INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech. 2017; 41:682–706 Published online 14 September 2016 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2572
Consolidation in spatially random unsaturated soils based on coupled flow-deformation simulation Y. Cheng1, L.L. Zhang2,*,†, J.H. Li3, L.M. Zhang4, J.H. Wang5 and D.Y. Wang6 1
State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Department of Civil Engineering, Shanghai Jiaotong University, 800 Dongchuan Road, Shanghai, China 2 State Key Laboratory of Ocean Engineering, Civil Engineering Department, Shanghai Jiaotong University, 800 Dongchuan Road, Shanghai, China 3 Department of Civil and Environmental Engineering, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen, China 4 Department of Civil and Environmental Engineering, Hong Kong University of Science and Technology, Hong Kong 5 State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, 800 Dongchuan Road, Shanghai, China 6 Department of Geotechnical Engineering, Southwest Jiaotong University, Chengdu, China
SUMMARY This paper integrates random field simulation of soil spatial variability with numerical modeling of coupled flow and deformation to investigate consolidation in spatially random unsaturated soil. The spatial variability of soil properties is simulated using the covariance matrix decomposition method. The random soil properties are imported into an interactive multiphysics software COMSOL to solve the governing partial differential equations. The effects of the spatial variability of Young’s modulus and saturated permeability together with unsaturated hydraulic parameters on the dissipation of excess pore water pressure and settlement are investigated using an example of consolidation in a saturated-unsaturated soil column because of loading. It is found that the surface settlement and the pore water pressure profile during the process of consolidation are significantly affected by the spatially varying Young’s modulus. The mean value of the settlement of the spatially random soil is more than 100% greater than that of the deterministic case, and the surface settlement is subject to large uncertainty, which implies that consolidation settlement is difficult to predict accurately based on the conventional deterministic approach. The uncertainty of the settlement increases with the scale of fluctuation because of the averaging effect of spatial variability. The effects of spatial variability of saturated permeability ksat and air entry parameters are much less significant than that of elastic modulus. The spatial variability of air entry value parameters affects the uncertainties of settlement and excess pore pressure mostly in the unsaturated zone. Copyright © 2016 John Wiley & Sons, Ltd. Received 24 May 2013; Revised 3 July 2015; Accepted 27 July 2016 KEY WORDS:
unsaturated soil; hydro-mechanical coupling, consolidation; spatial variability; random field; Monte Carlo simulation
1. INTRODUCTION In geotechnical engineering, deformation of soil is often coupled with the flow of water in soil pores. When a vertical load is applied on a layer of soil, it causes time-dependent soil deformation together with dissipation of excess pore water pressure. Such a process is called consolidation and is a typical process of coupled flow and deformation. There is a wide variety of engineering problems in which the consolidation of soils is important. For example, excessive consolidation settlement of a
*Correspondence to: L. L. Zhang, Professor, State Key Laboratory of Ocean Engineering, Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Civil Engineering Department, Shanghai Jiaotong University, 800 Dongchuan Road, Shanghai, China. † E-mail:
[email protected] Copyright © 2016 John Wiley & Sons, Ltd.
