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Equilizing of the Primary Stress State in the Rock Mass, Simulated by a Model of Layer in an Elastic-Viscous Medium

Equilizing of the Primary Stress State in the Rock Mass, Simulated by a Model of Layer in an... Arch. Min. Sci., Vol. 61 (2016), No 4, p. 853­873 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.1515/amsc-2016-0057 GRZEGORZ KORTAS* WYRÓWNYWANIE NAPRE PIERWOTNYCH W GÓROTWORZE SYMULOWANE MODELEM WARSTWY W ORODKU SPRYSTO-LEPKIM This paper is devoted to the analysis of the stress development process in the homogeneous and non-homogeneous rock mass. The rock-mass model consists of an elastic-viscous medium containing a layer (Fig. 1) that displays distinct geomechanical strain properties. When examining the process of stress equilizing in , the Norton-Bailey power creep law was applied in the numerical analysis. The relationship between fective stresses and , the modulus of elasticity, Poisson's coficient, and creep compliance were obtained. It was demonstrated that the relationship between fective stress and or creep compliance, for the assumed conditions in a homogeneous rock-mass, was approximated by hyperbolic functions 0 and 16). The process parameter included a certain value of creep compliance or of at which there occurred a half-way equilizing of primary stresses. An analogous function binds fective stresses with creep compliance. Our model studies indicated a number of relationships between bulk and shear strain with and creep compliance in the homogeneous and non-homogeneous rock mass, presented in Figs. 2-14, expressed by the functions of those specific parameters. The relationships obtained in this work resulted from our model assumptions. However, they demonstrated the influence of the geomechanical strain properties of rocks on the process of shaping the primary stress state in the rock mass and the tendency to reduce the principal stress differences in . Our research results suggested the necessity to simulate the primary stress state as an initial condition of the geomechanical numerical analysis concerning the rock-mass behaviour showing rheological properties. Keywords: primary stress state, non-homogeneous rock-mass, elastic-viscous medium, creep compliance, fective stress equilizing Wielowiekowe procesy geodynamiczne, doprowadzily do wyksztalcania si obecnego stanu napre w litosferze, który w zagadnieniach górniczych okrela si pierwotnym stanem napre górotworu. Praca powicona jest analizie dlugotrwalego procesu wyksztalcania si napre w jednorodnym i niejednorodnym górotworze, przeprowadzonej w trybie bada modelowych. Okrelano zmiany naprenia w orodku sprysto-lepkiego z warstw (Rys. 1), wykazujc odmienne odksztalceniowe wlaciwoci geomechaniczne. W obliczeniach numerycznych stosowano potgowe prawo pelzania Nortona-Bailey'a. * STRATA MECHANICS RESEARCH POLISH ACADEMY OF SCIENCES, UL. REYMONTA 27, 30-059 KRAKOW, POLAND Kwantyfikatorem zmieniajcych si z uplywem czasu rónic napre glównych bylo rednie naprenie ektywne w centralnym sektorze modelu geometrycznego badanego orodka. Otrzymano proste zwizki napre ektywnych z czasem, modulem sprystoci, wspólczynnikiem Poissona i podatnoci na pelzanie. Wykazano, e w jednorodnym górotworze zwizek naprenia ektywnego z czasem lub z podatnoci na pelzanie, aproksymuje funkcja hiperboliczna opisujca wyrównywanie napre glównych 0 i 16). Wykazano, e parametrem procesu wyrównywania si napre glównych jest pewna warto podatnoci na pelzanie Bc lub czasu c, przy których wystpuje polowiczne wyrównanie napre pocztkowych. Wykazano, e w orodku sprysto-lepkim z potgowym prawem pelzania wzgldny przyrost napre jest odwrotnie proporcjonalny do czasu lub do podatnoci na pelzanie. Z bada modelowych wynika szereg zwizków odksztalcalnoci objtociowej i postaciowej z czasem i podatnoci dla jednorodnego i niejednorodnego górotworu, przedstawionych na rysunkach 2-14 i wyraonych funkcjami tych parametrów. Stwierdzono, e funkcja wyrównywania si napre wyraa zasad, e wzgldny przyrost napre jest odwrotnie proporcjonalny do zmiennej wymiarowej, któr moe by czas lub podatno na pelzanie. W logarytmicznej skali czasu lub podatnoci wyodrbni mona trzy okresy lub przedzialy. W drugim okresie (Rys. 2) i drugim przedziale zmian podatnoci (Rys. 6) nastpuje intensywne wyrównywanie si napre ektywnych. Dla wartoci redniej geometrycznej czasów na kocach tego przedzialu c lub podatnoci na pelzanie Bc naprenie ektywne odpowiada polowie wartoci naprenia ektywnego na pocztku procesu. Stwierdzono, e wzrost wartoci wspólczynnika Poissona nie wplywa na warto c, natomiast wzrost modulu Younga orodka powoduje spadek wartoci c. Wzrost podatnoci na pelzanie nie wplywa na pocztkowe naprenia ektywne, ale powoduje wzrost c i okresu wyrównywania si napre. Stosunek c dwóch orodków rónicych si podatnoci na pelzanie odpowiada stosunkowi ich podatnoci, a stosunek Bc dwóch procesów, w których zmienn jest podatno na pelzanie, odpowiada stosunkowi czasów ich trwania. W orodku jednorodnym wykazujcym wlaciwoci reologiczne proces wyrównywania si napre glównych prowadzi do utworzenia hydrostatycznego stanu napre litostatycznych. Im podatno orodka na pelzanie jest wiksza, tym proces ten jest krótszy. Stwierdzono, e w modelu niejednorodnego górotworu o odmiennych wlaciwociach warstwy i jej otoczenia proces wyrównywania si napre jest tylko pocztkowo zgodny z funkcj hiperboliczn 1). Podobnie jak w orodku jednorodnym z uplywem czasu naprenia ektywne w warstwie d do zera. Pocztkowe naprenie ektywne ronie ze wzrostem modulu sprystoci warstwy oraz wydlueniu ulega drugi okres wyrównywania napre. Im warto modulu sprystoci warstwy jest mniejsza, czyli im mniejsza jest rónica modulów sprystoci objtociowej warstwy i jej otoczenia, tym proces wyrównywania napre jest dluszy. Rónicujc podatno warstwy i jej otoczenia, funkcja wyrównywania si napre w czasie ronie i osiga ekstremum, a czas, przy którym pojawia si to ekstremum zaley od stosunku podatnoci warstwy do podatnoci otoczenia. Zmiana podatnoci warstwy przy stalej podatnoci otoczenia prowadzi do równoleglego przesunicia w logarytmicznej skali czasu wykresu funkcji napre ektywnych, przez co proces wyrównywania si napre wydlua si. Wartoci napre ektywnych z lokalnym ekstremum, wystpujcym po krótkim okresie czasu, rosn prawie proporcjonalnie z glbokoci. Po bardzo dlugim czasie wplyw glbokoci zanika. Podobnie jak dla jednorodnego górotworu proces wyrównywania si napre mona przedstawi dla zmiennej podatnoci warstwy i czasu jako parametru przy zachowaniu stalej podatnoci otoczenia. Wyniki bada przedstawione w tej pracy pokazuj istotn rol czasu i wlaciwoci odksztalceniowych skal w procesie ksztaltowania si napre w orodku wykazujcym wlaciwoci reologiczne. Wskazuj take, e w obliczeniach numerycznych stanu napre wokól wyrobisk w orodkach niejednorodnych, napreniowe warunki pocztkowe w obszarach rónicych si podatnoci na pelzanie s odmienne. Taki stan napre pierwotnych moe by symulowany przez proces odwzorowujcy dlugotrwale oddzialywanie sil masowych. Slowa kluczowe: naprenia pierwotne, niejednorodno górotworu, orodek sprysto-lepki, podatno na pelzanie, wyrównywanie napre ektywnych 1. Introduction Long-term geodynamic processes have led to the development of the present-day stress state in the lithosphere, which is described as the primary stress state of the rock mass in mining research. Geomechanical calculations usually assume that the primary stresses resulted from mass forces and, consequently, the absolute values of horizontal stresses are smaller than those of vertical ones [e.g. Salustowicz 1965]. It is further assumed that Poisson's coficient is increasing with depth, and the stress differences are reduced (Brown & Hoeck, 1978). In situ measurements indicate that the observed distribution of stresses is considerably different from the distribution resulting from the present-day mass loads in certain areas (Amadei & Stephansson, 1997). It was found that, close to the continental plate motion and active orogenesis, horizontal stresses exceed vertical ones even several s at small depths. Such an primary stress state may essentially influence mining conditions and the method of deposit cuper excavation, e.g. in the Polish LGOM Basin (Butra et al., 2011). It is also known that non-homogeneous rock mass considerably influences the distribution of primary stresses; in particular, the rock density and rheological properties differences cause salt-dome uplifting, but with the occurrence of hydrostatic stress state in salt (Ode, 1968). The process of primary stress development on salt rocks can be simulated under model testing (Kortas, 2006). In the mathematical models used for the determination of stress and strain states around mine workings (Filcek et al., 1994), the primary stress state itutes the initial condition of the resolution of the systems of partial differential equations. The properties of rocks are usually studied by laboratory methods, with in situ examinations applied to the behaviour of the media made up of such rocks. Respective model tests with numerical analysis allow for the search for relationships between physical and -space variables which rlect the behaviour of the given medium in the assumed mathematical model. An example of that approach consists in the determination of the development of stress in , depending on the strain properties of particular stress areas, which is the subject-matter of the present research. The recognition of the primary stresses shaping process in that way is important for both knowledge and practice development. This paper presents model research results concerning the changes of stress in , in both homogeneous and non-homogeneous rock masses, with the rock mass being represented by a layer with separate strain properties in respect to the surrounding medium. 2. Research Method Our research on the rock mass behaviour was conducted by representation of the rock mass with elastic-viscous medium with the power creep law, similarly to e.g. (lizowski, 2006). When discussing the research method, the following were introduced: (i) description of the physical model, giving itutive relationships of stress and strain in , (ii) description of the assumed geometrical model, which represents a simple non-homogeneous form of the medium composed of a layer and its surroundings, and (iii) the properties of a certain hyperbolic function which was later applied to the analysis of the model study results. 