Non-uniform infiltration and subsurface flow in structured soils is observed in most natural settings. It arises from imperfect lateral mixing of fast advective flow in structures and diffusive flow in the soil matrix and remains one of the most challenging topics with respect to match observation and modelling of water and solutes at the plot scale.

This study extends the fundamental introduction of a space domain random walk
of water particles as an alternative approach to the Richards equation for
diffusive flow

The model proves to be a powerful alternative to existing dual-domain models, driven by experimental data and with self-controlled, dynamic macropore–matrix exchange from the topologically semi-explicitly defined structures.

Non-uniform subsurface flow is omnipresent in hydrology

Macropore settings may be very specific with respect to their topology, their
temporal dynamics and their interface characteristics in their
ecohydrological context: earthworm burrow configurations

All of these influences are rather complex and specific in detail. In addition, they challenge the model concepts since the advective processes take place in explicit structures with respective connectivity and spatial covariance and under conditions that are far from well-mixed. They extend across several scales in space and time.

Non-uniform flow arises from imperfect lateral mixing between a fast
advective fraction of water and solutes (travelling mainly driven by gravity
in large pores and soil structures) and a slow diffusive fraction (governed
by capillary forces in the soil matrix)

Despite the fact that there has been considerable progress in the
understanding of preferential flow and non-uniform infiltration, the topic
remains one of the most challenging in particular with respect to scale and
sub-scale representation of rapid subsurface flow and transport in
hydrological models

We thus propose a stochastic–physical model framework to jointly predict rapid advective water flows in soil structures and diffusive water flows when capillarity controls soil water dynamics, and the interaction between the two. The approach is developed for a representative plot domain with topologically explicit macropores. An overall goal of the model framework is to provide opportunities for virtual experiments on infiltration patterns and abiotic controls on specific niches for macro- and microbiota in structured subsurface domains.

The proposed model is a Lagrangian approach, treating water itself as
particles moving diffusively by means of a space domain random walk and
advectively as film flow in representative structures. Lagrangian approaches
to solute transport with unsaturated flow in heterogeneous media are well-established tools in hydrological modelling

We have shown in a previous study

The main objectives of this study are to (a) present the model theory, to (b) test the capability of the echoRD model to simulate the fingerprints' non-uniform infiltration and to (c) reveal whether advective and diffusive flow and the interactions between them may be represented in one consistent formulation. As the model shall allow for virtual experiments we base its parameterisation as much as possible on field observables or explicitly testable hypotheses. More specifically, we derive and test an energy-balance-based approach to control the exchange between the macropore domain and the surrounding matrix in a self-limiting manner.

The software developed and data used in this study are available under the
GNU General Public License (GPLv3) and the Creative Commons License
(CC BY-NC-SA 4.0), respectively, through a GitHub repository
(

Particle tracking is usually employed for simulating the advective dispersive
transport of solutes, but not for the water phase itself

In

Despite the successful application of the introduced particle model approach,
a 1-D version essentially lacks information about the lateral component of the
non-uniform distribution and resulting macropore–matrix exchange
characteristics in natural soils. One could be tempted to subsume an essence
of the recent model approaches for subsurface flow in discrete structures

The first is that macropore flow is much faster than saturated hydraulic conductivity. At the same time it is limited to a very small fraction of the soil column. This motivated the conceptualisation of multiple flow domains. However, the state of a specific flow path is substantially different from the averaged state of a elementary volume. Secondly, the topology of flow paths plays a role in this regard: macropores enable a quick vertical redistribution of event water. If the network of macropores is rather dense and lateral diffusion is not too slow, the resulting soil water dynamics can be uniformly described by some elevated, effective hydraulic conductivity. If the structures are sparse and lateral diffusion into the matrix is slower, lateral gradients in soil water potential and non-uniform flow fields are established.

As such the flow field depends on macropore topology, antecedent soil matrix state, macropore capacity and infiltration supply. In a 1-D approach such lateral gradients and their depletion cannot be described other than by some additional conceptual parameter or function and averaged matric potential states. The result would remain bound to a priori defined macropore–matrix exchange assumptions. Without proper control of the macropore–matrix interaction and thus control of the advective flow field, a fast fraction of particles would simply remain quick and drain from the domain, which contradicts the experimental findings.

