Introduction

In the Platte River valley of Nebraska, well fields for water supply for most cities are located near the river. Preventing the migration of potential contaminants from the river to nearby well fields requires a better understanding of the stream-aquifer connectedness. In addition, thousands of irrigation wells withdraw groundwater from the High Plains aquifer in this river valley, raising concerns about streamflow depletion. The Platte River segment between Kearney and Columbus (Fig. 1) dried up in the summers of 2002 to 2004. The drying river in this area meant river habitats significantly deteriorated for several federal endangered species. The Platte River was named as one of the nation’s most endangered rivers of 2003 by the conservation group American Rivers, which cited concerns that growing water demands will undermine a tri-state (Nebraska, Colorado, and Wyoming) agreement to protect the river. The Platte River was No. 7 on the annual list.

Fig. 1
figure 1

Map showing permeameter test sites along the Platte River in south-central and eastern Nebraska

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The vertical hydraulic conductivity (K v ) of a streambed is a key parameter in the determination of stream-aquifer hydrologic connections. Permeameter tests for measuring streambed hydraulic conductivity in the Platte River have been conducted by several investigators. Nguyen and Gilliland (1985) conducted permeameter tests in the Platte River channels near the Grand Island well field. They used a 5-cm diameter tube and pushed it vertically into a streambed to a depth of about 15 cm. The value of K v from 24 tests range from 27.7 to 159.6 m/d, with an average of 78.2 m/d. According to Chen (2000), a short penetration of tube into a streambed can introduce errors to the derived K v values. Landon et al. (2001) conducted permeameter tests on the Platte River near Brady, about 80 miles west to Kearney, and on several tributaries of this river, and provided results of K v values with some uncertainties due to a shallow penetration of the streambed (25 cm). The derived K v values show a trend with a change in tube diameters, and were seemingly overestimated by assuming an isotropic feature of the streambed (horizontal and vertical hydraulic conductivity was equal to 1) in the calculation of the K v values. Chen (2004) reported the K v values of the streambed along five transects at three sites on the Platte River in south-central Nebraska. In these permeameter tests, a 10-cm diameter tube was pushed into a streambed about 40 cm deep. A larger depth of streambed penetration can reduce errors in the calculation of K v values. Chen (2004) reported the values of K v fluctuate largely across the river channels. Rus et al. (2001) performed slug tests in the channels of the Platte River to measure the horizontal hydraulic conductivity (K h ). The mean values for four test sites (near North Platte, Brady, Lexington, and Grand Island) are 293, 140, 186, and 213 m/d, respectively, assuming the ratio of horizontal to vertical hydraulic conductivity to be 10. Slug tests, however, do not provide values of vertical hydraulic conductivity of the streambed.

To better understand the spatial variation of streambed hydraulic conductivity across the channels and from upstream to downstream along the river, more permeameter tests were conducted along the Platte River from south-central to eastern Nebraska (Fig. 1). Results of streambed hydraulic conductivities from these test sites were evaluated to determine statistical and geostatistical behaviors and analyze their relation to streamflow conditions. Experimental variograms are presented to show the spatial variations of K v across the channels of the Platte River.

River Hydrology of the Study Area

The Platte River, originating from the Rocky Mountains, is about 450 miles long within Nebraska, and joins the Missouri River at the state’s eastern border. The river is braided in south-central Nebraska, and sand/gravel bars divide the river into multiple channels. There are no large tributaries contributing flow to the Platte River in south-central Nebraska, but the Loup River and the Elkhorn River merge with the Platte River in eastern Nebraska (Fig. 1). The Loup River drains water from the Sand Hills area, which is covered mostly by sand dunes. The Elkhorn River drains an area of about 17,900 km2 in northeastern Nebraska, the eastern part of which is covered by glacial till of silty clay and loamy loess (Bentall 1971).

