Daniel Vogler Publications

daniel_vogler_234x323

Mailing Address
Dr. Daniel Vogler
Geothermal Energy & Geofluids
Institute of Geophysics
NO F 61
Sonneggstrasse 5
CH-8092 Zurich Switzerland

Contact
Phone +41 44 633 27 51
Email davogler(at)ethz.ch

Administration
Dominique Ballarin Dolfin
Phone +41 44 632 3465
Email ballarin(at)ethz.ch

Publications

A complete overview can be found here

REFEREED PUBLICATIONS IN JOURNALS

8.  Vogler, D., R. R. Settgast, C. Annavarapu, C. Madonna, P. Bayer, and F. Amann Experiments and Simulations of Fully Hydro-Mechanically coupled Response of Rough Fractures exposed to High Pressure Fluid Injection, Journal of Geophysical Research: Solid Earth, 123, pp. 1186-1200, 2018. Abstract
In this work, we present the application of a fully-coupled hydro-mechanical method to investigate the effect of fracture heterogeneity on fluid flow through fractures at the laboratory scale. Experimental and numerical studies of fracture closure behavior in the presence of heterogeneous mechanical and hydraulic properties are presented. We compare the results of two sets of laboratory experiments on granodiorite specimens against numerical simulations in order to investigate the mechanical fracture closure and the hydro-mechanical effects, respectively.

The model captures fracture closure behavior and predicts a non-linear increase in fluid injection pressure with loading. Results from this study indicate that the heterogeneous aperture distributions measured for experiment specimens can be used as model input for a local cubic law model in a heterogeneous fracture to capture fracture closure behavior and corresponding fluid pressure response.

