If you’ve ever done a step drawdown test, you know it’s a practical, field-ready tool for understanding well performance. But the conventional methods for analyzing the results – like the Jacob or Bierschenk solutions – can feel slow and cumbersome when you’re out in the field.
What if there were a quicker, easier way to estimate well and aquifer losses without diving into graphs or lengthy calculations? Let’s explore a method that lets you lean on your truck hood, calculator in hand, and get your results in minutes.
The Problem: Cumbersome Analysis Methods
Step drawdown tests are essential for assessing well efficiency, diagnosing performance issues, and determining optimal pump rates and depths. By analyzing the drawdown caused by aquifer losses (‘B’) and well hydraulics (‘C’), you can estimate the drawdown inside the well for any realistic discharge rate (Q) at a given time (t), accounting for the time-dependence of ‘B’. This relationship between drawdown and discharge provides a practical way to determine an optimum yield for the well or evaluate its condition and efficiency.
Yet, the most common analysis methods often require:
- Data transformation.
- Manual graphing.
- Trial-and-error data fitting.
These methods work, but they don’t fit the fast-paced reality of fieldwork. Spreadsheets and office-bound software can’t help you when you’re standing by the drill rig, deciding if a well is ready for production or calculating a sustainable pumping rate. In the field, you need answers now—not back at the office.
The Solution: A Simple, Algebraic Approach
Using Jacob’s (1946) famous well loss formula, S = BQ +CQ2, and a little algebra, the method I describe simplifies your well diagnosis – by a lot. With just two observations of drawdown at different discharge rates, you can quickly estimate both aquifer and well losses. The two formulae are provided below. I show how you can derive these formulae for yourself at the end of the article.
The Formulae:
With just two observations of drawdown at different discharge rates, you can calculate:
- Aquifer Losses (B):
\[
B = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_a^2 Q_b – Q_b^2 Q_a}
\]
- Well Losses (C):
\[
C = \frac{S_b Q_a – S_a Q_b}{Q_a Q_b^2 – Q_b Q_a^2}
\]
Where:
- Sa, Sb: Drawdown at two discharge rates
- Qa, Qb: Corresponding discharge rates
No graphs, no data transformations – simple and easy. I have been using this method for about 20 years and it will give you essentially the same answers as both the Jacob and Bierschenk solutions. You can use this algebraically derived solution anytime that you would use either the Jacob or Bierschenk solutions. Note that the exponent on Q can vary as suggested by Jacob (1946) and the formulae above can be calculated with any exponent. That being said, the exponent of 2 still appears to be the most commonly used in practice. I recommend that you compare the results with well calculations you have done in the past to see how it works.
Make Your Work Easier
Step drawdown tests are an indispensable part of well analysis, but they don’t have to be complicated. By using this straightforward method, you’ll get your answers with the calculator on your phone even before you make it back to your field truck.
At Anaqsim, we believe in removing unnecessary obstacles so you can focus on what really matters—understanding your groundwater systems and delivering results your clients will value.
Want the proof? Scroll down for the detailed derivation.
The Story Behind the Method
I didn’t invent this approach myself. About 20 years ago, a colleague suggested that Jacob’s foundational formula, (Jacob, 1946), could be algebraically manipulated to derive these results. Although they didn’t provide the actual formulae, the idea stuck with me.
After some thought and basic algebra, I was able to work out the solution. The process relies on substituting values and simplifying terms. For those interested in the mathematical proof, it follows these steps:
Starting with Jacob’s formula for two different pumping rates:
\begin{align}
S_a &= B Q_a + C Q_a^2 \\
S_b &= B Q_b + C Q_b^2
\end{align}
Where:
B = Aquifer losses
C = Well Losses
Sa or b = Drawdown in the discharge well (units in Length)
Qa or b = Discharge rate (units in Length cubed per Time)
Subscript a = Indicates the earlier value
Subscript b = Indicates the later value
Step 1: Solve for \( C \)
\[
C = \frac{S_a – B Q_a}{Q_a^2}
\]
Step 2: Solve for \( B \)
\[
B = \frac{S_b – C Q_b^2}{Q_b}
\]
Step 3: Substitute \( C \) into \( B \)
\[
B = \frac{S_b – \frac{Q_b^2 S_a}{Q_a^2} + \frac{Q_b^2 B Q_a}{Q_a^2}}{Q_b}
\]
Step 4: Isolate \( B \)
\[
B = \frac{S_b}{Q_b} – \frac{Q_b^2 S_a}{Q_b Q_a^2} + B \frac{Q_b^2 Q_a}{Q_a^2 Q_b}
\]
Simplify further:
\[
B = \frac{S_b Q_a^2}{Q_b Q_a^2} – \frac{Q_b^2 S_a}{Q_b Q_a^2} + B \frac{Q_b^2 Q_a}{Q_a^2 Q_b}
\]
Combine terms:
\[
B = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b Q_a^2} + B \frac{Q_b Q_a}{Q_a^2}
\]
Factor out \( B \) terms:
\[
B – B \frac{Q_b Q_a}{Q_a^2} = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b Q_a^2}
\]
Factor \( B \) on the left-hand side:
\[
B \left( \frac{Q_a^2}{Q_a^2} – \frac{Q_b Q_a}{Q_a^2} \right) = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b Q_a^2}
\]
Simplify the left-hand side:
\[
B \left( \frac{Q_a^2 – Q_b Q_a}{Q_a^2} \right) = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b Q_a^2}
\]
Isolate \( B \):
\[
B = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b Q_a^2} \cdot \frac{Q_a^2}{Q_a^2 – Q_b Q_a}
\]
Simplify further:
\[
B = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b (Q_a^2 – Q_b Q_a)}
\]
Factor out \( Q_a \) in the denominator:
\[
B = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b Q_a (Q_a – Q_b)}
\]
Final simplified solution:
\[
B = \frac{S_b Q_a^2 – Q_b^2 S_a}{Q_b Q_a (Q_a – Q_b)}
\]
Next, repeat the process by inserting B into C and isolating C.
Reference
Jacob CE. Drawdown test to determine effective radius of artesian well: American Society of Civil Engineers Proceedings, v. 72.