Thermal Pore Pressure Coupled Response

Note

The project file for this example is available to be viewed/run in FLAC3D.[1] The project’s main data files are shown at the end of this example.

Two simple examples are provided to illustrate a coupled thermal fluid-flow mechanical stress response. Both examples involve a single element that is saturated and undrained. In the first example the element is unconfined, and in the second the element is totally confined. The temperature in both cases is increased by a thermal heat source.

For the unconfined test, the element expands and the pore pressure decreases. The total mean stress remains at zero because the element boundaries are free.

The analytical solution for volumetric strain for the unconfined case is

(1)\[\Delta \epsilon _v\ =\ {{\alpha\ M \ \beta +\ 3K\ \alpha_t\ } \over {K +\ \alpha^2 \ M}} \Delta T\]

and for pore pressure change is

(2)\[\Delta p\ =\ M\ (- 3 \alpha\ \alpha_t +\ \beta ) \Delta T\]

where:

\(\alpha\) = Biot coefficient;

\(M\) = Biot modulus;

\(\alpha_t\) = drained linear thermal expansion coefficient;

\(\beta\) = undrained volumetric thermal expansion coefficient;

\(K\) = drained bulk modulus; and

\(\Delta T\) = temperature change.

In the confined test, the volume change is zero and the pore pressure increases due to the temperature increase. The total mean stress increases in this case because the element boundaries are fixed.

The analytical solution for total mean stress change is

(3)\[\Delta \sigma_{ii}\ =\ -\ (3K \alpha_t +\ \alpha M \beta) \Delta T\]

and for pore-pressure change is

(4)\[\Delta p\ =\ M \beta \Delta T\]

The parameters and their dimensionless values are:

density (\(\rho\))	1
bulk modulus (\(K\))	0.67
shear modulus (\(G\))	0.4
porosity (\(n\))	0.5
Biot modulus (\(M\))	4
Biot coefficient (\(\alpha\))	1
linear thermal expansion coefficient (\(\alpha_t\))	3
undrained volumetric thermal expansion coefficient (\(\beta\))	5
thermal conductivity (\(k\))	1
specific heat (\(C_p\))	1
temperature increment (\(\Delta T\))	0.4

For the unconfined test, the calculated values for \(\Delta \epsilon _v\) \((\Delta \epsilon _v = \Delta \epsilon _{11} + \Delta \epsilon _{22} + \Delta \epsilon _{33})\) and \(\Delta p\) from Equations (1) and (2) are 2.230 and -0.918, respectively. The results from FLAC3D agree with these values (< 0.01% difference).

For the confined test, the calculated values for the change in total mean stress, \(\Delta \sigma_{ii}\), and pore pressure, \(\Delta P\), are -10.412 and 8.000, respectively. Both values are within less than 0.01% of the analytical values from Equations (3) and (4).

Data File

model new
fish automatic-create off
model title 'Undrained thermo-poro-elastic unconfined and confined tests'
model large-strain off
model configure fluid thermal
program call 'fishFunctions'
; ------- main computation -------
zone create brick size 1 1 1 point 1 (1,0,0) point 2 (0,1,0) point 3 (0,0,1)
; --- mechanical model
zone cmodel assign elastic
zone property density [c_dens] bulk [c_k] sh [c_g]
; --- fluid flow model
zone fluid cmodel assign isotropic
zone fluid property undrained-thermal-coefficient [c_u_thc]
zone fluid biot on
zone gridpoint initialize biot [c_biom]
zone gridpoint initialize fluid-tension -1e10
; --- thermal model 
zone thermal cmodel isotropic
zone thermal property conductivity [c_cond] specific-heat [c_cv] ...
             expansion [c_thex]
;
zone apply source 1. range position-z -1 2
model save 'tpe_cpl-ini'

; --- unconfined test
; settings
model fluid active off
model thermal timestep fix 8e-4
; test
model mechanical active off 
model thermal active on
model step 500
model mechanical active on 
model thermal active off 
model fluid active off
model step 500
[ana_sol(0)]
model save 'tpe_cpl-unconfined'
fish list [v_strain] [delta_p]
fish list [err_vol] [err_pp]

; --- confined test
model restore 'tpe_cpl-ini'
; settings
zone gridpoint fix velocity ; <--- apply confinement
model fluid active off
model thermal timestep fix 8e-4
; test 
model mechanical active off 
model thermal active on
model step 500
model mechanical active on 
model thermal active off
model step 500
[ana_sol(1)]
model save 'tpe_cpl-confined'
fish list [delta_sxx] [delta_p]
fish list [err_sxx] [err_pp]

program return

Endnotes

[1]	To view this project in FLAC3D, use the program menu. Help ▼ Examples…   ⮡   FLAC     ⮡   Thermal       ⮡   ThermalPorePressureCoupledResponse         ⮡   ThermalPorePressureCoupledResponse.prj