7-1 7. and vadose zone

We have looked at two- dimensional flow, which assumes: B - horizontal - no vertical flow A When groundwater has a significant z-component, we need to extend our analysis. The flow equation would look like: ∂  ∂h  ∂  ∂h  ∂  ∂h  ∂h Kx  + Ky  + Kz  = Ss [7-1] ∂x  ∂x  ∂y  ∂y  ∂z  ∂z  ∂t This is fine in regions well below the water table (box A). In most environmental studies, however, we need to consider the region between the water table and the ground surface, which is called the vadose zone. The physics of the vadose zone is somewhat different from what we saw in Chapters 4-6. (1) Storage is controlled by the change in (θ) rather than elastic expansion and contraction (α and β). (2) (K) is dependent on θ.

The goal of this chapter is to understand the in regions near the ground surface (box B). 7-2 Storage of water in the vadose zone Below the water table, water is under positive pressure, the same way as water in a beaker. Above the water table, water is suspended between particles by some force. We need to extend the concept of pressure so that we can establish a θ-P relation that ties seamlessly with the elastic storage- pressure relation. First, we will examine how water is held by soil particles. + Electrostatic force + Water molecules are strongly polarized. - Mineral surface has uneven distribution of + and - charges, and it loves to hold water… hydrophilic.

Surface tension air Molecules near the air-water interface feel stronger force water inward than outward. A body of water tends to have the minimum surface area for a given volume. 7-3 We need to apply some force to increase the surface area of air- water interface. This force is called surface tension.

Unit of surface tension?

Capillary tube model 2r λ Fs

Fg a

Soil particles hold water in the same way as a capillary tube. Upward force Downward force 2 Fs = 2πrσ×cosλ Fg = πr ρwga σ : surface tension (≅ 0.07 N/m @ 20°C) λ : contact angle (usually small, i.e. cosλ ≅ 1)

2σ 1 Forces are balanced, Fs = Fg. ∴a = [7-2] ρw g r 7-4 Negative pressure Consider water in a beaker. Gauge pressure (P) increases downward. Note that P = 0 just below the water surface. a

elevation P = ρwga

0 gauge pres.

P = -ρwga

The same principle applies to a capillary tube inserted a to water in a beaker. elevation In this case, P = 0 at the water surface in the beaker. gauge pres. 0

Pressure head is defined by: ψ = P/ρwg P and ψ decrease (become more negative) along the tube. ψ = -a just below the air-water interface.

At the air-water interface in the tube, pressure gauge pressure abruptly changes from negative to zero. This is similar to the pressure difference between surface tension inside and outside of soap bubbles. 7-5 Soil tension and tensiometer Similarly, P and ψ in above the water table is negative. The magnitude of negative pressure is called soil tension. Soil particles are applying tension force to keep water suspended above the water table.

Soil tension is measured by a tensiometer. vacuum gauge Initially P = 0 in the air pocket. Water flows from inside the porous cup to the air pocket surrounding soil, which causes the air pocket to expand. As more water flows water out, P becomes smaller and smaller until equilibrium is reached. We can estimate soil tension from P (< 0) in the air pocket measured by a vacuum gauge. ceramic cup 7-6 New definition of the water table The water table is defined as the “surface on which ψ = 0”. Therefore, at the water table, h = ψ + z = 0 + z = z

Water-table wells and piezometers piezometer WT well Both are the devices to monitor groundwater, but each has a z = 950 different purpose. 947 945 A water-table (WT) well intersects the water table. The water level in WT wells represents the right at the water table. 933 In contrast, the water level in piezometers represents the hydraulic head at the screen. What is the vertical flow direction? Hydraulic gradient (dh/dz)? 7-7 Tube-bundle model In each tube, water rises to a height given by Eq. [7-2].

P3 P2

P1

The magnitude of P below the meniscus indicates the degree of tension. In a straw bundle, moisture content is determined by the proportion of tubes that have water at a given a (or P).

θ3 Note that; ψ = P/ρ g = -a P3 w and observe the relationship between ψ and θ.

