Category WATER IN ROAD STRUCTURES

The finite difference method doesn’t postulate explicitly any specific shape of the unknown field. As we are concerned with partial differential equations, exact deriva­tives are replaced by an approximation based on neighbouring values of the un­known (still denoted as p):

P,+1 – Pi-1

W/, 2L

where the subscript i denotes the cell number and L denotes the cell size. For an orthogonal mesh, such derivatives are easily generalised to variable cell dimen­sions. However, non-orthogonal meshes pose problems that are highly difficult to solve and are generally not used. Boundary conditions have then to be modelled by the juxtaposition of orthogonal cells, giving a kind of stepped edge for oblique or curved boundaries. Similarly, local refinement of the mesh induces irreducible global refinement. These aspects are the most prominent drawbacks of the finite difference method compared to the finite element one. On the other hand computing time is generally much lower with finite differences then with finite elements.

The basic idea of the finite element method is to divide the field to be analysed into sub-domains, the so-called finite elements, of simple shape: e. g. triangles, quadrilaterals with linear, parabolic or cubic sides for two-dimensional analysis. In each finite element, an analytically simple equation is postulated for the vari­able to be determined, i. e. the coordinate or displacement for solid mechanics, and the fluid pressure, temperature or concentration for diffusion problems. In order to obtain continuity, the unknown variable field has to be continuous at the limit between finite elements. This requirement is obtained thanks to common values of the field at specific points, the so-called nodes, which are linking the finite elements together. The field values at nodal points are the discretised problem unknowns.

For most solid mechanics and diffusion problems, isoparametric finite elements seem to be optimal (Zienkiewicz et al., 1988). The unknown field l (here represent­ing a set of displacements in all directions) may then be written, for solid mechanics cases1 as:

l = Nl g, n)xL L = 1, m (11.14)

where m is the number of nodes in the model. This unknown field, l_, then depends on the nodal unknowns xL (not only referring to the x – direction) and on shape functions Nl, which, themselves, depend on the isoparametric coordinates, f, n, defined on a reference, normalised, space. The strain rate and the spin may then be derived thanks to Eq. 11.5, the stress rate is obtained by Eqs. 11.3 and 11.4 and is time integrated. Eventually, equilibrium (Eq. 11.1) has to be checked.

For scalar diffusion or advection-diffusion problems, the unknown field, p, rep­resenting a general pressure (which could be pore pressure, u (the use here), tem­perature, T, or concentration, C, by appropriately changing the notation) may then be written:

p = Nl d, n) Pl L = 1, m (11.15)

Where p depends on the nodal unknowns, pL, and on the shape functions, Nl. Then Darcy’s fluid velocity and the storage changes may be derived thanks to Eqs. 11.7 and 11.8 (or, respectively, Eqs. 11.9 and 11.10). No time integration is required here. Finally, the balance equation (Eq. 11.6) has to be checked.

The finite element method allows an accurate modelling of the boundary con­dition, thanks to an easily adapted finite element shape. Internal boundaries of any shape between different geological layers or different solids can be modelled. Spe­cific finite elements for interfacial behaviour or for unilateral boundaries have also been developed (e. g. Charlier & Habraken, 1990). Variations of the finite element

size and density over the mesh are also easy to manage, with the help of modern mesh generators.

11.2.1 Introduction

An approximated solution of most problems described by a set of partial differential equations may be obtained by numerical methods like the finite element method (FEM), the discrete element method (DEM), the finite difference method (FDM), the finite volume method (FVM), or the boundary element method (BEM). For the problems concerned here, the most commonly used methods are the finite element and the finite difference procedures. Commonly, non-linear solid mechanics is better solved using the finite element method. Boundary element methods have strong limitation in the non-linear field. Finite difference methods are not easy to apply to tensorial equations (with the exception of the FLAC code, developed by Itasca).

Diffusion and advection-diffusion problems are often solved by finite difference or finite element methods. Some finite difference (or finite volume) codes are very popular for fluid flow, like e. g. MODFLOW, TOUGH2 (Pruess et al., 1999) for aquifer modelling or ECLIPSE (Schlumberger 2000) for oil reservoir modelling. These codes have been developed over a number of years and possess a number of specific features allowing users to take numerous effects into account. However, they suffer from some drawbacks, which limit their potential for modelling coupled phenomena. Therefore only a little information will be included here concerning finite difference approaches.