According to U. S. Geological Survey’s streamflow records (http://water.usgs.gov/waterwatch/), the annual streamflow rate was 44 m3 /s between 1935 and 2002 at the gauging station near Grand Island, 50 m3 /s between 1929 and 2002 at the Duncan gauge about 11 km southwest of Columbus, and increased to 159 m3 /s at the Ashland gauge between 1929 and 1959 and between 1989–2002. A streamflow record was not available between 1960 and 1988 for the Ashland gauging station. The higher flow rate of the Platte River near Ashland is basically the contribution of water from the Loup River and the Elkhorn River. According to Bentall (1971), the Elkhorn and Loup rivers together contributed 63.7 percent of the flow of the Platte at its mouth during the 1959–68 decade. That so large a percentage of total flow was derived from about a quarter of the Platte’s total drainage basin is due mostly to greater precipitation in the eastern part of the basin (Bentall 1971) and the constant baseflow discharging to the rivers in the Sand Hills.

The width of Platte River channels varies from one test site to another. The width of the main channel (south channel) in south-central Nebraska can be as wide as 250 m. The river is not braided at the Ashland test sites, and its width is about 400 m. The Platte River is characterized by its shallow water depth. During the permeameter tests, the water depth across most of the river was shallower than 60 cm.

Nebraska, after California and Texas, ranks third nationally in groundwater use per state. According to Hutson and others (2004), total groundwater withdrawals in Nebraska were 7,860 million gallons per day in 2000, the largest groundwater use per person in the nation. There are nearly 100,000 registered wells, most for gravity or center-pivot irrigation, in Nebraska. In the central Platte valley of Nebraska, high capacity irrigation wells occur densely. In some counties, the average number of irrigation wells is more than 3.4 per square kilometers . Irrigation wells constructed in the alluvial aquifer often have a high-yield capacity, ranging from 500 to 1,500 gallons per minute. The streambed sediments consist mainly of sand and gravel, which provide a good connection between the river and the surrounding alluvial aquifers. The combination of shallow water depth and coarse streambed sediments potentially makes the streamflow of the Platte River vulnerable to depletion due to groundwater withdrawals in the nearby alluvial aquifer. A small decline of the water table in the aquifer can potentially deplete a significant amount of streamflow or even dry up the river.

Methods

Permeameter tests

Figure 2 is a schematic diagram showing a permeameter test in a streambed. The tube is vertically pushed into the top part of the streambed and a sediment column is formed at the lower part of the tube. Water is added to fill the rest of the tube. The hydraulic head inside the tube begins to fall as soon as pouring water stops. The rate of falling head inside the tube depends on the hydraulic conductivity of the sediment column in the tube and the head difference between the streambed and the tube. During a permeameter test, hydraulic heads at a given time were recorded for the derivation of K v values.

Fig. 2
figure 2

Schematic showing a permeameter test for measuring streambed hydraulic conductivity in river channels

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Figure 1 shows the sites of permeameter tests conducted on the channels of the Platte River from Kearney in south-central Nebraska to Ashland in eastern Nebraska. The hydraulic heads were measured manually for the test sites near Kearney, I279, I285, and I305, and were measured using pressure transducers for the sites near Doniphan, Chapman, Central City, and Ashland. The number of head readings for manual measurements ranges from 6 to 15, and often exceeds 80 using pressure transducers. Figure 3 shows the decline of hydraulic head with time for one test at the Doniphan site on the south channel of the Platte River and another test at the Ashland site in eastern Nebraska. The diameter of the tube for the permeameter test was 10 cm. The length of sediment column for each test was about 40 cm, and test duration varied from about 10 to 20 minutes for a sand and gravel streambed. The tests for I279, I285, and I305 were conducted in October 2001, and the results of streambed hydraulic conductivity were presented in Chen (2004). The permeameter tests for the Kearney site were conducted in October 2004. The tests for the other sites were performed in 2002. There were 48 tests along four transects at the Ashland site. During the permeameter tests, the water depth at each test location was measured to determine its relationship to streambed hydraulic conductivities.

