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Time of concentration

Time of concentration (Tc), a fundamental concept in hydrology, refers to the duration required for stormwater runoff to travel from the hydraulically most distant point in a watershed to the outlet or a designated point of interest.^[1] This time is calculated as the sum of travel times across different flow regimes, including sheet flow, shallow concentrated flow, and open channel flow, and it represents the moment when the entire drainage area contributes to peak runoff at the outlet.^[2] The importance of time of concentration lies in its role as a key parameter for predicting the timing and magnitude of flood peaks, which is essential for stormwater management, flood control, and hydraulic infrastructure design.^[3] It directly influences the selection of rainfall intensity in peak discharge calculations, particularly in methods like the Rational Method and the Natural Resources Conservation Service (NRCS) TR-55 procedure, where shorter Tc values—often resulting from urbanization—increase peak flows and flood risks.^[4] Accurate estimation of Tc helps engineers design culverts, storm sewers, and detention basins to mitigate erosion, prevent flooding, and comply with regulatory standards for water quality and quantity control.^[5] Several empirical and semi-empirical methods are used to compute time of concentration, tailored to watershed characteristics such as slope, land use, and soil type.^[1] Common approaches include the NRCS Velocity Method, which divides flow into segments and applies Manning's equation for velocities (e.g., sheet flow travel time t_{sh} = \frac{0.007 (n L)^{0.8}}{(P_2^{0.5} S^{0.4})} in hours, where n is Manning's roughness coefficient, L is flow length in feet, P_2 is 2-year 24-hour rainfall in inches, and S is slope), and the NRCS Lag Method for basin lag time T_l = \frac{L^{0.8} (S + 1)^{0.7}}{1900 Y^{0.5}} in hours (where L is flow length in feet, S is retention parameter, Y is average watershed slope in percent), with T_c \approx T_l / 0.6.^[2] Other widely applied formulas are the Kirpich equation for channel flow (t_c = 0.0078 \left( \frac{L}{S^{0.5}} \right)^{0.77} in minutes) and the Kerby method for overland flow, often combined for comprehensive estimates.^[1] These methods emphasize the need for site-specific data, with adjustments for impervious surfaces in urban settings to reflect accelerated runoff.^[6]

Definition and Conceptual Basis

Definition

The time of concentration, denoted as T_c, is defined as the time required for runoff to travel from the hydraulically most distant point in a watershed to the outlet or point of interest.^[1] This parameter captures the duration for a parcel of water to propagate through the drainage system under equilibrium flow conditions.^[2] Conceptually, T_c represents the watershed's response time to a rainfall event, marking the moment when the entire contributing area is actively supplying flow to achieve peak discharge at the outlet.^[6] It reflects the time for the equilibrium between rainfall excess and runoff to establish across the basin, influencing the timing and magnitude of hydrograph peaks.^[7] The concept originated in late 19th-century hydrology, with Emil Kuichling providing the first explicit definition in 1889 as part of the Rational Method for urban sewer design, where it denotes the travel time for runoff from the farthest impervious surfaces under uniform rainfall intensity.^[7] Kuichling's formulation emphasized the duration needed for rainfall to produce maximum discharge, building on earlier work by Mulvany in 1851.^[8] Typically expressed in minutes for small urban watersheds or hours for larger rural basins, T_c scales with drainage area and topography.^[1]

Hydrologic Role

The time of concentration (Tc) serves as a fundamental parameter in hydrologic modeling by defining the duration of rainfall required to achieve peak runoff rates in a watershed. Specifically, it represents the time necessary for runoff from the entire contributing area to reach the outlet, such that rainfall durations shorter than Tc result in only partial watershed contribution to the peak flow, while durations equal to or exceeding Tc allow the system to reach a state of equilibrium where the full area contributes simultaneously.^[9]^[10] In the context of intensity-duration-frequency (IDF) curves, Tc plays a critical role by establishing the appropriate storm duration for selecting design rainfall intensities used in peak flow computations. Hydrologic methods, such as the Rational Method, rely on Tc to index IDF curves and derive the rainfall intensity that corresponds to the time for runoff to equilibrate across the watershed, ensuring accurate estimation of design storms for various return periods.^[11]^[12] Tc significantly influences the shape and characteristics of the runoff hydrograph, with shorter Tc values producing hydrographs featuring steeper rising limbs and higher peak discharges due to rapid concentration of flows, whereas longer Tc values lead to more gradual rises and attenuated peaks as runoff disperses over time. This variability in hydrograph form underscores Tc's importance in predicting watershed response to precipitation events.^[10]^[9] In urban hydrology, Tc is essential for designing infrastructure to manage peak flows, including the sizing of culverts, storm drains, and detention basins, where accurate Tc estimation ensures systems can accommodate equilibrated runoff without overflow or erosion.^[12]^[1]

