Modified Maag’s Spherical Diffusion Model of Vacuum ...in geotechnical engineering. e sandy layer can cause severe damage to underground projects [– ]. Grouting is a common method

Research ArticleModified Maagrsquos Spherical Diffusion Model of VacuumPenetration Grouting

Feng Huang Jianguo Lyu He Gao and GuiheWang

School of Engineering amp Technology China University of Geosciences Beijing 100083 China

Correspondence should be addressed to Jianguo Lyu ljgcugbeducn

Received 13 August 2017 Revised 11 December 2017 Accepted 25 December 2017 Published 21 January 2018

Academic Editor Stefano de Miranda

Copyright copy 2018 Feng Huang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The penetration grouting is very widely used in geotechnical engineering nowadays but the slurry diffusion radius is not longenough because of low grouting pressure The vacuum grouting method is proposed to solve this problem However there isno diffusion theory model of vacuum grouting which makes the practical application lack scientific basis In this paper thedistribution law of vacuum negative pressure in soil is deducedThen the boundary conditions of Maagrsquos spherical diffusionmodelare modified by the vacuum pressure distribution law After that the vacuummodificationmodel is deduced Finally Maagrsquos modelandmodifiedmodel are analyzed according to a published experiment which proves that the vacuummodificationmodel is suitablefor predicting the slurry diffusion of vacuum groutingThe proposedmodel provides a reference for the theoretical study of vacuumgrouting

1 Introduction

The sand layer which has the characteristics of loose struc-ture no cohesion and poor self-stability is often encounteredin geotechnical engineering The sandy layer can causesevere damage to underground projects [1ndash3] Grouting isa common method of soil reinforcement in undergroundengineering and has been playing an important role intreating the sand layer [4ndash7] At present penetration groutingmethod is widely used because of low grouting pressureand little disturbance to soil structure [8] But at the sametime the low pressure leads to the short diffusion distance ofslurry However if the grouting pressure rises excessively itis easy to cause soil cracks excessive deformation and otherconsequences

Therefore researchers have been studying themechanismof grouting pressure of slurry diffusion and analyzed themechanical equilibriummechanism of diffusion termination[9ndash11] The researchers found that when the slurry wasinjected into the soil under constant grouting pressure theslurry flow would slow down because of the decrease ofpressure gradient and the increase of resistance which resultsin the decline of shear stress As soon as the shear stressdeclined to the yield stress of the slurry the diffusion would

be terminated [12] Vacuum grouting method generatesnegative pressure field in soil Under the condition of constantgrouting pressure the vacuum condition can increase thepressure gradient of the slurry so as to increase the shear rateof slurry in diffusion process and make the shear stress largerthan the yield stress which will lead to the longer diffusiondistance Vacuum grouting can change the current situationwhere the traditional penetration grouting diffusion distanceis insufficient

Researchers in recent years have analyzed a lot about thegrouting combined with vacuum in concrete structure fieldAssaad and Daou [13] evaluated the effect of vacuum on theamount of water extracted along with resulting changes ingrout properties including flowability static yield stress vis-cosity unit weight Wick-induced bleeding and compressivestrength Assaad et al [14] also presented that the partial orcomplete extraction of free mixing water due to vacuumingdecreases flowability In foundation treatment Han et al [15]presented a case study on the application of high pressure jetgrouting pile to prevent vacuum preloading from leaking inthe silty sand and silty clay layers Peng et al [16] proposedelectroosmotic grouting coupledwith vacuumdrainage at thecathode The experimental study shows that the treatment

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 1758651 7 pageshttpsdoiorg10115520181758651

2 Mathematical Problems in Engineering

effect of electroosmotic grouting with vacuum drainage isnot only better than general electroosmosis methods but alsomore uniform than such methods Shen [17] obtained therelationship between the degree of vacuum and the size ofthe grouting body and found that the diffusion radii of theslurry grown with the increase of vacuum degree (absolutevalue of vacuumpressure) and grouting pressureThe presentresearches provide a useful way to explore the slurry diffusionmechanism of vacuum penetration grouting However thepresent studies only obtained the macro and qualitative lawThey have not established the diffusion model of vacuumgrouting and the relationship between the grouting param-eters and the diffusion distance

