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Getting started with ZSOIL.PC

Chapter 6. Two-Phase Media 19

CHAPTER 6. TWO-PHASE MEDIA

Contents

6.1 One dimensional Terzaghi consolidation test 120

6.1.1 Data preparation 121

6.2 Theory 126

6.3 Stability of vertical cut with water table as SINGLE PHASE analysis 128

6.3.1 Total stress analysis (Ex_6_3_cut_wt.inp) 129

6.3.2 Effective stress analysis (Ex_6_4_cut_es.inp) 133

6.4 Stability of vertical cut with flow as TWO-PHASE analysis 135

6.4.1 Stability of vertical cut with permanent flow (Ex_6_5_cut_PermFlow.inp) 135

6.4.2 Stability of vertical cut with transient flow, uncoupled case 140

6.4.3 Stability of vertical cut with transient flow, coupled case 144

6.5 Validation test: Superficial foundation on a consolidating saturated

material 147

6.5.1 Analysis and drivers 147

6.5.2 Geometry, boundary conditions and load 147

6.5.3 Materials 147

6.5.4 Results 149

6.6 Instability due to rain 150

6.6.1 Analysis and drivers 150

6.6.2 Geometry, boundary conditions and load 151

6.6.3 Results 152

6.7 References 154

Appendix 6.1 Fluid material data required for different analysis types 155


Chapter 6. Two-Phase Media Page 120

Two-phase media behavior can be strongly or only weakly coupled. We start here with a

1-dimensional example of fully coupled flow-continuum interaction.

6.1 One dimensional Terzaghi consolidation test

A confined soil column is loaded instantaneously and load is maintained. Boundary

conditions are: atmospheric (zero pressure) at the top, no flow at bottom (for symmetry)

and lateral boundaries. The load is initially carried by an overpressure in the interstitial

fluid which diminishes with time until being completely transferred to the solid. An

analytical solution to this problem can be found in (Bowles, 1979) and in the Benchmark

section of ZSOIL manuals. Problem statement, analytical overpressure function of time

and numerical solution are shown in Fig. 6.1 (ZSOIL data:

Ex_6_1_Consolidation1D.inp).

Fig. 6.1 One-dimensional Terzaghi consolidation test, problem definition and

overpressure solution as a function of time, exact and ZSOIL solution



6.1.1 Data preparation

Drivers and analysis type for this problem are shown in Fig. 6.2. Axisymmetric or plane

strain analysis are both appropriate for this problem and yield the same solution. The

axisymmetric case is analyzed here. Problem is defined as Deformation + Flow,

Drivers are Initial State followed by Time Dependent/Consolidation. Consolidation

corresponds, we will see it later, to a fully coupled problem.

The initial state analysis generates the gravity pressure state in the sample. The

problem is linear and can be solved in 1 step. As a result, pressure varies linearly from 0

at the surface to -100 N/m2 at the bottom. After having run the analysis (Analysis/Run

Analysis), you may use Postprocessing/Graph Option/Maps, Time/Select current

time step = 0 and Settings/Graph Contents/Nodal Quantities/Pore Pressure to

visualize it, as shown in Fig. 6.6. A consolidation analysis is performed next with a time

step of 0.02 day (see units). The time step used is multiplied at each step by a factor of

1.05 which will progressively increase the size of the time-step as the consolidation

process slows down in amplitude, see Fig. 6.2.

Fig. 6.2 Analysis and Drivers for consolidation problem

Remark:

- There is a time-step lower bound, established by [Vermeer & Verruijt, 1981],

which must be satisfied in order to avoid pressure oscillations (see Table 6.1).

This time step limitation can be overcome in ZSOIL by activation of a stabilized

formulation [Truty & Zimmermann, 2006], see Fig. 6.3, but this formulation has a

computational cost.



Fig. 6.3 Activation of stabilization pressure oscillations for coupled 2-phase analysis

_______________________________________________________________________

Table 6.1 Critical time step for consolidation type computation

Δ𝑡 must satisfy: Δ𝑡 ≥ 𝛾𝑐𝑕2

𝛼𝐶𝑣

𝛼 = 1: 𝑓𝑜𝑟 𝑡𝑕𝑒 𝑡𝑖𝑚𝑒 𝑠𝑡𝑒𝑝𝑝𝑖𝑛𝑔 𝑠𝑐𝑕𝑒𝑚𝑒 𝑎𝑑𝑜𝑝𝑡𝑒𝑑 𝑖𝑛 𝐙𝐒𝐎𝐈𝐋𝛾𝑐 = 1/6 (𝑉𝑒𝑟𝑚𝑒𝑒𝑟); 𝛾𝑐 = 1/4 (𝑖𝑛 𝐙𝐒𝐎𝐈𝐋)𝑕: maximum element size𝐶𝑣 = (𝐸𝑜𝑒𝑑 𝑘)/𝛾𝐹