Flow Path Components

Overland Flow

Overland flow represents the initial phase of surface runoff, characterized as shallow, laminar sheet flow that spreads diffusely across impervious or pervious land surfaces without forming defined channels. This flow typically occurs in the headwaters of watersheds or on planar slopes, where rainfall excess begins to move downslope under the influence of gravity. Due to high surface friction, the flow depth remains shallow, generally less than 0.3 meters (1 foot), resulting in low velocities that prolong the travel time relative to subsequent flow regimes.^[10] The time of concentration for overland flow depends primarily on three key variables: surface roughness (quantified by Manning's n), slope (S, in ft/ft or m/m), and flow length (L). Surface roughness accounts for frictional resistance from vegetation, litter, or pavement texture, with typical n values ranging from 0.011 for smooth asphalt to 0.15 for dense grass. Steeper slopes accelerate flow, while longer paths increase travel time; however, overland flow is practically limited to the initial 100 to 300 feet (30 to 90 meters) before transitioning to more concentrated forms, as beyond this distance, rills or small channels begin to form.^[10]^[1] Basic estimation of overland flow travel time follows the principle that time equals distance divided by average velocity, where velocity is approximated using Manning's equation adapted for shallow sheet flow:

V = \frac{1.49}{n} R^{2/3} S^{1/2}

Here, V is the average velocity, n is Manning's roughness coefficient, S is the slope, and R is the hydraulic radius, often approximated as the flow depth for very shallow conditions (typically 0.001 to 0.01 m). This kinematic approximation neglects inertial effects, assuming uniform flow under gravity, and is suitable for slopes between 0.0005 and 0.10. Integrating velocity over the path length yields the travel time, though direct application requires iterative solutions for depth.^[10] For practical hydrologic design, an empirical adjustment derived from kinematic wave analysis is widely used, particularly the NRCS sheet flow equation:

T_c = \frac{0.007 (n L)^{0.8}}{P_2^{0.5} S^{0.4}}

In this formula (English units), T_c is the travel time in hours, L is the flow length in feet (maximum 100-300 ft depending on surface), n is Manning's n, S is the slope in ft/ft, and P_2 is the 2-year, 24-hour rainfall depth in inches, which empirically accounts for initial abstractions and infiltration effects. This equation, developed from field data and kinematic simulations, simplifies computation while maintaining accuracy for overland segments and forms part of the total time of concentration when summed with downstream components.^[13]

Shallow Concentrated Flow

Shallow concentrated flow occurs as an intermediate stage in runoff routing, where overland sheet flow converges into small, undefined channels such as rills, swales, or gullies, typically initiating beyond 30 to 90 meters (100 to 300 feet) from the hydraulically remote point in the watershed. This phase is characterized by flow depths of approximately 0.03 to 0.15 meters (0.1 to 0.5 feet) and lacks well-defined banks or cross-sections, distinguishing it from stable open channels. Velocities during this phase generally range from 0.5 to 2 meters per second (1.6 to 6.6 feet per second), influenced by slope and surface conditions, marking a transition to faster movement than the preceding sheet flow.^[14]^[10] Velocity estimation for shallow concentrated flow employs the velocity-area method, approximating flow in a wide rectangular section using empirical relations based on Manning's equation. In the widely adopted NRCS method, average velocity V is given by

V = k S^{0.5}

where S is the slope (dimensionless, ft/ft or m/m), and k is an intercept coefficient specific to surface type: 16.13 ft/s (4.92 m/s) for unpaved surfaces and 20.33 ft/s (6.20 m/s) for paved surfaces. These values derive from assumed hydraulic radii and roughness coefficients (Manning's n \approx 0.05 for unpaved, n \approx 0.013 for paved) and apply to slopes exceeding 0.005; for gentler slopes, direct application of Manning's equation with specified depths (e.g., 0.06 m for unpaved, 0.15 m for paved) is recommended.^[14]^[10] The time of concentration contribution from this phase, t_{sc}, is calculated as

t_{sc} = \frac{L}{V}

where L is the flow path length through the segment (in consistent units with V), often converted to hours by dividing by 3600 for English units or 1 for SI if V is in m/s and L in meters (yielding seconds, then converted as needed). In rural watersheds, this segment typically accounts for 10-20% of the total time of concentration due to moderate path lengths and elevated velocities relative to overland flow.^[14]^[10] This phase transitions from overland sheet flow when surface irregularities cause concentration, ending upon entry into defined open channels with stable banks and measurable cross-sections, often after path lengths of 100 to several hundred meters depending on terrain. Flow depths in this regime remain shallow, generally 0.03 to 0.15 meters, beyond which the assumptions of the velocity-area method no longer hold, signaling a shift to channel flow analysis. Brief integration with the prior overland flow phase occurs by chaining the endpoint of sheet flow as the starting point for length L.^[14]^[10]

