We propose an alternative model concept to represent rainfall-driven soil water dynamics and especially preferential water flow and solute transport in the vadose zone. Our LAST-Model (Lagrangian Soil Water and Solute Transport) is based on a Lagrangian perspective of the movement of water particles (Zehe and Jackisch, 2016) carrying a solute mass through the subsurface which is separated into a soil matrix domain and a preferential flow domain. The preferential flow domain relies on observable field data like the average number of macropores of a given diameter, their hydraulic properties and their vertical length distribution. These data may be derived either from field observations or by inverse modelling using tracer data. Parameterization of the soil matrix domain requires soil hydraulic functions which determine the parameters of the water particle movement and particularly the distribution of flow velocities in different pore sizes. Infiltration into the matrix and the macropores depends on their respective moisture state, and subsequently macropores are gradually filled. Macropores and matrix interact through diffusive mixing of water and solutes between the two flow domains, which again depends on their water content and matric potential at the considered depths.
The LAST-Model is evaluated using tracer profiles and macropore data obtained at four different study sites in the Weiherbach catchment in southern Germany and additionally compared against simulations using HYDRUS 1-D as a benchmark model. While both models show qual performance at two matrix-flow-dominated sites, simulations with LAST are in better accordance with the fingerprints of preferential flow at the two other sites compared to HYDRUS 1-D. These findings generally corroborate the feasibility of the model concept and particularly the implemented representation of macropore flow and macropore–matrix exchange. We thus conclude that the LAST-Model approach provides a useful and alternative framework for (a) simulating rainfall-driven soil water and solute dynamics and fingerprints of preferential flow as well as (b) linking model approaches and field experiments. We also suggest that the Lagrangian perspective offers promising opportunities to quantify water ages and to evaluate travel and residence times of water and solutes by a simple age tagging of particles entering and leaving the model domain.
Until now, the most commonly used hydrological models have followed an Eulerian perspective of the flow processes, with a stationary observer balancing dynamic changes in a control volume. The alternative Lagrangian perspective with a mobile observer travelling along the trajectory of a solute particle through the system (Currie, 2002) has up to now only been used to simulate advective–dispersive transport of solutes (Delay und Bodin, 2001; Zehe et al., 2001; Berkowitz et al., 2006; Koutsoyiannis, 2010; Klaus and Zehe, 2011). However, this particle tracking approach is mostly embedded in frameworks with Eulerian control volumes which still characterize the dynamics of the carrying fluid. Lagrangian descriptions of the fluid dynamics itself are only realized in a few models. But such a particle tracking framework may offer many advantages, especially in coping with the challenges induced by preferential water flow and solute transport in structured heterogeneous soils.
Preferential flow has become a major issue in hydrological research since the benchmark papers of, Flury et al. (1994), Uhlenbrook (2006) and Beven and Germann (2013). The term preferential flow is used to summarize a variety of mechanisms leading to a rapid water movement in soils. The most prominent one is the flow through non-capillary macropores (Beven and Germann, 2013) where water and solutes travel in a largely unimpeded manner due to the absence of capillary forces and bypass the soil matrix (Jarvis, 2007). Macropores can be classified into e.g. earth worm burrows, channels from degraded plant roots or shrinkage cracks, and all of them are non-static in space or time (e.g. Blouin et al., 2013; Nadezhdina et al., 2010; Palm et al., 2012; van Schaik et al., 2014; Schneider et al., 2018). Especially in rural areas and in combination with agrochemicals, macropore flow can be a dominant control on stream-water and groundwater pollution (e.g. Flury, 1996; Arias-Estévez et al., 2008). To understand such water and solute movements, a combination of plot-scale experiments and computer models is commonly used (Zehe et al., 2001; Šimůnek and van Genuchten, 2008; Radcliffe and Šimůnek, 2010; Klaus et al., 2013). One of the most frequently used approaches to simulate water flow dynamics and solute transport is to use the Darcy–Richards and advection–dispersion equations. Both equations fundamentally assume that solute transport is controlled by the interplay of advection and dispersion (Beven and Germann, 2013) and that the underlying soil water dynamics are dominated by capillary-driven diffusive flow. While the second assumption is well justified in homogeneous soils, it frequently fails in soils with macropores. Consequently, we separate at least two flow regimes in soils: the slow diffusive flow in the soil matrix and the rapid advective flow in the macropores. Partial mixing between these two flow regimes is non-trivial, as it depends on the hydraulic properties of the macropore walls, the water content of the surrounding soil, actual flow velocities, hydrophobicity of organic coatings and much more. The inability of the Richards equation to simulate partial mixing between both flow regimes is well known and a variety of different models have been proposed to address this problem (Šimůnek et al., 2003; Beven and Germann, 2013). But most of them are still fundamentally based on the Darcy–Richards equation, like the most prominent and well-established double-domain models, for instance the HYDRUS model of Šimůnek and van Genuchten (2008).
