Index Properties of Soil — Void Ratio, Porosity, Degree of Saturation

1. Three-Phase Diagram

Unlike steel or concrete, soil is not a continuous solid. It is a particulate material consisting of solid mineral grains with voids between them. The voids are filled partly or wholly with water. This three-component nature — solid, water, air — is captured in the three-phase diagram (also called the phase diagram or block diagram), which is the starting point for all index property relationships.

1.1 Notation

1.2 Phase Relationships

For a fully saturated soil: V_a = 0, so V_v = V_w.
For a completely dry soil: V_w = 0, so V_v = V_a.

2. Volume Ratios — Void Ratio & Porosity

2.1 Void Ratio (e)

Void ratio is preferred over porosity in geotechnical calculations because when the volume of soil changes (due to consolidation or compaction), the volume of solids V_s remains constant — making e easier to track mathematically.

2.2 Porosity (n)

2.3 Relationship Between e and n

Derivation: n = V_v/V = V_v/(V_s+V_v). Dividing numerator and denominator by V_s: n = (V_v/V_s) / (1 + V_v/V_s) = e/(1+e). ✓

2.4 Air Voids Ratio (a_v) and Air Content (A_c)

3. Water Content & Degree of Saturation

3.1 Water Content (w)

Water content is always expressed relative to the dry mass of solids, not total mass. This is because M_s is constant for a given soil sample regardless of wetting or drying, whereas total mass M changes.

3.2 Degree of Saturation (S)

4. Specific Gravity of Soil Solids (G_s)

G_s is determined by the pycnometer method (IS 2720 Part 3). Unless stated otherwise in problems, use G_s = 2.67 for sand and G_s = 2.70 for clay.

5. The Fundamental Relation: Se = wG_s

This is the single most useful equation in the index properties topic — it connects four key parameters.

Derivation

Special Cases

6. Unit Weights of Soil

Unit weight γ = W/V (weight per unit volume, kN/m³). All unit weight formulas are derived from the three-phase diagram using G_s, e, S, and γ_w = 9.81 kN/m³ ≈ 10 kN/m³.

6.1 Summary of Unit Weight Formulas

6.2 Relationship Between Bulk and Dry Unit Weight

6.3 Derivation of γ_sat

6.4 Zero Air Voids Line (ZAV)

7. Relative Density (D_r)

Relative density (also called density index I_D) describes how dense a coarse-grained soil (sand or gravel) is relative to its loosest and densest possible states.

e_max is measured by pouring dry sand loosely into a mould (minimum density test, IS 2720 Part 14). e_min is measured by vibrating saturated sand to maximum density (IS 2720 Part 14).

8. IS 2720 Test Methods

9. Worked Examples

Example 1 — Complete Phase Diagram (GATE CE type)

Problem: A soil sample has a bulk unit weight of 19.2 kN/m³, water content of 12 %, and G_s = 2.68. Find: (a) dry unit weight, (b) void ratio, (c) porosity, (d) degree of saturation.

(a) Dry Unit Weight

(b) Void Ratio

(c) Porosity

(d) Degree of Saturation

Example 2 — Saturated Unit Weight and Submerged Unit Weight

Problem: A clay sample has void ratio e = 0.80 and G_s = 2.72. Compute the saturated unit weight and submerged unit weight. Take γ_w = 9.81 kN/m³.

Alternatively: γ′ = (G_s−1)γ_w/(1+e) = (2.72−1)×9.81/1.80 = 1.72×9.81/1.80 = 9.37 kN/m³ ✓

Example 3 — GATE-Style: Find Missing Parameters

Problem (GATE CE 2018 type): A saturated soil sample (S = 100 %) has water content w = 40 % and G_s = 2.70. Find: (a) void ratio, (b) porosity, (c) saturated unit weight.

(a) Void Ratio

(b) Porosity

(c) Saturated Unit Weight

Note: High void ratio (e = 1.08) is typical for soft saturated clay — the result γ_sat ≈ 17.8 kN/m³ is consistent with soft clay behaviour.

10. Common Mistakes

Mistake 1: Expressing Water Content as M_w/M (Total Mass) Instead of M_w/M_s

What happens: Water content is computed as water mass divided by total mass, giving a value that is always less than 100 % regardless of how wet the soil is. This is sometimes called “moisture ratio” but is NOT what IS 2720 defines as water content.

