The determination of model dimension for an embankment to study soil atmosphere interaction with Finite Element Method

In order to study the interaction between soil and atmosphere, Héricourt experimental embankment was instrumented with different equipments for collecting the meteorological data, runoff, soil temperature and volumetric water content, etc. The corresponding water flux and heat flux boundary conditions of the Hydro-thermal coupled model of unsaturated soil can be calculated based on the energy and mass balance. However, in Finite Element Method, a suitable model dimension of experimental embankment needs to be set in advance, which will diminish the effect of bottom and lateral boundaries. Therefore, in this paper, four different dimension models are calculated respectively and compared in two cases with different boundary conditions. For Case 1, top water flux boundary condition is set as infiltration and evaporation happening on the soil surface alternately. In Case 2, top heat flux boundary condition is assumed based on daily solar radiation value. Comparing the results of four model dimensions, temperature and volumetric water content distributions in four dimensions lead nearly consistent results. Therefore, with consideration of computational time and accuracy, dimension 2 or 3 is suggested for Hydro-thermal study of Héricourt embankment


Introduction
For numerical calculation, it is important to set suitable dimension to describe the real situations of problem.In literature, a typical dimension value of embankment model for mechanical numerical calculation has been suggested by Azizi [1].Rujikiatkamjorn et al. [2] proposed a 3D model based on Finite Difference Method in Hexagonal Wire Mesh by FLAC 3D, which was applied to simulate the vertical settlement, excess pore pressures and lateral displacements of an embankment on soft Bangkok clay.With an experienced dimension, Yue [3] simulated the thermal regime of embankment and berm in permafrost based on 2D embankment model with finite element method.Kalliainen et al. [4] studied the effect of embankment dimensions on the mechanical behavior of railway track, showing that the embankment dimensions had a slight effect on the accumulated permanent deformations of the embankment on a very stiff subgrade.However, on more flexible subgrade, the rate of accumulated permanent deformation is remarkable and the effect of embankment dimension on accumulated permanent deformation of the embankment is also clearly visible.In the study of mechanical behavior of    Besides, the soil above groundwater table is in unsaturated state whose suction is expressed as function U 0 = -y (Fig. 4), where y is soil depth from top surface of embankment.

Case 2
In the second case, based on the real solar radiation, heat boundary condition is assumed as the top boundary condition, positive 45 W/m 2 s for 6 hours in day time and negative 15 W/m 2 s for 18 hours during night time (Fig. 9a et Fig. 9b).
The six points of two sections are nearly consistent in four different dimensions (Fig. 10 et Fig. 11).As shown in

Conclusions
In order to study the interaction between atmosphere and with consideration of less demanding of computational time and memory, dimension 2 or 3 are suggested to be adopted in the further study.
is subgrade stiffness that has a remarkable effect on its deformation behaviour, not the embankment dimension.With the experiences in literature, in most cases, model dimension was set for hydraulic or mechanical study, which asks for more attention on the variation of soil water content, strain and stress over time.However, for Hydro-thermal coupled model of unsaturated soil, no related suggestion of model dimension was proposed for numerical calculation.In this paper, in order to get an ideal model dimension for field Héricourt embankment, four different dimensions are compared.By adding water flux and heat flux boundary conditions respectively, the soil volumetric water content and temperature variations are calculated by Finite Element Method.Based on the comparison results, with considering computational accuracy and time consumed, the suitable dimension value is suggested to be applied in further study about the interaction between atmosphere and unsaturated soil.

2 Material and method 2 . 1
Soil parameters Studied Héricourt embankment was constructed by lime/cement treated silt.Relevant experimental work was conducted in order to get its parameters.The thermal conductivity of treated silt is set to be linear to volumetric water content as proposed by De Vries [5].Fig. 1 presents the variation of thermal conductivity as function of volumetric water content.Based on the test results, the fitting curve is expressed as, data shown in Fig. 2, the fitting water retention curve based on Van Genuchten model could be drawn.In the same model, accompanied with the saturated hydraulic conductivity measured, its unsaturated hydraulic conductivity curve is drawn in Fig. 3. Accordingly, soil water retention curve and hydraulic conductivity are written as equation (2) and (3) respectively, where saturated volumetric water content θ s = 0.4, residual volumetric water content θ r =0.004, parameter α = 0.003 kPa -1 , m = 0.18, n = 1.8, m 1 = 0.5, saturated hydraulic conductivity K s = 10 -9 m/s.

Figure 1 .
Figure 1.Thermal conductivity of treated silt

Figure 4 .
Figure 4. Embankment sketch and two section positions

Figure 5 .
Figure 5. Water flux top boundary condition Based on the Hydro-thermal coupled model of treated silt, only alternate infiltration and evaporation in six days are added as top boundary condition.With the volumetric water content variations of six points in section A and B recorded continuously.It can be seen that the values in six points are nearly consistent in four different dimensions (Fig. 6 et Fig. 7, 1-P1 represents point 1 in dimension 1, 2-P2 represents point 2 in dimension 2, etc.).However, the

Figure 6 .Figure 7 .Figure 8 .
Figure 6.Volumetric water content change of six points in section A in four dimensions

Fig. 10 ,
Fig. 10, in the first nine hours of nighttime when soil heat is negative as 15 W/m 2 s, surface point P1 decreases quickly to 8.69 °C and goes up to 12.65 °C in the daytime when receiving positive soil heat 45 W/m 2 s.During six days, it keeps nearly the same minimum and maximum temperature in each day.The similar situation exists in point P2, which is in temperature variation range 9.80~10.20 °C.Obvious time delay of 2.4 to 4.8 hours could be observed for points P2 compared with point P1 because of deeper position of point P2.Point P3, P4, P5 and P6 keep nearly constant temperature as the initial value, which means that they are located out of the influencing region of surface heat flux.This influencing depth is closely related to the thermal conductivity of treated soil.

Figure 9 .Figure 10 .UNSATFigure 11 .Figure 12 .
Figure 9. (a) Heat flux top boundary condition in six days (b) Details of heat flux variations in 24 hours embankment, the Hydro-thermal coupled unsaturated soil model is used to calculate the reaction of soil under instantaneous boundary conditions.With abundant meteorological data, the corresponding water flux and heat flux boundary condition could be obtained conveniently relying on the energy and mass balance.With Finite Element Method, a suitable dimension of this embankment needs to be set in advance, which takes the possible influencing region from atmosphere into consideration in the numerical calculation.In sum, four different dimension models are calculated respectively in two Cases.The first one is applied with water flux boundary condition, considering boundary heat flux as zero.Another one is applied with heat flux boundary condition based on the daily solar radiation values with no water flux exchange.Compared three dimensions with the results of the reliable model dimension (dimension 4), the temperature and volumetric water content values and variation patterns in these dimensions are nearly consistent.The contour lines of volumetric water content and temperature distribution are parallel to the surface line.It can be concluded that the effect of model dimension to field embankment is not significant for Hydro-thermal coupled unsaturated soil research.In dimension 1, the effect of left ground on embankment slope toe is not fully considered, which is not advised based on the analysis in this study.Therefore,

Table 1 .
The details of four dimensions