Hyporheic hydraulic geometry: Conceptualizing relationships among hyporheic exchange, storage, and water age.

Hyporheic exchange is now widely acknowledged as a key driver of ecosystem processes in many streams. Yet stream ecologists have been slow to adopt nuanced hydrologic frameworks developed and applied by engineers and hydrologists to describe the relationship between water storage, water age, and water balance in finite hydrosystems such as hyporheic zones. Here, in the context of hyporheic hydrology, we summarize a well-established mathematical framework useful for describing hyporheic hydrology, while also applying the framework heuristically to visualize the relationships between water age, rates of hyporheic exchange, and water volume within hyporheic zones. Building on this heuristic application, we discuss how improved accuracy in the conceptualization of hyporheic exchange can yield a deeper understanding of the role of the hyporheic zone in stream ecosystems. Although the equations presented here have been well-described for decades, our aim is to make the mathematical basis as accessible as possible and to encourage broader understanding among aquatic ecologists of the implications of tailed age distributions commonly observed in water discharged from and stored within hyporheic zones. Our quantitative description of "hyporheic hydraulic geometry," associated visualizations, and discussion offer a nuanced and realistic understanding of hyporheic hydrology to aid in considering hyporheic exchange in the context of river and stream ecosystem science and management.

Introduction

The magnitude and spatial extent of hyporheic exchange—the continuous bidirectional exchange of water between the stream channel and underlying sediments—is determined largely by the interactions between stream flows, channel form, and hydraulic properties of the streambed and alluvial aquifer sediments [1]. In fine-grained streambeds, the hyporheic zone may be restricted to the top few hundredths of a meter of the streambed [2, 3]. In rivers and montane streams where alluvial sediments consist primarily of coarse sands, pebbles, gravels, and/or cobbles, expansive coarse-grained alluvial aquifers may have hyporheic zones ranging from tenths to tens of meters thick and extending 1’s to 1000’s of meters laterally from the stream channel [4-8]. Regardless of extent, hyporheic zones are important drivers of stream biogeochemistry [9-11]. Expansive hyporheic zones may also exchange enough water with the channel to influence surface water temperature [12, 13]. At coarser spatial scales, the cumulative effect of hyporheic exchange can govern whole-ecosystem processes such as respiration and nutrient uptake in stream networks [14].

Hydrologists and stream ecologists have been expounding and refining conceptual models of hyporheic exchange for decades [15-19]. As a result of such work, most aquatic scientists accept that hyporheic exchange is an important driver of hydrologic and ecosystem processes in streams [20, 21], and some understand that hyporheic flow paths are nested hierarchically—that the majority of hyporheic exchange traverses relatively short flow paths in the hyporheic zone. Yet few scientists outside of hydrologists who study and model the hyporheic zone consider the inter-dependence among hyporheic hydrologic variables. Just as surface water exiting a stream reach has a distribution of transit time through the reach, emerging hyporheic water has a “water age distribution” describing the distribution of time that hyporheic discharge has spent within the hyporheic zone. Similarly, just as stream flow, water volume, and hydrologic residence time are inter-dependent in a stream reach, the rate of hyporheic exchange, magnitude of hyporheic water storage, and hyporheic water age distribution are inexorably linked. When considered alongside the shape of the water age distribution of hyporheic discharge (e.g., derived from tracer experiments), hydrologic inter-dependencies of hyporheic exchange rate, water volume, and water age can be used to infer important aspects of hyporheic hydrology in surprising detail.

The concept of interdependence among hydrologic variables in porous media hydrosystems is not new; the mathematics necessary to describe the relationships among flow rate, storage volume, and water age within steady-state porous media chemical reactors were published in a classic Chemical Engineering paper by Danckwerts [22]. In fact, these equations are included in basic Chemical Engineering textbooks that discuss the analysis, modeling, or design of chemical reactors [23]. Similar concepts have also been used to simulate transient storage (including hyporheic exchange) in streams [24, 25]. More recently, many of these same concepts have been extended to address non-steady state hydrosystems with arbitrarily complex water age distributions arising from transient (dynamic across time) rates of water flow [26-28]. The resulting “ranked StorAge Selection”, or rSAS framework has been applied to describe dynamic hydrology in both watersheds [29] and hyporheic zones [30].

Our goal in compiling this paper is to encourage consideration of water age distributions by aquatic scientists who may currently employ less rigorous conceptualizations of hyporheic hydrology. Although we acknowledge that transient hydrology (e.g., flood spates) plays a critical role in hyporheic dynamics and therefore stream ecosystems, we focus on steady-state assumptions in this paper, leaving aside the dynamic case. We believe starting with steady-state assumptions of hyporheic hydrology is appropriate to illustrate the ecological implications of asymmetrical, heavily “tailed” water age distributions typically observed in hyporheic zones, and we choose to present and apply modifications of the steady-state equations presented by Danckwerts [22]. In doing so, we trade the generality and flexibility of the dynamic rSAS approach in favor of the simplicity and relative transparency of steady-state assumptions. We believe that consideration of steady-state hydrology provides a useful initial case for incorporating improved hydrologic realism into conceptual models of hyporheic exchange, and provides a necessary foundation for subsequent exploration of non-steady-state conditions such as those inherent in the rSAS framework.

Water age distributions

Despite the fact that water exiting any hydrosystem is derived from water stored within the hydrosystem, the age distribution of exiting water is rarely equal to the age distribution of stored water. To understand this concept in the context of a hyporheic zone, consider the hyporheic zone as a collection of flow paths of different lengths. Each flow path begins at the channel, traverses some distance of the hyporheic zone, and re-enters the channel. The age distribution of hyporheic discharge represents water ages at the end of the hyporheic flow paths, while the age distribution of hyporheic water storage represents the water ages along the length of the flow paths. Thus, the paradox of different age distributions for hyporheic discharge and hyporheic storage is resolved by recognizing that exiting water is not a random sample of stored water.

Confusingly, the published terms used to describe the water age distribution of hydrosystem discharge vs. that of hydrosystem storage have become somewhat confounded. Specifically, one common convention refers to the water age distribution of hyporheic discharge as the hyporheic “residence time distribution,” e.g., [25, 31, 32]. Another convention uses “residence time distribution” to refer to the age distribution of water stored within a hydrosystem, while “transit time distribution” describes the water age distribution of hydrosystem discharge [27, 28, 30]. For the sake of clarity, we leave aside names for the two age distributions and choose to distinguish them with notation: AD (age distribution of hyporheic discharge) and AD (age distribution of hyporheic water storage). Use of such notation has the advantage of reinforcing that AD and AD describe the same metric (water age) as applied to different aspects of a hydrosystem (water discharge vs. internal water storage).

Methods

Conceptual model

Stream beds and alluvial aquifers host a mixture of waters sourced from the channel, catchment soils, deeper aquifers, and precipitation. As a strategic simplification that helps illuminate the dynamics of hyporheic exchange, our conceptual model of hyporheic hydrology considers only subsurface water that originates from the channel and that will ultimately return to the channel. We refer to this water as “hyporheic water” and largely leave aside the term “hyporheic zone”, except when used in the most general sense. Such hyporheic water comprises the majority of water found in the streambed, and often throughout coarse-grained alluvial aquifers [33].

We use the symbol τ to represent water age—the time elapsed since a parcel of hyporheic water left the stream channel and entered the alluvial aquifer. So defined, τ might vary from a fraction of a second to a year or more [34], depending on the size of the alluvial aquifer and the temporal scale of interest. τ0 and τ represent minimum and maximum water ages of interest, values which may or may not be zero and ∞, respectively, depending on the application of our conceptual model.