𝐸𝑜𝑒𝑑 = 𝐸(1 − 𝜈)/[(1 + 𝜈)(1 − 2𝜈)]

𝑘: 𝐷𝑎𝑟𝑐𝑦 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡; 𝛾𝐹 : 𝑓𝑙𝑢𝑖𝑑 𝑢𝑛𝑖𝑡 𝑤𝑒𝑖𝑔𝑕𝑡;𝐸, 𝜈:𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠

_______________________________________________________________

The critical t can easily be calculated for this particular application: 2 1 0.01

0.00254 1*1

crit c

v

ht t

C

, where h is taken where the pressure gradient is

steepest; this condition is satisfied by the chosen 0.02t



The load is applied in a single step at time step 1 and maintained, see Assembly/Load

function (Fig. 6.4), creating an instantaneous overpressure, uniform over the height of

the specimen, which can be observed in Results/Postprocessing/Maps, setting

Time/Select Reference time step to 0 and Time/Select Current time step to 0.02

or larger values, see Fig. 6.7.

Fig. 6.4 Load-time function (load q multiplier as a function of time)

Material data, under Assembly/Materials for the solid, assumed elastic, are E = 100

kN/m2, =0, =0. For the fluid, the bulk modulus (which must be high), Darcy’s

coefficients, e0 and the weight/unit volume are the data needed as shown in Fig 6.5.

As we have full saturation, and Sr are not needed.



Fig 6.5 Fluid material data specification

Results are illustrated in the next 2 figures. Fig. 6.6 illustrates the initial, gravity induced

pore pressure distribution; use Results/Postprocessing/Graph Option/Maps,

Time/Select current time step = 0 and Settings/Graph Contents/Nodal

Quantities/Pore Pressure.

Fig. 6.6 Initial, gravity induced pore pressure distribution



Fig. 6.7 illustrates the overpressure distribution as a function of time; use Time/Select

Reference time step = 0 and Time/Select Current time step = 0.02, then 80 to

visualize overpressures with respect to time t = 0.

Fig. 6.7 Overpressure distribution at time 0.02 (max) and 80 days (after consolidation)

Remark:

- Sensitivity of results to the initial void ratio is negligible



6.2 Theory

The equations of fully coupled 2-phase totally or partially saturated media are

summarized in Tables 6.2 and 6.3. Different degrees of coupling between the solid and

the fluid behavior can be distinguished and they are illustrated on the problem of

stability of a vertical cut in presence of a water table or flow. We will see that some

cases can be handled as single phase, while other require a two-phase analysis.

pw

p=γhw

hw

water table

pw

p=γhw

hwpw

p=γhw

hw

water table

sv

Fig. 6.8 Pore pressure pw and surface load sv induced by water table

The presence of a water table induces pore pressures, but also, when the water table is

located above the ground surface, a surface compression which applies as a total stress

on the surface of the two-phase medium and must be specified as boundary condition,

see Fig. 6.8. As a result effective stress at the soil surface will be “0”.

A simple two-phase (deformation+flow) test problem illustrates the case

(Ex_6_2_BoxWT.inp). Consider a box-shaped medium and the following data:

3 3: 2 ; ' : 0.2; : 22 / ; : 10 /w sat wwater table h m Poisson s ratio soil KN m water KN m

The stress state at a depth of 6 m below ground surface is, Fig. 6.9:

2

2 2

[ ] [ ] ( ) 72 /

18 / ; 80 /1

eff tot

vert vert w sat sat w w w w sat buoyant sat

eff eff

hor vert w

p h h h h h KN m

KN m p KN m

s s

s s



p= -80

hw=2m

hsat=6m

p= -80p= -80

hw=2m

hsat=6m

Fig. 6.9 Stress state Ex_6_2_BoxWT.inp

_______________________________________________________________________

Table 6.2 Equations of 2-phase media (a)

1. (σ′𝐢𝐣 + 𝐒𝐩δ𝐢𝐣),𝑗 + 𝐟𝐢 = 𝟎:𝐞𝐪𝐮𝐢𝐥𝐢𝐛𝐫𝐢𝐮𝐦

𝜎′𝑖𝑗 : 𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑠𝑡𝑟𝑒𝑠𝑠; 𝑆: 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜; 𝑝: 𝑝𝑜𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒, (+ in tension)

𝛿𝑖𝑗 : 𝐾𝑟𝑜𝑛𝑒𝑐ker 𝑑𝑒𝑙𝑡𝑎; 𝑓𝑖 : 𝑠𝑜𝑙𝑖𝑑 𝑏𝑜𝑑𝑦 𝑓𝑜𝑟𝑐𝑒

𝟐. 𝐒ε 𝐤𝐤 + 𝐯𝐤,𝐤𝐅 − 𝐜𝐩 = 𝟎: 𝐜𝐨𝐧𝐭𝐢𝐧𝐮𝐢𝐭𝐲

ε 𝐤𝐤: 𝑣𝑜𝑙𝑢𝑚𝑒𝑡𝑟𝑖𝑐 𝑠𝑡𝑟𝑎𝑖𝑛 𝑟𝑎𝑡𝑒 𝑖𝑛 𝑠𝑞𝑢𝑒𝑙𝑒𝑡𝑜𝑛; 𝑣𝑘 ,𝑘𝐹 : 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 𝑜𝑓 𝑓𝑙𝑢𝑖𝑑 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑖𝑒𝑠

𝑐: 𝑠𝑡𝑜𝑟𝑎𝑔𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 ∶ 𝑐 =𝑛𝑆

𝐾𝐹−

𝑑𝑆

𝑑𝑝

𝑛: 𝑝𝑜𝑟𝑜𝑠𝑖𝑡𝑦; 𝐾𝐹 : 𝑓𝑙𝑢𝑖𝑑 𝑏𝑢𝑙𝑘 modulus; 𝑝 : 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑡𝑖𝑚𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒𝑹𝒆𝒎𝒂𝒓𝒌𝒔:1. 𝑆𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑜𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑆 = 𝑆(𝑝)

2. 𝑃𝑜𝑟𝑜𝑠𝑖𝑡𝑦 𝑛 =𝑒

1 + 𝑒, 𝑒0𝑖𝑠 𝑢𝑠𝑒𝑟 𝑑𝑒𝑓𝑖𝑛𝑒𝑑

_______________________________________________________________________



_______________________________________________________________________

Table 6.3 Equations of 2-phase media (b) [Van Genuchten, 1980]

*

*

( )

( )

...

...

,

( )

...

i ij j

F

ij r ij

r

getfromATifneeded

get

S S p

pq k

fromATifneeded

z

k k S k

k

ij ij ij

Effective stresses

ζ' = ζ - Spδ

Darcy fl

FIGfrommanual

FIGfrommanual

ow

*

*

( )

( )

...

...

,

( )

...

i ij j

F

ij r ij

r

getfromATifneeded

get

S S p

pq k

fromATifneeded

z

k k S k

k

ij ij ij

Effective stresses

ζ' = ζ - Spδ

Darcy fl

FIGfrommanual

FIGfrommanual

ow

Sr: residual saturation

[1/m]: a measure of the thickness of transition from full to residual saturation

F: fluid specific weight

kij: Darcy coefficients

z: local altitude

______________________________________________________________

6.3 Stability of vertical cut with water table as SINGLE PHASE analysis

Either effective stress or total stress analysis can be performed. In both cases the

continuity equation is deactivated, along with certain terms in the equilibrium equation;

this is obtained by activation of Deformation under Analysis & Drivers; also, specific

weight data differ in both types of analysis.



_______________________________________________________________________

Table 6.4 Single phase (deformation) total stress analysis governing equations

(σ′𝐢𝐣 + 𝐒𝐩δ𝐢𝐣),𝑗 + 𝐟𝐢 = 𝟎:𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚

𝑠𝑝𝑒𝑐𝑖𝑓𝑦: (𝛾𝐷𝑅𝑌 )𝑏𝑖 ∶ 𝑎𝑏𝑜𝑣𝑒 𝑤𝑎𝑡𝑒𝑟 𝑡𝑎𝑏𝑙𝑒 (𝛾𝑆𝐴𝑇 )𝑏𝑖 : 𝑏𝑒𝑙𝑜𝑤 𝑤𝑎𝑡𝑒𝑟 𝑡𝑎𝑏𝑙𝑒 𝑏𝑖 :𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟

𝐒ε 𝐤𝐤 + 𝐯𝐤,𝐤𝐅 − 𝐜𝐩 = 𝟎: 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑖𝑡𝑦

𝐑𝐞𝐦𝐚𝐫𝐤: 1. 𝑃ore pressures must be explicitely specified in saturated domain,using superelements, in preprocessor2. Compression loads on submerged parts of the solid must be specified (see Fig. 6.11)

_______________________________________________________________________

6.3.1 Total stress analysis (Ex_6_3_cut_wt.inp)

6.3.1.1 Data Preparation

The problem is defined as Plane strain/Deformation (i.e. single phase). The Stability

driver is activated with SF varying from 1 to 3 with step 0.1 (Fig. 6.10).

