1 Introduction

The three-dimensional morphology of the pore space has a strong impact on effective macroscopic properties of porous media Sahimi (2011); Bear (2018); Blunt (2017). Among others, diffusive properties are of great interest in a wide spectrum of applications ranging from lithium-ion batteries via solid-oxide fuel cells to civil engineering Bear (2018); Newman and Thomas-Alyea (2004); Cooper et al. (2013); Das et al. (2018); Blunt (2017). Likewise, the 3D microstructure of the pore space greatly affects water, gas and carbon fluxes in soils Rabot et al. (2018). A typical example of a soil structure-mediated gas transport is the emission of greenhouse gases into the atmosphere Smith et al. (2003). Gas transport in soils is known to be primarily a diffusion-dominated process. More precisely, diffusion of gas in soils is mainly governed by the so-called soil gas diffusivity, for which empirical relationships, i.e., regression formulas (also called analytical prediction formulas), have been developed using easily available soil bulk properties like the air-filled porosity Millington (1959); Penman (1940); Marshall (1959) and then further improved by the integration of other soil properties such as soil bulk density Millington and Quirk (1961); Moldrup et al. (2000); Deepagoda et al. (2011) or specific surface area Moldrup et al. (2001). Moreover, it turned out that regression formulas of this type can be improved by incorporating further geometrical descriptors of the pore space Prifling et al. (2021). This motivates a more detailed characterization of the pore space.

Due to recent efforts and technological advancements in the field of soil imaging, notably in X-ray computed tomography (CT), it is possible to accurately visualize and characterize the soil pore space in three dimensions at high resolution. As a result, X-ray CT also offers the unprecedented opportunity to test the validity and relevance of incorporating physical descriptors of the pore space in empirical relationships aiming at predicting soil gas diffusivity. Previous studies relying on X-ray CT to derive gas transport parameters have shown promising results so far. In a study of no-tillage system in a subtropical climate, significant correlations between the porosity derived with X-ray CT and the logarithm of relative gas diffusivity and air permeability were found da Silva et al. (2021). In another study Katuwal et al. (2015), the porosity obtained by means of X-ray CT was also shown to be well correlated with air permeability. Furthermore, in Katuwal et al. (2015), it has been shown that there is a strong correlation of the lumped tortuosity-connectivity parameter obtained from standard gas measurements (as described in Buckingham (1904)) and the one obtained from X-ray CT. Thus, based on the literature, it seems that X-ray CT can be considered as a powerful tool to derive gas transport parameters. It is however still unclear which combination of geometrical descriptors can be used to best predict effective macroscopic properties as effective diffusivity.

Likewise, the progress in computational power and sophisticated numerical methods enables the computation of orientation-dependent diffusive properties from sound mathematical theory Ray et al. (2018). This is possible even on an extensive dataset of real and therefore quite complex geometries, which forms the basis for the quantitative investigation of microstructure–property relationships within this research. In particular, we consider the relationship between descriptors of 3D microstructure and diffusive properties.

In the present paper, we consider different kinds of regression formulas to predict diffusive properties from geometric descriptors of 3D pore space. Applying this approach to data from two different soil textures, we quantitatively investigate the performance and robustness of the considered regression formulas. We thoroughly investigate geometrical descriptors of the 3D microstructure of the samples’ pore space and analyze them in detail in the context of sand and loam. For this purpose, the loam and sand samples under consideration as well as the corresponding tomographic imaging techniques are explained in Sects. 2.1 and 2.2, respectively. Moreover, we provide details regarding the numerical computation of diffusive properties (Sect. 2.3) as well as various descriptors of the 3D pore space morphology (Sect. 2.4), which are used to characterize its 3D microstructure geometrically and with regard to its diffusive properties. Finally, in Sect. 2.5, our methods of model fitting and validation are stated, which are used for assessing the predictive power of microstructure–property relationships. Then, in Sect. 3.1, a statistical analysis of 4608 subsamples is carried out to quantitatively investigate differences between loam and sand by means of univariate as well as bivariate probability distributions. Moreover, microstructure–property relationships for soil gas diffusion are established in Sect. 3.2, where the goal is to derive accurate (and possibly general) prediction formulas, i.e., they do not require individual sets of fitting parameters for the two soil textures. In contrast to other methods of data science as, for example, neural networks, the empirically derived prediction formulas establish an interpretable relationship between the 3D morphology of the pore space and its diffusive properties. In particular, in Sect. 4, we compare the predictive power of conventional models discussed in the literature with that of more sophisticated models which additionally include further geometrical descriptors of the 3D pore space morphology. It turns out that the more accurate microstructure description achieved in this way allows us to surpass traditional prediction formulas with regard to the coefficient of determination as well as the mean absolute percentage error. Moreover, a quantitative comparison between two different notions of tortuosity, namely geodesic and geometric tortuosity, is carried out to investigate their suitability for establishing microstructure–property relationships. Section 5 concludes the paper and outlines possibilities for future research.

2 Materials and Methods

2.1 Sample Acquisition

The samples analyzed in this study were acquired within the framework of the plant growth experiments described in Lippold et al. (2021). In particular, cylindrical columns of \({23}\hbox { cm}\) in height and \({7} \hbox { cm}\) in diameter were packed with soil, where two different substrates (loam and sand) were used. The loam substrate was obtained from the upper \({50}\hbox { cm}\) of a haplic Phaeozem soil profile, dried to 10 % gravimetric water content and then sieved to \(< {1}\hbox { mm}\). The sand substrate constitutes a mix of 83.3 % quartz sand (WF 33, Quarzwerke Weferlingen, Germany) and 16.7 % of the sieved loam. Details on chemical and physical properties of these substrates are provided in Vetterlein et al. (2021). The loam pots were packed homogeneously to a bulk density of \({1.26}\,\hbox {g/cm}^{3}\), whereas the sand pots were packed to a bulk density of \({1.47}\,\hbox {g/cm}^{3}\). At the end of the growth period, a total of six samples (aluminum rings of \({1.6}\hbox { cm}\) in height and diameter) were taken from 5, 10 and \({15}\hbox { cm}\) depth (3 per substrate, one per depth) and were stored at \({4}^{\circ }\hbox {C}\) in sealed plastic bags prior to X-ray CT scanning.