Therefore this paper takes Newtonian fluid as theresearch object to deduce the diffusionmodel of vacuumpen-etration grouting on the basis of Maagrsquos spherical diffusionmodel The new diffusion model explains how the diffusiondistance be enlarged in mathematical way and predictsthe diffusion distance of the slurry by analysis formula byconsidering grouting coefficients The vacuum modificationmodel provides a reference to the scientific application ofvacuum penetration grouting

2 Maagrsquos Spherical Diffusion Model andVacuum Modification Model

Maag derived the diffusion formula of the grout in sand layerand proposed a spherical diffusion model of Newtonian fluid[17]The theoretical assumptions are that the grout is Newtonfluid when diffusing in sandy soil the diffusion of slurryfollows Darcyrsquos law the injected soil is homogeneous andisotropic there is no dynamic water pressure when groutingin groundwater the difference between the density of slurryand water is not considered the viscosity of the slurry isconstant during grouting and the slurry diffuses in sphericalform starting from the end of injecting pipe

Maagrsquos spherical diffusion model is shown in Figure 1According to Darcyrsquos law119876 = 119870119892119894119860119905 (1)

where 119894 = minus119889ℎ119889119903 119870119892 = 119870120573 and 119860 = 41205871199032According to the boundary conditions

119903 = 1199030ℎ = 119867119903 = 1199031ℎ = ℎ0

(2)

Equation (3) can be obtained as follows by submitting theboundary conditions into Darcyrsquos law

119867 minus ℎ0 = 1198761205734120587119870119905 ( 11199030 minus 11199031) (3)

then 119876 can be solved as

119876 = 4120587119870119905 (119867 minus ℎ0)120573 (11199030 minus 11199031) (4)

H

ℎ1

ℎ0

2r0

r1

dr

dℎ

Groundsurface

Figure 1 Schematic diagram of derivation of Maagrsquos model

Considering 1199031 minus 1199030 asymp 1199031 11199030 minus11199031 asymp 11199030 and119867minusℎ0 =ℎ1 as well as 119876 = (43)12058711990313119899 the expression of the diffusionradius of grouting can be derived as

1199031 = 3radic31198701199051199030ℎ1119899120573 (5)

119876mdashtotal amount of grouting (cm3)119860mdashslurry penetration area (cm2)119905mdashtotal time for grouting (s)119903mdashthe diffusion radius of the slurry at any moment(cm)1199031mdashthe final diffusion radius of the slurry (cm)1199030mdashradius of grouting pipe (cm)ℎ1mdashgrouting pressure head (cm)ℎ0mdashwater head of groundwater in the end of groutingpipe (cm)119867mdashthe sumof groundwater pressure head and grout-ing pressure head (cm)119899mdashporosity of soil120573mdashthe ratio of the viscosity of slurry to the viscosityof water 120583g120583w119870mdashpermeability coefficient of injected soil (cms)119870119892mdashpermeability coefficient for slurry (cms)

In order to study the influence of vacuum on the dif-fusion radius the distribution of vacuum pressure in soil isanalyzed firstly The plane seepage model is used to derivethe distribution formula of vacuum pressure It is assumedthat the seepage of air in soil is steady and follows DarcyrsquoslawThe vacuumpressure is converted into the correspondingwater head through the derivation of the model which canbe coupled with the grouting pressure head Finally the newboundary conditions are submitted into Maagrsquos sphericaldiffusion model and the expression of the diffusion radius ofthe slurry in vacuum condition is derived

Mathematical Problems in Engineering 3

a

a

b

AB

CD

a

Ss

SV

Figure 2 Soil particle and pore model

Figure 2 shows the simplified model of soil established byparticles and pores

The soil porosity ratio 119899 is119899 = 119881119886119881 = 11988621198871198862 (119886 + 119887) = 119887(119886 + 119887) (6)

while the porosity ratio 119899119904 of the soil section is

119899119904 = 119878119881119878119881 + 119878119904 = 119886119887119886 (119886 + 119887) = 119887(119886 + 119887) (7)

