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Radial penetration of cementitious grout – Laboratory verification of grout spread in a fracture model
Tunnelling and Underground Space Technology 72 (2018) 228–232
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Radial penetration of cementitious grout – Laboratory verification of grout spread in a fracture model
T
⁎
Johan Funehaga, , Johan Thörnb a b
Chalmers University of Technology and Tyréns AB, Gothenburg, Sweden Chalmers University of Technology and Bergab, Gothenburg, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: Bingham Grouting Verification Cement Radial penetration Hard rock
During the past two decades of research and development in the field of grouting in hard jointed rock, the design process has taken a number of significant leaps forward. A grouting design in hard rock can now be based on the penetration length of grout in individual rock fractures. For cementitious grouts, the most common rheological model used is the one for a Bingham fluid. The model is a conceptualisation of grout spread where two rheological properties of the grout – viscosity and yield stress – govern the penetration length along with the fracture aperture and applied grouting overpressure. This paper focuses on verification of radial Bingham flow of cementitious grout using a fracture model constructed from acrylic glass. Each test conducted using the fracture model was filmed, allowing the grout spread to be analysed as penetration length over time. The measured penetration lengths were then compared with analytical solutions derived for Bingham grout in a plane parallel fracture. The results indicate that the penetration of cementitious grout in fracture apertures of 125 μm and 200 μm is verified for up to 40% of the maximum possible penetration length. This can be compared to normal grouting, where the penetration lengths achieved are around 20% of the maximum penetration length.
1. Introduction During the past two decades of research and development in the field of grouting in hard jointed rock, the design process has taken a number of significant leaps forward. A grouting design in hard rock can now be based on the penetration length of grout in individual rock fractures. Gustafson and Stille (2005) show how a grouting design can be created. The design includes taking into account the apertures of the fractures in the rock mass, the type of grout used, the rheological properties and the grouting procedure, i.e. pressure and grouting times. In the case of cementitious grouts, the most common rheological model used is the one for a Bingham fluid. The model is a conceptualisation of grout spread where two rheological properties of the grout – viscosity and yield stress – govern the penetration length along with the fracture aperture and applied grouting overpressure. The shear stress (from the motion) must be greater than the yield stress of the grout for the fluid to move. The Bingham model is easy to apply when calculating the penetration length, which is probably why it is so widely used. The maximum possible penetration length for a cementitious grout depends on the applied pressure (Δp [Pa]), the fracture aperture (b [m]) and the yield stress of the grout (τ0 [Pa]) (Lombardi, 1985;
⁎
Hässler, 1991). One critical parameter that has been a source of debate is the hydraulic aperture. When conducting hydraulic tests in boreholes, a robust value for the hydraulic aperture can be derived by using the wellknown “cubic law”. The question of whether the hydraulic aperture represents the geometrical aperture field of a fracture entirely will not be discussed in this paper. However, it appears to be sufficient for the purposes of grout flow estimations, and it is used in this paper. Not all fractures are equal in size, Gustafson and Fransson (2006) suggest the use of the Pareto distribution to describe in statistical terms the hydraulic apertures of fractures encountered in core drilling. This is coupled with hydraulic tests performed in the borehole. Thörn et al. (2015) facilitated the analysis with a calculation tool designed to determine the fracture distribution and aperture distribution of the fractures that intersect a cored borehole. In Gustafson et al. (2013), a full derivation of a radial Bingham flow in a fracture is described. The grout used is a cement suspension with a yield stress and a viscosity. In the same paper, the formulation for onedimensional flow is verified using results from Håkansson (1993). This paper deals with verification of radial Bingham flow of cementitious grout using a fracture model constructed from acrylic glass.
Corresponding author. E-mail address: [email protected] (J. Funehag).
https://doi.org/10.1016/j.tust.2017.11.020 Received 19 January 2017; Received in revised form 23 August 2017; Accepted 5 November 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.
Tunnelling and Underground Space Technology 72 (2018) 228–232
J. Funehag, J. Thörn
Each test conducted using the fracture model was filmed, allowing the grout spread to be analysed as penetration length over time. The properties of the cementitious grout were checked using field tests (see e.g. Fransson et al., 2016) to measure density, yield stress, flowability and penetrability.
grouting. The plates are 1.8 m × 1.0 m (L × W) in size. The grouting hole (diameter = 50 mm) is placed equidistant to three sides, see Fig. 1. There is a rubber seal along the side of the acrylic glass. Water pressure can be applied to the shorter edges, referred to as the upstream and downstream sides. This enables grouting to take place in an artificial groundwater gradient and it makes it easier to clean the model. The upstream pressure is connected to a water container and the downstream pressure to a container used to collect wastewater and grout. When these two pressures are set at the same level, the water pressure in the model remains stationary. The grouting pressure is then set at the total pressure, i.e. the sum of the stationary pressure and the overpressure. Pressure regulators ensure that the upstream and downstream pressures remain constant. The water flow in the model is regulated by altering the pressure of the upstream and downstream pressures. Verification of the penetration length takes place by changing the aperture of the model and measuring the penetration length of the grout at different times. Each grouting test is recorded using an HD video camera. The recording is then used to determine the penetration length at different accumulated times and the grout front is drawn for each of the time steps. An example of a test is shown in Fig. 1. The penetration length is then plotted against time for four different axes (upstream, downstream and the two sides). Fig. 1 illustrates the difficulty in estimating the penetration length in closed boundary conditions (oval shaped final penetration lengths). The boundaries will affect the penetration length when it reaches closer to the walls. However, for shorter period, the spread of grout is almost circular and up to 30 cm of spread, estimation of the penetration length can be done. The grouting trials in this paper are optimised to reach a certain length over a certain time not to reach the boundaries in such a way that comparison between theory and experimental data can be compared.
