Groundwater travel time distributions (TTDs) provide a robust description of
the subsurface mixing behavior and hydrological response of a subsurface
system. Lagrangian particle tracking is often used to derive the groundwater
TTDs. The reliability of this approach is subjected to the uncertainty of
external forcings, internal hydraulic properties, and the interplay between
them. Here, we evaluate the uncertainty of catchment groundwater TTDs in an
agricultural catchment using a 3-D groundwater model with an overall focus on
revealing the relationship between external forcing, internal hydraulic
properties, and TTD predictions. Eight recharge realizations are sampled from
a high-resolution dataset of land surface fluxes and states.
Calibration-constrained hydraulic conductivity fields (

Travel/transit time distributions (TTDs) of groundwater provide a description
of how aquifers store and release water and pollutants under external forcing
conditions, which has significant implications for interdisciplinary
environmental studies. For example, remarkable time lags of the reaction of
streamflow with outer forcings and considerable amounts of “old water”
(i.e., water with an age of decades or longer) in streamflow have been
observed in many studies

The accurate quantification of groundwater travel time at a regional scale is
extremely challenging. A primary difficulty is that the complex geometric,
topographic, meteorologic, and hydraulic properties of hydrologic systems
control the flow and mixing processes and therefore define the unique shape
of the TTD

The techniques for determining groundwater TTDs can be categorized into two
groups: geochemical approaches and numerical modeling approaches

In contrast to such an analytical approach, physically based numerical models
can explicitly describe the geometry, topography, and geological structures,
and they can represent the flow paths of individual water particles.
Physically based numerical models are structurally complex and
computationally expensive and often have more parameters than lumped
parameter models do. These models can be classified as Eulerian approaches or
Lagrangian approaches

A reliable application of groundwater transport modeling is subject to many
sources of uncertainty, including measurement, model structural, and
parameter uncertainty

Biased characterization of the hydrodynamic system and oversimplified
assumptions will lead to a problematic prediction of TTDs. Many past studies
offer insights into the influence of recharge and hydrogeological
configuration on the prediction of TTDs. For example, some research studies
have been devoted to the development of analytical solutions for the
idealized catchment (or aquifer) under some essential assumptions and
simplifications

Although studies on catchment-scale groundwater TTDs are numerous, comprehensive uncertainty analysis that aims to unveil the different roles of external forcing and internal hydrostratigraphic structure using both a numerical model and SAS functions is scarce. In this regard, two important questions are the following. (1) How does the uncertainty of recharge (including its spatial nonuniformity) and hydraulic conductivities affect the TTD predictions in a mesoscale agricultural catchment, provided that the model is constrained to reality and groundwater head observations? (2) How does the uncertainty of inputs (forcings) and parameters influence the prediction of systematic preference for young/old water?

In this paper, we aim to answer these questions through a detailed
(uncertainty) analysis of an example application in a mesoscale real-world
catchment. In doing so, we establish a detailed groundwater model coupled to
a random walk particle tracking system for predicting groundwater TTDs. The
OpenGeoSys (OGS) groundwater model is used to simulate the groundwater flow,
while the input forcing is fed by the mesoscale hydrologic model (mHM) via
the mHM-OGS coupling interface

The candidate site in this paper is the Nägelstedt catchment, located in
central Germany (see Fig.

The Nägelstedt catchment used as the test catchment for this
study

The dominating sediment in the study area is the Muschelkalk (Middle
Triassic). The Muschelkalk has an overall thickness of about 220 m, and it
has been divided into three subgroups according to mineral composition: Upper
Muschelkalk (mo), Middle Muschelkalk (mm), and Lower Muschelkalk (mu). The
Upper Muschelkalk (mo) is mainly composed of limestone, marlstone, and
claystone, and it forms fractured aquifers

Eighteen monitoring wells distributed in this area are used to calibrate the
model (Fig.

