and (A.1). The integral of푔(푠)is obtained from an expansion
in partial fractions. We have
푔(푠)=
3푠^2
2푠^3 −3푠^2 +1
=
3푠^2
(2푠 + 1)(푠−1)^2
=
1
3
⋅
1
2푠 + 1
+
4
3
⋅
1
푠−1
+
1
(푠−1)^2
.
(A.16)
The integral of푔(푠)is readily determined. The function퐺(푠)
becomes from (A.10)and(A.16)
퐺(푠)=
1
6
⋅ln(2푠 + 1)+4
3
⋅ln(푠−1)−1
푠−1
−
3푠^3
2푠^3 −3푠^2 +1
,푠>1.
(A.17)
We w i l l u s e t h e f u n c t i o n f o r1<푠<∞.
The Inversẽ푠(푞).The inverse (A.9)is,forany푞≥0,the
solution of the cubic equation
2푠^3 −3푠^2 +1=3푞푠^2. (A.18)The solution is reported in detail in [ 14 ]. The cubic equation
has three real-valued solutions for positive푞-values, one of
which is larger than 1 (for푞=0there is a double root푠=1
and a third root푠 = −0.5,(A.16)). We need the solution푠>1.
It is given by
푠(푞)=̃
1
2 √1+푞⋅sin{(1/3)⋅arcsin[(1 + 푞)−1.5]},푞≥0.
(A.19)
Aplotshowsthat̃푠(푞)is an increasing function from̃푠(0) = 1
for푞≥0. It has the asymptote1.5 ⋅ (1 + 푞)for large푞.
We w i l l s h o w t h a t (A.19) is the inverse. We use the
notations
̃푠(푞)=푠=1
2 √1+푞⋅sin(휙/3),
휙=arcsin[(1 + 푞)−1.5
].(A.20)
In (A.18), we put3푞푠^2 on the left-hand side, divide by푠^3 ,and
insert푠=̃푠(푞)from (A.20). Then we have
(
1
푠
)
3
−3(1+푞)⋅1
푠
+2
=(2√1+푞⋅sin(휙
3
))
3
−3(1+푞)⋅2√1+푞⋅sin(휙
3
)+2
=2−2⋅(1+푞)1.5⋅[3⋅sin(휙
3
)−4⋅sin^3 (휙
3
)]
=2−2⋅(1+푞)
1.5
⋅sin(휙)=2−2⋅(1+푞)1.5⋅(1+푞)−1.5=0.
(A.21)On the third line we use a well-known trigonometric formula
relating sin(휙/3)to sin(휙).Wehaveshownthat(A.19)isthe
inverse.Symbols and Units
푏(m): Fracture aperture
퐼(m): Penetration length of injected grout
퐼max(m): Maximum penetration length of grout
퐼max,푝(m): Maximum penetration length of grout in a
pipe
퐼耠(—): Ratio between penetration and borehole
radius
퐼퐷(—): Relative penetration length
퐼퐷,푝(—): Relative penetration length in a pipe
퐿(m): Distance between grouting boreholes
푝(Pa): Pressure
푝퐷(—): Dimensionless pressure
푝푔(Pa): Grout pressure
푝푤(Pa): Water pressure
푄(m^3 /s): Grout injection flow rate
푟(m): Pipe radius, radial distance from borehole
centre
푟푏(m): Borehole radius
푟퐷(—): Dimensionless radius
푟푝(m): Grout plug radius
푟 0 (m): Pipe radius
푟耠(—): Ratio between distance from borehole centre
and borehole radius
푇(m^2 /s): Transmissivity
푡(s): Grouting time
푡 0 (s): Characteristic grouting time
푡퐷(—): Dimensionless grouting time
푉푔(m^3 ): Injected volume of grout
푉max(m^3 ): Maximum grout volume in a fracture
푉퐷(—): Dimensionless grout volume
푥(m): Length coordinate
푍(—): Bingham half-plug thickness
훾(—): Ratio between maximum penetration and
borehole radius
Δ푝(Pa): Driving pressure for grout
휇푔(Pas): Plastic viscosity of grout
휇푤(Pas): Viscosity of water
휌푤(kg/m^3 ): Density of water
휏 0 (Pa): Yield strength of grout.Acknowledgments
The authors would like to acknowledge the effort of Gunnar
Gustafsonwhodeceasedduringthestudy.References
[1] J. D. Osnes, A. Winberg, and J. E. Andersson, “Analysis of well
test data—application of probabilistic models to infer hydraulic
properties of fractures,” Topical Report RSI-0338, RE/SPEC,
Rapid City, Dakota, 1988.