3^rd Generation

Row

Column

Node

Appendix references

Concept

Depends on/

Related to

Contribution

3

1

PP_3,1

[35]

Total

porosity

ASTM_2,1

The intrinsic permeability (k_int) depends on & can be calculated by combining Hagen-Poiseuille flow equation to Darcy’s law and depends on the total porosity (Φ) according to equation (eq. 2, page 274):

$k_{int}=\frac{\Phi }{8\ast {\displaystyle {\int }_0^{\infty }f\left(r\right)\ast \text{d}r}}{\displaystyle {\int }_0^{\infty }f\left(r\right)\ast r^2\ast \text{d}r}$ ,

where the terms, definitions and dimensions of $\left(k_{int},\Phi ,f\left(r\right),r\right)$ are presented in detail in the terminology (Table A8) of the Appendix.

3

2

PP_3,2

[43]

Water

content

ASTM_2,2

Relative hydraulic conductivity (K_r) depends on water content (θ) according to the Mualem equation (1976), (eq.21, page 515):

$K_r={\Theta }^{1/2}\ast {\left[{\displaystyle {\int }_0^{\Theta }\frac{1}{h\left(x\right)}\ast \text{d}x}/{\displaystyle {\int }_0^1\frac{1}{h\left(x\right)}\ast \text{d}x}\right]}^2$ , $\Theta =\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}$

where the terms, definitions and dimensions of $\left(K_r,\Theta ,h,\theta \right)$ are presented in detail in the terminology (Table A8) of the Appendix.

3

3

ASTM_3,3

[5]

Hydraulic

conductivity

ASTM_2,3

Hydraulic diffusivity (D) depends on Hydraulic conductivity (K) according to the implicit equation:

$\begin{array}{l}D=\left(q\ast d_{inner}\right)\ast f(\left(\frac{K}{q}\right),\left(\frac{1}{Re}\right),\left(\frac{K^2\ast {\rho }_s}{\psi }\right),\left(\frac{K^2}{g\ast d_{inner}}\right),\left(\frac{K^2\ast d_{inner}^2\ast {\rho }_s^2}{{\rho }^2\ast v_z^2\ast d_{grain}^2}\right),\\ \left(\frac{d_{inner}^2}{k_{intr}}\right),\left(\frac{d_{inner}}{K\ast \left(t_i-t_p\right)}\right),\left(\frac{d_{inner}}{K\ast \left(t_p-t_f\right)}\right),\left(\frac{d_{inner}}{z}\right),\\ \left(\frac{K^2\ast d_{inner}\ast {\rho }_s}{\sigma }\right),\left(\frac{d_{inner}}{H}\right),\Phi ,\theta )\end{array}$

where the terms, definitions and dimensions of $\left(D,q,Re,d_{inner},K,{\rho }_s,\psi ,g,\rho ,v_z,d_{grain},k_{intr},t_i,t_p,t_f,z,\sigma ,H,\Phi ,\theta \right)$ are presented in detail in the terminology (Table A8) of the Appendix. For clarification purposes, (q) refers to infiltration rate, (Φ) is the total porosity, (θ) is the liquid content/soil-water content. Rayleigh’s method of indices was deployed along with the echelon matrix procedure as an additional confirmation method.

3

4

PP_3,4

[42]

Pressure

head

PP_2,4

Water retention (θ) is depended on pressure head (h) according to Van Genuchten model (1980) which predicts the hydraulic conductivity (Κ) from the water retention curve and is expressed as the following given relationship (eq. 2 & 3, page 892):

$\theta ={\left(\frac{1}{1+{\left(a\ast h\right)}^b}\right)}^c\left({\theta }_s-{\theta }_r\right)+{\theta }_r$ , where the terms, definitions and dimensions of $\left(\theta ,a,h,b,c,{\theta }_s,{\theta }_r\right)$ are presented in detail in the terminology (Table A8) of the Appendix. For clarification purposes, $\left(a,b,c\right)$ are parameters defined (ad hoc by Van Genuchten, 1980).