671017.pdf

0

0. 1

0 .2

0. 3

0. 4

0. 5

0. 6

0 .7

0. 8

0 .9

1

0 .00001 0.0001 0.001 0. 01 0. 1 1 10 100

tD=t/t 0

ID

=I/I

max

Figure 3: Relative penetration length as a function of dimensionless
time in horizontal fracture.

From ( 8 )andFigure 3 some interesting observations can
be drawn.

(i) The relative penetration is not a function of the

fracture aperture,푏. This means that the penetration

processhasthesametimescaleforallfractureswith

different apertures penetrated by a borehole.

(ii) The time scale is only a function of the grouting

pressure,Δ푝, and the grout properties,휇푔and휏 0 .Thus

the parameters are decided by choice of the grouter.

(iii) The time scale is determined by푡 0 =6휇푔Δ푝/휏 02 so

that at this grouting time about 80% of the possible

penetration length is reached in all fractures and after

5 푡 0 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 exper-
iments were published by H ̊akansson [ 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
modelscouldgivebetterfittodata,buttheBinghammodel
gave adequate results especially in the light of its simplicity.
The velocity of grout moving in a pipe of radius푟 0 can be
calculated to be [ 10 ]

푑퐼

푑푡

=−

푑푝

푑푥

⋅

(푟 0 )^2

8휇푔

[1 −

4

3

⋅

푍푝

푟 0

+

1

3

⋅(

푍푝

푟 0

)

4

],

푍푝=2휏 0 ⋅

儨儨儨

儨儨儨

儨儨

푑푝

푑푥

儨儨儨

儨儨儨

儨儨

−1

,푍푝<푟 0.

(10)

Here,푍푝is the radius of the plug flow in the pipe.
A force balance between the driving pressure,Δ푝,and
the resisting shear forces inside the pipe gives the maximum
grout penetration퐼max,푝:

퐼max,푝=

Δ푝 ⋅ 푟 0

2휏 0

. (11)

Table 1: Experimental data for grout penetration, from H ̊akansson

[ 10 ].

Experiment 푟 0 (m)Δ푝(kPa)휏 0 (Pa)휇푔(Pa s)퐼max,푝(m)푡 0 (s)

3mm 0.0015 50 6.75 0.292 5.55 1922

4mm 0.002^50 6.75 0.292 7.40^1922

Inserting ( 5 )and( 10 ), observing that푑푥/푑푡 = 푑퐼/푑푡,and

using the relative penetration depth퐼퐷,푝=퐼/퐼max,푝give after

simplifications:

푑퐼퐷,푝

푑푡

=

(휏 0 )

2

6휇푔Δ푝

⋅

3−4퐼퐷,푝+(퐼퐷,푝)

4

퐼퐷,푝

,

퐼퐷,푝=

퐼

퐼max,푝

.

(12)

Inserting푡퐷=푡/(6휇푔Δ푝/휏^20 ), the previous equation gives the

derivative푑퐼퐷,푟/푑푡퐷. The derivative of푡퐷as a function of퐼퐷,푝

is

푑푡퐷

푑퐼퐷,푝

=

퐼퐷,푝

3−4퐼퐷,푝+(퐼퐷,푝)

4

=

퐼퐷,푝

[1 − 퐼퐷,푝]

2

[3 + 2퐼퐷,푝+(퐼퐷,푝)

2

]

.

(12耠)

This equation may with some difficulty be integrated. We

obtain the following explicit equation for the푡퐷as a function

of퐼퐷,푝:

푡퐷=퐹푝(퐼퐷,푝),

퐹푝(푠)=

푠

6 (1−푠)

+

1

36

⋅ln[

3 (1−s)^2

3+2푠+푠^2

]

−

5 √ 2

36

⋅arctan(

푠√ 2

푠+3

).

(13)

A long, but straightforward calculation shows that the deriva-

tive satisfies ( 12 ). It is easy to see that푡퐷=0for퐼퐷,푝=푠=0.

In H ̊akansson [ 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 function퐼퐷,푝(푡퐷)

and experimental data is shown.

The experimental data follow the theoretical function

extremely well up to a value of푡퐷≈2. It shall also be borne in

mind that the grout properties were taken directly from lab-

oratory tests and no curve fitting was made. Hakansson [ ̊ 10 ],

whoassumedthemtobearesultfromdifferencesbetween

laboratory values and experiment conditions, also identified

the differences at the end of the curves. As predicted the

퐼퐷,푟−푡퐷-curves are almost identical for the two experiments.

Anotherstrikingfactisthatmorethan90%ofthepredicted

penetration is reached for푡퐷≈2.