2. Method The Bingham model is used in this context to characterise a cement suspension. Its characteristics are yield stress (τ0) and plastic viscosity (μg). Dai and Bird (1981) formulated the differential equation for the pressure distribution for Bingham flow and Hässler (1991) solved the equation numerically. According to Hässler (1991), the velocity (vg(r)) of the Bingham fluid in a circular disc under a pressure gradient (dp/dr) can be described as
vg (r ) = −
dp b2 ⎡ Z (r ) Z (r ) ⎞3⎤ · · 1−3· , rb ⩽ r ⩽ rb + I . + 4·⎛ dr 12μg ⎢ b ⎝ b ⎠⎥ ⎣ ⎦
(1)
where b is the aperture between the plates and Z(r) is the half Bingham plug thickness. Z is then a vertical coordinate starting from the centre of the motion (the centre of the open space between the plates). The plug thickness (Z(r)) depends on the yield stress (τg) of the grout according to:
Z (r ) =
τg dp dr
=
τg dp
, Z<
− dr
b . 2
(2)
It can be seen that the plug thickness can only vary over the aperture and that it varies according to the pressure gradient. When the pressure gradient is high, i.e. when grouting commences, the plug thickness is small. As long as the penetration length continues to increase, the pressure gradient will decrease and the plug thickness will increase. The negative value of the pressure gradient is the result of the pressure decreasing from the borehole to the grout front.
2.1.1. Grouting and framework design The physical aperture between the plates is set using washers of precise thickness. When the plate set-up is complete, a hydraulic test is carried out to compute the hydraulic aperture using the well-known “cubic law”. The flow through the fracture model is calculated by measuring the water collected in the waste container. The pressure through the packer is kept constant and the volume over time is derived. The fracture model is designed for use with maximum 0.2 MPa pressure. FEM calculations was used for scoping the material dimensions at acceptable levels of deformation under maximum pressure
2.1. Laboratory set-up The laboratory set-up was created to demonstrate the flow between two parallel plates. The fracture model is made up of two 40 mm thick acrylic glass sheets, fastened together using 35 bolts. The aperture can be changed by placing washers (shims) of a known thickness (0.100, 0.050 and 0.025 mm) between the plates, close to the bolts. The aim of the bolt pattern is to minimise the expansion of the aperture when
Fig. 1. Grout spread at 5, 10, 20, 30, 40, 50 and 60 s after grouting commenced.
229
Tunnelling and Underground Space Technology 72 (2018) 228–232
J. Funehag, J. Thörn
in a fracture. The formula is derived analytically and includes the grouting time. This means that if the grouting time is infinite, the penetration length is in accordance with Eq. (1). For a timeframe that is more realistic, i.e. 10–30 min, the actual penetration length (I) is a proportion of the maximum penetration length (Imax). The analytical formulation is solved by introducing a characteristic grouting time (t0) and two dimensionless parameters (tD and ID). In Gustafson and Stille (2005), it is shown how the analytical solution can be arrived at using an approximation of the numerical iteration and by introducing dimensionless parameters. The following approximation is used: The characteristic grouting time (t0) is
(approx. < 0.1 mm). This is in the same order of magnitude as the aperture and it should be pointed out that the precise, thin washers are for repeatability between runs. The aperture is retrieved from the hydraulic test, which is performed at the same pressure as the subsequent grouting. This procedure essentially cancels the effect of elastic expansion in the tightening bolts and bending of the plate between the bolts. As regards to the closed boundaries, the penetration length can be calculated up to approximately 0.3 m from the grouting borehole. With a maximum penetration length of 0.3 m, the pressures could range from 0.1 bar to 1 bar (1–10 m in head), depending to a certain extent on the yield stress and viscosity of the grout.
to = 2.2. Grout mixes
tD =
ID =
Run 1 Run 2
Marsh cone (s)
Filter pump (ml)
200 125
0.72 0.72
1.63 1.63
4 5
41 41
320 330
θ 2 + 4θ −θ
I = ID × Imax
(6)
(7)
(8)
It is can be clearly seen that if the grouting time is taken into account, the actual penetration length is always shorter than the maximum penetration length. 3. Results The results of two runs in two different hydraulic apertures, 200 μm and 125 μm, are shown in Table 2. The hydraulic aperture for run 2, the 125 μm, was computed from the measured volume of 0.1 l collected for 41 s. The overpressure was 0.2 bar. With a viscosity of water and a density of 0.0013 Pa s and 992 kg/m3 respectively results in a hydraulic aperture using the “cubic law” of 125 μm. The same setup, a specific volume was collected for a certain time was done for the aperture of 200 μm. Using the fixed volume and measure the time was estimated to give the best accuracy. For a volume of 0.5 l requires 50 s to be equal to the 200 μm. With error of 2 s in reading results in 2.5 μm in error in the calculated hydraulic aperture. The penetration lengths were calculated using the approximations in Eqs. (3)–(7) in a spreadsheet program. The calculated penetration lengths (Icalc) were compared with the measured penetration lengths (Imeas) and are shown in Fig. 2. The overpressure, Δp = 0.3 bar, was used for both runs. The measured penetration is along the axis that had a radial uniform shape (in these runs the up and down axis in Fig. 2). Fig. 2 shows an almost perfect fit for the measured penetration lengths using the calculated radial Bingham flow. The first penetration
Table 1 Properties of the grout mixes used in the fracture model. Yield stress (Pa)
tD tD and θ1D = 2(3 + tD ) 2(0.6 + tD )
The dimensionless penetration length is described using a value from 0 to 1 and is used to relate the dimensionless penetration length to the maximum possible penetration length (Imax) according to Eq. (1). The actual penetration length (I) is thus calculated as ID, with metres as the unit (Eq. (8));
where Δp is the overpressure applied, b is the aperture of the slot or fracture and τ0 is the yield stress of the cement suspension determined using the Bingham fluid rheological model. Gustafson et al. (2013) present the basic formula for Bingham flow
Density (g/ cm3)
(5)
The computational factor (θ) is used to calculate the dimensionless penetration length (ID) according to Eq. (7).
