• The flow of water in the unsaturated zone is complicated.
• The the pore spaces contain both soil gas and water.
• Hydraulic constants, such as conductivity and diffusion, become hydraulic functions.
• The transmissive/storage properties can change dramatically depending on the water content.

In the unsaturated (vadose) zone, pores contain both soil gas and water. Two core descriptors:

1. Volumetric water content:

V_w: volume of water, V_T: total sample volume, n: porosity.

2. Water saturation:

Key point In the unsaturated zone, typically \theta < n and s < 1.

• For a direct measurement of water content \theta_w as a weight fraction:
  1. Weight the soil in a covered container.
  2. Dry it in a convection oven for 24 hours.
  3. Weight it again.
  \theta_w = \frac{W_{wet - soil}}{W_{dry - soil}} -1

  Volumetric water content: \theta = \theta_w \,\dfrac{\rho_b}{\rho_w}

Indirect methods: measuring properties associated with water content.

• Electrical conductivity: Measuring the electrical resistivity of the soil: dependendent on water content.
• Time-domain reflectometry: Measuring dielectric constant of the soil: dependent on water content.
• gamma-ray attenuation: Water absorbs more gamma rays than soil.

Hydraulic head: determines the direction of fluid flow.
It combines elevation and pressure:

• Above the water table: \psi < 0 (suction/tension). Water pressure less than atmospheric pressure. Explains why water in unsaturated zon does not flow to a borehole.
• At the water table: \psi = 0.
• Below the water table (saturated): \psi > 0.

Tensiometer (ASTM 1992; Stannard 1986):

• Porous ceramic cup connected to a water column/manometer, vacuum gauge, or pressure transducer.
• Cup is water-filled → provides hydraulic connection to pore water.
• As soil water pressure changes, instrument records \psi (negative) to maintain equilibrium.

From the text example:

• Pressure head \psi: negative above the water table, positive below.
• Total head h = z + \psi: decreases with depth through the unsaturated zone → downward flow.

• Water content in the unsaturated zone varies with space and time.
• At early times → capillary forces dominate, increasing water content near the surface.
• Later → infiltration ceases, and water redistributes downward.
• Both capillary and gravitational forces act.
• A wetting front forms: narrow zone where water content sharply increases.

• After infiltration stops, the infiltrated pulse moves downward.
• The wetting front is clearly visible.
• Soil above the front gradually dries, while water below remains at higher content.
• This redistribution continues until equilibrium is reached or evapotranspiration intervenes.

• Near the surface, evapotranspiration removes water once infiltration ends.
• Eventually, infiltrated water reaches the saturated zone.
• The unsaturated zone returns toward its initial condition, with soil moisture gradually reduced.
• Figure illustrates soil water entering saturated regions.

• In unsaturated media, flow is complex due to two-phase flow (air + water).
• Both volumetric moisture content (\theta) and hydraulic conductivity (K) depend on water content.
• The pressure head/moisture content relationship describes how a sample behaves as water is added or removed.
• As soil dries, void spaces fill with air → harder for water to flow.

• As \theta decreases → \psi becomes more negative → capillary pressure increases.
• \theta is always on arithmetic scale, while pressure could be on log scale.
• Curves are nonlinear regardless of scaling.
• At extremes: small variation in \theta results in big variation in \psi.
• Minimum \theta approaches residual volumetric water content \theta_r.

To develop mathematical relationships, researchers measure different soil behaviors in the lab and fit equations to the data.

Effective saturation normalizes water content to be independent of soil type:

where:

• s_e = effective saturation (0 to 1)
• \theta_r = residual volumetric water content
• n = porosity (maximum \theta can be)

Both are written in terms of effective saturation s_e.

Brooks–Corey equation defines effective saturation as:

Parameters:

• \psi_b = bubbling or air-entry pressure head [L]
  • Pressure head at which air bubbles start entering the medium
  • Pressure needed to desaturate the largest pores in the medium
• \lambda = pore-size distribution index (dimensionless)

van Genuchten equation is expressed as:

Parameters:

• \alpha = scaling parameter [1/L]
• \beta = curve shape exponent (dimensionless)
• \gamma = 1 - \dfrac{1}{\beta} (constraint for van Genuchten)

Hysteresis: The actual behavior depends on whether air is entering (drying phase) or water is entering (wetting phase).

• Drying → air enters pores, displaces water
• Wetting → water enters, displaces air
• Shape is slightly different for each process

Soil type impact depends on pore-size distribution:

• Sand → uniform, large pores → sharp curves
• Fine sand → intermediate behavior
• Silt loam → broad pore size distribution → gradual curves, higher \theta at same \psi

As expected, for unsaturated media K is strongly dependent on the degree of saturation.

• As water content approaches zero → negative hydraulic pressure increases more and more.
• Close to residual volumetric content \theta_r → hydraulic conductivity becomes practically zero.
• Near saturation (\psi \approx 0) → K \approx maximum value.

To capture this behavior, we introduce the concept of relative hydraulic conductivity or permeability.

This shows us the ratio with respect to the saturation property:

where:

• K_r = relative hydraulic conductivity (0 to 1)
• K(\psi) = unsaturated hydraulic conductivity
• K_s = saturated hydraulic conductivity

Going with the same line of thought, the hydraulic conductivity can be written in terms of the negative pressure head:

Parameters:

• \psi_b = bubbling (air-entry) pressure head
• \lambda = pore-size distribution index

In the same way, we can describe the van Genuchten relationship as:

Parameters:

• \alpha = scaling parameter [1/L]
• \beta = curve steepness parameter
• \gamma = 1 - \dfrac{1}{\beta} (van Genuchten constraint)

Alternative form in terms of s_e:

with l \approx 0.5 (pore connectivity parameter).

The storage properties of an unsaturated soil can be represented by a parameter called specific moisture capacity c_m.

This is basically the derivative of water content with respect to pressure.

Physical meaning: Shows how much \theta changes for a given change in \psi.

Brooks–Corey model:

van Genuchten model:

Key models:

• Brooks-Corey: Uses bubbling pressure \psi_b and pore-size distribution \lambda
• van Genuchten: Uses scaling parameter \alpha, shape parameter \beta, and constraint \gamma = 1 - 1/\beta

All relationships are strongly nonlinear and depend on soil type and pore-size distribution.

Darcy’s law (unsaturated form):

• K(\psi): unsaturated conductivity
• h: hydraulic head

Relative conductivity form: K(\psi) = K_r(\psi) K_s

• K_r: relative conductivity (0–1)
• K_s: saturated conductivity

Combining Darcy + conservation of mass:

• General tensor form with K_{ij}.
• Inflow – outflow = storage change.
• Equivalent to Richards’ equation for unsaturated flow.

For vertical flow (z positive downward):

• Includes both pressure gradient and gravity term (+1).
• Common in infiltration models.

Linear groundwater (pore water) velocity:

• q: Darcy velocity
• \theta: volumetric water content
• s: saturation, n: porosity