It can be found that 119899 = 119899119904 according to (6) and (7)the soil porosity can be represented by the porosity in themodel Although the change of effective stress and structureof soil caused by vacuum exists the change is little for thereal grouting projects especially in sand soil Therefore it isassumed that the soil structure is rigid

The velocity of the air passing through a certain crosssection of soil can be expressed as

V = 1119903 1199021198611198922120587ℎ119899 (8)

According to Darcyrsquos law

119889119901119889119903 = 120583119892119896119892 V =120583119892119896119892 1119903

1199021198611198922120587ℎ119899 (9)

Suppose (120583119892119896119892)(1199021198611198922120587ℎ119899) = 119862 The control equationof the steady-state seepage in homogeneous porous media is

119889119901 = 1198621119903119889119903 (10)

Then the following result is obtained

119901 = 1198621198621 ln 119903 + 1198622 (11)

Groutingpipe

r1

R

rr

2r0

Vacuum pressuredistribution curve Vacuum

well

Figure 3 Schematic diagram of vacuum pressure distribution

The boundary conditions are assumed as

119903 = 119903119903119901 = 119901119903119903 = 119903119908119901 = 119901119908(12)

In the conditions the pressure119901119903 ismeasured at a distanceof 119903119903 from the vacuum well and 119901119908 of the vacuum well with aradius of 119903119908 is also measured Then (11) can be expressed as

119901119908 = 1198621198621 ln 119903119908 + 1198622119901119903 = 1198621198621 ln 119903119903 + 1198622 (13)

Finally the distribution of the pressure 119901 along thedistance 119903 caused by the vacuum well is

119901 = 119901119908 minus 119901119903ln (119903119908119903119903) ln( 119903119903119908) + 119901119908 (14)

It is known that the 101 kPa can be converted into thewater head of about 103m so the pressure difference Δℎcaused by vacuum can be also expressed by water head basedon the relationship between atmospheric pressure and waterhead The water head caused by vacuum pressure is

Δℎ = 1199010 minus 1199011199089806 minus (119901119908 minus 119901119903) ln (119903119903119908)9806 ln (119903119908119903119903) (15)

The vacuum distribution is shown in Figure 3Because the seepage radius of slurry is comparatively

much smaller than the range of vacuumpressure distributionit is assumed that the vacuum pressure on the left and rightside of the distance of 1199031 is the same shown in Figure 3

Assume

1199010 minus 1199011199089806 = 1198621119901119908 minus 1199011199039806 ln (119903119908119903119903) = 1198622

(16)


Then Maagrsquos spherical diffusion radius can be modifiedbased on the boundary conditions as

119903 = 1199030ℎ = 119867 minus Δℎ0Δℎ0 = 1198621 minus 1198622 ln( 119877119903119908)119903 = 1199031ℎ = 119867 minus Δℎ1Δℎ1 = 1198621 minus 1198622 ln(119877 minus 1199031119903119908 )

(17)

The new boundary conditions are submitted into Darcyrsquoslaw and the equations are integrated to obtain the diffusionradius under the vacuum effect

11990310158401 = 3radic31198701199051199030 (ℎ1 + Δℎ1015840)119899120573 (18)

where

Δℎ1015840 = 1198622 ln( 119877119877 minus 1199031) ge 0 (19)

It can be seen from (18) that the vacuum pressure canenlarge the slurry diffusion distance because the vacuumpressure makes the water head boundaries lower thanMaagrsquosdiffusion model

3 Verification Experiment of VacuumPenetration Grouting

According to the analysis above the final diffusion radiusof vacuum modification model will be influenced by manyfactors like vacuum degree soil conditions and so on Inthis paper the vacuummodification formula is verified by thepublished vacuum grouting experiment by Shen [17]

31 Experimental Device and Materials The vacuum grout-ing experiment system is presented in Figure 4

The medium sand fine sand and silty sand were used asthe injected stratumThe physical properties of experimentalsand soil are shown in Table 1

Acid sodium silicate solution was used in the experimentThe sodium silicate solution is 30∘ Bersquo and the concentrationof dilute sulphuric acid solution is 10 The volume ratio ofsodium silicate solution to sulphuric acid is 07 The gel timeof the slurry is about 350 s The slurry is Newtonian fluid andthe initial viscosity is 5mPasdots32 Results and Discussion In the experiment 119877 is 02m 119903119908is 5 cm and 1199030 is 1 cm The vacuum degree is 60 kPa 40 kPaand 20 kPa near the grouting pipe The grouting lasted for45min The radii of the vacuum modification model areshown in Table 2

