Stony soils that have a considerable amount of rock fragments (RFs) are
widespread around the world. However, experiments to determine the effective soil
hydraulic properties (SHPs) of stony soils, i.e., the water retention curve
(WRC) and hydraulic conductivity curve (HCC), are challenging. Installation
of measurement devices and sensors in these soils is difficult, and the data
are less reliable because of their high local heterogeneity. Therefore,
effective properties of stony soils especially under unsaturated hydraulic
conditions are still not well understood. An alternative approach to
evaluate the SHPs of these systems with internal structural heterogeneity is
numerical simulation. We used the Hydrus 2D/3D software to create virtual
stony soils in 3D and simulate water flow for different volumetric fractions of RFs,

Virtual stony soils with different rock fragment contents were generated in 3D using the Hydrus 2D/3D software.

Evaporation experiments and unit-gradient experiments were numerically simulated.

We used inverse modeling with the Richards equation to identify effective hydraulic properties of virtual stony soils.

The identified hydraulic properties were used to evaluate the scaling models of calculating hydraulic properties of stony soils.

Stony soils are soils with a considerable amount of rock fragments (RFs) and
are widespread in mountainous and forested watersheds around the world
(Ballabio et al., 2016; Novák and Hlaváčiková, 2019). RFs in
soil are particles with an effective diameter larger than 2 mm (Tetegan
et al., 2015; Zhang et al., 2016). Their existence in soil influences the
two constitutive soil water relationships known as soil hydraulic properties
(SHPs), i.e., the water retention curve (WRC) and the hydraulic conductivity curve
(HCC) (Russo, 1988; Durner and Flühler, 2006). The accurate
identification of SHPs is a prerequisite for the adequate prediction of water
flow in soil with the Richards equation (Farthing and Ogden, 2017; Haghverdi
et al., 2018). The SHPs depend on soil texture and structure (Kutilek, 2004;
Lehmann et al., 2020) and are influenced by the presence of RFs in soil. It
is generally accepted that RFs decrease the water storage capacity of soils
as well as their effective unsaturated hydraulic conductivity. In contrast, the
formation of macropores in the vicinity of embedded RFs may lead to an
increase in saturated hydraulic conductivity. While experimental evidence
and theoretical analyses show that the volumetric fraction of RFs,

Hlaváčiková and Novák (2014) proposed a model to scale the HCC of the background soil, parameterized with the van Genuchten–Mualem (van Genuchten, 1980) model, using the model of Bouwer and Rice (1984). Hlaváčiková et al. (2018) used the water content of RFs as an input parameter to scale the WRC of the background soil. Naseri et al. (2019) used the simplified evaporation method (Peters et al., 2015) to experimentally determine the effective SHPs of small soil samples containing various amounts of RFs. Their study criticizes the application of the scaling models developed for saturated stony soils to unsaturated conditions and emphasizes the need to develop approaches that consider more RF characteristics to calculate the SHPs of stony soils.

Recent advancements in computational hydrology and computing power suggest
the numerical simulation of soil water dynamics as a promising alternative
to the measurement of the effective SHPs of heterogeneous soils (Durner et al.,
2008; Lai and Ren, 2016; Radcliffe and Šimůnek, 2018). Numerical
simulations have several advantages: they do not demand strict experimental
setups; they are repeatable under a variety of initial and boundary conditions;
and, in contrast to laboratory experiments, spatial and temporal scales are not
restrictive factors in the simulations. These assets have made them a
favorable tool in water and solute transport modeling in heterogeneous soils
(Abbasi et al., 2003; Šimůnek et al., 2016). However, with few
exceptions, heterogeneous soils like stony soils have been simulated only
for simplified cases, i.e., either under fully saturated conditions or with
reduced dimensionality, i.e., simulations of stony soils in two spatial
dimensions (2D). Novák et al. (2011) calculated the effective saturated
hydraulic conductivity (

The inverse modeling approach has been applied to identify effective
hydraulic properties of soils in laboratory experiments (Ciollaro and
Romano, 1995; Hopmans et al., 2002; Nasta et al., 2011), using lysimeters and
field studies (Abbaspour et al., 1999, 2000), utilizing virtual lysimeters
with internal textural heterogeneity (Durner et al., 2008; Schelle et al.,
2013), and for the WRC of stony soils through field infiltration experiments
(Baetens et al., 2009). Although theoretical studies and laboratory
investigations on packed samples are insufficient to fully understand the
hydraulic processes in stony soils, they do lead the way to the improvement
and validation of effective models and their application at the field and
even larger scales. Inverse modeling is arguably the best approach to
achieve these aims because it allows one to validate effective models using
process modeling. Our aim in this study was to investigate the application
of inverse modeling to identify the effective SHPs of 3D virtual stony soils
and to explore its applicability to these soil systems as an example of
internal structural heterogeneity. We were interested in answering the
following questions:

Is it possible to describe the dynamics in the heterogeneous 3D system with the 1D Richards equation assuming a homogeneous soil?

