7.3 Time of concentration and travel time estimation

Time of concentration is crucial in understanding how watersheds respond to rainfall. It's the time water takes to flow from the farthest point to the outlet, influencing peak discharge and hydrograph shape. Accurate estimation is vital for flood risk assessment and hydraulic design.

Calculating time of concentration involves empirical formulas and hydraulic methods. Factors like watershed size, shape, slope, and surface roughness affect it. Travel time estimation considers flow velocity, channel geometry, and Manning's roughness coefficient, helping predict runoff hydrographs accurately.

Time of concentration and its significance

Definition and importance of time of concentration

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• Time of concentration (Tc) is the time required for water to flow from the most hydraulically distant point in a watershed to the outlet or point of interest
• Tc is a critical parameter in hydrological modeling determines the peak discharge and the shape of the runoff hydrograph
• Accurate estimation of Tc is essential for designing hydraulic structures (culverts, bridges), flood risk assessment, and stormwater management
• Tc helps in understanding the response time of a watershed to rainfall events and the potential for flooding

Factors influencing time of concentration

• Tc is influenced by various factors related to watershed characteristics and flow dynamics
• Watershed size affects Tc larger watersheds generally have longer flow paths and higher Tc values compared to smaller watersheds
• Watershed shape influences the distribution of flow paths and the concentration of runoff (elongated watersheds tend to have longer Tc than circular watersheds)
• Watershed slope impacts flow velocity and Tc steeper slopes result in faster flow and shorter Tc, while gentler slopes lead to slower flow and longer Tc
• Surface roughness, represented by Manning's roughness coefficient (n), affects flow resistance and Tc higher roughness values (dense vegetation, irregular surfaces) increase Tc, while smoother surfaces (paved areas, channels) reduce Tc
• Drainage network characteristics, such as the density, pattern, and sinuosity of streams, influence the efficiency of runoff conveyance and Tc

Calculating time of concentration

Empirical formulas for estimating time of concentration

• Empirical formulas provide simple and quick estimates of Tc based on watershed characteristics
• Kirpich formula: $Tc = 0.0078 × L^{0.77} × S^{-0.385}$, where L is the longest flow path (ft) and S is the average slope (ft/ft)
  • Developed for small agricultural watersheds in Tennessee
  • Suitable for watersheds with well-defined channels and steep slopes
• California Culverts Practice formula: $Tc = (0.87075 × L^3 / H)^{0.385}$, where L is the longest flow path (mi) and H is the elevation difference (ft)
  • Commonly used for designing culverts and small drainage structures in California
  • Considers the longest flow path and elevation difference in the watershed
• SCS lag equation: $Tc = L^{0.8} × (S + 1)^{0.7} / 1900 × Y^{0.5}$, where L is the hydraulic length (ft), S is the average watershed slope (%), and Y is the average runoff coefficient
  • Developed by the U.S. Soil Conservation Service (now NRCS)
  • Incorporates the hydraulic length, slope, and runoff characteristics of the watershed

Hydraulic methods for calculating time of concentration

• Hydraulic methods calculate Tc by considering the flow dynamics and channel characteristics
• Kinematic wave method: $Tc = Σ(Li / vi)$, where Li is the length of each flow segment and vi is the corresponding flow velocity
  • Accounts for the varying flow velocities along different segments of the flow path
  • Requires knowledge of flow velocities, which can be estimated using Manning's equation or other hydraulic relationships
• Muskingum-Cunge method: $Tc = K × (X + 0.5) / (1 - X)$, where K is the travel time of the flood wave and X is the weighting factor representing the relative importance of inflow and outflow
  • Considers the propagation of a flood wave through a channel reach
  • Accounts for the storage and attenuation effects in the channel
  • Suitable for channels with mild slopes and significant storage capacity

Factors affecting travel time

Flow velocity and its determinants

• Flow velocity is a primary factor influencing travel time higher velocities result in shorter travel times
• Flow velocity is determined by the balance between gravitational forces (driving flow) and resistance forces (friction, form drag)
• Gravitational forces are proportional to the slope of the flow path steeper slopes generate higher flow velocities
• Resistance forces are influenced by surface roughness, channel geometry, and flow characteristics
• Manning's equation relates flow velocity to channel slope, hydraulic radius, and roughness coefficient: $v = (1/n) × R^{2/3} × S^{1/2}$, where v is the flow velocity (m/s), n is Manning's roughness coefficient, R is the hydraulic radius (m), and S is the channel slope (m/m)