We adopt the aquifer-centric reference frame of hydrogeologists wherein water that enters the alluvial aquifer from the channel (“aquifer recharge”; q↓) is represented using a positive number and water returning to the channel from the aquifer (“aquifer discharge”; q↑) is represented as a negative number. We assume steady state conditions (q↓ = −q↑) typical of hyporheic hydrology when river discharge is stable. Our conceptual model (and the equations presented below) can be applied to a length, area, or volume of water, so long as the dimension of hyporheic water storage (s), q↑, and q↓ are consistent. Specifically, depending on the length dimensions chosen, q↓ and q↑ can describe a vertical water flux ([L][T-1]), a rate at which the cross- or longitudinal-section area of hyporheic water exchanges with the cross- or longitudinal-section area of the channel ([L2][T-1]), or a rate at which the volume of hyporheic water exchanges with the volume of water in a stream reach ([L3][T-1]). Correspondingly, s would be represented by a thickness of water beneath the wetted channel ([L]), a cross- or longitudinal-section area of hyporheic water ([L2]), or a volume ([L3]) of water stored beneath a channel reach. Therefore, we use the notation [Lx][T-1] to represent the dimensions of q↓ and q↑, and [Lx] to represent the dimensions of s, where x ∈ {1, 2, 3}.

Generally, conventional conceptual models of hyporheic exchange consider the hyporheic zone to be a single unit. To visualize the relationships among q↓, s, the AD, and the AD, we subdivide the hyporheic zone into multiple transient storage zones (TSZs; Fig 1). Each TSZ (i) is defined by a maximum water age (τ); the minimum water age of each TSZ is equal to τ. Therefore, the ranges of water age across the TSZs are serial and contiguous. Importantly, our use of the notation τ to denote the minimum water age of TSZ, τ to denote the maximum water age of TSZ, and n to denote the number of TSZs delineated in the hyporheic zone means that the symbol τ0 refers to both the minimum hyporheic water age of interest and the minimum hyporheic water age of TSZ1 (which are numerically equivalent). Similarly, τ refers to both the maximum hyporheic water age of interest and the maximum hyporheic water age of TSZ (values that are, again, numerically equivalent).

Fig 1

Simple graphical representation of transient storage zone hydrology within a hyporheic zone.

Grey boxes represent transient storage zones (TSZs) with associated hyporheic water storage (s) with water age between τ and τ. White arrows represent water recharging the aquifer (), flow between TSZs () or water discharging from the aquifer ().

Several important characteristic of such TSZs are worth noting, and arise from the fact that TSZs are delineated using water age:

All hyporheic recharge must enter the aquifer via TSZ1 because recharge water has τ = τ0.

Because hyporheic discharge ranges in age from τ0 to τ, every TSZ discharges water to the channel.

By definition, when the age of hyporheic water exceeds τ, the water becomes part of TSZ; over time, then, hyporheic water not discharged to the channel inhabits the TSZs serially, in order.

TSZs are delineated by time (water age). Yet because hyporheic water often moves along flowpaths as it ages, we discuss water as moving across space (flowing) between TSZs. As a result, TSZs are delineated spatially by isochronal surfaces—surfaces within the aquifer defined by equal τ (Fig 2). For instance, the isochronal surface representing τ = τ is the spatial boundary that delineates TSZ from TSZ.

Fig 2

Visualization of the non-contiguous nature of TSZs within floodplain aquifers.

Isochronal surfaces (“ribbons” in lower panel) demarcate TSZ’s within floodplain alluvial aquifers. Upper panel shows an aerial photo of the floodplain surface. Visualization was created using simulation results derived from an application of the HydroGeoSphere model [35] to the Meacham Creek (Oregon, USA) floodplain restoration conducted by the Confederated Tribes of the Umatilla Indian Reservation (Byron Amerson, Unpublished). Aerial imagery from the National Agriculture Imagery Program [36].

Fig 1 displays each TSZ as contiguous, yet TSZs can be disaggregated (e.g., red TSZ in Fig 2); all water aged τ < τ ≤ τ is part of TSZ regardless of location within the aquifer.

By visualizing the patterns of water storage within, exchange among, and discharge from such a collection of TSZs arrayed across water ages, we can create quantitative graphical depictions that reveal fundamental relationships among hyporheic water age, water storage, and water flow in hyporheic zones.

Quantifying hyporheic hydraulic geometry

Any TSZ is fully characterised by six metrics: maximum water age (τ), minimum water age (τ), flow rate from the prior TSZ (), hyporheic flow rate to the next TSZ (, discharge to the channel (), and associated water storage (Fig 1). To describe the equations necessary to estimate each of these metrics, we adopt the conventional Chemical Engineering notation [23] ultimately derived from [22]. Specifically, we use three functions: the exit age density function (E(τ)), the washout function (W(τ)), and the internal age density function (I(τ)).

E(τ) is a probability density function (PDF) with dimensions [T−1]. Given τ*, a particular water age of interest, E(τ*) returns the probability density that hyporheic water will return to the channel with a water age equal to τ*. Thus, E(τ) is a PDF representing the age distribution of hyporheic discharge (AD).

As with any probability density function, the area under the curve prescribed by E(τ) must be unity (so that the probability of hyporheic recharge returning to the channel at some time between τ0 and τ is 1.0):

If we chose a function, f(τ) [dimensionless] to represent the desired shape of the AD (and thus the shape of E(τ)), we can create a similarly shaped PDF by dividing values returned by f(τ*) by the integral of (area under) f(τ): (The constraint τ0 ≤ τ* ≤ τ carries through the rest of the equations in this paper. For brevity, we omit the constraints from the remaining equations.) Eq 2 can be restated as: where k [T-1] is the reciprocal integral or “normalizing constant” that converts f(τ) to a PDF.

Any definite integral of E(τ) (i.e., from τ to τ, where τ < τ) yields the probability [dimensionless] that hyporheic water will return to the channel with a water age τ ≤ τ ≤ τ. Considered another way, the finite integral of E(τ) is the fraction of q↓ that will return to the channel with τ ≤ τ ≤ τ: Thus, Eq 4 can be used to calculate (Fig 1) for each TSZ. Remembering that the integral of E(τ) from τ0 to τ is unity, Eq 4 also shows that , as would be expected under steady state flow.

The washout function, W(τ), is the complementary cumulative distribution of E(τ), which is the integral of E(τ) from any τ* to τ: W(τ) describes the probability [dimensionless] that hyporheic water will be discharged to the channel with an age greater than the specified value of τ*. Put another way, W(τ*) determines the fraction of q↓ that remains in the alluvial aquifer at water age τ*. Therefore, q↓ in Fig 1 can be calculated as:

Because W(τ*) returns the fraction of q↓ remaining in the aquifer at water age τ*, W(τ) provides the correct shape for a PDF describing the distribution of hyporheic water age, known as the “internal age density function” (I(τ)). Therefore:

Because I(τ) is a PDF representing the age distribution of hyporheic water in the alluvial aquifer, the integral of I(τ) from τ to τ yields the fraction of s (hyporheic water stored in the alluvial aquifer) having a water age τ ≤ τ ≤ τ.

Alternatively, we can think of water storage as the accumulation of flow over time. Since the integration of W(τ) represents accumulation of fractional remaining flow in the aquifer, we can also calculate the fraction of s having a water age τ ≤ τ ≤ τ as the product of the rate of exchange and the finite integral of W(τ): Thus, Eq 9 can be used to calculate (Fig 1) for each TSZ.

