Integrating X-ray CT Data into Models

Abstract

X-ray Computed Tomography (X-ray CT) offers important 4-D (i.e., 3-D scanning over time) structural information of the soil architecture. This imaging tool provides access to the 3-D morphological properties of the soil pore space such as the 3-D connectivity of pores that are essential to the understanding of water, solute, and gas transport processes. Other morphological properties such as pore-size distribution, specific surface area, or spatial heterogeneity of soil can be obtained from the X-ray CT images. Many studies have used this technique to better understand the evolution of macroscopic soil physical properties such as structural stability and relate it to spatial descriptors of soil pore space morphology when the soil undergoes wetting/drying cycles (e.g., Diel et al., 2019) or when it is subjected to different agricultural practices (e.g., Papadopoulos et al., 2009; Dal Ferro et al., 2013; Caplan et al., 2017). Non-equilibrium transfer processes, such as preferential transport, have also been related to the quantification of macropores in X-ray CT images (e.g., Larsbo et al., 2014; Katuwal et al., 2015; Soto-Gómez et al., 2018). In addition, X-ray CT data have proved particularly useful for reconstructing the skeletons of biopore networks, such as those burrowed by earthworms (Capowiez et al., 1998), and for monitoring their temporal dynamics (Joschko et al., 1993) (see Chap. 10). The role of air-filled soil pores and in particular their connectivity in 3-D in the transport of microbial-generated gaseous products (N₂O, CO₂) have been hypothesized (Rabot et al., 2015; Porre et al., 2016). X-ray CT data have also provided new knowledge about the 3-D architecture of root systems (e.g., Helliwell et al., 2013) and their impact on the 3-D soil architecture (see Chap. 9). For instance, root hairs were shown to modify the pore-size distribution and connectivity in the rhizosphere (e.g., Keyes et al., 2013; Koebernick et al., 2017, 2019). X-ray CT measurements have also allowed imaging aerenchymatous roots and the gas bubbles entrapped in the soil of rice paddies to explain transport of CO₂ and O₂ between roots and the atmosphere (Kirk et al., 2019). The dynamics of the spatial dispersion of soil microorganisms could be related to the 3-D description of the pore space obtained by X-ray CT (Juyal et al., 2020). The role of some pore-size classes could also be linked with soil carbon storage (Kravchenko et al., 2020) (see Chap. 10).

11.1 Introduction

X-ray Computed Tomography (X-ray CT) offers important 4-D (i.e., 3-D scanning over time) structural information of the soil architecture. This imaging tool provides access to the 3-D morphological properties of the soil pore space such as the 3-D connectivity of pores that are essential to the understanding of water, solute, and gas transport processes. Other morphological properties such as pore-size distribution, specific surface area, or spatial heterogeneity of soil can be obtained from the X-ray CT images. Many studies have used this technique to better understand the evolution of macroscopic soil physical properties such as structural stability and relate it to spatial descriptors of soil pore space morphology when the soil undergoes wetting/drying cycles (e.g., Diel et al., 2019) or when it is subjected to different agricultural practices (e.g., Papadopoulos et al., 2009; Dal Ferro et al., 2013; Caplan et al., 2017). Non-equilibrium transfer processes, such as preferential transport, have also been related to the quantification of macropores in X-ray CT images (e.g., Larsbo et al., 2014; Katuwal et al., 2015; Soto-Gómez et al., 2018). In addition, X-ray CT data have proved particularly useful for reconstructing the skeletons of biopore networks, such as those burrowed by earthworms (Capowiez et al., 1998), and for monitoring their temporal dynamics (Joschko et al., 1993) (see Chap. 10). The role of air-filled soil pores and in particular their connectivity in 3-D in the transport of microbial-generated gaseous products (N₂O, CO₂) have been hypothesized (Rabot et al., 2015; Porre et al., 2016). X-ray CT data have also provided new knowledge about the 3-D architecture of root systems (e.g., Helliwell et al., 2013) and their impact on the 3-D soil architecture (see Chap. 9). For instance, root hairs were shown to modify the pore-size distribution and connectivity in the rhizosphere (e.g., Keyes et al., 2013; Koebernick et al., 2017, 2019). X-ray CT measurements have also allowed imaging aerenchymatous roots and the gas bubbles entrapped in the soil of rice paddies to explain transport of CO₂ and O₂ between roots and the atmosphere (Kirk et al., 2019). The dynamics of the spatial dispersion of soil microorganisms could be related to the 3-D description of the pore space obtained by X-ray CT (Juyal et al., 2020). The role of some pore-size classes could also be linked with soil carbon storage (Kravchenko et al., 2020) (see Chap. 10).

Many experimental studies have highlighted the link between soil functioning and structural information of soil architecture extracted from X-ray CT. This is often performed by looking for correlations between measured soil functions and morphological properties of the pore space. From this vantage point, modelling offers an interesting and complementary methodological tool to these studies. In addition to their predictive character, models open the way to the testing of hypotheses about the role of the complex soil architecture on soil biological and physical processes, by carrying out structurally realistic in-silico experiments that can now be set up thanks to X-ray CT imaging. Development of spatially explicit pore-scale modelling in the 1990s, encompassed with a steep increase in desktop processors computing power and development of parallel computing algorithms, has made the routine use of X-ray CT images in modelling common. Early examples of the integration of X-ray CT data into spatially explicit models are found in the rock physics domain (Spanne et al., 1994; Hazlett, 1995). The inclusion of X-ray CT data into models focusing on soils occurred at a later stage (Falconer et al., 2005; Monga et al., 2008; Keyes et al., 2013). Consequently, the rock physics domain accounts for a large number of studies dealing with water flow, multiphase flow, and reactive transport (e.g., Blunt et al., 2013; Bultreys et al., 2016; de Paulo et al., 2020), whilst the soil domain has been the object of a lower number of studies, possibly because of its importance not being widely recognized and/or funding being more abundant for the problems dealing with oil and gas movement in the rocks. Synthetic porous media or packing of sands have also been imaged and processed into pore-scale models to study fundamental physical and biological processes in granular media (e.g., Schaap et al., 2007; Sukop et al., 2008; Peszynska et al., 2016). Such approaches are a valuable step forward to account for some of the soil complexity, but outputs obtained should be taken cautiously because the studied system can behave fundamentally differently from the real soil situation.

The main approach in using X-ray CT information in image-based models is the direct integration of the 3-D organization of the soil phases in the numerical grid of pore-scale models. Soil processes such as water flow, multiphase flow, diffusion, or reactive transport are subsequently computed in the 3-D pore space using Computational Fluid Dynamics (CFD) tools such as Finite-Element, Finite-Volume, or lattice-Boltzmann approach. In an alternative indirect approach, X-ray CT data is used to obtain key image-based structural qualities to parameterize soil features that are needed as input parameters into macroscopic soil models that ignore the explicit 3-D description of soil. One example of such a morphometric statistical approach is the extraction of critical pore diameters to be used in macroscopic analytical relations to predict saturated hydraulic conductivity. The structural morphometric parameters obtained from X-ray CT can also be used to generate an equivalent 3-D pore space, for instance extracting the Euler number (a number used for quantifying the connectivity of the pore space) to parameterize a pore network model (PNM). The latter approach is also classified as indirect. Soil structure can also be generated from explicit soil images by using more sophisticated statistical methods such as the Markov-chain Monte-Carlo method as performed by Wu et al. (2004). Although the images used in this latter work were 2-D thin sections, the method can be applied to X-ray CT data (Lin et al., 2020).

This chapter focuses on soil modelling approaches that make use of X-ray CT. Other porous media, such as carbonate rocks, sandstone, etc., have been thoroughly addressed in other reviews (e.g., Golparvar et al., 2021) to which readers are referred to. First, we focus on the practical aspects of the integration of X-ray CT data of the main soil phases into spatially explicit models highlighting the main approaches that can be used for such integration. We then provide an overview of how X-ray CT data has been integrated in soil modelling research distinguishing between work on a direct or an indirect use of X-ray CT.

11.2 Direct Integration of X-ray CT Data-Derived Soil Phases into Image-Based Models

In this section, we focus on the integration of X-ray CT data of the main soil phases, namely pore space (sect. 11.2.1), air and water (sect. 11.2.2), organic matter (sect. 11.2.3), and roots (sect. 11.2.4), highlighting the main modelling strategies that can be adopted.

A number of preliminary steps must be carried out before the integration of the soil phases in image-based models can be undertaken. These processes allow for the acquisition of a greyscale representation of the soil structural information that is usually segmented to produce binary images, i.e., black and white. Following this, the different soil phases are then labelled by different colours identifying solid particles and non-solid particles (pores), which are normally used by image-based models despite providing less signal information than greyscale images. Readers interested in segmentation and post-treatment processes are referred to Chaps. 5 and 6 along with the reviews of Iassonov et al. (2009) and Schlüter et al. (2014).