θ2 Note also that the bundle is P 2 saturated at level #1, even a 3 though it is above the water surface in the bucket. θ1 P1 This saturated zone above the water table is called capillary fringe. 7-8 The ψ-θ relationship in real soils Real soils are not quite the same as a bundle of tubes, but the magnitude of tension applied to water by soil particles is inversely proportional to pore size. Below the water table, water experiences positive pressure (ψ > 0). Water is under negative pressure (ψ < 0) in the capillary fringe even though the soil is still saturated. Above the capillary fringe, the largest pores in the soil can no longer keep water and the soil becomes unsaturated. As water content decreases upward, water is stored in smaller pores that can provide stronger tension.

unsaturated zone vadose zone capillary fringe ψ = 0

Energy state of water Remember that pressure (N/m2) also represents potential (J/m3). The same principle holds for P < 0. As the soil becomes drier, pressure potential decreases and the energy state of water becomes lower. This concept is very important for understanding the flow of soil water. 7-9 Examples of soil water characteristic curve The relationship between ψ and θ is called the soil-water characteristics. See the example below. -106 -100 clay-rich soil

5 ) -10 -10 3

4

-10 (m) - 1

ψ sandy soil 3 energy state (J/m -10 - 0.1

-102 - 0.01 00.20.4 θ The magnitude of ψ indicates the degree of soil tension. Higher tension corresponds to lower energy state; water is more strongly held in pore spaces. Compare θ of the two soils at ψ = -1 m.

Why are they different? Small-size pores in clay-rich soil can hold water more tightly than large-size pores in sandy soil. 7-10 Storage of water in unconfined aquifer Suppose a vertical column of unconfined aquifer (see below). What are the required conditions for having no flow in the column?

This condition is called hydrostatic. dψ Recall: h = ψ + z ∴ = dz

In this example, the water table is located at 948 m above the mean sea level, and 2 m below the ground surface. -2 ψ (m) 0 950 h = z + ψ

no flow (m) z ψ z

948

948 950 h (m), z (m)

Study this diagram carefully and observe that the magnitude of ψ is equal to the distance above the water table. 7-11 Under hydrostatic condition, θ decreases monotonically from the water table to the ground surface. The shape of θ-z curve is identical to that of θ-ψ curve. (see page 7-9)

0 θ n -2 ψ (m) 0 950

no flow (m) z

948

Suppose we drain the column to lower the water table and reestablish hydrostatic condition. 0 θ n This area represents the total amount of water drained by lowering the water table. 7-12 Specific yield When the water table (WT) is lowered in a sediment column, significant amount of water is retained in the sediment.

gravel silt The amount of water drained per unit drop a a of WT is referred to as specific yield (Sy).

Sy = b/a

For gravel, S ≅ n b b y

Typical values of Sy for different types of sediments are listed in page 4-12.

Caution: The above definition of Sy assumes hydrostatic condition and strictly one-dimensional system. These assumptions are hardly met in real field situations and recent

studies suggest that Sy should not be used in rigorous scientific studies. Nevertheless Sy is still widely used by environmental consultants and water resources engineers. 7-13 Hydraulic conductivity in the vadose zone Remember that the hydraulic conductivity (K) of saturated sediments is roughly proportional to (pore radius)2. As the soil becomes unsaturated, larger pores drain first. The remaining pores have smaller radii and higher resistance to flow. As a result, K decreases dramatically as water content decreases. This diagram demonstrates this effect for a typical prairie soil having a high clay content. 10-7

10-8 (m/s)

K 10-9

10-10 0.2 0.3 0.4 0.5 θ ∂h Darcy’s law needs an adjustment: q = −K (θ ) etc. z z ∂z Noting that θ is a function of ψ, we can also write: ∂h q = −K (ψ ) etc. z z ∂z Therefore, K is dependent on the variable h ( = ψ + z) itself. This is called a non-linear dependence of K on h, and has a major influence on evaporation and transpiration. 7-14 Evapotranspiration and subsurface water flow When evapotranspiration takes water away, the soil becomes dry and θ decreases, which causes the soil tension to increase and ψ to decrease (see page 7-9).

θ -2 ψ 950 hydrostatic hydrostatic

z

948

As ψ decreases near the surface, h also decreases. This creates a negative hydraulic gradient, which drives water to flow upward from the water table to the root zone. -2 ψ 0 950

h = z + ψ

z ψ z 948

948z, h 950 7-15 K as the rate-limiting factor for evapotranspiration The K of unsaturated soil decreases dramatically as the soil water is taken away. To overcome the increased flow resistance, hydraulic gradient (dh/dz) becomes steeper and h (and ψ) becomes lower. This causes K to decrease even more!

The K eventually becomes so small that the flow of liquid water stops. Vapor dry flow in dry soil is very small compared moist to liquid water flow in moist soil.

Actual evaporation from the dry soil surface is much smaller than potential evaporation. Plants can still transpire a significant amount of water, as long as soil water is accessible to the root system. However, the soil in the root zone also becomes dry after a prolonged period of no precipitation.

