Using Automated Seepage Meters to Quantify the Spatial Variability and Net Flux of Groundwater to a Stream
Abstract
We utilized 251 measurements from a recently developed automated seepage meter (ASM) in streambeds in the Nebraska Sand Hills, USA to investigate the small-scale spatial variability of groundwater seepage flux (q) and the ability of the ASM to estimate mean q at larger scales. Small-scale spatial variability of q was analyzed in five dense arrays, each covering an area of 13.5–28.0 m2 (169 total point measurements). Streambed vertical hydraulic conductivity (K) was also measured. Results provided: (a) high-resolution contour plots of q and K, (b) anisotropic semi-variograms demonstrating greater correlation scales of q and K along the stream length than across the stream width, and (c) the number of rows of points (perpendicular to streamflow) needed to represent the groundwater flux of areas up to 28.0 m2. The findings suggest that representative streambed measurements are best conducted perpendicular to streamflow to accommodate larger seepage flux heterogeneity in this direction and minimize sampling redundancy. To investigate the ASM's ability to produce accurate mean q at larger scales, seepage meters were deployed in four stream reaches (170–890 m), arranged in three to six transects (three to eight points each) per reach across the channel. In each reach, the mean seepage flux from ASMs was compared to the seepage flux from bromide tracer dilution. Agreement between the two methods indicates the viability of a modest number of seepage meter measurements to determine the overall groundwater flux to the stream and can guide sampling for solutes and environmental tracers.
Key Points
-
A recently developed seepage meter produced high-resolution maps of groundwater flux through streambeds in the Nebraska Sand Hills, USA
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Anisotropic semi-variograms suggest representative point sampling is best conducted in rows of evenly spaced points across the stream width
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Groundwater flux measured at 3–6 channel transects of seepage meters (3–8 points each) estimated total flux in reaches up to 890 m long
Plain Language Summary
Surface waters can be windows into groundwater aquifers. When conditions are right, groundwater is pushed up through the streambed and into the stream bringing with it natural and artificial solutes that are used as tracers for aquifer evaluation. Therefore, measurements of groundwater seepage fluxes are important. Utilizing a new tool (Solomon et al., 2020, https://doi.org/10.1029/2019WR026983) for point measurements of groundwater flux (over an area of about 10 cm2), this study makes an unprecedented number of direct discharge measurements in the Sand Hills, Nebraska, USA in both (a) small areas (13.5–28.0 m2) of streambed and (b) along a much larger (170–890 m) stream segments. The results indicate that a modest number of discharge measurements in transects perpendicular to streamflow may accurately represent the groundwater flux in both small sections of stream and even in larger stream sections up to 890 m long. This means that if groundwater is discharging into the stream, the average seepage flux from the groundwater aquifer over the entire stream segment can be obtained through sampling a relatively small number of streambed locations. This potentially provides water resource managers valuable information on fluxes of both groundwater contaminants and age-dating tracers without installing resource-demanding wells.
1 Introduction
Investigating groundwater-surface water interconnection is critical to evaluating the sustainability of aquatic ecosystems through quantifying water and solute fluxes between aquifers and riparian zones (Burkholder et al., 2008; Hancock et al., 2005; Paerl et al., 2006). Quantifying these fluxes is also crucial for water resource managers to estimate aquifer volumes and surface water resiliency to drought conditions (Alley et al., 1999; Rushton & Tomlinson, 1979; Winter et al., 1998). These investigations often rely on the quantification of groundwater seepage flux (q) using field data. Groundwater seepage can be quantified in aggregate at the stream reach scale using methods such as differential stream discharge measurements (Harvey & Wagner, 2000) or dilution tracer tests (Kimball et al., 2001; Payn et al., 2009); and at the point-scale using Darcian approaches (summarized by Gilmore, Genereux, Solomon, Solder, et al. (2016)) or seepage meters (Kalbus et al., 2006; Rosenberry et al., 2020). Point scale (<100 cm2) measurements of q are particularly important for flow-weighting chemical fluxes and groundwater ages for aquifer transit time distributions (Cook & Herczeg, 2000; Gilmore, Genereux, Solomon & Solder 2016; Gilmore, Genereux, Solomon, Solder, Kimball, et al., 2016; Solomon et al., 2015; Stolp et al., 2010). Such flow-weighting techniques depend on a representative distribution of flux measurements which can be difficult to resolve due to high spatial heterogeneity and variability of q.
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What is the small-scale spatial variability of q and at what separation distance is spatial auto-correlation of q lost?
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Is spatial auto-correlation more prevalent along certain streambed directions?
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How many point-measurements of q are needed to represent groundwater flux at varying spatial scales?
Relatively little work has been conducted on these specific questions despite substantial evolution of methods to investigate q and its spatial distribution in streambeds and lakes (Kalbus et al., 2006; Rosenberry et al., 2020). Questions 1 and 2 have been examined in the Coastal Plain of North Carolina at densities of 0.03–0.13 measurements per m2 and by application of the Darcian approach (Kennedy et al., 2008, 2009), but higher measurement densities may provide additional insight into the spatial distributions of q. Examination of question 3 using direct, point-scale measurements of q (e.g., Hatch et al., 2010; Solder et al., 2016) remains largely untested or inconclusive when combined with other reach-scale flux measurement methods (Brodie et al., 2007; González-Pinzón et al., 2015; Kikuchi et al., 2012). Gilmore, Genereux, Solomon, Solder, Kimball, et al. (2016) compared reach-scale seepage flux measurements from averaged points and “seepage blankets” (a method based on collecting groundwater seepage beneath a weighted mat ∼1 m2) to dilution tracer tests with varied results.