11.2.1 Integration of X-ray CT Data of Pore Space Geometry

Different strategies can be adopted to integrate the structural information of pore space geometry provided by the segmented X-ray CT images into image-based models. We will distinguish four strategies differing in the degree of simplification of the soil structural information and the immediacy of their use: voxel-based, finite-element meshing, morphological, and irregular PNM approaches. The main characteristics of these strategies are summarized in Fig. 11.1.

Fig. 11.1

Overview of four main methods used to integrate X-ray CT 3-D structural data in image-based models. In the figure, points that can be seen as advantages are in green while disadvantages are depicted in red. PNMs stands for pore network models. In the figure, C_G is an indication of the computing time required to generate the numerical grid and C_S is an indication of the computing time required to simulate diffusion of a chemical species through the network for 10 days. An X-ray CT image of 512³ voxels and an average desktop computer (e.g., AMD Ryzen 7 or Intel i7 processor, and 16/32 GB of RAM) have been assumed on providing these indicative numbers

The most straightforward strategy is to make use of all the structural information elements provided by the image to reconstruct the numerical grid. Depending on the CFD tools used, two approaches can be distinguished, the so-called ‘voxel-based’ and ‘finite-element’ modelling techniques. Voxel-based models directly map the voxels of the segmented X-ray CT image to the model’s numerical grid, as is typically done when using lattice-Boltzmann models. Depending on the desired degree of accuracy of the soil processes, the lattice-Boltzmann grid can be further refined (e.g., Sukop et al., 2008). Finite-Element solvers require the reconstruction of the numerical grid using more or less sophisticated meshing software (e.g., iso2mesh, Tran et al., 2020).

Another strategy is to reduce the number of computational elements of the segmented X-ray CT image while keeping key structural information, as performed by the so-called ‘morphological’ and ‘irregular Pore Networks’ models. In these cases, the integration of X-ray CT data requires a number of operations specific for each modelling approach. The main steps required are reported in Fig. 11.2. Morphological models approximate the complex pore space morphology with a set of simplified geometrical primitives such as balls (e.g., Hilpert & Miller, 2001; Vogel et al., 2005; Lehmann et al., 2006; Monga et al., 2008), ellipsoids or generalized cylinders (Ngom et al., 2012; Kemgue et al., 2019), located along the median axes of the pore space. This set of maximal balls recover the skeleton of the pore space. These balls also function as the nodes of an adjacency graph, which considers the neighbourhood information of each geometric primitive (e.g., balls) where the soil processes are locally computed (Monga et al., 2007). They can also be used to construct generalized cylinders or ellipsoids (Ngom et al., 2012; Kemgue et al., 2019). Like morphological models, irregular PNMs construct networks of nodes located along the median axes of the imaged pore space (e.g., Ebrahimi et al., 2013). Irregular PNMs use the same operations than the morphological approach to extract clusters of maximal balls. Then these clusters of maximal balls are converted to pore-throat network (Fig. 11.2). In both approaches, the size of the given geometrical primitives (in morphological models) and the bonds (in irregular PNMs) are locally computed from the measured area of inscribed circles or balls within the pores (e.g., Perez-Reche et al., 2012). PNMs classically use cylindrical or angular bonds (throats). There are standalone open software tools (http://openpnm.org, Gostick et al., 2016; http://porespy.org, Gostick et al., 2019) that allow constructions of these irregular pore networks.

Fig. 11.2

Overview of the operations required to integrate X-ray CT images of the soil architecture into image-based models. For illustrative purpose, we sketch a binary segmented image showing pore and solid space. Note that other soil phases may be added (e.g., water phase, solid organic matter). (a) The segmented X-ray CT image is used directly by voxel-based approaches. Note that only the pore phase is represented in the figure. (b) Finite-element approaches use a mesh approaching where the segmented pore space is located. (c) The morphological approach requires the skeletonization of the segmented pore space image, which is used as a base to construct a minimal set of maximal balls that approach the soil pore space. Once identified, the relationships and connectivity of the maximal balls are written in an adjacency graph, which is valuated by attaching to each ball a set of attributes (e.g., air or water, organic matter mass). (d) In addition to the steps described by the morphological approach, irregular pore networking models extract clusters of maximal balls that are converted to pore-throat networks. This requires sorting the maximal balls by decreasing sizes, with larger balls ranked as ancestors and small ones as daughters, the identification of common element between families (bonds between pores), and the assignment of geometrical properties to the bonds identified (e.g., angularity, surface area, shape factors). Further details of the substeps required by the morphological and irregular pore network approaches can be found in Table S11.1

In PNMs, the choice of the bond shape can have important consequences on the model behaviour. Microscopic soil images revealed that natural soil pores consist of heterogeneous and angular cross-sections rather than cylindrical shapes (Blank & Fosberg, 1989; Hamamoto et al., 2016). Quantifying such angular form of pores is not only crucial for a realistic representation of soil microscale structure, but it also has a significant impact on water configuration and soil hydraulic conductivity (Raoof & Hassanizadeh, 2012). For instance, using fast tomography imaging in simplified porous media made of packing of glass beads and in sands, Hoogland et al. (2016) characterized corner flow and conductivity dynamics during soil drainage process. As pores are drained or dried, a fraction of water remained in the pores’ angular corner (Tuller et al., 1999). Many studies have shown that the assumption of cylindrical pore cross-section that only allows for single-phase occupancy was insufficient to represent water configuration and hydraulic continuity (Zhou et al., 2000; Tuller & Or, 2001). On the other hand, angular pores allow retaining two or more fluids simultaneously and thus better represent liquid configuration in soil. Joekar Niasar et al. (2009) developed a geometry-based approach for pore network modelling to characterize and produce various ranges of pore cross-sections to describe natural pore geometry, including irregular hyperbolic triangles and regular hyperbolic polygons. The pore geometry and angularity are also essential factors controlling colloid and microbial transport in unsaturated soil (Bradford & Torkzaban, 2008; Saiers & Lenhart, 2003). For instance, Ebrahimi and Or (2014) showed that pore angularity impacts effective water film thickness necessary for bacterial motility in unsaturated soil. In addition, soil macropores and their orientation are known to significantly impact hydraulic conductivity by allowing the rapid flow of water in the soil. Micropores are created by worm holes, plant roots, soil aggregation, and soil wetting/drying. Recent advances using X-ray CT imaging allowed for characterization and quantification of such soil pores at higher resolution compared to traditional methods that used dye tracing, spectral image analysis, and sectioning soil into thin layers (Katuwal et al., 2015; Li et al., 2016; Hu et al., 2020). It is worth mentioning that as with all imaging and image-based techniques, there is a constant temptation to include more and more detail on different physical processes into models. Novel technologies provide a detailed view of soil pore structure however, the challenge remains in finding the right balance in terms of the number of measurable details that need to be captured in the models and observations to effectively describe soil function while reducing the inherent complexity of soil. Not all details matter but direct simulation alone or in combination with homogenization approaches (discussed later) is needed to determine which features matter and when; this process allows for systematic classification of processes/phenomena into important and unimportant functional traits.

The advantage of morphological models and irregular PNMs is that they provide a close approximation of the X-ray CT image-based soil pore space as they retain the spatial heterogeneity of the clustering of pores and connectivity in addition to the pore-size distribution (Blunt, 2001; Xiong et al., 2016; Perez-Reche et al., 2012). These methods compact the data information provided by the X-ray CT images and represent an interesting alternative to estimate soil processes in larger-scale simulations than can do the voxel-based models.

