# WIPP-Salt Model

A crushed-salt constitutive model is implemented in FLAC3D/3DEC to simulate volumetric and deviatoric creep compaction behaviors. The model is a variation of the WIPP model, and is based on the model described by Sjaardema and Krieg (1987), with an added deviatoric component as proposed by Callahan and DeVries (1991).

Formulations

In the crushed-salt constitutive model, the material density, $$\rho$$, is a variable that evolves as a function of compressive volumetric strain, $$\epsilon_v$$, from the initial crushed-salt emplacement value, $${\rho}_i$$, to the ultimate intact salt density, $${\rho}_f$$. The relation between the rate of change of volumetric strain and density for use in the FLAC3D/3DEC incremental Lagrangian formulation may be outlined as follows. (Remember that, as a convention, stresses and strains are negative in compression.)

Consider a given material domain of mass, $$m$$ (which at time, $$t$$, has volume, $$V_{\circ}$$, and density, $${\rho}_{\circ}$$), and let the volumetric strain increment, $$\Delta \epsilon_v$$, correspond to a change in volume, $$\Delta V$$, and in density from $$\rho_{\circ}$$ to $$\rho$$ during the time interval, $$\Delta t$$. By virtue of mass conservation, we have

(1)$\rho_{\circ} V_{\circ} = \rho (V_{\circ} + \Delta V)$

and, by definition of volumetric strain, we obtain

(2)$\rho = {{\rho_{\circ}}\over{1 + \Delta \epsilon_v}}$

Also, from a continuum approach, we may write

(3)$\rho = {{m}\over{V}}$

and the rate-of-change of density of the given mass is

(4)$\dot{\rho} = -{{m}\over{V^2}} \dot{V}$

Using $$\dot{\epsilon_v} = \dot{V} / V$$ together with Equation (3), after some manipulation we obtain

(5)$\dot{\epsilon_v} = -{{\dot{\rho}}\over{\rho}}$

A measure of the crushed-salt compaction is given by the fractional density, $$F_d$$, defined as the ratio between actual and ultimate salt densities:

(6)$F_d = {{\rho}\over{\rho_f}}$

In the model implementation, it is assumed that the creep-compaction mechanism is irreversible (the density can only increase and cannot decrease) and bounded (no further compaction occurs after the intact salt value has been reached).

Constitutive Equations

In the crushed-salt model, elastic stress and strain rates are related by means of the incremental expression of Hooke’s law:

(7)$\dot{\sigma}_{ij} = 2G \left[\dot{\epsilon}_{ij}^e -{{\dot{\epsilon}_{kk}^e}\over{3}} \delta_{ij} \right] + K \dot{\epsilon}_{kk}^e \delta_{ij}$

where $${\delta}_{ij}$$ is the Kronecker delta.

In this expression, the bulk modulus, $$K$$, and shear modulus, $$G$$, are related to the density by a nonlinear empirical law of the form

(8)$K = K_f e^{K_1(\rho - \rho_f)}$
(9)$G = G_f e^{G_1(\rho - \rho_f)}$

where $$\rho_f$$, $$K_f$$, and $$G_f$$ are properties of the intact salt, and $$K_1$$, $$G_1$$ are two constants determined from the condition that bulk and shear must take their initial values at the initial value of the density.

It is assumed that, for density values below that of the intact salt, the total strain-rate $$\dot\epsilon_{ij}$$ can be expressed as the sum of three contributions: nonlinear elastic, $$\dot\epsilon_{ij}^e$$; viscous compaction, $$\dot\epsilon_{ij}^c$$; and viscous shear, $$\dot\epsilon_{ij}^v$$. The elastic strain-rate takes the form

(10)$\dot\epsilon_{ij}^e = \dot\epsilon_{ij} - \dot\epsilon_{ij}^c - \dot\epsilon_{ij}^v$

The viscous compaction term is based on an experimental compaction-rate law of the form

(11)$\dot{\rho}^c = -B_0 \left[ 1 - e^{-B_1 \sigma} \right] e^{B_2 \rho}$

where $$\sigma = \sigma_{kk} / 3$$ is the mean stress, and $$B_0$$, $$B_1$$, $$B_2$$ are constants determined experimentally from results of isotropic compaction tests.

