The different available root architecture models (see e.g., Dunbabin et al., 2013) are partly different in the way they represent the growth processes, such that the equations describing these processes are very different. For example, branch emergence is a function of apical zone length in CRootBox (Schnepf et al., 2018b) while it is a function of delay time in Root Typ (Pagès et al., 2004). Root radius is a time-dependent function scaled to distance along the root in OpenSimRoot (Postma et al., 2017) while it is computed according to the pipe model in ArchiSimple (Pagès et al., 2014). Thus, we are looking at process-level differences between the different models, and each participating RSA model will have a different set of parameters that drive root growth. This is the reason why, in this benchmark, we do not prescribe a parameter set as in e.g., M2 or M3, but we let each participating model derive its respective model parameters based on a reference dataset. In this first benchmark (M1.1), modelers simulate root systems for the same duration as the age of the root systems in the reference dataset.

2.2.1.1.1. Reference data set. Although the parameterization of 3D models using a set of parameters derived from 2D images has some limitations, it has been shown to be a simple and efficient strategy allowing the simulation of realistic 3D root systems (Landl et al., 2018). Our reference dataset contains two distinct sets of images: (1) images of lupin roots grown for 11 days in an aeroponic setup (Lobet et al., 2011), and (2) images of maize roots grown for 8 days on filter papers (Hund et al., 2009). All images were analyzed using the semi-automated root image analysis software SmartRoot (Lobet et al., 2011) and root tracings were saved as RSML files for further analysis (Figure 1). These RSML files were then processed using functions of the R package archiDART developed to compute root system- and single root-level metrics (Delory et al., 2016, 2018). These metrics have been made open-access (https://github.com/RSA-benchmarks/collaborative-comparison/tree/master/root_architecture/data) and should help modelers to parameterize their respective RSA model.

2.2.1.1.2. Required output. The following results are to be uploaded via pull requests to this path on the github repository: M1 Root architecture development/M1.1 RSA calibration/M1.1 Numerical results.

2.2.1.1.3. Reference data analysis and automated model comparison. Statistical evaluation of a root architecture model has for example been done by Delory et al. (2018); Schnepf et al. (2018a). This motivated the creation of two data analysis pipelines for the first benchmark (M1.1) that will be used to compare simulation outputs with reference experimental data (reference root systems) (Figure 2A). These two data analysis pipelines are implemented in the Jupyter Notebook RSA calibration.ipynb that can be found on the github repository that contains code that will automatically include every model output in the analysis that is available in the prescribed folder. The analysis relies on the functions available in the R package archiDART (Delory et al., 2016, 2018). In the first pipeline, traits computed at the root system level (e.g., total root system length, number of roots per branching order) are compared between all simulated and reference root systems. This comparison takes place in three steps: (1) identifying the key morphological, architectural, and topological (Fitter indices, Fitter, 1987; Fitter and Stickland, 1991) traits explaining differences between simulated and reference root systems using multivariate data analysis techniques (e.g., discriminant analysis and principal component analysis), (2) looking at the point in time, beyond the time period for which there are measurements, when simulated and reference root systems start to diverge/converge with regard to the key root system traits identified in the previous step and how large these differences are, and (3) assess the degree of dissimilarity between simulated and reference root systems using dissimilarity metrics based on the raw data (Janssen and Heuberger, 1995).

In the second pipeline, dissimilarities in architecture between reference and simulated root systems are compared using persistent homology. Persistent homology is a topological framework that has proven to be a very powerful tool for capturing variations in plant morphology at different spatial scales (Li et al., 2017, 2018). The main output of a persistent homology analysis is a persistence barcode recording the appearance and disappearance of each root branch when a distance function traverses the branching structure (see Figure 1 in Delory et al., 2018). The degree of similarity between different root system topologies can be assessed by computing a pairwise distance matrix to compare persistence barcodes. In addition, Delory et al. (2018) showed that both trait-based and persistent homology approaches nicely complement each other and allow root researchers to more accurately describe differences in root system architecture (Delory et al., 2018). In our data analysis pipeline, a persistent homology analysis comprises the following steps: (1) computing a persistence barcode for each simulated and reference root system using a geodesic distance function, (2) computing dissimilarities between persistence barcodes using a bottleneck distance, (3) visualize dissimilarities between root systems using multidimensional scaling, and (4) test specific hypotheses using permutational multivariate analysis of variance (PERMANOVA) (Anderson, 2001).

