Formula Sheet: Hydrology

Precipitation Analysis

Rainfall Intensity-Duration-Frequency (IDF) Relationships

General IDF Formula: \[i = \frac{a}{(t_d + b)^c}\]

• i = rainfall intensity (in/hr or mm/hr)
• a, b, c = empirical coefficients specific to location and return period
• t_d = storm duration (min)

Alternative Form: \[i = \frac{KT^x}{(t_d + b)^n}\]

• K, x, b, n = empirical coefficients
• T = return period (years)

Average Precipitation Over an Area

Arithmetic Mean Method: \[P_{avg} = \frac{1}{n}\sum_{i=1}^{n}P_i\]

• P_avg = average precipitation over the area
• P_i = precipitation at station i
• n = number of rain gauge stations
• Note: Best when stations are uniformly distributed and terrain is relatively flat

Thiessen Polygon Method: \[P_{avg} = \frac{\sum_{i=1}^{n}A_i P_i}{\sum_{i=1}^{n}A_i}\]

• A_i = area of influence for station i (Thiessen polygon area)
• P_i = precipitation at station i
• Note: Accounts for non-uniform distribution of gauges

Isohyetal Method: \[P_{avg} = \frac{\sum_{i=1}^{n}A_i P_{i,avg}}{\sum_{i=1}^{n}A_i}\]

• A_i = area between adjacent isohyets
• P_i,avg = average precipitation between two isohyets (arithmetic mean of the two isohyet values)
• Note: Most accurate method, especially for mountainous terrain

Missing Data Estimation

Normal Ratio Method: \[P_x = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{N_x}{N_i}\right)P_i\]

• P_x = estimated precipitation at station x
• N_x = normal annual precipitation at station x
• N_i = normal annual precipitation at surrounding station i
• P_i = precipitation at surrounding station i for the period in question
• n = number of surrounding stations (typically 3)
• Condition: Use when any N_i differs from N_x by more than 10%

Simple Arithmetic Average Method: \[P_x = \frac{1}{n}\sum_{i=1}^{n}P_i\]

• Condition: Use only when all N_i are within ±10% of N_x

Runoff and Hydrograph Analysis

Rational Method

Peak Discharge Formula: \[Q = CiA\] For US Customary Units: \[Q = \frac{CiA}{K}\] where K = 1.008 ≈ 1.0 for practical purposes

• Q = peak runoff rate (ft³/s or cfs)
• C = dimensionless runoff coefficient (0 to 1)
• i = rainfall intensity (in/hr) for duration equal to time of concentration
• A = drainage area (acres)
• Limitation: Applicable for small watersheds (typically < 200="" acres="" or="" 80="">
• Assumption: Rainfall intensity is uniform over entire watershed duration

SI Units: \[Q = \frac{1}{360}CiA\]

• Q = peak runoff rate (m³/s)
• i = rainfall intensity (mm/hr)
• A = drainage area (hectares)

Composite Runoff Coefficient: \[C_{composite} = \frac{\sum_{i=1}^{n}C_i A_i}{\sum_{i=1}^{n}A_i}\]

• C_i = runoff coefficient for sub-area i
• A_i = sub-area i

Time of Concentration

Kirpich Equation: \[t_c = 0.0078 L^{0.77} S^{-0.385}\]

• t_c = time of concentration (min)
• L = maximum flow length (ft)
• S = slope of watershed (ft/ft)
• Application: Small agricultural watersheds

Upland Method (Overland Flow): \[t_o = \frac{1.8(1.1-C)\sqrt{L}}{\sqrt[3]{S}}\]

• t_o = overland flow time (min)
• C = rational method runoff coefficient
• L = overland flow length (ft), typically ≤ 300 ft
• S = slope (percent)

Velocity Method: \[t_c = \sum_{i=1}^{n}\frac{L_i}{60V_i}\]

• t_c = time of concentration (min)
• L_i = length of flow segment i (ft)
• V_i = velocity in segment i (ft/s)
• Note: Sum overland flow, shallow concentrated flow, and channel flow times

Manning's Equation for Velocity: \[V = \frac{1.49}{n}R^{2/3}S^{1/2}\]

