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The closure behavior of rock fractures subject to normal loading is essential for deformation analysis of fractured rock masses and the modeling of associated coupled processes. Previous studies have mainly focused on characterizing the closure behavior of mated fractures, and the applicability of existing models to unmated fractures and characteristics of involved parameters have not been comprehensively investigated. We conducted normal loading tests on three tensile granite fractures with different mating degrees by dislocating the fracture surfaces with three offsets. A high-resolution contact model was used to simulate the fracture closure behavior, which was validated by comparing the simulated closure curves and the surface damage areas with experimental measurements. After that, it was applied to simulate the closure behavior for nine numerically generated surfaces with different surface roughness characteristics. The experimental and numerical results exhibit a downward bending trend in the stiffness versus stress curves, which existing models cannot accommodate. An extended hyperbolic model was established by introducing an exponent parameter into the classical Barton–Bandis hyperbolic model, which can better represent the experimental data than previous models. Via regression analysis, we found that the exponent parameter has an approximately constant value of 0.3. The maximum normal displacement is about two times of the mean aperture. The initial stiffness is positively correlated with the elastic modulus and the correlation length and negatively correlated with the mean aperture. Parameters involved in the extended model have precise physical meanings, and they are mathematically predictable based on measurable mechanical and geometrical properties of the rock fractures.

Solutions for radial flow of a Bingham fluid are analyzed in this paper. It aims to eliminate confusions in the literature concerning the plug flow region in different solutions for analysis and design of grouting in rock fractures. The analyses based on the force balance equation reveal that the plug flow region in Bingham radial flow is independent of the fracture radius, and is not a growth function adapted from the solution of one-dimensional (1D) slit flow according to ‘similarity’. Based on the shear stress distribution, we analytically proposed that a non-uniform plug flow region cannot exist. The Bingham fluid (grout) penetration and flowrate evolution as functions of grouting time are given using the correct expression for the plug flow region. The radius-independent plug flow region and the presented flowrate evolution equation are also verified numerically. For radial flow, the relative penetration length is equal to the relative width of plug flow region, which is the same as that for 1D channel flow. Discrepancies in analytical solutions for grout penetration and flowrate evolution were also illustrated. The clarification of the plug flow region and evaluation of discrepancies in analytical solutions presented in this work could simplify modeling and design of grouting in rock engineering applications.

In rock grouting, idealized 2D-radial laminar flow of yield stress fluids (YSF) is a fundamental flow configuration that is used for cement grout spread estimation. A limited amount of works have presented analytical and numerical solutions on the radial velocity profiles between parallel disks. However, to the best of our knowledge, there has been no experimental work that has presented measured velocity profiles for this geometry. In this paper, we present velocity profiles of Carbopol (a simple YSF), measured by pulsed ultrasound velocimetry within a radial flow model. We describe the design of the physical model and then present the measured velocity profiles while highlighting the plug-flow region and slip effects observed for three different apertures and volumetric flow rates. Although the measured velocity profiles exhibited wall slip, there was a reasonably good agreement with the analytical solution. We then discuss the major implications of our work on radial flow.

Analysis of rock fracture deformation by normal stress is important for quantifying hydromechanical properties of fractured rocks that are related to a large number of geophysical problems and geoengineering applications. Experimental and numerical results for the closure of crystalline rock fractures subject to normal stress are presented in this study. An efficient high-resolution, half-space elastic–plastic contact model for analyzing the closure of crystalline rock fractures based on the Boussinesq’s solution is validated by high-precision and high-resolution experimental data. Using the validated elastic–plastic model, we investigate the correlation between fracture-specific stiffness and multi-scale surface roughness. The wavelet analysis method and the extended averaged slope magnitude for asperity heights (referred to as $$ Z_{2}^{{3{\text{D}}}} $$) are introduced to characterize the multi-scale surface roughness. The results show that the elastic–plastic contact model is effective and precise in modeling the closure of crystalline rock fractures, which matches better with the test results than the elastic model. The multi-scale features of surface roughness can be well characterized by the wavelet analysis and the extended roughness parameter $$ Z_{2}^{{3{\text{D}}}} $$. The specific stiffness is nonlinearly correlated with the multi-scale surface roughness that possibly follows a power law. The validated elastic–plastic contact model and the multi-scale surface roughness characterization methods, as well as the nonlinear correlation between the specific stiffness and the multi-scale surface roughness presented in this study, are helpful for evaluating the dependence of mechanical behaviors of rock fractures on its multi-scale surface roughness.

