# Steering Parameters for Rock Grouting

Authors: Gunnar Gustafson; Johan Claesson; Åsa Fransson
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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.Figure1 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 distanceL, grout penetration I.The transmissivity is coupled to the hydraulic aperture of the fracture by the cubic law [4, 5]:
(1)T=ρwgb312μw,
where μw is the viscosity,  ρw 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 μg. 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=pg-pw, and the shear stress gives the maximum grout penetration, Imax, for a fracture of aperture b (e.g. [9, 11]):
(2)Imax=Δp·b2τ0.
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

LetI(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
(3)dIdt=-dpdx·b212μg[1-3·Zb+4·(Zb)3],
where
(4)Z=τ0·|dpdx|-1,Z<b2.
Assuming parallel flow and a viscosity of the grout much higher than for water, the pressure gradient can be simplified to be
(5)dpdx=-ΔpI.
Equations (4), (5) and (2), give 2Z/b=I/Imax. The equation for the relative penetration depth ID=I/Imax becomes from (3) after simplifications
(6)dIDdt=(τ0)26μgΔp·2-3ID+(ID)3ID,ID=IImax=2Zb.
We define the characteristic time t0 and the dimensionless time tD:
(7)t0=6μgΔp(τ0)2,tD=tt0.
Equation (6) gives the derivative dID/dtD. The derivative of tD as a function of ID is
(8)dtDdID=ID2-3ID+(ID)3=ID(2+ID)(1-ID)2.
The right-hand function of ID 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 tD as a function of ID:
(9)tD=F1(ID),F1(s)=s3(1-s)+29·ln[2(1-s)2+s].
It is straightforward to verify that derivative of (9) is given by (8) and that ID=0 for tD=0.A plot ofID=I/Imax as a function of tD=t/(6μgΔp/τ02) 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, μg and τ0. Thus the parameters are decided by choice of the grouter.(iii)
The time scale is determined byt0=6μgΔp/τ02 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 radiusr0 can be calculated to be [10]
(10)dIdt=-dpdx·(r0)28μg[1-43·Zpr0+13·(Zpr0)4],Zp=2τ0·|dpdx|-1,Zp<r0.
Here, Zp 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 Imax,p:
(11)Imax,p=Δp·r02τ0.
Inserting (5) and (10), observing that dx/dt=dI/dt, and using the relative penetration depth ID,p=I/Imax,p give after simplifications:
(12)dID,pdt=(τ0)26μgΔp·3-4ID,p+(ID,p)4ID,p,ID,p=IImax,p.
Inserting tD=t/(6μgΔp/τ02), the previous equation gives the derivative dID,r/dtD. The derivative of tD as a function of ID,p is
(12′)dtDdID,p=ID,p3-4ID,p+(ID,p)4=ID,p[1-ID,p]2[3+2ID,p+(ID,p)2].
This equation may with some difficulty be integrated. We obtain the following explicit equation for the tD as a function of ID,p:
(13)tD=Fp(ID,p),Fp(s)=s6(1-s)+136·ln[3(1-s)23+2s+s2]-5236·arctan(s2s+3).
A long, but straightforward calculation shows that the derivative satisfies (12). It is easy to see that tD=0 for ID,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 ID,p(tD) and experimental data is shown.Table 1
Experimental data for grout penetration, from Håkansson [10].
Experiment
r
0 (m)
Δ
p (kPa)
τ
0 (Pa)
μ
g (Pa s)
I
max
,
p (m)
t
0 (s)
3 mm
0.0015
50
6.75
0.292
5.55
1922
4 mm
0.002
50
6.75
0.292
7.40
1922Figure 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 oftD≈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 ID,r-tD-curves are almost identical for the two experiments. Another striking fact is that more than 90% of the predicted penetration is reached for tD≈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 vg(m/s) decreases as 1/r, [16]. Let rb be the radius of the injection borehole, and let rb+I be the radius of the grout injection front at any particular time t. We have
(14)vg=-dpdr·b212μg[1-3·Zb+4·(Zb)3],rb≤r≤rb+I,
where
(15)Z=τ0·|dpdr|-1,Z<b2.
Let the grout injection rate be Q(m3/s). The total grout flow is the same for all r:
(16)Q=2πrb·vg,rb≤r≤rb+I.
Combing (14) and (16), we get after some calculation the following implicit differential equation for the pressure as a function of the radius:
(17)6μgQπb2τ0·1r=s·[2-3·s-1+s-3],s=b2Z=b2τ0·|dpdr|
or
(18)r=2μgQπb2τ0·3s22s3-3s2+1,s=-b2τ0·dpdr,rb≤r≤rb+I.
The injection excess pressure is Δp. We have the boundary condition
(19)p(rb)-p(rb+I)=Δp.
Here, we neglect a pressure fall in the ground water, since the viscosity of grout is much larger than that of water.The solutionp(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=rb+I(t) is equal to the time derivative of I(t). We have from (16)
(20)Q(I)=2πb·[rb+I(t)]·dIdt,I(0)=0.
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 forp(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 functionQ(I), we may determine the motion of the grout front from (20) by integration.The front positionI 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:
(21)Q=0,s=1⟹-dpdr=2τ0b⟹p=K-2τ0b·r.
Here, K is a constant. The boundary condition (19) determines the maximum value of I:
(22)p(rb)-p(rb+Imax)=2τ0b·(-rb+rb+Imax)=Δp⟹Imax=bΔp2τ0.
We get the same value (2) as in the plain case.The complete solution in the radial case involves the following constants:(23)Imax=bΔp2τ0,γ=Imaxrb=bΔp2rbτ0,t0=6μgΔp(τ0)2,Q0=6πb(Imax)2t0=πb3Δp4μg.


