Delmarva Unit Hydrograph

Delmarva Unit Hydrograph

The NRCS dimensionless unit hydrograph and the NRCS triangular unit hydrograph are widely utilized in the United States to develop storm hydrographs for evaluating and designing soil and water resources management practices. These unit hydrographs are based on a standard set of assumptions and are commonly used to model water flow in various hydrologic systems. However, in some regions of the country, such as the Delmarva Peninsula, the local topography is relatively flat, and there is a large amount of surface storage available. As a result, the shape of the observed storm hydrographs in these areas can differ significantly from those generated using the NRCS unit hydrographs.

To address this issue, a unit hydrograph known as the Delmarva unit hydrograph has been developed and used by utilities in Delaware, Maryland, Virginia, and some parts of New Jersey. This unit hydrograph is similar to the NRCS dimensionless unit hydrograph, but it has been modified to better represent the runoff characteristics of the Delmarva Peninsula. By using the Delmarva unit hydrograph, utilities in these states can more accurately model the flow of water in their systems and better understand the impacts of various soil and water resources management practices. Overall, the Delmarva unit hydrograph is an important tool for helping to improve the management and conservation of water resources in this region.

The Delmarva unit hydrograph uses the following equation to estimate the peak flow rate.

Qp = 284 * A / Tp

where

Qp = peak flow rate in cfs.

A = area of the watershed, in square miles, draining to the location of the unit hydrograph.

Tp = time to peak of the unit hydrograph in hour

Time to peak, and lag time are calculated according to Equations 97 and 98, respectively. When compared with the NRCS methods, the Delmarva unit hydrograph produces lower peak flow rate but yields the same flow volume.

3.16 Groundwater Flow in Innovyze H20Calc

 

3.16 Groundwater Flow

Groundwater has always been one of the most important sources of water supply. Virtually all parts of the earth are underlain by water, and wells have been constructed to provide a water supply when surface water was not readily available. Groundwater-bearing formations which are sufficiently permeable to yield usable quantities of water are called aquifers. When an aquifer is not overlain by an impermeable layer it is said to be unconfined. Confined aquifers consist of a water-bearing layer contained between two less permeable layers. This section presents the calculation of the steady flow in the confined or unconfined aquifers. It also presents the well hydraulics in confined or unconfined steady flow.

Steady Flow in a Confined Aquifer

 

If there is the steady movement of ground water in a confined aquifer, there will be a gradient or slope to the potentiometric surface of the aquifer. A portion of a confined aquifer of uniform thickness is shown in the following figure. The potentiometric surface has a linear gradient. There are two observation wells where the hydraulic head can be measured.

 

 

Steady Flow in a Confined Aquifer

 

For flow of this type, Darcy’s law may be used directly.

                                                                                                   

where q          =          flow rate per unit width of aquifer (m²/d, ft²/d)

            K         =          hydraulic conductivity of aquifer (m/d, ft/d) (see Table 3-13)

            dh/dl    =          hydraulic gradient

            B          =          aquifer thickness (m, ft) 

            L          =          flow length (m, ft) 

            h₂         =          head at length L (m, ft) 

            h₁         =          head at the origin (m, ft) 

 

Table 3-13: Soil Properties

Soil Type	Hydraulic Conductivity, K (cm/s)
Clayey	10-9 - 10-6
Silty	10-7 - 10-3
Sandy	10-5 - 10-1
Gravelly	10-1 - 102

The aquifer transmissivity is defined as a measure of the amount of water that can be transmitted horizontally through a unit width by the fully saturated thickness of the aquifer under a hydraulic gradient of 1. The transmissivity, T, (m²/d, ft²/d) is the product of the hydraulic conductivity (K) and the saturated thickness of the aquifer (B) as

 

T = BK

Steady Flow in an Unconfined Aquifer

 

 

 

Steady Flow in an Unconfined Aquifer

 

In an unconfined aquifer, the fact that the water table is also the upper boundary of the region of flow complicates flow determinations. This problem was solved by Dupuit who made the following simplifying assumptions: (1) the hydraulic gradient is equal to the slope of the water table and (2) for small water-table gradients, the streamlines are horizontal and the equipotential lines are vertical. Using these assumptions, Depuit developed Equation (102), commonly known as the Dupuit equation.

                                                                                          

 

where q          =          flow rate per unit width of aquifer (m²/d, ft²/d)

            K         =          hydraulic conductivity of aquifer (m/d, ft/d) (see Table 3-13)

            L          =          flow length (m, ft) 

            h₂         =          head at length L (m, ft) 

            h₁         =          head at the origin (m, ft) 

Well Hydraulics of Confined Steady Flow

 

Darcy’s equation can be used to analyze axially symmetric flow into a well. If a well pumps long enough, the water level may reach a state of equilibrium; that is, there is no further drawdown with time. The following figure illustrates the flow conditions for a well in a confined aquifer.