We applied the ASM in the Nebraska Sand Hills to advance the understanding of small-scale spatial variability of q and evaluate the ability of point-scale measurements of seepage flux to estimate the total groundwater flux at larger scales. We addressed questions 1 and 2 by measuring q and hydraulic conductivity (K) in five dense arrays of regularly spaced points (13.5–28.0 m2) at spatial-resolutions of 0.5–1.0 m and at measurement densities of 1.3–2.6 measurements per m2. From these dense arrays, we created contour maps of q and K for visual inspection and analyzed the small-scale spatial variability and auto-correlation of q and K using anisotropic semi-variograms. We addressed question 3 at small spatial scales by evaluating how many rows of points (perpendicular to streamflow) are needed to represent the groundwater flux of an entire dense array. We addressed question 3 at larger spatial scales (170–890 m stream length) by comparing averaged ASM streambed flux measurements to flux estimated by dilution tracer tests.
2 Site Description
The Sand Hills region is the largest grass-stabilized dune region in the western hemisphere and serves as the main recharge zone for the High Plains Aquifer (HPA) (Bleed & Flowerday, 1998; Gosselin et al., 1999; Winter 1986). The study region is located in the northern region of the HPA, where saturated thickness is estimated to be about 300 m (Gilmore et al., 2019). The underlying aquifer is primarily Eolian sand and loamy/sandy alluvium (Loope & Swinehart, 2000). Studies of three-dimensional hydraulic conductivity of surficial sediments (soils and streambeds) in the Sand Hills show varying degrees of saturated zone heterogeneity (Cardenas & Zlotnik, 2003; Wang et al., 2008), but to a lesser degree than spatial variability of streambed properties at sites outside of the Sand Hills (e.g., Genereux et al., 2008; Hess et al., 2002; Rosenberry & Pitlick, 2009).
The regional climate is semi-arid and can be categorized into four geographic land cover types: lakes and wetlands, subirrigated meadows, dry valleys, and upland dunes (Billesbach & Arkebauer, 2012; Healey et al., 2018). Mean annual precipitation and evapotranspiration in the research area is estimated to be 460–520 mm yr−1 and 440–480 mm yr−1 respectively (Szilagyi et al., 2003). Recharge rate throughout the Sand Hills is variable by ecosystem, and is estimated to be between 1 and 44 mm yr−1 in the subirrigated meadows most of which occurs during the rainy season (April–October) (Billesbach & Arkebauer, 2012; Szilagyi et al., 2003; Szilagyi & Jozsa, 2013).
The study stream is the South Branch of the Middle Loup (SBML) River which is located in the subirrigated meadow between dunes to the north and south (Healey et al., 2018) (Figure 1). The water table is often near the land surface (Gosselin et al., 2006) and the meadow is often flooded during high precipitation events. The SBML predominantly flows west to east before feeding the Middle Loup River. The Upper Middle Loup Watershed has relatively uniform geology and hydrology (Hobza & Schepers, 2018). The baseflow-dominated stream has an annual discharge (Q) of 13.2 m3s−1 and a Q95/Q5 of 1.41 with a Q95 of 16.3 m3s−1 and Q5 of 11.5 m3s−1 measured downstream of the study site at Dunning, Nebraska (Hobza & Schepers, 2018).
Site map and location of the Upper Middle Loup (UML) Watershed within the Sand Hills, Nebraska, USA and the South Branch of the Middle Loup (SBML) River. Sites 6000, 13000, 20000, and 40500 are named after distance downstream from an arbitrary point near the Gudmundsen Sandhills Research Laboratory.
This study was conducted during two field campaigns: 17–30 May 2019, and 29 July–4 August 2019. According to a weather station in Whitman, Nebraska, 15.4 cm of rainfall occurred in May 2019 and 11.9 cm of rainfall occurred during the May field campaign (7.9 cm is typical for May), whereas 0.3 cm of rainfall occurred during the site visit between 29 July and 4 August 2019 (US Climate Data, 2021). At site 13000 in August the streamflow was 0.071 m3 s−1, while the same site in May was flowing at 0.328 m3 s−1.
3 Methods
3.1 Study Design
Seepage flux measurements were collected at four study sites along the SBML: 6000, 13000, 20000, and 40500. “Study site” refers to the general location along the stream and is named by length in meters downstream of a point (101°28’43” W, 42°04’35” N) arbitrarily selected east of a culvert near Gudmundsen Sandhills Research Laboratory. For example, study site 40500 is located 40,500 m downstream from the zero point, as measured along the channel. At each study site, measurements of q were made by the ASM in discrete locations called “points” which were organized in the streambed according to the spatial scale of the research questions addressed in this study.
To investigate the research questions at small spatial scales, points were organized in “dense arrays” covering 13.5–28.0 m2 of streambed area. Each dense array contained 35–42 points arranged in a regular grid pattern (Figure 2), generally with a spacing between adjacent measurement points of 0.5–1.0 m in both the lateral and longitudinal direction (perpendicular and parallel to streamflow, respectively). Grid spacing was somewhat dependent on stream morphology and points were placed where stream depth was at least 5 cm, the minimum needed for the ASM. Dense arrays were conducted at sites 13235 and 20460 in May 2019, and 13235, 20460, and 40550 in August 2019 (Figure 3). The exact array locations were selected based on areas of high seepage flux from previous reconnaissance field work using temperature methods (Gilmore et al., 2019). In each dense array, seepage flux measurements were conducted first followed by falling head slug tests to measure K as described below.
