Modelling of Climate Induced Moisture Variations and Subsequent Ground Movements in Expansive Soils

Abstract

Expansive behaviour of soil in response to moisture changes is a significant issue for lightly loaded structures. Recent reports have shown more than 4000 houses in Victoria, Australia have been damaged due to abnormal moisture changes beneath footings. For design purposes, the moisture change due to climate is crucial. This paper provides details of modelling of climate induced soil moisture changes and subsequent ground movements. The soil moisture variation due to climate was modelled using Vadose/w for two sites in Melbourne, Australia. The model was validated against the regular measurements from the field. The predicted soil moistures from the Vadose/w model were used to predict the possible ground movement using FLAC3D. The predicted ground movements were also validated using the field monitored ground movements at the sites. Further, the model was used to determine the possible ground movements due to long-term climate conditions. The model results demonstrate the reductions of soil moisture and shrinkage movements during the millennium drought. The model predictions also suggest that the soils have not been able to fully recover that moisture deficit through the drought breaking rains in 2010 and 2011. This model can be used to observe expectable ground movements due to climate changes and hence can greatly assess the footing performance for different climate scenarios.

Appendix

Vadose/w is developed to accommodate actual evaporation using Wilson’s (1990) observations. The actual evaporation is calculated from Penman-Wilson formulation as shown in Eq. 3.

$${\text{AE}} = \frac{{\Gamma{\text{Q}} + {{\upupsilon{\text{E}}}}_{\text{a}} }}{{{{\upupsilon}{\text{A}}} + {{\Gamma }}}}$$

(3)

where, AE = actual vertical evaporative flux (mm/day), Γ = slope of the saturation vapour pressure versus temperature curve at the mean temperature of the air (kPa/°C), Q = net radiant energy available at the surface (mm/day), υ = psychometric constant, E_a = f(υ)P_a (B-A), f(υ) = function dependent on wind speed, surface roughness and eddy diffusion = 0.35(1 + 0.15U_a), U_a = wind speed (km/hr), P_a = vapour pressure in the air above the evaporating surface (kPa), B = inverse of the relative humidity = 1/hA, A = inverse of the relative humidity at the soil surface = 1/hr.

This formulation requires wind speed, relative humidity and net radiation from climate parameters. However, if the net radiation data is not available, the evaporative flux can also be calculated using Eq. 4.

$${\text{E}} = {\text{PE}}\left( {\frac{{{\text{h}}_{\text{r}} - \left( {\frac{{{\text{V}}_{{{\text{p}}.{\text{sat}}.{\text{air}}}} }}{{{\text{V}}_{{{\text{p}}.{\text{sat}}.{\text{soil}}}} }}} \right){\text{h}}_{\text{A}} }}{{1 - \left( {\frac{{{\text{V}}_{{{\text{p}}.{\text{sat}}.{\text{air}}}} }}{{{\text{V}}_{{{\text{p}}.{\text{sat}}.{\text{soil}}}} }}} \right){\text{h}}_{\text{A}} }}} \right)$$

(4)

where, E = evaporative flux (mm/day), PE = user supplied potential evaporation (mm/day), h_r = relative humidity at the soil surface, V_p.sat.air = saturated vapour pressure of air, V_p.sat.soil = saturated vapour pressure of soil surface, h_A = relative humidity of air above the soil surface.

The temperature of the soil surface can be calculated using

$${\text{T}}_{\text{s}} = {\text{T}}_{\text{a}} + \frac{1}{{{{\upupsilon f}}\left( {{\upupsilon }} \right)}}({\text{Q}} - {\text{E}})$$

(5)

(Wilson 1990) for “no snow” conditions.

$${\text{T}}_{\text{s}} = {\text{T}}_{\text{a}} + \frac{1}{{\upupsilon{\text{ f}}\left( {{\upupsilon }} \right)}}\left( {{\text{Q}} - {\text{E}}} \right)$$

(6)

T_s = temperature at the soil surface (°C), T_a = temperature of air above the soil surface (°C), υ = psychometric constant, AE = actual vertical evaporative flux (mm/day), Q = net radiant energy available at the surface (mm/day).

The effects of the vegetation on surface soils can be included in Vadose/w using three different formulations. The variations of leaf area index and depth of the roots must be specified as functions of the time period of the analysis. In addition, the moisture limiting factor, which is dependent on the ability of vegetation to suck moisture from soil at different sections, has to be specified.

The actual evaporation is limited by the water absorption from the vegetation and it is calculated from Eq. 6 (Vadose 2013).

$$AE = AE^{*} \left[ {1 - \left( { - 0.21 + 0.7 \times \sqrt {LAI} } \right)} \right]$$

(7)

where, AE^* = actual vertical evaporative flux (mm/day), AE = modified actual vertical evaporative flux (mm/day), LAI = leaf area index.

The potential transpiration is dependent on potential evaporation as defined in Eq. 8.

$$PT = PE\left( { - 0.21 + 0.7 \times \sqrt {LAI} } \right)$$

(8)

where, PE = potential evaporation (mm/day), PT = potential transpiration (mm/day).

The actual transpiration depends on the ability of tree roots to suck moisture from the soil. Therefore, it is related to the plant moisture limiting factor and root depth, as shown in Eq. 9.

$$AT = PRU \times PML$$

(9)

where, AT = actual transpiration (mm/day), PML = plant moisture limiting factor, $PRU = \frac{2PT}{{R_{T} }}\left( {1 - \frac{{R_{n} }}{{R_{T} }}} \right)A_{n}$, R_T = total thickness of root zone, R_n = the depth of the node in question, A_n = the nodal contributing area of the node in question.

Then, the modified actual evaporative flux is used in governing flow equations. Equations 10 and 11 show the governing differential equations for the 1D coupled process of moisture and heat flow.

$$\frac{\partial }{\rho \partial z}\left( {D_{V} \frac{{\partial P_{V} }}{\partial z}} \right) + \frac{\partial }{\partial z}\left( {k_{z} \frac{{\partial \left[ {\frac{P}{\rho g} + z} \right]}}{\partial z}} \right) + Q = \lambda \frac{\partial P}{\partial t}$$

(10)

$$L_{V} \frac{\partial }{\partial z}\left( {D_{V} \frac{{\partial P_{V} }}{\partial z}} \right) + \frac{\partial }{\partial z}\left( {k_{tz} \frac{\partial T}{\partial z}} \right) + Q_{t} + \rho cV_{z} \frac{\partial T}{\partial z} = \lambda_{t} \frac{\partial T}{\partial t}$$

(11)

where, ρ = water density, z = elevation head, D_v = vapour diffusivity coefficient, P_v = vapour pressure of soil moisture, k_z = hydraulic conductivity in the z (vertical) direction, P = water pressure, g = acceleration due to gravity, Q = applied boundary flux, Λ = slope of the volumetric water content function, t = time, L_v = latent heat of vaporization, k_tz = thermal conductivity in the z-direction, T = soil temperature, Q_t = applied thermal boundary flux, ρc = volumetric specific heat value, V_z = Darcy’s water velocity in vertical direction, λ_t = volumetric specific heat value.

The above governing equations are solved in finite element method as programmed in Vadose/w software. The detailed solving procedure can be found in the software manual (Vadose 2013).