Research Project B04

Hybrid discrete-continuum models for fractured porous media: model choice and random field/network generation


Data sets published by project B04 can be found on the DaRUS website

Publications in scientific journals

  1. (Journal-) Articles

    1. Wagner, A., Sonntag, A., Reuschen, S., Nowak, W., & Ehlers, W. (2023). Hydraulically induced fracturing in heterogeneous porous media using a TPM-phase-field model and geostatistics. PAMM, 23(1), Article 1.
    2. Gonzalez-Nicolas, A., Bilgic, D., Kröker, I., Mayar, A., Trevisan, L., Steeb, H., Wieprecht, S., & Nowak, W. (2022). Optimal Exposure Time in Gamma-Ray Attenuation Experiments for Monitoring Time-Dependent Densities. Transport in Porous Media.
    3. Cheng, K., Lu, Z., Xiao, S., Oladyshkin, S., & Nowak, W. (2022). Mixed covariance function kriging model for uncertainty quantification. International Journal for Uncertainty Quantification, 12(3), Article 3.
    4. Xiao, S., Xu, T., Reuschen, S., Nowak, W., & Hendricks Franssen, H.-J. (2021). Bayesian Inversion of Multi-Gaussian Log-Conductivity Fields With Uncertain Hyperparameters: An Extension of Preconditioned Crank-Nicolson Markov Chain Monte Carlo With Parallel Tempering. Water Resources Research, 57(9), Article 9.
    5. Reuschen, S., Jobst, F., & Nowak, W. (2021). Efficient discretization-independent Bayesian inversion of high-dimensional multi-Gaussian priors using a hybrid MCMC. Water Resources Research.
    6. Reuschen, S., Nowak, W., & Guthke, A. (2021). The Four Ways to Consider Measurement Noise in Bayesian Model Selection—And Which One to Choose. Water Resources Research, 57(11), Article 11.
    7. Höge, M., Guthke, A., & Nowak, W. (2020). Bayesian Model Weighting: The Many Faces of Model Averaging. Water, 12(2), Article 2.
    8. Xu, T., Reuschen, S., Nowak, W., & Franssen, H.-J. H. (2020). Preconditioned Crank-Nicolson Markov Chain Monte Carlo Coupled With Parallel Tempering: An Efficient Method for Bayesian Inversion of Multi-Gaussian Log-Hydraulic Conductivity Fields. Water Resources Research, 56(8), Article 8.
    9. Reuschen, S., Xu, T., & Nowak, W. (2020). Bayesian inversion of hierarchical geostatistical models using a parallel-tempering sequential Gibbs MCMC. Advances in Water Resources, 141, 103614.
    10. Xiao, S., Reuschen, S., Köse, G., Oladyshkin, S., & Nowak, W. (2019). Estimation of small failure probabilities based on thermodynamic integration and parallel tempering. Mechanical Systems and Signal Processing, 133, 106248.


About this Project

Fractured porous media are geometrically very complex. There are many competing model concepts to represent their structure in flow simulations; these models differ drastically in their level of geometric detail and in their level of simplification and abstraction. Systematically choosing between these vastly different models, calibrating chosen models and specifying their predictive uncertainty is far from trivial. The goal of this project is to tackle the interlocked model-selection-and-calibration problem. Achieving this goal requires a list of algorithmic developments in the field of simulation-based Bayesian statistics. 


Selected Results for Work Package 1

Radon transform (right) of fracture image (left).
Randomly generated fracture image.

We developed a novel statistical analysis of fracture images based on the Radon transform (see Figure 1). Lead by Prof. Bárdossy, who was co-PI of in the first funding period (but is now about to retire), we extended the Radon transform to additionally extract the correlation structure of fracture aperture along identified fracture lines, and to extract information on connectivity of fracture networks (see Figure 2). Then, selected parts of the extended Radon transform are being randomised to generate randomised images that match the original statistics of fracture length, aperture (including roughness), density, angles and connectivity. Compared to other data-driven methods to random field generation, it generalises better than multi-point geostatistical methods commonly called “training images”. The manuscript for publication is currently in preparation.

Selected Results for Work Package 2

Groundwater models need information about heterogeneity of soil properties in aquifers to predict flow and transport adequately. We developed efficient Markov Chain Monte Carlo (MCMC) methods for Bayesian Inversion of groundwater models to infer these properties in those cases where imaging data (such as in work package 1) cannot be made available.

Inversion results of aquifer with mean (left) and standard derivation (right) of hydraulic conductivity.
Independent samples of MCMC.

First, we developed the parallel-tempering sequential Gibbs MCMC that enables us to predict the spatial distribution and internal heterogeneities of channelized porous media. This algorithm is able to (a) predict the spatial position of channels (large fractures), (b) quantify uncertainty in position, width and soil properties of channels and surrounding porous media (see Figure 3 left and right). Further, many inverse problems lead to so-called multimodal posteriors. This means, in our problem, that different possible channel geometries exist. In a collaboration with the visualization group, we visualize and cluster all geometries and show the probability of occurrence for all of them. Figure 4 shows a small selection of equally likely solutions to the inverse problem. The material is published in Water Resources Research.

Proposal step of the sequential pcn-MCMC.