As an illustrative example, we have used a set of synchrotron X-ray CT images of soil aggregates from Peth et al. (2014) to show how pore space is integrated in three selected image-based model approaches: a lattice-Boltzmann Method (LBM), a morphological model, and an irregular PNM. Before segmenting the image, we reduced the noise of the image using a non-local filter mean and passed an unsharp mask to enhance contrast between phases (Schlüter et al., 2014). We then applied a local segmentation using a watershed algorithm (Marker-Controlled Watershed in MorpholibJ plugin of Fiji/ImageJ, Legland et al., 2016) to obtain a binary image, with white voxels corresponding to the solid phase and black voxels corresponding to the pore phase. Post-processing of the binary image was further performed with a majority filter to remove segmentation noise. The segmented image was then integrated in the LBM which is a voxel-based model by making a direct correspondence between each X-ray CT voxels and the nodes of the model numerical grid. The same segmented image was then integrated in the morphological model of Monga et al. (2014) and the irregular PNM of Ebrahimi et al. (2013). Figure 11.3 displays the same region of interest (ROI) of the image to better highlight the differences in pore space recovering by the three approaches. The porosity obtained with the three approaches was 0.1233, 0.1054, and 0.0986 for the voxel-based, morphological, and irregular PNM approaches, respectively. Comparison of the pore-size distribution obtained with the three methods can be seen in Fig. 11.4. A priori, underestimations of the pore space volume recovered by the irregular PNM and morphological model can be expected due to the approximations these methods assume. Nonetheless, for this image, these approximations did not translate into a substantial loss of the pore-size classes of the pore volume. The morphological approach, which consists of a minimal set of maximal balls that barely overlap, reproduced the pore-size distribution of the ROI computed with the morphological opening released in the MorpholibJ plugin of Fiji/ImageJ (Legland et al., 2016) well. The slight underestimation of pore volume in PNM is associated with the throat assignment step in which the neighbouring maximal balls to a pore are converted to a throat with a constant size. The throat size is chosen based on the average size of maximal balls connecting two pores that tends to overrepresent small balls.

Fig. 11.3

3-D view of a pore geometry integrated in the voxel-based, morphological, and PNM approaches. In the morphological approach, balls are used as geometric primitives. In the irregular PNM approach, cylindrical bonds are used. The size of the ROI is 56x56x56 voxels. The voxel resolution is 9.77 μm (Peth et al., 2014)

Fig. 11.4

Pore-size distribution of the ROI (56×56×56 voxels) of Peth et al. (2014) image, obtained with a voxel-based (purple), morphological (green), and the irregular PNM (orange) depicted in Fig. 11.3

11.2.2 Integration of X-ray CT Data of Air–Water Distribution

The fact that attenuation of X-rays by water is comparable to that of organic matter hinders distinguishing these two phases in soil X-ray CT images. Imaging protocols are not always straightforward (e.g., Rezanezhad et al. (2010) did not distinguish water from organic matrix of peat soils). In the case of peat soils, distinguishing water from organic matrix relies on the equilibration time between peat and water, which results in some solutes being incorporated to the water surrounding peat, blurring the distinction between phases. If water in the saturated peat is replaced with deionized water, contrast is enhanced and distinction of the two phases is possible (Iain Young, personal communication). High quality images and a resolution targeted to the visualization of air–water interfaces in well resolved pores are thus required. For instance, air-filled and water-filled pores were successfully segmented in X-ray CT images of soil columns (e.g., Mooney, 2002; Daly et al., 2015). Imaging the spatial distribution of water and air phases in the soil pore space can be more easily measured using synchrotron radiation X-ray CT (e.g., Carminati et al., 2008; Pot et al., 2015; Daly et al., 2017). Another option is to stain water by heavy contrast agents. Li and Tang (2019) used an iodine-based contrast medium to visualize water films in glass beads and sand, but they could not observe them in a clayey soil. Our previous experiences have shown that the drawback of using contrast agents, particularly in structured clayey soils, is the massive change in electrolyte concentration in the soil solution brought by adding contrast agent solution (such as KI), which could lead to strong soil dispersion (Pot et al., 2011). Scotson et al. (2019) investigated the use of gold nanoparticles for X-ray CT contrast enhancement, which might well provide a promising avenue, especially for soil biological processes, as gold is inert.

The water distribution and movement can be more easily measured using neutron imaging. Neutrons interact strongly with light elements such as hydrogen allowing for better distinction of soil water at the expense of a poorer representation of soil minerals. Combining X-ray and Neutron CT images using data registration techniques allows exploiting this complementarity and obtaining the solid, air, and water phase distributions experimentally (Clark et al., 2019). This process requires de-resolving the higher resolution image (usually X-ray CT) to match the lower resolution data, which can be done, for instance using Fiji/ImageJ. Despite that in principle, the whole range of soil water content can be resolved, the accuracy is largely influenced by the spatial and temporal resolution achievable by the tomographic method (Tötzke et al., 2021). For instance, Moradi et al. (2011) reported rhizosphere volumetric water content profiles of a sandy soil sieved to <2 mm that ranged from 0.05 to 0.4, approximately.

To date, only a few image-based models have integrated X-ray CT imaging of water and air distributions directly (Keyes et al., 2013; Daly et al., 2015, 2016). Rather, these distributions have been computed using drainage and imbibition algorithms that calculate the typical capillary pressure head that allows air or water invasion in the pore network. The sequence of pore invasion by air or water is determined by pore size and pore connectivity only and ignoring the viscous effects and gravity. In the case of drainage, the network of pores is initially assumed to be water filled. Then pores connected to the network entry or already invaded neighbouring pores are drained in a sequence of decreasing pore size. Each pore is evaluated for air invasion based on its size and experienced pressure head according to the Young–Laplace equation. This is adopted in modelling frameworks using morphological models and PNMs (e.g., Carminati et al., 2008; Monga et al., 2014; Ebrahimi & Or, 2015; Pot et al., 2015; Hoogland et al., 2016; Kemgue et al., 2019). Another option is to apply interfacial (van der Waals) and capillary forces to the simulated fluid to generate a phase separation between regions of high fluid density and low fluid density, respectively, attributed to water and air. This approach is adopted in the voxel-based LBM (e.g., Falconer et al., 2012; Genty & Pot, 2013) and in finite-element solvers such as the commercial software COMSOL Multiphysics (Cooper et al., 2017, 2018). LBM solvers with free licenses for research such as PALABOS (Latt et al., 2021) and DL_MESO (Seaton et al., 2013) are available. When interfacial and capillary forces are computed there is a remarkable agreement of the position of the air–water menisci to that measured in X-ray CT images (Pot et al., 2015). Good agreement can also be obtained using morphological models and PNMs (Carminati et al., 2008; Pot et al., 2015) even if the air–water interface presents as an abstracted geometry of the real pore space using Euclidean geometries with defined angularities (Hoogland et al., 2016).

The success of numerical methods to compute water and air distributions in pore space geometries extracted from X-ray CT images and the difficulty in obtaining X-ray CT imaging of water and air distribution in complex soil samples explain the wide adoption of these methods. As an example, Fig. 11.5 displays a comparison between synchrotron X-ray CT imaging and air–water interfaces computed using the LBM and the morphological approach. The mean absolute error between simulations and measurements of the position of the menisci were 5% and 18% for the LBM and morphological approach, respectively. As can be seen in the picture, performance of the methods is remarkable, especially for the LBM approach.

Fig. 11.5

Images of air–water interfaces in a disturbed soil sample, as measured by synchrotron X-ray CT imaging and as produced by a voxel-based approach and a morphological model (Pot et al., 2015). Cyan is the iso-contour of the air–water interface and dark blue is the liquid phase

11.2.3 Integration of X-ray CT Data of Solid Organic Matter Distribution

The similar attenuation of X-rays by water and solid organic matter also hinders the imaging of soil organic matter in X-ray CT images (see Chap. 10). However, large fragments of organic matter, the particulate organic matter (POM), can be rather easily visually identified due to their recognizable shape, as was shown when comparing X-ray CT with micromorphological study of organo-mineral assemblages in thin sections (Elyeznasni et al., 2012). Multi-modal segmentation tools can be used to correctly segment POM and avoid false coatings of organic matter around large pores, due to partial volume effect (e.g., Elyeznasni et al., 2012; Schlüter et al., 2014). However, the identification of small organic matter embedded in the soil matrix remains a challenge and needs the adoption of organic matter staining (Peth et al., 2014; Van Loo et al., 2014; Quigley et al., 2018; Piccoli et al., 2019a; Zheng et al., 2020; Maenhout et al., 2021). Despite that some measured X-ray CT organic matter distributions are available in the literature, this information has not been used in modelling studies yet. Instead, scenarios of organic matter placements in the segmented X-ray CT images have been used (e.g., Falconer et al., 2015; Portell et al., 2018).

To initiate integration of realistic 3-D X-ray CT images of spatial distribution of organic matter, we take the example of the soil image obtained by Peth et al. (2014) that we partially displayed in Fig. 11.3 and extracted the structural information required by image-based modelling (Fig. 11.6). Peth et al. (2014) irreversibly fixed osmium to the organic compounds of the soil. To locate the stained organic matter, the soil sample was scanned at a synchrotron at two photon energies above and below the absorption edge of X-ray by osmium. To obtain Fig. 11.6, we first applied a noise reduction pre-processing on both images, using a median 3-D filter and passed an unsharp mask to enhance contrast between phases (Schlüter et al., 2014). Following the procedure in Peth et al. (2014), we subtracted these images and obtained a 3-D image where the highest grey levels correspond to the highest concentrations of osmium associated with large fragments of organic matter (POM) and lower grey levels correspond to organic matter coatings of pore walls or organic matter residing in smaller pores within the soil matrix (Peth et al., 2014). A global segmentation using three thresholds whose values were initially chosen using Otsu and C-means segmentation algorithms was carried out. This resulted in a multi-modal image where the three types of organic matter identified in Peth et al. (2014), associated to different levels of osmium concentration, were distinguished. A majority filter was passed on the resulting image to remove noise associated with the segmentation step. To integrate this structural information on organic matter in an image-based model, we merged this information with the former binary image containing information on the pore and solid phase as discussed in sect. 11.2.1. To do so we simply added both images to get a multi-modal image. This structural information can now be directly integrated in an image-based model, e.g., the lattice-Boltzmann model of Pot et al. (2010), where different reactivity of the solid–pore interface is included.