Eqs 4, 6 and 9 provide a means of quantifying our entire conceptual model (Fig 1) for any number (n) of TSZs defined in the hyporheic zone. Because W(τ) is derived from E(τ) and, in turn, E(τ) arises from f(τ), the only requirements to quantify our conceptual model are: 1) a choice of f(τ), which describes the desired shape of the AD; 2) a choice of τ for each desired TSZ (including τ0 and τ); and 3) an estimate of q↓. Because values of τ are chosen to define each TSZ, only q↓ and f(τ) are unknown. Rearranging Eq 9 reveals that q↓ can be estimated by: Thus, by assuming a shape for f(τ), the only value required to describe the hydraulic geometry (Fig 1) of the hyporheic zone is s, the thickness, cross-sectional area, or volume of hyporheic water beneath a stream.

Visualizing hyporheic hydraulic geometry

In some cases, simple field observations provide an estimate of s. For instance, bedrock-confined hyporheic zones common to montane systems often permeate the entire alluvial aquifer [33]. In these systems, values of s can be approximated as the product of various aquifer dimensions (length, width, and/or depth, depending on the dimensions of s) and aquifer porosity. In other systems, values for s may be more difficult to ascertain. Regardless, in the absence of an estimate of s for a particular system, hydraulic geometry for a representative unit (RU) of the aquifer can be visualized.

We define an RU as an idealized hyporheic zone with unit volume (s = 1, [Lx]) which has, by definition, the same AD and AD as any larger hyporheic zone it represents. Usefully, q↓ scales linearly with s (Eq 10) and the hydrologic metrics of any TSZ scale linearly with q↓ (Eqs 4, 6 and 9). Therefore, s is a factor that converts , , or for an RU to the same value for the associated hyporheic zone. Further, ratios among , , and are identical for an RU and the associated hyporheic zone.

Specifying f(τ)

Both a power-law [6, 25, 37] and exponential function [38] are commonly used to represent the AD of hyporheic exchange in streams. In S1 Appendix, we present solutions for E(τ), W(τ), and for both a power-law and exponential representation of an AD. Importantly, either representation yields a heavily tailed distribution where the median age of emerging water is often substantially lower than the mean; i.e., most hyporheic water is discharged back to the channel with an age somewhat younger than the mean age, while a small portion of the water exits with an age far greater than the mean. Thus, the choice of either a power law or exponential function yields a sufficiently skewed distribution for the purpose of conceptualizing and visualizing important aspects of hyporheic hydrology. Below, we illustrate hyporheic hydrology using a power law with a negative exponent to represent the shape of the AD: although an appropriately parameterized exponential distribution would yield substantively similar results given our purpose.

Because such a power law approaches infinity as τ approaches zero, Expression 11 implies infinite hyporheic exchange rates at vanishingly small τ. Application of Expression 11 is therefore eased by the choice of a suitably small minimum water age of interest (τ0) that is greater than zero. Additionally, hyporheic zones are not infinite flow systems. Therefore, we also set a maximum water age of interest (τ) less than infinity. Values of τ0 = 60 s and τ = 3.1536 x 107 s (1 y) provided reasonable values to approximate and visualize the hydraulic geometry of a hyporheic zone similar to systems we study, e.g., the Nyack Floodplain of the Middle Fork Flathead River, Montana, USA [6, 39] and the main-stem Umatilla River, Oregon, USA [6, 33]. Thus, for our purposes here, the definition of “hyporheic water” excludes water that enters the stream bed but returns to the channel with τ < 60 s, as well as water “lost” to any deeper groundwater flow system (i.e., with τ > 1 y). In establishing τ0 and τ, we are not arguing that the excluded water ages have no hydrologic or ecological importance. Rather, we chose values of τ0 and τ that are inclusive of water ages typical of hyporheic water in the expansive alluvial aquifers we study [39]. Graphically, selection of τ0 and τ results in a PDF that describes the shaded region in Fig 3. The function describing the height of the shaded region for any τ is: Thus, Eq 12 provides the basis for E(τ) (Eq 2) in our application.

Fig 3

Graphical representation of a probability density function proportional to a power-law.

The area (shaded) is used to determine a probability density function (PDF) defined by τ0 to τ, assuming the PDF is proportional to a power law.

To visualize the hydraulic geometry of a hyporheic zone, we calculated hyporheic geometry metrics for an RU. Specifically, we subdivided the RU into 50 TSZs, individually denoted as TSZ. Values of τ (maximum water age in each TSZ) were determined such that hyporheic water storage was equal across TSZs (i.e., s = 0.02 mx for each of the 50 TSZs). Note that, by definition, notation referencing characteristics of each TSZ is synonymous with previously used notation; specifically q↓ equates to , q↑ equates to , and s equates to .

We surveyed the literature for empirical observations of α from tracer release experiments, finding that α ranges from approximately 1.3 to 1.9 (Table 1). To consider hyporheic zones across the observed range of α, we calculated values of q↓ and q↑ for each TSZ using four different values of α: α ∈ {1.3, 1.5, 1.7, 1.9}.

Table 1

Some reported values of α based on experimental tracer releases.

Source	α	Stream Name	Description
Haggerty et al., 2002 [25]	1.28	(not reported)1	2nd-order mountain stream
Gooseff et al., 2005 [40]	1.28	WS031	2nd-order stream reach with extensive colluvial and entrained alluvial fill and step-pool morphology
Gooseff et al., 2003 [41]	1.30	WS031	2nd-order, single-thread, tightly spaced pool-step morphology, 12.6% avg. gradient
	1.53	LO4101	4th-order, single-thread alluvial channel with widely-spaced step-pool/step-riffle morphology, 4.84% avg. gradient
	1.58	LO4111	4th-order, braided alluvial channel that terminates at a channel-spanning bedrock outcrop, 4% avg. gradient
Gooseff et al., 2007 [42]	1.87	Headquarters Stream2	agricultural stream flowing through irrigated grazing land with grassy banks
	1.74	Ditch Creek2	natural
	1.89	Two Ocean Creek2	natural
Drummond et al., 2012 [43]	1.35	Säva Stream3	markedly vegetated with emerged and submerged macrophytes, sediment consists mainly of clay, but at the upper part sediment consists of silt, gravel, and detritus particles of differing sizes

1H.J. Andrews Experimental Forest, Oregon, USA

2Jackson Hole, Wyoming, USA

3Uppsala, Uppsala County, Sweden

For each value of α, we plotted cumulative hyporheic discharge across water ages, marking τ for each TSZ on the plot (Fig 4). Thus, the x-axis range between adjacent marks on each curve represents the range of water age associated with individual TSZs. The y-axis range between adjacent marks on each curve represents the the amount of water discharged from each TSZ to the channel.

Fig 4

Cumulative hyporheic discharge to the channel (log scale) by water age (log scale) for a representative unit (RU) of water stored in a hyporheic zone.

Each curve is associated with a different value of α, the negative power-law exponent used to describe the hyporheic water age distribution of hyporheic discharge. Units of hyporheic exchange (y-axis) can be interpreted in terms of length (e.g., m day−1), area (e.g., m2 day−1), or volume (e.g., m3 day−1) depending on whether the RU is one dimensional (e.g., 1 m of hyporheic water thickness), two dimensional (e.g., 1 m2 of hyporheic water cross-sectional area), or three dimensional (e.g., 1 m3 of hyporheic water). Hash marks on each curve demarcate the maximum water age (τ) for each of 50 transient storage zones (TSZs) in the RU; each TSZ contains 2% (0.02 mx) of water stored in the RU.