In the preceding section, differential equations were given for three types of prob­lems. In order to solve these equations, we need to define boundary and initial con­ditions. Classical boundary conditions may be considered: imposed displacements or forces for solid mechanics problems and imposed fluid pressures, temperatures, concentrations or fluxes for diffusion and advection-diffusion problems.

However, it may be useful to consider much more complex boundary conditions. For example, in solid mechanics, unilateral contact with friction or interface be­haviour is often to be considered.

When coupling phenomena, the question of boundary conditions increases in complexity and has to be discussed.

Transport of pollutant or of heat in porous media is governed by a combination of advection and diffusion (Chapter 6, Section 6.3.1). The advection phenomenon is related to the transport (noted as a flow f ) of any substance by a fluid flow,

described by the fluid’s velocity, f df/:

fad. = Cf (11.11)

The substance concentration, C, is generally supposed to be small enough not to influence the fluid flow. In porous media, due to the tortuosity of the pore network, and due to the friction, advection is always associated with a diffusion characterised by the diffusion-dispersion tensor, D. Therefore, the total flux of substance is:

Lad. = Cf – iD9’C <1U2>

Balance equations and storage equations may be written in a similar way to the one for diffusion problems Eqs. 11.6, 11.8 and 11.10.

Compared to the diffusion constitutive law, Eqs. 11.7 and 11.9, here an advection term appears which doesn’t depend on the concentration gradient, but directly on the concentration. This is modifying completely the nature of the equations to be solved. Problems dominated by advection are very difficult to solve numerically (Charlier & Radu, 2001). In order to evaluate the relative advection effect, it is useful to evaluate the Peclet’s number, Pe, which is the ratio between the diffusive and advective effects:

fluid j-

Pe = fdiff— (11.13)

2 Dh V ‘

where L is an element dimension and Dh is the hydrodynamic dispersion coefficient (see Chapter 6, Section 6.3.1).

Thermal conduction exchanges (Chapter 4) in solids and diffusion of contaminants (Chapter 6) are modelled by similar diffusion equations.

The balance equation is written:

дг/г + Q — S (11.6)

where / represents a flux of fluid or heat, Q represents a sink term and S represents the storage of fluid or of heat. When modelling a diffusion problem with the finite element method, the most often used formulation is based on fluid pore pressure, u, or on temperature, T.

Then the Darcy’s law for fluid flow in porous media gives the fluid flux (this equation has been presented in a slightly different form in Chapter 1 (Eq. 1.2) and in Chapter 2, Eqs. 2.15 and 2.16):

/і — — (дги + дг pgz) (11.7)

д

with the intrinsic permeability K (possibly depending on the saturation degree), the dynamic viscosity, д, the density, p, the fluid pressure, u, the altitude, z, and the gravitational acceleration, g. The fluid storage term, S, depends on the saturation degree, Sr, and on the fluid pressure (see Chapter 2, Section 2.7):

S — /n(u, Sr)

For thermal conduction one obtains Fourier’s law – see Eq. 4.1, rewritten here as:

fi = – kdiT (11.9)

with the conductivity coefficient, X. The heat storage (enthalpy) term depends on the temperature, T (Chapter 4, Section 4.4):

£ = fn(T) (11.10)

Diffusion of contaminant follows a similar law (Chapter 6, Section 6.3.1). The diffusion problem is non-linear when:

• the permeability depends (directly or indirectly) on the fluid pore pressure;

• the fluid storage is a non-linear function of the pore pressure;

• partial saturation occurs;

• the conductivity coefficient depends on the temperature; and

• the enthalpy is a non-linear function of the temperature.

When the storage term is considered, the time dimension of the problem has to be addressed.

On the one hand, solid mechanics can be modelled on the following basis. The equilibrium equation is:

дг <jjj + Pj = 0 (11.1)

Where Pj is a member of P, the vector of volume forces, ji;- is a member of j, the Cauchy stress tensor, and д represents the spatial partial derivative operator:

di = (11.2)

dXi

The stress tensor is obtained by the time integration of an (elastic, elasto-plastic or elasto-visco-plastic) constitutive equation (see Chapter 9, Section 9.4.2; Laloui, 2001; Coussy & Ulm, 2001):

Jij = fn(j, e, Z) (11.3)

where (jij is the stress rate, e is the strain rate and Z is a set of history parameters (state variables, like e. g. the preconsolidation stress). In the most classical case of elasto-plasticity, this equation reduces to:

jч = hi

where Eejkl is a member of the elasto-plastic constitutive (stress-strain) tensor, Eep. Most constitutive relationships for geomaterials are non-linear ones and not as pre­viously introduced in Eq. 9.1 in a linear version.