We use the mHM-OGS coupled model, proposed by

The catchment water storage is conceptually partitioned into soil zone
storage and deep groundwater storage; the two corresponding components are
computed by mHM and OGS, respectively. The soil-zone dynamics of TTD has been
well studied using mHM in a previous work

In this study, we focus on the travel times in the saturated zone. Saturated
groundwater flow is characterized by the continuity equation and Darcy's law:

We use the RWPT method to track the particle movement. The RWPT method is
embedded in the source codes of OGS

The steady-state model configuration is achieved using a temporally averaged
recharge of the simulated daily recharges over a long period (1955–2005).
The gridded recharges estimated by mHM are interpolated and then assigned to
each grid node on the upper surface of the OGS mesh using a bilinear
interpolation approach. No-flow boundaries are assumed at the outer edges
that are defined by catchment divides, except for the northwestern and
northeastern edges, where fixed-head boundaries are applied

Adjustable ranges of the hydraulic parameters.

Recharge realizations used in this study (unit: mm). They were
sampled from a high-resolution dataset of land surface fluxes for Germany

The numerical experiment to explore the uncertainty of TTDs is performed
through the following workflow.

Eight spatially distributed recharge realizations are sampled from a high-resolution dataset of land surface fluxes for Germany, in which mHM is used to simulate land surface hydrological processes. The details of the dataset and the sampling method are described in the following section.

For each recharge realization, a series of equally probable realizations
of

The NSMC method takes advantage of the hybrid
Tikhonov–TSVD (truncated singular value decomposition) method in the PEST parameter
estimation code to produce Monte Carlo realizations of parameters

In each parameter realization, a large number of particles are injected through the top surface of the groundwater model. The spatial density of particles is proportional to the spatially distributed recharge rates.

To accurately interpret the travel time distribution, a large number of
particles (e.g., approximately 80 000 particles in the case study) is
released into the top surface of the groundwater model. The released
particles serve as samples of water parcels for deriving their travel time
distributions. In doing so, the density of particles is set proportional to
the recharge at the corresponding grid cell (Fig. ^{3} year

An ensemble of forward simulations using the RWPT method is performed over
all realizations of

In each realization of the ensemble parameter sets, forward simulations of particle tracking are performed. In this study, we focus on the predictive uncertainty within the convection process. Therefore, the molecular diffusion coefficients are universally set to 0 for all ensemble simulations. The porosity of the study domain is set to 0.2 universally. Through the above procedures, the flow paths and the corresponding residence times can be fully traced in the model at random times and locations, facilitating the detailed characterization of TTDs.

In parallel to this analysis, a sensitivity analysis for the spatial variability of recharge is also performed. Two different recharge scenarios are compared for this purpose: (1) the spatially distributed recharge generated by mHM and (2) the uniform recharge that is equal to the spatial average of the distributed recharge. Other parameters, including the porosity and the hydraulic conductivity, remain identical in these two recharge scenarios.

Two
different spatial distributions of particle tracers for the RWPT
method.

A high-resolution dataset of land surface fluxes and states across Germany is
used for sampling recharge scenarios. This dataset was established on the
basis of a daily simulation with 4 km spatial resolution using mHM for a
time span of 60 years (1951–2010)

Eight representative recharge realizations (R1–R8) are sampled from 100 realizations for this study to save computational time. To enhance the representativeness of the samples, the 100 recharge realizations are sorted in an ascending order by their spatial averages. The selected recharge realizations are uniformly sampled from the sorted recharge realizations. In doing so, the maximum and minimum recharges are included in the samples such that the whole range of recharge realizations is fully covered.

A 3-D stratigraphic mesh is established on the basis of hydrogeological
characterizations elaborated in Sect.

Box-plot of stochastically generated hydraulic conductivity
(

Observed and simulated groundwater heads for each parameter and recharge realization. The results of 400 realizations (R1K1–R8K50) are categorized by recharge realization and shown in different panels.