• V = velocity (ft/s)
• n = Manning's roughness coefficient
• R = hydraulic radius (ft)
• S = slope (ft/ft)

NRCS Lag Time: \[t_{lag} = \frac{L^{0.8}(S+1)^{0.7}}{1900\sqrt{Y}}\]

• t_lag = lag time (hr)
• L = hydraulic length of watershed (ft)
• S = potential maximum retention after runoff begins = (1000/CN) - 10 (inches)
• Y = average watershed land slope (percent)
• CN = curve number

Relationship between Time of Concentration and Lag Time: \[t_{lag} = 0.6 \times t_c\]

• Note: This is an approximation; t_lag is time from centroid of rainfall to peak of hydrograph

NRCS (SCS) Curve Number Method

Direct Runoff Equation: \[Q = \frac{(P - I_a)^2}{P - I_a + S}\]

• Q = accumulated direct runoff (in or mm)
• P = accumulated rainfall (precipitation) depth (in or mm)
• I_a = initial abstraction (in or mm)
• S = potential maximum retention after runoff begins (in or mm)
• Condition: P > I_a; if P ≤ I_a, then Q = 0

Initial Abstraction Approximation: \[I_a = 0.2S\] Simplified Runoff Equation (using I_a = 0.2S): \[Q = \frac{(P - 0.2S)^2}{P + 0.8S}\]

• Condition: Valid only when P ≥ 0.2S

Potential Maximum Retention (US Customary): \[S = \frac{1000}{CN} - 10\]

• S = potential maximum retention (inches)
• CN = curve number (dimensionless, range 0-100)

Potential Maximum Retention (SI): \[S = \frac{25400}{CN} - 254\]

• S = potential maximum retention (mm)

Composite Curve Number: \[CN_{composite} = \frac{\sum_{i=1}^{n}CN_i \times A_i}{\sum_{i=1}^{n}A_i}\]

• CN_i = curve number for sub-area i
• A_i = sub-area i

Unit Hydrograph Theory

Unit Hydrograph Definition:

• Hydrograph resulting from 1 inch (or 1 cm) of direct runoff generated uniformly over the drainage area at a constant rate for an effective duration

Direct Runoff Hydrograph by Convolution: \[Q_n = \sum_{m=1}^{M}P_m U_{n-m+1}\]

• Q_n = total direct runoff at time n
• P_m = excess rainfall during time period m
• U_n-m+1 = unit hydrograph ordinate
• M = number of excess rainfall periods

S-Curve Method (for changing unit hydrograph duration):

• S-curve = summation of infinite series of unit hydrographs offset by one time unit
• Used to derive unit hydrographs of different durations from an existing unit hydrograph

NRCS (SCS) Dimensionless Unit Hydrograph

Peak Discharge: \[q_p = \frac{484 A}{t_p}\]

• q_p = peak discharge per unit drainage area (cfs/mi²/in)
• A = drainage area (mi²)
• t_p = time to peak (hr)
• 484 = constant for US Customary units

Alternative Peak Discharge Formula: \[Q_p = \frac{484 A Q}{t_p}\]

• Q_p = peak discharge (cfs)
• Q = direct runoff depth (inches)

Time to Peak: \[t_p = \frac{D}{2} + t_{lag}\]

• t_p = time to peak (hr)
• D = duration of excess rainfall (hr)
• t_lag = lag time (hr)

Base Time: \[T_b = 2.67 t_p\]

• T_b = base time of unit hydrograph (hr)
• t_p = time to peak (hr)

Snyder's Synthetic Unit Hydrograph

Standard Lag Time: \[t_{p,R} = C_t(L L_{ca})^{0.3}\]

• t_p,R = basin lag time for standard duration t_R (hr)
• C_t = coefficient representing watershed slope and storage (0.4 to 0.8, typical 0.6)
• L = length of main stream from outlet to divide (mi)
• L_ca = length along main stream to point nearest centroid of watershed (mi)

Standard Duration: \[t_R = \frac{t_{p,R}}{5.5}\]

• t_R = standard duration of excess rainfall (hr)