Open Channel Flow

Open channel flow represents the channelized phase of runoff travel within defined streams, ditches, or pipes, where water is conveyed in a stable, banked path toward the watershed outlet. This phase typically occurs after overland and shallow concentrated flow, serving as the primary conveyance mechanism in the lower portions of the flow path. In large watersheds, open channel flow often constitutes the longest segment of the time of concentration due to extended channel lengths, with typical velocities ranging from 1 to 5 m/s depending on channel characteristics and flow conditions.^[15] The time of concentration component for open channel flow, t_{ch}, is estimated as the channel length L divided by the average velocity V, or t_{ch} = \frac{L}{V}. Velocity V is commonly computed using Manning's equation:

V = \frac{1.49}{n} R^{2/3} S^{1/2}

where n is Manning's roughness coefficient, R is the hydraulic radius (cross-sectional area divided by wetted perimeter), and S is the channel slope, all in US customary units (ft/s for V, ft for R, ft/ft for S). For SI units, the constant 1.49 is replaced by 1.0 (m/s). The roughness coefficient n varies by channel type; for example, it is approximately 0.013 for smooth concrete channels and 0.035 for straight natural streams with no rifts or pools.^[15]^[16] The flow path length L for open channel flow is measured along the thalweg (deepest part of the channel) or the centerline of the main channel, which is particularly critical in large basins where this segment dominates the total time of concentration. To account for channel sinuosity or meanders, the actual curvilinear distance along the channel is used rather than a straight-line valley length, effectively reducing the average velocity compared to a hypothetical straight path. This open channel contribution is summed with upstream flow path times to yield the overall watershed time of concentration.^[15]

Estimation Methods

Empirical Formulas

Empirical formulas provide simple, data-driven estimates of time of concentration (Tc) based on observed hydrologic data from small watersheds, making them suitable for preliminary design in engineering applications where detailed topographic or hydraulic data are unavailable. These methods typically rely on key parameters such as flow path length (L) and watershed slope (S), calibrated from field measurements in specific conditions like agricultural or rural settings. They are widely used due to their ease of application, though their accuracy diminishes for larger or more complex watersheds. The Kirpich formula, developed in 1940, estimates Tc for steep, small agricultural watersheds under 100 acres, where channel flow predominates. It is expressed as:

T_c = 0.0078 \, L^{0.77} \, S^{-0.385}

where T_c is in minutes, L is the maximum flow path length in feet, and S is the average watershed slope in ft/ft. This equation was calibrated using data from experimental watersheds in Tennessee, emphasizing its utility for rural, ungaged areas with slopes greater than 2%. ^[1]^[10] The Kerby formula, introduced in 1959, estimates overland flow time of concentration and is often combined with other methods for shallow concentrated and channel flow, making it appropriate for unpaved, rural watersheds where sheet flow transitions to rills. The overland flow component is calculated as:

t_o = 0.828 \left( \frac{L_o}{S_o^{0.5}} \right)^{0.467} n_o^{0.376}

where t_o is in minutes, L_o is the overland flow length in feet, S_o is the overland slope in ft/ft, and n_o is the retardance coefficient (related to Manning's roughness; e.g., 0.011 for smooth asphalt, 0.15 for dense grass). For shallow concentrated flow, velocities are typically assumed at 1.8 ft/s for unpaved and 3.3 ft/s for paved surfaces, or use complementary methods like Kirpich for channels. Developed from limited data on small, grassed watersheds, it performs best for overland lengths under 400 feet and areas under 200 acres with minimal channel development. ^[1]^[10] The NRCS (formerly SCS) lag method, refined in the mid-20th century, derives Tc from basin lag time (T_lag), which represents the time from the center of mass of excess rainfall to the peak of the unit hydrograph. The lag time is given by:

T_{lag} = \frac{L^{0.8} (S + 1)^{0.7}}{1900 Y^{0.5}}

where T_{lag} is in hours, L is the flow length in feet, S is the retention parameter (related to curve number CN as S = \frac{1000}{CN} - 10), and Y is the average watershed slope in percent; Tc is then approximated as T_c = \frac{T_{lag}}{0.6}. This approach, applicable to agricultural and developing watersheds up to several thousand acres, integrates land use via CN and was calibrated from diverse U.S. data for use in flood peak estimation. ^[10]^[17] In the Rational Method, Tc is estimated to determine the rainfall intensity duration, assuming equilibrium runoff occurs when rainfall duration equals Tc, valid for small urban or rural areas under 200 acres. Tc is typically computed using velocity-based approximations along the flow path, such as average overland velocities (1-3 ft/s depending on surface) or channel velocities (via Manning's equation), summed for total travel time; empirical aids like nomographs or the above formulas are often employed for simplicity. This method prioritizes quick calculations for stormwater design in impervious settings. ^[18]^[19]

