MICROMECHANICAL ROCK MODELS

Transcription

1 MICROMECHANICAL ROCK MODELS HEINZ KONIETZKY PROF. DR.-ING. HABIL. GEOTECHNICAL INSTITUT, TU BERGAKADEMIE FREIBERG, GERMANY 1 INTRODUCTION Micromechanical behavior of rocks is dependent on microstructure. Microstructure in turn is characterized by grain size and grain shape as well as pore size and shape. Also, orientation of grains and pores have influence. In addition we have to consider the physical strength and deformation properties of the grains including binder material and existing microcracks. All these components together define the macroscopic stress-strain relations, the damage evolution and the failure behavior. Numerical simulations at the grain size level can help to understand the complex behavior of rocks and to detect, which components have significant effect on the overall behavior. This article describes research which was performed at the Chair for Rock Mechanics at the Geotechnical Institute of TU Bergakademie Freiberg within the last few years using discontinuum approaches to simulate rocks and rock-like material at the grain size level. The performed research work concentrates on the following topics: Microcrack development (inter- and intra-granular fracturing) Lifetime prediction of rocks under static longterm loading Techniques to duplicate microstructure of rocks (stochastic approaches and digital image processing) Influence of microstructure on poroelasticity Micromechanical backanalysis of laboratory tests Topological analysis of micro structure This article gives an insight into some of the developed procedures and modelling results. 2 GEOMETRICAL MODEL SET-UP Three in principle different approaches were developed to set-up two-dimensional model geometries at the grain size level: Geometrical model set-up for Discrete Element models based on digital image processing of rock surfaces based on thin-section analysis Geometrical model set-up for Discrete Element models based on stochastic evaluation of rock structure at the grain size level Implementation of stochastically distributed microcracks into continuum models (not discussed within this paper) and Discrete Element models Figure 1 illustrates a procedure based on digital image processing. This procedure starts with digital color images of a rock surface, followed by a conversion into a gray-scale image. Different grayscale windows refer to different mineral components (grains) or pores. This digital grayscale image is superimposed by a regular mesh or Voronoi based polyhedra. Depending on the mesh resolution each grain or pore is covered by several or

2 many elements or Voronoi bodies. Exemplary, Figure 2 shows a Voronoi body based numerical model for granite, where each grain is composed of several Voronoi blocks. Each of these blocks can be either stiff or internally meshed. In case of internal meshing, the grains are deformable and will behave according to the specified visco-elasto-plastic constitutive law. Nevertheless, such a model is restricted in such a way, that crack propagation can occur only along the Voronoi block edges. Therefore, sufficient model resolution is necessary. An alternative to a deterministic approach is a stochastic one. In that case grains and/or pores are analyzed in terms of their size, shape and orientation. Based on these data corresponding distribution functions and correlations are obtained. This information is finally used to set-up stochastic equivalent models. Due to the stochastic (random) nature an arbitrary number of equivalent deterministic models can be created. Such a procedure allows to determine robustness and sensitivity of models. Figures 3 shows an example of grain shape and size characterization proposed by Heilbronner & Barett (2013). Based on this proposed scheme, the following procedure was developed. First a huge number of clumps or clusters is created in a fully random fashion based on a pre-defined number of spheres as illustrated in Figure 4. Different overlapping of balls at different angles lead to an unlimited number of clumps or clusters. Then, the parameters according to Heilbronner & Barett (2013) are used to classify all these clumps or clusters. Then, they are compared with grain shapes of the rock and those clumps and clusters, respectively, are selected which fit into the classification. Finally, they are adjusted in size and placed into the model specimen. Afterwards a correction in terms of adequate pore space distribution is necessary. Figure 5 illustrates a specific procedure, which starts with placing spheres inside the specimen according to number and size of grains. In a second step spheres are replaced by clumps to duplicate the grain shape according to the rock analysis. Figure 1. Scheme of model set-up based on digital image (right: gray-scale image, left: numerical model). Finally, this particle arrangement is superimposed to a Voronoi body structure with high resolution which gives the final grain structure. Figure 6 documents a particle based approach with clusters. In that case, the specimen is first filled with spheres. In a second phase, ellipses (2D) or ellipsoids (3D) according to the desired distribution functions of grain size and shape were placed inside the specimen in a random fashion. All spheres within one ellipse/ellipsoid were bonded and form a grain. Figure 2. Exemplary model set-up of a granite by means of Voronoi based clusters.

