Technical Papers
Nov 4, 2022

Numerical Simulation of Rock Fracture and Permeability Characteristics under Stress–Seepage–Damage Coupling Action

Publication: International Journal of Geomechanics
Volume 23, Issue 1

Abstract

In the field of rock mechanics, the stress–seepage–damage coupling numerical simulation of rock has always been a hot topic but a difficult problem. Based on this background, we newly derive the smoothed particle hydrodynamics (SPH) form of the seepage equation and a two-dimensional (2D) stress–seepage–damage coupling constitutive model. The proposed coupling model considers the heterogeneity of the engineering rock mass and overcomes the difficulty of the conventional SPH algorithm in stress–seepage–damage calculation. Stable one-dimensional (1D) seepage is first simulated to verify the correctness of the seepage equation and that the simulation results coincide with traditional analytical solutions. Then two triaxial compression experiments considering seepage and no-seepage conditions are simulated to show that the coupled model can well simulate the progressive rock failure process and the change in permeability. The correctness of the coupling model is verified by comparing the simulated results with experimental results. The existence of a seepage field advances the initial cracking time and weakens the peak strength of the rock. Finally, the progressive damage processes of surrounding rock excavation under unloading action are simulated, showing that stress–seepage–damage coupling model has application prospects in rock engineering. The research results may provide some references for the application of the SPH method in the stress–seepage–damage coupling simulation of rock.

Get full access to this article

View all available purchase options and get full access to this article.

Acknowledgments

This research was supported by the Chongqing Natural Science Fund General Project (Grant Number cstc2020jcyj-msxmX0904), the Chongqing Talent Plan (Grant Number CQYC2020058263), the Chongqing Technology Innovation and Application Development Project (Grant Number cstc2021ycjh-bgzxm0246), the China Postdoctoral Science Foundation General Project (Grant Number 2021M693739), and special funding for postdoctoral research projects in Chongqing (Grant Number 2021XM2019).