Kinematic Wave Approach

The kinematic wave approach offers a physically based method for estimating the time of concentration (Tc) in hydrologic modeling, focusing on the propagation of unsteady flow waves without requiring the full solution of dynamic hydrodynamic equations. This method approximates the Saint-Venant equations, which govern one-dimensional unsteady open-channel flow through conservation of mass and momentum, by omitting the local and convective acceleration terms as well as the pressure gradient in the momentum equation. The resulting simplification assumes that gravity and bed friction dominate flow behavior, with discharge Q expressed as a function of flow cross-sectional area A (or depth for overland flow), enabling the kinematic wave celerity to be defined as c = \frac{dQ}{dA}.^[20] Within this framework, Tc represents the travel time for a kinematic shock—the leading front of the disturbance caused by rainfall excess—to advance from the upstream boundary of the flow path to the outlet. For a path length L, this yields the approximation T_c \approx \frac{L}{c}, where c is evaluated at the equilibrium depth corresponding to the design rainfall intensity. When Manning's equation is employed to relate flow to depth, slope, and roughness, the celerity simplifies to c = \frac{5}{3} V for wide channels, with V denoting the mean velocity; Manning's equation provides the velocity-depth relationship fundamental to open-channel flow computations. This derivation is particularly suited to overland or shallow concentrated flow segments, capturing the nonlinear advancement of the wave front.^[21]^[20] The approach excels in solving for the time to reach flow equilibrium in overland or channel segments, accounting for depth-dependent wave speeds that vary with rainfall intensity and surface conditions. It proves valuable in urban settings with heterogeneous roughness, such as pavements transitioning to vegetated areas, where variable friction influences wave propagation and enhances Tc accuracy over uniform-flow assumptions.^[22] Implementation of the kinematic wave method is widespread in software like HEC-HMS, a model from the U.S. Army Corps of Engineers, which applies kinematic routing to subbasins and channels; here, Tc emerges as an output from the simulated wave travel time during hydrograph transformation.^[23]

Hydrograph-Based Methods

Hydrograph-based methods for estimating the time of concentration (Tc) derive this parameter from the analysis of observed or synthetic hydrographs, leveraging the temporal distribution of runoff to infer watershed response times. These approaches are particularly useful in data-rich environments or for complex watersheds where spatial variability in flow paths influences the hydrograph shape. By examining the timing of hydrograph peaks, inflections, or equilibrium points, Tc can be related to the duration over which the entire contributing area delivers runoff to the outlet.^[24] The time-area method conceptualizes the watershed as divided into isochrones—contour lines of equal travel time from the outlet—and estimates Tc as the time required for runoff from the farthest isochrone to reach the outlet, effectively summing travel times across subareas that cumulatively cover the entire watershed. In this approach, a cumulative area-time diagram is constructed by ordering subareas by their travel times, and the hydrograph is generated by convolving rainfall excess with the time-area histogram; Tc corresponds to the time at which the cumulative area equals the total watershed area, marking full hydrograph equilibrium. This method assumes uniform rainfall excess and neglects storage effects, focusing on translation of the rainfall hyetograph.^[25] Unit hydrograph analysis infers Tc from the characteristics of a unit hydrograph, typically by relating the time to peak (Tp) to basin lag and concentration time, where Tp ≈ D/2 + 0.6 Tc, with D as the unit duration, allowing backward estimation of Tc from observed or synthetic unit hydrograph shapes. The unit hydrograph represents the runoff response to one unit of excess rainfall over duration D, and Tc is often taken as the time from excess rainfall cessation to the hydrograph inflection point on the recession limb, derived through S-curve construction or convolution of multiple unit hydrographs to match observed events. This method is widely applied in synthetic hydrograph development, such as the NRCS dimensionless unit hydrograph, where Tc shapes the rising limb duration.^[26] The geomorphic instantaneous unit hydrograph (GIUH) derives Tc from Horton's laws of drainage networks, which describe stream order branching ratios and lengths, linking geomorphic parameters like bifurcation ratio (Rb), stream length ratio (Rl), and area ratio (Ra) to the probability density function of travel times across the network. In this framework, the GIUH is modeled as a cascade of nonlinear reservoirs representing stream orders, with Tc expressed as the integral of travel times scaled by network topology and a dynamic velocity parameter; specifically, Tc ≈ (L_ω / v) * f(Rb, Rl), where L_ω is the length of the main stream, v is flow velocity, and f incorporates geomorphic ratios to capture basin response scaling. Developed for ungauged basins, GIUH emphasizes the fractal-like structure of river networks to predict hydrograph timing without direct flow path measurements.^[27]^[28] Calibration of Tc using hydrograph-based methods involves adjusting initial estimates to match observed streamflow peaks and timings in gauged basins, often by iteratively modifying travel times or lag parameters until simulated hydrographs align with recorded events in terms of peak discharge and time to peak. This process, applied in models like the Clark unit hydrograph, uses observed data to refine Tc such that the basin storage coefficient and response function reproduce hydrograph volumes and shapes, improving accuracy for flood prediction in calibrated watersheds. Such adjustments account for unmodeled factors like variable infiltration, ensuring Tc reflects site-specific hydrologic behavior.^[29]^[24]

