# 3 Hydraulic Models

Variably saturated flow through a deforming porous medium can be described by the mass balance equation:

S\boldsymbol{m}^{\textsf{{\tiny T}}}\frac{\partial \boldsymbol{\varepsilon}}{\partial t} + \frac{e}{1+e}\left(\frac{\partial S}{\partial t} + \frac{S}{K_w}\frac{\partial p_w}{\partial t}\right) + \nabla^{\textsf{{\tiny T}}}\boldsymbol{v} = 0 \tag{3.1}

which is to be supplemented with the usual mechanical governing equation (momentum balance and strain-displacement relations) and the generalized Darcy’s law:

\boldsymbol{v} = -K_r\boldsymbol{K}\nabla h_s = -K_r\boldsymbol{K}\nabla \left(\frac{p_w}{\gamma_w}+z\right) \tag{3.2}

where:

• \boldsymbol{\varepsilon} = Strain
• \boldsymbol{m} = (1,1,1,0,0,0)^{\textsf{{\tiny T}}}, making \boldsymbol{m}^{\textsf{{\tiny T}}}\boldsymbol{\varepsilon} the volumetric strain
• K_w = Bulk modulus of water (= 2,200 MPa)
• e = Void ratio
• S = Degree of saturation
• \boldsymbol{v} = (v_x,v_y,v_z)^{\textsf{{\tiny T}}} = Fluid velocity [m/s]
• \boldsymbol{K} = Saturated hydraulic conductivity modulus [m/s]
• K_r = Relative hydraulic conductivity (a function of degree of saturation)
• z = Vertical coordinate
• \gamma_w = Unit weight of water (= 9.8 kN/m³)
• p_s = Pressure [kN/m²]
• h_s = p_s/\gamma_w + z = Head

In OPTUM GX, the effects of deformation as a result of seepage are neglected, making the governing equation:

\frac{e}{1+e}\left(\frac{\partial S}{\partial t} + \frac{S}{K_w}\frac{\partial p_s}{\partial t}\right) = \nabla^{\textsf{{\tiny T}}}\left[K_r\boldsymbol{K}\nabla \left(\frac{p_s}{\gamma_w}+z\right)\right] \tag{3.3}

Typical values of hydraulic conductivity for different materials are shown in Figure 3.1.

Besides the constants n and \boldsymbol{K}, the solution of this equation requires the relative hydraulic conductivity relation and the saturation-pressure relation (also known as the water retention curve or the soil water characteristic curve).

In OPTUM GX, the saturated hydraulic conductivity modulus is always given by:

\boldsymbol{K} =
\begin{bmatrix}
K_x &  &  \\
 & K_y &  \\
 &  & K_z
\end{bmatrix} \tag{3.4}

where K_x, K_y, and K_z are the saturated hydraulic conductivities in the x, y, and z directions, respectively.

The van Genuchten model is the most widely used hydraulic model in soil science. It relates degree of saturation to pressure head by:

S =
\begin{cases}
S_r + (S_s - S_r)\left(1 + |\alpha \hat{h}|^n\right)^{-m} & \text{for } \hat{h} \le 0 \\
S_s & \text{for } \hat{h} > 0
\end{cases} \tag{3.5}

where \hat{h} = -p/\gamma_w, m = 1 - 1/n, and:

• S_r = Residual degree of saturation (may be slightly greater than 0)
• S_s = Fraction of water-filled pores at full saturation (may be slightly less than 1)
• \alpha [m⁻¹] = Model parameter related to air entry pressure
• n = Model parameter related to the rate at which water is extracted from the soil once the air entry pressure has been exceeded

The relative hydraulic conductivity is related to the effective saturation S_e as:

K_r =
\begin{cases}
S_e^{1/2}\left[1 - \left(1 - S_e^{1/m}\right)^m\right]^2 & \text{for } S_e < 1 \\
1 & \text{for } S_e = 1
\end{cases} \tag{3.6}

where

S_e = \frac{S - S_r}{S_s - S_r} \tag{3.7}

Alternatively, K_r can be expressed in terms of \hat{h} as:

K_r =
\begin{cases}
\displaystyle\frac{\left[1 - |\alpha \hat{h}|^{n-1}\left(1 + |\alpha \hat{h}|^n\right)^{-m}\right]^2}{\left(1 + |\alpha \hat{h}|^n\right)^{m/2}} & \text{for } \hat{h} \le 0 \\
1 & \text{for } \hat{h} > 0
\end{cases} \tag{3.8}

Typical values of the parameters n and \alpha are given in the table below, and typical retention and relative hydraulic conductivity curves are shown in Figure 3.2.

The Basic model, which is the default model for all materials, is a van Genuchten model with \alpha = 4, \quad n = 4, \quad S_r = 0, \quad S_s = 1 and \quad e = 1. This gives a relatively steep pressure-saturation curve corresponding to a very coarse material.