The focus of the work presented here has been concentrated on climate impact analysis. A hydrological model, SHETRAN [1, 27], capable of simulating a wide range of processes affecting slope pore water pressures (including hourly meteorological inputs, the capability to model infiltration, evapo-transpiration and overland flow) was used to derive daily near surface pore water pressures, which were used as the surface boundary condition in the fully coupled hydro-mechanical finite difference code FLAC Two-Phase Flow [37], leading to (un)saturated flow and pore pressure changes causing mechanical deformation. Flow model verification and validation of the pore pressure transfer method are detailed within Davies et al. [16] and use in modelling a natural slope failure is summarised in Davies et al. [17]. In summary this method enables the simulation of temporal deformations of a slope as a response to weather and, over long time periods, climate.
Hydrological model
In SHETRAN the subsurface is modelled as a porous, heterogeneous medium with variable saturation [26]. Vegetation behaviour in SHETRAN is simulated using a modified version of the Rutter et al. [58] model for rainfall interception. This controls canopy water storage, evaporation from the stored water on the canopy, as well as precipitation throughfall and stemflow and is described in Abbot et al. [1]. A root water uptake model is also included where user specified root density and depth along with a relationship between pore water suction and transpiration, controls vegetation root water uptake behaviour.
SHETRAN uses the extended Richard’s equation to describe variably saturated flow. This is given in its pressure form for two dimensions in terms of saturated conductivity, \((k_{x}^{\mathrm{w}}, k_{z}^{\mathrm{w}})\), relative conductivity, \(k_{\mathrm{r}}\), pressure potential, \(\psi\), and the volumetric flow rate for flow out of the medium \(q_{\mathrm{t}}\) (actual transpiration) as follows: [1, 54]:
$$\begin{aligned} \eta \frac{\partial \psi }{\partial t}=\frac{\partial }{\partial x}\left[ k_{x}^{\mathrm{w}}k_{\mathrm{r}}\frac{\partial \psi }{\partial x}\right] +\frac{\partial }{\partial z}\left[ k_{z}^{w}k_{\mathrm{r}}\frac{\partial \psi }{\partial z}\right] +\frac{\partial (k_{\mathrm{r}}k_{z}^{\mathrm{w}})}{\partial z}-{q_{\mathrm{t}}} \end{aligned}$$
(1)
where \(\eta\) is the storage coefficient defined using specific storage, \(S_{\mathrm{s}}\), porosity (assumed to be the same as saturated volumetric water content), \(\theta _{\mathrm{s}}\), volumetric water content, \(\theta\) and the pressure potential as:
$$\begin{aligned} \eta = \frac{\theta S_{\mathrm{s}}}{\theta _{\mathrm{s}}}+\frac{\mathrm{d}\theta }{\mathrm{d}\psi } \end{aligned}$$
(2)
\(S_{\mathrm{s}}\) is a function of the compressibility of the soil and is derived as follows [74]:
$$\begin{aligned} S_{\mathrm{s}} = \rho _{\mathrm{w}}g\left( \frac{1}{K}+\theta _{\mathrm{s}}\frac{1}{K_{\mathrm{w}}}\right) \end{aligned}$$
(3)
where \(\rho _{\mathrm{w}}\) is the fluid density, g is gravatational acceleration, K is the bulk modulus of the soil and \(K_{\mathrm{w}}\) is the fluid bulk modulus (2.2 GPa). The surface boundary flux due to surface evaporation (\(q_{\mathrm{e}}\)) and rainfall reaching the ground surface (\(q_{\mathrm{p}}\)) where no surface ponding has occurred is given by the following:
$$\begin{aligned} k_{\mathrm{r}}k^{\mathrm{w}}_{z}\left( \frac{\partial \psi }{\partial z}+1\right) =q_{\mathrm{p}}-q_{\mathrm{e}} \end{aligned}$$
(4)
Whenever ponding has occurred, the surface boundary flux switches to a head condition based on the depth of the ponded water.