Dense array with seepage meters measuring groundwater seepage fluxes (left) Schematic layout of points to measure small-scale variability of seepage (q) and hydraulic conductivity (K) (top) Half of the total array of seepage meters at the site 13235, August 2019 with streamflow direction shown. At each point except one on the far left, an automated seepage meter (ASM) sits atop a thin-walled tube that penetrates the streambed.
(Top) Locations of all dense arrays and reach-scale sites in the South Branch of the Middle Loup River. Reach-scale seepage flux measurements include automated seepage meter (ASM) transects and Br tracer dilution. (Middle) Locations of the dense array, ASM transects, and Br tracer sampling locations in the 20000 sampling reach. (Bottom) Example of a single transect of ASM measurement points.
To investigate the ability of averaged q point measurements to estimate seepage flux at larger spatial scales, a sampling reach (170–890 m stream length) was defined at each study site. Within the sampling reach, three to six transects were spaced 20–320 m apart along the stream channel. Each transect contained three to eight points evenly spaced perpendicular to streamflow (Figure 3, Table 1). Average seepage flux from the seepage meter transects was compared to the flux estimated by measuring stream discharge change per unit stream length (dQ/dL) along the same stream reaches. Change in stream discharge was based on measurements from a bromide salt tracer injected into the stream at a steady rate in May and August 2019 (Table 1).
| Transects | Bromide tracer | |||||||
|---|---|---|---|---|---|---|---|---|
| Study site | Date (2019) | Transect location | Sampling reach length (m) | Points per transect | Point spacing(m) | Date (2019) | Measurement locations | Sampling reach length (m) |
| 6000 | 30 May | 6300 | 290 | 3 | 0.50 | 27 May | 5600, 5700, 6000, 6500, 7000 | 1,400 |
| 6400 | 3 | 0.60 | ||||||
| 6590 | 3 | 0.60 | ||||||
| 13,000 | 18 May | 13230 | 470 | 4 | 1.00 | 25 May | 12500, 13000, 13500, 14000,14500 | 2,000 |
| 13235 | 8 | 1.00 | ||||||
| 13460 | 4 | 0.75 | ||||||
| 13480 | 4 | 0.50 | ||||||
| 13580 | 4 | 1.00 | ||||||
| 13700 | 4 | 1.00 | ||||||
| 20000 | 17 May | 20300 | 890 | 5 | 1.00 | 25 May | 20000, 20500, 21000, 21200, 21500, 22000, 22500 | 2,500 |
| 20460 | 5 | 0.75 | ||||||
| 20780 | 5 | 0.85 | ||||||
| 20910 | 5 | 0.75 | ||||||
| 21190 | 5 | 0.90 | ||||||
| 40500 | 1 August | 40500 | 170 | 6 | 0.80 | 2 August | 40300, 40500, 40600, 40700, 41000 | 700 |
| 40550 | 5 | 0.80 | ||||||
| 40600 | 5 | 1.00 | ||||||
| 40670 | 5 | 1.00 | ||||||
3.2 Measuring Groundwater Seepage Flux (q)
Groundwater seepage flux (q) was measured using a novel ASM (Solomon et al., 2020) at discrete streambed points. Each point is the location at which a 7.6 cm inner diameter hollow PVC tube was inserted into the streambed and affixed with a seepage meter to measure q. The tube insertion depth ranged from 30 to 50 cm and was determined based on streamflow and tube stability. The tubes were inserted using an SDI Vibecore (https://specialtydevices.com/product/vibecore-mini/) which is widely used for sediment coring (Manheim & Hayes, 2002) and will minimize the effects of friction between the tube wall and surrounding sediment which can cause significant disturbance to the sediment column (Wright, 1991).
A submerged electric valve was located in a hole drilled into the side of each tube. To begin the test, the valve was open and the initial water level inside the tube was equivalent to the stream level outside the tube. The seepage measurement period began when the valve closed and the water level inside the tube moved toward equilibrium with the hydraulic head at the bottom of the tube. The seepage flux was equal to the time rate of water level change in the tube immediately after the valve was closed (Solomon et al., 2020).
The rate of water level change in the tube was measured by two thin metal probes, one submerged in the water in the tube and the other connected to a precision linear actuator and a conductance circuit. The linear actuator lowers the probe until it contacts the water and the water level is determined with a precision of ±0.05 mm. The seepage flux measured from a single ASM point is the mean of three tests conducted over a two-hour period and the uncertainty (error) for a given point is the error of all three tests propagated using standard methods (Meyer, 1975). The seepage meter and the uncertainty of calculated q is discussed in detail in Solomon et al. (2020). Supporting information on negligible effects of streambed disturbance and quality assessment and control is given in Text S1 and S2, and Figure S1 in Supporting Information S1.
3.3 Measuring Hydraulic Conductivity
The spatial variability of vertical hydraulic conductivity (K) and hydraulic gradient (J) at point-measurement scales in a streambed has been widely reported (Abimbola et al., 2020; Cardenas & Zlotnik, 2003; Chen, 2005; Cheng et al., 2011; Genereux et al., 2008; Kelly & Murdoch, 2003; Kennedy et al., 2009; Leahy, 2007; Springer et al., 1999). We determined K in the dense array points using falling head slug tests (e.g., Chen, 2000, 2004; Genereux et al., 2008; Hvorslev, 1951; Leahy, 2007). In this study, a thin-walled tube was already installed into the streambed for the ASM. After ASM measurements at each point were completed, the ASM was removed from the tube and a clear, graduated polycarbonate tube 61 cm long was coupled to the top of the PVC tube in the sediment to provide a larger head gradient, and thus a shorter measurement period, during the falling head test. The side hole was plugged with a rubber stopper, the tube was filled with stream water, and the water level decline was measured along the clear, graduated tube a minimum of five times during a single falling head test. Tests generally lasted for approximately 10 min, but up to 1 hr for points with lower K. The uncertainty (error) of K was evaluated using standard methods considering error propagation (Kline, 1985; Meyer, 1975; Peters et al., 1974) and discussed by Genereux et al. (2008) and Leahy (2007).
