# Soil swelling and mass/volume relationships

Given moisture content, Specific gravity of solids, initial volume, and weight. I'm asked to compute the moist unit weight, dry unit weight, and degree of saturation of this compacted soil. This is already done. This compacted soil sample was then submerged in water. After 2 weeks, it was found that the sample had swelled and its total volume had increased by 5%. Then I'm asked to compute the new unit weight and moisture content of the soil sample after 2 weeks of submersion in water.

Moisture content and total volume are known to change, but what properties remain constant during the submersion? Can S(r) be taken as 1?

• Okay, so now i know Sr, V(total) and Specific gravity, is this enough to solve the problem?
– Gon
Feb 4 '15 at 15:14
• I'm a CE/geotech and I think there is enough info here (I'll know after I start drafting an answer). Also: I believe this question is a good example of the kind of homework type questions we actually do want on the site, as the bottom line is a request for explanation of a concept and not "please do my work for me." Feb 4 '15 at 16:36
• Its a question on theory, if you've taken the class before it should be clear to you exactly what i am asking. but ill rephrase it once more
– Gon
Feb 5 '15 at 2:04
• At risk of adding more complications, is it possible that bacteria or chemical reactions could have changed the chemical makeup of the soil? Feb 10 '15 at 21:58
• @Adam Miller sure it's possible but extremely unlikely. Most soil is primarily silica unless it's some kind of peat, and silica is chemically inert for the most part. It's just not something you see happen in the real world very often. Feb 16 '15 at 1:55

The given information describing a compacted soil sample is as follows:

• initial moisture content, $\omega_{init}$
• specific gravity, $G_s$
• initial volume, $V_{init}$
• initial weight, $W_{init}$

For completeness: the following information has already been determined:

• moist unit weight, $\gamma_{wet}$ using the relationship $\gamma_{wet}=\frac{W_{init}}{V_{init}}$
• dry unit weight, $\gamma_{d-init}$ using the relationship $\gamma_{d-init}=\frac{\gamma_{wet}}{1+\omega_{init}}$
• saturation, $S$ using the relationship $S=\frac{V_{water}}{V_{voids}}=\frac{V_{water}}{V_{init}-V_{solids}}=\frac{\frac{W_{init}\omega_{init}}{\gamma_w}}{V_{init}-\frac{\gamma_{d}V_{init}}{G_s\gamma_w}}$

(where $\gamma_w$ is the unit weight of water)

## Problem

The problem is to determine the unit weight and the moisture content after the soil sample has been submerged and allowed to swell 5%.

The key detail for this problem is:

This compacted soil sample was then submerged in water.... After two weeks...

A soil sample that has been submerged in water for two weeks can/should be assumed** to have become saturated ($S=100\%$); i.e., all of the air in the void spaces has escaped, and the void space is now 100% filled with water.

The list of soil sample properties that can be assumed to remain constant after submersion is fairly short:

• Specific gravity, $G_s$
• Weight of solids, $W_s$

All of the other properties, such as saturation, unit weight, dry unit weight, moisture/water content, void ratio, etc. are dependent on the volume of voids and the amount of water in the soil. Both the amount of water (it was submerged) and the volume (it has swelled) have changed, so ALL of these properties will also change.

Once all of this has been recognized, the remaining portion of the problem is trivial:

• New wet unit weight: $\gamma_{new}=\gamma_{sat-new}=\frac{W_s+W_{w-new}}{V_{new}}=\frac{\gamma_{d-init}V_{init}+\gamma_w(V_{new}-V_{solids})}{V_{vew}}=\frac{\gamma_{d-init}V_{init}+\gamma_w(V_{new}-\frac{\gamma_{d}V_{init}}{G_s\gamma_w})}{V_{init}(1+5\%)}$
• New moisture content: $\omega_{new}=\frac{W_{w-new}}{W_{solids}}=\frac{\gamma_w(V_{new}-V_{solids})}{W_{solids}}=\frac{\gamma_w(V_{init}(1+5\%)-\frac{\gamma_{d}V_{init}}{G_s\gamma_w})}{\gamma_{d-init}V_{init}}$

## Mechanism of Soil Swelling Behavior

The simplified effective stress equation is as follows:

$\sigma^{\prime}=\sigma-u$

Where $\sigma^{\prime}$ is the effective stress, $\sigma$ is the total stress, and $u$ is the pore water pressure.

The above equation assumes a static condition. However, when the simplified effective stress equation is imbalanced, a dynamic condition occurs and the soil must either consolidate (i.e, "shrink"), or swell. Swelling of soil occurs when the two sides of the simplified effective stress equation are not balanced, and:

1. There is positive pore water pressure inside the void space of the soil, and
2. the effective stress inside of the soil matrix is greater than the externally applied total stress minus the pore water pressure.

Said another way: when a soil is compacted, some amount of total stress is applied. Once equilibrium has been achieved, this total stress is associated with some combination of effective stress and pore water pressure. If the total stress changes, the previous combination of effective stress and pore water pressure within the soil matrix initially remains, but the imbalance this causes must dissipate over time. In order for the imbalance to dissipate, the voids must either increase in volume (swelling), or decrease in volume (consolidation), depending on the nature of the imbalance.

In this case, the total stress has been removed/reduced. The pore water pressure is "pushing" against the "walls" of the soil matrix pores (as always happens when $u>0$ - even when the simplified effective stress equation is balanced). Due to the reduction of total stress, there is too much internal stress (i.e., effective stress) being applied, and it must be relieved by a decrease in *pore water pressure * (i.e., an increase in volume). Or said another way, the applied total stress is not enough to stop the pores from expanding due to the pushing of the internal pore water pressure. Therefore the soil will swell until this imbalanced condition is resolved.

**The reasons for this assumption are somewhat complicated, and the assumption may not always be accurate. However, in general, the most conservative assumption for most soil mechanics/geotechnical problems is for the soil to be saturated. Therefore, if there is reason to believe the soil may be saturated, even if there is uncertainty, we almost always assume the soil is in fact saturated.

Look at the typical diagram of soil showing soil/water/air:

Simple

Thinking simplistically about it the items that could change:

• Soil mass can't change. No soil was added. It would be good to assume that no major chemical reactions occurred either.
• Water mass can change. It was sitting in water.
• Air can't increase if the sample was submerged. Once again ignore any major chemical reactions that might create gas.
• Mass and volume have a well defined ratio for each substance.

From these items, the only way that volume could increase would be if the volume of water increased. This would mean an increase in the volume of voids.

That is the simple (maybe naive) way to think about it.

This is also where the Atterberg Limits come into play. They define the water contents where the physical properties of the soil change.

Complicated

The more complicated way to think about the system would be to consider chemical changes to the soil. Without getting too specific into items that I'm not qualified to explain, it is possible that chemical reactions could occur that cause the soil volume to increase by itself. Think about how rust is a chemical reaction that effectively causes steel to increase in volume. This would also change the mass.

Including chemical reactions into the mix creates questions like:

• Does it make any sense to compare the properties of this new soil compound to the old soil compound?
• Is the reaction reversible? e.g. Does drying the sample cause everything to go back to the original masses and volumes?

Without more constraints on what we are working with, it is hard to give a definitive answer.

• I'd get rid of the Complicated part of the answer. ...it was found that the sample had swelled... Swelling is well defined soil mechanics terminology that means the soil sample volume has increased by virtue of purely physical processes. There's nothing chemical going on here. Feb 9 '15 at 14:42