Fig. 6.10 Driver for Ex_6_3_cut_wt.inp



6.3.1.2 Geometry/boundary conditions/initial conditions

Fig. 6.11 Geometry, boundary conditions, initial conditions and loads

Mesh and box solid boundary conditions are generated as usual. The presence of the

water table requires:

1. Specification of the initial water pressure field. Use FE Model/Initial

Condition/Initial Pressure/Create… On 4 Nodes option, and define consecutively

two subdomains (Fig. 6.11), with pressure values from -24 at nodes 1 & 4 to -64 at

nodes 2 & 3 (orange subdomain) and from 0 at nodes 1 & 4 to -64 at nodes 2 & 3

(yellow subdomain),

2. Introduction of surface loads corresponding to external water pressure through option

FE Model/Loads/Surface Load/Variable/2 nodes,

3. Introduction of different values of specific weight above and below the water surface.

6.3.1.3 Materials

Fluid data needed for this problem are: fluid pressure field and surface pressure on

submerged surface.

The introduction of different values of specific weight above (dry) and below (sat) the

water surface requires specification of 2 materials. Both correspond to the same Mohr-

Coulomb material, but with different weights (Fig. 6.12).


Chapter 6. Two-Phase Media 1

below water table: = 18

above w.t

below water table:

above w.t.: =13

Fig. 6.12 Materials for Ex_6_3_cut_wt.inp

6.3.1.4 Results

Instability is detected for a safety factor of 2.1 and last converged step of 2.0, which

leads to SF = 2.0, see Fig. 6.13.

SF = 2.0

Fig. 6.13 Displacement intensities increment and safety factor at onset of instability



Remark:

- Total stress analysis with initial state. An initial state analysis could be

activated first in order establish first the initial stress state on the undeformed

configuration, but it does not influence the final safety factor (Fig 6.14). In order

to do that, open Analysis & Drivers screen, click on Stability driver then

Insert, select Initial State driver with default parameters, exit drivers’ screen,

File/Save under name EX_6-3_cut_wt_is.inp and run. Observe that the safety

factor obtained is the same as before.

SF = 2.0

Fig. 6.14 Safety factor for total stress analysis with water table and initial state



6.3.2 Effective stress analysis (Ex_6_4_cut_es.inp)

Governing equations

_______________________________________________________________________

Table 6.5 Single phase (deformation) effective stress analysis gov. equations

(σ′𝐢𝐣 + 𝐒𝐩δ𝐢𝐣),𝑗 + 𝐟𝐢 = 𝟎: 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚

𝑠𝑝𝑒𝑐𝑖𝑓𝑦: (𝛾𝐷𝑅𝑌 )𝑏𝑖 ∶ 𝑎𝑏𝑜𝑣𝑒 𝑤𝑎𝑡𝑒𝑟 𝑡𝑎𝑏𝑙𝑒 (𝛾𝑆𝐴𝑇 − 𝛾𝑓𝑙𝑢𝑖𝑑 )𝑏𝑖 : 𝑏𝑒𝑙𝑜𝑤 𝑤𝑎𝑡𝑒𝑟 𝑡𝑎𝑏𝑙𝑒

𝑏𝑖 :𝑔𝑟𝑎𝑣𝑖𝑡𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟


_______________________________________________________________________

6.3.2.1 Data Preparation

There is no change in the Analysis/Problem definition with respect to total stress

formulation. The problem is defined as Plane strain/Deformation (i.e. single phase).

The Stability driver is activated with SF varying from 1 to 3 with step 0.1.


Effective stress analysis does not require specification of the initial water pressure field,

and specific weights must be adapted above (dry) above and below (buoyant) the water

table, see Fig. 6.15.



= 13

= 8

Fig. 6.15 Specific weights for effective stress analysis

6.3.2.3 Material

Only solid data are needed.

6.3.2.4 Results

The safety factor obtained is again 2.0.



6.4 Stability of vertical cut with flow as TWO-PHASE analysis

Either Uncoupled or Coupled analysis can be performed under mode Deformation

+Flow. In both cases the continuity equation is activated. The active terms in the

governing equations are specified in Table 6.5 for uncoupled analysis and Table 6.6 for

coupled analysis.

6.4.1 Stability of vertical cut with permanent flow (Ex_6_5_cut_PermFlow.inp)

Pore pressures are evaluated from second equation and substituted into equilibrium; this

case is uncoupled (or weakly coupled).

