# Transient Analysis of Grout Penetration With Time-Dependent Viscosity Inside 3D Fractured Rock Mass by Unified Pipe-Network Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- UPM uses pipes for capturing transport processes but ignores mechanical behaviors, while LEM uses trusses for capturing the mechanical behaviors;
- UPM introduces pipes parallel to the fracture planes to capture the transport processes in fractures. On the other hand, in LEM the breakages of trusses represent damage/fractures which originally intersect with the fracture planes, but are not parallel to;
- Unlike LEM, UPM cannot simulate fracture propagations and nucleations, which is suitable for simulating transport processes in naturally highly fractured rock mass.

## 2. Methodology

#### 2.1. Rheological Models of Grout

#### 2.2. Unified Pipe-Network Method (UPM) Discrete Model

#### 2.3. Considering the Time-Dependent Viscosity in UPM

## 3. Model Verification

#### 3.1. Verifying the Rheological Models

#### 3.2. Transient Flow Considering the Time-Dependent Viscosity of Grout

## 4. Application in Simulating Grouting Process in Rock Mass with Fracture Networks

#### 4.1. The Influence of Viscosity

#### 4.2. The Influence of Fractures and Grout Operation Method

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

UPM | Unified Pipe-Network Method |

LEM | Lattice Elements Method |

## References

- Mohajerani, S.; Baghbanan, A.; Wang, G.; Forouhandeh, S. An efficient algorithm for simulating grout propagation in 2d discrete fracture networks. Int. J. Rock Mech. Min. Sci.
**2017**, 98, 67–77. [Google Scholar] [CrossRef] - Jeannin, P.; Malard, A.; Rickerl, D.; Weber, E. Assessing karst-hydraulic hazards in tunneling: the brunnmühle spring system bernese jura, switzerland. Environ. Earth Sci.
**2015**, 74, 7655–7670. [Google Scholar] [CrossRef] - Funehag, J.; Gustafson, G. Design of grouting with silica sol in hard rock–New methods for calculation of penetration length, Part I. Tunn. Undergr. Space Technol.
**2008**, 23, 1–8. [Google Scholar] [CrossRef] - Bezuijen, A.; Grotenhuis, R.T.; van Tol, A.; Bosch, J.; Haasnoot, J. Analytical model for fracture grouting in sand. J. Geotech. Geoenviron. Eng.
**2010**, 137, 611–620. [Google Scholar] [CrossRef] - Hernqvist, L.; Fransson, Å.; Gustafson, G.; Emmelin, A.; Eriksson, M.; Stille, H. Analyses of the grouting results for a section of the apse tunnel at äspö hard rock laboratory. Int. J. Rock Mech. Min. Sci.
**2009**, 46, 439–449. [Google Scholar] [CrossRef] - Lisa, H.; Christian, B.; Åsa, F.; Gunnar, G.; Johan, F. A hard rock tunnel case study: Characterization of the water-bearing fracture system for tunnel grouting. Tunn. Undergr. Space Technol.
**2012**, 30, 132–144. [Google Scholar] [CrossRef] - Seo, H.; Choi, H.; Lee, I. Numerical and experimental investigation of pillar reinforcement with pressurized grouting and pre-stress. Tunn. Undergr. Space Technol.
**2016**, 54, 135–144. [Google Scholar] [CrossRef] - Gustafson, G.; Stille, H. Prediction of groutability from grout properties and hydrogeological data. Tunn. Undergr. Space Technol.
**1996**, 11, 325–332. [Google Scholar] [CrossRef] - Amadei, B.; Savage, W. An analytical solution for transient flow of bingham viscoplastic materials in rock fractures. Int. J. Rock Mech. Min. Sci.
**2001**, 38, 285–296. [Google Scholar] [CrossRef] - Xu, Y.; Peng, W.H. Research on multiple holes grouting of fractured rock Mass. Appl. Mech. Mater.
**2013**, 256, 547–551. [Google Scholar] [CrossRef] - Foyo, A.; Sánchez, M.A.; Tomillo, C. A proposal for a secondary permeability index obtained from water pressure tests in dam foundations. Eng. Geol.
**2005**, 77, 69–82. [Google Scholar] [CrossRef] - Stille, H.; Gustafson, G.; Hassler, L. Application of new theories and technology for grouting of dams and foundations on rock. Geotech. Geol. Eng.