If so, what are the effective SHPs of stony soils and how do they relate to the SHPs of the background soil?

The Hydrus 2D/3D software was used to generate virtual stony soils and
simulate the water flow in the created 3D geometries. Water
flow in Hydrus 2D/3D is modeled by the Richards equation (Šimůnek
et al., 2006, 2008), which is the standard model for variably saturated
water flow in porous media. The Hydrus 2D/3D software solves the mixed form
of the Richards equation numerically using the finite-element method and an
implicit scheme in time (Celia et al., 1990; Šimůnek et al., 2008,
2016; Radcliffe and Šimůnek, 2018). The 3D form of
the Richards equation under isothermal conditions, without sinks/sources,
and assuming an isotropic hydraulic conductivity is as follows:

The virtual stony soils in 3D were created by placing spherical inclusions
in a background soil. In accordance with real laboratory experiments (not
reported here), we generated virtual soil columns as cylinders with a
diameter of 16 cm, a height of 10 cm, and a total volume of

Visualization of the generated stony soils in 3D, including the
dimension of the RFs and the soil cylinder as well as the location of the observation points

We simulated evaporation (EVA) (Peters and Durner, 2008) and multistep unit-gradient (MSUG) experiments (Sarkar et al., 2019). For EVA, a linear
distribution of the pressure head (

In the virtual MSUG experiment, the soil column was initially fully
saturated with a constant pressure head of 0 cm. A sequence of stepwise
decreasing constant pressure heads was assigned to the upper and lower
boundaries of the column. The duration of the virtual MSUG experiment was
100 d, and the pressure head in the upper and lower boundaries was
simultaneously decreased stepwise to a pressure head of

The converging and diverging flow fields around obstacles produce spatially different pressure heads, even under unit-gradient conditions; as opposed to saturated conditions, these different pressure heads are associated with different water saturations and different local hydraulic conductivities under unsaturated conditions. We were interested in the extent to which this could lead to nonlinear effects in the derivation of the effective hydraulic properties, in particular the effective HCC. Furthermore, as the flow field for a given volume fraction of obstacles depends on dimensionality, i.e., is different in a 2D simulation compared with a 3D simulation, studying the effects in the unsaturated region was one of the main motivations for performing this numerical analysis in 3D.

A 10 d EVA experiment in 1D was simulated with the
Hydrus-1D software package (Šimůnek et al., 2006, 2008) to obtain the SHP parameters
using inverse modeling. The generated data from the EVA and MSUG forward
simulations in 3D were used as input to the 1D inverse simulations. Time
series of the pressure heads at three observation depths, mean volumetric
water contents in the column during the virtual EVA experiment, and the data
points of the effective HCC from the virtual MSUG experiment were used in
the objective function. The time series of the mean volumetric water content
was calculated from the initial water content, cumulative evaporation, and
soil volume. The measurement range for pressure heads used in the objective
function was from saturation down to

The SHPs of stony soils obtained by inverse modeling were compared to SHPs
that were predicted by available scaling models and used for their
evaluation. Considering that

In a more recent approach, Novák et al. (2011) developed a linear
relationship based on the 2D numerical simulation results as a first
approximation to scale the saturated hydraulic conductivity of stony soils:

Another model that has been developed for mixtures with spherical inclusions
is the Maxwell model (Maxwell, 1873; Corring and Churchill, 1961; Peck and
Watson, 1979; Zimmermann and Bodvarsson, 1995). It takes the value of

It should be noted that all approaches apply at any pressure head

Figure 2 visualizes the pressure head (cm), water content (cm

Visualization of the pressure head (cm), water content (cm

The dependency of the relative saturated hydraulic conductivity
(

Comparison of the values of

Obviously, the results of our simulations confirm a linear reduction in

We note that these results may differ in natural soils, where an increase in the saturated hydraulic conductivity might be expected because of macropore flow in lacunar pores at the interface between the background soil and RFs (Beckers et al., 2016; Hlaváčiková et al., 2016; Arias et al., 2019). We have not included such a process in our 3D simulations.

