Examples 2

Example 1 – Full depth crack
A full-depth crack allows the water to reach the top of the granular layer, saturating it. In order to model how the moisture spreads from this point throughout the layer, we keep the top left element of the structure saturated (Figure 1 to Figure 3).
Figure 1: Moisture infiltration through a full-depth crack – t=0
Figure 2: Moisture infiltration through a full-depth crack – t=4
Figure 3: Moisture infiltration through a full-depth crack – t=8
Example 2 – Pipe leak
Similarly to the case of a full depth crack, the effect of a leaking pipe can be simulated by keeping a node internal to the structure constantly saturated (Figure 4 to Figure 6).
Figure 4: Moisture infiltration from a leaking pipe – t=0
Figure 5: Moisture infiltration from a leaking pipe – t=4
Figure 6: Moisture infiltration from a leaking pipe – t=8
Example 3 – Micro-cracks
The presence of micro-cracks over a certain length of pavement could be modelled by keeping the degree of saturation of some nodes underneath the asphalt layer constant but lower than 1 (Figure 7 to Figure 9).
Figure 7: Moisture infiltration through micro-cracks– t=0
Figure 8: Moisture infiltration through micro-cracks– t=4
Figure 9: Moisture infiltration through micro-cracks– t=8

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Variably Saturated flow

Numerous numerical models have been developed for simulating water movements in materials with various degrees of saturation. Clement et al (1994) reviewed many of them underlining the most common shortcomings. In most applications, the pressure-based form of the variably saturated flow equation through homogeneous and isotropic porous media is used (Cooley, 1983; Huyakorn et al., 1984):
                                                                                        (1)
where:
S_s is the specific storage of the medium [L^-1]
Ψ is the pressure head [L]
θ is the water content
η is the porosity
K(Ψ) is the hydraulic conductivity [L T^-1]
t is time [T]
This equation describes the movement of water in soils under the following assumptions:
1. The dynamics of the air phase do not affect those of the water phase;
2. The density of water is only a function of pressure;
3. The spatial gradient of the water density is negligible.
Finite elements solutions to this approach often incur mass-balance problems in unsaturated media (Celia et al., 1990).
The finite-difference algorithm developed by Clement et al. has the advantages of being computationally simple while remaining capable of modelling a wide variety of problems, including infiltration into dry soils. In this approach, a mixed form of Equation 1 is solved by calculating the pressure heads Ψ at each node for a given time step (n+1) using a modified Picard iteration, where at each iteration (m+1) we solve a system of linear equations defined as:
                                                                                                                                                                                                                                   (2)
where
 
 
 
                                                                                                              (3)
where
Θ is the moisture content
K is the hydraulic conductivity
C is the water capacity
Δx is the horizontal step
Δz is the vertical step
Δt is the time step
S_s is the specific storage
η is the porosity
As can be seen, all the coefficients used for the current iteration (m+1) are calculated from the results of the previous iteration (m). Θ, K and C vary for each node and are functions of Ψ, therefore need to be re-calculated after each iteration.
As discussed earlier, the link between Θ and Ψ is given by the Soil-Water Characteristic Curve (SWCC). The hydraulic conductivity of the materials can be expressed as a function of the pressure heads (Van Genuchten, 1980) by:
                                                                                               (4)
Finally, by definition, the water capacity represents the slope of the SWCC and therefore can be expressed as:
                                                                                                                                               (5)
Equation 2 applies to all interior nodes, while at boundary nodes it needs to be modified to consider the appropriate boundary conditions.
In the examples shown hereafter we consider a pavement structure like the one in Figure 1, where the water table depth is 2m from the top of the unbound layer. In normal conditions this system is in equilibrium and the water content does not vary with time.
Figure 1: Variably saturated flow – pavement structure
The boundaries considered are as follows (see also Figure 2):
► Top and right-hand side nodes: Neumann boundaries (no flow);
► Bottom nodes: Dirichlet boundaries (constant moisture content);
► Left-hand side nodes: symmetry.
Figure 2: Variably saturated flow – boundary conditions