**2012**, 30, 603–624. [Google Scholar] [CrossRef] - Brantberger, M.; Stille, H.; Eriksson, M. Controlling grout spreading in tunnel grouting: Analyses and developments of the gin-method. Tunn. Undergr. Space Technol.
**2000**, 15, 343–352. [Google Scholar] [CrossRef] - Axelsson, M.; Gustafson, G. A robust method to determine the shear strength of cement-based injection grouts in the field. Tunn. Undergr. Space Technol.
**2006**, 21, 499–503. [Google Scholar] [CrossRef] - Draganović, A.; Stille, H. Filtration and penetrability of cement-based grout: Study performed with a short slot. Tunn. Undergr. Space Technol.
**2011**, 26, 548–559. [Google Scholar] [CrossRef] - Lee, J.; Bang, C.; Mok, Y.; Joh, S. Numerical and experimental analysis of penetration grouting in jointed rock masses. Int. J. Rock Mech. Min. Sci.
**2000**, 37, 1027–1037. [Google Scholar] [CrossRef] - Neuman, S.P. Trends, prospects and challenges in quantifying flow and transport through fractured rocks. Hydrogeol. J.
**2005**, 13, 124–147. [Google Scholar] [CrossRef] - Moës, N.; Bolbow, J.; Belytschko, T. A finite element method for crack growth without remeshing. Int. J. Numer. Methods Eng.
**1999**, 46, 131–150. [Google Scholar] [CrossRef] - Moës, N.; Belytschko, T. Extended finite element method for cohesive crack growth. Eng. Fract. Mech.
**2002**, 69, 813–833. [Google Scholar] [CrossRef] - Song, J.-H.; Areias, P.; Belytschko, T. A method for dynamic crack and shear band propagation with phantom nodes. Int. J. Numer. Methods Eng.
**2006**, 67, 868–893. [Google Scholar] [CrossRef] - Areias, P.; Song, J.-H.; Belytschko, T. Analysis of fracture in thin shells by overlapping paired elements. Comput. Methods Appl. Mech. Eng.
**2006**, 195, 41–43. [Google Scholar] [CrossRef] - Wu, J.-Y.; Li, F.-B. An improved stable XFEM (Is-XFEM) with a novel enrichment function for the computational modeling of cohesive cracks. Comput. Methods Appl. Mech. Eng.
**2015**, 295, 77–107. [Google Scholar] [CrossRef] - Simo, J.; Oliver, J.; Armero, F. An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Comput. Mech.
**1993**, 12, 277–296. [Google Scholar] [CrossRef] - Saloustros, S.; Pelà, L.; Cervera, M.; Roca, P. Finite element modelling of internal and multiple localized cracks. Comput. Mech.
**2017**, 59, 299–316. [Google Scholar] [CrossRef] - Saloustros, S.; Cervera, M.; Pelà, L. Tracking multi-directional intersecting cracks in numerical modelling of masonry shear walls under cyclic loading. Meccanica
**2018**, 53, 1757–1776. [Google Scholar] [CrossRef] - Saloustros, S.; Cervera, M.; Pelà, L. Challenges, tools and applications of tracking algorithms in the numerical modelling of cracks in concrete and masonry structures. Arch. Comput. Methods Eng.
**2018**. [Google Scholar] [CrossRef] - Nikolić, M.; Ibrahimbegovic, A.; Miscevic, P. Discrete element model for the analysis of fluid-saturated fractured poro-plastic medium based on sharp crack representation with embedded strong discontinuities. Comput. Methods Appl. Mech. Eng.
**2016**, 298, 407–427. [Google Scholar] [CrossRef] - Nikolić, M.; Ibrahimbegovic, A.; Miscevic, P. Brittle and ductile failure of rocks: Embedded discontinuity approach for representing mode i and mode ii failure mechanisms. Int. J. Numer. Methods Eng.
**2015**, 102, 1507–1526. [Google Scholar] [CrossRef] - Nikolić, M.; Ibrahimbegovic, A. Rock mechanics model capable of representing initial heterogeneities and full set of 3d failure mechanisms. Comput. Methods Appl. Mech. Eng.
**2015**, 290, 209–227. [Google Scholar] [CrossRef] - Zhang, Y.; Lackner, R.; Zeiml, M.; Mang, H. Strong discontinuity embedded approach with standard SOS formulation: Element formulation, energy-based crack-tracking strategy, and validations. Comput. Methods Appl. Mech. Eng.
