Physical Hydrogeology Unsaturated Flow (brief lecture) Why study the unsaturated zone? Evapotranspiration Infiltration Toxic Waste Leak Irrigation UNSATURATAED ZONE Aquifer Important to: Agriculture (most research) Toxic waste Runoff generation Interface w/atmosphere (GCM) Points regarding Unsaturated or Vadose Zone or Zone of Aeration: occurs above the water table pores are only partially filled with water; the moisture is content less than the porosity; zone consists of air, water, and rock fluid pressure, p, is less than atmospheric pressure, and fluid pressure head,, is less than zero; under tension hydraulic head is measured with aid of a tensiometer hydraulic conductivity, K, and moisture content,, are functions of the pressure head [or, K( ), ( )] I. Definitions: 1
= volumetric moisture content (decimal fraction) = volume of water / bulk volume of soil (L3/L3) = pressure head, p / g, where pressure, p, is the gauge pressure (that above atmospheric); is the fluid density and g is the gravitational constant. At water table = 0, and because we always have h = + z, the head at the water table is equal to z, or h = z. Our quantities in (L) units are: h = hydraulic head (energy per unit weight) z = elevation head = pressure head pressure above water table --- < 0 pressure below the water table --- >0 When < 0, this is called the tension head or suction head. Some call it pressure head, but its value is negative. Unsaturated Zone when a = ( ) K = K( ) Saturated Zone when > a = n K = Ksat when a there is a capillary fringe; saturated under tension. 2
Vol. moisture content unsaturated zone capillary fringe h=z Z=0 saturated zone Saturation moisture content equals porosity II. How to measure Because pressure head is negative in the unsaturated zone, we can't use piezometer to measure head; no standing water in the well above water table. Tensiometer: Directly measures vacuum gage A porous cup is attached to an airtight water-filled tube. Porous cup inserted into soil at a known depth where it comes into contact with the soil and reaches hydraulic equilibrium. The vacuum (negative pressure) created at the top of the tube is measured as 3
III. Hydraulic Conductivity in the Unsaturated Zone Ultimately hydraulic conductivity, K, depends on moisture content,, and pressure head, ---- K=K( ) & K=K( ) What would you expect intuitively for K in unsaturated case? Water Solid Air Compared to saturated media, when air exists K will be lower. The more air, the smaller the cross-sectional area for flow of water. Large pores are empty and only the smaller pores remain filled. What will happen to K for declining water content (more air)? K is lower for smaller because: 1) cross-sectional area remaining reflects only water-filled smaller pores. What do you know about K in media w/ large pores vs. small pores? Large connected pores high K; small pores low K here is greater capillary tension because for a lower there is a more negative fluid pressure head,. Water is held tighter, and K is therefore lower, because tension is greater (adsorption becomes more dominant). Note that for saturation, there is no tension. water solid 4
K (saturated conditions) K (unsaturated conditions) positive negative The K( ) relationship differs for coarse vs. fine media. This is important when considering unsat. flow through layered media. S a n d l o g K C l a y - 350 c m - 100 c m L o g o f p r e s s u r e h e a d o f w a t e r IV. Calculating Flow in the Unsaturated Zone Complexity with unsaturated flow: Darcy's law is now nonlinear because K is a function of the pressure head. h q x K x called the Darcy- Buckingham eqn. Equilibrium Flow Analysis 5
Simple use of Darcy's Law for local flow ground surface z (cm) A B 400 200 0 Datum Total head is driving force, h = + z so Location Measured Elevation Total Pressure Head Head Head A -100 300 200 B - 90 200 110 What is the direction of flow? pressure head decreases upward, so flow appears to be upward, BUT, gravitational (position) gradient is added to pressure gradient to get total gradient and flow is downward when considering total head. (defined as value < 0, opposite to sign of gradient) dh dz = H A - H B Z A - Z B = 200-110 100 = 0.9 Compute unsaturated conductivity, use exponential model -- semilog plot of ln K( ) vs. is a straight line with slope c and intercept K s K( ) = K s exp (c ) 6
Saturated value of K, K s = 1.0 cm/d for a silty sand, c = -0.02 cm 1, and the average pressure head A + B 2 K( ) = 1.0 x exp (-0.02 x 95) = 0.15 q z = -K dh dz = -95 cm = (0.15)(0.9) = -0.13 cm/d Negative (downward) q corresponds to a positive gradient dh/dz. V. WATER STORAGE in the Unsaturated Zone How does capillarity affect water storage? Consider drainage and rewetting to see how moisture content is a function of, or Capillary Rise and Water Retention: Storage depends on arrangement of pore sizes and whether soil is draining or wetting. During drainage - (on right) small-diameter pore throats do not fully dewater until tension increases (pressure decreases, e.g., due to water table decline). some underlying large pores remain full to compute pressure necessary to drain these pores use capillary rise 7
formula Upon wetting of a dry soil - (on left) large pores remain empty and small pores fill under capillary (negative) pressure filling large pore requires threshold pressure associated with larger diameter at a given pressure, water content (storage) is greater during drainage (large pores full) than during wetting (large pores empty) VI. Richards Equation for unsaturated flo (1931) x K( ) x + y K( ) y + z K( ) + 1 = C( ) z t Solution (x, y, z, t) requires relationships K( ), and C( ) C( ) is the soil moisture capacity or the specific soil-water capacity (the unsaturated storage property of a medium) --(units of 1/L) how much water one gets for unit drop in pressure head; like our specific storage in saturated flow Two Characteristic Curves : K( ) vs. and C( ) vs. relationships used to describe storage and transmissive properties of unsaturated media as a function of pressure relationships show hysteresis (non-unique parameter values as a function of wetting history) 8
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Appendix A: Capillary Rise (idealized capillary fringe tension saturated zone) P air 2r P water capillary rise, h c contact angle, h c 2 cos / gr given that contact angle near zero gives cos 1 h c 2 / gr and at 18 o C this reduces to surface Surface tension of between liquid gas and liquid (M/T 2 ) h c = (2 x 73 g/s 2 )/ (0.999 g/cm 3 x 980 cm/s 2 x r cm) = 0.15/r cm So a 2 mm capillary (a fine sand) will show a rise of h c = 0.15/0.1 = 1.5 cm, and a silt will give perhaps 15 cm. 11
Appendix B: Darcy's law in 3D (for isotropic conditions) h h q x K( ) q y K( ) x y Recall that p = g or h = p g + z or and h = p g h q z K( ) z = + z so for the vertical flow component, q z = -K( ) ( p g + z) z or q z = -K( ) ( + z) z q z = -K( ) z + 1 At a constant elevation along the x, y plane we have horizontal flow and h x ψ x and h y ψ y h z = ( +z) z = z + z z (second term appears only for vertical flow component) 12