Mechanical model
In the SHETRAN model used in this work, potential evapotranspiration (PET) is converted into actual evapotranspiration (AET) based on the magnitude of soil suction. To disaggregate potential transpiration (PT) from potential evaporation (PE), assumptions must be made about the ground vegetation cover. As such PT is a function of the area of ground surface covered by vegetation in plan view, known as the plant leaf area index (\(p_1\)). The proportion of PET assigned to transpiration is equal to \(p_1\) and that to PE is equal to the area of bare ground (\(1-p_1\)). PT is distributed across the defined rooting depth (\(d_{\mathrm{r}}\)), with proportions allocated to differing depths within the rooting zone, equal to the proportion of the total root mass (\(n_{\mathrm{r}}\)) at depth \(d_{\mathrm{r}}\). PT is in turn scaled to actual transpiration (\(q_{\mathrm{t}}\)) by SHETRAN as a function of soil suction for each defined section of the root zone (see Table 2) and PE is scaled to actual evaporation (\(q_{\mathrm{e}}\)) based on the suction at the soil surface. This is based on the approach of Feddes et al. [28]:
$$\begin{aligned} q_{\mathrm{t}}(d_{\mathrm{r}})=\alpha _{\mathrm{r}}p_1PT \end{aligned}$$
(5)
where \(\alpha _{\mathrm{r}}\) varies as a function of the matric suction as per Fig. 1. From this, four specific values of matric suction can be identified. \(\psi _{\mathrm{m}}^\mathrm{a}\) which is the anaerobiosis suction, below this suction root water uptake does not occur (\(\alpha _{\mathrm{r}}\) = 0), at suctions greater than this value, \(\alpha _{\mathrm{r}}\) increases, reaching a maximum at \(\psi _{\mathrm{m}}^{1}\). This represents the lower limit of the readily available water (RAW) range, the upper limit being at \(\psi _{\mathrm{m}}^{2}\). Within this range \(\alpha _{\mathrm{r}}\) = 1 and evapotranspiration occurs at the potential rate. Above this suction value (\(\psi _{\mathrm{m}}^{2}\)), plant stress begins to occur where with increasing suction, \(\alpha _{\mathrm{r}}\) begins to fall, reaching zero at the permanent wilting point (\(\psi _{\mathrm{m}}^{\mathrm{w}}\)). \(\psi _{\mathrm{m}}^\mathrm{a}\) to \(\psi _{\mathrm{m}}^{\mathrm{w}}\) represents the totally available water (TAW) range. "Appendix 1" gives the vegetation model and parameters used for the canopy model used in SHETRAN. For further information see Abbot et al. [1].
Fluid flow in FLAC is a function of the pore water, \(p_{\mathrm{w}}\) and pore air, \(p_{\mathrm{g}}\), pressures. These are related to the matric suction, \(p_{\mathrm{m}}\), whereby \(p_{\mathrm{m}}=p_{\mathrm{g}}-p_{\mathrm{w}}\). To describe the soil water retention curve (SWRC) which is the relationship between \(p_{\mathrm{m}}\) and the effective saturation, \(\theta _{\mathrm{e}}\), SHETRAN and FLAC use the closed form version of the van Genuchten equation [27, 71]:
$$\begin{aligned} p_{\mathrm{c}}(S_{\mathrm{w}})= \frac{\rho _{\mathrm{w}}g}{\alpha }\left[ \theta _{\mathrm{e}}^{-1/m}-1\right] ^{1-m} \end{aligned}$$
(6)
The coefficients \(\alpha\) and m were derived here by curve fitting. In turn, \(\theta _{\mathrm{e}}\) can be related to \(\theta\) or the degree of water saturation, \(S_{\mathrm{w}}\), as follows:
$$\begin{aligned} \theta _{\mathrm{e}} =\frac{\theta -\theta _{\mathrm{r}}}{\theta _{\mathrm{s}}-\theta _{\mathrm{r}}}=\frac{S_{\mathrm{w}}-S_{\mathrm{r}}}{1-S_{\mathrm{r}}} \end{aligned}$$
(7)