Figure 4 The experimental system [17]

Table 1 Physical properties of sand samples

Sand soil 120588 (gcm3) 119870 (cms) 119899 () 120596 () 119866 (gcm3)Medium 161 145 times 10minus2 378 107 259Fine 171 559 times 10minus3 345 125 261Silty 179 975 times 10minus4 310 153 260Where 120588 is natural density 119870 is permeability coefficient 119899 is porosity 120596 ismoisture content 119866 is specific gravity

Table 2 Diffusion radii of Maagrsquos spherical model and vacuummodification model

119875119903 (kPa) Sand soil Δℎ1015840 (cm) 11990310158401 (mm) 1199031 (mm) Δ119903 (mm)

60Medium 2515 136 859 501Fine 1340 91 644 266Silty 606 48 36 12


20Medium 587 110 859 241Finer 370 79 644 146Silty 183 44 36 8

Where 119901119903 is vacuum degree Δℎ1015840 is head change caused by vacuum 11990310158401 isradius of vacuummodificationmodel 1199031 is radius ofMaagrsquos spherical modelΔ119903 is absolute difference between 11990310158401 and 1199031

The comparison of diffusion radii in Maagrsquos and vacuummodification models in vacuum condition is shown in Fig-ures 5ndash7

It is revealed from Figures 5ndash7 that under the effect ofvacuum negative pressure the radii of Maagrsquos spherical dif-fusion model are smaller than those of vacuum modificationmodel It illustrates that the vacuumgrouting can enhance thediffusion ability of slurry by enlarging the diffusion radiusHowever Maagrsquos spherical diffusionmodel does not considerthe effect of vacuumpressure and it can not be used to predictthe slurry diffusion distance in vacuum condition

The radii differences under different vacuum degrees areshown in Figure 8

Figure 8 shows that the radii differences of vacuummod-ification model and Maagrsquos model grow with the increases ofthe vacuum degree It also can be seen that the larger thepermeability coefficient of the soil the larger the value ofradius difference


0

20

40

60

80

100

120

140

160

Medium sand Fine sand Silty sand

Diff

usio

n ra

dius

(mm

)

Radius of vacuum modification modelRadius of Maagrsquos spherical modelThe differences

Figure 5 Diffusion radii under vacuum degree of 60 kPa

0

20

40

60

80

100

120

140


Diff

usio

n ra

dius

(mm

)

Radius of vacuum modification modelRadius of Maagrsquos spherical diffusion modelThe differences


In Shenrsquos [17] experiment the slurry was injected into3 kinds of sand under vacuum degrees of 20 kPa 40 kPaand 60 kPa respectively The theoretical values obtained byvacuummodificationmodel and the correspondingmeasure-ment values in experiments are presented in Table 3

The variation of experimental values and theoreticalvalues is shown in Figures 9ndash11

It can be obtained from Figures 9ndash11 that the trend ofvariation of the experimental diffusion radii is the same asthat of the theoretical results The diffusion radii of the grout

0

20

40

60

80

100

120


Diff

usio

n ra

dius

(mm

)



0

10

20

30

40

50

60

20 30 40 50 60

Radi

us d

iffer

ence

(mm

)

Vacuum degree (kPa)

Medium sand layerFine sand layerSilt layer

Figure 8 Radii differences under different vacuum degrees

grow with the increase of the vacuum degree The growthrate increases with the increase of permeability coefficientThe larger the vacuum degree the larger the differenceof the slurry diffusion radius The curves of modificationmodel are closer to the experiment values than the curvesof Maagrsquos model especially in the silty layer It also canbe seen that the greater the vacuum degree the smallerthe deviations between the experimental values and thevalues of vacuum modification model At the same time thedeviations between the experimental values and the values


Table 3Theoretical and experimental values of the diffusion radiusunder different vacuum degrees and soil conditions

