Green and Ampt Infiltration CEVE 412 January 30, 2018

Natural vs developed watershed Developed watersheds typically have higher imperviousness Affects infiltration and surface runoff

Factors affecting infiltration

Factors affecting infiltration Soil types have a wide range of hydraulic conductivity (clay: lowest and sand: highest) Level of compaction affects porosity

Various Soil types Sand Loam Clay

Factors affecting infiltration Rainfall intensity and magnitude can overwhelm infiltration rate and capacity

Horton vs Green and Ampt 𝑓= 𝑓 𝑐 + 𝑓 0 − 𝑓 𝑐 𝑒 −𝑘𝑡 𝑓= 𝐾 𝑠 (1 − 𝑀 𝑑 ψ 𝐹 ) Empirically derived to describe the exponential decay of infiltration rate over time Decay coefficients difficult to determine Assumes ponding regardless of actual rainfall intensity Based on theoretical derivation of Darcy’s Law Parameters can be determined from measurable soil parameters Allows for calculation of ponding time

Principles (assumptions): Soil is homogeneous and stable. The supply of ponded water at the surface is not limited Capillary suction is uniform and constant A sharp wetting front exists and advances at the same rate as water infiltrates The soil is uniformly saturated above the wetting front, the vol. water contents remain constant and below the advancing wetting front.

Green and Ampt Method (1911) Used to predict cumulative infiltration as a function of time and readily available soil parameters.

Soil Properties - Terminology F(t) = L (η - θi) = L Md Moisture content (θ) = ratio of the volume of water to the total volume of a unit of porous media. Md = the difference between the porosity and the initial moisture content Porosity (η) = ratio of interconnected void volume to total sample volume. Hydraulic conductivity (K) = volume of water that will flow through a unit soil column in a given time. Capillary suction head (ψ) = measure of the combined adhesive forces that bind the water molecules to solid walls and the cohesive forces that attract water molecules to each other. Md η

Similarly, f = Ks 𝜕ℎ 𝜕𝑧 Darcy’s Law q = -K(θ) ∂h/∂z q = Darcy velocity z = depth below surface h = potential or head = z + ψ K(θ) = unsaturated K Similarly, f = Ks 𝜕ℎ 𝜕𝑧 f = Ks ℎ1−ℎ2 𝑧1−𝑧2 = Ks ℎ0−(−𝐿−ψ) 𝐿 ≈ Ks 𝐿+ψ 𝐿 Substitute 𝐹 𝑀𝑑 for L, then: f = Ks ψ𝑀𝑑+𝐹 𝐹 = Ks ψ𝑀𝑑 𝐹 +1 1 2

Equation for Green & Ampt Infiltration f(t) = i for t ≤ tp f(t) = Ks(1 + Mdψ/F) for t > tp where: f(t) = infiltration rate at t Ks = vertically saturated K Md = moisture deficit Ψ = capillary suction F = cumulative depth of water infiltrated into the soil i = rainfall rate or intensity tp = the time it takes to have water begin to pond at the surface

Important Note When Ψ is given as positive, use Formula A B When Ψ is given as positive, use Formula A When Ψ is given as negative, use Formula B

three Infiltration Scenarios Case 1: i < Ks (line A) f = i. Runoff will never occur and all rainfall will infiltrate regardless of the duration Case 2: Ks < i < f (line B) f = i until it reaches ponding time, tp Case 3: Ks < i and i > f (curve C) Runoff can occur. f = Ks ψ𝑀𝑑 𝐹 +1 Ponding Time

Green and Ampt Parameters Parameters Ks, ψ, and Md can be determined from physical measurements in the soil. Data Sources: U.S. Natural Resources Conservation Service (NRCS) Soil Data Mart. Note: There are two different data sets available: State Soil Geographic (STATSGO) and Soil Survey Geographic (SSURGO). SSURGO is more detailed. See Rawls’ work to convert soil info to Green-Ampt parameters (Rawls, Brakensiek, & Saxton, 1982; Rawls, Brakensiek, & Miller, 1983).

Green and Ampt Parameters

Porosity & soil moisture content Md η θe θi θr 1.0 θr = residual water content Se = effective saturation

Porosity & soil moisture content 𝑆 𝑒 = θ 𝑖 − θ 𝑟 η− θ 𝑟 = θ 𝑖 − η− θ 𝑒 θ 𝑒 𝑆 𝑒 θ 𝑒 = θ 𝑖 −η + θ 𝑒 Since 𝑀 𝑑 = η− θ 𝑖 , then rearranging the above equation yields: 𝑀 𝑑 = θ 𝑒 1− 𝑆 𝑒

Double ring infiltrometer Measures infiltration rate or hydraulic conductivity based on constant or falling head Outer ring forms a buffer zone to ensure a 1-D vertical flow within the inner ring for accurate infiltration measurements

applications Agriculture: irrigation Urban planning: green infrastructure Hydrology: watershed/drainage studies, hydrologic models

Example 2-12 (Hydrology and Floodplain Analysis, 6th ed.) Since i > Ks  Cases 2 and 3 f = i for 0 ≤F ≤Fp, and f = Ks ψ𝑀𝑑 𝐹 +1 for F > Fp where Fp = amount of water that infiltrates the soil before surface ponding occurs Note: Md = θs - θi

Example 2-12 (Hydrology and Floodplain Analysis, 5th ed.) Rearrange equation f = Ks ψ𝑀𝑑 𝐹 +1 to obtain Fp: Fp = ψ𝑀𝑑 1−𝑖/𝐾𝑠 = −(9.37 𝑖𝑛)(0.518−0.318) 1−(7.858 𝑖𝑛 ℎ𝑟 )/(1.97 𝑖𝑛 ℎ𝑟 ) = 0.625 in. For 0 ≤F ≤Fp, f = i = 7.88 in/hr. For F >Fp, f = Ks ψ𝑀𝑑 𝐹 +1 f = i f = Ks ψ𝑀𝑑 𝐹 +1 Fp

Example 2-12 (modified) How long does it take before the surface starts ponding? tp = Fp / i = 0.625 in / 7.88 in/hr = 0.0793 hr ≅ 4.8 mins Also, how do we determine the infiltration depth, F after a certain time, t has elapsed? Chow et al., 1988 Use goal seek in Excel to determine F.