Multiple calibration-constrained

The travel time is defined as the time spent by a moving element (either a
water particle or a solute) in a control volume of a hydrologic system. In
principle, the control volume can be defined at arbitrary spatial scales
(i.e., from the molecular scale to the regional scale). Considering a
hydrologic system in which the input flux (

The SAS function describes the fraction of water parcels leaving the control
volume at time

Three instances of SAS functions using gamma distribution are shown in
Fig.

In the idealized saturated groundwater aquifer, Eq. (

The theoretical framework of predictive uncertainty in this paper is based on

While the computationally expensive Bayesian approach offers a complete
theoretical framework for predictive uncertainty evaluation, practical
modeling efforts are often based on model calibration and a subsequent
analysis of error or uncertainty in post-calibration predictions

For the sake of clarity, we number the recharge realizations from R1 (with
the lowest recharge rate) to R8 (with the highest recharge rate). For each
recharge realization, 50

3-D view of flow pathlines of some particles in realization R5K1. Note that only a limited number of particle pathlines are displayed here.

Flow paths of particle tracers in a random parameter realization (R5K1) are
displayed in Fig.

Figure

The exponential model under the RS assumption is fit to the ensemble-averaged
TTD of numerical solutions (see the black lines in Fig.

Based on Eq. (

Moreover, we assess the propagation to the MTT predictions from input and
parameter uncertainty yielded by the eight recharge realizations and the
Monte Carlo realizations of hydraulic conductivities.
Figure

Travel time distributions of ensemble simulations and analytical
solutions categorized by recharge realization. The orange lines show the
simulated TTDs of all realizations of

Effective groundwater storages related to the transport process for each recharge realization.

Uncertainty quantification: Monte Carlo simulations of MTT
predictions categorized by recharge realization.

Cumulative rank SAS functions as a function of normalized age-ranked
storage.

Figure

Sensitivity of

Figure

The difference in TTDs and SAS functions is not induced by the variability in
internal hydraulic properties, since the two simulations share the same

In the idealized aquifers where groundwater flow is of the
Dupuit–Forchheimer type, the recharge is uniform and the aquifer is locally
homogeneous. TTD is controlled
by recharge, saturated aquifer thickness, and porosity, and it is independent of
hydraulic conductivity

The mechanisms behind the effects of the recharge rate and the

We also underline the value of observational data in reducing predictive
uncertainty in simulated TTDs. In this study, the majority of model
parameters can be adequately conditioned by spatially distributed groundwater
head observations (Fig.

The analytical solution of TTD, assuming a random sampling of water, cannot
properly replicate the TTD of numerical simulation in the study domain. In
the stratigraphic aquifer with complex topography and diffuse recharge, the
analytical solution using Eq. (

It is obvious that the analytical solution under the RS assumption cannot
explicitly include the impact of the distributed hydraulic properties of
stratigraphic aquifers and the spatially nonuniform recharge. The above
limitations of analytical models may introduce a significant predictive error
for the TTD predictions, as shown in Fig.

The SAS function provides a good interpretability for simulation results
using the fully distributed model in terms of characterizing the preference
for releasing water of different ages. We find that the SAS functions are
weakly dependent on the

The sensitivity of the TTDs and SAS functions to the spatial pattern of recharge forcings can be explained mainly by the different flow paths of particle tracers, resulting primarily from the spatially heterogeneous fields of recharge across the study catchment. For the regional groundwater system, the spatial variation of recharge determines the distribution of starting points of the flow pathlines of tracer particles. For example, more particles will be injected from recharge zones that are typically located in high-elevation regions, resulting in a higher weight of flowlines starting from high-elevation regions. The pronounced spatial variability of recharge also controls the systematic (water age) preference for particles existing from the system (to river discharge) that originated from different regions and therefore exerts a strong control on the shape of the SAS function.