_____________________________________________________________________

Table 6.5 Two- phase (deformation+flow) uncoupled total stress analysis

governing equations

(σ′𝐢𝐣 + 𝐒𝐩δ𝐢𝐣),𝑗 + 𝐟𝐢 = 𝟎: 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚

𝑠𝑝𝑒𝑐𝑖𝑓𝑦: 𝛾𝐷𝑅𝑌 and 𝛾𝐹 , fi will be computed as 𝛾 = (𝛾𝐷𝑅𝑌 + nS𝛾𝐹)bi


𝑹𝒆𝒎𝒂𝒓𝒌𝒔:1. 𝑆𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑜𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑆 = 𝑆 𝑝

2. 𝑃𝑜𝑟𝑜𝑠𝑖𝑡𝑦 𝑛 =𝑒

1 + 𝑒, 𝑒0 𝑖𝑠 𝑢𝑠𝑒𝑟 𝑑𝑒𝑓𝑖𝑛𝑒𝑑

Remarks:

𝐷𝑎𝑡𝑎 𝑛𝑒𝑒𝑑𝑒𝑑 𝑓𝑜𝑟 𝑡𝑕𝑒 𝑓𝑙𝑢𝑖𝑑 𝑎𝑟𝑒: 𝛾𝐹 , 𝑡𝑕𝑒 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑤𝑒𝑖𝑔𝑕𝑡 𝑜𝑓 𝑡𝑕𝑒 𝑓𝑙𝑢𝑖𝑑, 𝑒0, 𝑡𝑕𝑒 𝑣𝑜𝑖𝑑 𝑟𝑎𝑡𝑖𝑜, 𝛽𝐹 ,𝑡𝑕𝑒 𝑓𝑙𝑢𝑖𝑑 𝑏𝑢𝑙𝑘 mod𝑢𝑙𝑢𝑠, 𝐾𝐷𝑎𝑟𝑐𝑦 , 𝑡𝑕𝑒 𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦, 𝑆𝑟 , 𝑡𝑕𝑒 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜

𝑎𝑛𝑑 𝛼 , 𝑎 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑡𝑕𝑒 𝑡𝑕𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑧𝑜𝑛𝑒 𝑓𝑟𝑜𝑚 𝑓𝑢𝑙𝑙 𝑡𝑜 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛.

Values adopted for 0 F,e are needed but have no influence in the steady flow case.

Dry must be specified for the solid.

_____________________________________________________________________



6.4.1.1 Data preparation

The initial state analysis will compute the steady-state flow pattern and hence the

pressure field, from the fluid boundary conditions, and the gravity stress state. The

stability analysis will then be performed as above, in total stress, using pressures and

saturation ratios computed during initial state analysis.

Fig 6.16 Drivers for Ex_6_5_cut_PermFlow.inp


Mesh and solid boundary conditions are defined as usual. As the two-phase flow problem

activates the continuity equation, boundary conditions for the fluid will be needed (Fig.

6.17). These include:

- Imposed pressures on both sides, compatible with water tables: use FE

Model/Boundary Conditions/Pressure BC/Create...2 Nodes option in

preprocessor, click on the two end nodes and give pressure values, negative in

compression. For instance, on the right side, ptop = 0 and pbottom = -64 kPa

- A seepage boundary condition along the vertical face of the cut, as we do not

know where the flow will hit the face and therefore the height of seepage. For

this, use the FE Model/Seepage/Create...2 Nodes option

- Also, very important: imposed pressures due to water depth at the bottom of the

cut (FE Model/Boundary Conditions/Pressure BC/Create...2 Nodes) and

corresponding surface loads on both the horizontal and vertical faces (FE

Model/Loads/Surface Load/Variable/2 Nodes).



water

tablessurface load due to

water table

seepage BCpressure BC

pressure BC water

tablessurface load due to

water table

seepage BCpressure BC

pressure BC

Fig. 6.17 Geometry/boundary conditions for EX_6-5_cut_PermFlow.inp

6.4.1.3 Materials

Two materials are needed: one for the continuum, one for seepage, see Fig. 6.18.

The continuum is a standard Mohr-Coulomb material. For two-phase media

(Deformation+Flow), body forces are computed automatically by the code, given dry

and e0, the initial void ratio.

Seepage boundary were discussed in the chapter on flow, the default value of the

permeability multiplier is used here, Kv = 1.

Data needed for the fluid are: 𝛾𝐹 = 10, 𝑒0 = 1, 𝛽F = 10𝑒10 , 𝐾𝐷𝑎𝑟𝑐𝑦 = 10𝑒−6, 𝑆𝑟 = 0 𝑎𝑛𝑑 𝛼 = 5

Remark:

- Values given to 0 F&e are needed but unimportant as flow is permanent.