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structure founded on soil can lead to functional problems or even failure of the structure. Distresses in highway pavements or earth dams are often caused by large consolidation settlement. In the above problems, soils can be unsaturated. For example, earth fill dams and highway embankments are usually at an unsaturated state initially because the fill materials are compacted to a certain in-place density at a controlled moisture content. A large portion of land surface is covered with residual soils that are unsaturated and the ground water table is at some depth below the ground surface. Therefore, it is necessary to include both saturated and unsaturated soils in consolidation problems. Some researchers have investigated the coupled flow and deformation in unsaturated soils [1–9]. Wong et al. [1] illustrated numerical solutions to coupled consolidation for an unsaturated soil column. Kim [3] presented a coupled numerical model for land deformation in partially saturated soils because of surface loading. Following the formulation developed by Dakashanamurthy et al. [4], Conte [5] presented a solution for coupled consolidation in unsaturated soils in which both the air phase and water phase are continuous. Wu et al. [6] derived an analytical solution to 1D coupled water infiltration and deformation using a Fourier integral transform. Qin et al. [7] and Shan et al. [8] developed an analytical solution to one-dimensional consolidation in unsaturated soils. Khoshghalb and Khalili [9] proposed a meshfree algorithm for the fully coupled analysis of flow and deformation in unsaturated poro-elastic media. In these previous studies, the soil is assumed to be homogenous. In reality, soil properties vary spatially even within the same soil layer as a result of deposition and post-deposition processes [10–13]. The effect of inherent spatial variability of soil properties on the performance of geotechnical works has received considerable attention in recent years. Paice et al. [12], Fenton and Griffiths [14] combined random field theory with the finite-element method to investigate the effect of randomly and spatially correlated soil stiffness on total and differential settlement of spread footings. El Gonnouni et al. [15] assessed the influence of soil inherent variability on soil displacement from underground urban construction. Tartakovsky et al. [16], Le et al. [17], and Mousavi Nezhad et al. [18] analyzed unsaturated flow in heterogeneous soils with spatially distributed uncertain hydraulic parameters. Cho [19], Le [20], Li et al. [21, 22], and Jiang et al. [23, 24] assessed the effect of inherent variability on slope stability. Srivastava et al. [25], Zhu et al. [26], and Santoso et al. [27] simulated permeability as a spatially correlated log-normally distributed random variable and studied its influence on water flow and slope stability. Meftah et al. [28] developed a 3D numerical tool to analyze the effect of random distributions of key parameters on the behavior of a partially saturated medium. Mousavi Nezhad et al. [29] simulated contaminant transport through unsaturated soils considering the effects of soil heterogeneity using a stochastic finite element (FE) model. Ching and Phoon [30] compared the spatial average discretization method and the midpoint discretization method and discussed tolerable maximum element size to achieve a prescribed accuracy in random field FE analysis. For consolidation related problems, Ferronato [31] developed a three dimensional hydromechanical coupled FE model of the Northern Adriatic basin and addressed the impact of spatial variability of rock compressibility on the land subsidence using stochastic simulation. Teatini et al. [32] investigated the land uplift caused by seawater injection considering the spatial random heterogeneity of permeability. Badaoui et al. [33] and Huang et al. [34] conducted probabilistic analyses of uncoupled and coupled soil consolidation in saturated soils, respectively. The results indicated that the consolidation process is very slow for relatively small values of the vertical scale of fluctuation [33]. The use of the average degree of consolidation defined by excess pore pressure to predict settlement may lead to misleading results which could either overestimate or underestimate the time rate of settlement [34]. From the above literature review, the coupled flow and deformation in unsaturated soils and spatial variability of soil properties have been addressed separately. Very limited research has been carried out on the effects of the spatial variability of deformable unsaturated soils. Recently, Le et al. [35] investigated rainfall-induced differential settlement of a strip foundation on an unsaturated soil with spatially varying values of either preconsolidation stress or porosity. They adopted the elasto-plasic Barcelona Basic Model (BBM) [36] for representing the mechanical behavior of unsaturated soils and used a fully coupled hydro-mechanical code for performing the numerical solutions. The present paper will expand the knowledge on the coupled flow and deformation in spatially random unsaturated soil, focusing on the problem of consolidation. The objective of this study is to investigate the effect of spatial varied soil properties on the dissipation of excess pore pressure and the Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2017; 41:682–706 DOI: 10.1002/nag
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Y. CHENG ET AL.
corresponding settlement and compare the stochastic analysis with the deterministic analysis. Spatial variability of soil properties are simulated using the covariance matrix decomposition method based on the random field theory. The simulated spatial random soil properties are imported into an interactive software environment COMSOL to solve the governing partial differential equations (PDEs) for coupled flow and deformation in spatially random unsaturated soils. The numerical modeling and Monte Carlo simulations (MCSs) are integrated to investigate the consolidation of a spatially random unsaturated soil using an example problem. In the illustrated example, the influences of the coefficients of variation and scales of fluctuations of Young’s modulus and saturated permeability, together with the parameters in soil water characteristic curve and permeability function, on the surface settlement and dissipation of excess pore water pressure are presented and discussed.