Fig. 11.6

3-D views of soil fragment showing osmium-stained soil organic matter. (a) Total organic matter with high (red), medium (cyan), and low (green) relative osmium concentration, (b) Piece of organic matter showing a high relative osmium concentration, (c) Pore phase (dark blue) and the same organic matter stained with a high relative osmium concentration. The size of the ROI is 56×56×56 voxels. The voxel resolution is 9.77 μm (Peth et al., 2014)

11.2.4 Integration of X-ray CT Data of Roots

Despite the recent popularity for rhizosphere related research, few rhizosphere image-based modelling approaches are available to date, probably due to the rhizosphere complexity (both in terms of processes and soil phases involved) and to the challenges faced by the structural imaging of plant–soil interactions such as the identification of soil phases and difficulty of imaging fluxes (see Roose et al., 2016). Distinction among root and other soil phases is increasingly being performed yet it remains challenging, suffering from the limitations such as partial volume effects already highlighted previously. For instance, the pioneering work of Keyes et al. (2013) used Synchrotron X-ray CT and successfully identified root hairs, soil, pore space, and root surface regions. Also, along with the characterization of mucilage and root hairs, and water sorption of chickpea roots, Rabbi et al. (2018) computed the radial and axial water permeability using lattice-Boltzmann modelling based on X-ray CT images of sieved and repacked soil. Given that they are usually filled with water, roots can also be identified using neutrons. For instance, Zarebanadkouki et al. (2012, 2014) used neutron radiography to visualize the transport of deuterated water and developed a method to reconstruct flow into roots. Segmentation of the reconstructed images is complex, often involving semiautomatic methods (Keyes et al., 2013; Daly et al., 2016). Open-source semiautomated tools to follow root growth such as RooTrak (Mairhofer et al., 2016) or the Fiji/ImageJ plugin Smart Root (Lobet et al., 2011) as well as commercial tools such as WinRHIZO are available (see Chap. 9).

Once segmented, the root structural information is used to set up the processes of interest. To date, the most studied processes are nutrient uptake (mainly P), uptake of water in saturated and unsaturated conditions, and root growth. Unlike root architecture-based approaches, which use volumetric reactions/uptake terms, spatially explicit image-based approaches use the root structural information to define the boundary conditions suitable to the process of interest (Ruiz et al., 2020b). Keyes et al. (2013) used Michaelis–Menten nutrient uptake conditions to simulate P uptake by root and root hairs. McKay Fletcher et al. (2019) also assumed a Michaelis–Menten nutrient uptake boundary condition on all active roots but added a constant citrate exudation rate per surface on root tips to study organic-acid enhanced P uptake. Daly et al. (2018) assumed a water uptake rate proportional to the radial conductivity of water through root epidermis and the pressure drop between root internal xylem pressure and the water soil potential. The internal xylem pressure adopted was 1 MPa during the day and 0 MPa at night. For an in-depth explanation of the mathematical expressions adopted by current approaches, readers are referred to the reviews of Ruiz et al., (2020b) and Daly et al. (2016). So far, assumptions about the specific uptake/exudation regions need to be tackled using expert knowledge due to the lack of structurally explicit data. Root (or root hair) growth is currently being studied using time-dependent boundary conditions. A common technique to do so is assuming a given geometry and activating the appropriate boundary condition according to a (measured) growth rate (e.g., McKay Fletcher et al., 2020). In this approach, the developed root segment is used, but the uptake/exudation properties are activated at the appropriate time after having started the simulation.

11.3 Overview of Image-Based Modelling Works

11.3.1 Modelling of Soil Physical Processes

11.3.1.1 Saturated Water Conditions

The macroscopic properties of transport in soils depend on the 3-D organization of the pores and their size (Dullien, 1992). In that regard, the direct integration into transport models of X-ray CT images accurately describing soil cavities and their connectivity is a unique opportunity to explain soil hydrodynamic properties. Several works have numerically calculated 3-D water velocity fields in X-ray CT images of soil samples (e.g., Daly et al., 2015; Dal Ferro et al., 2015; Pot et al., 2020; Zhang et al., 2021a). To carry out image-based modelling, the images are segmented into binary information representing the soil pores and the solid phase (see Sect. 11.2.1). Mathematically, rigorous up-scaling for fluid flow computation, where Darcy’s law is derived from Stokes’ equations solved at the microscopic scale (Daly & Roose, 2014), enables the calculation of the hydraulic conductivity and soil water retention curve, based on a representative element volume (REV)(see Chap.3). The periodicity of the soil architecture of the REV is also required. However, the search for an image size that is representative of the physical properties of the soil such as porosity and/or hydraulic conductivity has been rarely carried out (Daly et al., 2015; Gharedaghloo et al., 2018; Gerke & Karsanina, 2020). In other cases, the aim is to understand how flow paths emerge in the complex pore space (e.g., Pot et al., 2020) and how the macroscopic property of interest (calculated at the ROI scale), such as permeability, emerges from the 3-D water velocity fields calculated at the microscopic scale (pore organization within the ROI, e.g., Menon et al., 2011; Deurer et al., 2012; Khan et al., 2012; Zhang et al., 2021b).

In almost all the studies above, the voxels of the segmented image were directly mapped to grid nodes and the Stokes’ water flow was computed in the complex pore geometry using either the Lattice-Boltzmann approach (Menon et al., 2011, 2015; Hyväluoma et al., 2012; Zhou et al., 2018; Pot et al., 2020; Zhang et al., 2021b), finite-element solvers (e.g., Daly et al., 2015, 2017; Gerke et al., 2018; Gerke & Karsanina, 2020) or a finite-volume solver (Gackiewicz et al., 2019). Other approaches such as smoothed particle hydrodynamics (Dal Ferro et al., 2015) have also been used. Gharedaghloo et al. (2018) assumed an irregular PNM model. The network of pores extracted from segmented X-ray CT images consisted of cylindrical throats and volumeless nodes (see Sect. 11.2.1) and the water flow through the throats was calculated by solving a system of linear water balance equations (Valvatne & Blunt, 2004). To carry out the flow calculation, periodic conditions at the edge of the images are applied and a pressure gradient or force field (gravity) imposed. Once the steady state is reached, the 3-D water velocity field, which is the velocity at each node of the computational grid, is saved. Visualization of these 3-D fields reveals preferential flow paths. As a matter of example, visualization and characterization of flow pathways evidenced that pores of similar size contributed unequally to flow resistance in biological crusts (Menon et al., 2011). This work also provided qualitative insights on soil surface seal functioning by showing how only a few flow channels penetrated through the surface seal and revealed strong lateral flows below these seal structures (Hyväluoma et al., 2012). Permeability (or saturated hydraulic conductivity) is then calculated by integrating the 3-D fluid velocity fields over the ROI. In that respect, image-based modelling outputs have highlighted the major role that large conductive pores have on water flow (e.g., Dal Ferro et al., 2015). They have also confirmed that: (i) variability in permeability is largely affected by size and by a few key pore throats (e.g., Gharedaghloo et al., 2018; Pot et al., 2020; Zhang et al., 2021a) and (ii) that porosity is not the sole morphological parameter explaining such permeability, suggesting that pore-size distribution and the pore network connectivity and tortuosity must be considered as well (e.g., Zhou et al., 2018).

The outputs of image-based models have also been compared to X-ray CT-derived pore space morphological parameters such as porosity, pore-size distribution, connectivity (Euler number), tortuosity, and critical diameter of the percolating pores. Many of these parameters can already be calculated with existing open-source free image analysis software (e.g., Fiji/ImageJ, QuantIm), while others, including tortuosity, remain more complicated to obtain (e.g., Gommes et al., 2009; Roque & Costa, 2020). Tortuosity can be estimated from spectral dimension (e.g., Crawford et al., 1993) or by computed 3-D fluid velocity fields or solute diffusion fields in 3-D images (e.g., Koponen et al., 1996; Berg & Held, 2016; Zhang et al., 2021b). Correlations between these morphological parameters and outputs of the image-based models are commonly investigated in the literature. For instance, prediction of a reduction of flow rate when biological crusts and repacked soil aggregates are submitted to compaction was related not only to the reduction in total porosity but also to a modified pore-size distribution (Menon et al., 2011, 2015).

