Stress Capillarity and Permeability

Stress on soil doesn't depend on the area as long as any external loading is not applied.
Total Stress is a physical parameter which can be measured by pressure cell.
Effective Stress is not a measurable quantity, in fact, it's not even real but many soil properties such as compressiblity, consolidation, settlement, shear stress and bearing capacity depend on it. These parameters do not directly depend on Total Stress.

The soil above the water table is not completely dry but there is water up to a certain height above the water table, this water is called capillary water. It happens because water in the pores is not only subjected to gravitational pull but also to a force of adhesion between the soil particles and water molecules inside the soil pores.

$\overline{\sigma }=\sigma -u$

$h=\frac{0.03}{d}=\frac{0.03}{0.2{{D}_{10}}}$

${{h}_{c}}=\frac{C}{e{{D}_{10}}}$

$Seepage Force ={{\gamma }_{w}}\Delta hb$

$\frac {Seepage Force}{Volume}=i{{\gamma }_{w}}$

$i=\frac{\Delta h}{b}$

${{i}_{cr}}=\frac{{{\gamma }_{sub}}}{{{\gamma }_{w}}}=\frac{G-1}{1+e}$

$\overline{\sigma }=$ Effective Stress
${\sigma }=$ Total Stress
$u=$ Pore Pressure
$i=$ Hydraulic Gradient
$h=$ Capillary Rise
$C=$ Empirical Constant $=0.1-0.5$ ${cm^2}$
${\Delta h}=$ Drop in total Head
${{i}_{cr}}=$ Critical Hydraulic Gradient

Permeability

Permeability is the ease with which water can flow through any medium. It is a very important property of soil. Knowledge of permeability is essential to solve many soil engineering problems such as settlement of buidling, yield of wells, seepage through the earth structures etc.

$q=kiA$

$V=ki$

${{V}_{s}}=\frac{V}{n}$

${{K}_{p}}=\frac{K}{n}$

Constant Head Permeabilty Test
$K=\frac{qL}{Ah}$

Falling Head Pearmabilty Test
$K=\frac{aL}{At}{{\log }_{e}}\left( \frac{{{h}_{1}}}{{{h}_{2}}} \right)$

Unconfined Aquifer Test
$K=\frac{q{{\log }_{e}}\frac{{{r}_{2}}}{{{r}_{1}}}}{\pi (h_{2}^{2}-h_{1}^{2})}$

$R=3000d\sqrt{K}$

Confined aquifer Test
$K=\frac{q{{\log }_{e}}\frac{{{r}_{2}}}{{{r}_{1}}}}{2\pi D({{h}_{2}}-{{h}_{1}})}$

$K=\frac{1}{{{C}_{S}}}\times \frac{{{\gamma }_{w}}}{\mu }\times \frac{1}{S_{A}^{2}}\times \frac{{{e}^{3}}}{1+e}$

$K=CD_{10}^{2}$

$K={{C}_{V}}{{\gamma }_{w}}{{m}_{v}}$

Permeability of Stratified Soil
${{K}_{H}}=\frac{{{K}_{1}}{{H}_{1}}+{{K}_{2}}{{H}_{2}}+{{K}_{3}}{{H}_{3}}...}{{{H}_{1}}+{{H}_{2}}+{{H}_{3}}...}$

${{K}_{V}}=\frac{{{H}_{1}}+{{H}_{2}}+{{H}_{3}}...}{\frac{{{H}_{1}}}{{{K}_{1}}}+\frac{{{H}_{2}}}{{{K}_{2}}}+\frac{{{H}_{3}}}{{{K}_{3}}}...}$

Unconfined Aquifer
Source: NPTEL

$q=$ Dischagre per unit time
$K=$ Permeabilty of soil
$i=$ Hydraulic gradient $=$ loss of head per unit lenth
$V=$ Superficial Velocity of flow
${{V}_{s}}=$ Seepage Velocity
${{K}_{p}}=$ Coefficient of Percolation
${\mu}=$ Coefficient of Viscosity
${{C}_{S}}=$ Shape factor Cofficient
${{S}_{A}}=$ Specific Surface Area
${{C}_{v}}=$ Coefficient of Consolidation
${{m}_{v}}=$ Coefficient of Volume Compressiblity

Confined Aquifer
Source: NPTEL

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