z = -50 cm, ψ = -100 cm, h = z + ψ = -50cm cm = -150 cm Which direction will water flow? 25 cm define z = 0 at soil surface h = z + ψ = cm = -200cm z = -25 cm, ψ = -100 cm h = z + ψ = -25 cm cm = -125 cm 25 cm

Water Balance of a soil element Input - Output =  Storage  w  q z’   x   y   t -  w  (q z’ +  q z’ )   x   y   t =  w     x   y   t   z  z’  t  q z’ =   z’  t

Darcy’s Law describing the rate of water movement in the vertical (z) direction through an unsaturated porous medium, upward flow is negative, downward flow is positive q z = K h (  ) d(z+ ψ (  )) = K h (  ) dz + d ψ (  ) = K h (  ) + K h (  ) d ψ (  ) dz dz dz dz Where q z = rate of water movement per unit cross sectional area (length/time) K h (  ) = unsaturated hydraulic conductivity (length/time), a function of soil moisture content (  ) z = elevation above an arbitrary benchmark (length) ψ (  ) = matric potential (length), a function of soil moisture content(  )  q z =  K h (  ) +  K h (  )  ψ (  )  z  z  z  q z’ =   z’  t

Richards Equation of soil moisture change and movement over time   =  K h (  ) +  ψ (  ) -  K h (  )  t  z´  z´  z´  = volumetric moisture content t = time K h (  ) = hydraulic conductivity at the current moisture content ψ (  )= soil matric potential at the current moisture content z´ = positive distance in the downward flow direction z´ = -z, where z = elevation, so that  z = -1  z´

Richards Equation, difference approximation for small changes     K h (  ) +  ψ (  ) -  K h (  )  t  z´  z´  z´  = volumetric moisture content t = time K h (  ) = hydraulic conductivity at the current moisture content ψ (  )= soil matric potential at the current moisture content z´ = positive distance in the downward flow direction z´ = -z, where z = elevation, so that  z = -1  z´

One Dimensional Soil Profile Continuous variation in soil properties can be represented by nodes or points that represent the center of a soil layer For each node, the elevation and soil properties ( , K(  ), ψ (  )) are defined to reflect the actual conditions, and the Richards Equation used to describe how soil moisture at each level will change as function of the points above and below for a small increment of time. The resulting equations (one for each node) are solved simultaneously for each time step. The results for each time step are used to calculate how soil moisture will move in the next time increment.

Green and Ampt Approach to simulating infiltration of water into the soil surface Initial assumption is soil water content is uniform in the profile at  =  o Stage 1: Infiltration rate = water input rate Stage 2: Infiltration rate < water input rate, soil surface becomes saturated (  =  ), and the wetting front moves into the soil profile. Above the wetting front, the soil is saturated, below the wetting front the soil is at  o. The depth of water infiltrated at a given time will be F(t) = (  -  o)z’ f (t) where z’ f (t) = the depth that the wetting front has penetrated into the soil

To apply Darcy’s law, the saturated soil at the surface has zero tension, but it will have a positive pressure if there is water ponded on the surface. At the leading edge of the wetting front, the water is being drawn into the soil by the soil water tension at the wetting front  f which is considered to be a function of the soil properties. h = depth of ponding = H z = z f (t),  =  f h = z f (t)+  Soil surface Wetting front  = 0 K = saturated hyd. Cond.

Applying Darcy’s law in difference formulation: q z = K h (  )  (z+ ψ (  ))  z K h (  ) can be replaced with saturated hydraulic conductivity K h * And also recognizing that  z = z f ‘(t) where z’ f (t) = the depth that the wetting front has penetrated into the soil And recalling that F(t) = (  -  o)z’ f (t) And doing a number of substitutions and reorganizations leads too..

Green and Ampt Infiltration Equation after ponding at the soil surface f(t) = f*(t) = K h * 1 + |ψ f | (  -  0 ) F(t) Where: f(t) = rate of water infiltration into the soil (cm/sec) f*(t) = rate of water infiltration into the soil after ponding (cm/sec) K h * = saturated hydraulic conductivity (cm/sec) ψ f = matric potential at the wetting front  = porosity of the soil  0 = initial soil moisture content F(t) = cumulative water infiltrated into the soil.

Influence of water input rate (e.g., rainfall or irrigation)

Influence of initial water content on infiltration rates

In practice many hydraulic parameters such as K h (  ) = hydraulic conductivity as a function of moisture content ψ (  )= soil matric potential as a function of moisture content K h * = saturated hydraulic conductivity (cm/sec) ψ f = matric potential at the wetting front  = porosity of the soil are estimated from pedotransfer functions which relate the above quantities, which are unmeasured in most soils, to some characteristics that are commonly measured. So “known” characteristics, such as clay content, silt content, organic matter content, or soil depth may be used to estimate these difficult to measure and often unmeasured characteristics such as K h (  ), the hydraulic conductivity as a function of moisture content.

Limitations of applying Darcy’s Law in unsaturated soils with vegetation (e.g., Richards Eq. And Green and Ampt Eq.): These are data intensive, and there are uncertainties about variations in soil properties with depth and time. Even when soil properties are reasonably well known, there is uncertainty about how K h (  ) and ψ(  ) vary with soil properties. Existence of large “macro-pores” caused by living, dead and decaying plant roots; insects, worm, and other animal burrows; cracks due to drying or freezing, which can transmit large volumes of water rapidly may be non-Darcian flow (rapid and independent of soil matrix) Nonetheless, Richards equation and the Green and Ampt equations can simulate average water flow rates by adjusting K h (  ) and ψ(  ) to fit observed water flow, but difficulties are often encountered when these approaches are used to simulate water quality if macro-pores are not taken into account. Flow in macropores carries the contaminants much, much faster than the simulated average flow through the soil matrix.

Examples of spatial heterogeneity in infiltration Fingered Flow in sandy laboratory medium

Richard’s Equation: Small time steps are used (generally less than 1 hour), so the approach can be computationally intensive, and requires hourly precipitation data. This can be valuable for simulating surface runoff generated by intense storms. While such events are important, they tend to be rare in well vegetated landscapes.

Simulated and measured stream flow using a single soil profile water balance to represent the Vermilion River watershed, without using Richard’s or Green-Ampt equations.

Where does surface runoff come from?

Distribution of Saturated Soils near Danville, VT March 21, 1973 Source: Dunne and Leopold

Distribution of Saturated Soils Near Danville, VT August 25, 1973 Source: Dunne and Leopold, 1978 Variable Source Area Concept