# Infiltration (hydrology)

Cross-section of a hillslope depicting the vadose zone, capillary fringe, water table, and phreatic or saturated zone. (Source: United States Geological Survey.)

Infiltration is the process by which water on the ground surface enters the soil. Infiltration rate in soil science is a measure of the rate at which soil is able to absorb rainfall or irrigation. It is measured in inches per hour or millimeters per hour. The rate decreases as the soil becomes saturated. If the precipitation rate exceeds the infiltration rate, runoff will usually occur unless there is some physical barrier. It is related to the saturated hydraulic conductivity of the near-surface soil. The rate of infiltration can be measured using an infiltrometer.

## Introduction

Infiltration is governed by two forces: gravity and capillary action. While smaller pores offer greater resistance to gravity, very small pores pull water through capillary action in addition to and even against the force of gravity.

The rate of infiltration is determined by soil characteristics including ease of entry, storage capacity, and transmission rate through the soil. The soil texture and structure, vegetation types and cover, water content of the soil, soil temperature, and rainfall intensity all play a role in controlling infiltration rate and capacity. For example, coarse-grained sandy soils have large spaces between each grain and allow water to infiltrate quickly. Vegetation creates more porous soils by both protecting the soil from pounding rainfall, which can close natural gaps between soil particles, and loosening soil through root action. This is why forested areas have the highest infiltration rates of any vegetative types.

The top layer of leaf litter that is not decomposed protects the soil from the pounding action of rain; without this the soil can become far less permeable. In chaparral vegetated areas, the hydrophobic oils in the succulent leaves can be spread over the soil surface with fire, creating large areas of hydrophobic soil. Other conditions that can lower infiltration rates or block them include dry plant litter that resists re-wetting, or frost. If soil is saturated at the time of an intense freezing period, the soil can become a concrete frost on which almost no infiltration would occur. Over an entire watershed, there are likely to be gaps in the concrete frost or hygroscopic soil where water can infiltrate.

Once water has infiltrated the soil it remains in the soil, percolates down to the ground water table, or becomes part of the subsurface runoff process.

### Process

The process of infiltration can continue only if there is room available for additional water at the soil surface. The available volume for additional water in the soil depends on the porosity of the soil[1] and the rate at which previously infiltrated water can move away from the surface through the soil. The maximum rate that water can enter a soil in a given condition is the infiltration capacity. If the arrival of the water at the soil surface is less than the infiltration capacity, is sometimes analyzed using hydrology transport models, mathematical models that consider infiltration, runoff and channel flow to predict river flow rates and stream water quality.

## Research findings

Robert E. Horton[2] suggested that infiltration capacity rapidly declines during the early part of a storm and then tends towards an approximately constant value after a couple of hours for the remainder of the event. Previously infiltrated water fills the available storage spaces and reduces the capillary forces drawing water into the pores. Clay particles in the soil may swell as they become wet and thereby reduce the size of the pores. In areas where the ground is not protected by a layer of forest litter, raindrops can detach soil particles from the surface and wash fine particles into surface pores where they can impede the infiltration process.

## Infiltration in wastewater collection

Wastewater collection systems consist of a set of lines, junctions and lift stations to convey sewage to a wastewater treatment plant. When these Herr lines are compromised by rupture, cracking or tree root invasion, infiltration/inflow of stormwater often occurs. This circumstance can lead to a sanitary sewer overflow, or discharge of untreated sewage to the environment.

## Infiltration calculation methods

Infiltration is a component of the general mass balance hydrologic budget. There are several ways to estimate the volume and/or the rate of infiltration of water into a soil. Three excellent estimation methods are the Green-Ampt method, SCS method, Horton's method, and Darcy's law.

### General hydrologic budget

The general hydrologic budget, with all the components, with respect to infiltration F. Given all the other variables and infiltration is the only unknown, simple algebra solves the infiltration question.

$F=B_I+P-E-T-ET-S-I_A-B_O$

where

F is infiltration, which can be measured as a volume or length;
$B_I$ is the boundary input, which is essentially the output watershed from adjacent, directly connected impervious areas;
$B_O$ is the boundary output, which is also related to surface runoff, R, depending on where one chooses to define the exit point or points for the boundary output;
P is precipitation;
E is evaporation;
T is transpiration;
ET is evapotranspiration;
S is the storage through either retention or detention areas;
$I_A$ is the initial abstraction, which is the short term surface storage such as puddles or even possibly detention ponds depending on size;
R is surface runoff.