2.1. Physical Model The numerical analysis relied on linear Hooke's law which can be transcribed in the form of two tensor equations ), binding stress and strain, of which one describes the shape change law and the other one the bulk change law: 1 2 E ) Where strain deviator: D = ­ Am ( ­ Kronecker's delta), stress deviator: D = ­ Am, strain: Am = 1/3 1 + 22 + 33) and stress: Am = 1/3 1 + 22 + 33). To simulate viscous behaviour, we used the Norton-Bailey power creep law, determined by the dependence of strain rate on the stress state and the t: Q 3 mA exp 2 RT n 1 t m 1 (2) where Q is the activation energy, R is the gas ant, T is temperature in Kelvin, and A is the material ant. The dependence of strain on the values of n and m in the Norton-Bailey power creep law is discussed in (Maj, 2012); in our calculations, the following assumptions were made: · n = 2 and m = 1. For those ants, equation (2) has the form of = 3/2 B . The function of the material ants A, Q, and R in (4), in the specific temperature T, dines the material's creep compliance B, determined by equation: B A exp Q RT (3) Practically, creep process activation energy Q is not determined for rocks, and R is a universal physical ant equal to the work fected by heating 1 mol of perfect gas by 1 degree Kelvin, being equal to 8.314462 J/(mol*K). For that reason, the determination of the value B(T) is the actual result of laboratory tests. In the strain process of the medium under a ant load, the fective stress s, expressed by principal stresses s1, s2, and s3 in (4), is changing in : (t ) 1 (t ) 2 (t )] 2 (t ) 3 (t )] 3 (t ) 1 (t )] (4) The process of stress equilizing and medium strain development, initiated by mass load, occurs only if (t ) > 0 and B 0 in certain medium zones. The process is concluded when the principal stress components are identical, that is = 0. Therore, fective stress may be a quantifier of the stress equilizing process. In the medium with the power creep law, multiplication, division, involution, or logarithming in itutive equations correspond to addition, substraction, multiplication, and division in a linear medium, respectively. Therore, let the value determine the power change of the value of viscous properties from 1B to 2B: log( 2B /1B) (5) Then, each increase of by 1.0 corresponds to the ten increase of the creep compliance B. Similarly, one can dine h as a power change of the bulk rigidity modulus. At the specific value of Poisson's coficient v, the diversity of the bulk rigidity modulus (Helmholtz modulus) E/[3 ­ 2v)] can be reduced to the function of Young's elongation strain modulus E: log( 2E /1E ) for (6) In that approach, the values of h and b are the parameters of shear and bulk strain changes in the medium. 2.2. Geometric Model and the Method of Study The primary geometric stress model adopted for stress equalizes testing was a medium composed of a layer and its surroundings, each with different physical properties. The same rockmass geometric model, presented in Fig. 1, was adopted for all calculations. In that model, a layer sloping at 20°, with ant vertical thickness of 15 m, was different than its surroundings. The layer's centre was located at the depth of 234 m, within a rectangle of the horizontal dimension of 330 m and the vertical one of 300 m. Within the small rectangle MNOP (Fig. 1), average stress values were calculated. The occurrence of an anhydrite layer on the southern edge of the ForeSudeten Monocline can be an example of the corresponding conditions (Kortas & Maj, 2012). Surroundings 300 15 m er Lay 150 m M 330 m Fig. 1. Geometric Model Surroundings Numerical calculations were conducted by the finite element method, with the assumption of the plain state of strain, with the zero displacement boundary condition in horizontal direction. The values of (t ), changing in , were the searched study results, calculated here as the average value of the fective stresses obtained in part of the layer, in the horizontal distance of ±75 m from the centre of the model (Fig. 1). The sets of tasks, parameterized with the variables, created series of tasks. A graphic presentation of the obtained functional relationships for several series of tasks revealed the relationships between stresses, strain properties, and that were looked for under model studies. 2.3. The Function of Stress Equilizing in In the rock mass that is susceptible to creep, the initiation of the diversity of fective stresses occurs, resulting from sudden changes in loads caused for example by a tectonic motion, with the drop of fective stresses resulting from long-term rheological processes. The process of changes in stress can be simulated in the first approximation by studying, after applying preliminary load, the behaviour of a model of medium in the . When simulating a lithostatic process in rock salt (e.g. Kortas, 2006), it was found that the stresses existing in a medium that had viscous properties were subjected to equalization or equilizing after the initial elastic reaction ( = 0) at the ( ): min = 1 < 2 < 3 = max lithostatic process 1 = 2 = 3 = max (). During the simulation of the equilizing of stresses in the t, we can distinguish three periods: (i) in the first short period (until the p), the principal stresses are slightly different from the initial stresses resulting from elastic reaction, (ii) the second period (p k) is the basic period of stress equilizing, (iii) in the third period (k < ), the differences in the principal stress values disappear, and, after a very long , principal stresses are almost the same. Let the non-nominal value parameter , with the conventional value belonging to the range of 0 < < 0.01 serve the determination of the boundaries of the three periods of principal stress equilizing. Then, through parameter , the s of the beginning p and of the end k of the second stress balance period are correlated with the boundary values of the fective stresses by equation: d ( 0) p) ( k) (7) That concerns the plain state of strain 2() = 3(). In the homogeneous rock-mass model, the elastic reaction to the mass load pz causes the development of principal stresses that are directly proportional to the depth H (in the direction Y in Fig. 1) and the density of the medium , that is: 1 2 min 2 H g max pz pz [ / )] (8) In the plain state of strain, the fective stress corresponds to the difference between the maximum and minimum principal stresses: max min 1 2 pz 1 (9) During the under discussion process, the stresses () change from the initial value of s to the final value of () = 0. Let that change in the medium with a power creep law be described by decreasing hyperbolic function 0), with three ants Cp, Ck, and c, in the form of: Cp 1 / Ck 0) If in 0), for the average stresses of = m and the maximum stresses of = max, the , then () min and Ck = min. For the fective stresses of = , reducing to zero with , the ant Ck = 0. At the beginning of the process = 0, the stresses: = m = m , = max = max and = = . Let (c) = /2 in the c. Then ( c) [Cp ­ Ck]/2, and for fective stresses: (c) = /2. Upon introduction of the values of those ants to 0), we can determine the particular functions max(), min (), and m(). Function 0) will have the following form for fective stresses: / 1) Transformations of equations (7) and 1) will produce simple relations 2), binding p, k, and c with : p c p k p k 2) With those assumptions, for a small value of , the ant of the stress equilizing process c, called here the period of half-way equilizes of stresses, is the geometric average of the s p and k, determined by functions (7) and 1). Function 1) was applied to the approximation of the results of the model studies concerning the stress state equilizing processes in the layer and its surroundings. 3. Research Results and Result Analysis The results of the completed model studies are presented in two sections below. In the first case, with the assumption of a homogeneous rock mass, the layer and its surroundings displayed the same geomechanical properties, and in the second case, the properties of the layer and of its surroundings were diverse. The selected values of the shear and bulk strain parameters of the layer and its surroundings, as well as the calculated average values of the fective stresses, were tabularized and presented graphically as a function of . 3.1. Research Results for a Homogeneous Medium The purpose of the initial studies was to determine the relationship between the stress equilizing process and the elastic and viscous properties of a homogeneous rock mass. The preliminary series of tasks assumed the following: E = 17 GPa, v = 0.15, and B = 5*10­28 Pa­2s­1. Stress equilizing was simulated during the period of 3.16*1013 s, or 100,000 years. Table 1 contains the values of average principal, fective, and average stresses, calculated in several periods, and Fig. 2 shows the respective graphs in the function. For the = 0, min = ­5.05 , max = ­0.89 , and = 4.16 were obtained in the central sector of the layer. In the second and basic period of the studied process, the minimum principal stress was ant: min() = ; however, with the increase of , the maximum principal stresses were dropping: max() min and () = max() ­ min() 0 (Fig. 2). Assuming = 0.005, the boundary values of the fective stresses of the second period of stress equilizing amounted to ­ (p) = 0.025 , (k) ­ () = 0.025 , tending to 1() = 2() = 3 = min after the conclusion of the process. TABLE 1 Average values of stresses within the layer s 3*100 3*10­7 3*10­10 3*10­13 3*10­16 1 i 2 i 3 i i sr (i 3) ­0.884 ­0.887 ­2.612 ­5.006 ­5.009 ­0.884 ­0.891 ­2.612 ­5.006 ­5.009 ­5.0096 ­5.0096 ­5.0096 ­5.0096 ­5.0096 4.1250 4.1229 2.3972 0.0035 0.0004 ­2.2592 ­2.2610 ­4.4115 ­5.0073 ­5.0095 First period of stress equalizing Stress [] -1 -2 -3 -4 -5 -6 1E+005 = 2= max Second period of stress equalizing + 2+ 3)/3 Third period of stress equalizing min 1E+012 1E+013 1E+014 1E+009 Fig. 2. Average principal and fective stresses in the layer, in the function of The examples of the distribution of the average fective stresses at the beginning and at the end of the simulated stress equilizing process are illustrated in Fig. 3. In the initial series of tasks, the stress equilizing process was simulated with the ants values of the medium's viscous properties and diverse elastic property values. The assumed parameters are presented in Table 2 (series 1-4) and the results in Fig 4. Function 1) was applied to the approximation of the calculation results presented in Fig. 2, marking with upper index the task series numbers in Table 2. 