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Mine slope stability analysis by coupled finite element modelling
To cite this paper: Int. J. Rock Mech. & Min. Sci. 34:3-4, paper No. 242.
Copyright © 1997 Elsevier Science Ltd
Copyright © 1997 Elsevier Science Ltd Int. J. Rock Mech. & Min. Sci. Vol. 34, No. 3-4, 1997 ISSN 0148-9062 To cite this paper: Int. J. RockMech. &Min. Sci. 34:3-4, Paper No. 242
MINE S L O P E S T A B I L I T Y A N A L Y S I S B Y C O U P L E D FINITE ELEMENT MODELLING W.C~ P a r i s e a u ; S.C. S c h m e l t e r ; A . K . S h e i k
Department of Mining Engineering University of Utah, 315 WBB Salt Lake City, Utah, 84112-1183 USA ABSTRACT
The potentially destabilizing effect of water pressure on rock slope stability is examined assuming coupled poroelastic/plastic behavior and compared with previous poroelastic results. A coupled finite element code that accounts for the simultaneous effects of rock mass deformation and transient fluid flow is used for this purpose. Rock mass behavior is based on the concept of effective stress, Hooke's law, Darcy's law, associated plasticity and a parabolic yield condition appropriate to rock masses. The main effect of plasticity, which limits the range of purely elastic behavior by rock mass strength, is greater displacements and persistent yielding. Yielding anticipated in poroelastic analyses, where the ratio of strength to stress is less than one, is initially somewhat more extensive than in the poroelastic/plastic case, but diminishes considerably with time. In the poroelastic/plastic case, yielding that occurs during a slope cut persists in time and space despite depressurization. Applicability of poroelastic/plastic finite element analysis to actual open pit mine slopes is demonstrated. Copyright @ 1997 Elsevier Science Ltd
KEYWORDS Slope Stability • C o u p l e d M o d e l l i n g • Poroelasticity • Plasticity • R o c k Masses • O p e n Pit M i n e s • Seepage BACKGROUND
Detournay, Cheng 1993 present an excellent discussion ofporoelastic theory. Cui et al. 1994 present a finite element formulation. In soil mechanics, poroelastic/plastic analyses (e.g, Aubury, Hujeux 1979, Siriwardane, Desai 1981, Chang, Duncan 1983, Yamagami et al. 1985, Oka et al. 1986, Borja, Lee 1990) closely followed the intense development of poroelastic analyses during the 1980s and focused mainly on the consolidation problem using the popular Cam-Clay material model and critical state concept (Schofield, Wroth 1968). In rock mechanics, poroelasticity is experiencing a revival that is based on the fundamental work of Biot 1941,1955, which also underpins much work in soil mechanics, and is intended for application to petroleum engineering. Application of coupled models to mining engineering problems are few and generally use improvised material models to account for the effect of joints on the rock mass response (e.g., Ouyang, Elsworth 1993, Bai, Elsworth 1994, Mohammed, Mitri 1996). Modelling relatively large rock masses associated with surface mines often precludes accounting for individual joints, fractures, faults and so forth, so some type of equivalent properties approach is usually needed. Most equivalent properties models depend on the assumption of a representative volume
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element (RVE) that in jointed rock masses implies a test volume many times greater in linear dimension than joint spacing. Such models have limited utility in numerical analyses because the smallest element or cell size must be no smaller than the RVE. An approach to equivalent properties of porous, jointed rock masses that does not depend on the assumption of a RVE is outlined by Pariseau 1991, 1993. Once the equivalent elastic moduli, hydraulic conductivities and strengths are known, application of conventional coupled finite element technology is possible. Recent coupled poroelastic mine slope stability analyses (Pariseau, Schmelter 1995, Schmelter, Pariseau 1996, 1997) under: (1) dry conditions (no water pressure), (2) wet conditions with fixed water pressures (semi-coupled) and (3) wet conditions where water pressures changed with time and rock mass deformation (fully coupled) resulted in significant differences in the distribution of the local factor of safety, which is defined as the ratio of strength to stress at a point. The same preexcavation total stresses, slope geometry, elastic moduli, rock mass strengths and hydraulic conductivities were used in each case. The role of fluid (water) compressibility was also examined in the fully coupled case. An outline of the poroelastic model formulation and the finite element form of the governing system of equations used here is given by Pariseau 1996. Additional discussions relative to possible application to underground as well as surface mining are presented by Pariseau 1994, 1995. Figure 1 shows the model mesh used to examine the main features ofporoelastic analysis of rock slope stability. The mesh in Fig. 1 contains 2100 elements. Slope angle is 45 degrees; slope height is 300 m (1,000 ft) and bench height is 15 m (50 ft). The slope is mechanically and hydraulically homogenous and isotropic. The premining stress is due to gravity alone; gravity is switched on and the material allowed to settle under the weight of solid and fluid contributions to total weight before mining. Material properties are listed in Table 1. These input data are considered reasonable for rock masses in contrast to laboratoryscale rock properties. The coupling parameters were set to one for normal stresses and zero for shear stresses. Fluid (water) compressibility is the reciprocal of the bulk modulus and is zero in the incompressible fluid case. Normal displacement and fluid flow are prevented at the mesh sides and bottom ("rollered" and "impermeable"). Fluid pressure is zero at the mesh top. Fluid pressure is also set to zero along the slope face during and after a cut. Loading occurs during a cut. The loads are surface tractions along the cut face and are equal in magnitude but opposite in sense to the premining total stresses. Cut loads are applied over a short time step when the fluid is assumed to be incompressible, but may be applied instantaneously if the fluid is considered compressible. A consolidation phase follows loading as time passes and the slope approaches a new equilibrium position. The consolidation phase requires an increasing time step that ranged in dimensionless time t* from 10 -7 to 10 3. Dimensionless time t* = cvt/H2 where cv, t and H are consolidation coefficient, real time and slope height, respectively. The local factor of safety in the pit region for the wet semi-coupled case after the slope is cut is shown in Figure 2. For this run, the water table was assumed to coincide with the slope profile. Water pressure was fixed as the product of specific weight and depth below the water table. This is a common approach to including water pressure in simplified slope stability analyses. In Figure 2, there is an extensive region behind the slope which has a factor of safety less than one. Despite very high safety factors in the crest of the slope, the slope would certainly be considered unstable. Clearly, the actual distribution of pressure is important to realistic stability analyses. Figure 3a shows contours of the local factor of safety immediately after mining for the fully coupled case under the assumption of an incompressible fluid. For comparison, Figure 3b shows the compressible
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fluid case at the same time. The local factor of safety distribution in the incompressible fluid case indicates a stable slope, while in the compressible fluid case, the slope would be considered unstable, immediately after excavation. Incompressibility leads to an overly optimistic assessment of slope stability. The reason is the generation of unrealistically high suctions near the slope face immediately after mining. In the compressible fluid case, unrealistic suctions are not seen. The results in Figure 3 show that the local factor of safety is sensitive to fluid compressibility. Water has a bulk modulus of about 1.7 GPa (2.4x105 psi). A cemented sand fill ("soil") has a modulus of about 0.17 GPa (2.4x104), while a rock mass modulus of 17 GPa (2.4x 106 psi) is reasonable. Thus, in soil mechanics fluid compressibility (reciprocal of bulk modulus) is an order of magnitude greater than the solid, while in rock mechanics the modulus contrast is reversed and the fluid is much more compressible than the solid. For these reasons, fluid incompressibility is not a valid assumption in rock mechanics. Interestingly, the long-term distributions of poroelastic safety factor (and fluid pressure) approach the same values, from below in the compressible fluid case, from above in the incompressible case.
COUPLED ELASTIC/PLASTIC FINITE E L E M E N T M O D E L A two dimensional coupled finite element code, UT2CP, was developed from a three-dimensional code UTAH4 and used in all analyses. This code has the option of a linear or quadratic yield criterion for determining a local factor of safety. In principal stress space, the isotropic, quadratic yield criterion has the form of a paraboloid of revolution about the hydrostatic axis. The linear criterion in isotropic form is the well-known Drucker-Prager yield condition. A quadratic yield criterion, more suitable for mine-scale rock masses, was used here. Anisotropic elastic moduli, strengths, hydraulic conductivities and effective stress coefficients are allowed. Initial effective stresses, fluid pressures and velocities may be specified, so a sequence of cuts and fills may be followed. The program also has the option of using an elimination or an iterative equation solver; the former is essential for small time step stability, while the latter is more economical at larger time steps. At present, a tangent stiffness approach is used in the elastic/plastic domain. A suite of poroelastic validation problems insures code reliability in the elastic range of deformation (Pariseau 1994). Comparisons with known solutions in the plastic domain have not been done. Figure 4 shows the poroelastic/plastic distribution of the local factor of safety in the case of a compressible fluid immediately after mining. Comparison with the poroelastic result in Figure 3b shows that the extent of yielding is slightly greater. Figure 5 shows the pressure distribution immediatley after mining and about 25 years later. Figure 6 shows the long-term safety factor distributions in the poroelastic and/plastic compressible fluid cases. While the indicated extent of yielding in the purely elastic case diminishes in time, yielding in the elastic/plastic case persists even at very long times. Apparently, the material acquires a permanent "set", that is not reversed with diminished fluid pressure. The same phenomena was observed in one-dimensional consolidation analyses. If yielding does not occur, then depressurization tends to increase the local factor of safety, but after the onset of yielding, depressurization is to no avail in these example analyses.
APPLICATION TO AN OPEN PIT MINE A large open pit copper mine slope was also studied to determine the effects of water, time and mining activity on stability. Mining was simulated to a depth of about 1000 m (3000 ft) at a slope angle of about 45 degrees. The mine is the focus of a continuing study of the role of water on slope stability. The mine
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pit is developed in a mineralized stock that is mainly a quartz monzonite but includes skarns, hornfels and limestones. A two dimensional finite element model was generated for a cross-section (DD) through the north wall of the pit. Other sections are under study, but only one section is discussed here. Figure 7 shows the mesh, material boundaries, material number and coordinates for the DD model. Figure 8 is an enlargement of the slope region of the mesh connected to points A, B and C shown in Figure 7. A total of 11,248 elements was used in this model. Figure 8 also shows the boundaries of three sequential cuts that define (1) premining topography, (2) Pit 1 and (3) Pit 2. Table 2 shows the mechanical properties used in the model. The mine uses conventional engineering units, so the figures and Table 2 are also in engineering units. These properties imply mechanical isotropy. Anisotropic hydraulic conductivities ranged from 0.001 ft/day to 0.100 ft/day. The combination of 13 rock types and 11 hydro-zones results in the 43 unique sets of coupled property zones shown in Figure 7. Bulk modulus of the fluid (water) was 1.7 GPa (0.24x 106 psi) as in the simplified slope analysis. The coupling constants and boundary conditions were also the same as before. The in situ stress and initial fluid pressure distributions were given. The modeling strategy simulated excavation in a single step and then followed the evolution of the system in time using increasing time steps to 105 days. Figure 9 shows the poroelastic and/plastic distributions of the local factor of safety immediately after mining to Pit 2. A substantial region of low safety factor occurs in the toe of the slope. The slope would be unstable if no remedial measures were taken prior to reaching the Pit 2 depth. In actual practice, mining proceeds bench by bench over a period of many years. Depressurization and drainage programs are instituted as needed to maintain safe, stable slopes. Thus, the computer ouput shown in Figure 9 simply indicates the feasibility of carrying out a poroelastic/plastic computation. However, the analysis does underscore the need to take into account the role of water, the strength of the rock mass and the coupling between water pressure and rock mass deformation in wet mine slopes. CONCLUSION Comparison of two-dimensional poroelastic finite element slope stability analysis with poroelastic/plastic results of a generic homogenous, i sotropic rock slope cut to a depth of 300 m (1000 ft) at an angle of 45 degrees in a rock mass having reasonable values of elastic moduli, strengths and hydraulic conductivity shows that yield zones tend to persist in time despite depressurization and drainage. In the poroelastic case, strength does not limit stress and potential zones of yielding indicated by regions of low safety factor (high stress) diminish in time. The same does not occur in the more realistic elastic/plastic model, which in this case allowed neither hardening which would increase stability nor softening which would decrease stability. In view of the noticeable difference between the poroelastic and poroelastic/plastic results and the limited experience with the latter, additional study seems warranted. In this regard, the main conclusions reached in the earlier poroelastic study are worth reiterating. (1) fixing fluid pressures prior to a mechanical analysis (semi-coupled) gives an unrealistically large zone of potential failure in comparison with the fully coupled results and is therefore unduly pessimistic, (2) in fully coupled (elastic) analyses, the assumption of fluid incompressibility creates an unrealistic zone of high suction near the slope and a companion zone of high safety factors that diminishes in time, (3) accounting for fluid compressibility in fully coupled (elastic) analyses shows a zone of potential instability that decreases with time to a stable state in the long-term, and
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(4) no Mandel-Cryer effect of transitory but abnormally high fluid pressure was observed. Feasibility of practical application of fully coupled, two-dimensional poroelastic/plastic finite element analyses to large, open pit mining in wet ground was demonstrated. Complexities associated with a number of different rock types having a broad range of elastic moduli and strengths were readily handled. Hydraulic conductivities ranging over several orders of magnitude were also handled without difficulty. The advantage is a more realistic and technically sound computation of wet mine slope stability than is possible with conventional dry-code stability analysis even if supplemented by off-line flow net analyses for fluid pressure at steady state conditions. ACKNOWLEDGEMENT This research has been supported in part by the Department of the Interior's Mineral Institute program administered by the Bureau of Mines through the Generic Mineral Technology Center in Mine Safety and Environmental Engineering under grant number G1125251. Financial support of the Browning Scholarship Fund, Department of Mining Engineering, is gratefully acknowledged. The use of facilities provided by the Center for High Performance Computing at the University of Utah is appreciated.
FIGURES
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Paper 242, Figure 1. b~]O0
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Paper 242, Figure 2. 5302
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To cite this paper: Int. J. R o c k M e c h . & Min. Sci. 34:3-4, p a p e r No. 242.
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TABLES
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Paper 242, TABLE 1. TABLE I. STUDY SLOPE MATERIAL PROPERTIES
Property Young's modulus (GPa) Poisson's ratio (-) Shear modulus (GPa) Compressive strength (MPa) Tensile strength (MPa) Shear strength (MPa)
Property Specific gravity (solid) Specific gravity (fluid) Hydraulic conductivity (m/day) Coupling parameters (-) (effective stress coefficients) Fluid bulk modulus (GPa)
Value
16.6 0.2 6.9 10.0 1.0 1.9
Value 2.31 1.0 0.003 1.0 1.66
Paper 242, TABLE 2. TABLE 2. OPEN PIT MINE SLOPE ROCK PROPERTIES (I psi = 6.89 kPa) Geologic Unit 1 2 3 4 5 6 7 8 9 10 11 12 13
Monzonite - Upper Monzonite- Lower Latite Dike Quartzite I- Upper Quartzite I- Lower Quartzite II Quartzite III Quartzite IV- Upper Quartzite IV - Lower Limestone - Gametite Limestone - UMS Waste Rock Overburden
Young's Modulus (106 psi) 0.88 1.60 1.90 0.90 1.20 1.70 2.10 3.00 6.10 5.40 0.30 0,0036 1.00
Poisson's Ratio
0.38 0.38 0.35 0.30 0.30 0.28 0.28 0.25 0.20 0.20 0.32 0.25 0.20
Shear Modulus (10 s psi) 0.32 0.59 0.71 0.35 0.45 0.66 0.84 1.20 2.50 2.30 0.11 0.0014 0,42
Compressive Strength (psi)
217 413 627 327 402 637 992 1911 3380 2315 1064 25 992
Tensile Strength (psi) 70 109 186 93 112 169 252 449 702 391 295 6 252
Shear Strength (psi) 72 122 197 100 122 189 289 535 889 550 323 7 289
Specific Gravity 2.53 2.53 2.58 2.58 2.58 2,58 2.58 2.58 2.58 3.53 3.16 2.16 2.31
References References
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Aubry D., J.C.Hujeux 1979 Special Algorithms for Elastoplastic Consolidation with Finite Elements. Prec. Third Int. Conf. Numer. Meth. Geomechanics. Vol. 1, Balkema, pp 133-141. Bai M., D.Elsworth 1994 Model ing of Subsidence and Stress-Dependent Hydraulic Conductivity for Intact and Fractured Porous Media. RockMech. RockEngng. Vol. 27, pp 189-208. Biot M.A. 1941 General Theory of Three-Dimensional Consolidaton. J. AppI. Phys. Vol 12, pp 155-164. Biot M.A. 1955 Theory of Elasticity and Consolidation for a Porous Anisotropic Solid. J. AppI. Phys. Vol. 26, pp 182-185. Borja R.I., S.R.Lee 1990 Cam-Clay Plasticity, Part I: Implicit Integration of Elasto-Plastic Constitutive Relations. Comp. Meth. AppI. Mech. Engnrg. Vol 78, pp 49-72. Chang C.S., J.M.Duncan 1983 Consolidation Analysis for Partly Saturated Clay by Using an Elastic-Plastic Effective Stress-Strain Model. Int. J. Numen Anal. Methods' Geomech. Vol 7, pp 39-55. Cui L., A.H.-D.Cheng, V.Kaliakin, Y.Abousleiman, J.-C.Roegiers 1994 Finite Element Analyses of Anisotropic Poroelastic Problems. Proc. Eighth Int. Conf. on Computer Methods and Advances in Geomechanics. Balkema, pp 1567-1572. Detournay E., A.H.-D.Cheng 1993 Fundamentals of Poroelasticity, Comprehensive Rock Engineering, Vol. 2, Pergamon, pp 113-171. Mohammed M.M., H.S.Mitri 1996 Slope Stability Analysis in Wet Ground. Proc. 2ndNorth American Rock Mechanics Symposium. Balekma, pp 543-550. Oka F., T.Adachi, Y.Okano 1986 Two-dimensional Consolidation Analysis Using an Elasto-viscoplastic Constitutive Equation. Int. J. Numen Anal. Methods' Geemech. Vol 10, pp 1-16. Ouyang Z., D.Elsworth 1993 Evaluation of Groundwater Flow into Mined Panels. Int. J. Rock Mech. Min. Sci. & Geemech. Abst~ Vol. 30, pp 71-79. Pariseau W.G. 1991 Estimation of Permeability in Well-Jointed Rock Masses. Prec. Eighth Int. Conf. on Computer Methods and Advances in Geomechanics. Balkema, pp 1567-1572. Pariseau W.G. 1993 Equivalent Properties of a Jointed Biot Material. Int.J. ReckMech. Min. Sci. & Geemech. Abs~ Vol. 30, No. 7, pp. 1151-1157. Pariseau W.G. 1994 Design Considerations for Stopes in Wet Mines. Prec. 12th Annual GMTC Workshop Mine Systems Design and Ground Control. Virginia Polytechnic Institute and State University, Blacksburg, pp 37-48. Pariseau W.G. 1995 Coupled 3D FE Modelling of Mining in Wet Ground. Prec. of the Third Canadian Conference on Computer Applications in the Mineral Industry. McGill University, Montreal, pp 283-292. Pariseau W.G. 1996 Finite Element Analysis of Water Pressure and Flow on Shaft and Tunnel Stability. Trans. SME. Vol. 298, pp 1839-1846. Pariseau W.G., S.C.Schmelter 1995 Progress in Wet Mine Measurements for Stability. Prec. 13th Annual GMTC Workshop Mine Safety and Environmental Engineering. Virginia Polytechnic Institute and State University, Blacksburg, pp 71-81. -
Schmelter S.C., Pariseau W.G. 1996 Case Study of Safety and Wet Mine Slope Stability. Prec. 14th Annual GMTC Workshop - Mine Safety and Environmental Engineering. Virginia Polytechnic Institute and State University and Pittsburgh Research Center, Pittsburgh, Oct. 28-29. Schmelter S.C., Pariseau W.G. 1997 Coupled Finite Element Modelling of Slope Stability. SME Annual Meeting, Denver, Feb. 24-27. Preprint 97-3. Schofield A., RWroth 1968 Critical State Soil Mechanics. McGraw-Hill, London, pp 310. Siriwardane H.J., C.S.Desai 1981 Two Numerical Schemes for Nonlinear Consolidation. Int. J. Num. Meth.
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Engrng, Vol 17, pp 405-426. Yamagami T. Y. Ueta, S.Harumoto 1985 Elasto-plastic Uncoupled Procedure for Consolidation Analysis and its Comparison with Coupled Procedures. Proc. Fifth Int. Conf. Num. Meth. Geomech. Balkema, pp 605-612.
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Copyright © 1997 Elsevier Science Ltd Int. J. Rock Mech. & Min. Sci. Vol. 34, No. 3-4, 1997 ISSN 0148-9062 To cite this paper: Int. J. RockMech. &Min. Sci. 34:3-4, Paper No. 242
MINE S L O P E S T A B I L I T Y A N A L Y S I S B Y C O U P L E D FINITE ELEMENT MODELLING W.C~ P a r i s e a u ; S.C. S c h m e l t e r ; A . K . S h e i k
Department of Mining Engineering University of Utah, 315 WBB Salt Lake City, Utah, 84112-1183 USA ABSTRACT
The potentially destabilizing effect of water pressure on rock slope stability is examined assuming coupled poroelastic/plastic behavior and compared with previous poroelastic results. A coupled finite element code that accounts for the simultaneous effects of rock mass deformation and transient fluid flow is used for this purpose. Rock mass behavior is based on the concept of effective stress, Hooke's law, Darcy's law, associated plasticity and a parabolic yield condition appropriate to rock masses. The main effect of plasticity, which limits the range of purely elastic behavior by rock mass strength, is greater displacements and persistent yielding. Yielding anticipated in poroelastic analyses, where the ratio of strength to stress is less than one, is initially somewhat more extensive than in the poroelastic/plastic case, but diminishes considerably with time. In the poroelastic/plastic case, yielding that occurs during a slope cut persists in time and space despite depressurization. Applicability of poroelastic/plastic finite element analysis to actual open pit mine slopes is demonstrated. Copyright @ 1997 Elsevier Science Ltd
KEYWORDS Slope Stability • C o u p l e d M o d e l l i n g • Poroelasticity • Plasticity • R o c k Masses • O p e n Pit M i n e s • Seepage BACKGROUND
Detournay, Cheng 1993 present an excellent discussion ofporoelastic theory. Cui et al. 1994 present a finite element formulation. In soil mechanics, poroelastic/plastic analyses (e.g, Aubury, Hujeux 1979, Siriwardane, Desai 1981, Chang, Duncan 1983, Yamagami et al. 1985, Oka et al. 1986, Borja, Lee 1990) closely followed the intense development of poroelastic analyses during the 1980s and focused mainly on the consolidation problem using the popular Cam-Clay material model and critical state concept (Schofield, Wroth 1968). In rock mechanics, poroelasticity is experiencing a revival that is based on the fundamental work of Biot 1941,1955, which also underpins much work in soil mechanics, and is intended for application to petroleum engineering. Application of coupled models to mining engineering problems are few and generally use improvised material models to account for the effect of joints on the rock mass response (e.g., Ouyang, Elsworth 1993, Bai, Elsworth 1994, Mohammed, Mitri 1996). Modelling relatively large rock masses associated with surface mines often precludes accounting for individual joints, fractures, faults and so forth, so some type of equivalent properties approach is usually needed. Most equivalent properties models depend on the assumption of a representative volume
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element (RVE) that in jointed rock masses implies a test volume many times greater in linear dimension than joint spacing. Such models have limited utility in numerical analyses because the smallest element or cell size must be no smaller than the RVE. An approach to equivalent properties of porous, jointed rock masses that does not depend on the assumption of a RVE is outlined by Pariseau 1991, 1993. Once the equivalent elastic moduli, hydraulic conductivities and strengths are known, application of conventional coupled finite element technology is possible. Recent coupled poroelastic mine slope stability analyses (Pariseau, Schmelter 1995, Schmelter, Pariseau 1996, 1997) under: (1) dry conditions (no water pressure), (2) wet conditions with fixed water pressures (semi-coupled) and (3) wet conditions where water pressures changed with time and rock mass deformation (fully coupled) resulted in significant differences in the distribution of the local factor of safety, which is defined as the ratio of strength to stress at a point. The same preexcavation total stresses, slope geometry, elastic moduli, rock mass strengths and hydraulic conductivities were used in each case. The role of fluid (water) compressibility was also examined in the fully coupled case. An outline of the poroelastic model formulation and the finite element form of the governing system of equations used here is given by Pariseau 1996. Additional discussions relative to possible application to underground as well as surface mining are presented by Pariseau 1994, 1995. Figure 1 shows the model mesh used to examine the main features ofporoelastic analysis of rock slope stability. The mesh in Fig. 1 contains 2100 elements. Slope angle is 45 degrees; slope height is 300 m (1,000 ft) and bench height is 15 m (50 ft). The slope is mechanically and hydraulically homogenous and isotropic. The premining stress is due to gravity alone; gravity is switched on and the material allowed to settle under the weight of solid and fluid contributions to total weight before mining. Material properties are listed in Table 1. These input data are considered reasonable for rock masses in contrast to laboratoryscale rock properties. The coupling parameters were set to one for normal stresses and zero for shear stresses. Fluid (water) compressibility is the reciprocal of the bulk modulus and is zero in the incompressible fluid case. Normal displacement and fluid flow are prevented at the mesh sides and bottom ("rollered" and "impermeable"). Fluid pressure is zero at the mesh top. Fluid pressure is also set to zero along the slope face during and after a cut. Loading occurs during a cut. The loads are surface tractions along the cut face and are equal in magnitude but opposite in sense to the premining total stresses. Cut loads are applied over a short time step when the fluid is assumed to be incompressible, but may be applied instantaneously if the fluid is considered compressible. A consolidation phase follows loading as time passes and the slope approaches a new equilibrium position. The consolidation phase requires an increasing time step that ranged in dimensionless time t* from 10 -7 to 10 3. Dimensionless time t* = cvt/H2 where cv, t and H are consolidation coefficient, real time and slope height, respectively. The local factor of safety in the pit region for the wet semi-coupled case after the slope is cut is shown in Figure 2. For this run, the water table was assumed to coincide with the slope profile. Water pressure was fixed as the product of specific weight and depth below the water table. This is a common approach to including water pressure in simplified slope stability analyses. In Figure 2, there is an extensive region behind the slope which has a factor of safety less than one. Despite very high safety factors in the crest of the slope, the slope would certainly be considered unstable. Clearly, the actual distribution of pressure is important to realistic stability analyses. Figure 3a shows contours of the local factor of safety immediately after mining for the fully coupled case under the assumption of an incompressible fluid. For comparison, Figure 3b shows the compressible
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fluid case at the same time. The local factor of safety distribution in the incompressible fluid case indicates a stable slope, while in the compressible fluid case, the slope would be considered unstable, immediately after excavation. Incompressibility leads to an overly optimistic assessment of slope stability. The reason is the generation of unrealistically high suctions near the slope face immediately after mining. In the compressible fluid case, unrealistic suctions are not seen. The results in Figure 3 show that the local factor of safety is sensitive to fluid compressibility. Water has a bulk modulus of about 1.7 GPa (2.4x105 psi). A cemented sand fill ("soil") has a modulus of about 0.17 GPa (2.4x104), while a rock mass modulus of 17 GPa (2.4x 106 psi) is reasonable. Thus, in soil mechanics fluid compressibility (reciprocal of bulk modulus) is an order of magnitude greater than the solid, while in rock mechanics the modulus contrast is reversed and the fluid is much more compressible than the solid. For these reasons, fluid incompressibility is not a valid assumption in rock mechanics. Interestingly, the long-term distributions of poroelastic safety factor (and fluid pressure) approach the same values, from below in the compressible fluid case, from above in the incompressible case.
COUPLED ELASTIC/PLASTIC FINITE E L E M E N T M O D E L A two dimensional coupled finite element code, UT2CP, was developed from a three-dimensional code UTAH4 and used in all analyses. This code has the option of a linear or quadratic yield criterion for determining a local factor of safety. In principal stress space, the isotropic, quadratic yield criterion has the form of a paraboloid of revolution about the hydrostatic axis. The linear criterion in isotropic form is the well-known Drucker-Prager yield condition. A quadratic yield criterion, more suitable for mine-scale rock masses, was used here. Anisotropic elastic moduli, strengths, hydraulic conductivities and effective stress coefficients are allowed. Initial effective stresses, fluid pressures and velocities may be specified, so a sequence of cuts and fills may be followed. The program also has the option of using an elimination or an iterative equation solver; the former is essential for small time step stability, while the latter is more economical at larger time steps. At present, a tangent stiffness approach is used in the elastic/plastic domain. A suite of poroelastic validation problems insures code reliability in the elastic range of deformation (Pariseau 1994). Comparisons with known solutions in the plastic domain have not been done. Figure 4 shows the poroelastic/plastic distribution of the local factor of safety in the case of a compressible fluid immediately after mining. Comparison with the poroelastic result in Figure 3b shows that the extent of yielding is slightly greater. Figure 5 shows the pressure distribution immediatley after mining and about 25 years later. Figure 6 shows the long-term safety factor distributions in the poroelastic and/plastic compressible fluid cases. While the indicated extent of yielding in the purely elastic case diminishes in time, yielding in the elastic/plastic case persists even at very long times. Apparently, the material acquires a permanent "set", that is not reversed with diminished fluid pressure. The same phenomena was observed in one-dimensional consolidation analyses. If yielding does not occur, then depressurization tends to increase the local factor of safety, but after the onset of yielding, depressurization is to no avail in these example analyses.
APPLICATION TO AN OPEN PIT MINE A large open pit copper mine slope was also studied to determine the effects of water, time and mining activity on stability. Mining was simulated to a depth of about 1000 m (3000 ft) at a slope angle of about 45 degrees. The mine is the focus of a continuing study of the role of water on slope stability. The mine
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pit is developed in a mineralized stock that is mainly a quartz monzonite but includes skarns, hornfels and limestones. A two dimensional finite element model was generated for a cross-section (DD) through the north wall of the pit. Other sections are under study, but only one section is discussed here. Figure 7 shows the mesh, material boundaries, material number and coordinates for the DD model. Figure 8 is an enlargement of the slope region of the mesh connected to points A, B and C shown in Figure 7. A total of 11,248 elements was used in this model. Figure 8 also shows the boundaries of three sequential cuts that define (1) premining topography, (2) Pit 1 and (3) Pit 2. Table 2 shows the mechanical properties used in the model. The mine uses conventional engineering units, so the figures and Table 2 are also in engineering units. These properties imply mechanical isotropy. Anisotropic hydraulic conductivities ranged from 0.001 ft/day to 0.100 ft/day. The combination of 13 rock types and 11 hydro-zones results in the 43 unique sets of coupled property zones shown in Figure 7. Bulk modulus of the fluid (water) was 1.7 GPa (0.24x 106 psi) as in the simplified slope analysis. The coupling constants and boundary conditions were also the same as before. The in situ stress and initial fluid pressure distributions were given. The modeling strategy simulated excavation in a single step and then followed the evolution of the system in time using increasing time steps to 105 days. Figure 9 shows the poroelastic and/plastic distributions of the local factor of safety immediately after mining to Pit 2. A substantial region of low safety factor occurs in the toe of the slope. The slope would be unstable if no remedial measures were taken prior to reaching the Pit 2 depth. In actual practice, mining proceeds bench by bench over a period of many years. Depressurization and drainage programs are instituted as needed to maintain safe, stable slopes. Thus, the computer ouput shown in Figure 9 simply indicates the feasibility of carrying out a poroelastic/plastic computation. However, the analysis does underscore the need to take into account the role of water, the strength of the rock mass and the coupling between water pressure and rock mass deformation in wet mine slopes. CONCLUSION Comparison of two-dimensional poroelastic finite element slope stability analysis with poroelastic/plastic results of a generic homogenous, i sotropic rock slope cut to a depth of 300 m (1000 ft) at an angle of 45 degrees in a rock mass having reasonable values of elastic moduli, strengths and hydraulic conductivity shows that yield zones tend to persist in time despite depressurization and drainage. In the poroelastic case, strength does not limit stress and potential zones of yielding indicated by regions of low safety factor (high stress) diminish in time. The same does not occur in the more realistic elastic/plastic model, which in this case allowed neither hardening which would increase stability nor softening which would decrease stability. In view of the noticeable difference between the poroelastic and poroelastic/plastic results and the limited experience with the latter, additional study seems warranted. In this regard, the main conclusions reached in the earlier poroelastic study are worth reiterating. (1) fixing fluid pressures prior to a mechanical analysis (semi-coupled) gives an unrealistically large zone of potential failure in comparison with the fully coupled results and is therefore unduly pessimistic, (2) in fully coupled (elastic) analyses, the assumption of fluid incompressibility creates an unrealistic zone of high suction near the slope and a companion zone of high safety factors that diminishes in time, (3) accounting for fluid compressibility in fully coupled (elastic) analyses shows a zone of potential instability that decreases with time to a stable state in the long-term, and
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(4) no Mandel-Cryer effect of transitory but abnormally high fluid pressure was observed. Feasibility of practical application of fully coupled, two-dimensional poroelastic/plastic finite element analyses to large, open pit mining in wet ground was demonstrated. Complexities associated with a number of different rock types having a broad range of elastic moduli and strengths were readily handled. Hydraulic conductivities ranging over several orders of magnitude were also handled without difficulty. The advantage is a more realistic and technically sound computation of wet mine slope stability than is possible with conventional dry-code stability analysis even if supplemented by off-line flow net analyses for fluid pressure at steady state conditions. ACKNOWLEDGEMENT This research has been supported in part by the Department of the Interior's Mineral Institute program administered by the Bureau of Mines through the Generic Mineral Technology Center in Mine Safety and Environmental Engineering under grant number G1125251. Financial support of the Browning Scholarship Fund, Department of Mining Engineering, is gratefully acknowledged. The use of facilities provided by the Center for High Performance Computing at the University of Utah is appreciated.
