Evolution of shear bands and cracks in multi-stage triaxial tests with water-saturated sandstone: A study of micro-tectonics with a fractal perspective

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Highlights

  • Mohr-Coulomb-condition as a limit state is confirmed for fractal rock structures.

  • Multi-step triaxial tests are analogue models for crustal tectonic processes.

  • No causal relation was found between pore fluid pressure and solid pattern changes.

  • Varying pore-and solid-pressures cause generation and dwindling of critical phenomena.

  • Quasi-static and kinematic critical phenomena form random and chaotic sequences.

Abstract

A series of multi-stage triaxial compression tests revealed quite chaotic reactions of water-saturated Permotriassic sandstones on changing pore water pressures. Not before focussing the various test stages with respect to occurring critical phenomena, which strike the fabric structures due to their fractal nature during any triaxial testing, eventually an approach became tractable: Pore systems with cracks and shear bands can be widened in a quasi-static way, but only up to a stability limit. Thereafter the fabric collapses with a seismogenic contraction of the pore system causing a rise of the pore water pressure. Changes of structural patterns occur as formation either of fissures up to collapse with acoustic emission, or of a rather fractal pattern of crossing shear bands generated without audible noise but with fine grit due to comminution. Dilation and contraction of the pore system are visible from overall volume changes. Both modes of pattern formation can be understood as critical phenomena of the mineral solid. Scale-independent features of these changes of fabric patterns can be transferred to the meso- and macro-scales in the field and in the lithosphere, respectively. However, questions of scale-dependent features related to cohesion should be further investigated.

Introduction

Changes of pore water pressure are often related causally with seismic activity in the earth's upper crust (Segall, 1989). Hydraulic fracturing operations in boreholes are connected with seismicity of mostly small magnitudes (Pearson, 1981). The depletion of oil and gas reservoirs generates seismic activity with a delay of months, years or tens of years and with small to medium magnitudes (Geertsma, 1973; Teufel, 1996; Dahm et al., 2015; Hettema et al., 2000). In both cases a change of pore water pressure apparently induces seismic events, also called man-made earthquakes (comp. “HiQuake”: inducedearthquakes.org). These events are not yet predictable, neither their intensity nor their temporal evolution. Therefore, cause and effect of pressure changes of pore water in rocks should be studied by means of laboratory tests in order to obtain a better understanding of interactions between pore water and solid. Thus sandstone samples in triaxial setups act as analogue models of brittle crustal rocks.

Rock formations in the lithosphere display patterns of scale-independent features, particularly joints and faults, so that the description as a continuum is not appropriate. They have a pore system (πορος = passage) so that spatial and temporal distributions of mechanical quantities are not differentiable or even discontinuous. Therefore, while rock is matter, it is not a material (Gudehus, 2019) and should not be considered as a continuum. Müller (1963) called a rock formation a discontinuum because of its differently oriented planar structural elements (joints). The discontinuous character impedes the understanding of fracture processes because any natural rock is already more or less fractured. Consequently, failure is an inadequate notion for the lithosphere whose ongoing fracture is a characteristic natural feature. The permanent fracturing processes within the lithosphere are linked with critical phenomena which cause features of spatial and temporal fractality. Moreover, it is impossible to define strength for a natural (i.e. discontinuous) rock formation, as this would require arbitrary representative volume elements and boundary conditions.

In his overview “Dynamics of Multiscale Earth Systems” Neugebauer (2003) questions the representativeness of averages for rock formations and asks for new kinds of constitutive relations. The present paper is an attempt in this direction. Constitutive relations of rock cannot be local like those of materials so that the notion of a representative volume element is questionable. Therefore, testing results with rock samples are not likewise representative as with materials. This feature also concerns the pore water (fluid) and its interaction with the mineral solid. The randomness of spatial fluctuations in rocks is not mild (Gaussian) but wild with ‘fat tails’ so that one extreme case matters as much as all the smaller ones together (Mandelbrot, 1982; Sornette, 2000; Gudehus, 2019).

The world stress map project WSM (Heidbach et al., 2018) provides a detailed picture of the stress distribution in the earth's crust, which exhibits a dependence on the chosen mesh size. Nevertheless, there are constitutive features that are scale-independent and so can be detected in the lab. Such scale-independent characteristics concern lithosphere sections independently of their spatial extension, and they govern the evolution of tectonic pore systems, i.e. of patterns of cracks and shear bands in samples or joints and faults in the lithosphere. With its wild randomness the causal dependence of 'reactions' on 'actions' remains fuzzy and can be expressed only by means of probabilistic correlations (Sornette, 2000).