Sand stratum 119901119903 (kPa) 119903101584010158401 (mm) 1199031015840 (mm) Δ1199031015840 (mm)

Medium60 145 136 minus6240 90 122 35520 70 110 571

Fine60 110 91 minus17240 80 86 7520 55 79 436

Silty60 75 48 minus3640 60 46 minus23320 45 44 minus22

Where 119901119903 is vacuum degree 119903101584010158401 is experimental value 1199031015840 is theoretical valueof modification model Δ1199031015840 is difference of 119903101584010158401 and 1199031

0

20

40

60

80

100

120

140

160

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)

Experimental valuesValues of modification modelValues of Maagrsquos spherical model

Figure 9 The variation of theoretical and experimental values inmedium sand layer

of Maagrsquos model grow with the increases of vacuum degreeThe differences between the vacuummodification values andthe experimental values fall in the range of minus36 to 57The theoretical model assumes that the soil is isotropic andthe soil structure remains unchanged but the soil used inexperiment can not reach the ideal conditions Due to thesize limitation of the model box the boundary conditions ofexperimental and theoretical models can not be completelyconsistent Some studies show that the deviations betweenthe actual measured values and the theoretical values areacceptable [18] Therefore the vacuum modification modelis reasonable to predict the diffusion radius of vacuumpenetration grouting

0

20

40

60

80

100

120

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)


Figure 10 The variation of theoretical and experimental values infine sand layer

0

10

20

30

40

50

60

70

80

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)


Figure 11 The variation of theoretical and experimental values insilt layer

4 Conclusions

(1) Vacuum grouting method can enlarge the grout diffusiondistance compared to the traditionalmethodwithout vacuumpressure

(2) It can be concluded from the theoretical analysisand experiment that the modified Maagrsquos spherical diffusionmodel is more reasonable to predict the diffusion of groutthan the traditional Maagrsquos model At the same time it isshown in the vacuummodification model that the higher the


absolute value of vacuum pressure the larger the diffusionradius

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This study is supported by ldquoTheFundamental Research Fundsfor the Central Universities of Chinardquo (no 2-9-2015-082)

References

[1] G Zhang L Song and L Chen ldquoA case study on treatmentmeasures for a rock tunnel with inrush mud and sand gushingdisasterrdquo Disaster Advances vol 5 no 4 pp 489ndash493 2012

[2] C A Anagnostopoulos ldquoLaboratory study of an injectedgranular soil with polymer groutsrdquoTunnelling andUndergroundSpace Technology vol 20 no 6 pp 525ndash533 2005

[3] G Zheng T Cui X Cheng et al ldquoStudy of the collapsemechanism of shield tunnels due to the failure of segments insandy groundrdquo Engineering Failure Analysis vol 79 pp 464ndash490 2017

[4] R W I Brachman C D Martin and S A Gilliss ldquoGrout fieldtrials in outwash sandsrdquo Canadian Geotechnical Journal vol 41no 1 pp 1ndash11 2004

[5] W Sui J Liu W Hu J Qi and K Zhan ldquoExperimentalinvestigation on sealing efficiency of chemical grouting in rockfracture with flowing waterrdquo Tunnelling and Underground SpaceTechnology vol 50 pp 239ndash249 2015

[6] D Neupane H Yasuhara N Kinoshita and H Putra ldquoDis-tribution of grout material within 1-m sand column in insitucalcite precipitation techniquerdquo Soils and Foundations vol 55no 6 pp 1512ndash1518 2015

[7] Y S Kim and A J Whittle ldquoParticle network model forsimulating the filtration of a microfine cement grout in sandrdquoJournal of Geotechnical and Geoenvironmental Engineering vol135 no 2 pp 224ndash236 2009

[8] J Yoon and C El Mohtar ldquoGroutability of granular soilsusing sodium pyrophosphate modified bentonite suspensionsrdquoTunnelling and Underground Space Technology vol 37 pp 135ndash145 2013

[9] C S El Mohtar J Yoon and M El-Khattab ldquoExperimentalstudy on penetration of bentonite grout through granular soilsrdquoCanadian Geotechnical Journal vol 52 no 11 pp 1850ndash18602015