We also summarize the relationships between τ, s, and q↑ for the hyporheic RU in quantitative depictions associated with each value of α (Fig 5). The depictions use a color map to show the age distribution of hyporheic water stored within an RU of a hyporheic zone. Superimposed on the color map is the volume of water that is discharged from each of the 50 hyporheic TSZs over a period of one hour. Whether for a single TSZ or for the entire hyporheic zone, the ratio of the colored area to the area of the grey overlay represents the ratio of stored hyporheic water to discharged hyporheic water over a period of 1 h. Note that the depicted time-scale of discharge, in this case 1 h, is discernible from the plot; by definition, the grey overlay crosses the circle perimeter at the depicted time-scale.

Fig 5

Quantitative depiction of hyporheic hydraulic geometry for different values of α.

Pie chart represents 50 transient storage zones (TSZs) within a representative unit (RU; 1 mx) of hyporheic water stored within the hyporheic zone. Each TSZ contains 2% (0.02 mx) of the RU’s water storage; color represents the mean water age of each TSZ. The area of superimposed grey wedges is proportional to the water units discharged to the channel from each TSZ in a 1 h period. The 1 h time-scale of depicted discharge can be inferred from plot because, at τ = 1 h, the storage and discharge wedges are equal in area. The nautilus shaped distribution of grey wedges describes the relative water discharge from each TSZ, while the color distribution represents water age across TSZs within the hyporheic zone.

Illustrative applications

A visualization from conservative tracer data

To create a visualization of patterns of exchange in a simple, real-world hyporheic system, we constructed an annular flume by nesting a cylindrical 38 cm diameter food-grade polyethylene tank within a similar 56 cm diameter tank, thus creating a 9 cm wide circular “raceway” approximately 40 cm deep (Fig 6a). We added glass beads (1.2–1.6 mm in diameter) to a depth of 15 cm within the raceway and then graded the beads around the raceway to form a single sinusoidal dune, 20 cm thick on one side of the flume and 10 cm deep on the opposite side. We then filled the raceway with 25 l of water, yielding to a combined water- and glass-bead depth of 30 cm. Flow around the raceway (velocity of ∼10 cm s−1) was induced using by the jet from a submersible impeller-driven aquarium pump.

Fig 6

Visualization of hyporheic water exchange in an annular flume.

(a) Cut-away diagram of the annular flume. (b) Observed (points) and modeled (line) surface water specific conductance. Modeled data derived by fitting values of τ and α to observed data using Eqs 15 and 16. (c) Visualized relationship among water age, hyporheic exchange, and interstitial water storage in the flume. Grey wedges represent aquifer discharge rates for a period of 10 seconds.

We removed 100 ml of water from the flume, added 1g NaCl to the sample, and reintroduced the salt solution to the flume as a slug. We monitored specific conductance in the “channel” (surface) water (K) of the flume until K came into equilibrium with the the specific conductance of “hyporheic” (interstitial) water (K) within the glass beads. The rate a which K reached equilibrium was mediated by hyporheic exchange induced by water flow over the dune. The observed K prior to the addition of the salt slug (K) was 248.4 μS cm−1, the K immediately following the salt slug addition (K0) was 383.6 μS cm−1, and the K at equilibrium (K∞) was 356.9 μS cm−1.

We assumed that specific conductance was proportional to salt concentration; therefore, we treated values of specific conductance as the concentration of a non-reactive solute. Considering conservation of mass, the average value of K and K, weighted by V and V respectively, is at all times after the slug injection equal to K∞. At the time of slug addition, this equality can be represented as: where V is the total volume of water added to the flume (25 l). We calculated the volume of surface water in the flume (V) empirically, by rearranging Eq 13: The volume of hyporheic water (V) is simply V − V. Using Eq 14, V = 20.06 l and V = 4.94 l.

Given estimates of V and V and again considering conservation of mass, the value of K for any elapsed time since the slug release (t) can be determined from the mean K of hyporheic water: Hyporheic geometry—specifically the internal age density function (I(τ))—can be used to estimate the mean value of as surface water mixes with hyporheic water by assuming that hyporheic water maintains the same conservative tracer concentration it had when it entered the hyporheic zone: Note that Eq 16 neglects the effects of dispersion within the hyporheic zone, a simplification that becomes increasingly problematic in natural streams, where hyporheic systems have greater water ages.

We wrote code in the R statistical computing environment [44] to solve Eq 15 and a finite difference approximation of Eq 16 via iteration over time, with a time step representing 1/1000 of the duration between τ0 and τ. We assumed a value of 1 s for τ0 and used the code to fit values of τ and α (nested within I(τ) in Eq 16) to the observed values of K from the annular flume (Fig 6b). The agreement between modeled and observed data suggested a power law was a useful approximation of the shape of the AD of the dune in the flume. Resulting parameter estimates were τ = 4337 s and α = 1.70. Based on τ0 = 1 s and fitted values of τ and α, the relationship between storage, water age, and hyporheic discharge in the flume is shown in Fig 6c.

Scaling hyporheic effects on water temperature

To further illustrate the utility of considering hyporheic geometry, we use E(τ) and I(τ) to consider how variation in α might influence temperature dynamics in water stored within the hyporheic zone vs. in water discharged from the hyporheic zone. For this simple application, we offer the concept of a “representative hyporheic flow path”—conceptually, a flow path that reflects how water temperature typically varies with water age in the aquifer (Fig 7). As water traverses the representative flow path and water age increases, daily and seasonal variation typical of the channel are lagged and damped relative to the stream channel [45, 46]. At the beginning of the flow path, high-frequency diel temperature patterns damp quickly. Farther along the flow path, low-frequency annual temperature signals are damped.

Fig 7

Characteristic temperature damping and lagging with water age in an expansive hyporehic zone based on relationships presented by Helton et al. [47].

Lines represent idealized patterns of temperature along a “representative flow path” through an expansive coarse-grained alluvial aquifer for four different dates (approximate annual maximum, minimum, and mean stream temperatures) as a function of water age. Each line on the daily temperature plot (left) represents the expected pattern of temperature variation for a different hour of the day. Daily temperature variation damps quickly with water age. Seasonal variation in temperature (right) is visible at greater water ages.

We calculated the mean temperature of water stored within the hyporheic zone () as: and the mean temperature of hyporheic discharge as: where T(t, τ) is a function that returns hyporheic water temperature at a specified time of the year (t*) for a specified water age (τ*)—specifically the hyporheic temperature dynamics represented in Fig 7. Because both I(τ) and E(τ) are PDFs, Eqs 17 and 18 simply calculate a weighted average of the temperatures along the flow path, where the PDFs provide the weights. I(τ) represents the AD—the relative amount of hyporheic water storage across water ages. E(τ) represents the AD—the relative amount of hyporheic discharge across water ages. Fig 8 shows the results of a simple model representing Eqs 17 and 18, plotted atop the channel temperature pattern used to drive the model.

Fig 8

Simulated patterns of mean water temperature within the channel, the alluvial aquifer and upwelling from the aquifer.

Simulated patterns of mean temperature for water discharged from the alluvial aquifer (light grey) and stored within an idealized alluvial aquifer (black) plotted with channel water temperature (dark grey; dashed) over time for different values of α.

