Research Article Transport Model of Underground Sediment

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 367918 7 pageshttpdxdoiorg1011552013367918

Research ArticleTransport Model of Underground Sediment in Soils

Sun Jichao1 and Wang Guangqian2

1 School of Water Resources and Environment China University of Geosciences Beijing 100083 China2 State Key Laboratory of Hydroscience and Engineering Tsinghua University Beijing 100084 China

Correspondence should be addressed to Sun Jichao sjc00126com

Received 23 June 2013 Accepted 2 September 2013

Academic Editors S Kaewunruen and H C Wu

Copyright copy 2013 S Jichao and W Guangqian This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Studies about sediment erosion were mainly concentrated on the river channel sediment the terrestrial sediment and theunderground sediment The transport process of underground sediment is studied in the paper The concept of the flush potentialsediment is founded The transport equation with stable saturated seepage is set up and the relations between the flush potentialsediment and water sediment are discussed Flushing of underground sediment begins with small particles and large particleswill be taken away later The pore ratio of the soil increases gradually The flow ultimately becomes direct water seepage and thesediment concentration at the same position in the water decreases over time The concentration of maximal flushing potentialsediment decreases along the path The underground sediment flushing model reflects the flushing mechanism of undergroundsediment

1 Introduction

The transport of the sediment in the soil driven by under-ground water force is known as underground sedimenttransport which is shown in Figure 1

Previous studies about sediment were mainly concen-trated on the river sediment marine sediment and terrestrialsediment However these studies focused on sediment onthe surface generally [1] A new research direction is theunderground sediment [2 3] the sediment inside the soiland the new study enhances the level of research from theexisting surface sediment to underground sediment

The existence and transport of underground sedimentwill cause a lot of engineering problems Sediment trans-port within the soil of the lower river course of changingwater level results in sand draining and may eventuallylead to the collapse of the dam embankment changes ofgroundwater level could lead to surface collapse caused byunderground sediment transport sediment in faults andfissures is transported driven by external disturbances (suchas earthquakes and water level fluctuations) and leads tothe activation of faults and fractures underground sedimentis also transported with the movement of water and oiltransport causes blockage of oil fissures and decreases the

speed of oil production All these problems may lead tomajor accidents or increase costs and they are related withunderground sediment transport directly

2 Research Statuses

Research of the underground sediment transport involvesin sediment movement mechanics and is interdisciplinaryThere has been a lot of research on these two aspects

Previous studies on sediment movement mechanicscould be roughly divided into two areas the mechanism ofphysical phenomena and engineering applicationsThere hasbeen an extensive research on the first area which relatesto settling starting moving and bedload movement ofsediment particles the sediment carrying capacity nonequi-librium sediment transport gravity flow river simulationreservoir sedimentation river evolution and so forth [4]

These studies of the mechanism of physical phenomenonmechanism and engineering applications of sediment move-ment mechanics concentrate on sediment start transportand saturation and their methods are indoor and outdoorexperiments and mechanical analysis carrying on sedimentof the coast river river basin and surface These research

2 The Scientific World Journal

Fixed soils

Potential flowing

sediment

Flow waterFlow sediment

Flus

hing

chan

ge

Flow waterFixed sedimentFlow sediment

Fixed water

Figure 1 Schematics of particle exchange and transport of under-ground sediment

papers have not been involved in underground sediment[2 3]

Studies of permeation fluid mechanics in this area focuson the seepage of flow in saturated [5 6] and unsaturatedsoils [7 8] Accompanied by the study of sediment trans-port defining of the state point is conducted based on theexperimental and numerical computation [9ndash12] There areonly a few studies on related theories of sediment transport ingroundwater especially in terms of the quantitative transportprocess

The two processes of subsurface erosion and volumeuplift [13] have been observed in the experiment of thetwo processes with the conclusion that the irreversibledisplacement of each soil particle corresponds to a criticalhydraulic gradient [14]These results of the experiments havesome practical values but they lack necessary and sufficientproof of relevant theory Therefore they are not so rationaland applicable in practical application

