Topographic Retention Basin Drainage (Environmental & Civil Engineering) Solution

An environmental and water resources engineer studies a high-resolution topographic survey contour map of an alluvial valley in the Las Vegas wash basin to design a regional stormwater detention basin. The contour lines represent lines of constant ground surface elevation $E(x, y)$ marked in meters ($\text{m}$) above sea level.

At a prospective retention inlet located at survey coordinate point $P(x_0, y_0)$:

• The contour lines run perfectly horizontal (straight east-to-west, parallel to the $x$-axis).
• Moving vertically northward (in the positive $y$-direction) from point $P$, the contour elevation labels decrease steadily: $105\text{ m} \to 100\text{ m} \to 95\text{ m}$.
• The contour lines representing $5\text{ m}$ elevation drops are spaced tightly at $\Delta y = 10\text{ m}$ intervals on the ground.

1. State the visual rules for determining the sign of partial derivatives $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$ from a contour plot.
2. Determine the signs and instantaneous numerical values (or approximations) of $\frac{\partial E}{\partial x}$ and $\frac{\partial E}{\partial y}$ at point $P$.
3. Explain why the horizontal orientation of the contour line forces $\frac{\partial E}{\partial x} = 0$.
4. In hydrological engineering, stormwater runoff velocity $V$ is proportional to the hydraulic ground gradient $S = \sqrt{(\partial E/\partial x)^2 + (\partial E/\partial y)^2}$. Calculate $S$ at point $P$ and state the direction in which surface water will flow.

Step-by-Step Solution:

1. State the Visual Step-and-Observe Rules:

   • For $\frac{\partial E}{\partial x}$: Step to the right (east, $+x$) holding $y$ constant. Crossing into higher contours means $\frac{\partial E}{\partial x} > 0$; crossing into lower contours means $\frac{\partial E}{\partial x} < 0$; traveling tangent along a contour means $\frac{\partial E}{\partial x} = 0$.
   • For $\frac{\partial E}{\partial y}$: Step upward (north, $+y$) holding $x$ constant. Crossing into higher contours means $\frac{\partial E}{\partial y} > 0$; crossing into lower contours means $\frac{\partial E}{\partial y} < 0$. (Defined in Visual Estimation of Partial Derivatives from Contour Maps.)

2. Evaluate the Signs and Magnitudes:

   • East-West Derivative $\frac{\partial E}{\partial x}$: Moving along the $+x$ axis travels directly along the horizontal contour line. Because elevation does not change along the contour, the instantaneous rate of change is zero:
     $$\frac{\partial E}{\partial x} = 0$$
   • North-South Derivative $\frac{\partial E}{\partial y}$: Moving along the $+y$ axis crosses from $100\text{ m}$ to $95\text{ m}$ over a distance of $\Delta y = 10\text{ m}$.
     $$\Delta E = 95 - 100 = -5\text{ m}$$
     $$\frac{\partial E}{\partial y} \approx \frac{\Delta E}{\Delta y} = \frac{-5\text{ m}}{10\text{ m}} = -0.50\text{ m/m} = -50\%$$
     Thus, $\frac{\partial E}{\partial y}$ is negative, with a magnitude of $0.50\text{ m/m}$.

3. Geometric Reason for Vanishing East-West Derivative:

   • A contour line $E(x, y) = c$ represents an isoline of constant height. Because the contour runs east-west, the tangent vector to the contour is $\vec{T} = \langle 1, 0 \rangle = \hat{i}$.
   • Differentiating in the direction of the contour tangent measures zero change in elevation. Hence, the ground is completely level along the east-west axis at point $P$.

4. Calculate Hydraulic Gradient and Flow Direction:

   • State the Hydraulic Ground Gradient Formula:
     $$S = \sqrt{\left(\frac{\partial E}{\partial x}\right)^2 + \left(\frac{\partial E}{\partial y}\right)^2} = \sqrt{0^2 + (-0.50)^2} = \sqrt{0.25} = 0.50\text{ m/m}$$
   • Flow Direction: Under gravity, water flows down the steepest descent gradient ($-\nabla E$):
     $$-\nabla E = -\left\langle \frac{\partial E}{\partial x}, \frac{\partial E}{\partial y} \right\rangle = -\langle 0, -0.50 \rangle = \langle 0, 0.50 \rangle$$
   • Surface runoff flows due north (straight down the fall line into the detention inlet) at a steep $50\%$ drop.