Second, we developed efficient MCMC methods for aquifers where the heterogeneity of aquifers can be parametrized with multi-Gaussian fields. This means, that either no fractures are present in the aquifer or many fractures exist and the position of each single fracture is negligible for the global behavior of the aquifer (instead, only fracture densities are modelled as effective porous medium). This work showed great synergy with a DFG-funded benchmarking project within workgroup. We joined forces and were able to (a) enhance posterior exploration by combining pCN-MCMC with parallel tempering (b) create a new hybrid MCMC method for more efficient posterior sampling in “simple” inverse problems and (c) use these algorithms to create a benchmark dataset with Bayesian inverse problems for the community. Figure 5 shows the conceptualization of the hybrid MCMC method (center), which is a hybrid between pCN-MCMC (left) and Gibbs MCMC (right). From this work package, one paper is accepted and one is submitted in Water resources research.

The same collaboration currently develops an MCMC that can perform joint estimation of multi-Gaussian fields and of their statistical structure (e.g. their variance and characteristic length scales), with manuscript close to be submitted.

Marginal posterior (colored bars) and prior (horizontal lines) distribution oft mean.
Marginal posterior (colored bars) and prior (horizontal lines) distribution of standard derivation.

Third, on a side project, we applied our knowledge of MCMC methods to estimate small failure probabilities. This approximation is important in many mechanical systems where the failure of one of many parts may lead to the failure of the whole system (results not shown here, but available as publication).

Selected Results and Ongoing Work for Work Package 3

In recent work, we develop a framework on how to consistently handle measurement noise in Bayesian model selection (BMS). We show that the results of BMS change drastically dependent on where noise is modeled and give guidance on how to perform BMS in the presence of measurement noise.

We currently work on Bayesian model selection to quantify which groundwater model is best suited for which application. We focus on where measurement noise is modeled and how we can use this knowledge to better understand which question Bayesian model selection answers.

International Cooperation

"We initiated a collaboration with Prof. Niklas Linde at the University of Lausanne, who is an internationally well-known expert on geostatistical inversion and geophysical inversion. A mutual visit was impossible under Corona conditions. He will serve on the PhD committee of Sebastian Reuschen."

"Related to our works on statistical generation of fractured porous media, we entered a collaboration with Prof. Beth Parker at the University of Guelph. Prof. Parker has unique datasets on fracture distributions in a rock formation directly under a river, and together we modelled the exchange of water and solutes between the river and the fractured rock (hyporheic exchange in fractured-bedrock rivers). In this model, fractures are represented as discrete elements with high permeability, and fracture networks are randomly generated based on statistics on fracture density, length, aperture and orientation. Fractures observed directly at the river bed are enforced during random generation. A manuscript is under revision with Water Resources Research."

Reynold, C., Parker, B., Steelman, C., Thoms, A., Nowak, W. (2020). Can hyporheic exchange occur within a bedrock river?. Water Resources Research.

Outlook for the second funding phase

In the second funding phase, we plan to move from statistical generation of static fracture networks (or their geostatistical representation on basis of conductivity fields) to randomization of hydraulic initiation, propagation and stimulation of fractures. We will rely on a dense collaboration with neighboring projects in our Project Area, as these projects operate and develop the models that we will randomize. With the randomized models, we will address the following hypotheses and research questions:

Our key hypothesis is that the effect of spatial randomness across scales can be conceptually lumped at a single scale, i.e. at the REV scale as resolved by numerical grids of REV-scale models. As first hypothesis (H1), we will reduce the required randomisation from spatial heterogeneity (possibly at several scales) down to a random time series. Thus, we will develop a stochastification of phasefield models by randomly perturbing the crack decisions at every time step. As contrasting second hypothesis (H2), we will lump multi-scalic spatial heterogeneity at the REV scale, and thus pursue a random spatial representation of heterogeneity on the numerical grid. We anticipate that the temporal stochastification is conceptually more elegant and has advantages in experimental data demand, spatial resolution and computational efficiency. However, the highly non-linear interactions in hydraulic stimulation may be more interpretable when treated with spatial randomisation.

Our central research questions focus on stochastic modelling (RQ1), methods in computational statistics (RQ2, RQ4) and process understanding (RQ3):

  • RQ1 For temporal stochastification, what is the best point of attack in the phase-field model?
  • RQ2 How can we infer the hyperparameters of the stochastification (H1) or randomisation (H2)?
  • RQ3 What types of heterogeneity (strength, characteristic length) lead to substantial deviations in fracture initiation/propagation/stimulation from macroscopically homogeneous expectations? How much is the initiation/propagation/stimulation of a specific fracture dominated by the plain hydraulics or heterogeneity? Is hydraulic heterogeneity (e.g. in hydraulic conductivity) or mechanical heterogeneity (e.g. in Lamé parameters or in critical strength) more relevant? How strong are global, non-linear interactions along stimulated preferential flow paths, and how much can they be triggered or suppressed by heterogeneity?
  • RQ4 When is H1 or H2 better? How can we test and validate them statistically?

For further information please contact

This image shows Wolfgang Nowak

Wolfgang Nowak

Prof. Dr.-Ing.

Principal Investigator, Research Projects B04 and B05

This image shows András Bárdossy

András Bárdossy

Prof. Dr. rer. nat. Dr.-Ing.

Principal Investigator, Research Project B04

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