FIGURES
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Paper 242, Figure 1. b~]O0
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TABLES
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Paper 242, TABLE 1. TABLE I. STUDY SLOPE MATERIAL PROPERTIES
Property Young's modulus (GPa) Poisson's ratio (-) Shear modulus (GPa) Compressive strength (MPa) Tensile strength (MPa) Shear strength (MPa)
Property Specific gravity (solid) Specific gravity (fluid) Hydraulic conductivity (m/day) Coupling parameters (-) (effective stress coefficients) Fluid bulk modulus (GPa)
Value
16.6 0.2 6.9 10.0 1.0 1.9
Value 2.31 1.0 0.003 1.0 1.66
Paper 242, TABLE 2. TABLE 2. OPEN PIT MINE SLOPE ROCK PROPERTIES (I psi = 6.89 kPa) Geologic Unit 1 2 3 4 5 6 7 8 9 10 11 12 13
Monzonite - Upper Monzonite- Lower Latite Dike Quartzite I- Upper Quartzite I- Lower Quartzite II Quartzite III Quartzite IV- Upper Quartzite IV - Lower Limestone - Gametite Limestone - UMS Waste Rock Overburden
Young's Modulus (106 psi) 0.88 1.60 1.90 0.90 1.20 1.70 2.10 3.00 6.10 5.40 0.30 0,0036 1.00
Poisson's Ratio
0.38 0.38 0.35 0.30 0.30 0.28 0.28 0.25 0.20 0.20 0.32 0.25 0.20
Shear Modulus (10 s psi) 0.32 0.59 0.71 0.35 0.45 0.66 0.84 1.20 2.50 2.30 0.11 0.0014 0,42
Compressive Strength (psi)
217 413 627 327 402 637 992 1911 3380 2315 1064 25 992
Tensile Strength (psi) 70 109 186 93 112 169 252 449 702 391 295 6 252
Shear Strength (psi) 72 122 197 100 122 189 289 535 889 550 323 7 289
Specific Gravity 2.53 2.53 2.58 2.58 2.58 2,58 2.58 2.58 2.58 3.53 3.16 2.16 2.31
References References
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Aubry D., J.C.Hujeux 1979 Special Algorithms for Elastoplastic Consolidation with Finite Elements. Prec. Third Int. Conf. Numer. Meth. Geomechanics. Vol. 1, Balkema, pp 133-141. Bai M., D.Elsworth 1994 Model ing of Subsidence and Stress-Dependent Hydraulic Conductivity for Intact and Fractured Porous Media. RockMech. RockEngng. Vol. 27, pp 189-208. Biot M.A. 1941 General Theory of Three-Dimensional Consolidaton. J. AppI. Phys. Vol 12, pp 155-164. Biot M.A. 1955 Theory of Elasticity and Consolidation for a Porous Anisotropic Solid. J. AppI. Phys. Vol. 26, pp 182-185. Borja R.I., S.R.Lee 1990 Cam-Clay Plasticity, Part I: Implicit Integration of Elasto-Plastic Constitutive Relations. Comp. Meth. AppI. Mech. Engnrg. Vol 78, pp 49-72. Chang C.S., J.M.Duncan 1983 Consolidation Analysis for Partly Saturated Clay by Using an Elastic-Plastic Effective Stress-Strain Model. Int. J. Numen Anal. Methods' Geomech. Vol 7, pp 39-55. Cui L., A.H.-D.Cheng, V.Kaliakin, Y.Abousleiman, J.-C.Roegiers 1994 Finite Element Analyses of Anisotropic Poroelastic Problems. Proc. Eighth Int. Conf. on Computer Methods and Advances in Geomechanics. Balkema, pp 1567-1572. Detournay E., A.H.-D.Cheng 1993 Fundamentals of Poroelasticity, Comprehensive Rock Engineering, Vol. 2, Pergamon, pp 113-171. Mohammed M.M., H.S.Mitri 1996 Slope Stability Analysis in Wet Ground. Proc. 2ndNorth American Rock Mechanics Symposium. Balekma, pp 543-550. Oka F., T.Adachi, Y.Okano 1986 Two-dimensional Consolidation Analysis Using an Elasto-viscoplastic Constitutive Equation. Int. J. Numen Anal. Methods' Geemech. Vol 10, pp 1-16. Ouyang Z., D.Elsworth 1993 Evaluation of Groundwater Flow into Mined Panels. Int. J. Rock Mech. Min. Sci. & Geemech. Abst~ Vol. 30, pp 71-79. Pariseau W.G. 1991 Estimation of Permeability in Well-Jointed Rock Masses. Prec. Eighth Int. Conf. on Computer Methods and Advances in Geomechanics. Balkema, pp 1567-1572. Pariseau W.G. 1993 Equivalent Properties of a Jointed Biot Material. Int.J. ReckMech. Min. Sci. & Geemech. Abs~ Vol. 30, No. 7, pp. 1151-1157. Pariseau W.G. 1994 Design Considerations for Stopes in Wet Mines. Prec. 12th Annual GMTC Workshop Mine Systems Design and Ground Control. Virginia Polytechnic Institute and State University, Blacksburg, pp 37-48. Pariseau W.G. 1995 Coupled 3D FE Modelling of Mining in Wet Ground. Prec. of the Third Canadian Conference on Computer Applications in the Mineral Industry. McGill University, Montreal, pp 283-292. Pariseau W.G. 1996 Finite Element Analysis of Water Pressure and Flow on Shaft and Tunnel Stability. Trans. SME. Vol. 298, pp 1839-1846. Pariseau W.G., S.C.Schmelter 1995 Progress in Wet Mine Measurements for Stability. Prec. 13th Annual GMTC Workshop Mine Safety and Environmental Engineering. Virginia Polytechnic Institute and State University, Blacksburg, pp 71-81. -
Schmelter S.C., Pariseau W.G. 1996 Case Study of Safety and Wet Mine Slope Stability. Prec. 14th Annual GMTC Workshop - Mine Safety and Environmental Engineering. Virginia Polytechnic Institute and State University and Pittsburgh Research Center, Pittsburgh, Oct. 28-29. Schmelter S.C., Pariseau W.G. 1997 Coupled Finite Element Modelling of Slope Stability. SME Annual Meeting, Denver, Feb. 24-27. Preprint 97-3. Schofield A., RWroth 1968 Critical State Soil Mechanics. McGraw-Hill, London, pp 310. Siriwardane H.J., C.S.Desai 1981 Two Numerical Schemes for Nonlinear Consolidation. Int. J. Num. Meth.
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To cite this paper: Int. J. Rock Mech. & Mm. Sci. 34:3-4, paper No. 242.
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Engrng, Vol 17, pp 405-426. Yamagami T. Y. Ueta, S.Harumoto 1985 Elasto-plastic Uncoupled Procedure for Consolidation Analysis and its Comparison with Coupled Procedures. Proc. Fifth Int. Conf. Num. Meth. Geomech. Balkema, pp 605-612.
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