2. THEORY OF COUPLED FLOW AND DEFORMATION IN UNSATURATED SOIL 2.1. Governing equations Mathematical equations for flow and deformation in unsaturated soils can be derived from physical principles. A number of formulations have been presented in the literature [37–46]. The main differences of various formulations are: (i) stress state variables, (ii) constitutive models for soil skeleton, (iii) volume change of soil skeleton, (iv) changes in the storage of unsaturated soil, and (v) hydraulic properties of unsaturated soil. Here the formulation presented by [3], which is basically along the line of Biot’s (1941) poro-elastic theory of consolidation and uses effective stress and pore water pressure (negative value for soil suction) as two independent stress state variables, is adopted. It is assumed the air phrase is continuous and the air pressure is atmospheric. The constitutive model for unsaturated soil is elastic. The governing equations for the coupled flow and deformation in unsaturated soils are as follows [3]: dS w ∂h ∂ ∂uk ¼0 (1) þ nS w βw γw þ αc S w ∇½u þ n dh ∂t ∂t ∂xk ∂ ∂ui ∂uj ∂uk þλ δij ðαc S w γw hÞe δij þ ½nS w ρw þ ð1 nÞρs e gi ¼ 0; μ þ ∂xj ∂xi ∂xk ∂xj
i; j ¼ x; y; z (2)
where u denotes the water flow velocity tensor; h is the pore water pressure head; n is the porosity; Sw is the degree of water saturation (0 ≤ Sw ≤ 1); dSw/dh is the specific water saturation capacity; γw is the unit weight of water (9.806 × 103 N/m3); βw is the compressibility of water (5.0 × 1010 m2/N); αc is Biot’s hydromechanical coupling coefficient or the effective stress coefficient (0 ≤ αc ≤ 1); uk is the displacement in the k direction; xk is the coordinate in the k direction; t is time; λ and μ are often referred to as Lamé’s constants, with λ = Eν/(1 + ν)(1 2ν) and μ = G = E/2(1 + ν), in which G is the shear modulus, E is the Young’s modulus, and ν is the Poisson’s ratio; δij is Kronecker’s delta; ρs is the solid density; ρw is the density of water; gi is the component of gravitational acceleration g in the i direction; the superscript, e, denotes the incremental values of physical quantities, that is the difference of a soil variable between the current state and the initial state. For example, he = h h0 is the incremental pressure head in which h and h0 are the current and initial pressure heads, respectively. According to Darcy’s law, the water flow velocity, u, can be written as: u ¼ k∇H
(3)
where k = ksatkr is the effective permeability tensor, kr is the relative permeability (0 ≤ kr ≤ 1), and ksat is the saturated permeability tensor; H = h + z is the total hydraulic head, z is the vertical axis and elevation head. Under a plane strain condition, the above governing equations, Eqs (1) and (2), can be simplified as follows: ∂ ∂ ∂ ∂ dS w ∂h ∂ ∂u ∂v þ nS w βw γw k x ðh þ y Þ þ k y ðh þ yÞ ¼ n þ αc S w þ (4) dh ∂x ∂x ∂y ∂y ∂t ∂t ∂x ∂y Copyright © 2016 John Wiley & Sons, Ltd.
Int. J. Numer. Anal. Meth. Geomech. 2017; 41:682–706 DOI: 10.1002/nag
COUPLED FLOW AND DEFORMATION IN SPATIALLY RANDOM UNSATURATED SOILS
∂ ∂u ∂u ∂u ∂v ∂ ∂u ∂v e G þ þλ þ ðα c S w γw h Þ þ G þ ¼0 ∂x ∂y ∂x ∂x ∂y ∂y ∂y ∂x
685
(5)
∂ ∂v ∂v ∂u ∂v ∂ ∂u ∂v G þ þλ þ ðαc S w γw hÞe þ G þ þ ½nS w ρw þ ð1 nÞρs e g ¼ 0 ∂y ∂y ∂y ∂x ∂y ∂x ∂y ∂x (6) where kx and ky are the hydraulic conductivities in the x and y directions, respectively; u, v denote the displacements in the x and y directions, respectively. It should be noted that for an unsaturated soil, pressure head h is negative and both the degree of saturation, Sw, and the permeability are dependent upon h. The function between the degree of saturation and the negative pressure head (Sw h) or between the volumetric water content θw and soil suction ψ (i.e. θw ψ) is defined as the soil-water characteristic curve (SWCC). The function of permeability k with respect to soil suction (k–ψ) or negative pressure head (k–h) is defined as the permeability function. In this study, the functions of the degree of saturation and the relative permeability of the unsaturated soil are defined based on the van Genuchten model [47] as follows: 8 1 S wr