Force analysis of particles in the pore channels is con-ducted in the literature [15] obtaining the critical flowvelocities formula when sediment particles start to moveHowever sediment particles are nonsticky sediment with thesame diameter which might be inconsistent with the actualsituation as actual sediment Thus its application has certainlimitations

3 Underground Sediment TransportEquations with Stable Saturated Seepage

The assumption in the paper is that the diameter of under-ground sediment particles which can be transported is muchsmaller than the diameter of the rest of the soil particles

A small unit along the path of the underground sedimenttransport is studied which is shown in Figure 2

dy

swSdtdy

dx

nSdxdy

nSdxdy +120597(nS)

120597tdtdxdy

swSdt dy +120597(swS)

120597xdxdtdy

qsdt dx dy

Figure 2 Unit body of underground sediment transport

31 The Sediment Transport Equation Within unit body119889119909119889119910 and time 119889119905 the sediment volume carried by the flow

120597 (V119904119908119878)

120597119909119889119909119889119905119889119910 (1)

Within unit body 119889119909119889119910 and time 119889119905 the sediment volumein water increases as follows

120597 (119899119878)

120597119905119889119905119889119909119889119910 (2)

Sediment in the119910direction has no concentration gradientnamely there is no sediment diffusion in this directionthe sediment volume in water supplied by flush potentialsediment is

119902119904119889119905119889119909119889119910 (3)

wherein 119902119904is the volume of sediment in the soil entering the

water per unit time per unit volumeTherefore within the unit body 119889119909119889119910 and time 119889119905 the

increased sediment equation is

120597 (119899119878)

120597119905119889119905119889119909119889119910 = minus

120597 (V119904119908119878)

120597119909119889119909119889119905119889119910 + 119902

119904119889119905119889119909119889119910 (4)

The following equation is obtained after simplification

120597 (119899119878)

120597119905= minus120597 (V119904119908119878)

120597119909+ 119902119904 (5)

32 Relations between the Flush Potential Sediment andWaterSediment The sediment in water supplied by flush potentialsediment is shown in Figure 2 and the volume is

119902119904119889119905119889119909119889119910 (6)

wherein when 119902119904is positive it indicates the sediment volume

in soil entering in the water per unit time per unit volumewhen it is negative it indicates the speed of the sedimentin cementing to the soil which can be called the solventingspeed

The sediment will start to flow into the seepage water Asthey are saturated underground sediments the flow rate isconstant But at the same time a part of the fixed sedimentflows into the seepagewater with the end result that exchangehappens between this part of sediment and flow water of thesame volume namely the fixed sediment changes into flow

The Scientific World Journal 3

x

Diss

olut

ion

Prec

ipita

tes

qs

S lt Smaxn lt nmax

S gt Smaxn gt nmax

Figure 3 Relationship between solventing speed and greatestsediment concentration

sediment and the flow water changes into fixed water of thesame volume

A new parameter the flush pore ratio of the fixedsediment 119899

1 is then introduced which meets

1205971198991

120597119905= 119902119904 (7)

wherein 1198991is the volume potential flush sediment changing

from resting to moving and the overall volume of thepotential flush sediment is 119899

1max Consider

1198991max = 1198991 (119905 997888rarr +infin) (8)

The entry speed of potential sediment is directional thatis on the one hand the flush potential sediment in the soilflows into the seepage and on the other hand sediment inthe seepage cements into the soil

The speed is related to the particle size distributionparticle composition the speed of seepage water and seepagewater sediment concentration of sediment in the soil Thespecific relationship is as follows the higher the seepagespeed the bigger the 119902

119904 the greater the seepage water

sediment concentration the smaller the 119902119904 the greater the

potential flush force of the sediment the bigger the 119902119904 The

graphical representation is as in Figure 3The solventing speed 119902

119904shown in Figure 3 complies with

the relevant laws as follows

119902119904= 119870(119878max minus 119878)