 

 

Steady Flow to a Well in a Confined Aquifer

 

In the case of steady radial flow in a confined aquifer, the steady pumping rate is

 

                                                                 

 

where Q         =          pumping rate (m³/d, ft³/d)      

            K         =          hydraulic conductivity of aquifer (m/d, ft/d) 

            B          =          aquifer thickness (m, ft) 

            T          =          hydraulic transmissivity (m²/d, ft²/d)

            h₂         =          head at distance r₂ from the pumping well (m, ft)

            h₁         =          head at distance r₁ from the pumping well (m, ft)

            d₂         =          drawdown at distance r₂ from the pumping well (m, ft)

            d₁         =          drawdown at distance r₁ from the pumping well (m, ft)

 

Well Hydraulics of Unconfined Steady Flow

 

The following figure illustrates the flow conditions for a well in an unconfined aquifer. In the case of steady radial flow in an unconfined aquifer, the steady pumping rate is

 

                                                                                                                 

 

or

 

                                                                                                        

 

All the terms are consistent with the definitions given above.

 

 

Steady Flow to a Well in an Unconfined Aquifer

 

Moody Friction Factor Calculator from Innovyze H20Calc

 

3.3 Moody Friction Factor Calculator

Darcy-Weisbach friction factor, f, can be evaluated in terms of equivalent sand grain roughness, e, and Reynolds number, Re. Reynolds number is a dimensionless ratio of inertial forces to viscous forces acting on flow and is defined for any cross-sectional shape as

          

For Re < 2,000, flow is referred to as laminar; if Re > 4,000, flow is generally turbulent. If Re is between 2,000 and 4,000, the flow is in a transitional region.

 

For laminar flows, the friction factor, f, is defined as

                                                                                                                                 

Numerous formulas exist to determine the friction factor. The two most popular equations are the Colebrook-White (implicit) and the Swamee-Jain (explicit). The Colebrook-White equation is

                                                                                    

which must be solved iteratively. Swamee and Jain (1976) developed an explicit formula of the friction factor, f, for 4000 ≤ Re ≤ 10⁸ (turbulent flow region) and 10^-6 ≤ e/D ≤ 10^-2 as

 

A cubic interpolation from the Moody diagram can be applied for the transitional flow range (2000 ≤ Re ≤ 4000) as

Procedure to find friction factor f

 

First, the relative roughness (e/D) and Reynolds number must be calculated. The Reynolds number is a function of kinematic viscosity of the fluid at the fluid’s temperature. Table 3-2 lists the kinematic viscosity for water over a range of temperature. Then, determine relative roughness of the pipe. Table 3-1 can be used as a guide to estimate equivalent sand-grain roughness for various types of pipes. Then, calculate the friction factor using either of the equations described above depending on the flow regime (i.e. laminar, transitional, or turbulent) based on the Reynolds number.

 

Table 3-2: Kinematic Viscosity of Water

Temperature		SI (m2/s × 10-7)	English (ft2/s × 10-5)
T(oC)	T(oF)
0	32	17.7	1.91
10	50	13.0	1.40
20	68	10.1	1.09
30	86	8.03	0.86
40	104	6.58	0.71
50	122	5.52	0.59
60	140	4.72	0.51
70	158	4.13	0.44
80	176	3.65	0.39
90	194	3.25	0.35
100	212	2.95	0.32

              Source: Boulos et al. (2006)

Head Loss in Innovyze H2OCalc

 

3.7 Head loss due to Transitions and Fittings (Local loss)

Whenever flow velocity changes direction or magnitude in a conduit (e.g., at fittings, bends, and other appurtenances) added turbulence is induced. The energy associated with that turbulence is eventually dissipated into heat that produces a minor head loss, or local (or form) loss. The local (minor) loss associated with a particular fitting can be evaluated by

                                                                                                      

where   V         =          mean velocity in the conduit (m/s, ft/s)

                K         =          loss coefficient for the particular fitting involved.

The table given below provides the loss coefficients (K) for various transitions and fittings.

 

Table 3-3: Typical Minor Loss Coefficients

Type of form loss			K
Expansion	Sudden	D1 < D2
	Gradual	D1/D2 = 0.8	0.03
		D1/D2 = 0.5	0.08
		D1/D2 = 0.2	0.13
Contraction	Sudden	D1 > D2
	Gradual	D2/D1 = 0.8	0.05
		D2/D1 = 0.5	0.065
		D2/D1 = 0.2	0.08
Pipe entrance	Square-edge		0.5
	Rounded		0.25
	Projecting		0.8
Pipe exit	Submerged pipe to still water		1.0
Tee	Flow through run		0.6
	Flow through side outlet		1.8
Orifice	(Pipe diameter  /orifice diameter)	D/d = 4	4.8
		D/d = 2	1.0
		D/d = 1.33	0.24
Venturi (long-tube)	(Pipe diameter  /throat diameter)	D/d = 3	1.1
		D/d = 2	0.5
		D/d = 1.33	0.2
Bend	90o miter bend with vanes		0.2
	90o miter bend without vanes		1.1
	45o miter bend		0.2
Type of form loss (continued)			K
Bend	45o smooth bend:      (bend radius  /pipe diameter)	r/D = 1	0.37
		r/D = 2	0.22
		r/D = 4	0.2
	90o smooth bend	r/D = 1	0.5
		r/D = 2	0.3
		r/D = 4	0.25
	Closed return bend		2.2
Sluice	Submerged port in wall		0.8
	As conduit contraction		0.5
	Without top submergence		0.2
Valve	Globe valve, fully open		10
	Angel valve, fully open		5.0
	Swing check valve, fully open		2.5
	Gate valve, fully open		0.2
	Gate valve, half open		5.6
	Butterfly valve, fully open		1.2
	Ball valve, fully open		0.1

