Underground Contaminant Dispersion Plume Solution

In environmental and hydrological engineering, subterranean chemical spills threaten regional drinking water aquifers. When an industrial solvent leaks through porous underground gravel and fractured limestone, environmental remediation engineers construct 3D spatial dispersion models to estimate pollutant concentrations, design containment grouting walls, and drill monitoring recovery wells.

An environmental engineering firm in Nevada models the chemical pollutant concentration $C$ (in parts per million, $\text{ppm}$) migrating from a buried ruptured storage pipe using the 3D rational function:

where $x, y, z$ represent 3D spatial coordinates (measured in meters) relative to an injection test wellhead $(0, 0, 0)$.

A deep monitoring sensor is installed at the coordinate position $(2, 1, 2)$.

1. State the algebraic domain condition for $C(x, y, z)$ and write the mathematical domain $D$ in set-builder notation. Describe the 3D surface that is physically and mathematically excluded from the domain.
2. Determine the mathematical range $R$ of the concentration function over its valid domain, analyzing the behavior both inside and outside the excluded boundary surface.
3. Evaluate the contaminant concentration at the sensor location $(2, 1, 2)$. Explain what this value means for environmental safety zones and groundwater monitoring.

Step-by-Step Solution:

1. State the Denominator Condition and Identify Excluded Surface: State the Three-Variable Non-Zero Denominator Condition:

   $$x^2 + y^2 + z^2 - 9 \ne 0 \implies x^2 + y^2 + z^2 \ne 9$$
   • Set-Builder Notation:
     $$D = \{(x, y, z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 \ne 9\}$$
   • Geometric Description: The mathematical domain consists of all points in 3D space except for those lying exactly on a hollow spherical shell of radius $\rho = \sqrt{9} = 3\text{ meters}$ centered at the wellhead $(0, 0, 0)$. (Defined in Domain and Range of Multivariable Functions.)

2. Determine the Concentration Range: State the Radial Distance Coordinate Substitution: Let $\rho^2 = x^2 + y^2 + z^2$, where $\rho \ge 0$. The denominator is $d = \rho^2 - 9$.

   • Inside the Excluded Shell ($\rho \in [0, 3)$): As $\rho^2$ ranges from $0$ to $9$, the denominator $d \in [-9, 0)$.
     • At the center wellhead $(0, 0, 0)$, $C = \frac{100}{-9} \approx -11.11\text{ ppm}$.
     • As $\rho \to 3^-$, $d \to 0^- \implies C \to -\infty$.
     • Inner range component: $(-\infty, -100/9]$.
   • Outside the Excluded Shell ($\rho \in (3, \infty)$): As $\rho^2$ ranges from $9$ to $\infty$, the denominator $d \in (0, \infty)$.
     • As $\rho \to 3^+$, $d \to 0^+ \implies C \to +\infty$.
     • As $\rho \to \infty$, $d \to \infty \implies C \to 0^+$.
     • Outer range component: $(0, \infty)$.
   • Full Mathematical Range:
     $$R = \left(-\infty, -\frac{100}{9}\right] \cup (0, \infty)$$

3. Evaluate at Sensor Coordinate $(2, 1, 2)$: Compute the squared distance from the wellhead:

   $$\rho^2 = 2^2 + 1^2 + 2^2 = 4 + 1 + 4 = 9$$
   Notice that $\rho^2 = 9 \implies \rho = 3\text{ meters}$! Substitute into the concentration expression:
   $$C(2, 1, 2) = \frac{100}{9 - 9} = \frac{100}{0} \implies \textbf{Undefined (Singularity)}$$

Meaning of the Answer & Real-Life Application:

The sensor at $(2, 1, 2)$ sits precisely on the mathematical boundary sphere $\rho = 3\text{ m}$. In physical hydrology, this radius corresponds to the outer perimeter of an impermeable compacted grout containment barrier where the fluid flow model exhibits an asymptotic boundary singularity. Because the theoretical equation diverges at $\rho = 3\text{ m}$, field engineers must reposition monitoring sensors slightly beyond the containment shell (e.g., at $\rho = 4\text{ m}$, where $C = \frac{100}{16 - 9} = 14.29\text{ ppm}$) to obtain reliable, finite sensor readings without digital telemetry overflow.