120572

(9)

wherein 119870 is the Solventing speed factor 120572 is the coefficientas shown in Figure 3 taking 1 3 5 and so forth theodd function 119878max is the maximum seepage water sedimentconcentration which can be called the greatest potentialsediment concentration or potential sediment concentrationfor short 119878 is the seepage water sediment concentration

The flushing process must satisfy the conditions of thepore ratio which are as follows

Dissolution 119878 lt 119878max and it must satisfy 1198991lt 1198991max

Cementing 119878 gt 119878max and it must satisfy 1198991gt 0 The opposite

situation is as follows when 119878 lt 119878max but 1198991 ge 1198991max thereis no dissolution but 119878 flush is maintained when 119878 gt 119878maxbut 1198991= 0 or 119899

1= 1198991min there is no cementing and 119878 flush is

maintainedThis could not happen because it refers to the factthat blocking occurs on the soil and the seepage has stopped

The maximum sediment concentration of the seepagewater 119878max is directly related to the speed of the soil seepagewaterThe smaller the seepage speed of the water the smallerthe 119878max There is an exponential relationship between thetwo and relation [16] between the sediment concentrationand flow rate is cubic namely

119878max = 119870119904V3

119904119908 (10)

The surface of the sediment particles is affected by thelocal vortex of low near the microcontact surface of the soilwith water thus is carried away by the flow Though there islocal variation of the flow speed it is very low in terms ofthe overall seepage flow Thus based on the potential sandconcentration of river sediment carrying force the potentialsediment concentration formula of the underground sedi-ment transport is determined as follows

119878max = 119870119904V119904119908 (11)

wherein119870119904is the coefficient of themaximum amount of sand

carrying sedimentThe maximum pore ratio 119899max is related to the seepage

water speed and the sediment particles gradation distributionin the soil In view of the complicated sediment particledistribution it is assumed that the gradation distributiondoes not affect the underground sediment transport so onlythe speed of the seepage water is taken into account

Referring to formula (9) themaximumpore ratio is takenas

1198991max = 1198701198991V119904119908 (12)

In the actual calculation it can also be considered that themaximum pore ratio is constant value

According to the formulas (7) and (9) it is obtained that

1205971198991

120597119905= 119870(119870

119904V119904119908minus 119878)120572

(13)

Simultaneous of (5) (7) and (13) helps in the formation ofthe underground sediment transport equation

The above underground sediment transport is saturatedsediment flushing thus the hydroscience speed V

119904119908remains

unchanged the pore ratio n is the effective porosity ofthe underground sediment delivery ratio which means thatthe pore ratio 119899 involved in the sediment transport is alsounchanged the above equation can be simplified as

119899120597119878

120597119905= minusV119904119908

120597119878

120597119909+ 119870(119870

119904V119904119908minus 119878)120572

(14)

When 119889119905 = 119889119909V119904119908 the above equation becomes

V119904119908(119899 + 1)120597119878

120597119909= 119870(119870

119904V119904119908minus 119878)120572

(15)


Further convert formula (15) to

V119904119908(119899 + 1)119889119878

119889119909= 119870(119870

119904V119904119908minus 119878)120572

(16)

This is the underground sediment transport equation whichis also underground sediment distribution equation along thepath

4 Equation Solving

In terms of (16) solving difficulties differs largely variablevalue of the parameter 120572 Equation (9) is of an odd functiongraphics with the value of 120572 being 1 3 5 7 thus theanalytical calculation results of (16) are

when 120572 = 1

119878 = minus(V119904119908(119899 + 1)

119870 (120572 minus 1) (119909 + 1198621))

1(120572minus1)

+ 119870119904V119904119908

(17)

when 120572 = 1

119878 = 119870119904V119904119908+ 119890minus(119870119909V

119904119908(119899+1))

