Hydraulic conductivity input in PLAXIS using van Genuchten functional forms in a groundwater flow analysis

The soil-water characteristic curve relates suction to saturation (S), volumetric water content (θ) and hydraulic conductivity (k) and the corresponding functional forms are θ(Ψ), k(Ψ) and S(Ψ) respectively. In PLAXIS, the van Genuchten functional forms S(Ψ) and k(Ψ) can be entered in every soil/rock material data set in two ways. One way to generate these functional forms is to directly input the van Genuchten parameters and the second way is to use a Spline function to fit smooth curves to tabulated data. The van Genuchten parameters or the Spline function for any soil is input in the Groundwater tab of the Material properties and selecting User-defined for the data set.

To get parameters which are suitable for input in PLAXIS a step-wise procedure is provided to input the van Genuchten functional forms for direct input or as a spline function. The procedure converts the functional form θ(Ψ) and k(Ψ) to the functional form S(Ψ) and k_rel(Ψ) or vice versa.

Often within the realm of geotechnical analysis, it is required to convert measured data concerning the degree of saturation to volumetric water content (or vice versa) for its use in either Finite Element or Limit Equilibrium Analysis in PLAXIS or SEEP/W or PLAXIS LE. To interconvert between the functional forms we need to understand porosity and its relationship with the degree of saturation. The relationship between the degree of saturation and volumetric water content is defined by the following equation:

θ = S ƞ

where S is the degree of saturation, ƞ is the soil porosity and θ is the volumetric water content.

If we know S at any suction value and we know ƞ, we can then calculate θ. If ƞ remains constant throughout a transient analysis, we can convert saturation to volumetric water content at any suction value. This is because in a flow-only transient analysis the porosity does not change as described below.

The following phase diagram shows the different phases in a soil. Fig 1a shows an unsaturated soil comprising three phases which are soil solids, water and air.
The volume of soil solids, water and air are denoted by V_s, V_w and V_a, respectively.
The total volume V_t = V_a + V_w + V_s.
Volume of voids V_v = V_a + V_w.

On the other hand, Fig 1b shows the phase diagram for a fully saturated soil in which there are only two phases of water and soil solids because the volume of air gets filled by water.

Porosity is defined as (V_a+V_w )/V_total

In a flow-only transient analysis, the volume of voids remains the same at any point in time and hence the porosity remains the same whether the soil is unsaturated or saturated as seen in both Fig 1a and Fig 1b.

1. Van Genuchten parameters

The closed-form relationship for the van Genuchten functional form θ(Ψ) is given by the following equation: 

where θ_res is the residual volumetric water content, θ_sat is the saturated volumetric water content.
α', n and m are curve-fitting parameters.
α' is the reciprocal of the air-entry value with units of 1/kPa
n controls the shape of the θ(Ψ) plot after the air-entry value.
m = 1-1/n

Similarly, the closed-form relationship for the functional form k(Ψ) is given by the following equation:

where k_sat is the saturated hydraulic conductivity.

The input fitting parameters for the functional form S(Ψ) and k_rel(Ψ) are g_a, g_n, g_c, g_l and k_rel and k_sat which are explained below.

The closed-form relationships for the functional forms S(Ψ) and k_rel(Ψ) are given below:

where S_res is the residual degree of saturation, S_sat is the saturated degree of saturation and k_rel is the relative hydraulic conductivity.
g_a is reciprocal of air-entry value similar to α' with units of 1/m so its value is determined by dividing α' by the unit weight of water.
g_n controls the shape of the S(Ψ) plot after the air-entry value so g_n = n.
g_c = 1- 1/g_n 
g_l is a fitting parameter controlling the shape of the k_rel(Ψ) plot and affects the effective degree of saturation S_eff and can be taken as 0 for some soils.

Based on the relationships provided above, the van Genuchten’s parameters can be interconverted from the functional form k(Ψ) and θ(Ψ) to the functional form S(Ψ) and k_rel(Ψ) and vice versa.

2. Spline function

The Spline function requires direct input of the capillary height Ψ, relative hydraulic conductivity k_rel, and the degree of saturation which are all positive values. Data for the Spline function can be entered in the PLAXIS Groundwater tab of the Material properties and selecting User-defined for the data set and then selecting Spline for the Model.  

2.1 Converting the functional form k(Ψ) and θ(Ψ) to k_rel(Ψ) and S(Ψ) 

1. Note the values for θ_res and θ_sat.
2. Use θ = S η to determine, porosity then S_res and S_sat. Porosity remains constant in a flow-only analysis
3. Get k_rel using k_rel= k/k_sat for each data point from the k(Ψ) plot
4. Get data points from k(Ψ) and θ(Ψ) plots
5. Convert suction pressure to suction head by dividing with the unit weight of water
6. Get the degree of saturation using θ = S η against the data points for θ

This gives the data for spline fit in the functional forms S(Ψ) and k_rel(Ψ)

Note: If the suction head data points for the k(Ψ) and θ(Ψ) plots do not coincide then manually read the S values at the suction values corresponding to the k(Ψ) data points.

2.2 Converting the functional form S(Ψ) and k_rel(Ψ) to θ(Ψ) and k(Ψ)   

1. Note the values for S_res and S_sat.
2. Use θ = S η to determine, porosity then θ_res and θ_sat. Porosity remains constant in a flow-only analysis
3. Get k_rel using k_rel= k/k_sat for each data point from the k_rel(Ψ) plot
4. Get data points from k_rel(Ψ) and S(Ψ) plots
5. Convert suction head to suction pressure by multiplying with the unit weight of water
6. Get volumetric water content using θ = S η against the data points for S

This gives the data for spline fit in the functional forms θ(Ψ) and k(Ψ)

Note: If the suction head data points for the k_rel(Ψ) and S(Ψ) plots do not coincide then manually read the θ values at the suction values corresponding to the k_rel(Ψ) data points.