In an alternative indirect approach, X-ray CT images are first analysed to deliver morphological parameters such as the ones described above. These parameters are subsequently used in macroscopic empirical models that predict water content, hydraulic conductivity, and solute diffusivity. Examples of these empirical models are the well-known empirical Kozeny–Carman equation and other relations based on percolation theory and critical path analysis (Katz & Thompson, 1986) which relate the saturated hydraulic conductivity with specific surface area, tortuosity, and a characteristic length of the pore size. As found in works dealing with porous rocks (e.g., Nishiyama & Yokoyama, 2017), this indirect integration of 3-D X-ray CT data of soils has allowed to demonstrate that the critical path diameter is a better parameter to be used in the Kozeny–Carman equation as compared to the mean hydraulic radius (Koestel et al., 2018; Pot et al., 2020). Luo and Lin (2009) used X-ray CT imaging to show that fractal dimension alone was insufficient to characterize soil macropore structure and flow patterns including a lacunarity function that reflects size distribution of macropores as a diagnostic value to better describe soil structural properties. Packages included in existing image analysis software can also be used to compute diffusion processes. The QuantIm software (Vogel, 2008) solves Fick’s law of diffusion in 3-D with a finite difference approach. Using this package, Deurer et al. (2012) calculated the apparent diffusion coefficient of macropore networks around soil aggregates. They extracted small ROIs from segmented X-ray CT images of soil cores sampled under tree-rows and wheel-tracks and computed the diffusive flux within them. The size of the ROI was chosen to encompass a sufficiently large number of soil aggregates (almost 100). They attributed the absence of significant differences in the gas diffusion coefficient between ROIs to the similarity in the volume of pores larger than 1 mm and to the fact that they were well connected. Lattice-Boltzmann model (Zhou et al., 2019) and PNM (Gharedaghloo et al., 2018) were also used to compute solute transport properties. According to theory of transport in porous media, Zhou et al. (2019) predicted an increase of the solute dispersion coefficient in soil macroaggregates amended with biochar resulting from an increased saturated hydraulic conductivity. For peat soils, the diffusion coefficient decreased with depth driven by an increased tortuosity whereas dispersivity did not vary considerably (Gharedaghloo et al., 2018).

11.3.1.2 Unsaturated Water Conditions

Soil water movement has been modelled classically using Richards’ equation, which consists of a mass conservation combined with two equilibrium measurements for the partially saturated hydraulic conductivity and soil suction characteristic that relates the level of relative soil saturation to the soil water pore pressure. As such, both measurements can take a long time and hence the need for intelligent estimation of these values based on various morphological and textural information. In addition, both hydraulic conductivity and suction characteristic exhibit hysteresis, i.e., the shape of the curve as a function of saturation is different depending on if the measurement is done whilst drying or wetting the sample. Currently, this compounds the problem for measurements as to capture the hysteresis one would need to be doing numerous long wetting and drying experiments (weeks and months) very accurately so the locations of the Haine’s jumps are accurately captured for hysteresis; not to mention the extra sets to capture scanning curves between the wetting and drying at the intermediate points. Nonetheless, as suggested by Berg et al. (2013), who were able to directly image Haine’s jumps in sandstone samples using X-ray CT, future developments may allow to speed up the process.

Currently, Haine’s jumps can be predicted using modelling. The mathematical foundations for this that were used on X-ray CT images were established by Daly and Roose (2015). Cooper et al. (2018) expanded the previous work of Daly and Roose (2015) and Cooper et al. (2017) to include the effect of plant exudates on the water release curve (and hence Haine’s jumps) and the hydraulic conductivity. We next discuss how this was undertaken by giving a flavour of mathematical analysis involved.

We have already discussed above the ways X-ray CT measurements can be used to capture some of the statistics for estimating saturated hydraulic conductivity. An essentially similar approach can be employed to capture the partially saturated hydraulic conductivity curves if one scans the soil samples whilst wetting and drying and segments out the water–mineral and water–air interfaces. However, this is a very intensive image analysis endeavour. Rather, it would be much better to be able to predict computationally in combination with soil pore space imaging how the wetting and drying influence partially saturated hydraulic conductivity and soil suction characteristics. The fundamental science/mathematics concerning this was established by Daly and Roose (2014, 2015) and implemented directly on images by Cooper et al. (2017) and expanded to include root exudates that change the water wetting and drying properties in the pore space by Cooper et al. (2018). All this work was possible due to the rigorous application of the homogenization theory. Homogenization theory is a perturbation approach which assumes that a composite material, i.e., soil in this case, is comprised of many small things. This means that there is a microscale x and macroscale y and that the ratio of the two x/y= ε is small. The next step in homogenization assumes that the solution depends on each of these independently and upon application of the perturbation in ε, a sequence of analytic mathematical problems can be found that determine what microscale features matter at the macroscopic scale. For example, after Daly and Roose (2014, 2015) and Cooper et al. (2017, 2018), this approach gave a computational analytic approach to calculating partially the saturated hydraulic conductivity tensor and soil suction characteristic, including wetting and drying Haine’s jumps, based on the images of the pore space.

Constrained by the intensive image analysis tasks involved, very few works have attempted to use successive X-ray CT scans of soil samples at different water contents. Using an indirect integration of X-ray CT data, Rezanezhad et al. (2009, 2010) proposed a modified Kozeny–Carman equation taking into account the soil air-filled porosity. The parameters of the air-filled fraction were extracted from X-ray CT images of peat soils scanned at different water contents. Reasonably good agreement was obtained between the predicted conductivity and the experimentally measured one. The increasing number, volume, and tortuosity of the air-filled pores imaged were key parameters in these organic soils to explain the decrease of the unsaturated hydraulic conductivity. Fast scanning X-ray CT data of soil columns subjected to rainfall infiltration were also used to extract two key parameters to be used in a 1-D macroscopic preferential water flow model coupling a soil-matrix capillary flow and a gravity-driven macropore flow: the percolating macroporosity and the characteristic size of the porous medium, which was defined as the average half-distance between percolating macropores (Lissy et al., 2020). The latter parameter controls the lateral flow from macropores to soil matrix and was calculated using a Voronoï diagram of the X-ray CT image. Fixing these parameter values in the model strongly improved the mechanistic understanding of water exchanges between macropores and soil matrix. In particular, as it could be expected, the model was able to generate large water exchanges between macropores and the soil matrix for initially drier soil conditions.

Indirect integration of X-ray CT data has also been used to compute the partially saturated hydraulic conductivity and the soil suction characteristic. Knowledge of the pore-size distribution was delivered by 3-D X-ray CT imaging and the water release curve was calculated from it using the Young–Laplace equation (Beckers et al., 2014). Computation of the soil hydraulic function was subsequently carried out using the macroscopic numerical unimodal (van Genuchten, 1980) and dual porosity (Durner, 1994) models calibrated with the water release curve predicted from the pore-size distribution. The addition of supplementary X-ray CT-image-based data in the water retention curve allowed an improved calibration of the dual porosity model and enhanced predictions of the near-saturated hydraulic function.

Pore-scale imaging can also be used to characterize fractal geometry of pore space and water distribution with implications to determine gas transport and permeability of porous media (Yu & Liu, 2004; Liu et al., 2019; Piccoli et al., 2019b; San José Martínez et al., 2010). For instance, Liu et al. (2019) used X-ray CT imaging to show that the water and gas distributions of quartzitic sands are fractal. Using an indirect approach, they extracted the area and tortuosity fractal dimensions from X-ray CT images in order to develop a fractal theory-based relative permeability model to describe water and gas distributions in pores. Others have developed a number of fractal based models to describe soil unsaturated hydraulic conductivity using a fractal model for the water retention curve and the pore-size distributions (Tyler & Wheatcraft, 1990; Xu, 2004; Alfaro Soto et al., 2017; Ghanbarian & Hunt, 2017).

