Steering Parameters for Rock Grouting

Abstract

In Swedish tunnel grouting practice normally a fan of boreholes is drilled ahead of the tunnel front where cement grout is injected in order to create a low permeability zone around the tunnel. Demands on tunnel tightness have increased substantially in Sweden, and this has led to a drastic increase of grouting costs. Based on the flow equations for a Bingham fluid, the penetration of grout as a function of grouting time is calculated. This shows that the time scale of grouting in a borehole is only determined by grouting overpressure and the rheological properties of the grout, thus parameters that the grouter can choose. Pressure, grout properties, and the fracture aperture determine the maximum penetration of the grout. The smallest fracture aperture that requires to be sealed thus also governs the effective borehole distance. Based on the identified parameters that define the grouting time-scale and grout penetration, an effective design of grouting operations can be set up. The solution for time as a function of penetration depth is obtained in a closed form for parallel and pipe flow. The new, more intricate, solution for the radial case is presented.

1. Introduction

In Swedish tunnelling pregrouting is normally used when considered necessary for the reduction of groundwater inflows. Cement grout, occasionally with plasticisers added, is preferred for economical and environmental reasons. Recently, the increased demands on tunnel tightness have led to an approach to pregrouting where the whole tunnel is systematically pregrouted according to a few predetermined standard strategies. This has led to a massive increase of performed grouting, and subsequently there is a strong need for effective design methods and steering parameters for the grouting activities.

In pregrouting a fan of boreholes is drilled around the tunnel periphery ahead of the tunnel front, grout is injected through the boreholes in order to create a low permeability zone around the tunnel, and finally the tunnel is excavated by the drill and blast method within the zone until the next cycle starts with drilling of the grouting fan. Normally grouting boreholes, 15–18 m long, are used which give 3-4 blasting rounds per cycle.

Figure 1 shows the grouting fan and some fractures as a background for the design problem. Through the borehole grout is injected, which spreads through the fractures. At any time the grout has penetrated a distance, 𝐼, from the borehole, which is individual for each fracture. For a successful grouting the penetration between the boreholes should bridge the distance between the boreholes, 𝐿, for water-bearing fractures having a transmissivity, 𝑇, above a critical value determined by their frequency and the demands on tunnel tightness. Recent investigations of the transmissivity distributions of fractures in Swedish Precambrian crystalline rocks [1–3] have shown that only a small portion of the fractures and joints, 5–15% at a threshold level of 𝑇=10⁻⁹m²/s, are pervious and that the statistical distribution of the transmissivities of the conductive fractures is approximately lognormal.

Figure 1 

Grouting fan and grout penetration. Borehole distance 𝐿, grout penetration 𝐼.

The transmissivity is coupled to the hydraulic aperture of the fracture by the cubic law [4, 5]: 𝜌𝑇=^𝑤𝑔𝑏³12𝜇_𝑤, (1) where 𝜇_𝑤 is the viscosity, 𝜌_𝑤 is the density of water, and 𝑏 is the so-called hydraulic aperture of the fracture. The hydraulic aperture determined by the cubic law has shown to be a good estimate for the grouting aperture [6, 7].

From this it follows that in a borehole to be grouted, only a few fractures are pervious and only a small number of these contribute significantly to the groundwater flow through the rock because of the large skewness of the transmissivity distribution.

The normally used cement grouts can reasonably well be characterised as Bingham fluids [8–10]. They are thus characterised by a yield strength, 𝜏₀, and a plastic viscosity 𝜇_𝑔. From the Bingham model it follows that flow can only take place in the parts of the fluid where the internal shear stresses exceed the yield strength. This means that a stiff plug is formed in the centre of the flow channel surrounded by plastic flow zones; see Figure 2. The advance of the grout front ceases when the shear stresses at the walls of the fracture equal the yield strength of the grout. A simple force balance of the difference between the grouting and the resisting water pressures, Δ𝑝=𝑝_𝑔−𝑝_𝑤, and the shear stress gives the maximum grout penetration, 𝐼_max, for a fracture of aperture 𝑏 (e.g. [9, 11]): 𝐼_max=Δ𝑝⋅𝑏2𝜏₀. (2) The relevant design question is thus how to make sure that the penetration length is long enough to bridge the distance between the grouting boreholes for the critical fractures and the length of time it takes to reach the maximum penetration or a significant portion of it.

Figure 2 

Grout penetrating a fracture.

In order to obtain an analytical solution, the problem has to be simplified. In particular, it is assumed that the aperture is constant, not varying along the fracture. The grout properties are assumed to be constant in time. These limitations should be kept in mind when these analytical solutions are used.