Discussion

Fig 4 reveals several important aspects of the expected change in hyporheic hydrology associated with variation in α. Remembering that our conceptual model assumes steady state flow—i.e., recharge of the hyporheic zone from the channel (q↓) is equal in magnitude to the sum of hyporheic discharge from all TSZs (q↑)—the terminus of each cumulative distribution in Fig 4 represents the rate of hyporheic exchange (magnitude of q↓ and q↑) associated with each value of α. Thus, Fig 4 shows how each increase in α yields a substantial increase in the rate of hyporheic exchange through the RU; as values of alpha increase from 1.3 to 1.9, total hyporheic exchange per unit hyporheic water storage spans more than 3 orders of magnitude. Additionally, Fig 4 shows that α influences the fraction of the RU from which water of different ages emerges. For instance, remembering that all TSZs were defined to contain 0.02 mx (2%) of the water storage in the RU, Fig 4 shows that the first TSZ (TSZ1) discharges water with an age between 60 s to 2.4 h when α = 1.3. When α = 1.9, water with an age between 60 s to 2.4 h flows through roughly 15 TSZs (∼ 30% of the aquifer).

Our graphical depictions in Fig 5 illustrate this same concept in a different way. The rate of hyporheic exchange per unit volume of hyporheic water storage (proportional to the size of grey wedges in Fig 5) increases markedly as α increases from 1.3 to 1.9. Logically, this pattern must be true. As α increases from 1.3 to 1.9, the water age of hyporheic discharge, AD, skews toward younger water, and thus a hyporheic zone with a younger mean water age. In order for the mean of the AD for an RU to skew younger, flow through the fixed volume must increase. Importantly, however, regardless of the value of α and the associated rate of hyporheic exchange, the bulk of hyporheic exchange is associated with younger water age.

Fig 5 also illustrates that, regardless of the value of α, the AD (τ of water discharged to the channel) is more heavily skewed toward young water ages than the AD (τ of stored hyporheic water). In other words, there is a greater proportion of older hyporheic water stored than the proportion that is discharged. Such a difference in the distribution of τ for the AD vs. the AD has several important implications for ecosystem dynamics in streams. Specifically, the AD is important for understanding how hyporheic discharge influences channel water characteristics. In contrast, the AD of hyporheic water storage governs the relative contribution of hyporheic processes (e.g., productivity, metabolism, nutrient cycling) to the whole stream ecosystem. Therefore, the differences between the AD and the AD suggest that processes occurring in young hyporheic water are likely to drive the influence of hyporheic exchange on surface water. In contrast, processes associated with advanced water age will be important drivers of ecological dynamics occurring within hyporheic zones themselves.

Importantly, the discussion to this point has considered a hyporheic zone with fixed storage—a representative unit of the hyporheic zone. Yet, water storage in a hyporheic zone is not static, but changes with variation in surface discharge [5, 48]. Water storage in hyporheic zones is a function of channel stage, sediment properties (e.g, hydraulic conductivity and porosity) and alluvial aquifer size. Eq 10 highlights the importance of the size of the hyporheic zone in determining the magnitude of q↓. Specifically, for any given value of α, q↓ is directly proportional to hyporheic storage volume.

Our work suggests that research efforts to identify stream characteristics correlated with values of α and s would expand opportunities for rapid characterization of hyporheic hydrology across stream networks [49, 50], especially if such correlates were obtainable from readily available spatial data sets (digital elevation models, LIDAR, aerial photography, etc.). Prior research [1, 18, 37] has shown that streams with self-organized geomorphic patterns at multiple scales (e.g., streambed dunes, well organized pool-riffle sequences, side channels, greater sinuosity—hereafter “geomorphic complexity”) are associated with higher rates of hyporheic exchange, suggesting that geomorphic analysis may be an important starting point in attempts to estimate α and s. For instance, if more complex channels have higher rates of hyporheic exchange, such channels are likely to be associated with larger values of α (assuming s is similar, which is a generous assumption). Similarly, processes or management actions that reduce sinuosity, side channels, and the complexity of bed-forms alter hyporheic water age distributions. Such actions generally reduce hyporheic exchange rates and increase hyporheic water ages, yielding smaller values of α with associated implications for water temperature, nutrient dynamics, and other aspects of hyporheic ecology.

Complicating this picture, of course, is the fact that changes in river stage are likely to yield values of α and s that are dynamic over time within a single stream reach [48]. Thus, the internal distribution of water age is apt to vary with river stage [30], perhaps especially in systems with seasonally inundated side-channels [5]. Although we discuss α and s somewhat independently, the two values may be interdependent. Large hyporheic zones (with higher s) are likely to allow greater opportunity for development of long flow paths, which may be associated with an increase in the frequency of long flow paths and an associated reduction in α. Although logical when considering the equations presented here, our suggestion of a relationship between α and s is speculative. Yet if such relationships exist, the challenge of estimating s and α across basins may be somewhat more tractable than if the two values are relatively independent of one another.

The results from our simple flume experiment (Fig 6) yielded detailed information about the nature of hyporheic exchange within the glass-bead dune in the flume. Using Eq 10 and fitted values of τ and α, we can conclude that the rate of hyporheic exchange within the flume for water age 1 s < τ < 4337 s was approximately 0.23 l s−1. Dividing the 4.93 l of hyporheic water storage by the exchange rate yields ∼21 s as the mean water age of hyporheic discharge. Yet the empirically derived estimate of maximum water age in the system (τ = 4337 s) indicated that some of the flume’s hyporheic water required more than an hour to exchange with surface water. Such discrepancies between mean and maximum water age underscore the tailed shape of the flume’s exit function (E(τ)). In fact, applying E(τ) with fit values of τ and α reveal that only 10% of water entering the hyporheic zone remained in the flume’s hyporheic zone for longer than than 23 s, 40% exited the hyporheic with a water age between 2.6s and 23 s, and fully 50% of hyporheic exchange exited with a water age between 1 and 2.6 s. These water age values may seem surprisingly skewed, yet the fitted value of α (1.70) for the flume is well within the range of observed values from field experiments in natural streams (Table 1).

Our analysis of stream temperature illustrates the importance and benefits of considering hyporheic hydraulic geometry. Field studies reveal that hyporheic zones contain habitats with exceptionally diverse thermal, biogeochemical, and biological conditions [47, 51, 52]. Considering the age distributions of water discharge and storage within the hyporheic zone provides the potential for a quantitatively rigorous but practical mechanism for scaling hyporheic heterogeneity to whole stream networks. Importantly, however, the results in Fig 8 are intended to be illustrative rather than predictive. Our application of Eqs 17 and 18 is oversimplified. First, T(t*, τ) is held constant across values of α when, in fact, changing the rate of hyporheic exchange will alter the patterns of temperature damping and lagging in the aquifer [45]. Further, our application shows patterns of hyporheic temperatures for a fixed stream water temperature regime (plotted in Fig 8); we do not consider the feedback of returning hyporheic water altering stream channel temperature, which in turn would affect hyporheic temperature. Regardless, Eqs 17 and 18 illustrate conceptually how considering the hydraulic geometry of the hyporheic zone (in this case, E(τ) and I(τ)) can be used to scale hyporheic water characteristics from flow-paths to floodplains, given a value for α and any linear or non-linear empirical or mechanistic relationship between τ and a water characteristic of interest (e.g., temperature (Fig 7), nutrient concentrations, microbial community composition) in the hyporheic zone.