X-ray CT image-based information has also been indirectly integrated into 3-D macroscopic numerical models of water and solute transport. Unlike a direct integration, a computation mesh covering the whole domain, not only the pore space, is generated in this approach. Values of the parameters of the water release curve and hydraulic conductivity function are locally calculated as a function of either the type of material (denser regions defined as high grey values of the X-ray CT image and less dense regions defined as low grey values, Kasteel et al., 2000) or as a function of the local porosity obtained after segmentation of the X-ray CT images (Dohnal et al., 2013). These studies confirmed that solute spreading simulated at the column scale was sensitive to the difference in the local saturated hydraulic conductivity but not to the local dispersivities (Kasteel et al., 2000). Additionally, Dohnal et al. (2013) complemented X-ray CT data with 3-D data of water content obtained by Magnetic Resonance Imaging (MRI) to determine maps of entrapped air that was added to the 3-D macroscopic model as no-flow regions. Their modelling results supported the assumption that the observed decrease in outflow rate during repeated infiltration experiments could be caused by entrapped air in large pores. Moreover, the spatial distribution of the entrapped air had a greater impact than its volume.

Comparing predictions of an image-based model relying on accurate representations of 3-D air–water interfaces to long established semi-empirical equations from experimental determinations was used to better understand solute transport in unsaturated conditions (Genty & Pot, 2014). At the same time, this allowed testing the robustness of the direct image-based models used. Genty and Pot (2014) integrate X-ray CT pore geometries in an image-based model that computes the phase field equation (described in Sect. 11.2.2) obtaining explicit air–water interfaces at different water contents and at equilibrium. The air–water distributions were subsequently used by a second image-based model of solute transport. The 3-D solute diffusion fields at specific water contents produced by this second model allowed the estimation of the effective diffusion coefficient at the scale of the ROI through an inversion of a macroscopic analytical solution of the Fick’s law. A good agreement with the macroscopic semi-empirical equation based on Archie’s law, which relates the effective solute diffusion coefficient and the water saturation, confirmed the classic value of the exponent of Archie’s law used for soils (Genty & Pot, 2014). Hu et al. (2018) also used knowledge of equilibrated air and water distributions, this time calculated by a morphological image-based approach, and directly integrated this 3-D structural information in a simplified free-surface model considering static equilibrium to predict unsaturated water flow.

The final approach to simulating the unsaturated fluid flow in soils found in the literature is based on an indirect use of X-ray CT data to generate simpler spatially explicit models where the soil processes of interest are subsequently simulated. This was first done by Vogel and Roth (2001), who integrated soil pore-size distribution and topology data on the pore connectivity obtained from a 3-D reconstruction of serial thin sections of soil to produce a large number of equivalent pore networks, using a regular PNM. These authors demonstrated that this structural information was sufficient to predict the soil suction characteristic and hydraulic properties. Later, Köhne et al. (2011) applied the procedure described by Vogel and Roth (2001) to obtain an equivalent pore network based on X-ray CT images of two arable soils. It is worth noting that these works are following an indirect approach to the integration of X-ray CT data. Unlike irregular PNMs, regular PNMs are generated using regular assemblies of nodes interconnected by bonds of simplified geometry (e.g., cylinders). The length, radius, and number of bonds per nodes are parameters of the PNM. The modelled architecture reproduces an equivalent pore network of similar physical properties of given soil samples such as the pore geometry, size distribution, and degree of connectivity, which are extracted after image analysis of the target X-ray CT data.

11.3.2 Modelling of Biological Processes

11.3.2.1 Modelling of Microbial Activity

Image-based models are progressively moving towards the inclusion of soil biological activity, including fungal (e.g., Falconer et al., 2007), roots (e.g., Keyes et al., 2013), and bacterial (e.g., Vogel et al., 2015) processes, as it has been reported in recent reviews (Roose et al., 2016; König et al., 2020; Golparvar et al., 2021). Image-based modelling of soil microbial activity has essentially emphasized the role of soil architecture in establishing diverse spatial accessibility of organic substrates to decomposers that explain the spatial distribution and growth of fungi (Falconer et al., 2007, 2012, 2015; Pajor et al., 2010; Kravchenko et al., 2011; Cazelles et al., 2013), the bacterial diversity (Portell et al., 2018), and the microbial functions such as the decomposition of organic matter and CO₂ emissions (Ngom et al., 2011; Vogel et al., 2015, 2018; Monga et al., 2008, 2014). For instance, X-ray CT images of undisturbed sandy loam soil were directly integrated in an image-based model of fungal growth dynamics by Falconer et al. (2012). These authors simulated fungal biomass spreading in an X-ray CT pore space directly integrated in the numerical grid of the model. Fungal mycelia biomass was conceptualized by three biomass fractions: insulated (i.e., old inactive biomass), non-insulated biomass (i.e., active hyphal tips), and mobile (i.e., internal translocating resource) biomasses. Insulated biomass is immobile while non-insulated biomass and the internal resource spread through a diffusion process constrained by the pore space and existing fungal biomass subdomains, respectively. The system of partial differential equations was solved using the classical Crank-Nicholson scheme. Local fungal spread was modulated by a diffusion coefficient that is a function of the water content present in the numerical grid. The water content of the grid elements was computed by an LBM approach (as described in Sect. 11.2.2) implemented in the open-source PALABOS (Latt et al., 2021). Falconer et al. (2012) predicted a significant decrease in fungal growth by increased water saturation conditions. Water-filled pores alter the connectivity of the air phase, which slows down fungal colonization in line with the model assumptions. Nonetheless, the explicit integration of the 3-D distribution of air-water phases in the model suggested that it is the spatial location of water-filled pores rather than the water content itself that is responsible of this loss of connectivity and, therefore, that the bulk water content of the soil samples is insufficient to predict fungal growth and spatial expansion.

The explicit soil architecture imaged with X-ray CT can also be used to study survival and coexistence of different bacterial species in soil. Survival and coexistence of different bacterial species in simplified porous media have been shown to be controlled by the connectivity of the pores, the water saturation conditions, and the respective locations of the nutrients and bacteria (e.g., Long & Or, 2005; Ebrahimi & Or, 2015). As an example, Portell et al. (2018) investigated the role of soil architecture and location of organic matter in the coexistence of three bacterial strains showing competitive, generalist, and poorly competitive profiles. To do so, X-ray CT data of an undisturbed loamy soil was directly integrated in the numerical grid. The methodology coupled a lattice-Boltzmann model, accounting for the diffusion of dissolved organic carbon through the water phase (Genty & Pot, 2013, 2014), and an individual-based model, accounting for the activity of soil microorganisms. The approach also considers explicitly the distribution of organic matter hydrolysing over time following a first-order kinetic of constant rate. Hydrolysed organic carbon is released into the water phase neighbouring the particulate carbon, if there is any. The water–air distribution in the pore space was computed for three saturation levels (0.25, 0.5, and 1) using an LBM approach (see Sect. 11.2.2) described and parameterized in the literature (Genty & Pot, 2013; Pot et al., 2015). The lack of multi-modal data covering the organic matter distribution (see Sect. 11.2.3) and bacteria position was tackled here using simulation scenarios assuming a fixed bulk amount of organic matter initialized as solid fragments. Bacteria were positioned randomly on the interface between the pore and solid face, performing repetitions to account for the uncertainty of this distribution. Portell et al. (2018) predicted large effects of the spatial distribution of organic matter on the strain’s distribution but a largely unaffected total microbial biomass evolution. The poorly competitive strain was found to grow to the same extent of the other strains when located near large pieces of organic matter. Nonetheless, the presence of more competitive strains in the same microhabitat prevents the growth of the least competitive strain. These outputs are in agreement with predictions performed using simplified porous media that found that survival and coexistence of different bacterial species are controlled by the connectivity of the pores, the water saturation conditions, and the respective locations of the nutrients and bacteria (e.g., Long & Or, 2005; Ebrahimi & Or, 2015).

Integration of X-ray CT structural information into image-based models allowed the investigation of the role of differentiated soil porous architectures of a silt loam soil either subject to conventional tillage or under a grassland on organic matter decomposition (Ngom et al., 2011). In this study, the pore space of the segmented X-ray CT data was approached by means of a morphological approach based on spheres (see Sect. 11.2.1). The water retention curve of the ROI was simulated using the Young–Laplace law with a zero-contact angle between water and soil (Sect. 11.2.2). The biological sub-model described in Monga et al. (2008) was then used to simulate organic matter decomposition. This model simulates the growth, respiration, mortality, and dormancy of densities of microorganisms placed in its constituents’ maximal balls. To mimic extracellular enzymatic decomposition, the model assumes that soil organic matter within a ball can be decomposed by microorganisms placed in all balls connected through the water-filled path. Organic matter decomposition is modelled assuming an offer/demand approach comparing the maximal decomposable masses (offer) and the maximal microorganisms’ needs (demand). When the offer is less than the demand, the former is distributed according to the geodesic distance between the resources and the microorganisms using a polynomial repartition law. Simulation scenarios conducted by Ngom et al. (2011) used a temporal extent of 100 days and distributed organic matter and bacteria in pores of different sizes by placing organic matter in pores smaller than 20 μm and immobile bacteria in larger, more aerated pores. Ngom et al. (2011) suggested that small, isolated pores largely decreased organic matter availability to microbes, hindering mineralization in the conventional tillage treatment, whereas more connected pore space facilitated mineralization of most of the organic matter on the grassland treatment.

