Cement grouting is widely used in rock tunnelling to control groundwater inflow by sealing rock fractures. Accurately predicting grout propagation in rock fractures is crucial for the design, execution, and monitoring of rock grouting in engineering applications. Current methods rely on theoretical models, such as the real-time grouting control (RTGC) method, which is derived based on simplified fracture geometries like smooth parallel plates/disks. However, real rock fractures consist of rough surfaces with variable apertures. In this study, we present a computational model for theoretically predicting the propagation of non-Newtonian cement grout in variable fracture apertures. This model is validated with laboratory test data on grout propagation in a one-dimensional varying aperture long slot (VALS). We also analysed the impact of varying aperture on cement grout propagation processes. Our findings demonstrate that the presented computational model predicts the grout propagation process in this geometry with good accuracy. Moreover, we observed that varying aperture significantly affects the grout propagation process in fractures. The insights provided by our model and analysis results are potentially useful in rock tunnelling projects, specifically for the theoretical analysis of cement grout propagation in rock fractures.

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.

Modeling of cement grout propagation in fractured rocks is important for the design, execution and monitoring of rock grouting in practice. In this project, we studied non-Newtonian cement grouts propagation in rock fractures by theoretical analyses and numerical simulations. 

The analytical solutions for radial flow of a Bingham fluid, between parallel plates, are analyzed and existing disagreements in the literature regarding the two different analytical solutions that are used for analysis of grouting in rock fractures is investigated. The analyses reveal that the two solutions are both zero-order approximation solutions based on different assumptions, that is with or without consideration of the vertical velocity component across the aperture. The one without considering the vertical velocity yields to a solution with radius-independent plug flow region, which largely simplifies the calculations. By using the solution with radius-independent plug flow region, Bingham grout penetration and flowrate (at the injection borehole) evolution as functions of grouting time are given for the first time. Discrepancies in the two approximation solutions for grout penetration and flowrate evolution are illustrated, showing negligible differences. The clarification of the plug flow region and evaluation of discrepancies in the two solutions presented in this work improves our confidence and simplifies modeling and design of grouting in rock engineering applications. 

In reality, rock fracture grouting process involves non-Newtonian fluid flow in fractures with rough surfaces, which is rarely studied in the literature. To investigate the impact of fracture surface roughness on rock fracture grouting, we presented direct numerical simulations of non-Newtonian grouts flow in single rough-walled fractures, using a regularized method (i.e., the Bingham-Papanastasiou model) to approximate the yield-stress. The rough-walled rock fracture models are created from a laser-scanned surface of a granite rock sample, to represent realistic features of natural rock fractures. The numerical results show nonlinear behaviors of non-Newtonian fluid flow in rough-walled fractures caused by non-Newtonian rheological properties and enhanced by the fracture surface roughness. The surface roughness significantly reduces the effective transmissivity (defined as the ratio between the flowrate and the pressure gradient) when Reynolds number (Re) is relatively large, i.e., Re >10.

A mathematical model based on the Reynolds flow equation for cement grout propagation in a homogeneous water-saturated rock fracture is presented. 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 flow 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 propagation 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. Applying an analytical solution based on single-phase flow, i.e., neglecting the impact of groundwater flow, for modeling of rock fracture grouting, will generally overestimate the propagation length.

The two-phase flow model for single fractures is extended to simulate non-Newtonian cement grouts propagation in water-saturated fracture networks. We verified the two-phase flow model by comparing numerical simulation results of two-phase flow of cement grouts propagation in fracture networks with the benchmark data in Håkansson (1987). Using this extended model for numerical simulations on the grout propagation the impacts of network geometry, hydraulic aperture distribution and the rheological properties (yield stress and plastic viscosity) are investigated. Cement grout propagation in randomly generated two-dimensional discrete fracture network (2D DFN) are simulated with different cases of hydraulic aperture variability, i.e., constant aperture, uncorrelated and length-correlated heterogeneous apertures following a truncated lognormal distribution. The results indicate that network structure and hydraulic aperture variability significantly affect the grout propagation negatively in 2D DFN systems. The randomized network structure and uncorrelated heterogeneous apertures significantly delay the propagation rate and largely increase the variability range of the penetration volume fraction (the ratio between penetrated volume and total volume of fractures). In contrast, in the case with length-correlated heterogeneous apertures, the propagation rate increases, while the variability range and rate of change of the penetration volume fraction decreases. The rheological properties of cement grouts, i.e., yield stress and plastic viscosity, also significantly affect cement grouts propagation in fracture networks. The propagation rate in the fracture networks reduces with the increase of the yield stress and the plastic viscosity of the cement grouts. The results presented in this report will be helpful in the design and prediction of rock grouting.