**2015**, 287, 335–366. [Google Scholar] [CrossRef] - Zhang, Y.; Zhuang, X. Cracking elements: A self-propagating strong discontinuity embedded approach for quasi-brittle fracture. Finite Elem. Anal. Des.
**2018**, 144, 84–100. [Google Scholar] [CrossRef] - Zhang, Y.; Zhuang, X. A softening-healing law for self-healing quasi-brittle materials: analyzing with strong discontinuity embedded approach. Eng. Fract. Mech.
**2018**, 192, 290–306. [Google Scholar] [CrossRef] - Zhou, S.; Zhuang, X.; Zhu, H.; Rabczuk, T. Phase field modelling of crack propagation, branching and coalescence in rocks. Theor. Appl. Fract. Mech.
**2018**, 96, 174–192. [Google Scholar] [CrossRef] - Wu, J.-Y.; Nguyen, V.-P. A length scale insensitive phase-field damage model for brittle fracture. J. Mech. Phys. Solids
**2018**, 119, 20–42. [Google Scholar] [CrossRef] - Wu, J.-Y. Robust numerical implementation of non-standard phase-field damage models for failure in solids article. Comput. Methods Appl. Mech. Eng.
**2018**, 340, 767–797. [Google Scholar] [CrossRef] - Zhou, S.; Rabczuk, T.; Zhuang, X. Phase field modeling of quasi-static and dynamic crack propagation: Comsol implementation and case studies. Adv. Eng. Softw.
**2018**, 122, 31–49. [Google Scholar] [CrossRef] - Zhou, S.; Zhuang, X.; Rabczuk, T. A phase-field modeling approach of fracture propagation in poroelastic media. Eng. Geol.
**2018**, 240, 189–203. [Google Scholar] [CrossRef] - Areias, P.; Reinoso, J.; Camanho, P.; César de Sá, J.; Rabczuk, T. Effective 2d and 3d crack propagation with local mesh refinement and the screened poisson equation. Eng. Fract. Mech.
**2018**, 189, 339–360. [Google Scholar] [CrossRef] - Areias, P.; Msekh, M.; Rabczuk, T. Damage and fracture algorithm using the screened Poisson equation and local remeshing. Eng. Fract. Mech.
**2016**, 158, 116–143. [Google Scholar] [CrossRef] - Areias, P.; Rabczuk, T.; Msekh, M. Phase-field analysis of finite-strain plates and shells including element subdivision. Comput. Methods Appl. Mech. Eng.
**2016**, 312, 322–350. [Google Scholar] [CrossRef] - Areias, P.; Rabczuk, T.; Dias-da-Costa, D. Element-wise fracture algorithm based on rotation of edges. Eng. Fract. Mech.
**2013**, 110, 113–137. [Google Scholar] [CrossRef] [Green Version] - Areias, P.; Rabczuk, T. Finite strain fracture of plates and shells with configurational forces and edge rotations. Int. J. Numer. Methods Eng.
**2013**, 94, 1099–1122. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y. Multi-slicing strategy for the three-dimensional discontinuity layout optimization (3D DLO). Int. J. Numer. Anal. Methods Geomech.
**2017**, 41, 488–507. [Google Scholar] [CrossRef] [PubMed] - Zhang, Y.; Zhuang, X. Stability analysis of shotcrete supported crown of NATM tunnels with discontinuity layout optimization. Int. J. Numer. Anal. Methods Geomech.
**2018**, 42, 1199–1216. [Google Scholar] [CrossRef] - Min, K.; Rutqvist, J.; Tsang, C.; Jing, L. Stress-dependent permeability of fractured rock masses: A numerical study. Int. J. Rock Mech. Min. Sci.
**2004**, 41, 1191–1210. [Google Scholar] [CrossRef] - Baghbanan, A.; Jing, L. Stress effects on permeability in a fractured rock mass with correlated fracture length and aperture. Int. J. Rock Mech. Min. Sci.
**2008**, 45, 1320–1334. [Google Scholar] [CrossRef] - Saeidi, O.; Stille, H.; Torabi, S.R. Numerical and analytical analyses of the effects of different joint and grout properties on the rock mass groutability. Tunn. Undergr. Space Technol.
**2013**, 38, 11–25. [Google Scholar] [CrossRef] - Huang, D.; Cen, D.; Ma, G.; Huang, R. Step-path failure of rock slopes with intermittent joints. Landslides
**2015**, 12, 911–926. [Google Scholar] [CrossRef] - Huang, D.; Song, Y.; Cen, D.; Fu, G. Numerical modeling of earthquake-induced landslide using an improved discontinuous deformation analysis considering dynamic friction degradation of joints. Rock Mech. Rock Eng.
