Seasonal Variations of Hydraulic Exchange Between Surface Water and Groundwater in an Alluvial Plain Setting Using ²²²Rn

Abstract

Understanding dynamic groundwater–surface water interactions in alluvial plains is critical for sustainable water resource management, yet seasonal variability and spatial heterogeneity of these exchanges remain imprecisely quantified. Here, we present an improved ²²²Rn mass balance model for evaluating the seasonal hydraulic exchange between groundwater and the Xintongyang Canal in the Taizhou alluvial plain over the course of a hydrologic year. To reduce the model uncertainty, the “background” ²²²Rn for non-groundwater sources was incorporated into the model to replace the influence of hyporheic exchange. The results indicate that the hydraulic exchange process of surface water and groundwater has significant spatiotemporal differences. Based on the calculations from the ²²²Rn mass balance model, the canal leakage flux follows the order of summer > autumn > winter > spring over the course of a hydrologic year. In contrast, the groundwater discharge flux follows the order of summer > spring > autumn > winter. During a hydrological year, summer demonstrated the most intense water exchange dynamics, with peak fluxes reaching 0.0455 m³/(s·m) for surface water leakage and 0.0013 m³/(s·m) for groundwater discharge, revealing pronounced spatial heterogeneity in dominant exchange processes. ²²²Rn activity in canal and groundwater varies significantly across different regions, with canal leakage being the dominant mode of hydraulic exchange within the study area. The change of the hydraulic exchange process was mainly affected by factors such as rainfall. In the process of promoting surface water leakage, precipitation will also strengthen the supplement of groundwater and contribute to the groundwater discharge in most of the canal sections. This study offers insight into the seasonal variations of groundwater and surface water interaction within an alluvial plain.

1. Introduction

Surface water and groundwater are an integral part of any hydrological system, where rivers and lakes often exchange water and solutes with adjacent aquifers. However, with the impact of climate change and human activities, the interaction between surface water and groundwater has changed. This transformation is particularly pronounced in alluvial plains, where the heterogeneity of aquifers and complex hydraulic exchanges create formidable challenges in quantifying surface water leakage and groundwater discharge fluxes. A comprehensive understanding of the interaction between these hydrological systems is vital to tracing pollutant migration routes and the management of water resources. The knowledge of this interaction promotes the efficient utilization of water resources and the prediction of environmental changes that provides important feedback for land surface hydrology and ecosystem dynamics [1,2].

Most studies of groundwater and surface water interactions range in time scales from hours to months [3,4,5]. Short-term studies on small scales can provide a high spatial and temporal resolution of groundwater and surface water interactions [6] and reveal the groundwater-driven interaction between hydrodynamics and hydrochemical processes [7,8,9,10]. Due to hydrological time lags and sampling time biases, short-term studies have large margins of error in determining groundwater contributions to the total water balance and are limited in promptly responding to unexpected events such as floods and extreme precipitation. Long-term annual-to-decadal observations provide insights into groundwater generation and its contribution to the hydrological system’s water balance [11,12,13]. In contrast, seasonal observations better describe the response of hydraulic exchange processes to changes in seasonal characteristics.

In general, groundwater flow follows Darcy’s law. According to Darcy’s law, groundwater flow is mainly determined by the hydraulic gradient and hydraulic conductivity of an aquifer [14]. Groundwater flow paths are principally governed by hydraulic gradients derived from spatial variations in hydraulic head, which may exhibit indirect correlations with surface topography in specific hydrogeologic settings (e.g., unconfined aquifers with homogeneous permeability). However, this relationship becomes decoupled in systems with anisotropic permeability or structural controls [15]. Due to the heterogeneous anisotropy of natural aquifers, the values of hydraulic conductivity and hydraulic gradient can span several orders of magnitude. The wide range and uncertainty in hydraulic conductivity and hydraulic gradient across different spatial scales make it difficult to estimate the hydraulic exchange flux between groundwater and surface water [16,17].

Because the hydraulic exchange between rivers, lakes, and groundwater is usually discrete and uneven, the detailed exchange processes often are not assessed thoroughly. The extent of interaction between groundwater and surface water depends on several factors, including topography, underlying lithology, the hydraulic properties of aquifers, temporal variations in precipitation, and local groundwater flow patterns [18]. Therefore, understanding the interaction between groundwater and surface water remains a unique challenge [19], especially in alluvial plains with relatively flat terrain, where the influence of topography on the hydraulic exchange is reduced. Moreover, the seasonal variations in precipitation and the fluctuations in rivers and groundwater complicate the hydraulic exchange between rivers and adjacent aquifers.

The hydrologic flow state of rivers on an alluvial plain is controlled by different sources of water, including the inflow of tributaries, runoff, and groundwater discharge. All these sources are ultimately driven by precipitation, topography, and aquifer properties. The water balance in the alluvial plain is highly dynamic due to the complex interactions between groundwater and surface water at different spatial and temporal scales. This problem is faced in the estimation of the interaction between groundwater and the Xintongyang Canal in the Taizhou alluvial plain, China. To better understand hyporheic exchange processes, widely used hydrological tracers such as radon (²²²Rn) and stable isotopes of hydrogen (δ²H) and oxygen (δ¹⁸O) are employed to identify and quantify water exchange fluxes between surface water and groundwater systems. These tracers combine complex and variable groundwater discharge patterns and reflect the average net groundwater discharge at each scale [20,21]. An important advantage of the tracer method is that it can synthesize groundwater tracer signals to provide a more representative estimate of the overall groundwater flow over a large area [22].

Among the many natural tracers used for estimating flux exchange between groundwater and surface water, radon (refer to ²²²Rn) is increasingly adopted [23,24,25,26,27,28,29,30,31,32]. This radioactive isotope (²²²Rn, T_1/2 = 3.82 days) is produced through the decay chain of uranium in aquifers, with an activity value in groundwater about 2–3 orders of magnitude higher than in surface water [33,34]. This activity difference makes it an ideal tracer for estimating water exchange between groundwater and surface water. The conservative behavior of ²²²Rn in the hydrologic system gives it an advantage over other geochemical groundwater tracers, such as anions, cations, methane (CH₄), and stable isotopes, which can be affected by evaporation, condensation, and biological and chemical transformations [35,36]. The relatively short half-life of ²²²Rn provides potential in the study of rapid mixing processes that occur between the groundwater and surface on time scales of hours to several days [37], especially during a short response to rainfall events. This potential can help in using detailed and relevant datasets for hydrological water budget modeling.