When modelling a solid mechanics problem with the finite element method, the most commonly used formulation is based on displacements that make up the vector l or on actual coordinates that make up the vector x. If one considers only small strains and small displacements, the strain rate reduces to the well-known Cauchy’s strain rate:

Sij — 2 (djlj + djli) (11.5)

The time dimension is not addressed for solid mechanics problems, except when a viscous term is considered in the constitutive model. Generally, the time that ap­pears in the time derivatives inEqs. 11.3, 11.4 and 11.5 is only aformal one.

When trying to replicate in-situ behaviour by computational techniques, a number

of different physical phenomena (Gens, 2001) need to be considered, including:

• The non-linear solid mechanics and especially granular unbound or bound mate­rial mechanics: we consider the relations between displacements, strains, stresses and forces within solids. The material behaviour is described by a constitutive model, which can take into account elasto-plasticity or elasto-visco-plasticity;

• The fluid flow within porous media: fluid can be a single phase of various na­tures (water, air,…) or it can be an association of two fluids, leading to unsat­
urated media (water and air,…). In the second case, partial saturation leads to permeability and storage terms depending on the saturation degree or on the suction level, involving non-linear aspects;

• The thermal transfers within porous media: conduction is the leading process in a solid (in the geomaterial matrix), but convection can also occur in the porous volume, as a consequence of the fluid flow. Radiation transfer could also occur inside the pores, but it will be neglected here. Conduction coefficients and latent heat may depend on the temperature; and

• The pollutant transport or any spatial transfer of substance due to the fluid flow: the pollutant concentration may be high enough to modify the densities, involv­ing non-linear effects.

All these problems are non-linear ones, and can be formulated with sets of partial

differential equations. However, only three types of differential equations have to be

considered, concerning respectively:

i) solid mechanics;

ii) diffusion; and

iii) advection-diffusion problems.

Robert Charlier[25], Lyesse Laloui, Mihael Brencic, SigurSur Erlingsson, Klas Hansson and Pierre Hornych

Abstract Different physical problems have been analysed in the preceding chapters: they relate to water transfer, to heat transfer, to pollutant transfer and to mechanical equilibrium. All these problems are governed by differential equations and boundary conditions but analytical solutions are, in general, unobtainable because of the com­plex interaction of the various aspects which are always present in real-world situ­ations. In such circumstances, numerical modelling can give a valuable alternative methodology for solving such highly coupled problems. The first part of this chapter is dedicated to a brief statement of the finite element method for highly coupled phenomena. In the second part, a number of numerical simulations are summarised as an illustration of what could be done with modern tools. The chapter shows that it is possible to achieve realistic results although, at present, some simplification is often required to do so.

Keywords Finite element method ■ multi-physics coupling ■ partial differential equation ■ examples of applications

This Chapter presents in-situ and laboratory experimental techniques used to de­scribe mechanical behaviour of pavement materials (soils and aggregates) at dif­ferent saturation stages. Repeated triaxial load testing can be applied to obtain both stiffness characteristics and assessments of the ability of the material to with­stand accumulation of permanent deformation during cyclic loading. For unsatu­rated soils, in addition to mechanical variables, a moisture/suction control should be added, which can be imposed by several techniques as explained in the chapter. A brief presentation of the model parameters and tests needed for model calibration was introduced with particular reference to the modelling approaches described in Chapter 9. Evaluations of pavement structural capacity based on deflection mea­surements with non-destructive testing equipment have also been presented. Finally, some examples of laboratory and in-situ measurement are shown.

Based on experimental results presented above it can be concluded as follows:

i) Bearing capacity and unconfined compressive strength decrease with increase in moisture content.

ii) The permanent axial strain increases when water content approaches to wOPM.

iii) Resilient modulus decreases as the water content approaches to wOPM. The resilient modulus of soils decreases by a factor 4-5, for realistic (temperate) seasonal variation of moisture contents.

iv) Reduction in resilient modulus with suction depends also on the grading coeffi­cient: lower grading parameters (i. e. more fine particles) yields larger modulus reductions as saturation is approached.

v) In-situ experimental data confirms that resilient moduli decrease with decrease in suction. The soil-water characteristic curve depends on the grading of tested material meaning that the modulus-suction relationship is likely to be very soil- specific.