2.2 X-ray Computed Tomography Scanning and Binarization

X-ray CT scanning was performed with an industrial \(\mu\)CT scanner (X-TEK XTH 225, Nikon Metrology) including an Elmer-Perkin 1620 detector panel (\(1750 \times 2000\) px). The scanner was operated at \({115}\hbox { kV}\) and a current of \({85}\,\mu A\). A total of 2748 projections with an exposure time of \({708}\,\hbox {ms}\) each were acquired during a full rotation of the samples. The obtained images were reconstructed into a three-dimensional tomogram by means of the filtered back projection algorithm with the CT Pro 3D software (Nikon Metrology) Banhart (2008). During reconstruction, the grayscale range was normalized with a percentile stretching method, which sets the darkest and brightest 0.2 % voxels to 0 and 255, respectively, and performs a linear stretching in between. The voxel size equals \({10}\,\mu m\).

Denoising and contrast enhancement of the grayscale images were performed using a 2D non-local means filter (\(\sigma = 15\)) and an unsharp mask filter (radius = 1 and weight = 0.6) Buades et al. (2005); Burger and Burge (2016), respectively. Subsequent binarization of the images was performed to distinguish between pore space and the soil matrix. The automatic threshold detection method presented in Otsu (1979) was used for an adequate and objective choice of the global threshold.

For each texture and depth, the samples were truncated into stacks of 1536 slices with each \(1024 \times 1024\) voxels. This coincides with cuboid of \(15.36\,\textrm{mm}\) height, \(10.24\,\textrm{mm}\) width and \(10.24\,\textrm{mm}\) length. For the purpose of the study, we further divided the cuboid into non-overlapping cubic cutouts with \(128 \times 128 \times 128\) voxels (\(1.28 \,\textrm{mm}\times 1.28 \,\textrm{mm}\times 1.28 \,\textrm{mm}\)) each. Thus, we obtained 768 cubic cutouts for each depth and soil texture, which led to a total number of 4608 subsamples. Selected subsamples are shown in Fig. 1.

Finally, isolated pores were removed, which led to only marginal changes with regard to the porosity, i.e., the 3D binary images contained only few unconnected pores. The overall difference in porosity for loam was smaller than 2 % of the initial porosity, for sand even less than 0.5 %.

Fig. 1
figure 1

Three-dimensional renderings of selected loam (top row) and sand (bottom row) subsamples, where the solid phase is depicted in gray. The three columns correspond to minimum (left, \(\dagger\)), mean (center, \(\circ\)) and maximum (right, \(\star\)) porosity

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2.3 Computation of Diffusive Properties

Diffusive transport of a chemical species with concentration field \(c:\mathbb {R}^{3} \rightarrow [0,\infty )\) through porous media is, in its simplest form, described by the Laplace equation

$$\begin{aligned} -\nabla \cdot (\mathbb {D}\nabla c) = 0, \end{aligned}$$
(1)

where \(\mathbb {D}\in \mathbb {R}^{3\times 3}\) denotes the effective diffusion tensor, which is the essential input to the transport model. It contains all the information that is specific to the porous medium under consideration. However, it is quite difficult to characterize \(\mathbb {D}\) for (natural) porous media. If the underlying geometry is available, e.g., from binary CT data, one possibility to determine the tensor \(\mathbb {D}\) is to apply the results of homogenization theory Hornung (1997); Ray et al. (2018), which was originally derived for periodic settings.

In order to carry out the numerical simulations for the effective diffusivities by homogenization within a reasonable time frame, the subsamples have been coarsened by combining eight voxels into one (super-) voxel. If four of the original voxels were solid and four void, the resulting (super-) voxel was at random turned into a void or solid voxel, otherwise the majority defines the type of the resulting voxel. Hence, after performing this procedure, the dimensions of the subsamples are \(64 \times 64 \times 64\) voxels (\(1.28 \,\textrm{mm}\times 1.28 \,\textrm{mm}\times 1.28 \,\textrm{mm}\)), where a voxel corresponds to \({20^{3}}\,\mu \hbox {m}^{3}= 8000\,\mu \hbox {m}^{3}\).

For a connected, sufficiently smooth pore space \(Y_p\subset Y = [0, 1.28 \,\textrm{mm}]^{3}\) and complementing solid phase \(Y_s=Y\backslash Y_p\), the entries of the diffusion tensor \(\mathbb {D}\) are given as

$$\begin{aligned} \mathbb {D}_{ij}&= \tfrac{1}{V(Y)}\int _{Y_p}D_m\bigl (\partial _{y_i}\zeta _j(y) + \delta _{ij}\bigr ) \,dy ,\qquad \text{ for } i,j \in \{1,2,3\}\text{, } \end{aligned}$$
(2)

with molecular diffusivity \(D_m > 0\) and the following supplementary cell problems in \(\zeta _j\), \(j \in \{1,2,3\}\):

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{rcll} - \nabla _y\cdot (\nabla _y \zeta _j + e_j) &{}=&{} 0 &{} \quad \text { in } Y_p, \\ (\nabla _y \zeta _j + e_j)\cdot \nu &{}=&{} 0 &{} \quad \text { on }\partial Y_s, \end{array}\\ \quad \zeta _j \text { is periodic in } y\text { and has vanishing mean} \end{array}\right. } \end{aligned}$$
(3)

Here, V(Y) denotes the volume of V, \(\nu\) is the unit outer normal of the solid phase, \(e_j\) the unit vector in direction j, and \(\delta _{ij}\) the Kronecker delta. Note that in the present setting, we assume zero diffusivity within the solid phase.

The partial differential equations given in (3) are solved within the connected pore space and discretized using the local discontinuous Galerkin (LDG) method Aizinger et al. (2018); Rupp and Knabner (2017), which is implemented within the parallel finite element software, M++ Wieners (2005). The resulting linear systems are solved using a combination of the BiCGstab solver and the ILUT preconditioner Saad (2003) through Trilinos Heroux and Willenbring (2003).

