Application of the generalised-α method in dynamic ...

Computer Methods and Recent Advances in Geomechanics – Oka, Murakami, Uzuoka & Kimoto (Eds.) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-00148-0

Application of the generalised-α method in dynamic analysis of partially saturated media J. Ghorbani, M. Nazem & J.P. Carter Australian Research Council Centre of Excellence for Geotechnical Science and Engineering, The University of Newcastle, NSW, Australia

ABSTRACT: The main aim of this study is to apply a robust time integration technique for simulating the behaviour of unsaturated soils subjected to dynamic loads. The governing equations of the three soil phases and their interactions are derived based on the mass conservation law, linear momentum balance and energy conservation of each phase in an isothermal environment. In addition, a number of experimental equations are employed to represent the hydraulic conductivity and the drainage characteristics of the soil, such as the suction-saturation relationship and the dependency of the hydraulic conductivity on suction. The global system of equations is then solved by using an implicit time-stepping algorithm based on the Generalised-α integration scheme. The accuracy of the numerical model and the finite element code is verified by comparing the numerical results with results obtained by an analytical solution.

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INTRODUCTION

During past decades, modelling multiphase flow through porous media has attracted researchers’ attention due to its wide range of application in the areas of underground natural resource recovery, waste storage, petroleum reservoirs, mitigation of liquefaction, earth dams and earthquake engineering. The equations regarding the interaction between the solid and fluid were first developed for quasi-static conditions by Biot (1941), and were then extended to dynamic problems by Biot (1956) Later, the introduction of so-called “mixture theory” by Truesdell (1957) and the subsequent extensions of this theory by other researchers provided the opportunity to establish a new basis for Thermo-Hydro-Mechanical-Chemical analysis of porous media which includes the behaviourof multiphase fluids. Among the numerical approaches for this type of analysis, the finite element method has received the greatest attention. Following the work of Li & Zienkiewicz (1992), a number of finite element simulations appeared in the literature, including Schrefler & Xiaoyong (1993), Schrefler et al. (1995), Li et al. (1999), Rahman & Lewis (1999), Ehlers et al. (2004), and Khoei & Mohammadnejad (2011). To enhance the convergence properties of this class of problems, the Generalized-α integration scheme by Chung & Hulbert (1993 ) is selected to integrate the global equations over time. For coupled analysis of geotechnical problems, Kontoe et al. (2008) showed that this method has a superior performance compared to the WBZ method proposed by Wood et al. (1980),

the HHT by Hilber et al. (1977), and the well-known Newmark (1959) method. In the Generalized-α method, different terms inthe equation of motion are calculated at different times within each time step. Similar to the HHT and WBZ methods, the Generalized-α method uses Newmark’s equations for the displacement and velocity variations. Also, two algorithmic parameters αm and αf are borrowed from the HHT method and the WBZ methods, respectively. A discussion of the properties of the Generalized-α method in a non-linear regime is given by Erlicher et al. (2002) Nazem et al. (2009) andSabetamal et al. (2014), who also employed the Generalized-α method to solve a few highly nonlinear problems in geomechanics. 2

FINITE ELEMENT DISCRETISATION OF THE GOVERNING EQUATIONS

The weak form of equations governing the behaviour of partially saturated mediaisbased on the mass conservation law and the linear momentum balance of solid, liquid, and gas phases in an isothermal environment (Li & Zienkiewicz 1992). The finite element discretisation can be achieved by considering the following shape functions for displacements u, pore water pressures Pw , and suction Pc (Khoei and Mohammadnejad 2011), such that

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Table 1.