Applications in Hydrologic Design

Peak Runoff Calculation

The time of concentration (Tc) plays a central role in peak runoff estimation through the Rational Method, a widely used approach for small urban and rural watersheds up to 200 acres. In this method, the peak discharge Q_p is calculated using the formula

Q_p = C \cdot I \cdot A

where C is the runoff coefficient representing the proportion of rainfall that contributes to direct runoff, I is the average rainfall intensity (in inches per hour) for a storm duration equal to Tc selected from intensity-duration-frequency (IDF) curves, and A is the contributing drainage area in acres, yielding Q_p in cubic feet per second (cfs). This duration choice for I assumes that the peak occurs when the entire watershed contributes runoff, which aligns with the time required for water from the farthest point to reach the outlet. The method's simplicity relies on steady, uniform rainfall over the Tc period, making it suitable for preliminary design of stormwater infrastructure like culverts and pipes.^[18] For more complex sites involving multiple subareas with differing flow paths, the Modified Rational Method extends the basic approach to handle unbalanced hydrographs where peaks from individual subareas do not coincide due to varying Tc values. Here, triangular or trapezoidal hydrographs are generated for each subarea using its specific Tc to determine the time to peak and base duration (typically 2 × Tc), with rainfall intensity adjusted based on the relationship between storm duration and Tc. These hydrographs are then temporally shifted according to travel times and superimposed at the downstream junction to identify the overall peak discharge, preserving volume while accounting for timing differences that could otherwise overestimate or underestimate combined flows. This technique is particularly valuable in urban developments with heterogeneous land uses, ensuring accurate routing for detention or conveyance systems.^[30] A practical example illustrates the application: Consider a 50-acre urban watershed with a Tc of 15 minutes, a runoff coefficient C of 0.75 for impervious surfaces, and a 10-year storm event where the 15-minute IDF intensity is 4.5 inches per hour. The peak discharge is then Q_p = 0.75 \times 4.5 \times 50 = 168.75 cfs, which informs pipe sizing to prevent surcharging during design storms. Such calculations directly guide hydraulic design by linking Tc-derived intensities to infrastructure capacity. The sensitivity of peak runoff to Tc underscores the need for accurate estimation, as even modest variations can significantly impact results in small basins. For instance, a ±10% change in Tc can alter Q_p by 20-30% due to the nonlinear decay of rainfall intensities with duration on IDF curves, amplifying errors in undersized or oversized systems. Studies on urban watersheds confirm significant differences in estimated peak discharge from varying Tc methods like Kerby versus NRCS approaches.

Stormwater Management

In stormwater management, the time of concentration (Tc) plays a pivotal role in the sizing of culverts and inlets, particularly in urban settings where impervious surfaces shorten Tc, leading to higher rainfall intensities for the critical storm duration. The Rational Method, commonly applied for small urban watersheds, uses Tc to select the rainfall intensity from intensity-duration-frequency curves, as the storm duration equal to Tc produces the maximum peak flow at the outlet. In urban areas, reduced Tc values—often below 10 minutes due to paved surfaces—result in steeper intensity-duration curves, necessitating larger culvert and inlet capacities to handle elevated design discharges and prevent localized flooding.^[19] For detention basin design, Tc is integral to routing inflow hydrographs and controlling outflows to mitigate peak flows downstream. Inflow hydrographs are generated using Tc to define the timing of runoff arrival, with methods like the Modified Rational Hydrograph extending the storm duration to multiples of Tc (e.g., 3 to 25 times Tc) for accurate volume estimation and peak attenuation. Outflow structures, such as orifices or weirs, are sized to release stored water at rates that effectively lengthen the basin's Tc equivalent, ensuring post-development hydrographs do not exceed pre-development peaks for design storms (e.g., 2-year to 100-year events). This routing approach balances storage volume against release timing, reducing erosion and surcharge risks.^[31]^[32] Low-impact development (LID) practices leverage Tc to replicate natural hydrologic conditions by increasing travel times through permeable surfaces and distributed controls. Techniques such as bioretention cells and vegetated swales extend overland flow paths, raising Tc by enhancing surface roughness and infiltration rates (e.g., up to 3 inches per hour in permeable media), which disperses runoff and lowers peak flows by 20-50% compared to conventional impervious designs. For instance, disconnecting impervious areas and incorporating permeable pavements can increase Tc from under 5 minutes to over 15 minutes in urban sites, promoting volume reduction and delaying hydrograph peaks.^[33]^[32] Regulatory frameworks incorporate Tc to guide permissible discharge rates and ensure sustainable stormwater control. State manuals, such as New Jersey's Stormwater Best Management Practices, mandate Tc calculations using the velocity method from USDA National Engineering Handbook Part 630 to verify that post-construction peaks remain at or below 50-80% of pre-construction levels for various storm frequencies, directly influencing outlet sizing and compliance. These guidelines emphasize Tc in hydrologic modeling to prevent downstream impacts, with similar principles applied in other jurisdictions to balance development with flood risk reduction.^[32]