As a result of the computations described above, the \(3\times 3\) diffusion tensor \(\mathbb {D}\) has been determined for all 4608 subsamples obtained from the X-ray CT scans of sand and loam, as described in Sect. 2.2. Further details regarding the analysis of these diffusion tensors are presented in Prechtel et al. (2022). In the following, the soil gas diffusion for the sand and loam subsamples will be described by the ratio of effective and intrinsic diffusivity which is sometimes called microstructure factor or M-factor Gaiselmann et al. (2014). In particular, we average over the three Cartesian axes directions such that the averaged M-factor is obtained from the diagonal entries of the diffusion tensor as \(M = \frac{1}{3 D_{m}} (\mathbb {D}_{11} + \mathbb {D}_{22} + \mathbb {D}_{33})\). This averaging is motivated by the fact that neither the loam nor the sand samples show pronounced anisotropy effects Prechtel et al. (2022).

2.4 Geometrical Descriptors of Pore Space

In this section, various geometrical descriptors of the 3D pore space morphology and methods for their estimation from voxelized image data are presented. In Fig. 2, two-dimensional versions of these descriptors are illustrated.

Fig. 2
figure 2

Schematic 2D illustration of microstructure descriptors: Exemplarily chosen 2D slice, where the pore space is depicted in black (a); surface between pore and solid phases, appearing as one-dimensional object in the 2D slice (b); chords in horizontal direction for two exemplarily chosen vertical height levels (c); subsets of pore space used in the definitions of \(\textsf{CPSD}\) (d) and \(\textsf{SMIP}\) (e), respectively; shortest distances from pore space to soil (f); two exemplarily chosen shortest paths (from left to right) (g); skeleton of pore space and shortest paths along the skeleton (from left to right) depicted in green and red color, respectively (h)

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2.4.1 Porosity \(\varepsilon\)

One of the most fundamental descriptors of the 3D pore space morphology is the porosity \(\varepsilon \in [0,1]\). The estimation of this geometrical descriptor from binarized 3D image data is straight-forward and carried out by computing the number of voxels assigned to the pore space (depicted in black in Fig. 2a), which is divided by the total number of voxels of the entire sampling window.

2.4.2 Specific Surface Area S

In addition to porosity, we consider the specific surface area of the pore space, denoted by S. It is defined as the surface area between the pore space and the solid phase divided by the volume of the sampling window. This quantity is then estimated from voxelized 3D image data using an approach presented in Schladitz et al. (2007), which is based on a convolution of the image with a \(2 \times 2 \times 2\) kernel avoiding the reconstruction of the actual surface. In Fig. 2b, the interface between pore and solid phases is depicted in red.

2.4.3 Mean Chord Length \(\mu (C)\)

A further descriptor of the 3D pore space morphology is its chord length distribution Matheron (1975); Serra (1982), where a chord is a line segment that is completely contained in a predefined phase and cannot be extended further without intersecting the complementary phase. Obviously, in general, the probability distribution of chord lengths depends on the orientation of the line segments. We compute the chord length distribution of the pore space for the three Cartesian axes directions. In particular, for each of these three directions, we compute the mean value of the corresponding chord length distribution. In the following, the average of these three mean values, denoted by \(\mu (C)\), is used for characterizing the 3D pore space morphology. In Fig. 2c, five different chords are shown in horizontal direction and depicted in five different colors, for two exemplarily chosen vertical height levels.

2.4.4 Constrictivity \(\beta\)

In order to explain this descriptor, we first recall the concept of the continuous pore size distribution (\(\textsf{CPSD}\)) as well as of simulated mercury intrusion porosimetry (\(\textsf{SMIP}\)). The continuous pore size distribution is a function \(\textsf{CPSD}:[0,\infty ) \rightarrow [0,1]\), where the value \(\textsf{CPSD}(r)\) is given by the volume fraction of the pore space which can be covered by (possibly overlapping) spheres with radius \(r\ge 0\) (such that the spheres are completely contained in the pore space) Serra (1982); Soille (2013). Furthermore, by \(r_{\textsf {max}}\) the maximum radius \(r>0\) is denoted such that \(\textsf{CPSD}(r) \ge \varepsilon /2\) where \(\varepsilon\) is the porosity. Note that the green shaded area in Fig. 2d corresponds to that part of the pore space which can be covered by potentially overlapping spheres having the same radius as the red circle in Fig. 2d.

The concept of \(\textsf{SMIP}\) is similar to that of \(\textsf{CPSD}\), with the only difference that \(\textsf{SMIP}(r)\) is now given by the volume fraction of the pore space which can be covered by (potentially overlapping) spheres with radius r forming an intrusion from a predefined direction, see Fig. 2e for an intrusion of spheres from left to right. Analogously to \(r_{\textsf {max}}\), by \(r_{\textsf {min}}\) the maximum radius \(r>0\) is denoted such that \(\textsf{SMIP}(r) \ge \varepsilon /2\). In general, \(r_{\textsf {min}}\) depends on the direction of the intrusion. However, as already mentioned above, the lack of pronounced anisotropy effects in the loam and sand samples motivates the following averaging. First, \(r_{\textsf {min}}\) is computed for each of the three axes directions separately and, subsequently, the average of the three obtained values is used.

The constrictivity of the pore space is then defined as \(\beta = (\frac{r_{\textsf{min}}}{r_{\textsf{max}}})^{2}\). It is a measure for the strength of bottleneck effects and has been originally introduced in Münch and Holzer (2008). Since \(r_{\textsf{min}} \le r_{\textsf{max}}\) by definition, it holds that \(\beta \in [0,1]\), where \(\beta \approx 1\) corresponds to a situation such that (almost) no constrictions within the pore space exist. In Prifling et al. (2021); Holzer et al. (2013); Neumann et al. (2020), it has been shown that this quantity has a significant impact on effective macroscopic properties of porous media such as effective diffusivity or permeability.