The third challenge refers to the matrix pore space and exchange/mixing of
rapid event water particles with the pre-event water to establish a local
thermodynamic equilibrium (LTE) – the well-organised distribution of water
particles in the respective smallest fractions of the available pore space, as
we further explained earlier

These issues led to the preliminary finding that a lumped 1-D version of the particle model could not succeed in reproducing the observed tracer distributions without thorough calibration to one specific antecedent state and one specific realisation of the advective flow field. The requirement of non-observable and non-static mixing parameters between the domains makes an application to predict behaviour under change challenging. Thus it is not very convincing if we desire to develop the model as a virtual laboratory.

In order to overcome the 1-D limitations without requirement for a pore-scale
determination of the macropore system or non-observable parameters, we define
a representative macropore–matrix domain with explicit topology
(Fig.

The 2-D soil matrix possesses a grid for the determination of soil properties and for particle density (and thus soil moisture) calculation. The 1-D macropore domains have an internal grid for film flow calculations, the lag distance is calculated as the projection of one water particle to the mean macropore diameter. In addition, the 1-D macropore domains have an interface area with the 2-D soil matrix domain. In this area particles are considered for exchange between the domains.

Representative macropore–matrix domain. A 2-D soil matrix with a periodic lateral boundary hosts several 1-D macropores with their respective capacities, interfaces and lateral distributions.

Similar to the use of particle tracking for simulating solute transport we conceptualise soil water as particles. Each particle represents a constant mass of water, defined by the set-up of the soil matrix calculation grid and the resolution of the porewater volume bins.

Diffusive soil water flow is simulated as a non-linear, space domain random
walk in the soil matrix, as presented in our previous study

In this form diffusivity

Example of delineation of the pore space into bins of equal volumes
or particles in our study. If the bins are organised in ascending order, this refers
to the LTE (local thermodynamic equilibrium) state of the pore
space

Alternatively to the

In addition, a counteracting stochastic process is introduced to handle the
effect of high diffusivity but the low number of open slots in the pore space
near saturation:

Numerically, the actual step of a particle is calculated in a
predictor–corrector approach, projecting the step of one particle,
anticipating an updated state/binning to update

In addition to the matrix domain the set-up contains several 1-D elements as
macropores (Fig.

The preferential flow network exhibits a large drainage capacity.

The second limit is given through the definition of initial maximum flow
velocity in the structures. Literature values in Table

Macropore flow is represented as 1-D film flow of particles along the pore
wall (Fig.

Macropore flow concept.

Direct experimental evidence of water dynamics at the macropore–matrix
interface hardly exists. Some orientation is given by findings of

Here we propose a thermodynamic approach for describing this key process on a
physical basis without introducing additional parameters based on the
Bernoulli equation:

Measured advective flow velocity values in earthworm pores range closely around
7.5

These measurements compare with a theoretical laminar flow velocity through a
pipe of the same cross section and with a unit pressure gradient by a factor
of about 500. A theoretical laminar flow velocity

Given its velocity, each particle in motion possesses an

Measured mean maximum advective velocity in burrows of the earthworm

Following

The projected infiltration rate

Now, the reduced advective velocity of a particle is estimated using friction
and exchange drag acting against

If the projected infiltration exceeds the particle radius

With the extension of the model to two dimensions, the partitioning of
infiltration into macropores and the soil matrix became an important aspect
of the model. As pointed out by

The parameterisation of the echoRD model based on observables is a key objective of this study. As pointed out previously, the required parameters for the model are retention characteristics (van Genuchten parameters) and a lateral and vertical density distribution of macropores. The retention properties of the soil matrix can be measured in standard pedo-physical analyses.

To derive macropore density distributions, horizontal panes of dye tracer
stains (e.g. Brilliant Blue experiments) can be analysed with the model
preprocessor. With this we make use of experimental data directly as
explained in Appendix

We rely on sequential calculation of the process domains:

infiltration at the top boundary into matrix and macropores;

diffusive matrix flux as a spatially explicit 2-D random walk;

film flow in the macropore;

macropore–matrix interaction (infiltration and exfiltration).

Checks for saturation and percolation below the lower boundary are performed
after step 2 and 3. The time step is controlled through
Courant and Neumann criteria based on the maximum possible diffusive and advective step at the
current max(

With regard to the representative domain, the interrelation of particle size and the numerical grid is noteworthy. The desired resolution and stochastic stability of the model is controlled by the grid size and the number of particles which represent saturation. Both are required model parameters. Obviously, this quickly leads to a large number of particles if we seek to resolve processes which locally change soil moisture by a few percent only. The following tests are realised with a relatively fine grid and a relatively large number of particles to avoid instabilities and artefacts.

