FLAC3D Theory and Background • Fluid-Mechanical Interaction

Unsteady Groundwater Flow in a Confined Layer

Note

To view this project in FLAC3D, use the menu command Help ► Examples…. Choose “Fluid/UnsteadyGroundwaterFlowConfinedLayer” and select “UnsteadyGroundwaterFlowConfinedLayer.prj” to load. The project’s main data files are shown at the end of this example.

A long embankment of width L = 100 m rests on a shallow saturated layer of soil. The width (L) of the embankment is large in comparison to the layer thickness, and its permeability is negligible when compared to the permeability, k = 10-12 m2/(Pa sec), of the soil. The Biot modulus for the soil is measured to be M = 10 GPa. Initial steady-state conditions are reached in the homogeneous layer. The purpose is to study the pore-pressure change in the layer as the water level is raised instantaneously upstream by an amount H0 = 2 m. This corresponds to a pore-pressure rise of p1=H0ρwg (with the water density ρw = 1000 kg/m3 and acceleration of gravity g = 10 m/s2) at the upstream side of the embankment. Figure 1 shows the geometry of the problem:

../../../../../_images/confinedlayer-profile.png

Figure 1: Confined flow in a soil layer.

The flow in the layer may be assumed to be one-dimensional. The model has width L. The excess pore pressure, p, initially zero, is raised suddenly to the value p1 at one end of the model. The corresponding analytical solution has the form

ˆp(ˆz,t) = 1ˆz2πn=1en2π2ˆt(sinnπˆzn)

where the z-axis is running along the embankment width and has its origin at the upstream side, ˆp=pp1, ˆz=zL, ˆt=ct/L2, and c=Mk.

In the FLAC3D model, the layer is defined as a column of 25 zones. The excess pore pressure is fixed at the value of 2 × 104 Pa at the face located at z = 0, and at zero at the face located at z = 100 m. The model grid is shown in Figure 2.

The analytical solution is programmed as a FISH function for direct comparison to the numerical results at selected fluid-flow times corresponding to ˆt = 0.05, 0.1, 0.2 and 1.0. The analytical and numerical pore-pressure results for these times are stored in tables.

../../../../../_images/confinedlayer-grid.png

Figure 2: FLAC3D grid for fluid flow in a confined soil layer.

UnsteadyGroundwaterFlowConfinedLayer-Explicit.dat contains the FLAC3D data file for this problem, using the explicit formulation to obtain the solution. UnsteadyGroundwaterFlowConfinedLayer-Implicit.dat contains the data file using the implicit formulation. The comparison of analytical and numerical excess pore pressures at four fluid-flow times for the explicit solution is shown in Figure 3, and for the implicit solution in Figure 4. Normalized excess pore pressure (p/p1) is plotted versus normalized distance (z/L) in the two figures, where Tables 2, 4, and 6 contain the analytical solution for excess pore pressures, and Tables 1, 3, and 5 contain the FLAC3D solutions. The four flow times are 5 × 104, 105, 2 × 105, and 106 seconds for both explicit and implicit solutions. Steady-state conditions are reached by the last time considered. For both solution formulations, the difference between analytical and numerical pore pressures at steady state is less than 0.1%.

break
../../../../../_images/confinedlayer-explicit-pp.png

Figure 3: Comparison of excess pore pressures for the explicit-solution algorithm (analytical values = lines; numerical values = crosses).

../../../../../_images/confinedlayer-implicit-pp.png

Figure 4: Comparison of excess pore pressures for the implicit-solution algorithm (analytical values = lines; numerical values = crosses).

Data File

UnsteadyGroundwaterFlowConfinedLayer-Explicit.dat

model new
model large-strain off
model title 'Unsteady groundwater flow in a confined layer: Explicit Method'
fish automatic-create off
model configure fluid
program call 'fishFunctions'
[constants]
zone create brick size 1 1 25 point 1 (10 0 0) point 2 (0 10 0) ...
                              point 3 (0 0 [length])
; --- fluid flow model ---
zone fluid cmodel assign isotropic
zone fluid biot on
zone fluid property permeability [c_cond] 
zone gridpoint initialize biot [c_biom]
zone face apply pore-pressure [dp1] range position-z 0
zone face apply pore-pressure 0    range position-z 100
; --- settings ---
model mechanical active off
model fluid active on
;
model solve fluid time-total  5e4
zone gridpoint list fluid range position-x -0 position-y 0
[num_sol]
[ana_sol]
model solve fluid time-total 10e4
[num_sol]
[ana_sol]
model solve fluid time-total 20e4
[num_sol]
[ana_sol]
model solve fluid time-total 100e4
[num_sol]
[ana_sol]
model save 'confinedlayer-explicit'

UnsteadyGroundwaterFlowConfinedLayer-Implicit.dat

model new
model large-strain off
model title 'Unsteady groundwater flow in a confined layer: Implicit Method'
fish automatic-create off
model configure fluid
program call 'fishFunctions'
[constants]
zone create brick size 1 1 25 point 1 (10 0 0) point 2 (0 10 0) ...
                              point 3 (0 0 [length])
; --- fluid flow model ---
zone fluid cmodel assign isotropic
zone fluid biot on
zone fluid property permeability [c_cond]
zone gridpoint initialize biot [c_biom]
zone face apply pore-pressure [dp1] range position-z 0
zone face apply pore-pressure 0    range position-z 100
; --- settings ---
model mechanical active off
model fluid active on
zone fluid implicit on
model fluid timestep fix 1e3
;
model solve fluid time-total 5e4
zone gridpoint list fluid range position-x 0 position-y 0
[num_sol]
[ana_sol]
model solve fluid time-total 10e4
[num_sol]
[ana_sol]
model solve fluid time-total 20e4
[num_sol]
[ana_sol]
model solve fluid time-total 100e4
[num_sol]
[ana_sol]
model save 'confinedlayer-implicit'

;; Now do again with an attach layer
model new
model large-strain off
model title 'Unsteady groundwater flow in a confined layer: Implicit Method'
fish automatic-create off
model configure fluid
program call 'fishFunctions'
[constants]
zone create brick size 1 1 25 point 1 (10 0 0) point 2 (0 10 0) ...
                              point 3 (0 0 [length])
zone separate by-face range position-z 52.0
zone attach by-face
; --- fluid flow model ---
zone fluid cmodel assign isotropic
zone fluid biot on
zone fluid property permeability [c_cond]
zone gridpoint initialize biot [c_biom]
zone face apply pore-pressure [dp1] range position-z 0
zone face apply pore-pressure 0    range position-z 100
; --- settings ---
model mechanical active off
model fluid active on
zone fluid implicit on
model fluid timestep fix 1e3
;
model solve fluid time-total 5e4
zone gridpoint list fluid range position-x 0 position-y 0
[num_sol]
[ana_sol]
model solve fluid time-total 10e4
[num_sol]
[ana_sol]
model solve fluid time-total 20e4
[num_sol]
[ana_sol]
model solve fluid time-total 100e4
[num_sol]
[ana_sol]
model save 'confinedlayer-implicit-att'