A promising alternative approach is provided by particle-based Lagrangian
models for subsurface fluid dynamics. The first implementation of such a
model for soil water dynamics is the SAMP model proposed by Ewen (1996a,
b). SAMP represents soil water by a large number of particles travelling in
an one-dimensional soil domain by means of a random walk which is based on
soil physics and soil water characteristics. A more recent example is the
two-dimensional MIP model of Davies et al. (2013) developed for
hillslopes. Fluid particles travel according to a distribution function of
flow velocities which needs to be estimated from tracer field experiments.
Exchange of particles among the different pathways is conceptualized as
a random process following an exponential distribution of mixing times.
Inspired by the SAMP model, Zehe and Jackisch (2016) conceptualized a
Lagrangian model describing soil water flow by means of a non-linear space
domain random walk. In line with Ewen (1996a, b), they estimated the
diffusivity and the gravity-driven drift term of the random walk based on
the soil water retention curve (
The particle-based Lagrangian model of Zehe and Jackisch (2016) initially assumed that all particles travel at the same diffusivity and velocity corresponding to the actual soil water content. But a comparison to a Richards solver revealed that this straightforward, naive random walk implementation highly overestimates infiltration and redistribution of water in the soil. The solution for this overestimation was to account for variable diffusive velocities. Now, particles in different pore sizes travel with various diffusivities, which are determined based on the shape of the soil hydraulic conductivity curve. This approach reflects the idea that the actual soil water content is the sum of volume fractions that are stored in different pore sizes and that the different pore sizes constitute flow paths which differ in both advective and diffusive velocities.
Recently, this model was advanced by Jackisch and Zehe (2018) with the implementation of a second dimension which contains spatially explicit macropores to simulate preferential flow. Within a macropore the velocity of each particle is described by interactions of driving and hindering forces. The driver is the potential energy of a particle, while energy dissipation due to friction at the macropore walls dissipates kinetic energy and accordingly reduces particle velocities. With this approach, Jackisch and Zehe (2018) tried to make maximum use of observables for model parameterization. The assets of their echoRD model are a self-controlling macropore film flow and its ability to represent two-dimensional infiltration patterns. The drawback of echoRD is the huge computational expense. The simulation time is about 10 to 200 times longer than real time.
The huge computational expense of the echoRD model is one main motivation
for us to develop a Lagrangian approach which balances necessary complexity
with greatest possible simplicity. The other motivation is the inability of
all the models mentioned above to simulate solute transport appropriately. This
is essential for a rigorous comparison of the model with tracer data and to
get closer to the simulation of reactive transport. Thus, the main
objectives of this study are to
present a new routine for solute transport and diffusive mixing for
well-mixed matrix flow conditions which are implemented in the model of
Zehe and Jackisch (2016) and to test this approach against tracer data
from plot-scale experiments carried out in the Weiherbach catchment (Zehe
and Flühler, 2001b); extend the model by implementing a macropore domain accounting for
preferential flow of water and solutes and related exchange with the matrix
domain. In contrast to the echorRD model, we maintain the one-dimensional
approach to keep the computational expense moderate.
The structure of our LAST-Model (Lagrangian Soil Water and
Solute Transport) is hence similar to a double-domain
approach. The main asset is that flow and transport in both domains and
their exchange are described by the same stochastic physics and that the
macropore domain can be parameterized by observable macropore geometries.
This fact may help to overcome the limiting assumptions of the
Darcy–Richards and advection–dispersion equations. The refined LAST-Model
is tested by extensive sensitivity analyses to corroborate its physical
validity. Further, it is also tested with four tracer infiltration
experiments at different study sites in the Weiherbach catchment which are
dominated by either well-mixed conditions (sites 23, 31) or preferential
flow in macropores (sites Spechtacker, 33). For comparison, these four
experiments are also simulated with HYDRUS 1-D.
The basis of our development is the Lagrangian model of Zehe and Jackisch (2016). It describes infiltration and water movement through a spatially
explicit one-dimensional soil domain dependent on the effects of gravity and capillarity
in combination with a spatial random walk concept. Water is represented by
particles with constant mass and volume. The density of soil water particles
in a grid element represents the actual soil water content
Concept of particle binning. All particles within a grid element are subdivided into bins (red rectangles) of different pore sizes. Depending on their related bin, the particles travel at different flow velocities.