Water bodies provide essential ecosystem services linked to morphometric features that might differ between natural lakes and reservoirs. We use the HydroLAKES global dataset to quantitatively compare large (area1 km

The impact of shear displacement under different mechanical boundary conditions on fluid flow and advective transport in a single fracture at the laboratory scale is demonstrated in the present study. The shear-induced changes of fracture aperture structures are determined by using the measured normal displacements and digitalized fracture surfaces from laboratory shear tests. Five shear tests on concrete replicas of the same fracture under different mechanical boundary conditions, including constant normal loading (CNL) and constant normal stiffness (CNS), are conducted to analyse the influence of mechanical boundary conditions on the shear-flow-transport processes. Fluid flow in the fracture with different shear displacements are simulated by solving the Reynolds equation. The Lagrangian particle tracking method is applied to model the advective transport in the fracture after shearing. The results generally show that the shear displacements and the normal loading conditions can significantly affect flow patterns and advective travel time distributions in the fracture. For mated fractures, the flow and transport will be enhanced by the increasing shear displacement because of shear dilation. For cases with the same shear displacement, the median advective travel time increases with the increasing boundary normal stiffness. The median advective travel time under the CNS boundary condition is generally longer than that under the CNL boundary condition. The results from this study can help to improve our understanding of stress-dependent solute transport processes in natural rock fractures.

Recently, Hoang et al. (2021) discussed our paper Zou et al. (2020). In our paper, we made a statement that Dai and Bird (1981)'s solution for two-dimensional (2D) radial Bingham fluid flow between parallel plates violates mass balance. Hoang et al. pointed out that Dai and Bird (1981)'s solution does not violate the mass balance because Dai and Bird (1981)'s solution and our analysis are based on different assumptions, i.e. with consideration of the vertical velocity component in the continuity equation or not, which leads to two different approximation models. In this sense, the mass balance of Dai and Bird (1981)'s solution should not be checked using our solution as a reference. In this reply, we add remarks on the two approximation models and their implication for rock grouting analysis. The discussion by Hoang et al. and this reply are helpful to thoroughly eliminate the existing confusion regarding the two solutions in the rock grouting research community.

Modeling of hydraulic fracturing processes is of great importance in computational geosciences. In this paper, a phase-field model is developed and applied for investigating the hydraulic fracturing propagation in saturated poroelastic rocks with pre-existing fractures. The phase-field model replaces discrete, discontinuous fractures by continuous diffused damage field, and thus is capable of simulating complex cracking phenomena such as crack branching and coalescence. Specifically, hydraulic fracturing propagation in a rock sample of a single pre-existing natural fracture or natural fracture networks is simulated using the proposed model. It is shown that distance between fractures plays a significant role in the determination of propagation direction of hydraulic fracture. While the rock permeability has a limited influence on the final crack topology induced by hydraulic fracturing, it considerably impacts the distribution of the fluid pressure in rocks. The propagation of hydraulic fractures driven by the injected fluid increases the connectivity of the natural fracture networks, which consequently enhances the effective permeability of the rocks.

There is a need for improved understanding of the mechanisms controlling solute transport in fractured crystalline rocks in order to address long-term safety analysis of repositories for spent nuclear fuel. In this contribution, flow and transport in three-dimensional discrete fracture networks with internal heterogeneity in aperture and permeability is investigated using a numerical DFN model. The fracture networks are obtained using field data of sparsely fractured crystalline rock from the Swedish candidate repository site for spent nuclear fuel. Then, heterogeneity textures with different correlation length and variance are created and mapped to each individual fracture of the network to represent internal fracture roughness. We demonstrate how the structure and variability of textures on the scale of individual fractures leads to different transport and dispersion behaviour at the scale of the network. Key thresholds for cases where flow dispersion is controlled by single-fracture heterogeneity versus network-scale heterogeneity are identified. Furthermore, we highlight enhanced flow channelling for cases where small-scale structure continues across intersections in a network, and highlight challenges for extension to large scale and site-specific modelling.

A statistical parametrization of transport combined with a new, general partition function for diffusive mass transfer (Cvetkovic, 2017, ) is here developed into a practical tool for evaluating tra ...