[10] F Bouchelaghem L Vulliet D Leroy L Laloui and FDescoeudres ldquoReal-scale miscible grout injection experimentand performance of advection-dispersion-filtration modelrdquoInternational Journal for Numerical and Analytical Methods inGeomechanics vol 25 no 12 pp 1149ndash1173 2001

[11] Z Saada J Canou L Dormieux J C Dupla and S MaghousldquoModelling of cement suspension flow in granular porousmediardquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 29 no 7 pp 691ndash711 2005

[12] J Yoon and C S El Mohtar ldquoA filtration model for evaluatingmaximum penetration distance of bentonite grout throughgranular soilsrdquo Computers amp Geosciences vol 65 pp 291ndash3012015

[13] J J Assaad and Y Daou ldquoCementitious grouts with adaptedrheological properties for injection by vacuum techniquesrdquoCement and Concrete Research vol 59 pp 43ndash54 2014

[14] J J Assaad Y Daou and J Harb ldquoInfluence of thixotropyon performance of grouts placed using vacuum injectiontechniquesrdquo ACI Materials Journal vol 112 no 2 pp 189ndash1982015

[15] W-J Han S-Y Liu D-W Zhang and G Du ldquoField behaviorof jet grouting pile under vacuum preloading of soft soils withdeep sand layerrdquo in Proceedings of the GeoCongress 2012 Stateof the Art and Practice in Geotechnical Engineering pp 70ndash77Geotechnical Special Publication USA March 2012

[16] J Peng H Ye and A N Alshawabkeh ldquoSoil improvementby electroosmotic grouting of saline solutions with vacuumdrainage at the cathoderdquo Applied Clay Science vol 114 pp 53ndash60 2015

[17] G L ShenResearch on simulation test of vacuum grouting underdifferent stratum Chinese) China University of GeosciencesBeijing China 2014

[18] W-J Ruan ldquoSpreading model of grouting in rock mass fissuresbased on time-dependent behavior of viscosity of cement-basedgroutsrdquoChinese Journal of RockMechanics and Engineering vol24 no 15 pp 2709ndash2714 2005

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effect of electroosmotic grouting with vacuum drainage isnot only better than general electroosmosis methods but alsomore uniform than such methods Shen [17] obtained therelationship between the degree of vacuum and the size ofthe grouting body and found that the diffusion radii of theslurry grown with the increase of vacuum degree (absolutevalue of vacuumpressure) and grouting pressureThe presentresearches provide a useful way to explore the slurry diffusionmechanism of vacuum penetration grouting However thepresent studies only obtained the macro and qualitative lawThey have not established the diffusion model of vacuumgrouting and the relationship between the grouting param-eters and the diffusion distance

Therefore this paper takes Newtonian fluid as theresearch object to deduce the diffusionmodel of vacuumpen-etration grouting on the basis of Maagrsquos spherical diffusionmodel The new diffusion model explains how the diffusiondistance be enlarged in mathematical way and predictsthe diffusion distance of the slurry by analysis formula byconsidering grouting coefficients The vacuum modificationmodel provides a reference to the scientific application ofvacuum penetration grouting

2 Maagrsquos Spherical Diffusion Model andVacuum Modification Model

Maag derived the diffusion formula of the grout in sand layerand proposed a spherical diffusion model of Newtonian fluid[17]The theoretical assumptions are that the grout is Newtonfluid when diffusing in sandy soil the diffusion of slurryfollows Darcyrsquos law the injected soil is homogeneous andisotropic there is no dynamic water pressure when groutingin groundwater the difference between the density of slurryand water is not considered the viscosity of the slurry isconstant during grouting and the slurry diffuses in sphericalform starting from the end of injecting pipe

Maagrsquos spherical diffusion model is shown in Figure 1According to Darcyrsquos law119876 = 119870119892119894119860119905 (1)

where 119894 = minus119889ℎ119889119903 119870119892 = 119870120573 and 119860 = 41205871199032According to the boundary conditions

119903 = 1199030ℎ = 119867119903 = 1199031ℎ = ℎ0

(2)

Equation (3) can be obtained as follows by submitting theboundary conditions into Darcyrsquos law

