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Thursday, December 26, 2013

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?












HORTON'S INFILTRATION MODEL

HORTON'S INFILTRATION MODEL

The infiltration process was thoroughly studied by Horton in the early 1930s [4]. An outgrowth of his work, shown graphically in Fig. 7.1, was the following relation for determining infiltration capacity:

         
(7.1)


where               = the infiltration capacity (depth/time) at some time t
k = a constant representing the rate of decrease in / capacity
 = a final or equilibrium capacity
 = the initial infiltration capacity


It indicates that if the rainfall supply exceeds the infiltration capacity, infiltration tends to decrease in an exponential manner. Although simple in form, difficulties in determining useful values for  and k restrict the use of this equation. The area under the curve for any time interval represents the depth of water infiltrated during that interval. The infiltration rate is usually given in inches per hour and the time t in minutes, although other time increments are used and the coefficient k is determined accordingly.

By observing the variation of infiltration with time and developing plots of f versus t as shown in Fig. 7.5, we can estimate  and k. Two sets of f and t are selected from the curve and entered in Eq. 7.1. Two equations having two unknowns are thus obtained; they can be solved by successive approximations for  and k.

Typical infiltration rates at the end of 1 hr ( ) are shown in Table 7.1. A typical relation between  and the infiltration rate throughout a rainfall period is shown graphically in Fig. 7.6a; Fig.7.6b shows an infiltration capacity curve for normal antecedent conditions on turf. The data given in Table 7.1 are for a turf area and must be multiplied by a suitable cover factor for other types of cover complexes. A range of cover factors is listed in Table 7.2.

Total volumes of infiltration and other abstractions from a given recorded rainfall are obtainable from a discharge hydro graph (plot of the stream flow rate versus time) if one is available. Separation of the base flow (dry weather flow) from the discharge hydrograph results in a direct runoff hydrograph (DRH), which accounts for the direct surface runoff, that is, rainfall less abstractions. Direct surface runoff or precipitation excess in inches uniformly distributed over a watershed can readily be calculated by picking values of DRH discharge at equal time increments through



FIGURE7.6

(a) Typical infiltration curve. (b) Infiltration capacity and mass curves for normal antecedent conditions of turf areas.



hydrograph and applying the formula [5]:

                                               
(7.2)    



Where              Pe = precipitation excess (in.)
      = DRH ordinates at equal time intervals (cfs)
   A = drainage area (mi2)
      =. Number of time intervals in a 24-hr period


For most cases the difference between the original rainfall and the direct runoff can be considered as infiltrated water. Exceptions may occur in areas of excessive subsurface drainage or tracts of intensive interception potential. The calculated value of infiltration can then be assumed as distributed according to an equation of the form of Eq. 7.1 or it may be uniformly spread over the storm period. Choice of the method employed depends on the accuracy requirements and size of the watershed.

To circumvent some of the problems associated with the use of Horton's infiltration model, some adjustments can be made [6]. Consider Fig. 7.5. Note that where the infiltration capacity curve is above the hyetograph, the actual rate of infiltration is equal to that of the rainfall intensity, adjusted for interception, evaporation, and other losses. Consequently, the actual infiltration is given by:

f(t) = min [fp(t), i(t)]                                           (7.3)


where f(t) is the actual infiltration into the soil and i(t) is the rainfall intensity. Thus the infiltration rate at any time is equal to the lesser of the infiltration capacity fp(t) or the rainfall intensity.



Commonly, the typical values of  and   are greater than the prevailing rainfall intensities during a storm. Thus, when Eq. 7.1 is solved for fp as a function of time alone, it shows a decrease in infiltration capacity even when rainfall intensities are much less than . Accordingly, a reduction in infiltration capacity is made regardless of the amount of water that enters the soil.

To adjust for this deficiency, the integrated form of Horton's equation may be ~
used: .


where F is the cumulative infiltration at time tp as shown in Fig. 7.7. In the figure, it is assumed that the actual infiltration has been equal to . As previously noted, this is not usually the case, and the true cumulative infiltration must be determined. This can be done using:


where f(t) is determined using Eq. 7.3.

Equations 7.4 and 7.5 may be used jointly to calculate the time tp, that is, the equivalent time for the actual infiltrated volume to equal the volume under the infiltration capacity curve (Fig. 7.7). The actual accumulated infiltration given by Eq. 7.5 is equated to the area under the Horton curve, Eq. 7.4, and the resulting expression is solved for tp. This equation:


cannot be solved explicitly for tp but an iterative solution can be obtained. It should be understood that the time tp is less than or equal to the actual elapsed time t. Thus the available infiltration capacity as shown in Fig. 7.7 is equal to or exceeds that given by



FIGURE7.7
Cumulative infiltration


Eq. 7.1. By making the adjustments described, fp becomes a function of the actual amount of water infiltrated and not just a variable with time as is assumed in the original Horton equation.
In selecting a model for use in infiltration calculations, it is important to know its limitations. In some cases a model can be adjusted to accommodate shortcomings; in other cases, if its assumptions are not realistic for the nature of the use proposed, the model should be discarded in favor of another that better fits the situation.
The first eight chapters of this book deal with the principal components of the hydrologic cyc1~.In later chapters; the emphasis is on putting these components together in various hydrologic modeling processes. When these models are designed for continuous simulation, the approach is to calculate the appropriate components of the hydrologic equation, Eq. 1.4, continuously over time. A discussion of how infiltration could be incorporated into a simulation model follows. It exemplifies the use of Horton's equation in a storm water management model (SWMM) [6].
First, an initial value of tp is determined. Then, considering that the value of fp depends on the actual amount of infiltration that has occurred up to that time, a value of the average infiltration capacity, ~, available over the next time step is calculated using:




Equation 4.3 is then used to find the average rate of infiltration, :




where i is the average rainfall intensity over the time step.
Following this, infiltration is incremented using the expression:




where F = t is the added cumulative infiltration (Fig. 7.7).
The next step is to find a new value of . This is done using Eq. 7.6. If
F= . But if the new  is less than tp +  (seeFig.7.7), Eq.7.6 must be solved by iteration for the new value of . This can be accomplished using the Newton-Raphson procedure [6].

  
Example 7.1

Given an initial infiltration capacity  of 2.9 in./hr and a time constant k of 0.28 in/hr derive an infiltration capacity versus time curve if the ultimate infiltration capacity is 0.50 in./hr. For the first 8 hours, estimate the total volume of water infiltrated in inches over the watershed.

Solution:
1.      Using Horton's equation (Eq. 7.1), values of infiltration can be computed for various times. The equation is:



2.      Substituting the appropriate values into the equation yields:



3.      For the times shown in Table 7.3,values of  f are computed and entered into the table. Using a spreadsheet graphics package, the curve of Fig. 7.8 is derived.





To find the volume of water infiltrated during the first 8 hours, Eq. 7.1 can be integrated over the range of 0-8:



            The volume over the watershed is thus 11.84 in.