Anti-slide pile is one of the most frequently used measures in landslide control globally. Pile-spacing has always been determined by the load capacity of single piles or according to engineering empirical experience. Many engineering practices and laboratory experiments show that the soil arching effect exists in landslide control with anti-slide piles. In this study, we aim to calculate pile-spacing in terms of the soil arching effect. We investigated the pile-soil interaction mechanism and propose that, at the limit, the pile-back soil arch resists landslide thrust only. According to Mohr–Coulomb strength theory and limit equilibrium theories, we derived a new pile-spacing calculation equation. We verified the derived pile-spacing calculation equation with real projects. The calculated results are similar to those of practical engineering designs, in which the difference is within 10%. The equation can be used in anti-slide pile preliminary design. This study can be a reference for pile-spacing calculation based on the soil arching effect.

Moisture dynamics in road systems significantly affect road structure design and maintenance. This study analysed moisture dynamics in a cross-section of motorway (the E18) in Sweden during a one-y ...

Modeling of cement grout flow in rock fractures is important for the design, monitoring and execution of rock grouting that is widely used in a variety of rock engineering applications. This study presents a mathematical model based on the Reynolds flow equation for cement grout flow in a homogeneous water-saturated rock fracture. The model is based on two-phase flow, i.e. grout as a Bingham fluid and groundwater as a Newtonian fluid, and is used for investigating the importance of the water phase in rock grouting. The modeling results for the two-phase flow generally show the importance of the water phase that can significantly affect the pressure distribution and grout penetration in the fracture, especially under the condition of grout hardening. Such effects depend on the viscosity ratio between the grout and groundwater, which becomes increasingly important for cases with smaller values of the viscosity ratio. The grout density also affects the grout penetration length. Applying an analytical solution based on single-phase flow, i.e. neglecting the impact of groundwater flow, for modeling grout injection, will generally overestimate the penetration length.

Measuring the radial flow velocity field of yield stress fluids (YSFs) between two parallel disks provides crucial data to understand the underlying flow phenomena. However, direct velocimetry of YSFs in the radial flow configuration remains a challenge, due to the complex fluid rheology and geometry constraints. In this paper, we present an experimental device for measuring YSF radial flow velocity profiles. Ultrasound Velocity Profiling (UVP) is used to non-intrusively measure the velocity profiles. The Tikhonov regularization method is implemented to obtain smooth velocity profiles, which are used to calculate the plug-flow region. Compared to our previous work on radial flow, the current contributions include: (i) additional structural frame members to maintain a constant aperture, (ii) wall slip reduction, and (iii) an improved velocity profile plug-detection algorithm. The results show that the experimental device and the measurement method are effective for further studying radial flow behavior of YSFs for industrial applications.

The assumption of cement grouts as ‘Bingham fluids’ has thus far facilitated the development of simplified analytical tools for rock grouting design. However, from a practical standpoint the use of the Bingham constitutive law throughout the design phase has several shortcomings that merit being more systematically addressed. For instance, the known key attributes of cementitious suspensions, e.g. wall-slip, thixotropy, shear rate and time dependency that influence the grout propagation especially at low shear rates are not considered when the Bingham model is used. As such, the consequences of overlooking such crucial phenomena whilst aiming for simplicity in design remain relatively unknown. In this work, we show how considering the shear rate aspects and stop criteria with the Bingham model can be better used for grouting design. Our design suggestions are based on fundamental analysis and theory of Bingham fluid radial flow and current knowledge on cement grout rheological behaviour, as reported from rheological experiments. Moreover, we contribute to the existing design procedure by presenting a nomogram that can readily be used for practical rock grouting design. The nomogram approach allows rapid computation of all key parameters, which in essence shows that unless wall slip is taken into account up to ~50% overestimation of grout travel results.

The processes of fluid flow and solute transport through rock fractures are of primary importance in environmental engineering and geosciences. This study presented numerical modeling results of fluid flow and solute transport in a 3D rock fracture-matrix system with an orthogonal intersection of two rough-walled rock fractures. The rough-walled fracture geometry models were built from laser-scanned data of a real rock surface, for a realistic representation of natural rock fracture surface roughness. The fluid flow in the two intersected fractures and solute transport in the fracture-matrix system were simulated by solving the Navier–Stokes equations (NSE) and transport equation in the entire system. The dependence of mixing on Peclet number (Pe) and flow directionality features were analyzed. The results directly visualized important channeling flow patterns that significantly enhanced the solute mixing process at the rough-walled fracture intersection. The illustrated channeling flow and associated impacts on mixing are particularly important in the prediction of solute transport in natural fractured rocks, especially when discrete fracture network (DFN) approach is applied.