### 2.4. Solution for the Pressure

In the dimensionless solution for the pressure, we use the borehole radius as scaling length:(24)r′=rrb,I′=Irb,rb≤r≤rb+I⟺1≤r′≤1+I′.
The pressure is scaled by Δp/γ. The variable s for the derivative of the pressure in (18) becomes
(25)p′=γ·(p-pw)Δp⟹s=b2τ0·(-dpdr)=-bΔp/γ2τ0rb·dp′dr′=-dp′dr′.
The dimensionless form of (18)-(19) becomes after some recalculations
(26)r′=Q′·g(-dp′dr′),g(s)=3s22s3-3s2+1,Q′=2μgQπb2τ0rb,p′(1)-p′(1+I′)=γ,pppppp1≤r′≤1+I′.
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
(27)p′(r′)=γ-Q′·[G~(Q′)-G~(Q′r′)],1≤r′≤1+I′.
The composite function G~(q), which is used for q=Q′ and q=Q′/r′, is defined by
(28)G~(q)=G(s~(q)),s~(q)=121+q·sin{(1/3)·arcsin[(1+q)-1.5]},G(s)=43·ln(s-1)+16·ln(2s+1)-1s-1-3s3(2s+1)(s-1)2.
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 factorQ′ has to be chosen so that the total pressure difference corresponds to the injection pressure, (26). This gives
(29)γ=Q′·[G~(Q′)-G~(Q′1+I′)].
This equation determines Q′ as a function of I′ and γ:
(30)Q′=f′(I′,γ),0≤I′≤γ,γ>0.
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 useImax as scaling length. We also use Q0 and t0 from (23)
(31)ID=IImax,I′=γID,QD=QQ0,tD=tt0.
The grout flux becomes from (23) and (26)
(32)Q0γ=πb2τ0rb2μg⟹Q=Q0γ·f′(I′,γ)=Q0·QD(ID,γ).
The dimensionless grout flux is then
(33)QD(ID,γ)=f′(γID,γ)γ,0≤ID≤1.
The dimensionless equation for the front motion is now from (32), (20), (31), and (23)
(34)Q0γ·f′(γID,γ)=2πb·(Imax)2t0·(1γ+ID)·dIDdtDordtDdID=γ3·1/γ+IDf(γID,γ).
By integration we get the time tD=t/t0 as an integral in ID:
(35)tD=13·∫0ID1+γID′f(γID′,γ)dID′,0≤ID<1.
We get tD as a function of the grout front position ID. 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 functionID=ID(tD,γ) for radial flow.A comparison of Figures3, 4, and 6 shows that the curves for ID(tD) 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(36)Vg(t)=πb[(rb+I(t))2-(rb)2]=πbI(t)2·[1+2rbI(t)].
Let Vg,max be maximum injection volume and VD the dimensionless volume of injected grout:
(36′)VD=VgVg,max,Vg,max=πb(Imax)2·[1+2γ].
Then we get, using (31), (24), (23), and the relation (35) between ID and tD,
(37)VD(tD,γ)=(ID)2·1+2/(γID)1+2/γ,ID=ID(tD,γ).
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].