2.4.5 Mean Spherical Contact Distance \(\mu (H)\)

Here we consider the distribution of the shortest (contact) distances from randomly chosen points within the pore space to the solid phase. It is estimated from voxelized image data by an algorithm presented in Mayer (2004), which is based on the computation of the Euclidean distance transform Soille (2013); Maurer et al. (2003). The mean value of this distribution will be denoted by \(\mu (H)\) in the following. In Fig. 2f, the shortest distances from points in the pore space to the solid phase are visualized by (brighter and darker) colors, where the increase of brightness corresponds to an increase of contact distance. In particular, the radii of the red spheres in Fig. 2f correspond to the shortest distances from their midpoints to the soil.

2.4.6 Mean value \(\mu (\tau _{\textsf{g}})\) and standard deviation \(\sigma (\tau _{\textsf{g}})\) of geodesic tortuosity

A further transport-relevant quantity is the so-called geodesic tortuosity of the pore space. It describes the lengths and windedness of transport paths, which are completely contained in the pore space. From 3D image data, the distribution of (local) geodesic tortuosity is determined by computing the lengths of shortest paths from randomly selected pore voxels on a predefined starting plane to a parallel target plane, divided by the distance between those two planes, where shortest paths are computed using Dijkstra’s algorithm Jungnickel (2008). In Fig. 2g, two exemplarily chosen shortest paths (from left to right) are shown in red, where the color of each pore space pixel corresponds to the extent of shortest distance to the target plane and the increase of brightness corresponds to an increase of path length. Usually, the starting and target planes are chosen orthogonal to the relevant transport direction. For the image data considered in the present paper, we compute the distribution of geodesic tortuosity with respect to each of the three Cartesian axes directions. The mean geodesic tortuosity \(\mu (\tau _{\textsf{g}})\) is then determined by averaging over all shortest path lengths divided by the distance between the starting and target planes. Analogously, the standard deviation \(\mu (\sigma _{\textsf{g}})\) is the empirical standard deviation of these normalized path lengths.

Note that in many applications, only \(\mu (\tau _{\textsf{g}})\) is considered in order to characterize the microstructure of porous media, despite the fact that \(\sigma (\tau _{\textsf{g}})\) contains further useful information about the 3D morphology of the pore space, see Sect. 3.2 for a detailed discussion. Furthermore, besides geodesic tortuosity, there exist several other concepts of tortuosity in the literature, see e.g., Holzer et al. (2022); Ghanbarian et al. (2013) for a comprehensive overview. An example of such alternative tortuosity concepts is the so-called geometric tortuosity, which is described in the next paragraph.

2.4.7 Mean value \(\mu (\tau _{\textsf{s}})\) and standard deviation \(\sigma (\tau _{\textsf{s}})\) of geometric tortuosity

In contrast to geodesic tortuosity described above, the concept of geometric tortuosity (sometimes called skeletonized tortuosity) is based on the computation of a skeleton of the pore space, see Fig. 2h for an illustration where the skeleton is depicted in green color. In the present study, we compute the skeleton of the pore space by means of the Lee thinning Lee et al. (1994); Saha et al. (2016). Similar to the concept of geodesic tortuosity, shortest paths along the transport direction are computed by Dijkstra’s algorithm Jungnickel (2008), where in case of geometric tortuosity only voxels that belong to the skeleton of the pore space are allowed to be visited. Two such shortest paths (from left to right) along the skeleton of the pore space are shown in Fig. 2h in red. Finally, we consider the distribution of the lengths of these shortest paths divided by the Euclidean distances between the starting and the target plane. Mean and standard deviation of this distribution are denoted by \(\mu (\tau _{\textsf{s}})\) and \(\sigma (\tau _{\textsf{s}})\), respectively. As in case of all direction-dependent microstructure characteristics considered in the present paper, these quantities are computed for each of the three Cartesian axis directions separately and then averaged.

Note that there exist various approaches for computing the skeleton of a 3D binary image, which in turn affects the computation of geometric tortuosity Soille (2013); Saha et al. (2016).

2.5 Model Fitting and Validation

We now present statistical methods which will be used in Sect. 3 for a comprehensive structural analysis of the loam and sand samples as well as for establishing microstructure–property relationships, which quantitatively characterize the connection between geometrical descriptors of 3D pore space morphology and soil gas diffusion. In particular, in Sect. 3.2, regression formulas for the prediction of the M-factor are considered, which are based on different sets of descriptors of the 3D pore space morphology. In general, this approach leads to results that are easier to interpret compared to other methods from data science such as neural networks.

The regression formulas considered in Sect. 3.2 contain a small number of regression parameters, which are fitted by means of training data, i.e., a fraction of the 4608 subsamples described in Sect. 2.2 is selected for determining “optimal” values for the regression parameters. In order to compute these parameters, the Levenberg–Marquardt algorithm Moré (1978) implemented in the python package scipy Virtanen et al. (2020) is used. This algorithm solves the unconstrained nonlinear least-squares problem of finding an optimal set of parameter values for a given regression formula and training data.

To evaluate the performance of regression formulas, we select \(k = 1024\) subsamples as test data, which have not been used for model fitting. Then, the goodness of fit is quantified by computing the coefficient of determination, denoted by \(R^{2}\), as well as the mean absolute percentage error, denoted by \(\textrm{MAPE}\). These two quantities are defined as

$$\begin{aligned} R^{2} = 1 - \frac{\sum \nolimits _{j=1}^{k}(M_{j} - \widehat{M}_{j})^{2} }{\sum \nolimits _{j=1}^{k} (M_{j} - \bar{M})^{2}} \qquad \text { and } \qquad \textrm{MAPE} = \frac{100}{k} \sum \limits _{j=1}^{k} \left| \frac{\widehat{M}_{j} - M_{j}}{M_{j}}\right| , \end{aligned}$$
(4)

where \(M_{1}, \ldots , M_{k}\) denote the M-factors computed by means of numerical simulations as described in Sect. 2.3 (ground truth), \(\bar{M}\) is the mean of \(M_{1}, \ldots , M_{k}\), and \(\widehat{M}_{1}, \ldots , \widehat{M}_{k}\) denote the M-factors obtained by an empirically derived regression formula.