References

Bi, J., and X. P. Zhou. 2017a. “A novel numerical algorithm for simulation of initiation, propagation and coalescence of flaws subject to internal fluid pressure and vertical stress in the framework of general particle dynamics.” Rock Mech. Rock Eng. 50 (7): 1833–1849. https://doi.org/10.1007/s00603-017-1204-4.
Bi, J., and X. P. Zhou. 2017b. “Numerical simulation of kinetic friction in the fracture process of rocks in the framework of general particle dynamics.” Comput. Geotech. 83: 1–15. https://doi.org/10.1016/j.compgeo.2016.10.019.
Bi, J., X. P. Zhou, and Q. H. Qian. 2016. “The 3D numerical simulation for the propagation process of multiple Pre-existing flaws in rock-like materials subjected to biaxial compressive loads.” Rock Mech. Rock Eng. 49 (5): 1611–1627. https://doi.org/10.1007/s00603-015-0867-y.
Biot, M. A. 1941. “General theory of three-dimensional consolidation.” J. Appl. Phys. 12 (2): 155–164. https://doi.org/10.1063/1.1712886.
Biot, M. A. 1955. “Theory of elasticity and consolidation for a porous anisotropic solid.” J. Appl. Phys. 26 (2): 182–185. https://doi.org/10.1063/1.1721956.
Biot, M. A., and G. Temple. 1972. “Theory of finite deformations of porous solids.” Indiana Univ. Math. J. 21 (7): 597–620. https://doi.org/10.1512/iumj.1972.21.21048.
Brace, W. F., J. B. Walsh, and W. T. Frangos. 1968. “Permeability of granite under high pressure.” J. Geophys. Res. 73 (6): 2225–2236. https://doi.org/10.1029/JB073i006p02225.
Bui, H. H., G. D. Nguyen, J. K. Kodikara, and M. Sanchez. 2015. “Soil cracking modelling using the mesh-free SPH method.” In Proc., 12th Australia New Zealand Conf. on Geomechanics. London: International Society for Soil Mechanics and Geotechnical Engineering (ISSMGE).
Chao, Z. M., G. T. Ma, and M. Wang. 2020. “Experimental and numerical modelling of the mechanical behaviour of low-permeability sandstone considering hydromechanics.” Mech. Mater. 148: 103454. https://doi.org/10.1016/j.mechmat.2020.103454.
Charlier, R., F. Collin, B. Pardoen, J. Talandier, J. P. Radu, and P. Gerard. 2013. “An unsaturated hydro-mechanical modelling of two in-situ experiments in Callovo-Oxfordian argillite.” Eng. Geol. 165: 46–63. https://doi.org/10.1016/j.enggeo.2013.05.021.
Chen, L., J. F. Liu, C. P. Wang, J. Liu, R. Su, and J. Wang. 2014. “Characterization of damage evolution in granite under compressive stress condition and its effect on permeability.” Int. J. Rock Mech. Min. Sci. 71: 340–349. https://doi.org/10.1016/j.ijrmms.2014.07.020.
Chen, X., J. Yu, C. A. Tang, H. Li, and S. Y. Wang. 2017. “Experimental and numerical investigation of permeability evolution with damage of sandstone under triaxial compression.” Rock Mech. Rock Eng. 50 (6): 1529–1549. https://doi.org/10.1007/s00603-017-1169-3.
Choo, L. Q., Z. Zhao, H. Chen, and Q. Tian. 2016. “Hydraulic fracturing modeling using the discontinuous deformation analysis (DDA) method.” Comput. Geotech. 76: 12–22. https://doi.org/10.1016/j.compgeo.2016.02.011.
Cook, N. G. 1992. “Natural joints in rock: Mechanical, hydraulic and seismic behavior and properties under normal stress.” Int. J. Rock Mech. Min. Sci. 29 (3): 198–223. https://doi.org/10.1016/0148-9062(92)91656-P.
Courant, R., K. Friedrichs, and H. Lewy. 1967. “On the partial difference equations of mathematical physics.” IBM J. Res. Dev. 11 (2): 215–234. https://doi.org/10.1147/rd.112.0215.
Davy, C. A., F. Skoczylas, J. D. Barnichon, and P. Lebon. 2007. “Permeability of macro-cracked argillite under confinement: Gas and water testing.” Phys. Chem. Earth Parts A/B/C 32 (8), 667–680. https://doi.org/10.1016/j.pce.2006.02.055.
Douillet-Grellier, T., B. D. Jones, R. Pramanik, K. Pan, A. Albaiz, and J. R. Williams. 2016. “Mixed-mode fracture modeling with smoothed particle hydrodynamics.” Comput. Geotech. 79: 73–85. https://doi.org/10.1016/j.compgeo.2016.06.002.
Faivre, M., B. Paul, F. Golfier, R. Giot, P. Massin, and D. Colombo. 2016. “2D coupled HM-XFEM modeling with cohesive zone model and applications to fluid-driven fracture network.” Eng. Fract. Mech. 159: 115–143. https://doi.org/10.1016/j.engfracmech.2016.03.029.
Feng, Q., J. Jin, S. Zhang, W. Liu, X. Yang, and W. Li. 2022. “Study on a damage model and uniaxial compression simulation method of frozen–thawed rock.” Rock Mech. Rock Eng. 55 (1): 187–211. https://doi.org/10.1007/s00603-021-02645-2.
Heiland, J. 2003. “Laboratory testing of coupled hydro-mechanical processes during rock deformation.” Hydrogeol. J. 11 (1): 122–141. https://doi.org/10.1007/s10040-002-0236-2.