Flood Routing

In flood routing, the time of concentration (Tc) serves as a critical parameter in the Muskingum method, a widely used hydrologic routing technique for channel reaches. The Muskingum storage equation, S = K [X I + (1 - X) O], incorporates the storage coefficient K, which represents the average travel time of the flood wave through the reach, and the weighting factor X, which characterizes the type of storage (X ≈ 0 for reservoir-like storage dominated by inflow, X ≈ 0.5 for channel translation with minimal storage). Tc of the upstream watershed informs the shape and timing of the inflow hydrograph to the reach, but K is estimated separately based on reach-specific characteristics such as length and flow velocities.^[34] For reservoir routing, the inflow hydrograph to the reservoir is shaped by the Tc of the upstream contributing watershed, which determines the duration and peak timing of the arriving flood wave based on the time for runoff to concentrate at the reservoir inlet. Storage within the reservoir then routes this inflow through level-pool or modified Puls methods, attenuating the peak discharge and extending the hydrograph base time, effectively prolonging the downstream Tc beyond the upstream value due to added delay from outflow structures like spillways. This process highlights Tc's role in scaling the temporal distribution of flood volumes for downstream impact assessment. In dam break or levee failure analysis, Tc influences the downstream flood wave travel time by governing the initial concentration and mobilization of breach outflow into the channel system, affecting wave celerity and dispersion. Hydrologic models incorporate Tc to estimate the propagation speed of the resultant flood wave, typically 2–10 miles per hour in alluvial valleys, where shorter Tc values accelerate peak arrival and increase inundation risks farther downstream. This integration ensures accurate routing of the non-steady flood hydrograph in tools like HEC-RAS unsteady flow simulations.^[35]

Influencing Factors and Limitations

Watershed and Rainfall Variables

Watershed area plays a primary role in determining the time of concentration (Tc), as larger areas typically involve longer flow paths from the hydraulically most distant point to the outlet, thereby extending Tc.^[36] For instance, in agricultural watersheds, Tc can range from minutes in small plots to hours in basins exceeding 100 km² due to increased travel distances.^[36] Conversely, steeper watershed slopes accelerate flow velocities, reducing Tc by promoting faster overland and channel conveyance.^[36] Land use significantly alters Tc through changes in surface roughness and impervious cover. Urbanization shortens Tc by replacing pervious surfaces with impervious ones, such as pavement and roofs, which minimize infiltration and shorten overland flow times while directing water into efficient drainage systems.^[2] This effect is pronounced in developing areas, where conversion to urban cover can reduce Tc by facilitating higher flow velocities in channels and pipes.^[36] Soil infiltration capacity also influences Tc, particularly for overland flow segments; soils with high infiltration rates, such as sandy types under vegetative cover, prolong Tc by absorbing more water and slowing surface runoff.^[36] In contrast, low-infiltration clay soils expedite runoff, shortening Tc in non-urban settings.^[4] Rainfall intensity affects Tc indirectly by influencing erosion processes that reshape flow paths over time. High-intensity storms can erode channels and overland surfaces, steepening slopes and reducing roughness, which ultimately shortens Tc in susceptible watersheds.^[37] Storm duration interacts with Tc by determining whether the rainfall reaches equilibrium at the outlet; durations shorter than Tc produce partial contributions, while longer ones allow full basin response.^[36] Antecedent moisture conditions modulate Tc through their impact on initial abstractions; wetter antecedent soils reduce infiltration capacity during the event, accelerating runoff and effectively shortening Tc compared to dry conditions where higher initial losses extend overland travel times.^[38] In urbanized watersheds, impervious surfaces largely eliminate antecedent moisture effects by bypassing infiltration altogether, making Tc more consistent regardless of prior wetness.^[39] Dry antecedent conditions, however, can increase overland flow times in rural or vegetated areas by enhancing initial storage in soils and depressions.^[40] Climate change projections indicate increased flood risks in warming regions, driven by more frequent intense, short-duration storms that accelerate runoff and amplify peak discharges through heightened erosion and saturation.^[41] These shifts amplify flood risks in both urban and rural contexts.^[41]