2. Derivation of Equations, Results, and Discussion

2.1. Grout Penetration

Let 𝐼(𝑡) be the position of the grout front at time 𝑡, Figure 2. The velocity of grout, 𝑑𝐼/𝑑𝑡, moving in a horizontal facture of aperture 𝑏 can according to Hässler [9] be calculated as 𝑑𝐼𝑑𝑡=−𝑑𝑝⋅𝑏𝑑𝑥12𝜇²_𝑔𝑍1−3⋅𝑏𝑍+4⋅𝑏³, (3) where 𝑍=𝜏₀⋅𝑑𝑝𝑑𝑥⁻¹𝑏,𝑍<2. (4) Assuming parallel flow and a viscosity of the grout much higher than for water, the pressure gradient can be simplified to be 𝑑𝑝𝑑𝑥=−Δ𝑝𝐼. (5) Equations (4), (5) and (2), give 2𝑍/𝑏=𝐼/𝐼_max. The equation for the relative penetration depth 𝐼_𝐷=𝐼/𝐼_max becomes from (3) after simplifications 𝑑𝐼_𝐷=𝜏𝑑𝑡6𝜇⁰²_𝑔⋅Δ𝑝2−3𝐼_𝐷+𝐼_𝐷³𝐼_𝐷,𝐼_𝐷=𝐼𝐼_max=2𝑍𝑏. (6) We define the characteristic time 𝑡₀ and the dimensionless time 𝑡_𝐷: 𝑡₀=6𝜇_𝑔Δ𝑝𝜏₀²,𝑡_𝐷=𝑡𝑡₀. (7) Equation (6) gives the derivative 𝑑𝐼_𝐷/𝑑𝑡_𝐷. The derivative of 𝑡_𝐷 as a function of 𝐼_𝐷 is 𝑑𝑡_𝐷𝑑𝐼_𝐷=𝐼_𝐷2−3𝐼_𝐷+𝐼_𝐷³=𝐼_𝐷2+𝐼_𝐷1−𝐼_𝐷². (8) The right-hand function of 𝐼_𝐷 is the ratio between two polynomials, which may be expanded in partial fractions. These are readily integrated. We obtain the following explicit equation for the 𝑡_𝐷 as a function of 𝐼_𝐷: 𝑡_𝐷=𝐹₁𝐼_𝐷,𝐹₁(𝑠)=𝑠3()+21−𝑠92()⋅ln1−𝑠.2+𝑠 (9) It is straightforward to verify that derivative of (9) is given by (8) and that 𝐼_𝐷=0 for 𝑡_𝐷=0.

A plot of 𝐼_𝐷=𝐼/𝐼_max as a function of 𝑡_𝐷=𝑡/(6𝜇_𝑔Δ𝑝/𝜏²₀) is shown in Figure 3.

Figure 3 

Relative penetration length as a function of dimensionless time in horizontal fracture.

From (8) and Figure 3 some interesting observations can be drawn.(i)The relative penetration is not a function of the fracture aperture, 𝑏. This means that the penetration process has the same time scale for all fractures with different apertures penetrated by a borehole.(ii)The time scale is only a function of the grouting pressure, Δ𝑝, and the grout properties, 𝜇_𝑔 and 𝜏₀. Thus the parameters are decided by choice of the grouter.(iii)The time scale is determined by 𝑡₀=6𝜇_𝑔Δ𝑝/𝜏²₀ so that at this grouting time about 80% of the possible penetration length is reached in all fractures and after 5𝑡₀ about 95% is reached. After that the growth is very slow and the economy of continued injection could be put in doubt.

2.2. Experimental Verification

A series of grouting experiments were published by Håkansson [10]. He used thin plastic pipes instead of a parallel slot for his experiments, and several constitutive grout flow models were tested against experimental data. As could be expected more complex models could give better fit to data, but the Bingham model gave adequate results especially in the light of its simplicity.

The velocity of grout moving in a pipe of radius 𝑟₀ can be calculated to be [10] 𝑑𝐼𝑑𝑡=−𝑑𝑝⋅𝑟𝑑𝑥8𝜇⁰²_𝑔41−3⋅𝑍_𝑝𝑟₀+13⋅𝑍_𝑝𝑟₀⁴,𝑍_𝑝=2𝜏₀⋅𝑑𝑝𝑑𝑥⁻¹,𝑍_𝑝<𝑟₀. (10) Here, 𝑍_𝑝 is the radius of the plug flow in the pipe.

