Green and Am-pt Infiltration

 Green and Am-pt Infiltration:

 The Horton equation captures the basic behavior of infiltration but the physical interpretation of the exponential constant is uncertain. Green and Ampt (1911) presented an approach that is based on fundamental physics and also gives results that match empirical observations. They use the following simplification of infiltration reality

                     

 Infiltrating water wetting front; metric suction pulls water into dry soil Dry soil saturated soil wetting front moves down into dry soil HZf

In reality, there is often not a sharp wetting front and/or the soil above the wetting front may not saturate. The equation to use if you need to consider the most realistic situation is the Richard’s equation; Richard’s equation is beyond the scope of this class but you should be aware of it.

The problem with all mechanistic infiltration equations is uncertainty about how to generalize to the field or landscape scale, especially with respect to the suction forces at the wetting front. None-the-less, many researchers are embracing these approaches and making good progress so you should have some rudimentary knowledge of, at least, the Green and Ampt concept.

Below is a summary of the relevant Green and Ampt infiltration equations. We will revisit these later in the semester after we are familiar with Darcy’s Law. The following equations come from Mein R.G. and C.L. Larson (1973) [Modeling infiltration during a steady rain. Water Resour. Res. 9(2): 384-394.] who reduced the Green and Ampt concept to something applicable.

In its simplest form the Green and Ampt equation for infiltration rate, f, can be written as:

The subscript “f” refers to the wetting front and “o” refers to the soil surface, e.g.,  is the hydraulic head at the wetting front (sum of matric forces at the wetting front and the weight of the water above), and ho is the hydraulic head at the surface (zero, unless there is water pounded on the surface). f = matric pressure at the wetting front [cm of water], Ks = saturated hydraulic conductivity [cm/hr]. The depth of the wetting front can be related to the cumulative amount of infiltrated water, F [cm], by:

where s = saturated moisture content and i = initial moisture content before infiltration began. Rearranging Eq. 2 to solve for Zf and substituting it into Eq. 1c, the infiltration rate, f(t), becomes:

where: P = rainfall rate [cm hr-1] and tp is the time when water begins to pond on the surface [hr]. Unfortunately, Eq. 3a does not have time as a variable but instead uses F, the cumulative amount of water that has infiltrated. Recognizing that f =dF/dt, we can solve Eq. 3 to get the following, somewhat complicated, expression for F(t):

where Fp = the amount of water that infiltrates before water begins to pond at the surface [cm] and tp = the time it takes to have water begin to pond at the surface [hr]. The following are expressions of these quantities.

To determine the amount of infiltration from a rain storm of duration, tr, and intensity P you will have to first determine the time at which surface ponding occurs (Eqs. 4 & 5). If tr < tp or P < Ks then the amount of infiltration, F = Ptr and the infiltration rate, f = P. If td > tp, then you will have to use Eq. 4 and find, by trial and error, the value F that gives t = tr. I usually set up an Excel spreadsheet with a column of F, incremented by small amounts, with adjacent columns for t (using Eq. 4) and f (using Eq. 3). Then I can make graphs of infiltration rate or amount verse time.

Example:

What’s the total runoff and infiltration [cm] from a 2-hour rainfall event with a 0.5 cm/hr intensity? When does runoff begin? The soil’s Ks 0.044 cm/hr, i = 0.25 and s = 0.50, and f = 22.4 cm (we could calculate Ksand f if we know the soil type). What’s the infiltration rate at the end of the storm? When you plot f vs t, does the curve look like anything we’ve seen before?