## 2.1. Grout Penetration

LetI(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
(3)dIdt=-dpdx·b212μg[1-3·Zb+4·(Zb)3],
where
(4)Z=τ0·|dpdx|-1,Z<b2.
Assuming parallel flow and a viscosity of the grout much higher than for water, the pressure gradient can be simplified to be
(5)dpdx=-ΔpI.
Equations (4), (5) and (2), give 2Z/b=I/Imax. The equation for the relative penetration depth ID=I/Imax becomes from (3) after simplifications
(6)dIDdt=(τ0)26μgΔp·2-3ID+(ID)3ID,ID=IImax=2Zb.
We define the characteristic time t0 and the dimensionless time tD:
(7)t0=6μgΔp(τ0)2,tD=tt0.
Equation (6) gives the derivative dID/dtD. The derivative of tD as a function of ID is
(8)dtDdID=ID2-3ID+(ID)3=ID(2+ID)(1-ID)2.
The right-hand function of ID 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 tD as a function of ID:
(9)tD=F1(ID),F1(s)=s3(1-s)+29·ln[2(1-s)2+s].
It is straightforward to verify that derivative of (9) is given by (8) and that ID=0 for tD=0.A plot ofID=I/Imax as a function of tD=t/(6μgΔp/τ02) 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, μg and τ0. Thus the parameters are decided by choice of the grouter.(iii)
The time scale is determined byt0=6μgΔp/τ02 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 radiusr0 can be calculated to be [10]
(10)dIdt=-dpdx·(r0)28μg[1-43·Zpr0+13·(Zpr0)4],Zp=2τ0·|dpdx|-1,Zp<r0.
Here, Zp 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 Imax,p:
(11)Imax,p=Δp·r02τ0.
Inserting (5) and (10), observing that dx/dt=dI/dt, and using the relative penetration depth ID,p=I/Imax,p give after simplifications:
(12)dID,pdt=(τ0)26μgΔp·3-4ID,p+(ID,p)4ID,p,ID,p=IImax,p.
Inserting tD=t/(6μgΔp/τ02), the previous equation gives the derivative dID,r/dtD. The derivative of tD as a function of ID,p is
(12′)dtDdID,p=ID,p3-4ID,p+(ID,p)4=ID,p[1-ID,p]2[3+2ID,p+(ID,p)2].
This equation may with some difficulty be integrated. We obtain the following explicit equation for the tD as a function of ID,p:
(13)tD=Fp(ID,p),Fp(s)=s6(1-s)+136·ln[3(1-s)23+2s+s2]-5236·arctan(s2s+3).
A long, but straightforward calculation shows that the derivative satisfies (12). It is easy to see that tD=0 for ID,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 ID,p(tD) and experimental data is shown.Table 1
Experimental data for grout penetration, from Håkansson [10].
Experiment
r
0 (m)
Δ
p (kPa)
τ
0 (Pa)
μ
g (Pa s)
I
max
,
p (m)
t
0 (s)
3 mm
0.0015
50
6.75
0.292
5.55
1922
4 mm
0.002
50
6.75
0.292
7.40
1922Figure 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 oftD≈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 ID,r-tD-curves are almost identical for the two experiments. Another striking fact is that more than 90% of the predicted penetration is reached for tD≈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 vg(m/s) decreases as 1/r, [16]. Let rb be the radius of the injection borehole, and let rb+I be the radius of the grout injection front at any particular time t. We have
(14)vg=-dpdr·b212μg[1-3·Zb+4·(Zb)3],rb≤r≤rb+I,
where
(15)Z=τ0·|dpdr|-1,Z<b2.
Let the grout injection rate be Q(m3/s). The total grout flow is the same for all r:
(16)Q=2πrb·vg,rb≤r≤rb+I.
Combing (14) and (16), we get after some calculation the following implicit differential equation for the pressure as a function of the radius:
(17)6μgQπb2τ0·1r=s·[2-3·s-1+s-3],s=b2Z=b2τ0·|dpdr|
or
(18)r=2μgQπb2τ0·3s22s3-3s2+1,s=-b2τ0·dpdr,rb≤r≤rb+I.
The injection excess pressure is Δp. We have the boundary condition
(19)p(rb)-p(rb+I)=Δp.
Here, we neglect a pressure fall in the ground water, since the viscosity of grout is much larger than that of water.The solutionp(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=rb+I(t) is equal to the time derivative of I(t). We have from (16)
(20)Q(I)=2πb·[rb+I(t)]·dIdt,I(0)=0.
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 forp(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 functionQ(I), we may determine the motion of the grout front from (20) by integration.The front positionI 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:
(21)Q=0,s=1⟹-dpdr=2τ0b⟹p=K-2τ0b·r.
Here, K is a constant. The boundary condition (19) determines the maximum value of I:
(22)p(rb)-p(rb+Imax)=2τ0b·(-rb+rb+Imax)=Δp⟹Imax=bΔp2τ0.
We get the same value (2) as in the plain case.The complete solution in the radial case involves the following constants:(23)Imax=bΔp2τ0,γ=Imaxrb=bΔp2rbτ0,t0=6μgΔp(τ0)2,Q0=6πb(Imax)2t0=πb3Δp4μg.