The only note on this method is one must be wise about which variables to use and which to omit, for doubles can easily be encountered. An easy example of double counting variables is when the evaporation, E, and the transpiration, T, are placed in the equation as well as the evapotranspiration, ET. ET has included in it T as well as a portion of E. Interception also needs to be accounted for, not just raw precipitation.

### Green-Ampt

Named for two men; Green and Ampt. The Green-Ampt[3] method of infiltration estimation accounts for many variables that other methods, such as Darcy's law, do not. It is a function of the soil suction head, porosity, hydraulic conductivity and time.

$\int_0^{F(t)} {F\over F+\psi\,\Delta\theta}\, dF = \int_0^t K\,dt$

where

${\psi}$ is wetting front soil suction head;
$\theta$ is water content;
$K$ is Hydraulic conductivity;
$F$ is the total volume already infiltrated.

Once integrated, one can easily choose to solve for either volume of infiltration or instantaneous infiltration rate:

$F(t)=Kt+\psi \, \Delta\theta \ln \left[1+{F(t)\over \psi \, \Delta\theta}\right].$

Using this model one can find the volume easily by solving for $F(t)$. However the variable being solved for is in the equation itself so when solving for this one must set the variable in question to converge on zero, or another appropriate constant. A good first guess for $F$ is the larger value of either $Kt$ or $\sqrt {2\psi \, \Delta\theta Kt}$. The only note on using this formula is that one must assume that $h_0$, the water head or the depth of ponded water above the surface, is negligible. Using the infiltration volume from this equation one may then substitute $F$ into the corresponding infiltration rate equation below to find the instantaneous infiltration rate at the time, $t$, $F$ was measured.

$f(t)=K\left[{\psi \, \Delta\theta\over F(t)}+1\right].$

### Horton's equation

Named after the same Robert E. Horton mentioned above, Horton's equation[3] is another viable option when measuring ground infiltration rates or volumes. It is an empirical formula that says that infiltration starts at a constant rate, $f_0$, and is decreasing exponentially with time, $t$. After some time when the soil saturation level reaches a certain value, the rate of infiltration will level off to the rate $f_c$.

$f_t=f_c+(f_0 - f_c)e^{-kt}$

Where

$f_t$ is the infiltration rate at time t;
$f_0$ is the initial infiltration rate or maximum infiltration rate;
$f_c$ is the constant or equilibrium infiltration rate after the soil has been saturated or minimum infiltration rate;
$k$ is the decay constant specific to the soil.

The other method of using Horton's equation is as below. It can be used to find the total volume of infiltration, F, after time t.

$F_t = f_ct+{(f_0 - f_c)\over k}(1-e^{-kt})$

### Kostiakov equation

Named after its founder Kostiakov[4] is an empirical equation which assumes that the intake rate declines over time according to a power function.

$f(t) = akt^{a-1}\!$

Where $a$ and $k$ are empirical parameters.

The major limitation of this expression is its reliance on the zero final intake rate. In most cases the infiltration rate instead approaches a finite steady value, which in some cases may occur after short periods of time. The Kostiakov-Lewis variant, also known as the "Modified Kostiakov" equation corrects for this by adding a steady intake term to the original equation.[5]

$f(t) = akt^{a-1}+f_0\!$

in integrated form the cumulative volume is expressed as:

$F(t) = kt^{a}+f_0t\!$

Where

$f_0$ approximates, but does not necessarily equate to the final infiltration rate of the soil.

### Darcy's law

This method used for infiltration is using a simplified version of Darcy's law.[3] In this model the ponded water is assumed to be equal to $h_0$ and the head of dry soil that exists below the depth of the wetting front soil suction head is assumed to be equal to $-\psi -L$.

$f=K\left[{h_0-(-\psi -L)\over L}\right]$

where

${\psi}$ is wetting front soil suction head
$h_0$ is the depth of ponded water above the ground surface;
$K$ is the hydraulic conductivity;
L is the total depth of subsurface ground in question.

or

$f=K\left[{L+S_f+h_0\over L}\right]$[6]
${f}$ Infiltration rate f (mm hour-1))
$K$ is the hydraulic conductivity (mm hour-1));
$L$ is the total depth of subsurface ground in question (mm).
${S_f}$ is wetting front soil suction head (${-\psi}$) = (${-\psi_f}$) (mm)
$h_0$ is the depth of ponded water above the ground surface (mm);

In summary all of these equations should provide a relatively accurate assessment of the infiltration characteristics of the soil in question.