A force balance between the driving pressure, Δ𝑝, and the resisting shear forces inside the pipe gives the maximum grout penetration 𝐼_max,𝑝: 𝐼_max,𝑝=Δ𝑝⋅𝑟₀2𝜏₀. (11) Inserting (5) and (10), observing that 𝑑𝑥/𝑑𝑡=𝑑𝐼/𝑑𝑡, and using the relative penetration depth 𝐼_𝐷,𝑝=𝐼/𝐼_max,𝑝 give after simplifications: 𝑑𝐼_𝐷,𝑝=𝜏𝑑𝑡6𝜇⁰²_𝑔⋅Δ𝑝3−4𝐼_𝐷,𝑝+𝐼_𝐷,𝑝⁴𝐼_𝐷,𝑝,𝐼_𝐷,𝑝=𝐼𝐼_max,𝑝. (12) Inserting 𝑡_𝐷=𝑡/(6𝜇_𝑔Δ𝑝/𝜏²₀), the previous equation gives the derivative 𝑑𝐼_𝐷,𝑟/𝑑𝑡_𝐷. The derivative of 𝑡_𝐷 as a function of 𝐼_𝐷,𝑝 is 𝑑𝑡_𝐷𝑑𝐼_𝐷,𝑝=𝐼_𝐷,𝑝3−4𝐼_𝐷,𝑝+𝐼_𝐷,𝑝⁴=𝐼_𝐷,𝑝1−𝐼_𝐷,𝑝²3+2𝐼_𝐷,𝑝+𝐼_𝐷,𝑝². 12^′ This equation may with some difficulty be integrated. We obtain the following explicit equation for the 𝑡_𝐷 as a function of 𝐼_𝐷,𝑝: 𝑡_𝐷=𝐹_𝑝𝐼_𝐷,𝑝,𝐹_𝑝(𝑠)=𝑠6()+11−𝑠3()36⋅ln1−𝑠²3+2𝑠+𝑠²−5√2𝑠√36⋅arctan2.𝑠+3 (13) A long, but straightforward calculation shows that the derivative satisfies (12). It is easy to see that 𝑡_𝐷=0 for 𝐼_𝐷,𝑝=𝑠=0.

In Håkansson [10] two grouting experiments in 3 and 4 mm pipes are reported. In Table 1, the relevant parameters for the experiments are shown based on the reported data. In Figure 4, a direct comparison between the function 𝐼_𝐷,𝑝(𝑡_𝐷) and experimental data is shown.

Table 1 

Experimental data for grout penetration, from Håkansson [10].

Figure 4 

Comparison of grout penetration function in a pipe with experimental data from Håkansson [10].

The experimental data follow the theoretical function extremely well up to a value of 𝑡_𝐷≈2. It shall also be borne in mind that the grout properties were taken directly from laboratory tests and no curve fitting was made. Håkansson [10], who assumed them to be a result from differences between laboratory values and experiment conditions, also identified the differences at the end of the curves. As predicted the 𝐼_𝐷,𝑟−𝑡_𝐷-curves are almost identical for the two experiments. Another striking fact is that more than 90% of the predicted penetration is reached for 𝑡_𝐷≈2.

2.3. Penetration in a Two-Dimensional Fracture

A more realistic model of a fracture to grout is perhaps a pseudo-plane with a system of conductive areas and flow channels [5]. If the transmissivity of the fracture is reasonably constant, a parallel plate model with constant aperture b can approximate it. If it is grouted through a borehole, there will be a radial, two-dimensional, flow of grout out from the borehole; see Figure 5. In reality, however, the flow will as for flow of water from a borehole be something in between a system of one-dimensional channels and radial flow [12].

Figure 5 

Radial penetration of grout in a fracture.