       Source: Nicklow and Boulos (2005)

Water Hammer from Innovyze H20Calc

 

3.14 Water Hammer

Surge analysis is important to estimate the worst-case events in the Water Distribution Systems (WDS). Transient regimes in WDS are inevitable and will normally occur as a result of action at pump stations and control valves. Regions that are particularly susceptible to transients are high elevation areas, locations with either low or high static pressures, and regions far removed from overhead storage. They are generally characterized by fluctuating pressures and velocities and are critical precisely because pressure variations can be of high magnitude, possibly large enough to break or damage pipes or other equipment, or to greatly disrupt delivery conditions.

 

This section presents the calculation of potential surge using Joukowski equation, which is widely applied as a simplified surge analysis, and wave speed calculation. In the end, it provides the calculation of the inertia of pumps and motors, which are important for transients caused by pump failure.

Joukowski Expression

 

The pressure rise for instantaneous closure is directly proportional to the fluid velocity at cutoff and to the velocity of the predicted surge wave. Thus, the relationship used for analysis is simply the well-known Joukowski expression for sudden closures in frictionless pipes

where  

              =          surge pressure (m , ft)

            =          velocity change of water in the pipeline (m/s, ft/s)

            c          =          wave speed (m/s, ft/s)

            A          =          cross-sectional area (m², ft² h)

            g          =          gravitational acceleration (9.81 m/s², 32.17 ft/s²)

Wave Speed

 

The wave speed, c, is influenced by the elasticity of the pipe wall. For a pipe system with some degree of axial restraint a good approximation for the wave propagation speed is obtained using

where  E_f         =          elastic modulus of the fluid (for water, 2.19 GN/m², 0.05 Glb/ft²)

            ρ          =          density of the fluid (for water, 998 kg/m³, 1.94 slug/ft³)

            E_c         =          elastic modulus of the conduit (GN/m², Glb/ft²)

            D         =          pipe diameter (mm, inch)

            t           =          pipe thickness (mm, inch)

            K_R        =          coefficient of restraint for longitudinal pipe movement.

 

The constant K_R takes into account the type of support provided for the pipeline. Typically, three cases are recognized with K_R defined for each as follows (m is the Poisson’s ratio for the pipe material):

 

Case a: The pipeline is anchored at the upstream end only.

 

K_R =  1 -  m / 2

Case b: The pipeline is anchored against longitudinal movement.

 

K_R =  1 - m²

Case c: The pipeline has expansion joints throughout.

 

K_R =  1

The following table provides physical properties of common pipe materials.

 

Table 3-8: Physical Properties of Common Pipe Materials

Material	Young’s Modulus (Ec)		Poisson’s Ratio, μ
	GN/m2	Glb/ft2
Asbestos Cement	23 - 24	0.53 - 0.55	-
Cast Iron	80 - 170	1.8 - 3.9	0.25 - 0.27
Concrete	14 - 30	0.32 - 0.68	0.1 - 0.15
Reinforced Concrete	30 - 60	0.68 - 1.4	-
Ductile Iron	172	3.93	0.3
PVC	2.4 - 3.5	0.055 - 0.08	0.46
Steel	200 - 207	4.57 - 4.73	0.30

 

Inertia of Pumps and Motors

 

The combined inertia of pumps and motors driving them, including the connecting shafts and couplings, is required for transient analysis associated with the starting and stopping of pumps. The equations provided below are intended to be used as an initial guide to the inertia values that may be used as a reasonable first approximation, when more accurate data is not available. The total inertia for the pump/motor unit is the sum of both pump and motor inertias. The following inertia calculations are based on Thorley (2004).

Pump Inertias

 

From the linear regression analysis of 300 pump inertia data, two equations were developed for predicting the inertia I of pump impellers, including the entrained water and the shaft on which the impeller is mounted. The first equation represents the upper set of the data, and applies to single- and double-entry impellers, single and multistage, and horizontal and vertical, spindle machines.

                                                                                                   

where  I           =          pump inertia (kg m², lb ft²)

            C₁        =          coefficients (0.03768, 0.6674)

            P          =          power (kW, hp)

            N         =          pump speed (rev/min)

 

The second equation is for lower set of the data and represents relatively small, single-entry, radial flow impellers of lightweight design. This is applied to relatively small pumps of lightweight design.

                                                                                                      

where  C₂        =          coefficients (0.03407 in SI, 0.6244 in English)

Motor Inertias

 

Similar to pump inertia, linear regression of the motor inertia data yields the following equations.

                                                                                                       

where  C₃           =         coefficients (0.0043 in SI, 0.0648 in English).