These structural features evidently concern also the permeability. Darcy permeability requires laminar flow. Even in the linear range of the Navier-Stokes equations, the flow through fractal pore systems is not laminar due to successive hydraulic bifurcations. Therefore the diffusion process of pore water, i.e. its transition to an equilibrium after a disturbance (like with a sponge under water after placing a weight on top of it) is anomalous, i.e. not only determined by Gaussian thermal oscillations. Nevertheless, we use a measure of Darcy-like permeability to compare the overall behaviour of specific samples.

We state therefore hypothetically three constitutive features of sandstone by means of triaxial tests, which can be scaled up based on the fractal nature of geo-matter.

  • -

    Sandstone is not a continuum, but a discontinuum with non-differentiable distributions of mechanical properties; that is also valid for apparently homogenous core samples without visible cracks due to the fractal pore system.

  • -

    Sandstone displays scale-independent constitutive properties that may be transferred from the lab scale to local, regional or global scales, for which quasi-static and kinetic evolutions are known as slow and fast tectonics (τεκτων = builder and mover; e.g. tectonic plate movements/fault displacements).

  • -

    The flow of pore water (or other fluids) through tectonic pore systems cannot be captured with Darcy's law and the classical mass balance (often called continuity equation) as spatial and temporal derivatives do not exist in a classical sense with fractal distributions.

The fractal nature of rock can be attributed to critical phenomena; their occurrence is typical during evolutions of fractal processes (Sornette, 2000). Criticality means spontaneous critical phenomena, which generally are connected with fractality and wild randomness, i.e. stochastic fractality in MANDELBROT's (1982) sense. Differently from critical phenomena with gas or liquid those with rock leave back relics with spatial fractality, so successions of critical phenomena are complex and multi-fractal. That also means the absence of normal Gaussian fluctuations of properties, and the possibility of spontaneous changes of the system without additional energy supply. The relics of former critical phenomena in the fractal pore system of rock influence subsequent evolutions including pressure changes and flow of the pore water.

We argue that rock samples in triaxial devices constitute micro-tectonic analogues of lithosphere sections (field scale). In other words, using adequate laws of similarity an upscaling from micro-tectonics to macro-tectonics can be achieved by means of scale-independent constitutive features despite or just because of fractality. These are working hypotheses, which we endeavour to justify in the present paper.

We explain our triaxial testing procedure by means of notions and definitions that deviate from those of material sciences (Section 2). The results (Section 3) are subdivided in experiments with pore water flow (Section 3.1), controlled changes of injection pressure of pore water (Section 3.2) and an evaluation of injection pressure changes during unchanged total pressures (Section 3.3). Hydraulically induced fabric changes, apparently initiated by changes of pore water pressure, are evaluated by means of externally detectable evolutions (Section 3.4). Different types of fabric pattern formation are characterized by volume changes and visible changes at the cylindrical surface (Section 3.5). As a result, an overall collapse criterion is proposed for repeatedly stressed sandstone samples (Section 3.6). The conclusion with discussion (Section 4) is an approach towards a comprehensive view on sandstone with fractal pore systems and with an evolution of patterns that are related with critical phenomena in this kind of geo-matter.

Section snippets

Triaxial testing setup and procedure, overall quantities, samples

The triaxial testing program was carried out with fluid-saturated rock samples in a 5 MN load frame (Fig. 1). The test system contains an axial force measurement cell with 1800 kN capacity in a vessel with 70 MPa confining pressure capacity. The axial force, the confining pressure and the injection pressure of pore water at the endplates are servo-hydraulically controlled. The servo-hydraulic testing system Dion-Pro® enables alternatively the control of force or displacement of the axial piston

Pore water flow through sandstone samples

The permeability of sandstone samples is investigated by permeability tests of BT sandstones in the triaxial device. The pore water flow occurs between the endplates during their triaxial testing at specified overall pressure conditions. The test stages with permeability tests are specified by σ̂3 and p̂f (Menezes, 2019). As indicated in Section 2, the overall permeability k̂f is determined as the asymptote of the apparent permeability k̃f, which exceeds k̂f initially due to compression and

Conclusion, discussion and outlook

Rock is matter and not material in a continuum. Spatial, temporal and episodic fractality is a basic property of geo-matter. Their fractal nature is due to critical phenomena and relics of the same (Gudehus, 2019). The fractal nature of geo-matter implies scale-independent features. Natural faults, joints, shear bands and fissures display scale-independent characteristics, as they occur in outcrops, boreholes or regional sections. They can be investigated by means of triaxial tests with

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We thank Gerd Gudehus, Karlsruhe Institute of Technology, for intensive discussions, helpful explanations and abundance of patience during the development of this paper.

For discussions and theoretical support we also thank our collaborating teams in the research projects 776 of DGMK and SUBI of the BMBF program GEO:N.

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