General material parameters (Li & Schanz 2011)

where Nu , Nw and Nc are shape functions for displacement, pore water pressure and suction, respectively. Based on the these interpolation functions the following system of equations is achieved

where B is the strain-displacement matrix, and C is the damping matrix. The definitions of the matrices and vectors are given in Appendix 1. 3 TIME INTEGRATION In the Generalised-α method it is assumed that all variables are known at time t and it is desired to find the values of unknowns at time t + t, where t denotes the size of time step. In this method the inertia forces in the momentum equation are measured at time t + (1 − m )t, whereas the other terms are measured at time t + (1 − f )t, where αm and αf are two algorithmic parameters that should satisfy the following relations In order to solve the nonlinear system of coupled equations in (4)-(6) at each time step, an iterative procedure based on the Newton-Raphson’s scheme is employed to guarantee the residual values are less than a prescribed tolerancein each iteration. This procedure is presented in Appendix 2. 4 in which ρ∞ is the desirable value of spectral radius at infinity, and β, γ, and θ are the Newmark’s parameters. The unconditional stability of the method is guaranteed whenever

In addition, the displacements and velocities are updated by Newmark’s equation according to (Newmark, 1959)

IMPLEMENTATION AND VALIDATION

The Generalised-α method described in the previous section has been implemented into SNAC, a finite element code developed by the Geotechnical group at the University of Newcastle. In order to validate the code, an elastic and unsaturated sand column subjected to a dynamic load is investigated numerically. An analytical solution for this problem was presented by Li & Schanz (2011). The length of the one-dimensional soil column is 10 m. It is assumed that the side walls of the column are rigid, frictionless and impermeable. The bottom boundary is impermeable and fixed. At the top of the column, a vertical pressure, σ = 1.0 Pa, is prescribed (See Figure 1). The material properties are presented in Table 1. In this study, a 6-noded isoparametric element for the displacement is coupled with a sub-parametric 3-noded element for pore pressure and suction (See Figure 2).

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Figure 3. Comparison of finite element results of vibration at the top of column in saturation degrees of 0.9 and 0.5 with results obtained by analytical solution by Li & Schanz (2011).

Figure 1. Applied pressure at top of the column.

Figure 2. The coupled 6-noded element used in analysis.

4.1

Figure 4. Suction variation at the bottom of column fora saturation degree of 0.8.

Soil-water characteristic curve

The following equation proposed by Brooks & Corey (1964) is used for the soil-water characteristic curve in this problem

where Pc is the capillary pressure, P d is the nonwetting fluid entry pressure, ν is the pore size distribution index, and Se denotes the effective wetting degree of fluid saturation given by

where Srw is the residual wetting fluid saturation and Sra is the non-wetting fluid entry saturation. The relative permeabilities of the water phase krw and air phase kra are obtained from

The derivative of the fluid saturation with respect to the capillary pressure is expressed as

The pore size distribution index ν is set to 1.5, the residual water saturation Srw is set to 0, the air entry saturation Sra is set to 1, and P d is equal to 50 kPa. The vertical displacements at the top of the column versus time obtained by the finite element method presented here as well as by the analytical approach given by Li & Schanz (2011) are plotted in Figure 3. There is a good agreement between the results obtained by the two approaches, and the amplitude and wave length are more or less identical in both simulations. However, as Li & Schanz (2011) pointed out, the magnitude of the displacements is independent of s2 because the effective bulk modulus does not change for the range of water saturations considered here. The change of suction with time at the bottom of the column is also presented in Figure 4.

4.2

Consolidation analysis

The intrinsic permeability of the soil was increased 1000 times to facilitate a faster completion of the consolidation phase. The dissipation of excess pore water pressure at the bottom of the column for different degrees of saturation is shown in Figure 5.As expected, the results indicate that the excess pore water pressure dissipates at a faster rate as the degree of saturation is reduced. Figure 6 also demonstrates the displacement at the top of column after applying the load and during the consolidation phase.

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Figure 5. Dissipation of excess pore water pressure with time at the bottom of column in different degrees of saturation.

Figure 6. Displacement at the top of column versus time.

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CONCLUSION

The Generalized-α method for dynamic analysis of partially saturated porous media was formulated and has been implemented into a finite element framework to enhance the convergence properties of the dynamic analysis of unsaturated soils. The results obtained by application of this technique were compared to those obtained by a semi-analytical approach and good agreement was achieved. In addition, the Generalized-α shows good performance in simulating the consolidation phase following the early stage of dynamic excitation. REFERENCES Biot, M. A. 1941. General theory of three-dimensional consolidation. Journal of Applied Physics 12(2): 155-164. Biot, M. A. 1956. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low-frequency range. The Journal of the Acoustical Society of America 28: 168. Brooks, R. H. & Corey, A. T. 1964. Hydraulic Properties of Porous Media. Hydrology Papers 3.