Assumptions and Sources of Error

Estimation of time of concentration (Tc) relies on several key assumptions that simplify complex hydrologic processes. Common assumptions include uniform rainfall excess across the watershed, steady uniform flow without backwater effects, and negligible storage within the drainage area, which are particularly embedded in methods like the Rational Method. These models also presume that the entire basin contributes to peak flow when rainfall duration equals Tc and ignore spatial variability in rainfall and infiltration, especially in larger basins exceeding 200 acres. Additionally, no backwater or storage effects from downstream controls are considered, limiting applicability to ungauged, rural settings without significant detention features.^[42]^[43]^[19] Sources of error in Tc estimation often stem from terrain and land-use characteristics. Empirical formulas, such as the Kirpich or SCS methods, tend to overestimate Tc in flat terrains because they are calibrated primarily on steeper slopes, leading to conservative but inaccurate peak flow predictions. In urbanizing watersheds, failure to recalibrate Tc after development can result in underestimation of runoff peaks, as impervious surfaces accelerate flow and shorten Tc, but legacy rural estimates inflate travel times and reduce calculated intensities. Measurement errors, including unsynchronized timing in field data or single-point rainfall gauging that misses areal variability, further contribute to inaccuracies, particularly in complex storms.^[42]^[44] Uncertainty in Tc estimates typically ranges from ±20% to 30%, varying with basin characteristics and method; for example, measured Tc values in Appalachian watersheds showed standard deviations of 7–24 minutes for Tc around 18–35 minutes. Sensitivity to parameters like Manning's roughness (n) and slope (S) is high, and Monte Carlo simulations incorporating variability in these inputs are recommended to quantify propagation of errors into peak discharge predictions.^[42] To mitigate these limitations, advancements include GIS-based delineation of flow paths for more accurate longest-hydraulic-path identification and integration of field measurements like dye tracing to validate travel times in channels and overland segments. Hybrid models combining empirical formulas with physics-based kinematic wave approximations can reduce estimation errors by up to 15% compared to purely empirical approaches, particularly in heterogeneous terrains.^[42]^[45]^[46]