3 Figure 3. Grain characterization procedure developed by Heilbronner & Barett (2013) with indication of parameters. Figure 7 shows a further approach, which considers in addition to the grain structure also microcracks with different orientation and length at certain points along the edges of the Voronoi bodies. Figure 8 shows two models with focus on porosity. It is remarkable, that under compressive loading such models with high porosity show huge stress concentrations and pronounced force chains. In that case the internal structure is responsible also for strong local tensile stresses even if the outer loading is compressive. Shape, size and orientation of pores has also pronounced effect on poroelastic constants, like Biot s coefficient (Tan & Konietzky 2014). Figure 4. Randomly created clumps based on 5 spheres each. All these different procedures can also be combined to set-up sophisticated models considering complex microstructure. Extensive calibration of model parameters is necessary. This could be performed by inverse optimization on the basis of several laboratory tests in combination with additional measurements or use of already available data for minerals and grains, respectively. Calibration should comprise quiet different loading situations and should involve both, analysis of fracture pattern and macroscopic stress-strain relations. Also, it should be taken into account, that several parameters, like calculation of stress intensity factors, are meshdependent.

4 Figure 5. Specific procedure for set-up of Voronoi based DEM model in four steps as explained inside the text.

5 Figure 6. Sphere based model with clusters representing grains (Groh et al. 2011). Figure 7. Voronoi-based grain structure with randomly placed micro-cracks (white color).

6 Figure 8. Low and high porous specimen with randomly created pores according to pore analysis. 3 2-DIMENSIONAL TIME-DEPENDENT SIMULATIONS The models described above can be used to simulate time-independent and time-dependent behavior. If subcritical crack growth is taken into account, life-time prediction can be performed and time-dependent damage evolution can be investigated. Discontinuum based approaches consider crack propagation along boundaries of discrete elements. In case of subcritical crack growth the strength degradation equation for each element and microcrack i, respectively, based on Charles equation is expressed as: dj i P dt = λ i P [A I ( K Ii K Ic ) n I + AII ( K IIi K IIc ) n II ] (3) where J p is the strength parameter p (e.g. cohesion, tensile strength, friction coefficient etc.); λ P is a degradation constant; AI and AII are related to shares of tensile and shear fracture. The Charles-Hillig approach does not assume the existence of discrete microcracks, but describes a timedependent weakening of contacts: dj i P dt = (λ p i v 0 e u/rt )e Vσ i/rt (4) where V is the molar volume and σi is either the normal or shear stress depending on the considered fracture mode. Figure 15 shows several damage stages as function of time for a granite sample under uniaxial load and Figure 16 shows the fracture pattern in more detail in a zoomed window of the sample for several points in time. Crack propagation and fracture opening is visible.

7 Figure 9. Damage pattern of granite specimen under constant uniaxial loading with ongoing time (Chen & Konietzky 2014). Figure 10. Detailed view on damage development (see figure 9). Figure 11. Observed fracture pattern on slotted discs tested under uniaxial compression, but different orientation. Figure 12. Simulated fracture pattern on slotted discs tested under uniaxial compression, but different orientation (compare with Figure 11).

8 Figure 13. Lifetime versus driving stress ratio: laboratory test results from Lajtai & Bielus (1986) and own simulation results (Chen & Konietzky 2014). Figure 17 shows fracture pattern of slotted Brazilian granite samples (discs). Different orientation of slot against direction of uniaxial compression leads to quite different fracture pattern involving pure tensile fracturing and mixed-mode fracturing including wing-crack propagation. Figure 18 shows the corresponding numerical simulations based on a Discrete Element approach at the grain size level. This procedure was extended by incorporation of time-dependent failure modelling as explained above to predict lifetime as shown in Figure 19. More detailed explanations are given by Chen et al. (2014, 2015, 2016). 4 THREE-DIMENSIONAL MODELLING AND STRUCTURAL CHARACTERIZATION Increasing computational power including parallel processing allows three-dimensional simulations. The above mentioned techniques to set-up grain based models were extended to third dimension. Exemplary, Figures 14 and 15 show some three-dimensional DEM-models. To set-up Voronoi based models an interesting technique was developed, which starts with packages of spherical objects (weighted point process) followed by the construction of Voronoi bodies around this points (spheres). Figures 16 and 17 illustrate the principal procedure to construct such Voronoi bodies. Based on manipulation of input data inhomogeneous and anisotropic Voronoi structures can be created as shown in Figure 18. Experience has shown, that detailed microstructure has enormous influence on strength and deformability of material. Often more general values like density, porosity or coordination number are not sufficient to characterize the material. In case of sphere-based models more complex functions which considers linear and spherical contact distribution functions, pair correlation functions or Ripley s K-function are useful tools to explain differences in material behavior (Wagner et al. 2010).

9 Figure 14. Three-dimensional models based on spherical elements. Figure 15. Three-dimensional models based on polyhedral.