1E+006 1E+007 1E+008 1E+010 1E+011 Fig. 3. Distribution of the fective stress (x, y, ) [Pa]: Lt: ( = 0 s) Max = 5.96 ; Right: = 3*1013 s) Max = 0.004 TABLE 2 The values of variables and ants of the medium in task series from 1 to 6 Series No. E [GPa] v [1] B [Pa­2s­1] [] Bc [Pa­2s­1] c 5*10­28 5*10­28 5*10­28 5*10­28 5*10­25 5*10­22 from 5*10­29 to 5*10­23 from 5*10­29 to 5*10­18 from 3.15*105 to 3.15*1013 from 3.15*105 to 3.15*1014 from 3.15*105 to 3.15*1013 from 3.15*105 to 3.15*1014 from 1.00*103 to 1.00*1011 from 3.15*102 to 3.15*106 1*105 s 1*1010 s ­ ­ ­ ­ ­ ­ 4*10­26 4*10­21 0.34*1012 8.33*1012 8.33*1012 0.34*1012 0.34*109 0.34*106 ­ ­ The test results indicated that, with the ant creep compliance B, the following were obtained in task series 1 and 4: ­ For series 1 and 4 (v = 0.15), identical values of = 4.16 and identical s of half-way fective stress equilizing: 1c = 4c = 0.357*1010s. ­ For series 2 and 3 (v = 0.30), identical values of = 2.89 and identical s of half-way fective stress equilizing: 2c = 3c = 8.33*1011 s. ­ The change of in series 1 and 3, as well as 2 and 4, resulting from the change of the Poisson coficient value, in accordance with equation (9). For the conditions B = and v = the relationship of the two c (E) values is approximately inversely proportional to the value E and proportional to : 2 2 c ( E) 1 1 c ( E) 2 1 ( 0) , B 3) = 4 Average fective Stress [] v = 0,15 2 v = 0,30 =3 /2= /2 E=1 GPa /2 =3 /2 E=17 GPa 2 1 3 c= c =4 1.0E+005 1.0E+006 1.0E+007 2 1 c c 1 2 1.0E+008 1.0E+009 Fig. 4. The dependence of the layer's (, E, ) on the stress equalizing ; parameters: E and Therore, with the drop of the modulus E, the of half-way stress equilizing is increasing. Subsequent tests of task series 5 and 6 (Table 2) consisted in checking the influence of the changes in the creep compliance of the medium B on the stress equilizing process. The respective calculations were conducted with the same values of elastic properties. The simulation results are presented in Fig. 5. The simulations (, B) indicated that, at the specific values of elastic properties, the string of applied values of the parameter B led to the generation of similar graphs (Fig. 5). The value of does not depend on the medium's creep compliance. With the decrease of creep compliance value, the s p, c, and k are increasing. The relationship between creep compliance and the half-way stress equilizing c is then expressed by this proportion: B B , or c c 1 10 , E The influence of the change of the creep compliance 1B 2B can be reduced to equation: 1.0E+010 1.0E+011 1.0E+012 1.0E+013 1.0E+014 4) 1.0E+015 ( 2 c , 2B) B / 2B, 1B) 5) Average fective Stress [] 1.0E+003 4 4B = 5*10-28Pa-2s-1 5 5B = 5*10-25 Pa-2 s-1 6 6B = 5*10-22Pa-2s-1 /2 1.0E+004 1.0E+005 1.0E+014 1.0E+015 1.0E+007 1.0E+008 1.0E+009 1.0E+010 1.0E+011 1.0E+012 1.0E+013 Fig. 5. The dependence of the layer's (, B) on the stress equalizing ; parameter: B Since for the specific z the of the conclusion of the second stress equilizing period k is increasing with the value of c 1), together with the decrease of creep compliance; therore, the lower the rock mass's creep compliance the longer the stress equilizing period. The duration of the second stress equilizing period is then inversely proportional to creep compliance. And thus, for series 6: c 108 s and for series 4: c 1014 s (Fig. 5). Model studies demonstrated that the influence of creep compliance on the stress equilizing is similar as process depends on duration. Consequently, in function 1), instead of the , we can use the variable B, an will be a process parameter as a ant, that is: ( B) 1 B / Bc Consequently, instead of the ant c in 1), there is Bc in 6): the creep compliance at which half-way fective stress equilizing occurs, in the = adopted in the task series. Irrigation of silt formation, or rock fracturing and crushing, are the examples of the process leading to the increase of the rock's creep compliance. Similarly to the influence of on stresses, shown in Fig. 2, we can identify three periods of the influence of creep compliance on fective stress equilizing (Fig. 6). The creep boundaries in 1.0E+006 , where lim ( B) 6) the second period are associated with the small conventional value of , dined as in formula (7), but with p and k replaced by Bp and Bk, respectively. Average fective Stress [] First zone of creep compliance Second zone of creep compliance Bp Bc Bk Creep Compliance B (in logaritmic scale) Fig. 6. The zones of the influence of creep compliance on stress equalizing, for the k = In zone (i), B < Bp, the creep compliance value does not significantly influence the change of the maximum values of , in zone (ii), Bp < B < Bk, the increase of creep compliance essentially influences stress equilizing, and in zone (iii), Bk < B, the increase of the creep compliance value B does not significantly influence the change of the values of which are close to zero. The value of Bc determines the creep compliance at which half-way fective stress equilizing will occur in the assumed , or (Bc) = /2. For the specific values of elastic properties, the parameter k is associated with Bc by proportion: Bc Bc Third zone of creep compliance or 10 , E 7) Fig. 7 presents the results of (B) calculations in two task series, 7 and 8 (Table 2.), for two s: = k 105 s and 1010 s. According to 7), we received log( 8 / 7) = log010 s /105 s) = log( 8Bc,7Bc) = log(4*10­21 ­2 ­1 Pa s /4*10­26 Pa­2s­1) = 5. Relationships 8) occur between the creep compliance values of Bp, Bk, and Bc, depending on z, similarly to 2): Bp Bc 1 , ) Bc Bp Bk )2 Bc B p Bk 8) =4,16 GPa Average fective Stress [] =2,08 GPa = 105 s = 1010 s 1.0E-029 1.0E-028 1.0E-027 1.0E-026 1.0E-025 1.0E-024 1.0E-023 1.0E-022 1.0E-021 1.0E-020 1.0E-019 1.0E-018 Bc=4*10-26Pa-2s-1 Bc=4*10-21Pa-2s-1 Creep Compilance Bwar [Pa-2s-1] Fig. 7. The dependence of the layer's (B, ) on the compliance of the medium B, parameter: Relationships 8) characterized the equilizing of stresses resulting from the increase of creep compliance of the elastic-viscous medium, with power creep law. We should, however, mention that those relationships will be different if a change of bulk strain occurs, together with a change of the medium's creep compliance. 3.2. Research Results for a Non-Homogeneous Medium The non-homogeneous medium is characterized by diverse properties of its components, i.e. the layer and its surroundings in the model under discussion. Another series of model studies concerned first the influence of the t on the average fective stresses () in the layer with the creep compliance of warB = 10­28 Pa­2s­1, and otoB = 10­25 Pa­2s­1. In those studies, the creep compliance of the surroundings is higher than that of the layer, while the diversity of the surroundings's creep compliance otoB and of the layer warB is characterized by = 3. Consequently, the layer's creep compliance is by 3 orders lower than that of the surroundings. In the first series of tests (Table 3, series 11-15), the analysts were studying the dependence of fective stresses on (, warE), at specific properties, parameterizing the layer's elasticity modulus war E [5..100 GPa]. The calculation results are presented in Fig. 8. Similarly to the homogeneous medium model, the dependence of (, E) is a decreasing function, with asymptote (, E) = 0, determining the status after the principal stresses were equalized in the layer. The value of the c, for which (c) equals /2, is increasing in this series of tasks, with the decrease of the modulus E of the layer. So, for E = 100 GPa, c ~ 109 s, and for E = 5 GPa, c ~ 1011 s (Fig. 8). The increase of the gradient (, E) /E bore the curve TABLE 3 Properties, parameters, and variables in the task series concerning the homogeneous medium Series No. Variable E Layer v 1 B Pa­2s­1 E Medium v 1 B Pa­2s­1 3 Compliance war B 5*10 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 = 3.2*109 ­28 5*10­25 5*10­25 5*10­25 5*10­25 5*10­25 5*10­25 5*10­23 5*10­21 5*10­19 5*10­17 5*10­23 5*10­21 5*10­19 5*10­24 5*10­21 Average fective Stress [] 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1E+004 11: Ewar=100 GPa Inversion of the gradient p 12: Ewar=50 GPa (E) 13: Ewar=25 GPa 14: Ewar=17 GPa 15: Ewar=5 GPa Ew ar = 5 GPa Ewa r= 17 GPa Ewar = 25 GPa Creep Compliance 1E+005 1E+006 1E+007 1E+008 1E+009 1E+010 1E+011 Ewa r= 50 GPa Ew ar=100 GPa Fig. 8. The dependence of the layer's (, E) at ant of stress equalizing, parameter: E 1E+012 1E+013 1E+014 bending zone is positive and it is negative afterwards. In the first half of the second period, with the increase of the layer's modulus E (in the range of warE > 5 GPa) and with the increase warE, the fective stress is increasing in the layer; however, in the second half, the fective stress is decreasing (Fig. 8). The juxtaposition of fective stresses in the layer's elastic modulus function for = 0, corresponding to the immediate reaction and the same for the long-term reaction, is shown in Fig. 9. Average fective Stress [] E = 5 GPa Immidiate reaction, =0 (E, ) = 0,76 (E/1GPa) 0,663 Long term reaction =3.2e10 (E, s) = 4,0 (E/1GPa)-0.34 Young's Modulus E [GPa] Fig. 9. Inversion of the gradient (E,) /E in the [0..3.2*1010] Within the range of the specific values of the layer's elastic modulus E < 17 GPa, a local minimum of the function Min[ (, E)] = 2.2 is developing. Within the range of E > 17 GPa, the immediate reaction corresponding to ~ 0, (, E) is the increasing function warE, which is approximated by the increasing power function with the positive exponent of +0.663 (Fig. 9). However, the long-term reaction, after expiration of a long period of , is approximated by a decreasing power function, with a negative exponent of ­0.34. Our simulations indicated that, at the values of warE [17..25 GPa], within the range of [108..109s], there occurs a local extreme value for the e = 3.16*108. Subsequent calculations (Table 3, task series 16-20) were concentrating on the determination of the influence of the surroundings's creep compliance otoB on the position of the extreme value of (e) = Max[ ], or the determination of the e. In this task series, ant values of the elastic properties and the ant value of the layer's creep compliance warB = 5*10­24 Pa­2s­1 were preserved, with the change of the surroundings's creep compliance. It was found that Fig. 10. The distribution of fective stresses (x, y, ) [Pa]. Lt: E = 50 GPa and = 3.16*109, Max( = 17.8 ); Right: E = 5 GPa and = 3.16*1012, Max( =1.7 ) with the increase of = log( otoB/ warB), within the range of 5, the values of = 4.79 and (e) = Max () = 5.37 did not change, although the value of e was changing in a specific manner (Fig. 11). Within the range of < 5, together with the decrease of b down to 0, the fect of the extreme value occurrence is decreasing and it disappears at = 0, corresponding to medium's homogeneous model conditions. Average fective Stress [] 5 4 3 2 1 Bwar==5*10-28 Pa-2 s-1 0 1E+000 1E+001 1E+002 1E+003 1E+004 1E+005 1E+006 1E+007 ( e) = 5,37 = 4,79 1E+008 1E+009 1E+010 1E+011 1E+012 1E+013 1E+014 Fig. 11. The dependence of (, ) on the stress equalizing ; parameter: , with the indicator corresponding to the series number of Table 3 With the preservation of the specific value in the cycle of simulation, a similar fect is caused by the change of the creep compliance of the layer and its surroundings. Fig. 12 presents three curves whose parameters fulfill the condition = 5. It was found that the increase of the surroundings's creep compliance value 2otoB > 1otoB led to the