In a recent study, the impact of fertilization on the health of microbial communities was tackled using a direct finite-element based approach. This study (Ruiz et al., 2020a) examined how the soil structure and soil moisture content influence nitrogen movements around an ammonium nitrate fertilizer pellet and the subsequent impact on microorganisms’ activity. Nitrogen species accounted for in the model are ammonium, nitrate, and dissolved organic nitrogen, which were subjected to the processes of N immobilization, ammonification, nitrification as well as diffusion in the water phase. Samples were imaged with X-ray CT from where the soil phase, air phase, and the fertilizer pellet were segmented. The water distribution in the segmented pore space was computed using a two-phase water–air model (see Sect. 11.2.2). Release of nitrogen to the water phase is modelled using appropriate boundary conditions on the surface of the pellet. Boundary conditions around the soil phase were used to model the sorption and desorption of chemical species as well as chemical transformations mediated by microorganisms. The microbial population size responsible for these transformations was assumed to be fixed for the time length of the simulation. The reduction in the microbial activity due to cytotoxic concentrations of fertilizer was calibrated using preliminary experiments. Ruiz et al. (2020a) found that diffusion and concentration of nitrogen in the water phase is critically dependent upon soil moisture and chemical species considered and provides predictions for the maximum concentrations of ammonium and nitrate around the pellet under dry conditions.

In addition to the direct approach, X-ray CT data has also been indirectly used in modelling studies to understand the consequences of structural heterogeneities in soil microbial diversity and functioning. For instance, experimental studies have shown that low pore connectivity and heterogeneous water configuration in soil increase microbial diversity (Carson et al., 2010) and interaction (Tecon et al., 2018). Using statistical estimates of pore sizes and distances between interstitial sites obtained from existing X-ray CT images of soil samples, Resat et al. (2012) produced simplified soil aggregate structures with random, continuously-connected pores. They studied the dynamics of microorganisms using two distinct carbon obtention strategies: microorganisms with membrane-associated hydrolase and microorganisms releasing extracellular hydrolases. The authors found that coexistence of both strategies produced higher cellulose utilization efficiency and a reduced stochasticity. They also concluded that bacterial growth dynamics was the result of a balance between the substrate degradation kinetics, the dynamics of enzymes produced by the microorganisms, and the diffusive processes taking place in the modelled pore space. Their modelling work also supports previous evidence that bacteria tended to grow preferentially near the carbon source.

Burrowing activity of soil fauna makes the soil architecture a highly dynamic variable. A realistic consideration of the modification of soil architecture by its biological components is needed as it impacts not only soil microbial functions, but soil physical functions as well. The latter were investigated by Bastardie et al. (2002) who built simplified 3-D networks of earthworm burrows in which they simulated saturated water flow. The networks were made of a series of connected cylinders, which represented segments of earthworm burrow systems. Using an indirect approach, they extracted statistical estimates of length and orientation of these segments from X-ray CT images of a repacked soil core colonized by a single earthworm (Capowiez et al., 1998) to generate a large number of equivalent porous media. Earthworm trajectories were computed by predefined rules of worm movement obtained from experimental observations in 2-D terraria (Capowiez & Belzunces, 2001). Therefore, like regular PNMs, physical properties of these equivalent porous media such as length and orientation of the burrow segments are reproduced but their actual spatial arrangement is disregarded. Bastardie et al. (2002) showed parameters such as the volume of pore space and the interconnectedness of burrows were not relevant to explain the permeability of the porous media. Instead, the connectivity of the burrow systems (i.e., the number of independent paths) was found to be a more relevant parameter. So far, actual representation of temporal modification of soil architecture by earthworms has been undertaken using repeated X-ray CT scanning at a low temporal resolution, mainly to highlight modification caused by the treatment applied (Pagenkemper et al., 2015). Nonetheless, this information has not yet been directly used in a model. Instead, Blanchart et al. (2009) generated fractal soil geometry to investigate the role of earthworm burrowing including ingestion and production of casts in compacting and uncompacting soil using an equivalent porous medium. Along the same line, it is worth reporting modelling studies simulating temporal modification of soil architecture by soil microorganisms in which a simplified arrangement of solid particles on a regular numerical grid has been used by some authors (Crawford et al., 2012; Ray et al., 2017) to tackle the role of the production of gluing agents by microorganisms in gathering soil particles.

It is worth highlighting the role that image-based modelling of soil microbial activity can play in generating mechanistic knowledge of biologically driven processes in complex environments such as soil. One of the approaches used in that respect is the realization of sensitivity analysis where the soil architecture is explicitly taken into account using X-ray CT data (Cazelles et al., 2013; Vogel et al., 2015, 2018). Water–air distributions in the imaged pore space are then usually computed using image-based modelling for different water contents (see Sect. 11.2.2). Varied organic matter distributions (see Sect. 11.2.3) and positions of microorganisms are additionally considered for conducting simulation scenarios. Far from just being a mathematical exercise, such analysis provides valuable insights into how much soil functions and processes are affected by different soil architectures (provided by X-ray CT) and by changes on further abiotic (amount and distribution of organic matter, moisture content, etc.) and biotic (distribution of microorganisms, growth parameters, etc.) inputs. This allows testing of hypotheses about the role of soil architecture on soil microbial activity by unravelling, for instance, the relative importance of the biotic and abiotic components. As an example, analysing fungal physiological processes, soil architecture, and initial carbon distribution, Cazelles et al. (2013) found the spatial extension of hyphal tips and the fungal biomass yield were highly affected by the fungal physiological parameters associated with the growth and recycling processes. Also, Vogel et al. (2018) found that under low accessibility of organic matter to soil bacteria, conditioned by the local structural heterogeneities in the bacteria micro-habitats, the decomposition of organic matter was no longer dependent on the physiological parameters. A general point related with this kind of analysis is that the model assumptions and limitations need to be properly communicated and understood so the results can be appropriately interpreted.

11.3.2.2 Modelling of Root Processes

The prospective of steering rhizosphere management to boost plant productivity while reducing agrochemicals has attracted much attention of the scientific community, including the development of many rhizosphere models. Nonetheless, the number of image-based models existing to date is still very limited (Ruiz et al., 2020b). If we consider models where the soil architecture is considered explicitly, the list of approaches is even more reduced. Keyes et al. (2013) used Synchrotron X-ray CT and developed a finite-element image-based model of phosphate uptake by wheat roots and root hairs and implemented it in COMSOL. The volume mesh used by the model includes root hairs, soil, root surface, and water defined separately. Different grey levels were used to identify the meshed elements thanks to the high quality of the synchrotron light. Potential heterogeneity on the P transporters along the root was not considered due to the lack of data and uptake properties were assumed to be the same over the surface. These authors found that the root surface was responsible for a slightly superior P uptake (1%) than the root hairs, despite their closest proximity to the soil surfaces, where most of the phosphate was located. In a follow-up study, Daly et al. (2016) expanded the previous model to include both bulk and rhizosphere soil. The soil near the root (i.e., rhizosphere soil) was explicitly described through the integration of X-ray CT data and surrounded by a large/infinite bulk soil (i.e., away from the root). The latter was achieved with the use of a boundary condition obtained by means of homogenization theory. Root-hair growth was mimicked by considering the structure and a time-dependent uptake boundary condition. The study found that the key criterion for convergence of the P uptake was the size of the rhizosphere region considered.

Further work on P uptake by roots focuses on complexifying the P uptake mechanism but also simplifying the structural description of the soil phase. McKay Fletcher et al. (2020) added the exudation of citrate, which can enhance P uptake and study the effect of root system architecture on organic-acid enhanced P uptake of 12-day-old wheat plants. Time-lapse X-ray CT was used to define the root architectures. Boundary conditions were placed as dictated by the measured X-ray CT structures and growth simulated by activating the appropriate conditions mimicking the X-ray CT root growth measured. The soil volume was identified in the X-ray CT images, but the soil architecture was disregarded and a homogenous soil, including a parameter to account for an approximate volumetric water content, adopted. McKay Fletcher et al. (2019) further evolved the approach to investigate how changes in root system architectures of barley (Hordeum vulgare) affect P uptake from a fertilizer pellet. The study elegantly combines modelling, X-ray CT, Scanning Electron Microscopy with Electron Dispersive X-ray Spectroscopy (SEM-EDS), and soil thin sections. As in McKay Fletcher et al. (2020), X-ray CT identified solid, water, and pores were assumed as a single homogeneous volume and modelled using a single finite-element mesh. Appropriate boundary conditions are then placed on the boundaries of the root and fertilizer pellet, which were also identifiable in the X-ray CT images.