To clarify the complex hydraulic exchange methods between rivers containing tributaries and groundwater in the alluvial plain, here, we quantify the contribution of groundwater discharge and surface water leakage to the long-term water budget of the Xintongyang Canal in the Taizhou alluvial plain by monitoring ²²²Rn for one hydrologic year. The data are used to evaluate the seasonal hydrological exchange between the canal and the adjacent aquifer in the alluvial plain and to specify in which season the intense hydraulic exchange is occurring. Notably, the Xintongyang Canal functions as a third-class navigable waterway rather than an irrigation channel, exhibiting fundamental divergences in hydrological design principles and operational protocols compared to agricultural water delivery systems. The results will provide a reasonable basis for the regional water resource assessment.

2. Data

2.1. Site Description

The 21.7 km long Xintongyang Canal crosses the Taizhou alluvial plain (32°27′–32°34′ N and 119°48′–119°59′ E) in Jiangsu Province (Figure 1). The canal’s width and depth vary, at 80~190 m and 3.2~8.2 m, respectively. The canal, constructed in the 1950s with brick-lined sidewalls and a channel bed composed of sandy-clay soils, exhibits severe structural deterioration in its revetments due to six decades of hydrological aging and environmental stress. The canal is located at the confluence of the Yangtze and the Huai River systems and slopes from the southwest to the northeast, with an altitude of 5.5 to 7 m below sea level. The canal in the study area consists of five tributaries, of which three are inflow tributaries (T1, T2, and T4) and two are outflow tributaries (T3 and T5) (Figure 1). The flow of the tributaries is small, with an average velocity of 0.12 m/s. The directions of the tributaries are marked by arrows in Figure 1. A dense river network, which has complex interactions between groundwater and surface water, transects the alluvial plain area. There are few hydrologic studies related to the canal, and there is no long sequence flow dataset. Data on hydraulic gradient and hydraulic conductivity of aquifers in the area are also sparse. The alluvial plain is composed of Quaternary clastic materials (dominated by sand and mud) overlying weathered bedrocks. These aquifers control the hydrogeological conditions.

The study area’s hydrogeological architecture is controlled by deltaic facies associations, comprising interbedded sands and clays that form a multi-aquifer system with anisotropic permeability (Figure 2). The groundwater hydraulic gradient is typically within the range of 1 × 10⁻⁴–3.9 × 10⁻⁴ in similar deltaic aquifers, and the aquifer permeability is relatively low (e.g., hydraulic conductivity K ≈ 10⁻⁷–10⁻⁵ m/s for silt–clay mixtures), such that the groundwater flow is slow [38,39]. The thickness of the unconfined aquifer ranges from 20 to 50 m, and the buried depth of the groundwater is generally between 0.5 and 2.0 m from the ground surface. The bottom of the canal is connected with the adjacent unconfined aquifer and has a close hydraulic exchange. Therefore, this study focuses on the discussion of hydrology exchange between the canal and adjacent unconfined aquifer.

The climate of the study area belongs to the north subtropical humid monsoon climate zone, with four distinct seasons and abundant precipitation. The average annual temperature in the study area ranged from 13.9 °C to 15.7 °C, and the average precipitation and evaporation were 1027.6 mm and 848.53 mm, respectively. Southeasterly winds prevail throughout the year, with a perennial average wind speed of 3.4 m/s. Monthly precipitation in the study area during the sampling period is shown in Figure 3. In general, there is a peak of rainfall from May to September, with the maximum rainfall in July being 241 mm.

2.2. Sampling Methodology

Field measurements and water sampling campaigns along the canal and its tributaries (Figure 1) were conducted on four dates in 2022 (13 January, 31 March, 20 July, and 7 September) to capture seasonal variations in hydrological characteristics. Nineteen surface water samples and six groundwater samples were collected in each season (

Table 1. Data on ²²²Rn activity in the canal and groundwater measured by several surveys. ±σ refers to the standard deviation of the ²²²Rn activity.

		222Rn (Bq/m3)	±σ	222Rn (Bq/m3)	±σ	222Rn (Bq/m3)	±σ	222Rn (Bq/m3)	±σ
	Seasons	Autumn		Winter		Spring		Summer
Location
R1		2039	271	2157	352	2255	340	745	216
R2		430	75	2521	740	2118	240	2366	270
R3		1287	213	2049	226	3435	416	1381	315
R4		1755	376	1833	440	2250	540	681	240
R5		1696	256	1386	237	3384	492	265.1	48
R6		415	75	1484	410	2574	360	1731	410
R7		429	114	1741	537	1714	430	1837	282
R8		1246	166	291	102	1660	402	2166	402
R9		2140	393	898	313	3265	413	2626	413
R10		2425	693	2013	440	919	240	1536	440
R11		1399	372	1725	196	847	120	511	144
R12		2399	792	1725	340	847	310	511	190
R13		1740	536	2356	438	2029	556	1385	198
R14		429	147	1890	295	2162	512	307	86
R15		862	278	875	169	2593	769	586	169
R16		1282	221	1700	455	3350	676	2789	327
R17		418	102	2912	540	2211	540	286	90
R18		874	246	1430	238	2368	403	1898	210
R19		847	263	2335	610	2982	610	260	90
G1		36,155	2005	14,530	1241	34,825	2480	12,789	2239
G2		31,684	3398	21,207	2275	45,985	2341	16,827	1766
G3		27,755	1969	15,202	1240	37,620	2430	12,983	949
G4		30,447	2456	20,683	1422	45,339	2426	11,366	966
G5		29,122	2609	19,892	1576	45,794	2554	15,632	1711
G6		29,651	1592	17,512	669	38,912	2701	19,092	1488

). Due to the complicated field conditions, the surface water sampling interval is about 2 to 3 km. In order to reduce the randomness and uncertainty of sampling, we conducted sampling experiments at multiple sites and selected some representative sampling points for calculation and discussion. At each sampling site, water samples were collected through a sampler at about 50 cm below the water surface, near the center of the canal cross-section (denoted as R samples in Figure 1). Here we assumed that all sources of ²²²Rn can be mixed instantaneously, and the ²²²Rn activity in the canal does not change over a short timescale, which means it reaches a stable state. Water samples collected at any point in the canal can represent the average of the section [36]. A similar procedure was used for the tributary water samples. Groundwater was sampled from wells in the unconfined aquifer (denoted as G samples in Figure 1). Groundwater sampling points are located within 2 km of the canal. All monitoring wells are fully penetrating, with screen tops positioned ≥1.5 m below the mean water table and screen lengths covering 75% of the aquifer thickness. Before sampling, about three times the well volume of groundwater was pumped by a submersible pump to ensure that all of the “old” groundwater in the well was removed. Therefore, the sampled groundwater from wells was actually “fresh” groundwater and hence representative for the aquifer. The groundwater samples were stored in 1000 mL PVC bottles, which were overflowed and tightened quickly to prevent contact with the air after sampling. These groundwater samples from unconfined aquifer wells nearby canal and tributaries were used to calculate the mass balance model.