For fitting the model parameters, we use all data of subsamples from \({5}\hbox { cm}\) and \({10}\hbox { cm}\) depth from both loam and sand (3072 subsamples). Model validation by means of \(R^{2}\) and \(\textrm{MAPE}\) is performed on subsamples from \({15}\hbox { cm}\) depth from both loam and sand (1536 subsamples). The parameters of all regression formulas considered in this study are fitted on the same (training) dataset, while validation of the fitted regression formulas is performed on subsamples from a depth which is not used for fitting. By performing model fitting and validation on different types of soil (loam and sand), we can show whether the best fitting regression formula is applicable for a general soil microstructure or only for structures very similar to those on which the model was fitted.

3 Results

3.1 Statistical Analysis of 3D Pore Space Morphology

The geometrical descriptors stated in Sect. 2.4 are used to quantitatively characterize the 3D microstructure of the soil subsamples considered in this paper. Figure 3 shows the histograms which have been obtained for various geometrical descriptors of subsamples from loam and sand, respectively. It turned out that some descriptors, such as porosity \(\varepsilon\) and constrictivity \(\beta\), can vary a lot between subsamples from the same soil texture at the chosen scale, whereas on average they differ only little between loam and sand, see Fig. 3a and f. Thus, such descriptors might be less useful to distinguish between the textures of both soil textures, but rather useful for describing variations of individual samples from a given soil texture. On the other hand, descriptors like the specific surface area S and mean chord length \(\mu (C)\) behave rather different for the two soil textures, see Fig. 3b and c. Therefore, they seem to capture the morphological differences between the textures much better than \(\varepsilon\) and \(\beta\).

Fig. 3
figure 3

Histograms of geometrical descriptors—namely porosity (a), specific surface area (b), mean chord length (c), bottleneck radius \(r_{\textsf{min}}\) (d), characteristic pore radius \(r_{\textsf{max}}\) (e), constrictivity (f), mean and standard deviation of geodesic tortuosity (g, h), mean distance to solid phase (i), mean and standard deviation of geometric tortuosity (j)—and the M-factor (l) computed from tomographic image data for the \(2 \cdot 3 \cdot 768 = 4608\) subsamples of loam (blue) and sand (orange), respectively. The symbols \(\dagger\), \(\circ\) and \(\star\) indicate the values obtained for the subsamples shown in Fig. 1

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A more detailed characterization of soil textures can be achieved by considering bivariate probability densities for pairs of descriptors, see Fig. 4. While the univariate histograms of the mean chord length \(\mu (C)\) obtained for subsamples of loam and sand show a certain overlapping, see Fig. 3c, the joint (bivariate) probability densities of \(\mu (C)\) and the mean spherical contact distance \(\mu (H)\) allow to reliably distinguish between loam and sand, see Fig. 4b.

When looking at Fig. 3g and j, see also Fig. 4e, we observe that the results obtained for the two notions of (geodesic and geometric) tortuosity are rather different. In particular, the mean geodesic tortuosity \(\mu (\tau _{\textsf{g}})\) is smaller for sand in comparison with the corresponding value obtained for loam (i.e., paths in the pore space of sand are shorter than those of loam), whereas the mean geometric tortuosity \(\mu (\tau _{\textsf{s}})\) is greater for sand. This is likely to be caused by the fact that sand comprises of larger pores than loam such that the paths along the coarse skeleton become comparatively long, whereas the finer pore structure observed in loam leads to a skeleton with a large number of nodes, which is closer to the actual axes of the pore space. The two notions of (geodesic and geometric) tortuosity will be further discussed in Sect. 4 with regard to their usefulness as geometrical descriptors for microstructure–property relationships.

Fig. 4
figure 4

Bivariate probability densities for pairs of geometrical descriptors, visualized as contour plots based on kernel density estimation, which have been obtained for the \(2 \cdot 3 \cdot 768 = 4608\) cubic subsamples of either loam (blue) or sand (orange). Furthermore, the values of Pearson’s correlation coefficient \(\rho\) are given for each descriptor pair, separately computed for loam and sand

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A full analysis of the correlation coefficients between all pairs of geometrical descriptors considered in this paper is provided in the Appendix, see Fig. 7. For selected pairs of geometrical descriptors, Fig. 4 shows bivariate probability densities obtained by kernel density estimation as well as the corresponding Pearson correlation coefficients. With the exception of \((\varepsilon , \mu (\tau _{\textsf{g}}))\), the joint densities of all other pairs of geometrical descriptor shown in Fig. 4 strongly depend on the considered soil texture, which once again highlights the structural differences between loam and sand. This suggests that any regression formula expressing the M-factor in terms of one of these pairs of geometrical descriptors would need to be adapted specifically to the distinct textures of sand and loam. Vice versa, regression formulas based on \(\varepsilon\) and \(\mu (\tau _{\textsf{g}})\) may be valid independent of the considered soil texture.

Fig. 5
figure 5

Scatter plots visualizing the dependence of the M-factor on various pairs of geometrical descriptors. The (brighter or darker) colors of the data points indicate whether the value of the corresponding M-factor is high or low. In this figure we do not distinguish between the two soil textures. For clarity of presentation, only randomly selected \(10\ \%\) of the 4608 cubic subsamples have been used to generate the 461 data points in each subfigure

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3.2 Regression Formulas for the Quantification of Microstructure–Property Relationships

The scatter plots shown in Fig. 5 illustrate the dependence of the M-factor on various pairs of geometrical descriptors. In this section, we consider several empirically derived regression formulas which will be used for the quantification of microstructure–property relationships and, in particular, for predicting the M-factor from the knowledge of an appropriately chosen vector of geometrical descriptors of the 3D pore space morphology.

A simple regression formula, which only involves the porosity \(\varepsilon\), is given by

$$\begin{aligned} \widehat{M}_{1} = \varepsilon ^{c_{1}},\qquad \text { for some }c_{1}\ge 1. \end{aligned}$$
(5)

This power-law type of a regression formula is frequently used in the literature, where the most well-known examples are probably the Buckingham formula (\(c_{1}=2\)) Buckingham (1904), the Millington–Quirk formula (\(c_{1} = \frac{4}{3}\)) Millington and Quirk (1961), the formula derived by Marshall (\(c_{1} = \frac{3}{2}\)) Marshall (1959), and the work of Lai et al. \((c_{1} = \frac{5}{3})\) Lai et al. (1976). The data considered in the present paper leads to \(c_{1} = 1.71\), which is right in between the values used in the Buckingham and Millington–Quirk formulas, but closest to that of Lai et al. Lai et al. (1976).

