On the Stability of Underground Caves in Calcareous Rocks Due to Long-Term Weathering
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Acta Geotechnica manuscript No. (will be inserted by the editor) On the stability of underground caves in calcareous rocks due to long-term weathering Miguel A. M´anica · Matteo O. Ciantia · Antonio Gens Received: date / Accepted: date Abstract This paper addresses the problem of the sta- Abbreviations bility of structures on calcareous rocks due to long- term weathering processes. The case study consists of a BVP Boundary value problem building resting on a calcarenite rock formation where CHM Chemo-hydro-mechanical two abandoned man-made caves exist directly under GDP Grain dissolution process the structure. The boundaries of the caves were ex- HMC Hyperbolic Mohr-Coulomb posed to a slightly acidic environment inducing time- LTD Long-term debonding dependent weathering. Analyses were performed follow- STD Short-term debonding ing a semi-decoupled approach, where the weathering process, driven by a reactive transport mechanism, was first solved and its results were fed to the mechani- List of symbols cal problem which hence accounted for the spatial and B Parameter in f temporal evolution or rock damage. For the mechanical d CaCO Calcium carbonate species in the solid phase problem, a nonlocal constitutive model was employed 3(s) D Isotropic diffusion coefficient for the objective simulation of localised deformations. Relevant outcomes are obtained regarding the evolution Dij Diffusion tensor of the structure's stability and about the importance of E Young's modulus + regularising the finite element solution in the presence H3O(aq) Acid ions of brittle materials. J2 Second invariant of the deviatoric stress ten- sor K Darcy isotropic permeability Keywords weathering · calcareous rock · stability · K1, K2 Reaction rate constants nonlocal plasticity · localisation M Mass NG Number of Gauss points Sr Degree of saturation Miguel A. M´anica Sr,cr Minimum Sr for all the depositional bonds Institute of Engineering, National Autonomous University of to suspend Mexico, Mexico City, Mexico Y Weathering function E-mail: [email protected] Ydis Component of the weathering function for Matteo O. Ciantia the LTD process School of Science and Engineering, University of Dundee, Y Component of the weathering function for Dundee, UK sus E-mail: [email protected] the STD process a Constant controlling the curvature of the Antonio Gens φ Department of Civil and Environmental Engineering, Univer- hyperbolic hardening function sitat Polit`ecnica de Catalunya, Barcelona Tech, Spain bc Softening rate for the cohesion and tensile E-mail: [email protected] strength 2 Miguel A. M´anicaet al. bφ Softening rate for the friction angle 1 Introduction c∗ Asymptotic cohesion ∗ cini Initial asymptotic cohesion In contact with atmospheric conditions, rocks are sub- ∗ cini,uw Initial asymptotic cohesion for ξdis = 0 jected to weathering: a complex physicochemical pro- f Yield function cess inducing continuous changes in their properties fd Function defining the shape of f in the de- and composition and, ultimately, leading to the com- viatoric plane plete disintegration of the rock. From an engineering ls Length scale parameter standpoint, weathering induce a continuous reduction m Parameter in fd of the mechanical properties having major implications n Effective porosity on the stability of structures (Castellanza et al., 2018; p Mean stress Ciantia et al., 2018). pH Potential of the hydrogen Calcarenites are particularly susceptible to weath- pt Isotropic tensile strength ering. They can lose instantly up to 60% of their dry ptini Initial isotropic tensile strength strength when their pores are filled with water and, if they remain saturated, their strength continues de- ptini,uw Initial isotropic tensile strength for ξdis = 0 creasing due to the dissolution of the bonds and grains qi Component of the real seepage velocity along the i axis in the long-term (Ciantia and Hueckel, 2013; Ciantia et al., 2013). Consequently, a structure resting on cal- rkl Radial distance between the kth and lth Gauss points careous rocks, exposed to atmospheric conditions, may experience large displacements, or even collapse, due to sij Deviatoric stress tensor w Normalised averaging factor this degradation of the mechanical properties (Parise and Lollino, 2011). Therefore, the prediction of this w Weighting function 0 phenomenon, as well as its implications on the mechan- α Parameter in f d ical behaviour, are of uttermost importance. In this re- δ Kronecker delta ij gard, considerable progress has been made in recent Major principal strain 1 years in studying the interactions between soil and rock 3 Minor principal strain p masses and the environment (Gens, 2010). The classi- eq Equivalent plastic