119867 minus ℎ0 = 1198761205734120587119870119905 ( 11199030 minus 11199031) (3)

then 119876 can be solved as

119876 = 4120587119870119905 (119867 minus ℎ0)120573 (11199030 minus 11199031) (4)

H

ℎ1

ℎ0

2r0

r1

dr

dℎ

Groundsurface

Figure 1 Schematic diagram of derivation of Maagrsquos model

Considering 1199031 minus 1199030 asymp 1199031 11199030 minus11199031 asymp 11199030 and119867minusℎ0 =ℎ1 as well as 119876 = (43)12058711990313119899 the expression of the diffusionradius of grouting can be derived as

1199031 = 3radic31198701199051199030ℎ1119899120573 (5)

119876mdashtotal amount of grouting (cm3)119860mdashslurry penetration area (cm2)119905mdashtotal time for grouting (s)119903mdashthe diffusion radius of the slurry at any moment(cm)1199031mdashthe final diffusion radius of the slurry (cm)1199030mdashradius of grouting pipe (cm)ℎ1mdashgrouting pressure head (cm)ℎ0mdashwater head of groundwater in the end of groutingpipe (cm)119867mdashthe sumof groundwater pressure head and grout-ing pressure head (cm)119899mdashporosity of soil120573mdashthe ratio of the viscosity of slurry to the viscosityof water 120583g120583w119870mdashpermeability coefficient of injected soil (cms)119870119892mdashpermeability coefficient for slurry (cms)

In order to study the influence of vacuum on the dif-fusion radius the distribution of vacuum pressure in soil isanalyzed firstly The plane seepage model is used to derivethe distribution formula of vacuum pressure It is assumedthat the seepage of air in soil is steady and follows DarcyrsquoslawThe vacuumpressure is converted into the correspondingwater head through the derivation of the model which canbe coupled with the grouting pressure head Finally the newboundary conditions are submitted into Maagrsquos sphericaldiffusion model and the expression of the diffusion radius ofthe slurry in vacuum condition is derived


a

a

b

AB

CD

a

Ss

SV





119899119904 = 119878119881119878119881 + 119878119904 = 119886119887119886 (119886 + 119887) = 119887(119886 + 119887) (7)



V = 1119903 1199021198611198922120587ℎ119899 (8)


119889119901119889119903 = 120583119892119896119892 V =120583119892119896119892 1119903

1199021198611198922120587ℎ119899 (9)


119889119901 = 1198621119903119889119903 (10)


119901 = 1198621198621 ln 119903 + 1198622 (11)

Groutingpipe

r1

R

rr

2r0


well



119903 = 119903119903119901 = 119901119903119903 = 119903119908119901 = 119901119908(12)


119901119908 = 1198621198621 ln 119903119908 + 1198622119901119903 = 1198621198621 ln 119903119903 + 1198622 (13)


119901 = 119901119908 minus 119901119903ln (119903119908119903119903) ln( 119903119903119908) + 119901119908 (14)





Assume

1199010 minus 1199011199089806 = 1198621119901119908 minus 1199011199039806 ln (119903119908119903119903) = 1198622

(16)




(17)


11990310158401 = 3radic31198701199051199030 (ℎ1 + Δℎ1015840)119899120573 (18)

where

Δℎ1015840 = 1198622 ln( 119877119877 minus 1199031) ge 0 (19)





















0

20

40

60

80

100

120

140

160


Diff

usio

n ra

dius

(mm

)



0

20

40

60

80

100

120

140


Diff

usio

n ra

dius

(mm

)






0

20

40

60

80

100

120


Diff

usio

n ra

dius

(mm

)



0

10

20

30

40

50

60

20 30 40 50 60

Radi

us d

iffer

ence

(mm

)

Vacuum degree (kPa)







Medium60 145 136 minus6240 90 122 35520 70 110 571

Fine60 110 91 minus17240 80 86 7520 55 79 436



0

20

40

60

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100

120

140

160

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)




0

20

40

60

80

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20 30 40 50 60

Diff

usio

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(mm

)

Vacuum degree (kPa)



0

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20 30 40 50 60

Diff

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(mm

)

Vacuum degree (kPa)