The quantification of dissolved ²²²Rn in aqueous samples was performed using a RAD7 portable radon detector equipped with the RAD H₂O accessory (Durridge Company, Billerica, MA, USA), which facilitates efficient radon extraction via a closed-loop aeration system. The core function of the system is to transfer radon from the water phase to the gas phase by dynamic aeration, which is then detected by a semiconductor alpha spectrometer. The detection range of RAD7 is 3.7–750,000 Bq/m³ (0.1–20,000 pCi/L) with a relative uncertainty of ±0.5 cpm, and its sensitivity is comparable to or better than that of the liquid scintillation method. All ²²²Rn activities reported herein represent water-phase activities expressed in Bq/m³, unless otherwise specified. Since the half-life of ²²²Rn is only 3.8 days, most samples are analyzed within a few hours after collection to reach an acceptable activity value. The mean delay time between sampling and measurement was about 6~12 h. Decay correction factor (DCF) was used to cover the period between sampling and measurement and is given by the formula DCF = EXP (T/132.4), where T is the decay time in hours [40].

During the sampling process, the longitude and latitude of the sampling points were recorded through the handheld GPS (eTrex 20, Garmin International Inc., Olathe, KS, USA). Parameters such as conductivity (EC), pH value, and water temperature were measured in situ (PHH-7200, OMEGA Engineering Inc., Norwalk, CT, USA). Data such as wind speed, temperature and relative humidity (RH) were collected on-site by the OMEGA multi-functional environmental meter (RH87, OMEGA Engineering Inc., Norwalk, CT, USA). The water flow velocity was measured by ADCP (Teledyne RD Instruments, Poway, CA, USA). The water depth of the canal was measured using a hydrological echo sounder (Syqwest Inc., North Falmouth, MA, USA). The geometric dimensions of each sampled section were estimated using remote sensing images from Landsat (http://www.gscloud.cn/, accessed on 13 January 2022). Other details of the canal (slope ratio and bottom width) were obtained from the government’s water department. The dataset on rainfall in the study area was obtained from the website of the National Meteorological Science Data Center (http://data.cma.cn/site/index.html, accessed on 19 March 2023).

3. Modelling Methods

The flux of hydraulic exchange between groundwater and surface water was calculated by an improved ²²²Rn mass balance method. It is a mass balance model for radon in a river section, which takes into account all sink and source terms of ²²²Rn. A certain section of the canal is taken as the case object (Figure 4) to establish a ²²²Rn mass balance model with all source and sink terms required to be determined. ²²²Rn sources consist of several parts: (1) ²²²Rn activity from upstream inflow; (2) ²²²Rn activity due to in situ production from dissolved radium (here refer to ²²⁶Ra) and diffusion from sediment of the canal bed; (3) ²²²Rn activity derived from potential groundwater discharge; (4) ²²²Rn activity from hyporheic exchange of the canal bed sediments; and (5) ²²²Rn activity from inflow tributaries. ²²²Rn sinks include several pathways: (1) loss of ²²²Rn activity due to downstream outflow; (2) loss of ²²²Rn activity by atmospheric evasion; (3) loss of ²²²Rn activity from potential canal leakage; (4) ²²²Rn decay; and (5) loss of ²²²Rn activity due to outflow tributaries.

The detection of abnormally high ²²²Rn levels in the canal water is usually considered as a sign of groundwater discharge into the canal [41]. Due to the rapid mixing of ²²²Rn in water bodies, we consider that discharge of groundwater is rapidly mixed with the canal water, which increases the ²²²Rn activity in the canal. The vertical stratification and cross-section distribution of ²²²Rn activity are ignored here. The measured value of the collected water samples represents the ²²²Rn activity of the river section. It is assumed that the ²²²Rn inventory in river sections is a specific value without considering the influence of rainfall events. The sources and sinks of the section reach a stable state.

According to the mass conservation assumption of ²²²Rn in the canal, a ²²²Rn mass balance model in a one-dimensional stable state can be established to calculate the interaction between groundwater and canal water [34,42,43]. However, the model constructed by Su et al. [42] does not consider the impact of tributaries on the hydraulic exchange, especially when multiple tributaries cross each other. Yi et al. [43] analyzed the impact of tributaries on rivers; however, the model is only applicable to the situation of groundwater discharge into rivers. Moreover, Yi’s model does not consider the influence of hyporheic exchange, which may cause a large groundwater discharge or a small canal leakage estimate. In contrast to that model, other models do incorporate the hyporheic exchange [31].

Hyporheic exchange occurs between the water flow and the sediments in the canal. The area where hyporheic exchange occurs is called the hyporheic zone. The hyporheic zone will release a lot of ²²²Rn activity with canal water passing through; this process increases ²²²Rn activity in the canal water but does not affect the flux [36]. It is rather difficult to accurately estimate the effects of hyporheic exchange [42]. However, the relevant experiments about hyporheic exchange in this study were not completed, and thus we will use the “background” ²²²Rn activity to accommodate the influence of the hyporheic exchange [44,45].

In the absence of groundwater discharge, ²²²Rn activity in the canal water is generally close to a small value. Non-groundwater sources of ²²²Rn include sediment diffusion, ²²⁶Ra decay, and hyporheic exchange in the ²²²Rn mass balance model. Therefore, the lowest value of ²²²Rn activity in a section of a tributary or the canal in different seasons can be used as a “background” indicator and is expected to represent ²²²Rn from sources other than groundwater. Here, we assume that the difference of contribution from riverbed to overlying water on the spatial scale is ignored, and the “background” activity of ²²²Rn (C_b) in the canal to be constant in different seasons. In the absence of groundwater discharge, the ²²²Rn activity in surface water reaches its minimum value, corresponding to the background level. Therefore, our model calculations subtract this “background” activity from measured values to quantify both canal leakage and groundwater discharge.

Considering the influence of high-flow tributaries on the canal, we have added the effects of tributaries to the ²²²Rn balance model as follows:

When ${Q_u+{\sum }_{i=1}^n{Q}_{I_i}<Q}_d+{\sum }_{j=1}^n{Q}_{O_j}, C_u<C_d,$ assuming that the major type of interaction is groundwater discharge, then the mass balance model of ²²²Rn can be approximated as

$$Q_d(C_d-C_b)=Q_u{(C}_u-C_b)e^{\left(-\alpha L\right)}+{\int }_0^L{q}_g{C}_g{e}^{\left(-\alpha x\right)}dx+∆T$$

(1)

where $q_g$ represents the calculated average flux per unit length of groundwater discharge to the canal [m³/(s·m)]. $Q_u$ and $Q_d$ stand for the water fluxes at the upstream and downstream sampling sites, respectively [m³/s]; $C_u$, $C_d$, and $C_g$ represent the ²²²Rn activity measured at the upstream and downstream sampling sites and groundwater, respectively [Bq/m³]; $C_b$ denotes the “background” activity, defined as the equilibrium ²²²Rn activity without groundwater discharge [Bq/m³]; and α refers to the total loss coefficient [m⁻¹]. L stands for the distance between upstream and downstream sampling sites [m]; x represents the distance from the site where exchange occurs at the downstream sampling site, which is a continuous variable [m]; $∆T$ refers to the variation of ²²²Rn activity in the canal that is caused by all tributaries’ input between the upstream and downstream [Bq/s]; $Q_{I_i}$ and $Q_{Oj}$ stand for the water fluxes at the No.i inflow and No.j outflow tributary, respectively [m³/s]; $C_{I_i}$ is the ²²²Rn activity at the inflow tributary [Bq/m³].

