Obtaining accurate porosity information of fluvial sediment deposits is helpful and desirable for many tasks of river engineers. Besides direct measurements of single samples and empirical formulas specialized for specific cases, packing models promise efficient predictions due to their theoretical and extensible foundation. The objective of this work is thus to investigate the usability of three such models in order to obtain a suitable porosity prediction method for the challenging case of fluvial sediment packing. There, the complexity originates from wide continuous size distributions, from silt to gravel, and different grain shapes. We use data obtained from extensive numerical packing simulations to determine the required model parameters and to verify the models' accuracy for moderate size ratios. This study reveals systematic deficits in one of the models, which can be attributed to the absence of a built-in mixture packing model. By combining these findings with data from laboratory measurements and extending the model to include cohesive effects, we exemplify that reasonable porosity predictions can be obtained with the Compressible Packing Model for the Rhine river in Germany. Through an additional comparison with data from French rivers, guidelines for a successful prediction in cases with limited prior knowledge of the model parameters are developed. Future model enhancements of the packing models directly, and by incorporating more effects that are known to influence porosity, are expected to improve the predictive performance.

Porosity is one of the key structural properties of sediment deposits, as it determines the hydraulic conductivity of the soil

Porosity of sediment packing can be determined in various ways, which all come with individual benefits and challenges. The straightforward way is to carry out field or laboratory measurements for which different techniques have been developed

On the other hand, empirical porosity predictors can be an effective alternative. They commonly rely on statistical descriptors of the grain properties, like the mean grain size

Such a framework is offered by so-called packing, or mixture, models. Pioneered by

Packing configurations considered and modeled by packing models that have been exemplified for a binary packing.

For fluvial sediment, an immediate challenge arises due to the large grain size range, spanning from silt, over sand, to gravel, and there is thus 4 orders of magnitude between the diameter of the largest and the smallest grains. The naturally occurring grain size distributions cover the whole aforementioned range, in stark contrast to strictly binary or ternary packing for which these packing models have been primarily designed and used. Only a small subset of the abovementioned packing models was developed for such multi-sized cases

The main objective of the present work is thus to fill this gap by providing a systematic and comparative study of multi-sized packing models in order to derive a suitable porosity predictor. The models are parameterized with data from laboratory experiments and numerical simulations with fluvial sediment and subsequently verified for different grain size distributions. In line with this idea, the paper is structured as follows: Sect.

This section presents several packing models that have been proposed for multi-size packing. For all of them, we provide a concise but complete summary, followed by a direct comparison of the models.

Depending on the scientific community, the packing of solids is commonly described by different quantities like porosity, the solid packing fraction, the void ratio, or the specific volume. As a consequence, the available packing models are formulated differently. However, all of them can be directly converted to provide an estimate for porosity, which is the focus of this work.

Considering a system of total volume

Generally, the solid volume contains grains of various sizes and shapes, as commonly encountered in fluvial sediment deposits. Analogous to sieving the grains with a hierarchy of sieves of mesh sizes

The partial volume fraction of size class

Sketch of a packing model and its parameterization. The brown arrows denote the influence of a property on one of the model components, as considered in the present work. Blue arrows indicate the flow of data within the model.

We provide a sketch of the input and output of a packing model in Fig.

The previously defined quantities

Since it is not known a priori which of the

For very fine grains typically encountered in fluvial sediment deposits, porosity is affected by cohesive effects which then have to be included in the model. This extension is discussed in Sect.

Next, we present three different packing models. We stick mostly to the original formulation and notation, with slight adaptions for clarity and consistency. While an in-depth discussion of the derivation and specifics of each model is beyond the scope of our paper and can be found in the original work, we mention the resulting model and mention the underlying assumptions.

For all models, the simplified binary versions are given in Appendix

The Linear Mixture Packing Model (LMPM) was originally proposed by

By introducing the critical size ratio of entrance

Finally, the specific volume of the packing is obtained as

By introducing more parameters, the LPM has recently been extended to take into account additional interaction mechanisms, like the wedging effect

Graphical representation of the packing models considered in this study (in Sect.