**2016**, 49, 4767–4786. [Google Scholar] [CrossRef] - Ren, H.; Zhuang, X.; Rabczuk, T. Dual-horizon peridynamics: A stable solution to varying horizons. Comput. Meth. Appl. Mech. Eng.
**2017**, 318, 762–782. [Google Scholar] [CrossRef] [Green Version] - Ren, H.; Zhuang, X.; Cai, Y.; Rabczuk, T. Dual-horizon peridynamics. Int. J. Numer. Methods Eng.
**2016**, 108, 1451–1476. [Google Scholar] [CrossRef] [Green Version] - Rabczuk, T.; Zi, G.; Bordas, S.; Nguyen-Xuan, H. A simple and robust three-dimensional cracking-particle method without enrichment. Comput. Methods Appl. Mech. Eng.
**2010**, 199, 2437–2455. [Google Scholar] [CrossRef] - Rabczuk, T.; Belytschko, T. Cracking particles: A simplified meshfree method for arbitrary evolving cracks. Int. J. Numer. Meth. Eng.
**2004**, 61, 2316–2343. [Google Scholar] [CrossRef] - Han, F.; Lubineau, G.; Azdoud, Y. Adaptive coupling between damage mechanics and peridynamics: A route for objective simulation of material degradation up to complete failure. J. Mech. Phys. Solids
**2016**, 94, 453–472. [Google Scholar] [CrossRef] [Green Version] - Han, F.; Lubineau, G.; Azdoud, Y.; Askari, A. A morphing approach to couple state-based peridynamics with classical continuum mechanics. Comput. Methods Appl. Mech. Eng.
**2016**, 301, 336–358. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Pichler, C.; Yuan, Y.; Zeiml, M.; Lackner, R. Micromechanics-based multifield framework for early-age concrete. Eng. Struct.
**2013**, 47, 16–24. [Google Scholar] [CrossRef] - Zhang, Y.; Zeiml, M.; Pichler, C.; Lackner, R. Model-based risk assessment of concrete spalling in tunnel linings under fire loading. Eng. Struct.
**2014**, 77, 207–215. [Google Scholar] [CrossRef] - Zhang, Y.; Zeiml, M.; Maier, M.; Yuan, Y.; Lackner, R. Fast assessing spalling risk of tunnel linings under RABT fire: From a coupled thermo-hydro-chemo-mechanical model towards an estimation method. Eng. Struct.
**2017**, 142, 1–19. [Google Scholar] [CrossRef] - Yang, M.; Yue, Z.; Lee, P.K.; Su, B.; Tham, L. Prediction of grout penetration in fractured rocks by numerical simulation. Can. Geotech. J.
**2002**, 39, 1384–1394. [Google Scholar] [CrossRef] [Green Version] - Hässler, L.; Håkansson, U.; Stille, H. Computer-simulated flow of grouts in jointed rock. Tunn. Undergr. Space Technol.
**1992**, 7, 441–446. [Google Scholar] [CrossRef] - Rahmani, H. Estimation of Grout Distribution in a Fractured Rock By Numerical Modeling. Ph.D. Thesis, University of British Columbia, Vancouver, BC, Canada, 2009. [Google Scholar]
- Fidelibus, C.; Lenti, V. The propagation of grout in pipe networks. Comput. Geosci.
**2012**, 45, 331–336. [Google Scholar] [CrossRef] - Cacas, M.-C.; Ledoux, E.; Marsily, G.D.; Tillie, B.; Barbreau, A.; Durand, E.; Feuga, B.; Peaudecerf, P. Modeling fracture flow with a stochastic discrete fracture network: Calibration and validation: 1. the flow model. Water Resour. Res.
**1990**, 26, 479–489. [Google Scholar] [CrossRef] - Dershowitz, W.; Einstein, H. Characterizing rock joint geometry with joint system models. Rock Mech. Rock Eng.
**1988**, 21, 21–51. [Google Scholar] [CrossRef] - Ma, G.; Wang, H.; Fan, L.; Wang, B. Simulation of two-phase flow in horizontal fracture networks with numerical manifold method. Adv. Water Resour.
**2017**, 108, 293–309. [Google Scholar] [CrossRef] - Ren, F.; Ma, G.; Fu, G.; Zhang, K. Investigation of the permeability anisotropy of 2d fractured rock masses. Eng. Geol.
**2015**, 196, 171–182. [Google Scholar] [CrossRef] - Ren, F.; Ma, G.; Wang, Y.; Fan, L. Pipe network model for unconfined seepage analysis in fractured rock masses. Int. J. Rock Mech. Min. Sci.
**2016**, 88, 183–196. [Google Scholar] [CrossRef] - Ren, F.; Ma, G.; Wang, Y.; Li, T.; Zhu, H. Unified pipe network method for simulation of water flow in fractured porous rock. J. Hydrol.