(3)
WCR
t I , and ID = t0 Imax
θ2D =
Studying the force equilibrium of a Bingham fluid using a stiff kernel/plug shows that the grout front continues to move until the shear stress is equal to the yield stress of the Bingham fluid. In Gustafson and Stille (2005), the distance to this point is denoted as the maximum penetration length (Imax) according to
Hydraulic aperture (μm)
(4)
The dimensionless time is determined by setting a desired grouting time (t). This is then used in Eq. (6) below to calculate the computational factor (θ) The factor is derived for one-dimensional (1D) or twodimensional (2D) flow.
2.3. Calculation of the penetration length
Δp × b 2τ0
τo2
where the viscosity of the grout is μg. The characteristic grouting time involves the applied overpressure and the grout properties, yield stress and viscosity, and with seconds as the unit. These properties are adjusted and changed, which involves adapting the grout, grouting pressure and grouting time to the grouting operation in question. The corresponding dimensionless time (tD) and penetration (ID) are
The cement used is an ordinary Portland cement – INJ30 (manufactured by Sika). It has a d95 of 30 μm. The rheological parameters modelled are the viscosity (μ [Pa s]) and the yield stress (τ [Pa]). The grout properties used to fit the maximum penetration length and pressures were a viscosity of around 30 mPa s and a yield stress of 2–4 Pa. Prior to the run, several mixes of the grout were measured using a rotational viscometer to calculate the viscosity and yield stress. The concept of measuring the grout properties follows the methodology described in Fransson et al. (2016). Together with the mixing test, various methods suitable for field use were employed. Mud balance was used to measure the grout density, yield stick to measure the yield stress of the grout (see Axelsson and Gustafson, 2006) and Marsh cone to measure the flowability and to confirm and compare with the viscosity determined using a rheometer. The filter pump test was used to confirm that the mixing was correct. The mixing procedure is as follows: Pour in 80% of the total water volume, turn on the mixer and slowly pour in the cement. Mix for 30 s and then pour in the rest of the water. After mixing for one minute, add 2% Set Control II (manufacturer SIKA) of the cement weight and then mix for one minute. The mixer used was a highspeed vortex flow rotational mixer (up to 30,000 rpm). The grout properties achieved for the two test set-ups are presented in Table 1. The WCR is the water to cement ratio and is in weight. The hydraulic aperture of the fracture model was determined by measuring the outflow from the fracture during constant-head injection of water. The aperture could be computed using the “cubic law”.
Imax =
6μg × Δp
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Tunnelling and Underground Space Technology 72 (2018) 228–232
J. Funehag, J. Thörn
Table 2 Results from the comparison of the measured penetration lengths with the calculated lengths. Time (s)
5 10 20 30 40 50 60 70 80 90 100
Run 1, bhyd = 200 μm Δp = 0.3 bar
Run 2, bhyd = 125 μm Δp = 0.3 bar
Imeas (m)
Icalc (m)
Imeas (m)
Icalc (m)
0.10 0.13 0.16 0.19 0.22 0.24 0.26 0.26 0.28 0.28 0.29
0.07 0.10 0.15 0.18 0.20 0.23 0.25 0.26 0.28 0.30 0.31
0.03 0.05 0.08 0.12 0.15 0.16 0.17 0.18 0.19 0.20 0.20
0.05 0.07 0.10 0.12 0.13 0.15 0.16 0.17 0.18 0.19 0.20
is difficult to measure as there is a small amount of dilution at the front and it is not easy to distinguish the actual grout front. In addition, a number of air bubbles were trapped, which made the grout front jagged (Fig. 2, right). The measured penetration interpreted from the video captures. The captures are in real time and each time step it is paused and a photo is taken from the video. The photo is edited with a disc fitted to the longest penetration along one of the axis and adjusted due to the angle of the video capture. The measured penetration length with this methodology is giving a measuring accuracy of some 5–10%.
Fig. 3. Relationship between ID and tD for radial Bingham flow.
factors ID and tD. The interpretation can then be used for guidance when making a choice of grouting pressure and grouting times when grouting in hard rock. A summary of how the calculations can be used is shown in Fig. 4 and Table 3. Fig. 4 above shows typical values for the grouting/sealing of fractures in hard rock in a shallow tunnel (< 100 m of overburden). The grouting times are often less than 40 min and the maximum overpressure used is 3 MPa. The relationship (ID) is a measure of how far the penetration has reached in relation to the possible maximum penetration (Imax). The typical penetration length, e.g. for grout type 3 and a fracture aperture of 125 μm, is 6 m after 20 min of grouting. This means that the penetration has reached 26% (6 m/23.4 m). For the longest penetration, with grout type 1 and an aperture of 200 μm, the figure is only 5% or so after 20 min. This illustrates that the verification of the penetration lengths in the laboratory set-up is valid for typical grout values given the set-ups used.