In this benchmark, modelers use the same input parameter set as in M1.1, but simulate root system growth and development for a longer time period (60 days). The aim of this second benchmarking exercise is to assess if the different models diverge (or converge) if simulations are run for a longer time period and extrapolate beyond the provided data set (Figure 2B). This is of great importance, as parameterization of RSA models is often based on relatively young plants, whereas knowledge of RSA of older root systems is scarce. Therefore, for this M1.2 scenario, experimental data are not used as the basis of comparison anymore. It has to be noted that these two benchmark problems focus on root architecture dynamics modeling only, thus effect of soil properties on root growth is not explicitly modeled.

2.2.1.2.1. Required output. The following results are to be uploaded via pull requests to this path on the github repository: M1 Root architecture development/M1.2 RSA simulation/M1.2 Numerical results.

2.2.1.2.2. Analysis pipeline for M1.2. For the second benchmark (M1.2), three data analysis pipelines are used to compare simulation outputs given by different root architecture models. For this benchmark, the reference experimental data cannot be used as a reference as data of 60 days old plants is not available. The first two data analysis pipelines for M1.2 are very similar to the ones described earlier for the M1.1 benchmark. First, model outputs are compared using morphological, architectural, and topological traits computed at the root system level. Second, differences in root system morphology are analyzed using persistent homology. In addition to these two analysis pipelines, we included a third one to analyse differences in vertical root distribution between root systems simulated with different root architecture models. To do so, we use the modeling approach described in Oram et al. (2018). Briefly, relative cumulative root length density [Y(d)] is computed using Equation (1).

Equation (2) is fitted to the computed Y(d) using a non-linear least square means fitting procedure. The fitting constant β is used to compare modeled rooting depth, with high β corresponding to deep rooting.

This benchmark scenario is taken from Vanderborght et al. (2005). All parameters, initial and boundary conditions are given in Table 2 and are described below. For each of the soil types, sand, loam and clay, we consider the rate of infiltration into a soil with an initial homogeneous soil water pressure head of ψ_s = −400 cm. All profiles are 200 cm deep, at the top boundary we prescribe a constant influx of 100 cm d⁻¹ as long as the soil is still unsaturated, and a Dirichlet boundary condition of ψ_s = 0 cm as soon as the soil is fully saturated. Note that the prescribed infiltration value is high, such that in most scenarios, the boundary condition will switch to Dirichlet very soon. At the bottom boundary, we prescribe free drainage. Since this problem only produces gradients in the vertical direction, we compare numerical model results with the 1D analytical solution described in Vanderborght et al. (2005).

2.2.2.1.1. Reference solution. The analytical solution is given by the traveling wave equation

where D_w is the water diffusivity (defined as Dw=K(θ)∂ψs∂θ), θ_sur is the water content at the soil surface, θ_i is the initial water content, θ_a is a reference water content (taken to be θ_a = (θ_sur + θ_i)/2), η=|z|-[K(θsur)-K(θi)]tθsur-θi and Δη(θ) is the distance of the front to the position of the reference water content. The implementation of this analytical solution, implemented in the Jupyter Notebook M2.1 Benchmark problem.ipynb, reproduces Figures 4a–c from Vanderborght et al. (2005), where the water content is plotted after 0.1, 0.2, and 0.3 days for the sand scenario; 0.2, 0.5, and 1 days for the loam scenario; and 0.1, 0.2, and 0.5 days for the clay scenario (see Figure 3).

2.2.2.1.2. Required output. The following simulation results of participating models are to be uploaded via pull requests to this path on the github repository: M2 Water flow in soil/M2.1 Infiltration/M2.1 Numerical results.

Note that we do not prescribe spatial or temporal resolution of the outputs, as that may depend on the individual numerical schemes.

This benchmark reproduces Figure 5 of Vanderborght et al. (2005). We consider four scenarios (sand, loam 1, loam 2, clay) in which we are interested in the actual evaporation over time from an initially moist soil (ψ_i = −40 cm for the sand scenario and ψ_i = −200 cm for all other scenarios). The domain is 100 cm deep with a width and length of 10 cm. At the top boundary, we prescribe a constant efflux of J_s,pot = 0.1 cm d⁻¹ for the sand and loam 1 scenario, and 0.3 cm/day for the loam 2 and clay scenarios, at the bottom we prescribe zero-flux. When the soil reaches a critical soil water pressure head of −10,000 cm at the surface, we switch to a Dirichlet boundary condition with ψ_s = −10,000 cm.