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7.  Vogler, D., S. Ostvar, R. Paustian, and B.D. Wood A Hierarchy of Models for Simulating Experimental Results from a 3D Heterogeneous Porous Medium, Advances in Water Resources, 114, pp. 149-163, 2018. Abstract
In this work we examine the dispersion of conservative tracers (bromide and fluorescein) in an experimentally-constructed three-dimensional dual-porosity porous medium. The medium is highly heterogeneous ($\sigma_Y^2=5.7$), and consists of spherical, low-hydraulic-conductivity inclusions embedded in a high-hydraulic-conductivity matrix. The bi-modal medium was saturated with tracers, and then flushed with tracer-free fluid while the effluent breakthrough curves were measured. The focus for this work is to examine a hierarchy of four models (in the absence of adjustable parameters) with decreasing complexity to assess their ability to accurately represent the measured breakthrough curves. The most information-rich model was (1) a direct numerical simulation of the system in which the geometry, boundary and initial conditions, and medium properties were fully independently characterized experimentally with high fidelity. The reduced models included; (2) a simplified numerical model identical to the fully-resolved direct numerical simulation (DNS) model, but using a domain that was one-tenth the size; (3) an upscaled mobile-immobile model that allowed for a time-dependent mass-transfer coefficient; and, (4) an upscaled mobile-immobile model that assumed a space-time constant mass-transfer coefficient. The results illustrated that all four models provided accurate representations of the experimental breakthrough curves as measured by global RMS error. The primary component of error induced in the upscaled models appeared to arise from the neglect of convection within the inclusions. We discuss the necessity to assign value (via a utility function or other similar method) to outcomes if one is to further select from among model options. Interestingly, these results suggested that the conventional convection-dispersion equation, when applied in a way that resolves the heterogeneities, yields models with high fidelity without requiring the imposition of a more complex non-Fickian model.
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6.  Kling, T., D. Vogler, L. Pastewka, F. Amann, and P. Blum Numerical simulations and validation of contact mechanics in a granodiorite fracture, Rock Mechanics and Rock Engineering, 51/9, pp. 2805-2824, 2018. Abstract
Numerous rock engineering applications require a reliable estimation of fracture permeabilities to predict fluid flow and transport processes. Since measurements of fracture properties at great depth are extremely elaborate, representative fracture geometries typically are obtained from outcrops or core drillings. Thus, physically valid numerical approaches are required to compute the actual fracture geometries under in situ stress conditions. Hence, the objective of this study is the validation of a fast Fourier transform (FFT)-based numerical approach for a circular granodiorite fracture considering stress-dependent normal closure. The numerical approach employs both purely elastic and elastic–plastic contact deformation models, which are based on high-resolution fracture scans and representative mechanical properties, which were measured in laboratory experiments. The numerical approaches are validated by comparing the simulated results with uniaxial laboratory tests. The normal stresses applied in the axial direction of the cylindrical specimen vary between 0.25 and 10 MPa. The simulations indicate the best performance for the elastic–plastic model, which fits well with experimentally derived normal closure data (root-mean-squared error $=9 \mu m$). The validity of the elastic–plastic model is emphasized by a more realistic reproduction of aperture distributions, local stresses and contact areas along the fracture. Although there are differences in simulated closure for the elastic and elastic–plastic models, only slight differences in the resulting aperture distributions are observed. In contrast to alternative interpenetration models or analytical models such as the Barton–Bandis models and the ”exponential repulsion model”, the numerical simulations reproduce heterogeneous local closure as well as low-contact areas (<2\%) even at high normal stresses (10 MPa), which coincides with findings of former experimental studies. Additionally, a relative hardness value of 0.14 for granitic rocks, which defines the general resistance to non-elastic deformation of the contacts, is introduced and successfully applied for the elastic--plastic model.
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5.  Perras, M.A., and D. Vogler Compressive and Tensile Behavior of 3D-Printed and Natural Sandstones, Transport in Porous Media, pp. 1-23, 2018. Abstract
The presented work compares the mechanical behavior from standard unconfined compressive strength and indirect tensile strength tests of natural sandstone and artificial sand-based specimens created by 3D additive manufacturing. Three natural sandstones of varying strength and stiffness were tested to capture a wide range of behavior for comparison with the 3D-printed specimens. Sand grains with furan and silicate binders, as well as, ceramic beads with silicate binder were 3D-printed by commercial suppliers.
The tensile and compressive strength, the stiffness, the crack initiation and the crack damage thresholds and the strain behavior were examined to determine if the mechanical behavior of the 3D-printed specimens is similar to natural sandstones. The Sand-Furan 3D-prints behaved the closest to the weak natural sandstone. The compressive strength to stiffness ratio, also known as the modulus ratio, and the compressive to tensile strength ratio of the 3D-printed Sand-Furan specimens were found to be similar to the natural sandstones tested in this study and literature values. The failed specimens composed of ceramic beads with silicate binder, both in compression and tension, showed fracture growth not commonly observed in natural specimens. The other 3D-printed specimens generally fractured in a similar manner to natural specimens, although several of the quartz sand with furan binder specimens showed fracturing behavior similar to high porosity natural specimens. Over all, using the commercially available quartz sand with furan binder 3D-print materials showed promise to be able to replicate natural rock specimen behavior.
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4.  Hobé, A., D. Vogler, M.P. Seybold, A. Ebigbo, R.R. Settgast, and M.O. Saar Estimating Fluid Flow Rates through Fracture Networks using Combinatorial Optimization, Advances in Water Resources, 122, pp. 85-97, 2018. Abstract
To enable fast uncertainty quantification of fluid flow in a discrete fracture network (DFN), we present two approaches to quickly compute fluid flow in DFNs using combinatorial optimization algorithms. Specifically, the presented Hanan Shortest Path Maxflow (HSPM) and Intersection Shortest Path Maxflow (ISPM) methods translate DFN geometries and properties to a graph on which a max flow algorithm computes a combinatorial flow, from which an overall fluid flow rate is estimated using a shortest path decomposition of this flow. The two approaches are assessed by comparing their predictions with results from explicit numerical simulations of simple test cases as well as stochastic DFN realizations covering a range of fracture densities. Both methods have a high accuracy and very low computational cost, which can facilitate much-needed in-depth analyses of the propagation of uncertainty in fracture and fracture-network properties to fluid flow rates.
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3.  Vogler, D., S.D.C. Walsh, E. Dombrovski, and M.A. Perras A Comparison of Tensile Failure in 3D-Printed and Natural Sandstone, Engineering Geology, 226, pp. 221-235, 2017. Abstract
This work investigates the possibility of replication of natural rock specimens, which can be used to analyze rock mechanical behavior by subjecting a number of identical specimens to tensile tests and a variety of analysis methods. We compare the properties of fractures generated in artificial sandstone specimens to those generated in natural sandstone specimens. Artificial sandstone specimens, created using 3D additive manufacturing printing processes, were subject to tensile failure using the Brazilian test method and the results from these tests were compared to results from Brazilian tests conducted on natural sandstones. The specimens included two distinct types of synthetic rock, unaltered from the manufacturers typical process, and three natural sandstones. For each test, the loading history to failure of the specimens were recorded and the failure mode was confirmed using digital imaging techniques. In addition, three dimensional images were taken of the fracture surfaces, which were then used to compare the geometric characteristics of all materials tested. The indirect tensile strength of the artificial sandstone specimens ranged between 1.0 and 2.8 MPa. Natural sandstone specimens with a wide range of indirect tensile strengths were tested for comparison. These included a strong sandstone, an intermediate sandstone, and a weak sandstone; which were found to have indirect tensile strength ranges of 10.5-25.5 MPa, 4.4-6.4 MPa, and 0.9-1.1 MPa, respectively. Digital image correlation confirmed that the artificial specimens generally failed in a tensile (mode I) fracture, similar to the natural specimens. Likewise, fracture surface roughness measures showed no clear distinction between weak natural and artificial sandstones. This indicates that there are distinct similarities between the fractures generated in the natural and artificial sandstones of comparable indirect tensile strengths. The three dimensionally printed sandstone specimens are shown to exhibit indirect tensile strength, surface roughness and crack propagation behavior which resembles a weak natural sandstone.
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2.  Vogler, D., S.D.C. Walsh, P. Bayer, and F. Amann Comparison of Surface Properties in Natural and Artificially Generated Fractures in a Crystalline Rock, Rock Mechanics and Rock Engineering, 50/11, pp. 2891-2909, 2017. Abstract
This work studies the roughness characteristics of fracture surfaces from a crystalline rock by analyzing differences in surface roughness between fractures of various types and sizes. We compare the surface properties of natural fractures sampled in situ and artificial (i.e., man-made) fractures created in the same source rock under laboratory conditions. The topography of the various fracture types is compared and characterized using a range of different measures of surface roughness. Both natural and artificial, and tensile and shear fractures are considered, along with the effects of specimen size on both the geometry of the fracture and its surface characterization. The analysis shows that fracture characteristics are substantially different between natural shear and artificial tensile fractures, while natural tensile fracture often spans the whole result domain of the two other fracture types. Specimen size effects are also evident, not only as scale sensitivity in the roughness metrics, but also as a by-product of the physical processes used to generate the fractures. Results from fractures generated with Brazilian tests show that fracture roughness at small scales differentiates fractures from different specimen sizes and stresses at failure.
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1.  Vogler, D., F. Amann, P. Bayer, and D. Elsworth Permeability Evolution in Natural Fractures Subject to Cyclic Loading and Gouge Formation, Rock Mechanics and Rock Engineering, 49/9, pp. 3463-3479, 2016. Abstract
Increasing fracture aperture by lowering effective normal stress and by inducing dilatant shearing and thermo-elastic effects is essential for transmissivity increase in enhanced geothermal systems. This study investigates transmissivity evolution for fluid flow through natural fractures in granodiorite at the laboratory scale. Processes that influence transmissivity are changing normal loads, surface deformation, the formation of gouge and fracture offset. Normal loads were varied in cycles between 1 and 68 MPa and cause transmissivity changes of up to three orders of magnitude. Similarly, small offsets of fracture surfaces of the order of millimeters induced changes in transmissivity of up to three orders of magnitude. During normal load cycling, the fractures experienced significant surface deformation, which did not lead to increased matedness for most experiments, especially for offset fractures. The resulting gouge material production may have caused clogging of the main fluid flow channels with progressing loading cycles, resulting in reductions of transmissivity by up to one order of magnitude. During one load cycle, from low to high normal loads, the majority of tests show hysteretic behavior of the transmissivity. This effect is stronger for early load cycles, most likely when surface deformation occurs, and becomes less pronounced in later cycles when asperities with low asperity strength failed. The influence of repeated load cycling on surface deformation is investigated by scanning the specimen surfaces before and after testing. This allows one to study asperity height distribution and surface deformation by evaluating the changes of the standard deviation of the height, distribution of asperities and matedness of the fractures. Surface roughness, as expressed by the standard deviation of the asperity height distribution, increased during testing. Specimen surfaces that were tested in a mated configuration were better mated after testing, than specimens tested in shear offset configuration. The fracture surface deformation of specimen surfaces that were tested in an offset configuration was dominated by the breaking of individual asperities and grains, which did not result in better mated surfaces.
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PROCEEDINGS REFEREED

