the 300 CCDFs.
The WIPP PA modeling system consists of a set of loosely coupled deterministic models (BRAGFLO, PANEL, NUTS, SECOTP2D, and CUTTINGS_S) that provide scenariospecific results to the code CCDFGF (Figure PA2). CCDFGF is, in contrast, a stochastic simulation model used to simulate potential futures of repository performance where drilling and mining intrusions can impact the state of the repository and produce release events. CCDFGF implements intrusions as stochastic events, thus incorporating the aleatory uncertainty associated with projections of future events. This section describes how aleatory uncertainty is implemented in PA. Epistemic uncertainty is discussed in Section PA6.0.
As discussed in Section PA2.2.2, aleatory uncertainty is defined by the possible futures x_{st,i} conditional on the set i of parameters used in Equation (PA.2). Section PA3.2, Section PA3.3, Section PA3.4, Section PA3.5, Section PA3.6, Section PA3.7, Section PA3.8, and Section PA3.9 describe the individual components t_{j}, e_{j}, l_{j}, b_{j}, p_{j}, a_{j}, and t_{min} of x_{st,i} and their associated probability distributions. The concept of a scenario as a subset of the sample space of x_{st,i} is discussed in Section PA3.10. The procedure used to sample the individual elements x_{st,i} is described in Section PA6.5.
AICs and PICs will be implemented at the WIPP site to deter human activity detrimental to repository performance. AICs and PICs are described in detail in the CRA2004, Chapter 7.0 and in appendices referenced in Chapter 7.0. In this section, the impact of AICs and PICs on PA is described.
AICs will be implemented at the WIPP after final facility closure to control site access and ensure that activities detrimental to disposal system performance do not occur within the controlled area. The AICs will preclude human intrusion in the disposal system. A 100year limit on the effectiveness of AICs in PA is established in 40 CFR § 191.14(a). Because of the regulatory restrictions and the nature of the AICs that will be implemented, PA assumes there are no inadvertent human intrusions or mining in the controlled area for 100 years following repository closure.
PICs are designed to deter inadvertent human intrusion into the disposal system. Only minimal assumptions were made about the nature of future society when designing the PICs to comply with the assurance requirements. The preamble to Part 194 limits any credit for PICs in deterring human intrusion to 700 years after disposal (U.S. Environmental Protection Agency 1996a, p. 5231). Although the DOE originally took credit for PICs in the CCA PA, but has not taken credit since. Not including PICs is a conservative implementation, as no credit is taken for a beneficial process.
As described in Section PA2.3.2.2, drilling intrusions in PA are assumed to occur randomly in time and space following a Poisson process. Specifically, the drilling rate considered within the area marked by a berm as part of the system for PICs (Fox 2008, Table 46) is 5.85 ´ 10^{3} intrusions per square kilometer per year (km^{2}/yr). AICs are assumed to prevent any drilling intrusions for the first 100 years after the decommissioning of the WIPP (Section PA3.2). In the computational implementation of PA, it is convenient to represent the Poisson process for drilling intrusions by its corresponding rate term l_{d}( t) for intrusions into the area marked by the berm. Specifically,
where 0.6285 km^{2} is the area enclosed by the berm (Fox 2008, Table 45) and t is elapsed time since decommissioning the WIPP.
The function l_{d}( t) defines the parameter of the exponential distribution that gives rise to the times of intrusions, t_{j} of Equation (PA.2). In the computational implementation of the analysis, the exponential distribution is sampled to define the times between successive drilling intrusions (Figure PA13, Section PA6.5). A key assumption of the exponential distribution is that events are independent of each other, so the occurrence of one has no effect on the occurrence of the next event. The process giving rise to such events is sometimes called a Poisson process because the distribution of such events over a fixed interval of time is a Poisson distribution. Due to the 10,000year regulatory period specified in section 191.13, t_{j} is assumed to be bounded above by 10,000 years in the definition of x_{st,i}. Further, t_{j} is bounded below by 100 years as defined in Equation (PA.6).
Figure PA13. CDF for Time Between Drilling Intrusions
The variable e_{j} is a designator for whether or not the j^{th} drilling intrusion penetrates an excavated, wastefilled area of the repository: e_{j} = 0 or 1 implies penetration of nonexcavated or excavated area, respectively. The corresponding probabilities P[e_{j} = 0] and P[e_{j} = 1] for e_{j} = 0 and e_{j} = 1 are
where 0.1273 km^{2} and 0.6285 km^{2} are the excavated area of the repository and the area of the berm, respectively (Fox 2008, Table 45).
Locations of drilling intrusions through the excavated, wastefilled area of the repository are discretized to the 144 locations in Figure PA14. Assuming that a drilling intrusion occurs within the excavated area, it is assumed to be equally likely to occur at each of these 144 locations. Thus, the probability pL_{k} that drilling intrusion j will occur at location l_{k}, k = 1, 2, ¼, 144 in Figure PA14 is
Figure PA14. Discretized Locations for Drilling Intrusions
The conceptual models for the Castile include the possibility that pressurized brine reservoirs underlie the repository (Section PA4.2.10). The variable b_{j} is a designator for whether or not the j^{th} drilling intrusion penetrates pressurized brine, where b_{j}_{ }= 0 signifies nonpenetration and b_{j} = 1 signifies penetration of pressurized brine. In PA, the probability pB_{1} = P[b_{j }= 1] is sampled from a uniform distribution ranging from 0.01 to 0.60 (see GLOBAL:PBRINE in Table PA19).
Three borehole plugging patterns, p_{k}, are considered in PA: (1) p_{1}, a full concrete plug through the Salado to the Bell Canyon Formation (hereafter referred to as Bell Canyon), (2) p_{2}, a twoplug configuration with concrete plugs at the Rustler/Salado interface and the Castile/Bell Canyon interface, and (3) p_{3}, a threeplug configuration with concrete plugs at the Rustler/ Salado, Salado/Castile, and Castile/Bell Canyon interfaces. The probability that a given drilling intrusion will be sealed with plugging pattern p_{k}, k= 1, 2, 3, is given by pPL_{k}, where pPL_{1} = P[k = 1] = 0.015, pPL_{2} = P[k = 2] = 0.696, pPL_{3} = P[k = 3] = 0.289 (Fox 2008, Table 46).
The waste intended for disposal at the WIPP is represented by 767 distinct waste streams, with 690 of these waste streams designated as CHTRU waste and 77 designated as RHTRU waste (Leigh, Trone, and Fox 2005, Section 4.4). For the CRA2009 PA, the 77 separate RHTRU waste streams are represented by a single, combined RHTRU waste stream. The activity levels for the waste streams are given in Fox (2008, Table 48 and Table 49). Each waste container emplaced in the repository contains waste from a single CHTRU waste stream. Waste packaged in 55gallon (gal) drums is stacked 3 drums high within the repository. Although waste in other packages (e.g., standard waste boxes, 10drum overpacks, etc.) may not be stacked 3 high, PA assumes that each drilling intrusion into CHTRU waste intersects 3 different waste streams. In contrast, all RHTRU waste is represented by a single waste stream, and so each drilling intrusion through RHTRU waste is assumed to intersect this single waste stream. Appendix MASS2009, Section MASS21.0 examines the sensitivity of PA results to the assumption that three waste streams are intersected by each drilling intrusion into CHTRU waste.
The vector a_{j} characterizes the type of waste penetrated by the j^{th} drilling intrusion. Specifically,
a_{j} = 0 if e_{j} = 0 (PA.10)
(i.e., if the i^{th} drilling intrusion does not penetrate an excavated area of the repository);
a_{j} = 1 if e_{j} = 1 and RHTRU is penetrated (PA.11)
a_{j} = [iCH_{j1}, iCH_{j2}, iCH_{j3}] if e_{j} = 1 and CHTRU is penetrated (PA.12)
where iCH_{j1}, iCH_{j2}, and iCH_{j3} are integer designators for the CHTRU waste streams intersected by the j^{th} drilling intrusion (i.e., each of iCH_{j1}, iCH_{j2}, and iCH_{j3} is an integer between 1 and 690).
Whether the j^{th} intrusion penetrates a nonexcavated or excavated area is determined by the probabilities pE_{0} and pE_{1} discussed in Section PA3.4. The type of waste penetrated is determined by the probabilities pCH and pRH. The excavated area used for disposal of CHTRU waste (aCH) is 1.115 ´ 10^{5} square meters (m^{2}) and the area used for disposal of RHTRU waste (aRH) is 1.576 ´ 10^{4} m^{2} (Fox 2008, Table 45), for a total disposal area of aEX = aCH + aRH = 1.273 ´ 10^{5} m^{2}. Given that the j^{th} intrusion penetrates an excavated area, the probabilities pCH and pRH of penetrating CHTRU and RHTRU waste are given by
As indicated in this section, the probabilistic characterization of a_{j} depends on a number of individual probabilities. Specifically, pEx_{0} and pEx_{1} determine whether a nonexcavated or excavated area is penetrated (Section PA3.5); pCH and pRH determine whether CHTRU or RHTRU waste is encountered, given penetration of an excavated area; and the individual waste stream probabilities in Fox (2008, Table 48 and Table 49), determine the specific waste streams iCH_{j1}, iCH_{j2}, and iCH_{j3} encountered given a penetration of CHTRU waste. The probability of encountering a particular CHTRU waste stream is computed as the ratio of the volume of that waste stream to the volume of CHTRU waste.
Full mining of known potash reserves within the LWB is assumed to occur at time t_{min}. The occurrence of mining within the LWB in the absence of institutional controls is specified as following a Poisson process with a rate of l_{m} = 1 ´ 10^{ }^{4} yr^{ }^{1} (Fox 2008, Table 46). However, this rate can be reduced by AICs and PICs. Specifically, AICs are assumed to result in no possibility of mining for the first 100 years after decommissioning of the WIPP. In PA, PICs do not affect the mining rate. Thus, the mining rate l_{m}( t) is
where t is the elapsed time since decommissioning of the WIPP.
In the computational implementation of the analysis, l_{m}( t) is used to define the distribution of time to mining. The use of l_{m}( t) to characterize t_{min} is analogous to the use of l_{d} to characterize the t_{j}, except that only one mining event is assumed to occur (i.e., x_{st,}_{i} contains only one value for t_{min}) in order to be consistent with guidance given in Part 194 that mining within the LWB should be assumed to remove all economically viable potash reserves. Due to the 10,000year regulatory period specified in section 191.13, t_{min} is assumed to be bounded above by 10,000 years in the definition of x_{st,i}.
A scenario is a subset of the sample space for aleatory uncertainty. The underlying goal of scenario definition is to define the state of repository conditions prior to and following intrusion events. Scenarios are specific cases of inputs or system states that are selected to cover the range of possible cases. Given the complexity of the futures x_{st,i} (see Equation (PA.2)), many different scenarios can be defined. The computational complexity of the function f (x_{st}v_{su}) in Section PA2.2.3 limits evaluation to only a few intrusion scenarios. As presented in Section PA2.3.2, PA considers four fundamental intrusion scenarios:
E0 = no drilling intrusion through an excavated area of the repository
E1 = a drilling intrusion through an excavated area of the repository that penetrates pressurized brine in the Castile
E2 = a drilling intrusion through an excavated area of the repository that does not penetrate pressurized brine in the Castile
E1E2 = two or more previous intrusions, at least one of which is an E1 intrusion
These definitions of intrusion scenarios capture the most important events impacting the state of the repository: whether or not the repository is inundated by the penetration of a brine pocket, and whether or not there exists a possible route of release upward via a borehole. The state of the repository is also designated as E0, E1, E2, or E1E2. Scenarios for some of the processlevel models consist of a single intrusion scenario occurring at specific times. CCDFGF is used to simulate multiple intrusions over 10,000 years.
If only the intrusion scenarios controlled the state of the repository, then the state would be defined by the sequence of drilling events alone. However, CCDFGF also considers the impact of plugging pattern on boreholes. A borehole with a full plugging pattern that penetrates the waste area is also assumed to have no impact, and leaves the repository in its previous state, including the undisturbed state (see Section PA6.8.4.1 and Figure PA41 for more details). Thus, an E2 intrusion event into an E0 repository will result in an E0 state if a full plugging pattern is used, or an E2 state otherwise. An E1 intrusion subsequent to an E2 intrusion will leave the repository in an E1E2 state, where it will remain, regardless of subsequent intrusions. It is therefore important to distinguish between the type of intrusion, listed above, and the state of the repository.
The probability that no excavated area will be penetrated during the 10,000year interval can be computed using a distribution of the number of penetration events and the probability that a drilling event will penetrate the excavated area. For the Poisson distribution of drilling events, the probability of there being n events in the 10,000year history is
where l_{d} is the mean drilling rate per year in the period following the period of AICs, 9,900 is the number of years in which drilling can occur after the institutional control period of 100 years, and n is the number of drilling events. The probability of having n events all within the nonexcavated area is pEx_{0}^{n}, or specifically 0.797^{n}. Thus, the probability of having only events in the nonexcavated area over 10,000 years, i.e., having no drilling intrusions into the excavated area, is just the sum across all n of the products of the probability of having exactly n drilling events and the probability that all n events penetrate the unexcavated area:
The calculated probability becomes
This probability is the lower bound on the probability of the repository being in an E0 state, given that it does not include the consideration of the plugging pattern.
The probability of a single E1, E2, or E1E2 intrusion over 10,000 years is relatively small. Assuming that pB_{1} takes on its mean value of 0.305 (see Section PA3.6), and ignoring the impact of the plugging pattern, for a constant rate of drilling, l_{d}, these equations are
and
respectively, where (pEx_{1} × l_{d})_{}represents the annual rate of drilling into the excavated region of the repository which is multiplied by 9900 to give the rate per 9,900 years. The probability of an intrusion into the excavated area is subsequently multiplied by the probability of hitting or missing a brine pocket. In this form, it can be seen that the term for the probability for intrusion is equivalent to the PDF of the Poisson distribution for n = 1:
The expressions defining the probability of being in the E0 state after 10,000 years and of having a single E1 or E2 intrusion event after 10,000 years are relatively simple because the scenarios E0, E1, and E2 are relatively simple. The scenario E1E2 is more complex and, as a result, computing its probability is also more complex. Closedform formulas for the probabilities of quite complex scenarios can be derived, but they are very complicated and involve large numbers of iterated integrals (Helton 1993). These probabilities of single E1 and E2 intrusions are relevant to the scenarios used by the processlevel models.
CCDFGF simulates histories that can have many intrusion events (WIPP Performance Assessment 2003a). The processlevel models evaluate the releases at a small number of specific times for each of the four intrusion scenarios. Releases from the repository are calculated using results from these fundamental scenarios (Section PA6.7 and Section PA6.8). Releases for an arbitrary future are estimated from the results of these fundamental scenarios (Section PA6.8); these releases are used to construct CCDFs by Equation (PA.4).
Previous WIPP PAs have used the Monte Carlo approach to construct the CCDF indicated in Equation (PA.4). The Monte Carlo approach generates releases for 10,000 possible futures. CCDFs are constructed by treating the 10,000 releases values as order statistics; each release is assigned a probability of 1 ´ 10^{4}, and the CCDF can be constructed by plotting the complement of the sum of the probabilities ordered by the release value. The CRA2009 PA uses the same approach as the CRA2004 PA.
This section describes how releases to the accessible environment are estimated for a particular future in PA.
The function f(x_{st,i}) estimates the radionuclide releases to the accessible environment associated with each of the possible futures (x_{st,i}) that could occur at the WIPP site over the next 10,000 years. In practice, f(x_{st,i}) is quite complex and is constructed by the models implemented in computer programs used to simulate important processes and releases at the WIPP. In the context of these models, f(x_{st,i}) has the form
where
_{ } x_{st,i} ~ particular future under consideration
_{ } ~ future involving no drilling intrusions but a mining event at the same time t_{min} as in x_{st}
_{ } f_{C}(x_{st,i}) ~ cuttings and cavings release to accessible environment for x_{st,i} calculated with CUTTINGS_S
_{ } f_{B}(x_{st,i}) ~ twophase flow in and around the repository calculated for x_{st,i} with BRAGFLO; in practice, f_{B}(x_{st,i}) is a vector containing a large amount of information, including pressure and brine saturation in various geologic members
_{} ~ spallings release to accessible environment for x_{st,i} calculated with the spallings model contained in DRSPALL and CUTTINGS_S; this calculation requires repository conditions calculated by f_{B}(x_{st,i}) as input
_{ } ~ DBR to accessible environment for x_{st,i} also calculated with BRAGFLO; this calculation requires repository conditions calculated by f_{B}(x_{st,i}) as input
_{ } _{} ~ release through anhydrite MBs to accessible environment for x_{st,i} calculated with NUTS; this calculation requires flows in and around the repository calculated by f_{B}(x_{st,i}) as input
_{ } ~ release through Dewey Lake to accessible environment for x_{st,i} calculated with NUTS; this calculation requires flows in and around the repository calculated by f_{B}(x_{st,i}) as input
_{ } _{} ~ release to land surface due to brine flow up a plugged borehole for x_{st,i} calculated with NUTS; this calculation requires flows in and around the repository calculated by f_{B}(x_{st,i}) as input
_{ } _{} ~ flow field in the Culebra calculated for x_{st,}_{0} with MODFLOW; x_{st,}_{0} is used as an argument to f_{MF} because drilling intrusions are assumed to cause no perturbations to the flow field in the Culebra
_{ } _{} ~ release to Culebra for x_{st,i} calculated with NUTS or PANEL as appropriate; this calculation requires flows in and around the repository calculated by f_{B}(x_{st,i}) as input
_{} ~ groundwater transport release through Culebra to accessible environment calculated with SECOTP2D. This calculation requires MODFLOW results (i.e., f_{MF}(x_{st,i})) and NUTS or PANEL results (i.e., _{}) as input
The remainder of this section describes the mathematical structure of the mechanistic models that underlie the component functions of f(x_{st,i}) in Equation (PA.23).
The Monte Carlo CCDF construction procedure, implemented in the code CCDFGF (WIPP Performance Assessment 2003a), uses a sample of size nS = 10,000 in PA. The individual programs that estimate releases do not run fast enough to allow this many evaluations of f. As a result, a twostep procedure is being used to evaluate f in calculating the summation in Equation (PA.23). First, f and its component functions are evaluated with the procedures (i.e., models) described in this section for a group of preselected futures. Second, values of f(x_{st}) for the randomly selected futures x_{st,i} used in the numerical evaluation of the summation in Equation (PA.23) are then constructed from results obtained in the first step. These constructions are described in Section PA6.7 and Section PA6.8, and produce the evaluations of f(x_{st}) that are actually used in Equation (PA.23).
For notational simplicity, the functions on the righthand side of Equation (PA.23) will typically be written with only x_{st} as an argument (e.g., f_{SP}(x_{st}) will be used instead of f_{SP}[x_{st}, f_{B}(x_{st})]). However, the underlying dependency on the other arguments will still be present.
The major topics considered in this chapter are twophase flow in the vicinity of the repository as modeled by BRAGFLO (i.e., f_{B}) (Section PA4.2), radionuclide transport in the vicinity of the repository as modeled by NUTS (i.e., f_{MB}, f_{DL}, f_{S}, f_{NP}) (Section PA4.3), radionuclide transport in the vicinity of the repository as modeled by PANEL (i.e., f_{NP}) (Section PA4.4), cuttings and cavings releases to the surface as modeled by CUTTINGS_S (i.e., f_{C}) (Section PA4.5), spallings releases to the surface as modeled by DRSPALL and CUTTINGS_S (i.e., f_{SP}) (Section PA4.6), DBRs to the surface as modeled by BRAGFLO (i.e., f_{DBR}) (Section PA4.7), brine flow in the Culebra as modeled by MODFLOW (i.e., f_{MF}) (Section PA4.8), and radionuclide transport in the Culebra as modeled by SECOTP2D (i.e., f_{ST}) (Section PA4.9).
Quantifying the effects of gas and brine flow on radionuclide transport from the repository requires a twophase (brine and gas) flow code. The twophase flow code BRAGFLO is used to simulate gas and brine flow in and around the repository (Nemer 2007b and 2007c). Additionally, the BRAGFLO code incorporates the effects of disposal room consolidation and closure, gas generation, and rock fracturing in response to gas pressure. This section describes the mathematical models on which BRAGFLO is based, the representation of the repository in the model, and the numerical techniques employed in the solution.
Twophase flow in the vicinity of the repository is represented by the following system of two conservation equations, two constraint equations, and three equations of state:
Gas Conservation
Brine Conservation
Saturation Constraint
Capillary Pressure Constraint
Gas Density
r_{g} (determined
by RedlichKwongSoave (RKS) equation of state; see Equation (PA.51))
(PA.28)
Brine Density
Formation Porosity
where
g = acceleration due to gravity (meters per second squared [m])
h = vertical distance from a reference location (m)
k_{rl} = relative permeability (dimensionless) to fluid l, l = b (brine), g (gas)
P_{c} = capillary pressure in Pascals (Pa)
P_{l} = pressure of fluid l (Pa)
q_{rl} = rate of production (or consumption, if negative) of fluid l due to chemical reaction (kilograms per cubic meter per seconds [kg/m^{3}/s])
q_{l} = rate of injection (or removal, if negative) of fluid l (kg/m^{3}/s)
S_{l} = saturation of fluid l (dimensionless)
t = time (s)
a = geometry factor (m)
r_{l} = density of fluid l (kg/m^{3})
m_{l} = viscosity of fluid l (Pa s)
f = porosity (dimensionless)
f_{0} = reference (i.e., initial) porosity (dimensionless)
P_{b}_{0 } = reference (i.e., initial) brine pressure (Pa), constant in Equation (PA.29) and spatially variable in Equation (PA.30)
r_{0} = reference (i.e., initial) brine density (kg/m^{3})
c_{ f} = pore compressibility (Pa^{1})
c_{b} = brine compressibility (Pa^{1})
K = permeability of the material (m^{2}), isotropic for PA (Howarth and ChristianFrear 1997)
For the brine transport Equation (PA.25), the intrinsic permeability of the material is used. For the gas transport Equation (PA.24), the permeability K is modified to account for the Klinkenberg effect (Klinkenberg 1941). Specifically,
where a and b are gas and formationdependent constants. Values of a = 0.3410 and b = 0.2710 were determined from data obtained for MB 139 (ChristianFrear 1996), with these values used for all regions in Figure PA15.
The conservation equations are valid in one (i.e., Ñ = [¶/¶x]), two (i.e., Ñ = [¶ /¶ x, ¶ /¶ y]), and three (i.e., Ñ = [¶ /¶ x, ¶ /¶ y, ¶ /¶ z]) dimensions. In PA, the preceding system of equations is used to model twophase fluid flow within the twodimensional region shown in Figure PA15. The details of this system are now discussed.
The a term in Equation (PA.24) and Equation (PA.25) is a dimensiondependent geometry factor and is specified by
a = area
normal to flow direction in onedimensional flow (i.e., DyDz; units =
m^{2})
= thickness normal to flow plane in
twodiemensional flow (i.e., Dz; units = m)
= 1 in threedimensional flow
(dimensionless)
(PA.32)
PA uses a twodimensional geometry to compute twophase flow in the vicinity of the repository, and as a result, a is the thickness of the modeled region (i.e., Dz) normal to the flow plane (Figure PA15). Due to the use of the twodimensional grid in Figure PA15, a is spatially dependent, with the values used for a defined in the column labeled “Dz.” Specifically, a increases with distance away from the repository edge in both directions to incorporate the increasing pore volume through which fluid flow occurs. The method used in PA, called rectangular flaring, is illustrated in Figure PA16 and ensures that the total volume surrounding the repository is conserved in the numerical grid. The equations and method used to determine a for the grid shown in Figure PA15 are described in detail by Stein (2002).
The h term in Equation (PA.24) and Equation (PA.25) defines vertical distance from a reference point. In PA, this reference point is taken to be the center of MB 139 at the location of the shaft (i.e., (x_{ref}, y_{ref}) = (23664.9 m, 378.685 m), which is the center of cell 1266 in Figure PA17). Specifically, h is defined by
where q is the inclination of the formation in which the point (x, y) is located. In PA, the Salado is modeled as having an inclination of 1 degree from north to south, and all other formations are modeled as being horizontal. Thus, q = 1 degree for points within the Salado, and q = 0 degrees otherwise. Treating the Salado as an inclined formation and treating the Castile, Castile brine reservoir, Rustler, and overlying units as horizontal creates discontinuities in the grid at the lower and upper boundaries of the Salado. However, this treatment does not create a computational problem, since the Salado is isolated from vertical flow; its upper boundary adjoins the impermeable Los Medaños Member (formerly referred to as the Unnamed Member) at the base of the Rustler, and its lower boundary adjoins the impermeable Castile.
In the solution of Equations (PA.24) through (PA.30), S_{b} and S_{g} are functions of location and time. Thus, P_{c}, k_{rb}, and k_{rg} are functions of the form P_{c}( x, y, t), k_{rb}( x, y, t), and k_{rg}( x, y, t). In the computational implementation of the solution of the preceding equations, flow of phase l out of a computational cell (Figure PA17) cannot occur when S_{l}( x, y, t) £ S_{lr}( x, y, t), where S_{lr} denotes the residual saturation for phase l. The values used for S_{lr}, l = b, g are summarized in Table PA3.
Values for f_{0} and c_{ f} (Equation (PA.30)) are also given in Table PA3. Initial porosity f_{0} for the DRZ is a function of the uncertain parameter for initial halite porosity f_{0}_{H} (HALPOR; see Table PA19) and is given by Martell (1996a) and Bean et al. (1996), Section 4):
f_{0} = f_{0}_{H} + 0.0029 (PA.34)
Initial porosity f_{0} of the Castile brine reservoir is calculated from the uncertain sampled parameter for the bulk Castile rock compressibility (BPCOMP; see Table PA19), according to the following relationship:
where 1.0860 ´ 10^{10} is a scaling constant that ensures that the productivity ratio, PR, remains constant at 2.0 ´ 10^{3} m^{3}/Pa. The productivity ratio PR is computed by
where V is the volume of the grid block representing the Castile brine reservoir in Figure PA15. Because of this relationship, the initial porosity of the brine reservoir ranges from 0.1842 to 0.9208. This range of porosity is not meant to represent an actual reservoir, but rather allows a reservoir to supply a volume of brine to the repository in the event of an E1 intrusion consistent with observed brine flows in the Delaware Basin.
The compressibility c_{ f} in Equation (PA.30) and Table PA3 is pore compressibility. Compressibility is treated as uncertain for Salado anhydrite, Salado halite, and regions of pressurized brine in the Castile. However, the sampled value for each of these variables corresponds to bulk compressibility rather than to the pore compressibility actually used in the calculation. Assuming all of the change in volume during compression occurs in the pore volume, the conversion from bulk compressibility c_{r} to pore compressibility c_{ f} is approximated by
where f_{0} is the initial porosity in the region under consideration.
The primary model used in PA for capillary pressure P_{c} and relative permeability k_{rl} is a modification of the BrooksCorey model (Brooks and Corey 1964). In this model, P_{c}, k_{rb}, and k_{rg} are defined by
Figure PA17. Identification of Individual Cells in BRAGFLO Grid
Table PA3. Parameter Values Used in Representation of TwoPhase Flow 