4. Discussion The maximum penetration length (Imax) (Gustafson et al., 2013) is given in Eq. (3). Using this as a reference, if the grouting had continued for an infinite period of time the penetration could have been 0.75 m in a 200 μm aperture and 0.48 m in a 125 μm aperture. The final penetration lengths shown in the verification for the two hydraulic apertures (200 μm = 0.29 m and 125 μm = 0.2 m) are then scaled via ID to almost 40% of the maximum possible penetration lengths. According to Gustafson and Stille (2005), the most efficient level of grouting achieved is up to a maximum of around 60% of ID for a radial flow (2D), and relates to a tD of 1. The relationship between ID and tD for radial Bingham flow is shown in Fig. 3. For the same grout mix set-up (μ = 30 mPa s and τ0 = 4 Pa) and a fracture aperture of 200 μm but with a more realistic overpressure of 2 MPa, the maximum penetration is Imax = 50 m (Eq. (3)). The characteristic time (t0) will be 375 min (Eq. (4)). With a grouting time of 20 min, a penetration length of 8.9 m is achieved. This corresponds to almost 20% of the maximum possible penetration (Imax). The laboratory verification suggests that a comparison between measured and calculated penetration lengths can be made if the penetration is scaled using
5. Conclusion The study of penetration lengths of Bingham grouts in the laboratory set-up verifies the results for two fracture apertures, 125 μm and 200 μm and one type of grout. The grout type is a typical stable grout and the fracture apertures are seen as normal apertures apparent in crystalline hard rock at shallow depths, up to 50 m. The statement of normal apertures down to 50 m of depth bases on experience from evaluation waterloss measurements of grouting boreholes in several tunnel projects. Apertures can for sure be larger and smaller for a heterogeneous rock mass than the 125–200 μm. Fig. 2. Left: A plot of the measured penetration lengths and the calculated lengths in two different hydraulic apertures. The overpressure was 0.3 bar and the grout properties are as shown in Table 1. Right: a photo showing the radial grout flow in the fracture model. It should be noted that the photo is not taken vertically above the borehole, which makes it a bit tedious to predict the penetration length.
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Tunnelling and Underground Space Technology 72 (2018) 228–232
J. Funehag, J. Thörn
Fig. 4. Calculated radial penetration of three different grout types in two different fracture apertures, small = 125 μm and large = 200 μm, at an overpressure of Δp = 3 MPa.
Table 3 Tabulated values for the calculated penetration lengths shown in Fig. 4. Grout properties
Hydraulic apertures
Calculated values
Grout type
Yield stress [Pa]
Viscosity [Pa s]
bmin [μm]
bmax [μm]
t0 [min]
Imax (bmin) [m]
Imax (bmax) [m]
(1) (2) (3)
1 3 8
0.02 0.025 0.04
125 125 125
200 200 200
6000 833.3 187.5
187.5 62.5 23.4
300.0 100.0 37.50
Foundation (BeFo) and the Swedish construction industry’s organisation for research development (SBUF), gave financial support. Help with the runs in the fracture model by Sebastian Hortberg, Tyréns AB is highly appreciated.
The grout properties and the pressure were chosen to suit the laboratory set-up, enabling a maximum radial penetration length of 0.5 m to be achieved within a timeframe of minutes. The work in this paper does not consider the physical and chemical changes that the cement undergoes. It is the author’s belief that this does not affect the penetration length within the short time frames that the grouting is commenced. The pressures were relatively small to normal practice for grouting in hard rock, i.e. less than 0.2 MPa. Doing tests with other types of properties of the grout in the fracture model is limited to the possible penetration lengths and the pressures applied. However, with the scaling factor (ID) this results in 40% of the maximum penetration length, which is longer than what is normally the case for grouting fractures in hard rock. The verification of the penetration lengths using the Bingham fluid model can be used for design of grouting. Within the framework of this paper with fracture apertures of 125 and 200 μm with a stable grout having a yield shear strength of 3–4 Pa will give penetration length of some 6–14 m for a grouting time of 15–20 min and a overpressure of 3 MPa. For shallow tunnels in hard crystalline rock this is close where the design ends up with. The equations used for verification are based on the principle that the penetration length can be computed if the fracture aperture, rheological grout properties and applied overpressure are known.
References Axelsson, M., Gustafson, G., 2006. A robust method to determine the shear strength of cement-based injection grouts in the field. Tunn. Undergr. Space Technol. 21 (5), 499–503 Elsevier. Dai, G., Bird, B.R., 1981. Radial flow of a Bingham fluid between two fixed circular disks. J. Non-Newtonian Fluid Mech. 8, 349–355. Gustafson, G., Claesson, J., Fransson, Å., 2013. Steering parameters for rock grouting. J. Appl. Math. 2013, 269594. Gustafson, G., Fransson, Å., 2006. The use of the Pareto distribution for fracture transmissivity assessment. Hydrogeol. J. 14, 15–20. http://dx.doi.org/10.1007/s10040005-0440-y. Gustafson, G., Stille, H., 2005. Stop criteria for cement grouting. Felsbau 3, 62–68. Håkansson, U., 1993. Rheology of fresh cement-based grouts. Doctoral Thesis. Department of Infrastructure and Environmental Engineering, Royal Institute of Technology, Stockholm, Sweden. Hässler, L., 1991. Grouting of rock—simulation and classification. Doctoral Thesis. Department of Soil and Rock Mechanics, Royal Institute of Technology, Sweden, pp. 159. Fransson, Å., Funehag, J., Thörn, J., 2016. Swedish grouting design: hydraulic testing and grout selection. Proc. Inst. Civ. Eng. - Ground Improvement 169, 275–285. http://dx. doi.org/10.1680/jgrim.15.00020. Lombardi, G., 1985. The Role of Cohesion in Cement Grouting of Rock. Commission Internationale des Grands Barrages, Lausanne. Thörn, J., Kvartsberg, S., Runslätt, E., Almfeldt, S., Fransson, Å., 2015. Calculation tool for rock characterisation during grouting design –theory and users guide. BeFo, Stockholm (in Swedish).
