Soil phase relations

phase diagram soilvoid ratioporositydegree of saturationdry unit weightsoil weight volume relationships

Void ratio, porosity, saturation, water content and the dry, saturated and submerged unit weights of a three-phase soil.

Void Ratio and Porosity (e = n/(1 − n))

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

Converts between void ratio, the void volume per unit of solid, and porosity, the void volume per unit of total soil volume.

Water (Moisture) Content

w = \frac{M_w}{M_s}\times 100

Gravimetric water content of a soil as the mass of pore water divided by the mass of oven-dry solids, expressed as a percentage.

Degree of Saturation (Se = wGs)

S = \frac{w\,G_s}{e}

Degree of saturation from water content, specific gravity of solids and void ratio, using the phase identity Se = wGs.

Dry Unit Weight from Moist Unit Weight

\gamma_d = \frac{\gamma}{1 + \dfrac{w}{100}}

Strips the pore water out of a measured bulk unit weight to give the dry unit weight used for compaction control and phase work.

Dry Unit Weight from Gs and Void Ratio

\gamma_d = \frac{G_s\,\gamma_w}{1 + e}

Dry unit weight of a soil from the specific gravity of its solids and its void ratio, the phase-diagram route used to back out e in the lab.

Saturated Unit Weight

\gamma_{sat} = \frac{(G_s + e)\,\gamma_w}{1 + e}

Unit weight of a soil whose voids are completely full of water, from the specific gravity of the solids and the void ratio.

Submerged (Buoyant) Unit Weight

\gamma' = \gamma_{sat} - \gamma_w

Effective or buoyant unit weight of soil below the water table, the saturated unit weight less the uplift of the water it displaces.

Relative Density of a Granular Soil

D_r = \frac{e_{max} - e}{e_{max} - e_{min}}\times 100

Relative density of a sand or gravel, placing its in-situ void ratio on the scale between its loosest and densest laboratory states.

How they fit together

Soil is three phases in one volume — solids, water and air — and every relation here is bookkeeping on that phase diagram. Fix any two of void ratio, water content, specific gravity and degree of saturation, and the rest follow, which is why Se = wGs turns up in nearly every derivation. The unit weights are the same diagram weighed instead of measured: dry counts solids only, moist adds whatever water is present, saturated fills every void, and submerged subtracts the buoyancy of the water the soil displaces.

Choose by which denominator your data uses, and this is where people go wrong: void ratio divides voids by solids and can exceed 1, while porosity divides voids by total volume and never can. Water content likewise divides by dry mass, not wet, so a soil at 100% water content is half water by mass and is perfectly ordinary — organic soils run several hundred percent. Relative density is a different animal again: it compares a granular soil's void ratio to its own loosest and densest states, so it is only meaningful for sands and gravels, and it is not interchangeable with relative compaction from a Proctor test. Submerged unit weight is roughly half the saturated value, and forgetting to use it below the water table is the most expensive arithmetic mistake in the list.