Equation for Well Function Series to Number of 4 Digits Calculator

Well Function Equation:

\[ W(u) = -0.577216 - \ln(u) + u - \frac{u^2}{2 \cdot 2!} + \frac{u^3}{3 \cdot 3!} \]

Well Function W(u):

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1. What is the Well Function Equation?

The Well Function equation, also known as the exponential integral, is used in groundwater hydrology to analyze drawdown in confined aquifers during pumping tests. It represents the theoretical response of an ideal aquifer to pumping.

2. How Does the Calculator Work?

The calculator uses the Well Function series expansion:

\[ W(u) = -0.577216 - \ln(u) + u - \frac{u^2}{2 \cdot 2!} + \frac{u^3}{3 \cdot 3!} \]

Where:

\( u \) — Well parameter (dimensionless)
\( \ln(u) \) — Natural logarithm of u
Constants derived from the series expansion of the exponential integral

Explanation: This series expansion provides an approximation of the well function for small values of u, which is essential for analyzing early-time drawdown data in aquifer tests.

3. Importance of Well Function Calculation

Details: Accurate calculation of the well function is crucial for determining aquifer properties such as transmissivity and storativity from pumping test data. It helps hydrogeologists characterize aquifer behavior and predict groundwater flow.

4. Using the Calculator

Tips: Enter the well parameter (u) as a positive dimensionless value. The calculator will compute the well function W(u) rounded to 4 decimal digits as specified.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of validity for this series expansion?
A: This series expansion is most accurate for small values of u (typically u < 1). For larger values, additional terms may be needed or alternative methods should be used.

Q2: What does the well parameter 'u' represent?
A: The well parameter u is a dimensionless quantity that combines aquifer properties, pumping rate, and time: \( u = \frac{r^2S}{4Tt} \), where r is distance, S is storativity, T is transmissivity, and t is time.

Q3: Why is the Euler-Mascheroni constant (0.577216) used?
A: The Euler-Mascheroni constant appears in the series expansion of the exponential integral and ensures the mathematical correctness of the approximation.

Q4: How many terms are typically needed for accurate results?
A: For most practical applications, 4-6 terms provide sufficient accuracy. This calculator uses the first four significant terms of the series expansion.

Q5: What are typical values of W(u) in aquifer testing?
A: W(u) values typically range from 0.1 to 10+ depending on aquifer conditions, with smaller u values corresponding to larger W(u) values.