Hoek, E. 1983. “Strength of jointed rock masses.” Geotechnique 33 (3): 187–223. https://doi.org/10.1680/geot.1983.33.3.187.
Hoek, E. 1990. “Estimating Mohr-Coulomb friction and cohesion values from the Hoek-Brown failure criterion.” Int. J. Rock Mech. Min. Sci. 27 (3): 227–229. https://doi.org/10.1016/0148-9062(90)94333-O.
Hoek, E., and E. T. Brown. 1980. “Empirical strength criterion for rock masses.” J. Geotech. Eng. Div. 106 (9): 1013–1035. https://doi.org/10.1061/AJGEB6.0001029.
Hoek, E., and E. T. Brown. 1997. “Practical estimates of rock mass strength.” Int. J. Rock Mech. Min. Sci. 34 (8): 1165–1186. https://doi.org/10.1016/S0148-9062(97)00305-7.
Huang, L. K., J. J. Liu, F. S. Zhang, E. Dontsov, and B. Damjanac. 2019. “Exploring the influence of rock inherent heterogeneity and grain size on hydraulic fracturing using discrete element modeling.” Int. J. Solids Struct. 176: 207–220. https://doi.org/10.1016/j.ijsolstr.2019.06.018.
Huang, S. Y., Y. Y. Wu, X. B. Meng, L. W. Liu, and W. Ji. 2018. “Recent advances on microscopic pore characteristics of low permeability sandstone reservoirs.” Adv. Geo-Energy Res. 2 (2): 122–134. https://doi.org/10.26804/ager.2018.02.02.
Jiao, Y. Y., H. Q. Zhang, H. M. Tang, X. L. Zhang, A. C. Adoko, and H. N. Tian. 2014. “Simulating the process of reservoir-impoundment-induced landslide using the extended DDA method.” Eng. Geol. 182: 37–48. https://doi.org/10.1016/j.enggeo.2014.08.016.
Johnson, D. H., F. Vahedifard, B. Jelinek, and J. F. Peters. 2017. “Micromechanics of undrained response of dilative granular media using a coupled DEM-LBM model: A case of biaxial test.” Comput. Geotech. 89: 103–112. https://doi.org/10.1016/j.compgeo.2017.04.011.
Karimian, E., and M. Oliaei. 2019. “Application of smoothed finite element method in coupled hydro-mechanical analyses.” Sci. Iran. 26 (1): 234–245. https://doi.org/10.24200/sci.2017.4235.
Kilmer, N. H., N. R. Morrow, and J. K. Pitman. 1987. “Pressure sensitivity of low permeability sandstones.” J. Pet. Sci. Eng. 1 (1): 65–81. https://doi.org/10.1016/0920-4105(87)90015-5.
Kristinof, R., P. G. Ranjith, and S. K. Choi. 2010. “Finite element simulation of fluid flow in fractured rock media.” Environ. Earth Sci. 60 (4): 765–773. https://doi.org/10.1007/s12665-009-0214-2.
Kumar, D., S. Kundu, R. Kumhar, and S. Gupta. 2020. “Vibrational analysis of Love waves in a viscoelastic composite multilayered structure.” Acta Mech. 231 (10): 4199–4215. https://doi.org/10.1007/s00707-020-02767-8.
Kumari, R., and A. K. Singh. 2021. “Dispersion and attenuation of shear wave in couple stress stratum due to point source.” J. Vib. Control. 28 (13–14): 1754–1768. https://doi.org/10.1177/1077546321998880.
Li, G., C. A. Tang, L. C. Li, and H. Li. 2016. “An unconditionally stable explicit and precise multiple timescale finite element modeling scheme for the fully coupled hydro-mechanical analysis of saturated poroelastic media.” Comput. Geotech. 71: 69–81. https://doi.org/10.1016/j.compgeo.2015.09.003.
Li, S. P., Y. S. Li, Y. Li, Z. Y. Wu, and G. Zhou. 1994. “Permeability-strain equations corresponding to the complete stress—strain path of Yinzhuang Sandstone.” Int. J. Rock Mech. Min. Sci. 31 (4): 383–391. https://doi.org/10.1016/0148-9062(94)90906-7.
Lian, Y., H. H. Bui, G. D. Nguyen, H. T. Tran, and A. Haque. 2021. “A general SPH framework for transient seepage flows through unsaturated porous media considering anisotropic diffusion.” Comput. Methods Appl. Mech. Eng. 387: 114169. https://doi.org/10.1016/j.cma.2021.114169.
Libersky, L. D., and A. G. Petschek. 1991. “Smooth particle hydrodynamics with strength of materials.” In Vol. 395 of Advances in the free Lagrange method, edited by H. E. Trease, M. F. Fritts, and W. P. Crowley, 248–257. Berlin: Springer.
Libersky, L. D., A. G. Petschek, T. C. Carney, J. R. Hipp, and F. A. Allahdadi. 1993. “High strain Lagrangian hydrodynamics: A three-dimensional SPH code for dynamic material response.” J. Comput. Phys. 109 (1): 67–75. https://doi.org/10.1006/jcph.1993.1199.
Liu, G. R., and M. B. Liu. 2003. Smoothed particle hydrodynamics: A meshfree particle method. Singapore: World Scientific.
Lomize, G. M. 1951. Vol. 127 of Flow in fractured rocks. Moscow: Gesenergoizdat.
Louis, C. 1972. “Rock hydraulics.” In Vol. 165 of Int. Centre for Mechanical Sciences, edited by L. Müller, 299–387. Vienna: Springer.
Luege, M., J. Lucero, C. Torrijos, and A. Orlando. 2016. “Coupled mechanical and fluid flow analysis in fractured saturated porous media using the XFEM.” Appl. Math. Modell. 40 (7–8): 4480–4504. https://doi.org/10.1016/j.apm.2015.11.032.