Region 
Material 
Material Description 
BrooksCorey Pore Distribution
(PORE_DIS)^{a} 
Threshold Pressure Linear Parameter
(PCT_A)^{a} 
Threshold Pressure Exponential
Parameter 
Residual Brine Saturation 
Residual Gas Saturation 
Porosity 
Pore Compressibility^{a} 
Intrinsic Permeability 
Salado 
S_HALITE 
Undisturbed halite 
0.7 
0.56 
0.346 
0.3 
0.2 
HALPOR^{b} 
f(HALCOMP)^{b,d} 
10^{x}, x = HALPRM^{b} 
DRZ 
DRZ_0 
DRZ, 5 to 0 years 
0.7 
0.0 
0.0 
0.0 
0.0 
f(HALPOR)^{b,c} 
f(HALCOMP)^{b,d} 
1.0 ´ 1017 

DRZ_1 
DRZ, 0 to 10,000 years 
0.7 
0.0 
0.0 
0.0 
0.0 
f(HALPOR)^{b,c} 
f(HALCOMP)^{b,d} 
10^{x}, x = DRZPRM^{b} 
MB 138 
S_MB138 
Anhydrite MB in Salado 
ANHBCEXP^{b} 
0.26 
0.348 
ANRBSAT^{b} 
ANRGSSAT^{b} 
0.011 
f(ANHCOMP)^{b,d} 
10^{x}, x = ANHPRM^{b} 
Anhydrite AB 
S_ANH_AB 
Anhydrite layers A and B in Salado 
ANHBCEXP^{b} 
0.26 
0.348 
ANRBSAT^{b} 
ANRGSSAT^{b} 
0.011 
f(ANHCOMP)^{b,d} 
10^{x}, x = ANHPRM^{b} 
MB 139 
S_MB139 
Anhydrite MB in Salado 
ANHBCEXP^{b} 
0.26 
0.348 
ANRBSAT^{b} 
ANRGSSAT^{b} 
0.011 
f(ANHCOMP)^{b,d} 
10^{x}, x = ANHPRM^{b} 
Waste Panel 
CAVITY_1 
Single waste panel, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
WAS_AREA 
Single waste panel, 0 to 10,000 years 
2.89 
0.0 
0.0 
WRBRNSAT^{b} 
WRGSSAT^{b} 
0.848^{f} 
0.0 
2.4 ´ 10^{13} 
Rest of Repository (RoR) 
CAVITY_2 
RoR, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
REPOSIT 
RoR, 0 to 10,000 years 
2.89 
0.0 
0.0 
WRBRNSAT^{b} 
WRGSSAT^{b} 
0.848^{f} 
0.0 
2.4 ´ 10^{13} 
Ops and Exp 
CAVITY_3 
Operations area, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
OPS_AREA 
Operations area, 0 to 10,000 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
0.18 
0.0 
1.0 ´ 10^{11} 
Exp 
CAVITY_3 
Experimental area, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
a Parenthetical parameter names are property names for the corresponding material, as indicated in Table PA19. b Uncertain variable; see Table PA19. c See Equation (PA.34). d See Equation (PA.37); f_{0} can also be defined by an uncertain variable. e These materials are using relative permeability model = 11; see Table PA4. f Initial value of porosity f_{0}; porosity changes dynamically to account for creep closure (see Section PA4.2.3). g See Equation (PA.35). 
Table PA3. Parameter Values Used in Representation of TwoPhase Flow (Continued) 

Region 
Material 
Material Description 
BrooksCorey Pore Distribution
(PORE_DIS)^{a} 
Threshold Pressure Linear Parameter
(PCT_A)^{a} 
Threshold Pressure Exponential
Parameter 
Residual Brine Saturation 
Residual Gas Saturation 
Porosity 
Pore Compressibility^{a} 
Intrinsic Permeability 
— 
EXP_AREA 
Experimental area, 0 to 10,000 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
0.18 
0.0 
1.0 ´ 10^{11} 
Castile 
IMPERM_Z 
Castile 
0.7 
0.0 
0.0 
0.0 
0.0 
0.005 
0.0 
1.0 ´ 10^{35} 
Castile Brine Reservoir 
CASTILER 
Brine Reservoir in Castile 
0.7 
0.56 
0.346 
0.2 
0.2 
f(BPCOMP)^{b,g} 
f(BPCOMP)^{b,d} 
10^{x}, x = BPPRM^{b} 
Culebra 
CULEBRA 
Culebra Member of Rustler 
0.6436 
0.26 
0.348 
0.08363 
0.07711 
0.151 
6.622517 ´ 10^{10} 
7.72681 ´ 10^{14} 
Magenta 
MAGENTA 
Magenta Member of Rustler 
0.6436 
0.26 
0.348 
0.08363 
0.07711 
0.138 
1.915942 ´ 10^{9} 
6.309576 ´ 10^{16} 
Dewey Lake 
DEWYLAKE 
Dewey Lake Redbeds 
0.6436 
0.0 
0.0 
0.08363 
0.07711 
0.143 
6.993007 ´ 10^{8} 
5.011881 ´ 10^{17} 
Santa Rosa 
SANTAROS 
Santa Rosa Formation 
0.6436 
0.0 
0.0 
0.08363 
0.07711 
0.175 
5.714286 ´ 10^{8} 
1.0 ´ 10^{10} 
Los Medaños 
UNNAMED 
Los Medaños Member of Rustler 
0.7 
0.0 
0.0 
0.2 
0.2 
0.181 
0.0 
1.0 ´ 10^{35} 
Tamarisk 
TAMARISK 
Tamarisk Member of Rustler 
0.7 
0.0 
0.0 
0.2 
0.2 
0.064 
0.0 
1.0 ´ 10^{35} 
Fortyniner 
FORTYNIN 
Fortyniner Member of Rustler 
0.7 
0.0 
0.0 
0.2 
0.2 
0.082 
0.0 
1.0 ´ 10^{35} 
DRZ_PCS 
DRZ_0 
DRZ, 5 to 0 years 
0.7 
0.0 
0.0 
0.0 
0.0 
f(HALPOR)^{b,c} 
f(HALCOMP)^{b,d} 
1.0 ´ 10^{17} 
DRZ_PCS 
DRZ_PCS 
DRZ above the panel closures, 0 to 10,000 years 
0.7 
0.0 
0.0 
0.0 
0.0 
f(HALPOR)^{b,c} 
f(HALCOMP)^{b,d} 
10^{x}, x = DRZPCPRM^{b} 
a Parenthetical parameter names are property names for the corresponding material, as indicated in Table PA19. b Uncertain variable; see Table PA19. c See Equation (PA.34). d See Equation (PA.37); f_{0} can also be defined by an uncertain variable. e These materials are using relative permeability model = 11; see Table PA4. f Initial value of porosity f_{0}; porosity changes dynamically to account for creep closure (see Section PA4.2.3). g See Equation (PA.35). 
Table PA3. Parameter Values Used in Representation of TwoPhase Flow (Continued) 

Region 
Material 
Material Description 
BrooksCorey Pore Distribution
(PORE_DIS)^{a} 
Threshold Pressure Linear Parameter
(PCT_A)^{a} 
Threshold Pressure Exponential
Parameter 
Residual Brine Saturation 
Residual Gas Saturation 
Porosity 
Pore Compressibility^{a} 
Intrinsic Permeability 
CONC_PCS 
CAVITY_4 
Concrete portion of panel closures, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
CONC_PCS 
Concrete portion of panel closures, 0 to 10,000 years 
CONBCEXP^{b} 
0.0 
0.0 
CONBRSAT^{b} 
CONGSSAT^{b} 
0.05 
1.2 ´ 10^{9} 
10^{x}, x = CONPRM^{b} 
DRF_PCS 
CAVITY_4 
Drift adjacent to panel closures, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
DRF_PCS 
Drift adjacent to panel closures, 0 to 10,000 years 
2.89 
0.0 
0.0 
WRBRNSAT^{b} 
WRGSSAT^{b} 
0.848^{f} 
0.0 
2.4 ´ 10^{13} 
CONC_MON 
CAVITY_4 
Concrete monolith portion of shaft seals, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
CONC_MON 
Concrete monolith portion of shaft seals, 0 to 10,000 years 
0.94 
0.0 
0.0 
SHURBRN^{b} 
SHURGAS^{b} 
0.05 
1.2 ´ 10^{9} 
1.0 ´ 10^{14} 
Upper Shaft 
CAVITY_4 
Upper portion of shaft seals, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
SHFTU 
Upper portion of shaft seals, 0 to 10,000 years 
CONBCEXP^{b} 
0.0 
0.0 
SHURBRN^{b} 
SHURGAS^{b} 
0.005 
2.05 ´ 10^{8} 
10^{x}, x = SHUPRM^{b} 
a Parenthetical parameter names are property names for the corresponding material, as indicated in Table PA19. b Uncertain variable; see Table PA19. c See Equation (PA.34). d See Equation (PA.37); f_{0} can also be defined by an uncertain variable. e These materials are using relative permeability model = 11; see Table PA4. f Initial value of porosity f_{0}; porosity changes dynamically to account for creep closure (see Section PA4.2.3). g See Equation (PA.35). 
Table PA3. Parameter Values Used in Representation of TwoPhase Flow (Continued) 