Equations (3) and (4) give the grout flow in the plane case. The grout flow velocity is constant (in𝑥) and equal to the front velocity 𝑑𝐼/𝑑𝑡. In the radial case we replace 𝑥 by 𝑟. The grout flow velocity 𝑣_𝑔(m/s) decreases as 1/𝑟, [16]. Let 𝑟_𝑏 be the radius of the injection borehole, and let 𝑟_𝑏+𝐼 be the radius of the grout injection front at any particular time 𝑡. We have 𝑣_𝑔=−𝑑𝑝⋅𝑏𝑑𝑟12𝜇²_𝑔𝑍1−3⋅𝑏𝑍+4⋅𝑏³,𝑟_𝑏≤𝑟≤𝑟_𝑏+𝐼, (14) where 𝑍=𝜏₀⋅𝑑𝑝𝑑𝑟⁻¹𝑏,𝑍<2. (15) Let the grout injection rate be 𝑄(m³/s). The total grout flow is the same for all 𝑟: 𝑄=2𝜋𝑟𝑏⋅𝑣_𝑔,𝑟_𝑏≤𝑟≤𝑟_𝑏+𝐼. (16) Combing (14) and (16), we get after some calculation the following implicit differential equation for the pressure as a function of the radius: 6𝜇_𝑔𝑄𝜋𝑏²𝜏₀⋅1𝑟=𝑠⋅2−3⋅𝑠⁻¹+𝑠⁻³,𝑏𝑠==𝑏2𝑍2𝜏₀⋅𝑑𝑝𝑑𝑟 (17) or 𝑟=2𝜇_𝑔𝑄𝜋𝑏²𝜏₀⋅3𝑠²2𝑠³−3𝑠²𝑏+1,𝑠=−2𝜏₀⋅𝑑𝑝,𝑟𝑑𝑟_𝑏≤𝑟≤𝑟_𝑏+𝐼. (18) The injection excess pressure is Δ𝑝. We have the boundary condition 𝑝𝑟_𝑏−𝑝𝑟_𝑏+𝐼=Δ𝑝. (19) Here, we neglect a pressure fall in the ground water, since the viscosity of grout is much larger than that of water.

The solution 𝑝(𝑟) of (18)-(19) has the front position 𝐼 as parameter. The value of 𝑄 has to be adjusted so that the pressure difference Δ𝑝 is obtained in accordance with (19). The front position 𝐼=𝐼(𝑡) increases with time. The flow velocity at the grout front 𝑟=𝑟_𝑏+𝐼(𝑡) is equal to the time derivative of 𝐼(𝑡). We have from (16) 𝑄(𝐼)=2𝜋𝑏⋅𝑟_𝑏+𝐼(𝑡)⋅𝑑𝐼(0)𝑑𝑡,𝐼=0. (20) This equation determines the motion of the grout front. It depends on the required grout injection rate 𝑄(𝐼), which is obtained from the solution of (18)-(19) for each front position 𝐼.

The solution for radial grout flow is much more complicated than for the plain case and the pipe case. We must first solve the implicit differential equation for 𝑝(𝑟). This involves the solution of a cubic equation in order to get the derivative 𝑑𝑝/𝑑𝑟 and an intricate integration in order to get 𝑝(𝑟). From the solution, we get the required grout flux for any front position 𝐼.

With known function 𝑄(𝐼), we may determine the motion of the grout front from (20) by integration.

The front position 𝐼 increases from zero at 𝑡=0 to a maximum value for infinite time. Then the flux 𝑄 must be zero. Equation (18) gives 𝑄=0 for 𝑠=1. Then we have a linear pressure variation: 𝑄=0,𝑠=1⟹−𝑑𝑝=𝑑𝑟2𝜏₀𝑏⟹𝑝=𝐾−2𝜏₀𝑏⋅𝑟. (21) Here, 𝐾 is a constant. The boundary condition (19) determines the maximum value of 𝐼: 𝑝𝑟_𝑏−𝑝𝑟_𝑏+𝐼_max=2𝜏₀𝑏⋅−𝑟_𝑏+𝑟_𝑏+𝐼_max=Δ𝑝⟹𝐼_max=𝑏Δ𝑝2𝜏₀. (22) We get the same value (2) as in the plain case.

The complete solution in the radial case involves the following constants: 𝐼_max=𝑏Δ𝑝2𝜏₀𝐼,𝛾=^max𝑟_𝑏=𝑏Δ𝑝2𝑟_𝑏𝜏₀,𝑡₀=6𝜇_𝑔Δ𝑝𝜏₀²,𝑄₀=6𝜋𝑏𝐼_max²𝑡₀=𝜋𝑏³Δ𝑝4𝜇_𝑔. (23)

2.4. Solution for the Pressure

In the dimensionless solution for the pressure, we use the borehole radius as scaling length: 𝑟^′=𝑟𝑟_𝑏,𝐼^′=𝐼𝑟_𝑏,𝑟_𝑏≤𝑟≤𝑟_𝑏+𝐼⟺1≤𝑟^′≤1+𝐼^′. (24) The pressure is scaled by Δ𝑝/𝛾. The variable 𝑠 for the derivative of the pressure in (18) becomes 𝑝^′=𝛾⋅𝑝−𝑝_𝑤𝑏Δ𝑝⟹𝑠=2𝜏₀⋅−𝑑𝑝𝑑𝑟=−𝑏Δ𝑝/𝛾2𝜏₀𝑟_𝑏⋅𝑑𝑝^′𝑑𝑟^′=−𝑑𝑝^′𝑑𝑟^′. (25) The dimensionless form of (18)-(19) becomes after some recalculations 𝑟^′=𝑄^′⋅𝑔−𝑑𝑝^′𝑑𝑟^′,𝑔(𝑠)=3𝑠²2𝑠³−3𝑠²,𝑄+1_′=2𝜇_𝑔𝑄𝜋𝑏²𝜏₀𝑟_𝑏,𝑝^′(1)−𝑝^′1+𝐼^′=𝛾,1≤𝑟^′≤1+𝐼^′. (26) This is the basic equation to solve for the pressure distribution. It is to be solved for 0<𝐼^′<𝛾 for positive values of the parameter 𝛾.

