3.6 Hydrology

3.6 Hydrology

Introduction to Hydrology

• Hydrology is the science concerned with the occurrence, distribution, movement, and properties of water on Earth.

• It forms the basis for water resources management, flood control, drought mitigation, and environmental protection.

• This section covers the fundamental concepts of the hydrologic cycle, methods for measuring and analyzing streamflow, techniques for relating rainfall to runoff, statistical methods for flood estimation, and an introduction to groundwater systems.

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1. Hydrologic Cycle and Water Balance Components

1.1 The Hydrologic Cycle

• A conceptual model describing the continuous movement of water on, above, and below the surface of the Earth, driven by solar energy and gravity.

• Key Processes:

  • Precipitation (P): Liquid (rain) or solid (snow, hail) water falling from the atmosphere.

  • Interception: Precipitation caught by vegetation before reaching the ground.

  • Evaporation (E): Conversion of liquid water to vapor from water surfaces and soil.

  • Transpiration (T): Release of water vapor from plants.

  • Evapotranspiration (ET): Combined process of evaporation and transpiration.

  • Infiltration (I): Movement of water from the ground surface into the soil.

  • Percolation: Downward movement of infiltrated water through soil and rock to recharge groundwater.

  • Runoff (Q): Surface and subsurface flow of water toward streams and rivers.

  • Condensation: Process by which water vapor changes to liquid, forming clouds.

1.2 Water Balance Equation (Hydrologic Budget)

• A fundamental application of the conservation of mass to a hydrologic system over a specified time period.

• General Form for a Drainage Basin (Watershed):

  $\boldsymbol{P - (Q + ET + G + \Delta S) = 0}$

  where,

  • $\boldsymbol{P}$ is precipitation.

  • $\boldsymbol{Q}$ is streamflow (runoff) leaving the basin.

  • $\boldsymbol{ET}$ is evapotranspiration.

  • $\boldsymbol{G}$ is net groundwater outflow (outflow minus inflow).

  • $\boldsymbol{\Delta S}$ is the change in storage within the basin (in soil, groundwater, surface water, snowpack).

• Simplified Form (Ignoring groundwater flow):

  $\boldsymbol{P = Q + ET + \Delta S}$

• Applications:

  • Estimating missing components (e.g., ET from known P, Q, and ΔS).

  • Water resources planning for reservoirs and aquifers.

  • Assessing long-term climatic impacts on water availability.

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2. Flow Measurement and Rating Curves

2.1 Direct Flow Measurement

• Velocity-Area Method:

  • Principle: Discharge is the integral of velocity over the cross-sectional area.

    $\boldsymbol{Q = \int_{0}^{B} \int_{0}^{h} v(y,z) \ dy \ dz}$

    In practice, the area is divided into vertical subsections.

  • Procedure:

    1. Measure water depth (stage) and establish cross-section geometry.

    2. Measure point velocities at specified depths in each subsection using a current meter (cup or propeller type) or Acoustic Doppler Current Profiler (ADCP).

    3. Compute mean velocity in each subsection.

    4. Compute subsection discharge = Mean Velocity × Subsection Area.

    5. Sum subsection discharges to get total $\boldsymbol{Q}$.

• Dilution Gauging (Tracer Method):

  • Useful for turbulent streams where cross-section is difficult to define.

  • Constant Rate Injection: A known concentration tracer is injected at a constant rate. Downstream concentration at equilibrium is measured.

    $\boldsymbol{Q = q \left( \frac{C_1 - C_2}{C_2 - C_0} \right)}$

    where, $\boldsymbol{q}$ is injection rate, $\boldsymbol{C_1}$ is tracer concentration injected, $\boldsymbol{C_2}$ is downstream concentration, $\boldsymbol{C_0}$ is background concentration.

2.2 Stage Measurement and Rating Curves

• Stage: The water surface elevation measured relative to a fixed datum (zero of the gauge).

• Measurement Devices: Staff gauges, bubble gauges, pressure transducers, float gauges, radar/laser sensors.

• Stage-Discharge Relationship (Rating Curve):

  • An empirical, site-specific relationship between measured stage ($\boldsymbol{h}$) and discharge ($\boldsymbol{Q}$).