Region 
Material 
Material Description 
BrooksCorey Pore Distribution
(PORE_DIS)^{a} 
Threshold Pressure Linear Parameter
(PCT_A)^{a} 
Threshold Pressure Exponential
Parameter 
Residual Brine Saturation 
Residual Gas Saturation 
Porosity 
Pore Compressibility^{a} 
Intrinsic Permeability 
Lower Shaft 
CAVITY_4 
Lower portion of shaft seals, 5 to 0 years 
NA^{e} 
NA^{e} 
NA^{e} 
0.0 
0.0 
1.0 
0.0 
1.0 ´ 10^{10} 
— 
SHFTL_T1 
Lower portion of shaft seals, 0 to 200 years 
CONBCEXP^{b} 
0.0 
0.0 
SHURBRN^{b} 
SHURGAS^{b} 
0.005 
4.28 ´ 10^{9} 
10^{x}, x = SHLPRM1^{b} 
— 
SHFTL_T2 
Lower portion of shaft seals, 200 to 10,000 years 
CONBCEXP^{b} 
0.0 
0.0 
SHURBRN^{b} 
SHURGAS^{b} 
0.005 
4.28 ´ 10^{9} 
10^{x}, x = SHLPRM2^{b} 
Borehole plugs 
CONC_PLG 
Concrete borehole plug, before plug degradation 
0.94 
0.0 
0.0 
0.0 
0.0 
0.32 
1.1875 ´ 10^{9} 
10^{x}, x = PLGPRM^{b} 
— 
BH_SAND 
Borehole after plug degradation, 200 years after intrusion 
0.94 
0.0 
0.0 
0.0 
0.0 
0.32 
0.0 
10^{x}, x = BHPRM^{b} 
Upper Borehole 
BH_OPEN 
Borehole above repository before plug degradation 
0.7 
0.0 
0.0 
0.0 
0.0 
0.32 
0.0 
1.0 ´ 10^{9} 
— 
BH_SAND 
Borehole after plug degradation, 200 years after intrusion 
0.94 
0.0 
0.0 
0.0 
0.0 
0.32 
0.0 
10^{x}, x = BHPRM^{b} 
Lower Borehole 
BH_OPEN 
Borehole below repository before creep closure 
0.7 
0.0 
0.0 
0.0 
0.0 
0.32 
0.0 
1.0 ´ 10^{9} 
— 
BH_CREEP 
Borehole below repository after creep closure, 1,000 years after intrusion 
0.94 
0.0 
0.0 
0.0 
0.0 
0.32 
0.0 
10^{x}/10, x = BHPRM^{a} 
a Parenthetical parameter names are property names for the corresponding material, as indicated in Table PA19. b Uncertain variable; see Table PA19. c See Equation (PA.34). d See Equation (PA.37); f_{0} can also be defined by an uncertain variable. e These materials are using relative permeability model = 11; see Table PA4. f Initial value of porosity f_{0}; porosity changes dynamically to account for creep closure (see Section PA4.2.3). g See Equation (PA.35). 
where
l = pore distribution parameter (dimensionless)
P_{t}( k) = capillary threshold pressure (Pa) as a function of intrinsic permeability k (Webb 1992)
_{} = effective brine saturation (dimensionless) without correction for residual gas saturation
_{} = effective brine saturation (dimensionless) with correction for residual gas saturation
The values used for l, a, h, S_{br}, S_{gr}, and k are summarized in Table PA3. The statement that the BrooksCorey model is in use means that P_{c}, k_{rb}, and k_{rg} are defined by Equation (PA.38) through Equation (PA.40).
In the anhydrite MBs, either the BrooksCorey model or the van GenuchtenParker model is used as determined by the subjectively uncertain parameter ANHBCVGP (see Table PA19). A linear model is used to represent twophase flow in an open borehole (i.e., for the first 200 years after a drilling intrusion for boreholes with twoplug or threeplug configurations, in the open cavities [CAVITY_1, . . , CAVITY_4], and for the experimental and operations areas). This is discussed further below.
In the van GenuchtenParker model, P_{c}, k_{rb}, and k_{rg} are defined by (van Genuchten 1978)
where m = l/(1 + l) and the capillary pressure parameter P_{VGP} is determined by requiring that the capillary pressures defined in Equation (PA.38) and Equation (PA.44) are equal at an effective brine saturation of S_{e}_{2} = 0.5 (Webb 1992). The van GenuchtenParker model is only used for the anhydrite MBs in the Salado and uses the same values for l, S_{br}, and S_{gr} as the BrooksCorey model (Table PA3).
In the linear model used for the open borehole (RELP_MOD = 5), P_{c}, k_{rb}, and k_{rg} are defined by
P_{c} = 0, k_{rb} = S_{e}_{1}, k_{rg} = 1 – S_{e}_{1} (PA.47)
Another linear model (RELP_MOD = 11) is used for the open cavities (CAVITY_1, . . . , CAVITY_4) for the −5 to 0 year portion of the simulation (see Section PA4.2.2) and the experimental and operations areas (t = 0 to 10,000 years) which, in PA, are modeled without a timedependent creep closure:
where l = gas or brine and tol is a tolerance (slope) over which the relative permeability changes linearly from 0 to 1. In PA, tol = 1 ´ 10^{2} (dimensionless). Thus, the relative permeabilities are ~ 1 for saturations away from residual saturation.
Capillary pressure P_{c} for both the van GenuchtenParker and BrooksCorey models becomes unbounded as brine saturation S_{b} approaches the residual brine saturation, S_{br}. To avoid unbounded values, P_{c} is capped at 1 ´ 10^{8} Pa in selected regions (Table PA4).
Gas density is computed using the RKS equation of state, with the gas assumed to be pure H_{2}. For a pure gas, the RKS equation of state has the form (Walas 1985, pp. 43−54)
where
R = gas constant = 8.31451 Joules (J) mole (mol)^{ }^{1} K^{ }^{1}
T = temperature (K) = 300.15 K (= 30 °C; 81 °F)
V = molar volume (m^{3} mol^{ }^{1})
a = 0.42747_{}/P_{crit}
b = 0.08664 RT_{crit}_{ }/P_{crit}
Table PA4. Models for Relative Permeability and Capillary Pressure in TwoPhase Flow
Material 
Relative Permeability^{a } (RELP_MOD) 
Capillary Pressure^{b } (CAP_MOD) 
Material 
Relative Permeability^{a } (RELP_MOD) 
Capillary Pressure^{b } (CAP_MOD) 
S_HALITE 
4 
2 
WAS_AREA 
12 
1 
DRZ_0 
4 
1 
DRZ_1 
4 
1 
S_MB139 
ANHBCVGP^{c} 
2 
DRZ_PCS 
Sampled 
1 
S_ANH_AB 
ANHBCVGP^{c} 
2 
CONC_PCS 
4 
2 
S_MB138 
ANHBCVGP^{c} 
2 
UNNAMED 
4 
1 
CAVITY_1 
11 
1 
TAMARISK 
4 
1 
CAVITY_2 
11 
1 
FORTYNIN 
4 
1 
CAVITY_3 
11 
1 
DRF_PCS 
12 
1 
CAVITY_4 
11 
1 
REPOSIT 
12 
1 
IMPERM_Z 
4 
1 
CONC_MON 
4 
2 
CASTILER 
4 
2 
SHFTU 
4 
1 
OPS_AREA 
11 
1 
SHFTL_T1 
4 
1 
EXP_AREA 
11 
1 
SHFTL_T2 
4 
1 
CULEBRA 
4 
2 
CONC_PLG 
4 
1 
MAGENTA 
4 
2 
BH_OPEN 
5 
1 
DEWYLAKE 
4 
2 
BH_SAND 
4 
1 
SANTAROS 
4 
1 
BH_CREEP 
4 
1 
^{a} Relative permeability model, where 4 = BrooksCorey model given by Equation (PA.38) through Equation (PA.40), 5 = linear model given by Equation (PA.47), 11 = linear model given by Equation (PA.48) through Equation (PA.50), 12 = modified BrooksCorey model to account for cutoff saturation (Nemer 2007c), and ANHBCVGP ~ use of BrooksCorey or van GenuchtenParker model treated as a subjective uncertainty. ^{b} Capillary pressure model, where 1 = capillary pressure is unbounded, 2 = P_{c} bounded above by 1 ´ 10^{8} Pa as S_{b} approaches S_{br}. ^{c} See ANHBCVGP in Table PA19. 
a = _{}
» _{} for H_{2} (Graboski and Daubert 1979)
T_{crit} = critical temperature (K)
P_{crit} = critical pressure (Pa)
T_{r} = T / T_{crit} = reduced temperature
w = acentric factor
= 0 for H_{2} (Graboski and Daubert 1979)
In order to account for quantum effects in H_{2}, effective critical temperature and pressure values of T_{crit} = 43.6 K and P_{crit} = 2.047 ´ 10^{6} Pa are used instead of the true values for these properties (Prausnitz 1969). Equation (PA.51) is solved for molar volume V. The gas density r_{g} then is given by
where M_{w,H}_{ 2} is the molecular weight of H_{2} (i.e., 2.01588 ´ 10^{ }^{3} kg/mol; see Weast 1969, p. B26).
Brine density r_{b} is defined by Equation (PA.29), with r_{b}_{0}= 1230.0 kg/m^{3} at a pressure of P_{b}_{0} = 1.0132 ´ 10^{5}^{ }Pa and c_{b} = 2.5 ´ 10^{ }^{10} Pa^{ }^{1} (Roberts 1996). Porosity, f, is used as defined by Equation (PA.30) with two exceptions: in the repository (see Section PA4.2.3) and in the DRZ and MBs subsequent to fracturing (see Section PA4.2.4). The values of f_{0} and c_{ f} used in conjunction with Equation (PA.30) are listed in Table PA3. The reference pressure P_{b}_{0} in Equation (PA.30) is spatially variable and corresponds to the initial pressures P_{b}( x, y, −5) (here, −5 means at time equal to −5 years; see Section PA4.2.2). The gas and brine viscosities m_{l}, l = g, b in Equation (PA.24) and Equation (PA.25) were assumed to have values of m_{g} = 8.93 ´ 10^{ }^{6} Pa s (H2:VISCO; see Vargaftik 1975) and m_{b} = 2.1 ´ 10^{ }^{3} Pa s (BRINESAL:VISCO; see McTigue 1993).
The terms q_{g}, q_{rg}, q_{b}, and q_{rb} in Equation (PA.24) and Equation (PA.25) relate to well injection or removal (i.e., q_{g}, q_{b}) and reaction, production, or consumption (i.e., q_{rg}, q_{rb}) of gas and brine, with positive signs corresponding to injection or production and negative signs corresponding to removal or consumption. In the longterm Salado flow calculations, no injection or removal of gas or brine is calculated using q_{g} and q_{b}. Thus, q_{g} and q_{b} are equal to zero. That is, after an intrusion, the borehole is treated as a porous media, rather than a point source or sink of brine and gas. Furthermore, the mass and pressure lost to a DBR during the intrusion is conservatively ignored in the BRAGFLO calculations. In the DBR calculations discussed in Section PA4.7, q_{g} and q_{b} are used to describe injection and production wells in the DBR grid.
In PA, no gas consumption occurs through the term q_{rg} (see below), and gas production has the potential to occur (due to corrosion of steel or microbial degradation of CPR materials) only in the waste disposal regions of the repository (i.e., Waste Panel, South RoR, and North RoR in Figure PA15). Thus,
q_{rg} ³ 0 in waste disposal regions of Figure PA15
Gas consumption occurs due to the reaction of CO_{2} with MgO in the waste panels, and potentially from the sulfidation of steel. This gas consumption is not modeled using q_{rg}, but is accounted for by reducing the gas generation rate q_{rg}, as discussed in Section PA4.2.5. Finally, no brine production occurs, and brine consumption has the potential to occur (due to the consumption of brine during steel corrosion) only in the waste disposal regions of the repository. Thus,
q_{rb}_{ } £ 0 in waste disposal regions of Figure PA15
More detail on the definition of q_{rg} and q_{rb} is provided in Section PA4.2.5.
In each twophase flow simulation, a short period of time representing disposal operations is simulated. This period of time is called the startup period, and covers 5 years from t = 5 years to 0 years, corresponding to the amount of time a typical panel is expected to be open during disposal operations. All grid locations require initial brine pressure and gas saturation at the beginning of the simulation (t = 5 years).
The Rustler and overlying units (except in the shaft) are modeled as horizontal with spatially constant initial pressure in each layer (see Figure PA15). Table PA5 lists the initial brine pressure, P_{b}, and gas saturation, S_{g}, for the Rustler.
The Salado (Mesh Rows 3–24 in Figure PA15) is assumed to dip uniformly q = 1 degree downward from north to south (right to left in Figure PA15). Except in the repository excavations and the shaft, brine is initially assumed (i.e., at 5 years) to be in hydrostatic equilibrium relative to an uncertain initial pressure P_{b,ref} (SALPRES; see Table PA19) at a reference point located at shaft center at the elevation of the midpoint of MB 139, which is the center of Cell 1266 in Figure PA17. This gives rise to the condition
Table PA5. Initial Conditions in the Rustler
Name 
Mesh Row 
_{Pb}_{ (x, y, 5), Pa} 
_{Sg}_{ (x, y, 5)} 
Santa Rosa ^{c} 
33 
1.013250 ´ 10^{5} 
1  S_{b} =
0.916 
Santa Rosa ^{c} 
32 
1.013250 ´ 10^{5} 
1  S_{b} =
0.916 
Dewey Lake^{c} 
31 
1.013250 ´ 10^{5} 
1  S_{b} =
0.916 
Dewey Lake^{c} 
30 
7.355092 ´ 10^{5} 
1  S_{b} =
0.916 
Fortyniner^{c} 
29 
1.47328 ´ 10^{6} 
0^{b} 
Magenta 
28 
9.465 ´ 10^{5 } (MAGENTA:PRESSURE) 
0^{b} 
Tamarisk^{c} 
27 
1.82709 ´ 10^{6} 
0^{b} 
Culebra 
26 
9.141 ´ 10^{5} ^{(CULEBRA:PRESSURE)} 
0^{b} 
Los Medaños ^{c} 
25 
2.28346 ´ 10^{6} 
0^{b} 
^{a} The names in parenthesis are parameters in the WIPP PA Parameter Database. ^{b} The Rustler is assumed to be fully saturated. This initial condition is set in the program ICSET. See Nemer and Clayton (2008, Section 3.2). ^{c} These pressures are calculated in the ALGEBRA1 step analogously to Equation (PA.55) below, using the brine density of 1220 kg/m^{3}. See subsequent discussion taking θ = 0 and the reference point (x_{ref}, y_{ref}) at the top of the Dewey Lake. See the ALGEBRA input file ALG1_BF_CRA09.INP in library LIBCRA09_BF, class CRA091 on the WIPP PA cluster for details. See Nemer and Clayton (2008, Section 4.1.7) for details on the ALGEBRA1 step. 
where
h(x, y) is defined in Equation (PA.33)
r_{b}_{0} = 1220 kg/m^{3} (BRINESAL:DNSFLUID)
c_{b} = 3.1 ´ 10^{ }^{10} Pa^{ }^{1} (BRINESAL:COMPRES)
g = 9.80665 m/s^{2}
P_{b,ref} = 1.01325 ´ 10^{5} Pa (BRINESAL:REF_PRES)
P_{b}_{0} = sampled farfield pressure in the undisturbed halite (S_HALITE:PRESSURE)
In the Salado, initial gas saturation S_{g}( x, y, 5) = 0 (see Nemer and Clayton 2008, Section 4.1.6).
The Castile (Mesh Rows 1 and 2) is modeled as horizontal, and initial brine pressure is spatially constant within each layer (no dip), except that the brine reservoir is treated as a different material from rest of Castile and has a different initial pressure which is a sampled parameter. Specifically, outside the brine reservoir, pressure is calculated using Equation (PA.55) above with no dip (q = 0) in the ALGEBRA1 step. Within the reservoir, P_{b}( x, y, 5) = BPINTPRS, the uncertain initial pressure in the reservoir (see Table PA19). Initial gas saturation S_{g}( x, y, 5) = 0.
Within the shaft (areas Upper Shaft, Lower Shaft, and CONC_MON) and panel closures (areas CONC_PCS and DRF_PCS), P_{b}( x, y, 5) = 1.01325 ´ 10^{5} Pa and S_{g}( x, y, 5) = 0. Within the excavated areas (Waste Panel, South RoR, and North RoR, Ops and Exp), P_{b}( x, y, 5) = 1.01325 ´ 10^{5} Pa and S_{g}( x, y, 5) = 0.
At the end of the initial fiveyear startup period and the beginning of the regulatory period (t = 0 years), brine pressure and gas saturation are reset in the shaft, panel closures, and excavated areas. In the shaft (areas Upper Shaft, Lower Shaft, and CONC_MON) and panel closures (areas CONC_PCS and DRF_PCS), P_{b}( x, y, 0) = 1.01325 ´ 10^{5} Pa and S_{g}( x, y, 0) = 1 ´ 10^{ }^{7} (see CONC_MON:SAT_IBRN). In the waste disposal regions (areas Waste Panel, South RoR, and North RoR), P_{b}( x, y, 0) = 1.01325 ´ 10^{5} Pa and S_{g}( x, y, 05) = 0.985 (see WAS_AREA:SAT_IBRN). In the other excavated areas, P_{b}( x, y, 0) = 1.01325 ´ 10^{5} Paand S_{g}( x, y, 0) = 1.0.
Salt creep occurs naturally in the Salado halite in response to deviatoric stress. Inward creep of rock is generally referred to as creep closure. Creep closure of excavated regions begins immediately from excavationinduced deviatoric stress. If the rooms were empty, closure would proceed to the point where the void volume created by the excavation would be eliminated as the surrounding formation returned to a uniform stress state. In the waste disposal region, inward creep of salt causes consolidation of the waste, and this waste consolidation continues until loading in the surrounding rock is uniform, at which point salt creep and waste consolidation ceases. The amount of waste consolidation that occurs and the time it takes to consolidate are governed by the waste properties (e.g., waste strength, modulus, etc.), the surrounding rock properties, the dimensions and location of the room, and relative quantities of brine and gas present.
The porosity of the waste disposal regions and neighboring access drifts (i.e., Waste Panel, South RoR, North RoR, and DRF_PCS in Figure PA15) is assumed to change through time due to creep closure of the halite surrounding the excavations. The equations on which BRAGFLO is based do not incorporate this type of deformation. Therefore, the changes in repository porosity due to halite deformation are modeled in a separate analysis with the geomechanical program SANTOS, which implements a quasistatic, largedeformation, finiteelement procedure (Stone 1997). Interpolation procedures are then used with the SANTOS results to define porosity (f) within the repository as a function of time, pressure, and gas generation rate.
For more information on the generation of the porosity surface for BRAGFLO in PA, see Appendix PORSURF2009.
Fracturing within the anhydrite MBs (i.e., regions MB 138, Anhydrite AB, and MB 139 in Figure PA15) and in the DRZ (region DRZ in Figure PA15) is assumed to occur at pressures slightly above lithostatic pressure, and is implemented through a pressuredependent compressibility c_{r}( P_{b}) (Mendenhall and Gerstle 1995). Specifically, MB fracturing begins at a brine pressure of
where P_{bi} and P_{b}_{0} are spatially dependent (i.e., P_{b}_{0} = P(x, y, 0) as in Section PA4.2.2) and DP_{i} = 2 ´ 10^{5} Pa (see S_MB138:PI_DELTA in Fox 2008, Table 30)
Fracturing ceases at a pressure of
and a fully fractured porosity of
where DP_{a} = 3.8 ´ 10^{6} Pa (see S_MB138:PF_DELTA in Fox 2008, Table 30), f_{0} is spatially dependent (Table PA3), and Df_{a} = 0.04, 0.24, and 0.04 for anhydrite materials S_MB138, S_ANH_AB, and S_MB139, respectively (see S_MB138:DPHIMAX in Fox 2008, Table 30).
Once fractured, compressibility c_{r} becomes a linear function
of brine pressure for P_{bi} £ P_{b} £ P_{ba}, with c_{ra} defined so that the solution f of
satisfies f (P_{ba}) = f_{a}; specifically, c_{ra} is given by
The permeability k_{f}( P_{b}) of fractured material at brine pressure P_{b} is related to the permeability of unfractured material at brine pressure P_{bi} by
where k is the permeability of unfractured material (i.e., at P_{bi}) and n is defined so that k_{f}( P_{ba}) = 1 ´ 10^{ }^{9} m^{2} (i.e., n is a function of k, which is an uncertain input to the analysis; see ANHPRM in Table PA19). When fracturing occurs, k_{f}( P_{b}) is used instead of k in the definition of the permeability for the fractured areas of the anhydrite MBs.
Fracturing is also modeled in the DRZ region in Figure PA15. The fracture model implementation is the same as for the anhydrite materials. In this case, fracturing would be in halite rather than anhydrite, but because of the limited extent of the DRZ and the proximity of the nearby interbeds, this representation was deemed acceptable by the Salado Flow Peer Review panel (Caporuscio, Gibbons, and Oswald 2003).
Gas production is assumed to result from anoxic corrosion of steel and the microbial degradation of CPR materials. Thus, the gas generation rate q_{rg} in Equation (PA.24) is of the form
where q_{rgc} is the rate of gas production per unit volume of waste (kg/m^{3}/s) due to anoxic corrosion of Febase metals, and q_{rgm} is the rate of gas production per unit volume of waste (kg/m^{3}/s) due to microbial degradation of CPR materials. Furthermore, q_{rb} in Equation (PA.25) is used to describe the consumption of brine during the corrosion process.
Gas generation takes place only within the waste disposal regions (i.e., Waste Panel, South RoR, and North RoR in Figure PA15) and all the generated gas is assumed to have the same properties as H_{2} (see discussion in Appendix MASS2009, Section MASS3.2). In PA, the consumable materials are assumed to be homogeneously distributed throughout the waste disposal regions (i.e., the concentration of Febase metals and CPR materials in the waste is not spatially dependent). A separate analysis examined the potential effects on PA results of spatially varying Febase metal and CPR material concentrations, and concluded that PA results are not affected by representing these materials with spatially varying concentrations (see Appendix MASS2009, Section MASS21.0).
The rates q_{rgc}, q_{rb}, and q_{rgm} (kg/m^{3}/s) are defined by
gas generation by corrosion
brine consumption by corrosion
and microbial gas generation
where
D_{s} _{ } = surface area concentration of steel in the repository (m^{2}_{ }surface area steel/ m^{3} disposal volume)
D_{c} = mass concentration of cellulosics in the repository (kg biodegradable material/m^{3} disposal volume)
_{} = molecular weight of H_{2} (kg H_{2}/mol H_{2})
_{} = molecular weight of water (H_{2}O) (kg H_{2}O/mol H_{2}O)
R_{ci} = corrosion rate under inundated conditions (m/s)
R_{ch} = corrosion rate under humid conditions (m/s)
R_{mi} = rate of cellulose biodegradation under inundated conditions (mol C_{6}H_{10}O_{5}/kg C_{6}H_{10}O_{5}/s)
R_{mh} = rate of cellulose biodegradation under humid conditions (mol C_{6}H_{10}O_{5}/kg C_{6}H_{10}O_{5}/s)
S_{b,eff} = effective brine saturation due to capillary action in the waste materials (see Equation (PA.90) in Section PA4.2.6)
_{} = _{}
_{} = stoichiometric coefficient for gas generation due to corrosion of steel, i.e., moles of H_{2} produced by the corrosion of 1 mole of Fe (mol H_{2}/mol Fe)
_{} = stoichiometric coefficient for brine consumption due to corrosion of steel, i.e., moles of H_{2}O consumed per mole of H_{2} generated by corrosion (mol H_{2}O/mol H_{2})
_{} = average stoichiometric factor for microbial degradation of cellulose, i.e., the moles of H_{2} generated per mole of carbon consumed by microbial action (mol H_{2}/mol C)
r_{Fe} = molar density of steel (mol/m^{3})
B_{fc} = parameter (WAS_AREA:BIOGENFC; discussed in detail later in this section) uniformly sampled from 0 to 1, used to account for the uncertainty in whether microbial gas generation could be realized in the WIPP at experimentally measured rates.
The products R_{ci} D_{s} r_{Fe} X_{c} , R_{ch} D_{s} r_{Fe} X_{c}, R_{mi} D_{c} y, and R_{mh} in Equation (PA.68) and Equation (PA.70) define constant rates of gas generation (mol/m^{3}/s) that continue until the associated substrate (i.e., steel or cellulose) is exhausted (i.e., zero order kinetics are assumed). The terms S_{b,eff} and _{} in Equation (PA.68) and Equation (PA.70), which are functions of location and time, correct for the amount of substrate exposed to inundated and humid conditions, respectively. All the corrosion and microbial action is assumed to cease when no brine is present, which is the reason that 0 replaces S_{g} = 1 in the definition of _{}. In PA, R_{ch} = 0 and R_{ci}, R_{mh}, and R_{mi} are defined by uncertain variables (see WGRCOR, WGRMICH, WGRMICI in Table PA19). However, R_{mh} is now sampled based on the sampled value of R_{mi}: see Nemer and Clayton (2008, Section 5.1.3). Further, M_{H}_{ 2} = 2.02 ´10^{ }^{3} kg/mol (Lide 1991, pp. 17, 18), M_{H}_{ 2}_{O} = 1.80 ´ 10^{ }^{2} kg/mol (Lide 1991, pp. 17, 18), r_{Fe} = 1.41 ´ 10^{5} mol/m^{3} (Telander and Westerman 1993), and D_{s}, D_{c}, X_{c}(H_{2}OH_{2}), X_{c}(H_{2}Fe), and y(H_{2} C) are discussed below.
The concentration D_{s} in Equation (PA.68) is defined by
where
A_{d} = surface area of steel associated with a waste disposal drum (m^{2}/drum)
V_{R} = initial volume of a single room in the repository (m^{3})
n_{d} = ideal number of waste drums that can be closepacked into a single room
In PA, A_{d} = 6 m^{2}/drum (REFCON:ASDRUM), V_{R} = 3,644 m^{3} (REFCON:VROOM), and n_{d} = 6804 drums (REFCON:DRROOM).
The biodegradable materials to be disposed at the WIPP consist of cellulosic materials, plastics, and rubbers. Cellulosics have been demonstrated experimentally to be the most biodegradable of these materials (Francis, Gillow, and Giles 1997). The occurrence of significant microbial gas generation in the repository will depend on whether (1) microbes capable of consuming the emplaced organic materials will be present and active, (2) sufficient electron acceptors will be present and available, and (3) enough nutrients will be present and available.
During the CRA2004 PABC, the EPA (Cotsworth 2005) indicated that the probability that microbial
gas generation could occur (WMICDFLG) should be set equal to 1 in
PA calculations. In the CRA2004, the probability that
microbial gas generation could occur was assigned a value of
0.5. To comply with the EPA’s letter, in the CRA2004
PABC and the CRA2009 PA the parameter WMICDFLG was changed so
that the probability that microbial gas generation could occur
was set to 1 while preserving the previous probability
distribution on whether CPR could be degraded. This is
summarized in Table PA6, and is
discussed further in Nemer and Stein
Table PA6. Probabilities for Biodegradation of Different Organic Materials (WAS_AREA:PROBDEG) in the CRA2009 PA and the CRA2004 PA
WAS_AREA:PROBDEG 
Meaning 
Probability CRA2004 
Probability CRA2009 
0 
No microbial degradation can occur 
0.5 
0.0 
1 
Biodegradation of only cellulose can occur 
0.25 
0.75 
2 
Biodegradation of CPR materials can occur 
0.25 
0.25 
(2005, Section 5.4). Because there are significant uncertainties in whether the experimentally observed gasgeneration rates could be realized in the WIPP repository, during the CRA2004 PABC the EPA agreed to allow the DOE to multiply the sampled microbial rates by a parameter (WAS_AREA:BIOGENFC) uniformly sampled from 0 to 1. This is discussed further in Nemer, Stein, and Zelinski (2005, Section 4.2.2).
In cases where biodegradation of rubbers and plastics occur, rubbers and plastics are converted to an equivalent quantity of cellulosics based on their carbon equivalence (Wang and Brush 1996a). This produces the density calculation
for biodegradation of cellulosics only 