Fig. 3
figure 3

Curves showing decline of hydraulic head in the tube of permeameter tests in the Platte River

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Estimation of streambed hydraulic conductivities

Hvorslev (1951) provided a formula to calculate the K v of the sediment column in the pipe such that

$$K_{v} = \frac{{\frac{{\pi D}}{{11\,m}} + L_{v}}}{{(t_{2} - t_{1})}}\ln (h_{1} /h_{2})$$
(1)

where L v [L] is the length of the sediment column in the pipe, D [L] is the inner diameter of the pipe, h 1 [L] and h 2 [L] are the hydraulic heads measured inside the pipe at time t 1 and t 2 since a permeameter test began, and \(m = {\sqrt {K_{h} /K_{v}}},\) which is often unknown at the time of computation. Assuming an isotropic streambed (m =1) will simplify Eq. (1). This simplification leads to an overestimation of the K v value for anisotropic stream sediments with K h > K v .

Chen (2000) used a simplified version of Eq. (1) by neglecting the term πD/11m such that

$$K_{v} = \frac{{L_{v}}}{{(t_{2} - t_{1})}}\ln (h_{1} /h_{2}).$$
(2)

This simplification implies a very strong anisotropic streambed where K h ≫ K v . Chen (2000) noticed that this application can, however, underestimate the K v value for stream sediments of smaller anisotropy, for example, 1 < m < 5.

Any pair of head readings at given times can be used to calculate a value of K v . The values of K v , however, may vary slightly from one calculation to another for the same test. This inconsistency comes mainly from measurement noise. In order to eliminate it, the least squares method was used in this study to take into account all the head readings simultaneously and inversely calculate K v values.

Equation (1) can be rewritten such that

$$h = \frac{{K_{v} \tau}}{B}$$
(3)

where h=ln (h 1 /h 2) (scaled hydraulic head), τ=(t 2 − t 1), and B=(πD/11m+L v ). The inverse method minimizes the squared differences between the observed and calculated h

$$E = {\sum\limits_{i = 1}^N {(h_{{oi}} - h^{*}_{i})^{2}}}$$
(4)

where h oi and \( h^{ * }_{i} \) are the observed and calculated scaled hydraulic heads, respectively, at scaled time τ i (i = 1, 2, 3, ..., N). N is the number of paired head readings used in the computation. The value of K v is determined by iterations such that

$$K^{{j + 1}}_{v} = K^{j}_{v} + \Delta K_{v} $$
(5)

where \(\Delta K_{v} = \frac{{B{\sum\limits_{i = 1}^N {(h_{{oi}} - h^{*}_{i})(\tau_{1}, \tau_{2}, ...,\tau _{N})^{T}}}}}{{{\sum\limits_{i = 1}^N {(\tau _{i})^{2}}}}}\) with (τ1, τ2,...,τ N )T= a column vector, and K j v is the value calculated from previous iteration. The initial value K 0 v was given prior to the first iteration.

The approach was also implemented using Eq. (2) to calculate the K v values. For each permeameter test, a value of K v was calculated based on Eqs. 1 and 2, respectively.

Statistical and geostatistical analysis of the streambed hydraulic conductivities

Kruskal-Wallis test was used to determine the similarity of the K v values of the streambed between each test site. The hypotheses tested by the Kruskal-Wallis test include: H0: all of the n population distribution functions are identical; H1: at least one of the populations tends to yield larger observations than at least one of the other populations (Conover 1980; Gilbert 1987).

At each test location, water depth was measured. The correlation coefficients between K v and water depth were computed for individual transects to evaluate their relationship.

Experimental variograms were computed for the Doniphan and Ashland test sites. Because the measurements of K v along each transect at the two sites were at regular sampling intervals, one-dimensional variograms (or semi-variograms) were calculated using (Webster and Oliver, 2000)

$$\gamma (p) = \frac{1}{{2(M - p)}}{\sum\limits_{i = 1}^{M - p} {{\left( {K_{i} - K_{{i + p}}} \right)}^{2}}}$$
(6)

where i = 1, 2, 3, ..., M denote the locations of permeameter tests, and p is the number of lags. Variogram models were derived by fitting the experimental variograms.