In a first step we implement a routine for solute transport into the
particle model by assigning a solute concentration
The second and main model extension is the implementation of a one-dimensional
preferential flow domain considering the influence of macropores on water
and solute dynamics. This requires four main steps.
Design of a physically based structure of the preferential flow domain Conceptualization of the infiltration and partitioning of water into the two
domains Description of advective flow in the macropores Conceptualization of water and tracer exchange between the macropore and the
matrix domain
We define a one-dimensional macropore or preferential flow domain (pfd) which is surrounded by a one-dimensional soil matrix domain with vertically distinct boundaries. In line with other Lagrangian models, we represent water as particles with constant mass and volume corresponding to their domain affiliation. As the vertical extent and volume of the pfd are much smaller than those of the matrix domain, the corresponding particles must be much smaller to ensure that an adequate number of particles travel within the pfd for a valid stochastic approach.
Conceptual visualization of
The pfd comprises a certain amount of macropores. Each macropore has the
shape and structure of a straight circular cylinder with a predefined length
Our one-dimensional approach can of course not account for the lateral positions of the
macropores, but the pfd allows a depth distribution of macropores, which is
important for calculating the depth-dependent exchange with the matrix
(Sect. 2.3.4). To calculate the water content and tracer concentrations,
the macropores of the pfd are vertically subdivided into grid elements of
a certain length dz
In a cubic packing the particles are arranged in the way that the centres of
the particles form the corners of a cube. The concept of cubic packing
facilitates the calculation of the proportion of particles having contact with
the lateral surface of a grid element. The rectangle in Fig. 2a describes
such a lateral surface of a grid element, with a height corresponding to the
grid element length dz
As a one-dimensional approach does not allow an explicit, spatial distribution of the incoming precipitation water over the soil surface, we use an implicit, effective infiltration concept. The infiltration and distribution of water are controlled by the actual soil moisture and the flux densities driven by the hydraulic conductivity and the hydraulic potential gradient of the soil matrix as well as by friction and gravity within the macropores (Weiler, 2005; Nimmo, 2016; Jackisch and Zehe, 2018). For example, a soil matrix with a low hydraulic conductivity increases the proportion of water infiltrating the macropores as it preferentially uses pathways of low flow resistance.
In our model, we use a variable flux condition at the upper boundary of the
soil domain dependent on the precipitation intensity. Incoming precipitation
water accumulates in an initially empty fictive surface storage from which
infiltrating water masses and related particle numbers are calculated. To
this end, we distinguish several cases. In Case 1, the top soil grid
elements of the soil matrix and the pfd are initially unsaturated and the
infiltration capacity of the soil matrix is smaller than the incoming
precipitation flux density. Water infiltrates the soil matrix and the excess
water is redistributed to the pfd and infiltrates it with a
macropore-specific infiltration capacity. Case 2 applies when the top matrix
grid element is saturated and water exclusively infiltrates the pfd until
all macropores are also saturated. Case 3 occurs when both the top matrix
layer and the pfd are saturated, leading to an accumulation of precipitation
water in the surface storage. As soon as the water contents in the first
soil matrix grid element and in the pfd are subsequently decreasing due to
downward water flow or drainage of the macropores, infiltration again occurs
according to Case 1. The incoming precipitation mass (
According to Eq. (3), the infiltration rate into the matrix is based on Darcy's law, and thus we are generally able to account for an extra pressure due to a ponded surface, e.g. in Case 3. But in our simulation cases, ponding heights are small and have only a marginal effect. After the precipitation water has infiltrated into the two domains, the masses are converted to particles which are initially stored in the first grid elements of the matrix and pfd. They are now ready for the displacement process.
In the pfd, we assume a steady-state balance between gravity and dissipative
energy loss at the macropore walls. This implies purely advective flow
characterized by a flow velocity
Commonly, macropore–matrix interactions are challenging to observe within field experiments. One approach is to evaluate the isotopic composition of water in the two domains (Klaus et al., 2013). In theory it is often assumed that the interactions and water dynamics at the interface between macropores and the matrix are mainly controlled by the matric head gradients and the hydraulic conductivity of both domains which depend on an exchange length and the respective flow velocities (Beven and Germann, 1981; Gerke, 2006).