Adjusted Lag Time (for non-standard duration): \[t_{lag} = t_{p,R} + 0.25(t_D - t_R)\]

• t_lag = adjusted lag time (hr)
• t_D = desired duration of excess rainfall (hr)

Peak Flow Rate: \[Q_p = \frac{640 C_p A}{t_{lag}}\]

• Q_p = peak discharge (cfs) per inch of runoff
• C_p = peaking coefficient (0.4 to 0.8, function of watershed characteristics)
• A = drainage area (mi²)
• 640 = conversion constant

Base Time (Snyder): \[T_b = \frac{3 + \frac{t_{lag}}{8}}{24}\] or \[T_b = 3 + \frac{3t_{lag}}{24}\]

• T_b = base time (days)
• t_lag = lag time (hr)

Hydrograph Separation and Analysis

Baseflow Separation Methods

Straight-Line Method:

• Connect point of rise on recession limb to point where direct runoff ends on falling limb
• Time to end of direct runoff: N days after peak

\[N = A^{0.2}\]

• N = time interval (days)
• A = drainage area (mi²)

Constant Discharge Method:

• Extend pre-storm baseflow rate horizontally until it intersects recession curve

Concave Method:

• Project pre-storm recession curve to point beneath peak, then straight line to end of direct runoff

Hydrograph Recession

Exponential Recession: \[Q_t = Q_0 K^t\] or \[Q_t = Q_0 e^{-\alpha t}\]

• Q_t = discharge at time t
• Q₀ = discharge at time t = 0 (start of recession)
• K = recession constant (<>
• α = recession coefficient
• t = time since start of recession

Relationship between K and α: \[K = e^{-\alpha}\]

Infiltration

Horton's Infiltration Equation

\[f_t = f_c + (f_0 - f_c)e^{-kt}\]

• f_t = infiltration capacity at time t (in/hr or mm/hr)
• f_c = final (constant) infiltration capacity (in/hr or mm/hr)
• f₀ = initial infiltration capacity (in/hr or mm/hr)
• k = decay constant (hr⁻¹)
• t = time from beginning of storm (hr)

Cumulative Infiltration: \[F = f_c t + \frac{f_0 - f_c}{k}(1 - e^{-kt})\]

• F = cumulative infiltration depth (in or mm)

Green-Ampt Infiltration Method

Infiltration Rate: \[f = K\left(1 + \frac{\psi \Delta\theta}{F}\right)\]

• f = infiltration rate (in/hr or mm/hr)
• K = hydraulic conductivity of soil (in/hr or mm/hr)
• ψ = wetting front soil suction head (in or mm)
• Δθ = change in moisture content = (porosity - initial moisture content) (dimensionless)
• F = cumulative infiltration depth (in or mm)

Cumulative Infiltration (implicit): \[F = Kt + \psi\Delta\theta \ln\left(1 + \frac{F}{\psi\Delta\theta}\right)\] Time to Ponding: \[t_p = \frac{K\psi\Delta\theta}{i(i-K)}\]

• t_p = time to ponding (hr)
• i = rainfall intensity (in/hr or mm/hr)
• Condition: Valid when i > K

Philip's Infiltration Equation

\[f = \frac{1}{2}St^{-1/2} + A\]

• f = infiltration rate (in/hr or mm/hr)
• S = sorptivity (in/hr^1/2 or mm/hr^1/2)
• A = constant approximately equal to hydraulic conductivity
• t = time (hr)

Cumulative Infiltration: \[F = St^{1/2} + At\]

Kostiakov Equation

\[F = at^b\]

• F = cumulative infiltration (in or mm)
• a, b = empirical constants
• t = time (min or hr)

Infiltration Rate: \[f = \frac{dF}{dt} = abt^{b-1}\]

Flow Duration and Frequency Analysis

Flow Duration Curve

Exceedance Probability: \[P = \frac{m}{n+1} \times 100\%\]

• P = percent of time discharge is equaled or exceeded
• m = rank of discharge value (1 = highest)
• n = total number of values

Flood Frequency Analysis

Return Period (Recurrence Interval): \[T = \frac{1}{P}\]