Fig. 6.18 Material data for EX_6-5_cut_PermFlow.inp

6.4.1.4 Results

The displacement intensity field at failure, use Graph Option/Maps and Settings/Graph

Contents/Displacement/ABS is plotted and corresponding safety factor of SF = 1.7

are shown in Fig. 6.19.

The water pressure field with free surface identification is shown in Fig. 6.20, use Graph

option/Maps and Settings/Graph Contents/Pore pressure and scale: Min = -64

and Max = 0.



SF=1.7SF=1.7

Fig. 6.19 Safety factor and displacement intensities at failure for

Ex_6_5_cut_PermFlow.inp

free surfacefree surface

Fig. 6.20 Water pressure field for Ex_6_5_cut_PermFlow.inp



6.4.2 Stability of vertical cut with transient flow, uncoupled case

When transient inflow boundary conditions are present, either Uncoupled or Coupled

analyses are still possible. The driver associated with Uncoupled analyses is Time

dependent/Driven load, flow will be handled as a succession of steady state (or

permanent) flows; the driver associated with Coupled analyses is Time

Dependent/Consolidation, the flow inside the medium may then have a delayed

reaction, we have seen that in the chapter on flow problems. Initial state and Stability

drivers can be associated with either coupled or uncoupled analyses.

For the Uncoupled case the same governing equations as for the previous steady state

case apply, the transient behavior is computed as a sequence of steady states.

6.4.2.1 Data preparation for Ex_6_6_cut_trbc_unc.inp

Fig.6.21 Drivers for Ex_6_6_cut_trbc_unc.inp



An initial state analysis is performed first which will define the initial stress state and the

initial water conditions, see results.

A stability analysis follows which yields a safety factor of SF = 1.9.

A driven load analysis follows with time dependent flow boundary conditions; this is

performed as a succession of steady states, in this case in one step.

Stability is evaluated again at the end of the procedure and yields SF = 1.85

6.4.2.2 Geometry/Boundary conditions/Loads

Mesh and solid boundary conditions are defined as usual.

Fluid boundary conditions are input using fluid heads (Fig. 6.22), coupled with load

functions to manage the level of the water head (to do that: select boundary edges and

apply Preprocessing/FE Model/Boundary Condition/Pressure BC/Fluid head on

selected edges) and associated with seepage elements, a must since the type of

boundary condition cannot be anticipated (we have seen that in the chapter dedicated to

Flow).

LF1

6.4 4LF2

6.4

6.4

4

LF1

6.4 4

LF1

6.4 4LF2

6.4

LF2

6.4

6.4

4

Fig. 6.22 Fluid heads, associated surface pressure, load functions LF1 for left fluid head,

LF2 for right fluid head



Observe that fluid head on the right side is at level 1 m, associated with load function

(and level multiplier) number 2, constant and equal to 6.4. This could also be handled

with a fixed pressure BC. The fluid head on the left side is at level 1 m, associated with

load function 1, varying from 6.4 at t = 0 to 4 at t = 0.1, this fluid head is also

associated with a surface pressure (a total stress “head”, in red in Fig 6.22, check

Create load when using Preprocessing/FE Model/Boundary Condition/Pressure

BC/Fluid head on selected edges) induced by the water load and acting on

submerged surfaces, observe that the direction of this load must be specified (here:

opposite to external normal).

6.4.2.3 Material


Also, dry = 13 must be specified for the solid; = dry + n S F, with n = e/(1+e) will be

calculated automatically.

6.4.2.4 Results

Fig. 6.23 Initial water pressure field: use Graph Option/Maps & Settings/Graph

Contents/Pore pressure and scale Min = -64, Max = 0. Set also Time = 0



-74-74

Fig 6.24 Initial effective vertical stress state: Graph option/Maps and Settings/Graph

Contents/Continuum/Effective stresses/YY. Set also Time = 0

Stability evaluated at the end of the procedure yields SF = 1.85, the steady state

solution (with Ex_6_5_cut_PermFlow.inp) gave SF = 1.70 previously.



6.4.3 Stability of vertical cut with transient flow, coupled case

For the coupled flow case, governing equations are given in Table 6.6.

_______________________________________________________________________

Table 6.6 Two-phase (Deformation+Flow) coupled, consolidation, total stress

analysis governing equations

_______________________________________________________________

6.4.3.1 Data preparation for Ex_6_7_cut_trbc_co.inp

The only change with respect to the previous analysis is the Time

Dependent/Consolidation driver which replaces Driven load.

Fig.6.25 Drivers and units for coupled analysis: Ex_6_7_cut_trbc_co.inp

Remark:

- Time is real for consolidation analysis, we select intuitively t = 0.01.

