Infiltration (hydrology)

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 and the rate at which previously infiltrated water can move away from the surface through the soil. The maximum rate at that water can enter 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 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 into 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 water infiltration rate into the 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 1-D 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 solution of the Soil Moisture Velocity Equation. In the case of uniform initial soil water content and deep, well-drained soil, some excellent approximate methods exist to solve the infiltration flux for a single rainfall event. Among these are the Green and Ampt (1911) method, Parlange et al. (1982). 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.

Finite water-content vadose zone flow method

This method approximates Richards' (1931) partial differential equation that de-emphasizes soil water diffusion. This was established by comparing the solution of the advection-like term of the Soil Moisture Velocity Equation and comparing against exact analytical solutions of infiltration using special forms of the soil constitutive relations. Results showed that this approximation does not affect the calculated infiltration flux because the diffusive flux is small and that the finite water-content vadose zone flow method is a valid solution of the equation is a set of three ordinary differential equations, is guaranteed to converge and to conserve mass. It requires the assumption that the flow occurs in the vertical direction only (1-dimensional) and that the soil is uniform within layers.

Green and Ampt

The name was derived from two men: Green and Ampt. The Green-Ampt 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.

${\displaystyle {\int }_0^{F(t)}\frac{F}{F+\psi \mathrm{\Delta }\theta }dF={\int }_0^{t}Kdt}$

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:

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

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 ${\displaystyle \sqrt{2\psi \mathrm{\Delta }\theta Kt}}$. These values can be obtained by solving the model with a log replaced with its Taylor-Expansion around one, of the zeroth and second order respectively. 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.

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

Horton's equation

Named after the same Robert E. Horton mentioned above, Horton's equation 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 the 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.

${\displaystyle F_t=f_{c}t+\frac{(f_0-f_c)}{k}(1-e^{-kt})}$

Kostiakov equation

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

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

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 this by adding a steady intake term to the original equation.

${\displaystyle f(t)=ak{t}^{a-1}+f_0}$

in integrated form, the cumulative volume is expressed as:

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

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. 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 ${\displaystyle -\psi -L}$.

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

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

${\displaystyle f=K\left[\frac{L+S_f+h_0}{L}\right]}$

${\displaystyle f}$ 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);