where \(\theta _{\mathrm{r}}\) and \(S_{\mathrm{r}}\) are the residual volumetric moisture content and residual water saturation, respectively. In FLAC, the flow velocities for the wetting (\(q_\mathrm{i}^{\mathrm{w}}\)) and non-wetting (\(q_\mathrm{i}^{\mathrm{g}}\)) fluid are calculated as follows [37]:
$$\begin{aligned} q_\mathrm{i}^{\mathrm{w}}&= -\frac{k_{\mathrm{r}}^{\mathrm{w}}}{g\rho _{\mathrm{w}}}\frac{\partial }{\partial x_\mathrm{j}}(p_{\mathrm{w}}-\rho _{\mathrm{w}}g_\mathrm{k}x_\mathrm{k}) \end{aligned}$$
(8)
$$\begin{aligned} q_\mathrm{i}^{\mathrm{g}}&= -\frac{k_{\mathrm{r}}^{\mathrm{g}}}{g\rho _{\mathrm{w}}}\frac{\partial }{\partial x_\mathrm{j}}(p_{\mathrm{g}}-\rho _{\mathrm{g}}g_\mathrm{k}x_\mathrm{k}) \end{aligned}$$
(9)
where \(\mu _{\mathrm{w}}/\mu _{\mathrm{g}}\) are the fluid and gas viscosities defining the viscosity ratio and \(\rho _{\mathrm{w}}\) and \(\rho _{\mathrm{g}}\) are fluid and gas densities. In FLAC and SHETRAN, the unsaturated hydraulic conductivity of the soil to the fluid, \(k^{\mathrm{w}}_{\mathrm{r}}\), and the gas phases, \(k^{\mathrm{g}}_{\mathrm{r}}\), at a given effective saturation, \(\theta _{\mathrm{e}}\), are derived using the van Genuchten-Maulem equation [71]:
$$\begin{aligned} k_{\mathrm{r}}^{\mathrm{w}}&= k^{\mathrm{w}}(\theta _{\mathrm{e}})^{0.5}\left[ 1-\left( 1-\theta _{\mathrm{e}}^{\frac{1}{m}}\right) ^m\right] ^2 \end{aligned}$$
(10)
$$\begin{aligned} k_{\mathrm{r}}^{\mathrm{g}}&= k^{\mathrm{w}}\frac{\mu _{\mathrm{w}}}{\mu _{\mathrm{g}}}(1-\theta _{\mathrm{e}})^{0.5}\left[ 1-\theta _{\mathrm{e}}^{\frac{1}{m}}\right] ^{2m} \end{aligned}$$
(11)
The surface non-wetting pore pressure was fixed at atmospheric in the modelling undertaken here, however wetting and non-wetting pressures were free to develop within the model and in turn influence the capillary pressure and hence effective stress (see the mechanical model section for more information).
Recent field work undertaken at Newbury to measure in-situ hydraulic conductivity [21] has demonstrated that the near surface values (<1m depth) are significantly higher than at greater depths. This region has been incorporated into the model as a zone of elevated hydraulic conductivity (1.0\(\times\)10\(^{-8}\) m/s) in the uppermost elements. Below this region, a mean effective stress (\(p'\)) dependent relation was used to define the saturated hydraulic conductivity (\(k^{w}\)) distribution with depth within the model taking the following form [73]:
$$\begin{aligned} k^{\mathrm{w}}=k_\mathrm{ref}\times \exp ^{\mathrm{ap}'} \end{aligned}$$
(12)
where \(k_\mathrm{ref}\) is the reference hydraulic conductivity at zero mean effective stress and a is a parameter describing the change in conductivity with \(p'\) (\(a=-0.003\) [44]).
In this work SHETRAN has been used to model the meteorologically driven seasonal fluctuations in pore water pressure and FLAC Two-Phase Flow has been used to create the fully coupled hydro-mechanical models. The primary mechanism of deterioration examined in this work is progressive failure attributed to strain softening of the soil mass. Previous numerical modelling studies of slope stability in London Clay adopted the use of a strain softening Mohr–Coulomb constitutive model with effective stress dependent elastic parameters to model the mechanical response and potential progressive failure of infrastructure slopes and cuttings [25, 43, 51, 53, 55, 57]. A similar constitutive model is adopted in this work (see Fig. 2).