10 Figure 16. Example: Voronoi diagram with Laguerre decomposition (principle in 2D). Figure 17. Example: Voronoi diagram with Laguerre decomposition (principle in 3D). Figure 18. Example: Voronoi bodies with inhomogeneous (above) and anisotropic (below) structure.

11 Figure 19. Stress-strain, permeability and damage evolution for a salt sample under uniaxial compressive stress using a three-dimensional Voronoi based model (Wagner et al. 2012) Exemplary, Figure 19 illustrates the complex hydro-mechanical behavior of a coarse-grained salt sample under uniaxial compression. The diagram above shows vertical stress vs. vertical strain and volumetric strain, respectively. The volumetric strain curve indicates volumetric compaction at the beginning followed by dilation caused by increasing microcracking at a later stage. The diagram below shows evolution of damage and permeability. On the right hand side the damage pattern in form of developed microcracks are shown for 3 different stages in parallel to the hydraulic pressure distribution inside the sample. 5 CONCLUSIONS Depending on the modelling task, scale and material either continuum (e.g. Konietzky et al. 2009; Li & Konietzky 2014, 2015) and/or discontinuum based approaches are suitable to simulate time-dependent or timeindependent damage development of rocks at the grain size level. The modelling reveals, that grain size distribution, grain shape distribution, grain orientation distribution as well as mineral content and binder between grains in terms of their mechanical properties play important role for lifetime and mechanical behavior in general. The same holds for microcracks and pores, respectively. Especially, Discrete Element based micromechanical models using Voronoi bodies or clusters/clumps based on spheres are very promising to consider micromechanical structure in an explicit manner. Both, inter-granular as well as intra-granular fracturing can be modelled. Use of modelling strategies, which consider grain, pore and microcracks in an explicit manner avoids complicated constitutive relations and reduces the number of model parameters. On the other side it allows to apply fundamental physical relations to describe complex material behavior, like explained for instance by Stahl & Konietzky (2011) for broken rock material (ballast). New mathematical approaches which consider internal network structure of solids are promising tools to explain the material behavior at the microscale.

12 ACKNOWLEDGMENT Several scientific co-workers and PhD students (see also references) working at the Chair for Rock Mechanics at TU Bergakademie Freiberg have been intensively involved into the above mentioned scientific work within the last few years. Their contributions are highly appreciated. REFERENCES Chen, W. & H. Konietzky (2014). Simulation of heterogeneity, creep, damage and lifetime for loaded brittle rocks. Tectonophysics, 633, Chen, W., H. Konietzky, S.M. Abbas (2015). Numerical simulation of time-independent and time-dependent fracturing in sandstone. Engineering Geology, 193(2), Chen, W., H. Konietzky, X. Tan & T. Frühwirt (2016). Pre-failure damage analysis for brittle rocks under triaxial compression. Computers and Geotechnics,74, Groh, U., H. Konietzky, K. Walter & M. Herbst (2011). Damage simulation of brittle heterogeneous materials at the grain size level. Theoretical and Applied Fracture Mechanics, 55, Heilbronner, R. & S. Barett (2013). Image analysis in earth sciences microstructure and textures of earth materials. Springer, Heidelberg, Berlin. Konietzky, H., A. Heftenberger & M. Feige (2009). Life time prediction for rocks under static compressive and tensile loads a new simulation approach. Acta Geotechnica, 4, Lajtai, Z.E. & P.L. Bielus (1986). Stress corrosion cracking of Luc du Bonnet granite in tension and compression. Rock Mechanics Rock Engineering, 19, Li, X. & H. Konietzky (2014). Simulation of time-dependent crack growth in brittle rocks under constant loading conditions. Engi neering Fracture Mechanics, 119, Li, X. & H. Konietzky (2015). Numerical simulation schemes for time-dependent crack growth in hard brittle rock. Acta Geotechnica, 10, Stahl, M. & H. Konietzky (2011). Discrete element simulation of ballast and gravel under special consideration of grainshape, grain size and relative density. Granular Matter, 4, Tan, X. & H. Konietzky (2014). Numerical study of variation in Biot s coefficient with respect to microstructure of rocks. Tectonophysics, 610, Wagner, A., M. Hütter & D. Stoyan (2010). More on the microstructural characterization of dense particle gels. Journal of the Euro pean Ceramic Society, 30, Wagner, A., H. Konietzky, K. Otparlik, C. Müller & C. Lerch, (2012). Microstructure simulation based on discrete element models theory and application in salt mechanics. Publ. Geotechnical Institute of TU Bergakademie Freiberg, H. Konietzky (Ed.), ,