## 2.4. Solution for the Pressure

In the dimensionless solution for the pressure, we use the borehole radius as scaling length:(24)r′=rrb,I′=Irb,rb≤r≤rb+I⟺1≤r′≤1+I′.
The pressure is scaled by Δp/γ. The variable s for the derivative of the pressure in (18) becomes
(25)p′=γ·(p-pw)Δp⟹s=b2τ0·(-dpdr)=-bΔp/γ2τ0rb·dp′dr′=-dp′dr′.
The dimensionless form of (18)-(19) becomes after some recalculations
(26)r′=Q′·g(-dp′dr′),g(s)=3s22s3-3s2+1,Q′=2μgQπb2τ0rb,p′(1)-p′(1+I′)=γ,pppppp1≤r′≤1+I′.
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
(27)p′(r′)=γ-Q′·[G~(Q′)-G~(Q′r′)],1≤r′≤1+I′.
The composite function G~(q), which is used for q=Q′ and q=Q′/r′, is defined by
(28)G~(q)=G(s~(q)),s~(q)=121+q·sin{(1/3)·arcsin[(1+q)-1.5]},G(s)=43·ln(s-1)+16·ln(2s+1)-1s-1-3s3(2s+1)(s-1)2.
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 factorQ′ has to be chosen so that the total pressure difference corresponds to the injection pressure, (26). This gives
(29)γ=Q′·[G~(Q′)-G~(Q′1+I′)].
This equation determines Q′ as a function of I′ and γ:
(30)Q′=f′(I′,γ),0≤I′≤γ,γ>0.
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 useImax as scaling length. We also use Q0 and t0 from (23)
(31)ID=IImax,I′=γID,QD=QQ0,tD=tt0.
The grout flux becomes from (23) and (26)
(32)Q0γ=πb2τ0rb2μg⟹Q=Q0γ·f′(I′,γ)=Q0·QD(ID,γ).
The dimensionless grout flux is then
(33)QD(ID,γ)=f′(γID,γ)γ,0≤ID≤1.
The dimensionless equation for the front motion is now from (32), (20), (31), and (23)
(34)Q0γ·f′(γID,γ)=2πb·(Imax)2t0·(1γ+ID)·dIDdtDordtDdID=γ3·1/γ+IDf(γID,γ).
By integration we get the time tD=t/t0 as an integral in ID:
(35)tD=13·∫0ID1+γID′f(γID′,γ)dID′,0≤ID<1.
We get tD as a function of the grout front position ID. 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 functionID=ID(tD,γ) for radial flow.A comparison of Figures3, 4, and 6 shows that the curves for ID(tD) 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(36)Vg(t)=πb[(rb+I(t))2-(rb)2]=πbI(t)2·[1+2rbI(t)].
Let Vg,max be maximum injection volume and VD the dimensionless volume of injected grout:
(36′)VD=VgVg,max,Vg,max=πb(Imax)2·[1+2γ].
Then we get, using (31), (24), (23), and the relation (35) between ID and tD,
(37)VD(tD,γ)=(ID)2·1+2/(γID)1+2/γ,ID=ID(tD,γ).
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, Imax=Δp·τ0/2b, and a time-dependent scaling factor, ID(tD), 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, tD=t/t0, is the ratio between the actual grouting time, t, and a time scaling factor, t0=6μgΔp/τ02, the characteristic grouting time. Here μg 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 about5t0 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.



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*Source: 269594-2013-11-17.xml*