3.4 Stream Discharge From Bromide Tracer Dilution
To quantify reach-scale seepage flux by tracer dilution, three constant rate injections of sodium bromide (NaBr) and potassium bromide (KBr) were conducted in May and August 2019 using methods described by Kimball et al. (2001, 2002) and Solomon et al. (2015). Stream water Br− concentration was measured at ∼500 m intervals downstream of the injection point, and at tributaries. This sampling for stream Br− concentration was conducted at least 48 hr after the start of each injection to ensure Br− concentrations as near as possible to steady-state in the stream which was supported by Br− breakthrough curves from ISCO autosamplers. Sampling was conducted on 25 May, 27 May, and 2 August 2019 (Table 1).
(1)3.5 Anisotropic Semi-Variogram
The semi-variogram is a statistical tool used to characterize the spatial continuity of a variable (Deutsch, 2003) and is used in this study to measure the spatial distance in the streambed at which autocorrelation of q and K (separately) is zero. To normalize the semi-variograms, a Gaussian transform to standard normal was applied to all data sets which sets the variance and therefore the sill equal to one (Journel, 1989; Spiegel & Stephens, 2008). Previous studies show that streambed K may follow normal or log-normal distributions (e.g., Cardenas & Zlotnik, 2003; Genereux et al., 2008; Hess et al., 1992; Leahy, 2007; Rehfeldt et al., 1992; Springer et al., 1999). Parameters q and K in this study follow Gaussian distributions at approximately half the sites, other sites exhibit positive skewness.
(2)
is the semi-variogram or variance at a certain lag distance h which is the spacing between two measurement points (e.g., Hess et al., 1992), z is a property (e.g., q or K) at any x location, and 
is the number of data pairs in the data set for the property in question for given h. The semi-variograms were computed using the Statios Gslib geostatistical software (Deutsch & Journel, 1998).Anisotropic (directional) semi-variograms were computed in the direction perpendicular to streamflow (referred to as “stream width”) and the direction parallel to streamflow (“stream length”). Other studies of spatial autocorrelation of K values fit an exponential model to the semi-variogram to identify the range (Cardenas & Zlotnik, 2003; Chen, 2005; Genereux et al., 2008; Kitanidis, 1997; Leahy, 2007) or choose a sine hole effect model to describe damped oscillation or periodic behaviors of the variogram around the sill (Kitanidis, 1997; Olea, 1999; Webster, 1977). In this study, Gaussian, spherical, exponential, and linear models were all applied to the experimental variograms without a nugget effect and the results were compared both visually and by residual sum of squares (RSS).
(3)
and considered the horizontal distance in the streambed at which autocorrelation is zero and measurements are considered to be independent. The ratio of stream length correlation scale (
) to stream width correlation scale (
) was used to examine the autocorrelation of q and K in the principal directions of stream anisotropy. Anisotropic variogram analysis gives the spatial variability of q and K in the principal streambed directions addressing research questions 1 and 2. Additional details on variogram parameters can be found in Humphrey (2021) and supporting information (Data Set S1, Data Set S3).3.6 Sensitivity of Mean q to the Number Rows in a Dense Array
We investigated how well the mean q from a subset of measurement points within a dense array (
) could estimate the mean q for all the points in the array (
). The subsets of points were always chosen from "rows” perpendicular to streamflow, so that each subset would include the potential variation across the channel. This was based on the anisotropic variogram results which indicate larger spatial variability across the stream channel than parallel to streamflow. The smallest subset was always one complete row (5–8 points) and subsets always included whole numbers of rows. The number of rows in each subset is denoted in the subscript (e.g., 
indicates a subset of four complete rows of points from a dense array).
First, the 90% confidence interval (C.I.) of 
was computed for each dense array using the MATLAB “bootstrp” function with replacement (n = 10,000). Second, the bootstrapped (n = 10,000) means of a given subset of rows were compiled into a single vector (
). This vector was then filtered for mean values inside the 90% confidence interval (5% and 95% confidence limits for 
). Finally, the fraction of these 
values that fall within the 90% confidence interval of 
is considered the probability, P(q), that subsampling the points successfully estimated the mean groundwater seepage flux at the dense array (after Kennedy et al., 2008). This addresses research question 3 at small spatial scales. More detail and a useful illustration of P(q) can be found in supporting information (Text S4 and Figure S2 in Supporting Information S1).
3.7 Comparison of Seepage Meter and Stream Discharge Results at the Reach-Scale
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Seepage flux from ASM transects.
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Stream discharge measurements based on Br- tracer dilution in stream water.
To appropriately compare seepage flux values from each method, the average seepage flux (m d−1) calculated by seepage meters in each reach-scale transect was multiplied by the average stream width (m) and all transects in a given reach were averaged to yield the volume of groundwater input per unit length of stream [m2 d−1]. Stream discharge change (i.e., groundwater flux) per unit stream length was calculated as the slope of a linear regression through at least five Br− measurement stations along a given study reach using Q (m3 d−1) as the y component and stream distance, L (m), as the x component such that the slope is equal to dQ/dL [m3 d−1 m−1]. The uncertainty (error) of seepage flux from ASM transects was evaluated using standard methods considering error propagation (Kline, 1985; Meyer, 1975; Peters et al., 1974). The uncertainty (error) of seepage flux from stream discharge measurements was calculated as the uncertainty of the slope using the LINEST function in Excel (Morrison, 2014).