for biodegradation of CPR materials 
where m_{cel} is the mass of cellulosics (kg), m_{r} is the mass of rubbers (kg), and m_{p} is the mass of plastics (kg).
In the CRA2009 PA, the emplacement materials (cellulose and plastic) have been added to m_{cel} and m_{p}. This is discussed further in Nemer and Clayton (2008, Section 5.1.1). Density values for CPR materials can be found in Fox (2008, Table 34).
The most plausible corrosion reactions after closure of the WIPP are believed to be (Wang and Brush 1996a)
Fe + 2H_{2}O = Fe(OH)_{2} + H_{2} (PA.73)
and
3Fe + 4H_{2}O = Fe_{3}O_{4} + 4H_{2} (PA.74)
When normalized to 1 mole of Fe and linearly weighted by the factors x and _{}, the two preceding reactions become
where x and _{} are the fractions of Fe consumed in the reactions in Equation (PA.73) and Equation (PA.74), respectively. Although magnetite (Fe_{3}O_{4}) has been observed to form on Fe as a corrosion product in lowMg anoxic brines at elevated temperatures (Telander and Westerman 1997) and in oxic brine (Haberman and Frydrych 1988), there is no evidence that it will form at WIPP repository temperatures. If Fe_{3}O_{4} were to form, H_{2} would be produced (on a molar basis) in excess of the amount of Fe consumed. However, anoxic corrosion experiments (Telander and Westerman 1993) did not indicate the production of H_{2} in excess of the amount of Fe consumed. Therefore, the stoichiometric factor x in Reaction (PA.75) is set to 1.0 (i.e., x = 1), which implies that Reaction (PA.73) represents corrosion. Thus, the stoichiometric factor for corrosion is
which implies that one mole of H_{2} is produced for each mole of Fe consumed, and the stoichiometric factor for brine consumption is
which implies that two moles of H_{2}O are consumed for each mole of H_{2} produced.
The most plausible biodegradation reactions after closure of the WIPP are believed to be (Wang and Brush 1996a)
denitrification C_{6}H_{1O}O_{5} + 4.8H^{+} + 4.8NO_{3}^{ } = 7.4H_{2}O + 6CO_{2} + 2.4N_{2} (PA.78)
sulfate reduction C_{6}H_{1O}O_{5} + 6H^{+} + 3SO_{4}^{2}^{ } = 5H_{2}O + 6CO_{2} + 3H_{2}S (PA.79)
methanogenesis C_{6}H_{1O}O_{5} + H_{2}O = 3CH_{4} + 3CO_{2} (PA.80)
Accumulation of CO_{2} produced by the above reactions could decrease pH and add carbonate to increase An solubility in the repository (Wang and Brush 1996b).
However, in the CRA2004 PABC, the EPA (Cotsworth 2005) directed the DOE to remove methanogenesis (Equation (PA.80)) from PA. The EPA cited the presence of calcium sulfate as gypsum and anhydrite in the bedded salt surrounding the repository as possible sources of sulfate. These sources of sulfate would, if accessible, promote sulfate reduction (Equation (PA.82)), which is energetically and kinetically favored over methanogenesis. In response, the DOE removed methanogenesis from PA. Additionally, the DOE removed Fe sulfidation from PA as a gasconsuming reaction because sulfidation of steel produces one mole of H_{2} for every mole of H_{2}S consumed,
Fe + H_{2}S = FeS + H_{2} (PA.81)
This and the removal of methanogenesis are discussed fully in Nemer and Zelinski (2005).
To provide added assurance of WIPP performance, a sufficient amount of MgO is added to the repository to remove CO_{2} (Bynum et al. 1997). MgO in polypropylene “supersacks” is emplaced on top of the threelayer waste stacks to create conditions that reduce An solubilities in the repository (see Appendix MgO2009 and Appendix SOTERM2009, Section SOTERM2.3). If brine flows into the repository, MgO will react with water in brine and in the gaseous phase to produce brucite (Mg[OH]_{2}). MgO will react with essentially all of the CO_{2} that could be produced by complete microbial consumption of the CPR materials in the waste, and will create hydromagnesite with the composition Mg_{5}(CO_{3})_{4}(OH)_{2}·4H_{2}O (Appendix MgO2009; Appendix SOTERM2009, Section SOTERM2.0). The most important MgO hydration and carbonation reactions that will occur in the WIPP are
MgO + H_{2}O(aq and/or g) → Mg(OH)_{2} (PA.82)
5Mg(OH)_{2} + 4CO_{2}(aq and/or g) → Mg_{5}(CO_{3})_{4}(OH)_{2}×4H_{2}O (PA.83)
In these equations, “aq and/or g” indicates that the H_{2}O or CO_{2} that reacts with MgO and/or brucite could be present in the aqueous phase (brine) and/or the gaseous phase. The removal of CO_{2} by MgO is not explicitly modeled as a separate reaction in PA. Rather, the effect of CO_{2} consumption is accounted for by modifying the stoichiometry of Reaction (PA.78), Reaction (PA.79), and Reaction (PA.80) to remove the CO_{2} from the mass of gas produced by microbial action.
The average stoichiometry of Reaction (PA.78) , Reaction (PA.79), and Reaction (PA.80), is
where the average stoichiometric factor y in Reaction (PA.84) represents the number of moles of gas produced and retained in the repository from each mole of carbon consumed. This factor y depends on the extent of the individual biodegradation pathways in Reaction (PA.84), and the consumption of CO_{2} by MgO. A range of values for y is estimated by considering the maximum mass of gas that can be produced from consumption of cellulosics (M_{cel}) and Febase metals (M_{Fe}), and is derived as follows (Wang and Brush 1996b). Estimates of the maximum quantities M_{cel} and M_{Fe} (mol) of cellulosics (i.e., C_{6}H_{10}O_{5}) and steels that can be potentially consumed in 10,000 years are given by
where m_{cel} and m_{Fe} are the masses (kg) of cellulosics (see Equation (PA.72) for definition) and steels initially present in the repository. The mass of cellulosics that can be consumed is determined by the uncertain parameter WMICDFLG (see Table PA19). The mass of steels m_{Fe} = 5.16 ´ 10^{7} kg; this value is calculated as
where V_{CH} and V_{RH} are the volumes of CHTRU and RHTRU waste, r_{WCH} and r_{WRH} are the Fe densities in CHTRU and RHTRU waste, and r_{CCH} and r_{CRH} are the Fe densities of the containers of CHTRU and RHTRU waste (see Fox 2008, Table 34). The terms 6000 m_{cel}/162 and 1000 m_{Fe}/56 in Equation (PA.85) and Equation (PA.86) equal the inventories in moles of cellulosics and steel, respectively. The terms 3.2 ´ 10^{11} R_{m} m_{cel} and 4.4 ´ 10^{16} R_{ci} A_{d} n_{d} equal the maximum amounts of cellulosics and steel that could be consumed over 10,000 years. In Equation (PA.85), R_{m} = max{R_{mh}, R_{mi}}, where R_{mh} and R_{mi} are defined by uncertain variables (see WGRMICH and WGRMICI in Table PA19, respectively), and 3.2 ´ 10^{11} = (3.15569 ´ 10^{7} s/yr) (10^{4} yr). In Equation (PA.86), A_{d} n_{d} is the total surface area of all drums (m^{2}) and the factor 4.4 ´ 10^{16} = (3.15569 ´ 10^{7} s/yr) (10^{4} yr) (1.41 ´ 10^{5} mol/ m^{3}), where r_{Fe} = 1.41 ´ 10^{5} mol/m^{3} (see Equation (PA.72)) (Telander and Westerman 1993), converts the corrosion rate from m/s to mol/m^{2}/s.
A range of possible values for the average stoichiometric factor y in Reaction (PA.84) can be obtained by considering individual biodegradation pathways involving M_{cel} and accounting for the removal of CO_{2} by the MgO.
In the absence of methanogenesis and steel sulfidation, y from Equation (PA.84) becomes
where _{} is the quantity of NO_{3}^{ } (mols) initially present in the repository. Specifically, _{}= 4.31 ´ 10^{7} mol (Fox 2008, Table 39).
Capillary action (wicking) is the ability of a material to carry a fluid by capillary forces above the level it would normally seek in response to gravity. In the current analysis, this phenomena is accounted for by defining an effective saturation given by
where
S_{b,eff} = effective brine saturation
S_{b} = brine saturation
S_{wick} = wicking saturation
S_{min}_{ } = minimum brine saturation at which code can run in the wastefilled areas
α = smoothing parameter = 1000
The effective saturation given by Equation (PA.90) differs from that in the CRA2004 PA in that S_{b,eff} now approaches zero as S_{b} approaches a small value S_{min}. In simulations where Fe corrosion dried out the repository, the time required to complete the simulation could be quite long. In order to speed up the code and increase robustness, the parameter S_{min} was added. For PA, S_{min} = 0.015, which was small enough to not affect the results, while greatly reducing run time. This is explained fully in Nemer and Clayton (2008, Section 5.2.2).
The effective saturation is used on a grid block basis within all waste regions (Waste Panel, South RoR, and North RoR in Figure PA15). The wicking saturation, S_{wick}, is treated as an uncertain variable (see WASTWICK in Table PA19). The effective brine saturation S_{b,eff} is currently used only to calculate the corrosion of steel (Equation (PA.68)) and the microbial degradation of cellulose (Equation (PA.70)), and does not directly affect the twophase flow calculations indicated.
The WIPP excavation includes four shafts that connect the repository region to the surface: the air intake shaft, salt handling shaft, waste handling shaft, and exhaust shaft. In PA, these four shafts are modeled as a single shaft. The rationale for this modeling treatment is set forth in SNL (Sandia National Laboratories 1992, Volume 5, Section 2.3).
The shaft seal model included in the PA grid (Column 43 in Figure PA15) is the simplified shaft model used in the CRA2004 PA and the CRA2004 PABC. The simplified shaft seal model used in PA is described by Stein and Zelinski (2003) and is briefly discussed below; this model was approved by the Salado Flow Peer Review Panel (Caporuscio, Gibbons, and Oswald 2003).
The planned design of the shaft seals involves numerous materials, including earth, crushed salt, clay, asphalt, and Salado Mass Concrete (SMC) (see the CCA, Appendix SEAL). The design is intended to control both shortterm and longterm fluid flow through the Salado portion of the shafts. For the CCA PA, each material in the shaft seal was represented in the BRAGFLO grid. Analysis of the flow results from the CCA PA and the subsequent CCA PAVT (Sandia National Laboratories 1997 and U.S. Department of Energy 1997) indicated that no significant flows of brine or gas occurred in the shaft during the 10,000year regulatory period. As a result of these analyses, a simplified shaft seal model was developed for the CRA2004 PA.
A conceptual representation of the simplified shaft seal system used in PA is shown in Figure PA18. The simplified model divides the shaft into three sections: an upper section (shaft seal above the Salado), a lower section (within the Salado), and a concrete monolith section within the repository horizon. A detailed discussion of how the material properties were assigned for the simplified shaft seal model is included in James and Stein (2003). The permeability value used to represent the upper and lower sections is defined as the harmonic mean of the component materials’ permeability in the detailed shaft seal model (including permeability adjustments made for the DRZ assumed to surround the lower shaft seal section within the Salado). Porosity is defined as the thicknessweighted mean porosity of the component materials. Other material properties are described in James and Stein (2003).
The lower section of the shaft experiences a change in material properties at 200 years. This change simulates the consolidation of seal materials within the Salado and significantly decreases permeability. This time was chosen as a conservative overestimate of the amount of time expected for this section of the shaft to become consolidated. The concrete monolith section of the shaft is unchanged from the CCA PA and is represented as being highly permeable for 10,000 years to ensure that fluids can access the north end (operations and experimental areas) in the model. In three thin regions at the stratigraphic position of the anhydrite MBs, the shaft seal is modeled as MB material (Figure PA18). This model feature is included so that fluids flowing in the DRZ and MB fractures can access the interbeds to the north of the repository “around” the shaft seals. Because these layers are so thin, they have virtually no effect on the effective permeability of the shaft seal itself.
The simplified shaft model was tested in the AP106 analysis (Stein and Zelinski 2003), which supports the Salado Flow Peer Review (Caporuscio, Gibbons, and Oswald 2003). The results of the AP106 analysis demonstrate that vertical brine flow through the simplified shaft model is comparable to brine flows seen through the detailed shaft model used in the CCA PA and subsequent CCA PAVT calculations.
PA includes panel closures models that represent the Option D panel closure design (see the CRA2004, Chapter 6.0, Section 6.4.3), which are unchanged from the CRA2004. Option D closures (Figure PA19) are designed to allow minimal fluid flow between panels. PA explicitly represents selected Option D panel closures in the computational grid using a model approved by the Salado Flow Peer Review Panel (Caporuscio, Gibbons, and Oswald 2003). The Option D panel closure design has several components: an SMC monolith, which extends into the DRZ in all directions; an empty drift section; and a block and mortar explosion wall (Figure PA19). Each set of panel closures are represented in the BRAGFLO grid by 4 materials in 13 grid cells (Figure PA20):
18. Schematic View of the Simplified Shaft Model (numbers on right indicate dimensions in meters) PA
Figure PA19. Schematic Side View of Option D Panel Closure
· Six cells of panel closure concrete (area CONC_PCS, material CONC_PCS)
· One cell above and one cell below the concrete material consisting of MB anhydrite (areas MB 139 and Anhydrite AB, materials S_MB139 and S_ANH_AB, respectively)
· Two cells of healed DRZ above Anhydrite AB and the panel closure system (PCS) (area DRZ_PCS, material DRZ_PCS)
· Three cells of empty drift and explosion wall (area DRF_PCS, material DRF_PCS)
Figure PA20. Representation of Option D Panel Closures in the BRAGFLO Grid
Properties for the materials making up the panel closure system are listed in Table PA3.
The Option D panel closure design requires the use of a saltsaturated concrete, identified as SMC, as specified for the shaft seal system. The design of the shaft seal system and the properties of SMC are described in the CCA, Appendix SEAL. The BRAGFLO grid incorporates the material CONC_PCS, which is assigned the material properties of undegraded SMC and is used to represent the concrete portion of the Option D panel closure system (Figure PA15). A doublethickness concrete segment represents the northernmost set of panel closures (between the north RoR and the operations area). This feature represents the two sets of panel closures in series that will be emplaced between the wastefilled repository and the shaft.
In the BRAGFLO grid, regions where the Option D panel closures intersect the MBs are represented as blocks of MB material (Figure PA15). This representation is warranted for two reasons:
1. The MB material has a very similar permeability distribution (10^{ }^{21} to 10^{ }^{17.1} m^{2}) to the concrete portion of the Option D panel closures (10^{ }^{20.699}^{ }to 10^{ }^{17} m^{2}), and thus, assigning this material as anhydrite MB in the model has essentially the same effect as calling it concrete, as long as pressures are below the fracture initiation pressure.
2. In the case of high pressures, it is expected that fracturing may occur in the anhydrite MBs and flow could go “around” the panel closures and out of the twodimensional plane considered in the model grid. In this case, the flow would be through the MB material, which incorporates a fracture model, as described above.
After constructing the concrete portion of the panel closure, the salt surrounding the monolith will be subjected to compressive stresses, which will facilitate the rapid healing of disturbed halite. The rounded configuration of the monolith creates a situation very favorable for concrete durability: high compressive stresses and low stress differences. In turn, the compressive stresses developed within the salt will quickly heal any damage caused by construction excavation, thereby eliminating the DRZ along this portion of the panel closure. The permeability of the salt immediately above and below the rigid concrete monolith component of Option D will approach the intrinsic permeability of the undisturbed Salado halite.
To represent the DRZ above the monoliths, PA uses the material DRZ_PCS in the BRAGFLO grid (Figure PA15). The values assigned to DRZ_PCS are the same as those used for the DRZ above the excavated areas (material DRZ_1, see Table PA3), except for the properties PRMX_LOG, PRMY_LOG, and PRMZ_LOG, the logarithms of permeability in the x, y, and z directions, respectively. These permeability values are assigned the same distributions used for the material CONC_PCS. In this instance, the values are based on the nature of the model setup, not directly on experimental data (although the general range of the distribution agrees with experimental observations of healed salt). The use of these permeabilities ensures that any fluid flow is equally probable through or around the Option D panel closures, and represents the range of uncertainty that exists in the performance of the panel closure system.
The DRF_PCS is the material representing the empty drift and explosion wall. For simplicity, this material is assumed to have hydrologic properties equivalent to the material representing the waste panel and is used for the three sets of panel closures represented in the grid (Figure PA15). The creep closure model is applied to this material to be consistent with the neighboring materials. The assignment of a high permeability to this region for the PA calculations, which contains the explosion wall, is justified because the explosion wall is not designed to withstand the stresses imposed by creep closure beyond the operational period and will be highly permeable following rapid room closure.
The major disruptive event in PA is the penetration of the repository by a drilling intrusion. In the undisturbed scenario (see Section PA6.7.1), these blocks have the material properties of the neighboring stratigraphic or excavated modeling unit, and there is no designation in the borehole grid except for the reduced lateral dimensions of this particular column of grid blocks.
In the scenarios simulating drilling disturbance, these cells start out with the same material properties as in the undisturbed scenario, but at the time of intrusion, the borehole grid blocks are reassigned to borehole material properties. The drilling intrusion is modeled by modifying the permeability of the grid blocks in Column 26 of Figure PA15 (values listed in Table PA7). Furthermore, the drilling intrusion is assumed to produce a borehole with a diameter of 12.25 in. (0.31 m) (Vaughn 1996, Howard 1996), borehole fill is assumed to be incompressible, capillary effects are ignored, residual gas and brine saturations are set to zero, and porosity is set to 0.32 (see materials CONC_PLG, BH_OPEN, BH_SAND, and BH_CREEP in Table PA3). When a borehole that penetrates pressurized brine in the Castile is simulated (i.e., an E1 intrusion), the permeability modifications indicated in Table PA7 extend from the ground surface (i.e., Grid Cell 2155 in Figure PA17) to the base of the pressurized brine (i.e., Grid Cell 2225 in Figure PA17). When a borehole that does not penetrate pressurized brine in the Castile is under consideration (i.e., an E2 intrusion), the permeability modifications indicated in Table PA7 stop at the bottom of the lower DRZ (i.e., Grid Cell 1111 in Figure PA17).
Highpressure Castile brine was encountered in several WIPParea boreholes, including the WIPP12 borehole within the controlled area and the ERDA6 borehole northeast of the site. Consequently, the conceptual model for the Castile includes the possibility that brine reservoirs underlie the repository. The E1 and E1E2 scenarios include borehole penetration of both the repository and a brine reservoir in the Castile.
Unless a borehole penetrates both the
repository and a brine reservoir in the Castile, the Castile
is conceptually unimportant to PA because of its expected low
permeability. Two regions are specified in the disposal
system geometry of the Castile horizon: the Castile (Rows 1
and 2 in
Table PA7. Permeabilities for Drilling Intrusions Through the Repository
Time After Intrusion 
Assigned Permeabilities 
0–200 years 
Concrete plugs are assumed to be emplaced at the Santa Rosa (i.e., a surface plug with a length of 15.76 m; corresponds to Grid Cells 2113, 2155 in Figure PA17) and the Los Medaños Member of the Rustler (i.e., a plug at the top of the Salado with a length of 36 m; corresponds to Grid Cell 1644 in Figure PA17). Concrete plugs are assumed to have a permeability loguniformly sampled between 10^{19} m^{2} to 10^{17}m^{2} (see material CONC_PLG in Table PA4). The open portions of the borehole are assumed to have a permeability of 1 ´ 10^{9} m^{2}. 
200–1200 years 
Concrete plugs are assumed to fail after 200 years (U.S. Department of Energy 1995). An entire borehole is assigned a permeability typical of silty sand loguniformly sampled between 10^{16.3} m^{2} and 10^{11} m^{2} (see parameter BHPRM in Table PA19 and material BH_SAND in Table PA4). 
> 1200 years 
Permeability of borehole reduced by one order of magnitude in the Salado beneath the repository due to creep closure of borehole (Thompson et al. 1996) (i.e., k = 10^{x}/10, x = BHPRM, in Grid Cells 2225, 1576, 26, 94, 162, 230 of Figure PA17). No changes are made within and above the lower DRZ (see material BH_CREEP in Table PA4). 
Figure PA15) and a reservoir (Row 1, Columns 23 to 45 in Figure PA15). The Castile region has an extremely low permeability, which prevents it from participating in fluid flow processes.
It is unknown whether a brine reservoir exists below the repository. As a result, the conceptual model for the brine reservoirs is somewhat different from those for known major properties of the natural barrier system, such as stratigraphy. The principal difference is that a reasonable treatment of the uncertainty of the existence of a brine reservoir requires assumptions about the spatial distribution of such reservoir and the probability of intersection (see Attachment MASS2009, Section MASS.18.1). A range of probabilities for a borehole hitting a brine reservoir is used (see Section PA3.6).
In addition to the stochastic uncertainty in the location and hence in the probability of intersecting reservoirs, there is also uncertainty in the properties of reservoirs. The manner in which brine reservoirs would behave if penetrated is captured by parameter ranges and is incorporated in the BRAGFLO calculations of disposal system performance. The conceptual model for the behavior of such a brine reservoir is discussed below. The properties specified for brine reservoirs are pressure, permeability, compressibility, and porosity, and are sampled from parameter ranges (see Table PA19).
Where they exist, Castile brine reservoirs in the northern Delaware Basin are believed to be fractured systems, with highangle fractures spaced widely enough that a borehole can penetrate through a volume of rock containing a brine reservoir without intersecting any fractures, and therefore not producing brine. Castile brine reservoirs occur in the upper portion of the Castile (Popielak et al. 1983). Appreciable volumes of brine have been produced from several reservoirs in the Delaware Basin, but there is little direct information on the areal extent of the reservoirs or the existence of the interconnection between them. Data from WIPP12 and ERDA6 indicate that fractures have a variety of apertures and permeabilities, and they deplete at different rates. Brine occurrences in the Castile behave as reservoirs; that is, they are bounded systems.
Determining gas and brine flow in the vicinity of the repository requires solving the two nonlinear PDEs in Equation (PA.24) through Equation (PA.30) on the computational domain in Figure PA15, along with evaluating appropriate auxiliary conditions. The actual unknown functions in this solution are P_{b} and S_{g}, although the constraint conditions also give rise to values for P_{g} and S_{b}. As two dimensions in space and one dimension in time are in use, P_{b}, P_{g}, S_{b}, and S_{g} are functions of the form P_{b}( x, y, t), P_{g}( x, y, t), S_{b}( x, y, t), and S_{g}( x, y, t).
Solving Equation (PA.24) through Equation (PA.30)
requires both initial value and boundary value conditions for
P_{b} and
S_{g}.
The initial value conditions for P_{b} and
S_{g} are
given in Section PA4.2.2. As
indicated there, the calculation starts at time t =
−5 years, with a possible resetting of values at t =
0 years, which corresponds to final waste emplacement and sealing
of the repository. The boundary conditions are such that no
brine or gas moves across the exterior grid boundary (Table PA8). This Neumanntype
boundary condition is maintained for all time. Further,
BRAGFLO allows the user to maintain a specified pressure and/or
saturation at any grid
Table PA8. Boundary Value Conditions for P_{g} and P_{b}
Boundaries below (Row 1, y = 0 m) and above (Row 33, y = 1039 m) system for 0 £ x £ 46630 m (Columns 168) and 5 yr £ t. Below, j refers to the unit normal vector in the positive y direction. 

