#### 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, , 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, , for water-bearing fractures having a transmissivity, , 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 , are pervious and that the statistical distribution of the transmissivities of the conductive fractures is approximately lognormal.

The transmissivity is coupled to the hydraulic aperture of the fracture by the cubic law [4, 5]: where is the viscosity, is the density of water, and 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, , and a plastic viscosity . 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, , and the shear stress gives the maximum grout penetration, , for a fracture of aperture (e.g. [9, 11]): 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.

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 be the position of the grout front at time , Figure 2. The velocity of grout, , moving in a horizontal facture of aperture can according to Hässler [9] be calculated as where Assuming parallel flow and a viscosity of the grout much higher than for water, the pressure gradient can be simplified to be Equations (4), (5) and (2), give . The equation for the relative penetration depth becomes from (3) after simplifications We define the characteristic time and the dimensionless time : Equation (6) gives the derivative . The derivative of as a function of is The right-hand function of 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 as a function of : It is straightforward to verify that derivative of (9) is given by (8) and that for .

A plot of as a function of is shown in Figure 3.

From (8) and Figure 3 some interesting observations can be drawn.(i)The relative penetration is not a function of the fracture aperture, . 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, , and the grout properties, and . Thus the parameters are decided by choice of the grouter.(iii)The time scale is determined by so that at this grouting time about 80% of the possible penetration length is reached in all fractures and after 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 can be calculated to be [10] Here, is the radius of the plug flow in the pipe.

A force balance between the driving pressure, , and the resisting shear forces inside the pipe gives the maximum grout penetration : Inserting (5) and (10), observing that , and using the relative penetration depth give after simplifications: Inserting , the previous equation gives the derivative . The derivative of as a function of is This equation may with some difficulty be integrated. We obtain the following explicit equation for the as a function of : A long, but straightforward calculation shows that the derivative satisfies (12). It is easy to see that for .

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 and experimental data is shown.

The experimental data follow the theoretical function extremely well up to a value of . 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 -curves are almost identical for the two experiments. Another striking fact is that more than 90% of the predicted penetration is reached for .

##### 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].

Equations (3) and (4) give the grout flow in the plane case. The grout flow velocity is constant () and equal to the front velocity . In the radial case we replace by . The grout flow velocity decreases as , [16]. Let be the radius of the injection borehole, and let be the radius of the grout injection front at any particular time . We have where Let the grout injection rate be . The total grout flow is the same for all : Combing (14) and (16), we get after some calculation the following implicit differential equation for the pressure as a function of the radius: or The injection excess pressure is . We have the boundary condition Here, we neglect a pressure fall in the ground water, since the viscosity of grout is much larger than that of water.

The solution of (18)-(19) has the front position as parameter. The value of has to be adjusted so that the pressure difference is obtained in accordance with (19). The front position increases with time. The flow velocity at the grout front is equal to the time derivative of . We have from (16) This equation determines the motion of the grout front. It depends on the required grout injection rate , which is obtained from the solution of (18)-(19) for each front position .

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 . This involves the solution of a cubic equation in order to get the derivative and an intricate integration in order to get . From the solution, we get the required grout flux for any front position .

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

The front position increases from zero at to a maximum value for infinite time. Then the flux must be zero. Equation (18) gives for . Then we have a linear pressure variation: Here, is a constant. The boundary condition (19) determines the maximum value of : We get the same value (2) as in the plain case.

The complete solution in the radial case involves the following constants:

##### 2.4. Solution for the Pressure

In the dimensionless solution for the pressure, we use the borehole radius as scaling length: The pressure is scaled by . The variable for the derivative of the pressure in (18) becomes The dimensionless form of (18)-(19) becomes after some recalculations This is the basic equation to solve for the pressure distribution. It is to be solved for for positive values of the parameter .

The solution is derived in detail in [14]. A brief derivation is presented in the appendix. The dimensionless pressure is given by The composite function , which is used for and , is defined by The function is the root to the cubic equation for . The function is an integral of .

The value of the factor has to be chosen so that the total pressure difference corresponds to the injection pressure, (26). This gives This equation determines as a function of and : The value of for is zero in accordance with (21)-(22): .