Another simple regression formula is given by

$$\begin{aligned} \widehat{M}_{2} = c_{1} \varepsilon ^{c_{2}}, \qquad \text {for some } c_{1}, c_{2} > 0, \end{aligned}$$
(6)

which is used, for example, in Currie (1960) (\(c_{1} = 1.9, c_{2} = 1.4\)), Currie (1961) (\(c_{1} = 1.75, c_{2} = 2.1\)) and Grable and Siemer (1968) (\(c_{1} = 5.25, c_{2} = 3.36\)). Moreover, the formula

$$\begin{aligned} \widehat{M_{3}} = c_{1} e^{c_{2} \varepsilon }, \end{aligned}$$
(7)

which has been introduced in Richter and Großgebauer (1978) with \(c_{1} = 0.0085\) and \(c_{2} = 6.8\), is considered in the following. Besides, there are many other prediction formulas that only involve porosity, see e.g., Holzer et al. (2022); Shen and Chen (2007) for further details. However, a further consideration of this type of formulas is beyond the scope of the present paper, where a particular focus is now put on empirically derived regression formulas that—besides porosity—involve further (more sophisticated) descriptors of 3D pore space morphology such as the mean geodesic tortuosity \(\mu (\tau _{\textsf{g}})\) or the constrictivity \(\beta\). In particular, we consider the formula

$$\begin{aligned} \widehat{M}_{4} = \varepsilon ^{c_{1}} \beta ^{c_{2}} \mu (\tau _{\textsf{g}})^{c_{3}}, \qquad \text { for some } c_{1}>, c_{2}\ge 0, c_{3}\le 0, \end{aligned}$$
(8)

which has been originally introduced in Stenzel et al. (2016) (with \(c_{1} = 1.15, c_{2} = 0.37, c_{3} = -4.39\)), and refitted to 63,000 virtually generated microstructures in Prifling et al. (2021) (with \(c_{1} = 1, c_{2} = 0, c_{3} = -8.45\)).

Since Equation (8) does not hold in the dilute limit (i.e. for \(\varepsilon \rightarrow 1\)), the modified formula

$$\begin{aligned} \widehat{M}_{5} = \varepsilon ^{c_{1} + c_{2}\beta } \mu (\tau _{\textsf{g}})^{c_{3}},\qquad \text { where } c_{1}+c_{2}\ge 0, c_{3}\le 0, \end{aligned}$$
(9)

has been used in Neumann et al. (2020) (with \(c_{1} = 1.67, c_{2} = -0.48, c_{3} = -5.18\)) and in Prifling et al. (2021) (with \(c_{1} = 1.25, c_{2} = -1.25, c_{3} = -7.82\)), where the constrictivity \(\beta\) appears now in the exponent of the porosity \(\varepsilon\).

Besides formulas based on porosity, constrictivity and mean geodesic tortuosity, further geometrical descriptors of 3D pore space morphology can be taken into account. For example, the formula

$$\begin{aligned} \widehat{M}_{6,j} = c_{1} \mu (\tau _j)^{c_{2}} \sigma (\tau _j)^{c_{3}} \varepsilon ^{c_{4}}, \qquad \text{ where } j \in \{\textsf{g},\textsf{s}\}, \end{aligned}$$
(10)

which has been introduced in Barman et al. (2019) for \(j=\textsf{g}\) (with \(c_{1} = 0.06, c_{2} = -2, c_{3} = -0.6, c_{4} = 1\)) and also considered in Prifling et al. (2021) (with \(c_{1} = 1.18, c_{2} = -9.17, c_{3} = 0.03, c_{4} = 1.02\)), uses the standard deviation \(\sigma (\tau _j)\) of (geodesic or geometric) tortuosity as an additional geometrical descriptor.

Moreover, we consider the formula

$$\begin{aligned} \widehat{M}_{7} = \varepsilon ^{c_{1}} S^{c_{2}} r_{\textsf {max}}^{c_{3}}, \end{aligned}$$
(11)

which has not yet been considered in the literature. In general, the specific surface area S has only rarely been used in the literature for the prediction of diffusive properties. However, a particular case of such a regression formula is presented in Moldrup et al. (2001), which uses the formula \(\widehat{M} = 1.1 (\varepsilon - 0.039 S^{0.52})\). However, this kind of regression formula can lead to negative values for the M-factor. In contrast, Equation (11) ensures that \(\widehat{M}_{7} \ge 0\).

Finally, we predict the M-factor by means of

$$\begin{aligned} \widehat{M}_{8} = \varepsilon ^{c_{1}}\left( \frac{\mu (C)}{\mu (H)}\right) ^{c_{2}}, \end{aligned}$$
(12)

which—besides porosity—contains the ratio of mean chord length \(\mu (C)\) and the mean spherical contact distance \(\mu (H)\) of a randomly chosen point within the pore space to the solid phase. The latter quantity can be regarded as some kind of shape information since this quotient in Equation (12) is larger for more elongated pores.

Fig. 6
figure 6

Scatter plots visualizing the values obtained for the M-factor by numerical simulations versus those of the predicted M-factors \(\widehat{M}_i, \, i=1,\ldots ,8\) obtained for the test data of either loam (blue) or sand (orange). The parameters appearing in the prediction formulas given in Eqs. (5)−(12) have been fitted to the entire training data for loam and sand, see Table 1. In particular, Eq. (10), i.e. \(\widehat{M}_{6,j}\), occurs twice since the concepts of both geodesic and geometric tortuosities are considered

Full size image

4 Discussion

First, we consider the case, where data gained for both loam as well as sand samples (taken in the depths of 5 and \({10}\hbox { cm}\)) are used for training. Scatter plots visualizing the values obtained for the M-factor by numerical simulations versus those of the predicted M-factors \(\widehat{M}_i,\, i=1,\ldots ,8\) obtained for the test data of either loam or sand is shown in Fig. 6, where the parameters appearing in the prediction formulas given by Eqs. (5)–(12) have been fitted to the entire training data for loam and sand, see Table 1. The resulting values obtained for the coefficient of determination (\(R^{2}\)) and the mean absolute percentage error (\(\textrm{MAPE}\)) can also be found in Fig. 6. Furthermore, we compare the performance of the regression formulas given by Eqs. (5) −(12), where the model parameters are fitted to data considered in the present paper, with that of regression formulas which have been proposed previously in the literature, see Table 2. Bear in mind that when using the latter formulas we do not refit the model parameters, but use the same parameter values as they have been derived—based on different kinds of data and using different methodological approaches—in the original papers.

