36.7.2. Theis Equation

Theis Equation:

Q s= W(u)

4πT Where:

• s = Drawdown
• Q = Discharge
• T = Transmissibility
• W(u) = Well function

r²S u=

4Tt

The Theis Equation is used to describe the behavior of groundwater when it is extracted from a well under unsteady flow conditions. This equation relates several key variables: 's' is the drawdown, which is the reduction in the water level in the well caused by pumping; 'Q' is the discharge, representing the rate at which water is being pumped from the well; 'T' is the transmissibility, which measures how easily groundwater can move through aquifers; and 'W(u)' is the well function, which accounts for the effects of time and distance on the drawdown. Additionally, 'u' is calculated using the radius of the well, the storage coefficient, and the elapsed time, allowing for an evaluation of drawdown over time.

Imagine your favorite drink is served in a glass that, when sipped from, causes the level of the drink to drop. If you drink faster (higher 'Q'), the level drops more quickly ('s'), and the type of glass affects how quickly you can sip (this relates to 'T'). The Theis Equation is like a recipe that tells you how much drink you can have based on how fast you sip (discharge), how high the drink was (drawdown), and how easily the drink flows from the glass (transmissibility).

• s = Drawdown

In the context of groundwater, drawdown ('s') refers to the decrease in water level within a well that occurs when water is pumped out. This phenomenon happens because water is being removed from the aquifer, causing a drop in pressure and level. Understanding drawdown is crucial because it indicates how much water is being extracted and impacts the aquifer's ability to replenish that water.

Think of drawdown as the bathtub water level falling when you pull the plug. The more water you drain out of the tub (pumping), the lower the water level becomes (drawdown). Like the bathtub that needs more water to refill, aquifers need time to recharge their water levels after pumping.

• Q = Discharge

Discharge ('Q') represents the volume of water that is extracted per unit time from the well. It is an important metric since it helps determine how much groundwater can be safely pumped without harming the aquifer or surrounding environment. High discharge rates can benefit water supply systems but may also lead to depletion if not managed properly.

Imagine you have a large bottle of soda with a straw. If you suck the soda out quickly (high discharge), the soda level in the bottle drops fast. If you drain it faster than the store can refill it, eventually, you end up with an empty bottle. Similarly, when pumping groundwater, it’s crucial to balance how quickly we extract it to prevent drawing it down faster than it can be replaced.

• T = Transmissibility

Transmissibility ('T') evaluates how easily water can flow through an aquifer. It combines the coefficient of permeability (the ability of the material to transmit water) and the thickness of the aquifer. Higher transmissibility means water can flow more freely, which is essential for efficient pumping from the well.

Imagine a sponge and a dense piece of rubber. Water flows quickly through the sponge (high transmissibility) but barely seeps through the rubber (low transmissibility). Just like sponges retain water more easily, aquifers with high transmissibility can supply groundwater more effectively when we need it.

• W(u) = Well function

The well function ('W(u)') is a mathematical representation that considers how drawdown varies with time and distance from the well. This function is critical for analyzing how the effects of pumping diminish over time and as one moves further from the well. Understanding 'W(u)' allows engineers to predict water levels based on real conditions.

Imagine blowing up a balloon; when you first blow, it expands rapidly in your hands, but as you continue, the rate of expansion slows down. The well function captures this early large impact of pumping and how it lessens with time and distance from the well, much like the balloon's diminishing expansion effect.

• u = r²S / 4Tt

The variable 'u' is used in the context of the Theis Equation to incorporate radial distance from the well ('r'), the storage coefficient ('S'), the transmissibility ('T'), and the time since pumping started ('t'). This equation helps understand the extent to which the aquifer responds over time as water is pumped out, providing critical data for managing groundwater resources effectively.

Think of 'u' as a measuring tool that calculates how far effects from pumping extend over time and distance. If you drop a pebble into a pond, ripples (changes in water level) spread out, but they become weaker as they travel further from where you dropped the pebble. The 'u' calculation gives a way to forecast how those ripples of drawdown diminish, helping manage the water supply accordingly.

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