**2017**, 547, 80–96. [Google Scholar] [CrossRef] - Ren, F.; Ma, G.; Wang, Y.; Fan, L.; Zhu, H. Two-phase flow pipe network method for simulation of CO2 sequestration in fractured saline aquifers. Int. J. Rock Mech. Min. Sci.
**2017**, 98, 39–53. [Google Scholar] [CrossRef] - Nikolić, M.; Karavelić, E.; Ibrahimbegovic, A.; Miščević, P. Lattice element models and their peculiarities. Arch. Comput. Methods Eng.
**2018**, 25, 753–784. [Google Scholar] [CrossRef] - Grassl, P. A lattice approach to model flow in cracked concrete. Cement Concr. Compos.
**2009**, 31, 454–460. [Google Scholar] [CrossRef] [Green Version] - Nikolić, M.; Do, X.N.; Ibrahimbegovic, A.; Nikolić, Ż. Crack propagation in dynamics by embedded strong discontinuity approach: Enhanced solid versus discrete lattice model. Comput. Methods Appl. Mech. Eng.
**2018**, 340, 480–499. [Google Scholar] - Kazemian, S.; Prasad, A.; Huat, B.; Bazaz, J.B.; Mohammed, T.; Aziz, F.A. Effect of aggressive ph media on peat treated by cement and sodium silicate grout. J. Central South Univ. Technol.
**2011**, 18, 840–847. [Google Scholar] [CrossRef] - Kazemian, S.; Prasad, A.; Huat, B.B.; Ghiasi, V.; Ghareh, S. Effects of cement–sodium silicate system grout on tropical organic soils. Arab. J. Sci. Eng.
**2012**, 37, 2137–2148. [Google Scholar] [CrossRef] - Porcino, D.; Marcianò, V.; Granata, R. Static and dynamic properties of a lightly cemented silicate-grouted sand. Can. Geotech. J.
**2012**, 49, 1117–1133. [Google Scholar] [CrossRef] - Sun, Z.; Li, S.; Liu, R.; Zhang, Q.; Zhang, L.; Zheng, Z. Fracture defusing mechanism and pressure characteristic tests of rapid setting cement-based grouts. Yantu Lixue (Rock Soil Mech.)
**2014**, 35, 2219–2225. (In chinese) [Google Scholar] - Zhang, Q.; Zhang, L.; Liu, R.; Li, S.; Zhang, Q. Grouting mechanism of quick setting slurry in rock fissure with consideration of viscosity variation with space. Tunnel. Underg. Space Technol.
**2017**, 70, 262–273. [Google Scholar] [CrossRef] - Mohajerani, S.; Baghbanan, A.; Bagherpour, R.; Hashemolhosseini, H. Grout penetration in fractured rock mass using a new developed explicit algorithm. Int. J. Rock Mech. Mining Sci.
**2015**, 80, 412–417. [Google Scholar] [CrossRef] - Ruan, W. Research on diffusion of grouting and basic properties of grouts. J. Geotech. Eng.
**2005**, 27, 69–73. (In Chinese) [Google Scholar] - Bear, J. Dynamics of Fluids in Porous Media (Dover Civil and Mechanical Engineering); Courier Corporation: ChelmsfordNorth Chelmsford, MA, USA, 1988; ISBN 978-0486656755. [Google Scholar]
- Bear, J. Hydraulics of Groundwater (Dover Books on Engineering); Courier Corporation: ChelmsfordNorth Chelmsford, MA, USA, 2007; ISBN 978-0486453552. [Google Scholar]
- Li, S.; Han, W.; Zhang, Q.; Liu, R.; Weng, X. Research on time-dependent behavior of viscosity of fast curing grouts in underground construction grouting. Chin. J. Rock Mech. Eng.
**2013**, 32, 1–7. [Google Scholar] - Wang, Y.; Ma, G.; Ren, F.; Li, T. A constrained delaunay discretization method for adaptively meshing highly discontinuous geological media. Comput. Geosci.
**2017**, 109, 134–148. [Google Scholar] [CrossRef] - Gustafson, G.; Stille, H. Stop criteria for cement grouting. Felsbau: Zeitschrift für Geomechanik und Ingenieurgeologie im Bauwesen und Bergbau
**2005**, 25, 62–68. [Google Scholar] - Liu, R. Study on Diffusion and Plugging Mechanism of Quick Setting Cement Based Slurry in Underground Dynamic Water Grouting and Its Application; Shandong University: Jinan, China, 2012; pp. 84–86. [Google Scholar]
- Bandis, S.; Lumsden, A.; Barton, N. Fundamentals of rock joint deformation. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1983**, 20, 249–268. [Google Scholar] [CrossRef]

