1.3 Hydrology & Groundwater Hydraulics

Hydrology & Groundwater Hydraulics

Environmental engineers must evaluate the movement, distribution, and quality of water above and below the earth's surface. Surface hydrology focuses on rainfall and runoff modeling, while groundwater hydraulics (hydrogeology) focuses on flow through porous media and well pumping dynamics.

Surface Hydrology

The Rational Method

The Rational Method is widely used for estimating peak surface runoff from small, primarily urban watersheds (typically less than 200 acres). The governing equation is: $Q = C \cdot i \cdot A$ Where:

• $Q$ = Peak runoff rate (cfs)
• $C$ = Runoff coefficient (dimensionless, representing the fraction of rainfall that becomes runoff. E.g., asphalt might have $C=0.9$, while a lawn has $C=0.2$)
• $i$ = Rainfall intensity (inches/hour), determined from an Intensity-Duration-Frequency (IDF) curve for a duration equal to the Time of Concentration ($t_c$)
• $A$ = Watershed area (acres)

(Note: While units appear mixed, 1 acre-inch/hour perfectly equates to 1.008 cubic feet per second, allowing the conversion factor to be effectively 1.0)

SCS Curve Number Method

Developed by the Soil Conservation Service (now NRCS), the Curve Number (CN) method estimates total direct runoff volume from a storm event. The CN is an empirical parameter (ranging from 30 to 100) based on hydrologic soil group, land use, treatment, and antecedent moisture condition. $Q = \frac{(P - I_a)^2}{(P - I_a) + S}$ Where:

• $Q$ = Runoff depth (inches)
• $P$ = Total rainfall depth (inches)
• $S$ = Potential maximum retention after runoff begins (inches), calculated as $S = \frac{1000}{CN} - 10$
• $I_a$ = Initial abstraction (inches), commonly estimated as $0.2S$

Hydrograph Analysis

A hydrograph plots discharge ($Q$) versus time. A unit hydrograph represents the direct runoff resulting from one inch of excess rainfall generated uniformly over a watershed at a constant rate for a specified duration. It serves as a linear transfer function to predict runoff from storms of any magnitude.

Groundwater Hydraulics (Hydrogeology)

Darcy's Law

The fundamental equation describing laminar fluid flow through a porous medium is Darcy's Law: $Q = -K \cdot A \cdot \frac{dh}{dl}$ Where:

• $Q$ = Groundwater flow rate
• $K$ = Hydraulic conductivity (length/time, representing the permeability of the soil and properties of the fluid)
• $A$ = Cross-sectional area perpendicular to flow
• $\frac{dh}{dl}$ = Hydraulic gradient (change in head over a specific distance)

Aquifer Properties

1. Hydraulic Conductivity ($K$): A measure of the ease with which water can move through pore spaces.
2. Transmissivity ($T$): The rate at which water is transmitted through a unit width of an aquifer under a unit hydraulic gradient. For a confined aquifer of thickness $b$, $T = K \cdot b$.
3. Storativity ($S$): The volume of water an aquifer releases from or takes into storage per unit surface area of the aquifer per unit change in head. In unconfined aquifers, this is essentially the Specific Yield ($S_y$).

Well Drawdown and Pumping Analysis

When a well is pumped, a cone of depression forms in the potentiometric surface (or water table) around the well.

1. Steady-State Flow (Thiem Equation): When pumping reaches equilibrium and the cone of depression stabilizes, drawdown can be calculated. For a confined aquifer: $Q = \frac{2 \pi T (h_2 - h_1)}{\ln(r_2/r_1)}$ Where $h_1$ and $h_2$ are the hydraulic heads at radial distances $r_1$ and $r_2$ from the well.

2. Unconfined Aquifer Drawdown (Dupuit-Forchheimer): In unconfined aquifers, the saturated thickness decreases near the well, complicating the mathematics. The steady-state equation becomes: $Q = \frac{\pi K (h_2^2 - h_1^2)}{\ln(r_2/r_1)}$

3. Transient Flow (Theis Equation): Before steady-state is reached, the drawdown ($s$) at a radial distance ($r$) and time ($t$) in a confined aquifer is given by the Theis equation: $s = \frac{Q}{4 \pi T} W(u)$ Where $W(u)$ is the well function, and $u = \frac{r^2 S}{4 T t}$.

4. Cooper-Jacob Approximation: For small values of $u$ (typically $u < 0.01$, occurring at large times or small distances), the Theis well function can be approximated using a logarithmic expansion: $s = \frac{2.3 Q}{4 \pi T} \log_{10}\left(\frac{2.25 T t}{r^2 S}\right)$ This linearizes the drawdown against the logarithm of time, making it highly useful for determining aquifer transmissivity and storativity from pump test data by plotting time vs. drawdown on semi-logarithmic paper.