Hydrology and Earth Science Calculation Reference

Posted on Feb 4, 2026 in Geology

Hydrological Calculations and Formulas

Lysimeter Evapotranspiration (ET) Calculation

The Lysimeter method calculates Evapotranspiration (ET) based on mass change:

  1. Total Mass Change: (Initial Mass + Water Added) – (Final Mass + Water Lost) = 10 kg
  2. Daily Weight Change: 10 kg / 5 days = 2 kg/day
  3. Convert to Volume: 2 kg/day = 0.002 m³/day (Assuming 1 L = 1 kg and 1 L = 0.001 m³)
  4. Convert to Depth (m/d): 0.002 m³/day / Area of Bucket = 0.0002 m/day
  5. Convert to mm/d: 0.0002 m/day × 1000 = 0.2 mm/day

Groundwater Level Fluctuations (ET Estimation)

Evapotranspiration (ET) can be estimated using groundwater level changes:

$$ET = S_y (24a + b)$$

  • $S_y$: Specific Yield (dimensionless)
  • $a$: Rate of rise (e.g., 12 AM to 4 AM)
  • $b$: Total daily fluctuation (measured 24 hours apart)

Example Calculation

  1. Calculate ‘a’ value (Rate of Rise): (4 AM level – 12 AM level) / Time interval. Example: (0.6 m – 0.5 m) / 4 hours = 0.025 m/hr
  2. Calculate ‘b’ value (24-hour Fluctuation): Difference between levels measured 24 hours apart. Example: (6 AM Day 2 = 0.53 m) – (6 AM Day 1 = 0.62 m) = -0.09 m (a drop)
  3. Substitute Values: Use $S_y$, $a$, and $b$ in the ET formula.

Water Balance Equation (Storage Change)

The change in storage ($S$) for a system (like a lake or watershed) is the difference between inputs and outputs:

$$S = Input – Output$$

$$S = (Q_{in} + P + G_{in}) – (Q_{out} + ET + G_{out})$$

If the lake level does not change, $S=0$.

  • $Q_{in}, Q_{out}$: Surface Inflow/Outflow
  • $P$: Precipitation
  • $ET$: Evapotranspiration
  • $G_{in}, G_{out}$: Groundwater Inflow/Outflow

Snow Hydrology Formulas

  • Snow Liquid Ratio (SLR): $$SLR = \frac{\text{Snow Depth}}{\text{Water Equivalent}}$$
  • Snow Depth: Snow Depth = SLR × Water Equivalent
  • Water Equivalent: Water Equivalent = Snow Depth / SLR
  • Density of Snow: $$\rho_{\text{snow}} = \rho_{\text{water}} \times \frac{\text{Water Depth}}{\text{Snow Depth}}$$ (Where $\rho_{\text{water}} = 1000 \text{ kg/m}^3$)

Evaporation Measurement (Pan Method)

Actual Evaporation ($E_a$) from a surface is calculated using a measured pan evaporation ($E_{pan}$):

$$E_a = K_p \times E_{pan}$$

  • $K_p$: Pan Coefficient (dimensionless)
  • To convert to a daily rate: $E_a$ / Number of Days

Hydrological Unit Conversions

  • Volume Rate Conversion: $\text{m}^3/\text{day} = \text{m}^3/\text{s} \times 86,400 \text{ s/day}$
  • Depth to Volume Conversion: $\text{m}^3/\text{day} = (\text{mm}/\text{day} \div 1000) \times \text{Area}$

Note on Watershed Inflow: A watershed may have no surface inflow ($Q_{in} = 0$) if it is defined as the area where all water drains out to a single outlet (e.g., a headwater basin).

Soil Properties and Porosity

Porosity ($\eta$) of Soil: The ratio of void volume to total volume.

$$\eta = \frac{\text{Volume Void}}{\text{Total Volume}} \times 100\%$$

Where Volume Void = Total Volume – Solid Volume.

Atmospheric Humidity and Vapor Pressure

  • $e_s$: Saturated Vapor Pressure
  • $e_a$: Air Vapor Pressure

Evaporation/Condensation Conditions:

  • If $e_s > e_a$: Evaporation occurs.
  • If $e_s = e_a$: Equilibrium (Saturation).
  • If $e_s < e_a$: Condensation occurs.