Nonlinear flow in 3D rough-walled rock fractures is simulated by solving the Navier-Stokes equations. The emphasis is on the impact of shear-caused aperture changes (variable apertures and asperity contacts) and flow conditions (inertial term) upon nonlinear flow behavior. In order to compare shear effects, two 3D fracture models, with and without shear, were established with identical initial rough-walled surfaces topographies of a realistic rock sample. Five groups of simulations with different inflow boundary conditions of flowrates/Reynolds numbers ( Re ) were conducted to demonstrate shear-enhanced nonlinearity of flow fields and limitations of local cubic law (LCL) approach. The flow results clearly show channeling flow along the preferential paths, transverse flow around the contact spots, and eddy flows behind contact spots with increasing Re , which cannot be observed in 2D models. The effective transmissivity of the 3D fracture model was calculated from the modeling results of velocity and pressure fields. The results showed that the effective transmissivity is a function of local apertures with important uncertainties even when Re is small (i.e. Re = 0.4 in this study), thus the validity of the transmissivity evaluation using LCL approach for nonlinear flow in 3D rough-walled rock fractures is questionable.

In this study, we present an analytical model for modelling two-phase flow of silica sol grouting in a single rock fracture. This model considers the time-dependent rheological property of the silica sol. The impact of groundwater flow in silica sol grouting is investigated by comparing the results of the present two-phase flow model with that of the single-phase flow model ignoring the groundwater flow. It generally shows that the groundwater flow significantly affects the pressure distribution and grouts penetration in the fracture. Ignoring the groundwater flow in single-phase flow models will underestimate the pressure at the propagation front and overestimate the propagation length. The analytical model presented in this study is able to describe the more realistic two-phase processes in rock grouting, which can be readily used in practice to reduce the potential uncertainties in the application of previous simplified analytical models.

This study focuses on numerical modeling of fluid flow and solute transport in rough-walled rock fractures and fracture-matrix systems, with the main aim to investigate the impacts of fracture surface roughness on flow and transport processes in rock fractures. Both 2D and 3D fracture models were built from laser-scanned surface tomography of a real granite rock sample, to consider realistic features of surface tomography and potential asperity contacts. The flow was simulated by directly solving the Navier-Stokes equations (NSE) and the transport was modeled by solving the advection-dispersion equation (ADE) in the entire domain of fracture-matrix system, including matrix diffusion process. Such direct simulations provided detailed flow and concentration fields for quantitatively analysis of flow and transport behavior. The detailed analysis of surface roughness decomposition, complex flow patterns (i.e., channeling, transverse and eddy flows), effective advective flow apertures, effective transmissivity, effective dispersivity, residence time, transport resistance and specific surface area demonstrated significant impacts of realistic fracture surface roughness on fluid flow and solute transport processes in rock fractures. The results show that the surface roughness and shear displacement caused asperity contacts significantly enhance nonlinearity and complexity of flow and transport processes in rough-walled fractures and fracture-matrix systems. The surface roughness also causes invasion flows in intersected fractures which enhance solute mixing at fracture intersections. Therefore, the fracture surface roughness is an important source of uncertainty in application of such simplified models like cubic law (CL) for fluid flow and analytical solutions for solute transport in rock fractures. The research conducted advances our understanding of realistic flow and transport processes in natural fractured rocks. The results are useful for model validation/extension, uncertainty analysis/quantification and laboratory experiments design in the context of various applications related to fracture flow and transport.

This study presents analytical solutions for single-phase radial flow of yield-power-law fluids between homogeneous fractures (smooth parallel plates), which covers the special cases of yield-power-law fluids, i.e., Herschel-Bulkley, Bingham and power-law fluids, as well as Newtonian fluids. The analytical solution for radial flow of Herschel-Bulkley fluids is given for the first time. A formula for the plug flow region created by the yield stress is deduced, showing that this is constant and independent of the radius for radial flow of yield-stress fluids, which is also verified by numerical simulations. For modeling of the rock grouting process where the yield-power-law grouts are injected into water-saturated fractures, we also established a general mathematical model for the two-phase (one yield-power-law grout and one Newtonian fluid) radial flow in homogeneous fractures using the Reynolds equation based on the derived analytical solutions for single-phase radial flow of yield-power-law grouts. By solving the two-phase radial flow problem, the evolution of pressure distribution and grout propagation length is illustrated. The results generally show the high sensitivity of the rheological parameters and the potentially important impact of water flow and time-dependent rheological properties on the radial propagation of yield power-law grouts.