When ${Q_u+{\sum }_{i=1}^n{Q}_{I_i}>Q}_d+{\sum }_{j=1}^n{Q}_{O_j}, C_u>C_d$, it is regarded as a canal leakage, then the mass balance model of ²²²Rn can be approximated as

$$Q_d(C_d-C_b)=Q_u{(C}_u-C_b)e^{\left(-\alpha L\right)}-\underset{0}{\overset{L}{\int }}{q}_r{(C}_u-C_b)e^{\left(-\alpha \left(L-x\right)\right)}dx+∆T$$

(2)

where $q_r$ represents the calculated average flux per unit length of the canal leakage [m³/(s·m)].

When ${Q_u+{\sum }_{i=1}^n{Q}_{I_i}>Q}_d+{\sum }_{j=1}^n{Q}_{O_j},C_u<C_d \mathrm{o}\mathrm{r} {Q_u+{\sum }_{i=1}^n{Q}_{I_i}<Q}_d+{\sum }_{j=1}^n{Q}_{O_j},C_u>C_d$, the interaction between the canal and the groundwater is complex, where canal leakage and groundwater discharge occur simultaneously, leading to a mass balance model of ²²²Rn approximation as

$$\begin{array}{cc}{Q}_d(C_d-C_b)=& Q_u{(C}_u-C_b)e^{\left(-\alpha L\right)}+\underset{0}{\overset{L}{\int }}{q}_g{C}_g{e}^{\left(-\alpha x\right)}dx\hfill \\ & -\underset{0}{\overset{L}{\int }}{q}_r{\displaystyle \frac{{C_u+C}_d-2C_b}{2}}{e}^{\left(-\alpha \left(L-x\right)\right)}dx+∆T\hfill \end{array}$$

(3)

Because Equation (3) is difficult to solve, it is necessary to combine the water balance model as follows:

$$Q_d=Q_u+{(q}_g-q_r)L$$

(4)

In Equations (1)–(3), ∆T refers to the effect of all the confluence of inflow and outflow tributaries on the canal, such that

$$∆T=\sum _{i=1}^n{Q}_{I_i}{C}_{I_i}{e}^{\left(-\alpha x_i\right)}-\sum _{j=1}^n{Q}_{O_j}{C}_u{e}^{\left(-\alpha \left(L-x_j\right)\right)}$$

(5)

where $x_i$ and $x_j$ refer to the distance between the confluence of the No.i inflow and No.j outflow tributary and the downstream sampling site [m]. For instance, sampling points such as R14 and R18 of tributaries are directly involved in the calculation, and R15 and R19 of tributaries are for reference.

The left side of Equations (1)–(3) represents the ²²²Rn activity detected downstream without groundwater discharge. The right-hand terms in Equations (1)–(3) characterize the ²²²Rn activity components as follows: the first term represents the upstream-originated ²²²Rn activity unaffected by groundwater discharge; the second term in Equations (1) and (2) corresponds to groundwater discharge into tributaries and lateral leakage loss, respectively, while in Equation (3) the second and third terms describe groundwater discharge and canal leakage loss; the final term (ΔT, common to all equations) quantifies tributary inflow/outflow effects on canal ²²²Rn activity.

As the conceptual model of the actual case of the canal and groundwater interaction is visualized, several hypotheses are considered in the model, including the following: (1) the change of water flux and ²²²Rn activity in the canal were used to examine the change of source and sink; (2) sampling days with precipitation were avoided, which would have increased the difficulty of sampling and the uncertainty of ²²²Rn activity in water; and (3) due to the short sampling period, the effect of evaporation on the system was ignored.

Equations (1)–(3) correspond to three different types of surface water and groundwater interaction, respectively. When the solute (²²²Rn) is characterized by decay and degassing, the total solute loss due to radioactive decay and degassing needs to be considered [46]. The representation of this assumption in the ²²²Rn mass balance model is shown by Equation (6), where α refers to the total ²²²Rn loss coefficient, including radioactive decay and degassing:

$$\alpha =\beta +\gamma$$

(6)

whereas β refers to the decay coefficient related to the average velocity of the canal (m⁻¹), and γ represents the loss coefficient caused by degassing (m⁻¹). β can be calculated as follows [42]:

$$\beta ={\displaystyle \frac{{\lambda }_{Rn}}{v}}$$

(7)

The decay of ²²²Rn per unit time is related to the average flow velocity because the actual velocity is variable. Therefore, the upstream and downstream average velocity is generally taken to be a constant in the canal. ${\lambda }_{Rn}$ is the radioactive decay coefficient of ²²²Rn,${\lambda }_{Rn}=2.08\times {10}^{-6} {\mathrm{s}}^{-1}$; v is the average flow velocity (m/s) of the canal, which determines the time from upstream to downstream.

Many models can be used to evaluate the ²²²Rn exchange between air and canal water due to degassing [47,48]. We consider here the surface renewal model to calculate the loss of ²²²Rn activity caused by the turbulent gas exchange in the canal. The degassing loss coefficient can be expressed as follows [49]:

$$\gamma ={\displaystyle \frac{D^{0.5}}{v^{0.5}{h}^{1.5}}}$$

(8)

$$-logD=\left({\displaystyle \frac{980}{T_{air}+273.15}}\right)+1.59$$

(9)

where D means the molecular diffusion coefficient for gas (m²/s); v is the average flow velocity (m/s); h refers to the average depth of the canal (m); and $T_{air}$ represents the air temperature (°C).