_{}×j = 0 Pa / m 
No gas flow condition 
_{}×j = 0 Pa / m 
No brine flow condition 
Boundaries at left (Column 1, x = 0 m) and right (Column 68, x = 46630 m) of system for 0 £ y £ 1039 m (Rows 133) and 5 yr £ t. Below, i refers to the unit normal vector in the positive x direction. 

_{} ×i = 0 Pa / m 
No gas flow condition 
_{}×i = 0 Pa / m 
No brine flow condition 
block. This is not a boundary condition and is not required to close the problem. This feature is used to specify Dirichlettype conditions at the surface grid blocks (Columns 168, Row 33, Figure PA15) and at the farfield locations in the Culebra and Magenta (Columns 1 and 68, Row 26, and Columns 1 and 68, Row 28, Figure PA15). These auxiliary conditions are summarized in Table PA9.
A fully implicit finitedifference procedure is used to solve Equation (PA.24) through Equation (PA.30). The associated discretization of the gas mass balance equation is given by
Table PA9. Auxiliary Dirichlet Conditions for S_{g} and P_{b}
Surface Grid Blocks 

_{} = 0.08363 
Columns 1–42, 44–68, Row 33, 5 yr £ t Saturation is not forced at the shaft cell on the surface because its saturation is reset to 1.0 at t = 0 yr. 
_{} = 1.01 ´ 10^{5} Pa 
Columns 1–68, row 33, –5 yr £ t 
Culebra and Magenta Far Field 

_{} = 9.14 ´ 10^{5} Pa 
i = 1 and 68, j = 26, –5 yr £ t (Culebra) 
_{} = 9.47 ´ 10^{5} Pa 
i = 1 and 68, j = 28, –5 yr £ t (Magenta) 
where F represents the phase potentials given by
and
the subscripts are defined by
i = xdirection grid index
j = ydirection grid index
_{} = xdirection grid block interface
_{} = ydirection grid block interface
x_{i} = grid block center in the xcoordinate direction (m)
y_{j} = grid block center in the ycoordinate direction (m)
_{} = grid block length in the xcoordinate direction (m)
_{} = grid block length in the ycoordinate direction (m)
the superscripts are defined by
n = index in the time discretization, known solution time level
n+1 = index in the time discretization, unknown solution time level
and the interblock densities are defined by
The interface values of k_{rg} in Equation (PA.91) are evaluated using upstream weighted values (i.e., the relative permeabilities at each grid block interface are defined to be the relative permeabilities at the center of the adjacent grid block with the highest potential). Further, interface values for ar_{g} k_{x}/ m_{g} and ar_{g} k_{y}/ m_{g} are obtained by harmonic averaging of adjacent grid block values for these expressions. Currently all materials are isotropic, i.e. k_{x} = k_{y} = k_{z}.
The discretization of the brine mass balance equation is obtained by replacing the subscript for gas, g, by the subscript for brine, b. As a reminder, P_{g} and S_{b} are replaced in the numerical implementation with the substitutions indicated by Equation (PA.27) and Equation (PA.26), respectively. Wells are not used in the conceptual model for longterm Salado flow calculations, but they are used for DBR calculations. Thus, for longterm Salado flow calculations, the terms q_{g} and q_{b} are zero. For longterm Salado flow calculations, the wellbore is not treated by a well model, but rather is explicitly modeled within the grid as a distinct material region (i.e., Upper Borehole and Lower Borehole in Figure PA15).
The resultant coupled system of nonlinear brine and gas mass balance equations is integrated in time using the NewtonRaphson method with upstream weighting of the relative permeabilities, as previously indicated. The primary unknowns at each computational cell center are brine pressure and gas saturation.
The Darcy velocity vectors v_{g}( x, y, t) and v_{b}( x, y, t) for gas and brine flow (m^{3}/m^{2}/s = m/s) are defined by the expressions
and
Values for v_{g} and v_{b} are obtained and saved as the numerical solution of Equation (PA.24) through Equation (PA.30) is carried out. Cumulative flows of gas, C_{g}( t, B), and brine, C_{b}( t, B), from time 0 to time t across an arbitrary boundary B in the domain of (Figure PA15) is then given by
for l = g, b, where a(x, y) is the geometry factor defined in Equation (PA.32), n(x, y) is an outwardpointing unit normal vector, and _{} denotes a line integral. As an example, B could correspond to the boundary of the waste disposal regions in Figure PA15. The integrals defining C_{g}( t, B) and C_{b}( t, B) are evaluated using the Darcy velocities defined by Equation (PA.92) and Equation (PA.93). Due to the dependence of gas volume on pressure, C_{g}( t, B) is typically calculated in moles or cubic meters at standard temperature and pressure, which requires an appropriate change of units for v_{g} in Equation (PA.95).
Additional information on BRAGFLO and its use in the CRA2009 PA can be found in the BRAGFLO user’s manual (Nemer 2007c), the BRAGFLO design document (Nemer 2007b) and the analysis package for the Salado flow calculations in the CRA2009 PA (Nemer and Clayton 2008).
The NUTS code is used to model radionuclide transport in the Salado. NUTS models radionuclide transport within all regions for which BRAGFLO computes brine and gas flow, and for each realization uses as input the corresponding BRAGFLO velocity field, pressures, porosities, saturations, and other model parameters, including, for example, the geometrical grid, residual saturation, material map, and compressibility. Of the radionuclides that are transported vertically due to an intrusion or up shaft, it is assumed that the lateral radionuclide transport is in the most transmissive unit, the Culebra. Therefore, the radionuclide transport through the Dewey Lake to the accessible environment (f_{DL} in Equation [PA.33]) and to the land surface due to longterm flow (f_{S} in Equation [PA.33]) are set to zero for consistency.
The PA uses NUTS in two different modes. First, the code is used in a computationally fast screening mode to identify those BRAGFLO realizations for which it is unnecessary to do full transport calculations because the amount of contaminated brine that reaches the Culebra or the LWB within the Salado is insufficient to significantly contribute to the total integrated release of radionuclides from the disposal system. For the remaining realizations, which have the possibility of consequential release, a more computationally intensive calculation of each radionuclide’s full transport is performed (see Section PA6.7.2).
This section describes the model used to compute radionuclide transport in the Salado for E0, E1, and E2 scenarios (defined in Section PA2.3.2). The model for transport in the E1E2 scenario, which is computed using the PANEL code, is described in Section PA4.4.
NUTS models radionuclide transport by advection (see Attachment MASS2009, Section MASS13.5). NUTS disregards sorptive and other retarding effects throughout the entire flow region. Physically, some degree of retardation must occur at locations within the repository and the geologic media; it is therefore conservative to ignore retardation processes. NUTS also ignores reactionrate aspects of dissolution and colloid formation processes, and mobilization is assumed to occur instantaneously. Neither molecular nor mechanical dispersion is modeled in NUTS. These processes are assumed to be insignificant compared to advection, as discussed further in Attachment MASS2009, Section MASS13.5.
Colloidal actinides are subject to retardation by chemical interaction between colloids and solid surfaces and by clogging of small pore throats (i.e., by sieving). There will be some interaction of colloids with solid surfaces in the anhydrite interbeds. Given the low permeability of intact interbeds, it is likely that pore apertures will be small and some sieving will occur. However, colloidal particles, if not retarded, are transported slightly more rapidly than the average velocity of the bulk liquid flow. Because the effects on transport of slightly increased average pore velocity and retarded interactions with solid surfaces and sieving offset one another, the DOE assumes residual effects of these opposing processes will be either small or beneficial, and does not incorporate them when modeling An transport in the Salado interbeds.
If brine in the repository moves into interbeds, it is likely that mineral precipitation reactions will occur. Precipitated minerals may contain actinides as trace constituents. Furthermore, colloidalsized precipitates will behave like mineralfragment colloids, which are destabilized by brines, quickly agglomerating and settling by gravity. The beneficial effects of precipitation and coprecipitation are neglected in PA.
Fractures, channeling, and viscous fingering may also impact transport in Salado interbeds, which contain natural fractures. Because of the low permeability of unfractured anhydrite, most fluid flow in interbeds will occur in fractures. Even though some properties of naturally fractured interbeds are characterized by in situ tests, other uncertainty exists in the characteristics of the fracture network that may be created with high gas pressure in the repository. The PA modeling system accounts for the possible effects on porosity and permeability of fracturing by using a fracturing model (see Section PA4.2.4). The processes and effects associated with fracture dilation or fracture propagation not already captured by the PA fracture model are negligible (see the CCA, Appendix MASS, Section MASS.13.3 and Appendix MASS Attachment 13.2). Of those processes not already incorporated, channeling has the greatest potential effect.
Channeling is the movement of fluid through the largeraperture sections of a fracture network with locally high permeabilities. It could locally enhance An transport. However, it is assumed that the effects of channeled flow in existing or altered fractures will be negligible for the length and time scales associated with the disposal system. The DOE believes this assumption is reasonable because processes are likely to occur that limit the effectiveness of channels or the dispersion of actinides in them. First, if gas is present in the fracture network, it will be present as a nonwetting phase and will occupy the portions of the fracture network with relatively large apertures, where the highest local permeabilities will exist. The presence of gas thus removes the most rapid transport pathways from the contaminated brine and decreases the impact of channeling. Second, brine penetrating the Salado from the repository is likely to be completely miscible with in situ brine. Because of miscibility, diffusion or other local mixing processes will probably broaden fingers (reduce concentration gradients) until the propagating fingers are indistinguishable from the advancing front.
Gas will likely penetrate the liquidsaturated interbeds as a fingered front, rather than a uniform front. Fingers form when there is a difference in viscosity between the invading fluid (gas) and the resident fluid (liquid brine), and because of channeling effects. This process does not affect An transport, however, because actinides of interest are transported only in the liquid phase, which will not displace gas in the relatively highpermeability regions due to capillary effects.
The following system of PDEs is used to model radionuclide transport in the Salado:
for l = 1, 2, …, n_{R}, where
_{} = Darcy velocity vector (m^{3}/m^{2}/s = m/s) for brine (supplied by BRAGFLO from solution of Equation (PA.93))
C_{bl} = concentration (kg/m^{3}) of radionuclide l in brine
C_{sl} = concentration (kg/m^{3}) of radionuclide l in solid phase (i.e., not in brine), with concentration defined with respect to total (i.e., bulk) formation volume (only used in repository; see Figure PA15)
S_{l} = linkage term (kg/m^{3}/s) due to dissolution/precipitation between radionuclide l in brine and in solid phase (see Equation (PA.97))
f = porosity (supplied by BRAGFLO from solution of Equation (PA.24) through Equation (PA.30))
S_{b} = brine saturation (supplied by BRAGFLO from solution of Equation (PA.24) through Equation (PA.30))
l_{l} = decay constant (s^{ }^{1}) for radionuclide l
P(l) = {p: radionuclide p is a parent of radionuclide l}
n_{R} = number of radionuclides,
and a is the dimensiondependent geometry factor in Equation (PA.32). PA uses a twodimensional representation for fluid flow and radionuclide transport in the vicinity of the repository, with a defined by the element depths in Figure PA15. Although omitted for brevity, the terms a, v_{b}, C_{bl}, C_{sl}, S_{l}, S_{b}, and f are functions a(x, y), v_{b}( x, y, t), C_{bl}( x, y, t), C_{sl}( x, y, t), S_{l}( x, y, t), S_{b}( x, y, t), and f(x, y, t) of time t and the spatial variables x and y. Equation (PA.95) and Equation (PA.96) are defined and solved on the same computational grid (Figure PA15) used by BRAGFLO for the solution of Equation (PA.24) through Equation (PA.30).
Radionuclides are assumed to be present in both brine (Equation (PA.95)) and in an immobile solid phase (Equation (PA.96)), although radionuclide transport takes place only by brine flow (Equation (PA.95)). The maximum radionuclide concentration is assumed to equilibrate instantly for each element (Section PA4.3.2). Then each individual radionuclide equilibrates between the brine and solid phases based on the maximum concentration of its associated element and the mole fractions of other isotopes included in the calculation. The linkage between the brine and solid phases in Equation (PA.95) and Equation (PA.96) is accomplished by the term S_{l}, where
where
_{ } = maximum concentration (kg/m^{3}) of element El(l) in oxidation state Ox(l) in brine type Br(t), where El(l) denotes the element of which radionuclide l is an isotope, Ox(l) denotes the oxidation state in which element El(l) is present, and Br(t) denotes the type of brine present in the repository at time t (see Section PA4.3.2 for definition of _{}).
_{ } = concentration (kg/m^{3}) of element El(l) in brine (p = b) or solid (p = s), which is equal to the sum of concentrations of radionuclides that are isotopes of same element as radionuclide l, where k Î El(l) only if k is an isotope of element El(l):
_{ } = difference (kg/m^{3}) between maximum concentration of element El(l) in brine and existing concentration of element El(l) in brine
MF_{pl}
= mole fraction of radionuclide l in phase p, where
p = b (brine) or
p = s (solid)
CM_{l} = conversion factor (mol/kg) from kilograms to moles for radionuclide l
_{ } = Dirac delta function (s^{ }^{1}) (d(t  t) = 0 if t ¹ t and _{})
In the CCA and the CRA2004, the function S_{T} had an additional parameter, Mi, representing the presence or absence of microbial activity. However, during preparations for the CRA2004 PABC, the EPA directed the DOE to change the probability of microbial activity to 1.0 (Cotsworth 2005). The DOE therefore revised the probability distribution so that all vectors would have the possibility for microbial degradation of cellulose, while retaining the percentage of vectors showing microbial degradation of rubbers and plastic at 0.25 (Nemer 2005). This, in effect, sets Mi equal to “Yes” for all vectors. Mi is therefore suppressed as an argument of S_{T} in the remainder of this section.
The terms S_{l}, S_{b}, C_{p,El(l)}, MF_{pl} , and f are functions of time t and the spatial variables x and y, although the dependencies are omitted for brevity. The Dirac delta function, d(t – t), appears in Equation (PA.97) to indicate that the adjustments to concentration are implemented instantaneously within the numerical solution of Equation (PA.95) and Equation (PA.96) whenever a concentration imbalance is observed.
The velocity vectorv_{b} in Equation (PA.95) and Equation (PA.96) is defined in Equation (PA.93) and is obtained from the numerical solution of Equation (PA.24) through Equation (PA.30). If B denotes an arbitrary boundary (e.g., the LWB) in the domain of Equation (PA.95) and Equation (PA.96) (as shown in Figure PA15), the cumulative transport of C_{l}( t, B) of radionuclide l from time 0 to time t across B is given by
where n(x, y) is an outwardpointing unit normal vector and _{} denotes a line integral over B.
Equation (PA.95) and Equation (PA.96) models advective radionuclide transport due to the velocity vector v_{b}. Although the effects of solubility limits are considered, no chemical or physical retardation is included in the model. Molecular diffusion is not included in the model, because the radionuclides under consideration have molecular diffusion coefficients on the order of 10^{ }^{10} m^{2}/s, and thus can be expected to move only approximately 10 m over 10,000 years due to molecular diffusion. Mechanical dispersion is also omitted in NUTS, as the uniform initial radionuclide concentrations assumed within the repository and the use of timeintegrated releases in assessing compliance with section 191.13 mean that mechanical dispersion will have negligible impact on overall performance.
A maximum concentration S_{T}( Br, Ox, El) (mol/liter [L]) is calculated for each brine type (Br Î {Salado, Castile}), oxidation state (Ox Î {III, IV, V, VI}), and element (El Î {Am, Pu, U, Th}). The maximum concentration is given by
where S_{D}( Br, Ox) is the dissolved solubility (mol/L) and S_{C}( Br, Ox, El) is the concentration (mol/L) of the element sorbed to colloids.
The dissolved solubility S_{D}( Br, Ox) is given by
where
S_{FMT}( Br, Ox) = dissolved solubility (mol/L) calculated by FractureMatrix Transport (FMT) model (WIPP Performance Assessment 1998a) for brine type Br and oxidation state Ox
_{ } = logarithm (base 10) of uncertainty factor for solubilities calculated by FMT expressed as a function of oxidation state Ox
Table PA10 lists the calculated values of S_{FMT}( Br, Ox); details of the calculation are provided in Appendix SOTERM2009 and Brush (2005). The uncertainty factors UF(Ox) are determined by the uncertain parameters WSOLVAR3 for the III oxidation state and WSOLVAR4 for the IV oxidation state; definitions of these parameters are provided in Table PA19, and further details regarding the calculation of the solubilities is given in Appendix SOTERM2009, Section SOTERM5.1.3.
Table PA10. Calculated Values for Dissolved Solubility (see Appendix SOTERM2009, Table SOTERM16)
Brine 
Oxidation State 

