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, \(\dot\epsilon_{ij}^d\), into elastic and viscous parts (\(\dot\epsilon_{ij}^{de}\) and \(\dot\epsilon_{ij}^{dv}\), respectively):

(1)\[\dot\epsilon_{ij}^d =\dot\epsilon_{ij}^{de}+\dot\epsilon_{ij}^{dv}\]

where the deviatoric strain-rate is obtained:

(2)\[\dot\epsilon_{ij}^d=\dot\epsilon_{ij} - {\dot\epsilon_{kk}\delta_{ij}\over3}\]

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

(3)\[\dot\epsilon_{ij}^{de}={\dot\sigma_{ij}^d\over2G}\]

where \(G\) is the elastic shear modulus, and

(4)\[\dot\sigma_{ij}^d=\dot\sigma_{ij} - {\dot\sigma_{kk}\delta_{ij}\over3}\]

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

(5)\[\dot\epsilon_{ij}^{dv}={3\over2}\,\Bigl\{ {\sigma_{ij}^d\over\bar\sigma} \Bigr\}\dot\epsilon\]

where the scalar strain-rate, \(\dot\epsilon\), is composed of two parts, \(\dot\epsilon_p\) and \(\dot\epsilon_s\), corresponding to primary and secondary creep, respectively,

(6)\[\dot\epsilon=\dot\epsilon_p+\dot\epsilon_s\]

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

(7)\[\begin{split}\dot\epsilon_p = \begin{cases} (A-B\epsilon_p)\dot{\epsilon}_s & \text{if } \dot{\epsilon}_s \ge \dot{\epsilon}_{ss}^* \\ & \\ [A-B(\dot{\epsilon}_{ss}^*/\dot{\epsilon}_s) \epsilon_p] \dot{\epsilon}_s & \text{if } \dot{\epsilon}_s < \dot{\epsilon}_{ss}^* \\ \end{cases}\end{split}\]

The secondary creep rate is

(8)\[\dot\epsilon_s=D\,\bar\sigma^n e^{(-Q/RT)}\]

where \(D\), \(n\), \(A\), \(B\), and \(\dot\epsilon_{ss}^*\) are material constants, \(R\) is the universal gas constant, \(Q\) is the activation energy, \(T\) is the temperature in degrees Kelvin, and \(\bar\sigma\) is the von Mises stress:

(9)\[\bar\sigma=\sqrt{3\,\sigma_{ij}^d\sigma_{ij}^d\over2}\]

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

(10)\[\dot\epsilon_{kk}={\dot\sigma_{kk}\over3K}\]

where \(K\) is the bulk modulus.

An iterative approach is used to apply the preceding equations, because the constitutive models in FLAC3D 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, \(\sigma_{ij}^d\), are taken to be the current ones; creep strain-rates are computed according to Equation (5). New deviatoric stress components, \(\sigma_{ij}^{d'}\), are then computed on the basis of Equations (1), (3), and (5):

(11)\[\sigma_{ij}^{d'}=\sigma_{ij}^{d^\circ}+2G\Delta t (\dot\epsilon_{ij}^d-\dot\epsilon_{ij}^{dv})\]

where \(\sigma_{ij}^{d^\circ}\) are the stress components that exist on entry to the constitutive model, and \(\Delta t\) 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)\[\sigma_{ij}^d=(\sigma_{ij}^{d^\circ}+\sigma_{ij}^{d'})/2\]

Further, the mean primary creep-strain, \(\epsilon_p\), is determined during every iteration:

(13)\[\epsilon_p=\epsilon_p^\circ+\dot\epsilon_p\Delta t/2\]

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

(14)\[\epsilon_p^\circ:=\epsilon_p^\circ+\dot\epsilon_p\Delta t\]

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
\(\dot \epsilon_{ss}^*\)	s-1	5.39 × 10-8

wipp Model Properties

Use the following keywords with the zone property 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, ε̇^*ss

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

[read only] accumulated primary creep strain, ε̇s

creep-rate-primary f

[read only] accumulated primary creep strain rate, εs

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.

Footnote

Read only properties cannot be set by the user. However, they may be listed, plotted, or accessed through FISH.