Pressurized Cylindrical Cavern (FLAC3D)
Problem Statement
Note
The project file for this example is available to be viewed/run in FLAC3D.[1] The project’s main data file is shown at the end of this example.
A pressurized cylindrical cavern is excavated in bedded salt.[2] The salt bed is sandwiched between 15m thick layers of a stiff elastic material (Material 2) above and below the salt bed. The farfield insitu state of stress is anisotropic. The objective of the analysis is to determine the closure of the cavern and the change in the state of stress in Material 2 along the interface with the salt.
The salt is 60 m thick and is assumed to be infinite in horizontal extent. The cylindrical cavern has a diameter of 90 m, a height of 30 m, and is located in the middle of the salt bed (i.e., a 15 m layer of salt exists between the top and bottom of the cavern and the stiff elastic layers).
The farfield insitu state of stress is (compressive stresses are negative):
\(σ_{xx}\) = 56 MPa (eastwest)
\(σ_{yy}\) = 28 MPa (northsouth)
\(σ_{zz}\) = 35 MPa (vertical)
The salt exhibits a creep behavior that is characterized by a singlecomponent power law. The rock mass properties for the salt, the stiff elastic layer (Material 2), and the overburden rock (Material 3) are summarized in Table 1.
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Salt 
Material 2 
Material 3 


Elastic Properties: 

Young’s Modulus in psi 
10e6 
30e6 
10e6 
Young’s Modulus in MPa 
68.9475 
206.8428 
68.9475 
Poisson’s Ratio 
0.3 
0.3 
0.3 
Creep Properties: 