The solution is derived in detail in [14]. A brief derivation is presented in the appendix. The dimensionless pressure is given by 𝑝^′𝑟^′=𝛾−𝑄^′⋅𝐺𝑄^′−𝐺𝑄^′𝑟^′,1≤𝑟^′≤1+𝐼^′. (27) The composite function 𝐺(𝑞), which is used for 𝑞=𝑄^′ and 𝑞=𝑄^′/𝑟^′, is defined by 𝐺𝑞𝑠̃𝑞,𝑠̃𝑞=1=𝐺21+𝑞⋅sin(1/3)⋅arcsin1+𝑞^−1.5,𝐺(𝑠)=43()+1⋅ln𝑠−16()−1⋅ln2𝑠+1−𝑠−13𝑠³(2𝑠+1)(𝑠−1)². (28) The function 𝑠(̃𝑞) is the root to the cubic equation 𝑞⋅𝑔(𝑠)=1 for 𝑠>1. The function 𝐺(𝑠) is an integral of 𝑠⋅𝑑𝑔/𝑑𝑠.

The value of the factor 𝑄^′ has to be chosen so that the total pressure difference corresponds to the injection pressure, (26). This gives 𝛾=𝑄^′⋅𝐺𝑄^′−𝐺𝑄^′1+𝐼^′. (29) This equation determines 𝑄^′ as a function of 𝐼^′ and 𝛾: 𝑄^′=𝑓^′𝐼^′,𝛾,0≤𝐼^′≤𝛾,𝛾>0. (30) The value of 𝑄^′ for 𝐼^′=𝛾 is zero in accordance with (21)-(22): 𝑓^′(𝛾,𝛾)=0.

2.5. Motion of Grout Front

In the dimensionless formulation of the equation for the motion of the grout front, we use 𝐼_max as scaling length. We also use 𝑄₀ and 𝑡₀ from (23) 𝐼_𝐷=𝐼𝐼_max,𝐼^′=𝛾𝐼_𝐷,𝑄_𝐷=𝑄𝑄₀,𝑡_𝐷=𝑡𝑡₀. (31) The grout flux becomes from (23) and (26) 𝑄₀𝛾=𝜋𝑏²𝜏₀𝑟_𝑏2𝜇_𝑔𝑄⟹𝑄=⁰𝛾⋅𝑓^′𝐼^′,𝛾=𝑄₀⋅𝑄_𝐷𝐼_𝐷,𝛾. (32) The dimensionless grout flux is then 𝑄_𝐷𝐼_𝐷=𝑓,𝛾^′𝛾𝐼_𝐷,𝛾𝛾,0≤𝐼_𝐷≤1. (33) The dimensionless equation for the front motion is now from (32), (20), (31), and (23) 𝑄₀𝛾⋅𝑓^′𝛾𝐼_𝐷𝐼,𝛾=2𝜋𝑏⋅^max2𝑡₀⋅1𝛾+𝐼_𝐷⋅𝑑𝐼_𝐷𝑑𝑡_𝐷or𝑑𝑡_𝐷𝑑𝐼_𝐷=𝛾3⋅1/𝛾+𝐼_𝐷𝑓𝛾𝐼_𝐷.,𝛾 (34) By integration we get the time 𝑡_𝐷=𝑡/𝑡₀ as an integral in 𝐼_𝐷: 𝑡_𝐷=13⋅^𝐼𝐷₀1+𝛾𝐼^′_𝐷𝑓𝛾𝐼^′_𝐷,𝛾𝑑𝐼^′_𝐷,0≤𝐼_𝐷<1. (35) We get 𝑡_𝐷 as a function of the grout front position 𝐼_𝐷. Also in this case the inverse function describes the relative penetration as a function of the dimensionless grouting time. Figure 6 shows this relation for a few 𝛾-values.

Figure 6 

Grout penetration function 𝐼𝐷=𝐼𝐷(𝑡𝐷,𝛾) for radial flow.