The Compressible Packing Model (CPM) was presented by

The ideal case is obtained by using another LPM, similar to the LPM above, and assuming a single dominant size class

The ideal packing density of the complete packing would then be obtained as

The actual packing density

To be consistent with our notion of

The CPM has been extended, based on geometrical considerations for the loosening effect by an additional parameter

Recently,

Assuming

We corrected a mistake in the original work in Eq. (18)

The void ratio of the packing is then obtained as

The model is sketched in Fig.

Comparison of the three different packing models for binary cases with

While an in-depth comparison of the models and their performance for the specific case of fluvial sediment will be given in the upcoming sections, it is worthwhile to compare them more generally for the parameterization commonly applied in the literature. A direct comparison in terms of predicted porosity for binary packing with varying volume fractions and size ratios is shown in Fig.

Comparison of interaction functions proposed in the literature for spherical grains for the three different packing models.

Apart from the general model itself, the differences originate from the applied interaction functions. All models feature the parameters

In this section, the binary packing of sedimentary grains was simulated, and their porosity was evaluated in order to obtain the interaction parameters required for the packing models presented in Sect.

For this study, we made use of the numerical method, the simulation setup, and the evaluation approach that had been validated extensively in a similar recent study by

The packing was simulated in a horizontally periodic domain sized

Sieve mesh sizes and correspondingly considered mean sizes

The binary packing was here generated for grains with sizes between 2.8 and 31.5 mm. The extent of the size classes and their mean sizes, according to Eq. (

Visualization of three different binary packing for the case

For gravel-sized sediment with non-cohesive interactions, the actual grain size of uniform packing should not influence porosity

Comparison of different model predictions (lines) for the simulated binary setups with Rhine sediment (markers).

Then, for each of the combinations of two different size classes, the binary variants of the packing models, as given in Appendix

Interaction parameters of the different prediction models for the simulated binary setups with Rhine sediment.

The obtained interaction parameters for each model are shown in Fig.

This step finalized the calibration of the interaction functions for all models, which we kept, as here determined, for the remainder of this work.

Interaction function and fitted coefficients, as shown in Fig.

This section applies the packing models with just the calibrated interaction functions for continuous size distributions that contain coarse sand to gravel. The reference results were again obtained by means of numerical simulations. The purpose of this study is the validation of the packing models, given conditions comparable to the ones used in their calibration in Sect.

The packing was simulated and evaluated with the same method as for the binary packing in Sect.

Packing simulations become increasingly challenging for larger size differences, as the number of grains grows cubically, and numerical stability has to be ensured at the cost of smaller time steps. To limit the computational cost of the present simulations, we restricted this validation study to a narrow size range that corresponds to coarse sand and gravel. For the first set of setups, we applied the same seven size fractions as in Table

Thus, in total, 20 size distributions were simulated, and we provide their size distributions and evaluated porosity values

To obtain the porosity prediction, the coefficients of all interaction functions were taken as found via the previous calibration study and given in Table

Measured (from simulations) vs. predicted porosity for the different packing models.

The evaluation of the obtained predictions for the three different models for the 20 size distributions is shown in Fig.

These findings revealed that the NPM does not permit accurate porosity predictions for continuous size distributions, whereas this is seemingly the case for LMPM and CPM. One possibly decisive difference between these two groups is that the LMPM and CPM both have built-in mixture models for the size class

As a final step, we applied the prediction models to laboratory measurement data obtained for sediment from different rivers. These data contained samples in the size range of 0.02 to 200 mm, thus ranging from silt to coarse gravel. This evaluation significantly extends the size range compared to the previous validation in Sect.