Luo, W., Y. P. Qin, M. M. Zhao, C. X. Wang, and Y. R. Wang. 2011. “Test study on permeability properties of the sandstone specimen under triaxial stress condition.” Procedia Eng. 26 (1): 173–178. https://doi.org/10.1016/j.proeng.2011.11.2156.
Ma, H., J. Wang, J. Qian, X. Tan, L. Chen, Y. Deng, Z. Lu, and L. Ma. 2021. “Two-dimensional SPH analysis of seepage with water injection process for different crack morphologies.” KSCE J. Civ. Eng. 25: 1909–1917. https://doi.org/10.1007/s12205-021-1202-7.
Massart, T. J., and A. P. S. Selvadurai. 2014. “Computational modelling of crack-induced permeability evolution in granite with dilatant cracks.” Int. J. Rock Mech. Min. Sci. 70: 593–604. https://doi.org/10.1016/j.ijrmms.2014.06.006.
Meng, T., R. Liu, X. Meng, D. Zhang, and Y. Hu. 2019. “Evolution of the permeability and pore structure of transversely isotropic calcareous sediments subjected to triaxial pressure and high temperature.” Eng. Geol. 253: 27–35. https://doi.org/10.1016/j.enggeo.2019.03.007.
Meng, T., Y. C. You, J. Chen, and Y. Q. Hu. 2017. “Investigation on the permeability evolution of gypsum interlayer under high temperature and triaxial pressure.” Rock Mech. Rock Eng. 50 (8): 2059–2069. https://doi.org/10.1007/s00603-017-1222-2.
Mitchell, T. M., and D. R. Faulkner. 2008. “Experimental measurements of permeability evolution during triaxial compression of initially intact crystalline rocks and implications for fluid flow in fault zones.” J. Geophys. Res.: Solid Earth 113 (B11): B11412. https://doi.org/10.1029/2008JB005588.
Monaghan, J. J., and J. C. Lattanzio. 1985. “A refined particle method for astrophysical problems.” Astron. Astrophys. 149 (1): 135–143.
Neuzil, C. E. 2003. “Hydromechanical coupling in geologic processes.” Hydrogeol. J. 11 (1): 41–83. https://doi.org/10.1007/s10040-002-0230-8.
Nguyen, V. P., T. Rabczuk, S. Bordas, and M. Duflot. 2008. “Meshless methods: A review and computer implementation aspects.” Math. Comput. Simul. 79 (3): 763–813. https://doi.org/10.1016/j.matcom.2008.01.003.
Oda, M., T. Takemura, and T. Aoki. 2002. “Damage growth and permeability change in triaxial compression tests of Inada granite.” Mech. Mater. 34 (6): 313–331. https://doi.org/10.1016/S0167-6636(02)00115-1.
Pirnia, P., F. Duhaime, Y. Ethier, and J. S. Dube. 2019. “Drag force calculations in polydisperse DEM simulations with the coarse-grid method: Influence of the weighting method and improved predictions through artificial neural networks.” Transp. Porous Media 129 (3): 837–853. https://doi.org/10.1007/s11242-019-01308-9.
Pramanik, R., and D. Deb. 2015. “Implementation of smoothed particle hydrodynamics for detonation of explosive with application to rock fragmentation.” Rock Mech. Rock Eng. 48 (4): 1683–1698. https://doi.org/10.1007/s00603-014-0657-y.
Rutqvist, J., and O. Stephansson. 2003. “The role of hydromechanical coupling in fractured rock engineering.” Hydrogeol. J. 11 (1): 7–40. https://doi.org/10.1007/s10040-002-0241-5.
Schulze, O., T. Popp, and H. Kern. 2001. “Development of damage and permeability in deforming rock salt.” Eng. Geol. 61 (2): 163–180. https://doi.org/10.1016/S0013-7952(01)00051-5.
Shaw, A., D. Roy, and S. R. Reid. 2011. “Optimised form of acceleration correction algorithm within SPH-based simulations of impact mechanics.” Int. J. Solids Struct. 48 (25–26): 3484–3498. https://doi.org/10.1016/j.ijsolstr.2011.09.003.
Shou, Y. D., and X. P. Zhou. 2021. “A coupled hydro-mechanical non-ordinary state-based peridynamics for the fissured porous rocks.” Eng. Anal. Boundary Elem. 123: 133–146. https://doi.org/10.1016/j.enganabound.2020.12.001.
Souley, M., F. Homand, S. Pepa, and D. Hoxha. 2001. “Damage-induced permeability changes in granite: A case example at the URL in Canada.” Int. J. Rock Mech. Min. Sci. 38 (2): 297–310. https://doi.org/10.1016/S1365-1609(01)00002-8.
Tang, C. A., H. Liu, P. K. K. Lee, Y. Tsui, and L. G. Tham. 2000. “Numerical studies of the influence of microstructure on rock failure in uniaxial compression — Part I: Effect of heterogeneity.” Int. J. Rock Mech. Min. Sci. 37 (4): 555–569. https://doi.org/10.1016/S1365-1609(99)00121-5.
Tang, C. A., L. G. Tham, P. K. K. Lee, T. H. Yang, and L. C. Li. 2002. “Coupled analysis of flow, stress and damage (FSD) in rock failure.” Int. J. Rock Mech. Min. Sci. 39 (4): 477–489. https://doi.org/10.1016/S1365-1609(02)00023-0.
Tang, S. B., and C. A. Tang. 2012. “Numerical studies on tunnel floor heave in swelling ground under humid conditions.” Int. J. Rock Mech. Min. Sci. 55: 139–150. https://doi.org/10.1016/j.ijrmms.2012.07.007.
VonNeumann, J., and R. D. Richtmyer. 1950. “A method for the numerical calculation of hydrodynamic shocks.” J. Appl. Phys. 21 (3): 232–237. https://doi.org/10.1063/1.1699639.
Wang, H. L., W. Y. Xu, and S. Q. Yang. 2006. “Experimental investigation on permeability evolution law during course of deformation and failure of rock specimen.” Rock Soil Mech. 27 (10): 1703–1708. https://doi.org/10.1016/S1005-8885(07)60042-9.
Wang, J. A., and H. D. Park. 2002. “Fluid permeability of sedimentary rocks in a complete stress–strain process.” Eng. Geol. 63 (3–4): 291–300. https://doi.org/10.1016/S0013-7952(01)00088-6.
Wang, Q., X. L. Hu, W. B. Zheng, L. X. Li, C. Zhou, and C. Y. Ying. 2021. “Mechanical properties and permeability evolution of red sandstone subjected to hydro-mechanical coupling: Experiment and discrete element modelling.” Rock Mech. Rock Eng. 54 (5): 2405–2423. https://doi.org/10.1007/s00603-021-02396-0.
Wang, S. G., D. Elsworth, and J. S. Liu. 2013. “Permeability evolution during progressive deformation of intact coal and implications for instability in underground coal seams.” Int. J. Rock Mech. Min. Sci. 58: 34–45. https://doi.org/10.1016/j.ijrmms.2012.09.005.
Wang, S. Y., S. W. Sloan, D. C. Sheng, and C. A. Tang. 2012. “Numerical analysis of the failure process around a circular opening in rock.” Comput. Geotech. 39: 8–16. https://doi.org/10.1016/j.compgeo.2011.08.004.
Wu, Z. J., and L. N. Y. Wong. 2014. “Extension of numerical manifold method for coupled fluid flow and fracturing problems.” Int. J. Numer. Anal. Methods Geomech. 38 (18): 1990–2008. https://doi.org/10.1002/nag.2293.
Yang, T. H., L. G. Tham, C. A. Tang, Z. Z. Liang, and Y. J. Tsui. 2004. “Influence of heterogeneity of mechanical properties on hydraulic fracturing in permeable rocks.” Rock Mech. Rock Eng. 37 (4): 251–275. https://doi.org/10.1007/s00603-003-0022-z.
Yu, S. Y., X. H. Ren, J. X. Zhang, H. J. Wang, and Z. H. Sun. 2021. “An improved form of smoothed particle hydrodynamics method for crack propagation simulation applied in rock mechanics.” Int. J. Min. Sci. Technol. 31 (3): 421–428. https://doi.org/10.1016/j.ijmst.2021.01.009.
Zeng, K., J. Xu, P. He, and C. Wang. 2011. “Experimental study on permeability of coal sample subjected to triaxial stresses.” Procedia Eng. 26: 1051–1057. https://doi.org/10.1016/j.proeng.2011.11.2273.
Zhang, C. Q., X. T. Feng, H. Zhou, S. L. Qiu, and W. P. Wu. 2012. “Case histories of four extremely intense rockbursts in deep tunnels.” Rock Mech. Rock Eng. 45 (3): 275–288. https://doi.org/10.1007/s00603-011-0218-6.
Zhang, C. L. 2016. “The stress–strain–permeability behavior of clay rock during damage and recompaction.” J. Rock Mech. Geotech. Eng. 8 (1): 16–26. https://doi.org/10.1016/j.jrmge.2015.10.001.
Zhang, W., K. Maeda, H. Saito, Z. Li, and Y. Huang. 2016. “Numerical analysis on seepage failures of dike due to water level-up and rainfall using a water–soil-coupled smoothed particle hydrodynamics model.” Acta Geotech. 11 (6): 1–18. https://doi.org/10.1007/s11440-016-0488-y.
Zhang, Z. Y., X. G. Jin, and W. Luo. 2019. “Numerical study on the collapse behaviors of shallow tunnel faces under open-face excavation condition using mesh-free method.” J. Eng. Mech. 145 (11): 04019085. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001661.
Zhao, J. H., L. M. Yin, and W. J. Guo. 2018a. “Stress-Seepage coupling of cataclastic rock masses based on digital image technologies.” Rock Mech. Rock Eng. 51 (8): 2355–2372. https://doi.org/10.1007/s00603-018-1474-5.
Zhao, Y. J., F. G. Wang, C. S. Li, Y. Q. Cao, and H. L. Tian. 2018b. “Study of the corrosion characteristics of tunnel fissures in a karst area in southwest China.” Geofluids 2018: 1–19. https://doi.org/10.1155/2018/6234932.
Zhao, Y., K. Zheng, C. L. Wang, J. Bi, and H. Zhang. 2022. “Investigation On model-I fracture toughness of sandstone with the structure of typical bedding inclination angles subjected to three-point bending.” Theor. Appl. Fract. Mech. 119: 103327. https://doi.org/10.1016/j.tafmec.2022.103327.
Zhou, X. P., Y. X. Xie, J. Bi, and F. Berto. 2019. “Numerical simulation of supershear ruptures in rock mass based on general particle dynamics.” Fatigue Fract. Eng. Mater. 42 (4): 905–918. https://doi.org/10.1111/ffe.12959.
Zoback, M. D., and J. D. Byerlee. 1975. “The effect of microcrack dilatancy on the permeability of westerly granite.” J. Geophys. Res. 80 (5): 752–755. https://doi.org/10.1029/JB080i005p00752.

