WIPP Model

An empirical creep law, known as the WIPP-reference creep law, has been developed to describe the time- and temperature-dependent creep of natural rock salt, specifically for nuclear waste isolation studies. The model is described by Herrmann et al. (1980a and b); a different expression of the same creep law is also given by Senseny (1985).

The WIPP-reference creep law, as implemented in FLAC3D, partitions the deviatoric strain-rate tensor, ˙ϵdij, into elastic and viscous parts (˙ϵdeij and ˙ϵdvij, respectively):

(1)˙ϵdij=˙ϵdeij+˙ϵdvij

where the deviatoric strain-rate is obtained:

(2)˙ϵdij=˙ϵij˙ϵkkδij3

The elastic part is related to the deviatoric stress-rate,

(3)˙ϵdeij=˙σdij2G

where G is the elastic shear modulus, and

(4)˙σdij=˙σij˙σkkδij3

The viscous part of the deviatoric strain-rate is coaxial with the deviatoric stress tensor (normalized by its magnitude, ˉσ, defined in Equation (9), and is given by

(5)˙ϵdvij=32{σdijˉσ}˙ϵ

where the scalar strain-rate, ˙ϵ, is composed of two parts, ˙ϵp and ˙ϵs, corresponding to primary and secondary creep, respectively,

(6)˙ϵ=˙ϵp+˙ϵs

The formulation for the primary creep rate depends on the magnitude of the secondary creep rate:

(7)˙ϵp={(ABϵp)˙ϵsif ˙ϵs˙ϵss[AB(˙ϵss/˙ϵs)ϵp]˙ϵsif ˙ϵs<˙ϵss

The secondary creep rate is

(8)˙ϵs=Dˉσne(Q/RT)

where D, n, A, B, and ˙ϵss are material constants, R is the universal gas constant, Q is the activation energy, T is the temperature in degrees Kelvin, and ˉσ is the von Mises stress:

(9)ˉσ=3σdijσdij2

The volumetric response of the model is purely elastic and is given by

(10)˙ϵkk=˙σkk3K

where K is the bulk modulus.

An iterative approach is used to apply the preceding equations, because the constitutive models in FLAC3D or 3DEC take the components of strain rate as independent variables. A model must supply the new stress tensor, on the assumption of constant strain increments. On the first iteration, the stress components, σdij, are taken to be the current ones; creep strain-rates are computed according to Equation (5). New deviatoric stress components, σdij, are then computed on the basis of Equations (1), (3), and (5):

(11)σdij=σdij+2GΔt(˙ϵdij˙ϵdvij)

where σdij are the stress components that exist on entry to the constitutive model, and Δt is the creep timestep.

On the next and subsequent iterations, the averages of the new and old stress components are used in the creep equations:

(12)σdij=(σdij+σdij)/2

Further, the mean primary creep-strain, ϵp, is determined during every iteration:

(13)ϵp=ϵp+˙ϵpΔt/2

and used in Equation (13). The quantity ϵp is the primary creep-strain on entry to the constitutive model; it is updated on exit:

(14)ϵp:=ϵp+˙ϵpΔt

The WIPP-model notation is summarized and typical values are listed in Table 1.

Table 1: Notation for the WIPP formulation
WIPP notation Units Typical Value
A 4.56
B 127
D Pa-ns-1 5.79 × 10-36
n 4.9
Q cal/mol 12,000
R cal/mol K 1.987
˙ϵss s-1 5.39 × 10-8

References

Herrmann, W., W.R. Wawersik and H. S. Lauson. Analysis of Steady State Creep of Southeastern New Mexico Bedded Salt, Sandia National Laboratories, SAND80-0558 (1980a).

Herrmann, W., W.R. Wawersik and H. S. Lauson. Creep Curves and Fitting Parameters for Southeastern New Mexico Rock Salt, Sandia National Laboratories, SAND80-0087 (1980b).

Senseny, P.E. “Determination of a Constitutive Law for Salt at Elevated Temperature and Pressure,” American Society for Testing and Materials, Reprint 869 (1985).



wipp Model Properties

Use the following keywords with the zone property (FLAC3D) or zone property (3DEC) command to set these properties of the WIPP model.

wipp
activation-energy f

activation energy, Q

bulk f

bulk modulus, K

constant-a f

WIPP model constant, A

constant-b f

WIPP model constant, B

constant-d f

WIPP model constant, D

constant-gas f

gas constant, R

creep-rate-critical f

critical steady-state creep rate, ε̇ss

exponent f

WIPP model exponent, n

poisson f

Poisson’s ratio, v

shear f

shear modulus, G

temperature f

zone temperature, T

young f

Young’s modulus, E

creep-strain-primary f (r)

accumulated primary creep strain, ε̇s

creep-rate-primary f (r)

accumulated primary creep strain rate, εs

Key

(r) Read-only property.
This property cannot be set by the user. Instead, it can be listed, plotted, or accessed through FISH.

Notes

  • Only one of the two options is required to define the elasticity: bulk modulus K and shear modulus G, or Young’s modulus E and Poisson’s ratio v.
  • The creep behavior is triggered by deviatoric stress, while the volumetric behavior does not consider creep.