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Modeling saturated porous media with elasto-plastic behavior and

non-Darcy flow law considering different permeability coefficients

R. Taslimian, Assadollah NoorzadSchool of Civil Engineering, University College of Engineering, University of Tehran, Tehran,

Iran

Ali NoorzadFaculty of Water Engineering, Power and Water University of Technology, Tehran, Iran

SUMMARY:In analysis of saturated porous media under earthquake loading, the seepage velocity may probably be

sufficiently high, that viscous drag forces cannot follow from Darcy seepage law. In this situation, the flow law

governing the porous media is non-Darcy. However, this issue has not been focused in the field of soil-pore fluid

interaction and even in conventional soil mechanics. In the present research, a fully explicit dynamic finite

element method has been considered. Also the extension of Biot formulation is developed to compute the fluid

movement based on the non-Darcy relationship. The uw formulation is used for the governing equations that

describe the coupled problem in terms of solid skeleton displacement ( ) and relative fluid displacement () asprimary variables. Finally, the effect of different permeability coefficients on the liquefaction potential isdiscussed for the Darcy and non-Darcy flow laws and a comparison of the results has been made.

Keywords: Finite element method; Liquefaction; Soil-pore fluid interaction;Non-Darcy flow; permeabilitycoefficient

1. INTRODUCTION

An important issue of concern in earthquake engineering is soil liquefaction phenomenon.

Liquefaction is caused by cyclic shear strain reversals in loose cohesionless soils. As the materials

begin to densify, excess pore water pressures are generated. If these excess pore water pressures arenot allowed to dissipate, there is an accompanying reduction in effective stress. Therefore, drainage

plays a key role in the beginning of liquefaction. Fluid transient movement suppresses variations in the

excess pore pressure. Hence, drainage may prevent or delay the triggering of liquefaction. The most

significant parameter affecting drainage is the permeability coefficient. A large value of the

permeability coefficient leads to a rapid dissipation of excess pore water pressure. On the other hand,

it causes a large amount of the seepage velocity and consequently the flow turbulence restrains

dissipation of excess pore water pressure. Actually, this transient pressure gradient is governed bynonlinear-Darcy law in the turbulence flow.

The flow law in porous media was proposed by many researchers such as Chai et al. (2010) based on

empirical and numerical results. It is well known that the inertial effect in larger Reynolds number is

more significant than viscous effect in the flow law. To observe the inertial effect clearly, streamlines

Figure 1.1. Streamline for laminar flow Figure 1.2. Streamline for transition to turbulence flow

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Figure 1.3. Streamline for turbulence flow

in the vicinity of a sphere (idealized model of a sand grain) for the laminar, transition to turbulenceand turbulence flow are displayed in Figs 1.1, 1.2 and 1.3, respectively. A linear relation between the

average seepage velocity and the pressure gradient is governed in laminar flow. As the Reynolds

number increases up to a critical value, the flow will become transition to turbulence and turbulence,

and consequently the relationship will be nonlinear.

In the present research, the space discretization of governing equations with Non-Darcy flow is

presented for finite elements solution. Therefore that is required to develop the extended Biot

formulation (Biot, 1941, 1955, 1956, 1962; Biot and Willis, 1957) for Non-Darcy flow. The Biot

formulation has been extended by Zienkiewicz (1982), Zienkiewicz and Shiomi (1984), Chan (1998)

and Zienkiewicz et al. (1990, 1999) among others.

2. GOVERNING EQUATIONS

Among the fully coupled formulations for the saturated porous medium (momentum balance for the

soil-fluid (mixture), momentum balance of the fluid, effective stress within soil mass, and

conservation of fluid mass) that were originally formulated by Biot, momentum balance of the fluid is

only altered for the nonlinear flow as:

where

In which is the pore pressure for the fluid in the pores, is the density of the pore fluid, is the

body acceleration, is displacement of the solid skeleton, is average relative displacement of thefluid, is the porosity of the porous media, is the magnitude of the gravitational acceleration, isthe permeability tensor equal to in the isotropic case, is the Kronecker delta and is theaverage diameter of grains. In this study, the elasto-plastic behavior of soil during seismic event is

simulated using a generalized plasticity composing of a yield surface together with non-associated

flow law developed by Pastor and Zienkiewicz (1986) called the PZ mark III model.