In this section, we outline our application tests of the echoRD model and a reference to real-world conditions in order to examine the capability of the chosen simplifications. In order to focus on the proposed concept and hypothesised process descriptions, the following tests are realised with an underlying grid resolution for particle density calculation of 5 mm. The water particles are set to a size of 0.002 times a grid cell (equivalent to 0.33 mg).

With the extension to two dimensions and the introduction of representative
macropores, the test applications shall especially address the following aspects:

2-D diffusive, non-uniform soil water redistribution;

interaction of 1-D advective paths with the 2-D soil matrix;

sensitivity to state variables and model parameters;

robustness of the representative macropore setting;

reproduction of a real-world irrigation experiment.

The central benchmark of the model is a series of generic test applications
with different soil types, precipitation intensities and antecedent soil
moisture. The aim is to examine the consistency and capability of the model
and the self-controlled non-uniform flow with regard to points a–c. The test
matrix is spanned by

soil water retention parameters for a sandy soil, a loamy soil and a
loess soil (Table

two different antecedent moisture states at 0.15 and 0.31 m

precipitation intensities at 10, 40 and 60 mm h

We conducted a series of plot-scale irrigation experiments in different soil
landscapes

Soil matrix retention parameters used in the application tests. Loamy
sand and silty loam according to

Because the model is intended as an exploration tool extending real-world
experiments, a further test of the model aims at reproducing one experiment
in the Weiherbach basin in south-west Germany with loess soils on a fallow
plot (49.13517

Weiherbach irrigation experiment as model reference. Brilliant Blue dye stains in excavation horizons.

The echoRD model set-up is based on a stochastic matrix definition of seven
equally valid ensembles of measurement and literature references
(

Simulated soil moisture dynamics in generic application tests of
loess soil. The marginal plots give the distribution of all particles (blue)
and newly infiltrated particles (red).

The generic application tests show the capability of the model to calculate
self-controlled, non-uniform infiltration patterns (Figs.

The simulations of 40 mm irrigation in 0.5 h on loess silt with different
antecedent soil water content show the development of a non-uniform flow
field conditioned by the representative macropores
(Fig.

Table of simulated soil moisture dynamics in generic application tests for loess. Marginal plots give the distribution of all particles (blue) and newly infiltrated particles (red).

Comparing the different soil types of loamy sand, silty loam and loess silt, the
two respective antecedent moisture states and three irrigation intensities,
more insight into the simulated soil water dynamics is given
(Fig.

Moreover, the larger the supply sustaining the advective fraction, the greater the depth or breakthrough reached. When analysing the simulated dynamics this also led to different apparent velocities in the respective macropores (see video in the Supplement). This behaviour is consistent with field observations and our expectations. As such, the model proves to fulfil the required objectives a–c.

A more quantitative reference is obtained when comparing the depth
distribution of new particles of the application tests directly
(Fig.

Simulated depth distribution of new particles in generic application
tests.

Dynamics of the moments of the breakthrough curves (BTCs) of loamy sand
simulations.

It is noteworthy to regard the development of the corresponding moments of
the depth distribution of infiltrating particles
(Fig.

In addition, we performed model parameter-related tests drawing different
realisations of the macropore setting from the same ensemble (Fig.

Tests with different particle resolutions showed that definitions that are too coarse can result in local averaging, which underestimates the actual depth distribution of the infiltrating water. A similar effect was observed with very coarse internal calculation grid definitions, which could no longer represent local state changes due to infiltration from the macropores into the surrounding matrix.

The last benchmark is the reproduction of observed tracer profiles based on measured parameters (test aspect e).

Simulated particle distribution in mimicry of the Weiherbach, 40 min after irrigation onset. A video of the simulated dynamics is given in the Supplement.

The simulation depicts the observed stain patterns and concentration profiles
very well (Fig.

Changes in soil water content are accumulated to the integration volume of
the TDR sensor for better comparison (Fig.

A closer look at the outcrops in Fig.

Moreover, it can be noted that the modelled depth distribution of new particles coincides with the observed tracer breakthrough. This is especially interesting because the macropores are defined as reaching through the full domain as earthworm burrows are reported to reach depths below 2 m. Hence the self-controlled limitation of advective flow in the macropores appears to be capable of reproducing the true process.