4.  von Planta, C., D. Vogler, M. Nestola, P. Zulian, and R. Krause Variational Parallel Information Transfer between Unstructured Grids in Geophysics – Applications and Solutions Methods, PROCEEDINGS, 43rd Workshop on Geothermal Reservoir Engineering Stanford University, 2018. Abstract
State of the art simulations of enhanced geothermal systems are multi-physics simulations, where different physical properties are often being modeled on different geometries or grids. For example, for the simulation of fluids many prefer finite difference or volume formulations on structured grids, whereas in mechanics finite element formulations on unstructured grids are often preferred. To transfer information between these various geometries, we use a generic variational transfer operator, which has previously been introduced as pseudo-L2-projection. In this paper, we demonstrate that this transfer operator can be particularly useful for geophysics with three applications: First, in contact simulations we often have non-matching surfaces at the contact boundary. Here, the transfer operator acts as a mortar projection and we show with high resolution rock fracture geometries from the Grimsel Test Site in Switzerland how the variational transfer operator is used to formulate the contact problem. Secondly, we present a three-dimensional fluid-structure simulation, computing water flow between two rock surfaces and simultaneous deformation with an immersed boundary approach. In this method the solid, which is formulated on an unstructured grid, interacts with the fluid, formulated on a structured grid, by means of weakly enforced velocity constraints at the interface between fluid and solid. And lastly, we show how the transfer operator can be used to effectively solve contact problem between rock bodies. Using our operator, we can generate nested multilevel hierarchies, enabling us to solve the problem with optimal complexity, thus extending the possible size of simulations immensely.
3.  Deb, P., D. Vogler, S. Dueber, P. Siebert, S. Reiche, C. Clauser, R.R. Settgast, and K. Willbrand Laboratory Fracking Experiments for Verifying Numerical Simulation Codes, 80th EAGE Conference and Exhibition, pp. 1-4, 2018. Abstract
Carefully designed and well monitored experiments are irreplaceable when it comes to producing reliable data sets for a detailed understanding of physical processes, such as hydraulic fracturing. While such experiments provide insight into the governing physical processes, numerical simulations provide additional information on system behaviour by enabling a straightforward study of parameter sensitivity. In this study, we focus on both these aspects. We report on results from (1) a benchmark experimental facility for performing hydraulic fracturing experiments on large rock samples in the laboratory under controlled conditions and (2) numerical simulations of these experiments using programs, which, in future, may be used for designing hydraulic stimulation layouts. We conduct series of experiments in order to ensure reproducibility and accuracy of the measurements. This experimental data set is then shared with several research institutes to be used for verifying their simulation software. Results from the simulation provide further insight regarding parameters, which contribute to uncertainties during measurements. Detailed study of the sensitive parameters help us to improve our experimental set up further and to perform future experiments under even better controlled conditions.
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2.  Vogler, D., R.R. Settgast, C.S. Sherman, V.S. Gischig, R. Jalali, J.A. Doetsch, B. Valley, K.F. Evans, F. Amann, and M.O. Saar Modeling the Hydraulic Fracture Stimulation performed for Reservoir Permeability Enhancement at the Grimsel Test Site, Switzerland, Proceedings of the 42nd Workshop on Geothermal Reservoir Engineering Stanford University, 2017. Abstract
In-situ hydraulic fracturing has been performed on the decameter scale in the Deep Underground rock Laboratory (DUG Lab) at the Grimsel Test Site (GTS) in Switzerland in order to measure the minimum principal stress magnitude and orientation. Conducted tests were performed in a number of boreholes, with 3–4 packer intervals in each borehole subjected to repeated injection. During each test, fluid injection pressure, injection flow rate and microseismic events were recorded amongst others. Fully coupled 3D simulations have been performed with the LLNL’s GEOS simulation framework. The methods applied in the simulation of the experiments address physical processes such as rock deformation/stress, LEFM fracture mechanics, fluid flow in the fracture and matrix, and the generation of micro-seismic events. This allows to estimate the distance of fracture penetration during the injection phase and correlate the simulated injection pressure with experimental data during injection, as well as post shut-in. Additionally, the extent of the fracture resulting from simulations of fracture propagation and microseismic events are compared with the spatial distribution of the microseismic events recorded in the experiment.
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1.  Vogler, D., R. Settgast, C. Annavarapu, P. Bayer, and F. Amann Hydro-Mechanically Coupled Flow through Heterogeneous Fractures, Proceedings of the 41st Workshop on Geothermal Reservoir Engineering Stanford University, pp. SGP-TR-209, 2016. Abstract
Heterogeneous aperture distributions are an intrinsic characteristic of natural fractures. The presence of highly heterogeneous aperture distributions can lead to flow channeling, thus influencing the macroscopic behavior of the fluid flow. High-fidelity numerical simulation tools are needed for realistic simulation of fracture flow when such features are present. Here, focus is set on the role of mechanical fracture closure for fluid flow and appropriate simulation by a fully hydro-mechanically (HM) coupled numerical model. In a laboratory experiment, an artificial fracture in a granodiorite sample is created. During different sequential loading cycles, the development of fracture closure, contact area and contact stress are examined. Constant fluid flow rate injection into the center of the rough fracture is modelled to investigate the impact of fracture closure on the flow field and injection pressure. Results show that the numerical framework for heterogeneous fracture surfaces allows for reproducing experimental data of dry, mechanical tests at the laboratory scale, and it may offer advanced understanding and prediction of the behavior of reservoirs that are subject to high-pressure fluid injections.
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THESES