Daly et al. (2018) used time resolved X-ray CT and modelling to test the accuracy of spatially averaging the water uptake of young wheat plants grown in pots of 50 mm diameter x 80 mm height. Roots, solid, and pore space were identified in the X-ray CT images. The water–air interfaces were computed by implementing Richards’ equation for partially saturated flow parameterized with the water release characteristics and the saturation-dependent hydraulic conductivity measured. A finite-element mesh describing the root surface, the soil architecture, and the air phase was created and model outputs for a number of root systems were compared to the averaged model of Roose and Fowler (2004) to account for the effect of explicitly including the root architecture. Comparison of these approaches for a number of root system architectures and two soil types found less than 2% divergence. Nonetheless, the simplified model could not capture the heterogeneous water distribution seen in the image-based approach. Since Daly et al. (2018) performed a number of image treatments averaging many soil particles, it has been suggested that this may actually be masking some nuances at the pore scale not captured in this study (Ruiz et al., 2020b). A more comprehensive review about the use of image-based modelling to describe water dynamics in the rhizosphere is provided by Daly et al. (2017).

Landl et al. (2019) simulated the impact of naturally created biopores originated from decayed plant roots and earthworm activity on root growth, soil water flow, and root water uptake. In this work, soil properties are defined at the nodes of regular cubic grids, each having a hydraulic state and soil properties. A soil element (volume) was subsequently defined by the properties of the 8 surrounding nodes. They assumed two regular cubic grids, a coarse grid (1 cm) used to compute water flow and a finer one (0.5 cm) to compute growth development affected by soil local conditions. Biopores were modelled as a soil material with a very low penetration resistance to the roots and assumed to be air-filled, thus, not contributing to the water flow. An X-ray CT image stack (231 μm thick) was initially used to set up the position of the biopores directly. A downgrading procedure involving an initial skeletonization of the X-ray CT data was used to assign biopore voxels to the fine grid. The method was found to produce an increased bioporosity while keeping biopore length densities fairly constant. Model parameters were calibrated via the optimization of the root architecture parameters so that simulated and experimentally observed root length distributions were in agreement with each other. Simulation scenarios with the calibrated model showed biopores sustainedly mitigated transpiration deficits by permitting roots to take up water from deeper soil layers. This was found also when a reduced root water uptake in biopores due to limited root–soil contact was assumed.

Assuming an indirect integration approach, X-ray CT data of root architectures growing in soil can be used to parameterize existing root system models. Root system models are approaches simulating the structure, topology, and 3-D placements of roots. Root growth is simulated assuming a set of rules regarding elongation, branching, and death and a discrete approach defining individual segments of root that may have attributes such as radius, age, or hydraulic properties associated is usually adopted (Schnepf et al., 2020). A set of benchmarks, including reference calibration data sets, have recently been proposed by Schnepf et al. (2020), which suggest that methodologies may converge in the near future. As a matter of example of the methodology, Landl et al. (2021) coupled a 3-D root architecture with a rhizodeposition model that accounts for the transport of rhizodeposition. The coupled model is used to investigate spatio-temporal distribution patterns of citrate and mucilage in soil. Root system architectures of Vicia faba and Zea mays were extracted from X-ray CT images and saved as Root System Markup Language (Lobet et al., 2015). An in-house computer code was then used to extract the model parameters of the root architecture model (e.g., internodal distance, root radius, branching angle, maximal root length). Soil architecture was not explicitly accounted for in this approach. After running a number of simulation scenarios, Landl et al. (2021) found that: (i) rhizodeposition hotspot volumes around roots reached maximum values at intermediate root growth rates, (ii) root branching led to an increased volume of rhizodeposition hotspot zones due to the rhizosphere of individual roots overlapping, and (iii) that the rhizodeposition hotspot values were smaller around the Zea mays root system as compared to the one predicted by Vicia faba.

11.4 Conclusions

Whether direct or indirect, the combination of X-ray CT and image-based modelling is a powerful tool to understand the consequences of the soil structural heterogeneity on soil functions. Indirect modelling methods allow consideration of this structural heterogeneity in an implicit manner. Direct approaches explicitly describe structural heterogeneity. The power of the direct approach is that it allows focus on particular aspects of soil, which often are difficult, or simply impossible, to manipulate experimentally. This strength comes at the cost of requiring higher computing resources which, in practice, limit the volume of soil simulated. These detailed microscale description of soil processes can then be extrapolated to the macroscale using mathematical homogenization.

Despite providing new insights on soil functioning, the robustness of image-based modelling frameworks relies on the quality of the X-ray CT data, as well as the mathematical and computational tools available. Caution must be made because computation tools may introduce their own errors into the problem being tackled, which can depend on the particular process or components being considered. Another critique often made regarding spatially explicit image-based models, mainly those focused on the soil microscale, is the unfeasibility of ‘validating’ the model outputs. This is due to the considerable heterogeneity found in soils and to our lack of knowledge of the soil functioning at this scale, which often requires using inverse modelling techniques and educated guesses. In simpler cases of validating image-based models simulating saturated water flow, some discrepancies between the predicted saturated hydraulic conductivity calculated in the X-ray CT images and experimental measures have been reported (Elliot et al., 2010; Dal Ferro et al., 2015; Koestel et al., 2018). This relatively poor performance has been suggested to be related to an inadequate imaging resolution that hinders the visualization of smaller pores in the X-ray CT images that still contribute to the water flow (Dal Ferro et al., 2015; Koestel et al., 2018). The first point to make here is that, in this context, models are first a tool to test and develop alternative hypotheses that can be subsequently tested experimentally. The second point is that when the purpose of modelling is prediction, not being able to obtain a specific experimental measure does not necessarily mean that the model cannot be validated. Confidence in model outputs can be obtained using multiple experimental data or knowledge data measured at different scales and/or for related scenarios (e.g., Grimm et al., 2005). Being able to reproduce situations and outputs for which the model was not explicitly calibrated or designed for suggests that the mechanisms implemented are robust. A closer exchange and collaboration among experimentalist and modellers would revert in the identification of experimental approaches and measurable (indirect) metrics allowing validating current modelling approaches.

A limitation of image-based modelling is inherent to trade-off between sample size and image resolution. The larger the soil sample, the higher the unresolved porosity that we are missing. For some processes such as water flow, this may not be a big deal but for others, such as the ones related to the diffusion of solutes (Zhang et al., 2016) and the biological activity of bacteria and viruses, it may not be acceptable. Decreasing sample size complicates experimental manipulation and may require increasing the number of repetitions. The unresolved porosity issue could be, in principle, partially tackled using greyscale images instead of segmented X-ray CT (Baveye et al., 2017). Reviewing the current work, it is self-evident that very few studies attempt to use this greyscale information which, in our view, should be increasingly explored in the near future. Further opportunities to tackle the resolution limit can also rely on the combination of methods. The representation of reactive species transport in soils, challenging at the time, offers examples of this method combination. In reactive transport, the bottleneck is often associated with the low resolution of X-ray CT images to estimate pore surface area. Some studies have suggested to use a sorption correction function which accounts for the deviation between the complex pore surface area measured in the X-ray CT images and the simpler one generated in an equivalent pore network (Köhne et al., 2011). Such correction functions allow for incorporating the information on the pore surface area into pore network models. Using this strategy, Köhne et al. (2011) improved the transport prediction of a reactive species through intact soil samples. Overall, further studies are required to systematically combine submicron and nanometre scale roughness of soil pore surfaces for accurate prediction of solute and colloid transport. Nanoscale surface heterogeneities have been shown to both increase and decrease colloid attachment in porous surfaces resulted from different influences of roughness under energy barrier absent or present, respectively (Shen et al., 2011; Rasmuson et al., 2017; Torkzaban & Bradford, 2016). As a matter of example, Lin et al. (2021) used a PNM and showed that nanoscale roughness alters the sticking efficiency and colloid attachment non-monotonically with tendency to increase with roughness.

The chapter has also highlighted that the simultaneous use of multi-modal image-based description of the soil is, with few exceptions, still to be translated into modelling exercises. This is the case for imaged organic carbon but, to a certain extent, also for soil microorganisms, although methods are being developed (Baveye et al., 2018). The most common method to date seems to rely on thin slicing of resin impregnated soil cores combined with a specific (FISH) or unspecific (calcofluor white) staining method to visualize microorganisms. These 2-D maps are then aligned with the 3-D structural X-ray CT data following complex alignment procedures (Hapca et al., 2015; Schlüter et al., 2019).