The packing of very fine grains, like clay or fine silt, differs from larger grains, as additional grain interactions, like van der Waals or electrostatic forces, can become dominant over all other friction-based or gravitational forces. As a result, such grains often form aggregates

Effect of cohesion on the porosity of a uniform packing of a given mean diameter. Data from laboratory experiments with sediment from the Rhine river

By considering uniform packing,

For fluvial sediment, we carried out an analogous evaluation to determine suitable coefficients of Eq. (

In addition to affecting the porosity of uniform packing,

A total of 10 examples of the size distribution of the Rhine sediment samples shown as a cumulative distribution.

As part of their porosity measurement campaign in the Rhine river,

To set up the porosity predictors for this data set, we assumed that the sediment form was generally similar to the ones we used for calibration in Sect.

Comparison of porosity prediction performance (

Measured (from laboratory) vs. predicted porosity for the different packing models with fluvial sediment from the Rhine river

The outcome of this comparative study is stated in Table

To investigate the generalizability of the packing models and their parameterizations to other rivers, we made use of a second extensive data set obtained by

Therefore, in the context of the packing models, no calibration of the interaction and cohesion models was possible as in Sects.

Comparison of porosity prediction performance (

The results of this study are given in Table

By construction, the CPM allows for another option to account for loose rather than dense packing, i.e., via the compaction index

Applying the packing models as porosity predictors to actual fluvial sediment data in the previous section has revealed their capabilities and also some challenges. The latter primarily arose in the determination or estimation of the various input parameters and quantities; i.e., the initial porosity and the interaction functions that depend on the grain shape. To facilitate the following discussion, we refer again to the overview sketch in Fig.

Two of the main model parameters, the volume fractions

Regarding the other relevant parameters, the above study with the Rhine data can be considered to be the ideal case. Since we had some digital scans of similar sedimentary grains available, simulations were used to calibrate the interaction functions to binary packing (see Sect.

For the comparison to the data from the two French rivers, neither detailed information about the sediment form nor dedicated measurements of uniform packing were available. This lack of information made it necessary to assume that the Rhine-based parameterization of the interaction functions and the cohesion model were still applicable here. This might be justified, since the actual sediment form is probably similar to the one of the Rhine, and thus, no large differences in the calibrated parameters can be expected. A major challenge, however, was to determine the initial porosity for each size class. The subsequent sensitivity study revealed its large influence on the resulting predictions. In practice, the initial porosity that resulted in the best predictions for the available data could be used to essentially calibrate this input to the cases at hand, i.e., specific to the fluvial environment.

Generally, if form information like the average elongation or flatness is available, then the porosity predictions for dense uniform packing derived by

If the actual form is strongly different from the grains considered here, then porosity measurements for uniform and binary packing should be carried out to verify or recalibrate the parameters of the interaction functions. This analysis can be achieved either in the laboratory or via simulations

Furthermore, by using the same initial porosity for all size classes and the interaction functions calibrated through measurements for gravel-sized sediment, we implicitly assumed that the grain shape is similar for all size classes. Such an assumption was found to be applicable for gravel-sized grains from the Rhine by

Moving towards field measurements, the challenges associated with the determination of the input variables grow further. There, the underlying model assumption that the packing is a random grain mixture could be violated by selective deposition conditions that might have led to imbrication or stratification inside the sediment bed. Additionally, it is usually not possible to obtain field measurements for uniform packing, as those do not occur naturally. These effects lead to additional uncertainties in the obtained predictions and might essentially limit the applicability of the models considered here. Advanced simulation studies that closely resemble the deposition conditions in actual rivers, could be used there as a promising yet demanding alternative to determine the required model input and promote further model improvements.

All in all, the intrinsic complexity of grain packing with the large number of influence factors directly carries over to corresponding prediction models, where all of them should be taken into account, i.e., modeled. The more these factors can be quantified or excluded, the better the expected outcome using the prediction models considered here will be.

