PUBLS. INST. GEOPHYS. POL. ACAD. SC., M-9 (395), 006 Estimation of Dimension of a Regular-Type Sinkhole Activated by Abandoned Shafts Zenon PILECKI 1, and Adam BARANOWSKI 1 AGH University of Science and Technology Mickiewicza 30, 30-05 Kraków, Poland e-mail: pilecki@min-pan.krakow.pl Mineral and Energy Economy Research Institute Polish Academy of Sciences Wybickiego 7, 31-61 Kraków, Poland Abstract Sinkholes are natural phenomena in the landscape that cover the majority of old mining sites of shallow hard coal and zinc-lead exploitation in the Silesian Coal Basin and in the Olkusz area. Each sinkhole has unique characteristics due to the mechanism of its occurrence. Some of them are related to old, abandoned shafts, often not quite well closed. Dependence of sinkhole diameter upon overburden soil parameters has been established by numerical modelling. This dependence correlates quite well with the hypothesis of Professor Chudek, which has not been proved yet. The paper discusses some theoretical aspects of sinkhole occurrence in the conditions of the Silesian Coal Basin. In the core part, basic assumptions and results of numerical calculations are presented. Finally, the calculated dependence and the resultant graphs are compared to the analytical ones. 1. Introduction The origin of discontinuous surface deformations can be traced back not only to post-extraction voids, but also to inadequately safeguarded or abandoned shafts collapsing. The knowledge of the old small mine shafts used in the long past to extract shallow coal and ore deposits in the Upper Silesian Coal Basin (GZW) and in the Olkusz area is poor. The information on their locations is incomplete and the preserved old mine plans do not warrant finding their outlets on the surface.
Only a small portion, mere 5%, of all discontinuous surface deformations observed in the GZW area occurred due to improper closure practices of shafts and small pits (Chudek 00). This does not mean that the hazard they pose, mainly for civil engineering, is insignificant. The areas threatened by the possibility of sinkhole formation can be identified, but it is impossible to forecast the time of their occurrence. For instance, the probability of destruction of a shaft closed to the relatively modern standard, by being covered by a reinforced concrete plug, increases with time due to developing degradation of both its lining and the surrounding rockmass. It could also happen that the material filling a closed shaft shifted, creating voids. The fill material may be drawn off into the excavations adjacent to the shaft if the seal separating them from the shaft pipe is damaged or was not installed. The surface area around the shaft may gradually settle down or form a sinkhole in an abrupt manner. The following can be named as the major factors influencing the process of formation of shaft-related sinkholes: Unfavourable groundwater conditions resulting from heavy rainfalls, which may cause shaft fill to be washed away, suffosion of the medium surrounding the shaft barrel or physical or chemical corrosion of the shaft lining; Progressing disintegration of the shaft or pit lining; Washing out of the shaft fill material; Increased load on the ground surface from newly erected structures; Dynamic loads caused by traffic-related vibrations or seismicity. One of the most important parameters determining the level of risk for the land surface users is the maximal dimension of the sinkhole area. This work describes one of the methods, based on numerical modelling, of determining sinkhole size. The method may be used to forecast the diameter of a sinkhole not only when it is shaftrelated but also when it stems from other underground voids of more or less known dimensions. Available are also analytical methods, described in the works of Chudek (00) or Bell (1988), allowing to calculate the extent of a sinkhole and linking it to the properties of loose overburden. In this paper the authors compare the results obtained by numerical modelling with those calculated analytically.. Analytical Method of Calculating the Ground Level Size of a Sinkhole Discontinuous surface deformations are defined by Chudek (00) as such changes to the structure of the surface that break its continuity. To avoid misunderstanding, because of numerous meanings and understandings of the term, let us accept that in this paper a sinkhole shall mean such discontinuous deformation of the surface that is characterized by sinking of the ground surface. The rockmass prone to form sinkholes has to be divided into two distinct zones directly influencing the type and extent of discontinuous deformations (Chudek 00): The strata of competent rock (base rock), containing the void, The layer of loose overburden.