The volumetric compaction strain-rate $$\dot\epsilon_v^c$$ may be derived after substitution of Equation (11) for $$\dot{\rho}$$ in Equation (5):

(12)$\dot{\epsilon_v^c} = {{1}\over{\rho}} B_0 \left[ 1 - e^{-B_1 \sigma} \right] e^{B_2 \rho}$

In the this implementation, it is assumed that volumetric compaction can only take place if the mean stress is compressive. Furthermore, a cap is assumed for the preceding expression so that no further compaction arises once the intact salt density has been reached.

Viscous Compaction

The total compaction strain rate has the expression

(13)$\dot\epsilon_{ij}^c = \dot\epsilon_v^c \left[ {{\delta_{ij}}\over{3}} - \beta {{{\sigma_{ik}^d}{\delta_{kj}}}\over{\bar{\sigma}}} \right]$

where $$\sigma_{ij}^d$$ is the deviatoric stress tensor, $$\bar\sigma = \sqrt{3J_2}$$ is the von Mises stress, and $$J_2 = \sigma_{ij}^d\sigma_{ij}^d / 2$$. In this formula, the parameter $$\beta$$ is a constant set equal to one, so that in a uniaxial compression test, the lateral compaction strain-rate components vanish.

Viscous Shear

The viscous shear strain rate corresponds to that of the WIPP model. The primary creep strain rate is the same as that given in the WIPP model, but the secondary creep strain-rate has the deviatoric stress magnitude, $$\bar\sigma$$, divided by the fractional density (Equation (6)). It has the form

(14)$\dot\epsilon_s=D\,{\left( {{\bar\sigma}\over{F_d}} \right)}^n e^{(-Q/RT)}$

where the parameters are as previously defined.

As the material approaches full compaction, the fractional density approaches one. Because a cap is introduced to eliminate further creep compaction when the intact salt density is reached, the viscous shear behavior evolves toward that of the intact salt. Note that in the framework of the WIPP model, the intact salt creep behavior is triggered by deviatoric stresses, while the volumetric behavior is elastic.

Implementation

In the this implementation of the crushed-salt model, the total stresses and strain rates are decomposed into volumetric and deviatoric components. The incremental equations governing the volumetric behavior are linearized and solved explicitly for the mean stress increment. The creep compaction strain-rate is then derived and used in the expression for the deviatoric behavior whose implementation otherwise closely follows that adopted for the WIPP model. Finally, total stresses for the step are evaluated from the updated volumetric and deviatoric components.

References

Callahan, G.D., and K.L. DeVries. Analysis of Backfilled Transuranic Waste Storage Rooms, RE/SPEC, Inc., report to Sandia National Laboratories SAND91-7052 (1991).

Sjaardema, G.D., and R.D. Krieg. A Constitutive Model for the Consolidation of WIPP Crushed Salt and Its Use in Analyses of Backfilled Shaft and Drift Configurations, Sandia National Laboratories, SAND87-1977 (1987).

wipp-salt Model Properties

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

wipp-salt
activation-energy f

activation energy, $$Q$$

bulk f

bulk modulus, $$K$$

bulk-final f

final, intact salt, bulk modulus, $$K_f$$

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, $$R$$

compaction-0 f

creep compaction parameter, $$B_0$$

compaction-1 f

creep compaction parameter, $$B_1$$

compaction-2 f

creep compaction parameter, $$B_2$$

creep-rate-critical f

critical steady-state creep rate, $$ε̇^*_{ss}$$

density-final f

final, intact salt, density, $$ρ_f$$

density-salt f

density, $$ρ$$

exponent f

WIPP model exponent, $$n$$

poisson f

Poisson’s ratio, $$v$$

shear f

elastic shear modulus, $$G$$

shear-final f

final, intact salt, shear modulus, $$G_f$$

temperature f

zone temperature, $$T$$

young f

Young’s modulus, $$E$$

compaction-bulk f (r)

creep compaction parameter, $$K$$

compaction-shear f (r)

creep compaction parameter, $$G$$

density-fractional f (r)

current fractional density, $$F_d$$

Key

• 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 two densities: density-salt $$ρ$$, and density-final $$ρ_f$$ , should be considered as internal variables for the constitutive level calculation only, which should not be confused with the real material density.