As expected, due to its simplicity, Equation (5) performs worst among all prediction formulas stated in Sect. 3.2. In this case, important structural details of the 3D pore space morphology are disregarded since only the porosity \(\varepsilon\) is used to characterize the morphology of the pore space. In case of Equation (5), one obtains \(c_1=1.71\), when fitting it to the entire set of training data considered in the present paper. This value is roughly in the middle between 1.5 Marshall (1959) and 2 Buckingham (1904), but quite close to the values of \(c_1\) derived in Millington and Quirk (1961) and Lai et al. (1976). When introducing an additional model parameter, as in Eqs. (6) and (7), the \(\textrm{MAPE}\) can be reduced to \(7.2 \%\) and \(7.5 \%\), respectively. However, fitting Equation (6) to our soil data leads to quite different values for \(c_1\) and \(c_2\) compared to those obtained in Grable and Siemer (1968). Due to the large difference in the values of the exponent \(c_2\) of the porosity \(\varepsilon\) obtained in the present study and in Currie (1961); Grable and Siemer (1968), respectively, it is not surprising that the predictive power of Equation (6) with \(c_{1} = 1.92\) and \(c_{2} = 2.29\), as derived from our soil data, is significantly better. On the other hand, note that the value which we obtained for \(c_{1}\) in Equation (6) is similar to the one obtained in Currie (1960). In case of results derived in Currie (1961), applying an already existing regression formula directly to the present soil data performs reasonably well (leading to a \(\textrm{MAPE}\) of \(13.12 \%\)), even though refitting the regression parameters to the soil data considered in the present paper leads—as expected—to better results. With regard to Equation (7), it is interesting to point out that choosing \(c_{1} = 0.02\) and \(c_{2} = 6.88\) leads to a significantly better predictive power compared to that obtained in Richter and Großgebauer (1978), even though their parameter values of \(c_{1} = 0.01\) and \(c_{2} = 6.8\) are quite close to the ones derived in the present paper.

A significant improvement of predictive power can be achieved by including further geometrical descriptors of the 3D pore space morphology into the regression formulas. However, fitting the model parameters in Equation (8) to our soil data leads to \(c_{2} = 0.04\), i.e., the constrictivity \(\beta\) only plays a negligible role in the prediction of the M-factor. This effect has already been observed in Prifling et al. (2021), where 63,000 microstructures have been used for model fitting. Furthermore, directly applying the regression formulas derived in Prifling et al. (2021) and Stenzel et al. (2016) to the present soil data leads to the best results among all (previously derived) formulas from the literature listed in Table 2. In general, the validation scores \(R^2\) and \(\textrm{MAPE}\) can be significantly improved by using Equation (8) compared to the prediction formulas given in Eqs. (5)−(7), which only use the porosity \(\varepsilon\). This emphasizes the impact of the mean geodesic tortuosity \(\mu (\tau _{\textsf{g}})\) with regard to the prediction of diffusive mass transport in porous media. Interestingly, the value which we obtained in Equation (9) for the coefficient \(c_2\) of constrictivity \(\beta\) is not equal to zero, see Table 1. This matches with the results derived in Prifling et al. (2021) and Neumann et al. (2020). Anyhow, it turned out that the predictive power of the regression formula given in Equation (9) is similar to that of the formula given in Equation (8), even though the value which we obtained for the exponent \(c_2\) of \(\beta\) in Equation (8) is close to zero, i.e., practically Equation (8) does not include the constrictivity \(\beta\) at all.

Note that the predictive power can be further increased when using Equation (10) for predicting the M-factor, where the regression formula given in Equation (10) is based on porosity, mean geodesic tortuosity as well as the standard deviation of geodesic tortuosity. In this case, for the validation scores \(R^2\) and \(\textrm{MAPE}\) we obtained that \(R^{2}=0.9\) and \(\textrm{MAPE}=5.4 \%\), which turns out to be the best result among all prediction formulas considered in the present paper. This indicates that not only the mean value \(\mu (\tau _{\textsf{g}})\) of shortest path lengths is of interest with regard to the prediction of diffusive mass transport, but also the corresponding standard deviation \(\sigma (\tau _{\textsf{g}})\). However, directly applying Equation (10) with the values of the regression parameters derived in Prifling et al. (2021) and Barman et al. (2019) leads to a \(\textrm{MAPE}\) of \(11.2\%\) and \(46.7\%\), respectively. This large difference can be explained by the corresponding values of the regression parameters, where in particular the values obtained for the exponents of \(\mu (\tau _{\textsf{g}})\) and \(\sigma (\tau _{\textsf{g}})\) differ significantly between the results of Prifling et al. (2021) (\(c_{2} = -9.17, c_{3} = 0.03\)), Barman et al. (2019) (\(c_{2} = -2, c_{3} = 0.6\)) and our results (\(c_{2} = -6.07, c_{3} = 0.07\)) obtained in the present paper.

We also remark that Equation (11), which uses porosity, specific surface area as well as the characteristic pore radius \(r_{\textsf{max}}\), leads to similar results as Eqs. (8) and (9). Interestingly, using \(r_{\textsf{max}}\) for predicting the M-factor turns out to be useful, even though Equation (8) fitted to our soil data does practically not make use of the constrictivity \(\beta\), which is closely linked to \(r_{\textsf{max}}\). Finally, the M-factor can be predicted by means of Equation (12). This leads to \(R^{2} = 0.87\) and \(\textrm{MAPE}=6.2\%\), which are similar values to those obtained for Eqs. (8), (9) and (11), see Table 1. Thus, the dimensionless quantity in Equation (11) given by the ratio of the mean chord length \(\mu (C)\) and the mean spherical contact distance \(\mu (H)\), which characterizes the shape of pores, is—in combination with porosity—well suited for predicting diffusive properties.