Relative Humidity ($RH$):

$$RH = \frac{\text{Vapor Pressure (Actual)}}{\text{Saturated Vapor Pressure (at that temperature)}}$$

Precipitation Estimation Methods

Interpolation (Normal Ratio Method)

Used to estimate missing precipitation ($P_4$) data based on nearby stations:

$$P_4 = \frac{N_4}{3} \left[ \frac{P_1}{N_1} + \frac{P_2}{N_2} + \frac{P_3}{N_3} \right]$$

  • $N$: Average annual precipitation (PPT)
  • $P$: Precipitation in that specific month

Average Precipitation Estimation

Arithmetic Mean Method:

$$P_{\text{avg}} = \frac{P_1 + P_2 + P_3 + \dots (\text{Sum of all gauges})}{\text{Total number of gauges}}$$

Isohyetal and Thiessen Polygon Methods

  • Isohyetal Method: Volume = Area × Depth (Requires drawing lines of equal precipitation).
  • Thiessen Polygon Method Steps:
    1. Calculate Precipitation × Area for each polygon.
    2. Sum the (Precipitation × Area) products.
    3. Sum the total Area.
    4. Average Precipitation = (Sum of PPT × Area) / (Sum of Area)

Water Balance Example: Lake Storage Change

Problem: Calculate the change in storage ($S$) for a lake over 1 day. Determine if storage increased or decreased. Lake Area = 2 km².

  1. Initial Substitution (S = Input – Output):

    $$S = (P + Q_{in} + G_{in}) – (E + Q_{out} + G_{out})$$

    $$S = (10 \text{ mm/d} + 25 \text{ m}^3/\text{d} + 1 \text{ m}^3/\text{d}) – (2 \text{ mm/d} + 24 \text{ m}^3/\text{d} + 1 \text{ m}^3/\text{d})$$

  2. Simplify (Cancel Groundwater): Assuming groundwater flows are equal and opposite ($G_{in} = G_{out} = 1 \text{ m}^3/\text{d}$):

    $$S = (10 \text{ mm/d} + 25 \text{ m}^3/\text{d}) – (2 \text{ mm/d} + 24 \text{ m}^3/\text{d})$$

  3. Convert Depth Units (P and E) to Volume Units:
    • Convert mm/d to m/d: $10 \text{ mm/d} = 0.01 \text{ m/d}$; $2 \text{ mm/d} = 0.002 \text{ m/d}$.
    • Convert Area: $2 \text{ km}^2 = 2 \times 10^6 \text{ m}^2$.

    $$S = (0.01 \text{ m/d} \times 2,000,000 \text{ m}^2 + 25 \text{ m}^3/\text{d}) – (0.002 \text{ m/d} \times 2,000,000 \text{ m}^2 + 24 \text{ m}^3/\text{d})$$

    $$S = (20,000 \text{ m}^3/\text{d} + 25 \text{ m}^3/\text{d}) – (4,000 \text{ m}^3/\text{d} + 24 \text{ m}^3/\text{d})$$

    $$S = 20,025 \text{ m}^3/\text{d} – 4,024 \text{ m}^3/\text{d}$$

  4. Final Storage Change:

    $$S = 16,001 \text{ m}^3/\text{d}$$

    Since $S$ is positive, the storage Increased.

  5. Vertical Change in Lake Elevation (Depth Change, $d$):

    $$d = \frac{S}{\text{Area}}$$

    $$d = \frac{16,001 \text{ m}^3/\text{d}}{2,000,000 \text{ m}^2} = \mathbf{0.0080005 \text{ m/d}}$$

Groundwater Systems and Flow

Types of Aquifers

  1. Unconfined Aquifer: An open aquifer where rainwater directly recharges the water table.
  2. Confined Aquifer: Groundwater trapped between impermeable layers (aquitards) and under pressure.
  3. Semi-confined (Leaky) Aquifer: A mostly confined aquifer that slowly gains or loses water through a leaky layer.
  4. Karst Aquifer: A limestone aquifer with caves and cracks where water flows very fast and is easily polluted.

Hydraulic Head and Pressure

Hydraulic Head ($h$): The total energy per unit weight of fluid.

$$h = h_p + h_z$$

  • $h_p$: Pressure Head
  • $h_z$: Elevation Head

Hydraulic Pressure ($P$):

$$P = \rho \times g \times h$$

  • $\rho$: Density of liquid (e.g., $1000 \text{ kg/m}^3$ for water)
  • $g$: Acceleration due to gravity ($9.81 \text{ m/s}^2$)
  • $h$: Hydraulic head (m)
  • Unit: Pascals (Pa)

Measuring Hydraulic Head: Measured from the top down using a water level tape:

$$h = \text{Elevation of Surface} – \text{Depth to Water Level}$$

Hydraulic Gradient ($i$): The change in head over a distance.