4 Results
4.1 q and K in the Dense Arrays
Mean q and K in the dense arrays were 0.02–0.38 and 3.13–13.43 m d−1, respectively (Table 2). Positive q signifies groundwater discharging vertically into the stream. Measured q was positive for 157/169 total points. Where q was negative (12 points), the apparent downward seepage was small (
= −0.02 m d−1) and q error were relatively large for negative values (mean error for negative values = 0.51 m d−1). Box and whisker plots indicate a large range of both q and K at most sites with the exception of q at site 13235 in August (Figure 4). Frequency histograms of q and K data show some sites with a normal (Gaussian) distribution (q and K at 13235 in May and K 20460 in August) while the data at other sites exhibit positive skewness (Figure 4). The relationship between q and K at all dense array sites is discussed in supporting information (Text S5, Figures S3 and S4 in Supporting Information S1). Error associated with individual q measurements are given in Figure S5 in Supporting Information S1. The mean q and K values at site 13235 decreased by 90% and 69% from May to August and these values at site 20460 decreased by 40% and 35%. Streamflow fluctuated considerably between field work conducted in May and August 2019. For example, at site 8960 the streamflow was 2.68 m3 s−1 on 26 May 2019 and 0.83 m3 s−1 on 31 August 2019.
| Seepage flux (q) | Hydraulic conductivity (K) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Site | Test month | Area m2 | n | Mean (m d−1) | Error | Std. dev. | C.V. | n | Mean (m d−1) | Error | Std. dev. | C.V. |
| 13235 | May | 28.00 | 35 | 0.22 | 2.66 | 0.12 | 55.31 | 40 | 9.97 | 0.43 | 4.21 | 42.18 |
| August | 15.75 | 34 | 0.02 | 0.49 | 0.03 | 125.97 | 39 | 3.13 | 0.20 | 3.48 | 111.21 | |
| 20460 | May | 13.50 | 23 | 0.38 | 1.15 | 0.22 | 58.33 | 37 | 13.43 | 0.54 | 5.62 | 41.83 |
| August | 14.50 | 38 | 0.23 | 0.66 | 0.15 | 64.54 | 40 | 8.72 | 0.45 | 3.26 | 37.39 | |
| 40550 | August | 20.00 | 39 | 0.16 | 0.43 | 0.12 | 77.38 | 42 | 4.26 | 0.32 | 3.11 | 72.99 |
- Note. The area shown is the streambed area covered by the seepage meter array, n is the number of measurement points. For each measured parameter (q and K), the arithmetic mean, uncertainty, standard deviation (σ) and coefficient of variation (C.V) are given.
Boxplots and histograms of seepage flux (q) and hydraulic conductivity (K). (top) The central red line indicates the median, the bottom and top edges of the box indicate the first and third quartiles, respectively. The black dashed lines extend to approximately ±2.7σ and the red markers signify data points beyond these limits. (bottom) Histograms (nbins = 6) of q and K show varied distribution: note the unique x and y scale for q at site 13235 August. All units are m d−1.
Contour plots of the small-scale spatial variability of q and K (Figures 5, S6 and S7 in Supporting Information S1) were created using cubic interpolation in MATLAB (Humphrey, 2021) (Data Set S2). These plots were used to (a) qualitatively examine the relationship between q and K, (b) observe spatial relationships among q, K, and stream morphology (e.g., point bar—cutbank relationships, similar to Cardenas and Zlotnik, 2003), and (c) relate the spatial change of groundwater seepage flux to temporal decrease in streamflow between May and August for sites 13235 and 20460 (site 40550 was inaccessible in May due to dangerous stream conditions).
Summary of data of in situ seepage at site 40550: (a) distribution of seepage flux q with black dots and red circles denoting q measurements and missing data (Text S2 in Supporting Information S1); (b) distribution of hydraulic conductivity (K); (c) google earth satellite site view with q contour overlay; (d) field photo with q contour overlay.
4.2 Semi-Variograms at Dense Arrays
Anisotropic semi-variograms were used to analyze the small-scale spatial variability at dense array sites and the correlation scales (
) were used to determine the horizontal distance in the streambed at which autocorrelation is zero. For q data in the dense arrays, the stream width correlation scales (
) are 1.01–1.91 m and the stream length correlation scales (
) are 1.35–2.50 m (Table 3). The values of 
and 
for K data were slightly smaller on average than for q (Table 3). Mean 
and 
of q are 1.57 and 1.97 m respectively while mean 
and 
of K are 1.08 and 1.61 m respectively. For both q and K, 
< 
at each site with the exception of q at dense array 13235 in May and K at dense array 20460 in May, that is, the ratio, 
/ 
is greater than one at all but two sites. In several cases, the correlation scale was very close to or less than the smallest lag distance indicating that estimated λ should be seen as an upper limit (closer sampling might have led to a smaller estimated λ).
| Site | Test month | Seepage flux (q) | Hydraulic conductivity (K) | ||||
|---|---|---|---|---|---|---|---|
| λL (m) | λW (m) | λL/λW | λL (m) | λW (m) | λL/λW | ||
| 13235 | May | 1.63 | 1.91 | 0.85 | 0.72 | 0.30 | 2.37 |
| August | 2.50 | 1.56 | 1.60 | 2.33 | 1.78 | 1.31 | |
| 20460 | May | 1.35 | 1.01 | 1.34 | 0.71 | 0.91 | 0.78 |
| August | 2.43 | 1.78 | 1.36 | 1.67 | 1.11 | 1.50 | |
| 40550 | August | 1.92 | 1.60 | 1.20 | 2.63 | 1.27 | 2.07 |
| Mean | 1.97 | 1.57 | 1.27 | 1.61 | 1.08 | 1.60 | |
Temporal variation in λ is exhibited in both sampling areas studied in May and August (Table 3). In 7 of 8 cases including both q and K, λ values increased between May and August; the only exception was sampling area 13235, where λw of q decreased from 1.91 m in May to 1.56 m in August which may be explained by the decrease in stream width.