Acknowledgement Sven Tyréns Foundation (STS), Rock Engineering Research
232
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology journal homepage: www.elsevier.com/locate/tust
Radial penetration of cementitious grout – Laboratory verification of grout spread in a fracture model
T
⁎
Johan Funehaga, , Johan Thörnb a b
Chalmers University of Technology and Tyréns AB, Gothenburg, Sweden Chalmers University of Technology and Bergab, Gothenburg, Sweden
A R T I C L E I N F O
A B S T R A C T
Keywords: Bingham Grouting Verification Cement Radial penetration Hard rock
During the past two decades of research and development in the field of grouting in hard jointed rock, the design process has taken a number of significant leaps forward. A grouting design in hard rock can now be based on the penetration length of grout in individual rock fractures. For cementitious grouts, the most common rheological model used is the one for a Bingham fluid. The model is a conceptualisation of grout spread where two rheological properties of the grout – viscosity and yield stress – govern the penetration length along with the fracture aperture and applied grouting overpressure. This paper focuses on verification of radial Bingham flow of cementitious grout using a fracture model constructed from acrylic glass. Each test conducted using the fracture model was filmed, allowing the grout spread to be analysed as penetration length over time. The measured penetration lengths were then compared with analytical solutions derived for Bingham grout in a plane parallel fracture. The results indicate that the penetration of cementitious grout in fracture apertures of 125 μm and 200 μm is verified for up to 40% of the maximum possible penetration length. This can be compared to normal grouting, where the penetration lengths achieved are around 20% of the maximum penetration length.
1. Introduction During the past two decades of research and development in the field of grouting in hard jointed rock, the design process has taken a number of significant leaps forward. A grouting design in hard rock can now be based on the penetration length of grout in individual rock fractures. Gustafson and Stille (2005) show how a grouting design can be created. The design includes taking into account the apertures of the fractures in the rock mass, the type of grout used, the rheological properties and the grouting procedure, i.e. pressure and grouting times. In the case of cementitious grouts, the most common rheological model used is the one for a Bingham fluid. The model is a conceptualisation of grout spread where two rheological properties of the grout – viscosity and yield stress – govern the penetration length along with the fracture aperture and applied grouting overpressure. The shear stress (from the motion) must be greater than the yield stress of the grout for the fluid to move. The Bingham model is easy to apply when calculating the penetration length, which is probably why it is so widely used. The maximum possible penetration length for a cementitious grout depends on the applied pressure (Δp [Pa]), the fracture aperture (b [m]) and the yield stress of the grout (τ0 [Pa]) (Lombardi, 1985;
⁎
Hässler, 1991). One critical parameter that has been a source of debate is the hydraulic aperture. When conducting hydraulic tests in boreholes, a robust value for the hydraulic aperture can be derived by using the wellknown “cubic law”. The question of whether the hydraulic aperture represents the geometrical aperture field of a fracture entirely will not be discussed in this paper. However, it appears to be sufficient for the purposes of grout flow estimations, and it is used in this paper. Not all fractures are equal in size, Gustafson and Fransson (2006) suggest the use of the Pareto distribution to describe in statistical terms the hydraulic apertures of fractures encountered in core drilling. This is coupled with hydraulic tests performed in the borehole. Thörn et al. (2015) facilitated the analysis with a calculation tool designed to determine the fracture distribution and aperture distribution of the fractures that intersect a cored borehole. In Gustafson et al. (2013), a full derivation of a radial Bingham flow in a fracture is described. The grout used is a cement suspension with a yield stress and a viscosity. In the same paper, the formulation for onedimensional flow is verified using results from Håkansson (1993). This paper deals with verification of radial Bingham flow of cementitious grout using a fracture model constructed from acrylic glass.
Corresponding author. E-mail address: [email protected] (J. Funehag).
https://doi.org/10.1016/j.tust.2017.11.020 Received 19 January 2017; Received in revised form 23 August 2017; Accepted 5 November 2017 0886-7798/ © 2017 Elsevier Ltd. All rights reserved.
Tunnelling and Underground Space Technology 72 (2018) 228–232
J. Funehag, J. Thörn
Each test conducted using the fracture model was filmed, allowing the grout spread to be analysed as penetration length over time. The properties of the cementitious grout were checked using field tests (see e.g. Fransson et al., 2016) to measure density, yield stress, flowability and penetrability.
grouting. The plates are 1.8 m × 1.0 m (L × W) in size. The grouting hole (diameter = 50 mm) is placed equidistant to three sides, see Fig. 1. There is a rubber seal along the side of the acrylic glass. Water pressure can be applied to the shorter edges, referred to as the upstream and downstream sides. This enables grouting to take place in an artificial groundwater gradient and it makes it easier to clean the model. The upstream pressure is connected to a water container and the downstream pressure to a container used to collect wastewater and grout. When these two pressures are set at the same level, the water pressure in the model remains stationary. The grouting pressure is then set at the total pressure, i.e. the sum of the stationary pressure and the overpressure. Pressure regulators ensure that the upstream and downstream pressures remain constant. The water flow in the model is regulated by altering the pressure of the upstream and downstream pressures. Verification of the penetration length takes place by changing the aperture of the model and measuring the penetration length of the grout at different times. Each grouting test is recorded using an HD video camera. The recording is then used to determine the penetration length at different accumulated times and the grout front is drawn for each of the time steps. An example of a test is shown in Fig. 1. The penetration length is then plotted against time for four different axes (upstream, downstream and the two sides). Fig. 1 illustrates the difficulty in estimating the penetration length in closed boundary conditions (oval shaped final penetration lengths). The boundaries will affect the penetration length when it reaches closer to the walls. However, for shorter period, the spread of grout is almost circular and up to 30 cm of spread, estimation of the penetration length can be done. The grouting trials in this paper are optimised to reach a certain length over a certain time not to reach the boundaries in such a way that comparison between theory and experimental data can be compared.
