Green-Ampt Equation 🌿💦: and Kostiakov Equation 📐💧:

 Kostiakov Equation 📐💧:

The Kostiakov equation is an empirical formula used to model the rate of water infiltration into soil over time. The formula is:

$�(�)=��{�}^{(�-1)}$

where:

• $�(�)$ is the infiltration rate at time $�$ 🕒,
• $�$ and $�$ are empirical constants 📊,
• $�$ is the time since infiltration began ⏰.

This equation assumes the intake rate declines over time according to a power function. To address its limitation of assuming a zero final intake rate, a variant known as the Kostiakov-Lewis equation is used:

$�(�)=��{�}^{(�-1)}+{�}_0$

where ${�}_0$ is the steady-state infiltration rate 🔄.

Green-Ampt Equation 🌿💦: The Green-Ampt equation is a method for estimating infiltration into soils during rainfall events 🌧️, given by the formula:

$�(�)=��+�\mathrm{\Delta }�\ln (1+\frac{�(�)}{�\mathrm{\Delta }�})$

where:

• $�(�)$ is cumulative infiltration at time $�$ 🕒,
• $�$ is hydraulic conductivity of the soil 🌱,
• $�$ is time since infiltration began ⏰,
• $�$ is the wetting front soil suction head 🌀,
• $\mathrm{\Delta }�$ is the change in volumetric water content 🌊.

The assumptions of this equation include homogeneous soil, constant hydraulic conductivity, and ponding conditions on the soil surface 💧. By rearranging and integrating, one can solve for either cumulative infiltration $�(�)$ or the infiltration rate $�(�)$:

$�(�)=�[\frac{�\mathrm{\Delta }�}{�(�)}+1]$

Different methods of infiltration calculation (modified from Wikipedia)

Different methods of infiltration calculation (modified from Wikipedia)

Method 📚	Description 📝	Emoji Representation 😃
Richards' Equation (1931)	A rigorous standard for coupling groundwater to surface water through a non-homogeneous soil using a numerical solution.	🌊➕🌏
Finite Water-Content Vadose Zone Flow Method	Approximation of Richards' Equation emphasizing 1-D flow in homogeneous soil layers.	1️⃣➡️🌧️
Green and Ampt (1911)	Method for estimating infiltration flux for a single rainfall event in uniform, well-drained soil.	💧💹
Horton's Equation	Empirical method describing infiltration rate declining exponentially with time till soil saturation.	📉💧
Kostiakov Equation	Empirical method with intake rate declining over time according to a power function.	⏲️➖💧
Darcy's Law (Simplified)	Simplified method for infiltration calculation, often seen as too basic compared to Green and Ampt.	💧⚖️

1. Richards' Equation (1931) 🌊➕🌏:

   • Provides a standard rigorous approach for calculating infiltration into soils. It's computationally intensive and sometimes has difficulty with mass conservation.

2. Finite Water-Content Vadose Zone Flow Method 1️⃣➡️🌧️:

   • An approximation method that allows 1-D groundwater and surface water coupling in homogeneous soil layers.

3. Green and Ampt (1911) 💧💹:

   • Provides an excellent approximate method to solve the infiltration flux for a single rainfall event, especially in uniform, well-drained soil.

4. Horton's Equation 📉💧:

   • An empirical method describing how infiltration starts at a constant rate and decreases exponentially with time until the soil saturation level reaches a certain value.

5. Kostiakov Equation ⏲️➖💧:

   • Assumes that the intake rate declines over time according to a power function, with a variant that adds a steady intake term to correct for zero final intake rate assumption.

6. Darcy's Law (Simplified) 💧⚖️:

   • A simplified method for infiltration calculation, often criticized for being too basic and missing the cumulative infiltration depth.

These methods provide various ways to estimate the volume and/or rate of infiltration of water into the soil, each with its own set of assumptions, complexities, and use cases.

