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Contaminant Dispersion Wellhead Flux (Environmental & Civil Engineering)

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In groundwater remediation and subsurface hydrogeology across the Las Vegas Valley aquifer basin, environmental engineers model the transport of industrial contaminants (such as perchlorate or volatile organic compounds) migrating toward remediation extraction wells. Under idealized radial groundwater flow toward a central recovery borehole, the radial advective-dispersive solute mass flux J(r)J(r) as a function of radial distance rr (in meters) from the well axis is governed by:

J(r)=−D0rln⁡(rR0)J(r) = -D_0 r \ln\left(\frac{r}{R_0}\right)
where D0=1.80 g/(m⋅day)D_0 = 1.80\text{ g}/(\text{m}\cdot\text{day}) represents the aquifer dispersion-coupling coefficient, and R0=100.0 metersR_0 = 100.0\text{ meters} is the maximum hydrodynamic capture radius of the extraction cone of depression.

As groundwater converges directly toward the borehole centerline (r→0+r \to 0^+), direct substitution into the flux expression yields the indeterminate product form:

lim⁡r→0+J(r)=−D0⋅(0)⋅ln⁡(0+)=0⋅(−∞)\lim_{r \to 0^+} J(r) = -D_0 \cdot (0) \cdot \ln(0^+) = 0 \cdot (-\infty)

If the contaminant mass flux were to diverge to infinity as r→0+r \to 0^+, chemical concentration gradients would induce severe mineral precipitation, encrustation, and catastrophic clogging of the slotted stainless-steel well screen. To verify the physical viability of the remediation well and design gravel filter packs, the engineering team must determine the boundary behavior of J(r)J(r).

  1. Classify the indeterminate form of lim⁡r→0+J(r)\lim_{r \to 0^+} J(r), rewrite the algebraic expression as a quotient suitable for L'Hôpital's Rule, and compute the exact limiting boundary flux Jboundary=lim⁡r→0+J(r)J_{\text{boundary}} = \lim_{r \to 0^+} J(r) in g/(m⋅day)\text{g}/(\text{m}\cdot\text{day}).
  2. Using the Product Rule for differentiation, find the critical radial distance r∗r^* where the contaminant flux reaches its maximum value JmaxJ_{\text{max}}, and calculate J(r∗)J(r^*).
  3. Compare the boundary behavior at r→0+r \to 0^+ with the peak flux at r∗r^*, and explain how environmental engineers utilize this result to select slot sizes for well screens and schedule acid-wash maintenance cycles without risking uncontained aquifer breakthrough.