4.3 Probability of Estimating Dense Array 
From Subsets of Rows
We evaluated the probabilities of estimating the average seepage flux from a dense array by using a smaller number of measurements. We compared the measured average groundwater seepage flux over a streambed area of 13.5–28.0 m2 (
) to the distribution of average seepage fluxes from a subset of rows (
). While median 
values generally align with the median 
, a subset consisting of a single row of points (
) slightly overestimated the median 
by as much as 8% (Figure S9 in Supporting Information S1). The 90% confidence intervals (C.I.) of 
are on average 50% greater than the 90% C.I for 
, indicating that a single row of points will yield a reasonable estimate of the small-scale mean groundwater flux. The 90% C.I. distribution of likely mean values for a given subset of rows converges toward the 90% C.I. of 
as more rows are incorporated and do so with diminishing returns.
We examined the probability, P(q), that 
will estimate 
(Table 4, Figure 6). For 
the probability of estimating 
within the 90% C.I. is between 18.0% (site 20460 May) and 48.9% (site 13235 August). Adding an additional row in the subset increases the probability 
to 43.3%–62.6%. If roughly half the transects are added, the probability that the 
exceeds 61% for all dense array sites (Figure 7).
From a Subset of Rows| Dense array all rows | P(q) Number of rows in a subset | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Site | Month | LCL | q̅all | UCL | q̅sub1 | q̅sub2 | q̅sub3 | q̅sub4 | q̅sub5 | q̅sub6 | q̅sub7 | q̅sub8 |
| 13235 | May | 0.19 | 0.22 | 0.25 | 36.5% | 51.9% | 67.5% | 81.7% | 100% | - | - | - |
| August | 0.01 | 0.02 | 0.03 | 48.9% | 59.4% | 70.2% | 78.8% | 85.7% | 91.3% | 95.7% | 100% | |
| 20460 | May | 0.31 | 0.38 | 0.46 | 18.0% | 43.3% | 61.7% | 77.8% | 100% | - | - | - |
| August | 0.19 | 0.23 | 0.27 | 32.9% | 47.9% | 61.9% | 75.2% | 87.4% | 100% | - | - | |
| 40550 | August | 0.13 | 0.16 | 0.19 | 43.5% | 62.6% | 75.6% | 85.5% | 92.6% | 100% | - | - |
-
Note. The mean seepage flux of each dense array (
) and the 5% lower confidence limit (LCL) and 95% upper confidence limit (UCL) are given in m d−1 alongside P(q) as a function of the number of rows in a subset. P(q) is the probability that the mean q from a given subset of rows will estimate 
.
Probability P(q) of estimating the mean seepage flux (
) as a function of the subsets of rows. The x-axis represents a standardized view of the subset of rows (
) analyzed in each dense array which have varying number of total rows.
4.4 Seepage Flux in Sampling Reaches
The ability to estimate the groundwater seepage flux over a reach-scale stream length (170–890 m) with a small number of discrete seepage measurements was evaluated by comparing seepage meter point measurements to the change of stream discharge from Br− tracer dilution (dQ/dL). Transect seepage fluxes are within the uncertainty of seepage flux from Br− stream discharge at reaches 13000, 20000, and 40500 and are nearly within the uncertainty at reach 6000 (Table 5, Figures 7, S10 and S11 in Supporting Information S1). Both methods indicate that seepage flux increases downstream and the results from both methods are highly correlated (R2 = 0.97).
| ASM transects | Bromide tracer | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Site | Test month | Stream length(m) | Points (n) | Points per 100 m stream | Transects per 100 m stream | q̅ * W (m2d−1) | Error (m2d−1) | Stream length (m) | # of stations | dQ/dL (m3d−1m−1) | Error (m3d−1m−1) |
| 6000 | May | 290 | 9 | 3.1 | 1.0 | 0.12 | 0.18 | 1,400 | 5 | 0.35 | 0.02 |
| 13000 | May | 470 | 27 | 5.7 | 1.3 | 0.41 | 0.33 | 2,000 | 5 | 0.70 | 0.23 |
| 20000 | May | 890 | 25 | 2.8 | 0.6 | 1.06 | 0.09 | 2,500 | 7 | 1.33 | 0.61 |
| 40500 | August | 170 | 21 | 12.4 | 2.4 | 1.44 | 0.14 | 700 | 5 | 1.63 | 0.33 |
- Note. Reach-scale areal seepage flux for each quantification method. Seepage flux (q) from transects is multiplied by the stream width (W) of the transect and summed. Q is the stream discharge from bromide tracer dilution and L is stream length. The seepage flux from the bromide tracer is calculated as the slope of a linear regression through at least five measurement stations along the study reach.
Seepage flux per unit length of channel from automated seepage meter (ASM) transects and bromide dilution tracer. The one-to-one line is presented as a dotted line and the linear regression and 95% confidence intervals (C.I.) are presented as dashed lines with an R2 = 0.97.
