Module 5 · Lesson 5.1

The three-phase model

Volumes and masses of solids, water and air, and the ratios that describe them.

Why this matters

Almost everything a soil does — how much it settles, how strong it is, how fast water drains through it — depends on how much void space it has and how much of that space is filled with water. Phase relationships are the bookkeeping that turns a few simple measurements (a mass, a volume, an oven-dried mass) into the quantities the rest of soil mechanics needs.

What you should already know

Comfortable with ratios and rearranging simple formulae
The idea of density and unit weight (mass or weight per unit volume)

A soil is a mixture of three phases: mineral solids, water and air. We picture them separated into a phase diagram — a single column with the solids at the bottom, water above them and air at the top — so we can talk about the volume and mass of each phase separately.

Let the volumes be $V_{s}$ (solids), $V_{w}$ (water) and $V_{a}$ (air), and the masses be $M_{s}$ and $M_{w}$ (air has negligible mass). The voids are everything that is not solid:

V_{v} = V_{w} + V_{a}, V = V_{s} + V_{w} + V_{a} .

Every phase relationship below is just a ratio of these quantities. Nothing here is empirical — it is definition and geometry.

Void ratio

e = \frac{V _{v}}{V _{s}}

What it calculates: the volume of voids per unit volume of solids

e: void ratio (dimensionless)
V_v: volume of voids (m³)
V_s: volume of solids (m³)

In plain terms: Void ratio measures voids against the solids only, so it does not change when the soil is compressed unless the grains rearrange. It can exceed 1 for a very open soil.

Porosity

n = \frac{V _{v}}{V}

What it calculates: the fraction of the total volume that is voids

n: porosity (fraction, or ×100 for %)
V: total volume (m³)

In plain terms: Porosity measures voids against the total volume, so it is always between 0 and 1.

From first principles

Relating porosity and void ratio

We want to show: Show that porosity and void ratio are two views of the same thing: n = e/(1+e).

Both measure void space; they only differ in what they compare it against — porosity against the whole, void ratio against the solids. Fixing the solids volume at 1 makes the algebra fall out.

Two more ratios describe how much water is present:

Water content $w = M_{w} / M_{s}$ — the mass of water per mass of solids, found by oven-drying. Often quoted as a percentage; a soft clay might have $w = 45%$ .
Degree of saturation $S = V_{w} / V_{v}$ — the fraction of the void space filled with water. $S = 0$ is dry, $S = 1$ is fully saturated.

and one describes the solids themselves:

Specific gravity $G_{s} = ρ_{s} / ρ_{w}$ — the density of the mineral grains relative to water. Most soil minerals give $G_{s}$ between about 2.6 and 2.75, so 2.65–2.70 is a reasonable default when a measured value is not available.

Summary

A soil has three phases: solids, water and air; voids = water + air.
Void ratio e = Vv/Vs references the solids; porosity n = Vv/V references the whole.
n = e/(1+e) and e = n/(1−n) — the same information, two forms.
Water content w = Mw/Ms; degree of saturation S = Vw/Vv; specific gravity Gs = ρs/ρw ≈ 2.65–2.70.

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This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint