A 3D Numerical Approach to Assess the Temporal Evolution of Settlement Damage to Buildings on Cavities Subject to Weathering

Sinkholes, of both natural and anthropogenic origin, represent an important geological hazard in Europe and, worldwide and every year, cause significant property losses and causalities. Sinkhole formation is often the end result of complex physical hydraulic chemical phenomena (weathering) inducing continuous changes of the soils and rocks forming the earth crust causing a progressive reduction in mechanical properties including strength and deformability and variations of permeability. From an engineering perspective, the weathering-induced degradation of mechanical properties makes such phenomenon quite relevant as it generates deformation in the considered geotechnical systems and can compromise its mechanical stability in the worst case. For instance, leaching chemical-aggressive species can degrade the bond strength of cemented soils, causing settlements of structures founded on these materials (Lollino et al. 2013). In evaporites, the interaction of water with the gypsum pillars of an abandoned mine can transform this rock into a much weaker and compressible material, which can eventually collapse under the weight of the overburden (Castellanza et al. 2010). Also the simple saturation of a dry porous carbonate rock, limestone or other karstic formations leads to considerable changes in strength and stiffness that may cause large surface settlements unacceptable for the service life of existing structures, or even collapse in the form of sinkholes (Parise and Lollino 2011; Parise and Vennari 2017; Sowers 1996).

The substantial increase in sinkhole formation in the Apulian region in the last decades (Parise and Lollino 2011; Parise and Vennari 2017; Sowers 1996) has pushed the regional authorities to introduce new regulations regarding the procedures and requirements needed for risk assessment (L.R. 2002/19). In particular, in the annex introduced in 2006 specifically regarding the risk associated with underground mines (A.d.b. Puglia 2006), it is specified that for all existing structures in high-risk regions (PG3) two options are possible: the first is to fill the cavity with materials with similar mechanical properties of the host Calcarenite, the second is to perform numerical analyses to assess and predict the mechanical behaviour of the geosystem and propose eventual consolidation measures to fulfil required levels of serviceability and safety. Also a simple planning permission to change the use class of a construction now requires the application of one of these two options. When a building is resting on the top of an abandoned cavity (Fig. 1), the engineering question to be addressed is: how long will the building remain in serviceability conditions?

Despite considerable progress made in recent years in the study of the interactions between the soil (or rock) mass and the environment (including hydraulic, thermal and chemical coupling phenomena) (Alonso et al. 1990; Cekerevac and Laloui 2004; Della Vecchia and Romero 2013; Gajo et al. 2015; Gens 2010; Hueckel and Borsetto 1990; Jommi 2000; Uchaipichat and Khalili 2009), the application of advanced models to perform 3D numerical simulations to address such question is rare. In particular, chemo-hydro-mechanical (CHM)-coupled advanced numerical simulations are usually performed for 2D idealized problems to show the robustness of integration schemes or to simulate laboratory-scale boundary value problems (Fernandez-Merodo et al. 2007; Tamagnini et al. 2002; Tamagnini and Ciantia 2016).

In engineering practice, according to Eurocode 7 (EC7-BSI 2004), it is necessary to demonstrate that building serviceability requirements can be met at the serviceability limit state (SLS). Typically, building settlements (either gross or distortional) have to be smaller than limiting values which depend on the building type, ground type, and design approach. The choice of the analytical or numerical method used to assess the expected values of settlements is left to the engineer, and to this end, elastic methods, Winkler-based analyses and standard finite element (FE) analyses may typically be used (Knappett and Craig 2012). However, the effect of weathering on serviceability of buildings is not contemplated by the design code, and in particular, no specific requirements for considering this are given in EC7. As shown by Conte et al. (2010), a simple way to model weathering in FE analyses is to assume that a portion of the domain consists of a formation with lower mechanical characteristics. As described therein, the authors show how weathering changes stability of a weathered slope modelled in 2D. However, limit state analyses do not provide reliable deformation results as the deformations associated with c–ϕ analysis are not representative of reality.

