Application |
PLAXIS 2D PLAXIS 3D |

Version |
PLAXIS 2D CONNECT Edition PLAXIS 3D CONNECT Edition |

Original Author |
Faseel Khan - Technical Support Group |

Date created |
02 November 2022 |

Date modified |
02 November 2022 |

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)

- Note the values for θ
_{res}and θ_{sat}. - Use θ = S η to determine, porosity then S
_{res}and S_{sat}. Porosity remains constant in a flow-only analysis - Get k
_{rel }using k_{rel}= k/k_{sat }for each data point from the k(Ψ) plot - Get data points from k(Ψ) and θ(Ψ) plots
- Convert suction pressure to suction head by dividing with the unit weight of water
- 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)

- Note the values for S
_{res}and S_{sat}. - Use θ = S η to determine, porosity then θ
_{res}and θ_{sat}. Porosity remains constant in a flow-only analysis - Get k
_{rel }using k_{rel}= k/k_{sat }for each data point from the k_{rel}(Ψ) plot - Get data points from k
_{rel}(Ψ) and S(Ψ) plots - Convert suction head to suction pressure by multiplying with the unit weight of water
- 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.