Substituting Equation (4) into Equations (1)–(3) and the integral method were used to solve the problem, respectively. The following analytical solutions can be obtained:

$$\begin{gathered} q_g={\displaystyle \frac{Q_d(C_d-C_b)-Q_u{(C}_u-C_b)e^{\left(-\alpha L\right)}-{\sum }_{i=1}^n{Q}_{I_i}{C}_{I_i}{e}^{\left(-\alpha x_i\right)}+{\sum }_{j=1}^n{Q}_{O_j}{(C}_u-C_b)e^{\left(-\alpha \left(L-x_j\right)\right)}}{C_g}} \\ ·{\displaystyle \frac{\alpha }{1-e^{\left(-\alpha L\right)}}} \end{gathered}$$

(10)

$$\begin{gathered} q_r={\displaystyle \frac{Q_d(C_d-C_b)-Q_u{(C}_u-C_b)e^{\left(-\alpha L\right)}-{\sum }_{i=1}^n{Q}_{I_i}{C}_{I_i}{e}^{\left(-\alpha x_i\right)}+{\sum }_{j=1}^n{Q}_{O_j}{(C}_u-C_b)e^{\left(-\alpha \left(L-x_j\right)\right)}}{C_u-C_b}} \\ ·{\displaystyle \frac{\alpha }{e^{\left(-\alpha L\right)}-1}} \end{gathered}$$

(11)

$$\left\{\begin{array}{c}{q}_g={\displaystyle \frac{2\alpha (Q_d(C_d-C_b)-Q_u{(C}_u-C_b)e^{\left(-\alpha L\right)}-{\sum }_{i=1}^n{Q}_{I_i}{C}_{I_i}{e}^{\left(-\alpha x_i\right)}+{\sum }_{j=1}^n{Q}_{O_j}{(C}_u-C_b)e^{\left(-\alpha \left(L-x_j\right)\right)})}{(1-e^{\left(-\alpha L\right)})(2C_g-C_u-C_d+{2C}_b)}}-{\displaystyle \frac{\left(C_u+C_d{-2C}_b\right)\left(Q_d-Q_u\right)}{L\left(2C_g-C_u-C_d+{2C}_b\right)}}\\ q_r={\displaystyle \frac{2\alpha (Q_d(C_d-C_b)-Q_u{(C}_u-C_b)e^{\left(-\alpha L\right)}-{\sum }_{i=1}^n{Q}_{I_i}{C}_{I_i}{e}^{\left(-\alpha x_i\right)}+{\sum }_{j=1}^n{Q}_{O_j}{(C}_u-C_b)e^{\left(-\alpha \left(L-x_j\right)\right)})}{(1-e^{\left(-\alpha L\right)})(2C_g-C_u-C_d+{2C}_b)}}-{\displaystyle \frac{2C_g(Q_d-Q_u)}{L(2C_g-C_u-C_d+{2C}_b)}}\end{array}\right.$$

(12)

To cross-validate the ²²²Rn mass balance model, a flow balance analysis was implemented to quantify river–groundwater interactions. The net water flux (Equation (13)) determines the magnitude of exchange, with its sign (+/−) indicating the direction of water flux (groundwater discharge to the river or recharge to the aquifer).

$$q={\displaystyle \frac{Q_u-Q_d+{\sum }_{i=1}^n{Q}_{I_i}-{\sum }_{j=1}^n{Q}_{O_j}}{L}}$$

(13)

4. Results and Discussion

4.1. ²²²Rn Activity in the Canal and Groundwater

The ²²²Rn activity data of the canal and groundwater measured by multiple sampling are shown in Table 1. The four sampling times correspond to autumn, winter, spring, and summer, respectively. The range of ²²²Rn activity measured in the canal during the four sampling periods was 415~2425 Bq/m³, 291~2912 Bq/m³, 847~3435 Bq/m³, and 260~2789 Bq/m³, respectively. The boxplot of ²²²Rn activity of canal water and groundwater in the study area is shown in Figure 5. This reflects the dispersion degree of surface water and groundwater in the different sampling periods. ²²²Rn activity of the canal reached the highest value in spring, which was 3435 Bq/m³. The water ²²²Rn activity was generally lower in summer and autumn; this is attributed to dilution by precipitation and degassing by high temperature [48].

According to the source and sink terms in the radon balance model, the ²²²Rn activity in surface water is mainly affected by factors such as precipitation and groundwater discharge. Seasonal analysis reveals surface water ²²²Rn activities follow spring > winter > autumn > summer. The monthly precipitation variation within the year shows the greatest in summer and the least in winter. Therefore, the abundance of water in summer and autumn dilutes the radon in canal water, resulting in a low ²²²Rn activity. Notably, early September sampling coincided with the transitional period between summer and autumn, resulting in no significant difference in ²²²Rn activity between these seasons. In contrast, spring exhibits markedly higher ²²²Rn activities compared to autumn despite reduced rainfall, demonstrating that groundwater discharge contributions dominate over precipitation-driven dilution during this season.

The ²²²Rn activity of groundwater within one hydrologic year indicates a variation range from 11,366 Bq/m³ to 45,985 Bq/m³ (Table 1), which is 1–2 orders of magnitude higher than that of surface water. The reasons for the low ²²²Rn activity of groundwater in summer may come from two aspects. On the one hand, rainfall in summer accounts for 48.95% of the annual rainfall, and sufficient rainfall diluted ²²²Rn activity in the canal water and groundwater. On the other hand, the frequent exploitation of groundwater by residents in summer accelerates the regeneration rate and shortens the water–rock interaction time, which is also one of the reasons for the low ²²²Rn activity in groundwater in summer. Groundwater is the carrier of ²²²Rn in aquifers. The ²²²Rn activity of groundwater is lower in winter, which is primarily attributed to dilution from canal water recharge, with a secondary contribution from colder temperatures enhancing radon solubility and suppressing its degassing from water to air. In spring and autumn, as the recharge of surface water to groundwater is weakened, ²²²Rn activity is relatively high.

The dynamic variation of ²²²Rn activity in the canal water arises from continuous changes of local hydrological sources and sinks along the canal section, reflecting the hydraulic exchange process between surface water and adjacent aquifers. Therefore, defining the variation of ²²²Rn activity in the canal can indirectly reflect the change of sources and sinks. In the calculation of the ²²²Rn mass balance model, the whole canal is divided into four sections according to the distribution of sampling points.

The length of the first section (from the sampling points R1 to R3) is about 4650 m. The first section is connected with a T1, which is an inflow tributary. The second section (from sampling points R3 to R5) is 4510 m in length and contains two tributaries, T2 and T3. Among them, tributary T2 is the source of this section, and tributary T3 is the sink. The third section (from sampling points R5 to R7) with a length of 4890 m contains two tributaries, T4 and T5. Among them, T4 is the source of this section, and T5 is the sink. The length of the last section (from sampling points R7 to R9) is 3520 m.

The variation of ²²²Rn activity along the canal is shown in Figure 6. It can be seen from Figure 6 that ²²²Rn activity along the canal varies greatly in different seasons, indicating that there are frequent interactions between groundwater and canal water. The results of four seasons show that the ²²²Rn activity of the canal water in spring was higher than that in other seasons. The highest ²²²Rn activity (3335 Bq/m³) occurs at the sampling site R3, and the lowest (260 Bq/m³) was in summer and also at sampling site R5.