Results and Discussions

Variation of K values across the river channels

Most permeameter tests were conducted to measure K v of the streambed, and a few tests were for K h . At the Ashland site (Fig. 1), permeameter tests were conducted along four transects (Fig. 4) on the west half of the Platte River, three transects approximately perpendicular to the river and one parallel to it. The average K v values for transects I through IV were 9.1, 19.4, 21.8, and 22.3 m/d, respectively. The values of K v show a spatial variation across each transect (Fig. 5). The values of K v from transect I are the smallest among the four transects. The K v values from transects I, II, and III (Fig. 5a) have an increased tendency toward locations farther from the river bank. These permeameter tests were conducted on June 19 and 20, 2002, and during that time, the daily streamflow rate was 121 and 147 m3 /s at the Ashland gauging station (http://water.usgs.gov/waterwatch/). The averaged water depth of the test locations along transects I, II, III, and IV was 8.9, 14.2, 23.1, and 7.8 cm, respectively.

Fig. 4
figure 4

Locations of permeameter tests along four transects in the Platte River, the Ashland test site, eastern Nebraska

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Fig. 5
figure 5

Variation of K v values along transects I, II, and III (a) and transect IV (b) at the Ashland site

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One permeameter test was conducted to measure K h , and it was 58.5 m/d. Another test was conducted to measure the hydraulic conductivity along a 45-degree angle from the streambed, and the value was 24.8 m/d. The two tests were conducted near the middle of transect IV. Chen (2000) described the methods of measuring the K values along arbitrary directions in streambeds.

A total of 10 permeameter tests were conducted across the main channel of the Platte River near Central City (Fig. 1). Figure 6a shows the variation of eight K v values, from 19 to 64.3 m/d, across the Platte River channel, about 236 m wide at the test site. The average K v values along this transect were 43.7 m/d. A pair of tests was conducted 15 m north of the transect and about 75 m from the north bank of the river. One test was for K h (60.3 m/d) and the other was for the K v (53 m/d). The value of K h is slightly larger than the value of K v . These tests were conducted on September 18, 2002, and the daily streamflow at the Grand Island gauge was 14.3 m3 /s. The averaged water depth of the test locations was 16.3 cm.

Fig. 6
figure 6

Variation of K v values across the Platte River at the Central City test site (a) and at the Doniphan test site (b).

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Four permeameter tests 10 m apart were conducted in the south channel of the Platte River near Chapman (Fig. 1). The K v values were 61.9, 41.2, 41.5, and 42.4 m/d, respectively. The channel was 80 m wide. These tests were conducted on September 18, 2002.

Twenty-one permeameter tests were conducted in the south channel of the Platte River at the Doniphan test site (Fig. 1). The channel was 156 m wide and divided by a sand bar of 62 m into a south and north part. Eighteen tests were conducted in the north part at 3 m intervals. Figure 6b shows the K v values varying along the transect ranging from 5.8 to 68.4 m/d, with an average value of 42.4 m/d. These tests were conducted on September 17, 2002; the average water depth of the 21 test locations in the south channel was 13.8 cm.

Figure 7 shows locations of 11 permeameter tests in the Platte River near Kearney. The tests were conducted on October 15, 2004, and the streamflow, recorded in the nearby gauging station, was about 6 m3 /s, 1/6 of the averaged rate of this day between 1982 and 2002. As shown in this figure, large sand bars appeared in the river channel and occupied a large width of this channel. Among the 11 tests, eight (tests 1 through 8, see Fig. 7) were conducted to measure the vertical hydraulic conductivity and three (tests 9, 10, and 11) for horizontal hydraulic conductivity. The hydraulic conductivity values are shown in Figure 7. The vertical hydraulic conductivity also varies largely in space with an average value of 32.5 m/d. The horizontal hydraulic conductivity varies from 98.8 to 148.4 m/d. These few tests for K h are not sufficient to characterize the spatial variation of K h . These K h values are smaller than those determined by slug tests (Rus et al. 2001) for other parts of the Platte River channel and those determined by a pumping test in an island on the Platte River (Chen and Chen 2003).