Chung, J. & Hulbert, G. 1993. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method. Journal of applied mechanics 60(2): 371–375. Ehlers, W., Graf, T. & Ammann, M. 2004. Deformation and localization analysis of partially saturated soil. Computer methods in applied mechanics and engineering 193(27): 2885–2910. Erlicher, S., Bonaventura, L. & Bursi, O. S. 2002. The analysis of the Generalized-α method for non-linear dynamic problems. Computational mechanics 28(2): 83–104. Hilber, H. M., Hughes, T. J. & Taylor, R. L. 1977. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering & Structural Dynamics 5(3): 283–292. Khoei, A. & Mohammadnejad, T. 2011. Numerical modeling of multiphase fluid flow in deforming porous media: A comparison between two-and three-phase models for seismic analysis of earth and rockfill dams. Computers and Geotechnics 38(2): 142–166. Kontoe, S., Zdravkovic, L. & Potts, D. M. 2008. An assessment of time integration schemes for dynamic geotechnical problems. Computers and Geotechnics 35(2): 253–264. Li, P. & Schanz, M. 2011. Wave propagation in a 1-D partially saturated poroelastic column. Geophysical Journal International 184(3): 1341–1353. Li, X., Thomas, H. & Fan, Y. 1999. Finite element method and constitutive modelling and computation for unsaturated soils. Computer methods in applied mechanics and engineering 169(1): 135–159. Li, X. & Zienkiewicz, O. 1992. Multiphase flow in deforming porous media and finite element solutions. Computers & structures 45(2): 211–227. Nazem, M., Carter, J. P. & Airey, D. W. 2009. Arbitrary Lagrangian–Eulerian method for dynamic analysis of geotechnical problems. Computers and Geotechnics 36(4): 549–557. Newmark, N., M. 1959. A Method of Computation for Structural Dynamics. Journal of the Engineering Mechanics Division 85(3): 67–94. Rahman, N. A. & Lewis, R. W. 1999. Finite element modelling of multiphase immiscible flow in deforming porous media for subsurface systems. Computers and Geotechnics 24(1): 41–63. Sabetamal, H., Nazem, M., Carter, J. P. & Sloan, S. W. 2014. Large deformation dynamic analysis of saturated porous media with applications to penetration problems. Computers and Geotechnics 55(0): 117–131. Schrefler, B. & Xiaoyong, Z. 1993. A fully coupled model for water flow and airflow in deformable porous media. Water Resources Research 29(1): 155–167. Schrefler, B. A., Zhan, X. & Simoni, L. 1995. A coupled model for water flow, airflow and heat flow in deformable porous media. International Journal of Numerical Methods for Heat & Fluid Flow 5(6): 531–547. Truesdell 1957. Sulle Basi Della Thermomeccanica. C. Rend. Lincei 22: 33–38. Wood, W. L., Bossak, M. & Zienkiewicz, O. C. 1980. An alpha modification of Newmark’s method. International Journal for Numerical Methods in Engineering 15(10): 1562–1566.

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7 APPENDIX 2

6 APPENDIX 1

In the Generalised-α method acceleration term at time t+(1-αm )t is defined by

while the other terms at time t+(1-αf )t are obtained by

In addition, the time derivatives of pore water pressure and suction are obtained b

The vector of residuals at two consecutive iterations are related to each other by

where k g and k w are hydraulic conductivities of gas and water phases which are defined by

where J is the Jacobian matrix and is defined by

in which k int is the intrinsic permeability matrix of the soil, and µ denotes the dynamic viscosity of each fluid phase. Sw is the degree of saturation, ρw and ρg are densities of water and gas. The average density of mixture ρ is obtained by:

and after simplification

where n is the porosity of soil K, Kw and Kg are the bulk modulus of the solid grains, water and gas. Biot coefficient, α, is determined by

where Kt is the bulk modulus of porous medium and w ˙ α represents Darcy velocity of each fluid. Finally, mis for a three-dimensional case

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