5 Discussion
5.1 Small-Scale Spatial Variability of q and K
Mean q and K values from the dense array at site 13000 are similar to those reported in Gilmore et al. (2019) for the same study reach (
and 
). The mean K values in this study are lower than mean K values found at other sites >300 km downstream in the Loup River (14.7–27.57 m d−1) which is reasonable considering these studies report larger streambed grain sizes (Cheng et al., 2011; Korus et al., 2020). The spatial variability of K in this study is smaller than in other sandy environments (Chen, 2004; Genereux et al., 2008; Korus et al., 2020) which is a testament to the relative homogeneity of Sand Hills sediments. Seepage rates and the spatial variability of q are within the range of reported values for some sandy streambed environments (Craig, 2005; Gilmore, Genereux, Solomon, & Solder, 2016; Gilmore, Genereux, Solomon, Solder, Kimball, et al., 2016) and a sandy lakebed environment (Kishel & Gerla, 2002). These studies report both greater heterogeneity of K and similar or smaller heterogeneity of q to this study which suggests factors other than K (e.g., surrounding topography or riverbed geomorphology) are contributing to the heterogeneity of q as has been reported (Fleckenstein et al., 2006; Shanafield et al., 2010). A similar study of small-scale spatial variability of q and K conducted in the sandy streambeds of the Elkhorn River and its tributary in northeastern Nebraska reported slightly larger ranges of q and K (−1.2–0.9 and 0.06–39.39 m d−1, respectively) (Chen et al., 2009). The similarity of this study site (i.e., large meandering channels in a sandy stream bottom) and relative ranges of q and K further indicates that relatively high spatial variability of q can occur even in relatively homogenous environments.
Spatial patterns of q and K appear to be somewhat correlated. For example, at site 40550 in August, a two-m-wide area of low K extends from the upstream left bank diagonally to the downstream right bank, similar to the spatial pattern of low q (Figures 5a and 5b). Areas where q and K are not correlated could be the result of a variable hydraulic gradient in the riverbed (i.e., the difference between the head at the bottom of the tube and the head in the stream divided by the installation depth), measurement errors (particularly in q), or “shifts” between the measurement location of q and the location where that groundwater is actually discharged into the stream. In other words, correlating spatial patterns of q and K may be complicated by the installation depth of the ASM which measures q at a depth 30–50 cm within the streambed while vertical K is measured for the entire tube length installed in the streambed (30–50 cm total vertical length). In the case that q is perfectly vertical, actual groundwater seepage flux at the stream water interface will be equal to seepage flux measured at the tube installation depth. In the case that q is not perfectly vertical, the measured q at the tube installation depth will not cross the surface water interface until some distance downstream of the measurement location resulting in a shift between measured q and actual q.
Although such a shift is possible, it is likely to be small at this site. Gilmore et al. (2019) investigated seepage flux in the SBML based on: (a) inferred high-flux from distributed temperature sensing (DTS) measured by a fiber optic cable placed on the streambed and (b) the Darcian method of estimating q from measured vertical K in a falling head test (similar to this study) and vertical hydraulic gradient (i) from a piezometer. K and i were measured at 30 cm depth in the streambed both at pre-determined streambed locations and at the inferred high-flux zones. The study found vertical seepage fluxes (measured at 30 cm depth) were significantly higher at the inferred high-flux zones (detected by DTS at the streambed surface), suggesting a strong correlation (Gilmore et al., 2019). Finally, the focus of this study is spatial patterns and net groundwater exchange with the stream, therefore q was intentionally measured at a depth below the hyporheic zone.
Previous studies have shown that K varies in relationship to stream morphology with generally higher K values correlated with areas of larger stream velocities and thus deposition of larger sediments (Hatch et al., 2010; Käser et al., 2009; Sebok et al., 2015). Site 40550 is located on a meander and the spatial patterns of q and K conform with flow regimes of stream bend deposits (Bridge, 1992; Cardenas & Zlotnik, 2003) where higher q and K values are along the cut-bank where stream velocity is greatest. The area of high q (0.2–0.4 m d−1) and K (8–12 m d−1) lies directly in the left cutbank of the stream (Figures 5c and 5d). Juxtaposed to this area of high q and K in the cutbank is an area of relatively low q and K at the point-bar of the stream (located near the right side of Figures 5a and 5b).
Site 20460 is unique in that the stream widens (33% in May and 40% in August) at the downstream end of the dense array and more points were installed to accommodate the wider streambed. The widening stream disrupts the surface flow velocity field and may contribute to shorter correlation scales at 20460. Site 13235 does not lie in a meander like site 40550 and the channel width does not change like site 20460. Under baseflow conditions, this site exhibits the largest autocorrelation of q parallel to streamflow and relatively large autocorrelation of K which may be explained by more uniformly distributed velocity fields. At all August sites, higher q values are located near the left bank and for all sites the left bank is situated near a hillslope. The higher q values in these cases indicate that adjacent stream topography may have a larger effect on the hydraulic gradient in the stream under baseflow conditions.
Variogram analysis reveals significant distinction in autocorrelation of streambed parameters in perpendicular directions. With two exceptions, the ratio of stream length correlation scale to stream width correlation scale 
is greater than one which suggests stronger autocorrelation of streambed parameters parallel to streamflow (Table 3). The high mean 
ratio suggests that representative point sampling is best conducted in evenly spaced points perpendicular to streamflow to capture the mean parameter value with a minimum number of samples (i.e., eliminate redundancy that would come from closely spaced, correlated samples).
