Present-day stress inversion from a single near-surface fault: A novel mathematical approach
Introduction
Stress is a fundamental factor controlling rock deformation and, therefore, knowledge of stress states is essential for a complete understanding of past and contemporary tectonic processes. Crustal stress in the geological past can be interpreted using routine methods of palaeostress analyses which rely on the inversion of sets of fault slip data (Fry, 1999; Yamaji, 2000; Melichar and Kernstocková, 2010; Tranos, 2015). Palaeostress analyses are most commonly used to understand orogen-scale deformation (Vojtko et al., 2010) although the same approaches can also be applied to other phenomena such as gravitational slope failures (Baroň et al., 2011, 2013, 2017). Comprehensive reviews on different types of paleostress analyses have been presented by, for example, Angelier (1994) and Celerier et al. (2012). In contrast, contemporary tectonic stress is usually estimated on the basis of earthquake focal mechanisms (e.g. Hardebeck and Michael, 2006) or on the basis of strain recorded during mining, construction, tunneling, and drilling operations (Zoback et al., 2003). It is also possible to estimate recent stress states in the shallow crust on the basis of, for example, overcoring, borehole breakouts, and hydraulic fracturing (Zang and Stephansson, 2010). Much information about recent global tectonic stresses has been compiled in the framework of the World Stress Map Project (Heidbach et al., 2016).
Here presented novel numerical approach uses high resolution three dimensional displacement data from a single fault near the ground surface in order to determine the contemporary stress states. The stimulus for developing this method was provided by our fault activity observations at seven sites in the Eastern Alps since 2013 (Baroň et al., 2016). The whole mathematical approach is demonstrated on a distinct fault reactivation event on 7 November 2014 in Obir Cave near the Periadriatic Fault.
Section snippets
Input data
The input data for the proposed method are orientation of the fault surface represented by the down directed fault normal n = [nx, ny, nz]T and the vector of displacement between the particular fault blocks P = [Px, Py, Pz]T. The vector P reflects displacement of the hanging wall relative to the footwall (Fig. 1a). Three dimensional displacement between two fault planes in all three orthogonal directions can be measured using a range of instruments spanning from traditional ones such as the
Hypothesis and theoretical background
The stress state at a given point is described by the second order stress tensor, [Tσ], which can be expressed either by the orientation and magnitudes of the principal stresses or in the form of a symmetric matrix:where σxx, σyy, σzz are the normal stresses acting on planes perpendicular to the x, y, and z axes and σxy, σxz, σyz are the components of the shear stresses acting on these surfaces. Our hypothesis states that it is possible to compute the reduced
Mathematical solution
The second and the third conditions simplify then the stress tensor to the form:
Eq. (1) can be modified using the simplified form of the stress tensor (Equation (4)) to the matrix form:and in combination with Eq. (2) it is possible to derive the equations for all non-zero components of the stress tensor:
Numerical example
To demonstrate the numerical approach outlined above, an example from the real fault-displacement event is presented using input data recorded in Obir Cave (Fig. 3) near the Periadriatic Fault in the Eastern Alps. The monitored sinistral NNE-SSW striking fault is conjugated to the major dextral WNW-ESE striking Periadriatic Fault within a broad transpressional duplex. The investigated fault is exposed for approximately fifty meters at a shallow depth beneath the ground surface. The orientation
Discussion and conclusions
The proposed numerical approach enables computation of the stress tensor on the basis of a single fault reactivation event using only the orientation of the fault surface and the three dimensional vector of the fault movement as the input. To determine the stress tensor with six independent variables from the fault displacement data, six independent equations should be defined. For this purpose, we adopted the classical approaches of the reduced stress tensor and Anderson's theory (Equations (2)
Acknowledgements
We thank Matt Rowberry, Christa Pfarr and Bernhard Grasemann for their constructive reviews and proofreading of this manuscript. We are also indebted to Lukas Plan, Ivanka Mitrovic, Christian Varch, Norbert Kucher, and Harald Langer for their assistance in Obir Cave. Last but not least, we would like to thank Toru Takeshita for editorial assistance and anonymous reviewers for their constructive reviews. Data collection was financed by the Austrian Science Fund (FWF Project P25884-N29
References (35)
- et al.
Perturbation of stress and oceanic rift extension across transform faults shown by earthquake focal mechanisms in Iceland
Earth Planet Sci. Lett.