This work proposes to exploit a CHM-coupled numerical model to study the serviceability conditions of buildings lying on cavities affected by environmental weathering by introducing a practical scalar index: the building damage index (BDI). To answer to the question ‘how long will the building remain in safety conditions’, instead of assuming a strength profile and then performing a strength reduction analysis, hydro-chemical weathering scenarios are postulated and, by means of the reactive transport of contaminants, they are simulated. In this way, the spatial–temporal evolution of rock strength and stiffness will develop, and the surface settlements will induce damage to the structure until it is no longer safe. The method hence requires the use of both a coupled CHM constitutive model and a FE code which solves the hydro-mechanical problem and the transport of chemical species in porous medium. Small-scale coupled weathering experiments (needed to calibrate and validate the model) are then a prerequisite of fundamental importance for the proposed approach. In fact, in order to make reliable future predictions of the BDI, the model must first capture the temporal evolution of weathering at the laboratory scale.

In the following, after describing the method used to calculate the temporal evolution of the BDI, a brief overview of 3D contaminant transport modelling concepts is presented. Subsequently, the main elements of a recently developed advanced multiscale coupled CHM constitutive model (Ciantia and di Prisco 2015) used to describe the degradation of the soft carbonate rock induced by changes in degree of saturation (short-term debonding) and chemical dissolution (long-term debonding) (Ciantia et al. 2015a, b; Ciantia and Hueckel 2013) are described. Standard mechanical and advanced CHM experimental tests on the calcarenite from site are also performed and used to calibrate the model. Experimental results of two boundary value experiments reproducing weathering scenarios at the laboratory scale are also presented and used for validation. Finally, 2D and 3D coupled FE analyses simulating long-term effect of the weathering-induced settlements and the temporal evolution of BDI of a building in Canosa di Puglia built on top of an abandoned cavity (Fig. 1) are presented. Attention is placed on comparing how the BDI evolves with time for the 2 and 3D cases.

Despite the fact that final stages of sinkhole formation in soft rocks is usually characterized by a rapid brittle type of global failure (Lollino and Andriani 2017), recent interferometric synthetic aperture radar (inSAR) measurements show that ground movements start to accumulate with time several months before the collapse (Chang and Hanssen 2014; Intrieri et al. 2015; Jones and Blom 2014; Nof et al. 2013). In the Marsala Panatletico 2011 sinkhole, signs of instability, such as open fractures and detachment blocks, were observed already 1 year prior to collapse (Fazio et al. 2017). Leucci et al. (2002) show that due to leakage of water into the ground can cause differential surface settlements causing the opening of fractures on the overlying buildings. The formation of cracks on buildings, implying ground level subsidence, months before the Gallipoli sinkhole of 2007 (Fig. 2) were also observed (Delle Rose 2007). These surface settlements, if identified within time, can avoid disasters and allow for actions for risk mitigation (De Giorgi and Leucci 2014).

All these observations lead to the need of introducing an approach able to estimate the level of safety of the buildings lying on rocks susceptible to weathering prior to the brittle collapse of the geosystem, namely the building damage index, BDI. The formulation of the BDI is inspired by the Eurocode 7 serviceability limit states (which are defined as states that correspond to conditions beyond which specified service requirements for a structure or structural member are no longer met) and the work of Boscardin and Cording (1989) that related the level of damage of buildings to surface settlements. Accordingly, the surface differential settlements obtained by FE analyses are here used to assess how far a building is from a non-acceptable service condition. By modelling the reactive transport of chemical species and using a coupled CHM constitutive model where physical time is an intrinsic variable and rock damage is accounted for, it is possible to simulate weathering scenarios and monitor the temporal evolution of surface settlements and hence a BDI function of time. The BDI(t) is clearly strongly dependent on the weathering scenario assumed and on the constitutive model used to reproduce the mechanical behaviour. It gains in reliability as both the weathering mechanism reproduced, and the coupled constitutive models become more realistic. While advanced constitute relationships for calcarenites are now well established (Ciantia and di Prisco 2015), realistic hydro-chemical weathering scenarios for underground cavities such as the ones reported by Ghabezloo and Pouya (2006) are harder to validate due to complexity of the processes involved. Nevertheless, in the absence of in situ experimental data on the hydro-chemical initial and boundary conditions, one may assume worst-case scenarios, that would in any case result more realistic than a classical stability analysis trough simultaneous homogenous strength reduction in all the whole domains (Castellanza et al. 2008). Temporal evolution of surface settlements are then used to determine the corresponding evolution of horizontal strain and angular distortion at the base of the building used to employ the well-known Boscardin and Cording abacus and to calculate BDI(t) that, referring to Fig. 3, is defined as:where ||OA(t)|| is the distance of the current state from the origin, while ||OB|| is the distance from level of damage 3 to the origin obtained as an extension of OA. When the BDI < 1, the settlement-induced damage on the building is slight or negligible, while if the BDI > 1, the damage is moderate or severe.

In order to employ Boscardin and Cording’s abacus to define the subsidence induced damage to the building and use Eq. (1), the following assumptions are required:

The first assumption is deemed to be reasonable as the bell-shaped subsidence basin that characterizes sinkholes (Gutiérrez et al. 2008) has 2D sections which are very similar to the typical Gaussian-shaped transverse settlement observed in tunnelling excavation (Peck 1969). The latter two also limit the use of this approach to a certain typology of buildings, the one used to construct the abacus.

Despite the fact that this work uses a simplified method to define the level of structural damage, the methodological approach would remain unchanged also when running full soil–structure interaction simulations and can also be applied using other abaci or building damage criteria (e.g. Burland 1995; Burland and Wroth 1974; Franzius et al. 2006; Potts and Addenbrooke 1997; Son and Cording 2005).

The key aspect needed to introduce the BDI(t) is to link the evolution of surface settlements to the hydro-chemical time-dependent weathering processes, and this is only possible if the mechanical behaviour is described by means of advanced coupled CHM constitutive and numerical models.

The FEM code used in this work is the GeHoMadrid code (GHM) developed by prof. M. Pastor and co-workers (Fernandez-Merodo et al. 2007). GHM is a non-commercial research code which solves the hydro-mechanical problem, following the u − p_w Biot coupled formulation (Zienkiewicz and Shiomi 1984), and hence the hydraulic problem (new seepage velocity, pore-pressure and saturation variations) and the mechanical response are solved simultaneously.

The material tested in the laboratory was retrieved from the man-made cave of the selected case study (represented in Fig. 1) that was excavated in a calcarenite deposit in the urban area of Canosa di Puglia. The calcarenite is characterized by a porosity of 59% and a unit weight in a dry condition of 11.1 kN/m³. The experimental tests performed to adequately characterize the soft rock and calibrate the constitutive model are:

Following Ciantia et al. (2015a), by analysing the thin sections with both a polarized light microscope and SEM, it was possible to identify the presence of both depositional and diagenetic bonds (Fig. 6). Punctual EDS analyses (Fig. 6) performed during SEM imaging of the thin sections confirm that both type of bonds are formed by CaCO₃ microcrystals. While UC and BT tests are very useful to identify the yield locus at low confinement pressures, the TXD test are needed to identify the yield locus shape at high pressures (Fig. 7). In Fig. 8a typical axial stress–strain curves at different degrees of saturation are shown; the strength and stiffness reduction associated with higher saturation degrees are clearly visible. Figure 8b, c shows the normalized trend of σ_c and E versus S_r. These results have been obtained by performing more than 40 UC tests on specimen with 38 mm diameter using a displacement-controlled loading frame at 0.05 mm/min. The LTD process has been analysed by means of a two-step experimental test: following the same procedure of Castellanza and Nova (2004), the samples were first weathered by means of an acid solution (pH =  2.8) in order to accelerate the weathering process. In this initial phase, the dissolved mass ΔM is the measured variable and ξ_dis is evaluated by means of Eq. (26). Subsequently, the samples are tested, and in Fig. 9b, c, the dependence of both strength and stiffness on ξ_dis is illustrated, while in Fig. 9a, a selection of axial stress and strain curves at different values of ξ_dis is shown. With the experimental tests presented, the constitutive model is calibrated, and the corresponding parameters and model fitting curves are reported in Tables 1, 2, 3, 4 and Figs. 7a, 8a, 9a.

As it was already mentioned, the reliability of the approach is strongly affected by various aspects, but the most critical is the correct numerical reproduction of weathering mechanism taking place. The capability of the model to capture rock weathering from a quantitative point of view at the laboratory scale is then a prerequisite. As the CHM initial and boundary conditions in the laboratory can be controlled with reasonable accuracy, before using the approach as a predicting tool, the calibrated model should first capture the laboratory-scale response. For this reason, two BVP laboratory tests were performed and simulated with the already calibrated model. The first (BVP #1) is a model footing test on dry and wet calcarenite, while the second (BVP #2) is a loaded small-scale pillar subject to both the STD and LTD weathering mechanisms. For this latter the experimental results are taken from Ciantia et al. (2015b). The calcarenite samples of the two BVP’s were prepared from a Calcarenite di Gravina Formation block extracted few miles from the analysed cavity, hence have different initial strength conditions (p_m0), but the rest of the model parameters are the same.

The BDI(t) of the complex 3D geometry concerning the interaction of a cave (excavated about two centuries ago) and two buildings in a PG3 zone (Fig. 1) is here addressed. It should be noted that such situations are extremely common in the Apulian region (Parise and Vennari 2017), but this particular case study is considered as the authors were directly involved with the risk assessment of this cavity (Ciantia et al. 2015c) and hence disposed of in situ material needed for the model calibration (see Sect. 4). The current global stability of the same geosystem determined by means of the standard c‐ϕ reduction method (Potts et al. 1990) resulted to be 3.4 and 1.5 for the dry and saturated states, respectively (Ciantia et al. 2015c). Despite this current stable condition in both dry and saturated conditions, it is known that the long-term weathering of the rock could lead to subsidence and eventually to a failure mechanism such as the one in Fig. 2 (Ghabezloo and Pouya 2006). The BDI(t) as introduced in Sect. 2 is hence used to tentatively predict the evolution of the damage on the buildings under study. The weathering scenario assumed consists of two phases:

The laboratory tests on the calcarenite collected in situ were used to calibrate the initial mechanical properties of the material which, assuming them to be the same for all the domains, were assigned as the model’s initial state. During phase 1, simulated by setting S_r = 1 and hence using the ‘wet’ mechanical properties for the calcarenite, the geosystem remains in equilibrium. For this reason, in what follows, only the long-term chemical damage is considered (phase 2). In the absence of environmental hydraulic field data and boundary conditions, inspired by Ghabezloo and Pouya (2006), the cave boundaries were exposed to a different acidic pH environment. Such boundary conditions trigger the development of a coupled chemo-mechanical diffusion and deformation process. To consider the real geometry of the problem and realistic building loads, a 3D numerical analysis is performed, and the mesh and problem definition are shown in Fig. 15a, b. To highlight the importance of the three-dimensionality of the problem, a 2D plain strain analysis in correspondence of section A-A′ is also performed. The mesh of section A-A′ and the problem definition are shown in Fig. 15c. For the 3D FEM model discretization, 46,961 quadratic tetrahedral H10P4C4 elements (67,581 nodes and 169,270 degrees of freedom) are used, while the 2D model is meshed with 1879 triangular T6P3C3 elements (3920 nodes and 8642 degrees of freedom). Standard displacement boundary conditions have been applied to the external boundaries, with fixed horizontal and vertical displacements at the bottom and fixed horizontal displacements on the lateral boundaries. The external loads (i.e. the foundation loads) acting at the ground surface are kept constant, while the LTD process evolves from the inner boundary of the cave. To represent building A (see Fig. 1), a pressure of 150 kPa is imposed under each of the 9 square footings, while a pressure to represent building B a uniform pressure of 40 kPa is imposed under the area covered by it. For the 2D analysis, the same pressure of building A is applied despite the 2D geometrical limitations (plain strain hypothesis). On the cavity surfaces, a fixed constant concentration of [H₃O⁺] ions, corresponding to a pH of 6.7, is imposed. They will propagate into the rock by means of a diffusive phenomenon causing the long-term dissolution of CaCO₃. The reactive transport model parameters used for the simulations are those of BVP#2 and are summarized in Table 5. To highlight the importance of the cavity environmental conditions, two further 3D simulations were run: one imposing a pH of 7.5, while in the other, the pH imposed at the cavity boundaries was set to 6.

