Three-Phase System & Index Properties of Soil Notes

Last reviewed 16 Sept 2026 · 7 min read

The three-phase system

A soil mass consists of solids, water and air. For analysis the phases are separated in a phase diagram:

Volumes: $V = V_{s} + V_{v}$ , $V_{v} = V_{w} + V_{a}$
Masses (weights): $M = M_{s} + M_{w}$ (air has negligible mass)

A dry soil has two phases (solids + air); a saturated soil has two phases (solids + water).

Basic definitions

DefinitionVolume and mass relationships

Water content $w = \frac{M _{w}}{M _{s}}$ (on dry mass; can exceed 100% in soft clays and peat)
Void ratio $e = \frac{V _{v}}{V _{s}}$ (can exceed 1)
Porosity $n = \frac{V _{v}}{V}$ (always < 1)
Degree of saturation $S = \frac{V _{w}}{V _{v}}$ (0 dry, 1 saturated)
Air content $a_{c} = \frac{V _{a}}{V _{v}} = 1 - S$ ; percentage air voids $n_{a} = \frac{V _{a}}{V} = n a_{c}$
Specific gravity of solids $G = \frac{ρ _{s}}{ρ _{w}}$ (sands ≈ 2.65; clays 2.7–2.8; organic soils lower)

DefinitionUnit weights

Bulk unit weight $γ = W / V$
Dry unit weight $γ_{d} = W_{s} / V$
Saturated unit weight $γ_{s a t}$ (all voids full of water)
Submerged (buoyant) unit weight $γ^{'} = γ_{s a t} - γ_{w}$
Unit weight of solids $γ_{s} = G γ_{w}$

Interrelations

FormulaMost-used relations

n = \frac{e}{1 + e}, e = \frac{n}{1 - n}

S e = w G

γ = \frac{( G + S e ) γ _{w}}{1 + e}, γ_{d} = \frac{G γ _{w}}{1 + e} = \frac{γ}{1 + w}

γ_{s a t} = \frac{( G + e ) γ _{w}}{1 + e}, γ^{'} = \frac{( G - 1 ) γ _{w}}{1 + e}

γ_{d} = \frac{( 1 - n _{a} ) G γ _{w}}{1 + w G} (zero-air-voids line when n_{a} = 0)

Determination in the laboratory and field

Water content

Oven-drying method (standard): dry at 105–110 °C for about 24 hours. Organic and gypsum-bearing soils are dried at lower temperature (about 60 °C) to avoid burning or losing water of crystallisation.
Rapid methods: calcium carbide (moisture meter — acetylene gas pressure), sand bath, torsion balance (infrared), pycnometer (for coarse soils), alcohol method.

Specific gravity (pycnometer / density bottle)

G = \frac{M _{2} - M _{1}}{( M _{2} - M _{1} ) - ( M _{3} - M _{4} )}

$M_{1}$ = empty bottle, $M_{2}$ = bottle + dry soil, $M_{3}$ = bottle + soil + water, $M_{4}$ = bottle + water. Density bottle (50 ml) for fine soils; pycnometer (about 1 litre) for coarser soils. Entrapped air must be removed (vacuum or boiling).

Field density

Core cutter method — soft, cohesive soils free of gravel.
Sand replacement method — gravelly and hard soils; the volume of the excavated hole is found from calibrated sand.
Also: water displacement, rubber balloon, nuclear density gauge.

Relative density (density index)

For coarse-grained soils, the state of packing matters more than the water content:

I_{D} = \frac{e _{ma x} - e}{e _{ma x} - e _{min}} \times 100 = \frac{γ _{d, ma x}}{γ _{d}} \cdot \frac{γ _{d} - γ _{d, min}}{γ _{d, ma x} - γ _{d, min}} \times 100

$I_{D}$ (%)	Description
0–15	Very loose
15–35	Loose
35–65	Medium dense
65–85	Dense
85–100	Very dense

Relative compaction $= \frac{γ _{d}}{γ _{d, ma x}} \times 100$ (used in compaction control).

Grain size analysis

Sieve analysis

For particles coarser than 75 µm: shake through a stack of sieves, plot percentage finer (log scale of size) — the particle size distribution curve.

Sedimentation (hydrometer or pipette) analysis

For particles finer than 75 µm, based on Stokes' law for the terminal velocity of a sphere settling in a fluid:

v = \frac{( γ _{s} - γ _{w} ) D ^{2}}{18 μ}

Valid for particles roughly 0.2 µm to 0.2 mm; assumptions: spherical particles, no interference between particles, laminar settling. A dispersing agent (sodium hexametaphosphate) prevents flocculation.

Grading coefficients

C_{u} = \frac{D _{60}}{D _{10}}, C_{c} = \frac{D _{30}^{2}}{D _{10} D _{60}}

$D_{10}$ = effective size (10% finer). A soil is well graded when

gravel: $C_{u} > 4$ and $1 \leq C_{c} \leq 3$ ,
sand: $C_{u} > 6$ and $1 \leq C_{c} \leq 3$ .

Otherwise it is poorly graded — uniform ( $C_{u} \approx 1$ ) or gap-graded (missing sizes; $C_{c}$ outside 1–3).

Particle sizes (IS 1498)	Range
Boulder	> 300 mm
Cobble	80–300 mm
Gravel	4.75–80 mm
Sand	0.075–4.75 mm (coarse 2–4.75, medium 0.425–2, fine 0.075–0.425)
Silt	0.002–0.075 mm
Clay	< 0.002 mm

Consistency (Atterberg) limits

Fine-grained soils pass through states as water content falls: liquid → plastic → semi-solid → solid. The boundaries are:

Liquid limit (LL, $w_{L}$ ) — water content at which a groove in the Casagrande cup closes over 12.7 mm (½ inch) in 25 blows; or the cone penetrometer gives 20 mm penetration.
Plastic limit (PL, $w_{P}$ ) — water content at which a soil thread just crumbles when rolled to 3 mm diameter.
Shrinkage limit (SL, $w_{S}$ ) — the water content below which further drying causes no further volume decrease (the soil is saturated at this point).