3.  Vogler, D. Hydro-mechanically coupled processes in heterogeneous fractures: experiments and numerical simulations, Dissertation ETH Zurich, 169 pp., 2016. Abstract
Enhanced Geothermal Systems (EGS), CO2-sequestration, oil- and gas reservoirs rely on an in-depth understanding of geomechanics and fluid flow in the subsurface to achieve production targets. In Switzerland, EGS are commonly targeted for deep basement formations of crystalline rock, as these are deep enough underground to provide high temperatures. In crystalline rock, fluid flow through fractures dominates transport processes, while mechanical behavior strongly depends on fracture topography and strength. This work focusses on fracture behavior in crystalline rock, such as granite and granodiorite, by investigating: (1) Differences in fracture topography linked to fracture size and nature; (2) Hydromechanically coupled processes in heterogeneous fractures in experiments on the laboratory scale; and (3) Hydro-mechanically coupled processes in heterogeneous fractures in simulations on the laboratory and field scale, supported by laboratory experiments. All rock specimens in this work are granite or granodiorite specimens obtained from the Grimsel Test Site (GTS), Switzerland. Fracture topography is studied by overcoring mode I and mode II fractures from core material and by subjecting intact specimens to Brazilian tests. This yields a range of fractures of various nature with sizes between 1 to 30 cm. Fracture topography is compared with the JRC, Z2 measure, fractal dimensions (Hausdorff and Box count dimension) and correlation functions (Two point correlation function and lineal path function) to quantify and compare roughness with a large range of parameters. Additionally, surface roughness is compared to specimen tensile strengths. Results show a clear distinction of natural shear and artificial tensile fractures, as measured with the Z2 measure. Fracture roughness appears to be linked to specimen size when comparing whole fracture sizes. Computing local roughness on small surface patches (e.g. 1 cm x 1 cm) yields smoother surfaces for large fractures, further indicating that fracture roughness is scale dependent and that this scale dependency can be traced down to scales significantly smaller than the whole fracture. The scale of the specimen has an influence on the probable fracture propagation path and therefore the tensile strength, which leads to different surface roughnesses of the induced tensile fracture. As specimen sizes increase, the tensile strength decreases and the fracture roughness increases. In summary, fractures of different nature and size can be distinguished by surface roughness measures, indicating that fracture origin has significant influence on surface topography. This is especially important, as fracture topography is linked to fracture conductivity and strength. Laboratory tests on granodiorite specimen were performed to investigate the relation of fluid flow rate, injection pressure, confining stress and fracture aperture during testing. Cylindrical specimen were overcored from natural tensile and shear fractures and subjected to a fluid pressure gradient across the fracture to sustain a constant flow rate. The specimens were tested in mated configuration and with shear offset in the fracture between 1 and 6 mm. Additionally, specimen fracture surfaces were scanned before and after testing to study the relationship of fracture transmissivity evolution during testing and surface deformation. Confining stress varied between 1 and 68 MPa for 5 to 10 cycles, yielding changes in transmissivity of up to three orders of magnitude. Shear offset of specimens lead to transmissivity increase of up to three orders of magnitude. Specimens experienced strongly damaged fracture surface and gouge production, which reduced transmissivity up to one order of magnitude for subsequent load cycles. While fracture surface roughness increased during testing, this effect was especially pronounced for specimens with shear offset. Almost all tests show hysteretic behavior during individual load cycles, indicating stress path dependent behavior of transmissivity. The experimental results qualitatively demonstrate and quantify mechanisms commonly encountered in EGS reservoir fractures. To further system understanding and predictive capabilities, a novel numerical model was tied into the GEOS framework to compute fully hydro-mechanically coupled processes in heterogeneous fractures. The model is compared against three experimental test sets investigating cylindrical granodiorite specimens with axial loads between 0.25 and 10 MPa for: (i) dry fracture closure; (ii) contact stress evolution in fractures during normal loading; and (iii) constant fluid flow rate injection into the fracture center. The non-linear behavior or fracture normal closure and fluid injection pressure increase with increasing axial load is replicated by the numerical model, by using the fracture aperture fields obtained from photogrammetry scans as model input. The numerical model captures contact stress evolution with axial load increase and shows a linear increase in contact area with axial load. Study of flow field simulations show an early onset of channeling, for axial loads as low as 2 MPa. Additionally, simulations of a field scale domain (100 m x 100 m x 40 m), with a 100 x 100 m fracture plane are performed. Pre-existing natural fractures were scanned, to use their aperture field to generate a synthetic aperture field for the fracture plane. In the next step, vertical stresses of 8.3 MPa, corresponding to the host rock of the fracture origin at the GTS are applied to the system. This yields the unique aperture field corresponding to the given stress state. Fluid is subsequently injected with constant pressure head into the fracture center with pressures between 0.01 and 8.7 MPa. While heterogeneous flow paths and pressure diffusion can be observed, the model additionally allows to observe heterogeneous fracture opening due to lowered effective normal stresses during injection. Further, the hydro-mechanically coupled analysis of the velocity and pressure field shows a deviation of the pressure distribution from linear diffusion for increased injection pressures, once hydro-mechanical contact between the fluid and the rock mass is established. Fluid pressure induced fracture opening is shown to strongly depend on aperture magnitude before injection and aperture magnitudes of the surrounding fracture region. Thereby, the model captures mechanical and hydraulic behavior of the laboratory tests, while providing unique insights for heterogeneous fracture behavior under compression and high pressure fluid injection. In summary, this work attempts to scrutinize heterogeneous fractures, and especially related hydro-mechanical processes. This is done by investigating possible bias by specimen fracture nature and size selection for testing. Hydro-mechanical processes are studied in experiments, which aim to replicate reservoir conditions, and showcase the impact of specific fractures, stress paths and gouge production. Finally, this work presents an approach to incorporate the observed phenomena in a numerical framework, which is tested against specifically designed laboratory tests. This work combines laboratory scale investigations by employing the framework to perform fully hydro-mechanically coupled simulations of a field scale fracture with heterogeneous aperture distribution, which yields quantitative results of fracture opening during high-pressure injection. The presented work thereby contributes to further understanding of fracture processes, which characterize behavior of Enhanced Geothermal Systems and other subsurface phenomena.
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2.  Vogler, D. A comparison of different model reduction techniques for model calibration and risk assessment, MSc Thesis University of Stuttgart, 62 pp., 2013. Abstract
Many engineering systems represent challenging classes of complex dynamic systems. Lacking information about their systems properties leads to model uncertainties up to a level where quantification of uncertainties may become the dominant question in modeling, simulation andapplication tasks. Uncertainty quantification is the prerequisite for probabilistic risk assessment and related tasks. The current work will present recent approaches for these challenges based on response surface techniques, which reduce massively the initial complex model. The reduction is achieved by a regression-like analysis of model output with orthonormal polynomials that depend on the model input parameters. This way, the model response to changes in uncertain parameters, design or control variables is represented by polynomials for each model prediction of interest. This technique is known as polynomial chaos expansion (PCE) in the field of stochastic PDE solutions. The reduced model represented by the response surface is vastly faster than the original complex one, and thus provides a promising starting point for follow-up tasks: uncertainty quantification, model calibration and probabilistic risk assessment. Obviously, a response surface can be constructed in different ways. Methods for constructing the response surface can demand only a minimum number of model evaluations, but as well may ask for many model evaluations to achieve a better quality of the involved projection integrals. The scope of the current work is to test and compare different integration rules, i.e., methods to choose the sets of parameter values for which the model has to be evaluated. To test and compare the different methods, their accuracy in uncertainty quantification, model calibration and risk assessment will be measured against brute-force reference computations based on the original model. As illustrative example, we consider a study from the field of CO2 storage in the subsurface.
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1.  Vogler, D. Investigation of transport phenomena in a highly heterogeneous porous medium, MSc Thesis Oregon State University, 70 pp., 2012. Abstract
This work focuses on solute mass transport in a highly heterogeneous two-region porous medium consisting of spherical low-hydraulic conductivity inclusions, embedded in a high-hydraulic conductivity matrix. The transport processes occuring in the system are described by three distinct time scales. The first time scale reflects the characteristic time for convective transport in the high-conductivity matrix. The second time scale reflects the characteristic time for diffusive transport in the low-conductivity inclusions. The third time scale reflects the characteristic time for convection within the inclusions. Two Péclet numbers can be defined that compare the time scales and provide qualitative insight into the net transport behavior in two-region media. To model this system, four different representations were developed: (1) a Darcy-scale model that involved direct microscale computation over the entire domain of the experimental system, (2) a direct microscale simulation computed on a simplified domain that had similar geometric parameters (e.g. volume fraction of inclusions) as the complete domain for the experimental system, (3) a volume averaged model (after Chastanet and Wood [2008]) which uses a constant mass transfer coefficient and (4) a volume averaged model which employs a time-dependent mass transfer coefficient. Two different experimental conditions were investigated: a high flow rate, and a low flow rate. Detailed understanding of the experimental system was developed, which led to accurate prediction of the system’s behavior for the higher flow rate. Accurate early time fit of the data was achieved for the experiment with the lower flow rate, while late time behavior between the models and experimental data diverged. Further investigations of the experimental system were conducted to examine possible sources of errors that could lead to an inaccurate description of the system’s properties. Additional mixing within the system, inhomogeneous distribution of the effective diffusion coefficient and imprecise initial estimates of the hydraulic parameters are all possible explanations for the inaccurate model representation of the system’s behavior for the lower flow rate case.
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REFEREED PUBLICATIONS IN JOURNALS