A comparison of Figures 3, 4, and 6 shows that the curves for 𝐼_𝐷(𝑡_𝐷) are similar for the three flow cases. The main difference to parallel flow is that penetration is somewhat slower for the radial case. Around 80% of maximum penetration is reached after 3𝑡₀ and to reach 90% takes about 7𝑡₀. The principle is, however, the same and the curves could be used in the same way.

2.6. Injected Volume of Grout

The injected volume of grout as a function of time is of interest. The volume is 𝑉_𝑔(𝑡)=𝜋𝑏𝑟_𝑏+𝐼(𝑡)²−𝑟_𝑏²=𝜋𝑏𝐼(𝑡)²⋅1+2𝑟_𝑏𝐼(𝑡). (36) Let 𝑉_𝑔,max be maximum injection volume and 𝑉_𝐷 the dimensionless volume of injected grout: 𝑉_𝐷=𝑉_𝑔𝑉_𝑔,max,𝑉_𝑔,max=𝜋𝑏𝐼_max²⋅21+𝛾. 36^′ Then we get, using (31), (24), (23), and the relation (35) between 𝐼_𝐷 and 𝑡_𝐷, 𝑉_𝐷𝑡_𝐷,𝛾=𝐼_𝐷²⋅1+2/𝛾𝐼_𝐷1+2/𝛾,𝐼_𝐷=𝐼_𝐷𝑡_𝐷,𝛾. (37) Equations presented in this paper have been used in Gustafson and Stille [15] when considering stop criteria for grouting. Grouting projects where estimates of penetration length have been made are, for example, [13, 15, 16]. Penetration length has also been a key to presenting a concept for estimation of deformation and stiffness of fractures based on grouting data [13]. In addition to grouting of tunnels, theories have also been applied for grouting of dams [18].

3. Conclusions

The theoretical investigation of grout spread in one-dimensional conduits and radial spread in plane parallel fractures have shown very similar behavior for all the investigated cases. The penetration, 𝐼, can be described as a product of the maximum penetration, 𝐼_max=Δ𝑝⋅𝜏₀/2𝑏, and a time-dependent scaling factor, 𝐼_𝐷(𝑡_𝐷), the relative penetration length. Here Δ𝑝 is the driving pressure, 𝜏₀ is the yield strength of the grout, and 𝑏 is the aperture of the penetrated fracture. The time factor or dimensionless grouting time, 𝑡_𝐷=𝑡/𝑡₀, is the ratio between the actual grouting time, 𝑡, and a time scaling factor, 𝑡₀=6𝜇_𝑔Δ𝑝/𝜏₀², the characteristic grouting time. Here 𝜇_𝑔 is the Bingham viscosity of the grout. The relative penetration depth has a value of 70–90% for 𝑡=𝑡₀ and reaches a value of more than 90% for 𝑡>7𝑡₀ for all fractures.

From this a number of important conclusions can be drawn.(i)The relative penetration is the same in all fractures that a grouted borehole cuts. This means that given the same grout and pressure the grouting time should be the same in high and low yielding boreholes in order to get the same degree of tightening of all fractures. This means that the tendency in practice to grout for a shorter time in tight boreholes will give poor results for sealing of fine fractures.(ii)The maximum penetration is governed by the fracture aperture and pressure and yield strength of the grout. The latter are at the choice of the grouter.(iii)The relative penetration, which governs much of the final result, is determined by the grouting time.(iv)The pressure and the grout properties determine the desired grouting time. These are the choice of the grouter alone.(v)It is poor economy to grout for a longer time than about 5𝑡₀ since the growth of the penetration is very slow for a time longer than that. On the other hand, if the borehole takes significant amounts of grout after 5𝑡₀, there is reason to stop since it indicates an unrestricted outflow of grout somewhere in the system.

The significance of this for grouting design is as follows.(i)The conventional stop criteria based on volume or grout flow can be replaced by a minimum time criterion based only on the parameters that the grouter can chose, that is, grouting pressure and yield strength of the grout.(ii)Based on an assessment of how fine fractures it is necessary to seal, a maximum effective borehole distance can be predicted given the pressure and the properties of the grout.(iii)The time needed for effective grouting operations can be estimated with better accuracy.(iv)In order to avoid unrestricted grout pumping also a maximum grouting time can be given, where further injection of grout will be unnecessary.

Appendix

Derivation of the Solution for the Pressure

We seek the solution 𝑝^′(𝑟^′) to (26): 𝑟^′=𝑄^′⋅𝑔−𝑑𝑝^′𝑑𝑟^′,1≤𝑟^′≤1+𝐼^′,𝑔(𝑠)=3𝑠²2𝑠³−3𝑠²+1,0≤𝐼^′≤𝛾. (A.1) Here, 1+𝐼^′ is the position of the grout front. The parameter 𝛾 is positive. Taking zero pressure at the grout front, the boundary conditions for the dimensionless pressure become 𝑝^′(1)=𝛾,𝑝^′1+𝐼^′=0. (A.2) The dimensionless grout flux 𝑄^′ is to be chosen so that the previous boundary conditions are fulfilled. The value of 𝑄^′ will depend on the front position 𝐼^′.

Solution in Parameter Form. In order to see more directly the character of the equation, we make the following change of notation: 𝑥⟷𝑟^′,𝑦⟷−𝑝^′,𝑓(𝑠)=𝑄^′⋅𝑔(𝑠). (A.3) The equation is then of the following type: 𝑥=𝑓𝑑𝑦.𝑑𝑥 (A.4) There is a general solution in a certain parameter form to this type of implicit ordinary differential equation [19]. The solution is 𝑥(𝑠)=𝑓(𝑠),𝑦(𝑠)=𝑠⋅𝑓(𝑠)−^𝑠𝑓𝑠^′𝑑𝑠^′. (A.5) We have to show that this is indeed the solution. We have 𝑑𝑥=𝑑𝑠𝑑𝑓,𝑑𝑠𝑑𝑦(𝑠)𝑑𝑠=1⋅𝑓+𝑠⋅𝑑𝑓(𝑠)𝑑𝑠−𝑓=𝑠⋅𝑑𝑓.𝑑𝑠 (A.6) The ratio between these equations gives that 𝑠 is equal to the derivative 𝑑𝑦/𝑑𝑥. We have 𝑑𝑦=𝑑𝑥𝑑𝑦/𝑑𝑠𝑑𝑥/𝑑𝑠=𝑠⟹𝑓𝑑𝑦(𝑠)𝑑𝑥=𝑓=𝑥. (A.7) The right-hand equation shows that (A.5) is the solution to (A.4).

Explicit Solution. Applying this technique to (A.1), we get the solution 𝑟^′=𝑄^′⋅𝑔(𝑠),−𝑝^′(𝑠)=𝑠⋅𝑄^′⋅𝑔(𝑠)−𝑄^′⋅^𝑠𝑔𝑠^′𝑑𝑠^′. (A.8) We introduce the inverse to 𝑔(𝑠) in the following way: 1=𝑞⋅𝑔(𝑠)⟺𝑠=𝑔⁻¹1𝑞=𝑠̃𝑞⟺1=𝑞⋅𝑔𝑠̃𝑞. (A.9) The pressure with a free constant 𝐾 for the pressure level may now be written as 𝑝^′(𝑠)=𝑄^′⋅𝐺(𝑠)+𝐾,𝐺(𝑠)=^𝑠𝑔𝑠^′𝑑𝑠^′−𝑠⋅𝑔(𝑠). (A.10) The solution is then from (A.8)–(A.10) (with 𝑞=𝑄^′/𝑟^′) 𝑝^′(𝑠)=𝑄^′(𝑠)𝑠̃𝑄⋅𝐺+𝐾,𝑠=^′𝑟^′ (A.11) or, introducing the composite function 𝐺(𝑞), 𝐺𝑞=𝐺𝑠̃𝑞,𝑝^′𝑟^′=𝑄^′⋅𝐺𝑄^′𝑟^′+𝐾. (A.12) The boundary condition (A.2) at 𝑟^′=1 is fulfilled for a certain choice of 𝐾. The explicit solution is 𝑝^′𝑟^′=𝛾−𝑄^′⋅𝐺𝑄^′−𝐺𝑄^′𝑟^′,1≤𝑟^′≤1+𝐼^′. (A.13) The other boundary condition (A.2) at 𝑟^′=1+𝐼^′ is fulfilled when 𝑄^′ satisfies the equation 𝛾=𝑄^′⋅𝐺𝑄^′−𝐺𝑄^′1+𝐼^′. (A.14) We note that the derivative −𝑑𝑝^′/𝑑𝑟^′ is given by 𝑠: 𝑠=−𝑑𝑝^′𝑑𝑟^′. (A.15) The pressure derivative is equal to –1 for zero flux, (21) and (25), in the final stagnant position 𝐼^′=𝛾. The magnitude of this derivative is larger than 1 for all preceding positions 𝐼^′<𝛾. This means that 𝑠 is larger than (or equal to) 1 in the solution.