1198621 (18)

wherein 1198621 is a constant and its specific value needs to bedetermined by the boundary condition

During the flushing at the moment of time 119905 threesoil pore ratios are relative to the overall volume of thesoil (including volume of the flowing water moving sandthe potential static water the potential static sand and thefixed sand) Among it the volume ratio of the flowing watersediment is 119899 when the time is 0 the volume ratio of theflowing water sediment is the initial pore ratio of the soil119899 = 119899

0 the volume of water of the flowing water sediment

namely the flowing water ratio is 119899 times (1 minus 119878) the pore ratioof the potential flowing sediment namely the volume ratio ofthe potential flowing sediment 119899

1max is

1198991max = 1198991 (119905 997888rarr + prop)

= int

+prop

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(19)

The overall ratio of the soil according formula (13) as theflowing sand cannot support the externalmechanical effect ofthe soil is also counted as the overall pore volume as follows

119899119905= 1198990+ 1198991(119905)

= 1198990+ int

119905

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(20)

Under real-time condition

119899119905(119894+1)= 119899119905(119894)+ 1198991119905(119894+1)

= 119899119905(119894)+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908minus 119878)120572

119889119905

(21)

The boundary conditions areWhen 119909 = 0 119878 = 0 and therefore a constant term of 119865

is 119870119904V119904119908

When 120572 = 1

1198621 =(119899 + 1)119870

119904V2119904119908

119870119890120572 ln(119870119904V119904119908) (120572 minus 1) (22)

After simplification it is

1198621 =(119899 + 1)119870

119904V2119904119908

119870(119870119904V119904119908)120572

(120572 minus 1) (23)

When 120572 = 1

1198621 = minus119870119904V119904119908 (24)

5 The Termination Condition of UndergroundSediment Flushing

Flushing of the underground sediment is not always goingfollowing this rule unrestrictedly During the flushing asthe pore ratio increases the sediment decreases and will bewashed away by the flow eventually If there is no collapsein the soil the pore ratio of this process will remain thesame and underground sediment transport will be replacedby groundwater seepage Variable factors and parametersof the process can be ascribed as the potential sedimentconcentration 119878max could not simply adopt the formula119878max = 119870119904V

3

119904119908 but only complies with it at the beginning of the

flushing and decreases later on It is directly related with poreratio and themaximal ratio adopts the ratio 119899

119908formed finally

at this flowing speed In actual calculation the maximal poreratio can be deemed as a constant

Therefore the potential sediment concentration 119878maxduring the flushing is amended in the following formula

119878max =119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) (25)

The initial pore ratio is 1198991199080= 1198990

6 Calculation Examples

61The Initial State Thecalculated parameters are as follows119870119904= 2 V

119904119908= 005ms 119870 = 1 119899 = 02 119909 = 0 rarr 1

and 1198991max = 06 According to (17) and (18) the sediment

concentration is calculatedwith the results shown in Figure 4Furthermore calculate the underground sediment pro-

cess in accordance with (20) and (25) The calculationindicates that since the 119870

119904V119904119908

of formula (16) is replaced by((119870119904V119904119908)(1198991maxminus1198990))(1198991maxminus119899119908) and also replaced according

to (20) while (20) relates to the calculation of sedimentcontent concentration 119878 the solution is to conduct iterativecalculation in accordance with results of the initial state Thetime variable 119905 is introduced in (20) to obtain the resultsvariation law during the flushing process of the undergroundsediment


10

9

8

7

6

5

4

3

2

1

00 01 02 03 04 05 06 07 08 09 1

x distance (m)

yse

dim

ent c

once

ntra

tion

()

120572 = 10

120572 = 15

120572 = 20

120572 = 25120572 = 30

Figure 4 Sediment concentration distribution of initial transportalong the path 119909 (m) 119910 sediment concentration ()

Equation (20) becomes

119899119905(119894+1)= 119899119905(119894)