Finally, our assumption of a power-law AD [25, 31, 37, 53] underscores a subtle and important fact: the rate of hyporheic exchange reported for any given stream is dependent on the range of water ages considered. In our equations, values of τ0 and τ create distinct although admittedly artificial demarcations between channel-, hyporheic-, and ground-water where no such clear distinction exists. The choice of τ0 > 0, for instance, suggests that water spending less time than τ0 in the streambed is considered to have never left the channel. While use of τ0 in this manner might seem unappealing (interpreted as yielding an underestimate of the “true” hyporheic exchange), the opposite assumption (τ0 ≈ 0) is no less cumbersome. As τ0 becomes vanishingly small, the volume of water that merely contacts the streambed surface for a fraction of a second would constitute a very high if not inflated rate of “hyporheic exchange”. We therefore suggest that—regardless of the assumed AD shape—the minimum duration of interaction with the streambed that constitutes “hyporheic water” may be a non-trivial consideration.

Consider, for instance, the fact that power-law scaling may break down for very low water ages [54]. Such a result might be expected, given the fractal nature of nested hyporheic flow paths [55]. Just as the measured length of any coastline is dependent upon the scale of measurement, the magnitude of any empirical or modeled estimate of q↓ must be dependent upon some inherent minimum time-scale of water age. It follows, then that reported estimates of hyporheic exchange rates in the literature are dependent upon the time-scale of water ages considered or measured, although this is seldom acknowledged and the timescale that applies to a given estimate of hyporheic exchange typically is neither pondered nor reported. For instance, three-dimensional finite element or finite difference floodplain models can simulate hyporheic exchange as driven by stream discharge regime and stream morphology. The choice of cell size or node spacing within such a model sets the lower limit on the length of the flow path (and therefore the lower limit of water age) that can be considered within the model. Thus, any estimate of hyporheic exchange rate from such a model is bounded by the water age associated with the finest spatial scale at which geomorphic features can be represented within the model. Similarly, a seepage meter [56] placed to estimate rates of exchange can not account for exchange that would otherwise occur within the bed area encompassed by the meter. As another example, rates of hyporheic exchange from tracer experiments are typically estimated from the tail of the breakthrough curve; short time-scale hyporheic exchange is largely indistinguishable from in-channel transient storage that occurs on the same time scale.

Thus, we offer caution against the assumption that there is a single, time-scale independent rate of hyporheic exchange within any stream channel. We argue that any reported rate of hyporheic exchange is associated with an implicit range of water age time-scales, likely to range from > 0 to < ∞. In essence, given the tailed distribution of water-age typical of hyporheic zones, any two methods of estimating q↓ that consider two different time-scales of water age will yield two different estimates of q↓, even if applied contemporaneously to the same stream.

While the time-scale dependence of hyporheic exchange may seem somewhat intractable, acknowledging this aspect of hyporheic exchange can be rather straightforward. In empirical studies, we encourage researchers to think carefully about the minimum and maximum length- or time-scales captured by their experiments, and report them. In modeling experiments, the choice of τ0 (for 1-D models) or the density of landscape tessellation (for 3-D models) can be carefully considered in relation to the model’s purpose. As an example, we can build on the water temperature application we presented earlier. Specifically, the magnitude of water temperature change at a hyporheic water age of 60 s will seldom be measurable. Therefore, τ0 = 60 s is an appropriate value for applications concerned with understanding hyporheic water temperature. We believe that future experiments designed to characterize hyporheic hydrology and associated physical, chemical, and biological properties will benefit from more mindful and rigorous consideration of the inter-dependency between hyporheic exchange magnitude and the range of water age timescales inherent in the design of simulation or empirical studies of hyporheic hydrology.

Conclusion

Assuming we accept the assumption that the AD of hyporheic zones is asymmetrical and tailed, the mathematical linkage between s, τ, and q↓ in hyporheic zones is defined by the shape of the AD. In the case of a power-law representation, this shape is determined by the value of α, the exponent of the power law (Eq 11). As α increases, the shape of a power law plot (e.g., Fig 3) becomes more concave and the AD of water exiting the hyporheic zone becomes more skewed toward younger water ages. If the volume of the hyporheic zone is held constant, a skew toward younger water ages requires an increase in the rate of flow through the hyporheic zone (e.g., the rate of hyporheic exchange). Conceptually, then, the inherent inter-relationships among s, τ, and q↓ per unit volume of hyporheic zone yield the patterns shown in Figs 4 and 5. Specifically, when α increases, the rate of hyporheic exchange increases, shifting the AD and AD toward younger water.

The equations in this paper, from which our conclusions are drawn, are presented in a manner intended to be as transparent as possible. While grasping the math improves understanding, the visualizations alone are sufficient to illustrate the following key points:

the AD is not the same as the AD because the AD represents τ at the ends of hyporheic flow paths while AD represents τ along the length of hyporheic flow paths;

the AD and AD are linked mathematically—each can be derived from the other;

the AD (and to a lesser extent, the AD) is heavily skewed toward young water ages and thus the majority of hyporheic exchange traverses brief flow paths;

the AD is important for predicting hyporheic influences on surface water while the AD is useful for characterizing and scaling heterogeneity within the hyporheic zone itself;

with our simplified approach and an assumption of a power-law, one variable (α) is sufficient to characterize the AD and AD while addition of a second variable (s) allows estimates of hyporheic exchange rates; and

τ0 and τ are not arbitrary factors that can cause over- or under-estimates some “true” value of q↓. Rather, they represent the bounds of the water age timescales to which any empirical or modeled estimate of q↓ applies.

By employing the established mathematical notation of chemical engineers, we intend to promote a view of streams and their associated hyporheic zones as “natural bioreactors.” Adopting such a view provides a simple mechanism for scaling field observations of hyporheic metabolism, biogeochemistry, and temperature to whole stream ecosystems. Finally, we hope our characterization of hyporheic hydraulic geometry will help engender broader appreciation for the shape of hyporheic water age distributions and associated implications for the ecology of running waters.

(TEX)

Click here for additional data file.

24 Jun 2021

PONE-D-21-04355

Hyporheic hydraulic geometry

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Reviewer #1: This manuscript analyzes travel-time distributions in the hyporheic zone under the assumptions of steady state and negligible net gain or loss of groundwater. The authors emphasize that the age distribution of the water returning to the stream differs from the age distribution within the hyporheic zone. However, this has already been known. I had really hard times figuring out what's new in the manuscript.

The authors motivate their study by the analysis of expansive coarse-grained alluvial aquifers (for which they introduce the completely unnecessary acronym ECGAA). I doubt that the hydrology of these large gravel bodies is ever at steady state, and I also doubt that neglecting any net groundwater upwelling or permanant loss of streamwater to groundwater is appropriate. However, under the given restrictions, the relationship between travel times (also denoted transit times) and groundwater age can be looked up in Liao & Cirpka (2011, doi: 10.1029/2010WR009927, Appendix A), which is most likely not the first such derivation. Note that a formulation of hyporheic exchange that are based on transit/travel-times have already been proposed by Wörman et al. (2002, doi:10.1029/2001WR000769) if not much earlier, whereas formulations using the age distribution/memory function are at the heart of multi- (or single-)rate mass transfer formulations. Under the given conditions, you get one from the other by taking derivatives or integration. To complete the description of exchange, you either need the volume of the hyporheic zone (relative to the volume of the stream), or the exchange-rate coefficient. Period.