2. Method The Bingham model is used in this context to characterise a cement suspension. Its characteristics are yield stress (τ0) and plastic viscosity (μg). Dai and Bird (1981) formulated the differential equation for the pressure distribution for Bingham flow and Hässler (1991) solved the equation numerically. According to Hässler (1991), the velocity (vg(r)) of the Bingham fluid in a circular disc under a pressure gradient (dp/dr) can be described as
vg (r ) = −
dp b2 ⎡ Z (r ) Z (r ) ⎞3⎤ · · 1−3· , rb ⩽ r ⩽ rb + I . + 4·⎛ dr 12μg ⎢ b ⎝ b ⎠⎥ ⎣ ⎦
(1)
where b is the aperture between the plates and Z(r) is the half Bingham plug thickness. Z is then a vertical coordinate starting from the centre of the motion (the centre of the open space between the plates). The plug thickness (Z(r)) depends on the yield stress (τg) of the grout according to:
Z (r ) =
τg dp dr
=
τg dp
, Z<
− dr
b . 2
(2)
It can be seen that the plug thickness can only vary over the aperture and that it varies according to the pressure gradient. When the pressure gradient is high, i.e. when grouting commences, the plug thickness is small. As long as the penetration length continues to increase, the pressure gradient will decrease and the plug thickness will increase. The negative value of the pressure gradient is the result of the pressure decreasing from the borehole to the grout front.
2.1.1. Grouting and framework design The physical aperture between the plates is set using washers of precise thickness. When the plate set-up is complete, a hydraulic test is carried out to compute the hydraulic aperture using the well-known “cubic law”. The flow through the fracture model is calculated by measuring the water collected in the waste container. The pressure through the packer is kept constant and the volume over time is derived. The fracture model is designed for use with maximum 0.2 MPa pressure. FEM calculations was used for scoping the material dimensions at acceptable levels of deformation under maximum pressure
2.1. Laboratory set-up The laboratory set-up was created to demonstrate the flow between two parallel plates. The fracture model is made up of two 40 mm thick acrylic glass sheets, fastened together using 35 bolts. The aperture can be changed by placing washers (shims) of a known thickness (0.100, 0.050 and 0.025 mm) between the plates, close to the bolts. The aim of the bolt pattern is to minimise the expansion of the aperture when
Fig. 1. Grout spread at 5, 10, 20, 30, 40, 50 and 60 s after grouting commenced.
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in a fracture. The formula is derived analytically and includes the grouting time. This means that if the grouting time is infinite, the penetration length is in accordance with Eq. (1). For a timeframe that is more realistic, i.e. 10–30 min, the actual penetration length (I) is a proportion of the maximum penetration length (Imax). The analytical formulation is solved by introducing a characteristic grouting time (t0) and two dimensionless parameters (tD and ID). In Gustafson and Stille (2005), it is shown how the analytical solution can be arrived at using an approximation of the numerical iteration and by introducing dimensionless parameters. The following approximation is used: The characteristic grouting time (t0) is
(approx. < 0.1 mm). This is in the same order of magnitude as the aperture and it should be pointed out that the precise, thin washers are for repeatability between runs. The aperture is retrieved from the hydraulic test, which is performed at the same pressure as the subsequent grouting. This procedure essentially cancels the effect of elastic expansion in the tightening bolts and bending of the plate between the bolts. As regards to the closed boundaries, the penetration length can be calculated up to approximately 0.3 m from the grouting borehole. With a maximum penetration length of 0.3 m, the pressures could range from 0.1 bar to 1 bar (1–10 m in head), depending to a certain extent on the yield stress and viscosity of the grout.
to = 2.2. Grout mixes
tD =
ID =
Run 1 Run 2
Marsh cone (s)
Filter pump (ml)
200 125
0.72 0.72
1.63 1.63
4 5
41 41
320 330
θ 2 + 4θ −θ
I = ID × Imax
(6)
(7)
(8)
It is can be clearly seen that if the grouting time is taken into account, the actual penetration length is always shorter than the maximum penetration length. 3. Results The results of two runs in two different hydraulic apertures, 200 μm and 125 μm, are shown in Table 2. The hydraulic aperture for run 2, the 125 μm, was computed from the measured volume of 0.1 l collected for 41 s. The overpressure was 0.2 bar. With a viscosity of water and a density of 0.0013 Pa s and 992 kg/m3 respectively results in a hydraulic aperture using the “cubic law” of 125 μm. The same setup, a specific volume was collected for a certain time was done for the aperture of 200 μm. Using the fixed volume and measure the time was estimated to give the best accuracy. For a volume of 0.5 l requires 50 s to be equal to the 200 μm. With error of 2 s in reading results in 2.5 μm in error in the calculated hydraulic aperture. The penetration lengths were calculated using the approximations in Eqs. (3)–(7) in a spreadsheet program. The calculated penetration lengths (Icalc) were compared with the measured penetration lengths (Imeas) and are shown in Fig. 2. The overpressure, Δp = 0.3 bar, was used for both runs. The measured penetration is along the axis that had a radial uniform shape (in these runs the up and down axis in Fig. 2). Fig. 2 shows an almost perfect fit for the measured penetration lengths using the calculated radial Bingham flow. The first penetration
Table 1 Properties of the grout mixes used in the fracture model. Yield stress (Pa)
tD tD and θ1D = 2(3 + tD ) 2(0.6 + tD )
The dimensionless penetration length is described using a value from 0 to 1 and is used to relate the dimensionless penetration length to the maximum possible penetration length (Imax) according to Eq. (1). The actual penetration length (I) is thus calculated as ID, with metres as the unit (Eq. (8));
where Δp is the overpressure applied, b is the aperture of the slot or fracture and τ0 is the yield stress of the cement suspension determined using the Bingham fluid rheological model. Gustafson et al. (2013) present the basic formula for Bingham flow
Density (g/ cm3)
(5)
The computational factor (θ) is used to calculate the dimensionless penetration length (ID) according to Eq. (7).
(3)
WCR
t I , and ID = t0 Imax
θ2D =
Studying the force equilibrium of a Bingham fluid using a stiff kernel/plug shows that the grout front continues to move until the shear stress is equal to the yield stress of the Bingham fluid. In Gustafson and Stille (2005), the distance to this point is denoted as the maximum penetration length (Imax) according to
Hydraulic aperture (μm)
(4)
The dimensionless time is determined by setting a desired grouting time (t). This is then used in Eq. (6) below to calculate the computational factor (θ) The factor is derived for one-dimensional (1D) or twodimensional (2D) flow.
2.3. Calculation of the penetration length
Δp × b 2τ0
τo2
where the viscosity of the grout is μg. The characteristic grouting time involves the applied overpressure and the grout properties, yield stress and viscosity, and with seconds as the unit. These properties are adjusted and changed, which involves adapting the grout, grouting pressure and grouting time to the grouting operation in question. The corresponding dimensionless time (tD) and penetration (ID) are
The cement used is an ordinary Portland cement – INJ30 (manufactured by Sika). It has a d95 of 30 μm. The rheological parameters modelled are the viscosity (μ [Pa s]) and the yield stress (τ [Pa]). The grout properties used to fit the maximum penetration length and pressures were a viscosity of around 30 mPa s and a yield stress of 2–4 Pa. Prior to the run, several mixes of the grout were measured using a rotational viscometer to calculate the viscosity and yield stress. The concept of measuring the grout properties follows the methodology described in Fransson et al. (2016). Together with the mixing test, various methods suitable for field use were employed. Mud balance was used to measure the grout density, yield stick to measure the yield stress of the grout (see Axelsson and Gustafson, 2006) and Marsh cone to measure the flowability and to confirm and compare with the viscosity determined using a rheometer. The filter pump test was used to confirm that the mixing was correct. The mixing procedure is as follows: Pour in 80% of the total water volume, turn on the mixer and slowly pour in the cement. Mix for 30 s and then pour in the rest of the water. After mixing for one minute, add 2% Set Control II (manufacturer SIKA) of the cement weight and then mix for one minute. The mixer used was a highspeed vortex flow rotational mixer (up to 30,000 rpm). The grout properties achieved for the two test set-ups are presented in Table 1. The WCR is the water to cement ratio and is in weight. The hydraulic aperture of the fracture model was determined by measuring the outflow from the fracture during constant-head injection of water. The aperture could be computed using the “cubic law”.
Imax =
6μg × Δp
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Table 2 Results from the comparison of the measured penetration lengths with the calculated lengths. Time (s)
5 10 20 30 40 50 60 70 80 90 100
Run 1, bhyd = 200 μm Δp = 0.3 bar
Run 2, bhyd = 125 μm Δp = 0.3 bar
Imeas (m)
Icalc (m)
Imeas (m)
Icalc (m)
0.10 0.13 0.16 0.19 0.22 0.24 0.26 0.26 0.28 0.28 0.29
0.07 0.10 0.15 0.18 0.20 0.23 0.25 0.26 0.28 0.30 0.31
0.03 0.05 0.08 0.12 0.15 0.16 0.17 0.18 0.19 0.20 0.20
0.05 0.07 0.10 0.12 0.13 0.15 0.16 0.17 0.18 0.19 0.20
is difficult to measure as there is a small amount of dilution at the front and it is not easy to distinguish the actual grout front. In addition, a number of air bubbles were trapped, which made the grout front jagged (Fig. 2, right). The measured penetration interpreted from the video captures. The captures are in real time and each time step it is paused and a photo is taken from the video. The photo is edited with a disc fitted to the longest penetration along one of the axis and adjusted due to the angle of the video capture. The measured penetration length with this methodology is giving a measuring accuracy of some 5–10%.
Fig. 3. Relationship between ID and tD for radial Bingham flow.
factors ID and tD. The interpretation can then be used for guidance when making a choice of grouting pressure and grouting times when grouting in hard rock. A summary of how the calculations can be used is shown in Fig. 4 and Table 3. Fig. 4 above shows typical values for the grouting/sealing of fractures in hard rock in a shallow tunnel (< 100 m of overburden). The grouting times are often less than 40 min and the maximum overpressure used is 3 MPa. The relationship (ID) is a measure of how far the penetration has reached in relation to the possible maximum penetration (Imax). The typical penetration length, e.g. for grout type 3 and a fracture aperture of 125 μm, is 6 m after 20 min of grouting. This means that the penetration has reached 26% (6 m/23.4 m). For the longest penetration, with grout type 1 and an aperture of 200 μm, the figure is only 5% or so after 20 min. This illustrates that the verification of the penetration lengths in the laboratory set-up is valid for typical grout values given the set-ups used.