Figures 16 and 17 report the main results as a function of physical time (days) from the beginning of the weathering process assumed (i.e. the LTD process) for the 2D and 3D simulations, respectively, for the 6.7 pH simulations. The dissolution process that evolves from the inner boundary is shown in the first column of Fig. 16 by the contour in time and space of the contaminant ion [H₃O⁺]. As the contaminant concentration increases, the dissolution process evolves causing a mass reduction and a consequent increase in the scalar ξ_dis leading to a reduction in the bonding-related variables p_m and p_t (2nd column of Figs. 16 and 17). This strength reduction causes yielding of the rock to concentrate in correspondence of the partition wall for both the 2D and 3D simulations. As the exposure time with the [H₃O⁺] ions increases, the plastic strains localize into a shear band (3rd column) and the displacements increase (4th column). Nevertheless, before plastic strains localize into a shear band, as clearly shown in Fig. 18a, material degradation developing from the wall boundaries induces a stress redistribution in the partition wall Fig. 18b. Although non-local approaches (Conte et al. 2010) should be adopted to avoid mesh dependency when looking at the failure mechanism, it was observed that further mesh refinement did not change the results of the simulation in terms of surface displacement with time. Interestingly, the failure of the partition wall does not cause a global failure mechanism, and this is due to the residual frictional strength within the shear band combined with the rooftop geometry. In fact, as represented in Fig. 18c, d, the stress corresponding to a point within the shear band starts from an initial elastic state, yields after about 120 days of acid exposure and evolves until critical state is reached. Figure 19 shows comparison of the displacement field for the 2D and 3D numerical simulations used to calculate the BDI for the 6.7 pH environment. The chemo-mechanical coupled induced displacements are well documented in terms of the displacement contours (Fig. 19a) for the 3D simulation. The vertical displacement profiles with time for the 2D and 3D (section A-B) simulations are in Fig. 19b, c. The displacement time curves for the point with maximum surface displacement and the buildings rigid rotation evolution with time for the simulations run are reported in Fig. 19d, e. As expected, the 2D simulation deforms more than the 3D one thanks to the lateral support provided by the rock not intersected by the tunnel. The change in curvature of the displacement time curve (more visible for the 2D case) can be identified as the sinkhole inception time as from this point onward the displacement will accelerate until the simulations stops converging. Interestingly, the surface subsidence observed in the 3D simulation acquires a circular shape, and it is caused by the failure of the partition wall. Finally, in Fig. 20a the angular distortion horizontal strain paths for the 2 and 3D analyses with different pH conditions are reported. By means of Eq. (1), in Fig. 20b, the BDI evolving with time is also reported. The time needed to attain a BDI of one results to be 1080, 200 and 60 days for the 7.5, 6.7 and 6 pH case, respectively. The plot also shows how the three-dimensional effect delays the BDI attaining a value of unity from 165 to 200 days for the 6.7 pH simulation. In this contribution, only two sections were considered for the BDI assessment of building A; however, any section can be analysed and the final BDI(t) would be the first one (section A-A′ in this simulation) reaching a value of 1.

Many abandoned man-made caves induce sinkholes because of the degradation of the constituent soft rock. In this paper, the problem superficial subsidence prior to sinkhole formation in urban area is tackled and in particular:

Limitations of the BDI include aspects inherent to most deterministic numerical models of geotechnical problems. In particular the reliability of the approach will depend on: (i) the amount (and associated uncertainty) of information required concerning the geometry of the problem, (ii) the CHM material properties, the chemical reactions and processes, (iii) local heterogeneities and (iv) the assumption of the weathering scenarios. Many of these inconveniences may be addressed by employing a probabilistic approach (Griffiths et al. 2002; Griffiths and Fenton 2004; Zhang and Ng 2005). We also believe that in a near future remote sensing techniques such as the interferometric synthetic aperture radar (inSAR) from satellite images will provide temporal evolution of subsidence that will allow to test and improve the reliability of the proposed methodology.