Capillary action is the rise or depression of a liquid in a small tube or pore due to surface tension, cohesion within the liquid, and adhesion between the liquid and the solid surface.
Capillary rise is the movement of water above the groundwater table against gravity while still being connected to the water table as its source.
Soil moisture is water in soil above the water table. Capillary moisture is the water held above the water table by capillary action. The vadose zone is the unsaturated zone above the groundwater table.
Attraction Forces: Cohesion and Adhesion
Cohesion is attraction between molecules of the same liquid. In water, hydrogen bonding gives strong cohesion and helps create surface tension.
Adhesion is attraction between the liquid and a solid surface. If adhesion to the wall is stronger than liquid cohesion, water wets the surface and climbs, forming a concave meniscus.
For mercury in glass, cohesion within mercury is stronger than adhesion to glass, so the meniscus is convex and the liquid surface is depressed instead of raised.
Surface Tension and Meniscus
Surface tension is the tendency of a liquid surface to act like a stretched membrane. It is one reason small insects can stand on water. In soil, surface tension helps create suction in partially saturated pores.
Meniscus is the curved liquid surface in a pore or capillary tube. The curvature of the meniscus is what creates capillary pressure.
$$\sigma \approx 0.064 \text{ N/m at normal room temperature}$$
$$\sigma \approx 0.067 \text{ N/m at freezing temperature}$$
The surface tension of clean water at about 20 degrees Celsius is often taken as about $0.0728 \text{ N/m}$ in physics-style capillary tube problems.
Contact Angle and Wetting
The contact angle $\theta$ measures how the liquid meets the solid surface. Small contact angles indicate wetting and capillary rise. Large contact angles indicate non-wetting behavior and possible capillary depression.
For a circular tube, smaller diameter means larger capillary rise.
Capillary Pressure and Soil Suction
Capillary rise can also be viewed as a suction problem. The curved meniscus causes water pressure to be lower than air pressure above the water table.
$$u_a-u_w = \frac{2\sigma\cos\theta}{r}$$
$$u_a-u_w = \frac{4\sigma\cos\theta}{d}$$
$$u_a-u_w = \gamma_w h_c$$
This suction gives partially saturated granular soils an apparent cohesion. It is not true cementation; it can disappear when the soil becomes submerged or fully saturated.
Capillary Rise in Soils
Soil pores behave like many irregular capillary tubes. Fine-grained soils have smaller pores, so they can pull water higher. Coarse sands and gravels have larger pores, so capillary rise is smaller.
Capillary water does not exist below the groundwater table because the soil there is submerged. Capillary action ceases where submergence occurs.
Capillary conductivity or capillary permeability describes the rate of capillary rise, not just the final height.
Concept: Apparent Cohesion
In moist sand, water menisci form between grains. Surface tension pulls grains together, creating negative pore water pressure and a temporary strength gain called apparent cohesion.
$$\sigma' = \sigma - u_w$$
When $u_w$ is negative due to suction, effective stress increases. This explains why damp sand can stand temporarily, while dry sand or submerged sand loses that apparent bond.
Problem: Capillary Rise in Clean Glass Tube
A clean glass capillary tube with diameter 0.2 mm is inserted into water with surface tension $0.0728 \text{ N/m}$. Calculate the height of capillary rise in the tube. Assume water wets clean glass, so $\theta=0^\circ$.
$$h = \frac{4\sigma\cos\theta}{\gamma_w d}$$
$$h = \frac{4(0.0728)\cos0^\circ}{9810(0.0002)}$$
$$h = 0.148 \text{ m}$$
Answer: The capillary rise is about 0.148 m, or 148 mm.
Problem: Mercury Capillary Depression
Determine the capillary depression of mercury in a glass capillary tube with diameter 2 mm when the contact-angle term is based on $\theta=40^\circ$. Surface tension is $0.514 \text{ N/m}$. Use $SG_{Hg}=13.6$.
Answer: The mercury level is depressed by about 5.90 mm.
Problem: Capillary Suction from Rise
If capillary water rises 0.80 m above the water table, estimate the capillary suction at the top of the water column.
$$u_a-u_w = \gamma_w h_c$$
$$u_a-u_w = 9.81(0.80)=7.85 \text{ kPa}$$
Answer: The capillary suction is about 7.85 kPa.
Problem: Equivalent Pore Diameter
Water rises 0.50 m in an idealized soil pore. If $\sigma=0.0728 \text{ N/m}$ and $\theta=0^\circ$, estimate the equivalent pore diameter.
$$d = \frac{4\sigma\cos\theta}{\gamma_w h}$$
$$d = \frac{4(0.0728)\cos0^\circ}{9810(0.50)}$$
$$d = 5.94\times10^{-5} \text{ m}$$
Answer: The equivalent pore diameter is about 0.059 mm.
Concept: Capillary Rise and Effective Stress in Soils
In partially saturated soils above the water table, the meniscus between soil grains creates a negative pore water pressure, or capillary suction. This suction increases the effective stress between grains beyond what gravity alone would cause. The effect is called apparent cohesion because it can make loose sand stand in steep cuts temporarily, but it vanishes when the soil becomes submerged or dries out completely and the menisci collapse.
The capillary zone above the water table can be divided into the saturated capillary fringe (pores completely filled with water under tension) and the partially saturated zone above it. In the fully saturated capillary fringe, $S=1$ but pore pressure is negative. The effective stress in this zone is higher than in an equivalent submerged layer.
Problem: Capillary Rise in Two Tubes of Different Diameter
Two clean glass tubes are inserted into water. Tube A has a diameter of 0.1 mm and Tube B has a diameter of 0.5 mm. The surface tension of water is 0.0728 N/m and the contact angle is zero. Find the height of capillary rise in each tube and compare.
The tube with the smaller diameter has 5 times more capillary rise. Fine-grained soils with small pores can pull water much higher than coarse sands.
Answer: $h_A=0.297$ m and $h_B=0.059$ m. Smaller diameter produces greater rise.
Problem: Surface Tension from Measured Rise Height
Water rises 220 mm in a clean glass capillary tube with a diameter of 0.136 mm. Assuming a contact angle of zero, back-calculate the surface tension of water.
Answer: The surface tension is approximately 0.0733 N/m, which is close to the accepted value of 0.0728 N/m at 20 degrees Celsius.
Problem: Effective Stress Increase from Capillary Suction
A fine sand deposit has a water table 2.5 m below the ground surface. The capillary fringe extends 0.8 m above the water table and is fully saturated in this zone. The saturated unit weight is 19.5 kN/m3 and the dry unit weight above the capillary fringe is 15.8 kN/m3. Compute the effective vertical stress at the top of the capillary fringe and at the water table.
The capillary fringe is from 1.7 m to 2.5 m below the surface. Above 1.7 m, the soil is dry.
$$\sigma_v\text{ at top of capillary fringe}=15.8(1.7)=26.86\text{ kPa}$$
Pore pressure at the top of the capillary fringe is negative (suction = $\gamma_w h_c$):
$$u = -9.81(0.8)=-7.85\text{ kPa}$$
$$\sigma'_v = 26.86-(-7.85)=34.71\text{ kPa at top of capillary fringe}$$
Answer: $\sigma'_v=34.71$ kPa at the top of the capillary fringe and $\sigma'_v=42.46$ kPa at the water table.
Problem: Required Tube Diameter for Target Rise Height
An engineer wants water to rise exactly 1.20 m in a capillary tube as part of a demonstration. The surface tension is 0.0728 N/m and the contact angle is zero. What tube diameter is required?