FormulaConsistency indices

Plasticity index $I_{P} = w_{L} - w_{P}$
Consistency index $I_{C} = \frac{w _{L} - w}{I _{P}}$ (1 at PL, 0 at LL; negative if $w > w_{L}$ )
Liquidity index $I_{L} = \frac{w - w _{P}}{I _{P}}$ , with $I_{C} + I_{L} = 1$
Flow index $I_{F} = \frac{w _{1} - w _{2}}{lo g _{10} ( N _{2} / N _{1} )}$ (slope of the flow curve)
Toughness index $I_{T} = I_{P} / I_{F}$
Shrinkage ratio $S R = \frac{M _{d}}{V _{d} ρ _{w}}$ (at SL); shrinkage limit $w_{S} = (\frac{1}{S R} - \frac{1}{G}) \times 100$
Activity $A = \frac{I _{P}}{% clay fraction ( < 2 μ m )}$

$I_{P}$	Plasticity
0	Non-plastic
< 7	Low
7–17	Medium
> 17	High

Activity	Clay	Typical mineral
< 0.75	Inactive	Kaolinite
0.75–1.25	Normal	Illite
> 1.25	Active	Montmorillonite

Sensitivity and thixotropy

S_{t} = \frac{q _{u} ( undisturbed )}{q _{u} ( remoulded, same water content )}

$S_{t}$	Description
1	Insensitive
1–2	Little sensitive
2–4	Moderately sensitive
4–8	Sensitive
8–16	Extra sensitive
> 16	Quick

Thixotropy — partial regain of strength with time after remoulding, at constant water content (e.g. drilling mud, pile driving in clay: capacity increases after a rest period).

Worked examples

Worked ExampleExample 1 — phase relations

A moist sample has mass 190 g, volume 100 cm³, and dry mass 160 g; $G$ = 2.70. Find $w$ , $ρ$ , $ρ_{d}$ , $e$ , $n$ , $S$ and percentage air voids.

Solution. $w = 30/160 = 18.75%$ ; $ρ = 1.90$ g/cm³; $ρ_{d} = 1.60$ g/cm³ $e = \frac{G ρ _{w}}{ρ _{d}} - 1 = \frac{2.70}{1.60} - 1 = 0.6875$ ; $n = \frac{0.6875}{1.6875} = 0.407$ $S = \frac{w G}{e} = \frac{0.1875 \times 2.70}{0.6875} = 0.736$ ; $n_{a} = n (1 - S) = 0.407 \times 0.264 = 10.7%$

Worked ExampleExample 2 — saturated clay

A saturated clay has $w$ = 40% and $G$ = 2.70. Find $e$ , $γ_{s a t}$ and $γ^{'}$ ( $γ_{w}$ = 9.81 kN/m³).

Solution. $e = w G = 1.08$ $γ_{s a t} = \frac{( 2.70 + 1.08 ) \times 9.81}{2.08} = 17.83 kN/ m^{3}$ ; $γ^{'} = 17.83 - 9.81 = 8.02 kN/ m^{3}$

Worked ExampleExample 3 — consistency and activity

A clay has LL = 55%, PL = 25%, natural water content 35% and 30% clay fraction. Find $I_{P}$ , $I_{C}$ , $I_{L}$ and activity.

Solution. $I_{P} = 30%$ ; $I_{C} = \frac{55 - 35}{30} = 0.667$ ; $I_{L} = \frac{35 - 25}{30} = 0.333$ ; $A = 30/30 = 1.0$ (normal clay).

Worked ExampleExample 4 — relative density

A sand has $e_{ma x}$ = 0.90, $e_{min}$ = 0.45 and in-situ $e$ = 0.60. Find $I_{D}$ .

Solution. $I_{D} = \frac{0.90 - 0.60}{0.90 - 0.45} \times 100 = 66.7%$ → dense.

Frequently tested points

$S e = w G$ ; $n = e / (1 + e)$ ; $γ_{d} = γ / (1 + w)$ ; $γ^{'} = (G - 1) γ_{w} / (1 + e)$ .
Water content by oven drying at 105–110 °C for 24 h.
$C_{u} = D_{60} / D_{10}$ , $C_{c} = D_{30}^{2} / (D_{10} D_{60})$ ; well graded sand $C_{u} > 6$ , gravel $C_{u} > 4$ , $1 \leq C_{c} \leq 3$ .
Stokes' law: $v \propto D^{2}$ .
LL at 25 blows (or 20 mm cone penetration); PL at 3 mm thread.
$I_{C} + I_{L} = 1$ ; activity > 1.25 active (montmorillonite).
Sensitivity > 16 → quick clay.

Common MistakeCommon mistakes

Using total mass instead of dry mass in water content.
Confusing porosity (on total volume) with void ratio (on solids volume).
Using $lo g_{e}$ in the flow index.

Revision SummaryChapter summary

Phase relations link water content, voids, saturation and unit weights.
Laboratory and field tests measure $w$ , $G$ and density.
Relative density describes coarse soils; grading curves describe size distribution.
Atterberg limits and indices describe fine soils; activity and sensitivity add mineral and structure information.

Downloaded from civilgyantests.com · Not for resale. Verify every code clause and date against the official source.

Three-Phase System & Index Properties of Soil