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3. DISCRETIZATION PROCEDURE

A suitable discretization process is required to obtain a numerical solution (Zienkiewicz and Taylor,

1991). The finite element method is employed for the spatial discretization while the generalized

Newmark time integration scheme (Katona and Zienkiewicz, 1985) in accordance to the original work

of Newmark (1959) is utilized for the time discretization.The governing equations based on nonlinearflow can be expressed into a set of algebraic equations in space using an appropriate Galerkin

statement as:

in which

where and are the shape functions for the solid and fluid phases, respectively. B is the strainmatrix and is the elastic constitutive matirix. is an artificial damping matrix (which may be of theRayleigh damping type) applied to the solid phase. are the applied forces from bodyforces on the mixture and on the fluid, external stresses and external pressures. Using nodal integration

of

, the matrices

are all diagonal. Therefore, as it is shown in the equilibrium

equation of (3), considering non-Darcy flow rule leads to damping in the system.

Based on the governing equations that are explained above, this paper is focused on the solution of

dynamic problem where high-frequency effects predominate. The code has been written based on

these equations which is fully explicit. It is called GLADYS-2E (for ) as it was developed byChan (1998) following the work described by Zienkiewicz and Shiomi (1984). However in the present

study, the computer program of GLADYS-2E has been modified and extended for the case of non-

Darcy flow rule (i.e. ).4. VERIFICATION

In order to verify the nonlinear flow, the results of numerical analyses are compared with the non-Darcy flow model for a column of soil. Fig. 2 shows the geometry of a 5m column of saturated soil

subjected to a vertical water pressure. The soil exhibits elastic behavior and there is free drainage at

the bottom and top surface of soil column. The numerical steady state solution for this problem is

performed to obtain the velocity of seeping fluid and also pressure gradient at the element A. The

analyses are carried out for three different vertical pressures. Fig. 3 shows the three points is

converged towards the line of non-Darcy flow law.

5. FINITE ELEMENT SIMULATIONS AND RESULTS

A 2D plane strain model is considered to compare the results under the Darcy flow and non-Darcy

flow conditions for the two permeability coefficients. The finite element model depicted in Fig. 4 issubjected to the excitation shown in Fig. 5.

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vertical pressure

A5(m)

Figure 2. Finite elements of the

verification problemFigure 3. Comparison of numerical results and flaw laws

The side and bottom of fluid boundaries are taken as impermeable. The top of the finite element model

has been considered a free boundary for the solid and zero pressure for the fluid phase . In order to

model the resistance of lateral boundary layers, lateral nodes are fixed together in the finite element

model. The displacement degrees of freedom on the lateral boundary are fixed ensuring that the

movements are identical. The material parameters of PZ mark-III model for loose sand are listed in

Table 1. In this model, the time step is set to 0.0002 s.

Table 1. P-Z mark-III model parameters

Figure 4. The finite element model

The time histories of excess pore pressure for the permeability coefficient of 0.3 and 0.6 (cm/s) at the

node A is shown in Fig. 6. Dash-dotted lines represent maximum pore pressures ratio ( ) atthe node A ( is excess pore pressure, and is the initial vertical effective stress). The indicates whether the generated pore pressure reaches the condition of zero effective stress or initiation

of liquefaction state when subjected to ground excitations.

Based on the numerical results it is observed that in the case of non-Darcy flow, the generated excess

pore pressure is more, rather than for the case of Darcy flow, and also in the larger permeability

coefficient the soil tends to liquefy less due to more drainage leading to a rapid reduction of excesspore pressure.