( ), :

: , ( )

:

Re

1. ( )

2.

j

DRY F i DRY F i

equilibrium

specify and f will becomputed as nS b

continuity

s

Saturation ratio is a function of pore pressure S S p

Data needed for the fluid

ij ij i

F

kk k,k

ζ' + Spδ +f = 0

Sε + v - cp = 0

mark :

0: , , , , ,

mod , , , ,

F F

Darcy r

are the specific weight of the fluid e thevoid ratio

the fluid bulk ulus K the permeability S the residual saturation ratio

and a measureof the thickness of the transition layer from full to residual saturatio

.

n



6.4.3.2 Materials


Also, dry = 13 must be specified for the solid; = dry + n S F, with n = e/(1+e) will be

calculated automatically.

The critical t can easily be calculated for this particular application:

𝐸𝑜𝑒𝑑 = 𝐸(1 − 𝜈)/[(1 + 𝜈)(1 − 2𝜈)] = 104(1 − 0.4)/[1.4 ∗ 0.2] = 2.14𝑒4

𝐶𝑣 = (𝐸𝑜𝑒𝑑 𝑘)/𝛾𝐹 = 2.14𝑒4 ∗ 1. 𝑒−6/10 = 2.14𝑒 − 3

𝑕𝑒𝑛𝑐𝑒 𝑡𝑕𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛:Δ𝑡 ≥ Δ𝑡𝑐𝑟𝑖𝑡 = 𝛾𝑐𝑕2

𝛼𝐶𝑣=

1

4

0.64

1 ∗ 2.14𝑒−3= 74.8, which cannot be met.

6.4.3.3 Results

The chosen Δt = 0.01 violates the lower bound condition on the time step.

As a consequence the pressure field presents oscillations see Fig. 6.26. These oscillations

can be eliminated using a stabilized formulation, activated, in ZSOIL, under

Control/Finite elements (in the Advanced version of ZSOIL), see Fig. 6.27.

The safety factors obtained are SF = 1.90 after initial state, and SF = 1.45 at the end

of consolidation with Ex_6_7_cut_trbc_co_Wst.inp



Fig. 6.26 Oscillatory pressure field for Ex_6_7_cut_trbc_co.inp

Fig. 6.27 Stabilized pressure field, and stabilization activation screen activation, for

Ex_6_7_cut_trbc_co_Wst.inp



6.5 Validation test: Superficial foundation on a consolidating saturated

material

This test is taken from the benchmarks of ZSOIL: Ex_6_8_Schiffmann.inp.

6.5.1 Analysis and drivers

The initial state driver will solve the steady state flow and establish hydrostatic pressure.

The time dependent driver will activate the loading and consolidation processes. Notice

that time is real time in a consolidation analysis, meaning time units are important (Fig.

6.28).

Fig. 6.28 Drivers for Ex_6_8_Schiffmann.inp

6.5.2 Geometry, boundary conditions and load

The problem is similar to the footing presented in chapter 4 as far as geometric

preprocessing is concerned, box boundary conditions are applied on the solid (fixed at

the bottom and sliding on both sides). Zero water pressure which corresponds to the

water table is imposed on the free surface at the top, through Preprocessing/FE

Model/Boundary Conditions/Pressure BC/2 Nodes option and no boundary

conditions are needed on the three other sides, which corresponds to “no flow” (q = 0)

conditions (Fig. 6.29).

A surface load associated with load function No 1 is applied on the footing, observe that

load will be rapidly increased from 0 to 100 MN/m2 between time 0 and 0.01. The load

time function, and hence the load, will then remain constant by default till the end of the

analysis at t=0.1.

6.5.3 Materials

The soil is considered elastic and weightless in this test; but all fluid parameters must be

specified (Fig. 6.30).



load water pressure=0

box solid BC

load water pressure=0

box solid BC

Fig.6.29 Geometry, boundary conditions, load and load function

Fig 6.30 Material data



6.5.4 Results

The load is initially carried as an overpressure (difference of pressure between time tref =

0 and tcurrent), then it transfers to the solid, see Fig. 6.31.

Fluid pressure

t=0.01

t=0.1

Vertical effective stress

blue=max neg.

red=max pos.

Fluid pressure

t=0.01

t=0.1

Vertical effective stress

blue=max neg.

red=max pos.

overpressure

Fig. 6.31 Redistribution of overpressure in time

Fig. 6.32 Distribution of excess pore water overpressure at t = 0.1 s. Vertical distribution

of pw at axis (left); horizontal distribution of pw at depth 0.5 m (right)

Remark:

- The introduction of more realistic soil parameters with e.g. sat = 20 kN/m3 and

=0.3 leads to only minor changes in the overpressures.