The observed and fitted time series of the pressure heads at the three
representative observation points are shown in Fig. 4 for the simulated
experiments of the four cases with different values of

The time series of the 3D-simulated (circles) and 1D-fitted (solid
lines) pressure heads at the observation points at three depths of the stony
soil columns with the

Table 1 shows the values of the root-mean-square error (RMSE) and mean
absolute error (MAE) between the observed and fitted time series of the
pressure heads at three observation points for values of

The RMSE and MAE values between the observed and fitted pressure
heads at three observation points for different values of

The WRCs and HCCs of the background soil (solid black line) and the
identified effective WRCs

The identified SHPs are presented in Fig. 5. The solid lines in the figure show the WRCs and HCCs of the virtual stony soils obtained by inverse
simulation (except the solid black lines, which are the WRCs and HCCs of the
background soil); the dashed lines represent the WRCs and HCCs scaled using Eq. (5) and Eq. (6), respectively; and the circles on the HCC plots represent the discrete data
points of hydraulic conductivity obtained by the virtual MSUG experiment.
The WRCs and HCCs are presented on a

The van Genuchten model parameters of the SHPs of the background
soil and of the inversely determined effective SHPs of the virtual stony
soils with different values of

According to Fig. 5 and Table 2, the value of the shape parameter

Similar to the WRC, an increase in

We had to include the data points of hydraulic conductivity from the virtual
MSUG experiment in the inverse objective function to get a precise
identification of the HCC obtained by inverse modeling near saturation. The
information content from the virtual EVA experiment gives a unique
identification only when the flux rate in the system reaches the magnitude
of the unsaturated hydraulic conductivity, which is around

As stated above, the model of Ravina and Magier (1984), which is a linear
scaling approach of the hydraulic conductivity (Eq. 6), underestimates the
reduction in conductivity in stony soils. We used the identified HCC as a
benchmark to evaluate and compare more advanced models for scaling HCCs,
namely the Novák, Maxwell, and GEM models (Eqs. 7, 8, and 9, respectively). Figure 6
illustrates the calculated HCCs of stony soils with different values of

Evaluation of the Novák, Maxwell, and GEM models for scaling HCC
of stony soils using the identified HCC as a benchmark. In each case, the HCCs
were obtained for

The HCCs calculated by the three models are generally in good agreement with
the identified HCC in the observed range of pressure heads. All three models
result in a more realistic estimate of the HCC compared with the simple linear
scaling approach. While the Novák model slightly underestimates the
identified HCC for all four RF contents, the GEM model, in contrast, overestimates the reduction in the hydraulic conductivity. The Maxwell model shows the same results as the GEM model except that it underestimates the HCC
for the stony soil with

In order to compare the performance of the three models, the average
deviation (

Performance of the Novák, Maxwell, and GEM models quantified by
the average deviation hydraulic conductivities (

Table 3 confirms the qualitative tendency toward an underestimation of the
conductivity reduction by the Novák model and toward overestimation by the
GEM and Maxwell models; however, Table 3 also shows that the difference between the three
models is not large and is probably not of relevance in practice (the GEM model
at the high value of

In this study, we created virtual stony soils with different volumetric
fractions of RFs in 3D and identified their effective SHPs from saturation up
to

Our results show that the boundary fluxes and the internal system states in
the virtual 3D EVA experiments, represented by the observed time series of
pressure heads at multiple depths, could be matched well by 1D simulations,
and the effective WRCs and HCCs of the considered stony soils were determined
accurately. Comparison with the scaling models showed that, by assuming a
homogeneous background soil and impermeable RFs, the effective WRC can be
calculated from the WRC of the background soil using a simple correction
factor equal to the volume fraction of background soil,

Care must be taken before generalizing these results to arbitrary conditions, e.g., highly dynamic boundary conditions with sequences of precipitation and higher and lower evaporation rates, which might yield different results due to the occurrence of nonequilibrium water dynamics and hysteresis. For real stony soils, changes in the pore size distribution of the background soil may result from the presence of RFs (Sekucia et al., 2020), with corresponding consequences for the effective SHPs. This influence was reported to be more common in compactible soils with a shrinkage–swelling potential (Fiès et al., 2002). In highly stony soils, where RFs are not embedded completely in the background soil, the existence of effective SHPs is still an open question. Finally, the impact of arrangement and size of RFs on evaporation dynamics and the effective SHPs needs to be understood. Tackling these problems requires a combination of experimental and modeling approaches.

The software packages used in this study are Hydrus-1D (public domain software) and Hydrus 2D/3D (proprietary). Further software information is available from

The Hydrus-3D simulation projects used in this study can be accessed on the research data repository at TU Braunschweig (

The study was designed by MN, SCI, and WD. MN performed the 3D simulations with Hydrus-3D and drafted the paper. SCI and MN did the inverse simulations. SCI and WD revised the paper and supervised the study.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors appreciate the insightful and constructive comments and remarks from David Dunkerley.

This open-access publication was funded by Technische Universität Braunschweig.

This paper was edited by David Dunkerley and reviewed by two anonymous referees.