8.  Vogler, D., R. R. Settgast, C. Annavarapu, C. Madonna, P. Bayer, and F. Amann Experiments and Simulations of Fully Hydro-Mechanically coupled Response of Rough Fractures exposed to High Pressure Fluid Injection, Journal of Geophysical Research: Solid Earth, 123, pp. 1186-1200, 2018. Abstract
In this work, we present the application of a fully-coupled hydro-mechanical method to investigate the effect of fracture heterogeneity on fluid flow through fractures at the laboratory scale. Experimental and numerical studies of fracture closure behavior in the presence of heterogeneous mechanical and hydraulic properties are presented. We compare the results of two sets of laboratory experiments on granodiorite specimens against numerical simulations in order to investigate the mechanical fracture closure and the hydro-mechanical effects, respectively.

The model captures fracture closure behavior and predicts a non-linear increase in fluid injection pressure with loading. Results from this study indicate that the heterogeneous aperture distributions measured for experiment specimens can be used as model input for a local cubic law model in a heterogeneous fracture to capture fracture closure behavior and corresponding fluid pressure response.

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7.  Vogler, D., S. Ostvar, R. Paustian, and B.D. Wood A Hierarchy of Models for Simulating Experimental Results from a 3D Heterogeneous Porous Medium, Advances in Water Resources, 114, pp. 149-163, 2018. Abstract
In this work we examine the dispersion of conservative tracers (bromide and fluorescein) in an experimentally-constructed three-dimensional dual-porosity porous medium. The medium is highly heterogeneous ($\sigma_Y^2=5.7$), and consists of spherical, low-hydraulic-conductivity inclusions embedded in a high-hydraulic-conductivity matrix. The bi-modal medium was saturated with tracers, and then flushed with tracer-free fluid while the effluent breakthrough curves were measured. The focus for this work is to examine a hierarchy of four models (in the absence of adjustable parameters) with decreasing complexity to assess their ability to accurately represent the measured breakthrough curves. The most information-rich model was (1) a direct numerical simulation of the system in which the geometry, boundary and initial conditions, and medium properties were fully independently characterized experimentally with high fidelity. The reduced models included; (2) a simplified numerical model identical to the fully-resolved direct numerical simulation (DNS) model, but using a domain that was one-tenth the size; (3) an upscaled mobile-immobile model that allowed for a time-dependent mass-transfer coefficient; and, (4) an upscaled mobile-immobile model that assumed a space-time constant mass-transfer coefficient. The results illustrated that all four models provided accurate representations of the experimental breakthrough curves as measured by global RMS error. The primary component of error induced in the upscaled models appeared to arise from the neglect of convection within the inclusions. We discuss the necessity to assign value (via a utility function or other similar method) to outcomes if one is to further select from among model options. Interestingly, these results suggested that the conventional convection-dispersion equation, when applied in a way that resolves the heterogeneities, yields models with high fidelity without requiring the imposition of a more complex non-Fickian model.
/ Download
6.  Kling, T., D. Vogler, L. Pastewka, F. Amann, and P. Blum Numerical simulations and validation of contact mechanics in a granodiorite fracture, Rock Mechanics and Rock Engineering, 51/9, pp. 2805-2824, 2018. Abstract
Numerous rock engineering applications require a reliable estimation of fracture permeabilities to predict fluid flow and transport processes. Since measurements of fracture properties at great depth are extremely elaborate, representative fracture geometries typically are obtained from outcrops or core drillings. Thus, physically valid numerical approaches are required to compute the actual fracture geometries under in situ stress conditions. Hence, the objective of this study is the validation of a fast Fourier transform (FFT)-based numerical approach for a circular granodiorite fracture considering stress-dependent normal closure. The numerical approach employs both purely elastic and elastic–plastic contact deformation models, which are based on high-resolution fracture scans and representative mechanical properties, which were measured in laboratory experiments. The numerical approaches are validated by comparing the simulated results with uniaxial laboratory tests. The normal stresses applied in the axial direction of the cylindrical specimen vary between 0.25 and 10 MPa. The simulations indicate the best performance for the elastic–plastic model, which fits well with experimentally derived normal closure data (root-mean-squared error $=9 \mu m$). The validity of the elastic–plastic model is emphasized by a more realistic reproduction of aperture distributions, local stresses and contact areas along the fracture. Although there are differences in simulated closure for the elastic and elastic–plastic models, only slight differences in the resulting aperture distributions are observed. In contrast to alternative interpenetration models or analytical models such as the Barton–Bandis models and the ”exponential repulsion model”, the numerical simulations reproduce heterogeneous local closure as well as low-contact areas (<2\%) even at high normal stresses (10 MPa), which coincides with findings of former experimental studies. Additionally, a relative hardness value of 0.14 for granitic rocks, which defines the general resistance to non-elastic deformation of the contacts, is introduced and successfully applied for the elastic--plastic model.
/ Download
5.  Perras, M.A., and D. Vogler Compressive and Tensile Behavior of 3D-Printed and Natural Sandstones, Transport in Porous Media, pp. 1-23, 2018. Abstract
The presented work compares the mechanical behavior from standard unconfined compressive strength and indirect tensile strength tests of natural sandstone and artificial sand-based specimens created by 3D additive manufacturing. Three natural sandstones of varying strength and stiffness were tested to capture a wide range of behavior for comparison with the 3D-printed specimens. Sand grains with furan and silicate binders, as well as, ceramic beads with silicate binder were 3D-printed by commercial suppliers.
The tensile and compressive strength, the stiffness, the crack initiation and the crack damage thresholds and the strain behavior were examined to determine if the mechanical behavior of the 3D-printed specimens is similar to natural sandstones. The Sand-Furan 3D-prints behaved the closest to the weak natural sandstone. The compressive strength to stiffness ratio, also known as the modulus ratio, and the compressive to tensile strength ratio of the 3D-printed Sand-Furan specimens were found to be similar to the natural sandstones tested in this study and literature values. The failed specimens composed of ceramic beads with silicate binder, both in compression and tension, showed fracture growth not commonly observed in natural specimens. The other 3D-printed specimens generally fractured in a similar manner to natural specimens, although several of the quartz sand with furan binder specimens showed fracturing behavior similar to high porosity natural specimens. Over all, using the commercially available quartz sand with furan binder 3D-print materials showed promise to be able to replicate natural rock specimen behavior.