Loose sandParameters

1.15 1.030.450.45770 (kPa)1155 (kPa)4.20.26004000 (kPa)2

0 4 (kPa)

23 (m)

10(m)

Tied together

A

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

Velocityofpercolatin

gfluid

(cm/s)

Pressure gradient (kN/m)

Darcy flow law

Non-Darcy flow law

Numerical result

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Figure 5. Time history of input accelerations

Figure 6. Time history of excess pore pressure, (a) k= 0.3 cm/s, (b) k= 0.6 cm/s

-0.15

-0.05

0.05

0.15

Acceleration(g)

Input horizontal acceleration

-0.1

-0.05

0

0.05

0.1

0 5 10 15 20 25 30

Acceleration(g)

Time (sec)

Input vertical acceleration

-2

0

2

4

6

8

10

12

Excessporepressure(kPa)

(a)

Darcy flow

Non-Darcy flow

Liquefaction limit

-2

0

2

4

6

8

10

12

0 20 40 60 80 100

Excessporepressure(kPa)

Time (sec)

(b)

Darcy flow

Non-Darcy flow

Liquefaction limit

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Time history of Pore Pressure Increment (PPI) with non-Darcy flow comparing with Darcy flow at the

node A is illustrated in Fig. 7 for the two permeability coefficients. Actually, the PPI is an index

indicating the possibility of the flow turbulence. As displayed in Fig. 7, with larger permeabilitycoefficient, due to increasing flow velocity, the upper bound of difference between the Darcy and non-

Darcy flow is more pronounced. In other words, the increase of pore pressure in simulating with non-

Darcy flow with larger permeability coefficient is more in comparison to the situation when Darcy

flow governs. Also leads to reduction in duration that the difference appears between Darcy and non-

Darcy.

Figure 7. Time history of pore water pressure increment

The distribution of excess pore water pressures ratio contours are depicted in Fig. 8 for the four

different cases namely: (a) using the Darcy flow and coefficient of permeability 0.6 (cm/s) at 33.4 s,

(b) using the Darcy flow and coefficient of permeability 0.3 (cm/s) at 53.58 s, (c) using the Non-Darcy

flow and coefficient of permeability 0.6 (cm/s) at 33.4 s, (d) using the non-Darcy flow and coefficient

of permeability 0.3 (cm/s) at 53.58 s. As shown in Fig. 8, the difference of distribution between the

contour of (a) and the contour of (c) is more than the difference of distribution between the contour of

(b) and the contour of (d). When the coefficient of permeability is increased due to increasing of flow

velocity, the flow regime will be more turbulence. Therefore as expected, the difference between the

Darcy and non-Darcy flow is more pronounced for this case.

6. CONCLUSIONS

In the present paper, effects of different permeability coefficients on the liquefaction potential of

saturated loose sand are studied for the cases of considering Darcy and non-Darcy flow laws. Based on

the numerical results it is observed that in the case of non-Darcy flow, the generated excess pore

pressure is more, rather than for the case of Darcy flow, and also with larger permeability coefficient

the soil tends to liquefy less which leads to a rapid reduction of excess pore pressure. The progressive

build up of pore pressure in modeling with non-Darcy flow with larger permeability coefficient is

more in comparison to when Darcy flow governs.

0

5

10

15

20

25

30

0 20 40 60 80 100

PPI(%)

Time (sec)

k= 0.3 (cm/s)

k= 0.6 (cm/s)

t=33.4 s

t=53.58 s

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(a) (b)

(c) (d)

Figure 8. The distribution of excess pore pressures ratio contours, (a) using the Darcy flow and k= 0.6 (cm/s) at

33.4 s, (b) using the Darcy flow and k= 0.3 (cm/s) at 53.58 s, (c) using the non-Darcy flow and k= 0.6 (cm/s) at

33.4 s, (d) using the non-Darcy flow and k= 0.3 (cm/s) at 53.58 s

AKCNOWLEDGEMENTThe authors gratefully acknowledge the contributions of Professor Andrew H.C. Chan of the University of

Birmingham (U.K.) providing the computer program (GLADYS-2E of DIANA-SWANDYNE II) and for thisinvaluable comments and guidance.

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