2 The simplified underground stratigra phic model used in geomechanical numerical analyses since the CCA (Butcher 1997, Stone 1997b, Park and Holla nd 2007) was modified to include the panel intake drift and the distance between adjacent pane ls (Figure 1). Other than those two changes, the mesh is changed as little as possible from the original mesh used by Stone (1997b) to minimize possible errors introduced by the changes.

Figure 1. Simplified underground stratigraphic, half symmetry model and boundary conditions used in the JAS3D modeling of crushed salt panel closures.3 In order to be used in JAS3D, the model had to be made 3D by adding a depth of 0.5 m. Even though it is a 3D model, a plane st rain condition is assumed in the lengthwise direction of the panel closure. As shown in the verification and validation plan / validation document for JAS3D (WIPP PA 2009), modifying the simplified undergr ound mesh to 3D has very little effect on identical underground analyses made for porosity surface calculations. The plane strain assumption for the crushed salt panel closure wa s shown to be valid based on a simplified 3D analysis of a model panel closur e subjected to a constant stre ss loading condition (DOE 2012).

A plane of symmetry through the center of the pa nel closure is used to limit the number of elements in the mesh by having to discretize only ha lf of the model, a half symmetry model. This is accomplished by a no horizontal displacement boundary condition imposed on the left boundary of the mesh in Figure 1. The horizontal width of the model co rresponds to half the distance between the drifts of adjacent panels, which is 230 ft (70.10 m). Use of another symmetry boundary condition on the right side of th e mesh takes into account the influence an identical drift in an adjacent panel. On the top of the model is a traction boundary condition equivalent to the vertical in situ stress at that elevation. On the bo ttom of the model is a no displacement boundary condition in the vertical direction.

The drift where the panel closure will be is initially completely filled with run-of-mine salt. This salt is modeled using the volumetric creep model for crushed salt. Contact surfaces are invoked between the crushed salt of the pa nel closure and the intact salt surrounding the drift so that the materials can deform independently and separate if necessary. In addition, th e air pressure in the drift is assumed to be at one atmo sphere and obey the ideal gas law.

2 The Crushed Salt Constitutive Model

The run-of-mine salt making up the panel closure is modeled using the volumetric creep model in JAS3D. The volumetric creep model is base d on the work of Sjaardema and Krieg (1987). They developed their model based on the hydros tatic consolidation test s of crushed salt by Holcomb and Shields (1987). The following descript ion of the model is based on those found in Stone (1997a) who describes how the model was implemented into various Sandia structural mechanics codes. The model was used because of its similarity to the crushed salt model in FLAC3D (Itasca, 2003), used by RockSol Consu lting Group, Inc. for the panel closure design document.

The total strain rate is decomposed into volumet ric and deviatoric parts. Because intact salt creeps deviatorically when subjected to a devi atoric stress condition, crushed salt should logically be expected to also creep deviatorically. This expect ation becomes more reasonable as the density increases and the crushed salt approa ches an intact conditi on. The deviatoric salt creep model used in the volumetric creep mode l is based on the power-law creep model. The volumetric creep model is formulated such that th e material response of the consolidated crushed salt becomes identical to that of the intact materi al as the density of the crushed salt approaches that of intact salt. The FLAC/FLAC3D crushe d salt model used by RockSol differs from the Sandia model in that it appears to include prim ary creep as well as secondary creep in its implementation (Itasca, 2003). The solu tion technique is also different.4 The bulk, K , and shear, G , elastic moduli were found from Holcomb and Shields tests to depend on the density, ρ , of the material through re lationships of the form 

##   0 1

exp   KK K (1)


##   0 1

exp   GG G (2)

where K 0 , K 1 , G 0 , and G 1 are material constants.

