# 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. It is commonly used in both hydrology and soil sciences. The infiltration capacity is defined as the maximum rate of infiltration. It is most often measured in meters per day but can also be measured in other units of distance over time if necessary.[1]  The infiltration capacity decreases as the soil moisture content of soils surface layers increases. If the precipitation rate exceeds the infiltration rate, runoff will usually occur unless there is some physical barrier.

Infiltrometers, permeameters and rainfall simulators are all devices that can be used to measure infiltration rates.[2]

Infiltration is caused by multiple factors including; gravity, capillary forces, adsorption and osmosis. Many soil characteristics can also play a role in determining the rate at which infiltration occurs.

## Factors that affect infiltration

### Precipitation

Precipitation can impact infiltration in many ways. The amount, type and duration of precipitation all have an impact. Rainfall leads to faster infiltration rates than any other precipitation events, such as snow or sleet. In terms of amount, the more precipitation that occurs, the more infiltration will occur until the ground reaches saturation, at which point the infiltration capacity is reached. Duration of rainfall impacts the infiltration capacity as well. Initially when the precipitation event first starts the infiltration is occurring rapidly as the soil is unsaturated, but as time continues the infiltration rate slows as the soil becomes more saturated. This relationship between rainfall and infiltration capacity also determines how much runoff will occur. If rainfall occurs at a rate faster than the infiltration capacity runoff will occur.

### Soil characteristics

The porosity of soils is critical in determine the infiltration capacity. Soils that have smaller pore sizes, such as clay, have lower infiltration capacity and slower infiltration rates than soils that have large pore size, such as sands. One exception to this rule is when clay is present in dry conditions. In this case, the soil can develop large cracks which leads to higher infiltration capacity.[3]

Soil compaction is also impacts infiltration capacity. Compaction of soils results in decreased porosity within the soils, which decreases infiltration capacity.[4]

Hydrophobic soils can develop after wildfires have happened, which can greatly diminish or completely prevent infiltration from occurring.

### Soil moisture content

Soil that is already saturated has no more capacity to hold more water, therefore infiltration capacity has been reached and the rate cannot increase past this point. This leads to much more surface runoff. When soil is partially saturated then infiltration can occur at a moderate rate and fully unsaturated soils have the highest infiltration capacity.

### Organic materials in soils

Organic materials in the soil (including plants and animals) all increase the infiltration capacity. Vegetation contains roots that extent into the soil which create cracks and fissures in the soil, allowing for more rapid infiltration and increased capacity. Vegetation can also reduce surface compaction of the soil which again allows for increased infiltration. When no vegetation is present infiltration rates can be very low, which can lead to excessive runoff and increased erosion levels.[3] Similarly to vegetation, animals that burrow in the soil also create cracks in the soil structure.

### Land cover

If land is covered by impermeable surfaces, such as pavement, infiltration cannot occur as the water cannot infiltrate through an impermeable surface This relationship also leads to increased runoff. Areas that are impermeable often have storm drains which drain directly into water bodies, which means no infiltration occurs.[5]

Vegetative cover of the land also impacts the infiltration capacity. Vegetative cover can lead to more interception of precipitation, which can decrease intensity leading to less runoff, and more interception. Increased abundance of vegetation also leads to higher levels of evapotranspiration which can decrease the amount of infiltration rate.[5]  Debris from vegetation such as leaf cover can also increase infiltration rate by protecting the soils from intense precipitation events.

### Slope

When the slope of land is higher runoff occurs more readily which leads to lower infiltration rates.[5]

## 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[6] 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, it 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[7] 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 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. The rigorous standard that fully couples groundwater to surface water through a non-homogeneous soil is the numerical solution of Richards' equation. A newer method that allows full groundwater and surface water coupling in homogeneous soil layers, and that is related to the Richards equation is the Finite water-content vadose zone flow method. In the case of uniform initial soil water content and a deep well-drained soil, there are some excellent approximate methods to solve for the infiltration flux for a single rainfall event. Among these are the Green and Ampt (1911)[8] method, Parlange et al. (1982).[9] Beyond these methods there are a host of empirical methods such as, SCS method, Horton's method, etc., that are little more than curve fitting exercises.

