Interpreting Stream Hydrographs
Draft 3/26/2007
Page: 25
Lumped Parameter Analyses of Stream Hydrographs
A stream hydrograph is the electrocardiogram of a watershed, recording the hydrologic pulse of the region as it is stressed by drought or flood. It is no wonder that the analysis of stream hydrographs has fascinated hydrologists as a method for diagnosing processes within the watershed drained by the stream. This interest began more than a century ago with the work of Boussinesq in the late 1800s, and it continues today with both applications to field studies and the development of new methods.
The flow in a stream is generally recognized to have contributions from ground water and both overland flow and shallow subsurface stormflow. Overland and stormflow typically change over time periods of hours or days following precipitation events or snowmelts, whereas baseflow changes gradually over many days or weeks in response to seasonal or climatic variations, or to the drainage of water from an aquifer of finite volume. In some watersheds, drainage from the unsaturated zone may be also be an important contributor to the long-term discharge of streams (Hewitt ).
The basic process of analyzing a hydrograph begins by separating the components of stormflow and baseflow. This is done to facilitate analysis, but it also arises out of the different purposes that hydrograph analysis may serve. Investigators of flooding are primarily interested in the stormflow component, whereas investigators of ground water or long-term effects of water fluxes are more interested in the baseflow component. The separation process has received considerable attention but the first step of most of the proposed methods give similar results. This initial step is to go through the hydrograph record and identify days when the flow in the stream is primarily due to baseflow contributions. Such days are baseflow turning points. Typically, daily flows are only accepted as indicative of baseflow conditions if they occur during a period when the flow is gradually decreasing. Some methods use sophisticated filtering methods, whereas others use graphical or simple numerical procedures. They all identify turning points as days of low flows between pulses of high flow caused by storms, and most of them will pick similar days if implemented with appropriate parameters. Baseflow turning points identified by a method recommended by the Institute of Hydrology in England are shown in Figure 2.
The times identified as baseflow turning points may be days or weeks apart, particularly during a rainy season. In between the turning points the stream is flowing with contributions from both baseflow and stormflow. A complete hydrograph analysis requires baseflow and stormflow components to be identified throughout the record, so some method of interpolating between turning points is required. A variety of methods have been proposed, and herein lie the major differences between baseflow separation methods.
The baseflow component of many hydrographs has a distinctive behavior during dry weather. Notice how the minimal flows decrease during the summer months of the three years plotted on Figure 1. When plotted on semi-log axes, the minimum flows form a roughly straight line with a negative slope during the summer months. The baseflow component is not shown on the figure, but it corresponds to the minimal flow on the hydrograph. As a result, the baseflow component appears to decrease as a negative exponential during the summer months. In South Carolina, the summer months commonly have relatively small amounts of rainfall and evapotranspiration captures much of the rainfall that does occur. As a result, we expect that recharge to the aquifer that discharges to the Little River was negligible during summer months times when the baseflow decreased as a negative exponential. These periods of decreasing baseflow are called baseflow recessions. We infer that baseflow recessions occur when the recharge is negligible and the aquifer is continuously discharging to the stream.
Baseflow recessions are important to the management of both ground water and surface water resources during drought. The continuously diminishing baseflow during a recession is responsible for the diminishing stream flows during a drought. Understanding the controls of baseflow recession should help to understand to amount and rate of water that is available during a drought.
The objective of this analysis is to develop a method for predicting the baseflow to a stream as a function of time and recharge, and hopefully to gain some insight into baseflow recessions. This will allow us to interpolate between baseflow separation points, which will give the baseflow and stormflow components as continuous functions of time. This modest analysis can be extended to provide a method for estimating the major fluxes throughout a watershed as a function of time, which is a remarkable result.
Conceptual Model
The hydrogeology of nearly any watershed is extremely complicated when viewed in detail. It is possible to assemble a 3-D model that considers the spatial variations in material properties and processes present in a watershed, and considerable insights can be obtained from this type of model. These are called “distributed parameter” models because values of parameters, like hydraulic conductivity, are distributed and can vary across the model. Distributed parameter models can resemble the geometry of a watershed, and they can provide valuable insights when geometry is important. However, distributed models are time consuming to build and calibrate, and so the insights provided from such a model must be worth the effort of putting it together.
An alternative approach is to idealize the watershed as a simplified system that can be characterized using a small number of parameters. The many distributed parameters of the 3-D model are thereby lumped together to form one, or a few, effective parameters in a simplified model whose behavior resembles the actual watershed. This type of simplified analysis is called a “lumped parameter” model. Many of the models that we have described in the previous problems are types of lumped parameter models.
We will idealize a watershed as a sand-filled tank that is drained by a sand-filled hose held at constant head. This ignores the pattern of streams in the watershed, however, the tank analogy is a simplified approximation of the in the cross section in Figure 2; that is; the tank and sand-filled hose resembles the aquifer that discharges to a nearby stream. The discharge from the hose will decrease as the water level in the tank falls. We will equate the discharge from the hose to the contribution of baseflow to the discharge in a stream. The tank analogy can be justified on more theoretical grounds because the analytical solution for the problem shown in the cross-section consists of an infinite series. The first, and largest, term in the series is the same expression that we will derive for the tank. The higher order terms in the series account for the geometry of the aquifer.
