Contaminant Dispersion Wellhead Flux (Environmental & Civil Engineering) Solution

Problem Overview

We evaluate the boundary behavior of contaminant solute flux $J(r) = -D_0 r \ln(r / R_0)$ as radial distance $r \to 0^+$ where $D_0 = 1.80\text{ g}/(\text{m}\cdot\text{day})$ and $R_0 = 100.0\text{ m}$. Direct substitution yields the indeterminate product form $0 \cdot (-\infty)$. We convert the expression to a quotient, evaluate the limit using L'Hôpital's Rule, determine peak flux location $r^*$, and interpret results for groundwater filter pack design.

Step-by-Step Mathematical Solution

Step 1: Algebraic Inversion & L'Hôpital's Rule Evaluation

The governing formulation is L'Hôpital's Rule for Indeterminate Forms:

1. Classify the Indeterminate Form: As $r \to 0^+$:

   $$r \to 0, \qquad \frac{r}{100} \to 0^+ \implies \ln\left(\frac{r}{100}\right) \to -\infty$$
   Thus, the product $r \ln(r / 100)$ produces the indeterminate form $0 \cdot (-\infty)$.

2. Invert to a Fractional Quotient: To apply L'Hôpital's Rule, keep the logarithmic term in the numerator and push the linear factor $r$ to the denominator as $r^{-1} = \frac{1}{r}$:

   $$\lim_{r \to 0^+} J(r) = -D_0 \lim_{r \to 0^+} \frac{\ln(r / 100)}{r^{-1}}$$
   Direct substitution now yields the form $\frac{-\infty}{\infty}$, which grants a green light for L'Hôpital's Rule.

3. Differentiate Numerator and Denominator Independently: Numerator derivative using the Chain Rule:

   $$\frac{d}{dr}\left[\ln\left(\frac{r}{100}\right)\right] = \frac{1}{\frac{r}{100}} \cdot \frac{1}{100} = \frac{1}{r}$$
   Denominator derivative using the Power Rule:
   $$\frac{d}{dr}\left[r^{-1}\right] = -r^{-2} = -\frac{1}{r^2}$$

4. Simplify Algebraically and Evaluate:

   $$\lim_{r \to 0^+} \frac{\frac{1}{r}}{-\frac{1}{r^2}} = \lim_{r \to 0^+} \left[\frac{1}{r} \cdot \left(-r^2\right)\right] = \lim_{r \to 0^+} (-r) = 0$$
   Multiplying by the constant $-D_0$:
   $$J_{\text{boundary}} = \lim_{r \to 0^+} J(r) = -D_0 \cdot (0) = 0.00\text{ g}/(\text{m}\cdot\text{day})$$

Step 2: Peak Contaminant Flux Calculation ($r^*$)

To find the location of maximum radial flux, differentiate $J(r) = -D_0 [r \ln(r) - r \ln(R_0)]$ using the Product Rule:

Evaluating the maximum flux at $r^*$:

Step 3: Engineering Analysis & Filter Pack Sizing

• Boundary Behavior vs. Peak Flux: At the aquifer capture perimeter ($r = R_0 = 100\text{ m}$), $\ln(1) = 0 \implies J(100) = 0$. As groundwater travels inward, flux increases, reaching a maximum of $66.22\text{ g}/(\text{m}\cdot\text{day})$ at $r \approx 36.8\text{ m}$, and then decreases smoothly back to zero as $r \to 0^+$.
• Why $\lim_{r \to 0^+} J(r) = 0$ Matters: If $J(r)$ diverged to $\infty$ at the well screen ($r \to 0$), the massive concentration gradient would cause local supersaturation of heavy metals and dissolved solids, cementing the sand pack with mineral scale. Because the flux mathematically approaches $0$, the cylindrical surface area of the well screen ($A = 2\pi r L$) is sufficient to absorb incoming groundwater smoothly. Environmental engineers can use standard silica sand filter packs and schedule periodic mild surfactant flushes without fear of localized structural clogging.

Engineering Conclusion

In environmental hydrology, products of decaying distances and logarithmic driving potentials frequently generate $0 \cdot \infty$ indeterminate forms near point sinks and source boundaries. Relying on intuition might suggest the singularity $\ln(r) \to -\infty$ dominates, predicting runaway solute buildup.

By inverting the linear factor $r$ to $r^{-1}$ and applying L'Hôpital's Rule, engineers prove that the linear convergence of distance to zero ($r \to 0$) completely overpowers the weak logarithmic divergence ($\ln r \to -\infty$). This confirms that the contaminant flux vanishes at the borehole axis ($J_{\text{boundary}} = 0$). This guarantees that extraction well screens will not experience catastrophic chemical encrustation, providing rigorous mathematical validation for long-term municipal groundwater remediation designs.