2.2.2.2.1. Reference solution. The analytical solution to this problem is given by

where t′=Sw2(θsur,θi)4Jwpot2, tpot=Sw2(θsur,θi)2Jwpot2, Sw(θi,θsur)=(θi-θsur)4μ∫01Dw(Θ)dΘ, Θ=|θ-θsurθi-θsur|, μ=3β(1+{1-149[1-α(1-β)2]}0.5)2(1-β)[α(1-β)2-1], α=∫01(1-βΘ)2Dw(Θ)dΘ∫01Dw(Θ)dΘ, and β=[∫01ΘDw(Θ)dΘ∫01Dw(Θ)dΘ]2. Figure 4 shows the rate of evaporation over time for the four scenarios soil, loam 1, loam 2, clay.

2.2.2.2.2. Required output. The following simulation results of participating models are to be uploaded via pull requests to this path on the github repository: M2 Water flow in soil/M2.2 Evaporation/M2.2 Numerical results.

Note that we do not prescribe spatial or temporal resolution of the outputs, as that may depend on the individual numerical schemes. It is the responsibility of each participant, to upload the best possible solution.

In this benchmark problem, we assume a vertical single straight root segment surrounded by a soil with a constant and uniform soil water pressure head (i.e., the soil is not in hydrostatic equilibrium). We prescribe the root water pressure head at the root collar as ψ_x|_collar = ψ₀, and no axial flow at the root tips.

2.2.3.1.1. Reference solution. For constant k_r and k_x we can solve Equation (6) yielding

with c = 2r_rootπk_r/k_x. The integration constants d₁ and d₂ for above boundary conditions are given by

where l_seg is the segment length, and d is the determinant of above matrix

see Appendix A. Figure 5 shows the analytical solution to this benchmark using the parameters given in Table 4.

2.2.3.1.2. Required output. The following simulation results of participating models are to be uploaded via pull requests to this path on the github repository: M3 Water flow in roots/M3.1 Single root/M31 Numerical results/.

Note that we do not prescribe spatial resolution of the outputs, as that may depend on the individual numerical schemes.

In the following benchmark, we extend benchmark M3.1 from a single root to a root system. We consider water flow inside a small static root system of a lupine plant which was grown for 14 days in a soil-filled column of 20 cm depth and 7 cm diameter. The root system was imaged by MRI at Forschungszentrum Jülich; the segmented root structure is provided in RSML, DGF (Dune grid format) (Bastian et al., 2008) and RSWMS (Javaux et al., 2008) formats in the folder M3 Water flow in roots/M3.2 Root system/root_grid on the github repository. It is visualized in Figures 6A,B with colors denoting root order and root segment age.

2.2.3.2.1. Reference solution. The reference solution for this problem is given by the hybrid analytical-numerical solution of water flow in the root hydraulic architecture proposed by Meunier et al. (2017). The advantage of this solution is that it is independent of the spatial resolution of the root system (i.e., root segment length).

We consider two scenarios. The first one uses the same constant root hydraulic properties as given in Table 4, i.e., considering the same root hydraulic properties for each root segment. In the second scenario, we consider age-dependent root hydraulic properties for tap root and laterals of lupine as obtained by Zarebanadkouki et al. (2016, exponential function scenario) and converting distance from root tip to root age by assuming a root growth rate of 1 cm d⁻¹. This parameterization takes into account that roots get a higher axial conductivity and lower radial conductivity as they are becoming older (see Figure 7, a table with the actual values is provided on the github repository, in: M3 Water flow in roots/M3.2 Root system/M3.2 Benchmark problem.ipynb.

A sample 3-D visualization of the model output is shown in Figure 6C for the constant root hydraulic properties scenario. Figure 8 shows the effect of constant and age-dependent root hydraulic properties under otherwise same (soil and boundary) conditions.