$$i = \frac{H_2 – H_1}{L_2 – L_1}$$

Darcy’s Law

Describes the flow of fluid through a porous medium:

$$Q = -K A \left( \frac{dh}{dl} \right)$$

  • $Q$: Discharge (Volume flow rate)
  • $K$: Hydraulic Conductivity
  • $A$: Cross-sectional Area
  • $dh/dl$: Hydraulic Gradient

Atmospheric Processes and Weathering

Air Lifting Mechanisms

  1. Convergence Lifting: Air rises when winds move toward each other at the surface (converge on a low-pressure zone).
  2. Orographic Lifting: Air is forced upward when it encounters a mountain barrier.
  3. Convective Lifting: Warm air rises because it is less dense than the surrounding cold air.
  4. Frontal Lifting: Warm air is pushed upward over a colder, denser air mass.

The movement of air masses around Earth is mainly due to temperature gradients.

Adiabatic Temperature Change

Temperature changes without gaining or losing heat to the surrounding air. Air cools when rising and warms when sinking.

Soil Weathering Processes

  1. Physical Weathering: Rocks break into smaller pieces without chemical change (e.g., frost wedging).
  2. Chemical Weathering: Rocks change chemically (e.g., dissolving, rusting, hydrolysis).
  3. Biological Weathering: Plants, animals, and microbes break rocks apart (e.g., root growth).

Key Hydrometeorological Facts

  • The largest flux in the global water cycle is Evapotranspiration (ET).
  • Dew is not considered precipitation (PPT).
  • A surface water body with the greatest rate of evaporation is typically small, shallow, and freshwater.
  • The primary meteorological factor influencing evaporation is solar radiation.
  • In a stream, water velocity is fastest in the center and near the surface.

Sublimation

Sublimation occurs when a solid (like ice or snow) changes directly into a gas without first becoming a liquid.

Reasons why sublimation rates are often low:

  1. Saturated vapor pressure ($e_s$) is very low at cold temperatures.
  2. Most available energy is used to melt the snow/ice rather than sublimate it.

Cloud Formation Process

  1. ET Occurs: Water vapor is added to the air from a water body or surface.
  2. Air Rises and Cools: Warm, moist air rises through the atmosphere and cools adiabatically.
  3. Approaching Saturation: As the air rises and cools, it approaches saturation (it was not saturated at low levels).
  4. Condensation Begins: When saturation is reached, condensation begins, usually around condensation nuclei.
  5. Droplets Form and Grow: Tiny droplets form and grow by colliding with other droplets.
  6. Precipitation: When droplets become large and heavy, they fall due to gravity as precipitation.

Glaciology and Glacial Landforms

Glacial Landforms (Erosional and Depositional)

  • Hanging Valley: A small valley left high above a main valley after a glacier melts, often forming a waterfall. (Erosional)
  • Cirque: A bowl-shaped hollow carved into a mountain by a glacier where the glacier first formed. (Erosional)
  • Arête: A sharp, narrow ridge formed between two glacial valleys. (Erosional)
  • Horn: A pointed mountain peak formed where several cirques erode a mountain from different sides. (Erosional)
  • Esker: A long, winding ridge of stratified sand and gravel, often associated with kettle lakes. Formed by deposits in tunnels beneath the ice. (Depositional)
  • Drumlins: Elongated, streamlined hills formed when ice advances across a land surface, eroding sediment at the front into a large pile. (Erosional and Depositional)

Types and Anatomy of Glaciers

Types of Glaciers

  1. Alpine (Valley) Glacier: A glacier that flows down mountain valleys.
  2. Continental Glacier (Ice Sheet): A massive glacier that covers huge land areas (e.g., Greenland or Antarctica).

Anatomy of a Glacier

  1. Zone of Accumulation: The upper part of the glacier where snowfall is greater than melting (net gain).
  2. Zone of Ablation: The lower part of the glacier where melting is greater than snowfall (net loss).
  3. Equilibrium Line: The boundary between accumulation and ablation where gain equals loss.

Glacial Movement Factors

Glaciers move through two primary mechanisms:

  1. Basal Slip: Sliding over the underlying bedrock, often lubricated by meltwater.
  2. Plastic Flow: Internal deformation or slumping within the ice mass.

Factors influencing glacier velocity:

  • Slope: Steeper slopes result in faster velocity.
  • Total Ice Volume: Heavier mass generally leads to faster velocity.
  • Temperature: Warmer ice (near melting point) allows for faster and easier movement (more basal slip). Colder ice results in smaller movement.

Stream Flow Velocity

Stream flow velocity is fastest in the center of the stream and near the surface, where friction with the bed and banks is minimized.