• T = return period (years)
• P = annual exceedance probability (probability of event occurring in any given year)

Weibull Plotting Position: \[P = \frac{m}{n+1}\]

• P = exceedance probability
• m = rank (1 = largest value)
• n = number of years of record

Probability of at Least One Occurrence in n Years: \[P_r = 1 - (1-P)^n\] or \[P_r = 1 - \left(1 - \frac{1}{T}\right)^n\]

• P_r = risk or probability of at least one occurrence in n years
• P = annual exceedance probability
• n = time period (years)
• T = return period (years)

Log-Pearson Type III Distribution

General Equation: \[\log Q_T = \overline{\log Q} + K_T \cdot s\]

• Q_T = discharge for return period T
• log Q with overbar = mean of logarithms of annual peak discharges
• K_T = frequency factor (function of return period and skew coefficient)
• s = standard deviation of logarithms of annual peak discharges

Mean of Logarithms: \[\overline{\log Q} = \frac{1}{n}\sum_{i=1}^{n}\log Q_i\] Standard Deviation of Logarithms: \[s = \sqrt{\frac{\sum_{i=1}^{n}(\log Q_i - \overline{\log Q})^2}{n-1}}\] Skew Coefficient: \[G = \frac{n\sum_{i=1}^{n}(\log Q_i - \overline{\log Q})^3}{(n-1)(n-2)s^3}\]

• G = skew coefficient
• n = number of annual peaks in record
• Note: K_T values are obtained from tables based on G and return period T

Gumbel (Extreme Value Type I) Distribution

Discharge Equation: \[Q_T = \overline{Q} + K_T \cdot \sigma\]

• Q_T = discharge for return period T
• Q with overbar = mean of annual peak discharges
• K_T = frequency factor
• σ = standard deviation of annual peak discharges

Frequency Factor: \[K_T = -\frac{\sqrt{6}}{\pi}\left[0.5772 + \ln\left(\ln\frac{T}{T-1}\right)\right]\] or approximately: \[K_T = -0.45 + 0.7797\ln\left(-\ln\left(1-\frac{1}{T}\right)\right)\] Reduced Variate: \[y = -\ln\left(-\ln\left(1-\frac{1}{T}\right)\right)\] \[Q_T = \overline{Q} + \frac{\sigma}{s_n}(y - y_n)\]

• y = reduced variate
• y_n = mean of reduced variate (function of sample size n)
• s_n = standard deviation of reduced variate (function of sample size n)

Evaporation and Evapotranspiration

Water Budget Method for Reservoir

\[E = P + R_{in} - R_{out} - I - \Delta S\]

• E = evaporation
• P = precipitation on reservoir surface
• R_in = surface water inflow
• R_out = surface water outflow
• I = infiltration/seepage
• ΔS = change in storage
• Note: All terms in consistent units (volume or depth)

Energy Budget Method

\[E = \frac{Q_n - Q_h - Q_g}{\lambda \rho_w}\]

• E = evaporation rate
• Q_n = net radiation
• Q_h = sensible heat flux
• Q_g = heat flux into water body
• λ = latent heat of vaporization
• ρ_w = density of water

Mass Transfer (Aerodynamic) Method

\[E = (a + bu)(e_s - e_a)\]

• E = evaporation rate
• a, b = empirical coefficients
• u = wind speed
• e_s = saturation vapor pressure at water surface temperature
• e_a = vapor pressure of air

Meyer Equation

\[E = C(e_s - e_a)(1 + \frac{u}{10})\]

• E = evaporation (in/month)
• C = coefficient (approximately 15 for small, deep lakes; 11 for shallow lakes)
• e_s = saturation vapor pressure at water temperature (in Hg)
• e_a = vapor pressure of air (in Hg)
• u = monthly mean wind velocity (mph) at about 25 ft above ground

Penman Equation (Potential Evapotranspiration)

\[ET_p = \frac{\Delta R_n + \gamma E_a}{\Delta + \gamma}\]

• ET_p = potential evapotranspiration
• Δ = slope of saturation vapor pressure-temperature curve
• R_n = net radiation
• γ = psychrometric constant
• E_a = aerodynamic term (function of wind speed and vapor pressure deficit)