Fix: Water content w = M_w/M_s × 100 %. In very soft clays this can exceed 100 % — that is perfectly valid and means the mass of water exceeds the mass of dry solids.

Mistake 2: Confusing Void Ratio e with Porosity n

What happens: Both measure void space but relative to different references. Substituting e into a formula that requires n (or vice versa) gives wrong answers. For example, using e = 0.4 where n = 0.4 are very different states (n = 0.4 corresponds to e = 0.667).

Fix: e = V_v/V_s (always ≥ 0, can exceed 1); n = V_v/V (always between 0 and 1). Convert using e = n/(1−n) or n = e/(1+e) before substituting.

Mistake 3: Forgetting that γ_d = γ / (1+w) Requires w as a Decimal

What happens: Writing γ_d = γ / (1 + 12) instead of γ / (1 + 0.12) when w = 12 %. This gives an answer roughly 12 times too small.

Fix: In all formulas, w must be a decimal fraction (e.g., w = 12 % → w = 0.12). The × 100 % is only for reporting the final value of w.

Mistake 4: Using γ_w = 10 kN/m³ Without Checking Whether the Problem Uses 9.81

What happens: Many GATE problems specify γ_w = 9.81 kN/m³ in the data. Using 10 kN/m³ introduces a ~2 % error which can shift answers between options. Some problems explicitly use 10 kN/m³ — always check.

Fix: Use whatever γ_w the problem gives. When not stated, use 9.81 kN/m³ for precision or 10 kN/m³ for quick mental arithmetic — but be consistent throughout.

Mistake 5: Applying Se = wG_s With S as a Percentage Instead of a Decimal

What happens: Writing Se = wG_s with S = 60 and w = 0.12 gives 60 × e = 0.12 × 2.68, yielding e = 0.00536 — clearly wrong. Both S and w must be decimals in this formula.

Fix: In Se = wG_s, S is a decimal (e.g., 60 % → 0.60) and w is a decimal (e.g., 12 % → 0.12).

11. Frequently Asked Questions

Q1. Can void ratio e exceed 1.0? What does that mean physically?

Yes — void ratio can and does exceed 1.0 for many natural soils, particularly soft clays, silts, and organic soils. A void ratio of e = 1.0 simply means the volume of voids equals the volume of solid particles; e = 2.0 means the voids are twice the volume of solids. There is no theoretical upper limit for void ratio (peat has e values up to 10 or more). High void ratio indicates a loose, compressible soil that will undergo large settlements under load. Only for porosity n is the range physically restricted to 0–1, because n = V_v/V — you cannot have more void volume than total volume.

Q2. Why is void ratio e preferred over porosity n in geotechnical formulas?

When soil compresses (consolidates) or expands (swells), the volume of solid particles V_s remains essentially constant — only V_v changes. Because e = V_v/V_s and V_s is constant, tracking changes in e directly tracks changes in void volume: Δe = ΔV_v/V_s. In contrast, porosity n = V_v/V, and V itself changes during consolidation, making the mathematics more complex. For this reason, consolidation theory, void ratio–pressure relationships (e-log p curves), and all settlement calculations use e as the primary variable.

Q3. What is the practical significance of submerged (buoyant) unit weight?

When soil is submerged below the water table, each soil element experiences an upward buoyant force equal to the weight of water it displaces. The net downward unit weight is therefore γ′ = γ_sat − γ_w. This submerged unit weight is used whenever computing vertical effective stress below the water table: σ′_v = γ′ × z (for soil below water table). Using γ_sat in place of γ′ for submerged soil would compute total stress, not effective stress — a fundamental error that appears repeatedly in GATE problems.

Q4. How is relative density different from degree of compaction used in pavement design?

Relative density D_r is used exclusively for coarse-grained soils (sands and gravels) and describes how compact a deposit is relative to its possible loosest and densest states — it is based on void ratio. Degree of compaction (or relative compaction), used in pavement subgrade and embankment engineering, is the ratio of the field dry density to the maximum dry density from the Proctor test: RC = (γ_d,field/γ_d,max) × 100 %. It is applicable to any soil being compacted in a fill operation. The two measures are conceptually similar (both describe how compact the soil is relative to a reference state) but they use different reference states and different test procedures.