  • Typically expressed as a power-law equation:

    $\boldsymbol{Q = C (h - h_0)^n}$

    where, $\boldsymbol{C}$ and $\boldsymbol{n}$ are rating constants, and $\boldsymbol{h_0}$ is the gauge height at zero flow.

  • Development: Established by performing multiple direct discharge measurements (current meterings) at different stages.

  • Use: Once established, continuous stage records can be converted to a continuous discharge record.

• Rating Curve Shifts: Can occur due to changes in channel geometry (erosion, deposition, vegetation growth) and require periodic re-calibration.

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3. Hydrograph Analysis and Synthetic Unit Hydrographs

3.1 Streamflow Hydrograph

• A plot of discharge ($\boldsymbol{Q}$) against time ($\boldsymbol{t}$) at a specific location on a stream.

• Components of a Storm Hydrograph:

  • Rising Limb: Increase in discharge due to rainfall and resulting runoff.

  • Peak Discharge ($\boldsymbol{Q_p}$): Maximum flow rate.

  • Recession Limb (Falling Limb): Gradual decrease in discharge after the peak.

  • Baseflow: Sustained flow from groundwater seepage into the channel. Separated from the direct runoff caused by the storm.

3.2 Unit Hydrograph (UH) Theory

• Definition: The direct runoff hydrograph resulting from 1 unit (e.g., 1 cm, 1 inch) of effective rainfall (net rainfall) falling uniformly over a basin at a constant rate for a specified duration (D).

• Basic Assumptions:

  1. Linearity: Direct runoff is linearly proportional to effective rainfall excess.

  2. Time Invariance: The basin's response is constant for identical rain patterns.

  3. Spatial and Temporal Uniformity: Effective rainfall is uniformly distributed over the basin and constant during the duration $\boldsymbol{D}$.

3.3 Derivation of a Unit Hydrograph

• From a Single Storm:

  1. Separate baseflow from the total storm hydrograph to obtain the Direct Runoff Hydrograph (DRH).

  2. Compute the volume of direct runoff and convert it to depth of effective rainfall over the basin area.

  3. Divide the ordinates of the DRH by this effective rainfall depth (in consistent units) to obtain the unit hydrograph for storm duration $\boldsymbol{D}$.

• S-Hydrograph Method: Used to change the duration of a unit hydrograph (e.g., from a 2-hr UH to a 4-hr UH).

3.4 Synthetic Unit Hydrographs (SUH)

• Empirical or conceptual methods to derive a UH for ungauged basins using measurable basin characteristics.

• Common Methods:

  • Snyder's Method: Uses empirical formulas based on basin area, length, and a shape coefficient to determine UH peak flow ($\boldsymbol{Q_p}$), time to peak ($\boldsymbol{T_p}$), and base time ($\boldsymbol{T_b}$).

  • Soil Conservation Service (SCS) Dimensionless Unit Hydrograph: Provides a standardized, single-peaked UH shape defined by ratios of $\boldsymbol{Q/Q_p}$ and $\boldsymbol{t/T_p}$. The key parameter is the time of concentration ($\boldsymbol{T_c}$) or lag time ($\boldsymbol{T_l}$).

  • Clark's Method: Uses a time-area histogram and a linear reservoir to model runoff routing.

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4. Rainfall-Runoff Analysis

4.1 Effective Rainfall (Rainfall Excess)

• The portion of total rainfall that becomes direct surface runoff. It is total rainfall minus losses.

• Losses (Abstractions):

  • Initial Loss (Interception, Depression Storage): Losses before runoff begins.

  • Infiltration: Governed by models like:

    • Horton's Equation (Exponential Decay):

      $\boldsymbol{f_t = f_c + (f_0 - f_c) e^{-kt}}$

      where, $\boldsymbol{f_t}$ is infiltration capacity at time $\boldsymbol{t}$, $\boldsymbol{f_0}$ is initial capacity, $\boldsymbol{f_c}$ is final constant rate, $\boldsymbol{k}$ is decay constant.

    • Green-Ampt Method: Physically-based model using soil properties (porosity, suction head, conductivity).