In the study catchment, an oversimplified spatially uniform recharge results in a smaller MTT and a stronger preference for discharging young water compared to those taking the spatial variability of recharge. Such observations are conditioned to site-specific features of the study catchment, which is noticed only when the following apply: (a) a site is located in a headwater catchment under a humid climate condition; (b) the recharge rate in areas close to the drainage network is generally lower than that in areas far away from the drainage network; and (c) the system is under (near) natural conditions, meaning that artificial drainage and pumping do not dominate the groundwater budget.

The assumption of spatially uniform input forcing has been widely applied in
regional-scale subsurface hydrologic models

Uncertainty limits the applicability of groundwater models. Most of the
applied groundwater models are deterministic models that use direct values of
inputs and parameters instead of probabilistic distributions of them.
Specifically, both the model inputs and the inversion process are
deterministic, leading to a deterministic best-fit parameter set achieved
during model calibration. Our study reveals limitations of the above modeling
procedure and suggests that the probabilistic distribution of inputs and
parameters should be considered for the applied modeling. The main limitation
is that the single exclusive assignment of recharge is inadequate for the
simulation of transport processes because the error can be propagated from
inputs to the conditioned parameters (e.g., hydraulic conductivities) through
the calibration process (Fig.

The degree of predictive uncertainty is highly dependent on the
parameterization scheme. Some highly parameterized models are potentially ill
posed due to the paucity of data and therefore cannot be constrained by the
available calibration dataset. In this case, the predictive uncertainty of
TTDs is potentially high

In this study, we explore the relationship between the uncertainty of
recharge, calibration-constrained hydraulic conductivity realizations, and
predictions of groundwater TTDs. Using both a physically based numerical
model and a lumped analytical model, a comprehensive case study is performed
in an agricultural catchment (the Nägelstedt catchment). The RWPT method
is used to track the water samples through the modeling domain and compute
their travel times. Moreover, the analytical model is fit against the numerical solutions to
provide a reference for the effective storage and sampling behavior of the
system. Based on this study, the following conclusions are drawn.

In the Nägelstedt catchment model, the simulated MTTs are strongly dependent on the
recharge rate and weakly dependent on the postcalibrated

The framework of the SAS function provides a good interpretability of simulated TTDs in terms of characterizing the systematic preference for sampling young/old water as outflow. On the basis of this framework, we find that the ensemble simulations have a consistent preference for young water, despite the different recharge and hydraulic conductivity realizations. Our study provides a novel modeling framework to explore the effect of input uncertainty and parameter equifinality on TTDs and SAS functions through a combination of calibration-constrained Monte Carlo parameter generation, a numerical model, and a SAS function framework.

Both the shape and the breadth of catchment groundwater TTDs and SAS functions are sensitive to the spatial distribution of recharge. Therefore, a reasonable characterization of the spatial pattern of recharge is crucial for the reliable TTD prediction in the catchment-scale groundwater models.

The source code of the coupled model mHM-OGS v1.0 can be
freely accessed via the following on-line repository:

Random walk particle tracking solves a diffusion equation at local Lagrangian
coordinates rather than the classical advection–diffusion equation, which
can be expressed as

The velocity

The stochastic governing equation of 3-D RWPT can therefore be expressed as

Composite parameter sensitivities to the groundwater head observations.

The PEST algorithm calculates the sensitivity with respect to each parameter
of all observations (with the latter weighted as per user-assigned weights),
namely the “composite sensitivity”

The authors declare that they have no conflict of interest.

This study receives support from the Deutsche Forschungsgemeinschaft via Sonderforschungsbereich CRC 1076 AquaDiva. We kindly thank Sabine Sattler from the Thuringian State office for the Environment and Geology (TLUG) for providing basic geological data. We acknowledge the EVE Linux Cluster team at UFZ for their support for this study. We also acknowledge the Chinese Scholarship Council (CSC) for supporting Miao Jing's stay in Germany. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Brian Berkowitz Reviewed by: Erwin Zehe and one anonymous referee