The variation of ²²²Rn activity along the canal in different seasons suggests that hydraulic exchange between groundwater and canal water may vary seasonally, which will be quantitatively calculated later. Besides, part of the variation in ²²²Rn activity may also be related to changes in weather conditions (air temperature and wind speed) of the different seasons [42,48]. High temperatures and high wind speeds in summer and autumn increase the degassing of ²²²Rn in the canal, reducing ²²²Rn activity. This feature is shown by the relatively consistent ²²²Rn activity in autumn and spring. The higher ²²²Rn activity in spring suggests discharge of groundwater to the canal. However, the higher water temperature in September promoted the active release of ²²²Rn from the water surface and thereby reduced activity in the autumn season. The variation of ²²²Rn activity along the canal in January indicates a rather stable range. The low temperature had likely reduced groundwater inflow and gas releases from the water surface.

4.2. The Effect of Hyporheic Exchange

In the ²²²Rn mass balance model calculation, the hyporheic exchange is a factor affecting ²²²Rn activity but is not an easily measured parameter. ²²²Rn activity of the surface water is expected to be close to the minimum value when there is an insignificant groundwater effect. Therefore, we consider the lowest activity of the discrete ²²²Rn measurements to represent the isotope activity of the surface water without a groundwater source. This parameter is also referred to as the “background” activity of ²²²Rn and is considered to reflect the hyporheic exchange to the canal.

The sources of the “background” ²²²Rn activity include sediment diffusion, ²²⁶Ra decay, and hyporheic exchange. A “sediment equilibrated” test was performed to analyze the effect of ²²⁶Ra decay in the canal bed sediments (sediment diffusion) [50]. A quantity of 200 g of canal bed sediment and 300 mL of canal water were sealed in a 500 mL conical flask. The conical flask was placed on a cyclotron vibrator and cultured continuously for 4 weeks. Both ²²⁶Ra and ²²²Rn in pore water and overlying water reached radioactivity equilibrium in 4 weeks. The contribution of sediment diffusion can be obtained by measuring the ²²²Rn activity in the overlying water. Since the equilibrium value of ²²²Rn measured in the experiment was low (46 Bq/m³), the effect of sediment diffusion on the canal water was negligible. Due to the low activity of ²²⁶Ra dissolved in the canal, the effects of sediment diffusion and ²²⁶Ra decay on the canal water were negligible. Thus, we consider the hyporheic exchange flux is roughly equal to the “background” activity of ²²²Rn in the canal. It represents the contribution of the riverbed to the overlying water.

The “background” activity of about 260 Bq/m³ was obtained from the field experimental dataset. This value of “background” ²²²Rn activity is considered constant in the whole canal and at different seasons. The same assumption has also been applied in other studies that have used minimum values as a “background” indicator of ²²²Rn activity [44,45].

4.3. Estimation of Surface Water–Groundwater Interaction

In the studied sections of the Xintongyang Canal, both flow balance and ²²²Rn mass balance methods were applied to assess the hydraulic exchange between groundwater and surface water. Canal/tributary parameters used in the calculations are presented in

Table 2. ²²²Rn activity and other parameters of the sampling sites of the canal and tributaries. t₁ (autumn), t₂ (winter), t₃ (spring), and t₄ (summer) represent the four sampling seasons. ±σ refers to the standard deviation of the ²²²Rn activity.

No.	Date	Distance	Width	Depth	T	Velocity	Flux	222Rn	±σ
		(m)	(m)	(m)	(°C)	(m/s)	(m3/s)	(Bq/m3)
R1	t1	0	100	4.06	28.6	0.05	20.30	2039	271
R3	t1	4650	120	6.24	28.8	0.08	59.90	1287	213
R5	t1	9160	115	5.51	28.8	0.08	50.69	1696	256
R7	t1	14,050	84	5	27.8	0.05	21.00	429	114
R9	t1	17,570	87	4.35	28.3	0.06	22.71	2140	393
R11	t1	4262	120	4.85	28.1	0.12	69.77	1399	372
R13	t1	7980	76	4.63	30	0.09	31.68	1740	536
R14	t1	7980	92	4.75	27.2	0.08	34.94	429	147
R16	t1	10,611	66	4.5	27.1	0.15	44.55	1282	221
R18	t1	10,611	60	4.21	27	0.08	20.20	874	246
R1	t2	0	100	4.06	6.1	0.07	28.42	2157	352
R3	t2	4650	120	6.24	6.7	0.12	89.86	2049	226
R5	t2	9160	115	5.51	7.1	0.08	50.69	1386	237
R7	t2	14,050	84	5	7	0.11	46.20	1741	537
R9	t2	17,570	87	4.35	7	0.02	7.57	898	313
R11	t2	4262	120	4.85	8.4	0.17	96.90	1725	196
R13	t2	7980	76	4.63	7.2	0.08	28.16	2356	438
R14	t2	7980	92	4.75	7	0.09	39.31	1890	295
R16	t2	10,611	66	4.5	7	0.12	35.64	1700	455
R18	t2	10,611	60	4.21	7.4	0.14	36.07	1430	238
R1	t3	0	100	4.06	15.7	0.07	28.42	2255	340
R3	t3	4650	120	6.24	13.3	0.11	82.37	3435	416
R5	t3	9160	115	5.51	14.7	0.085	53.86	3384	492
R7	t3	14,050	84	5	16.1	0.05	21.00	1714	430
R9	t3	17,570	87	4.35	15.8	0.06	22.71	3265	413
R11	t3	4262	120	4.85	15	0.08	46.51	847	120
R13	t3	7980	76	4.63	14.9	0.13	45.76	2029	556
R14	t3	7980	92	4.75	15.2	0.1	43.68	2162	512
R16	t3	10,611	66	4.5	16.4	0.04	11.88	3350	676
R18	t3	10,611	60	4.21	15.6	0.05	12.62	2368	403
R1	t4	0	100	4.06	27.4	0.16	64.96	745	216
R3	t4	4650	120	6.24	27	0.12	89.86	1381	315
R5	t4	9160	115	5.51	27.6	0.16	101.38	265.1	48
R7	t4	14,050	84	5	28.4	0.2	84.00	1837	282
R9	t4	17,570	87	4.35	28.2	0.13	49.20	2626	413
R11	t4	4262	120	4.85	27.5	0.125	72.68	511	144
R13	t4	7980	76	4.63	27	0.32	112.63	1385	198
R14	t4	7980	92	4.75	27	0.11	48.05	307	86
R16	t4	10,611	66	4.5	28	0.16	47.52	2789	327
R18	t4	10,611	60	4.21	27.6	0.17	42.08	1898	210

, with model outputs summarized in

Table 3. The calculated results and detailed interaction processes for hydraulic exchange between the canal and groundwater obtained by the ²²²Rn mass balance model and flow model. q_r and q_g represent the calculated average flux per unit length of the canal leakage and the calculated average flux per unit length of groundwater discharge, respectively. Hydraulic exchange processes (a) and (b) refer to canal leakage and groundwater discharge, respectively.