Fig. 7
figure 7

The vertical and horizontal streambed hydraulic conductivities at the Kearney test site on the Platte River

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Figure 8 shows the histogram of the K v values from all the test sites in Figure 1, with the exclusion of three K v values for clay/silt at test site I305, the values of which range from 0.8 to 2.8 m/d (Chen 2004). The mean of K v from 124 measurements is 31.2 m/d and the standard deviation is 17.1 m/d. The skewness is 0.445. The Shapiro-Wilk test indicates that these K v values are not normally distributed but its distribution is close to normal.

Fig. 8
figure 8

Histogram showing the distribution of the K v values from all the test sites

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Kruskal-Wallis test

The Kruskal-Wallis test was used to determine the similarities of the K v values from these test sites presented in Figure 1. The K v values from 13 transects of the eight test sites, including five transects of three sites from Chen (2004), were treated as individual groups. The number of samples in these transects varies from four to twenty one. For meaningful results using the Kruskal-Wallis test, a total of three groups of data and three samples in each group are needed. The K v data meets this requirement. The results of the Kruskal-Wallis test at the α = 0.05 level indicate that

  1. 1.

    The K v values from the sites between Central City and Kearney, located in south-central Nebraska, have the same mean or statistically belong to the same population;

  2. 2.

    The mean of the K v values from each of the four transects at the Ashland site differs from the mean of the K v values from the sites above Central City;

  3. 3.

    At the Ashland site, the four groups have different means; K v values from transects II, III, and IV come from the same population, but the K v values of transect I belong to another population.

After exclusion of the 48 K v values from the Ashland site, the K v values of the Platte River in south-central Nebraska show a normal distribution (Fig. 9). The mean is 40.2 m/d, standard deviation is 14.8 m/d, and skewness is 0.16. Mean K v values for transects I, II, III, and IV at the Ashland site are only 9.1, 19.4, 21.8, and 22.8 m/d, which are about half or less than half of the mean K v in the other test sites. The lower streambed hydraulic conductivity at the Ashland site and its different statistical behavior are likely due to a supply of fine grains of sediments in the surrounding and adjacent watersheds, which were deposited in this shallow and wide reach of the Platte River.

Fig. 9
figure 9

Histogram showing a normal distribution of the 76  K v values from the Platte River in south-central Nebraska

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Correlation between water depth and K v values

Correlation coefficients between water depth and the vertical hydraulic conductivity were calculated for several transects. The correlation coefficients for transects I, II, III, and IV at the Ashland site were 0.61, 0.38, 0.6, and 0.23, respectively. The correlation coefficient was 0.43 for the Central City test site, and 0.79 for the Kearney test site. The correlation coefficient for the Doniphan test site was 0.43, excluding the K v from the test location closest to the northern bank, but became 0.16 including this value. This extreme value generates an erratic result compared to other correlation coefficients, and it is probably an outlier (Kitanidis 1997). All the correlation coefficients were positive. A larger K v value often occurred in the part of the channel with deeper water where the flow velocity is usually larger. The area of the channel where the water is deeper will likely have larger flow velocity, which can wash finer-grained particles off and carry them to areas of lower flow velocities.

Water depth in river channels represented only the present flow conditions during permeameter tests, but the sediment column tested for streambed hydraulic conductivity was deposited over a prolonged period, during which the flow conditions may have fluctuated. The correlation between the two variables may not be perfect. Nevertheless, these positive correlation coefficients suggest the influence of flow condition on the distribution of streambed hydraulic conductivities in the Platte River.

Spatial variation of K v across the channels

Spatial correlation of a geological variable often decreases with the increase in distance between two measurements. Variograms describe the spatial correlation of a geologic variable and increase with the lag distances of two tests. Variogram models such as spherical, exponential and Gaussian models have been commonly used to describe monotonic features of spatial correlation as the lag distance increases (Kitanidis 1997). Some variables, however, seem to fluctuate more or less periodically. A sine hole effect variogram model describes the damped oscillation around the sill (Olea 1999). Webster (1977) indicated periodic behaviors of some soil survey data.