Our model approach is also based on these assumptions as illustrated in Fig. 2c. We restrict exchange to the saturated parts of the pfd, assuming downward particle transport to be much larger than the lateral exchange, and we neglect diffusive exchange between solutes in the matrix and the pfd. We are aware that these simplifications might constrain the generality of our model. For instance, we also neglect the effect of a reverse diffusion from the matrix into the macropores. This effect can influence water and solute dynamics when the propagation of a pressure wave pushes matrix water into empty macropores, mainly in deeper-saturated matrix areas (Beven and Germann, 2013). We rely on those simplifications (a) to keep the model simple and efficient and (b) because the focus of our model is on unsaturated soil domains and during rainfall-driven conditions the macropores are most of the time largely filled due to their small storage volume.
The distribution of different macropore depths and the definition of
distribution factors can be derived from datasets containing information on
macropore networks observed in field experiments as described in Sect. 3.2. Based on these datasets, the current version of our model divides the
total amount of macropores nmac in the pfd into three depths. To this end, the
total number is multiplied by a distribution factor
The saturated grid elements (blue filled) of the largest macropores are
coupled to the respective grid elements of the medium and small macropores.
In this example, the red and black framed grid elements of the
three macropore sizes are coupled due to their saturation state and depth
order. This coupling ensures a simultaneous diffusive water flow out of the
respective grid elements of all three macropore depths. The mixing fluxes
(
The mixing masses are again converted into particle numbers with the two different particle masses. Due to the higher masses of the matrix particles a much lower number of particles enters the matrix. This has to be taken into account by choosing an adequate number of total particles present in the matrix, i.e. at least 1 million at moderately saturated hydraulic conductivities. In addition, it is ensured that the number of particles leaving a grid element of the pfd is lower than the maximum possible number of particles having contact with the lateral surface (cf. Sect. 2.3.1) dependent on its current soil water content. Please note that up to now our model has worked with a no-flow condition at the lower boundary of the pfd, but the model structure is generally capable of adding an additional diffusive drainage with particles leaving the macropores at their lower boundary.
The bases of the first evaluation of our solute transport and linear mixing
approach are data from tracer experiments conducted by Zehe and Flühler (2001b) in the Weiherbach catchment to investigate mechanisms controlling
flow patterns and solute transport. The Weiherbach Valley is located in the
southwest of Germany and has a total extent of 6.3 km
In this catchment, a series of irrigation experiments with bromide as tracer
were performed at 10 sites. At each site, a plot area of 1.4 m
Thus, in every 10 cm soil depth interval, 10 samples were taken and, for the
comparison with our one-dimensional simulation results, the bromide concentrations were
averaged over each sample depth. Note that the corresponding observations
provide the tracer concentration per dry mass of the soil
Simulation and tracer experiment parameters (average values) as
well as soil hydraulic parameters following Schäfer (1999) at sites
23, 31, Spechtacker and 33, where
The soil at the two sites can be classified as Calcaric Regosol (IUSS Working Group WRB, 2014). In line with the experiments, our model uses a spatial soil matrix discretization of 0.1 m and the soils initially contain in total 1 million water particles but with no tracer masses. Initial soil water contents and all further experimental and model parameters as well as the soil properties at these sites are listed in Table 1.
In a next step, our pfd model extension is again evaluated with the help of the results of two additional field tracer experiments of Zehe and Flühler (2001b). This time, we select study sites Spechtacker and 33, which show numerous worm burrows inducing preferential flow. The sites are also located in the Weiherbach catchment and the sprinkling experiments were equally conducted with the application of a block rainfall containing bromide on a soil plot. The soils can be classified as Colluvic Regosol (IUSS Working Group WRB, 2014).
Additionally, the patterns of the worm burrows were extensively examined at
these study sites. Horizontal layers at different depths of the vertical
soil profiles were excavated (cf. introduction of van Schaik et al., 2014)
and in each layer the amount of present macropores counted as well as the
diameters and depths measured. These detailed measurements provided an
extensive dataset of the macropore network at study sites Spechtacker
and 33. Based on this dataset, we can obtain those data we need for the
derivation of a mean macropore diameter, macropore depth distribution and
distribution factors. We focus on a mean macropore diameter of 5 mm at the
Spechtacker site because worm burrows with a diameter range of roughly 4–6 mm are dominant here, and at site 33 we select a mean diameter of 6 mm.
Figure 3 shows the mean number of macropores with these diameters at each
depth at both sites. Based on this distribution, we can identify and select
three considerable macropore depths at the Spechtacker site (0.5, 0.8
and 1.0 m) and two macropore depths at site 33 (0.6 and 1.0 m) (cf. Table 1). At these depths, we count circa 11, 3 and 2 macropores (nmac
Distribution of macropore numbers with average diameters of 5 mm (Spechtacker) and 6 mm (site 33) along the vertical soil profiles at the two study sites. The arrows highlight the derivation of the macropore numbers at different depths (cf. Sect. 3.2), whereby “Avg.” means that at these depths the macropore numbers are averaged because there was no clear macropore pattern observed.