The power-law creep model is cas t as a conventional power-law secondary creep model of the form



### exp     

       m 

### Q A R (3)

where 

###  is the effective deviatoric stress, A and m are material constants,  is the absolute temperature, R is the universal gas constant, and Q is the activation ener gy. Using forward Euler integration and a creep strain rate for von Mises flow, the expression for the deviatoric stress rate is given by:

1 3 2 exp 2     

  

           
 n m n ij ij ij Q SG S R (4)

The volumetric creep model is developed by extens ion of a one dimensional problem into three dimensions. A uniaxially compressed porous crus hed salt sample is envisioned as being composed of cylinders of salt, each of which ha s intact salt secondary creep behavior separated by areas of open space. The local stress acting on the salt cylinders is st ated in terms of the average stress on the porous sample . The cross-sectional area of the porous sample is expressed in terms of the net cross-sectional area of the salt cylinders. The final resulting continuum model for the rate of the deviatoric stre ss of the crushed salt is given as:

1 3 2 ( ) exp 2     

    

    

              m

n mn ij ij ij Q SG A S R (5)

where the constants A

, Q

, m , and ρ ∞ refer to values for the intact salt, ρ is the density.

The volumetric part of the model is divi ded into elastic and inelastic parts as



##  

   
c kk kk p d d K (6)5 where d kk

is the volumetric strain rate, 3 kk p 

 

 is the rate of cha nge of the pressure, c kk 

### d is the
volumetric creep strain rate, and K ( ρ ) is the density-dependent bulk modulus.

Laboratory consolidation tests have shown that the crushed salt volu metric creep rate can be fit by an expression of the form

1 0 1 1    

    B p c A kk d Be e (7)

where B 0 , B 1 , and A are material constants. The density is computed from the relationship

0 0 exp    

      



##  t

kk

t d dt (8)

where 0  is the density at time t 0
.
Equation (6) is solved for 

### p  and combined with the definition of the volumetric creep strain rate, Eq. (7), to yield



##   

#   1 0 1 1     

       B p A kk p K d Be e (9)

which is the expression that is integrated.

2.1 Input Parameters for the Crushed Salt Model

The input parameters used in the volumetric cr eep model and their sources are given in Table 1. The initial densities, 0  , corresponding to fractional dens ities of 65, 67, 70, 75, 80, and 85%, are 1404, 1440, 1512, 1620, 1728, and 1836 kg/m 3 . These represent, in order, initial porosities of 35, 33, 30, 25, 20, and 15%.6 Table 1. Input parameters used for the volumetric creep model panel closure.

Parameter Value Units Source
2G 0 2.12e+04 Pa Sjaardema and Krieg (1987), p. 59 K 0 1.76e+04 Pa Sjaardema and Krieg (1987), p. 59 D 5.79e-36 1/(Pa n ·s) Krieg (1984), p. 18

n 4.9 — Krieg (1984), p. 18 Q/RT 20.1422 — Krieg (1984), p. 18

G 1 6.53e-03 m 3 /kg Sjaardema and Krieg (1987), p. 59 K 1 6.53e-03 m 3 /kg Sjaardema and Krieg (1987), p. 59 B 0 1.30e+08 kg/(m 3 ·s) Sjaardema and Krieg (1987), pp. 14-15 B 1 8.20e-07 1/Pa Sjaardema and Krieg (1987), pp. 14-15 A -1.73e-02 m 3 /kg Sjaardema and Krieg (1987), p. 14
ρ ∞ 2160 kg/m 3 Krieg (1984), p. 14; Brodsky (1994), p. 2
ρ 0 1404 * kg/m 3 Initial fractional density of 65% of intact salt

* The value ρ 0 = 1404 kg/m 3 is for an initial fractional density of 65% intact, which corresponds to a porosity of 35%. Initial density values at time t= 0 vary depending on the predetermined porosity value being analyzed. The six initial porosities used were 15 , 20, 25, 30, 33, and 35% (or fractional densities 85, 80, 75, 70, 67, and 65%, respectively).

3 Intact Salt Constitutive Model

A combined transient-secondary creep constitu tive model for rock salt known as the multimechanism deformation (M-D) model proposed by Munson and Dawson (1979, 1982, 1984) and extended by Munson et al. (1989), was used for th e clean and argillaceous salt. The model is decomposed into an elastic volumet ric part and a de viatoric part.

