By three phase soil system show that the degree of saturation S (as ratio) in terms of mass, unit weight ,water content, specific gravity of soil grains G, and unit weight of water.

Degree of Saturation Derivation

Problem Statement

Using the three-phase soil system, show that the degree of saturation (\( S \)) in terms of mass unit weight (\( \gamma \)), water content (\( w \)), specific gravity of soil grains (\( G \)), and unit weight of water (\( \gamma_w \)) is given by:

\[ S = \frac{w}{\frac{\gamma (1 + w)}{\gamma_w G} – 1} \]

Three-Phase Soil System

Figure 1: Three-phase soil system showing the distribution of air, water, and solid phases

Solution

Step 1: Define the Three-Phase Soil System

The three-phase soil system consists of soil solids, water, and air. The total volume (\( V \)) is:

\[ V = V_s + V_w + V_a \]

Where:

\( V_s \) = volume of solids

\( V_w \) = volume of water

\( V_a \) = volume of air

Step 2: Express Mass and Volume Relationships

Using the definitions of water content (\( w \)), specific gravity (\( G \)), and mass unit weight (\( \gamma \)):

\[ w = \frac{M_w}{M_s}, \quad G = \frac{M_s}{V_s \cdot \gamma_w}, \quad \gamma = \frac{M_s + M_w}{V} \]

Where:

\( M_w \) = mass of water

\( M_s \) = mass of solids

\( \gamma_w \) = unit weight of water

Step 3: Derive the Expression for \( S \)

The degree of saturation (\( S \)) is defined as the ratio of volume of water to volume of voids:

\[ S = \frac{V_w}{V_v} = \frac{V_w}{V_w + V_a} \]

Combining the relationships and solving for \( S \), we arrive at: