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Inverse modeling of natural tracer transport in a granite massif with lumped-parameter and physically based models: case study of a tunnel in Czechia

Modélisation inverse du transport de traceurs naturels dans un massif granitique à l’aide de modèles globaux paramétriques et de modèles à base physique: étude de cas d’un tunnel en Tchéquie

Modelado inverso del transporte de trazadores naturales en un macizo de granito con modelos basados en parámetros físicos y agrupados: estudio de caso de un túnel en la República Checa

使用集总参数和物理机制的模型对花岗岩块中天然示踪剂运移进行逆向建模:以捷克隧道为例

Inverzní modelování transportu přirozených stopovačů v žulovém masivu fyzikálně založeným modelem a se sdruženými parametry: případová studie tunelu v České republice

Modelagem inversa do transporte de traçadores naturais em um maciço de granito com parâmetro concentrado e modelos fisicamente baseados: estudo de caso de um túnel na Tchéquia

Abstract

This study deals with numerical modelling of hydraulic and transport phenomena in granite of the Bohemian massif in Bedrichov, Czechia (Czech Republic). Natural tracers represented by stable isotopes δ18O and δ2H were collected at the tunnel outflow points and nearby catchment and their concentrations were monitored for seven years. The study compared transport simulations by a two-dimensional (2D) physically based model (advection-dispersion) developed in Flow123d software and a simpler lumped-parameter model, calculated with FLOWPC. Both variants were calibrated with UCODE software, either fitting the concentration data alone, or including the tunnel inflow rates in the case of the 2D model calibration (either in separate steps or within a single optimization problem). Since each of the models describes the tracer transport with different parameters, the models were compared based on the mean transit time as a postprocessed quantity. Besides this, two different options for processing the recharge data (input for both models) were evaluated. Calibration and data interpretation were possible for three of the four observed places in the tunnel, thus determining the depth limit of applicability of the stable isotopes. The estimates for discharge sampling at 25–35 m depth based on inverse modelling provide reasonable values of mean transit time (20–40 months) for the lumped parameter models, little revising the results of previous studies at the site. The resulting transport parameters of the advection-dispersion model (porosity and dispersivity) are in accordance with the hydrogeological structures present at the sampling sites.

Résumé

Cette étude porte sur la modélisation numérique des phénomènes hydrauliques et de transport dans le granite du massif Bohémien, à Bedrichov en Tchéquie (République tchèque). Des traceurs naturels, les isotopes stables δ18O et δ2H, ont été collectés aux exutoires du tunnel et dans le bassin versant voisin et leurs concentrations ont été suivies pendant sept ans. L’étude a comparé les simulations de transport par un modèle physique bidimensionnel (2D; advection-dispersion) développé avec le logiciel Flow123d et un modèle global paramétrique plus simple, implémenté sous FLOWPC. Les deux variantes ont été calées avec le logiciel UCODE, soit en ajustant les données de concentration seules, soit en incluant les débits des venues d’eau dans le tunnel dans le cas de la calibration du modèle 2D (soit en étapes séparées, soit au sein d’un seul problème d’optimisation). Comme chacun des modèles décrit le transport du traceur avec des paramètres différents, les modèles ont été comparés sur la base du temps de transit moyen en tant paramètre de post-traitement. En outre, deux options différentes ont été évaluées pour le traitement des données de recharge (entrée pour les deux modèles). Le calage et l’interprétation des données ont été possibles pour trois des quatre lieux d’observation dans le tunnel, déterminant ainsi la limite de profondeur d’applicabilité des isotopes stables. Les estimations pour les prélèvements effectués à 25–35 m de profondeur, basées sur la modélisation inverse, fournissent des valeurs raisonnables du temps de transit moyen (20–40 mois) pour les modèles globaux paramétriques, ce qui modifie peu les résultats des études antérieures sur le site. Les paramètres de transport résultants du modèle d’advection–dispersion (porosité et dispersivité) sont en accord avec les structures hydrogéologiques présentes sur les sites de prélèvement.

Resumen

Este estudio trata del modelado numérico de los fenómenos hidráulicos y de transporte en el granito del macizo de Bohemia en Bedrichov, República Checa. Se registraron trazadores naturales mediante isótopos estables δ18O y δ2H en los puntos de salida del túnel y en la cuenca de captación cercana, y se monitorearon sus concentraciones durante siete años. El estudio comparó las simulaciones de transporte mediante un modelo bidimensional (2D) basado en la física (advección–dispersión) desarrollado en el software Flow123d y un modelo más simple de parámetros agrupados, calculado con FLOWPC. Ambas modelos se calibraron con el software UCODE, ajustando únicamente los datos de concentración o incluyendo las tasas de entrada al túnel en el caso de la calibración del modelo 2D (ya sea en pasos separados o dentro de un único problema de optimización). Dado que cada uno de los modelos describe el transporte del trazador con diferentes parámetros, los modelos se compararon basándose en el tiempo medio de tránsito como cantidad postprocesada. Además de esto, se evaluaron dos opciones diferentes para el procesamiento de los datos de recarga (entrada para ambos modelos). La calibración e interpretación de los datos fue posible para tres de los cuatro sitios observados en el túnel, determinando así el límite de profundidad de aplicabilidad de los isótopos estables. Las estimaciones para el muestreo de la descarga a 25–35 m de profundidad basadas en el modelo inverso proporcionan valores razonables del tiempo medio de tránsito (20–40 meses) para los modelos de parámetros agrupados, revisando poco los resultados de estudios anteriores en el lugar. Los parámetros de transporte resultantes del modelo de advección–dispersión (porosidad y dispersividad) son acordes con las estructuras hidrogeológicas presentes en los sitios de muestreo.

摘要

本研究是关于捷克(捷克共和国)Bedrichov 的Bohemian地块花岗岩水力和运移的数值模拟。收集了在隧道出水点和附近集水区的稳定同位素18O和2H为代表天然示踪剂近7年的浓度监测数据。该研究通过Flow123d 软件开发了二维 (2D) 物理模型(对流-弥散)和使用 FLOWPC 计算的更简单的集总参数模型来比较运移模拟结果。变量都使用 UCODE 软件进行校准,或者单独拟合浓度数据,或者在 2D 模型校准的情况下包括隧道流入速率(或在单独的步骤中或在单个优化问题中)。由于每个模型都用不同的参数描述示踪剂运移,因此基于作为后处理的平均运移时间对模型进行了比较。除此之外,还评估了处理补给数据(两种模型的输入)的两种不同选项。可以对隧道中四个观测点中的三个点进行校准和数据解释,从而确定稳定同位素适用性的深度限制。基于逆模型对 25–35 m 深度的排泄采样的估计为集总参数模型提供了合理的平均运移时间(20–40月),几乎没有修改先前现场研究的结果。对流-弥散模型的最终运移参数(孔隙度和弥散度)与采样点的水文地质结构一致。

Abstrakt

Studie se zabývá numerickým modelováním proudění vody a transportu látek v granitu českého masivu v České republice. Ve vývěrech v tunelu a v blízkém povodí byly po dobu sedmi let monitorovány koncentrace stabilních isotopů δ18O a δ2H jako přirozených stopovačů. Tato práce porovnává simulace fyzikálně založeným 2D advekčně disperzním modelem pomocí softwaru Flow123d a jednodušším modelem se sdruženými parametry počítaným ve FLOWPC. Obě varianty byly kalibrovány v softwaru UCODE. Byly využity buď jen koncentrace stopovačů, nebo spolu s nimi také průtok v odebíraném vývěru v případě kalibrace 2D modelu (buď v samostatných krocích, nebo v rámci jedné optimalizační úlohy). Protože každý z modelů popisuje transport stopovačů rozdílnými parametry, byly modely porovnány na základě střední doby transportu (zdržení), získané postprocesingem. Kromě toho byly vyhodnoceny dvě různé varianty zpracování dat infiltrace srážek (vstup pro oba modely). Kalibraci a interpretaci výsledků bylo možné provést pro tři ze čtyř pozorovacích míst v tunelu, čímž byla stanovena mezní hloubka pro použití stabilních izotopů. Z inverzní úlohy modelu se sdruženými parametry pro vzorky v hloubce 25 až 35 metrů vychází rozumný odhad střední doby zdržení s hodnotou 20–40 měsíců, což mírně koriguje předchozí výsledky. Parametry transportu advekčně disperzního modelu (porozity a disperzivity) jsou ve shodě s hydrogeologickou strukturou na místech odběrů.

Resumo

Este estudo trata da modelagem numérica de fenômenos hidráulicos e de transporte em granito do maciço Boêmio em Bedrichov, na Tchéquia (República Tcheca). Traçadores naturais representados por isótopos estáveis δ18O e δ2H foram coletados nos pontos de saída do túnel e próximos da captação e suas concentrações foram monitoradas por sete anos. O estudo comparou simulações de transporte por um modelo bidimensional (2D) fisicamente baseado (advecção-dispersão) desenvolvido no software Flow123d e um modelo mais simples de parâmetros concentrados, calculado com FLOWPC. Ambas as variantes foram calibradas com software UCODE, seja encaixando os dados de concentração sozinhos, ou incluindo as taxas de entrada do túnel no caso da calibração do modelo 2D (em etapas separadas ou dentro de um único problema de otimização). Uma vez que cada um dos modelos descreve o transporte de traçadores com parâmetros diferentes, os modelos foram comparados com base no tempo médio de trânsito como uma quantidade pós-processada. Além disso, foram avaliadas duas opções diferentes para o processamento dos dados de recarga (entrada para ambos os modelos). A calibração e a interpretação dos dados foram possíveis para três dos quatro locais observados no túnel, determinando assim o limite de profundidade da aplicabilidade dos isótopos estáveis. As estimativas para amostragem de descarga a 25–35 m de profundidade com base na modelagem inversa fornecem valores razoáveis de tempo médio de trânsito (20–40 meses) para os modelos de parâmetros concentrados, pouco revisando os resultados de estudos anteriores no local. Os parâmetros de transporte resultantes do modelo de advecção-dispersão (porosidade e dispersividade) estão de acordo com as estruturas hidrogeológicas presentes nos locais de amostragem.

Introduction

Environmental isotope tracers are commonly used to estimate groundwater age, and thus improve understanding of solute transport and groundwater flow systems. They are particularly important for granite massif or fractured media, where the standard tracer tests may be difficult to apply due to low permeability of the rock mass. The isotopic composition of the rainfall creates a specific signature which can be observed at the monitored outflow sites. Quantitative interpretation is then necessary to evaluate the time series of isotope concentrations, including consideration of the choice of the model that can sufficiently capture the flow and transport processes based on the available data. Nowadays, granite massif environments are often considered as potential locations for nuclear waste repositories and the ability to predict the fate of solutes in such conditions is therefore extremely important.

Various numerical models, with different approaches to conceptualize the subsurface media, have been developed for isotope transport studies and applied for estimating water transit time in hydrological system (Holko 1995; Vitvar and Balderer 1997; Etcheverry and Perrochet 2000; Soulsby et al. 2000; Maloszewski et al. 2002; Asano et al. 2002; McGuire et al. 2002; Viville et al. 2006; Vincenzi et al. 2014). The context is quite wide, considering the focus of individual authors on particular tracers (conservative/nonconservative, natural/artificial) or on particular hydrology application (surface water/groundwater/both), and various purposes of study and data availability. Schilling et al. (2019) provides a review aimed at hydrological model calibration, where tracers were presented as one of the “unconventional” observations. McGuire and McDonnell (2006) reviewed catchment transit time studies and methods and considered stable isotopes as the main tracers available for catchment systems with young groundwater (i.e., <5 years old). Markovich et al. (2019) focused on mountain groundwater recharge quantification, with a specific conclusion on stable water isotopes, which are prone to strong bias due to the natural variability in the input signal (e.g. the altitude dependence). For a railway tunnel system in sedimentary rock in Northern Apennines, artificial multitracer tests were done to explain drainage impact on the surface streams (Vincenzi et al. 2009).

The lumped parameter model (LPM) is popular due to its simplicity as it treats the system as a whole homogeneous space and neglects the spatial variations (Małoszewski and Zuber 1982; Zuber 1986), which makes it an attractive option to predict tracer concentrations at a low modelling cost (Marçais et al. 2015). LPM is based on the assumption that the tracer at the outflow can be expressed as a convolution integral of input concentration and the system response function (or weighting function). Mostly piston flow or dispersion models are implemented, the latter leading to two parameters that need to be fitted: the mean transit time (MTT, τMT) and the dispersion parameter (Dp). Marçais et al. (2015) looked at the predictability of different types of LPM for synthetic data by implementing a three-dimensional (3D) groundwater model for a crystalline aquifer. Their results show that a hydrogeologically relevant LPM needs to be chosen for an unbiased estimation, while the quality of calibration does not guarantee the accuracy of the predictions. A lumped parameter model was also applied in the Klodzko basin in Poland (Mądrala et al. 2017) to determine transit times in the fractured aquifer. A conceptual model of the groundwater system was created based on the isotopic data together with the pumping test analysis that helped to locate the recharge zones. In the Mądrala et al. (2017) study, the fractures (fissures) together with rock matrix were considered as a homogeneous system. Applicability of the LPM for fractured aquifers was also confirmed in a study of pesticides transport by Farlin et al. (2013). A local study of precipitation breakthrough to a spring and a stream in granitic rock was done with stable isotopes for about 2 years residence time in particular using FLOWPC software (Viville et al. 2006).

Other approaches can be covered as a wider class of physically based models (PBM) or distributed parameter models and their principal feature is a link to groundwater flow simulation and the solution of governing flow and transport differential equations. One such case is sometimes being referred to as direct age models (DAM), in which the groundwater age comes as a direct result of the simulation (Turnadge and Smerdon 2014). The DAM approach goes back to the publication of Goode (1996), which introduced the approach of mass age, referring to the product of mean groundwater age and groundwater mass; this was later followed by various studies (e.g. Etcheverry and Perrochet 2000; Bethke and Johnson 2002, 2008). This is solved by an equation similar to the advection-dispersion equation, but with additional terms, either with the “age” variable derivative or with a unit artificial source term. As noted in Turnadge and Smerdon (2014), the direct age approaches are usually limited to qualitative studies of local scale due to numerical modeling requirements.

For general tracer transport models based on advection and hydrodynamic dispersion processes, the groundwater age can be estimated afterwards by postprocessing the computed results. Four different approaches for estimating the groundwater age with multiple tracers are explored in Sanford et al. (2017), specifically, the effect of dual domain (combination of fracture and matrix) in complex environments such as fractured aquifers. They compared single- and dual-component piston flow models with an advection-only and dual-domain flow model (developed in MODPATH). Among all of these conceptualizations, the dual-domain flow model had substantially better fit and was recommended by Sanford et al. (2017) for fractured-rock terrains.

Many authors point to a limitation of LPM use for hydrogeological complex systems (Turnadge and Smerdon 2014; Schilling et al. 2019), considering recent advances in computational tools and data collection which potentially offer more sophisticated evaluation. Even with approaches based on the transport differential solution, simplifications can be applied. Vincenzi et al. (2014) use a one-dimensional (1D) advection-dispersion equation for evaluation of the tracer breakthrough from a surface stream into a tunnel in relatively complex geological conditions. Such evaluation is in principal close to the LPM except it uses parameters with direct physical meaning.

Eberts et al. (2012) compared a lumped parameter and “distributed” model for four different production wells by evaluating geochemical data for natural tracers over a 4-year period. The distributed model was constructed in MODFLOW, using MODPATH for particle tracking. The paper stressed the importance of full age distribution rather than a model mean age to obtain consistent results for both types of models. Another example work comparing the LPM and PBM is McGuire and McDonnell (2006).

The use of optimization algorithms for model calibration is a common practice in hydrogeology (mentioned in the review of Schilling et al. (2019) as well); however, the case studies for natural tracers often lack detail on the calibration procedure, except studies such as those described in Krabbenhoft et al. 1990; Turner and Townley 2006; Zuber et al. 2011, where the authors use stable isotopes in relation to the calibration and validation of numerical models.

Therefore, in the context of the granite massif environment and the automatic calibration procedure, the comparison of the two modelling approaches by LPM and PBM for a long period of stable isotopes δ18O and δ2H measurements is very valuable and novel. Moreover, the site offers unique sampling conditions, where the tunnel inflow can be collected in quite a few individual water-bearing fractures along a several-hundred-meter compact granite which is otherwise dry, in an appropriate depth for the tracer time scale. A case study with transport from infiltration input to the tunnel discharge output can complement other more frequent hydrological and hydrogeological configurations.

This study follows from previous work and investigations at the excavated tunnel in Jizera Mountains in the Czech Republic, close to Bedrichov village (Klomínský and Woller 2011; Hokr et al. 2018). The Bedrichov tunnel was built during 1980–1981 as the water supply tunnel for the city of Liberec drilled in granite massif and it has also been used for various investigation projects, including the support of the nuclear waste repository programme in the Czech Republic. A 3D numerical model of groundwater flow that simulates the spatial distribution of tunnel inflow within the inhomogeneous rock subsurface was developed in Hokr et al. (2014). Additionally, models of natural tracer transport at four seepage sites were calibrated as a part of DECOVALEX-2015 project, applying a simplified 3D geometry that combines a single fracture and the rock matrix. A fictitious tracer pulse injection was simulated to determine groundwater age from the breakthrough curves (Hokr et al. 2016). Data for stable isotopes of water, tritium and chlorofluorocarbon (CFC) concentrations were evaluated with LPM as another part of the DECOVALEX-2015 project. The study (Gardner et al. 2016) was made with shorter data sequence than in this study and with a different choice of sampling points (only one was common to both studies and others were deeper). It was found that the estimated mean transit time is sensitive to the choice of seasonal infiltration weighting function as well as to the transit time distribution.

The aim of this study is to apply real tracer concentrations directly for the two-dimensional (2D) flow field model calibration, to estimate hydrogeological and transport parameters more precisely with the inverse modelling, and to improve the choice of tracer infiltration function. Regardless of the cited comments against the LPM use, it is a convenient starting step in evaluation. It can find coarse measures of residence time and overall dispersion, which can help in establishing and evaluation of other models. The next sections describe the study site, provide more details about the collected data, and outline the models used for isotope tracer transport. The results for both the LPM and 2D advection–dispersion model calibration are presented together with the corresponding discussion.

Site description and data collection

The excavated tunnel (Fig. 1) is located in the Jizera Mountains, in the north of the Czech Republic, which belongs to the Bohemian massif, specifically the Krkonose-Jizera Composite Massif (Klomínský and Woller 2011). The length of the tunnel is 2,600 m, the diameter is in a range of 3–4 m. There are irregularly distributed intervals of bare rock and shotcrete. The elevation of the lower (WSW) end is 657 m above sea level (asl), and the upper (ENE) end is 697 m asl giving a slope of approximately 1.5%. The highest surface elevation above the tunnel is at 820 m asl.

Fig. 1
figure1

a The site location, bc topography, and c hydrogeological tunnel conditions. The map (b) shows the position of the sampling section for this study in the context of expected faults based on lineaments (cyan and brown lines). The tunnel profile (c) contains the technical conditions, the geological and hydrogeological conceptual setting, and positions of inflow (rates) and natural tracer measurements. The four inflow points evaluated in this study are annotated in bold (W76–W226)

The primary purpose of the tunnel was to provide a water supply; besides that, the tunnel is used as an “underground laboratory” to collect various measurements (geological, seismic, hydrological, hydro-chemical etc.), which were carried out during the investigation projects at Bedrichov site. The time series of flow rate, temperature, pH, water electrical conductivity, and stable isotope concentrations were recorded and campaign sampling was made for 3H/3He content and concentrations of major ions.

Geological configuration

The whole area belongs to Jizera granite except the most SW part of Fig. 1b composed of Liberec granite. The scheme in Fig. 1c depicts the position of the tunnel within its geological and hydrogeological setting, which was discussed in several preceding works (e.g., Hokr et al. 2014). The site is composed of a shallow permeable zone (weathered rock) and deeper hard rock crossed by several subvertical fractures or faults. The interface position at 25–30 m depth is estimated from the tunnel wall observation, as the depth of the point of 100-m chainage, where a concrete panel or shotcrete covered wall of damaged rock changes to bare compact granite.

The total tectonic failure of granites within the tunnel is mild or moderate (Klomínský 2008; Žák et al. 2009). Long sections of markedly intact rock alternate with shorter intensively tectonically affected sections with strong mineral alterations. Clay and kaolinic material often fills cracks and cataclastic zones to a limited extent. They do not always match the zones with water inflow, which are much less frequent.

The fracture network is characterized by two distinct systems, which are subvertical and perpendicular to each other, in the NW–SE and NE–SW directions (Žák et al. 2009), corresponding also to a tectonic pattern of the landscape. Subhorizontal fracture systems and exfoliation are observed on the surface outcrops but not in the tunnel; thus, they are supposed to be restricted to the near-surface zone. None of them were observed in any part of the tunnel (Klomínský and Woller 2011).

Tunnel inflow, hydrogeological, and geochemical conditions

The visible and measurable inflow in the tunnel is concentrated at a few places or sections, while most of the tunnel wall elsewhere is dry. All the inflow is collected in the collection channel built in the tunnel floor. The cumulative inflow rate is monitored by V-shaped weirs embedded in the channel at the sections interface. Pressure or ultrasonic probes are used for continual logging of the water level to derive the outflow at each section. Selected localized inflows are monitored either by discharge vessels for medium flow springs with continuous flow, or by tipping buckets, suitable for small discharges, including dropping leakages. Independent water flow measurement via dilution of KBr solution was performed to verify the weir flow rate estimates (Rálek and Hokr 2013; Balvín et al. 2014).

The inflow of groundwater through the matrix and fractures of the rock formation is measured and sampled at 14 seepage points (main locations shown in Fig. 1c) twice or once per month. Measurements have been performed manually from February 2007 at regular intervals and automatic measurements with data loggers have been installed at the same locations in 2010–2012. The measurements are carried out manually by filling a scales vessel or automatically by means of water level measurement in a vessel with outflow by a hole. The measured points do not cover all the inflow; therefore, the inflow rates of tunnel intervals derived from the channel weirs are not comparable with the point inflow rates.

Most inflow to the tunnel is observed in the parts intersecting the shallow permeable zone, with flow rates of 0.5–1 L/s in the interval from 50 to 100 m from the entrance and 1.5 L/s in intervals 2,200–2,450 m (Fig. 1c). In the deeper part of the tunnel, across the fractured hard rock body, fully dry intervals occur, and there are short intervals with limited leakage (but not freely flowing water) and several places (faults/ fractures) with inflow of ones to tens of ml/s; the total inflow into the deeper part is below 0.5 L/s. The large temporal variations in the shallow section and relatively stable rate in the deep section also support the assumptions on the water table discussed in the following and limit the effects of the unsaturated zone to the shallow 100-m tunnel section.

The inflow water has low mineralization, neutral to slightly alkaline reaction, and is of Ca-HCO3 type. No distinct end members are detected except the water directly seeping from the reservoir. The mineralization varies mostly with depth from 50 to 200 mg/L. The lowest values correspond to the reservoir and shallow zone inflow; the highest values correspond to low-rate deeper inflows, with expected interaction with the rock minerals. The water chemistry is generally stable in time (Hokr et al. 2018).

Considering the topography (Fig. 1b,c), the whole area above the tunnel can be defined as an infiltration zone. Therefore, the contribution of meteoric water into the studied system is on the elevation range of 100 m. In addition, the part NE from the middle of the tunnel with a valley depression and a tectonic fault provides infiltration above the direct precipitation contribution (Fig. 1c); however, this is not relevant for this study tunnel section.

The presented configuration, derived from explicit observation of rock fracturing on the tunnel wall and from inflow rate and variability measurements, should lead to a topography-controlled flow with dominant presence of water in the shallow zone. The water table was not explicitly measured, but can be assumed to be a few meters below the terrain surface, which is a negligible distance for the purpose of models described in sections ‘Input data and problem formulation’ and ‘Boundary and initial conditions’, i.e. against the elevation differences controlling the hydraulic gradient, and in accordance with the occurrence of terrain discharges in the area.

Choice of seepage points

To model the tracer transport, four tunnel inflow observation points that appear suitable for the stable isotopes application were selected: in particular, their positions in the relatively shallow tunnel section implicate smaller “ages” of water, while the tracer concentration time series still keep some residue of the annual (seasonal) variations. In other words, only those cases where the age could be within the given tracer applicability are considered. For other parts of the tunnel, different natural tracers (like e.g. tritium, helium, and CFCs) must be considered and will be a subject of another publication.

Data were evaluated from the four seepage points at the beginning of the tunnel at 76 m (W76), 125 m (W125), 142 m (W142), and 226 m (W226) distance (Table 1). Each seepage point has specific conditions regarding the inflow of groundwater. W76 (inflow of 0–200 ml/s, average electrical conductivity of 80 μS/cm) is in the more permeable shallow weathered zone, while others are at water-bearing fractures in the much less permeable hard rock. Point W76 is 23 m under the surface; it has a quick reaction to precipitation and often dries out after a long period with no recharge. The seepage point W125 (35 m deep) is located at a single fracture with a very low dipping discharge of 0.02 ml/s and 140 μS/cm conductivity. Point W142 (41 m deep) is placed at a larger fracture zone, intersecting the tunnel in full circumference, and hence having continual outflow approximately 10 ml/s and conductivity of 110 μS/cm. The deepest (61 m) point W225 has continual discharge of 3 ml/s from a single fracture and average conductivity of 150 μS/cm. The positions of the four recorded discharge points are also shown in the scheme in Fig. 1c.

Table 1 List of studied seepage sites with geometric and inflow characteristics. The “recharge ratio” is the factor multiplying the real recharge to define the model boundary condition (explained in section ‘Boundary and initial conditions’)

Watershed measurements

Surface recharge and streamflow represent the input for the groundwater system. The data for this study are collected in the watershed Uhlirska operated by the Czech Technical University, located about 5 km to the north from the tunnel at a similar elevation (Šanda et al. 2014).

The Uhlirska catchment is an experimental watershed with long time series of measurements and experimental research (Šanda and Císlerová 2009; Hrnčíř et al. 2010; Dušek et al. 2012; Šanda et al. 2014, 2017; Jankovec et al. 2017; Votrubová et al. 2017). The data set spans the period from November 2005 to October 2016 and includes the precipitation amounts, outflow rates (based on stream flow rates), and precipitation tracer concentrations (Fig. 2). The balance between the precipitation and outflow is used to transform the tracer input from the precipitation into the recharge (especially the snow storage during most of the winter periods).

Fig. 2
figure2

Measured surface data at Uhlirska watershed for the model input: precipitation (cumulative per month), outflow (cumulative per month related to the whole watershed area), raw 18O isotope tracer concentration, and redistributed tracer concentration (effect of the snow storage)

In the vegetation-covered half of the year (May–October), the positive difference between the precipitation and the outflow (both expressed as the same quantities) is understood as being dominated by evaporation, with a contribution of the recharge (fixed ratio of 0.2). The tracer concentration in recharge water is unchanged.

In the nonvegetation half of the year (November–April), a positive balance is accounted for by the snow storage. If a negative balance occurs (outflow higher than precipitation), it is interpreted as thawing, and has contributed both to the outflow and to the infiltration (Fig. 3).

Fig. 3
figure3

Scheme of the hydrological balance for evaluation of model recharge rate and natural tracer concentration inputs affected by snow deposition (“redistribution” variant)

The balance is not sufficient for accurate determination of infiltration (as subtraction); therefore, as a generic assumption, the distribution of the rain and melt water together between the outflow and infiltration is considered also as the 80/20 ratio and, practically, the infiltration rate can be calculated by a 0.25 factor from the outflow rate.

The tracer concentration in the periods of thawing is calculated by a mixture of the actual precipitation and all previous accumulated precipitation considering their rates. Such sequence of concentrations and infiltration rates are denoted “redistributed”, contrary to a reference case with the infiltration ratio of 0.2 uniformly throughout the year without storage, denoted as “measured” in the results. (see also section ‘Lumped parameter model’).

Natural tracer data

The stable isotopes of hydrogen 2H and oxygen 18O are considered as natural tracers in this study. Their concentrations vary in the atmosphere and in the precipitation with dominant control by the temperature (seasonally) and with a potential bias through the altitude effect. The concentration is expressed as the relative difference against the standard (V-SMOW) in per mil. For simplicity, from here on this paper refers to the value only as “concentration” though in fact it has dimensionless units in parts per mil.

The isotope analysis has been performed by a liquid water isotope analyzer (LGR Inc.) at the Czech Technical University in Prague, with precision ±0.15 per mil V-SMOW (Penna et al. 2012). The reported error of the measurement from the laboratory (corresponding to the 95% confidence interval, resulting from repeated analyses of a single sample) is typically 0.5‰ for 2H and 0.07‰ for 18O (in the units used for concentration presentation).

In theory the 2H and 18O tracer concentrations are strongly correlated and are linked by a deterministic linear relation, the so-called “meteoric line” (a line in the plane with 2H and 18O concentration axes). The deviations from the meteoric line are caused both by atmospheric phenomena and by sampling or measurement error (typically the evaporation). The magnitude of 2H concentrations (in the delta SMOW units) is 8 times larger than that of 18O. To get comparable results (error values, fit criterion) for both tracers, the unit-dependent values of 2H were scaled down by a factor of 8.

For W76, W125, and W142, visible trends in the evolution of the concentrations were observed—increase and decrease over the periods longer than a year (Fig. 4). However, there is also a significant smaller-scale variation which is not simply identified with seasonal periods and is larger than the measurement error. In the W226 data, there is no clear trend or periodic component. It is difficult to decide if the variations are a result of sampling or analysis error or an unspecified process uncertainty (e.g. phenomena not captured by available flow and transport models). When evaluating the model-to-data fit, one must take into account that it is not known which features in the temporal evolution should be fitted; even fitting the average value does not mean the model provides any useful explanation of the phenomena or parameter values. Considering the similar year-average concentration across all sampling points for most of the time (Fig.4), the altitude effect is assumed to be negligible.

Fig. 4
figure4

Measured tracer concentration evolution in the evaluated tunnel sampling points with illustration of the error interval

Methods

To simulate the solute transport, two numerical conceptual models were chosen—a lumped parameter model (LPM) implemented in FLOWPC software (Maloszewski and Zuber 1996) and a 2D hydraulic and advection-dispersion model (also referred to as “2D model” in the following text) calculated by Flow123d software (Březina et al. 2017), which was developed by the research group of the authors. Although the 2D model can in some features be oversimplified, it is meant as a natural step from traditional LPM to more complex models, and it is expected to comprise all the features of LPM, but is built on flow and transport equation parameters.

Although the crystalline rock represents a more complex system for tracer evaluation in general, it was considered that the studied site configuration is appropriate for such approaches: the case of W76 is dominated by a single hydraulic system of the shallow zone and the remaining cases are represented by two “serial” systems of the shallow zone and the localized structure of either a planar fracture zone or a fracture/channel system with dominantly vertical flow in relatively short distance compared to the tunnel and topography scale (Fig. 1c and section ‘Input data and problem formulation’).

Both models were calibrated with the UCODE software (Poeter and Hill 1999) based on the time series of stable isotopes data from February 2010 to October 2016 collected at the four seepage points (W76, W125, W142, W226) and precipitation at the nearby catchment Uhlirska. For some 2D model variants, the tunnel inflow rates over the same period were used for calibration next to the isotope data. The two types of models have different parameters to be estimated in the inverse modelling process. The next two section present each model and their application for the tracer transport at the experimental site separately. Additionally, a postprocessing of the results is introduced that provides a single measure of the mean transit time (MTT) for both the models.

Lumped parameter model

Lumped parameter model (LPM) is based on the assumption that the transit time distribution function of the tracer particles through the groundwater system is known or can be assumed. The output and input concentrations can be then related via the following convolution integral (Maloszewski and Zuber 1996):

$$ {C}_{\mathrm{out}}(t)={\int}_0^{\infty }{C}_{\mathrm{in}}\left(t-\tau \right)g\left(\tau \right) d\tau $$
(1)

where Cout and Cin are the tracer output and input concentrations, τ is the transit time (TT) and g(τ) is the weighting function defining the transit time distribution.

The LPM in this work was implemented with the FLOWPC software (Maloszewski and Zuber 1996), in which the weight function can be chosen among piston, exponential, or dispersion distribution. In this work, a dispersion weighting function was chosen as it is a general model with no specific assumptions on the flow system configuration. In the dispersion weighting function, two parameters are estimated—mean transit time (MTT) τMT and dispersion parameter Dp, which is reciprocal of the Peclet number (mathematically equivalent case to the advection-diffusion equation). The MTT of a tracer is defined as (Maloszewski and Zuber 1996):

$$ {\tau}_{\mathrm{MT}}=\frac{\int_0^{\infty }t{C}_{\mathrm{I}}(t) dt}{\int_0^{\infty }{C}_{\mathrm{I}}(t) dt} $$
(2)

where CI(t) is the tracer concentration at the observation point as a result of an instantaneous injection at the entrance point at t = 0.

The precipitation tracer concentration and the precipitation rate are the required inputs in FLOWPC and the following formula (Maloszewski and Zuber 1996) is then applied for the input concentration Cin to the groundwater system:

$$ {C}_{\mathrm{in},i}(t)={\delta}_{\mathrm{gw}}+\frac{N{\alpha}_i{P}_i\left({\delta}_i-{\delta}_{\mathrm{gw}}\right)}{\sum_{i=1}^N\left({\alpha}_i{P}_i\right)} $$
(3)

where δi is the precipitation tracer concentration, δgw is the mean groundwater concentration of the tracer, N is the number of records indexed by i (e.g. months, preferred in complete hydrological cycles, i.e. 12, 24, 36 months etc.), Pi is monthly precipitation depth and αi is the monthly infiltration coefficient, which expresses the ratio between infiltration and precipitation. In this paper, two variants of the transformation of precipitation to infiltration are considered. The first one is with αi = 0.2, i.e. 20% of precipitation is infiltrated (introduced as “measured” in section ‘Watershed measurements’). The second type of infiltration assumes snow accumulation in winter (introduced as “redistributed”): the same αi factor is used but with Pi equal to the sum of meltwater and precipitation (back-calculated from the measured outflow, see Fig. 3).

Advection-dispersion model

General principle

The numerical model is based on the governing equations for steady-state or transient flow in the saturated zone – Darcy law and the fluid mass balance equation:

$$ \overrightarrow{u}=-K\nabla H,\kern1.25em S\frac{\partial H}{\partial t}+\nabla \cdotp \overrightarrow{u}=f\kern1.5em $$
(4)

where \( \overrightarrow{u} \) is the Darcy velocity (flux density) [m/s], H is the hydraulic head [m], K is the hydraulic conductivity [m/s], S is the specific storativity [m-1], and f is a source term [s-1].

The solute transport is governed by the advection-diffusion equation:

$$ \frac{\partial nc}{\partial t}+\nabla \cdotp \left(\overrightarrow{u}\ \mathrm{c}\right)-\nabla \cdotp \left( nD\nabla c\right)={f}_{\mathrm{c}} $$
(5)

where c is the solute concentration (dimensionless for the considered case of isotope tracers), n is the porosity [−], fc is a source term [s-1] and the diffusion-dispersion tensor coefficient D [m2/s] is defined by components as:

$$ {D}_{ij}={\delta}_{ij}{D}_{\mathrm{m}}{\tau}_{\mathrm{k}}+\left\Vert \overrightarrow{v}\right\Vert \left({\delta}_{ij}{\alpha}_{\mathrm{T}}+\left({\alpha}_{\mathrm{L}}-{\alpha}_{\mathrm{T}}\right)\frac{v_i{v}_j}{{\left\Vert \overrightarrow{v}\right\Vert}^2}\right) $$
(6)

where δij is Kronecker delta, Dm is the molecular diffusion coefficient [m2/s], τk is tortuosity [−], αL and αT are the longitudinal and transversal dispersivities respectively [m] and \( \overrightarrow{v} \) is the average pore velocity [m/s], which is related to the flux by the formula: \( \overrightarrow{v}=\frac{\overrightarrow{u}}{n} \).

The governing equations are solved for given parameters (K, S, n) and unknowns (H, u, c) with software Flow123d (Březina et al. 2017). Flow123d is an open-source code under active development, which is written in C++ and uses several established numerical libraries such as PETSc (Balay et al. 2019). The implemented processes include steady and transient flow in saturated or unsaturated media, multicomponent advective solute transport, nonequilibrium mobile–immobile exchange, sorption, single-component reaction, and an interface to geochemical codes. The numerical method used in the flow scenarios of this study was the mixed-hybrid finite element method (MHFEM; Maryška et al. 2008; Březina 2012).

Input data and problem formulation

In the conceptualization of solute transport towards the tunnel, a 2D model was used that represents a vertical slice perpendicular to the tunnel, assuming that the flow controlled by the tunnel drainage is dominant in the direction perpendicular to the tunnel axis (Figs. 5 and 6). It corresponds to the observed orientation of the water-bearing fractures intersecting the tunnel. This conceptualization may neglect a topography-controlled flow in the shallow zone with a head gradient along the tunnel direction; a possible effect on the observed data is discussed within the results. The 2D model has various interpretations with respect to the 3D hydrogeological structure and flow field and the paragraphs below elaborate on them. The rectangular 2D model has dimensions 300 m depth and 500 m length and the tunnel is represented by a circular hole (see Fig. 6 for the full geometry).

Fig. 5
figure5

The concept of hydrogeological structure, showing the necessary correction of the infiltration into the two types of 2D models (A and B) to balance the tunnel inflow rate (“recharge ratio”), together with explanation of “porosity” in the 2D model of a fracture (B) and with a sketch of the expected water table. Actually, case A of the infiltration balance is valid also for the W142 model with the fracture zone (case B structure)

Fig. 6
figure6

Two-dimensional model: a geometry, boundary conditions and calibrated parameters and b a concept of its embedding into large-scale spatial hydraulic conditions for determining boundary head values. Note that the vertical coordinate z is specific for each of the 2D models (origin at the surface)

To capture the inflow (tunnel drainage) at the cross-sections with various tunnel depths, four model variants are defined, for each measured seepage point at 76, 125, 142, and 225 m positions, with an appropriate depth location of the tunnel. The tunnel depths together with other observation site characteristics are given in Table 1.

The conceptual model consists of two horizontally divided subdomains, based on the description in section ‘Geological configuration’—the shallow permeable zone (SZ) and the lower hard-rock (HR) zone with vertical permeable structures (fractures or fracture zones). The depth is 25 m for W76 and 30 m for other model variants. The use of this concept for the 2D model is limited, due to the water balance between the subdomains influenced by the topography-controlled flow (see Fig. 5): For the fracture-like model variants, only a part of the infiltrated water flows further down from the shallow zone into the hard-rock zone and accesses the tunnel (case B), while for the shallow zone tunnel variant (possibly for more permeable fractures as well), the water drained by the tunnel is collected from a larger surface area than represented by the model geometry (1 m thickness—case A). The necessary correction (“recharge ratio”) is evaluated in Table 1 and commented in the following.

The two following physical meanings of the 2D slice are distinguished: (1) the shallow part is set as homogeneous media with the hydraulic conductivity (K, m/s) and porosity (n) for the respective rock type and this is equivalent to a 3D model setup, (2) in the lower subdomain with fractures, a transformation from the fracture data to the mathematically equivalent Darcy equation data is needed; for the D = 1 m model thickness, the model conductivity value is numerically understood as the transmissivity (T, m2/s):

$$ {K}_{\left(\mathrm{model}\right)}=\frac{K_{\left(\mathrm{real}\right)}\cdotp d}{D} = \frac{T}{D\ }\kern1.5em $$
(7)

where d (m) is the real structure thickness. Similarly, the “transport aperture” b (with units in meters) is represented as the 2D model porosity (dimensionless), i.e. the ratio to the unit thickness (Fig. 5)

$$ {n}_{\left(\mathrm{model}\right)}=\frac{b}{D}=\frac{n_{\left(\mathrm{real}\right)}\cdotp d}{D}\kern1.5em $$
(8)

In other words, the transport aperture gives the mobile water volume per 1 m2 of the planar permeable structure model domain.

Contrary to the parallel plane fracture model with the cubic law defining the transmissivity from the aperture, it is not considered that there is such a link between the two model parameters (namely the conductivity and the porosity in terms of the model input, explained in the preceding). This is because one cannot assume that the flow in the >10-m-scale model could be sufficiently close to the cubic law condition. The flow rate observation data determine the hydraulic parameters and, independently, the tracer observations determine the transport parameters, whose relation is also discussed for the results in the following.

In the case of the W70 profile, the tunnel hole intersects the upper subdomain, while in the other cases, the tunnel is located in the lower fractured subdomain. The respective subdomains therefore mainly impact the tunnel inflow, and are important in the inverse modelling estimation. However, in most of the variants for the inverse modelling, the model is conceptualized as homogeneous. Therefore, the respective environment around the tunnel (shallow zone/fracture zone) is then considered for the whole model.

Boundary and initial conditions

The boundary conditions are set to determine a flow field corresponding to gravitational flow with the tunnel drainage inside (Fig. 6a,b). The lateral sides are set as zero flux and at the tunnel wall the zero pressure head is set to represent atmospheric pressure. For the steady-state flow model, a fixed pressure head is prescribed on the bottom and top boundary. The vertical hydraulic difference is set as an appropriate part of the difference between the recharge boundary head and the discharge zone head in a lower terrain elevation (Fig. 6b). That is, the hydraulic head H =  –10 m at the bottom and H = 0 at the top (z = 0), assuming the water table follows the terrain variation (section ‘Tunnel inflow, hydrogeological, and geochemical conditions’).

It was not possible to prescribe the infiltration flux (either steady or variable) on the top boundary directly, due to the imbalance between the infiltration and tunnel inflow rate (Fig. 5). The proper value is derived from the steady-state model with the prescribed pressure difference. For the transient flow, the “recharge ratio” is at first evaluated (included to Table 1), between the total flux through the top boundary with H = 0 for the steady-state model and the “real” average infiltration (20% of measured precipitation). Then, the transient model used the top boundary condition of prescribed flux (Fig. 6a, second variant), with time-variable values obtained from the measured precipitation multiplied by that factor and by 0.2.

The solute transport boundary conditions are the following: zero mass flux is prescribed on the lateral sides, temporal concentration evolution is prescribed at the top boundary and the “outflow boundaries” are defined for the tunnel surface and the bottom boundary (i.e. the zero diffusive flux meaning the advection-only transport). This study considers the same infiltration data preprocessing as for the LPM (see section ‘Lumped parameter model’). Infiltration rates as well as the tracer concentrations are transformed to the monthly averaged values.

For the initial conditions, the simplest approximation is used of the hydraulic steady-state model with average infiltration rate (or for H = 0 top boundary) and a single value of tracer concentration corresponding to the tunnel inflow observed average. To reduce the impact of the initial conditions on the fitted parameters, the model was run for a sufficient time prior to the period used for data fitting. The output concentration in the tunnel Cout(t) is postprocessed as the mass flux to water flux ratio, i.e. the average of concentration over the tunnel circumference weighted by the flow rate.

Model parameters

In general, all the model parameters are unknown and could be a subject of inverse modelling; however, the available observed data likely do not bear enough information to determine all the parameters well. Therefore, in accordance with general model calibration guidelines the calibration starts with simple models with fewer parameters and only afterwards increase the complexity of the models.

The situation for the individual parameters is the following (refer to Table 2 and section ‘Calibration procedure’ with the particular inverse model variants):

  • Hydraulic conductivity K: determined always by inverse modelling (more options and details in section ‘Calibration procedure’)

  • Specific storativity: reference value S = 10−5 m−1 or by inverse modelling

  • Porosity n determined always by inverse modelling

  • Dispersivity: reference values 2 m longitudinal and 0.2 m transversal or determined by inverse modelling for the longitudinal and reference transversal

Table 2 List of variants used for the 2D model parameter calibration. The subscript “SZ” is for the shallow-zone (upper) subdomain and “HR” is for the hard-rock (lower) subdomain. For determining the measured data, the symbols for their model postprocessing counterparts are used—the tracer concentration Cout(t) and the flow rate Qtun(t) into the tunnel

The choice of dispersivity values is based on the concept of 1:10 ratio of dispersivity to the model characteristic distance (surface–tunnel).

Spatial and temporal discretization

To discretize the model, an unstructured mesh of nonuniform density was used, with higher density in the parts influenced by the tunnel drainage or where one expects significant transport pathways towards the tunnel. The triangular mesh has a step of 100 m at the distant boundaries, 0.5 m at the tunnel wall and either 2 or 5 m directly above the tunnel. There are actually two meshes—the first one with the coarser grid serves for the inverse modelling computations, while the second finer mesh is used to verify the fitted parameter (Fig. 7). No significant differences were observed in any case. The temporal discretisation used in this study was 0.1 month, i.e. 1/10 of the input data resolution.

Fig. 7
figure7

The mesh of seepage site a W76 (coarse) and b W226 (fine)

Transit time: postprocessing of the 2D model

Additionally to the flow and transport parameter estimation, the MTT of the groundwater system was evaluated (i.e. between the terrain surface input and the tunnel output), a counterpart to the same quantity for the lumped parameter model. This is made by solving a transport problem with a fictitious tracer with a Dirac pulse input boundary condition on the top. The breakthrough curve, i.e. the concentration evolution on the tunnel (output) boundary, equals the residence time distribution, up to a normalisation. Therefore, the MTT is calculated by approximating the first moment of the breakthrough function with the used temporal discretisation. The simulation time is 100 years, which is about ten times the period of observed data.

Calibration procedure

The purpose of solving the inverse problem (calibration task) is to find estimates of model parameters in order to obtain the best fit of the simulated data with their measured counterparts (called “observations” in this context). The parameters for the two different models (LPM and 2D) were outlined in the previous section and are: the mean transit time and the dispersion parameter for LPM and the hydraulic conductivity, storativity, porosity, and dispersivity for the 2D transport model (either homogeneous or heterogeneous with the two subdomains). The observations to be fitted are the temporal sequences of tracer concentration and in some cases also the tunnel inflow rates for the 2D model (the six-parameter calibration of W76).

The problem is solved as an optimization problem, i.e. the parameters are found by minimizing the cost function representing the model-measurement fit, in particular the sum of square weighted residuals (SSWR) as it is implemented in the UCODE software (Poeter and Hill 1999).

$$ \mathrm{SSWR}={\sum}_{i=1}^N{\left[{w}_i\left({C}_i^{\mathrm{meas}}-{C}_i^{\mathrm{mod}}\right)\right]}^2 $$
(9)

where N is the number of measurements, wi is a weight function, \( {C}_i^{\mathrm{mod}} \) is the modelled concentration and \( {C}_i^{\mathrm{meas}} \) is the measured concentration. The minimization algorithm is based on the gradient method, where the derivatives of the cost function with respect to the parameters are approximated by the perturbation method.

There are several other options to evaluate the fit of the model versus data; Eqs. (10) and (11) are used in the results evaluation. The mean square error:

$$ \sigma =\sqrt{\frac{\sum_{i=1}^N{\left({C}_i^{\mathrm{meas}}-{C}_i^{\mathrm{mod}}\right)}^2}{N}} $$
(10)

and is equivalent to the SSWR with unit weights for use in the optimization cost function. The N denominator and the square root give the physical meaning to σ as the “average deviation” in the measured quantity units, independent of the number of data. Unfortunately, the FLOWPC documentation (Maloszewski and Zuber 1996) and the built-in post-processing use the less practical alternative of the formula of σ, with N out of the root, which then loses these advantages.

Additionally, the quality of fit can be evaluated by model efficiency defined by Nash and Sutcliffe (1970) (NSE):

$$ \mathrm{NSE}=1-\left[\frac{\sum_{i=1}^N{\left({C}_i^{\mathrm{meas}}-{C}_i^{\mathrm{mod}}\right)}^2}{\sum_{i=1}^N{\left({C}_i^{\mathrm{meas}}-{C}^{\mathrm{mean}}\right)}^2}\right] $$
(11)

where Cmean is the mean concentration for all N measurements. It relates the model-observation deviation to the measured data variation and it ranges between minus infinity and one, with higher values indicating better agreement and negative values indicating that the observed mean is a better predictor than the model. Not only does it evaluate the deviation but it also expresses the predictive ability of the model and has been widely used to evaluate the performance of hydrologic models. The NSE criterion is not used for optimization but for the final model evaluation. Contrary to the original formulation of Nash and Sutcliffe (1970) and to the background idea, the FLOWPC uses another formula, although referring to the same source, but with variation of the model results instead of observations in the denominator. This is important to stress, since this study evaluates and presents both the criteria by means of the authors’ own computation of the formulas given in the literature for all models, i.e. instead of the outputs of FLOWPC software for LPM.

For the LPM calibration, two parameters were estimated with a full sequence of tracer concentration observations. The several optimization runs differ by the choice of input concentration (redistribution). For the 2D model calibration, the four variants listed in Table 2, starting from simpler to more complex ones, were evaluated. All of these variants were computed with redistribution of inflow concentration.

Only the last variant (4) in Table 2 proceeds directly with a coupled flow and solute transport model, while the three simpler variants (1–3) estimate the hydraulic conductivity first (manually from the linearity of the steady-state model with the zero pressure top boundary) and then calibrate the solute transport parameters. The obtained hydraulic conductivity values are presented in Table 3.

Table 3 Auxiliary data: Estimates of conductivity from the 2D steady-state hydraulic model (used for the solute transport model inversion variants 1–3) and the estimate of porosity from the Darcy velocity and the tracer velocity based on LPM results

The initial estimates for the 2D model were obtained from the transit time estimates by the LPM, converted to the velocity in 1D model of the tunnel-surface distance. One can obtain Darcy velocity from the known hydraulic conductivity value and the pressure difference and, consequently, the porosity from the ratio of the Darcy and tracer velocities (Table 3). It is noted that the presented porosity values are quite loose upper estimates, as the calculation combines the mean of transit time with the shortest input–output distance to get the transport velocity. The calibration was executed also with other initial estimates, to test the impact of the initial estimate.

Results

This section presents the estimates of model parameters and reports on the quality of fit between the simulated and observed times series. The estimates are computed for each of the two models—LPM and 2D distributed model. When evaluating the results, two factors are considered: first, how well the estimated parameters fit within the conceptual assumptions about the groundwater system at the field site, and second, the impact of the calibration procedure and measured data processing on the estimated parameters, their consistency between the two models, and the quality of fit. The results are presented with time in calendar dates; the start of observation period is in February 2010 and the end is in October 2016, with the total length of seven years. The MTT values are presented in months.

Calibration of the lumped parameter model

The results of calibration by UCODE are summarized in Table 4 and include the estimates of MTT and dispersion, and the fit criteria σ and NSE for the four model variants separately for each observation point (two different tracers, each with two input functions). The time evolution of the tracer concentration is plotted together with the observed time series and the laboratory analysis error (described in section ‘Watershed measurements’) in Fig. 8, separately for each calibrated variant.

Table 4 Summary of calibration results for LPM: Parameter estimates from the calibration and the data fit criteria. The σ parameter (in the units of concentration) is divided by 8 for the 2H tracer, to normalize the value against the 18O tracer (the ratio of concentrations is about 8, as expressed by the meteoric line)
Fig. 8
figure8

Results of the lumped parameter model simulations for the four individual inflow points—the tracer concentration evolution is plotted against the time (calendar dates)

Regarding the quality of the model fit, the simulated concentration versus time series at W76, W125, and W142 mostly follow the main features of the observed data (i.e. the trends in a larger time scale). The simulated data have only little seasonal variation; these variations are possibly also present in the observations, but are mixed with the noise resulting from the measurement error or process uncertainty. The rise of concentration values in the last 3 years of the observed period is now reproduced by the model, which was not the case in the previously published results with shorter data sets and a simpler model with uniform infiltration rate (Balvín et al. 2014).

When comparing the different seepage points, the fit is generally better for W76 and W142 which have larger concentration data variations. For the deepest seepage point (W226) though, the observations of concentration do not have visible trends, so it is impossible to decide if the variations in the model are related to the data. This is also reflected in the quantitative measures: the NSE is around 0.5 for W76, W125, and W142, while it is negative (or very low) for W226, which means that the model for W226 does not have better explanatory value than the average of the data, even though the deviation expressed by σ is similar to other cases. It is also noted that W125 has lower σ (better fit) than W76 and W142, while having also lower NSE (worse fit). This can be explained by smaller variations in the measured data of W125, reducing the value of NSE. The results for the measured and redistributed input appear visually similar, but the fit criteria are worse for W125 and W142 where the redistributed input is applied.

In this study, the MTT ranges from 22 to 34 months, and it is not simply proportional to the depth of the site as one could expect, but rather given by the different hydrogeological structures. Taking the W142 as a reference, the transit time of W76 is relatively larger compared to the depth, which can be explained by the flow field illustrated in Fig. 5. The outflow at the W76 is collected from a larger area uphill and follows the shallow zone/hard rock interface, i.e. the water does not flow only vertically, as it is assumed in the 2D perpendicular model. The transit time of W125 is larger than W142 even in absolute measure. This is related to the type of the structure, with the smaller fracture having lower transmissivity and smaller flow rate. The tracer velocity is directly proportional to hydraulic conductivity and indirectly to porosity. For the transit time evaluation (Table 4, between W125 and W142), the factor between flow rates is not necessarily compensated by the factor of mobile water volume (porosity, aperture). The calibration suggests that W226 has the largest MTT; however, based on the fit criteria and visual comparison mentioned previously, the particular value is not determined sufficiently, and will be discussed in section ‘Discussion’.

The dispersion coefficient is within the typical range mentioned in the literature (0.05–0.5 according to Maloszewski and Zuber 1996) without relation to a particular geology. The dispersion value for W76 and W142 is a little larger than for W125, which can be explained by the fact, that the flow in smaller fractures in W125 is more concentrated. The obtained results for W226 are again not relevant for comparison (due to poor model fit).

Calibration of the 2D advection–dispersion model

Similarly as for the LPM, all the parameter estimates, data fit criteria, and postprocessing of MTT (which will be discussed in the section ‘Discussion’) are summarised in Table 5 for the one-parameter and the two-parameter calibration with the steady-state or transient hydraulic model; the calibration for the six-parameter model (two subdomains model) is shown only for W76 in Table 6. Again, the table shows results for both the tracers (18O and 2H). The two-parameter calibration with porosity and dispersivity within one zone (variant 3 in Table 2 for W125 and W142) represents also the results of the variants 1 and 2 as the improvement of fit was small; these cases are explicitly discussed in section ‘Effect of the calibrated parameter choice’. The evolution of tracer concentration over time is plotted together with the measured time series in Fig. 9, which shows the two tracer breakthrough curves and the steady-state and transient model for each observation point.

Table 5 The 2D advection-dispersion model: summary of estimated transport parameters, fit criteria, and a postprocessing of transit time with a pulse tracer input
Table 6 Results of fit of the six-parameter inverse models at seepage point W76; the value in parentheses denotes a hard limit set in the optimization
Fig. 9
figure9

Results of the 2D advection-dispersion model for the four individual inflow points—temporal tracer concentration evolution

Here, there is comment on the data fit and then discussion regarding the actual parameter estimates. In general, visual examination suggests worse fit for the 2D model than for the LPM, and this is also confirmed by the fitting error criteria. However, the main source of misfit is the concentration shift in the second half of the observation period rather than a generally worse performance in capturing local minimum and maximum values. The 2D model calibration gives quite large σ and small NSE, even though the trends are often similar to the LPM with significantly better criteria. The σ value (in case of 2H, divided by the factor 8 to the common scale) is in the range 0.38–0.53 for W76 and W142 and 0.13–0.36 for W125 and W226. The NSE is often negative, always in case of oxygen calibration and also in some cases of hydrogen calibration. If positive, the NSE is low, in the range of 0.12 to 0.31 for W76, W125, and W142.

The trends in the concentration variations are well captured for all variants except in the case of W125 (steady-state flow and 2H tracer) and W226 (overall poor fit). The minimum of the measured data in 2011 and the peak followed by a small decrease or plateau in 2013 are generally well fitted for W76 and W142. The modelled drop of concentration after 2013 is much larger than the observed data for all models. The transient-flow models rise back to the concentration before the drop, while the steady-flow models rise gradually following the trend of the measured data. Consistently, the model with the transient flow compared to the steady-state flow produces better fit for the cases W76, W125 and W142. Since there is no clear trend in the W226 observation data, it is not easy to judge whether the model captures the measurement or not; although the technical criteria are similar to other cases, it gives a low credibility to estimated model parameters.

The estimated porosities are consistent between the two tracers and the two hydraulic model variants for W76, and the range from 0.06 to 0.1 is a reasonable value for the shallow zone of weathered granite. In both cases, the porosity resulting from the transient model is smaller than the porosity from the steady-state model. The estimated porosities for W125 are more spread, up to one order of magnitude. The mutual relation among the variants is similar to the W76 case: the lower values of porosity for the transient model and for 18O data compared to 2H data. On the other hand, the largest value for porosity (n = 0.001 at W125) is disregarded, which was actually for the case of fitting the data with a constant function. The meaning of the “porosity” in this case is the total width of the opening, which allows the magnitude in tenths of millimeters to be appropriate. The estimated porosity values for W142 are almost identical for both the tracers: 0.038 and 0.039 for the transient model and just below 0.1 for the steady-state model. This value means either several centimeters of total void space width, or a permeable structure like a fracture zone, with, for example, a 1 m width and 3–10% porosity (each with the same transmissivity). The porosities resulting from W226 calibration are in a very narrow range around 0.03, which is interpreted as a coincidence, due to different trends in the simulated concentration evolution and poor fit of the data (commented on in the preceding). There is no real trend to be captured and the steady and transient models produce quite different concentration evolution.

The calibrated dispersivities have meaningful values only for the transient models at the points W125 and W142, for which they are around 5 m, which is larger than the estimated ones from the surface–tunnel distance scale; however, it can be interpreted as an effect of actually longer transport pathways. For the steady-flow model, the optimization algorithm reached the chosen limit of 10 m. The model with the larger dispersion tends to produce a very flat curve which is not relevant for the data even with an improvement in the fit criteria.

The extremes in the concentration evolution can also be used for intuitive verification of the model-to-data fit: larger values of porosity lead to a slower transport and a later peak position. If the positions in the measured data were fitted, larger porosities would be obtained for the W125 transient model (both tracers), getting smaller differences for the porosities between the steady and unsteady flow models. Besides, larger dispersivity leads to a quicker breakthrough and earlier positions for the curve extremes. The combined effect of the porosity and the dispersivity leaves some freedom in the interpretation, although the fit-criteria minimization technically results in a unique choice of those two parameters. For example, it explains the difference between the calibrated parameters of the steady and unsteady flow models of W142 (Table 5).

Effect of the calibrated parameter choice

When one compares the data fit between the two-parameter and one-parameter variant, the improvement of calibrated porosity and dispersivity compared to the porosity-only case was much smaller than the improvement of the transient versus steady-state flow. The calibration with two different porosities for upper and lower subdomains (variant 2 in Table 2) led to either of the following two situations: there was a little change with respect to the initial estimate with the uniform porosity of one-parameter calibration, or there was a tendency to increase porosity for one subdomain and decrease it for the second one over reasonable limits (below 10−4, over 0.5) with a little effect on the fit criteria. Therefore, these results are not presented and they are considered as a case where the parameter estimates are not well supported by the observation data.

It is also noted that the use of two distinct hydraulic conductivities for each subdomain has a minor effect on the final estimates. The difference in their values is compensated by the hydraulic gradient to keep the water balance on the interface, and therefore the Darcy velocity field is almost unchanged.

Six-parameter calibration of W76

The six-parameter calibration was used as another option to improve the data fit and to estimate parameters for a more complex model. The calibration differs from the previous ones by the use of flow rate observations, as both the hydraulic and transport parameters were calibrated. The resulting concentration evolution and fit criteria seem promising. The rise of the concentration in 2015–2016 is captured much better, getting similar curves as for the LPM. The σ values are a little better compared to the transient homogeneous model with the two-parameter calibration, but the NSE makes a significant improvement to values around 0.5, which was the value obtained with the LPM. The problem lies in very inconsistent results across the variants and between the tracers. These differences are mainly in the parameter values, which often differ by orders of magnitude. Only the porosity stays within a quite small interval and with a clear relation between the upper and lower subdomains. The porosities are lower compared to the expected values for the rock types, which can be justified by a different behavior of the transient flow model that gives smaller flow rates.

Discussion

In this section, the results of the LPM and the 2D model are compared and discussed in terms of the relevance of the model, assumptions, the tracer time scale range, and estimated parameters in relation to other results in the literature.

For three observation points (W76, W125, W142), both models transform the seasonal variations in tracer concentration in a similar way and capture the longer time scale trends. The first minimum is fitted well in both the position and the value, but the second minimum (2013/2014) is much lower in most of the simulated curves. To illustrate the comparison, the graphs with compilation of tracer concentration evolution for three models for W76 are presented in Fig. 10. The difference between the models is rather in the detail of the evolution patterns, like particular concentration values and positions of extremes. The LPM predicts the rise in 2014 and 2015 much better than the 2D model. For W226, the model results are different: the LPM has a similar shape to other observation points but with more damping, while the 2D model produces quite a different evolution pattern with rather irregular variations. This can be related to a nonunique inverse model for a case with little information in the observation data.

Fig. 10
figure10

Comparison of three model variants of W76 and two tracers a δ18O and b δ2H: calibrated cases with different parameters on the background of the measured data series

Comparing the MTT estimates between the LPM and 2D model, there seemed to be a large disagreement at first, since the mean transit time obtained by postprocessing of the 2D model is much larger than the equivalent for the LPM. This must be examined through details of the transit time distribution curve: the first moment of the curve in the 2D model is much affected by the long paths, and typically its value is almost ten times larger than the position of the peak, unlike in the dispersion model where the first moment is about three times the peak position. By drawing the curves, one can observe that the rising fronts coincide, but the peak shapes and the declining parts differ. The measured data cover a smaller period (compared to the MTT model evaluation interval), so the fit should be evaluated through the partition of travel time paths in this range when accounting for the contributions to the mean. The distribution of the pathways’ transit times in the 2D model is different from the dispersion model; however, the results do not confirm if this is closer to reality. On the contrary, the TT distribution of the dispersion model appears to fit the data better, however, without any answers for the spatial flow patterns that control such transport processes.

The comparison of the MTT estimates between the sections is consistent with the raw data sequences: the trends in W76 and W142 data are similar, therefore one would expect similar water residence time values. The W125 data have similar features in the trend but with smaller amplitude, meaning either larger dispersion or larger residence time.

Apart from the depth influence, the MTT of the three points W76, W125, and W142 is affected by the type of hydrogeological structure and the model simplification. Neglecting the topography-controlled flow in the shallow zone means that the flow actually happens along longer paths than assumed in the models, which is most important for W76. For the 2D model, it means that the velocity is rather underestimated and the porosity rather overestimated. This effect works against the effect of neglecting the unsaturated zone. Considering a water-table depression above the shallow section of the tunnel, the actual water depth is smaller, but other water trajectories, starting at higher altitude, contribute along larger distance within the saturated zone. The relation between W125 and W142 is given by independent control parameters of hydraulics and transport—the velocity is proportional to the ratio of hydraulic conductivity and porosity for the same hydraulic gradient.

The estimates of MTT by LPM are quite consistent with the previously published results (Šanda 2013), which were in the range of 20–40 months, but in many cases the current estimate is up to 30% smaller. This is displayed in Fig. 11 as the ratio between previous and new modelled results, where all except W226 are under the equality line. Results of the MTT by use of various weight functions in Gardner et al. (2016) ranged from 15.24 to 86.04 months for W142, i.e. rather a large interval, and are included in Fig. 11. The MTT estimates fall into the stable isotopes range of 0.1–3 years (Suckow 2014) or 2–4 years (Zuber et al. 2011) close to the upper limit, which is mentioned in the preceding in the comments on the fit quality, which is worse for deeper points and behind the limit for W226.

Fig. 11
figure11

Comparison of the previous results of 18O with shorter data sequence (Šanda 2013) and the results of the dispersion model with shorter data sequence (Gardner et al. 2016) versus new results of the LPM calibration from this study

In most cases, the optimization algorithm converged uniquely to the presented value, but the sum of square residuals often decreased relatively little. The calibration with LPM gives quite consistent results of MTT and dispersion parameters between both the tracers (18O and 2H), implying that the part of the measurement error, which deviates the data from the meteoric line, does not have a significant impact on the fitted curves.

In this study, a redistribution is used for input concentration of stable isotopes to acknowledge the snow deposition; however, it cannot be concluded that it improves the model fit, as in most cases the two variants give comparable results. Gardner et al. (2016) confirmed a sensitivity of MTT to a recharge weighting function at the same site with the same seepage point W142 and three others points. This suggests that the delay of a few months for a part of the tracer is not so important, while the overall balance resulting from the nonuniform infiltration rate is critical. In particular, the individual month often differs in the precipitation or infiltration by a factor of up to 10. It seems that the larger concentration values in the period of 2015–2016 can be a result of the sequence of three seasons with larger infiltration in summer and smaller infiltration in winter. Therefore, although the effect of inaccurate infiltration on concentration evolution reduces the data fit, the model can still be meaningful for the purpose of the tracer transit time estimation.

Conclusions

This work presents a case study of natural tracer transport, using inverse modeling to estimate the fractured media parameters. The lumped parameter model was compared with a kind of physically based model (2D advection-dispersion equation), for which a transformation of recharge data was implemented. The evaluated data of stable isotopes of hydrogen and oxygen in the tunnel inflow at 23–61 m depth were at the limit of the tracer applicability. For the three shallower points, the seasonal variations are largely damped, while the longer-period variations can be captured by all the models. For the deepest point, there was no model that could uniquely fit the observed data. Most likely, the water residence time and the dispersion is too large at this profile and therefore other tracers than stable isotopes might be necessary.

Compared to the LPMs, the physically based models take into account more knowledge about the system. In particular, the 2D flow and advection-dispersion model with the theoretical flow pattern between the surface and the tunnel should represent better the tracer transport process but it did not improve the model fit of the data for the presented cases. It means that the actual flow pathways have a more complex character. The attempt to calibrate the model with separate domains of the shallow zone and the fracture-like structure was not successful, assuming the available data do not provide enough information to determine such a model well.

A simpler model such as LPM with the dispersion weighting function of TT distribution, can then approximate well the transport process, which integrates multiple uncertain geological effects of fractured rock, however, providing limited information for the media and flow spatial configuration. The main value of the 2D model is in the estimated parameters—instead of the MTT characterizing the whole system, one obtains parameters that characterize the individual hydrogeological structures. It is possible to distinguish different properties of the shallow zone, a small single fracture and a larger planar permeable structure (fracture zone), and the particular values are appropriate for the respective structures.

No improvement in the model was observed using the tracer input redistribution controlled by snow accumulation. It is challenging to estimate the nonuniform infiltration effect on tracer breakthrough curves. The tracer balance is well captured for a part of the data period, while the systematic deviation in another part suggests an error in balance of winter/summer recharge.

The inverse algorithm converged in most of the cases as long as the choice of fitted parameters was adequate. The option with two parameters produced good results, with the unique optimum consistent across the model variants (two tracers, steady/transient, and tracer input function). The coupled estimates of the MTT and dispersion parameter in the LPM and the porosity and dispersivity in the 2D model have a similar role; one controlling the general speed of the movement and second controlling the dispersion. Variants of the 2D model with two subdomains and with coupled calibration of hydraulics and transport were less successful; although the data-fit measures were better, there were many cases of nonunique parameter identification.

For efficient use of highly parameterized models, it would be necessary to improve the optimization problem’s formulation and to obtain more data. For example, some constraints could be prescribed based on the geological configuration, the observation weights set, or different observation types coupled into one optimization (e.g. two tracers). The configuration of flow could be also better captured with a 3D model, but with expected numerical difficulties of the shallow unsaturated zone representation.

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Acknowledgements

The authors thank the water management company SČVK, a.s. for allowing access to the tunnel.

Funding

This work was co-funded by the Radioactive Waste Repository Authority (as a partial result of project “Research support for Safety Evaluation of Deep Geological Repository”), the International Atomic Energy Agency within Research Contract CZ16335, and the Ministry of Education, Youth and Sports, and EU within the project RINGEN+, CZ.02.1.01/0.0/0.0/16_013/0001792.

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Correspondence to Aleš Balvín.

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This article is part of the topical collection “Progress in fractured-rock hydrogeology”

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Balvín, A., Hokr, M., Šteklová, K. et al. Inverse modeling of natural tracer transport in a granite massif with lumped-parameter and physically based models: case study of a tunnel in Czechia. Hydrogeol J 29, 2633–2654 (2021). https://doi.org/10.1007/s10040-021-02389-x

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Keywords

  • Stable isotopes
  • Inverse modeling
  • Lumped parameter model
  • Fractured rocks
  • Czechia