The elastic volumetric part is defined by

K kk kk 3    (10)

where, kk  = the total strain components

kk  = the total stress components
K = the elastic bulk modulus

The deviatoric part defined by

2
2 2 cos 2 3 sin 2 2 cos3 3 cos3 ij ij s ij ip pj ij s G e F s ss J J J                               (11)

where 3 kk ijij s    = the deviatoric stress7 G

= the elastic shear modulus

3 kk ijij e    = the deviatoric strain

1for
0 for ij i j i j   

    = Kronecker delta

J

2 and θ are the second invariant of the deviator stre ss and the Lode angle, respectively, and will be defined later.

The second term of Equation (11) represents the creep contribution. F is a multiplier on the steady-state creep rate to simulate the tran sient creep response according to the following equation

    

  



 

*

]/1[ * * ,]/1[

2* 2*

1

t t t

t t

e e F   

  (12)

where 

###  work-hardening parameter

  recovery parameter



###  *
t 

###  transient strain limit



###  is an internal state variable whose rate of change is determin ed by the following evolutionary equation

s 

### F    )1(

 (13)

In Equation (12), the work-hardening parameter 

###  is defined as



### )/log( G       (14)

where,  and 

###  are constants. The variable 

###  is the equivalent Tresca stress given by

  cos2 2 J  (15)

where 

###      


 2/3 2 3 

### )(2

33 arcsin 3 1
J J  is the Lode angle limited to the range ( 66     ).

qppq ssJ 2 1

2  = second invariant of the stress deviator

rpqrpq sssJ 3 1

3  = third invariant of the stress deviator8 The recovery parameter,  , is held constant. The transient strain limit is given by

M cT
t 

### GeK )/( 0 * 

###    (16)

where 0 K
, c , and M are constants.

The steady-state, or seco ndary creep strain rate,

s   , is given by

1 2

12 1 2 0) / / // 1 2 12 ( sinh n n Q R Q R QR Q R

s q Ae Ae H Be B e GG G                                  (17)

Where i A s and i B s are constants

i Q s are activation energies
 = the absolute temperature R = the universal gas constant

i n s are the stress exponents

q = the stress constant

0  = the stress limit of the dislocation slip mechanism

H = the Heaviside step function with the argument )( 0  



3.1 Input Parameters for the Intact Salt Model

The material constants used in the analyses for the clean and argillaceous salt are given in Table 2 and Table 3.

Table 2. Salt elastic properties (Butcher 1997)

Parameter Value Units
G E

 12,400 31,000

0.25 MPa MPa

–9 Table 3: Salt creep properties (Munson et al. 1989)

Parameter Clean Salt Argillaceous Salt Units
A 1 8.386e+22 1.407e+23 1/sec Q 1 25,000 25,000 cal/mole

n 1 5.5 5.5 n 1 B 1 6.086e+06 8.998e+06 1/sec A 2 9.672e+12 1.314e+13 1/sec Q 2 10,000 10,000 cal/mole

n 2 5.0 5.0

– B

2 3.034e-02 4.289e-02 1/sec



0 20.57 20.57 MPa
q 5335 5335

–
m 3.0 3.0

–

0 K 6.275e+05 2.470e+06

–

c 9.198e-03 9.198e-03 1/T

 -17.37 -14.96 –  -7.738 -7.738 –
 0.58 0.58 –

4 Run Control

Herrick 2012 contains the run control informati on for the JAS3D analyses in which the emplaced panel closures use a crushed salt fill with in itial porosities of 33, 25, 20, and 15% (or initial fractional densities of 67, 75, 80, and 85%, respec tively) which were presented in the DOE’s response to EPA questions (DOE 2012). This secti on contains the run control information for the additional analyses in which the emplaced panel closures use crushed salt fill with initial porosities of 35 and 30% (or initial fractional de nsities of 65 and 70%, re spectively). Of perhaps greatest importance is that the re sults module Jas3d/Output is the same for all analyses including those used in DOE (2012).