**Figure 1.**Schematic illustration of the grout flow in a fracture joint (

**a**) Newtonian grout flow, (

**b**) Bingham grout flow.

**Figure 2.**Transformation of the mass transport inside a fracture into the mass transport through artificial pipes on the fracture boundaries (regarding triangular mesh).

**Figure 4.**Transforming a 3D fracture network into an equivalent pipe network: (

**a**) domain with fractures (

**b**) domain with equivalent pipe networks.

**Figure 5.**Calculation of $\varphi $ in the unified pipe-network method (UPM) model: (

**a**) calculate $\varphi $ from injection point, (

**b**) determining the values of $\varphi $ from one node to another.

**Figure 7.**Comparison of the UPM model with the analytical solutions: (

**a**) Newtonian fluid, (

**b**) Bingham fluid.

**Figure 10.**Comparison of the UPM model with results obtained by Zhang [77].

**Figure 11.**Grout viscosity with time. Reference [77].

**Figure 15.**Sensitivity analyses with respect to the fracture aperture and joint roughness. (JRC: joint roughness coefficient.)

Parameters | Symbol | Unit | Value |
---|---|---|---|

Grout injection pressure | p | $\mathrm{Pa}$ | 30,000 |

Pore pressure of fracture | ${p}_{0}$ | $\mathrm{Pa}$ | 0 |

Grout density | $\rho $ | kg/m^{3} | 1400 |

Yield stress | ${\tau}_{0}$ | $\mathrm{Pa}$ | 1.0 |

Initial grout viscosity | ${\mu}_{0}$ | $\mathrm{Pa}\xb7\mathrm{s}$ | 0.04 |

Flow consistency index | k | $\mathrm{Pa}\xb7\mathrm{s}$ | 0.003182 |

Flow behavior index | n | − | 2.23 |

Fracture aperture | b | m | 0.005 |

Storage coefficient | S | [-] | 1.4 |

Gravitational acceleration | g | m/s^{2} | 9.8 |

Group | Fracture Number | Mean Length (m) | St Dev | Dip Angle (Degree) | Dip Direction (Degree) | |
---|---|---|---|---|---|---|

1 | 10 | x-axis | 5 | 2 | 16 | 40 |

y-axis | 3 | 2 | ||||

2 | 10 | x-axis | 3 | 2 | 56 | 256 |

y-axis | 3 | 2 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sun, Z.; Yan, X.; Liu, R.; Xu, Z.; Li, S.; Zhang, Y.
Transient Analysis of Grout Penetration With Time-Dependent Viscosity Inside 3D Fractured Rock Mass by Unified Pipe-Network Method. *Water* **2018**, *10*, 1122.
https://doi.org/10.3390/w10091122

**AMA Style**

Sun Z, Yan X, Liu R, Xu Z, Li S, Zhang Y.
Transient Analysis of Grout Penetration With Time-Dependent Viscosity Inside 3D Fractured Rock Mass by Unified Pipe-Network Method. *Water*. 2018; 10(9):1122.
https://doi.org/10.3390/w10091122

**Chicago/Turabian Style**

Sun, Zizheng, Xiao Yan, Rentai Liu, Zhenhao Xu, Shucai Li, and Yiming Zhang.
2018. "Transient Analysis of Grout Penetration With Time-Dependent Viscosity Inside 3D Fractured Rock Mass by Unified Pipe-Network Method" *Water* 10, no. 9: 1122.
https://doi.org/10.3390/w10091122