Moreover, Zehe and Flühler (2001b) measured saturated water flow through
a set of undisturbed soil samples containing macropores of different radii
at the Spechtacker study site with the assumption that flow through these
macropores dominated. In line with the law of Hagen and Poiseuille, they found a
strong proportionality of the flux through the macropores to the square of
the macropore radius, while frictional losses were 500 to 1000 times larger.
This dependence of the flux rate on the macropore radius can be described by
the linear regression shown in Fig. 4. Based on this linear regression,
the hydraulic conductivity of the macropores
Linear regression of the flux rate within the macropore
on the macropore radius (
The simulations with HYDRUS 1-D are performed with the same soil properties,
model setups and initial conditions introduced in Sect. 3.1 and 3.2
as well as shown in Table 1. The simulations of the well-mixed sites 23 and
31 are performed with a van Genuchten–Mualem single-porosity model for
water flow and an equilibrium model for solute transport. For the
simulations at the preferential flow sites Spechtacker and 33 we use
dual-porosity models for water flow (“Durner, dual van Genuchten–Mualem”) and solute transport (“Mobile–Immobile Water”). This means
HYDRUS assumes two differently mobile domains to account for preferential
flow. The theory of that approach describes preferential flow in the way
that the effective flow space is decreased due to the immobile fraction and
thus the same volume flux is forced to flow through this decreased flow
space, resulting in higher porewater velocities and consequently also in a
deeper percolation of water and solutes (Šimůnek and van Genuchten,
2008). For the parameterization of these two domains we select an immobile
soil water content ThImob. of 0.2 m
The sensitivity analyses of the model with the pfd extension are conducted
by varying several parameters describing the soil matrix and the pfd in a
realistic, evenly spaced value range. To this end, the saturated hydraulic
conductivity of the matrix
Parameter ranges of the sensitivity analyses and configurations of macropore depth distribution and distribution factors (cf. Fig. 10).
All model runs of the sensitivity analyses are performed at the Spechtacker site using 22 mm of rainfall in 140 min with a subsequent drainage duration of 1 d. Additional parameters like soil properties, antecedent moisture and concentration states, and bromide concentration of precipitation water remain constant (cf. Table 1).
The well-mixed sites 23 and 31 show a high similarity due to their spatial
proximity (Fig. 5a, b). The shape and courses of the simulated tracer mass
profiles coincide well with the observed ones over the entire soil domain,
with RMSE values of 0.23 and 0.28 g, respectively. The observed values are
within the uncertainty range, represented by the rose shaded areas. This
area reflects the uncertainty arising from a variation of
Final simulated and observed vertical bromide mass profiles of the
matrix at the two well-mixed sites 23 and 31
Note that in the experiments the tracer mass was not directly measured at
the soil surface, but the observations represent averages across 10 cm depth
increments, starting at a depth of 5 cm. A comparison of the simulated
masses close to the surface is thus not meaningful. This difference between
simulated and observed profiles near to the surface suggests that the coarse
resolution of the sampling grid is a likely reason for the relatively low
recovery rates of 77 % and 76 % at the two sites (cf.
Table 1). Overall, we conclude that manipulating
Our model with the new preferential flow domain is tested against two tracer
experiments on macroporous soils at sites Spechtacker and 33. At the
Spechtacker site, the simulated and observed tracer mass distributions are
generally in good accordance (Fig. 6a) with a RMSE of 0.3 g, and again the
values are within the uncertainty range. In this case, the rose area shows
the standard deviation of measured macropore numbers (
Final simulated and observed vertical bromide mass profiles of the
matrix at the two preferential flow sites Spechtacker and 33
In general, the simulated mass profile at site 33 corroborates the results
of the Spechtacker site (Fig. 6b). The simulated and observed tracer
masses are also in good accordance with a RMSE value of 0.15 g. In
contrast to the Spechtacker site, varying the macropore numbers and
diameters within the standard deviation (
The mass profiles at the well-mixed sites 23 and 31 simulated with HYDRUS
1-D show similar patterns and are in accordance with the observed profiles
with RMSE values of 0.1 g at site 23 and 0.37 g at site 31 (Fig. 5c, d).
