Groundwater Base Flow

Home , Discharge (hydrology), Streamflow

BSYSE 456/556 Lecture 8

A Streamflow Hydrograph portraits the rate of flow at all points in time during and after a storm or snowmelt event. Hydrologists rely on measured or computed (synthesized) hydrographs to provide peak flow rates in order to properly design hydraulic structures. As a hydrograph plots volumetric flow rates against time, integration of the area beneath a hydrograph between any two points in time gives the total volume of water passing the point of interest during the time interval. Therefore, in addition to peak flows, hydrographs provide information on volumes of runoff and can be used for various applications. For example, they allow analysis of sizes of reservoirs, storage tanks, detention ponds, and other facilities that deal with the total volumes of runoff. They may also be used for predicting the discharge of a stream during the dry seasons when flow is sustained almost entirely by the groundwater base flow. A knowledge of the magnitude and time distribution of streamflow is essential to many of the aspects of water management and environmental planning. (Based on King [1997] adapted from Viessman and Lewis [1995], and Watson and Burnett [1995].)

› Factors Affecting Hydrograph Shape

A hydrograph has five elements (Fig. 1): (1) direct surface runoff, direct surface-water inflow to the stream (2) groundwater base flow, composed of the water that percolates downward until it reaches the ground water reservoir and then flows to surface streams as ground-water discharge (3) interflow, the subsurface flow, in concentrated format, moving at shallow depths at the base of a weathered zone, or along a fault zone (4) throughflow, unsaturated flow in the vadose zone (5) channel precipitation, precipitation falling directly into the stream The rising portion of a hydrograph is known as the concentration curve; the region in the vicinity of the peak is called the crest segment; and the falling portion is the recession curve (Fig. 2). The shape of a hydrograph depends on precipitation pattern characteristics and basin properties. (Adapted from King [1997], Viessman and Lewis [1995], and Watson and Burnett [1995].)

› Base Flow Separation

Several techniques are used for base flow separation when the actual amount of base flow is not known. Most of these techniques are based on analyses of groundwater recession or depletion curves. The reference by King [1997] (adapted from Viessman and Lewis [1995]) shown on pages 3 and 4 describes in detail these techniques. It should be noted, though, the logarithm recession method (the third method) is based on theory and is preferred, yet no standard, universally accepted procedures exist.

› General Forms of Base Flow (Based on Hornberger et al. [1998])

If no inflow is added to the ground water during the groundwater recession, and if all groundwater discharge from the upstream area is intercepted at the stream-gauging point of interest, then the base flow can be described by

(1)

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Fig. 1 Portion of the hydrologic cycle pertinent to streamflow (Fig. 21-2, Watson and Burnett [1995])

Fig. 2 Hydrograph definition (Fig. 11.1, Viessman and Lewis [1995])

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4 BSYSE 456/556 Lecture 8 where = groundwater discharge at any time t after the start of the recession, [L3 T!1] = a specified initial discharge, [L3 T!1] = a recession constant, [T!1] = base of natural logarithms

The appropriateness of this equation can be investigated by applying the conservation of mass. During recession, as the inflow to the entire reservoir is assumed to be zero, we would expect that the outflow is a function of the head (averaged in certain way) of the ground water, h [L], i.e., the hydraulic head and the hydraulic gradient should decrease with time, and thereby reducing the driving force for discharge. We can write this relation as

(2) where is the discharge per unit width of stream [L2 T!1]. Conservation of mass requires the outflow to be balanced by the change in storage. The time rate of change of ground water stored should depend on h, the

storativity S for confined aquifer or specific yield Sy for unconfined aquifer (S or Sy is generally defined as: [volume of water released from (or taken into) storage]/[surface area of aquifer × decrease (or increase) in hydraulic head], and the length of the aquifer L (Fig. 3). The appropriate conservation equation would then be

(3)

Note that in Eq 3, is the approximated surface area of the aquifer, and so the left-hand side of the equation is the time rate of change of water stored in aquifer per unit width of stream. To integrate Eq 3, we assume the simplest form for such that

(4) where k is a proportionality constant [T!1]. Substituting Eq 4 into Eq 3 yields

6 (5)

Integrating Eq 5 gives

(6)

where h0 is the average hydraulic head at the initial time. Therefore, from Eq 4, we have

(7) or

(8)

Eq 8 is directly comparable with Eq 1. The comparison implies that the reservoir behaves as a linear reservoir, i.e., that outflow is directionally proportional to the amount of water stored.