A 
3.9 × 10^{7} MPa^{4.9} yr^{1} 

n 
4.9 
Modeling Procedure
The FLAC3D model is constructed using the 2D Extruder pane with the \(z\)axis as the axis of
rotation of the cylindrical cavern. The \(xy\)plane is a plane of symmetry because of the problem
geometry and anisotropic stress state. Figure Figure #caverngrid0
shows the FLAC3D grid for this
problem. Only oneeighth of the cavern is modeled because of the symmetry conditions. Three of the four
meshes in the figure displays various groups assigned interactively in the extruder. The last mesh in the
lower right shows the face groups assigned with the zone face skin
command.
The \(x\) = 0, \(y\) = 0, and \(z\) = 0 planes are symmetry planes and have rollered boundary conditions. The farfield boundary at \(x\) = 250 m is fixed in the \(x\)direction, and the boundary at \(y\) = 250 m is fixed in the \(y\)direction. A vertical stress of 35 MPa is applied at the top boundary (\(z\) = 250 m). The model is subjected to an initial insitu stressstate of \(σ_{xx}\) = 56 MPa, \(σ_{yy}\) = 28 MPa, \(σ_{zz}\) = 35 MPa. A gravitational stressgradient is not considered in this analysis.
The cavern is to be excavated over a period of 0.1 year. The excavation is simulated by reducing the normal pressure exerted along the cavern periphery from the insitu stress value to a value of 7 MPa after 0.1 year. The pressure is then held constant at 7 MPa for a total time of 1 year.
This analysis is divided into three stages. In the first stage, the model is brought to a preexcavation stress state that is considered to be compatible with the farfield insitu stresses and the creep behavior assumed for the salt. In the second stage the cavern is excavated by using a table, load, to reduce the pressure at the cavern wall linearly from the initial stress state to 7 MPa in 0.1 year. Finally, in the third stage, the creep response of the model is followed for a total time of 1 year.
Discussion
The salt will creep under the imposed anisotropic stress field, even without the presence of an excavation. This is because the creep rate in the powerlaw formulation is a function of deviatoric stress. For the first stage, in order to approach a stress state that is reasonably compatible with both the imposed stress conditions and the creep behavior, initial isotropic stresses of 35 MPa are specified for the zones representing the salt bed. As the contours of \(σ_{xx}\) in Figure 2 show, there is a stress discontinuity across the salt and Material 2 interface.
In the second stage, the effect of the excavation on the stresses within the stiff layer, Material 2, can be seen in Figure 3. This figure includes a plot of \(σ_{xx}\) contours and principal stress tensors on a vertical plane through the cavern at 0.1 year. The plot for \(σ_{xx}\) contours and principal stresses after 1 year of creep is also shown in Figure 4. The development of higher horizontal stresses across Material 2 directly above the excavation is observed in Figure 3; this stress is reduced with creep, as evidenced in Figure 4.
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Figure 5 shows a history of displacements of three locations around the cavern periphery after 1 year. Figure 6 and Figure 7 show displacement vectors on a vertical plane and a horizontal plane, respectively, through the center of the cavern. These three figures indicate that the greatest displacements are in the horizontal (\(x\) and \(y\)) direction. After 1 year, a maximum of approximately 0.41 m displacement occurs in the horizontal (negative \(x\) and \(y\)) direction, and approximately 0.05 m displacement occurs in the vertical (negative \(z\)) direction.
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Data File
PressurizedCylindricalCavern.dat
;
; Creep of Cylindrical Cavern
; anisotropic insitu stress field
; (units: m, MPa, year)
;
model new
model largestrain off
fish automaticcreate off
model title 'Pressurized cylindrical cavern'
model configure creep
; create model (quartersymmetry), created interactively
; and exported from State Pane
; Also assigns groups to salt layers, mat1 and mat2, cavern,
; and cavern walls + floor
program call 'geometry'
zone generate fromsketch
zone face skin ; Label model boundaries
; Assign material model and properties
zone cmodel assign power range group 'salt1' or 'salt2'
zone cmodel assign elastic range group 'mat2'
zone cmodel assign elastic range group 'mat3'
; material properties  E = 10e6 or 30e6 psi
; converted to MPa by factor 0.00689476
zone property young [10e6*0.00689476] poisson 0.3 ...
range group 'salt1' or 'salt2'
zone property constant1=3.9e7 exponent1=4.9 ...
range group 'salt1' or 'salt2'
zone property young [30e6*0.00689476] poisson 0.3 range group 'mat2'
zone property young [10e6*0.00689476] poisson 0.3 range group 'mat3'
; boundary and initial conditions
zone face apply velocitynormal 0 range group 'East' or 'West' ...
or 'North' or 'South' or 'Bottom'
zone face apply stressnormal 35 range group 'Top'
zone initialize stress xx 56 yy 28 zz 35
zone initialize stress xx 35 yy 35 zz 35 range group 'salt1' or 'salt2'
model creep active off
model solve ratiolocal 1e4
zone gridpoint initialize velocity (0,0,0)
model save 'caverninit'
; excavate cavern
zone null range group 'cavern' group 'salt1'
table 'load' add (0,35) (0.1,7) (1.0,7)
zone face apply stressnormal 1 table 'load' time creep ...
range group 'wall' or 'floor'
; histories
model history name 'ctime' creep timetotal
zone history name 'xclose' displacementx position (45, 0, 0)
zone history name 'yclose' displacementy position ( 0,45, 0)
zone history name 'zclose' displacementz position ( 0, 0,15)
;
zone gridpoint initialize displacement (0,0,0)
model creep active on
model creep timetotal = 0.0
model creep timestep starting 1.0e5
model creep timestep minimum 1.0e5
model creep timestep automatic
; solve for 0.1 yr excavation
model creep timestep maximum 5.0e5
model solve timetotal 0.1
model save 'cavern010'
; solve for 1 yr of creep
model creep timestep maximum 1.2e4
model solve timetotal 1.0
model save 'cavern100'
Endnotes
Endnote
⇐ Grid Generation for Intersecting Tunnels (FLAC3D)  Prediction of Borehole Closure in a Salt Formation (FLAC3D) ⇒
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