    • Φ-Index Method: A constant average loss rate. Effective rainfall = (Rainfall intensity - Φ) × Duration.

• Runoff Generation Mechanisms:

  • Hortonian Overland Flow: Occurs when rainfall intensity exceeds infiltration capacity (common in arid regions, impervious areas).

  • Saturation Overland Flow: Occurs when the soil becomes saturated from below, causing subsequent rainfall to run off (common in humid regions, near streams).

  • Subsurface Flow: Water that infiltrates and moves laterally through the soil to contribute to streamflow.

4.2 Rainfall-Runoff Models

• Lumped Models: Treat the watershed as a single unit. Examples: Rational Method, SCS Curve Number Method.

• Distributed Models: Account for spatial variability of rainfall, soil, land use, and topography. Examples: HEC-HMS, SWMM, MIKE SHE.

4.3 SCS Curve Number (CN) Method

• A widely used, empirical lumped method for estimating runoff volume from a storm.

• Runoff Depth Equation:

  (Otherwise $\boldsymbol{Q=0}$) where,

  • $\boldsymbol{Q}$ is accumulated direct runoff depth.

  • $\boldsymbol{P}$ is total storm rainfall depth.

  • $\boldsymbol{S}$ is the potential maximum retention.

  • $\boldsymbol{I_a}$ is the initial abstraction (estimated as $\boldsymbol{I_a = 0.2S}$).

• Curve Number (CN): A dimensionless index (range 0-100) that represents runoff potential based on soil type, land use, and antecedent moisture condition (AMC).

  • $\boldsymbol{S = \frac{1000}{CN} - 10}$ (with $\boldsymbol{S}$ in inches) or $\boldsymbol{S = \frac{25400}{CN} - 254}$ (with $\boldsymbol{S}$ in mm).

• Procedure: Determine CN from tables → Calculate $\boldsymbol{S}$ → Calculate $\boldsymbol{Q}$ for given $\boldsymbol{P}$.

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5. Flood Hydrology

5.1 Flood Frequency Analysis

• A statistical method to estimate the probability or average recurrence interval of floods of a given magnitude.

5.1.1 Key Concepts

• Annual Maximum Series (AMS): The highest peak discharge recorded each water year.

• Recurrence Interval (Return Period), $\boldsymbol{T_r}$: The average time interval between occurrences of a flood of a given magnitude or greater.

  $\boldsymbol{T_r = \frac{n+1}{m}}$

  where, $\boldsymbol{n}$ is number of years of record, $\boldsymbol{m}$ is the rank of the flood (1 for largest).

• Exceedance Probability ($\boldsymbol{P}$): The probability that a given flood magnitude will be exceeded in any given year.

  $\boldsymbol{P = \frac{1}{T_r}}$

• AEP (Annual Exceedance Probability): Same as $\boldsymbol{P}$. A 1% AEP flood has $\boldsymbol{T_r = 100}$ years.

5.1.2 Probability Distributions

• Data are fitted to a theoretical distribution to extrapolate to longer return periods.

• Common Distributions:

  • Log-Pearson Type III (LP3): Recommended for flood frequency analysis in the United States (USGS Bulletin 17B/C).

  • Gumbel (Extreme Value Type I): Simple, commonly used worldwide.

  • Log-Normal: Assumes the logarithms of the discharges are normally distributed.

• Procedure: Transform data (e.g., take logs) → Compute sample statistics (mean, standard deviation, skewness) → Use frequency factor $\boldsymbol{K_T}$ for chosen distribution and return period → Compute flood magnitude $\boldsymbol{Q_T = \bar{Q} + K_T s}$ (or in log space).

5.2 Design Flood

• The flood magnitude adopted for the design of a hydraulic structure (e.g., spillway, bridge, culvert).

• Selection Criteria:

  • Return Period: Based on the risk and consequences of failure.

    • Farm ponds, small culverts: $\boldsymbol{T_r = 10-25}$ years.

    • Urban drainage: $\boldsymbol{T_r = 25-100}$ years.

    • Dams (spillway design flood): $\boldsymbol{T_r = 100}$ to Probable Maximum Flood (PMF).