Blaney-Criddle Equation

\[ET = kpT\]

• ET = consumptive use (inches per month)
• k = empirical consumptive use coefficient (crop-dependent)
• p = monthly percentage of annual daytime hours (percent/100)
• T = mean monthly temperature (°F)

Thornthwaite Equation

\[ET_p = 16\left(\frac{10T}{I}\right)^a\]

• ET_p = unadjusted potential evapotranspiration (mm/month for 30-day month with 12-hr days)
• T = mean monthly temperature (°C)
• I = annual heat index = sum of 12 monthly values of i
• i = monthly heat index = (T/5)^1.514
• a = 0.000000675I³ - 0.0000771I² + 0.01792I + 0.49239

Reference Evapotranspiration (FAO Penman-Monteith)

\[ET_0 = \frac{0.408\Delta(R_n - G) + \gamma\frac{900}{T+273}u_2(e_s - e_a)}{\Delta + \gamma(1 + 0.34u_2)}\]

• ET₀ = reference evapotranspiration (mm/day)
• R_n = net radiation at crop surface (MJ/m²/day)
• G = soil heat flux density (MJ/m²/day)
• T = mean daily air temperature at 2 m height (°C)
• u₂ = wind speed at 2 m height (m/s)
• e_s = saturation vapor pressure (kPa)
• e_a = actual vapor pressure (kPa)
• Δ = slope of vapor pressure curve (kPa/°C)
• γ = psychrometric constant (kPa/°C)

Reservoir and Stream Routing

Continuity Equation

\[I - O = \frac{dS}{dt}\]

• I = inflow rate
• O = outflow rate
• S = storage
• t = time

Finite Difference Form: \[\frac{I_1 + I_2}{2} - \frac{O_1 + O_2}{2} = \frac{S_2 - S_1}{\Delta t}\]

• Subscript 1 = beginning of time interval
• Subscript 2 = end of time interval
• Δt = time interval

Modified Puls (Storage Indication) Method

Rearranged Continuity Equation: \[\frac{I_1 + I_2}{2}\Delta t + \left(\frac{S_1}{\Delta t} - \frac{O_1}{2}\right) = \frac{S_2}{\Delta t} + \frac{O_2}{2}\]

• Known: I₁, I₂, and (S₁/Δt - O₁/2)
• Unknown: (S₂/Δt + O₂/2)
• Solve using storage-indication curve: plot O vs. (S/Δt + O/2)

Storage-Outflow Relationship: \[S = f(O)\]

• Relationship depends on reservoir characteristics and outlet structures

Muskingum Method (Channel Routing)

Storage Equation: \[S = K[xI + (1-x)O]\]

• S = storage in reach
• K = travel time through reach (storage coefficient, hr)
• x = weighting factor (0 ≤ x ≤ 0.5); x = 0 for reservoir, x = 0.5 for pure translation
• I = inflow rate
• O = outflow rate

Routing Equation: \[O_2 = C_0I_2 + C_1I_1 + C_2O_1\]

• O₂ = outflow at end of time step
• I₂ = inflow at end of time step
• I₁ = inflow at beginning of time step
• O₁ = outflow at beginning of time step

Routing Coefficients: \[C_0 = \frac{-Kx + 0.5\Delta t}{K - Kx + 0.5\Delta t}\] \[C_1 = \frac{Kx + 0.5\Delta t}{K - Kx + 0.5\Delta t}\] \[C_2 = \frac{K - Kx - 0.5\Delta t}{K - Kx + 0.5\Delta t}\]

• Check: C₀ + C₁ + C₂ = 1
• Δt = routing time step
• Condition: K > Δt > 2Kx for numerical stability

Muskingum-Cunge Method

Parameter K: \[K = \frac{\Delta x}{c}\]

• Δx = length of river reach
• c = wave celerity

Wave Celerity: \[c = \frac{1}{B}\frac{dQ}{dy}\]

• B = top width of channel
• Q = discharge
• y = depth

For Wide Rectangular Channels: \[c = \frac{5}{3}V\]