decrease of e by the number of orders equal to the number of the orders of creep compliance's increase, without a change of the maximum value of (e). So, for example, in two series of tasks for warB = and otoB, the increase was by 2 orders higher: 3*10­23 Pa2s­1 to 3*10­21 Pa2s­1. However, for otoB = the layer's creep compliance warB decreased from 3*10­28 Pa2s­1 to 5*10­30 Pa2s­1 causing reduction of the e from 5*107 s to 5*105 s. =4,79 ( e)=5,37 =5,37 Average fective Stress [] warB=5*10-28Pa-2 s-1 =4,79 4 3 warB=5*10-26Pa-2 s-1 warB=5* 10-24 Pa-2 s-1 21,22,23 23 B2 / warB1=0,01 B2 / warB1=1 war B2 / warB1=100 war war 1E+008 1E+009 1E+010 21 1E+002 1E+003 1E+004 1E+004 1E+005 1E+006 1E+007 1E+008 1E+009 1E+010 1E+011 1E+012 1E+000 1E+001 1E+002 1E+003 B2/B1 1E+005 1E+006 1E+007 Fig. 12. The dependence of (, warB) on the stress equalizing ; parameter: creep compliance warB, with the condition = 5 The graphs of those curves are parallel in the logarithmic scale of (Fig. 12); therore, the shift of a curve owing to the change of the layer's creep compliance from warB1 to warB2 is directly proportional to the quotient of warB2 / warB1, and it is expressed by the following function in case of the ant value of the proportion otoB/ warB: ( war B1 , ) ( war B2 , war B2 / warB1 ), for log(oto B / warB) 9) That relationship expresses the principle that the of principal stress equilizing in the layer surrounded by the rocks that are more susceptible to creep is directly proportional to the creep compliance of that layer and inversely proportional to the surroundings's creep compliance. For the conditions of task series 21 (Table 3), we studied the influence of the depth on stress equilizing. With the increase of depth, the initial fective stress was increasing, together with the value of the difference of Max[ ()] ­ . After 5*1012 s, regardless of the assumed depths H, fective stresses were the same (Fig. 13) and close to zero. Therore, the influence of depth on the stress equilizing process disappeared after a long period. Average fective Stress [] 1E+003 Average fective Stress [] Zp =630 m (Zp, 0)]=12,05 Zp =430 m (Zp, 0)]=8,42 Zp =230 m (Zp, 0)]=4,79 Creep Compliance 1E+004 1E+005 1E+006 1E+007 1E+008 1E+009 1E+010 1E+011 1E+012 Fig. 13. The influence of depth on fective stresses (for the conditions of series 21, Table 3) Similarly to the homogeneous medium, fective stresses can be presented as a function of creep compliance and parameterize it with , using function 6) for approximation. Fig. 14 presents the results of the related calculation for the 3.16*109 s in two task series 24 and 25, with the specific values of elastic properties (Table 3). In those simulations, creep compliance was a parameter, otoB [5*10­24, 5*10­21 Pa2s­1] was the ant, and the layer's creep compliance was the variable. The relationships (t ) (Figs. 4 and 7) and (warB) are similar (Fig. 14). The initial fective stress is getting close to (warB = 0) = 4.82 , with the decrease war of B 0 for the conditions of series 24, and in the case of series 25, it is getting close to (warB = 0) = 3.39 . Therore, the surroundings's creep compliance otoB influences the initial value of the layer's fective stress, and the value of (warB = 0) is increasing with the increase of otoB. The approximation of the results indicated compliance with function 6); however, only in the range of the values B < Bc (Fig. 14). 4. Summary and Conclusions Our model studies present the development of the stresses in , in a layer with a small slope, which layer and their surroundings displays elastic-viscous properties that were simulated by the behaviour of the elastic-viscous medium with the power creep law. Our studies simulated, in a simplified manner, the stress equilizing process in the rock mass and the primary stress state in the rock mass, depending on the rock properties and . We should mention that the results of our studies rer to the conditions that correspond to our model assumptions. oto B= 5e-24 Pa-2s-1 = 4,82 GPa Average fective Stress [] oto B=5e-21 Pa-2s-1 = 3,39 GPa = =3,16 *109 s 1.0E-034 Creep Compliance 1.0E-033 1.0E-032 1.0E-031 war B [Pa-2s-1] 1.0E-029 1.0E-028 24 B c 25B c Fig. 14. The dependence of (warB, otoB) on the layer's creep compliance at = 3.16*109 = ; parameter: the medium's creep compliance otoB In our model studies, we were looking for the stress changes in the layer, as a function of bulk and shear strain of the layer and its surroundings, in which the principal stress differences are decreasing and reducing to zero with . It was assumed that fective stress was the measure of stress equilizing. Our model studies were conducted for homogeneous and non-homogeneous conditions of the medium. The following were found in the homogeneous model of the rock mass with uniform properties of the layer and its surroundings: · The stress equilizing process is well described by hyperbolic function 1) which expresses the principle that the relative increase of stresses is inversely proportional to the measurement variable, which can be either or creep compliance. The initial value of fective stress for ~ 0 or B ~ 0 is proportional to the depth and it depends on the value of the Poisson coficient. · In the logarithmic scale of or creep compliance, we can distinguish three periods and ranges. In the second period (Fig. 2) and the second range of changes in creep compliance (Fig. 6), intense fective stress equilizing occurs. For the value of the geometric average of s, at the end of that range c or of the creep compliance Bc, fective stress corresponds to a half of the fective stress recorded at the beginning of the process. · The increase of the Poisson coficient does not influence the value of c; however, the increase of Young's modulus of the medium causes the decrease of the value of c. The increase of creep compliance does not influence the initial fective stresses, but it causes the increase of c and of the stress equilizing period. 1.0E-030 1.0E-027 1.0E-026 1.0E-025 1.0E-024 1.0E-023 · The proportions of c of two media that are different in respect of creep compliance correspond to the proportions of their creep compliance rates, and the proportions of Bc of two processes in which creep compliance is the variable correspond to the proportions of respective durations. The following were found in the non-homogeneous model of the rock mass, with diverse properties of the layer and the surroundings: · The stress equilizing process only initially complies with hyperbolic function 1). Similarly to the homogeneous medium, the layer's fective stresses are decreasing to zero with . The initial fective stress is increasing with the increase of the layer's elasticity modulus, and the second stress equilizing period becomes extended. The lower the value of the elasticity modulus, or the lower the differences between the layer's bulk mudulus, the longer is the stress equilizing process. · With the changes in the creep compliance of the layer and its surroundings, the stress equilizing function is increasing in and obtains its extreme value. The at which the said extreme value appears depends on the proportion of the layer's creep compliance to the surroundings's creep compliance. Change in the layer's creep compliance at ant surroundings creep compliance leads to a parallel shift of the fective stress function in the logarithmic scale of by which the stress equilizing process becomes extended. · The values of fective stresses with a local extreme value, occurring after a short period, are increasing almost proportionally with depth. After a very long , the influence of depth disappears. · Similarly to the homogeneous rock mass, the stress equilizing process can be presented for a variable creep compliance of the layer and as the parameter, keeping the surroundings creep compliance as a ant. This study is intended to mention only the relationships of the stress equilizing process with the elastic-viscous medium and . Particular conclusions concern only the conditions specified in this study. A wider use of those conclusions, especially for the purpose of stress state modelling in the mine working surroundings, would require proper examination of the particular rock-mass properties. However, we can say that, despite such reservations, the results of our studies demonstrate an essential role of and strain properties in the stress development process in an elastic-viscous medium. The results also indicate that the ruction of a geometric rock-mass model, with rheological properties and workings, especially in the case of non-homogeneous media, should be preceded by a calculation analysis, leading to the stress equilizing state, by simulation of the primary lithostatic stress state in the rock mass in that way. Later that stress state will be affected by mine workings. I wish to thank Dr. Danuta Flisiak for her kind assistance in proof reading. 873 Rerence Amadei B., Stephansson O., 1997. Rock stress and its measurement. Chapman & Hall, Londyn-Weinheim-Nowy Jork-Tokio-Melbourne-Madras. Brown E.T., Hoek E., 1978. Trends in relationships between measured rock in situ stresses and depth. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 15, p. 211-215. Butra J., Dbkowski R., Pawelus D., Szpak M., 2011. Wplyw napre pierwotnych na stateczno wyrobisk górniczych. Cuprum, 58 ). Filcek H., Walaszczyk J., Tajdu A., 1994. Metody komputerowe w geomechanice górniczej. lskie Wydawnictwo Techniczne, Katowice. Kortas G. (red.), 2008. Ruch górotworu w rejonie wysadów solnych. Wydawnictwo Inst. Gosp. Surowcami Mineralnymi i Energi PAN, Kraków. Kortas G., 2006. Distributions of Convergence in a Modular Structure Projecting a Multi-Level Salt Mine. Archives of Mining Sciences, Vol. 51, Iss. 4. Kortas G., Maj A., 2012. Ekspertyza geotechniczna dotyczca gruboci pólek stropowych, spgowych oraz filarów dla eksploatacji górniczej zloa ,,Nowy Ld" pomidzy poziomami +35 a -30 m npm. GeoConsulting dla Kopalni Gipsu i Anhydrytu ,,Nowy Ld" w Niwnicach, Kraków, grudzie 2012 (praca niepublikowana). Kortas G., 2006. Distributions of Convergence in a Modular Structure Projecting a Multi-Level Salt Mine. Archives of Mining Sciences, Vol. 51, Iss. 4, p. 547-562. Maj A., 2012. Convergence of gallery workings in underground salt mines. Archives of Mining Sciences, Monograph, No. 14. Ode H., 1968. Review of Mechanical Properties of Salt Relating to Salt-Dome Genesis. Salin Deposits, Geological Society of America, Special Paper 88, s. 543-593. Salustowicz A., 1965. Zarys mechaniki górotworu. Wyd. lsk, Katowice, s. 19-22. lizowski J., 2006. Geomechaniczne podstawy projektowania komór magazynowych gazu ziemnego w zloach soli kamiennej. Studia. Rozprawy. Monografie Nr 137, Wyd. IGSMiE PAN, Kraków. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archives of Mining Sciences de Gruyter

Equilizing of the Primary Stress State in the Rock Mass, Simulated by a Model of Layer in an Elastic-Viscous Medium

Archives of Mining Sciences , Volume 61 (4) – Dec 1, 2016

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Abstract

Arch. Min. Sci., Vol. 61 (2016), No 4, p. 853­873 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.1515/amsc-2016-0057 GRZEGORZ KORTAS* WYRÓWNYWANIE NAPRE PIERWOTNYCH W GÓROTWORZE SYMULOWANE MODELEM WARSTWY W ORODKU SPRYSTO-LEPKIM This paper is devoted to the analysis of the stress development process in the homogeneous and non-homogeneous rock mass. The rock-mass model consists of an elastic-viscous medium containing a layer (Fig. 1) that displays distinct geomechanical strain properties. When examining the process of stress equilizing in , the Norton-Bailey power creep law was applied in the numerical analysis. The relationship between fective stresses and , the modulus of elasticity, Poisson's coficient, and creep compliance were obtained. It was demonstrated that the relationship between fective stress and or creep compliance, for the assumed conditions in a homogeneous rock-mass, was approximated by hyperbolic functions 0 and 16). The process parameter included a certain value of creep compliance or of at which there occurred a half-way equilizing of primary stresses. An analogous function binds fective stresses with creep compliance. Our model studies indicated a number of relationships between bulk and shear strain with and creep compliance in the homogeneous and non-homogeneous rock mass, presented in Figs. 2-14, expressed by the functions of those specific parameters. The relationships obtained in this work resulted from our model assumptions. However, they demonstrated the influence of the geomechanical strain properties of rocks on the process of shaping the primary stress state in the rock mass and the tendency to reduce the principal stress differences in . Our research results suggested the necessity to simulate the primary stress state as an initial condition of the geomechanical numerical analysis concerning the rock-mass behaviour showing rheological properties. Keywords: primary stress state, non-homogeneous rock-mass, elastic-viscous medium, creep compliance, fective stress equilizing Wielowiekowe procesy geodynamiczne, doprowadzily do wyksztalcania si obecnego stanu napre w litosferze, który w zagadnieniach górniczych okrela si pierwotnym stanem napre górotworu. Praca powicona jest analizie dlugotrwalego procesu wyksztalcania si napre w jednorodnym i niejednorodnym górotworze, przeprowadzonej w trybie bada modelowych. Okrelano zmiany naprenia w orodku sprysto-lepkiego z warstw (Rys. 1), wykazujc odmienne odksztalceniowe wlaciwoci geomechaniczne. W obliczeniach numerycznych stosowano potgowe prawo pelzania Nortona-Bailey'a. * STRATA MECHANICS RESEARCH POLISH ACADEMY OF SCIENCES, UL. REYMONTA 27, 30-059 KRAKOW, POLAND Kwantyfikatorem zmieniajcych si z uplywem czasu rónic napre glównych bylo rednie naprenie ektywne w centralnym sektorze modelu geometrycznego badanego orodka. Otrzymano proste zwizki napre ektywnych z czasem, modulem sprystoci, wspólczynnikiem Poissona i podatnoci na pelzanie. Wykazano, e w jednorodnym górotworze zwizek naprenia ektywnego z czasem lub z podatnoci na pelzanie, aproksymuje funkcja hiperboliczna opisujca wyrównywanie napre glównych 0 i 16). Wykazano, e parametrem procesu wyrównywania si napre glównych jest pewna warto podatnoci na pelzanie Bc lub czasu c, przy których wystpuje polowiczne wyrównanie napre pocztkowych. Wykazano, e w orodku sprysto-lepkim z potgowym prawem pelzania wzgldny przyrost napre jest odwrotnie proporcjonalny do czasu lub do podatnoci na pelzanie. Z bada modelowych wynika szereg zwizków odksztalcalnoci objtociowej i postaciowej z czasem i podatnoci dla jednorodnego i niejednorodnego górotworu, przedstawionych na rysunkach 2-14 i wyraonych funkcjami tych parametrów. Stwierdzono, e funkcja wyrównywania si napre wyraa zasad, e wzgldny przyrost napre jest odwrotnie proporcjonalny do zmiennej wymiarowej, któr moe by czas lub podatno na pelzanie. W logarytmicznej skali czasu lub podatnoci wyodrbni mona trzy okresy lub przedzialy. W drugim okresie (Rys. 2) i drugim przedziale zmian podatnoci (Rys. 6) nastpuje intensywne wyrównywanie si napre ektywnych. Dla wartoci redniej geometrycznej czasów na kocach tego przedzialu c lub podatnoci na pelzanie Bc naprenie ektywne odpowiada polowie wartoci naprenia ektywnego na pocztku procesu. Stwierdzono, e wzrost wartoci wspólczynnika Poissona nie wplywa na warto c, natomiast wzrost modulu Younga orodka powoduje spadek wartoci c. Wzrost podatnoci na pelzanie nie wplywa na pocztkowe naprenia ektywne, ale powoduje wzrost c i okresu wyrównywania si napre. Stosunek c dwóch orodków rónicych si podatnoci na pelzanie odpowiada stosunkowi ich podatnoci, a stosunek Bc dwóch procesów, w których zmienn jest podatno na pelzanie, odpowiada stosunkowi czasów ich trwania. W orodku jednorodnym wykazujcym wlaciwoci reologiczne proces wyrównywania si napre glównych prowadzi do utworzenia hydrostatycznego stanu napre litostatycznych. Im podatno orodka na pelzanie jest wiksza, tym proces ten jest krótszy. Stwierdzono, e w modelu niejednorodnego górotworu o odmiennych wlaciwociach warstwy i jej otoczenia proces wyrównywania si napre jest tylko pocztkowo zgodny z funkcj hiperboliczn 1). Podobnie jak w orodku jednorodnym z uplywem czasu naprenia ektywne w warstwie d do zera. Pocztkowe naprenie ektywne ronie ze wzrostem modulu sprystoci warstwy oraz wydlueniu ulega drugi okres wyrównywania napre. Im warto modulu sprystoci warstwy jest mniejsza, czyli im mniejsza jest rónica modulów sprystoci objtociowej warstwy i jej otoczenia, tym proces wyrównywania napre jest dluszy. Rónicujc podatno warstwy i jej otoczenia, funkcja wyrównywania si napre w czasie ronie i osiga ekstremum, a czas, przy którym pojawia si to ekstremum zaley od stosunku podatnoci warstwy do podatnoci otoczenia. Zmiana podatnoci warstwy przy stalej podatnoci otoczenia prowadzi do równoleglego przesunicia w logarytmicznej skali czasu wykresu funkcji napre ektywnych, przez co proces wyrównywania si napre wydlua si. Wartoci napre ektywnych z lokalnym ekstremum, wystpujcym po krótkim okresie czasu, rosn prawie proporcjonalnie z glbokoci. Po bardzo dlugim czasie wplyw glbokoci zanika. Podobnie jak dla jednorodnego górotworu proces wyrównywania si napre mona przedstawi dla zmiennej podatnoci warstwy i czasu jako parametru przy zachowaniu stalej podatnoci otoczenia. Wyniki bada przedstawione w tej pracy pokazuj istotn rol czasu i wlaciwoci odksztalceniowych skal w procesie ksztaltowania si napre w orodku wykazujcym wlaciwoci reologiczne. Wskazuj take, e w obliczeniach numerycznych stanu napre wokól wyrobisk w orodkach niejednorodnych, napreniowe warunki pocztkowe w obszarach rónicych si podatnoci na pelzanie s odmienne. Taki stan napre pierwotnych moe by symulowany przez proces odwzorowujcy dlugotrwale oddzialywanie sil masowych. Slowa kluczowe: naprenia pierwotne, niejednorodno górotworu, orodek sprysto-lepki, podatno na pelzanie, wyrównywanie napre ektywnych 1. Introduction Long-term geodynamic processes have led to the development of the present-day stress state in the lithosphere, which is described as the primary stress state of the rock mass in mining research. Geomechanical calculations usually assume that the primary stresses resulted from mass forces and, consequently, the absolute values of horizontal stresses are smaller than those of vertical ones [e.g. Salustowicz 1965]. It is further assumed that Poisson's coficient is increasing with depth, and the stress differences are reduced (Brown & Hoeck, 1978). In situ measurements indicate that the observed distribution of stresses is considerably different from the distribution resulting from the present-day mass loads in certain areas (Amadei & Stephansson, 1997). It was found that, close to the continental plate motion and active orogenesis, horizontal stresses exceed vertical ones even several s at small depths. Such an primary stress state may essentially influence mining conditions and the method of deposit cuper excavation, e.g. in the Polish LGOM Basin (Butra et al., 2011). It is also known that non-homogeneous rock mass considerably influences the distribution of primary stresses; in particular, the rock density and rheological properties differences cause salt-dome uplifting, but with the occurrence of hydrostatic stress state in salt (Ode, 1968). The process of primary stress development on salt rocks can be simulated under model testing (Kortas, 2006). In the mathematical models used for the determination of stress and strain states around mine workings (Filcek et al., 1994), the primary stress state itutes the initial condition of the resolution of the systems of partial differential equations. The properties of rocks are usually studied by laboratory methods, with in situ examinations applied to the behaviour of the media made up of such rocks. Respective model tests with numerical analysis allow for the search for relationships between physical and -space variables which rlect the behaviour of the given medium in the assumed mathematical model. An example of that approach consists in the determination of the development of stress in , depending on the strain properties of particular stress areas, which is the subject-matter of the present research. The recognition of the primary stresses shaping process in that way is important for both knowledge and practice development. This paper presents model research results concerning the changes of stress in , in both homogeneous and non-homogeneous rock masses, with the rock mass being represented by a layer with separate strain properties in respect to the surrounding medium. 2. Research Method Our research on the rock mass behaviour was conducted by representation of the rock mass with elastic-viscous medium with the power creep law, similarly to e.g. (lizowski, 2006). When discussing the research method, the following were introduced: (i) description of the physical model, giving itutive relationships of stress and strain in , (ii) description of the assumed geometrical model, which represents a simple non-homogeneous form of the medium composed of a layer and its surroundings, and (iii) the properties of a certain hyperbolic function which was later applied to the analysis of the model study results. 2.1. Physical Model The numerical analysis relied on linear Hooke's law which can be transcribed in the form of two tensor equations ), binding stress and strain, of which one describes the shape change law and the other one the bulk change law: 1 2 E ) Where strain deviator: D = ­ Am ( ­ Kronecker's delta), stress deviator: D = ­ Am, strain: Am = 1/3 1 + 22 + 33) and stress: Am = 1/3 1 + 22 + 33). To simulate viscous behaviour, we used the Norton-Bailey power creep law, determined by the dependence of strain rate on the stress state and the t: Q 3 mA exp 2 RT n 1 t m 1 (2) where Q is the activation energy, R is the gas ant, T is temperature in Kelvin, and A is the material ant. The dependence of strain on the values of n and m in the Norton-Bailey power creep law is discussed in (Maj, 2012); in our calculations, the following assumptions were made: · n = 2 and m = 1. For those ants, equation (2) has the form of = 3/2 B . The function of the material ants A, Q, and R in (4), in the specific temperature T, dines the material's creep compliance B, determined by equation: B A exp Q RT (3) Practically, creep process activation energy Q is not determined for rocks, and R is a universal physical ant equal to the work fected by heating 1 mol of perfect gas by 1 degree Kelvin, being equal to 8.314462 J/(mol*K). For that reason, the determination of the value B(T) is the actual result of laboratory tests. In the strain process of the medium under a ant load, the fective stress s, expressed by principal stresses s1, s2, and s3 in (4), is changing in : (t ) 1 (t ) 2 (t )] 2 (t ) 3 (t )] 3 (t ) 1 (t )] (4) The process of stress equilizing and medium strain development, initiated by mass load, occurs only if (t ) > 0 and B 0 in certain medium zones. The process is concluded when the principal stress components are identical, that is = 0. Therore, fective stress may be a quantifier of the stress equilizing process. In the medium with the power creep law, multiplication, division, involution, or logarithming in itutive equations correspond to addition, substraction, multiplication, and division in a linear medium, respectively. Therore, let the value determine the power change of the value of viscous properties from 1B to 2B: log( 2B /1B) (5) Then, each increase of by 1.0 corresponds to the ten increase of the creep compliance B. Similarly, one can dine h as a power change of the bulk rigidity modulus. At the specific value of Poisson's coficient v, the diversity of the bulk rigidity modulus (Helmholtz modulus) E/[3 ­ 2v)] can be reduced to the function of Young's elongation strain modulus E: log( 2E /1E ) for (6) In that approach, the values of h and b are the parameters of shear and bulk strain changes in the medium. 2.2. Geometric Model and the Method of Study The primary geometric stress model adopted for stress equalizes testing was a medium composed of a layer and its surroundings, each with different physical properties. The same rockmass geometric model, presented in Fig. 1, was adopted for all calculations. In that model, a layer sloping at 20°, with ant vertical thickness of 15 m, was different than its surroundings. The layer's centre was located at the depth of 234 m, within a rectangle of the horizontal dimension of 330 m and the vertical one of 300 m. Within the small rectangle MNOP (Fig. 1), average stress values were calculated. The occurrence of an anhydrite layer on the southern edge of the ForeSudeten Monocline can be an example of the corresponding conditions (Kortas & Maj, 2012). Surroundings 300 15 m er Lay 150 m M 330 m Fig. 1. Geometric Model Surroundings Numerical calculations were conducted by the finite element method, with the assumption of the plain state of strain, with the zero displacement boundary condition in horizontal direction. The values of (t ), changing in , were the searched study results, calculated here as the average value of the fective stresses obtained in part of the layer, in the horizontal distance of ±75 m from the centre of the model (Fig. 1). The sets of tasks, parameterized with the variables, created series of tasks. A graphic presentation of the obtained functional relationships for several series of tasks revealed the relationships between stresses, strain properties, and that were looked for under model studies. 2.3. The Function of Stress Equilizing in In the rock mass that is susceptible to creep, the initiation of the diversity of fective stresses occurs, resulting from sudden changes in loads caused for example by a tectonic motion, with the drop of fective stresses resulting from long-term rheological processes. The process of changes in stress can be simulated in the first approximation by studying, after applying preliminary load, the behaviour of a model of medium in the . When simulating a lithostatic process in rock salt (e.g. Kortas, 2006), it was found that the stresses existing in a medium that had viscous properties were subjected to equalization or equilizing after the initial elastic reaction ( = 0) at the ( ): min = 1 < 2 < 3 = max lithostatic process 1 = 2 = 3 = max (). During the simulation of the equilizing of stresses in the t, we can distinguish three periods: (i) in the first short period (until the p), the principal stresses are slightly different from the initial stresses resulting from elastic reaction, (ii) the second period (p k) is the basic period of stress equilizing, (iii) in the third period (k < ), the differences in the principal stress values disappear, and, after a very long , principal stresses are almost the same. Let the non-nominal value parameter , with the conventional value belonging to the range of 0 < < 0.01 serve the determination of the boundaries of the three periods of principal stress equilizing. Then, through parameter , the s of the beginning p and of the end k of the second stress balance period are correlated with the boundary values of the fective stresses by equation: d ( 0) p) ( k) (7) That concerns the plain state of strain 2() = 3(). In the homogeneous rock-mass model, the elastic reaction to the mass load pz causes the development of principal stresses that are directly proportional to the depth H (in the direction Y in Fig. 1) and the density of the medium , that is: 1 2 min 2 H g max pz pz [ / )] (8) In the plain state of strain, the fective stress corresponds to the difference between the maximum and minimum principal stresses: max min 1 2 pz 1 (9) During the under discussion process, the stresses () change from the initial value of s to the final value of () = 0. Let that change in the medium with a power creep law be described by decreasing hyperbolic function 0), with three ants Cp, Ck, and c, in the form of: Cp 1 / Ck 0) If in 0), for the average stresses of = m and the maximum stresses of = max, the , then () min and Ck = min. For the fective stresses of = , reducing to zero with , the ant Ck = 0. At the beginning of the process = 0, the stresses: = m = m , = max = max and = = . Let (c) = /2 in the c. Then ( c) [Cp ­ Ck]/2, and for fective stresses: (c) = /2. Upon introduction of the values of those ants to 0), we can determine the particular functions max(), min (), and m(). Function 0) will have the following form for fective stresses: / 1) Transformations of equations (7) and 1) will produce simple relations 2), binding p, k, and c with : p c p k p k 2) With those assumptions, for a small value of , the ant of the stress equilizing process c, called here the period of half-way equilizes of stresses, is the geometric average of the s p and k, determined by functions (7) and 1). Function 1) was applied to the approximation of the results of the model studies concerning the stress state equilizing processes in the layer and its surroundings. 3. Research Results and Result Analysis The results of the completed model studies are presented in two sections below. In the first case, with the assumption of a homogeneous rock mass, the layer and its surroundings displayed the same geomechanical properties, and in the second case, the properties of the layer and of its surroundings were diverse. The selected values of the shear and bulk strain parameters of the layer and its surroundings, as well as the calculated average values of the fective stresses, were tabularized and presented graphically as a function of . 3.1. Research Results for a Homogeneous Medium The purpose of the initial studies was to determine the relationship between the stress equilizing process and the elastic and viscous properties of a homogeneous rock mass. The preliminary series of tasks assumed the following: E = 17 GPa, v = 0.15, and B = 5*10­28 Pa­2s­1. Stress equilizing was simulated during the period of 3.16*1013 s, or 100,000 years. Table 1 contains the values of average principal, fective, and average stresses, calculated in several periods, and Fig. 2 shows the respective graphs in the function. For the = 0, min = ­5.05 , max = ­0.89 , and = 4.16 were obtained in the central sector of the layer. In the second and basic period of the studied process, the minimum principal stress was ant: min() = ; however, with the increase of , the maximum principal stresses were dropping: max() min and () = max() ­ min() 0 (Fig. 2). Assuming = 0.005, the boundary values of the fective stresses of the second period of stress equilizing amounted to ­ (p) = 0.025 , (k) ­ () = 0.025 , tending to 1() = 2() = 3 = min after the conclusion of the process. TABLE 1 Average values of stresses within the layer s 3*100 3*10­7 3*10­10 3*10­13 3*10­16 1 i 2 i 3 i i sr (i 3) ­0.884 ­0.887 ­2.612 ­5.006 ­5.009 ­0.884 ­0.891 ­2.612 ­5.006 ­5.009 ­5.0096 ­5.0096 ­5.0096 ­5.0096 ­5.0096 4.1250 4.1229 2.3972 0.0035 0.0004 ­2.2592 ­2.2610 ­4.4115 ­5.0073 ­5.0095 First period of stress equalizing Stress [] -1 -2 -3 -4 -5 -6 1E+005 = 2= max Second period of stress equalizing + 2+ 3)/3 Third period of stress equalizing min 1E+012 1E+013 1E+014 1E+009 Fig. 2. Average principal and fective stresses in the layer, in the function of The examples of the distribution of the average fective stresses at the beginning and at the end of the simulated stress equilizing process are illustrated in Fig. 3. In the initial series of tasks, the stress equilizing process was simulated with the ants values of the medium's viscous properties and diverse elastic property values. The assumed parameters are presented in Table 2 (series 1-4) and the results in Fig 4. Function 1) was applied to the approximation of the calculation results presented in Fig. 2, marking with upper index the task series numbers in Table 2. 1E+006 1E+007 1E+008 1E+010 1E+011 Fig. 3. Distribution of the fective stress (x, y, ) [Pa]: Lt: ( = 0 s) Max = 5.96 ; Right: = 3*1013 s) Max = 0.004 TABLE 2 The values of variables and ants of the medium in task series from 1 to 6 Series No. E [GPa] v [1] B [Pa­2s­1] [] Bc [Pa­2s­1] c 5*10­28 5*10­28 5*10­28 5*10­28 5*10­25 5*10­22 from 5*10­29 to 5*10­23 from 5*10­29 to 5*10­18 from 3.15*105 to 3.15*1013 from 3.15*105 to 3.15*1014 from 3.15*105 to 3.15*1013 from 3.15*105 to 3.15*1014 from 1.00*103 to 1.00*1011 from 3.15*102 to 3.15*106 1*105 s 1*1010 s ­ ­ ­ ­ ­ ­ 4*10­26 4*10­21 0.34*1012 8.33*1012 8.33*1012 0.34*1012 0.34*109 0.34*106 ­ ­ The test results indicated that, with the ant creep compliance B, the following were obtained in task series 1 and 4: ­ For series 1 and 4 (v = 0.15), identical values of = 4.16 and identical s of half-way fective stress equilizing: 1c = 4c = 0.357*1010s. ­ For series 2 and 3 (v = 0.30), identical values of = 2.89 and identical s of half-way fective stress equilizing: 2c = 3c = 8.33*1011 s. ­ The change of in series 1 and 3, as well as 2 and 4, resulting from the change of the Poisson coficient value, in accordance with equation (9). For the conditions B = and v = the relationship of the two c (E) values is approximately inversely proportional to the value E and proportional to : 2 2 c ( E) 1 1 c ( E) 2 1 ( 0) , B 3) = 4 Average fective Stress [] v = 0,15 2 v = 0,30 =3 /2= /2 E=1 GPa /2 =3 /2 E=17 GPa 2 1 3 c= c =4 1.0E+005 1.0E+006 1.0E+007 2 1 c c 1 2 1.0E+008 1.0E+009 Fig. 4. The dependence of the layer's (, E, ) on the stress equalizing ; parameters: E and Therore, with the drop of the modulus E, the of half-way stress equilizing is increasing. Subsequent tests of task series 5 and 6 (Table 2) consisted in checking the influence of the changes in the creep compliance of the medium B on the stress equilizing process. The respective calculations were conducted with the same values of elastic properties. The simulation results are presented in Fig. 5. The simulations (, B) indicated that, at the specific values of elastic properties, the string of applied values of the parameter B led to the generation of similar graphs (Fig. 5). The value of does not depend on the medium's creep compliance. With the decrease of creep compliance value, the s p, c, and k are increasing. The relationship between creep compliance and the half-way stress equilizing c is then expressed by this proportion: B B , or c c 1 10 , E The influence of the change of the creep compliance 1B 2B can be reduced to equation: 1.0E+010 1.0E+011 1.0E+012 1.0E+013 1.0E+014 4) 1.0E+015 ( 2 c , 2B) B / 2B, 1B) 5) Average fective Stress [] 1.0E+003 4 4B = 5*10-28Pa-2s-1 5 5B = 5*10-25 Pa-2 s-1 6 6B = 5*10-22Pa-2s-1 /2 1.0E+004 1.0E+005 1.0E+014 1.0E+015 1.0E+007 1.0E+008 1.0E+009 1.0E+010 1.0E+011 1.0E+012 1.0E+013 Fig. 5. The dependence of the layer's (, B) on the stress equalizing ; parameter: B Since for the specific z the of the conclusion of the second stress equilizing period k is increasing with the value of c 1), together with the decrease of creep compliance; therore, the lower the rock mass's creep compliance the longer the stress equilizing period. The duration of the second stress equilizing period is then inversely proportional to creep compliance. And thus, for series 6: c 108 s and for series 4: c 1014 s (Fig. 5). Model studies demonstrated that the influence of creep compliance on the stress equilizing is similar as process depends on duration. Consequently, in function 1), instead of the , we can use the variable B, an will be a process parameter as a ant, that is: ( B) 1 B / Bc Consequently, instead of the ant c in 1), there is Bc in 6): the creep compliance at which half-way fective stress equilizing occurs, in the = adopted in the task series. Irrigation of silt formation, or rock fracturing and crushing, are the examples of the process leading to the increase of the rock's creep compliance. Similarly to the influence of on stresses, shown in Fig. 2, we can identify three periods of the influence of creep compliance on fective stress equilizing (Fig. 6). The creep boundaries in 1.0E+006 , where lim ( B) 6) the second period are associated with the small conventional value of , dined as in formula (7), but with p and k replaced by Bp and Bk, respectively. Average fective Stress [] First zone of creep compliance Second zone of creep compliance Bp Bc Bk Creep Compliance B (in logaritmic scale) Fig. 6. The zones of the influence of creep compliance on stress equalizing, for the k = In zone (i), B < Bp, the creep compliance value does not significantly influence the change of the maximum values of , in zone (ii), Bp < B < Bk, the increase of creep compliance essentially influences stress equilizing, and in zone (iii), Bk < B, the increase of the creep compliance value B does not significantly influence the change of the values of which are close to zero. The value of Bc determines the creep compliance at which half-way fective stress equilizing will occur in the assumed , or (Bc) = /2. For the specific values of elastic properties, the parameter k is associated with Bc by proportion: Bc Bc Third zone of creep compliance or 10 , E 7) Fig. 7 presents the results of (B) calculations in two task series, 7 and 8 (Table 2.), for two s: = k 105 s and 1010 s. According to 7), we received log( 8 / 7) = log010 s /105 s) = log( 8Bc,7Bc) = log(4*10­21 ­2 ­1 Pa s /4*10­26 Pa­2s­1) = 5. Relationships 8) occur between the creep compliance values of Bp, Bk, and Bc, depending on z, similarly to 2): Bp Bc 1 , ) Bc Bp Bk )2 Bc B p Bk 8) =4,16 GPa Average fective Stress [] =2,08 GPa = 105 s = 1010 s 1.0E-029 1.0E-028 1.0E-027 1.0E-026 1.0E-025 1.0E-024 1.0E-023 1.0E-022 1.0E-021 1.0E-020 1.0E-019 1.0E-018 Bc=4*10-26Pa-2s-1 Bc=4*10-21Pa-2s-1 Creep Compilance Bwar [Pa-2s-1] Fig. 7. The dependence of the layer's (B, ) on the compliance of the medium B, parameter: Relationships 8) characterized the equilizing of stresses resulting from the increase of creep compliance of the elastic-viscous medium, with power creep law. We should, however, mention that those relationships will be different if a change of bulk strain occurs, together with a change of the medium's creep compliance. 3.2. Research Results for a Non-Homogeneous Medium The non-homogeneous medium is characterized by diverse properties of its components, i.e. the layer and its surroundings in the model under discussion. Another series of model studies concerned first the influence of the t on the average fective stresses () in the layer with the creep compliance of warB = 10­28 Pa­2s­1, and otoB = 10­25 Pa­2s­1. In those studies, the creep compliance of the surroundings is higher than that of the layer, while the diversity of the surroundings's creep compliance otoB and of the layer warB is characterized by = 3. Consequently, the layer's creep compliance is by 3 orders lower than that of the surroundings. In the first series of tests (Table 3, series 11-15), the analysts were studying the dependence of fective stresses on (, warE), at specific properties, parameterizing the layer's elasticity modulus war E [5..100 GPa]. The calculation results are presented in Fig. 8. Similarly to the homogeneous medium model, the dependence of (, E) is a decreasing function, with asymptote (, E) = 0, determining the status after the principal stresses were equalized in the layer. The value of the c, for which (c) equals /2, is increasing in this series of tasks, with the decrease of the modulus E of the layer. So, for E = 100 GPa, c ~ 109 s, and for E = 5 GPa, c ~ 1011 s (Fig. 8). The increase of the gradient (, E) /E bore the curve TABLE 3 Properties, parameters, and variables in the task series concerning the homogeneous medium Series No. Variable E Layer v 1 B Pa­2s­1 E Medium v 1 B Pa­2s­1 3 Compliance war B 5*10 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 5*10­28 = 3.2*109 ­28 5*10­25 5*10­25 5*10­25 5*10­25 5*10­25 5*10­25 5*10­23 5*10­21 5*10­19 5*10­17 5*10­23 5*10­21 5*10­19 5*10­24 5*10­21 Average fective Stress [] 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1E+004 11: Ewar=100 GPa Inversion of the gradient p 12: Ewar=50 GPa (E) 13: Ewar=25 GPa 14: Ewar=17 GPa 15: Ewar=5 GPa Ew ar = 5 GPa Ewa r= 17 GPa Ewar = 25 GPa Creep Compliance 1E+005 1E+006 1E+007 1E+008 1E+009 1E+010 1E+011 Ewa r= 50 GPa Ew ar=100 GPa Fig. 8. The dependence of the layer's (, E) at ant of stress equalizing, parameter: E 1E+012 1E+013 1E+014 bending zone is positive and it is negative afterwards. In the first half of the second period, with the increase of the layer's modulus E (in the range of warE > 5 GPa) and with the increase warE, the fective stress is increasing in the layer; however, in the second half, the fective stress is decreasing (Fig. 8). The juxtaposition of fective stresses in the layer's elastic modulus function for = 0, corresponding to the immediate reaction and the same for the long-term reaction, is shown in Fig. 9. Average fective Stress [] E = 5 GPa Immidiate reaction, =0 (E, ) = 0,76 (E/1GPa) 0,663 Long term reaction =3.2e10 (E, s) = 4,0 (E/1GPa)-0.34 Young's Modulus E [GPa] Fig. 9. Inversion of the gradient (E,) /E in the [0..3.2*1010] Within the range of the specific values of the layer's elastic modulus E < 17 GPa, a local minimum of the function Min[ (, E)] = 2.2 is developing. Within the range of E > 17 GPa, the immediate reaction corresponding to ~ 0, (, E) is the increasing function warE, which is approximated by the increasing power function with the positive exponent of +0.663 (Fig. 9). However, the long-term reaction, after expiration of a long period of , is approximated by a decreasing power function, with a negative exponent of ­0.34. Our simulations indicated that, at the values of warE [17..25 GPa], within the range of [108..109s], there occurs a local extreme value for the e = 3.16*108. Subsequent calculations (Table 3, task series 16-20) were concentrating on the determination of the influence of the surroundings's creep compliance otoB on the position of the extreme value of (e) = Max[ ], or the determination of the e. In this task series, ant values of the elastic properties and the ant value of the layer's creep compliance warB = 5*10­24 Pa­2s­1 were preserved, with the change of the surroundings's creep compliance. It was found that Fig. 10. The distribution of fective stresses (x, y, ) [Pa]. Lt: E = 50 GPa and = 3.16*109, Max( = 17.8 ); Right: E = 5 GPa and = 3.16*1012, Max( =1.7 ) with the increase of = log( otoB/ warB), within the range of 5, the values of = 4.79 and (e) = Max () = 5.37 did not change, although the value of e was changing in a specific manner (Fig. 11). Within the range of < 5, together with the decrease of b down to 0, the fect of the extreme value occurrence is decreasing and it disappears at = 0, corresponding to medium's homogeneous model conditions. Average fective Stress [] 5 4 3 2 1 Bwar==5*10-28 Pa-2 s-1 0 1E+000 1E+001 1E+002 1E+003 1E+004 1E+005 1E+006 1E+007 ( e) = 5,37 = 4,79 1E+008 1E+009 1E+010 1E+011 1E+012 1E+013 1E+014 Fig. 11. The dependence of (, ) on the stress equalizing ; parameter: , with the indicator corresponding to the series number of Table 3 With the preservation of the specific value in the cycle of simulation, a similar fect is caused by the change of the creep compliance of the layer and its surroundings. Fig. 12 presents three curves whose parameters fulfill the condition = 5. It was found that the increase of the surroundings's creep compliance value 2otoB > 1otoB led to the decrease of e by the number of orders equal to the number of the orders of creep compliance's increase, without a change of the maximum value of (e). So, for example, in two series of tasks for warB = and otoB, the increase was by 2 orders higher: 3*10­23 Pa2s­1 to 3*10­21 Pa2s­1. However, for otoB = the layer's creep compliance warB decreased from 3*10­28 Pa2s­1 to 5*10­30 Pa2s­1 causing reduction of the e from 5*107 s to 5*105 s. =4,79 ( e)=5,37 =5,37 Average fective Stress [] warB=5*10-28Pa-2 s-1 =4,79 4 3 warB=5*10-26Pa-2 s-1 warB=5* 10-24 Pa-2 s-1 21,22,23 23 B2 / warB1=0,01 B2 / warB1=1 war B2 / warB1=100 war war 1E+008 1E+009 1E+010 21 1E+002 1E+003 1E+004 1E+004 1E+005 1E+006 1E+007 1E+008 1E+009 1E+010 1E+011 1E+012 1E+000 1E+001 1E+002 1E+003 B2/B1 1E+005 1E+006 1E+007 Fig. 12. The dependence of (, warB) on the stress equalizing ; parameter: creep compliance warB, with the condition = 5 The graphs of those curves are parallel in the logarithmic scale of (Fig. 12); therore, the shift of a curve owing to the change of the layer's creep compliance from warB1 to warB2 is directly proportional to the quotient of warB2 / warB1, and it is expressed by the following function in case of the ant value of the proportion otoB/ warB: ( war B1 , ) ( war B2 , war B2 / warB1 ), for log(oto B / warB) 9) That relationship expresses the principle that the of principal stress equilizing in the layer surrounded by the rocks that are more susceptible to creep is directly proportional to the creep compliance of that layer and inversely proportional to the surroundings's creep compliance. For the conditions of task series 21 (Table 3), we studied the influence of the depth on stress equilizing. With the increase of depth, the initial fective stress was increasing, together with the value of the difference of Max[ ()] ­ . After 5*1012 s, regardless of the assumed depths H, fective stresses were the same (Fig. 13) and close to zero. Therore, the influence of depth on the stress equilizing process disappeared after a long period. Average fective Stress [] 1E+003 Average fective Stress [] Zp =630 m (Zp, 0)]=12,05 Zp =430 m (Zp, 0)]=8,42 Zp =230 m (Zp, 0)]=4,79 Creep Compliance 1E+004 1E+005 1E+006 1E+007 1E+008 1E+009 1E+010 1E+011 1E+012 Fig. 13. The influence of depth on fective stresses (for the conditions of series 21, Table 3) Similarly to the homogeneous medium, fective stresses can be presented as a function of creep compliance and parameterize it with , using function 6) for approximation. Fig. 14 presents the results of the related calculation for the 3.16*109 s in two task series 24 and 25, with the specific values of elastic properties (Table 3). In those simulations, creep compliance was a parameter, otoB [5*10­24, 5*10­21 Pa2s­1] was the ant, and the layer's creep compliance was the variable. The relationships (t ) (Figs. 4 and 7) and (warB) are similar (Fig. 14). The initial fective stress is getting close to (warB = 0) = 4.82 , with the decrease war of B 0 for the conditions of series 24, and in the case of series 25, it is getting close to (warB = 0) = 3.39 . Therore, the surroundings's creep compliance otoB influences the initial value of the layer's fective stress, and the value of (warB = 0) is increasing with the increase of otoB. The approximation of the results indicated compliance with function 6); however, only in the range of the values B < Bc (Fig. 14). 4. Summary and Conclusions Our model studies present the development of the stresses in , in a layer with a small slope, which layer and their surroundings displays elastic-viscous properties that were simulated by the behaviour of the elastic-viscous medium with the power creep law. Our studies simulated, in a simplified manner, the stress equilizing process in the rock mass and the primary stress state in the rock mass, depending on the rock properties and . We should mention that the results of our studies rer to the conditions that correspond to our model assumptions. oto B= 5e-24 Pa-2s-1 = 4,82 GPa Average fective Stress [] oto B=5e-21 Pa-2s-1 = 3,39 GPa = =3,16 *109 s 1.0E-034 Creep Compliance 1.0E-033 1.0E-032 1.0E-031 war B [Pa-2s-1] 1.0E-029 1.0E-028 24 B c 25B c Fig. 14. The dependence of (warB, otoB) on the layer's creep compliance at = 3.16*109 = ; parameter: the medium's creep compliance otoB In our model studies, we were looking for the stress changes in the layer, as a function of bulk and shear strain of the layer and its surroundings, in which the principal stress differences are decreasing and reducing to zero with . It was assumed that fective stress was the measure of stress equilizing. Our model studies were conducted for homogeneous and non-homogeneous conditions of the medium. The following were found in the homogeneous model of the rock mass with uniform properties of the layer and its surroundings: · The stress equilizing process is well described by hyperbolic function 1) which expresses the principle that the relative increase of stresses is inversely proportional to the measurement variable, which can be either or creep compliance. The initial value of fective stress for ~ 0 or B ~ 0 is proportional to the depth and it depends on the value of the Poisson coficient. · In the logarithmic scale of or creep compliance, we can distinguish three periods and ranges. In the second period (Fig. 2) and the second range of changes in creep compliance (Fig. 6), intense fective stress equilizing occurs. For the value of the geometric average of s, at the end of that range c or of the creep compliance Bc, fective stress corresponds to a half of the fective stress recorded at the beginning of the process. · The increase of the Poisson coficient does not influence the value of c; however, the increase of Young's modulus of the medium causes the decrease of the value of c. The increase of creep compliance does not influence the initial fective stresses, but it causes the increase of c and of the stress equilizing period. 1.0E-030 1.0E-027 1.0E-026 1.0E-025 1.0E-024 1.0E-023 · The proportions of c of two media that are different in respect of creep compliance correspond to the proportions of their creep compliance rates, and the proportions of Bc of two processes in which creep compliance is the variable correspond to the proportions of respective durations. The following were found in the non-homogeneous model of the rock mass, with diverse properties of the layer and the surroundings: · The stress equilizing process only initially complies with hyperbolic function 1). Similarly to the homogeneous medium, the layer's fective stresses are decreasing to zero with . The initial fective stress is increasing with the increase of the layer's elasticity modulus, and the second stress equilizing period becomes extended. The lower the value of the elasticity modulus, or the lower the differences between the layer's bulk mudulus, the longer is the stress equilizing process. · With the changes in the creep compliance of the layer and its surroundings, the stress equilizing function is increasing in and obtains its extreme value. The at which the said extreme value appears depends on the proportion of the layer's creep compliance to the surroundings's creep compliance. Change in the layer's creep compliance at ant surroundings creep compliance leads to a parallel shift of the fective stress function in the logarithmic scale of by which the stress equilizing process becomes extended. · The values of fective stresses with a local extreme value, occurring after a short period, are increasing almost proportionally with depth. After a very long , the influence of depth disappears. · Similarly to the homogeneous rock mass, the stress equilizing process can be presented for a variable creep compliance of the layer and as the parameter, keeping the surroundings creep compliance as a ant. This study is intended to mention only the relationships of the stress equilizing process with the elastic-viscous medium and . Particular conclusions concern only the conditions specified in this study. A wider use of those conclusions, especially for the purpose of stress state modelling in the mine working surroundings, would require proper examination of the particular rock-mass properties. However, we can say that, despite such reservations, the results of our studies demonstrate an essential role of and strain properties in the stress development process in an elastic-viscous medium. The results also indicate that the ruction of a geometric rock-mass model, with rheological properties and workings, especially in the case of non-homogeneous media, should be preceded by a calculation analysis, leading to the stress equilizing state, by simulation of the primary lithostatic stress state in the rock mass in that way. Later that stress state will be affected by mine workings. I wish to thank Dr. Danuta Flisiak for her kind assistance in proof reading. 873 Rerence Amadei B., Stephansson O., 1997. Rock stress and its measurement. Chapman & Hall, Londyn-Weinheim-Nowy Jork-Tokio-Melbourne-Madras. Brown E.T., Hoek E., 1978. Trends in relationships between measured rock in situ stresses and depth. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 15, p. 211-215. Butra J., Dbkowski R., Pawelus D., Szpak M., 2011. Wplyw napre pierwotnych na stateczno wyrobisk górniczych. Cuprum, 58 ). Filcek H., Walaszczyk J., Tajdu A., 1994. Metody komputerowe w geomechanice górniczej. lskie Wydawnictwo Techniczne, Katowice. Kortas G. (red.), 2008. Ruch górotworu w rejonie wysadów solnych. Wydawnictwo Inst. Gosp. Surowcami Mineralnymi i Energi PAN, Kraków. Kortas G., 2006. Distributions of Convergence in a Modular Structure Projecting a Multi-Level Salt Mine. Archives of Mining Sciences, Vol. 51, Iss. 4. Kortas G., Maj A., 2012. Ekspertyza geotechniczna dotyczca gruboci pólek stropowych, spgowych oraz filarów dla eksploatacji górniczej zloa ,,Nowy Ld" pomidzy poziomami +35 a -30 m npm. GeoConsulting dla Kopalni Gipsu i Anhydrytu ,,Nowy Ld" w Niwnicach, Kraków, grudzie 2012 (praca niepublikowana). Kortas G., 2006. Distributions of Convergence in a Modular Structure Projecting a Multi-Level Salt Mine. Archives of Mining Sciences, Vol. 51, Iss. 4, p. 547-562. Maj A., 2012. Convergence of gallery workings in underground salt mines. Archives of Mining Sciences, Monograph, No. 14. Ode H., 1968. Review of Mechanical Properties of Salt Relating to Salt-Dome Genesis. Salin Deposits, Geological Society of America, Special Paper 88, s. 543-593. Salustowicz A., 1965. Zarys mechaniki górotworu. Wyd. lsk, Katowice, s. 19-22. lizowski J., 2006. Geomechaniczne podstawy projektowania komór magazynowych gazu ziemnego w zloach soli kamiennej. Studia. Rozprawy. Monografie Nr 137, Wyd. IGSMiE PAN, Kraków.

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Archives of Mining Sciencesde Gruyter

Published: Dec 1, 2016

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