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4.  Hobé, A., D. Vogler, M.P. Seybold, A. Ebigbo, R.R. Settgast, and M.O. Saar Estimating Fluid Flow Rates through Fracture Networks using Combinatorial Optimization, Advances in Water Resources, 122, pp. 85-97, 2018. Abstract
To enable fast uncertainty quantification of fluid flow in a discrete fracture network (DFN), we present two approaches to quickly compute fluid flow in DFNs using combinatorial optimization algorithms. Specifically, the presented Hanan Shortest Path Maxflow (HSPM) and Intersection Shortest Path Maxflow (ISPM) methods translate DFN geometries and properties to a graph on which a max flow algorithm computes a combinatorial flow, from which an overall fluid flow rate is estimated using a shortest path decomposition of this flow. The two approaches are assessed by comparing their predictions with results from explicit numerical simulations of simple test cases as well as stochastic DFN realizations covering a range of fracture densities. Both methods have a high accuracy and very low computational cost, which can facilitate much-needed in-depth analyses of the propagation of uncertainty in fracture and fracture-network properties to fluid flow rates.
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3.  Vogler, D., S.D.C. Walsh, E. Dombrovski, and M.A. Perras A Comparison of Tensile Failure in 3D-Printed and Natural Sandstone, Engineering Geology, 226, pp. 221-235, 2017. Abstract
This work investigates the possibility of replication of natural rock specimens, which can be used to analyze rock mechanical behavior by subjecting a number of identical specimens to tensile tests and a variety of analysis methods. We compare the properties of fractures generated in artificial sandstone specimens to those generated in natural sandstone specimens. Artificial sandstone specimens, created using 3D additive manufacturing printing processes, were subject to tensile failure using the Brazilian test method and the results from these tests were compared to results from Brazilian tests conducted on natural sandstones. The specimens included two distinct types of synthetic rock, unaltered from the manufacturers typical process, and three natural sandstones. For each test, the loading history to failure of the specimens were recorded and the failure mode was confirmed using digital imaging techniques. In addition, three dimensional images were taken of the fracture surfaces, which were then used to compare the geometric characteristics of all materials tested. The indirect tensile strength of the artificial sandstone specimens ranged between 1.0 and 2.8 MPa. Natural sandstone specimens with a wide range of indirect tensile strengths were tested for comparison. These included a strong sandstone, an intermediate sandstone, and a weak sandstone; which were found to have indirect tensile strength ranges of 10.5-25.5 MPa, 4.4-6.4 MPa, and 0.9-1.1 MPa, respectively. Digital image correlation confirmed that the artificial specimens generally failed in a tensile (mode I) fracture, similar to the natural specimens. Likewise, fracture surface roughness measures showed no clear distinction between weak natural and artificial sandstones. This indicates that there are distinct similarities between the fractures generated in the natural and artificial sandstones of comparable indirect tensile strengths. The three dimensionally printed sandstone specimens are shown to exhibit indirect tensile strength, surface roughness and crack propagation behavior which resembles a weak natural sandstone.
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2.  Vogler, D., S.D.C. Walsh, P. Bayer, and F. Amann Comparison of Surface Properties in Natural and Artificially Generated Fractures in a Crystalline Rock, Rock Mechanics and Rock Engineering, 50/11, pp. 2891-2909, 2017. Abstract
This work studies the roughness characteristics of fracture surfaces from a crystalline rock by analyzing differences in surface roughness between fractures of various types and sizes. We compare the surface properties of natural fractures sampled in situ and artificial (i.e., man-made) fractures created in the same source rock under laboratory conditions. The topography of the various fracture types is compared and characterized using a range of different measures of surface roughness. Both natural and artificial, and tensile and shear fractures are considered, along with the effects of specimen size on both the geometry of the fracture and its surface characterization. The analysis shows that fracture characteristics are substantially different between natural shear and artificial tensile fractures, while natural tensile fracture often spans the whole result domain of the two other fracture types. Specimen size effects are also evident, not only as scale sensitivity in the roughness metrics, but also as a by-product of the physical processes used to generate the fractures. Results from fractures generated with Brazilian tests show that fracture roughness at small scales differentiates fractures from different specimen sizes and stresses at failure.
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1.  Vogler, D., F. Amann, P. Bayer, and D. Elsworth Permeability Evolution in Natural Fractures Subject to Cyclic Loading and Gouge Formation, Rock Mechanics and Rock Engineering, 49/9, pp. 3463-3479, 2016. Abstract
Increasing fracture aperture by lowering effective normal stress and by inducing dilatant shearing and thermo-elastic effects is essential for transmissivity increase in enhanced geothermal systems. This study investigates transmissivity evolution for fluid flow through natural fractures in granodiorite at the laboratory scale. Processes that influence transmissivity are changing normal loads, surface deformation, the formation of gouge and fracture offset. Normal loads were varied in cycles between 1 and 68 MPa and cause transmissivity changes of up to three orders of magnitude. Similarly, small offsets of fracture surfaces of the order of millimeters induced changes in transmissivity of up to three orders of magnitude. During normal load cycling, the fractures experienced significant surface deformation, which did not lead to increased matedness for most experiments, especially for offset fractures. The resulting gouge material production may have caused clogging of the main fluid flow channels with progressing loading cycles, resulting in reductions of transmissivity by up to one order of magnitude. During one load cycle, from low to high normal loads, the majority of tests show hysteretic behavior of the transmissivity. This effect is stronger for early load cycles, most likely when surface deformation occurs, and becomes less pronounced in later cycles when asperities with low asperity strength failed. The influence of repeated load cycling on surface deformation is investigated by scanning the specimen surfaces before and after testing. This allows one to study asperity height distribution and surface deformation by evaluating the changes of the standard deviation of the height, distribution of asperities and matedness of the fractures. Surface roughness, as expressed by the standard deviation of the asperity height distribution, increased during testing. Specimen surfaces that were tested in a mated configuration were better mated after testing, than specimens tested in shear offset configuration. The fracture surface deformation of specimen surfaces that were tested in an offset configuration was dominated by the breaking of individual asperities and grains, which did not result in better mated surfaces.
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PROCEEDINGS REFEREED