4 Conclusions







Acknowledgments


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Journal of












Journal of







Volume 2018






Volume 2018









a

a

b

AB

CD

a

Ss

SV





119899119904 = 119878119881119878119881 + 119878119904 = 119886119887119886 (119886 + 119887) = 119887(119886 + 119887) (7)



V = 1119903 1199021198611198922120587ℎ119899 (8)


119889119901119889119903 = 120583119892119896119892 V =120583119892119896119892 1119903

1199021198611198922120587ℎ119899 (9)


119889119901 = 1198621119903119889119903 (10)


119901 = 1198621198621 ln 119903 + 1198622 (11)

Groutingpipe

r1

R

rr

2r0


well



119903 = 119903119903119901 = 119901119903119903 = 119903119908119901 = 119901119908(12)


119901119908 = 1198621198621 ln 119903119908 + 1198622119901119903 = 1198621198621 ln 119903119903 + 1198622 (13)


119901 = 119901119908 minus 119901119903ln (119903119908119903119903) ln( 119903119903119908) + 119901119908 (14)





Assume

1199010 minus 1199011199089806 = 1198621119901119908 minus 1199011199039806 ln (119903119908119903119903) = 1198622

(16)




(17)


11990310158401 = 3radic31198701199051199030 (ℎ1 + Δℎ1015840)119899120573 (18)

where

Δℎ1015840 = 1198622 ln( 119877119877 minus 1199031) ge 0 (19)





















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Diff

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(mm

)



0

10

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30

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20 30 40 50 60

Radi

us d

iffer

ence

(mm

)

Vacuum degree (kPa)







Medium60 145 136 minus6240 90 122 35520 70 110 571

Fine60 110 91 minus17240 80 86 7520 55 79 436



0

20

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Diff

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Vacuum degree (kPa)




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0

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4 Conclusions







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Journal of












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Volume 2018






Volume 2018











(17)


11990310158401 = 3radic31198701199051199030 (ℎ1 + Δℎ1015840)119899120573 (18)

where

Δℎ1015840 = 1198622 ln( 119877119877 minus 1199031) ge 0 (19)





















0

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Diff

usio

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(mm

)



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Diff

usio

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(mm

)






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Diff

usio

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(mm

)



0

10

20

30

40

50

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20 30 40 50 60

Radi

us d

iffer

ence

(mm

)

Vacuum degree (kPa)







Medium60 145 136 minus6240 90 122 35520 70 110 571

Fine60 110 91 minus17240 80 86 7520 55 79 436



0

20

40

60

80

100

120

140

160

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)




0

20

40

60

80

100

120

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)



0

10

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30

40

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60

70

80

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)



4 Conclusions







Acknowledgments


References


























Journal of












Journal of







Volume 2018






Volume 2018









0

20

40

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80

100

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140

160


Diff

usio

n ra

dius

(mm

)



0

20

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60

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Diff

usio

n ra

dius

(mm

)






0

20

40

60

80

100

120


Diff

usio

n ra

dius

(mm

)



0

10

20

30

40

50

60

20 30 40 50 60

Radi

us d

iffer

ence

(mm

)

Vacuum degree (kPa)







Medium60 145 136 minus6240 90 122 35520 70 110 571

Fine60 110 91 minus17240 80 86 7520 55 79 436



0

20

40

60

80

100

120

140

160

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)




0

20

40

60

80

100

120

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)



0

10

20

30

40

50

60

70

80

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)



4 Conclusions







Acknowledgments


References


























Journal of












Journal of







Volume 2018






Volume 2018











Medium60 145 136 minus6240 90 122 35520 70 110 571

Fine60 110 91 minus17240 80 86 7520 55 79 436



0

20

40

60

80

100

120

140

160

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)




0

20

40

60

80

100

120

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)



0

10

20

30

40

50

60

70

80

20 30 40 50 60

Diff

usio

n ra

dius

(mm

)

Vacuum degree (kPa)



4 Conclusions







Acknowledgments


References


























Journal of












Journal of







Volume 2018






Volume 2018












Acknowledgments


References


























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Modified Maag’s Spherical Diffusion Model of Vacuum ...in geotechnical engineering. e sandy layer can cause severe damage to underground projects [– ]. Grouting is a common method