Download - CHAPTER 5. UNDERGROUND FLOW Contents · Chapter 5. Underground Flow Page 105 5.1 A two-dimensional steady state flow problem Dupuit flow towards a trench is a common problem in civil

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Chapter 5. Underground Flow 04

CHAPTER 5. UNDERGROUND FLOW

Contents

5.1 A two-dimensional steady state flow problem 105

5.2 Theory 109

5.2.1 Material data 110

5.2.2 Boundary conditions 111

5.3 An example of transient flow 112

5.3.1 Preprocessing: geometry and boundary conditions 113

5.3.2 Analysis and drivers 115

5.3.3 Material data 116

5.3.4 Results 117

5.4 References 118


Chapter 5. Underground Flow Page 105

5.1 A two-dimensional steady state flow problem

Dupuit flow towards a trench is a common problem in civil engineering. Let’s consider

the example shown in Fig. 5.1. It corresponds to a steady state flow with known

upstream and downstream head.

Fig. 5.1 Flow towards a trench, geometry

ZSOIL DATA: Ex_5_1_Dupuit_flow.inp

Let’s open the input file and examine the data (Fig. 5.2). Under Control/Analysis &

Drivers, we see that the problem is defined as Plane (strain) flow, with a Time

dependent/steady state driver. A single time step is sufficient in this case since flow is

steady, but iterations (within the step) will be needed to find the free surface.

Fig.5.2 Driver for flow analysis



Under Assembly/Preprocessing we discover the mesh and boundary conditions. There

are 3 types of boundary conditions:

- “no flow” on the upper and lower edge; this is a default boundary condition,

which does not need any input.

- Imposed pressure: select FE model/Boundary Condition/Pressure

BC/Update Parameters , click on the left boundary; if parameter screen does

not pop up, use scale, change sign, then diagram show reverse, then repeat:

update parameters and click again on pressure diagram, values can now be

updated.

- The third type of boundary condition is a seepage condition, it is implemented as

a seepage element: go back to FE model/Seepage/Direction/Show a set of

black line segments will point outwards of the seepage boundary (Fig. 5.3). This

type of boundary condition is used whenever a seepage out- or inflow can be

expected. The boundary condition will impose a flow proportional to the difference

of pressure between in- and outside the boundary, if the medium is saturated

inside, and a “no flow” condition if the medium is not saturated inside.

Exit preprocessor now without saving.

Seepage boundary condition

Pressure boundary condition

no flow

Flow % pressure difference



no flow




no flow


Fig. 5.3 Flow boundary conditions



Under Assembly/Materials we observe the presence of 2 materials:

Continuum/Elastic and Seepage/Seepage. The first one is there only because we

solve the flow problem on the whole (solid) domain using partial saturation, default

values of E , νννν and γγγγ can be left as is, they have no influence.

Flow under steady state requires the permeability KDarcy (which can be oriented is space

with angle ββββ) and γγγγF, the fluid unit weight to be specified. In addition, due to possible

partial saturation, (1/αααα) the thickness of the transition from full to residual saturation,

and Sr the residual saturation must be given. Other data, default or not, can be ignored.

Fig. 5.4 Material data for Continuum

Fig. 5.5 Material data for Seepage



Seepage requires specification of Kv, a penalty factor, multiplier of K [m3/(Ns)], an

internally optimized permeability factor which will regulate the outflow through the

seepage surface. Leave default value = 1, and see theory manual for details.

Analysis/Run Analysis can be performed in 1 step, but a few iterations are needed to

find the free surface.

Results/Postprocessing/Graph Option/Maps. After running the problem, the plot

of pressure maps indicates the position of the free surface, and settings can be changed

to improve the visibility. Settings/Graph Contents, with imposed scale (Min = 0,

Max = 1), indicates a free surface which corresponds to what is expected (Fig. 5.6).

Remarks:

- The free surface does not change too significantly in this problem, when Kv is

changed, but the outflow does. When Kv is increased permeability and therefore

outflow increases.

- The free surface corresponds to zero pressure, pressures above the free surface

are positive, but their contribution to total stress in the medium is multiplied by

the saturation, so that the effective water pressure above the free surface is in

fact zero: 'totij ij ijSpσ σ δ= + and 0Sp ≅ above the free surface.

Fig. 5.6 Color maps of pore pressure



5.2 Theory

Underground Darcy flow is governed by a diffusion equation, expressed here in terms of

pressure as the nodal unknown. The equations for partially saturated Darcy flow are

given in Table 5.1.

The approach adopted in ZSOIL considers that the flow domain coincides with the soil

domain, always, saturation will ultimately define the limits of flow. The formulation

accounts for partial saturation and the free surface corresponds to the p=0 line in Fig

5.6.