Surface roughness and Manning's roughness coefficient

• Surface roughness affects flow resistance and travel time higher roughness leads to slower velocities and longer travel times
• Manning's roughness coefficient (n) quantifies the surface roughness and its impact on flow
• Typical values of Manning's n range from 0.01 for smooth concrete to 0.1 for densely vegetated channels
• Estimating Manning's n requires field observations, published tables, or empirical relationships based on surface characteristics (grain size, vegetation type and density)
• Accurate estimation of Manning's n is crucial for reliable travel time calculations and hydrological modeling

Channel geometry and hydraulic characteristics

• Channel geometry, including cross-sectional shape, width, depth, and slope, influences flow hydraulics and travel time
• Hydraulic radius (R) is the ratio of the cross-sectional area (A) to the wetted perimeter (P): $R = A / P$
  • Larger hydraulic radii indicate more efficient flow conveyance and higher velocities
  • Channels with high width-to-depth ratios (wide and shallow) have smaller hydraulic radii compared to channels with low width-to-depth ratios (narrow and deep)
• Channel sinuosity, the degree of channelization, and the presence of obstructions or storage areas affect travel time
  • Sinuous channels have longer flow paths and higher travel times compared to straight channels
  • Channelization (straightening and deepening of channels) reduces travel time by increasing flow velocity and efficiency
  • Obstructions (boulders, woody debris) and storage areas (floodplains, wetlands) can slow down flow and increase travel time

Predicting runoff hydrographs

Time-area method for runoff hydrograph estimation

• The time-area method discretizes the watershed into subareas and calculates the travel time from each subarea to the outlet
• The runoff hydrograph is obtained by summing the contributions from each subarea, considering their respective travel times
• Steps in the time-area method:
  1. Divide the watershed into subareas based on travel time isochrones (lines of equal travel time)
  2. Estimate the travel time from each subarea to the outlet using empirical formulas or hydraulic methods
  3. Develop a time-area diagram representing the cumulative watershed area contributing runoff as a function of travel time
  4. Convert the excess rainfall hyetograph into a runoff hydrograph by multiplying the rainfall intensity by the contributing area for each time step
• The time-area method provides a simple and intuitive approach to predict runoff hydrographs based on the spatial distribution of travel times

Unit hydrograph approach and its application

• The unit hydrograph (UH) approach uses the principle of superposition to estimate the runoff hydrograph
• A unit hydrograph represents the direct runoff response to a unit depth (1 cm or 1 inch) of excess rainfall uniformly distributed over the watershed for a specified duration
• Characteristics of a unit hydrograph:
  • Time base is related to the time of concentration longer Tc results in a longer UH time base
  • Peak discharge is inversely proportional to the time of concentration shorter Tc leads to higher peak discharge
  • UH shape reflects the watershed's response to rainfall, with a steep rising limb and a gradual recession limb
• To estimate the runoff hydrograph for a given rainfall event:
  1. Derive the excess rainfall hyetograph by subtracting losses (infiltration, interception) from the total rainfall
  2. Convolve the excess rainfall hyetograph with the unit hydrograph using discrete convolution or matrix multiplication methods
  3. The resulting runoff hydrograph represents the direct runoff response to the specific rainfall event
• The unit hydrograph approach assumes linearity and time-invariance of the watershed response, which may not hold for extreme events or changing watershed conditions

Convolution techniques for runoff hydrograph computation

• Convolution techniques are used to compute the runoff hydrograph by combining the excess rainfall hyetograph with the unit hydrograph
• Discrete convolution method:
  • The excess rainfall hyetograph and unit hydrograph are discretized into time steps
  • The runoff hydrograph is computed by summing the products of the excess rainfall and the corresponding unit hydrograph ordinates for each time step: $Q_i = Σ(P_j × U_{i-j+1})$, where Q_i is the runoff at time step i, P_j is the excess rainfall at time step j, and U_{i-j+1} is the unit hydrograph ordinate at time step i-j+1
  • The convolution process accounts for the time-varying contribution of excess rainfall to the runoff hydrograph
• Matrix multiplication method:
  • The excess rainfall hyetograph and unit hydrograph are represented as vectors, and the convolution is performed through matrix multiplication
  • The runoff hydrograph is obtained by multiplying the excess rainfall vector by the unit hydrograph matrix: $Q = P × U$, where Q is the runoff hydrograph vector, P is the excess rainfall vector, and U is the unit hydrograph matrix
  • The matrix multiplication method provides a computationally efficient approach for runoff hydrograph estimation
• Convolution techniques enable the prediction of runoff hydrographs based on the principles of superposition and time-invariance, assuming a linear watershed response