Especially at site 23 the simulated mass profile is centred within the
uncertainty range of the measured
The concentration profile range of the matrix reveals a strong sensitivity
of the simulated profiles to
Final simulated bromide concentration (Cs) and soil moisture
(theta) profiles of the soil matrix
In contrast, at high
Final bromide concentration profiles at
Finally, the yellow curves in Fig. 8 show the proportion of solutes within
the matrix which originates from the macropores. In general, at all
The model results sensitively respond to a variation of macropore diameters. In the upper soil part, the solute concentrations and moisture are slightly higher, when macropores are small (Fig. 9a, b). In this case, the macropores collect only smaller amounts of water and solutes and the majority has directly infiltrated the soil matrix. Wider macropores transport larger amounts of water and solutes to greater depths, where they diffusively mix into the subsoil matrix. This deep redistribution is reflected by the characteristic profile shapes and the higher concentration and moisture values in the deep soil.
Final simulated bromide concentration (Cs) and soil moisture
(theta) profiles of the soil matrix at different macropore diameters (dmac)
Furthermore, the influence of different macropore numbers on the concentration and moisture profiles is marginal (Fig. 9c, d). This implies that the model does not respond to every geometrical parameter equally sensitively. The macropore number scales less than the diameter at the calculation of the further macropore measures. However, this could change when working with higher precipitation intensities.
Simulations with different macropore depth configurations again reveal a clear sensitivity of the model (Fig. 10a, b). A steady decrease in the deep redistribution of the concentration and moisture values from the deep (Configuration 1) to shallow depth configuration (Configuration 3) is obvious. Shallow macropores distribute the total amount of water and solutes mainly in the upper soil part, while deep macropores relocate this distribution to greater depths of down to 1 m. The results of the distribution factor configurations again corroborate the previous findings (Fig. 10c, d). Configuration B produces a homogeneous solute concentration profile from 0.2 m to the total depth. Both more realistic Configurations A and D comprise more small than big macropores. This increased number of small macropores ensures higher water and solute amounts in the first 0.5 m of the soil matrix due to an enhanced mixing in this area. Finally, the rather uncommon Configuration C with more big than small macropores shows converse results. Solute concentrations and moisture contents are strongly increased at great depths from 0.7 to 1 m because of increased diffusive mixing fluxes in these parts.
Final simulated bromide concentration (Cs) and soil moisture
(theta) profiles of the soil matrix at three different macropore depth
distribution configurations
We extend the Lagrangian model of Zehe and Jackisch (2016) with routines to consider transport and linear mixing of solutes within the soil matrix as well as preferential flow through macropores and related interactions with the soil matrix. The evaluation of the model with data of tracer field experiments, the comparison with results of HYDRUS 1-D and the sensitivity analyses reveal the feasibility and physical validity of the model structure as well as the robustness of the solute transport and linear mixing approach. The LAST-Model provides a promising framework to improve the linkage between field experiments and computer models to reduce working effort and to improve the understanding of preferential flow processes.
The initially performed simulations of the bromide mass profiles at the two well-mixed sites 23 and 31 support the validity of the straightforward assumptions of the underlying solute transport routine with its perfect mixing approach (Fig. 5a, b). In the presented version, our mixing routine works with a short mixing time to ensure an instantaneous mixing between event and pre-event particles to account for the well-mixed conditions at the selected sites. However, the model allows us to select longer mixing times or even a distribution of various mixing times to consider imperfect mixing among different flow paths.
The simulation results at the well-mixed sites 23 and 31 are confirmed by the commonly approved HYDRUS 1-D model. The simulated tracer mass profiles and RMSE values of both models are in good accordance at these sites (Fig. 5). The capability of predicting the solute dynamics is hence a big asset of our approach, and it is a solid base to realize the second model extension with the implementation of the preferential flow domain.
The results of the evaluation of the pfd extension show that our model is furthermore capable of simulating tracer experiments on macroporous soils and depicting well their observed one-dimensional tracer mass profiles with the typical fingerprint of preferential flow (Fig. 6a, b). Especially the tracer masses in the subsoil match well between simulated and observed data. This corroborates our assumptions concerning the macropore structure and the approach to describing macropore–matrix exchange which proved to be feasible for predicting solute distribution patterns due to preferential flow and related long transport lengths. In this context, we stress that the approach to simulating macropore–matrix exchange (cf. Fig. 2c) does not rely on an extra leakage parameter, but follows the theory of deriving an effective diffusive exchange between the domains (cf. Eq. 6).
In contrast, the HYDRUS 1-D model performance is clearly inferior and does not match the fingerprints of preferential flow in the mass profiles at sites Spechtacker and 33 (Fig. 6c, d). Especially the penetration of bromide through macropores into greater depths is ignored by HYDRUS 1-D, although we selected dual-porosity models for both water flow and solute transport (cf. Sect. 3.3). The better performance of our LAST-Model at the two preferential flow sites compared to HYDRUS is further reinforced by the RMSE values which are significantly different. The results imply that, when working with a dual-porosity approach, HYDRUS and the underlying theory of two differently mobile domains is indeed able to depict a generally deeper penetration of solutes, but it is not sufficient to exactly simulate the heterogeneous course and shape of the observed tracer mass profiles in preferential flow-dominated soil domains.