The notation preferred by the authors is confusing. The term residence time distribution is used by other authors as the age distribution. Please read the landmark paper of Botter et al. (2011, doi:10.1029/2011GL047666). This paper also discusses the impacts of transient flow (for catchments rather than the hyporheic zone, but their master equation could of course be adapted). It would be interesting to see how the concept of StorAgeSelection functions (Rinaldo et al., 2015, doi: 10.1002/2015WR017273) could be used to describe systems with extensive hyporheic exchange. But obviously, the authors are not aware of these concepts.

Another odd term used by the authors are "isotemporal surfaces". These guys are known as isochrones, a very established concept in hydrology.

As application, the authors chose the temperature regime of the extended gravel plains. Too bad, heat is the one extensive state variable in shallow groundwater for which I would definitely NOT rely on advective travel times. Conduction into the transverse direction leads to a comparably strong thermal exchange among streamtubes, causing solute travel times to be different from "temperature travel times", and even worse making heat exchange with the land surface a relevant process. The conceptual model of the authors is that the hyporheic zone is thermally isolated from the land surface, that transverse exchange can be neglected, and that the classical Stallman (JGR 1965) solution on the one-dimensional propagation of sinusoidal temperature variations holds. You'd better start with the 3-D heat balance equation with realistic boundary conditions. It is very likely that the diurnal signal of temperature cannot be explained by the travel times of solutes plus the Stallman solution.

Let me finish with my bewilderment on the statement "that integrals and probability density functions are typically presented more thoroughly in engineering curricula than in ecological curricula" (lines 42-43). What type of an argument is that? You can expect basic calculus and probability theory from any quantitative scientist (and I would add linear algebra, vector calculus, and differential equations to the mix). Otherwise it's not science. However, if the authors want to address readers without any math background, their equations won't help either.

Reviewer #2: In this study, Poole et al. use chemical engineering reactor theory to quantify residence time distribution (RTD) and the distribution of water ages for a conceptual, coarse sediment hyporheic zone. Their specific aims for this work are to introduce the theory to fluvial ecologists and resource managers, as well as highlight the relation between various metrics that describe the hyporheic zone and surface water-groundwater interactions. For the latter aim, the authors present various figures that demonstrate the relation between RTD, water age in the hyporheic zone, hyporheic exchange fluxes, and hyporheic zone storage volume for a simple alluvial aquifer parameterized with a power-law RTD. They discuss the assumptions underlying their modeling exercise.

There is no new mathematical theory introduced in the study, as its principal purpose is to broaden the application of canonical reactor theory. By discussing the relation between the various hyporheic zone river properties, the authors largely achieve this purpose, though I believe there are several concepts, clarfications, and theoretical advances (beyond Danckwerts) that would serve the reader if they were included in the discussion. I raise these points in the major comments section, as well as provide more specific comments, in the attached document.

Reviewer #3: The manuscript proposes the application of a series of concepts derived from chemical engineering to describe the main properties of hyporeic exchange in coarse-grained aquifers. Specifically, assuming a power law shape of the residence time distribution (RTD), it is discussed how hyporheic exchange can be summarized by two quantities, namely, the exponent of the power law (alpha) and the size of the hyporheic zone (s).

Even though the concepts expressed in the manuscript are not particularly novel (the fact that hyporheic exchange can be summarized by exchange flux and the RTD is in fact known), the way these concepts are summarized adapting an existing theory could represent a valid contribution. The limit of the manuscript is that the text could be more incisive, as the message is not well conveyed, if not unclear sometimes. More precisely, 1) the novel message should be better specified, 2) the application examples can be refined, and 3) some concepts should be clarified. All these issues are described in detail in the main comments below.

MAIN COMMENTS

1) NOVELTY SPECIFICATION: As I said, the manuscript does not present novel concepts (and this is correctly recognized in the text), so it is fundamental to state the key contribution of the work. In my view, the contribution is to choose a specific type of RTD (i.e., a one-parameter power law distribution) which is considered to be representative of hyporheic exchange, use this RTD to develop expressions for relevant quantities such as fluxes and age distributions (AD) as function of a few parameters (alpha and s), and suggest to employ this framework to classify streams by linking alpha and s to the characteristics of streams and catchments. If this was the intended aim of the manuscript, I recommend it to state more openly because it now gradually emerges from the text, and it is not evident to which extent the use of power law RTD is a mere example or a relevant assumption.

2) APPLICATION EXAMPLES: It is stated (line 359) that "chemical engineers have been applying these equations to bio-reactors (such as sewage treatment plants) to predict whole-system operation for decades [...]. Thus, a view of "streams as bioreactors" may provide a launchpad for potential collaborations between the engineering and ecological disciplines". It would then be extremely valuable to provide example of the applications. However, the applications of the theory that are suggested are not always very informative (e.g., temperature dynamics in fig 6; age of upwelling water in fig 4 works better), and I think that the reader is left out questioning how useful the proposed framework is. If a new approach is proposed, compelling examples should be provided. The same is true for Fig 7: most of the comments are about the limitations stemming from the assumptions, which is fine, but I would also stress the take-home message from the example.

3) CLARIFICATIONS: in a number of points, the text could be more straightforward and point out the aims of the concepts that are introduced, which are not always clear enough. Some examples follow:

- I think that the role of short residence times is overemphasized when it is said that "if we are willing to accept that hyporheic exchange scales according to a power law [...] our application of the equations assuming a power-law RTD reveals [that] the rate of hyporheic exchange in any given stream is dependent on the range of water ages considered" (lines 364-367). Starting from this point, almost a half of the discussion section deals with the implications of the power-law RTD; however, it is particularly the tails of RTDs that have been found to have power-law behavior, while this is not necessarily true for small times. In particular, the fractal behaviour of morphology (which leads to the fractal, nested system of hyporheic flow cells; line 387) should break up at small spatial scale due to physical constraints (e.g., at grain scale for topograpy-driven exchange, or at the Kolmogorov scale for turbulence-driven exchange). Moreover, at small scale different physical processes may prevail (i.e., diffusion rather than advection). What I am implying is that the role attributed to small residence times is likely to stop below some threshold time scale when the RTD may no longer be described by a power law. Because the manuscript is directed also to readers that are not familiar with hyporheic RTD, I think that the message "RTDs are always power-law distributed" could be misleading and the concepts expressed here may derive from the abnormal behaviour of the power-law distributions at small times when they could no longer be a good approximation of actual RT.

- The representative unit (RU) is defined at line 185 as "a conceptual unit of hyporheic water storage (s=1) which has, by definition, the same RTD and AD as the larger hyporheic zone it represents." This is probably not the best definition as it may lead to think that RUs are physical parts of the hyporheic zone (HZ), while it is unlikely that a portion of the hyporheic zone has the same RTD and AD than the total hyporheic zone. Later on it becomes clearer that the RU is introduced only to discuss the properties of HZ regardless of volume, i.e., an idealized HZ with unit volume. While the definition is formally correct, it does not make clear why the concept is introduced, leading to potential confusion for the reader.

- the manuscript "suggests that research efforts to identify stream characteristics correlated with values of alpha and s would expand opportunities for rapid characterization of hyporheic hydrology across stream networks [...], especially if such correlates were obtainable from readily available spatial data sets (digital elevation models, LIDAR, aerial photography, etc.)" (line 331-335). It is worth stressing the temporal variations of streamflow or groundwater flow also affect hyporheic exchange and should be factored when these data sets are built, or otherwise we would improperly attribute observed variations in hyporheic exchange to spatial rather than to temporal drivers. The picture is of course complex, but I think that it is important not to oversimplify it in this point.

- The notation is changed at line 221 by renaming many variables. Is this necessary? If this notation is simpler than the former one, why not using it throughout the manuscript?