In table 5.1, the continuity equation expresses that a local source induces a divergent

flow divergence and a local time-dependent pressure variation. The flow conditions can

be transient, in this case all terms of the continuity equation are active, or steady

state, in which case there is no pressure change in time and the corresponding term in

the equation, cp& , can be ignored.

_______________________________________________________________________

Table 5.1 Equations of partially saturated underground flow

_______________________________________________________________________

If the flow is confined, the medium fully saturated, and the flow condition steady, then

the problem is linear; it can be solved in a single step without iterations. But most of the

time the exact position of the free surface is unknown a priori, it is part of the problem to

be solved, the problem becomes therefore nonlinear and iterations are needed even for

the steady state flow, a fortiori within each time-step in the transient case.



5.2.1 Material data

Material data needed include Darcy’s coefficient K, and γγγγF, the fluid’s specific weight, in

the basic version; all other parameters are selected by default. In the advanced

version the fluid compressibility ββββF and two new coefficients are introduced, (1/αααα) [m], a

measure of the thickness of the transition from full saturation to residual saturation, and

Sr which is the second parameter.

_______________________________________________________________________

Table 5.2 Material data for flow problems

3

[ / ] : ' ,

[ / ] :

( . . ) :

[ ]:

m s Darcy s permeability coefficient often a scalar

N m fluid specific weight

e g free surface and

m thickness of transition from ful

ij

F

F

Steady state saturated flow

K

Steady state partially satu

K

γ

K,γ

(

rated flow

1/α)

2

[ ]

: constant

( . . ) :

[ / ] : mod

[ ] : ( . / . ),

/(1 ) :

l to residual saturation

seepage multiplier

e g free surface

N m fluid bulk ulus

void ratio vol of voids vol of solid from which

n e e po= +

r

v

Fr v

F

o

Transient partially saturated flow

S

K

K,γ ,α, S ,K

β

e

,

(1 ) / . 2

/ . ; / .w s

w s

rosity and

nS n mass unit vol of phase medium

mass unit vol of water mass unit vol of solid

ρ ρ ρρ ρ

= + − == =

_______________________________________________________________________

Material data αααα and Sr depend strongly on the granulometric structure of the medium,

values taken from literature are given in Table 5.3.

_______________________________________________________________________

Table 5.3 Flow parameters from [Yang & al. 2004]

Type of soil αααα Sr

Gravely Sand 100 0

Medium Sand 10 0

Fine Sand 8 0

Clayey Sand 1-1.7 0.23-0.09

[Yang & al. 2004] Factors affecting drying and wetting soil-water characteristic curves of

sandy soils. Canadian Geotech. J. 41, pp. 908-920, 2004.

_______________________________________________________________________



5.2.2 Boundary conditions

5.2.2.1 Pressure and flow boundary conditions

Three types of boundary conditions need to be considered: imposed pressure (in blue),

imposed flux (in black), and seepage (in red), as summarized in Figure 5.7.

Fig. 5.7 Flow boundary conditions

Pressure is prescribed at nodes where the pressure is known, it is used here to introduce

boundary conditions corresponding to the left free surface, a linear pressure distribution.

A “no flow” condition, q=0, is imposed at the bottom, this condition is natural, it is

applied “by default”, like the zero surface traction present by default in deformation

analysis.

5.2.2.2 Seepage

The seepage boundary condition, in red on the figure, enables us to handle seepage on

free surfaces, a situation which is frequently encountered, and it also serves to model

drains; it corresponds to a Darcy type equation over a pressure jump. The seepage

element has two layers, internal and external. External pressure, zero by default, is

prescribed on the external layer, and internal pressures are computed by the program on

the internal layer. The boundary condition will switch automatically from no flow to

pressure driven flow, depending on internal saturation, see Table 5.4.

_______________________________________________________________________

Table 5.4 Equation of seepage boundary condition [Aubry & Ozanam, 1988]



Referring to Figure 5.7 we observe that the water table intersects the soil surface above

the downstream water table. The position of the intersection is an unknown a priori; it

will depend on permeability, domain geometry and upstream head. We cannot therefore

anticipate which boundary condition to apply and where: pressure or “no flow”. This

situation will be handled by the “seepage surface element”. The seepage coefficient Kv

depends on the properties of the interface and is user defined. The condition will switch

automatically from “seepage” to “no flow” across the border when internal saturation is

detected to be less than 1, and vice-versa. As a rule, seepage boundary conditions will

be applied whenever boundary conditions are susceptible to change.

5.2.2.3 Fluid head

Fluid heads are a convenient way to introduce pressure boundary conditions which

change in time. Their use is illustrated in the following example

5.3 An example of transient flow

An example of transient flow with variable pressure head boundary conditions illustrated

in Fig. 5.8 is discussed next. A rectangular dam is subjected to a rapid filling followed by

a rapid drawdown on its upstream side. The analysis is carried out to determine the

time-evolution of internal pressures within the domain.