strain cal elastoplasticity framework has been extended to ac- p ¯eq Nonlocal equivalent plastic strain p commodate the mechanical and chemical degradation ij Plastic strain tensor of bonded soils and rocks (Ciantia and di Prisco, 2016; s Shear strains Gajo et al., 2019; Nova et al., 2003; Tamagnini et al., θ Lode's angle 2002; Tamagnini and Ciantia, 2016; Witteveen et al., ν Poisson's ratio 2013), and coupled chemo-hydro-mechanical (CHM) ξdis Normalised dissolved mass simulations have been performed to study the effect ξdis,cr Corresponds to ξdis when all diagenetic of weathering in engineering problems (Ciantia et al., bonds have been dissolved 2018). d σc0 Uniaxial compression strength under dry On the other hand, calcarenites might exhibit brit- conditions tle behaviour with a rapid decrease in their strength w σc0 Uniaxial compression strength under wet after peak under mechanical loading. Ductile behaviour conditions can also be observed depending on factors such as the σij Stress tensor confining pressure (Zimbardo, 2016; Lagioia and Nova, φ∗ Asymptotic friction angle 1995), the distribution of cement (Lollino and Andri- ∗ φini Initial asymptotic friction angle ani, 2017), temperature or saturation conditions (Nico- ∗ φpeak Peak asymptotic friction angle las et al., 2016), and their microstructure (Wong and ∗ φres Residual friction angle Baud, 2012). If brittle behaviour predominates in a par- p χ Value of eq separating the hardening and ticular engineering situation, strain-softening will gen- softening regimes erally result in a non-homogeneous strain field and de- Angle of dilation formations will tend to localise in the form of shear ! Constant controlling the volumetric compo- fractures or compaction bands (Raynaud et al., 2012; nent of plastic deformations Lollino and Andriani, 2017; Baxevanis et al., 2006). [·] Bulk fluid concentrations However, the numerical simulation of these phenomena, under the framework of continuum mechanics, embod- ies an intrinsic difficulty. In the presence of softening Stability of calcareous rocks due to long-term weathering 3 materials, continuum approaches deliver non-objective applied independently to each Gauss point of the me- results in the form of a strong dependence with the em- chanical model. For each weathering scenario, the struc- ployed mesh (de Borst et al., 1993). Vanishing energy ture was brought to failure, and the ultimate capacity dissipation and localisation into a region of vanishing was assessed in terms of the load-displacement curve. volume are obtained as the size of elements is reduced Since the failure phase involved the brittle behaviour (Pijaudier-Cabot and Baˇzant, 1987). They also exhibit of the rock under mechanical loading, the constitutive poor convergence properties (Summersgill et al., 2017) model employed was regularised following a nonlocal and sensitivity to the alignment of elements (Regueiro approach for avoiding mesh dependency issues. Rele- and Borja, 2001). Discrete methods (Utili and Crosta, vant outcomes are obtained regarding the evolution of 2011; Potyondy and Cundall, 2004; Kozicki and Donz´e, the structure's stability due to long-term weathering. 2008), or the combination of continuum and discrete ap- In addition, the importance of regularising the finite el- proaches (O~nateand Rojek, 2004; Morris et al., 2006; ement solution for stability analyses, in the presence of Mahabadi et al., 2012), are quite promising alterna- brittle materials, is demonstrated. tives to assess the localisation behaviour of brittle soils and rocks. However, their application to boundary value problems (BVP) of real engineering situations is still 2 Constitutive and numerical modelling of not generalised. Continuum approaches can still be used weathering and mechanical degradation if enriched with some characteristic length (Baˇzant and Jir´asek,2002). Different strategies, usually called reg- In the absence of a coupled CHM model regularised ularisation techniques, exist in the literature control- to simulate localised deformations objectively, the ef- ling the size of the localised region and preventing fects of weathering on the stability of the analysed case the pathological dependence with the employed mesh. are addressed following a semi-decoupled two-step ap- Among them, the nonlocal approach is an attractive al- proach. First, the reactive transport problem is solved ternative since it operates at a constitutive level and, to derive the spatial and temporal evolution of cal- therefore, it does not require the modification of the cium carbonate concentration. Then, by relating the formulation employed to solve the balance equations. In obtained concentration to the level of bonding, strength general, a nonlocal model is one where the behaviour of parameters are obtained for different times and used in a