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Infiltration

Dr. Huidae Cho
Department of Civil Engineering...New Mexico State University

1   Infiltration vs. percolation

infiltration-percolation-seepage.jpg

2   Infiltration

process by white precipitation moves downward through the surface of the earth and replenishes soil moisture, recharges aquifers, and ultimately supports streamflows during dry periods Viessman and Lewis (2003)

3   Infiltration rate

Influenced by

  • the type and extent of vegetation covers,
  • the condition of the surface crust,
  • temperature,
  • rainfall intensity,
  • soil properties, etc.

4   Capillary suction-relative conductivity-moisture content relation

See Figure 7.1.

5   Moisture profile development with a constant rainfall rate

See Figure 7.2.

6   Infiltration rate versus time for a given rainfall intensity

See Figure 7.3.

7   Horton’s infiltration model

\begin{equation} f_p=f_c+(f_0-f_c)e^{-kt} \label{eq:horton_fp} \end{equation}

  • $f_p$: the infiltration capacity (depth/time) at time $t$
  • $k$: the rate of decreasing infiltration capacity
  • $f_c$: a final or equilibrium capacity
  • $f_0$: the initial infiltration capacity

Given $f_c$, we can estimate $f_0$ and $k$ by measuring two sets of $t$ and $f_p$.

7.1   Estimation of the volume of infiltrated water using Horton’s model

We can estimate the total amount of infiltrated water by integrating Eq. (\ref{eq:horton_fp}).

\begin{equation} \begin{aligned} F(t_p)&=\int_0^{t_p}f_p(t)\,dt=\int_0^{t_p}f_c+(f_0-f_c)e^{-kt}\,dt\\ &=\left[f_ct-\frac{f_0-f_c}{k}e^{-kt}\right]_0^{t_p}=f_c(t_p-0)-\frac{f_0-f_c}{k}\left(e^{-kt_p}-e^{-k\cdot 0}\right)\\ &=f_ct_p+\frac{f_0-f_c}{k}\left(1-e^{-kt_p}\right) \end{aligned} \end{equation}

See Figure 7.7.

7.2   Numerical method for Horton’s infiltration model

  1. $f_0$, $f_c$, and $k$ are given
  2. Define $\Delta t$ for modeling
  3. At $t_p=0$, $f_p(t_p)=f_0$ and $F(t_p)=0$
  4. At $t_1=t_p+\Delta t$, $\bar{f_p}=\frac{1}{\Delta t}\int_{t_p}^{t_1}f_p(t)\,dt=\frac{F(t_1)-F(t_p)}{\Delta t}$
  5. $\bar{f}=\begin{cases}\bar{f_p}&\text{if }\bar{i}\ge\bar{f_p}\\\bar{i}&\text{otherwise}\end{cases}$
  6. $F(t_1)=F(t_p)+\bar{f}\Delta t$
  7. If $\bar{i}\ge\bar{f_p}$, the next $t_p=t_1$ and repeat from step 4.
  8. Otherwise, find $t$ such that $F(t)=F(t_1)$, let the next $t_p=t$, and repeat from step 4.

7.3   Exercise: Example 7.1

Given an initial infiltration capacity $f_0$ of 2.9 in/hr and a time constant $k$ of 0.28/hr, derive an infiltration capacity versus time curve if the ultimate infiltration capacity is 0.50 in/hr. Assume that the rainfall intensity is constantly greater than the infiltration rate. For the first 8 hours, estimate the total volume of water infiltrated in inches over the watershed.

7.4   Exercise: Numerical method for Horton’s infiltration model

Given an initial infiltration capacity $f_0$ of 2.9 in/hr and a time constant $k$ of 0.28/hr, derive an infiltration capacity versus time curve if the ultimate infiltration capacity is 0.50 in/hr. The rainfall intensity over time is $i(t)=2\chi_{[0,4)}(t)+1.5\chi_{[4,8]}(t)$ in/hr where the step function $\chi_A(x)=\begin{cases}1&\text{if }x\in A\\0&\text{otherwise}\end{cases}$. For the first 8 hours, estimate the total volume of water infiltrated in inches over the watershed. When does ponding or runoff start?

7.5   Homework: Numerical method for Horton’s infiltration model

  • $f_0$: 3.5 in/hr
  • $f_c$: 0.6 in/hr
  • $k$: 0.32/hr
  • $i$: 3 in/hr for $[0, 3)$ hours and 1 in/hr for $[3, 6]$ hours

Estimate the total volume of infiltrated water during the entire storm event using Horton’s model. Use a time step $\Delta t$ of 30 minutes. Show your full work either manually, programmatically, or in Excel.

8   Green-Ampt infiltration model

See Figure 7.9.

\begin{equation} f_p=\frac{K_s(L+S)}{L} \end{equation}

  • $K_s$: Hydraulic conductivity
  • $L$: Depth from the ground surface to the wetting front
  • $S$: Capillary suction at the wetting front

The cumulative infiltration \begin{equation} F=L\frac{V_{w,s}-V_{w,i}}{V_t}=L(\theta_s-\theta_i)=L\Delta\theta=L\times\text{IMD} \end{equation} where $\Delta\theta$ is the initial moisture deficit, IMD.

\begin{equation} f_p=K_s\left(1+\frac{S\times\text{IMD}}{F}\right) \end{equation}

8.1   Exercise: Green-Ampt model

Use the Green and Ampt model with a computational time step of 20 minutes to determine the direct runoff volume to result from the observed rainfall amounts tabulated as columns 1 and 2 of the table. The rain began at 6:00am on May 2023 and continued for 3 hours with a total cumulative depth of 108 mm. The watershed has a silt loam soil with an initial effective saturation $S_e$ ($\Delta\theta=(1-S_e)\theta_e$) of 30 percent. We will use these parameter values: $\theta_e=0.486$, $\psi=\SI{166.8}{mm}$, and $K=\SI{6.5}{mm/hr}$.