Modellering av cementbruks spridning i sprickist berg är viktigt för en ökad förståelse vid projektering, utförande och kontroll av injektering i praktiken. I detta projekt studerade vi icke-Newtonsk cementinjektering i bergssprickor genom teoretiska analyser och numeriska simuleringar.

De analytiska lösningarna för radiell strömning av en Bingham vätska studeras kritiskt och tvetydigheter i litteraturen beträffande pluggflödet i de två olika lösningar som används, för analys och design av injektering i bergssprickor, studeras. Analyserna baserade på en kraftbalans visar att pluggen vid radiell strömning av en Bingham vätska är oberoende av inträngningslängden. Bingham vätskans inträngning och flöde som funktion av injekteringstiden visas med användning av det konstanta pluggflödet. Skillnader i de två analytiska lösningarna och utveckling av flödet som funktion av tid illustreras. Förklaringen till pluggflödet och utvärderingen av skillnaderna i lösningarna som presenteras förbättrar vår kunskap och förenklar modellering och design av injektering i berg.

I praktiken utförs injektering med icke-Newtonska vätskor i råa sprickor, vilket dock sällan studeras. För att undersöka inverkan av en rå sprickyta vid injektering, presenteras numeriska beräkningar av icke-Newtonian strömning i enskilda råa sprickor, med hjälp av en regulariserad metod. De råa sprickmodellerna är skapade från en laserskannad yta av ett granitbergprov, för att representera realistiska egenskaper hos naturliga bergssprickor. De numeriska resultaten visar icke-linjära beteenden för flödet i råa sprickor orsakade av icke-Newtonska reologiska egenskaper förstärkta av sprickornas rånet. Råheten reducerar avsevärt den effektiva transmissiviteten när Reynolds tal (Re) är relativt stort, dvs Re> 10.

En matematisk modell baserad på Reynolds flödesekvation för inträngning av cementbruk i en slät, vattenmättad, bergspricka presenteras. Modellen är baserad på ett tvåfasflöde, dvs injektering som en Bingham-vätska och grundvatten som en Newtonsk vätska, vilka används för att undersöka påverkan av vattenfasen vid injektering. Modelleringsresultaten för tvåfasflödet visar i allmänhet på vikten av vattenfaten som väsentligen påverkar tryckfördelningen i sprickan, speciellt under härdning av bruket. Sådana effekter beror på viskositetsförhållandet mellan injekterings. Bruket och grundvattnet, vilka blir allt viktigare för fall en med mindre värden på viskositetsskillnaden. Att tillämpa en analytisk lösning baserad på ett enfasflöde, dvs att försumma påverkan av grundvatten vid modellering av ett injekteringförlopp, kommer att överskatta inträngningslängden.