The correlation scales of K (0.30–2.63 m) are similar or slightly shorter than other studies of sandy streambeds (e.g., Genereux et al. (2008) calculated λ between 1 and 8 m) and sand/silt/clay environments (e.g., Cardenas & Zlotnik (2003) calculated λ between 2 and 3.4 m). These values are slightly smaller than those found in sand/gravel aquifers (Hess et al., 1992; Rehfeldt et al., 1992; Sudicky, 1986) but within the range of fluvial structures in sandstone/siltstone outcrops (Davis et al., 1997). The slight differences in λ could in part be the result of differences in variogram modeling methods. For example, an exponential model generally yields slightly larger λ values than a gaussian model (Oliver & Webster, 2014). In contrast, these correlation scales are much shorter than those found in lacustrine environments (Bissell & Aichele, 2004). Correlation scales of seepage flux have been infrequently studied and values range from little or no correlation (Kennedy et al., 2008) to 5.2–109.8 m (Mohamed et al., 2021).
Site 13235 shows little to no temporal variation in spatial patterns from May to August. The pattern of high q parallel to streamflow remained in the same location and the area of relatively low q along the right bank was unchanged. Site 20460 showed higher degrees of temporal variation in spatial patterns between May and August. Temporal variations of the magnitude of λ, q, and K, and could be the result of fluvial morphological changes associated with higher streamflow during May fieldwork. Shorter λ measured in May could reflect a changing streambed in an unstable flow regime (Korus et al., 2020; Naganna et al., 2017; Shrivastava et al., 2020). Relatively higher stream velocities have been reported to scour finer-grained and deposit larger-grained sediments resulting in higher K values (e.g., Blom et al., 2017; Hempel, 2018; Surian et al., 2009; Tubino, 1991) which could explain both higher K and potentially higher q. Lower q in August could be at least partially attributed to the decrease in K although the magnitude of percent decrease in K from May to August (90% and 69%) was not matched by the magnitude of percent decrease in q from May to August (40% and 35%). This indicates that other factors may be contributing to an increase in the hydraulic gradient between the aquifer and the stream (e.g., stream stage decrease).
5.2 Reach-Scale Seepage Flux Representation
Seepage flux from ASM transects and Br− dilution agree on the following: (a) the general magnitude of seepage flux in each reach, (b) all reaches are gaining, and (c) seepage flux increases downstream in May. All ASM transect seepage fluxes are within ∼40% of the seepage fluxes from Br− dilution with the exception of reach 6,000 which is ∼66%. This could be due to higher heterogeneity in reach 6,000 combined with a relatively smaller seepage meter density.
There is modest correlation between both the density of stream points and transects per 100 m of stream length and the relative agreement between the two methods indicating that these point and transect densities may provide a guide for future studies that aim to quantify seepage or solute fluxes between surface water and groundwater.
Our reported seepage flux error from ASM transects is the result of propagating error from the mean of three tests at each streambed point and propagating the error from every point into the mean seepage flux for a given reach. Although error from a single seepage meter test is relatively small compared to the q value measured from a single test, the propagated error from all three tests becomes relatively large compared to the reported mean q value from a single point. The seepage flux error from Br− dilution becomes larger at downstream reaches (13000 and 20000) in May as high streamflow diluted the Br− closer to 3x the measurement detection limit, which is enough to greatly increase the uncertainty of Q. Error from ASM measurements may be a potential concern in environments where q is consistently smaller and the relative ASM error is much larger.
6 Summary and Conclusions
We applied a new automated seepage meter to (a) map small-scale streambed seepage patterns, (b) evaluate the autocorrelation of seepage fluxes, and (c) evaluate the number of ASM measurements needed to estimate small- and reach-scale seepage flux.
Small-scale seepage patterns and semi-variogram analysis of high-density point measurements indicate that independent samples should be taken in stream transects perpendicular to streamflow to accommodate seepage flux heterogeneity and minimize measurement/sample redundancy. Evidence from the dense arrays indicate that a small number of transects (1–3) can reasonably represent the mean seepage flux at small scales (13.5–28.0 m2) when points are spaced 0.5–1.0 m across the width of the stream, but this spacing depends on the correlation scales of q. Further analysis of reach-scale measurements indicates that seepage flux measured from 3 to 6 transects each containing 3–8 points (0.6–2.4 transects or 3.1–12.4 points per 100 m of stream length) generally agree with seepage estimates from bromide tracer tests within or near the uncertainty limits at spatial scales up to 890 m. This invites further investigation of the spatial variability of seepage flux at larger scales (1–10 km) as has been investigated for hydraulic conductivity (Rovey & Cherkauer, 1995).
These results indicate that a small number of point-transects can reasonably represent the groundwater seepage fluxes at the scale under 1 km at our study site. The implication is not that using discrete point measurements of q is the optimal method to estimate reach-scale flux. Rather, the ability to leverage a relatively small number of point measurements to represent reach-scale groundwater-surface water exchanges is critical for reach-scale environmental tracer studies where samples often need to be taken from discrete points in the streambed. Further work is necessary to determine whether this methodology can be effectively applied in other soft-bottom streambeds with slightly higher degrees of fluvial and geological heterogeneity such as in the Great Plains and eastern coastal plains of the U.S.
Acknowledgments
The field evaluation portion of this study was funded by the National Science Foundation Grant EAR-1744721. The authors thank Wil Mace for lab and field assistance and Yaser Kishawi, Nawaraj Shrestha, Jens Ammon for assisting in the collection of hundreds of data measurements. The authors thank the USGS for flow measurements.
Conflict of Interest
The authors declare no conflicts of interest relevant to this study.
Open Research
Data Availability Statement
Data used for generating the figures in this manuscript can be found in Supporting Information S1 and associated data files and are archived online at (https://doi.org/10.4211/hs.c169de42e6174baf842a319efe28a75e).