The tank is a model of an aquifer with inputs from recharge and outputs from baseflow. The water stored in the aquifer changes when the water level changes.
The objective of the analysis is to develop expressions for the average head in an aquifer and the baseflow discharge to a stream as a function of time and recharge. We want to use this analysis to understand baseflow recessions, and to predict the baseflow between any pair of baseflow turning points on a hydrograph.
Translation
1. Rate In The rate of water flowing into the aquifer is
Qin = RAw (1)
Where Aw is the area of the watershed in plan view.
2. Rate Out The rate being discharged by baseflow is
Qout = qoutAa (2)
Where Aw is the area of the aquifer along a cross-section parallel to the stream. From Darcy’s Law the baseflow flux is
qout = K (3)
so
Qout = (4)
3. Rate of change of storage. Water stored in the aquifer is
(5)
where n is the specific yield and the rate of change of storage is
(6)
Substituting (1) (4) and (6) into a volume balance gives
(7)
rearranging
(8)
Solution
Separating variables
and integrating
(9)
We will assume that we know the water level at some time and are interested in the water levels after that time, so the initial condition is that h = ho when t = to. This gives
(10)
and substituting into (9) gives
(11)
Introducing the lumped parameter
(12)
allows us to simplify
(13)
This gives the head in the aquifer as a function of time. We can determine the baseflow by substituting (4) into (13) to get ho = QoL/KAa and
(14)
so from (13)
(15)
where Qbo is the baseflow at t = 0 and Qb is the baseflow after that time. We should introduce another lumped parameter to characterize recharge
(16)
to get
(17)
or in dimensionless form
(18)
Baseflow is given as a function of time and recharge rate by (17) or (18). The response of the system is govern by the value of the watershed recession constant a. We probably should not take the geometric implication of a too literally—clearly the geometry of the aquifer will differ markedly from that of a tank and a hose. Nevertheless, a meaningful interpretation is to consider a as composed of terms related to geometry and terms related to hydrologic properties of the aquifer. This is important because we see that the expression in brackets depends only on the physical characteristics of the aquifer. Often this expression is simplified to a single value, the recession constant (Fetter, 2001).
The volume of water that is stored, or that can be released during a recession can be estimated from the analysis above. This type of estimate is important because it allows the volumetric flowrate during a drought to be forecasted. Moreover, the difference between the volume that is stored at the beginning of one recession compared to that from the previous recession is the volume that has been recharged to the aquifer.
Assuming the recharge is zero during a recession then (17) reduces to the baseflow during a recession
(18)
Note that the analysis predicts that the baseflow during a recession decreases as a negative exponential of time.
Calibration
The analysis of baseflow as a function of time can be calibrated by determining a value of a that best predicts hydrograph data from a particular watershed. Perhaps the easiest way to approach this is to determine the semi-log slopes of the recession portions of hydrographs during baseflow recessions. We assume that the recharge is negligible during a recession, so the semi-log slope should be equal to a. In practice the semi-log slopes of different baseflow recessions will vary somewhat, probably because some recharge occurs during recession. The usual practice is to measure the slopes from many recession events and determine the semi-log slope that either best represents these values. Some hydrologists recommend using an average slope, whereas others (like me) prefer that a should be slightly less than the greatest observed slope. In any case, we will assume that a can be obtained from hydrograph data describing recessions.
Volume of water in storage
The volume of water that could be released from storage in an aquifer will have applications in managing a water supply during drought. We can obtain an estimate for the volume of water that has been discharged from the aquifer from t = 0, the start of the recession, to t = t1 is obtained by integrating (18)
(19)
The maximum volume of water that can be released during the recession occurs when t1=¥.
(20)
so
(21)
It may be convenient to recast (19) as
(22)
thereby giving the volume of water released by baseflow at time t in terms of the difference between the initial discharge and the discharge at t. The volume stored in the aquifer that could be released is simply the difference between Vmax and V(t), or
(23)
This shows a simple relation between the volume of water stored in an aquifer, the baseflow, and the recession constant.
Recharge
The recharge rate between any pair of turning points can be estimated by rewriting (17) so that Q = Q1 is the baseflow that occurs at t = t1
(24)
Similarly, Q = Q2 is the baseflow that occurs at t = t2. Recharge to the aquifer that occurs from t1 to t2 can be determined by rearranging (24)
(25)
This simple analysis gives the average recharge flux in an aquifer between the times indicated by any pair of turning points, say on Figure 2. Applying (25) sequentially along a hydrograph will allow the recharge to be determined with time.
A recharge pulse followed by a recession: superposition in time
The analysis given above shows how the baseflow contribution to a stream changes during a recharge event of infinite duration, which is interesting but a bit unrealistic. We can use a simple trick to transform this result to give the baseflow during a recharge event where the recharge is constant and equal to R, followed by a recession event where the recharge is zero. We will assume the recharge event begins at t = 0 and ends at t = t1. During the recharge event, we simply use the original solution in (13). After the recharge even ends, however, the approach is to superimpose two solutions, one for a recharge event of magnitude R that begins at t=0 and is of infinite duration (the solution developed above), and another for a recharge event of magnitude –R that starts at t = t1 and continues indefinitely.
Baseflow for t<t1
(26)