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Knowledge of the in situ stress state of rock mass is fundamental for engineering, geological and geophysical applications. In situ stress state determination requires in principle the evaluation of the three principal stresses and the related principal directions, but it is widely recognized in the literature that the maximum horizontal stress is the most difficult component to accurately estimate. In the context of borehole methods, this paper proposes a step-by-step analytical procedure to estimate some bounds to the maximum horizontal stress, starting from a geomechanical description of the rock and relying on information generally available in the engineering practice. The procedure is divided in substeps, each one requiring additional information about the mechanical properties of the rock and on the geometrical properties of the failed portion of rock: more information available implies a lower uncertainty on in situ stress estimate. Furthermore, since the proposed procedure is analytical, it allows a complete and very easy implementation in a spreadsheet. The aim of the work is thus to provide a rigourous but simple analytical tool that can be used in engineering practicte to estimate some bounds to the maximum horizontal in situ stress state. The approach is finally validated by means of both numerical simulations, performed with a sophisticated numerical tool, and experimental field data coming from the literature.

We present a simplified model of damaging porous material, obtained through consistent linearization from a recursive-faulting material model described in (Pandolfi et al. 2016). The brittle damage material model is characterized by special planar micro-structures, consisting of nested families of equi-spaced frictional-cohesive faults in an otherwise elastic matrix material. The linear kinematics model preserves the main microstructural features of the finite kinematics one but offers a far better computational performance. Unlike models commonly employed in geo-mechanical applications, the proposed model contains a small number of parameters, to wit, two elastic moduli, three frictional-cohesive parameters, and three hydraulic response parameters, all of which having clear physical meanings and amenable to direct experimental measurement through standard material tests. The model is validated by comparison to triaxial hydro-mechanical experimental data. Despite the paucity of material constants, the salient aspects of the observed behavior are well captured by the model, qualitatively and quantitatively. As an example of application of the model, we simulate the excavation of a borehole in a rocky embankment.

Knowledge of the in situ stress state in rock and soil deposits is very important in many problems in civil, mining and petroleum engineering and energy development, as well as in geology and geophysics. The prediction of the response of rock masses interacting with underground structures is highly influenced by the stress field. For example, as pointed out in [1], in civil and mining engineering, in situ stresses control the distribution and magnitude of the stresses around underground openings such as tunnels, mines, shaft or caverns. Stress concentrations in the excavation walls may be large enough to overstress the rock, mobilize the strength of the rock mass and induce failure. On the other hand, tensile stresses in excavation walls may open existing fractures or create new ones which could result in block stability problems. An exact prediction of in situ stress acting over a rock mass, together with its spatial variation, is a very complex topic; the current stress state is a mixed consequence of tectonic conditions and of mechanical effects due to local thermochemo-hydraulic conditions. Due to the complex nature of rocks and rock masses, the stress field is rarely homogeneous and also its time evolution can be significant within a geological formation. Stress state is a symmetric second-order tensor and so it is defined by six independent components, e.g. the three principal stresses and the three principal directions. Stresses in rocks cannot be measured directly and can only be inferred by disturbing the rock. Amadei and Stephansonn [2] presented a detailed summary of the available sources of information from which it is possible to obtain the in situ stress state, involving hydraulic methods (i.e. hydraulic and sleeve fracturing), relief methods, jacking methods, strain recovery methods, borehole failure methods as well as fault-slip data analysis and earthquake focal mechanisms. Zoback [4] proposed an overview of a possible strategy to characterize the stress field: the vertical stress can be determined from the equilibrium in the vertical direction, i.e. by integration of the density logs, while observations of the geometrical arrangement of drilling-induced tensile fractures are an effective way to check whether the vertical stress is a principal stress. The orientation of the other principal stresses can be determined from wellbore observations, recent geologic observations and earthquake focal mechanisms, as well as from stress recovery methods. The magnitude of the minimum principal stress can be estimated from the analysis of hydraulic fracturing and leak-off tests, while the pore pressure can be either measured directly or estimated with some caution from geophysical logs or seismic data. In this paper the vertical stress is assumed to be a principal stress, so that the remaining two directions are supposed to be horizontal. This assumption is reliable for non-active regions or regions already relaxed from the previous tectonical stress. As pointed out by Bell [3], the free surface of sedimentary basins is generally horizontal, so that the principal stress directions are, to a good approximation, horizontal and vertical. From a practical point of view, if the vertical direction is assumed to be a principal one, it is sufficient to know just a horizontal principal direction, since the remaining one is orthogonal to the plane on which the other two are lying. Throughout the paper, Sv will represent the principal vertical in situ stress, SH represents the maximum horizontal in situ stress and Sh represents the minimum horizontal in situ stress, whose magnitude can be evaluated through hydraulic fracturing or leak off tests (except for reverse faulting regimes). Assuming that Sv and Sh are known, the aim of the note is to present a methodology for the definition of some boundaries for