The results of our LAST-Model mainly deviate from the observations in the upper soil parts. However, these deviations are within the uncertainty ranges revealed by the sensitivity analyses (Figs. 7, 9). Further, the model reveals difficulties in the simulation of bromide masses between 0.15 and 0.35 m soil depth at the Spechtacker site (Fig. 6a). Possible reasons could be the influence of (a) lateral endogeic worm burrows which are completely unknown and not represented in the model and (b) a nearby plough horizon. Both reasons result in a disturbance of the soil structure, leading to an increased uncertainty of soil properties in this region.
At site 33, our model is not able to sufficiently reproduce the hump of the
observed mass profile between 0.25 and 0.45 m soil depth (Fig. 6b). A
possible explanation for this issue could be the fact that the tracer
experiment and the examination of the macropore network were performed on
different dates. It is likely that uncertainties arise from this temporal
discrepancy with a mismatch between observed macropore geometries and
recovered tracer patterns due to natural soil processes as well as
anthropogenic soil cultivation during this time lapse. Another possible
explanation could be the fact that up to now the exchange has only been simulated
for saturated parts of the pfd (cf. Sect. 2.3.4) and hence the transport
of solute masses from the pfd into the matrix is delayed. A test of this
idea requires a refinement of the model in future research. Moreover,
varying macropore numbers and diameters in the range of the standard
deviation reveals just slight effects on the simulated mass profile at site
33 and is thus less sensitive compared to the results at the
Spechtacker site. The reason for this phenomenon is probably the higher total
number of macropores (nmac
Note that the conversion of solute masses into an integer number of particles results in small errors, leading to a small amount of solutes not entering the system and remaining in the fictive surface storage. To mitigate this model effect, a high number of total particles present in the matrix is necessary, at least 1 million. Besides many displacement steps of each particle, the total number of particles is important to render the random walk approach statistically valid (Uffink, 1990), although too high particle numbers will decrease the computational efficiency. Thus, we conclude that our extension of the Lagrangian particle model of Zehe and Jackisch (2016) is a promising tool for a straightforward one-dimensional estimation of non-uniform solute and water dynamics in macroporous soils. However, before the suitability of our model approach to simulate preferential flow of non-interacting tracers is generalized, further field experiments on a variety of differently structured soils are necessary. In the presented model version, we assume that a macropore distribution with maximally three different depths is a sufficient approximation of the observed macropore networks at study sites Spechtacker and 33 (cf. Sect. 3.2, Fig. 3). Nevertheless, as a variable macropore depth distribution might be observed at other sites, the implementation of the macropore depth distribution must be kept flexible for other soils in future model parameterizations. Besides the parameterization with experimental data, it is also possible to set up our model by using pedotransfer functions for the soil hydraulic properties and to vary the parameters of the pfd by inverse modelling, which needs prior knowledge of the depth of typical macropore systems (e.g. worm burrow networks) and literature data to parameterize macropore flow velocities. This method would reduce time and the amount of work, but it could result in equifinality as shown by Klaus and Zehe (2010) or Wienhöfer and Zehe (2014).
Some of our assumptions, like the macropore geometry, the simple volume filling or the depth distribution of macropores, were applied in a similar way in dual-porosity models before (Beven and Germann, 1981; Workman and Skaggs, 1990; van Dam et al., 2008), and a few previous studies even also worked with physically and geometrically separated domains (e.g. Russian et al., 2013). Thus, our model extension can be seen as an advancement of double-domain approaches by assuming simple volume-filling for macropore flow and particle tracking for matrix flow instead of relying on the Darcy–Richards equation. With these results, our model is one of the first which proves that simulations based on a Lagrangian perspective of both solute transport and dynamics of the carrying fluid itself are possible and very applicable. Also, the vertically distributed exchange between both domains seems feasible and does not rely on extra parameters like a leakage coefficient, e.g. as used in dual models (Gerke, 2006). The concept of cubic particle packing within the macropores (cf. Fig. 2a, Sect. 2.3.1) is strongly motivated by the hydraulic radius and can thus be transferred to flow in further kinds of macropore geometries, including flow between two parallel walls as occurs in soil cracks or corner flow in rills (Germann, 2018).