### 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.

${\displaystyle F=B_{I}+P-E-T-ET-S-I_{A}-R-B_{O}}$

where

F is infiltration, which can be measured as a volume or length;
${\displaystyle B_{I}}$ is the boundary input, which is essentially the output watershed from adjacent, directly connected impervious areas;
${\displaystyle B_{O}}$ 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;
${\displaystyle I_{A}}$ 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.

### Richards' equation (1931)

The standard rigorous approach for calculating infiltration into soils is Richards' equation, which is a partial differential equation with very nonlinear coefficients. The Richards equation is computationally expensive, not guaranteed to converge, and sometimes has difficulty with mass conservation.[10]

### Finite water-content vadose zone flow method

This method is an approximation of the Richards' (1931) partial differential equation that de-emphasized soil water diffusivity and emphasizes advection. This approximation does not affect the calculated infiltration flux because the diffusive flux has a mean of 0. The finite water-content vadose zone flow method[11] is a set of three ordinary differential equations, is guaranteed to converge and to conserve mass. It requires the assumption that soil is uniform within layers.

### Green and Ampt

Named for two men; Green and Ampt. The Green-Ampt[12] 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.

${\displaystyle \int _{0}^{F(t)}{F \over F+\psi \,\Delta \theta }\,dF=\int _{0}^{t}K\,dt}$

where

${\displaystyle {\psi }}$ is wetting front soil suction head (L);
${\displaystyle \theta }$ is water content (-);
${\displaystyle K}$ is hydraulic conductivity (L/T);
${\displaystyle F(t)}$ is the cumulative depth of infiltration (L).

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

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

Using this model one can find the volume easily by solving for ${\displaystyle F(t)}$. 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 ${\displaystyle F}$ is the larger value of either ${\displaystyle Kt}$ or ${\displaystyle {\sqrt {2\psi \,\Delta \theta Kt}}}$. The only note on using this formula is that one must assume that ${\displaystyle h_{0}}$, 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 ${\displaystyle F}$ into the corresponding infiltration rate equation below to find the instantaneous infiltration rate at the time, ${\displaystyle t}$, ${\displaystyle F}$ was measured.

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

### Horton's equation

Named after the same Robert E. Horton mentioned above, Horton's equation[12] 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, ${\displaystyle f_{0}}$, and is decreasing exponentially with time, ${\displaystyle t}$. After some time when the soil saturation level reaches a certain value, the rate of infiltration will level off to the rate ${\displaystyle f_{c}}$.

${\displaystyle f_{t}=f_{c}+(f_{0}-f_{c})e^{-kt}}$

Where

${\displaystyle f_{t}}$ is the infiltration rate at time t;
${\displaystyle f_{0}}$ is the initial infiltration rate or maximum infiltration rate;
${\displaystyle f_{c}}$ is the constant or equilibrium infiltration rate after the soil has been saturated or minimum infiltration rate;
${\displaystyle k}$ 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.

${\displaystyle F_{t}=f_{c}t+{(f_{0}-f_{c}) \over k}(1-e^{-kt})}$

### Kostiakov equation

Named after its founder Kostiakov[13] is an empirical equation which assumes that the intake rate declines over time according to a power function.

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

Where ${\displaystyle a}$ and ${\displaystyle k}$ 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.[14]

${\displaystyle f(t)=akt^{a-1}+f_{0}\!}$

in integrated form the cumulative volume is expressed as:

${\displaystyle F(t)=kt^{a}+f_{0}t\!}$

Where

${\displaystyle f_{0}}$ 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.[12] Many would argue that this method is too simple and should not be used. Compare it with the Green and Ampt (1911) solution mentioned previously. This method is similar to Green and Ampt, but missing the cumulative infiltration depth and is therefore incomplete because it assumes that the infiltration gradient occurs over some arbitrary length ${\displaystyle L}$. In this model the ponded water is assumed to be equal to ${\displaystyle h_{0}}$ and the head of dry soil that exists below the depth of the wetting front soil suction head is assumed to be equal to ${\displaystyle -\psi -L}$.

