Title: Estimating Transmission Losses in Ephemeral Stream Channels
Abstract: Procedures have been developed to estimate transmission loss volumes in abstracting (losing) ephemeral streams. A two -parameter linear regression equation relates outflow volume for a channel reach to the volume of inflow. A simplified two -parameter differential equation describes the transmission loss rate as a function of length and width of the wetted channel. Linkage relationships between the regression and differential equation parameters allow parameter (and thus, transmission loss) estimation for channels of arbitrary length and width. The procedure was applied using data from 10 channel reaches. Maximum loss rates were observed on Walnut Gulch, Arizona, and minimum loss rates were observed on Elm Fork of the Trinity River, Texas. All other data were between limits established at these two locations. Examples illustrate typical applications and show step -by -step procedures required to use the proposed method. Results and interpretations were summarized, and needs for additional research were specified. INTRODUCTION In much of the southwestern United States, watersheds are characterized as semiarid with broad alluviumfilled channels that abstract large quantities of streamflow (Babcock and Cushing, 1941; Burkham, 1970a, 1970b; and Renard, 1970). These abstractions or transmission losses are important because stream flow is lost as the flood wave travels downstream, and thus, runoff volumes are reduced. Although these abstractions are referred to as losses, they are an important part of the water balance. They diminish streamflow, support riparian vegetation, and recharge local aquifers and groundwater (Renard, 1970). Therefore, procedures are needed to estimate outflow volume, and, from that, transmission losses in abstracting streams. Various procedures have been developed to estimate transmission losses in ephemeral streams. These procedures range from simple regression equations to estimate outflow volumes (Lane, Diskin, and Renard, 1971) to simplified differential equations for loss rate as a function of channel length (Jordan, 1977). Contrasted with these simplified procedures dealing only with the volume of losses are the procedures used to route hydrographs through a losing channel. Lane (1972) proposed a storage -routing model as a cascade of leaky reservoirs, and Wu (1972) used the leaky reservoir concept in modeling surface irrigation. Smith (1972) used the kinematic wave model to route hydrographs in channels of ephemeral streams. Smith demonstrated that the wave front becomes steeper due to shock formation, and that the hydrograoh peak decreases in a downstream direction due to infiltration. Peebles (1975) modeled flow recession in ephemeral streams as the discharge from a single leaky reservoir with loss rates proportional to the depth of flow and storage volume proportional to the square of the depth. Therefore, there is a range of complexity in procedures for estimating transmission loss volumes and rates in ephemeral stream channels. In general, the simplified procedures require less information about physical features of the channels but are less general in the application. The more complex procedures may be more physically based, but they require correspondingly more data, and more complex computations. The purpose of this study was to develop a simplified procedure for estimating the volume of outflow, and from that the transmission losses, given an inflow volume at a point upstream. We also sought to develop procedures for estimating flow volume at any point along the stream between the inflow and outflow stations (Lane and Staff of the Southwest Rangeland Watershed Research Center, 1979). Since the study concerns streams where water is abstracted, the outflow volume must he less than the inflow volume. We assumed that for a particular channel reach,infiltration rates and other properties were uniform with channel length and width, so that the relationship between inflow and outflow volumes is unique, given the same antecedent conditions. We sought a simplified procedure with a minimum number of parameters and with reasonable hounds or limits on the estimates of transmission losses. Although all physical characteristics affecting transmission losses could not be explicitly incorporated in the equations, we sought a procedure that would directly account for channel size (length and width) and, thus, facilitate transfer of results from gaged to ungaged channel reaches. Finally, we sought a procedure that would be accurate for the average or 193 representative conditions for a particular channel reach. With these criteria, we expected to reproduce trends over a range of data. Relatively larger errors are expected for very small or very large inflows, or for events occurring under extreme conditions of antecedent moisture. ASSUMPTIONS AND LIMITATIONS For a given channel reach, we assumed infiltration and other channel properties were uniform with distance along the reach and with width across a channel cross -section. We assumed a unique relationship between inflow and outflow volume under given antecedent conditions. However, the procedure did not account for sediment concentration in the streamflow, temperature effects, seasonal trends, differences in peak flow rate, or hydrograph shape for the same inflow volume. Therefore, the procedure was designed to compute outflow volume and, from that, transmission loss volume, and did not compute flow rates or account for flow duration. Outflow and transmission loss rates were defined as functions of distance, not functions of time. Finally, we assumed a threshold volume or initial abstraction, and then a linear relation (above the threshold) between inflow volume and outflow volume. The assumption of a threshold volume made the outflow -inflow relation nonlinear in the systems theory sense, and outflow and loss rates were shown to be nonlinear functions of distance. DEFINITIONS AND UNITS Inflow volume, P, is the volume of inflow (acre -feet) at the upstream end of the channel reach, and outflow volume, Q, is the volume of outflow (acre -feet) at the downstream end of the channel reach. Transmission loss volume, P -Q, is the volume of losses (acre -feet) in the reach. The reach length, x, is the length of the channel (miles) between the upstream or inflow station and the downstream or outflow station. Channel width, w, is the average width of the channel (feet) for the reach. Ideally, w is the average width of channel wetted by the flood wave. In actual practice, the average width is the width of the channel between channel banks before out -ofbank flow occurs. Bank full discharge or average channel forming discharge can be used to estimate average channel width. Threshold volume, Po, is the inflow flow volume (acre -feet) required before outflow begins at the downstream station. Threshold volume can be interpreted as an initial abstraction or loss before outflow begins. DEVELOPMENT Two simple methods of analysis were used. The first is a linear regression procedure and the second is a simple differential equation expressing the rate of change to outflow volume with distance. LINEAR REGRESSION PROCEDURE In this procedure, the volume of outflow is assumed proportional to the volume of inflow (Lane, Diskin, and Renard, 1971): 0 Q =a + bP , P P0 (1) where Q = outflow volume, acre -ft P = inflow volume, acre -ft Po = threshold inflow volume, initial abstractions, acre -ft a = intercept, acre -ft, and h = slope. We assumed that for an abstracting channel a < 0.0 and 0.0 < h < 1.0 so that the threshold volume is Po = -a /h (2) If there are n pairs of (Pi, Qi) data for a reach, then linear regression or least squares analysis can be used to derive estimates of a and h in Eq. 1. The main disadvantage of the regression procedure described above is that the parameters a and h are unique to the particular reach and data set analyzed. That is, for a given channel reach of length x and width w, if we have values of a = a(x, w) and h = b(x, w), what are the values of a and h for different values of x and w? The traditional approach is to gage a large number of streams, and then try to relate a and b to channel properties, including x and w, to develop regional regression equations for a and h. The disadvantages of this procedure are: (1) observed data are required for a large number of channel reaches; (2) with small data sets, spurious correlations are common, and (3) arbitrary limits may he required so that the regression equations meet the constraints on a and h. The proposed alternative to the traditional approach is to construct a model directly incorporating x and w into the outflow -inflow (outflow as a function of inflow) equations. We followed this procedure using a differential equation to describe changes in outflow volume as a function of the rate of change 194
Publication Year: 1980
Publication Date: 1980-04-12
Language: en
Type: article
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Cited By Count: 11
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