Time6:006:206:407:007:207:408:008:208:409:00
$\Delta P$5361326172981

8.2   Homework: Derive the integrated form of the Green-Ampt model

Derive Eq. (7.13) in the text. Show your full work.

8.3   Exercise: Solve the same problem using Eq. (7.13)

\begin{equation} F-S\times\text{IMD}\times\ln\left(\frac{F+S\times\text{IMD}}{S\times\text{IMD}}\right)=K_st \label{eq:F} \end{equation}

  1. For $F<F_s$ ($f=i$), $F_s=\frac{S\times\text{IMD}}{i/K_s-1}\text{ for }i>K_s$; For $i\le K_s$, $\text{IMD}$ is updated and $F_s$ remains the same.
  2. For $F\ge F_s$ ($f=f_p$), $f_p=K_s\left(1+\frac{S\times\text{IMD}}{F}\right)$ and the surface is saturated.
  3. For an unsaturated condition, $\Delta F=i(t_2-t_1)$.
  4. For a saturated condition, $\Delta F=F_2-F_1$ using Eq. (\ref{eq:F}).

9   SCS runoff curve number method

This method developed by the Soil Conservation Service (SCS) is widely used for estimating runoff. It was developed empirically from studies of small agricultural watersheds.

9.1   Understanding the curve number (CN)

See Figure 7.14.

  • $I_a$: Initial abstraction in inches
  • $S$: Watershed storage including $I_a$ in inches
  • $S’$: Potential maximum retention exclusive of $I_a$ in inches, $S’=S-I_a$

What does $CN=100$ mean? The higher the CN, the higher the runoff!

Now see Table 7.8. Which hydrologic soil group has the highest infiltration rate?

9.2   Hydrologic soil groups

  • A: High infiltration rates greater than 0.76 cm/hr; deep well-drained sands and gravel
  • B: Moderate infiltration rates of 0.38–0.76 cm/hr; moderately fine to moderately coarse textured soils, such as loess and sandy loam
  • C: Low infiltration rates of 0.127–0.38 cm/hr; clay loam, shallow sandy loam, and clays
  • D: Low infiltration rates less than 0.127 cm/hr; clays with a high swelling potential, soils with a permanent high water table, or shallow soils over nearly impervious material

9.3   Antecedent moisture conditions (AMCs)

  • AMC I: Soils are dry, but not to the wilting point; Not suitable for design storms
  • AMC II: Average for annual floods
  • AMC III: Heavy or light rainfall have occurred in the past 5 days and the soil is nearly saturated

See Table 7.9.

9.4   Weighting CNs

\begin{equation} \text{CN}_w=\sum_if_i\cdot\text{CN}_i \end{equation}

  • $\text{CN}_w$: Weighted CN
  • $f_i$: Fraction of land cover $i$
  • $\text{CN}_i$: CN for land cover $i$

See Eq. (7.26) for example.

9.5   Limitations of the CN method

  • Generally homogeneous CN
  • If differences in CNs are greater than 5, better to subdivide the watershed and weight the runoff later
  • Should only be used when the CN exceeds 50 and the time of concentration is between 0.1 hr and 10 hr
  • $\frac{I_a}{P}$ should be between 0.1 and 0.5

9.6   Derivation of the CN method

Intuitively, \begin{equation} \frac{F}{S}=\frac{Q}{P-I_a} \label{eq:F_S} \end{equation}

At the beginning, both sides are 0. For a long rainfall event, both approach 1.

Plugging $F=P-I_a-Q$ into Eq. (\ref{eq:F_S}) and rearranging yields \begin{equation} Q=\frac{(P-I_a)^2}{P-I_a+S} \label{eq:Q} \end{equation}

$I_a=0.2S$ was experimentally determined from numerous gauged watersheds and the final equation becomes \begin{equation} Q=\frac{(P-0.2S)^2}{P+0.8S} \label{eq:Qfinal} \end{equation} where $S=\frac{1000}{CN}-10$ inches or $S=\frac{2540}{CN}-25.4$ cm.

9.7   Example 7.4

A watershed has a soil group C, with row crops on contoured and terraced land in good condition. For a 24-hr, 100-year precipitation of 8 inches, estimate the runoff using the SCS CN approach. Do this using Eqs. (7.27) and (7.28), and also make an estimate using Figure 7.14.

9.8   Exercise 1

Estimate the runoff that would result from 3.7 inches (9.4 cm) of rain falling on a watershed characterized by a CN of 78.

9.9   Exercise 2

A design storm is given in the table below. This storm occurs over a watershed characterized by a CN of 80. Estimate the runoff that results from each of the twelve 2-hour increments of rainfall.

Time (hr)024681012141618202224
Incremental
rainfall depth (cm)
00.480.580.720.951.5712.132.461.180.820.630.530.45

9.10   Homework: SCS runoff curve number method

A design storm is given in the table below. This storm occurs over a watershed characterized by a CN of 83. Estimate the runoff that results from each of the twelve 2-hour increments of rainfall.

Time (hr)024681012141618202224
Incremental
rainfall depth (cm)
00.580.620.730.851.4513.312.671.380.950.730.490.35