Sampling Period	Section	Canal Tract	Calculation by Radon Mass Balance			Calculation by Flow Balance
			qr (10−3·m3/ (s·m))	qg (10−3·m3/(s·m))	Interaction Processes	qr (10−3·m3/ (s·m))	qg (10−3·m3/(s·m))	Interaction Processes
autumn	1	R1–R3	8.5	0.0	(a)	6.5	0.0	(a)
	2	R3–R5	2.0	0.0	(a)	1.3	0.0	(a)
	3	R5–R7	16.5	0.0	(a)	11.1	0.0	(a)
	4	R7–R9	0.0	0.4	(b)	0.0	0.5	(b)
winter	1	R1–R3	6.0	0.0	(a)	7.6	0.0	(a)
	2	R3–R5	13.0	0.0	(a)	6.2	0.0	(a)
	3	R5–R7	1.1	0.2	(a)/(b)	0.8	0.0	(a)
	4	R7–R9	11.6	0.0	(a)	11.0	0.0	(a)
spring	1	R1–R3	0.0	1.0	(b)	0.0	1.6	(b)
	2	R3–R5	6.7	0.3	(a)/(b)	6.8	0.0	(a)
	3	R5–R7	8.6	0.0	(a)	6.6	0.0	(a)
	4	R7–R9	0.0	0.3	(b)	0.0	0.5	(b)
summer	1	R1–R3	0.0	0.5	(b)	10.3	0.0	(a)
	2	R3–R5	45.5	0.0	(a)	11.8	0.0	(a)
	3	R5–R7	4.4	0.8	(a)/(b)	4.7	0.0	(a)
	4	R7–R9	11.2	1.3	(a)/(b)	9.9	0.0	(a)

. While the flow balance method quantifies net water flux magnitudes (Equation (13)), it cannot identify the detailed exchange process of individual sections. Conversely, the ²²²Rn mass balance method describes the groundwater discharge and canal leakage of a specific canal section, resolving the direction of hydraulic exchange and the mixing mechanism.

As can be seen from Table 3, each canal section presents different changes in different sampling periods. The calculation results of the ²²²Rn one-dimensional steady flow model show that the hydraulic exchange process between canal and groundwater is very complex. Section 1 is characterized by canal leakage in autumn and winter, and groundwater discharge in spring and summer. In Section 2, canal leakage and groundwater discharge occur simultaneously in spring, while canal leakage is the main process in other seasons. Both canal leakage and groundwater leakage occurred in Section 3 in winter and summer, while only canal leakage occurred in spring and autumn. Section 4 is characterized by groundwater discharge in spring and autumn, canal leakage in winter, and both processes exist in summer. Thus, canal leakage and groundwater discharge occur simultaneously in some sections, such as Section 3 during sampling period 2, Section 2 during sampling period 3, and Section 3 and Section 4 during sampling period 4. Overall, the average canal leakage flux (0.0008 m³/(s·m) to 0.0045.5 m³/(s·m)) along the Xintongyang Canal is usually one order of magnitude larger than the average groundwater discharge flux (0.0002 m³/(s·m) to 0.0016 m³/(s·m)).

The fitting of the results calculated by the two models, respectively, is shown in Figure 7. The calculation results of two models in the same section are basically within an order of magnitude during the same season. As shown in Figure 7, most of the green and blue triangles in the same column are close together, which indicates that the quantitative calculation results of these two methods are similar, and it also verifies the reliability of the radon mass balance method. However, there are great differences in the calculation results of individual sections between the two methods. For example, in Section 2 during sampling period 2, the average flux of canal leakage calculated by the ²²²Rn model and flow model is 0.013 m³/(s·m) and 0.0062 m³/(s·m), respectively. The calculation results are in the same order of magnitude. In Section 2 during sampling period 4, the average flux of canal leakage calculated by the ²²²Rn model and flow model was 0.0455 m³/(s·m) and 0.0118 m³/(s·m), respectively. The field situation of Section 2 is complex, with the width of Section 2 being 120 m. The wind speed is relatively high at the crossing of tributaries (T2 and T3) and the canal. The degassing of ²²²Rn is unusually active under the influence of higher wind speeds. Thus, the escape of ²²²Rn is one of the main sources of uncertainty of the ²²²Rn mass balance model. It is necessary to reduce measurement error and improve model accuracy in field sampling and laboratory experiments.

When quantifying surface water–groundwater interactions, a comparison of the flow balance model and the ²²²Rn mass balance model reveals statistically consistent net exchange values between the two approaches. Furthermore, variations in ²²²Rn activity along the canal provide qualitative evidence of groundwater discharge dynamics. As exemplified by the significant ²²²Rn activity spikes observed in Reaches 1 and 4 during spring (Figure 6), these spatial anomalies directly indicate localized groundwater exfiltration hotspots, which is consistent with the results of our quantitative calculations. Therefore, through the mutual verification of various methods, the results show that the calculation results are reasonable. It is verified that the ²²²Rn one-dimensional stable model used in this paper has good operability and can more profoundly reveal the process of groundwater discharge and canal leakage of the alluvial plain.

4.4. Seasonal Hydraulic Exchange Between the Canal and Groundwater

Due to the dynamic changes of hydrological conditions [51,52] in different seasons, the interaction between the canal and groundwater is expected to be variable in the different seasons. A sampling period with precipitation was avoided, and ²²²Rn inventory was considered to be unaffected by rainfall and in a stable state. Consequently, to estimate the specific interaction process of each section of the canal, we compared the change of canal flow and ²²²Rn activity for the upstream and downstream sections. The parameters of each canal section and tributary are shown in Table 2. The hydraulic exchange between groundwater and the canal water was estimated by the ²²²Rn balance method, and the results are shown in Table 3.

The results show that the interaction between groundwater and surface water is very complex and has many influencing factors. The types and intensities of hydraulic exchange are different in different seasons and different sections of the canal. The calculation results in Table 3 show that the average flux of canal leakage and groundwater discharge in different sections has seasonal characteristics. It is reflected that the seasonal variation of river leakage is mainly controlled by rainfall and temperature [42,48]. During the process of runoff generation and confluence, rainfall events dilute the ²²²Rn activity of surface water and groundwater and change the hydraulic gradient of seepage. In winter, lower temperatures induce soil freezing, leading to pore blockage and reduced hydraulic conductivity. Studies demonstrate that frozen layers decrease the lateral hydraulic conductivity of sandy soils from 10⁻⁴ cm/s to 10⁻⁶ cm/s, thereby inhibiting surface water–groundwater interactions [53]. This process not only reduces river–groundwater interfacial exchange fluxes by over 80% [53], but also increases the response time of hydraulic exchange [53,54]. The calculation results also support this point.