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Fig. 3. A simple model of recession (Fig. 7.17, Hornberger et al., [1998]).

› Estimating the Recession Constant

If the hydrograph has been constructed on a semi-log paper, the approach described by Wanielista et al. [1997] can be employed to estimate the recession constants. In reality, the recession constant rarely takes on a single value over the entire recession curve. Ground water may enter the stream from different reservoirs at different times. Thus, it is often more accurate to represent the recession curve by a composite exponential where is the intercept for any segment “i” at t = 0, and is the slope for segment i. The s are called flow rate coefficients. Both the flow rate coefficients and recession constants can be estimated knowing the exponential form of the base flow. Thus by plotting , where is the peak discharge on the hydrograph, versus time, lines of constant slope can be determined that result in estimates of the recession constants and flow rate coefficients. This estimation procedure, which yields recession constants, inflection points, and estimates of base flow, is called the logarithm recession method. Fig. 4 shows an example of using this method. On the other hand, if the streamflow data is available, regression analysis can be used to obtain the estimates of the recession constants.

’ Mathematical Description of the Rising Limb (Based on Wanielista et al., 1997)

One method is to assume that watershed outflow rate is a linear function of storage. It appears to be appropriate for small watersheds and rainfall events of short duration. Based on mass balance, Input ! Output ± generation = accumulation.

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Fig. 4. An example showing the determination of multiple recession curves (Fig. 6.8., Wanielista et al.,1997)

(9) where r is rainfall excess rate, Q is outflow rate, S is storage, and t is time.

Assuming an approximate linear relationship exists between outflow rate and storage

(10) where K is a storage coefficient.

Eq 10 can be written in differential form as

(11)

Substituting Eq 11 into Eq. 9 results in

(12)

For the initial condition Q = 0 when t = 0, integrating Eq 12 yields

(13)

Eq 13 is valid only for the concentration curve of a hydrograph developed using a linear relationship between storage and outflow rate. As long as rainfall excess and stream flow data are available, the appropriate storage coefficient K can be estimated. Similarly, it is possible to use piecewise linear approximations of the storage-discharge relationship and thus have different K factors for different storage ranges.

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Study Questions:

1. What information important in water resources management and hydraulic engineering and construction can a hydrograph provide? What are the major elements of a hydrograph? What are the terms used to describe the rising and falling limbs and the peak area of a hydrograph?

2. Demonstrate and describe typically used techniques for hydrograph separation.

3. Derive the base flow equation (in an exponential form based on the conservation of mass law and Darcy’s law.)

4. Derive the stream discharge equation for the rising limb of a hydrograph.

5. How do you estimate the recession constant for the exponential form base flow equation?

References

Black, P.E., Watershed Hydrology, p. 231–236. 2nd ed., Ann Arbor Press Inc., Chelsea, 1996. Domenico, P.A., and F.W. Schwartz, Physical and Chemical Hydrogeology. p. 8–10. John Wiley & Sons, Inc., New York, 1998. Hornberger, G.M., J.P. Raffensperger, P.L. Wiberg, and K.N. Eshleman, Elements of Physical Hydrology, p. 163–166, The Johns Hopkins University Press, Baltimore, 1998. King, L.C., Handout 6, BSYSE 351 Fall 1997 Notes (adapted from Viessman, W., Jr. and G.L. Lewis,1995, Introduction to Hydrology, 4th ed., HaperCollins College Publishers), 1997 Viessman, W., Jr., and G.L. Lewis, Introduction to Hydrology, 4th ed., HaperCollins College Publishers, 1995. Wanielista, M., R. Kersten, and R. Eaglin, Hydrology, Water Quality and Quality Control, 2nd Ed., p. 183–193, John Wiley & Sons, Inc., 1997. Watson, I., A.D. Burnett, Hydrology: An Environmental Approach, p. 457–473, CRC Press, Inc., Boca Raton, 1995.

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Notes to Instructor