In FLAC the linear momentum balance and mass balance equations are discretised in space and time at each node of the finite deference mesh and solved simultaneously using explicit algorithms. At each time step, grid point quantities such as velocity, displacement, pore pressures and saturation are evaluated based on previous values. The Updated Lagrangian approach is adopted, whereby the grid point co-ordinates are changed as deformation progresses and the constitutive law is then invoked to update the effective stress based on the strain increment and the previous stress state. This cycle occurs within a single time step, \(\Delta t\), the size of which is calculated by the code as a function of the minimum element size, \(L_{\mathrm{e}}\), and the compressive wave velocity, \(V_{\mathrm{p}}\):
$$\begin{aligned} \Delta t=\frac{L_{\mathrm{e}}}{V_{\mathrm{p}}} \end{aligned}$$
(13)
where \(V_{\mathrm{p}}\) is a function of the density, \(\rho\), and the bulk, K, and shear, G, moduli of the material. During cycling, in the event of plasticity, the stress state is returned to the strain softening Mohr–Coulomb yield surface. For partially saturated materials, the plasticity models in FLAC Two-Phase Flow use a Bishop’s effective stress, \(\varvec{\sigma }'\), formulation with the following form:
$$\begin{aligned} \varvec{\sigma '}=\varvec{\sigma }-(S_{\mathrm{w}}p_{\mathrm{w}}+S_{\mathrm{g}}p_{\mathrm{g}})\varvec{I} \end{aligned}$$
(14)
where \(\varvec{\sigma }\) is total stress, \(S_{\mathrm{g}}\) is the degree of saturation of the air phase (\(S_{\mathrm{g}}=1-S_{\mathrm{w}}\)) and \(\varvec{I}\) is the identity vector. It has been recognised that the phenomenon of wetting-induced collapse is not well captured by the Bishop’s effective stress. However, it is generally accepted in the literature that the increase of shear strength with suction can be determined by this form of unsaturated effective stress and is also valid for high degree of saturations when the air phase is discontinuous, such as the conditions in this work. The selection of an unsaturated effective stress has been the subject of considerable debate [29, 40, 50].
To derive the stiffness of the London Clay used in the model, a constant value of Poisson’s ratio is adopted and the Young’s modulus, E, of the London Clay used in the model was dependent on \(p'\) based on the following relation [55]:
$$\begin{aligned} E=25\times (p'+100)\geqslant 4000\,\text {kPa} \end{aligned}$$
(15)
To describe the strength of the material the Mohr–Coulomb strain-softening constitutive model was utilised where a post failure reduction in strength occurs at user specified plastic shear strain increments (\(\Delta \varepsilon ^{ps}\)).
$$\begin{aligned} \Delta \varepsilon ^{ps}&= \frac{1}{6}\left[ (2\Delta \varepsilon _{1}^{\mathrm{ps}}-\Delta \varepsilon _{3}^{\mathrm{ps}})^2+(\Delta \varepsilon _{1}^{\mathrm{ps}}+\Delta \varepsilon _{3}^{\mathrm{ps}})^2\right. \nonumber \\&\quad\left. +(2\Delta \varepsilon _{3}^{\mathrm{ps}}-\Delta \varepsilon _{1}^{\mathrm{ps}})^2 \right] ^\frac{1}{2} \end{aligned}$$
(16)
in which \(\Delta \varepsilon _{j}^{\mathrm{ps}},\,(j=1,3)\) are the principal plastic shear strain increments. To quantify the degree of softening, the residual factor, \(R_\mathrm{f}\), for the failure surface is calculated as follows [60]:
$$\begin{aligned} R_\mathrm{f}=\frac{\tau _{\mathrm{p}}-\tau }{\tau _{\mathrm{p}}-\tau _{\mathrm{r}}} \end{aligned}$$
(17)
where \(\tau\) is the current shear strength available on the slip surface, \(\tau _{\mathrm{p}}\) and \(\tau _{\mathrm{r}}\) are the peak and residual shear strengths, respectively.