2.2.3.2.2. Required output. The following simulation results of participating models are to be uploaded via pull requests to this path on the github repository: M3 Water flow in roots/M3.2 Root system/M32a Numerical results and M3 Water flow in roots/M3.2 Root system/M32b Numerical results for the constant and age-dependent root hydraulic properties cases.

Note that we do not prescribe spatial resolution of the outputs, as that may depend on the individual numerical schemes.

This benchmark follows the paper of Schröder et al. (2008). Here we aim to see to what extent the different participating models can reproduce the hydraulic conductivity drop near the root surface under different soil conditions and transpiration demands. Thus, it requires the participating line-source based models to strongly increase the spatial resolution of the 3D soil domain. From this benchmark, we will learn, whether the spatial resolution required to reproduce radial soil water pressure head gradients would be in a feasible order of magnitude for larger soil-root systems or not. If not, there are approaches to estimate soil water pressure head drop at the root-soil interface from bulk soil values as e.g., in Schröder et al. (2009), Beudez et al. (2013), and Mai et al. (2019), see also benchmark C1.2.

The analytical solution is based on the analytical solutions of the 1D radially symmetric problem of water uptake by a single root, in which root water uptake is described as a boundary condition at the root-soil interface. We consider here two water uptake regimes, a non-stressed condition with maximum root uptake (q_root), and a stressed condition with a limiting plant root water potential constraining uptake. Based on the steady-rate assumption and using the matric flux potential Φ(hc)=∫-∞hcK(h)dh that linearizes the Richards equation, the radial soil water pressure head profiles for non-stressed and stressed conditions (stress conditions are given when the soil water pressure head at the root surface reaches −15,000 cm) are given by

and

where ρ=routrroot.

Given the soil water pressure head at the outer boundary, the solution computes the soil water pressure head profile toward the root. Due to the steady-rate assumption, the problem has become a stationary boundary value problem. However, under non-stressed conditions, we can calculate the time that corresponds to a given radial soil water pressure head profile by dividing the volume of water removed from the soil domain by the known water flow rate. The water remaining in a 1 cm long hollow cylinder around the root is given by

θ being the water content. The initially available water volume in the soil domain is given by

Thus, until the onset of stress, the corresponding time at which a given radial profile is reached is given by

For the three soils sand, loam, and clay (Table 3), we compute the analytical solution for two different values of q_root (q_root = 0.1 cm/d and q_root = 0.05 cm/d, alternatively), and with the following parameters: r_root = 0 02 cm, r_out = 1 cm, ψ_s,lim = −15,000 cm, q_out = 0.0 cm/d, for different soil water pressure heads at the outer end of the cylinder. Figure 9 shows the soil water pressure head gradients at the onset of stress (i.e., when the soil water pressure head at the root surface reached −15,000 cm) and the time of its occurrence. The value of the initial soil water pressure head is taken to be ψ_s,i = −100 cm. This analytical solution is for radial water flow in soil toward the root only, i.e., not considering gravity or water flow inside the roots. Ideally, in their numerical implementation of this benchmark, the different participating models will turn off gravity effects. The soil domain for this numerical implementation has a size of l × w × d = 1 × 1 × 1 cm. The horizontal spatial resolution is high enough such that hydraulic conductivity drop near root surface can be resolved. The axial and radial conductances are high, such that the pressure inside the root is everywhere the same and the uptake flux is uniform.

2.2.4.2.1. Required output. The following simulation results of participating models are to be uploaded via pull requests to this path on the github repository: M3 Water flow in roots/M3.2 Root system/M32a Numerical results and M3 Water flow in roots/M3.2 Root system/M32b Numerical results for the constant and age-dependent root hydraulic properties cases.

Note that we do not prescribe spatial or temporal resolution of the outputs, as that may depend on the individual numerical schemes.

To compare the results between the participating models, the desired outputs are

File names of the VTK files should indicate the simulator name, the state variable, the domain, and the output time in days, e.g., “DuMux_soil_theta_1d.vtk.” File names of the CSV files should indicate the simulator name and output time it days, e.g., “Dumux_1.csv.”

The required outputs for model intercomparison are

File names of the VTK files should indicate the simulator name, the state variable, the domain, and the output time in days, e.g., “DuMux_soil_theta_1d.vtk.” File names of the CSV files should indicate the simulator name and output time it days, e.g., “Dumux_1.csv.”