Fig. 5.8 Filling followed by drawdown behind a rectangular dam

Remark:

- You can either follow the description of the mesh and boundary conditions

creation below, or open ZSOIL data: Ex_5_2_FillDrawDown2D.inp.



5.3.1 Preprocessing: geometry and boundary conditions

Enter the geometrical preprocessor by selecting menu option

Assembly/Preprocessing. First, switch off the grid (press the G key) and the axes

(press the A key). Next, define the contour of the mesh. For this, move to Macro

Model/Point/Create/Point option, and create the following points, using the Apply

button: (0; 0) (0; 20) (10; 20) (10; 0). Leave third coordinate z = 0.

Press CTRL-F to optimize zoom with the newly created points. Now move to Macro

Model/Objects/Line/By 2 Points and define the contour of the mesh. Then click on

the Close button. Select option Macro Model/Subdomain/Create/Continuum inside

contour, and click inside of the domain. Now, select the Mesh/Create virtual mesh

method and click inside the subdomain. Structured mesh type is selected by default, as

this subdomain has four control points. Set splits to 10 x 20 and click on Create virtual

mesh.

Select Mesh/Virtual -> Real mesh method and click inside of the subdomain. Then,

press CTRL-M in order to hide the macro model, and to leave only the FE model.

Select the edges along the two vertical faces of the domain using the tool identified with

a red arrow in Fig. 5.9 and clicking successively on the top and bottom nodes of the

boundaries. Move to FE Model/Seepage/Create/On edge(s). Set seepage material

number = 2. Move to menu Selections/Unselect All.

Select option FE Model/Node/Outline/In zoom box and select all nodes belonging to

the left boundary of the domain. Move to FE Model/Boundary Condition/Pressure

BC/Fluid head on selected nodes. Set fluid head = 1 m and load function = 1 (see

Fig. 5.10). Move to menu Selections/Unselect All.

Finally move to FE Model/Boundary Condition/Pressure BC/Create…/On node

and set pore pressure = 0 to the node at the bottom-left corner of the mesh.

You may now exit the graphical preprocessor and save your work (File/Exit menu, and

answer Yes). Back in the principal Z_SOIL screen, select File/Save menu.


Chapter 5. Underground Flow 4

Fig. 5.9 Selecting edges and creating seepage elements

Fig.5.10 Pressure head definition and associated load function



5.3.2 Analysis and drivers

The solution procedure starts with an Initial State/Steady State analysis, this analysis

is performed at time t = 0, in one step, with some iterations, it will define the initial

steady state flow. This analysis is followed by a Time Dependent/Transient analysis

(Fig. 5.11), during which the upstream pressure head boundary condition is updated

using a load function (see Fig. 5.12). Time dependent pressure head boundary conditions

are a convenient way to introduce boundary conditions which vary in time; Observe that

pressure head boundary conditions are, as a rule, associated with seepage boundary

conditions. All other input are similar to the preceding example and can be followed

directly in ZSOIL.

Fig. 5.11 Drivers specification for fill-drawdown analysis

Remark:

- Observe that, for obvious reasons, pressure head boundary conditions are always

associated with seepage boundary conditions, since boundary conditions have to

be able to switch from imposed pressures to “no flow” condition q = 0.



Fig. 5.12 Load function associated with water level on the left boundary

5.3.3 Material data

Flow under transient state requires the permeability KDarcy (which can be oriented is

space), and γγγγF, the fluid unit weight, specification, in addition, due to possible partial

saturation, (1/αααα) [m] the thickness of the transition from full to residual saturation, and

Sr the residual saturation must be specified. Other data default or not, can again be

ignored.

Fig. 5.13 Material properties



5.3.4 Results

The evolution in pressure distribution as a function of time is illustrated in Figure 5.14.

Observe that the use of seepage boundary conditions allows to capture the strong

gradients along the boundaries, which result from the fast change in boundary

conditions.

Fig. 5.14 Pore pressure distribution in time

Remark:

- In Fig. 5.14, pore pressures scale has been set to min = -1 and max = 0 in order

to identify clearly the free surface, corresponding to zero pressure.



5.4 References

[Aubry & Ozanam, 1988] Free Surface tracking through non-saturated models. In

Numerical methods in geomechanics: 757-763, Swoboda (ed.), Balkema, Rotterdam

[Yang & al. 2004] Factors affecting drying and wetting soil-water characteristic curves of

sandy soils. Canadian Geotech. J. 41, pp. 908-920, 2004.

Download - CHAPTER 5. UNDERGROUND FLOW Contents · Chapter 5. Underground Flow Page 105 5.1 A two-dimensional steady state flow problem Dupuit flow towards a trench is a common problem in civil

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