References

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Section 11: Time of Concentration
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Time of concentration (Tc) is the time required for runoff to travel from the hydraulically most distant point in the watershed to the outlet.
[3]
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[7]
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### Summary of Historical Origin of Time of Concentration from https://hess.copernicus.org/articles/24/2655/2020/
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THE RELATION BETWEEN THE RAINFALL AND. THE DISCHARGE OF SEWERS IN POPULOUS. DISTRICTS. By EMIL KUICHLING, M. Am. Soc. C. E.. WITH DISCUSSION. The most important ...
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Time of concentration will vary, depending on the slope and character of the surface.
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Rainfall Intensity - Texas Department of Transportation
Software that facilitates Rational Method calculations often has IDF curves from rainfall data embedded into the software. ... time of concentration, small ...
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[13]
Sheet Flow Time Calculation - Texas Department of Transportation
t = sheet flow travel time (hr.) · n = overland flow roughness coefficient (provided in Table 4-6) · L = sheet flow length (ft) (100 ft. maximum) · P = 2-year, ...
[14]
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### Summary of Shallow Concentrated Flow for Time of Concentration (Tc)
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Below is a merged summary of shallow concentrated flow velocity formulas and related information from the provided segments, consolidating all details into a single, dense response. To maximize clarity and retain all information, I’ve organized key data into tables where appropriate (in CSV-like format for easy parsing) and included narrative text for explanations and additional context. The response avoids redundancy while ensuring all formulas, coefficients, examples, and useful URLs are included.
[16]
[PDF] Chapter 15 Time of Concentration - HydroCAD
The ve- locity method is the best method for calculating time of concentration for an urbanizing watershed or if hydrau- lic changes to the watercourse are ...
[17]
[PDF] Guide for Selecting Manning's Roughness Coefficients for Natural ...
This guide provides step-by-step procedures for selecting Manning's roughness coefficient (n) for natural channels and flood plains, using flood-plain photos ...
[18]
[PDF] Chapter 2 Estimating Runoff Volume and Peak Discharge - USDA
The WinTR-55 and WinTR-20 computer programs automate these procedures. B. The watershed should have only one main stream. If more than one exists, the branches.
[19]
Section 12: Rational Method - Texas Department of Transportation
The minimum duration to be used for computation of rainfall intensity is 10 minutes. If the time of concentration computed for the drainage area is less than 10 ...
[20]
[PDF] APPENDIX F – RATIONAL METHOD - Oregon.gov
Rainfall intensity at a duration equal to the time of concentration (Tc) is used to calculate the peak flow in the Rational. Method. The rainfall intensity can ...
[21]
[PDF] Basic Concepts of Kinematic-Wave Models
Equations 6 and 37 are sometimes called the Saint Venant equations. DEVELOPMENT OF. KINEMATIC-WAVE APPROXIMATIONS. Kinematic-wave approximations to equation 37 ...
[22]
Derivation of time of concentration - ScienceDirect.com
The time of concentration is derived from kinematic wave theory, considering rectangular and converging sections. Rainfall duration also influences it.
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Kinematic wave method for determination of road drainage inlet ...
Based on the kinematic wave theory, which is shown to be generally applicable to road drainage situations, a method has been developed for the determination ...<|control11|><|separator|>
[24]
Kinematic Wave Transform Model - Hydrologic Engineering Center
The kinematic wave method allows for the representation of variable scales of complexity within a watershed. A simple representation is shown in Figure below ...
[25]
[PDF] LITERATURE REVIEW ON TIMING PARAMETERS FOR ...
The time of concentration (Tc) is used in many procedures to develop runoff hydrographs or estimate peak discharges. TxDOT uses NRCS procedures for hydrologic ...
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chapter 10a: catchment routing, time-area method
The time-area method of hydrologic catchment routing transforms an effective storm hyetograph into a runoff hydrograph. The method accounts for translation only ...
[27]
[PDF] technical publications no. 85-5 a guide to SCS runoff procedures
= Time of concentration, hours t. = Overland flow travel time, hours t. = Time to peak of unit hydrograph, hours t. = Recession time of unit hydrograph, hours.
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https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/WR018i004p00877
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A geomorphoclimatic theory of the instantaneous unit hydrograph
The instantaneous unit hydrograph is conceived as a random function of climate and geomorphology varying with the characteristics of the rainfall excess.
[30]
[PDF] Technical Note 1 Pseudo-calibration of a Runoff Model by Adjusting ...
As described in Technical Note 1, the unit hydrograph parameters that can be adjusted in a rainfall-runoff model (in this case we will use an example of a model ...Missing: observed | Show results with:observed
[31]
[PDF] Detention Basin Design using Rational Hydrographs - Chesco.org
Mar 25, 2019 · In 1851, Mulvaney presented second principle relating the time of concentration to the storm event. The Rational Formula Explained. • Q = CI avg.<|separator|>
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[PDF] NJ DEP BMP Manual Chapter 5: Stormwater Calculations Revised ...
7:8-2.4(g)4, time of concentration is the time it takes for runoff to travel from the hydraulically most distant point of the drainage area to the point of ...
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### Summary of Role of Time of Concentration (Tc) in Low-Impact Development (LID) from https://www.waterboards.ca.gov/rwqcb2/water_issues/programs/stormwater/muni/nrdc/07%20lid%20hydrologic%20analysis.pdf
[34]
https://www.hec.usace.army.mil/confluence/hmsdocs/hmstrm/channel-flow/muskingum-model
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https://www.usbr.gov/ssle/damsafety/TechDev/DSOTechDev/DSO-98-04.pdf
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[PDF] Service Assessment Record Floods of Greater Nashville
Catastrophic flooding occurred in Greater Nashville, western Kentucky, and Middle. Tennessee May 1-4, 2010. The event began with heavy rain on Saturday, May 1.
[37]
[PDF] Chapter 15 Time of Concentration - HydroCAD
This chapter contains information on the watershed characteristics called travel time, lag, and time of concentration. These watershed characteristics influ-.
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Effect of rainfall intensity and duration on soil erosion on slopes with ...
Aug 15, 2021 · The objective of this study is to discuss how rainfall intensity and duration affect the role of TSM in soil erosion.
[39]
Estimating Time of Concentration of Overland Flow on Very Flat ...
Although researchers found that antecedent soil moisture has a significant effect on the time of concentration, changes of soil moisture were not monitored.
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Urbanisation impacts on storm runoff along a rural-urban gradient
Antecedent soil moisture alters the volume and timing of runoff generated in catchments with large rural areas, but was not found to affect the runoff response ...
[41]
Why continuous simulation? The role of antecedent moisture in ...
Jun 27, 2012 · The analysis shows that there is a consistent underestimation of the design flood events when antecedent moisture is not properly simulated, ...
[42]
Predicting the impact of climate change on urban drainage systems ...
... climate change impact therefore arises (Pagliara et al., 1998). These ... time of concentration. This assumption makes it possible to analytically ...
[43]
https://www.txdot.gov/manuals/des/hyd/chapter-4--hydrology/section-12--rational-method/assumptions-and-limitations.html
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Assumptions and Limitations - Texas Department of Transportation
If the time of concentration computed for the drainage area is less than 10 minutes, then 10 minutes should be adopted for rainfall intensity computations. The ...
[45]
[PDF] Stormwater - Iowa Statewide Urban Design and Specifications
Urbanization usually decreases time of concentration, thereby increasing the peak discharge. However, time of concentration can be increased as a result of ...
[46]
https://docs.lib.purdue.edu/context/open_access_dissertations/article/1742/viewcontent/Cho_Y.pdf
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[PDF] Development and evaluation of a watershed-scale hybrid hydrologic ...
Tc, time of concentration is obtained from a calibrated i derived time-area diagram for each watershed (HMS model adopts Tc with ratio of sub-basin area) ...<|control11|><|separator|>