COMSOL 6.4 - Transient Response of a Shallow Foundation on Unsaturated Soil

Transient Response of a Shallow Foundation on Unsaturated Soil

This example is an extension of the model Shallow Foundation on Unsaturated Soil. In the referred example, sudden changes in suction occur due to changes in the underground water level. In this example, the deformation of the clay stratum due to footing pressure and pore suction is analyzed after time-dependent changes in the boundary conditions. Two distinct natural events — a few days of rainfall and a few days of evaporation — are considered, after which the soil is subjected to a footing pressure.

The model is inspired by the example presented in Ref. 1. The Extended Barcelona Basic model (BBMx), which includes suction in its constitutive relationship, is used to demonstrate the behavior of a settlement on an unsaturated soil layer.

Model Definition

In this example, a 6 m wide and 3 m deep soil layer is studied. A 1 m wide footing is placed on top of the layer. The footing is modeled as a boundary load applied on the surface. The settlement analysis is carried out for an increasing footing pressure. Initially, in the first analysis, the ground water level is constant (3 m below the surface), and the footing pressure gradually increases to its final value. Once the final footing pressure is applied, a rainfall event with an infiltration rate of 10 cm/day is considered. The second scenario considers evaporation at a rate of 2 mm/day.

Figure 1: Dimensions, boundary conditions, pressure load, and flow rate for the unsaturated soil.

Soil Properties

The properties of the soil are given in Table 1.

Table 1: Material properties
Property	Variable	Value
Poisson’s ratio	ν	0.2
Soil density	ρs	1743 kg/m3
Water density	ρw	1000 kg/m3
Dynamic viscosity of water	μw	0.001 Pa.s
Swelling index	κ	0.006
Swelling index for changes in suction	κs	0.008
Compression index at saturation	λ	0.22
Compression index for changes in suction	λs	0.123
Slope of critical state line	M	1.24
Weight parameter	w	0.4
Soil stiffness parameter	m	60 kPa
Plastic potential smoothing parameter	bs	10
Tension to suction ratio	ks	0.6
Initial yield value for suction	sy	100 kPa
Initial void ratio	e0	0.666
Reference pressure	pref	10 kPa
Initial consolidation pressure	pc0	50 kPa
Initial porosity		0.4
Saturated hydraulic conductivity	ksat	1 m/day
Fitting parameter	α	2 1/m
Residual degree of saturation	Sres	0.23
Degree of saturation at full saturation	Ssat	1

The Richards’ Equation interface requires additional material properties, which are derived from Table 1 and based on data from Ref. 1.

Constraints and Loads

•

The soil layer is supported by a rigid and perfectly rough base. Apply a fixed constraint on the lower horizontal boundary.

•

Use a roller boundary condition on the left vertical boundary, symmetry on the right.

•

A hydraulic head of 0 m is initially assigned to the lower horizontal boundary in the Richards’ Equation interface.

•

The infiltration and evaporation rates are applied in the Richards’ Equation interface using the node. Infiltration is represented by a positive quantity in the node, and evaporation by a negative quantity.

•

The gravity load is applied using the node. The pore pressure in the saturated region of the layer is applied using an node.

•

A boundary load is applied on top of the soil layer to model the weight of the footing, see Figure 1.

Results and Discussion

Figure 2 shows the pressure head in the clay stratum after five days of rainfall (left) and five days of evaporation (right). The pressure head after five days of rainfall shows less variation across the layer due to the high infiltration rate. In contrast, with a low evaporation rate, the pressure head is almost linear with layer position and changes slightly at the top surface.

Figure 2: Pressure head after infiltration and evaporation.

The results presented in Figure 2 are emphasized in Figure 3 and Figure 4. With a high infiltration rate (Figure 3), the pressure head after 5 days of rainfall is within the range of 0–1 m instead of the initial range of 0–3 m. The pressure head after five days of rainfall matches the Gardener steady state solution quite well. Figure 4 shows changes in the pressure head with a low evaporation rate, with the opposite trend as compared to Figure 3. Evaporation decreases the water content, which lowers the pressure head. Due to the low evaporation rate, changes in the pressure head are much smaller as compared to the changes due to the rainfall event, and mostly affect the region near the top surface. Again, the pressure head after five days of evaporation matches the Gardener steady state solution.

The distribution of the von Mises stress due to footing pressure is shown in Figure 5. Five days of rainfall reduces the suction and increases the pore pressure. The water in the pores can bear more load, which reduces the stresses in the soil skeleton, see Figure 6. Vice versa, after five days of evaporation, the stress in the soil skeleton increases slightly due to the reduced amount of water in the pores, see Figure 7.

The footing pressure versus settlement after the rainfall and evaporation events is shown in Figure 8 and Figure 9, respectively. The collapse of the soil under the footing is expected with increasing infiltration. Collapse also occurs after evaporation (although barely visible in the figure). The vertical displacement of the top layer due to rainfall and evaporation is different (see Figure 10); with rainfall there is a large collapse under the footing, whereas the collapse due to evaporation is not that drastic.

The changes in pore suction due to rainfall and evaporation is shown in Figure 11. Pore suction decreases during rainfall event, which results in positive volumetric strains in the soil layer (expansion of pores). Pore suction increases when considering evaporation, which results in a negative volumetric strain in the soil (compression of pores).