Experimental variograms of K v were calculated for the Doniphan and Ashland test sites. For the Doniphan site, only the K v values north of the sand bar were used in the computation. A normality test indicates that these K v values are normally distributed. However, the analysis of correlation between K v and water depth suggested that the K v value from the location nearest to the north bank was an outlier and was excluded for the calculation of the experimental variogram. Figure 10a shows the curves of the experimental variogram and the fitted variogram model. A periodicity of spatial variation can be observed from this variogram. The fitted variogram model is written as follows:

$$\gamma (h) = 66.35 + 107.7[1 - \sin (\frac{{2\pi h}}{{13.45}} + 1.72)].$$
(7)
Fig. 10
figure 10

Experimental varigram and fitted variogram (solid line) models for the K v values at the Doniphan test site (a) and for the transect I of the Ashland test site (b).

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The variogram was fitted using the least squares methods in SAS (SAS Institute 2001). The nugget effect of this variogram is 66.35; the amplitude and length of wave are 107.7 and 13.45, respectively; the phase shift is 1.72. The length of wave indicates the horizontal distance beyond which the level of spatial correlation repeats. This experimental variogram shows three periodicities in about a 40-m lag distance.

An experimental variogram was also calculated using the K v values measured along Transect I at the Ashland test site. Figure 10b shows the curves of the experimental variogram and the fitted model. Periodicity also appears in this variogram. The fitted variogram model is

$$ \gamma (h) = 8.65 + 13.19{\left[ {1 - \sin {\left( {\frac{{2\pi h}} {{38.66}}} \right)} + 1.05} \right]}. $$
(8)

The nugget effect of this variogram is 8.65; the amplitude and length of wave are 13.19 and 38.66, respectively; the phase shift is 1.05. The length of wave for this variogram is about three times wider than that for the Doniphan site. This can probably be attributed to the wider channel at the Ashland test site. The channel width was about 400 m at the Ashland site but only 156 m at the Doniphan site in which a sand bar of 62 m existed.

The water depth, measured during permeameter tests, also fluctuated across the channels at both test sites. Figure 11a shows the water depth in the south channel of the Platte River at the Doniphan test site. As can be seen from this figure, the stream stage was relatively shallow (< 30 cm). This curve of water depth does show some periodicity, which may indicate its moderate impact on the periodicity occurring in the K v values. An experimental variogram was thus calculated to determine its possible periodicity. Figure 11b shows the experimental variogram of the water depth for the Doniphan site. It clearly displays the periodicity of the wave length, which is similar to that in the experimental variogram of K v .

Fig. 11
figure 11

Variation of water depth a across the south channel of the Platte River at the Doniphan test site and the experimental variogram of the water depth (b)

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Summary

The vertical hydraulic conductivity of a sandy streambed in the Platte River has a wide range, varying from about 3 to 65 m/d. The variation can occur in short distances, for example, 3 m apart, and is observed along transects across or parallel to channels of this river.

Statistically, the K v values from all the test sites between Kearney and Central City, south-central Nebraska, are the same but differ from the K v values from the Ashland site. This difference is mainly attributed to the addition of finer sediments in the Platte River in eastern Nebraska. The fine sediments carried by the Loup River and the Elkhorn River, as well as from other small creeks to the Platte River, result in a lower vertical K v value of the streambed for the Ashland site. At the Ashland site, the K v values belong to two populations. The K v values from Transect I are statistically different from K v values from the other three transects.

There is a positive correlation between K v and the water depth, ranging from 0.23 to 0.79 for the test sites. Larger K v often occurs in the part of channels with greater water depth. Periodic features of K v values across the channel were observed. This periodicity is probably produced by the variation of water depth across the channel, which controls the flow velocity.

Several permeameter tests conducted to measure the horizontal hydraulic conductivities of the streambed suggest that the horizontal hydraulic conductivity of streambed was larger than the vertical hydraulic conductivity at the same test locations. The streambed of the Platte River at the study sites consists mainly of sand and gravels; silt and clay layers occur only sporadically. No spatially continuous low-permeability layer was observed at the study sites.