- It is stated at line 197 that "For the remainder of this paper, we assume that the hyporheic RTD is proportional to a power law with a negative exponent" -> how critical is this assumption for the manuscript? See main comment 1.

OTHER COMMENTS

7 "expansive coarse-grained alluvial aquifers (ECGAAs)" -> I do not know what an expansive aquifer is. I recommend to explain it, even briefly.

73 it is said that q can represent, among other things, the "rate at which the cross-sectional area of hyporheic water exchanges with the cross-sectional area of the channel". This description is confusing (water flows through the cross section, and it is not straigthforward where it is exchanged betwenn channel and hyporheic zone). I suggest to describe it as the rate of water exchange per unit river length, which I think is a correct description.

122 "exit age density function" -> I think it is more coherent to refer it as residence time (RT) rather than age, since it has been already stated that RT denotes the time when water leaves the aquifer.

144 it would be useful to specify that W(t) is dimensionless, as the dimensions of other quantities have been reported.

153 missing "n" in "functioN"

177 this section is titled "Visualizing hydraulic geometry" and the nex one "Visualizing hyporheic hydraulic geometry". However, they both refer to the aquifer exchanging water with the channel, so the difference between the subject of the section is unclear. If there is no difference, titles should be changed.

181 "s can be approximated as the product of aquifer dimensions (length, width, and depth) and aquifer porosity." -> This is true only if s represent a volume, but as said before it can have different definitions. I suggest to better specify it here.

209 remove parentheses before and after "tau < 60 s".

Fig.3 I am not sure how informative this figure is.

240 "hyporehic" should be "hyporheic". Same at line 340.

250 "residence time" should be "water age"

311 "I(t) is not the same as, but can be derived from E(t)" -> is the other way around also true, as stated at line 445?

337 missing space after "hereafter"

340 "if more complex channels have higher rates of hyporehic exchange, such channels are likely to be associated with larger values of alpha." -> This is true only if the volume s is the same, while comparing river systems of different sizes could lead to very different results. I would rephrase slightly to avoid ambiguities.

355 "Considering multiple transient storage zones within the context of hyporheic hydraulic geometry provides the potential for a quantitatively rigorous but practical mechanism for scaling hyporheic heterogeneity to whole stream networks." -> the use of multiple transient storage zones is essentially a technical way to discretize water ages in finite classes. It is not very different than, e.g., using finite time steps in a particle tracking approach, or a discrete cells in a finite difference methods, and I would present it as such. Moreover, the link on how it can allow for upscaling is not well defined (see also main comment 2).

Reviewer #4: In this work the authors present a quantitative description of hydraulic geometry to visualize the interdependence among hydrologic variables such as age distribution and residence time distribution in the hyporheic zone. This is important because concepts related to AD and RTD are often misunderstood and misinterpreted and this paper will help to clarify and distinguish between these two variables. My own research pertains to hyporheic hydrology and I myself struggle with this distinction.

Overall, I very much enjoyed this paper and the rich development of equations describing the relationships between hyporheic flow, residence time, and storage. This paper is well-written, the theoretical constructs are well-developed and described, and the theme of the paper considers aspects of hyporheic zones that relate to scaling constructs that will be useful for non-modeling readers.

A main theme within this paper is to clarify differences and elucidate interdependencies between AD and the RTD. Lines 308-312 is one example where this distinction is highlighted and it has been noted elsewhere throughout the paper. Despite this prominent theme, after reading the paper, I still did not have a better understanding of how to conceptualize and understand the differences between the two. The authors are entrenched in the theoretical constructs and spend the majoring of the paper on this. Because of this, the conceptual development, and clear articulation of what these metrics actually mean in real life is lost. The temperature example did not help to resolve this issue for me. My suggestion is to resolve this using Figure 1. In Figure 1, you have already done the work showing TSZs in series, so can you provide a simple example of min/max ages for each TSZ within your conceptual model and associated residence times? And show how the pdf of the AD/RTD changes as a water particle moves through the TSZ. A simple example within your conceptual model would help distinguish between the two for the ecological/hydrological audience.

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Reviewer #1: No

Reviewer #2: Yes: Kevin R Roche

Reviewer #3: No

Reviewer #4: No

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Submitted filename: poole_review.pdf

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3 Sep 2021

We have attached a document entitled "Response_To_Reviewers.pdf" which provides in-depth responses to all reviewer comments.

Submitted filename: Response_To_Reviewers.pdf

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27 Oct 2021

PONE-D-21-04355R1Hyporheic hydraulic geometry: Ecological implications of relationships among hyporheic exchange, storage, and water agePLOS ONE

Dear Dr. Fogg,

Thank you for submitting your manuscript to PLOS ONE. After careful consideration, we feel that it has merit but does not fully meet PLOS ONE’s publication criteria as it currently stands. Therefore, we invite you to submit a revised version of the manuscript that addresses the points raised during the review process.

More specifically, I encourage you to take into consideration the concern of reviewer 3 on their second issue: lack of application examples. I wonder if another example could be provided to tackle this issue and make the manuscript less speculative.

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Reviewers' comments:

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Reviewer #3: (No Response)

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Reviewer #3: Partly

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Reviewer #3: (No Response)

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Reviewer #3: Yes

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Reviewer #3: MAIN COMMENTS

In my previous review I identified three issues that were in need of attention: (1) insufficient novely specifications, (2) lack of application examples, and (3) need to clarify some paragraphs. The revisions made by the authors have substantially improved the manuscript in terms of issues (1) and (3): the introduction is now better structured and helps the reader in understanding the aim of the concepts later described in the manuscript, and many passages are now clearer.

In terms of issue (2), I still think that a manuscript that aims to provide a framework to help interpreting natural processes would greatly benefit from an example of application of this framework that is not purely theoretical. At present, the provided examples (fig. 4-7) are obtained from model simulations, and the illustrative example of water temperature is no different. I am not sure if the case study from which fig.2 was derived could provide such an applicative example, but in that case (or with any field data) the potential impact of the manuscript would greatly increase. Otherwise the manuscript is much more speculative, and I leave to the Editor the choice if the manuscript can be considered without this application.

OTHER COMMENTS (line numbers refer to the track-change manuscript)

6 "the top few hundredths of a meter" -> I would say "top centimeters", but this is a matter of taste

46 "rSAS" is firstly introduced here, and the acronym should hence be defined.

95 "a hydrosystems" -> "a hydrosystem"

Fig.6: the last sentence of the caption is missing something ("greater water ages and ."). Remove "and" or complete.

493 "While this problem may seem somewhat intractable" -> I think that my previous comment to this part was not completely clear: I understand the point the authors are raising, i.e., the method of analysis may lead to underestimating the flux because of the chosen scale. What I meant is that a low scale probably exist in most situation, because the RTD is unlikely to assume infinite values for small values of residence times, as the discussion here seems to imply. This is true for different types of RTD (e.g., for both power law and exponential distributions): at small residence time, the increase in probability may break up. I simply suggest to avoid giving the impression that the "true" flux can never be found even if a very fine scale of analysis is employed.

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Reviewer #3: No

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15 Dec 2021

See response to reviewers document attached with submission.

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19 Dec 2021

Hyporheic hydraulic geometry: Conceptualizing relationships among hyporheic exchange, storage, and water age

PONE-D-21-04355R2

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30 Dec 2021

PONE-D-21-04355R2

Hyporheic hydraulic geometry: Conceptualizing relationships among hyporheic exchange, storage, and water age.

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