(2004) - et al.
Paleostress analysis of a gigantic gravitational mass movement in active tectonic setting: the Qoshadagh slope failure, Ahar, NW Iran
Tectonophysics
(2013) - et al.
Can deep seated gravitational slope deformations be activated by regional tectonic strain: first insights from displacement measurements in caves from the Eastern Alps
Geomorphology
(2016) - et al.
Stress field reconstruction in an active mudslide
Geomorphology
(2017) - et al.
Inferring stress from faulting: from early concepts to inverse methods
Tectonophysics
(2012) Striated faults: visual appreciation of their constraint on possible paleostress tensors
J. Struct. Geol.
(1999)- et al.
Favoured states of paleostress in the Earth's crust: evidence from fault-slip data
J. Struct. Geol.
(2006) - et al.
The instrumental resolution of a moiré extensometer in light of its recent automatization
Measurement
(2016) TR method (TRM): a separation and stress inversion method for heterogeneous fault-slip data driven by Andersonian extensional and compressional stress regimes
J. Struct. Geol.
(2015)The multiple inverse method: a new technique to separate stresses from heterogeneous fault-slip data
J. Struct. Geol.
(2000)
Determination of stress orientation and magnitude in deep wells
Int. J. Rock Mech. Min. Sci.
The Dynamics of Faulting
Sur l’analyse de mesures recueillies dans des sites faillés: L’utilité d’une confrontation entre les méthodes dynamiques et cinématiques. Comptes rendus hebdomadaires des séances de l'Académie des sciences. Série D
Sci. Nat.
Fault slip analysis and palaeostress reconstruction
Palaeostress analysis of a giant Holocene rockslide near Boaco and Santa Lucia (Nicaragua, Central America)
Large landslide stress states calculated following extreme climatic and tectonic events on El Hierro, Canary islands
Landslides
The mechanics of oblique slip faulting
Geol. Mag.
Cited by (9)
Three large prehistoric earthquakes in the Eastern Alps evidenced by cave rupture and speleothem damage
2022, GeomorphologyCitation Excerpt :A hypogene origin due to CO2-rich or thermal waters is assumed with the minimum age for the presumed origin of the cave system provided by the oldest speleothems, which formed about 400 ka ago; as the caves have formed most probably near the local erosional level about 500 m below their recent position, their age of few million years can be estimated (Spötl et al., 2017). The cave is an excellent natural laboratory with research focused on different aspects of speleothem geochemistry and petrology (Smith et al., 2009; Fairchild et al., 2010; Dredge et al., 2013; Wynn et al., 2018), present-day fault kinematic activity (Baroň et al., 2019a), present-day crustal stress variations (Sokol et al., 2018; Baroň et al., 2019b), and natural electromagnetic radiation associated with seismotectonic activity (Trčka et al., 2017). We used a 1 m resolution digital terrain model, i.e., a model without vegetation cover, of the examined cave surroundings derived from airborne LiDAR (courtesy of the Government of Carinthia) to understand the broader tectonic context of the cave-surface ruptures.
Gravitational and tectonic stress states within a deep-seated gravitational slope deformation near the seismogenic Periadriatic Line fault
2019, Engineering GeologyCitation Excerpt :Therefore, we are able to simplify the stress tensor and derive equations for the calculation of reduced stress tensor of Angelier (1994), i.e. the relative magnitudes of the principal stresses σ1, σ2 and σ3, their orientations and shape parameter of the stress ellipsoid. For full mathematical description of the method, we refer to the original paper by Sokol et al. (2018). We focused our analysis on two events, one in the summer of 2014, the second in winter of 2014/2015.
Present-day kinematic behaviour of active faults in the Eastern Alps
2019, TectonophysicsCitation Excerpt :All these approaches, however, cannot fully describe the character and nature of particular fault kinematic behavior in 3D. This may only be done using three dimensional extensometers with micrometric resolution to fully characterize the kinematic behaviour of the respective fault blocks over a longer interseismic period (e.g., Košťák and Avramova-Tačeva, 1988; Košťák et al., 1992; Stemberk et al., 2003; Šebela et al., 2010; Briestenský et al., 2015, 2018; Rinaldi-Montes et al., 2017; Blahůt et al., 2018; Sokol et al., 2018). This could be important especially for the fault behavior during an interseismic cycle, when displacements along faults are very small.