• V = average velocity

Parameter x: \[x = \frac{1}{2}\left(1 - \frac{Q}{BS_0c\Delta x}\right)\]

• S₀ = bed slope

Groundwater Hydrology

Darcy's Law

One-Dimensional Flow: \[Q = -KA\frac{dh}{dl}\] or \[Q = KiA\]

• Q = discharge (volume per time)
• K = hydraulic conductivity (length per time)
• A = cross-sectional area perpendicular to flow
• dh/dl = hydraulic gradient
• i = hydraulic gradient (positive in direction of flow)

Darcy Velocity (Specific Discharge): \[q = \frac{Q}{A} = Ki\]

• q = Darcy velocity or specific discharge (not actual pore velocity)

Seepage Velocity (Average Linear Velocity): \[v = \frac{q}{n} = \frac{Ki}{n}\]

• v = average linear (seepage) velocity
• n = porosity (dimensionless)

Aquifer Properties

Transmissivity: \[T = Kb\]

• T = transmissivity (length²/time)
• K = hydraulic conductivity
• b = saturated thickness of aquifer

Storativity (Storage Coefficient): \[S = S_s b\]

• S = storativity (dimensionless)
• S_s = specific storage (1/length)
• b = aquifer thickness
• Note: For confined aquifers, S typically 0.00001 to 0.001
• Note: For unconfined aquifers, S ≈ S_y (specific yield), typically 0.01 to 0.30

Specific Yield: \[S_y = \frac{\Delta V_d}{V_t}\]

• S_y = specific yield (dimensionless)
• ΔV_d = volume of water drained by gravity
• V_t = total volume of saturated material

Specific Retention: \[S_r = n - S_y\]

• S_r = specific retention
• n = porosity

Well Hydraulics - Steady State Flow

Confined Aquifer (Thiem Equation): \[Q = \frac{2\pi Kb(h_2 - h_1)}{\ln(r_2/r_1)}\] or \[Q = \frac{2\pi T(h_2 - h_1)}{\ln(r_2/r_1)}\]

• Q = pumping rate
• T = transmissivity
• K = hydraulic conductivity
• b = aquifer thickness
• h₂, h₁ = hydraulic heads at radial distances r₂ and r₁ from well
• r₂, r₁ = radial distances from pumping well

Drawdown in Confined Aquifer: \[s = \frac{Q}{2\pi T}\ln\frac{r_0}{r}\]

• s = drawdown at distance r from well
• r₀ = radius of influence (where s = 0)
• r = radial distance from well

Unconfined Aquifer (Dupuit Equation): \[Q = \frac{\pi K(h_2^2 - h_1^2)}{\ln(r_2/r_1)}\]

• h₂, h₁ = saturated thicknesses at distances r₂ and r₁

Drawdown in Unconfined Aquifer: \[s = h_0 - h = h_0 - \sqrt{h_0^2 - \frac{Q}{\pi K}\ln\frac{r_0}{r}}\]

• h₀ = initial saturated thickness
• h = saturated thickness at distance r

Well Hydraulics - Unsteady State Flow

Theis Equation (Non-Equilibrium): \[s = \frac{Q}{4\pi T}W(u)\]

• s = drawdown
• Q = constant pumping rate
• T = transmissivity
• W(u) = well function (dimensionless)

\[u = \frac{r^2S}{4Tt}\]

• u = dimensionless time parameter
• r = distance from pumping well
• S = storativity
• t = time since pumping began

Well Function (series expansion for small u): \[W(u) = -0.5772 - \ln(u) + u - \frac{u^2}{2 \cdot 2!} + \frac{u^3}{3 \cdot 3!} - \frac{u^4}{4 \cdot 4!} + ...\]

• Valid for u < 0.01="" (common="">

Cooper-Jacob Approximation (for small u): \[s = \frac{2.30Q}{4\pi T}\log\frac{2.25Tt}{r^2S}\] or \[s = \frac{0.183Q}{T}\log\frac{2.25Tt}{r^2S}\]

• Valid when u < 0.01,="" which="" occurs="" when="" t="" is="" large="" or="" r="" is="">
• Condition: r²S/(4Tt) <>