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fenv calculation 2

The inputs are organised in three hierarchical levels of detail, level 1 being the most accurate where most parameters are directly measured and level 3 being the simplest where only few parameters need to be known and are used to estimate everything else.
As noted before, the relationship between suction and degree of saturation for a soil is described by the Soil-Water Characteristic Curve (SWCC). Figure 1 shows various SWCCs for soils with different characteristics.
Figure 1: Soil-Water Characteristic Curves
The minimum inputs required (input level 3) to calculate the SWCC for a given soil are the plasticity index, PI, the percentage by weight passing the sieve #200, P200, and the effective grain size corresponding to 60% passing by weight, D60. As can be seen, these are all granulometric parameters of the material. P200 and PI are used for soils that contain a certain fraction of fines and have, therefore, a plasticity index larger than zero. They deliver the family of curves on the right hand side of Figure 1, where it is possible to see that thanks to the fine fraction the degree of saturation varies gradually with suction.
D60, instead, is used for coarser materials that have PI=0. In this case the SWCCs show a sharp change in degree of saturation with suction due to the lack of capillarity.
These curves and the determination of the parameters that define them are thoroughly described in the ME-PDG, Part 2 - Chapter 3, therefore won’t be reported in detail here.
The depth of the water table, which can be considered to vary seasonally or to be constant throughout the year, determines the values of suction at any point in the pavement, which are then used to calculate the degree of saturation by means of the SWCC. From the degree of saturation it is then possible to calculate F_u, the environmental factor for unfrozen or fully recovered material.
If the temperatures are below freezing point, the environmental factor takes the form of F_f, which has a constant value much larger than 1 since frozen materials are much stiffer.
Finally, when the soil is thawing the environmental factor becomes F_r (for recovering material) and is given a value lower than F_u. This value is at a minimum at the beginning of the recovering period, when the material is weakest, and tends gradually to F_u as the material returns to its normal, unfrozen state.
An example of how these factors appear with time for a number of nodes in a pavement is given in Figure 2.
Figure 2: Evolution of F_env with time
In Figure 2, it can be seed that on day 1 some nodes are frozen, some are thawing and the rest are unfrozen. As time passes, the coefficients for the frozen material remain constant for as long as the material stays frozen, while for the recovering material they increase up to the normal unfrozen value. Node 12, for instance, stays frozen until day 3 with F_f = 75, then starts thawing and the value of the environmental coefficient drops to 0.6. As the thawing goes on this value starts to increase and will eventually reach the value of F_u. Similarly, node 17 is already thawing at day 1 and at day 6 the thawing ends. At that point the material has completely recovered and the coefficient has reached a value of F_u of 0.9.

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Moisture - Introduction

While asphalt layers are heavily affected by temperature, the resilient and permanent behaviour of granular materials are a function of moisture content. A model, therefore, has to be provided to simulate how moisture varies in the pavement and how this affects the performance of the granular layers.

The moisture model presented hereafter is what is used in the ME-PDG, which seems to be most suitable thanks to its simplicity and flexibility. This approach takes into account the effect of moisture by multiplying the resilient modulus of the granular material at optimum moisture content by an environmental factor F_env, which can assume the three forms F_f (for frozen material), F_r (for thawing material) or F_u (for unfrozen material). These factors are function of the moisture content, which is calculated by means of a Soil-Water Characteristic Curve that defines, for a particular material, the relationship between suction and degree of saturation.

In general, water table and moisture contents are considered constant throughout the year if there is no water infiltration into the pavement layers. Nonetheless, there can be cases when the moisture content distribution changes, such as the appearance of full depth cracks or the bursting of a pipe. These are considered very traumatic events for a pavement and it can be very important to simulate how and when they might take place and the amount of damage they might cause. In order to take into account this type of events, a variably saturated flow model is also presented in this paper that allows estimating how moisture content might evolve in different case scenarios. This model is implements the two-dimensional finite difference algorithm discussed by Clement et al. and requires each time step to be solved iteratively by means of a Picard iteration, where each iteration consists in solving a system of linear equations.

Coupling this transient flow model with the F_env approach from the ME-PDG it is possible to estimate the mechanical properties of the granular layers for the critical cases discussed above, enabling the software to consider the presence of weak spots along the pavement that can lead to premature failure.

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