Simulated and observed tracer

The general adequacy of the echoRD model to represent non-uniform irrigation
water redistribution is outlined by the generic application tests. The water
particles move realistically in the conjugated domains under the tested
conditions. The mimicry of an irrigation experiment based on directly
measurable parameters also corroborates the proposed model framework with regard
to structural adequacy

During the development we followed

Despite the achievements, the echoRD model also has a number of limitations:
because the particles do not interact, any solute transport is governed by
the fluid movement alone. For the event scale this might be an acceptable
assumption. With a molecular diffusion coefficient of bromide in free water
(

Building on the idea of self-similarity in flow networks going back to the
works of

We explicitly avoid a direct and tortuous representation of a macropore
network as commonly observed

Representativity of the model domain for the plot scale is achieved when the integral of the dynamics is invariant to a larger domain extent under a given desired process resolution. For the mimicry of the irrigation experiment, we evaluated different domain size definitions for their respective BTC dynamics.

The combination of the particle approach with the connected domains avoids a number of implicit assumptions for the exchange between the domains. Our energy-balance approach to film flow in the macropores enables analyses of different infiltration patterns with self-controlled advection and diffusion. In addition to this process hypothesis, many alternative approaches to model the interfacial processes and the behaviour within the respective domains can be imagined. For this, the echoRD model allows for direct process hypothesis testing with the same objects.

We have shown that different infiltration patterns emerge based on different
antecedent conditions and forcing of the representative structured domain
(Sect.

The non-stationary and non-linear dispersivity underpins the limitations when
the processes during driven conditions are subsumed by explicit and universal
parameterisation. However, diffusive transport dominates quickly after the
supply ceases. This motivates a potential use of the full echoRD model to
derive state- and forcing-dependent distribution references for the advective
flow field, which can successively be used in more simple versions of the
particle model like our 1-D approach

Although the echoRD model possesses many degrees of freedom to adjust its
behaviour, it is not intended for parameter fitting. Instead, the model is
proposed as an exploration tool capable of extending real-world experiments. As
such, the model requires very few parameters, which can all be derived from
suitable experiments: soil matrix parameters are used for the determination of
the diffusive and storage properties of the soil and consist of soil water
retention parameters. If desired, the van Genuchten model can be replaced by
any other soil water retention model. Each calculation grid node of the
matrix domain can be assigned to a different soil matrix definition.
Macropores host the advective flow and are determined by the spatial
distributions (relative lateral distances and connected pore depth) and a
reference to maximum flow capacity. In addition some coating factor may be
defined for earthworm burrow coatings

There has been much debate about the derivation of effective parameters in
hydrological models

We envisage further use with dynamic macropore settings as the domain may update once it is empty and as a foundation to derive state- and forcing-dependent stochastic site properties which can be used in more lumped versions of the approach. Since the particle domain can always be converted into a Eulerian field of matric potential or soil water content and vice versa, the model can also be linked to a Richards model for periods when the diffusive flow assumption is valid.

In the application tests it was seen that the model is quite sensitive to antecedent conditions. Under hardly determinable state data this may lead to susceptibility of the model to uncertainty about the macropore–matrix exchange, which can be amplified through the non-linear retention properties. Moreover, the model has shown sensitivity to dead-ended macropores. Hence special care has to be given to provide valid data on the macropore distribution and vertical connectivity.

The simulation of soil water dynamics based on water itself as particles is generally very different from the common particle tracking for solutes. On the one hand there is no external drift and the activity of each particle depends on its neighbours. On the other hand a very large number of particles is needed to enable robust calculation of the low event signal against a rather high background or pre-event concentration. The reason for this is that the resolution of the process dynamics scales with the number of particles per volume reference (grid cell in our case). At the same time we require relatively small volume references to avoid integration over scales that are too large. All of these points demand a large number of particles which require frequent state updates about their relative concentration distributions and binning in the porewater space. Moreover, the calculation of film flow with many particles is similarly self-dependent.

The Courant and Neumann criterion for the time step control calculates a
global specification. Hence local wetting causes very small time steps for
the whole model. In combination with the previous concerns, this makes the
model computationally very expensive. Due to the self-dependent state, we
could not find any option to make use of the more efficient continuous time
random walk methodology

In the current state of experimental code, the model runs at about 10 to
200 times more slowly than the real time of the simulated case. Despite its
potential, we abandoned trials using grid-free methods to calculate the
particle density, e.g. by Voronoi polygon area calculation

One of the intended uses of the model is to overcome the limitations of
destructive irrigation experiments. So far it is impossible to repeat
tracer-based plot irrigation experiments as the site needs excavation for
sampling. Moreover the spatial and temporal scales of such experiments are
very difficult to observe

Figure

Besides the initial development of flow fingers and the evolution of the
skewness of the depth distribution of the event water (Sect.