Drawdown vs. Time (Cooper-Jacob): \[s = \frac{2.30Q}{4\pi T}\log t + C\]

• Plot of s vs. log(t) yields straight line
• Slope = 2.30Q/(4πT)

Transmissivity from Cooper-Jacob: \[T = \frac{2.30Q}{4\pi\Delta s}\]

• Δs = drawdown change per log cycle of time

Storativity from Cooper-Jacob: \[S = \frac{2.25Tt_0}{r^2}\]

• t₀ = time intercept where straight line projection intersects s = 0

Recovery Test

Residual Drawdown: \[s' = \frac{2.30Q}{4\pi T}\log\frac{t}{t'}\]

• s' = residual drawdown (difference between static level and recovered level)
• t = time since pumping started
• t' = time since pumping stopped

Detention and Retention Storage

Simplified Detention Basin Routing

Peak Discharge Reduction (Simplified): \[Q_{out} = Q_{in} - \frac{\Delta S}{\Delta t}\]

• Q_out = peak outflow
• Q_in = peak inflow
• ΔS = change in storage
• Δt = time interval

Required Storage Volume

Cumulative Inflow - Outflow Method: \[S_{required} = \max(\sum I\Delta t - \sum O\Delta t)\]

• S_required = required storage volume
• Sum over time intervals to find maximum difference between cumulative inflow and outflow

Water Quality and Mass Balance

First-Order Decay

\[C_t = C_0 e^{-kt}\]

• C_t = concentration at time t
• C₀ = initial concentration
• k = decay rate constant (1/time)
• t = time

Completely Mixed Reactor

Mass Balance: \[V\frac{dC}{dt} = QC_{in} - QC - kVC\]

• V = volume of reactor
• Q = flow rate
• C_in = inflow concentration
• C = concentration in reactor (and outflow)
• k = first-order decay constant

Steady-State Concentration: \[C = \frac{QC_{in}}{Q + kV}\]

Plug Flow Reactor

\[C_{out} = C_{in}e^{-kt}\]

• C_out = outlet concentration
• C_in = inlet concentration
• t = detention time = V/Q

Snowmelt

Degree-Day Method

\[M = C_m(T_a - T_b)\]

• M = daily snowmelt depth (in or mm)
• C_m = degree-day factor or melt coefficient (in/°F/day or mm/°C/day)
• T_a = mean daily air temperature (°F or °C)
• T_b = base temperature, typically 32°F or 0°C
• Condition: Valid only when T_a > T_b

Drainage Basin Characteristics

Drainage Density

\[D_d = \frac{L_t}{A}\]

• D_d = drainage density (length/area, e.g., mi/mi² or km/km²)
• L_t = total length of all streams
• A = drainage area

Basin Shape Factor

Form Factor: \[F_f = \frac{A}{L^2}\]

• F_f = form factor (dimensionless)
• A = basin area
• L = basin length

Compactness Coefficient: \[C_c = \frac{P}{2\sqrt{\pi A}}\]

• C_c = compactness coefficient (dimensionless, ≥ 1)
• P = perimeter of basin
• A = basin area
• Note: C_c = 1 for a circular basin

Equivalent Circular Diameter

\[D = 2\sqrt{\frac{A}{\pi}}\]

• D = equivalent circular diameter
• A = drainage area

Statistical and Probability Concepts

Basic Statistics

Mean: \[\overline{x} = \frac{1}{n}\sum_{i=1}^{n}x_i\] Variance: \[\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \overline{x})^2\] or (sample variance): \[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \overline{x})^2\] Standard Deviation: \[\sigma = \sqrt{\sigma^2}\] or \[s = \sqrt{s^2}\] Coefficient of Variation: \[C_v = \frac{\sigma}{\overline{x}}\]

• C_v = coefficient of variation (dimensionless)

Coefficient of Skewness: \[C_s = \frac{n\sum_{i=1}^{n}(x_i - \overline{x})^3}{(n-1)(n-2)s^3}\]

• C_s = coefficient of skewness
• Note: Also denoted as G in some formulations