The Function 𝐺(𝑠). The solution (A.13) and the composite function (A.12) involve the function 𝐺(𝑠) defined in (A.10) and (A.1). The integral of 𝑔(𝑠) is obtained from an expansion in partial fractions. We have 𝑔(𝑠)=3𝑠²2𝑠³−3𝑠²=+13𝑠²(2𝑠+1)(𝑠−1)²=13⋅1+42𝑠+13⋅1+1𝑠−1(𝑠−1)². (A.16) The integral of 𝑔(𝑠) is readily determined. The function 𝐺(𝑠) becomes from (A.10) and (A.16) 𝐺(𝑠)=16()+4⋅ln2𝑠+13()−1⋅ln𝑠−1−𝑠−13𝑠³2𝑠³−3𝑠²+1,𝑠>1. (A.17) We will use the function for 1<𝑠<∞.

The Inverse 𝑠(̃𝑞). The inverse (A.9) is, for any 𝑞≥0, the solution of the cubic equation 2𝑠³−3𝑠²+1=3𝑞𝑠². (A.18) The solution is reported in detail in [14]. The cubic equation has three real-valued solutions for positive 𝑞-values, one of which is larger than 1 (for 𝑞=0 there is a double root 𝑠=1 and a third root 𝑠=−0.5, (A.16)). We need the solution 𝑠>1. It is given by 𝑠̃𝑞=121+𝑞⋅sin(1/3)⋅arcsin1+𝑞^−1.5,𝑞≥0. (A.19) A plot shows that 𝑠(̃𝑞) is an increasing function from 𝑠(̃0)=1 for 𝑞≥0. It has the asymptote 1.5⋅(1+𝑞) for large 𝑞.

We will show that (A.19) is the inverse. We use the notations 𝑠̃𝑞1=𝑠=2,1+𝑞⋅sin𝜙/3𝜙=arcsin1+𝑞^−1.5. (A.20) In (A.18), we put 3𝑞𝑠² on the left-hand side, divide by 𝑠³, and insert 𝑠=𝑠(̃𝑞) from (A.20). Then we have 1𝑠³⋅1−31+𝑞𝑠=2+2𝜙1+𝑞⋅sin3³−31+𝑞⋅2𝜙1+𝑞⋅sin3+2=2−2⋅1+𝑞^1.5⋅𝜙3⋅sin3−4⋅sin³𝜙3=2−2⋅1+𝑞^1.5⋅sin𝜙=2−2⋅1+𝑞^1.5⋅1+𝑞^−1.5=0. (A.21) On the third line we use a well-known trigonometric formula relating sin(𝜙/3) to sin(𝜙). We have shown that (A.19) is the inverse.

Symbols and Units

𝑏(m):	Fracture aperture
𝐼 (m):	Penetration length of injected grout
𝐼max (m):	Maximum penetration length of grout
𝐼max,𝑝 (m):	Maximum penetration length of grout in a pipe
𝐼′ (—):	Ratio between penetration and borehole radius
𝐼𝐷 (—):	Relative penetration length
𝐼𝐷,𝑝 (—):	Relative penetration length in a pipe
𝐿 (m):	Distance between grouting boreholes
𝑝 (Pa):	Pressure
𝑝𝐷 (—):	Dimensionless pressure
𝑝𝑔 (Pa):	Grout pressure
𝑝𝑤 (Pa):	Water pressure
𝑄 (m3/s):	Grout injection flow rate
𝑟 (m):	Pipe radius, radial distance from borehole centre
𝑟𝑏 (m):	Borehole radius
𝑟𝐷 (—):	Dimensionless radius
𝑟𝑝 (m):	Grout plug radius
𝑟0 (m):	Pipe radius
𝑟′ (—):	Ratio between distance from borehole centre and borehole radius
𝑇 (m2/s):	Transmissivity
𝑡 (s):	Grouting time
𝑡0 (s):	Characteristic grouting time
𝑡𝐷 (—):	Dimensionless grouting time
𝑉𝑔 (m3):	Injected volume of grout
𝑉max (m3):	Maximum grout volume in a fracture
𝑉𝐷 (—):	Dimensionless grout volume
𝑥 (m):	Length coordinate
𝑍 (—):	Bingham half-plug thickness
𝛾 (—):	Ratio between maximum penetration and borehole radius
Δ𝑝 (Pa):	Driving pressure for grout
𝜇𝑔 (Pas):	Plastic viscosity of grout
𝜇𝑤 (Pas):	Viscosity of water
𝜌𝑤 (kg/m3):	Density of water
𝜏0 (Pa):	Yield strength of grout.

Acknowledgments

The authors would like to acknowledge the effort of Gunnar Gustafson who deceased during the study.