In a recent paper

In a series of application tests we showed the model's capability to represent (a) 2-D diffusive, non-uniform soil water redistribution, (b) self-controlled interaction of the 1-D advective paths with the 2-D soil matrix, (c) sensitivity to state variables and observable model parameters and (d) robustness of the representative macropore setting based on macropore depth distributions. Moreover, the model was successfully used to mimic a real-world irrigation experiment based on measured parameters.

This implies the structural adequacy of the model simulating advective flow as dynamic film flow in topologically explicit macropores and accounting for macropore–matrix exchange based on an energy-balance approach. The multi-domain interplay of advective and diffusive soil water redistribution exhibited a non-linear temporal evolution of the dispersivity. While the process description appears to be rather sophisticated, its parameterisation is very simple as the model relies on soil water retention properties for the soil matrix and data on the depth distribution of effective macropores.

As the model is intended to be a learning tool to extend real-world experiments, we
have shown its potential for virtual experiments under different antecedent
states, macropore settings and precipitation forcing. The model is also
envisaged to deliver a physically based foundation for infiltration
statistics, which can then inform Markov processes of higher orders in simpler 1-D
versions of the model

The echoRD model, reference data and the presented test
cases are accessible in a GitHub repository (

The echoRD model can be set up based on soil water retention data (as a table of van Genuchten parameters) for different soil layers and any sort of information about the macropore distribution. The easiest way is to provide images of horizontal outcrops of dye stain patterns to the preprocessor.

The rectified and cropped images with a defined resolution are read and
analysed for stained patches using scikit-image

In a next step these identified patches are analysed for distribution of topological parameters like total number, distance, size and diameter. Based on the least density among all horizons, the representative domain is scaled so that at least one effective macropore exists in the sparsest case. Thus, the fewer macropores, the larger the domain.

Subsequently, topological parameters are then resampled on the representative domain by allocating all representative macropores to a certain position on the 2-D matrix domain based on the observed lateral distance distribution. Moreover, contact areas are defined, depending on the circumference distribution of the patches.

An example is given in Fig.

Example of preprocessing of stain images, patch identification and statistics and resulting macropore positions in the representative domain.

This paper is accompanied by a repository at GitHub, in which the echoRD
model and the presented test cases are made publicly available:

All software and data are given under the GNU General Public License (GPLv3) and the Creative Commons License (CC BY-NC-SA 4.0) respectively. This is scientific, experimental code without any warranty or liability in any case. The code is not fully optimised yet and calculations are computationally demanding. However, you are invited to use, test and expand the model at your own risk. If you do so, please contact the first author and repository owner to keep them informed about bugs and modifications.

The repository holds the folder

Exfiltration time of a particle from the macropore wall into the adjoined soil matrix for different soils and soil moisture states.

In Sect.

Diffusive exfiltration from an irrigated artificial macropore.
Experiment by

To evaluate the capability of the model to simulate lateral diffusion and macropore matrix
exchange, we compared the macropore–matrix simulations and diffusive
redistribution of water particles to an experiment by

Figure

The videos of the modelled evolution of soil water content are given in the Supplement.

Simulated depth distribution of new particles in generic application tests. Different soil types after 1 h.

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

This study contributes to and greatly benefitted from the “Catchments As
Organized Systems” (CAOS) research unit. We sincerely thank the German
Research Foundation (Deutsche Forschungsgemeinschaft, DFG) for funding (FOR 1598,
ZE 533/9-1). Especially, we thank Loes van Schaik and Niklas Allroggen
for the initial discussion of macropore representation and joint experiments.
Moreover, this study greatly benefitted from the cooperation with the KIT
Engler Bunte Institute, analysing hundreds of samples for

Moreover, the realisation of the many test runs during the model development would have been impossible without the HPC infrastructure of the universities of Baden Württemberg (bwHPC). The authors acknowledge support by the DFG and the state of Baden Württemberg through bwHPC. We are very grateful for this unique opportunity and support. The publication processing charge was supported by the DFG and the Open Access Publishing Fund of Karlsruhe Institute of Technology. The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: Alberto Guadagnini Reviewed by: two anonymous referees