Information & Authors

Information

Published In

Go to International Journal of Geomechanics
International Journal of Geomechanics
Volume 23Issue 1January 2023

History

Received: Nov 18, 2021
Accepted: Jul 29, 2022
Published online: Nov 4, 2022
Published in print: Jan 1, 2023
Discussion open until: Apr 4, 2023

Permissions

Request permissions for this article.

ASCE Technical Topics:

Authors

Affiliations

Research Fellow, School of Civil Engineering, Chongqing Univ., Chongqing 400045, China; Research Fellow, Chongqing Univ. Industrial Technology Research Institute, Chongqing 401329, China; Research Fellow, Chongqing City Construction Investment (Group) Co., Ltd., Chongqing 400023, China. ORCID: https://orcid.org/0000-0003-0321-0567
Zhiqiang Zhou [email protected]
Ph.D. Student, College of Civil Engineering, Guizhou Univ., Guiyang, Guizhou 550025, China; Ph.D. Student, Guizhou Provincial Key Laboratory of Rock and Soil Mechanics and Engineering Safety, Guizhou Univ., Guiyang 550025, China (corresponding author). Email: [email protected]

Metrics & Citations

Metrics

Citations

Download citation

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Get Access

Access content

Please select your options to get access

Log in/Register Log in via your institution (Shibboleth)
ASCE Members: Please log in to see member pricing

Purchase

Save for later Information on ASCE Library Cards
ASCE Library Cards let you download journal articles, proceedings papers, and available book chapters across the entire ASCE Library platform. ASCE Library Cards remain active for 24 months or until all downloads are used. Note: This content will be debited as one download at time of checkout.

Terms of Use: ASCE Library Cards are for individual, personal use only. Reselling, republishing, or forwarding the materials to libraries or reading rooms is prohibited.
ASCE Library Card (5 downloads)
$105.00
Add to cart
ASCE Library Card (20 downloads)
$280.00
Add to cart
Buy Single Article
$35.00
Add to cart

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share