For the additional two analyses, JAS3D (versi on 2.4.C-WIPP) and its associated utilities (Algebra2 (version 1.28), Aprepro (version 2.06 ), BlotII2_cps (version 1.59A), FASTQ_cps (version 3.18), and GEN3DII (version 1.37a)) were run on April 20, 2012 on seals.sandia.gov using the Python run control scripts PC3R_3.py and PC3R_4.py and the Python module rc.py. The Postscript files generated by BlotII2 (version 1.59A) were c onverted to PDF files using the UNIX utility ps2pdf. All input files for the r uns were exported from the CVS repository $CVSLIB/Analyses/PC3R, modul e Jas3d/Input/Underground2.

The run control scripts and an associated C shell scripts, RunJas3d_3.csh and RunJas3d_4.csh, were exported from the RunContro l module of the same repository. JAS3D and associated utility codes and scripts were exported from code-specific CVS repositories (Table 4). The output files from JAS3D, Algebra, BlotII2, and ps2pdf were st ored in the Output module of the repository $CVSLIB/Analyses/PC3R. Additional data files (.dat) produced by BlotII2 were added to the Jas3d/Output module of the re pository after the PC3R.py scri pt completed. The log files10 PC3R_3.log and PC3R_4.log, and RT F versions of those files, are stored in the module RunControl/Log. All files were ta gged with the CVS tag PC3R.

Table 4. Run control code-s pecific CVS repositories.

File Repository Comment Builds/Linux64/algebra2 (Ver:1.28) /nfs/data/CVSLIB/ACCESS_WIPP/ALGEBR A2 Manipulates EXODUS finite element output data by evaluating expressions Builds/Linux64/aprepro (Ver:2.06) /nfs/data/CVSLIB/ACCESS_WIPP/APREPR O An algebraic preprocessing program Builds/Linux64/blotII2_cps (Ver:1.59A) /nfs/data/CVSLIB/ACCESS_WIPP/BLOTII2 Postscript driver for blot

Builds/Linux64/fastq_cps (Ver:3.18) /nfs/data/CVSLIB/ACCESS_WIPP/FASTQ Finite element 2-dimensional mesh generation program Builds/Linux64/gen3d2 (Ver:1.37a) /nfs/data/CVSLIB/ACCESS_WIPP/GEN3 DII Transforms a 2-dimensional GENESIS database into a 3- dimensional GENESIS database Builds/Linux64/jas3d (Ver:2.4.C-WIPP) /nfs/data/CVSLIB/ACCESS_WIPP/JAS3 D Finite element code for 3- dimensional analyses RunControl/algebra2.sh /n fs/data/CVSLIB/ACCESS_WIPP/ALGEBRA2 Script for manipulating EXODUS finite element output data by evaluating expressions RunControl/blotII2.sh /n fs/data/CVSLIB/ACCESS_WIPP/BLOTII2 Script for running graphics code for examining meshes RunControl/fastq.sh /nfs/data/CVSLIB/ACCESS_WI PP/FASTQ Script for running finite element 2-dimensional mesh generation program RunControl/gen3dII.sh /nfs/dat a/CVSLIB/ACCESS_WIPP/GEN3DII Script to transform a 2- dimensional GENESIS database into a 3-dimensional GENESIS database RunControl/jas3d.sh /nfs/data/CVSLIB/ACC ESS_WIPP/JAS3D Script for running jas3d.exe