We have compared three different porosity prediction models for the packing of fluvial sediment, which is characterized by a wide range of size ratios and complex grain shapes. These packing models, the Linear Mixture Packing Model (LMPM), Compressible Packing Model (CPM), and Nonlinear Packing Model (NPM), partition the grains into distinct size classes for which the respective average size, the mass fraction, and the porosity of a single class packing have to be provided. By considering the interaction between the contained size classes, modeled via interaction parameters and depending on the size ratios of the classes, they provide predictions of porosity.

These interaction parameters were calibrated by simulating the packing of several binary cases, using digital scans of actual sediment grains from the Rhine river in Germany. This study showed that all of these packing models are able to capture binary cases well if calibrated adequately, with a slightly better performance of the NPM. Based on the individually calibrated cases, the interaction parameters of the models could be formulated as functions of the size ratio, making them applicable for the general, multi-sized case.

We verified the models further by comparing them to 20 simulations, now featuring continuous size distributions between 1.4 and 31.5 mm. Keeping all other variables, like the packing process and porosity evaluation, the same, the LMPM and CPM predicted porosity reasonably well. On the other hand, the study revealed a systematic deficit of the NPM for these continuous size distributions, which we could attribute to a missing built-in mixture packing model for the dominant size class as opposed to the two other models.

Finally, we applied the LMPM and the CPM to two different data sets obtained from laboratory porosity measurements, with sediment from the Rhine river and two French rivers. Both featured size distributions ranging from 0.02 to 200 mm, thus covering the range from silt to coarse gravel. There, we demonstrated that the packing models should be augmented with a dedicated cohesion model, as it improved the predictions significantly. However, the models were less accurate than for the simulated packing, since not all the effects of the laboratory experiments could be quantified and were thus not included in the models. The comparison for the French rivers acted primarily as a sensitivity study regarding unknown input parameters for the packing models, as here information about the porosity of the size classes or about the sediment shape was not available. Given a sensible estimate of the single class porosity values, we could predict the porosity of mixtures reasonably well. Based on these findings, we discussed several ways that a suitable porosity predictor can be obtained for fluvial sediment deposits, with the general recommendation to make use of the CPM due to its flexibility.

Overall, we show the great potential that lies within these packing models, given a detailed knowledge of the packing process and some sediment properties. The observed discrepancy for the case of laboratory measurements and the expected even larger challenge for field measurements highlight that many different aspects of such a packing play a role that might not yet be properly accounted for in the present models or the provided model input. Further improvements are expected once the recent extensions of the LMPM and the CPM that aim to cover additional interaction effects among the size classes become available for multi-sized packing in combination with the here-found importance of a proper mixture packing model and a cohesion model. Simulations augmented by cohesive effects and featuring different deposition conditions can help to identify, quantify, and model these effects systematically here.

The range of size classes that defines the mixture for a specific class

The interaction parameters

The quadratic and cubic mixture parameters

The model's interaction parameters,

The two interaction functions,

In the following, we state the binary versions of the packing models presented in Sect.

As the LMPM

The packing densities of the two ideal packing models considered by the CPM

The void ratio of a binary packing according to the NPM

The simulation software, including the numerical method and the setups, is publicly available via the waLBerla main repository (

CR and RMF conceptualized the study. CR performed the simulations, developed the model implementation, and carried out the analysis. MT and RMF compiled and prepared the laboratory measurement data. CR prepared the paper, with contributions from all co-authors. RMF acquired the funding. UR provided access to the computational resources. SV administered the project.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors want to thank Axel Winterscheid and Wenjia Xu for fruitful discussions that helped to shape the research. The authors gratefully acknowledge the HPC resources provided by the Erlangen National High Performance Computing Center (NHR@FAU) of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU). NHR funding is provided by federal and Bavarian state authorities. NHR@FAU hardware is partially funded by the DFG (grant no. 440719683). Furthermore, the authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (

This research has been supported by the Deutsche Forschungsgemeinschaft (grant nos. FR 3509/4-2 and 433735254).

This paper was edited by Paola Passalacqua and reviewed by two anonymous referees.