Figure 3: Pressure head versus elevation for rainfall event.

Figure 4: Pressure head versus elevation for evaporation event.

Figure 5: von Mises stress due to footing pressure.

Figure 6: von Mises stress due to footing pressure after five days of rainfall.

Figure 7: von Mises stress due to footing pressure after five days of evaporation.

Figure 8: Footing pressure versus settlement after five days of rainfall.

Figure 9: Footing pressure versus settlement after five days of evaporation.

Figure 10: Vertical displacement of top layer of soil due to rainfall and evaporation.

Figure 11: Changes in pore suction due to rainfall and evaporation.

Notes About the COMSOL Implementation

The model setup represents a unidirectional multiphysics coupling where changes in pore pressure affect the soil deformation, but changes in the deformation have no effect on the pore pressure.

The Cam-Clay family of soil models, like the MCC or BBMx models, do not define any stiffness at zero stress. To avoid numerical instabilities, prescribe an initial mean stress equal to the reference pressure at zero strain.

Reference

1. A.A. Abed and P.A. Vermeer, “Numerical Simulation of Unsaturated Soil Behavior,” International Journal of Computer Applications in Technology, vol. 34, no. 1, 2009.

Geomechanics_Module/Soil/settlement_analysis_transient

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Right-click and choose .

In the tree, select > .

Right-click and choose .

Click

In the tree, select > .

Click

Geometry 1

Model parameters are available in the appended text file.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file settlement_analysis_transient_parameters.txt.

Footing Pressure

In the toolbar, click

and choose > .

In the window for , type Footing Pressure in the text field.

Locate the section. In the text field, type F_P.

In the table, enter the following settings:


t	f(t)
0	0
1	60

Locate the section. In the table, enter the following settings:


Argument	Unit
t	1

In the table, enter the following settings:


Function	Unit
F_P	kPa

Definitions

Variables 1

In the window, expand the > node.

Right-click and choose .

Model variables are available in the appended text file.

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file settlement_analysis_transient_variables.txt.

Create half of the geometry by exploiting symmetry.

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 3.

Add a line segment to represent the foundation.

Line Segment 1 (ls1)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the text field, type 2.5.

In the text field, type 3.

Locate the section. From the list, choose .

In the text field, type 3.

Click

Materials

Add a that contains information about the fluid and porous matrix properties together with the structural properties.

Porous Material 1 (pmat1)

In the window, under right-click and choose > .

Continue with setting up the physics. After that, the software automatically detects which material properties are required.

Richards’ Equation (dl)

Unsaturated Porous Medium 1

In the window for , locate the section.

From the list, choose .

Porous Matrix 1

In the window, click .

In the window for , locate the section.

From the list, choose . In the Se text field, type Se.

In the θl text field, type S_res*phi0+Se*(phi0-S_res*phi0).

In the Cm text field, type Cm.

In the κr text field, type k_rel.

In the θr text field, type S_res*phi0.

Initial Values 1

In the window, under > click .

In the window for , locate the section.

Click the button.

Hydraulic Head 1

In the toolbar, click

and choose .

Select Boundary 2 only.

Symmetry 1

In the toolbar, click

and choose .

Select Boundary 5 only.

Infiltration

In the toolbar, click

and choose .

In the window for , type Infiltration in the text field.

Select Boundaries 3 and 4 only.

Locate the section. In the P0 text field, type U_in.

Evaporation

Right-click and choose .

In the window for , type Evaporation in the text field.

Locate the section. In the P0 text field, type -U_out.

Continue with setting up the interface.

Solid Mechanics (solid)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Extended Barcelona Basic Model (BBMx)

In the toolbar, click

and choose .

In the window for , type Extended Barcelona Basic Model (BBMx) in the text field.

Locate the section. From the list, choose .

From the Γ(θ) list, choose .

Find the subsection. From the M list, choose .

In the s0 text field, type InitSuction.

In the s text field, type Suction.

In the pref text field, type pref.

In the pc0 text field, type pc0.

Select Domain 1 only.

External Stress 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the pA text field, type PorePressure.

In the pref text field, type 0.

From the αB list, choose . In the associated text field, type 1.

Go to the material node and assign the required material properties.

Materials

Porous Material 1 (pmat1)

In the window, under > click .

In the window for , locate the section.

In the εp text field, type phi0.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Plastic potential smoothing parameter	bb	bB	1	Barcelona Basic
Swelling index for changes in suction	kappaSwellingsu	kappa_s	1	Barcelona Basic
Density	rho	rhos	kg/m³	Basic
Compression index at saturation	lambdaComp0	lambda	1	Barcelona Basic
Poisson’s ratio	nu	Nu	1	Critical state model
Weight parameter	wb	wB	1	Barcelona Basic
Initial yield value for suction	sy0	syB0	Pa	Barcelona Basic
Slope of critical state line	M	MB	1	Critical state model
Soil stiffness parameter	mb	mB	Pa	Barcelona Basic
Initial void ratio	evoid0b	e0	1	Barcelona Basic
Swelling index at saturation	kappaSwelling0	kappa	1	Barcelona Basic
Tension to suction ratio	kb	kB	1	Barcelona Basic
Permeability	kappa_iso ; kappaii = kappa_iso, kappaij = 0	k	m²	Basic
Compression index for changes in suction	lambdaCompsu	lambda_s	1	Barcelona Basic