Summer fallow in the prairies Dry land agriculture requires leaving the land cultivated but unseeded every 3-4 years. This practice is called summer fallow. Very little evaporation occurs on the summer-fallow field and is conserved during the summer, which reduces the moisture deficit in the following year. 7-16 Suppose a soil column under hydrostatic condition. The water table is 5 m below the ground surface. θ ψ (m) h (m) 0 n -5 0 945 950 (m) z

945

rain θ ψ (m) h (m) 0 n -5 0 945 950

wetting ∂h slope = front ∂z

θ ψ (m) h (m) 0 n -5 0 945 950

∂h ∂z

∂h Recall Darcy’s law: q = −K(ψ) ∂z 7-17 K(ψ) decreases with ψ. The maximum value of K occurs when the soil is completely saturated (ψ = 0). This value is called saturated hydraulic conductivity (Ksat). Within the saturated zone above the wetting front, ∂h q ≅ −K sat ∂z The gradient is very steep at early time and decreases with time, so as the infiltration rate (-q). 6 Field School May 2000, Richmond, BC 4

2

infiltration rate (mm/min) 0 0 1020304050 time (min)

Recall: h = ψ + z ∂h  ∂ψ ∂z   ∂ψ  ∴−q ≅ Ksat = Ksat +  = Ksat +1 ∂z  ∂z ∂z   ∂z  ∂ψ Note that: ≅ 0 within the saturated zone. ∂z

At the late stage of infiltration tests, inf. rate = Ksat. 7-18 Infiltration and runoff

During major rain events Ksat of the top soil determines the infiltration capacity of the land. The rainfall intensity of most storm events is less than 70 mm/hr (= 2 × 10-5 m/s), and Ksat of most soils under natural condition is larger than -5 10 m/s. Therefore, Ksat is rarely exceeded by rainfall intensity. Alteration of the soil surface by compaction and paving may reduce Ksat by orders of magnitude and increase the chance of severe runoff and soil erosion.

Infiltration and Infiltration of rain or snowmelt water is followed by a period of evapotranspiration. The flow direction in the vadose zone changes frequently depending on the boundary condition, and the water table (WT) fluctuates accordingly. If the amount of infiltration exceeds evapotranspiration on average, there will be a net addition of water to the water table, which is called groundwater recharge. The depth of water table affects the rate of evapotranspiration. As the WT goes down the upper soil zone becomes drier, which reduces evaporation and root uptake. In contrast, the shallow WT helps to sustain high evapotranspiration rate. This “negative feedback” mechanism tends to keep the WT within a certain depth below the ground surface. 7-19 Water table and topography

The water table tends to stay within a certain depth of the ground surface.

In a region having homogeneous climate and geology, the water table forms a subdued replica of the ground surface. This is one of the most important guiding principle for understanding the occurrence of shallow groundwater. High water table under uplands drives groundwater to lowlands. Groundwater is recharged under uplands and discharged into streams and lakes. We will use this concept in the next chapter to study regional groundwater system and groundwater-stream interaction. 7-20 Flow equation in the vadose zone Recall the groundwater flow equation: ∂  ∂h  ∂  ∂h  ∂  ∂h  ∂h Kx  + Ky  + Kz  = Ss [7-1] ∂x  ∂x  ∂y  ∂y  ∂z  ∂z  ∂t In the vadose zone, the flow equation can be written as: ∂  ∂h  ∂  ∂h  ∂  ∂h  ∂θ Kx (ψ)  + Ky (ψ)  + Kz (ψ)  = ∂x  ∂x  ∂y  ∂y  ∂z  ∂z  ∂t

Noting that θ is a function of ψ; ∂θ ∂ dθ ∂ψ dθ ∂h ∂h = [θ(ψ)]= = = C (ψ) ∂t ∂t dψ ∂t dψ ∂t w ∂t where Cw ( = dθ/dψ) is called soil water capacity, which is a function of ψ.

The flow equation in the vadose zone is written as:

∂  ∂h  ∂  ∂h  ∂  ∂h  ∂h Kx (ψ)  + Ky (ψ)  + Kz (ψ)  = Cw(ψ) ∂x  ∂x  ∂y  ∂y  ∂z  ∂z  ∂t [7-3]

Eq. [7-3] is very similar to [7-1] except for the non-linear dependence of K and Cw on ψ. It is very difficult to solve non-linear differential equations like [7-3]. Interested students can take a higher-level course.