III 
IV 
V 
VI^{a} 

Salado 
3.87 ´ 10^{7} 
5.64 ´ 10^{8} 
3.55 ´ 10^{6} 
1.0 ´ 10^{3} 
Castile 
2.88 ´ 10^{7} 
6.79 ´ 10^{8} 
8.24 ´ 10^{7} 
1.0 ´ 10^{3} 
^{a} Values for the VI solubilities were mandated by the EPA (U.S. Environmental Protection Agency 2005). 
The concentration (mol/L) of the element sorbed to colloids S_{C} (Br, Ox, El) is given by
where
_{ } = solubility (concentration expressed in mol/L) in brine type Br of element El in oxidation state Ox resulting from humic colloid formation
= _{}
_{ } = scale factor used as a multiplier on S_{D}( Br, Ox) in definition of S_{Hum}( Br, Ox, El) (see Table PA11)
UB_{Hum} = upper bound on solubility (concentration expressed in mol/L) of individual An elements resulting from humic colloid formation
= 1.1 ´ 10^{}^{5} mol/L
_{ } = solubility (concentration expressed in mol/L) in brine type Br of element El in oxidation state Ox resulting from microbial colloid formation
=_{}
_{ } = scale factor used as multiplier on S_{D}( Br, Ox) in definition of S_{Mic}( Br, Ox, El) (see Table PA12)
_{ } = upper bound on solubility (concentration expressed in mol/L) of element El in oxidation state Ox resulting from microbial colloid formation (see Table PA12)
Table PA11. Scale Factor SF_{Hum}( Br, Ox, El) Used in Definition of S_{Hum}( Br, Ox, El) (see Appendix SOTERM2009, Table SOTERM21)
Brine 
Element (Oxidation State) 

Am(III) 
Pu(III) 
Pu(IV) 
U(IV) 
Th(IV) 
Np(IV) 
Np(V) 
U(VI) 

Salado 
0.19 
0.19 
6.3 
6.3 
6.3 
6.3 
9.1 ´ 10^{4} 
0.12 
Castile 
WPHUMOX3^{a} 
WPHUMOX3^{a} 
6.3 
6.3 
6.3 
6.3 
7.4 ´ 10^{3} 
0.51 
^{a} See Table PA19. 
Table PA12. Scale Factor SF_{Mic}( Ox, El) and Upper Bound UB_{Mic}( Ox, El) (mol/L) Used in Definition of S_{Mic}( Br, Ox, El) (see Appendix SOTERM2009, Table SOTERM21)
Parameter 
Element (Oxidation State) 

Am(III) 
Pu(III) 
Pu(IV) 
U(IV) 
Th(IV) 
Np(IV) 
Np(V) 
U(VI) 

SF_{Mic}(Ox, El) 
3.6 
0.3 
0.3 
0.0021 
3.1 
12.0 
12.0 
0.0021 
UB_{Mic}(Ox, El) 
1 
6.8 ´ 10^{}^{5} 
6.8 ´ 10^{}^{5} 
0.0021 
0.0019 
0.0027 
0.0027 
0.0021 
_{ } = solubility (concentration in mol/L) of element El resulting from An intrinsic colloid formation
= _{}
S_{Mn} = solubility (concentration in mol/L) of individual An element resulting from mineral fragment colloid formation
= 2.6 ´ 10^{ }^{8} mol/L
Since the solution of Equation (PA.95) and Equation (PA.96) for this many radionuclides and decay chains would be very timeconsuming, the number of radionuclides for direct inclusion in the analysis was reduced using the algorithm shown in Figure PA21. The CRA2009 PA uses the same reduction algorithm as the CCA PA (see the CCA, Appendix WCA); the algorithm was found to be acceptable in the CCA review (U.S. Environmental Protection Agency 1998a, Section 4.6.1.1).
Figure PA21. Selecting Radionuclides for the Release Pathways Conceptualized by PA
Using Figure PA21, the number of radionuclides considered was reduced from 135 to a total of 29 included in the decay calculations carried out by the PANEL code (Garner and Leigh 2005). These radionuclides belong to the following decay chains:
Radionuclides considered in the decay calculations that do not belong to one of the decay chains listed above are ^{147}Pm, ^{137}Cs, and ^{90}Sr. In addition, some intermediates with extremely short halflives, such as ^{240}U, were omitted from the decay chains.
Further simplification of the decay chains is possible based on the total inventories. Releases of radionuclides whose inventories total less than one EPA unit are essentially insignificant, as any release that transports essentially all of a given species outside the LWB will be dominated by the releases of other species with much larger inventories. In addition, ^{137}Cs and ^{90}Sr can be omitted because their concentrations drop to below 1 EPA unit within 150 years, which makes it improbable that a significant release of these radionuclides will occur. Isotopes such as ^{241}Pu, whose decay could affect the inventory of measurable isotopes, were reinstated.
After the reduction of radionuclides outlined inFigure PA21 and the above paragraph, the following 10 radionuclides remained from the decay chains shown above:
^{238}Pu does not significantly affect transport calculations because of its short halflife (87.8 years). The remaining nine radionuclides were then further reduced by combining those with similar decay and transport properties. In particular, ^{234}U, ^{230}Th, and ^{239}Pu were used as surrogates for the groups {^{234}U, ^{233}U}, {^{230}Th, ^{229}Th}, and {^{242}Pu, ^{240}Pu, ^{239}Pu}, with the initial inventories of ^{234}U, ^{230}Th, and ^{239}Pu being increased to account for the additional radionuclide(s) in each group.
In increasing the initial inventories, the individual radionuclides were combined on either a mole or curie basis (i.e., moles added and then converted back to curies, or curies added directly). In each case, the method that maximized the combined inventory was used; thus, ^{233}U was added to ^{234}U, ^{240}Pu to ^{239}Pu, and ^{229}Th to ^{230}Th by curies, while ^{242}Pu was added to ^{239}Pu by moles. In addition, ^{241}Pu was added to ^{241}Am by moles because ^{241}Pu has a halflife of 14 years and will quickly decay to ^{241}Am, and neglect of this ingrowth would underestimate the ^{241}Am inventory by about 4% (Table PA13). The outcome of this process was the following set of five radionuclides in three simplified decay chains:
which were then used with Equation (PA.95) and Equation (PA.96) for transport in the vicinity of the repository. As ^{238}Pu does not significantly affect transport calculations, only the remaining four radionuclides are used in the Culebra transport calculations (Section PA4.9). These radionuclides account for 99% of the EPA units in the waste after 2,000 years (Garner and Leigh 2005), and hence will dominate any releases by transport.
Table PA13. Combination of Radionuclides for Transport
Combination 
Isotope Initial Values 
Combination Procedure 
Combined Inventory 
^{233}U ® ^{234}U 
1.23 ´ 10^{3} Ci
^{233}U 
1.23 ´ 10^{3} Ci
^{233}U 
1.57 ´ 10^{3} Ci ^{234}U 
^{242}Pu ® ^{239}Pu
^{240}Pu ® ^{239}Pu 
1.27 ´ 10^{1} Ci
^{242}Pu 
1.27 ´ 10^{1} Ci
^{242}Pu 
6.78 ´ 10^{5} Ci ^{239}Pu 
^{229}Th ® ^{230}Th 
5.21 ´ 10^{0} Ci
^{229Th} 
5.21 ´ 10^{0} Ci
^{229Th} 
5.39 ´ 10^{0} Ci ^{230}Th 
^{241}Pu ® ^{241}Am 
4.48 ´ 10^{5} Ci
^{241}Pu 
5.38 ´ 10^{5} Ci
^{241}Pu 
5.33 ´ 10^{5} Ci ^{241}Am 
All BRAGFLO realizations are first evaluated using NUTS in a screening mode to identify those realizations for which a significant release of radionuclides to the LWB cannot occur. The screening simulations consider an infinitely soluble, nondecaying, nondispersive, and nonsorbing species as a tracer element. The tracer is given a unit concentration in all waste disposal areas of 1 kg/m^{3}. If the amount of tracer that reaches the selected boundaries (the top of the Salado and the LWB within the Salado) does not exceed a cumulative mass of 10^{–7} kg within 10,000 years, it is assumed there is no consequential release to these boundaries. If the cumulative mass outside the boundaries within 10,000 years exceeds 10^{–7} kg, a complete transport analysis is conducted. The value of 10^{ }^{7} kg is selected because, regardless of the isotopic composition of the release, it corresponds to a normalized release less than 10^{–6} EPA units, the smallest release displayed in CCDF construction (Stockman 1996). The largest normalized release would be 9.98 × 10^{–7} EPA units, corresponding to 10^{ }^{7} kg of ^{241}Am if the release was entirely ^{241}Am.
For BRAGFLO realizations with greater than 10^{ }^{7} kg reaching the boundaries in the tracer calculations, NUTS models the transport of five different radionuclide species (^{241}Am, ^{239}Pu, ^{238}Pu, ^{234}U, and ^{230}Th). These radionuclides represent a larger number of radionuclides: as discussed in Section PA4.3.3, radionuclides were grouped together based on similarities, such as isotopes of the same element and those with similar halflives, to simplify the calculations. For transport purposes, solubilities are lumped to represent both dissolved and colloidal forms. These groupings simplify and expedite calculations.
Equation (PA.95) and Equation (PA.96) are numerically solved by the NUTS program (WIPP Performance Assessment 1997a) on the same computational grid (Figure PA15) used by BRAGFLO for the solution of Equation (PA.24), Equation (PA.25), Equation (PA.26), Equation (PA.27), Equation (PA.28), Equation (PA.29), and Equation (PA.30). In the solution procedure, Equation (PA.95) and Equation (PA.96) are numerically solved with S_{l} = 0 for each time step, with the instantaneous updating of concentrations indicated in Equation (PA.97) and the appropriate modification to C_{sl} in Equation (PA.96) taking place after the time step. The solution is carried out for the five radionuclides indicated in Equation (PA.113).
The initial value and boundary value conditions used with Equation (PA.95) and Equation (PA.96) are given in Table PA14. At time t = 0 (corresponding to the year 2033), the total inventory of each radionuclide is assumed to be in brine; the solubility constraints associated with Equation (PA.97) then immediately adjust the values for C_{bl}( x, y, t) and C_{sl}( x, y, t) for consistency with the constraints imposed by S_{T}( Br, Ox, El) and available radionuclide inventory.
The n_{R} PDEs in Equation (PA.95) and Equation (PA.96) are discretized in two dimensions and then developed into a linear system of algebraic equations for numerical implementation. The following conventions are used in the representation of each discretized equation:
· The subscript b is dropped from C_{bl}, so that the unknown function is represented by C_{l}.
· A superscript n denotes time t_{n}, with the assumption that the solution C_{l} is known at time t_{n} and is to be propagated to time t_{n}_{ +1}.
· The grid indices are i in the xdirection and j in the ydirection, and are the same as the BRAGFLO grid indices.
Table PA14. Initial and Boundary Conditions for C_{bl}( x, y, t) and C_{sl}( x, y, t)
Initial Conditions for C_{bl}(x, y, t) and C_{sl}(x, y, t) 
_{} = _{} if (x, y) is a point in the repository (i.e., areas Waste Panel, South RoR and North RoR, in Figure PA15), where A_{l}(0) is the amount (kg) of radionuclide l present at time t = 0 (Table PA13) and V_{b}(0) is the amount (m^{3}) of brine in repository at time t = 0 (from solution of Equation (PA.24) through Equation (PA.30) with BRAGFLO) for all (x, y). = 0 otherwise. _{} = 0 if (x, y) is a point in the repository. 
Boundary Conditions for C_{bl}(x, y, t) 
_{ } = _{}, where B is any subset of the outer boundary of the computational grid in Figure PA15, _{} is the flux (kg/s) at time t of radionuclide l across B, v_{b}(x, y, t) is the Darcy velocity (m^{3}/m^{2}/s) of brine at (x, y) on B and is obtained from the solution of Equation (PA.24) through Equation (PA.30) by BRAGFLO, n(x, y) denotes an outwardpointing unit normal vector, and _{} denotes a line integral along B. 
· Fractional indices refer to quantities evaluated at grid block interfaces.
· Each time step by NUTS is equal to 20 BRAGFLO time steps because BRAGFLO stores results (here, v_{b}, f, and S_{b}) every 20 time steps.
The following finitedifference discretization is used for the l^{th} equation in each grid block (i, j):
where q_{b} is the grid block interfacial brine flow rate (m^{3}/s) and V_{R} is the grid block volume (m^{3}). The quantity q_{b} is based on _{} and a in Equation (PA.95) and Equation (PA.96), and the quantity V_{R} is based on grid block dimensions (Figure PA15) and a.
The interfacial values of concentration in Equation (PA.114) are discretized using the onepoint upstream weighting method (Aziz and Settari 1979), which results in
where w derives from the upstream weighting for flow between adjacent grid blocks and is defined by
By collecting similar terms, Equation (PA.115) can be represented by the linear equation
where
Given the form of Equation (PA.116), the solution of Equation (PA.95) and Equation (PA.96) has now been reduced to the solution of n_{R} ´ n_{G} linear algebraic equations in n_{R} ´ n_{G} unknowns, where n_{R} is the number of equations for each grid block (i.e., the number of radionuclides) and n_{G} is the number of grid blocks into which the spatial domain is discretized (Figure PA15).
The system of PDEs in Equation (PA.95) and Equation (PA.96) is strongly coupled because of the contribution from parental decay to the equation governing the immediate daughter. Consequently, a sequential method is used to solve for the radionuclide concentrations by starting at the top of a decay chain and working down from parent to daughter. This implies that when solving Equation (PA.116) for the l^{th} isotope concentration, all parent concentrations occurring in the righthandside term R are known. The system of equations is then linear in the concentrations of the l^{th} isotope. As a result, solving Equation (PA.95) and Equation (PA.96) is reduced from the solution of one algebraic equation at each time step with n_{R} ´ n_{G} unknowns to the solution of n_{R} algebraic equations each with n_{G} unknowns at each time step, which can result in a significant computational savings.
The matrix resulting from onepoint upstream weighting has the following structural form for a 3 ´ 3 system of grid blocks, and a similar structure for a larger number of grid blocks:

1 
2 
3 
4 
5 
6 
7 
8 
9 
1 
X 
X 
0 
X 





2 
X 
X 
X 
0 
X 




3 
0 
X 
X 
0 
0 
X 



4 
X 
0 
0 
X 
X 
0 
X 


5 

X 
0 
X 
X 
X 
0 
X 

6 


X 
0 
X 
X 
0 
0 
X 
7 



X 
0 
0 
X 
X 
0 
8 




X 
0 
X 
X 
X 
9 





X 
0 
X 
X 
where X designates possible nonzero matrix entries, and 0 designates zero entries within the banded structure. All entries outside of the banded structure are zero. Because of this structure, a banded direct elimination solver (Aziz and Settari 1979, Section 8.2.1) is used to solve the linear system for each radionuclide. The bandwidth is minimized by first indexing equations in the coordinate direction with the minimum number of grid blocks. The coefficient matrix is stored in this banded structure, and all infill coefficients calculated during the elimination procedure are contained within the band structure. Therefore, for the matrix system in two dimensions, a pentadiagonal matrix of dimension I_{BW} ´ n_{G} is inverted instead of a full n_{G} ´ n_{G} matrix, where I_{BW} is the bandwidth.
The numerical implementation of Equation (PA.96) enters the solution process through updates to the radionuclide concentrations in Equation (PA.115) between each time step, as indicated in Equation (PA.97). The numerical solution of Equation (PA.95) and Equation (PA.96) also generates the concentrations required to integrate numerically evaluating the integral that defines C_{l}( t, B) in Equation (PA.101).
Additional information on NUTS and its use in WIPP PA can be found in the NUTS users manual (WIPP Performance Assessment 1997a) and in the analysis package of Salado transport calculations for the CRA2009 PA (Ismail and Garner 2008). Furthermore, additional information on dissolved and colloidal actinides is given in Appendix SOTERM2009.
This section describes the model used to compute radionuclide transport in the Salado for the E1E2 scenario. The model for transport in E0, E1, and E2 scenarios is described in Section PA4.3.
A relatively simple mixedcell model is used for radionuclide transport in the vicinity of the repository after an E1E2 intrusion, when connecting flow between two drilling intrusions into the same waste panel is assumed to take place. With this model, the amount of radionuclide l contained in a waste panel is represented by
where
_{} = amount (mol) of radionuclide l in waste panel at time t
_{} = concentration (mol/m^{3}) of radionuclide l in brine in waste panel at time t (Equation (PA.118) and Equation (PA.119))
_{} = rate (m^{3}/s) at which brine flows out of the repository at time t (supplied by BRAGFLO from solution of Equation (PA.93))
and l_{l} and P(l) are defined in conjunction with Equation (PA.95) and Equation (PA.96).
The brine concentration C_{bl} in Equation (PA.117) is defined by
where
_{ } = mole fraction of radionuclide l in waste panel at time t
_{ } = volume (m^{3}) of brine in waste panel at time t (supplied by BRAGFLO from solution of Equation (PA.24), Equation (PA.25), Equation (PA.26), Equation (PA.27), Equation (PA.28), Equation (PA.29), and Equation (PA.30))
and S_{T}[ Br, Ox, El] is supplied by Equation (PA.102).
For Equation (PA.118) and Equation (PA.119), S_{T}[ Br, Ox, El] must be expressed in units of mol/L. In other words, C_{bl}( t) is defined to be the maximum concentration (S_{T} in Equation (PA.102)) if there is sufficient radionuclide inventory in the waste panel to generate this concentration (Equation (PA.118)); otherwise, C_{bl}( t) is defined by the concentration that results when all the relevant element in the waste panel is placed in solution (Equation (PA.119)). The dissolved and colloidal An equilibrate instantly for each element.
Given r_{b} and C_{bl}, evaluation of the integral
provides the cumulative release R_{l}( t) of radionuclide l from the waste panel through time t.
Equation (PA.117) is numerically evaluated by the PANEL model (WIPP Performance Assessment 1998b) using a discretization based on time steps of 50 years or less. Specifically, Equation (PA.117) is evaluated with the approximation
where
_{ } = gain in radionuclide l due to the decay of precursor radionuclides between t_{n}
and t_{n}_{+1} (see Equation (PA.123)), _{} = _{}.
As the solution progresses, values for C_{bl}( t_{n}) are updated in consistency with Equation (PA.118) and Equation (PA.119), and the products r_{b}( t_{n}) C_{bl}( t_{n}) are accumulated to provide an approximation to R_{l} in Equation (PA.121).
The term G_{l}( t_{n}, t_{n}_{+1}) in Equation (PA.122) is evaluated with the Bateman equations (Bateman 1910), with PANEL programmed to handle decay chains of up to five (four decay daughters for a given radionuclide). As a single example, if radionuclide l is the third radionuclide in a decay chain (i.e., l = 3) and the two preceding radionuclides in the decay chain are designated by l = 1 and l = 2, then
in Equation (PA.122).
The preceding model is used in two ways in PA. First, Equation (PA.121) estimates releases to the Culebra associated with E1E2 intrusion scenarios (see Section PA6.7.3). Second, the radionuclide concentrations are computed using the minimum brine volume for a significant release to estimate DBRs (see Section PA6.8.2.3). The calculation of the minimum brine volume used in the CRA2009 PA is found in Stein (2005). The calculated concentrations are the S_{l} term indicated in Equation (PA.97) which are used in the NUTS calculations discussed in Section PA4.3.
For E1E2 intrusions, the initial amount A_{l} of radionuclide l is the inventory of the decayed isotope at the time of the E1 intrusion. PANEL calculates the inventory of each of the 29 radioisotopes throughout the regulatory period. The initial concentration C_{bl} of radionuclide l is computed by Equation (PA.117), Equation (PA.118), and Equation (PA.119). For the DBR calculations, the initial amount A_{l} of radionuclide l is the inventory of the isotope at the time of repository closure.
Additional information on PANEL and its use in the CRA2009 PA calculations can be found in the PANEL user’s manual (WIPP Performance Assessment 2003b), the analysis package for PANEL calculations (Garner and Leigh 2005), and the analysis package for Salado transport calculations in the CRA2009 PA (Ismail and Garner 2008).
Cuttings are waste solids contained in the cylindrical volume created by the cutting action of the drill bit passing through the waste, while cavings are additional waste solids eroded from the borehole by the upwardflowing drilling fluid within the borehole. The releases associated with these processes are computed within the CUTTINGS_S code (WIPP Performance Assessment 2003c). The mathematical representations used for cuttings and cavings are described in this section.
The uncompacted volume of cuttings removed and transported to the surface in the drilling fluid, V_{cut}, is given by
where A is the drill bit area (m^{2}), H_{i} is the initial (or uncompacted) repository height (3.96 m), and D is the drillbit diameter (0.31115 m) (Fox 2008, Table 13). For drilling intrusions through RHTRU waste, H_{i} = 0.509 m is used (Fox 2008, Table 45).
The cavings component of the direct surface release is caused by the shearing action of the drilling fluid on the waste as it flows up the borehole annulus. Like the cuttings release, the cavings release is assumed to be independent of the conditions that exist in the repository during a drilling intrusion.
The final diameter of the borehole depends on the diameter of the drillbit and on the extent to which the actual borehole diameter exceeds the drillbit diameter. Although a number of factors affect erosion within a borehole (Chambre Syndicale de la Recherche et de la Production du Petrole et du Gaz Naturel 1982), the most important is the fluid shear stress on the borehole wall (i.e., the shearing force per unit area, N/m^{2}) resulting from circulating drilling fluids (Darley 1969, Walker and Holman 1971). As a result, PA estimates cavings removal with a model based on the effect of shear stress on the borehole diameter. In particular, the borehole diameter is assumed to grow until the shear stress on the borehole wall is equal to the shear strength of the waste, which is the limit below which waste erosion ceases.
The final eroded diameter D_{f} (m) of the borehole through the waste determines the total volume V (m^{3}) of uncompacted waste removed to the surface by circulating drilling fluid. Specifically,
where V_{cav} is the volume (m^{3}) of waste removed as cavings.
Most borehole erosion is believed to occur in the vicinity of the drill collar (Figure PA22) because of decreased flow area and consequent increased mud velocity (Rechard, Iuzzolino, and Sandha 1990, Letters 1a and 1b, App. A). An important determinant of the extent of this erosion is whether the flow of the drilling fluid in the vicinity of the collar is laminar or turbulent. PA uses Reynolds numbers to distinguish between the occurrence of laminar flow and turbulent flow. The Reynolds number is the ratio between inertial and viscous (or shear) forces in a fluid, and can be expressed as (Fox and McDonald 1985)
where Re is the Reynolds number (dimensionless), r_{f} is the fluid density (kg/m^{3}), D_{e} is the equivalent diameter (m), _{} is the fluid speed (m s^{ }^{1}), and h is the fluid viscosity (kg m^{ }^{1} s^{ }^{1}).
Typically, r_{f}, v, and h are averages over a control volume with an equivalent diameter of D_{e}, where r_{f} = 1.21 ´ 10^{3} kg/m^{3} (Fox 2008, Table 13), v = 0.7089 m s^{ }^{1} (based on 40 gal/min/in of drill diameter) (Berglund 1992), and D_{e} = 2 (R  R_{i}), as shown in Figure PA22. The diameter of the drill collar (i.e., 2R_{i} in Figure PA22) is 8.0 in = 0.2032 m (Ismail 2008). The determination of h is discussed below. PA assumes that Reynolds numbers less than 2100 are associated with laminar flow, while Reynolds numbers greater than 2100 are associated with turbulent flow (Walker 1976).
Drilling fluids are modeled as nonNewtonian, which means that the viscosity h is a function of the shear rate within the fluid (i.e., the rate at which the fluid velocity changes normal to the flow direction, m/s/m). PA uses a model proposed by Oldroyd (1958) to estimate the viscosity of drilling fluids. As discussed in the Drilling Mud and Cement Slurry Rheology Manual (Chambre Syndicale de la Recherche et de la Production du Petrole et du Gaz Naturel 1982), the Oldroyd model leads to the following expression for the Reynolds number associated with the helical flow of a drilling fluid within an annulus:
where r_{f}, D_{e}, and v are defined as in Equation (PA.126), and h_{ ¥} is the asymptotic value for the derivative of the shear stress (t , kg m^{ }^{1} s^{ }^{2}) with respect to the shear rate (G, s^{ }^{1}) obtained as the shear rate increases (i.e., _{} as _{}). PA uses Equation (PA.127) to determine whether drilling fluids in the area of the drill collar are undergoing laminar or turbulent flow.
The Oldroyd model assumes that the shear stress t is related to the shear rate G through the relationship
where h_{0} is the asymptotic value of the viscosity (kg m^{ }^{1} s^{ }^{1}) that results as the shear rate G approaches zero, and s_{1} and s_{2} are constants (s^{2}). The expression leads to
PA uses values of h_{0} = 1.834 ´ 10^{ }^{2} kg m^{ }^{1} s^{ }^{1}, s_{1} = 1.082 ´ 10^{ }^{6} s^{2}, and s_{2} = 5.410 ´ 10^{ }^{7} s^{2} (Berglund 1996), from which viscosity in the limit of infinite shear rate is found to be h_{ ¥} =
Figure PA22. Detail of Rotary Drill String Adjacent to Drill Bit
9.17 ´ 10^{ }^{3} kg m^{ }^{1} s^{ }^{1}. The quantity h_{ ¥} is comparable to the plastic viscosity of the fluid (Chambre Syndicale de la Recherche et de la Production du Petrole et du Gaz Naturel 1982).
As previously indicated, different models are used to determine the eroded diameter D_{f} of a borehole depending on whether flow in the vicinity of the drill collar is laminar or turbulent. The model for borehole erosion in the presence of laminar flow is described next, and then the model for borehole erosion in the presence of turbulent flow is described.
As shown by Savins and Wallick (1966), the shear stresses associated with the laminar helical flow of a nonNewtonian fluid, as a function of the normalized radius, r, can be expressed as
for R_{i}/ R £ r £ 1, where R_{i} and R are the inner and outer radii within which the flow occurs, as indicated in Figure PA22; t(R,r) is the shear stress (kg m^{ }^{1} s^{ }^{2}) at a radial distance DR beyond the inner boundary (i.e., at r = (R_{i} + DR)/R); and the variables C,J, and l depend on R and satisfy conditions Equation (PA.132) through Equation (PA.134) indicated below. The shear stress at the outer radius R is given by
As previously indicated, the borehole radius R is assumed to increase as a result of erosional processes until a value of R is reached at which t(R, 1) is equal to the shear strength of the waste. In PA, the shear strength of the waste is treated as an uncertain parameter (see WTAUFAIL in Table PA19). Computationally, determining the eroded borehole diameter R associated with a particular value of the waste shear strength requires repeated evaluation of t (R, 1), as indicated in Equation (PA.131), until a value of R is determined for which t(R, 1) equals the shear strength.
The quantities C, J, and l must satisfy the following three conditions (Savins and Wallick 1966) for Equation (PA.131) to be valid:
where h, the drilling fluid viscosity (kg m^{ }^{1} s^{ }^{1}), is a function of R and r; DW is the drill string angular velocity (rad s^{ }^{1}); and Q is the drilling fluid flow rate (m^{3} s^{ }^{1}).
The viscosity h in Equation (PA.132) through Equation (PA.134) is introduced into the analysis by assuming that the drilling fluid follows the Oldroyd model for shear stress in Equation (PA.128). By definition of the viscosity h,
and from Equation (PA.128)
thus the expression in Equation (PA.130) can be reformulated as
As discussed by Savins and Wallick (1966) and Berglund (1992), the expressions in Equation (PA.132) through Equation (PA.134) and Equation (PA.136) can be numerically evaluated to obtain C, J, and l for use in Equation (PA.130) and Equation (PA.131). In PA, the drill string angular velocity DW is treated as an uncertain parameter (see DOMEGA in Table PA19), and
where v = 0.7089 m s^{ }^{1} as used in Equation (PA.126), and h_{0}, s_{ 1}, and s_{ 2} are defined as in Equation (PA.128) and Equation (PA.129).
The model for borehole erosion in the presence of turbulent flow is now described. Unlike the theoretically derived relationship for erosion in the presence of laminar flow, the model for borehole erosion in the presence of turbulent flow is empirical. In particular, pressure loss for axial flow in an annulus under turbulent flow conditions can be approximated by (Chambre Syndicale de la Recherche et de la Production du Petrole et du Gaz Naturel 1982)
where DP is the pressure change (Pa), f is the Fanning friction factor (dimensionless), L is the distance (m) over which pressure change DP occurs, and r_{f} , v, and D_{e} are defined in Equation (PA.126).
For turbulent pipe flow, f is empirically related to the Reynolds number Re defined in Equation (PA.126) by (Whittaker 1985)
where D is the inside diameter (m) of the pipe and e is a “roughness term” equal to the average depth (m) of pipe wall irregularities. In the absence of a similar equation for flow in an annulus, Equation (PA.140) is used in PA to define f for use in Equation (PA.139), with D replaced by the effective diameter D_{e} = 2(R – R_{i}) and e equal to the average depth of irregularities in the wasteborehole interface. In the present analysis, e = 0.025 m (Fox 2008, Table 34), which exceeds the value often selected in calculations involving very rough concrete or riveted steel piping (Streeter 1958).
The pressure change DP in Equation (PA.139) and the corresponding shear stress t at the walls of the annulus are approximately related by
where _{} is the crosssectional area of the annulus (see Figure PA22) and 2pL(R + R_{i}) is the total surface area of the annulus. Rearranging Equation (PA.139) and using the relationship in Equation (PA.135) yields
which was used in the 1991 and 1992 WIPP PAs to define the shear stress at the surface of a borehole of radius R. The radius R enters into Equation (PA.132) through Equation (PA.134) through the use of D = 2(R – R_{i}) in the definition of f in Equation (PA.140). As with laminar flow, the borehole radius R is assumed to increase until a value of t(R) is reached that equals the sample value for the shear strength of the waste (i.e., the uncertain parameter WTAUFAIL in Table PA19). Computationally, the eroded borehole diameter is determined by solving Equation (PA.142) for R under the assumption that t(R) equals the assumed shear strength of the waste.
For the CRA2004 PA, a slight modification to the definition of t in Equation (PA.142) was made to account for drill string rotation when fluid flow in the vicinity of the drill collars is turbulent (Abdul Khader and Rao 1974; Bilgen, Boulos, and Akgungor 1973). Specifically, an axial flow velocity correction factor (i.e., a rotation factor), F_{r}, was introduced into the definition of t. The correction factor F_{r} is defined by
where v_{2100} is the norm of the flow velocity required for the eroded diameters to be the same for turbulent and laminar flow at a Reynolds number of Re = 2100, and is obtained by solving
for v_{2100} with D in the definition of f in Equation (PA.140) assigned the final diameter value that results for laminar flow at a Reynolds number of Re = 2100 (that is, the D in D_{e} = 2(R – R_{i}) = D – 2R_{i} obtained from Equation (PA.127) with Re = 2100). The modified definition of t is
and results in turbulent and laminar flow with the same eroded diameter at a Reynolds number of 2100, where PA assumes that the transition between turbulent and laminar flow takes place.
The following algorithm was used to determine the final eroded radius R_{f} of a borehole and incorporates a possible transition from turbulent to laminar fluid flow within a borehole:
Step 1. Use Equation (PA.127) to determine an initial Reynolds number Re, with R initially set to the drillbit radius, R_{0} = 0.31115 m (Fox 2008, Table 13).
Step 2. If Re < 2100, the flow is laminar and the procedure in Section PA4.5.2.1 is used to determine R_{f}. Because any increase in the borehole diameter will cause the Reynolds number to decrease, the flow will remain laminar and there is no need to consider the possibility of turbulent flow as the borehole diameter increases, with the result that R_{f} determined in this step is the final eroded radius of the borehole.
Step 3. If Re ³ 2100, then the flow is turbulent, and the procedure discussed in Section PA4.5.2.2 is used to determine R_{f} . Once R_{f} is determined, the associated Reynolds number Re is recalculated using Equation (PA.127) and R = R_{f} . If the recalculated Re > 2100, a transition from turbulent to laminar flow cannot take place, and the final eroded radius is R_{f} determined in this step. If not, go to Step 4.
Step 4. If the Reynolds number Re with the new R_{f} in Step 3 satisfies the inequality Re £ 2100, a transition from turbulent to laminar flow is assumed to have taken place. In this case, R_{f} is recalculated assuming laminar flow, with the outer borehole radius R initially defined to be the radius associated with Re = 2100. In particular, the initial value for R is given by the radius at which the transition from laminar to turbulent flow takes place:
which is obtained from Equation (PA.127) by solving for R with Re = 2100. A new value for R_{f} is then calculated with the procedure discussed in Section PA4.5.2.1 for laminar flow, with this value of R_{f} replacing the value from Step 3 as the final eroded diameter of the borehole.
Step 5. Once R_{f} is known, the amount of waste removed to the surface is determined using Equation (PA.125) with D_{f} = 2R_{f} .
Additional information on CUTTINGS_S and its use in the CRA2004 PA to determine cuttings and cavings releases can be found in the CUTTINGS_S user’s manual (WIPP Performance Assessment 2003c) and in the analysis package for cuttings and cavings releases (Ismail 2008).
Spallings are waste solids introduced into a borehole by the movement of wastegenerated gas towards the lowerpressure borehole. In engineering literature, the term “spalling” describes the dynamic fracture of a solid material, such as rock or metal (Antoun et al. 2003). In WIPP PA, the spallings model describes a series of processes, including tensile failure of solid waste, fluidization of failed material, entrainment into the wellbore flow, and transport up the wellbore to the land surface. Spallings releases could occur when pressure differences between the repository and the wellbore cause solid stresses in the waste exceeding the waste material strength and gas velocities sufficient to mobilize failed waste material.
The spallings model is described in the following sections. Presented first are the primary modeling assumptions used to build the conceptual model. Next, the mathematical model and its numerical implementation in the computer code DRSPALL are described. Finally, implementation of the spallings model in WIPP PA by means of the code CUTTINGS_S is discussed.
Assumptions underlying the spallings model include the future state of the waste, specifications of drilling equipment, and the driller’s actions at the time of intrusion. Consistent with the other PA models, the spallings model assumes massive degradation of the emplaced waste through mechanical compaction, corrosion, and biodegradation. Waste is modeled as a homogeneous, isotropic, weakly consolidated material with uniform particle size and shape. The rationale for selecting the spallings model material properties is addressed in detail by Hansen et al. (1997) and Hansen, Pfeifle, and Lord (2003).
Drilling equipment specifications, such as bit diameter and drilling mud density, are based on surveys of drillers in the Delaware Basin (Hansen, Pfeifle, and Lord 2003). Assumptions about the driller’s actions during the intrusion are conservative. Typically, the drilling mud density is controlled to maintain a slightly “overbalanced” condition so that the mud pressure is always slightly higher than the fluid pressures in the formation. If the borehole suddenly passes through a highpressure zone, the well can quickly become “underbalanced,” with a resulting fluid pressure gradient driving formation fluids into the wellbore. This situation is known as a kick and is of great concern to drillers because a violent kick can lead to a blowout of mud, gas, and oil from the wellbore, leading to equipment damage and worker injury. Standard drilling practice is to watch diligently for kicks. The first indicator of a kick is typically an increase in mud return rate, leading to an increase in mud pit volume (Frigaard and Humphries 1997). Downhole monitors detect whether the kick is air, H_{2}S, or brine. If the kick fluid is air, the standard procedure is to stop drilling and continue pumping mud in order to circulate the air pocket out. If the mud return rate continues to grow after drilling has stopped and the driller believes that the kick is sufficiently large to cause damage, the well may be shut in by closing the blowout preventer. Once shut in, the well pressure may be bled off slowly and mud weight eventually increased and circulated to offset the higher formation pressure before drilling continues. The spallings model simulates an underbalanced system in which a gas kick is assured, and the kick proceeds with no intervention from the drill operation. Therefore, drilling and pumping continue during the entire blowout event.