5 Results

The following figures, from the PostScript files (Jas3d/Output/*.ps), and associated data files (Jas3d/Output/*.dat), where * indicates the base filename, were used in support of parameter value selections for PCS-2012 PA (Camphouse 2012). Base filenaming for the panel closure analyses takes one of two forms: “pclo_vcXX _por-void” or “pclo_vcXX_ SIG-DY”. Here “plco” signifies that the analyses were of panel closures with the under ground stratigraphy included, “vcXX” indicates that the model used for the run-of-mine salt was the volumetric creep model beginning with an initial fractiona l density of XX percent, “por- void” means that the analyses considered porosity and void volume properties of the crushed salt, and “SIG-DY” means that the analyses looked at stresses and vertical displacements of the b ack of the drift or top of the crushed salt. For the latest two analyses, the file name “vc65” indicates that the run-of-mine salt was modeled with the volumetric creep model ha ving an initial fractional density of 65%, and filename “vc70” indicates that the run-of-mine salt was modeled as having an initial fractional density of 70%. The same filenaming convention is used for the previous set of analyses where the initial fractional densities 67, 75, 80, and 85% (Herrick 2012).11 5.1 Porosity History Plots

Starting with the smallest initial density (fract ional density = 65%) and pr ogressing to the highest (fractional density = 85%), th e porosity histories through th e first 500 years for all JAS3D underground analyses are given in Figure 2 through Figure 7.

Figure 2. Porosity history through 500 years for a crushed salt panel closur e in the intake drift having an initial porosity of 35%, which correspo nds to an initial fract ional density of 65%.12 Figure 3. Porosity history through 500 years for a crushed salt panel closur e in the intake drift having an initial porosity of 33%, which correspo nds to an initial fract ional density of 67%.

Figure 4. Porosity history through 500 years for a crushed salt panel closur e in the intake drift having an initial porosity of 30%, which correspo nds to an initial fract ional density of 70%.13 Figure 5. Porosity history through 500 years for a crushed salt panel closur e in the intake drift

having an initial porosity of 25%, which correspo nds to an initial fract ional density of 75%.

Figure 6. Porosity history through 500 years for a crushed salt panel closur e in the intake drift having an initial porosity of 20%, which correspo nds to an initial fract ional density of 80%.14 Figure 7. Porosity history through 500 years for a crushed salt panel closur e in the intake drift having an initial porosity of 15%, which correspo nds to an initial fract ional density of 85%.

As a check of the consistency of the result s obtained from JAS3D by the method described herein, a plot of the time at which the crushed salt in the panel closure co nsolidates to a porosity of 5% for the six different initial poros ities is shown in Figure 8. The high R 2 value demonstrates the consistency of the analytical method. The relationship was fit by a linear trend line using Excel.15 Figure 8. Plot of the time it takes for the crushe d salt fill in the panel closures to reach 5% porosity starting from the six different ini tial porosities used in the JAS3D analyses.

5.2 Backstress in the Back (Roof) of the Drift History Plots

The backstresses generated by the cr ushed salt in the back (roof) of the intake drift resulting from consolidation of the crushed sa lt filling a panel closure was estimated graphically. The next series of plots, Figure 9 through Fi gure 14, are histories of the vert ical stress in the back of the drift again through the first 500 year s. The location of the element used to obtain the data is in the center of the drift.16 Figure 9. Vertical stress history through 500 years in the back (roof ) of the intake drift produced by a run-of-mine salt panel closure ha ving an initial po rosity of 35%.

Figure 10. Vertical stress histor y through 500 years in the back (roof ) of the intake drift produced by a run-of-mine salt panel closure ha ving an initial po rosity of 33%.17 Figure 11. Vertical stress histor y through 500 years in the back (roof ) of the intake drift produced by a run-of-mine salt panel closure ha ving an initial po rosity of 30%.

Figure 12. Vertical stress histor y through 500 years in the back (roof ) of the intake drift produced by a run-of-mine salt panel closure ha ving an initial po rosity of 25%.18 Figure 13. Vertical stress histor y through 500 years in the back (roof ) of the intake drift produced by a run-of-mine salt panel closure ha ving an initial po rosity of 20%.