  • Probable Maximum Flood (PMF): The flood that may be expected from the most severe combination of critical meteorologic and hydrologic conditions. Used for high-hazard dams.

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6. Groundwater Hydrology

6.1 Aquifer Types and Properties

• Aquifer: A geologic formation that can store and transmit significant quantities of water under ordinary field conditions.

• Types:

  • Unconfined (Water Table) Aquifer: Has a water table that serves as its upper boundary. Water is under atmospheric pressure.

  • Confined (Artesian) Aquifer: Bounded above and below by impermeable layers (aquitards). Water is under pressure greater than atmospheric.

  • Leaky (Semi-confined) Aquifer: An aquifer that receives water from adjacent layers through leaky confining beds.

• Key Properties:

  • Porosity ($\boldsymbol{n}$): Ratio of void volume to total volume.

    $\boldsymbol{n = \frac{V_v}{V_T}}$

    Determines storage capacity.

  • Hydraulic Conductivity ($\boldsymbol{K}$): A measure of the ease with which water can move through porous material under a hydraulic gradient $\boldsymbol{[\mathrm{L/T}]}$.

  • Transmissivity ($\boldsymbol{T}$): The rate at which water is transmitted through a unit width of aquifer under a unit hydraulic gradient.

    $\boldsymbol{T = K \cdot b}$

    where, $\boldsymbol{b}$ is the saturated thickness of the aquifer.

  • Storage Coefficient:

    • Specific Yield ($\boldsymbol{S_y}$): For unconfined aquifers. Volume of water an aquifer releases from storage per unit surface area per unit decline in water table.

    • Storativity ($\boldsymbol{S}$): For confined aquifers. Volume of water an aquifer releases from storage per unit surface area per unit decline in hydraulic head. $\boldsymbol{S \ll S_y}$.

6.2 Groundwater Flow Equations

• Darcy's Law: Fundamental law for groundwater flow.

  $\boldsymbol{Q = -K A \frac{dh}{dl}}$

  or $\boldsymbol{v = -K \frac{dh}{dl}}$

  where, $\boldsymbol{v}$ is the Darcy velocity (specific discharge), $\boldsymbol{dh/dl}$ is the hydraulic gradient, and $\boldsymbol{A}$ is the cross-sectional area.

• Governing Equation (Groundwater Flow Equation): Combines Darcy's law with conservation of mass. For steady, homogeneous, isotropic confined flow in 2D:

  $\boldsymbol{\frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} = 0}$

  (Laplace's Equation).

6.3 Well Hydraulics

• Steady Radial Flow to a Well:

  • Confined Aquifer (Thiem Equation):

    $\boldsymbol{Q = \frac{2 \pi T (h_2 - h_1)}{\ln(r_2 / r_1)}}$

  • Unconfined Aquifer (Dupuit Equation):

    $\boldsymbol{Q = \frac{\pi K (h_2^2 - h_1^2)}{\ln(r_2 / r_1)}}$

  where, $\boldsymbol{h_1, h_2}$ are piezometric heads at distances $\boldsymbol{r_1, r_2}$ from the well.

• Unsteady (Transient) Flow: The Theis Equation models drawdown ($\boldsymbol{s}$) over time due to pumping in a confined aquifer.

  $\boldsymbol{s(r,t) = \frac{Q}{4 \pi T} W(u)}$

  where, $\boldsymbol{W(u)}$ is the well function, and $\boldsymbol{u = \frac{r^2 S}{4 T t}}$.

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Summary

• Measurement: Continuous streamflow records are derived from stage measurements and a calibrated rating curve.

• Modeling: Rainfall-runoff analysis (e.g., SCS-CN) provides the runoff volume, which is transformed into a streamflow hydrograph using a unit hydrograph.

• Design: Flood frequency analysis on historical streamflow data provides statistical estimates of the design flood for infrastructure.

• Systems View: The water balance equation integrates all components (P, Q, ET, ΔS, G) and links surface water hydrology with groundwater hydrology.

A comprehensive understanding of hydrology is essential for predicting water availability, managing flood risks, and designing sustainable water resource systems.

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