Figure 8 shows the average flux of canal leakage for different sampling periods. The average canal leakage flux of different sections within one hydrologic year indicates a variation range from 0.0011 m³/(s·m) to 0.0455 m³/(s·m). The largest leakage occurred in the second section of the canal in summer, with a maximum value of 0.0455 m³/(s·m). The minimum flux of canal leakage is 0.0011 m³/(s·m), which occurred in Section 3 of the canal in winter. Furthermore, the canal leakage reaches its maximum in summer and its minimum in spring, and the leakage flux within one hydrologic year is in the order of summer > autumn > winter > spring.

The canal leakage is more evident in summer than in other seasons. The rainy period is from mid-June to early July, and the accumulation of rainfall will lead to higher canal levels. Meanwhile, the infiltration of rainfall will also recharge the adjacent aquifer. Due to the influence of groundwater depth and soil permeability, there are evident differences between canal sections. The canal leakage in Section 2 in summer is particularly prominent, which is mainly caused by the inflow of tributary T2, with high ²²²Rn activity and high flow.

The groundwater discharge along the Xintongyang Canal is less evident than the water leakage. Figure 9 shows the average flux of groundwater discharge in different sampling periods. The average flux per unit length of groundwater discharge in different sections ranges from 0.0002 m³/(s·m) to 0.0013 m³/(s·m). The largest groundwater discharge occurred in Section 4 of the canal in summer, with a maximum value of 0.0013 m³/(s·m). The minimum groundwater discharge flux is 0.0002 m³/(s·m), which occurred in Section 3 of the canal in winter. The difference is that groundwater discharge reaches its maximum in summer and its minimum in winter. The groundwater discharge flux within one hydrologic year is in the order of summer > spring > autumn > winter. Increased precipitation in summer will strengthen the recharge of groundwater, which will promote groundwater discharge in most of the canal sections. In addition to being affected by rainfall, lower temperatures in winter may affect the flux of groundwater discharge. The canal sections where groundwater discharge occurs in autumn and winter were Section 4 and Section 3, respectively. With the increase of temperature in spring, the permeability of the aquifer changes, and the groundwater discharge increases evidently.

Overall, the seasonal hydraulic exchange between the canal and groundwater is complex, which depends on the season and hydrogeological conditions. Considering the heterogeneity of aquifer properties, the hydraulic exchange patterns between the canal and groundwater may vary several times.

4.5. Uncertainty Evaluation

The construction of the ²²²Rn mass balance model involved critical assumptions and simplifications regarding parameter selection and boundary conditions. For instance, the steady-state approach provides a temporally static representation of the system, which inherently fails to capture the dynamic nature of seasonal variations. While seasonal trends can be inferred, their discontinuous characterization requires iterative sampling campaigns to reconstruct transient hydrological processes across distinct time intervals. Meanwhile, the analysis avoids the influence of precipitation on the model. To improve the accuracy of the model, the disturbance items of the model can be reduced.

The uncertainty of hydraulic exchange flux between the canal and groundwater is determined by the propagation errors of different sources and sinks in the ²²²Rn mass balance model. The uncertainty of field measurement is an important aspect that affects the reliability of the calculated results.

The following statistical propagation error method is adopted to determine the uncertainty of hydraulic exchange flux calculated by the ²²²Rn mass balance model:

$${\sigma }^2={\sum }_{i=1}^k{\left({\sigma }_{X_i}{\displaystyle \frac{\partial f}{\partial X_i}}\right)}^2$$

(14)

where $\mathsf{\sigma }$ is the standard deviation of each variable, which is assumed to be a normal distribution. The variables taken into consideration include each of the parameters of the Formulas (10)–(12). ${\displaystyle \frac{\partial \mathrm{f}}{\partial X_i}}$ is the partial derivative of this equation with respect to $X_i$, and $X_i$ ($\mathrm{i}=1\dots \mathrm{k}$) is the variable that causes the calculation error. The standard deviation of ²²²Rn measurements was calculated by RAD7 portable radon detector. The results for uncertainty are presented in

Table 4. The uncertainty of hydraulic exchange flux calculated by the ²²²Rn mass balance model.

222Rn Samples for Surface Water	222Rn Samples for Groundwater	“Background” 222Rn	Hydraulic Exchange Flux
11–34.8%	3.8–17.5%	20%	4–39%

.

The parameters in the ²²²Rn mass balance model that are considered constant and have no or negligible effect include the decay constant and the width and depth of the canal. The analytical uncertainty for ²²²Rn samples ranged from 11 to 34.8% for surface water and from 3.8 to 17.5% for groundwater. The degassing of ²²²Rn is mainly controlled by water temperature and water velocity, but some uncertainty may come from ²²²Rn loss during sampling.

The results of the calculation in Equation (13) give about a 19% error estimate using the average value of several measurements of surface water samples. Thus, in these cases, the large uncertainty of ²²²Rn values in the end-member groundwater dominates the main source of the errors of calculation results. The estimation of “background” ²²²Rn has an error of about 20%. According to the feedback of the calculation results, the error of the hydraulic exchange flux between the canal water and groundwater ranges from 4% to 39%, which points out ²²²Rn activity as the main source of error.

5. Conclusions

The purpose of this study is to quantitatively evaluate the seasonal hydraulic exchange of groundwater and the Xintongyang Canal in the Taizhou alluvial plain using an improved ²²²Rn balance method. To reduce the model uncertainty, the “background” ²²²Rn for non-groundwater sources was introduced into the model to replace the influence of hyporheic exchange. At the same time, the flow model was used to verify the calculated results of the ²²²Rn mass balance model.

The results indicate that the interaction between groundwater and the canal has been transformed several times, especially in different seasons, and canal leakage is dominant. The interlacing and dense distribution of tributaries in the alluvial plain make the hydraulic exchange quite complex and varied. This was verified by the large variability of ²²²Rn activity along the canal in different seasons. The ²²²Rn mass balance method can quantitatively evaluate the interaction between groundwater and surface water and effectively consider the influence of each tributary. During a hydrological year, summer demonstrated the most intense water exchange dynamics, with peak fluxes reaching 0.0455 m³/(s·m) for surface water leakage (Section 2) and 0.0013 m³/(s·m) for groundwater discharge (Section 4), revealing pronounced spatial heterogeneity in dominant exchange processes.

Rainfall concentration makes canal leakage and groundwater discharge more frequent. The canal leakage volume of the four seasons is in the order of summer > autumn > winter > spring. However, the groundwater discharge volume of the four seasons is in the order of summer > spring > autumn > winter. This study offers insight into the seasonal variations of groundwater and surface water interactions within an alluvial plain. The results will provide a reasonable basis for the rational planning and utilization of water.