The process of deformation of the rockmass leading to the formation of a sinkhole on the surface has been described in numerous Polish and foreign publications, e.g. by Chudek (00), Fajklewicz et al. (004) or Popiołek and Pilecki (005). From these works the formation of a sinkhole may be outlined to proceed as follows: A zone of fractured rock is formed by tensile stresses occurring in the immediate roof of the void. Its shape is in general similar to that of a stress dome. Its height depends on the void s volume, the support and the properties of the surrounding rock (base rock), With time, the fragments of fractured and loosen rocks fall down, forming a fall at the bottom of the excavation. The position of the excavation s roof moves upward towards the surface. A secondary void is created between the roof and the collapsed rock, The developing rheological processes, which are usually intensified by water weathering, cause the fracture zone in the roof of the excavation to move towards the ground surface. At the same time, the volume of the void diminishes because the bulk volume of caved rock is greater than that of solid rock, If the thickness of the base rock strata through which the void travels on its way up is adequate, the secondary void may become completely filled. If it is too small for the void to be completely filled until its roof reaches the loose overburden, it is quite likely that a sinkhole will form on the land surface. If the overburden comprises loose soil, one has to expect the horizontal size of the sinkhole to be greater than that of the void propagating through the base rock. The size of the sinkhole will depend on the thickness of the loose soil overburden, soil strength properties, groundwater conditions and the criterion for the void to fill itself. In the case of old, closed shafts, sinkhole hazard exists if the shaft s lining installed in the loose soil layer disintegrates, or in the simplest cases if the plug closing the shaft s mouth collapses. The greatest diameter D of a sinkhole on the ground surface can be calculated from the equation (Bell 1988): o D Ztan(90 ) r = Θ +, (1) where Z is the thickness of the loose overburden, Θ is the angle of internal friction of the loose overburden material, and r is the radius of the shaft. The relationship (1) is illustrated in Fig. 1. The greater the thickness of the layer of loose overburden, the greater the area of the sinkhole. 3. Numerical Method of Calculating the Ground Level Size of a Sinkhole 3.1 Methodology of the calculations The quality of results obtained by numerical modelling to a great extent depends on a number of complex issues, beginning with the size and shape of the model, boundary and initial conditions, material property constants or the accepted calculation methodology. The numerical calculations were done for the small, old, closed
Z tan crown hole superficial deposits Z rock old open shaft Fig. 1. Method of calculating the diameter of a sinkhole formed in loose overburden above an old shaft (Bell 1988). shaft Andrzej situated in the Olkusz area. Normally, the input data collection consists in establishing the geomechanical properties of the rock medium. In the case of Andrzej shaft the material constants were obtained from the results of archival tests and complemented by the information available in the literature. The shaft was the cause of a medium-sized sinkhole (diameter of the funnel 6 m, depth m). The shaft s probable depth was 3 m and its cross-section was a square with a side of m. The calculations were made assuming plain strain conditions in an elasticplastic medium and the Mohr-Coulomb failure criterion. The modelling procedure consisted of the following basic stages: constructing the model, obtaining the equilibrium of forces in the field of primary stresses, calculating the model in several steps of shaft s constructing, obtaining in the end the equilibrium of forces in the field of secondary stresses, conducting a what-if analysis of stress and deformation for various overburden thickness and different types of overburden material. Figure shows the geometry of the model and its boundary and initial conditions. Axial symmetry of the model was assumed in order to increase the efficiency of the calculations. The values of the vertical and horizontal stress components were assumed as variable within the range defined by the model s depth, with the vertical to horizontal stress ratio assumed as λ = 0.4. The model had the shape of a 150 m high and 100 m wide rectangle. The computation zone size was 0.5 0.5 m. When analyzing the relationship between the diameter of the sinkhole and the depth of the overburden it was assumed, according to eq. (1), that the value of the shaft diameter does not influence the behaviour of the overburden material but only the diameter of the sinkhole.
Z superficial deposits shale dolomite ore dolomite limestone Fig.. Computational model with the boundary and initial conditions. Geological structure of the model was compiled from archival information. The base rock zone comprised four layers and the overburden one. The parameters of the overburden material were varied according to soil type. The model did not take ground water into account as the rockmass was assumed to be self-draining. The influence of the seeping water on the behaviour of the medium was also disregarded as a separate issue to be considered. The numerical modelling was done with the use of FLAC D v.4.0. developed by Itasca, USA, which utilizes the finite difference method. 3. Material constants It was assumed that the overburden was composed of one type of material and separate calculations were conducted for three types of soil (Table 1). The base rock was taken to comprise four strata (Table ). 3.3 Analysis of the results Figure 3 shows an example of calculation results of predicted behaviour of the overburden (sand and gravel) in the shaft s influence zone. The diameter of the formed funnel was approx. 6 m, which matched the field observations (Fig. 3a). Increasing the thickness of the overburden, the funnel s diameter increased up to a certain boundary value (critical), above which a chimney effect took place (Fig. 3b). This effect is illustrated by the graph of the dependence of normalized sinkhole diameter on the overburden thickness (Fig. 4). Character of the changes is bimodal. The dependence is
Soil type Table 1 The overburden material properties assumed for the calculations Density [kg/m 3 ] Bulk modulus Shear modulus Friction angle [deg] Cohesion Sand and gravel 1500 1.50E8 6.9E7 39 0 Clay 060.00E7 9.3E6 15.60E4 Loam 090 1.80E7 5.11E6 9 1.1E5 Stratum number Density [kg/m 3 ] Table The base rock material properties assumed for the calculations Bulk modulus Shear modulus Poisson ratio Friction angle [deg] Tensile strength Cohesion layer 1900 8.33E9 3.85E9 0.30 13.0 1.E+07 7E4 3 layer 550 7.11E9 5.33E9 0.0 36.3 6.E+06 9E6 4 layer 650 7.84E9 5.96E9 0.0 36.3 6.9E+06 9E6 5 layer 100.6E0 1.11E0 0.0 36.3 1.5E+06 9E6 strata: (Keuper series) serpentine silt and mudstone; 3 (Middle Triassic) dolomite; 4 (Middle Triassic) ore-bearing dolomite; 5 (Middle Triassic) limestone nonlinear up to critical depth and linear below it. For example, the behavior of the model in the nonlinear mode can be approximated by polynomial or logarithmic equations (Fig. 5): for sands and gravels for clays for loam y = 0.004x +.4355x 0.7549 ; R = 0.9984 () y = 9.4168 Ln( x) 1.4107 ; R = 0.9169 (3) y = 6.444 Ln( x) + 3.6114 ; R = 0.9068 (4) where y is the maximum sinkhole diameter within the analyzed range x, and x is the overburden thickness. For comparison Figs. 4 and 5 show the graphical presentation of sinkhole diameter obtained by means of the analytical method from eq. (1). It can be seen in Fig. 4 that after the thickness of the overburden comprising loose material sand and
a) 3 m b) 3,5 m X Displacement [m] interval [m] - contour 0 m X Displacement [m] interval [m] - contour 0 m Fig. 3. An example of the distribution of horizontal displacements in the area of sinkhole formation above the shaft pipe for overburden thickness of 3.5 m (a) and greater than the boundary value, which is variable and depends on material type (b). gravel exceeds 45 m, the diameter of the sinkhole remains constant. The results obtained for a specific sinkhole diameter by analytical and numerical methods are roughly the same up to the overburden width boundary value. Figure 5 shows the dependence of sinkhole diameter on the overburden thickness for all the types of overburden material considered (Table ). In the case of clay and loam the boundary condition values of overburden thickness is approx. 10 and 13 m and the sinkhole diameter does not exceed 5 m. 4. Conclusions The results of numerical calculations show that if the overburden comprises loose material and the base rock formations are competent, the factors determining sinkhole size are the overburden thickness and properties. For the overburden thickness greater than a certain (critical) boundary value, the diameter of the sinkhole remains more or less constant. For typical loose sand-and-gravel formations this limit value is approximately 45 m. It may be assumed that when the overburden thickness
exceeds the limit value, a chimney formation process begins. During the process a characteristic bell-shaped deformation appears in the overburden foot. Fig. 4. Dependence of normalized sinkhole diameter on the thickness of overburden comprising sand and gravel. Fig. 5. Dependence of sinkhole diameter on the overburden thickness for various types of material.
The presented numerical analysis was conducted on a relatively simple model of rock mass. Numerical calculations can be used to analyze models of much greater complexity, specifically, the overburdens comprising several layers made of loose and competent material. References Bell, F.G., 1988, Land development. State-of-the-art in the location of old mine shafts, Bull. of the Int. Ass. of Eng. Geology 37, 91-98. Chudek, M., 00, Geomechanika z podstawami ochrony środowiska górniczego i powierzchni terenu, Wyd. Politechniki Śl., Gliwice. Popiołek, E., and Z. Pilecki (red.), 005, Ocena przydatności do zabudowy terenów zagrozonych deformacjami nieciągłymi za pomoca metod geofizycznych, Wyd. IGSMiE PAN, Kraków. Accepted 10 April 006