Determining Infiltration

we might want to know how much water will soak into the ground over the next hour, etc, or how much water total has infiltrated in the last day, etc

can we predict what the infiltration rate will be for a particular area? if we can, the rate times time will give us a total volume infiltrated (and hence, not runoff); such information is necessary to use a technique like the stormwater management model (SWMM) for handling urban flooding

F = f * t

F might be in inches (equivalent depth, with f in inches per hour)

there are four main approaches to determining the value of the infiltration term of the hydrological equation

equations derived from the fluid mechanics of water moving into soil

empirically derived equations that seek to relate various physical parameters to each other

empirically derived equations with abstract parameters

actual measurements in place

if we assume that the soil is homogeneous and that rainfall is constant and uniform, we can make certain general statements about the infiltration process

the major factors which determine infiltration rate in this case are:

soil type

this tells us the size and number of pore or interconnected spaces in the soil

numerical values associated with this include the hydraulic conductivity, K (in/hr etc) and porosity, h (dimensionless, but may be given as ft/ft)

soil moisture

this ranges from some initial unsaturated value, q_{0},(like h) to the value for soil saturated with water, q_{s}

the difference between the present value and the final value is called Dq or IMD

note that q_{s} = h

infiltration rate decreases as rain continues

the initial infiltration rate is called f_{0}

the maximum possible rate that water can be taken up by a volume of soil at any given time is called the infiltration capacity (f_{MAX})

if the soil is not saturated with water, the infiltration capacity is higher than the saturaded conductivity (K)

as long as the rainfall rate is less than the infiltration capacity, then f_{0} will be the same as the rainfall rate

once water starts to infiltrate, there will be some saturated soil near the top

water is being drawn off the bottom of this so-called wetting zone by the capillary suction against the bottom (see Fig 3.10 on p 63)

when soil is unsaturated (as it is below the wetting zone), there is a driving force (called capillary suction) which pulls water into the soil

the number for this is y or S, units are ft, in, cm etc (this value is -1 times the capillary head, which is negative)

if there is some saturated soil near the top, water is being drawn off the bottom of this so-called wetting zone by the capillary suction against the bottom

to start with, the soil is at q _{0}, and infiltration will proceed at its highest rate, f_{0}

the maximum possible rate at any given time is called the infiltration capacity

as long as this is higher than the rainfall rate, then f_{0} will be the same as the rainfall rate

provided that the rainfall rate exceeds the saturated conductivity of the soil, the infiltration capacity will continue to decrease as the moisture content of the top layer of soil increases, until the soil surface is saturated with water

this happens when the infiltration capacity just equals the rainfall rate

the infiltration rate will then equal the infiltration capacity

as more water enters the soil, the infiltration capacity, and hence infiltration rate, will continue to decrease until both reach the saturated conductivity of the soil

Horton’s Model

Horton studied infiltration and came up with a simple mathematical model to describe the decrease in infiltration capacity as rainfall continues

the capacity starts at some initial capacity f_{0} and decays exponentially to some final or equilibrium value f_{c}

f = f_{c} + (f_{0}-f_{c})exp(-b t)

to calibrate this equation, we measure initial ponded infiltration rate (look at where the curve starts) to get f_{0}; then we look at infiltration rate after prolonged rain to get f_{c} (presumably, as an alternative we could set f_{c} = K_{s}); once we have those, we look at a third time point to calculate b :

(f -f_{c})

-ln -------

b = (f_{0}-f_{c})

-------------

t

in models that use this equation, there is an expression for the total water infiltrated up to time t

this can be obtained by integrating Horton’s equation over time to get a total volume infiltrated within the time interval:

F(t) = f_{c}t + [(f_{0}-f_{c})/b ](1-exp(-b t))

in practice, the rainfall intensity, i, is often less than f

hence we should really say

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

OK HERE’S A PROBLEM, Horton’s equation is based on the assumption the the amount of infiltrated water has been increasing at the rate f_{p}, but really it has been increasing no faster than i (the book summarizes this problem by pointing out that Horton’s equation only works for ponded conditions)

if we are going to use Horton’s equation to predict how much water the ground can soak up over the next time interval, we need to know what f will be (f_{av}*t gives us that amount)

Horton says that f is a function of time, but it is really a function of F, the amount of water that has soaked-in up to now

if we know F, we can back track to get a t_{p}, or time equivalent to the amount of time that would have been required if i ³ f

this t_{p} can then be plugged into the formula for f to give us a value to use for the next time interval

The Green-Ampt Model

Assume that infiltrating water follows Darcy's Law:

Flow = Driving Force ("head")/Length of Flow * Permeability

The driving force has three components: The head of water above the surface of the soil (H0), the head of water below the surface of the soil (L) and the capillary suction (S). the Length of Flow is L. And the permeability is the K value for the soil. So:

Assume the H0 ~ 0:

The length of the wetted soil will be longer than the amount of infiltrated water because some of the space is taken up by soil grains. In fact, in soil which is initially bone dry:

where h = porosity. For a soil with an initial water content of q _{i}, the initial moisture deficit (D q ) is given by:

The Green-Ampt equation can be rewritten:

If we define A=KS D q and B=K, then we get the form that Ward & Elliot use:

Measurement of Infiltration and Related Parameters

the infiltration capacity of a portion of soil can be simulated by doing a vertical permeability test or by simulating rainfall over a test plot

devices for doing these measurements are called infiltrometers

alternatively, infiltration rates can be inferred by comparing runoff and rainfall

Infiltration Indices

a rate of infiltration assumed to be constant across the length of the storm--though an initial abstraction may be taken out

the most common is j (in/hr)

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