##### 2.5. Motion of Grout Front

In the dimensionless formulation of the equation for the motion of the grout front, we use as scaling length. We also use and from (23) The grout flux becomes from (23) and (26) The dimensionless grout flux is then The dimensionless equation for the front motion is now from (32), (20), (31), and (23) By integration we get the time as an integral in : We get as a function of the grout front position . 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 -values.

A comparison of Figures 3, 4, and 6 shows that the curves for 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 and to reach 90% takes about . 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 Let be maximum injection volume and the dimensionless volume of injected grout: Then we get, using (31), (24), (23), and the relation (35) between and , 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, , can be described as a product of the maximum penetration, , and a time-dependent scaling factor, , the relative penetration length. Here is the driving pressure, is the yield strength of the grout, and is the aperture of the penetrated fracture. The time factor or dimensionless grouting time, , is the ratio between the actual grouting time, , and a time scaling factor, , the characteristic grouting time. Here is the Bingham viscosity of the grout. The relative penetration depth has a value of 70–90% for and reaches a value of more than 90% for 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 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 , 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 to (26): Here, is the position of the grout front. The parameter is positive. Taking zero pressure at the grout front, the boundary conditions for the dimensionless pressure become The dimensionless grout flux is to be chosen so that the previous boundary conditions are fulfilled. The value of will depend on the front position .

*Solution in Parameter Form.* In order to see more directly the character of the equation, we make the following change of notation:
The equation is then of the following type:
There is a general solution in a certain parameter form to this type of implicit ordinary differential equation [19]. The solution is
We have to show that this is indeed the solution. We have
The ratio between these equations gives that is equal to the derivative . We have
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
We introduce the inverse to in the following way:
The pressure with a free constant for the pressure level may now be written as
The solution is then from (A.8)–(A.10) (with )
or, introducing the composite function ,
The boundary condition (A.2) at is fulfilled for a certain choice of . The explicit solution is
The other boundary condition (A.2) at is fulfilled when satisfies the equation
We note that the derivative is given by :
The pressure derivative is equal to –1 for zero flux, (21) and (25), in the final stagnant position . The magnitude of this derivative is larger than 1 for all preceding positions . This means that is larger than (or equal to) 1 in the solution.

*The Function **.* The solution (A.13) and the composite function (A.12) involve the function defined in (A.10) and (A.1). The integral of is obtained from an expansion in partial fractions. We have
The integral of is readily determined. The function becomes from (A.10) and (A.16)
We will use the function for .

*The Inverse **.* The inverse (A.9) is, for any , the solution of the cubic equation
The solution is reported in detail in [14]. The cubic equation has three real-valued solutions for positive -values, one of which is larger than 1 (for there is a double root and a third root , (A.16)). We need the solution . It is given by
A plot shows that is an increasing function from for . It has the asymptote for large .

We will show that (A.19) is the inverse. We use the notations In (A.18), we put on the left-hand side, divide by , and insert from (A.20). Then we have On the third line we use a well-known trigonometric formula relating to . We have shown that (A.19) is the inverse.

#### Symbols and Units

(m): | Fracture aperture |

(m): | Penetration length of injected grout |

(m): | Maximum penetration length of grout |

(m): | Maximum penetration length of grout in a pipe |

(—): | Ratio between penetration and borehole radius |

(—): | Relative penetration length |

(—): | Relative penetration length in a pipe |

(m): | Distance between grouting boreholes |

(Pa): | Pressure |

(—): | Dimensionless pressure |

(Pa): | Grout pressure |

(Pa): | Water pressure |

(m^{3}/s): | Grout injection flow rate |

(m): | Pipe radius, radial distance from borehole centre |

(m): | Borehole radius |

(—): | Dimensionless radius |

(m): | Grout plug radius |

(m): | Pipe radius |

(—): | Ratio between distance from borehole centre and borehole radius |

(m^{2}/s): | Transmissivity |

(s): | Grouting time |

(s): | Characteristic grouting time |

(—): | Dimensionless grouting time |

(m^{3}): | Injected volume of grout |

(m^{3}): | Maximum grout volume in a fracture |

(—): | Dimensionless grout volume |

(m): | Length coordinate |

(—): | Bingham half-plug thickness |

(—): | Ratio between maximum penetration and borehole radius |

(Pa): | Driving pressure for grout |

(Pas): | Plastic viscosity of grout |

(Pas): | Viscosity of water |

(kg/m^{3}): | Density of water |

(Pa): | Yield strength of grout. |

#### Acknowledgments

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