Another remarkable result is the high model sensitivity towards the
saturated hydraulic conductivity
Our model shows further a remarkable sensitivity to the presence of a population of macropores, while differences in macropore properties comparatively have little impact. Generally, wider macropores collect and transport more water and solutes to greater depths than small ones (Fig. 9a, b). In contrast, high numbers of macropores do not necessarily result in a greater and deeper percolation of solutes (Fig. 9c, d). Jackisch and Zehe (2018) also reported this aspect and explain it with the distribution of the irrigation supply to all macropores, and this supply can drop below the diffusive mixing fluxes from the macropores into the matrix. However, this implies that the number of macropores becomes more sensitive at much larger irrigation rates.
Where and to which extent water and solutes are diffusively mixed from the macropores into the matrix clearly depend on the depth distribution of the macropores and the distribution of the mixing masses among the various depths (Table 2, Fig. 10). This concept of the distribution of macropore depths and mixing masses is important to meet the natural condition of a high spatial heterogeneity of the macropore network. Generally, the results of our sensitivity analyses are in line with the findings of Loritz et al. (2017) as they reveal a significant impact of the implementation of macropore flow on the model behaviour and its complexity.
Please note that we are aware of the fact that some results of the sensitivity analyses are straightforward and expectable. Nevertheless, we think that their presentation is necessary to allow the reader to check whether our Lagrangian approach with the macropore domain reproduces these results as the model concept is new. To this end, please also see further sensitivity analyses in the Appendix.
We overall conclude that the modified one-dimensional structure of our model is robust and provides a high computational efficiency with short simulation times, which is a big advantage of our model. In line with the underlying Lagrangian model of Zehe and Jackisch (2016), we also used the MATLAB programming language to develop the two model extensions. The model simulation at the Spechtacker site with the selected parameterization (cf. Table 1) only runs for about 5 min, even on a personal computer with moderate computing power. Without an active pfd, as is the case for the simulations at study sites 23 and 31, the model runs even faster. When performing these simulations on a high-performance computer or workstation, one could probably also run several model simulations in parallel within minutes.
Moreover, the efficiency allows for the implementation of further routines with as yet still appropriate simulation times. In this way, the model could prospectively consider retardation and adsorption effects as well as first-order reactions during the transport of non-conservative substances like pesticides. Until now, the solute movement of conservative tracers like bromide has only been determined by the water flow without any consideration of molecular diffusion or particle interactions, although some evidence suggests a non-conservative behaviour of bromide tracers under certain conditions (e.g. Whitmer et al., 2000; Dusek et al., 2015). In our case, we believe that the event scale and the short simulation times allow for the assumption of a conservative behaviour of bromide.
Moreover, the model can be extended to two-dimensional for simulations on hillslope or even catchment scales. In this regard, our model also offers the promising opportunity to quantify water ages and to evaluate travel and residence times of water and solutes by a simple age tagging of particles. This can shed light on the chemical composition and generation of runoff fluxes as well as on the inverse storage effect. This effect describes a greater discharge fraction of recent event water at a high catchment water storage than at low storage (Hrachowitz et al., 2013; Harman, 2015; Klaus et al., 2015; van der Velde et al., 2015; Sprenger et al., 2018).
The LAST-Model, reference data and presented test experiments are
available in a GitHub repository:
We performed additional sensitivity analyses to determine the effect of
different
Figure A1 generally confirms the findings of the sensitivity analyses with
different
Moreover, the temporal development of the concentrations is similar for all macropore diameters, with just marginal differences arising shortly after the rainfall event (Fig. A2). While the macropore diameter has a minor influence in the initial phase, stronger differences occur at the end of the simulation when the residual water and solute amounts of the fictive surface storage have finally infiltrated. Thus, mainly at the end of the simulations the influence of the macropores on the infiltration and the macropore–matrix mixing processes are remarkable, because the storage volume of the preferential flow domain is small and hence it can only collect small amounts of water and solutes in relation to the matrix domain. The centres of mass corroborate the results of Fig. 9a, b in a way that the big macropores have the tendency to transport more solute masses into the subsoil.
Time series of bromide tracer concentration profiles and
centres of mass at different
Time series of bromide tracer concentration profiles and centres
of mass at different macropore diameters (dmac) during the rainfall event
AS wrote the paper, did the main code developments and carried out the analysis. RL, WW and EZ contributed to the theoretical framework and helped with interpreting and editing.
The authors declare that they have no conflict of interest.
This research contributes to the Catchments As Organized Systems (CAOS) research group (FOR 1598) funded by the German Science Foundation (DFG).
The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.
This paper was edited by Alberto Guadagnini and reviewed by three anonymous referees.