It is worth mentioning, that—in general—the question of the best geometrical descriptors with regard to the prediction of the M-factor is hard to answer, since certain geometrical descriptors that do not seem to be correlated with diffusive mass transport can be well suited for establishing microstructure–property relationships when they are combined with further microstructure characteristics. Moreover, for some features of 3D microstructures, several variants of geometrical descriptors are considered in the literature for one and the same structural feature. For example, as already mentioned in Sect. 2.4, there exist different notions of tortuosity, where it turned out that geodesic tortuosity is superior to geometric tortuosity with regard to the prediction of diffusive mass transport in soil, see the validation scores stated in Table 1 for \(\widehat{M}_{6,\textsf{g}}\) and \(\widehat{M}_{6,\textsf{s}}\). This is likely to be caused by the fact that the skeleton of the pore space considered in the definition of \(\widehat{M}_{6,\textsf{s}}\) takes the complex morphology of the pore space only partially into account. On the other hand, not surprisingly, all regression formulas for predicting the M-factor, whose parameters have been fitted to the soil data considered in the present paper, perform better with respect to both validation scores \(R^2\) and \(\textrm{MAPE}\) than the corresponding regression formulas previously derived in the literature (and fitted to other types of image data), see Tables 1 and 2. This highlights the importance of using appropriately chosen data sets, which represent the nature of the 3D microstructure under consideration, for fitting the parameters of the regression formulas stated in Sect. 3.2.

Recall that so far we considered the case, where data gained from both loam as well as sand samples (taken in the depths of 5 and \({10}\hbox { cm}\)) were used for training. Results regarding microstructure–property relationships, where only a subset of the training data (either for loam or sand) is used for fitting the regression parameters, can be found in the Appendix, see Figs. 8, 9 and 10. As expected, the predictive power decreases in most cases when using only a subset of the training data for fitting compared to using the complete training data. However, using only data from sand for both fitting and validation often leads to even better results. This may be due to the fact that many descriptors as well as the M-factor show a smaller variation for sand than for loam. In addition, similar to the previously considered case, the concept of geodesic tortuosity leads to better results regarding the prediction of diffusive transport properties compared to geometric tortuosity. Moreover, it can be observed that Equation (10) seems to be quite robust in the sense that this regression formula leads to a large predictive power even if the corresponding regression parameters are fitted based on one soil texture and applied to the other soil texture. However, note that the regression formulas discussed in this section might only be valid to a limited extent if they are applied to other types of 3D microstructures, which are significantly different compared to the soil samples considered in the present paper.

In addition, the microstructure–property relationships in this study have been established on fairly small volumes (\(1.28 \,\textrm{mm}\times 1.28 \,\textrm{mm}\times 1.28 \,\textrm{mm}\)). This had the favorable effect that the variability of the M-factor across all subsamples was quite large for packed soils and hence suitable for testing the accuracy of different models. However, not only diffusion but also some geometrical descriptors of 3D pore space morphology like tortuosity are scale-dependent Ghanbarian (2022). Therefore, it will be tested in a forthcoming study to which extent these models hold for larger soil volumes.

5 Conclusion

In the present paper, we investigated the relationship between various geometrical descriptors of the 3D pore space morphology and diffusive transport properties for two soil different textures, namely loam and sand. For this purpose, sand and loam samples in 5, 10 and \({15}\hbox { cm}\) depth have been imaged by means of X-ray computed tomography and segmented into two phases (pores and solid). The six binary 3D images obtained in this way have been partitioned into 768 non-overlapping subsamples each, which resulted in a total number of 4608 subsamples. For each of these subsamples, the 3D morphology of the pore space has then been characterized by means of various geometrical descriptors including, among others, the porosity as well as the specific surface area, mean geodesic tortuosity, constrictivity and mean spherical contact distance of the pore space. Besides this geometrical characterization of the pore space, diffusive transport properties have been computed by numerically solving the Laplace equation with inhomogeneous flux boundary conditions.

A comprehensive statistical analysis of the 3D pore space morphology has been carried out to quantify structural differences between sand and loam by determining univariate as well as bivariate probability distributions for individual geometrical descriptors and for pairs of them. Among others, it turned out that specific surface area and mean chord length of the pore space can be used to distinguish between loam and sand.

Moreover, microstructure–property relationships have been investigated for both types of soil, loam and sand, by means of parametric regression formulas. By using sophisticated geometrical descriptors of the 3D pore space morphology, which have not yet been considered in the context of soil gas diffusion, the predictive power of the regression formulas could be increased significantly. Among others, the considered descriptors of 3D microstructure include mean and standard deviation of geodesic tortuosity, constrictivity and mean chord length. In particular, it has been shown that with regard to the prediction of diffusive properties, the concept of geodesic tortuosity is superior compared to geometric tortuosity, which is based on skeletonization of the pore space. Furthermore, the robustness of the regression formulas has been investigated by fitting the regression parameters to one soil texture and applying the resulting regression formulas to the other soil texture. It turned out that the best performing regression formula is based on porosity as well as mean and standard deviation of geodesic tortuosity, leading to a mean absolute percentage error of about 5 %. While, in general, the predictive power of parametric regression formulas depends on the specific soil texture under consideration, the best regression formula performed well also in the case when fitted on one soil texture and applied to the other soil texture.

In summary, by our analysis we obtained an improved understanding of the relevance of certain structural features of 3D soil microstructures—which we measured by means of geometrical descriptors—for diffusive transport properties and we were able to predict them much more accurately compared to the precision of conventional regression formulas from the literature. A possible topic for future research is extending this analysis to structures at different length scales. Developing an appropriate multi-scale approach would allow for the prediction of diffusive properties for very large soil structures. Moreover, the present approach can be applied to more realistically developed soils, where pronounced anisotropy effects are to be expected.