The spallings model calculates transient repository and wellbore fluid flow before, during, and after a drilling intrusion. To simplify the calculations, both the wellbore and the repository are modeled by onedimensional geometries. The wellbore assumes a compressible Newtonian fluid consisting of a mixture of mud, gas, salt, and waste solids; viscosity of the mixture varies with the fraction of waste solids in the flow. In the repository, flow is viscous, isothermal, compressible singlephase (gas) flow in a porous medium.
The wellbore and repository flows are coupled by a cylinder of porous media before penetration, and by a cavity representing the bottom of the borehole after penetration. Schematic diagrams of the flow geometry prior to and after penetration are shown in Figure PA23 and Figure PA24, respectively. The drill bit moves downward as a function of time, removing salt or waste material. After penetration, waste solids freed by drilling, tensile failure, and associated fluidization may enter the wellbore flow stream at the cavity forming the repositorywellbore boundary.
Figure PA23. Schematic Diagram of the Flow Geometry Prior to Repository Penetration
Figure PA24. Schematic Diagram of the Flow Geometry After Repository Penetration
Flow in the well is modeled as a onedimensional pipe flow with crosssectional areas corresponding to the appropriate flow area at a given position in the well, as shown in Figure PA25 and Figure PA26. This model is conceptually similar to that proposed by Podio and Yang (1986) for use in the oil and gas industry. Drilling mud is added at the wellbore entrance by the pump. Flow through the drill bit is treated as a choke with crosssectional area appropriate for the bit nozzle area. At the annulus output to the surface, the mixture is ejected at a constant atmospheric pressure. The gravitational body force acts in its appropriate direction based on position before or after the bit.
Figure PA25. Effective Wellbore Flow Geometry Before Bit Penetration
Figure PA26. Effective Wellbore Flow Geometry After Bit Penetration
Prior to drill bit penetration into the repository, gas from the repository can flow through drillingdamaged salt into the well. After penetration, the cavity at the bottom of the wellbore couples the wellbore flow and the repository flow models; gas and waste material can exit the repository domain into the cavity. The cavity radius increases as waste materials are moved into the wellbore.
The system of equations representing flow in the wellbore consists of four equations for mass conservation, one for each phase (salt, waste, mud, and gas); one equation for conservation of total momentum; two equations relating gas and mud density to pressure; the definition of density for the fluid mixture; and one constraint imposed by the fixed volume of the wellbore. The conservation of mass and momentum is described by
where
q = phase (w for waste, s for salt, m for mud, and g for gas)
V_{q}_{ } = volume (m^{3}) of phase q
V = total volume (m^{3})
r_{q}_{ } = density (kg/m^{3}) of phase q, constant for salt and waste (2,180 and 2,650 kg/m^{3}, respectively) and pressuredependent for gas and mud (see Equation (PA.149) and Equation (PA.150))
r = density of fluid mixture (kg/m^{3}) determined by Equation (PA.151)
u = velocity (m/s) of fluid mixture in wellbore
t = time (s)
z = distance (m) from inlet at top of well
S_{q}_{ } = rate of mass (kg/s) in phase q entering and exiting wellbore domain at position z (Equation (PA.162))
S_{mom}_{ } = rate of momentum (kg m/s^{2}) entering and exiting wellbore domain at position z (Equation (PA.165))
P = pressure (Pa) at position z
g = standard gravity (9.8067 kg/m/s^{2})
F = friction loss using pipe flow model (kg/m^{2}/s^{2}) determined by Equation (PA.153)
Gas is treated as isothermal and ideal, so the pressure and density are related by Boyle’s law:
where r_{g}_{,0}_{ } is the density of H_{2} gas at atmospheric pressure and 298 K (8.24182 ´ 10^{2} kg/m^{3}).
The mud is assumed to be a compressible fluid, so
where r_{m}_{,0} is the density of the mud at atmospheric pressure (1,210 kg/m^{3}) and c_{m} is the compressibility of the mud (3.1 ´ 10^{10} Pa^{1}).
The density of the fluid mixture is determined from the densities and volumes occupied by the phases:
The volume of each phase is constrained by the fixed total volume of the wellbore:
The friction loss is a standard formulation for pipe flow (Fox and McDonald 1985), where the head loss per unit length is given as
where the hydraulic diameter d_{h} is given by
In PA, D_{o} = 0.31115 m throughout the domain. From the bit to the top of the collar, D_{i} = 0.2032 m; above the collar, D_{i} = 0.1143 m. The area A is calculated as the area of the annulus between the outer and inner radii:
Thus, d_{h} = 0.108 m from the bit to the top of the collar, and d_{h} = 0.197 m above the collar.
The Darcy friction factor f in Equation (PA.153) is determined by the method of Colebrook (Fox and MacDonald 1985). In the laminar regime, which is assumed to be characterized by Reynolds numbers below 2100 (Walker 1976),
and in the turbulent regime (Re > 2100)
where _{} is the Reynolds number of the mixture, and h is the viscosity calculated in Equation (PA.158), below. As the wellbore mixture becomes particleladen, the viscosity of the mixture is determined from an empirical relationship developed for proppant slurry flows in channels for the oil and gas industry (Barree and Conway 1995). Viscosity is computed by an approximate slurry formula based on the volume fraction of waste solids:
where h_{0} is a base mixture viscosity (9.17 ´ 10^{ }^{3} Pa s), w = V_{w}/ V is the current volume fraction of waste solids, w_{max} is an empirically determined maximal volume fraction above which flow is choked (0.615), and s is an empirically determined constant (1.5) (Hansen, Pfeifle, and Lord 2003).
Initial conditions in the wellbore approximate mixture flow conditions just prior to waste penetration. The wellbore is assumed to contain only mud and salt. Initial conditions for the pressure, fluid density, volume fractions of mud and salt, and the mixture velocity are set by the following algorithm:
Step 1. Set pressure in the wellbore to hydrostatic: P(z) = P_{atm} – r_{m,}_{0} gz.
Step 2. Set mud density using Equation (PA.150).
Step 3. Set mixture velocity: u(z) = R_{m}/ A(z), where R_{m} is the volume flow rate of the pump (0.0202 m^{3}/s), and A(z) is the crosssectional area of the wellbore.
Step 4. Set volume of salt in each cell: V_{s,i}= R_{drill} A_{bit}Dz_{i}/ u_{i}, where R_{drill} is the rate of drilling (0.004445 m/s), _{} is the area of the bottom of the wellbore, and d_{bit} is the diameter of the bit (0.31115 m).
Step 5. Set volume fraction of mud in each cell: V_{m,i} = V_{i} – V_{s,i}.
Step 6. Recalculate mixture density using Equation (PA.151), assuming no waste or gas in the wellbore.
The initial conditions set by this algorithm approximate a solution to the wellbore flow (Equation (PA.147) and Equation (PA.148)) for constant flow of mud and salt in the well. The approximation rapidly converges to a solution for wellbore flow if steadystate conditions are maintained (WIPP Performance Assessment 2003d).
For simplicity, DRSPALL does not model flow of mud down the pipe to the bit. Mass can enter the wellbore below the drill bit and exit at the wellbore outlet. Below the bit, mud, salt, gas, and waste can enter the wellbore. PA assumes a constant volume of mud flow down the drilling pipe; therefore, the source term for mud, S_{m,in}, is set by the volumetric flow rate of the pump R_{m} (0.0202 m^{3}/s) and the density of the mud at the bottom of the wellbore:
Until the drill bit penetrates the repository, salt enters the wellbore at a constant rate:
Additional mass enters the wellbore by gas flow from the repository (S_{gas,in}) and spalling of waste material (S_{w,in}); these mass sources are discussed in Section PA4.6.2.3. The outlet of the wellbore is set to atmospheric pressure. Mass exiting the wellbore is determined from the mixture velocity, the area of the outlet A_{out} (0.066 m^{2}), and the density and volume fraction of each phase at the outlet of the wellbore:
Finally, the net change in mass and momentum for phase q is
The outlet of the wellbore is set to atmospheric pressure. Momentum exiting the wellbore is determined from the fluid velocity and the area of the outlet A_{out} (0.066 m^{2}):
No momentum is added by mass flow into the wellbore from the repository; thus
The repository is modeled as a radially symmetric domain. A spherical coordinate system is used for this presentation and for most DRSPALL calculations. In a few circumstances, cylindrical coordinates are used in PA calculations, where spall volumes are large enough that spherical coordinates are not representative of the physical process (Lord, Rudeen, and Hansen 2003). Cylindrical coordinates are also available; the design document for DRSPALL (WIPP Performance Assessment 2003e) provides details on implementing the repository flow model in cylindrical coordinates.
Flow in the repository is transient, compressible, viscous, and singlephase (gas) flow in a porous medium. Gas is treated as isothermal and ideal. The equations governing flow in the repository are the equation of state for ideal gases (written in the form of Boyle’s law for an ideal gas at constant temperature), conservation of mass, and Darcy’s law with the Forchheimer correction (Aronson 1986, Whitaker 1996):
where
P = pressure in pore space (Pa)
r_{g}_{ } = density of gas (kg/m^{3})
u = velocity of gas in pore space (m/s)
f = porosity of the solid (unitless)
h_{g}_{ } = gas viscosity (8.934 ´ 10^{6} Pa s)
k = permeability of waste solid (m^{2})
F = Forchheimer correction (unitless)
The Forchheimer correction is included in Equation (PA.168) to account for inertia in the flowing gas, which becomes important at high gas velocities (Ruth and Ma 1992). When the Forchheimer coefficient is zero, Equation (PA.168) reduces to Darcy’s law. A derivation of Equation (PA.168) from the NavierStokes equations is given by Whitaker (1996); the derivation suggests that F is a linear function of gas velocity for a wide range of Reynolds numbers.
In PA, the Forchheimer correction takes the form
where b_{nd} is the nonDarcy coefficient, which depends on material properties such as the tortuosity and area of internal flow channels, and is empirically determined (Belhaj et al. 2003). DRSPALL uses a value from Li et al. (2001) that measured highvelocity nitrogen flow through porous sandstone wafers, giving the result
Equation (PA.166) through Equation (PA.168) combines into a single equation for pressure in the porous solid:
where
and the Laplacian operator in a radially symmetric coordinate system is given by
where n = 2 and n = 3 for polar and spherical coordinates, respectively.
In DRSPALL, the permeability of the waste solid is a subjectively uncertain parameter that is constant for waste material that has not failed and fluidized. In a region of waste that has failed, the permeability increases as the waste fluidizes by a factor of 1 + F_{f}, where F_{f} is the fraction of failed material that has fluidized and is based on the fluidization relaxation time. This approximately accounts for the material bulking as it fluidizes.
Initial pressure in the repository is set to a constant value P_{ff}. A noflow boundary condition is imposed at the outer boundary (r = R):
At the inner boundary (r = r_{cav}), the pressure is specified as P(r_{cav}, t) = P_{cav}( t), where P_{cav}( t) is defined in the next section. The cavity radius r_{cav} increases as drilling progresses and waste material fails and moves into the wellbore; calculation of r_{cav} is described in Section PA4.6.2.3.3.
Prior to penetration, a cylinder of alteredpermeability salt material with diameter equal to the drill bit is assumed to connect the bottom of the wellbore to the repository. At the junction of the repository and this cylinder of salt, a small, artificial cavity is used to determine the boundary pressure for repository flow. After penetration, the cavity merges with the bottom of the wellbore to connect the wellbore to the repository.
The cylinder of salt connecting the wellbore to the repository is referred to as the DDZ in Figure PA23. The permeability of the DDZ, k_{DDZ}, is 1 ´ 10^{14} m^{2}. The spallings model starts with the bit 0.15 m above the repository; the bit advances at a rate of R_{drill} = 0.004445 m/s.
To couple the repository to the DDZ, the model uses an artificial pseudocavity in the small hemispherical region of the repository below the wellbore with the same surface area as the bottom of the wellbore (Figure PA26). The pseudocavity is a numerical device that smoothes the discontinuities in pressure and flow that would otherwise occur upon bit penetration of the repository. The pseudocavity contains only gas, and is initially at repository pressure. The mass of gas in the cavity m_{cav} is given by
where
S_{rep}_{ } = gas flow from repository into pseudocavity (kg/s); see Equation (PA.176)
S_{g, in}_{ } = gas flow from pseudocavity through DDZ into wellbore (kg/s); see Equation (PA.177)
Flow from the repository into the pseudocavity is given by
where
r_{g,rep} = gas density in repository at cavity surface (kg/m^{3}) = _{}
u_{rep} = gas velocity (m/s) in repository at cavity surface = _{}
f = porosity of waste (unitless)
A_{cav} = surface area of hemispherical part of the cavity (m^{2})
= _{}, where d_{bit} is the diameter of the bit (m)
Flow out of the pseudocavity through the DDZ and into the wellbore is modeled as steadystate using Darcy’s Law:
where
h_{g} = viscosity of H_{2} gas (8.934 ´ 10^{ }^{6} Pa s)
M_{w} = molecular weight of H_{2} gas (0.00202 kg / mol)
R = ideal gas constant (8.314 J/mol K)
T = repository temperature (constant at 300 K (27 ºC; 80 ºF))
L = length (m) of DDZ (from bottom of borehole to top of repository)
P_{cav} = pressure in pseudocavity (Pa)
P_{BH} = pressure at bottom of wellbore (Pa)
A justification for using this steadystate equation is provided in the design document for DRSPALL (WIPP Performance Assessment 2003e). The pseudocavity is initially filled with gas at a pressure of P_{ff}. The boundary pressure on the well side (P_{BH}) is the pressure immediately below the bit, determined by Equation (PA.147) and Equation (PA.148). The pressure in the pseudocavity (P_{cav}) is determined by the ideal gas law:
where m_{cav} is the number of moles of gas in the cavity and the cavity volume V_{cav} is given by
In PA, the drilling rate into the ground is assumed constant at 0.004445 m/s; thus L = L_{i} – 0.004445t until L = 0, at which time the bit penetrates the waste. The term L_{i} is the distance from the bit to the waste at the start of calculation (0.15 m).
After waste penetration, the bottom of the wellbore is modeled as a hemispherical cavity in the repository, the radius of which grows as drilling progresses and as material fails and moves into the cavity. Gas, drilling mud, and waste are assumed to thoroughly mix in this cavity; the resulting mixture flows around the drill collars and then up the annulus between the wellbore and the drill string. Gas flow from the repository into the cavity is given by Equation (PA.176); however, A_{cav} is now dependent on the increasing radius of the cavity (see Section PA4.6.2.3.3). Mudflow into the cavity from the wellbore is given by Equation (PA.159). Waste flow into the cavity is possible if the waste fails and fluidizes; these mechanisms are discussed in Section PA4.6.2.3.4 and Section PA4.6.2.3.5. Pressure in the cavity is equal to that at the bottom of the wellbore, and is computed by Equation (PA.178).
The cylindrical cavity of increasing depth created by drilling is mapped to a hemispherical volume at the bottom of the wellbore to form the cavity. This mapping maintains equal surface areas in order to preserve the gas flux from the repository to the wellbore. The cavity radius from drilling is thus
where DH is the depth of the drilled cylinder. In PA, the drilling rate into the ground is assumed constant at 0.004445 m/s; thus DH = 0.004445t until DH = H, the height of compacted waste (m). Since the initial height of the repository is 3.96 m, H is computed from the porosity f by _{}, where f_{0} is the initial porosity of a wastefilled room.
The cavity radius r_{cav} is increased by the radius of failed and fluidized material r_{fluid}, which is the depth to which fluidization has occurred beyond the drilled radius. That is,
Gas flow from the waste creates a pressure gradient within the waste, which induces elastic stresses in addition to the farfield confining stress. These stresses may lead to tensile failure of the waste material, an assumed prerequisite to spallings releases. While the fluid calculations using Equation (PA.166), Equation (PA.167), and Equation (PA.168) are fully transient, the elastic stress calculations are assumed to be quasistatic (i.e., soundspeed phenomena in the solid are ignored). Elastic effective stresses are (Timoshenko and Goodier 1970)
where b is Biot’s constant (assumed here to be 1.0) and s_{ff} is the confining farfield stress (assumed constant at 14.8 MPa).
The flowrelated radial and tangential stresses (s_{sr} and s_{s}_{ q} , respectively) are computed by equations analogous to differential thermal expansion (Timoshenko and Goodier 1970):
where P_{ff} is the initial repository pressure and u is Poisson’s ratio (0.38).
Since stresses are calculated as quasistatic, an initial stress reduction caused by an instantaneous pressure drop at the cavity face propagates instantaneously through the waste. The result of calculating Equation (PA.182) can be an instantaneous earlytime tensile failure of the entire repository if the boundary pressure is allowed to change suddenly. This is nonphysical and merely a result of the quasistatic stress assumption, combined with the true transient pore pressure and flowrelated stress equations. To prevent this nonphysical behavior, tensile failure propagation is limited by a tensile failure velocity (1000 m/s; see Hansen et al. 1997). This limit has no quantitative effect on results, other than to prevent nonphysical tensile failure.
At the cavity face, Equation (PA.182) and Equation (PA.184) evaluate to zero, consistent with the quasistatic stress assumption. This implies that the waste immediately at the cavity face cannot experience tensile failure; however, tensile failure may occur at some distance into the waste material. Consequently, the radial effective stress s_{r} is averaged from the cavity boundary into the waste over a characteristic length L_{t} (0.02 m). If this average radial stress _{} is tensile and its magnitude exceeds the material tensile strength (_{} > TENSLSTR), the waste is no longer capable of supporting radial stress and fails, permitting fluidization. The waste tensile strength is an uncertain parameter in the analysis (see TENSLSTR in Table PA15).
Equation (PA.183) and Equation (PA.185) evaluate shear stresses in the waste. DRSPALL does not use the waste shear stresses to calculate waste failure for spall releases. These stresses are included in this discussion for completeness.
Failed waste material is assumed to be disaggregated, but not in motion; it remains as a porous, bedded material lining the cavity face, and is treated as a continuous part of the repository from the perspective of the porous flow calculations. The bedded material may be mobilized and enter the wellbore if the gas velocity in the failed material (see Equation (PA.168)) exceeds a minimum fluidization velocity, U_{f}. The minimum fluidization velocity is determined by solving the following quadratic equation (Cherimisinoff and Cherimisinoff 1984, Ergun 1952)
where
a = particle shape factor (unitless)
d_{p}_{ } = particle diameter (m)
Fluidization occurs in the failed material to the depth at which gas velocity does not exceed the fluidization velocity; this depth is denoted by r_{fluid} and is used to determine cavity radius (Section PA4.6.2.3.3). If fluidization occurs, the gas and waste particles mix into the cavity at the bottom of the wellbore. Because this mixing cannot be instantaneous, which would be nonphysical (much as allowing instantaneous tensile failure propagation would be nonphysical), a small artificial relaxation time, equal to the cavity radius r_{cav} divided by the superficial gas velocity u(r_{cav}), is imposed upon the mixing phenomenon. The fluidized material is released into the cavity uniformly over the relaxation time.
The numerical model implements the conceptual and mathematical models described above (Section PA4.6.2). Both the wellbore and the repository domain calculations use timemarching finite differences. These are part of a single computational loop and therefore use the same time step. The differencing schemes for the wellbore and repository calculations are similar, but not identical.
The wellbore is zoned for finite differencing, as illustrated in Figure PA27, which shows zones, zone indices, grid boundaries, volumes, and interface areas. The method is Eulerian: zone boundaries are fixed, and fluid flows across the interfaces by advection. Quantities are zonecentered and integration is explicit in time.
To reduce computation time, an iterative scheme is employed to update the wellbore flow solution. The finitedifference scheme first solves Equation (PA.147) and Equation (PA.148) for the mass of each phase in each grid cell and the momentum in each grid cell.
The updated solution to Equation (PA.147) and Equation (PA.148) is then used to compute the volume of each phase, the pressure, and the mixture velocity in each grid cell.
All of the materials (mud, salt, gas, and waste) are assumed to move together as a mixture. Because fluid moves through the cell boundaries, the calculation requires a value for the flow through each cell boundary during a time step. These values are obtained by averaging the fluid velocities at the zone centers, given by
Figure PA27. FiniteDifference Zoning for Wellbore
The mass transport equation, prior to any volume change, becomes
Here, the source terms S_{m,i} correspond to material entering or exiting at the pump, cavity, and surface. The “upwind” zonecentered densities are used for the interfaces values, _{} and _{}.
Finally, any changed volumes are incorporated and numerical mass diffusion is added for stability:
where
and z_{q} is the diffusion coefficient for phase q. The density rf_{q} for phase q being diffused is calculated from the mixture density, r, and the mass fraction, f_{q} , of phase q in the referenced cell (f_{q} = rV_{q,i}/ rV_{i}). The numerical diffusion coefficient z_{q} is chosen empirically for stability. Separate diffusion coefficients could be used for the different materials (mud, gas, etc.); however, sufficient stability is obtained by diffusing only mud and salt using the same coefficient (z_{m} = z_{s} = 0.0001 and z_{w} = z_{g} = 0).
Momentum is differenced as