${\displaystyle f=K\left[{h_{0}-(-\psi -L) \over L}\right]}$

where

${\displaystyle {\psi }}$ is wetting front soil suction head
${\displaystyle h_{0}}$ is the depth of ponded water above the ground surface;
${\displaystyle K}$ is the hydraulic conductivity;
${\displaystyle L}$ is the vague total depth of subsurface ground in question. This vague definition explains why this method should be avoided.

or

${\displaystyle f=K\left[{L+S_{f}+h_{0} \over L}\right]}$[15]
${\displaystyle {f}}$ Infiltration rate f (mm hour−1))
${\displaystyle K}$ is the hydraulic conductivity (mm hour−1));
${\displaystyle L}$ is the vague total depth of subsurface ground in question (mm). This vague definition explains why this method should be avoided.
${\displaystyle {S_{f}}}$ is wetting front soil suction head (${\displaystyle {-\psi }}$) = (${\displaystyle {-\psi _{f}}}$) (mm)
${\displaystyle h_{0}}$ is the depth of ponded water above the ground surface (mm);

## References

1. ^ m.b, Kirkham (2014). "Preface to the Second Edition". Principles of Soil and Plant Water Relations. pp. xvii–xviii. doi:10.1016/B978-0-12-420022-7.05002-3. ISBN 9780124200227.
2. ^
3. ^ a b "Soil Infiltration" (PDF). United States Department of Agriculture. Retrieved 2019-03-20.
4. ^ Dadkhah, Manouchehr; Gifford, Gerald F. (1980). "Influence of Vegetation, Rock Cover, and Trampling on Infiltration Rates and Sediment Production1". JAWRA Journal of the American Water Resources Association. 16 (6): 979–986. Bibcode:1980JAWRA..16..979D. doi:10.1111/j.1752-1688.1980.tb02537.x. ISSN 1752-1688.
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6. ^ Hogan, C. Michael (2010). "Abiotic factor" Archived 2013-06-08 at the Wayback Machine in Encyclopedia of Earth. eds Emily Monosson and C. Cleveland. National Council for Science and the Environment. Washington DC
7. ^ Horton, Robert E. (1933). "The role of infiltration in the hydrologic cycle". Trans. Am. Geophys. Union. 14th Ann. Mtg (1): 446–460. Bibcode:1933TrAGU..14..446H. doi:10.1029/TR014i001p00446.
8. ^ Green, W. Heber; Ampt, G. A. (1911). "Studies on Soil Physics". The Journal of Agricultural Science. 4: 1–24. doi:10.1017/S0021859600001441.
9. ^ Parlange, J. -Y.; Lisle, I.; Braddock, R. D.; Smith, R. E. (1982). "The Three-Parameter Infiltration Equation". Soil Science. 133 (6): 337. Bibcode:1982SoilS.133..337P. doi:10.1097/00010694-198206000-00001.
10. ^ Richards, L. A. (1931). "Capillary Conduction of Liquids Through Porous Mediums". Physics. 1 (5): 318–333. Bibcode:1931Physi...1..318R. doi:10.1063/1.1745010.
11. ^ Ogden, F. L.; Lai, W.; Steinke, R.C.; Zhu, J.; Talbot, C.A.; Wilson, J.L. (2015). "A new general 1-D vadose zone flow solution method". Water Resour. Res. 51 (6): 4282–4300. Bibcode:2015WRR....51.4282O. doi:10.1002/2015WR017126.
12. ^ a b c Water Resources Engineering, 2005 Edition, John Wiley & Sons, Inc.
13. ^ Kostiakov, A.N. "On the dynamics of the coefficient of water-percolation in soils and on the necessity of studying it from a dynamic point of view for purposes of amelioration". Transactions of 6th Congress of International Soil Science Society. Moscow. pp. 17–21.
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15. ^ Hendriks, Martin R. (2010) Introduction to Physical Hydrology, Oxford University Press