4.  von Planta, C., D. Vogler, M. Nestola, P. Zulian, and R. Krause Variational Parallel Information Transfer between Unstructured Grids in Geophysics – Applications and Solutions Methods, PROCEEDINGS, 43rd Workshop on Geothermal Reservoir Engineering Stanford University, 2018. Abstract
State of the art simulations of enhanced geothermal systems are multi-physics simulations, where different physical properties are often being modeled on different geometries or grids. For example, for the simulation of fluids many prefer finite difference or volume formulations on structured grids, whereas in mechanics finite element formulations on unstructured grids are often preferred. To transfer information between these various geometries, we use a generic variational transfer operator, which has previously been introduced as pseudo-L2-projection. In this paper, we demonstrate that this transfer operator can be particularly useful for geophysics with three applications: First, in contact simulations we often have non-matching surfaces at the contact boundary. Here, the transfer operator acts as a mortar projection and we show with high resolution rock fracture geometries from the Grimsel Test Site in Switzerland how the variational transfer operator is used to formulate the contact problem. Secondly, we present a three-dimensional fluid-structure simulation, computing water flow between two rock surfaces and simultaneous deformation with an immersed boundary approach. In this method the solid, which is formulated on an unstructured grid, interacts with the fluid, formulated on a structured grid, by means of weakly enforced velocity constraints at the interface between fluid and solid. And lastly, we show how the transfer operator can be used to effectively solve contact problem between rock bodies. Using our operator, we can generate nested multilevel hierarchies, enabling us to solve the problem with optimal complexity, thus extending the possible size of simulations immensely.
3.  Deb, P., D. Vogler, S. Dueber, P. Siebert, S. Reiche, C. Clauser, R.R. Settgast, and K. Willbrand Laboratory Fracking Experiments for Verifying Numerical Simulation Codes, 80th EAGE Conference and Exhibition, pp. 1-4, 2018. Abstract
Carefully designed and well monitored experiments are irreplaceable when it comes to producing reliable data sets for a detailed understanding of physical processes, such as hydraulic fracturing. While such experiments provide insight into the governing physical processes, numerical simulations provide additional information on system behaviour by enabling a straightforward study of parameter sensitivity. In this study, we focus on both these aspects. We report on results from (1) a benchmark experimental facility for performing hydraulic fracturing experiments on large rock samples in the laboratory under controlled conditions and (2) numerical simulations of these experiments using programs, which, in future, may be used for designing hydraulic stimulation layouts. We conduct series of experiments in order to ensure reproducibility and accuracy of the measurements. This experimental data set is then shared with several research institutes to be used for verifying their simulation software. Results from the simulation provide further insight regarding parameters, which contribute to uncertainties during measurements. Detailed study of the sensitive parameters help us to improve our experimental set up further and to perform future experiments under even better controlled conditions.
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2.  Vogler, D., R.R. Settgast, C.S. Sherman, V.S. Gischig, R. Jalali, J.A. Doetsch, B. Valley, K.F. Evans, F. Amann, and M.O. Saar Modeling the Hydraulic Fracture Stimulation performed for Reservoir Permeability Enhancement at the Grimsel Test Site, Switzerland, Proceedings of the 42nd Workshop on Geothermal Reservoir Engineering Stanford University, 2017. Abstract
In-situ hydraulic fracturing has been performed on the decameter scale in the Deep Underground rock Laboratory (DUG Lab) at the Grimsel Test Site (GTS) in Switzerland in order to measure the minimum principal stress magnitude and orientation. Conducted tests were performed in a number of boreholes, with 3–4 packer intervals in each borehole subjected to repeated injection. During each test, fluid injection pressure, injection flow rate and microseismic events were recorded amongst others. Fully coupled 3D simulations have been performed with the LLNL’s GEOS simulation framework. The methods applied in the simulation of the experiments address physical processes such as rock deformation/stress, LEFM fracture mechanics, fluid flow in the fracture and matrix, and the generation of micro-seismic events. This allows to estimate the distance of fracture penetration during the injection phase and correlate the simulated injection pressure with experimental data during injection, as well as post shut-in. Additionally, the extent of the fracture resulting from simulations of fracture propagation and microseismic events are compared with the spatial distribution of the microseismic events recorded in the experiment.
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1.  Vogler, D., R. Settgast, C. Annavarapu, P. Bayer, and F. Amann Hydro-Mechanically Coupled Flow through Heterogeneous Fractures, Proceedings of the 41st Workshop on Geothermal Reservoir Engineering Stanford University, pp. SGP-TR-209, 2016. Abstract
Heterogeneous aperture distributions are an intrinsic characteristic of natural fractures. The presence of highly heterogeneous aperture distributions can lead to flow channeling, thus influencing the macroscopic behavior of the fluid flow. High-fidelity numerical simulation tools are needed for realistic simulation of fracture flow when such features are present. Here, focus is set on the role of mechanical fracture closure for fluid flow and appropriate simulation by a fully hydro-mechanically (HM) coupled numerical model. In a laboratory experiment, an artificial fracture in a granodiorite sample is created. During different sequential loading cycles, the development of fracture closure, contact area and contact stress are examined. Constant fluid flow rate injection into the center of the rough fracture is modelled to investigate the impact of fracture closure on the flow field and injection pressure. Results show that the numerical framework for heterogeneous fracture surfaces allows for reproducing experimental data of dry, mechanical tests at the laboratory scale, and it may offer advanced understanding and prediction of the behavior of reservoirs that are subject to high-pressure fluid injections.
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THESES