Singapore's efforts in rainwater harvesting and management

 

Here's an emoji-laden table summarizing Singapore's efforts in rainwater harvesting and management:

Topic 📚	Description 📝	Emoji Illustration 🌟
Rainwater Harvesting 🌧️	Singapore utilizes a comprehensive network of drains, canals, rivers, and stormwater collection ponds to channel rainwater into 17 reservoirs, making it one of the few countries to harvest urban stormwater on a large scale.	🌧️💦🔄
Reservoir Expansion 🏞️	The addition of Punggol and Serangoon Reservoirs, as well as the completion of Marina Reservoir, increased the water catchment area from half to two-thirds of Singapore's land surface by 2011.	🏞️⬆️🇸🇬
Future Water Catchment Goals 🎯	PUB aims to boost Singapore's water catchment area to 90% by 2060 by harnessing water from remaining streams and rivulets near the shoreline using technology to treat water of varying salinity.	🎯💧🔜
Reservoirs 🚰	List includes Pandan, Kranji, Jurong Lake, MacRitchie, Upper Peirce, Lower Peirce, Bedok, Upper Seletar, Lower Seletar, Poyan, Murai, Tengeh, Sarimbun, Pulau Tekong, Marina, Serangoon, and Punggol Reservoirs.	🚰📋💧
Rivers 🏞️	Extensive list of rivers including Singapore River, Sungei Kallang, Rochor River, and many others which play a role in the rainwater collection system.	🏞️💦🔄

This table encapsulates Singapore's robust approach towards maximizing rainwater harvesting amidst geographical constraints, its expansion of reservoirs to increase water catchment areas, and its ambitious goals for future water catchment, along with a mention of the numerous reservoirs and rivers that contribute to this system.

Lambda Calculus & Iteration Methods for SWMM5, InfoSWMM and ICM SWMM

Emoji-laden table reflecting the SWMM5, InfoSWMM, and ICM SWMM implementations:

Topic 📘	SWMM5 🌊	InfoSWMM 🔄	ICM SWMM 🌪️	Emoji Illustration 🎨
Successive Under-Relaxation (SUR) 🔄	Utilized for solving Node Continuity and Link Momentum/Continuity Equations with up to 8 iterations for convergence before moving to the next time step.	(Assumed similar unless specified by InfoSWMM documentation)	(Assumed similar unless specified by ICM SWMM documentation)	🔄➡️⌛
Lambda Calculus & Iteration Methods 💹	Various named iteration methods including successive approximation, fixed iteration or Picard Iteration, and Lambda Calculus anchor the study of broad topics in computer science.	(Assumed extension or enhancement in InfoSWMM's dynamic wave solution)	(Assumed extension or enhancement in ICM SWMM's dynamic wave solution)	💹🔄🧮
St. Venant Equation 📈	Utilized for simulating parameters from different sections of a link when the St. Venant Equation is applied.	(Assumed similar unless specified by InfoSWMM documentation)	(Assumed similar unless specified by ICM SWMM documentation)	📈🌊➡️
Normal Flow Equation 📉	Employed in supercritical flow scenarios or when water surface slope is less than the bed slope of the link, with simulated parameters only from the upstream end.	(Assumed similar unless specified by InfoSWMM documentation)	(Assumed similar unless specified by ICM SWMM documentation)	📉🌊⬆️
Non-Linear Term in Saint Venant Equation 🌀	Comprises six components that must balance at each time step for flow equation stability.	(Assumed similar unless specified by InfoSWMM documentation)	(Assumed similar unless specified by ICM SWMM documentation)	🌀⚖️⌛
Flow Equation Components 🧩	Includes terms like unsteady flow (dQ/dt), friction loss, bed slope (dz/dx), water surface slope (dy/dx), non-linear term (d(Q^2/A)/dx), and entrance/exit/other loss terms.	(Assumed similar unless specified by InfoSWMM documentation)	(Assumed similar unless specified by ICM SWMM documentation)	🧩💧📊

This table outlines the core concepts and methods used in the dynamic wave solution of SWMM5, InfoSWMM, and ICM SWMM, assuming a similar approach across the three unless otherwise specified by their respective documentation. Each version may have its own set of features, enhancements, or differences in the implementation of these methods and equations.