Modellen för tvåfasflöde i enskilda sprickor utvidgas för att simulera icke-Newtonsk strömning i vattenmättade sprick nätverk. Modellen verifieras genom att jämföra simuleringsresultat för utbredningen i spricknät verk med referensdata från Håkansson (1987). Med användning av denna utökade modell undersöktes effekterna på utbredningen av nätverksstruktur och hydraulisk variabilitet, dvs nätgeometri och fördelning av spricköppningar, samt reologiska egenskaper, d.v.s. flytgräns och viskositet. Injekteringens utbredning i slumpmässigt genererat tvådimensionellt diskret spricknätverk (2D DFN) simuleras med olika fall av variabilitet i spricköppning, d.v.s. konstant öppning, baserat på och längdkorrelerad heterogena öppningar, efter en trunkerad lognormal fördelning. Resultaten indikerar att både nätverksstruktur och hydraulisk variation har en stor påverkan för utbredningen i ett 2D DFN-system. Den slumpade nätverksstrukturen och de okorrigerade heterogena öppningarna minskar avsevärt utbredningshastigheten och ökar till stor del variationen i injekterad volym. För längdkorrelerade heterogena öppningar, ökar utbredningshastigheten, medan variabiliteten och förändringen av injekterad volym minskar. De reologiska egenskaperna hos cementbruk, d.v.s. flytgräns och viskositet, påverkar väsentligen utbredningen i ett spricknätverk. Utbredningshastigheten i spricknätverken minskar med en ökning av flyt gräns och viskositet hos cementbruket. Resultaten som presenteras i denna rapport kommer att vara till hjälp vid utformningen och förutsägelsen av bergsprutning.

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.

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.

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.

During the design phase in rock grouting applications (e.g. for tunnels), analytical and numerical techniques based on inputs from the rock mass characterization and grout flow properties are used to estimate the grout spread. The design process is complicated by the fact that the exact geometry (network of fractures) within the rock mass is not completely known. In addition, the rheological flow properties of commonly used cement-based grouts are complex due to thixotropy and hydration. In such cases, simplified one-dimensional (1D) and two-dimensional 2D fracture geometries are used as a basis for the design solution. As for cement grouts, their rheological behavior is normally described by simplified constitutive laws e.g. the Bingham model.  Several  analytical solutions  for 1D channel flow and 2D radial flow of cement grouts have been presented in the literature describing the spread of grouts in fractures. Experimentally, only a limited amount of work has been carried out to study idealized yield stress fluid (YSF) flow between stationary parallel disks. The importance of such tests is that they facilitate the verification of analytical solutions and their limitations. Thus, in order to investigate in principle, the nature of 2D Bingham fluid  velocity profiles in radial  flow, we carried out  for apparently the first time  ultrasound velocimetry measurements  within the constraints of an experimental model. The  radial  flow region was formed by the gap (aperture) between two parallel acrylic glass (Plexiglas) disks, each with a diameter of 1 meter and a thickness of 25 mm. The disk separation was attained from a variable height metallic spacer configuration. Ultrasound velocity profiling (UVP) was used for flow visualization through the measurement of velocity profiles of a model yield stress fluid (Carbopol) at different radial positions. The results are a comparison of the measured velocity profiles with those from analytical solutions. Of particular interest is the plug-flow region of the radial velocity profiles along the radial length (diameter) of the parallel disks. The current observations show a distinct plug region, coupled with wall slip effects for the Carbopol model YSF fluid that was used. The theoretically predicted velocity profiles are lower than  the measured ones, however within a reasonably similar magnitude range. The main discrepancies between the theoretical predictions and measured data are then discussed. Future studies would then be targeted at improving the current experimental setup, for detailed measurements of the  plug-flow region along the radial length, which remains a generally challenging issue for studies on YSFs and more specifically for rock grouting design.  Moreover,  considering  roughened walls to significantly reduce wall slip  that was  present in the current study will also be part of the project’s continuation.

Cement grouts propagation into a two-dimensional water-saturated fracture networks with different values of rheological properties are simulated by using an extended two-phase flow model. The cement grouts are typical non-Newtonian fluids that contain yield stress, which are often assumed as Bingham fluids. The aim of this study is to investigate the impact of Bingham rheological properties, i.e. yield stress and plastic viscosity, on cement gouts propagation in two-dimensional fracture networks. The results generally show that the rheological properties of cement grouts, i.e. yield stress and plastic viscosity, significantly affect cement grouts propagation in the fracture network. The propagation rate in the fracture networks reduces with the increase of the yield stress and the plastic viscosity of the cement grouts.