Calculator for Groundwater Flow using Radial Potential Flow

Dynamic Chart & Data Table

Chart showing hydraulic head vs. radial distance.

Hydraulic Conductivity (K) [m/day]	Resulting Flow Rate (Q) [m³/day]

Table showing sensitivity of Flow Rate to changes in Hydraulic Conductivity.

What is Calculating Groundwater Flow Using Radial Potential Flow?

Calculating groundwater flow using radial potential flow is a fundamental method in hydrogeology to determine the rate at which water moves towards a pumping well from a surrounding aquifer. This approach, often simplified to “radial flow,” assumes that water flows symmetrically from all directions towards the well, like spokes on a wheel. The term ‘potential’ refers to the hydraulic head, which is the energy that drives the water movement. The most common formula used for this calculation in a confined aquifer (an aquifer trapped between two impermeable layers) under steady-state conditions (where flow rates are stable over time) is the Thiem equation.

This calculation is crucial for water resource management, well design, and assessing the impact of pumping on the local water table. It helps engineers and hydrogeologists predict how much water a well can sustainably produce and how its operation might affect nearby wells or surface water bodies. The concept is a specific application of Darcy’s Law to a cylindrical coordinate system centered on the well.

The Thiem Formula and Explanation

For steady-state radial flow in a confined aquifer, the Thiem equation (1906) is used. It relates the pumping rate to the hydraulic properties of the aquifer and the observed water levels (hydraulic heads) in two observation wells.

The formula is:

Q = [2 * π * K * b * (h₂ - h₁)] / ln(r₂ / r₁)

Where:

Variable	Meaning	Unit (auto-inferred)	Typical Range
Q	Pumping Rate / Discharge	m³/day or ft³/day	Varies widely
K	Hydraulic Conductivity	m/day or ft/day	0.01 (silt) – 1000 (gravel)
b	Aquifer Saturated Thickness	m or ft	5 – 100+
h₁, h₂	Hydraulic Head at two points	m or ft	Site-specific
r₁, r₂	Radial Distance from well	m or ft	1 – 1000+
ln	Natural Logarithm	Unitless	N/A

Practical Examples

Example 1: Sand Aquifer

An engineer is testing a well in a confined sand aquifer. They want to calculate the expected pumping rate.

• Inputs:
  • Hydraulic Conductivity (K): 25 m/day
  • Aquifer Thickness (b): 30 m
  • Head at Well 1 (h₁): 48 m (measured at r₁ = 20 m)
  • Head at Well 2 (h₂): 50 m (measured at r₂ = 200 m)
• Calculation:
  • Transmissivity (T) = K * b = 25 * 30 = 750 m²/day
  • Q = [2 * π * 750 * (50 – 48)] / ln(200 / 20)
  • Q = [9424.78] / ln(10) ≈ 9424.78 / 2.3026
• Result: The calculated pumping rate (Q) is approximately 4093 m³/day.

Example 2: Siltier Aquifer (Imperial Units)

A second well is drilled in an area with finer material, and measurements are taken in Imperial units. For more on material properties, see our hydraulic conductivity calculator.

• Inputs:
  • Hydraulic Conductivity (K): 10 ft/day
  • Aquifer Thickness (b): 50 ft
  • Head at Well 1 (h₁): 145 ft (measured at r₁ = 30 ft)
  • Head at Well 2 (h₂): 150 ft (measured at r₂ = 300 ft)
• Calculation:
  • Transmissivity (T) = K * b = 10 * 50 = 500 ft²/day
  • Q = [2 * π * 500 * (150 – 145)] / ln(300 / 30)
  • Q = [15707.96] / ln(10) ≈ 15707.96 / 2.3026
• Result: The calculated pumping rate (Q) is approximately 6822 ft³/day.

How to Use This Groundwater Flow Calculator

Follow these steps for calculating groundwater flow using radial potential flow with our tool:

1. Select Unit System: Choose between ‘Metric’ (meters, days) or ‘Imperial’ (feet, days). The input labels will update automatically.
2. Enter Aquifer Properties:
   • Hydraulic Conductivity (K): Enter the K value for your aquifer material.
   • Aquifer Thickness (b): Input the saturated thickness of the confined aquifer.
3. Enter Observation Well Data: You need data from two piezometers (observation wells).
   • Head at Well 1 (h₁): Enter the measured water level elevation in the well closer to the pump.
   • Radial Distance to Well 1 (r₁): Enter the distance from the pumping well to this first observation well.
   • Head at Well 2 (h₂): Enter the water level in the farther observation well. This should be higher than h₁.
   • Radial Distance to Well 2 (r₂): Enter the distance from the pumping well to the second observation well. This must be greater than r₁.
4. Interpret the Results: The calculator instantly provides the steady-state pumping rate (Q). It also shows key intermediate values like Transmissivity (T = K * b), the Head Difference between the wells, and the average Hydraulic Gradient. This data is critical for any aquifer testing analysis.

Key Factors That Affect Radial Groundwater Flow

1. Hydraulic Conductivity (K)

This is the most sensitive parameter. A higher K value (e.g., gravel) allows water to flow much more easily than a low K value (e.g., clay), resulting in a higher pumping rate for the same head difference.

2. Aquifer Thickness (b)

A thicker aquifer provides a larger cross-sectional area for flow. Doubling the thickness will double the transmissivity and, therefore, the pumping rate (Q), all else being equal.

3. Hydraulic Gradient (Head Difference and Distance)

The “steepness” of the water table’s cone of depression. A larger difference in head (h₂ – h₁) over a shorter distance (r₂ – r₁) creates a steeper gradient, which drives flow more powerfully and increases Q.

4. Pumping Duration (Transient vs. Steady-State)

This calculator assumes steady-state (long-term, stable) conditions. In reality, when pumping starts (transient state), the cone of depression expands over time, and a property called the ‘Storage Coefficient’ is critical. A dedicated Theis solution calculator is used for transient analysis.

5. Well Efficiency

The Thiem equation assumes a 100% efficient well. In practice, well screen clogging or poor design can cause additional drawdown, making the actual achievable pumping rate lower than the theoretical calculation.

6. Aquifer Boundaries

The formula assumes an infinitely large aquifer. If the well is near a barrier boundary (like impermeable bedrock), the drawdown will be greater. If it’s near a recharge boundary (like a river), drawdown will be less.

Frequently Asked Questions (FAQ)

1. What’s the difference between this and a calculator for an unconfined aquifer?

In an unconfined aquifer, the water table itself is the upper boundary, and its thickness changes as it’s pumped. This requires a modification to the formula (using the Dupuit-Forchheimer assumptions) where head values are squared (h₂² – h₁²). This calculator is specifically for confined aquifers where thickness ‘b’ is constant.

2. Why is r₂/r₁ inside a natural logarithm (ln)?

The logarithmic term arises from integrating Darcy’s law across a cylindrical area. It reflects the physics of how flow converges from a large circumference (at r₂) to a small one (at r₁), causing head to drop more steeply closer to the well.

3. What happens if I enter h₁ greater than h₂?

The calculation will result in a negative flow rate (Q). This physically represents an injection well, where water is being forced into the aquifer, raising the hydraulic head near the well. Groundwater always flows from high head to low head.

4. What is Transmissivity (T)?

Transmissivity is a measure of how much water can be transmitted horizontally through the entire saturated thickness of the aquifer. It’s simply the Hydraulic Conductivity (K) multiplied by the Aquifer Thickness (b). It is a key parameter in all well-hydraulic calculations.

5. Can I use this for a single observation well?

No, the Thiem equation requires measurements from two observation wells to calculate the hydraulic gradient. To estimate properties from one pumping well and one observation well, you would typically use a transient method like the Theis or Cooper-Jacob analysis.

6. How do I choose the right units?

Always be consistent. If you use the ‘Metric’ setting, ensure all your inputs (K, b, h, r) are in meters and/or days. If ‘Imperial’, use feet and/or days. Mixing units is a common source of error.

7. What is “steady-state” flow?

Steady-state flow occurs when the water levels in the pumping and observation wells are no longer changing with time. The cone of depression has stabilized, and the amount of water being pumped is balanced by the amount of water flowing into the cone from the surrounding aquifer.

8. Where do I get the input values from?

These values are obtained from a “pumping test” or “aquifer test”. A central well is pumped at a constant rate, and the water level (drawdown) is monitored over time in one or more nearby observation wells (piezometers). This is a standard procedure in hydrogeological site investigation.