Figure 14. Vertical stress histor y through 500 years in the back (roof ) of the intake drift produced by a run-of-mine salt panel closure ha ving an initial po rosity of 15%.19 6 References:

Brodsky, N.S. 1994. Determination of the Intact Density of WIPP Salt Specimens. Calculation No. 325/03/02. Respec Inc., Rapid City, SD

Butcher, B.M. 1997. A Summary of the Sources of Input Parameter Values for the WIPP Final Porosity Surface Calculations. SAND97-0796. Sandi a National Laborator ies, Albuquerque, NM

Camphouse, R.C., M. Gross, C.G. Herrick, D.C. Kicker, B. Thompson. 2012. Recommendations and Justifications of Parameter Values for the Run-of-Mine Salt Panel Closure System Design Modeled in the PCS-2012 PA. Sandia Na tional Laboratories, Carlsbad, NM

DOE (U.S. Department of Energy). 2012. Respons e to First Set of EPA Questions to DOE, December 22, 2011. Letter from Mr. Juan Franco, Manager, Carlsbad Area Field Office, U.S. Department of Energy, to Mr. Jonathan Edwards, Director, Radiation Pr otection Division, U.S. Environmental Protection Agency, dated April 17, 2012.

Herrick, C.G. 2012. Calculations Performed in S upport of Reconsolidatio n of Crushed Salt in Panel Closures. Memorandum to the WIPP R ecords Center dated March 29, 2012. ERMS 557150. Sandia National Laborator ies, Carlsbad, NM.

Holcomb, D.J. and M.F. Shields. 1987. Hydrosta tic Consolidation of Crushed Salt with Added Water. SAND87-1990., Sandia National Laboratories, Albuquerque, NM

Itasca Consulting Group. 2003. FLAC3D: Fast La grangian Analysis of Continua in 3 Dimensions User’s Guide. Version 2.1, Optional Features, Section 2 – Creep Materials Models. Minneapolis, MN

Krieg, R. D. 1984. Reference Stratigraphy and Rock Properties for the Waste Isolation Pilot Plant (WIPP) Project. SAND83-1908. Sandia National Laboratories, Albuquerque, NM

Munson, D.E. and P.R. Dawson. 1979. Constitutive Model for the Low Temperature Creep of Salt (with Applicati on to WIPP). SAND79-1853. Sandia Nati onal Laboratories, Albuquerque, NM

Munson, D.E. and P.R. Dawson. 1982, A Transient Cr eep Model for Salt during Stress Loading and Unloading. SAND82-0962. Sandia Natio nal Laboratories, Albuquerque, NM

Munson, D.E. and P.R. Dawson. 1984. Salt Constitutive Modeling Using Mechanism Maps. The Mechanical Behavior of Salt, Proceedings of the First Conferen ce. Pennsylvania State University, University Park, PA. November 9-11, 1981. H.R. Hardy, Jr. and M. Langer, eds. Karl Distributors, Rockport, MA. pp. 717-737.

Munson, D.E., A.F. Fossum, and P.E. Senseny. 1989. Advances in Resolution of Discrepancies between Predicted and Measured in Situ Room Closures. SAND88-2948. Sandia National Laboratories, Albuquerque, NM20 Park, B.Y. and J.F. Holland. 2007. Structural Evalua tion of WIPP Disposal Room Raised to Clay Seam G. SAND2007-3334. Sandia Nationa l Laboratories, Albuquerque, NM

Sjaardema, G.D. and R.D. Krieg. 1987. A Constitu tive Model for the Consolidation of WIPP Crushed Salt and Its Use in Analyses of Back filled Shafts and Drift Configurations. SAND87- 1977. Sandia National Laborat ories, Albuquerque, NM

Stone, C.M. 1997a. SANTOS – A Two-Dimensional Finite Element Program for the Quasistatic, Large Deformation, Inelastic Response of So lids. SAND90-0543. Sandia Na tional Laboratories, Albuquerque, NM

Stone, C.M. 1997b. Final Disposal Room Stru ctural Response Calculations. SAND97-0795. Sandia National Laboratories, Albuquerque, NM

WIPP PA. 2009. Verification and Validation Plan / Validation Document for JAS3D Version 2.4.C-WIPP, document version 2.4.C-WIPP. ERMS 545606. Sandia National Laboratories, Carlsbad, NM.