4. Discussion The maximum penetration length (Imax) (Gustafson et al., 2013) is given in Eq. (3). Using this as a reference, if the grouting had continued for an infinite period of time the penetration could have been 0.75 m in a 200 μm aperture and 0.48 m in a 125 μm aperture. The final penetration lengths shown in the verification for the two hydraulic apertures (200 μm = 0.29 m and 125 μm = 0.2 m) are then scaled via ID to almost 40% of the maximum possible penetration lengths. According to Gustafson and Stille (2005), the most efficient level of grouting achieved is up to a maximum of around 60% of ID for a radial flow (2D), and relates to a tD of 1. The relationship between ID and tD for radial Bingham flow is shown in Fig. 3. For the same grout mix set-up (μ = 30 mPa s and τ0 = 4 Pa) and a fracture aperture of 200 μm but with a more realistic overpressure of 2 MPa, the maximum penetration is Imax = 50 m (Eq. (3)). The characteristic time (t0) will be 375 min (Eq. (4)). With a grouting time of 20 min, a penetration length of 8.9 m is achieved. This corresponds to almost 20% of the maximum possible penetration (Imax). The laboratory verification suggests that a comparison between measured and calculated penetration lengths can be made if the penetration is scaled using
5. Conclusion The study of penetration lengths of Bingham grouts in the laboratory set-up verifies the results for two fracture apertures, 125 μm and 200 μm and one type of grout. The grout type is a typical stable grout and the fracture apertures are seen as normal apertures apparent in crystalline hard rock at shallow depths, up to 50 m. The statement of normal apertures down to 50 m of depth bases on experience from evaluation waterloss measurements of grouting boreholes in several tunnel projects. Apertures can for sure be larger and smaller for a heterogeneous rock mass than the 125–200 μm. Fig. 2. Left: A plot of the measured penetration lengths and the calculated lengths in two different hydraulic apertures. The overpressure was 0.3 bar and the grout properties are as shown in Table 1. Right: a photo showing the radial grout flow in the fracture model. It should be noted that the photo is not taken vertically above the borehole, which makes it a bit tedious to predict the penetration length.
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Fig. 4. Calculated radial penetration of three different grout types in two different fracture apertures, small = 125 μm and large = 200 μm, at an overpressure of Δp = 3 MPa.
Table 3 Tabulated values for the calculated penetration lengths shown in Fig. 4. Grout properties
Hydraulic apertures
Calculated values
Grout type
Yield stress [Pa]
Viscosity [Pa s]
bmin [μm]
bmax [μm]
t0 [min]
Imax (bmin) [m]
Imax (bmax) [m]
(1) (2) (3)
1 3 8
0.02 0.025 0.04
125 125 125
200 200 200
6000 833.3 187.5
187.5 62.5 23.4
300.0 100.0 37.50
Foundation (BeFo) and the Swedish construction industry’s organisation for research development (SBUF), gave financial support. Help with the runs in the fracture model by Sebastian Hortberg, Tyréns AB is highly appreciated.
The grout properties and the pressure were chosen to suit the laboratory set-up, enabling a maximum radial penetration length of 0.5 m to be achieved within a timeframe of minutes. The work in this paper does not consider the physical and chemical changes that the cement undergoes. It is the author’s belief that this does not affect the penetration length within the short time frames that the grouting is commenced. The pressures were relatively small to normal practice for grouting in hard rock, i.e. less than 0.2 MPa. Doing tests with other types of properties of the grout in the fracture model is limited to the possible penetration lengths and the pressures applied. However, with the scaling factor (ID) this results in 40% of the maximum penetration length, which is longer than what is normally the case for grouting fractures in hard rock. The verification of the penetration lengths using the Bingham fluid model can be used for design of grouting. Within the framework of this paper with fracture apertures of 125 and 200 μm with a stable grout having a yield shear strength of 3–4 Pa will give penetration length of some 6–14 m for a grouting time of 15–20 min and a overpressure of 3 MPa. For shallow tunnels in hard crystalline rock this is close where the design ends up with. The equations used for verification are based on the principle that the penetration length can be computed if the fracture aperture, rheological grout properties and applied overpressure are known.
References Axelsson, M., Gustafson, G., 2006. A robust method to determine the shear strength of cement-based injection grouts in the field. Tunn. Undergr. Space Technol. 21 (5), 499–503 Elsevier. Dai, G., Bird, B.R., 1981. Radial flow of a Bingham fluid between two fixed circular disks. J. Non-Newtonian Fluid Mech. 8, 349–355. Gustafson, G., Claesson, J., Fransson, Å., 2013. Steering parameters for rock grouting. J. Appl. Math. 2013, 269594. Gustafson, G., Fransson, Å., 2006. The use of the Pareto distribution for fracture transmissivity assessment. Hydrogeol. J. 14, 15–20. http://dx.doi.org/10.1007/s10040005-0440-y. Gustafson, G., Stille, H., 2005. Stop criteria for cement grouting. Felsbau 3, 62–68. Håkansson, U., 1993. Rheology of fresh cement-based grouts. Doctoral Thesis. Department of Infrastructure and Environmental Engineering, Royal Institute of Technology, Stockholm, Sweden. Hässler, L., 1991. Grouting of rock—simulation and classification. Doctoral Thesis. Department of Soil and Rock Mechanics, Royal Institute of Technology, Sweden, pp. 159. Fransson, Å., Funehag, J., Thörn, J., 2016. Swedish grouting design: hydraulic testing and grout selection. Proc. Inst. Civ. Eng. - Ground Improvement 169, 275–285. http://dx. doi.org/10.1680/jgrim.15.00020. Lombardi, G., 1985. The Role of Cohesion in Cement Grouting of Rock. Commission Internationale des Grands Barrages, Lausanne. Thörn, J., Kvartsberg, S., Runslätt, E., Almfeldt, S., Fransson, Å., 2015. Calculation tool for rock characterisation during grouting design –theory and users guide. BeFo, Stockholm (in Swedish).
Acknowledgement Sven Tyréns Foundation (STS), Rock Engineering Research
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