6.6 Instability due to rain

Slope instabilities induced by rain have always happened. Simulation of such instabilities

with ZSOIL is easy but of course reliable material data are necessary if reliable results

are expected.

In this section, we will go through a simple academic example which illustrates what

data are needed for such analyses: Ex_6_9_rain2D.inp.

6.6.1 Analysis and drivers

The corresponding screen is shown in Figure 6.33. A fully coupled 2-phase consolidation

analysis is performed with repreated safety factor evaluation.

Fig. 6.33 Drivers sequence for continuous safety evaluation of slope under rain



6.6.2 Geometry, boundary conditions and load

Rain is simulated as an inflow boundary condition (see Fig. 6.34), the value of inflow,

resulting from the product of the inflow value “1” and associated load function (Fig.

6.35) varies from 0 to 0.04 [m/s] in time. To define the inflow in the preprocessor, first

select the edges on which the rain will be applied, and then use FE Model/Distributed

Fluxes/Fluid/Create...on Edge(s) option. A seepage boundary condition is needed

everywhere flow can cross a boundary (see Fig. 6.34).

Remark:

- This value needs to be smaller than the Darcy coefficient, 0.1 [m/s] here, for

inflow to be possible, otherwise the medium will perform as impermeable, surface

flow will occur and possibly erosion, but these phenomena are not simulated yet

in ZSOIL.

Fig. 6.34 Fluid boundary conditions, impermeable at bottom



Fig.6.35 Load function associated with rain inflow q

6.6.3 Results

Examining the saturation ratio distribution at time t = 45.5, Results/

Postprocessing/Maps and Settings/Graph contents/Continuum/Saturation ratio

plus Time/select current time step and choose t = 45.5, we observe a water table

identified by the fully saturated region (blue corresponds to saturation S = 0, red to S =

1), in addition, surface saturation starts increasing as inflow starts pouring through the

ground, as indicated by the top yellow layer, Fig. 6.36. The safety factor, resulting from

the successive stability analyses diminishes with time and ultimately failure occurs, as

indicated by the failure mechanism identified by displacement increment intensities (red

is max, blue is min, see Fig. 6.37), use Results/Postprocessing/Maps and

Settings/Graph contents/Nodal values/displacements/ABS plus Time/select

current time step t = 45.5 and Time/select reference time step t = 45.



Fig. 6.36 Saturation ratio distribution at time t = 45.5 (red S = 1, blue S = 0)

Fig 6.37 Displacement intensities, slope failure mechanism at time t = 45.5



6.7 References

[Bowles, 1979] Physical and Geotechnical Properties of Soil, Mc Graw-Hill.

[Schiffmann & al., 1960] R.L. Schiffmann, A.T. Chen, J.C. Jordan, An analysis of

consolidation theories, J. of Soil Mech. and Found. Div., Vol.95.

[Truty & Zimmermann, 2006] Stabilized mixed finite element formulations for materially

nonlinear partially saturated two-phase media, in Comput. Methods Appl. Mech. Engrg.

195 (2006), 1517-1546

[Van Genuchten, 1980] A closed form of the equation for predicting the hydraulic

conductivity of unsaturated soils. Soil Sciences Am. Soc., 44, 802-808.

[Vermeer & Verruijt, 1981] An accuracy condition for consolidation by finite elements.

Int. J. Num. Anal. Meth. Geomech., 5, pp 1-14.



Appendix 6.1 Fluid material data required for different analysis types

Driver/type γsoil γfluid KD βF e0 α4/ Sr

4/ Initial Cond. &

Boundary Cond.

Deformation anybut consol. γD 1/ no no no no no no fluid pressure field

Total stress γB2/ normal total stress

BC on solid

Deformation any but consol. γD 1/ no no no no no no normal total stress

Effective stress γSat2/ BC on solid

Deformation+flow anybut consol. γD5/ γF y any any 3/y y pressure+seepage

Permanent uncoupled normal total stressBC on solid

Deformation+flow anybut consol. γD5/ γF y y y y y fluid/solid head

Transient uncoupled +seepage

Deformation+flow consolidation γD5/ γF y y y y y fluid/solid head

Transient coupled +seepage

1/ above water table; 2/below water table; γD :dry, γB:buoyant, γSat: saturated γF:fluid

3/ value needed, but unimportant for permanent flow4/ required by partial saturation;

[1/m] is a measure of the thickness of the transition zone from full to residual saturation S r

5/ weight is computed as: ; /(1 )Dry FnS with porosityn e e

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