3.  Vogler, D. Hydro-mechanically coupled processes in heterogeneous fractures: experiments and numerical simulations, Dissertation ETH Zurich, 169 pp., 2016. Abstract
Enhanced Geothermal Systems (EGS), CO2-sequestration, oil- and gas reservoirs rely on an in-depth understanding of geomechanics and fluid flow in the subsurface to achieve production targets. In Switzerland, EGS are commonly targeted for deep basement formations of crystalline rock, as these are deep enough underground to provide high temperatures. In crystalline rock, fluid flow through fractures dominates transport processes, while mechanical behavior strongly depends on fracture topography and strength. This work focusses on fracture behavior in crystalline rock, such as granite and granodiorite, by investigating: (1) Differences in fracture topography linked to fracture size and nature; (2) Hydromechanically coupled processes in heterogeneous fractures in experiments on the laboratory scale; and (3) Hydro-mechanically coupled processes in heterogeneous fractures in simulations on the laboratory and field scale, supported by laboratory experiments. All rock specimens in this work are granite or granodiorite specimens obtained from the Grimsel Test Site (GTS), Switzerland. Fracture topography is studied by overcoring mode I and mode II fractures from core material and by subjecting intact specimens to Brazilian tests. This yields a range of fractures of various nature with sizes between 1 to 30 cm. Fracture topography is compared with the JRC, Z2 measure, fractal dimensions (Hausdorff and Box count dimension) and correlation functions (Two point correlation function and lineal path function) to quantify and compare roughness with a large range of parameters. Additionally, surface roughness is compared to specimen tensile strengths. Results show a clear distinction of natural shear and artificial tensile fractures, as measured with the Z2 measure. Fracture roughness appears to be linked to specimen size when comparing whole fracture sizes. Computing local roughness on small surface patches (e.g. 1 cm x 1 cm) yields smoother surfaces for large fractures, further indicating that fracture roughness is scale dependent and that this scale dependency can be traced down to scales significantly smaller than the whole fracture. The scale of the specimen has an influence on the probable fracture propagation path and therefore the tensile strength, which leads to different surface roughnesses of the induced tensile fracture. As specimen sizes increase, the tensile strength decreases and the fracture roughness increases. In summary, fractures of different nature and size can be distinguished by surface roughness measures, indicating that fracture origin has significant influence on surface topography. This is especially important, as fracture topography is linked to fracture conductivity and strength. Laboratory tests on granodiorite specimen were performed to investigate the relation of fluid flow rate, injection pressure, confining stress and fracture aperture during testing. Cylindrical specimen were overcored from natural tensile and shear fractures and subjected to a fluid pressure gradient across the fracture to sustain a constant flow rate. The specimens were tested in mated configuration and with shear offset in the fracture between 1 and 6 mm. Additionally, specimen fracture surfaces were scanned before and after testing to study the relationship of fracture transmissivity evolution during testing and surface deformation. Confining stress varied between 1 and 68 MPa for 5 to 10 cycles, yielding changes in transmissivity of up to three orders of magnitude. Shear offset of specimens lead to transmissivity increase of up to three orders of magnitude. Specimens experienced strongly damaged fracture surface and gouge production, which reduced transmissivity up to one order of magnitude for subsequent load cycles. While fracture surface roughness increased during testing, this effect was especially pronounced for specimens with shear offset. Almost all tests show hysteretic behavior during individual load cycles, indicating stress path dependent behavior of transmissivity. The experimental results qualitatively demonstrate and quantify mechanisms commonly encountered in EGS reservoir fractures. To further system understanding and predictive capabilities, a novel numerical model was tied into the GEOS framework to compute fully hydro-mechanically coupled processes in heterogeneous fractures. The model is compared against three experimental test sets investigating cylindrical granodiorite specimens with axial loads between 0.25 and 10 MPa for: (i) dry fracture closure; (ii) contact stress evolution in fractures during normal loading; and (iii) constant fluid flow rate injection into the fracture center. The non-linear behavior or fracture normal closure and fluid injection pressure increase with increasing axial load is replicated by the numerical model, by using the fracture aperture fields obtained from photogrammetry scans as model input. The numerical model captures contact stress evolution with axial load increase and shows a linear increase in contact area with axial load. Study of flow field simulations show an early onset of channeling, for axial loads as low as 2 MPa. Additionally, simulations of a field scale domain (100 m x 100 m x 40 m), with a 100 x 100 m fracture plane are performed. Pre-existing natural fractures were scanned, to use their aperture field to generate a synthetic aperture field for the fracture plane. In the next step, vertical stresses of 8.3 MPa, corresponding to the host rock of the fracture origin at the GTS are applied to the system. This yields the unique aperture field corresponding to the given stress state. Fluid is subsequently injected with constant pressure head into the fracture center with pressures between 0.01 and 8.7 MPa. While heterogeneous flow paths and pressure diffusion can be observed, the model additionally allows to observe heterogeneous fracture opening due to lowered effective normal stresses during injection. Further, the hydro-mechanically coupled analysis of the velocity and pressure field shows a deviation of the pressure distribution from linear diffusion for increased injection pressures, once hydro-mechanical contact between the fluid and the rock mass is established. Fluid pressure induced fracture opening is shown to strongly depend on aperture magnitude before injection and aperture magnitudes of the surrounding fracture region. Thereby, the model captures mechanical and hydraulic behavior of the laboratory tests, while providing unique insights for heterogeneous fracture behavior under compression and high pressure fluid injection. In summary, this work attempts to scrutinize heterogeneous fractures, and especially related hydro-mechanical processes. This is done by investigating possible bias by specimen fracture nature and size selection for testing. Hydro-mechanical processes are studied in experiments, which aim to replicate reservoir conditions, and showcase the impact of specific fractures, stress paths and gouge production. Finally, this work presents an approach to incorporate the observed phenomena in a numerical framework, which is tested against specifically designed laboratory tests. This work combines laboratory scale investigations by employing the framework to perform fully hydro-mechanically coupled simulations of a field scale fracture with heterogeneous aperture distribution, which yields quantitative results of fracture opening during high-pressure injection. The presented work thereby contributes to further understanding of fracture processes, which characterize behavior of Enhanced Geothermal Systems and other subsurface phenomena.
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2.  Vogler, D. A comparison of different model reduction techniques for model calibration and risk assessment, MSc Thesis University of Stuttgart, 62 pp., 2013. Abstract
Many engineering systems represent challenging classes of complex dynamic systems. Lacking information about their systems properties leads to model uncertainties up to a level where quantification of uncertainties may become the dominant question in modeling, simulation andapplication tasks. Uncertainty quantification is the prerequisite for probabilistic risk assessment and related tasks. The current work will present recent approaches for these challenges based on response surface techniques, which reduce massively the initial complex model. The reduction is achieved by a regression-like analysis of model output with orthonormal polynomials that depend on the model input parameters. This way, the model response to changes in uncertain parameters, design or control variables is represented by polynomials for each model prediction of interest. This technique is known as polynomial chaos expansion (PCE) in the field of stochastic PDE solutions. The reduced model represented by the response surface is vastly faster than the original complex one, and thus provides a promising starting point for follow-up tasks: uncertainty quantification, model calibration and probabilistic risk assessment. Obviously, a response surface can be constructed in different ways. Methods for constructing the response surface can demand only a minimum number of model evaluations, but as well may ask for many model evaluations to achieve a better quality of the involved projection integrals. The scope of the current work is to test and compare different integration rules, i.e., methods to choose the sets of parameter values for which the model has to be evaluated. To test and compare the different methods, their accuracy in uncertainty quantification, model calibration and risk assessment will be measured against brute-force reference computations based on the original model. As illustrative example, we consider a study from the field of CO2 storage in the subsurface.
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1.  Vogler, D. Investigation of transport phenomena in a highly heterogeneous porous medium, MSc Thesis Oregon State University, 70 pp., 2012. Abstract
This work focuses on solute mass transport in a highly heterogeneous two-region porous medium consisting of spherical low-hydraulic conductivity inclusions, embedded in a high-hydraulic conductivity matrix. The transport processes occuring in the system are described by three distinct time scales. The first time scale reflects the characteristic time for convective transport in the high-conductivity matrix. The second time scale reflects the characteristic time for diffusive transport in the low-conductivity inclusions. The third time scale reflects the characteristic time for convection within the inclusions. Two Péclet numbers can be defined that compare the time scales and provide qualitative insight into the net transport behavior in two-region media. To model this system, four different representations were developed: (1) a Darcy-scale model that involved direct microscale computation over the entire domain of the experimental system, (2) a direct microscale simulation computed on a simplified domain that had similar geometric parameters (e.g. volume fraction of inclusions) as the complete domain for the experimental system, (3) a volume averaged model (after Chastanet and Wood [2008]) which uses a constant mass transfer coefficient and (4) a volume averaged model which employs a time-dependent mass transfer coefficient. Two different experimental conditions were investigated: a high flow rate, and a low flow rate. Detailed understanding of the experimental system was developed, which led to accurate prediction of the system’s behavior for the higher flow rate. Accurate early time fit of the data was achieved for the experiment with the lower flow rate, while late time behavior between the models and experimental data diverged. Further investigations of the experimental system were conducted to examine possible sources of errors that could lead to an inaccurate description of the system’s properties. Additional mixing within the system, inhomogeneous distribution of the effective diffusion coefficient and imprecise initial estimates of the hydraulic parameters are all possible explanations for the inaccurate model representation of the system’s behavior for the lower flow rate case.
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