The influence of surface roughness on fluid flow and solute transport through three-dimensional (3D) crossed rock fractures are investigated by numerical simulations. Three crossed fracture models ...

Cement grouting is widely applied in rock tunneling and underground construction to reduce groundwater inflow and increase the tightness of rock masses. The rock grouting process involves complex non-Newtonian grouts propagation in fracture networks. In this study, a two-phase flow model extended for yield-power-law fluid (e.g., cement grout) propagation in water-saturated fracture networks is presented. The effective transmissivity is scaled from analytical solutions for single-phase yield-power-law fluids flow between a pair of smooth parallel plates. This extended two-phase flow model for fracture networks is verified based on a unique set of experimental data. The full experiment dataset is presented in this work for the first time. Impacts of rheological parameters and time-dependent rheological properties of injected yield-power-law fluids on propagation processes are investigated through numerical simulations. A measure referred to as the propagation volume fraction is defined as an indicator of the propagation process. The results generally show that the rheological properties significantly affect the evolution of the propagation volume fraction. The propagation rate reduces with increased yield stress, consistency index and flow index. The two-phase flow of yield-power-law fluid propagation in a heterogeneous fracture network is also simulated, showing that the heterogeneity of fracture apertures may significantly affect the propagation process. For the heterogeneous case, with two-point distribution of apertures, the propagation volume fraction can be represented by using the harmonic mean aperture. Since the yield-power-law constitutive model covers a wide range of non-Newtonian fluids, the results presented in this work can be used for studying non-Newtonian fluid propagation in a variety of homogeneous or heterogeneous fracture networks, which can be used for rock grouting design.

The objective of this paper is to investigate the effects of wall surface roughness on fluid flow through rock fractures. A wavelet analysis technique was developed to define a mathematical criteri ...

The mechanical behavior of sliding zone soils plays a significant role in landslide. In general, the sliding zone soils are basically in unsaturated state due to rainfall infiltration and reservoir water level fluctuation. Meanwhile, a large number of examples show that the deformation processes of landslides always take a long period of time, indicating that landslides exhibit a time-dependent property. Therefore, the deformation of unsaturated soils of landslide involves creep behaviors. In this paper, the Burgers creep model for unsaturated soils under triaxial stress state is considered based on the unsaturated soil mechanics. Then, by curve fitting using the least squares method, creep parameters in different matric suction states are obtained based on the creep test data of unsaturated soils in the sliding zones of Qianjiangping landslide. Results show that the predicted results are in good agreement with the experimental data. Finally, to further explore the creep characteristics of the unsaturated soils in sliding zones, the relationships between parameters of the model and matric suction are analyzed and a revised Burgers creep model is developed correspondingly. Simulations on another group of test data are performed by using the modified Burgers creep model and reasonable results are observed.

Fluid flow and solute transport in a 3D rough-walled fracture–matrix system were simulated by directly solving the Navier–Stokes equations for fracture flow and solving the transport equation for the whole domain of fracture and matrix with considering matrix diffusion. The rough-walled fracture–matrix model was built from laser-scanned surface tomography of a real rock sample, by considering realistic features of surfaces roughness and asperity contacts. The numerical modeling results were compared with both analytical solutions based on simplified fracture surface geometry and numerical results by particle tracking based on the Reynolds equation. The aim is to investigate impacts of surface roughness on solute transport in natural fracture–matrix systems and to quantify the uncertainties in application of simplified models. The results show that fracture surface roughness significantly increases heterogeneity of velocity field in the rough-walled fractures, which consequently cause complex transport behavior, especially the dispersive distributions of solute concentration in the fracture and complex concentration profiles in the matrix. Such complex transport behaviors caused by surface roughness are important sources of uncertainty that needs to be considered for modeling of solute transport processes in fractured rocks. The presented direct numerical simulations of fluid flow and solute transport serve as efficient numerical experiments that provide reliable results for the analysis of effective transmissivity as well as effective dispersion coefficient in rough-walled fracture–matrix systems. Such analysis is helpful in model verifications, uncertainty quantifications and design of laboratorial experiments.