+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

119889119905

= 119899119905(119894)+ [119905(119894+1)minus 119905(119894)]

times 119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

(26)

62 Model Calculations at 120572 = 1 The study area has a lengthof 100 cm

When 120572 = 1 and within the time of 2000 seconds thetransport process of underground sediment is calculated

From Figure 4 the following laws can be obtained

(1) The sediment concentration along the path increaseswith the increase of groundwater entrance distanceand the degree of its increase reduces with thedistance to the entrance the sediment concentrationultimately reaches a stable value

(2) The sediment concentration decreases with theincrease of the 120572 value

(3) During the initial period the sediment concentrationalong the path decreases with the increase of the 120572value

The following laws are obtained from Figure 5

(1) At the samemoment the pore ratio gradually reducesand finally approaches the minimum value

065

06

055

05

045

04

035

03

025

020 01 02 03 04 05 06 07 08 09 1

x distance (m)

ypo

re ra

tio

t = 200i (i = 1 2 3 10) unit s

Figure 5 Changing of pore ratio along the path over time 119909 (m) 119910pore ratio

0 01 02 03 04 05 06 07 08 09 1x distance (m)

10

9

8

7

6

5

4

3

2

1

0

yse

dim

ent c

once

ntra

tion

()

t = 200i (i = 1 2 3 10) unit s

SSmax

Figure 6 Changing of sediment concentration and potential sed-iment concentration along the path over time 119909 (m) 119910 sedimentconcentration ()

(2) At the same position the pore ratio increase over timeis increased and finally approaches the maximumvalue

(3) At the intermediate time reducing the pore ratioalong the path shows a law of increasing firstly anddecreasing later on in other words the absolutevalue of the curve slope or that of the first derivativeincreases firstly and then decreases


(1) Laws of the sediment concentration and potentialsediment concentration show strong consistency andsimilarity The potential sediment concentration of


same location at the same time is greater than thesediment concentration

(2) At the same moment the sediment (potential sed-iment concentration) concentration increases alongthe path gradually and eventually tends to be thepotential sediment concentration (maximum poten-tial sediment concentration)

(3) Sediment concentration (potential sediment concen-tration) of the same position decreases over time andtends to be the minimum value eventually

(4) At the intermediate time the increase of the sedimentconcentration (the potential sediment concentration)increases firstly and then decreases along the path inother words it shows a law of first increase and thendecrease of the curve of the slope or first derivative

63 Mechanistic Explanation When the groundwater flowsthrough the soil static sediment along the path enters intothe water and becomes flowing sediment during the entiredistance of the water course static sediment could enter thewater at any time thus the longer the flow path the more thesediment in the water and vice versa

Along with the flow path the sediment concentrationin the water increases and it is also more difficult forstatic sediment to enter into the water Therefore the netdifference between the sediment entering the water andthe original sediment in the water which becomes staticsediment later on gradually decreases which means that thequality difference of the sediment entering into the waterand sediment cementing in the water decreases From amacropoint of view the quality of static sediment enteringthe water is reducing and closing to zero thus the increase ofsediment concentration is decreasing along the path

A greater 120572 value indicates that groundwaterrsquos flushingon sediment is stronger and more serious At the veryshort period of the beginning the quicker the water flowsthe more the sediment is washed starting from this pointTherefore the higher the flowing speed the smaller thesediment concentration within the same distance along thepath namely the bigger the 120572 value the smaller greater thesediment concentration under the same condition Insteadthe smaller the 120572 value sediment within the same distancealong the path is less likely to be carried away thus the greaterthe sediment concentration in the water

With a certain distance from the entrance the smallerthe 120572 value the weaker the flushing force so time spenton passing the same distance of both the flowing water andthe flowing sediment is the same With the period morestatic sediments enter into the water Therefore in the initialperiod within the same distance the sediment concentrationincreases more rapidly than the case of large values of 120572 Inother words in the initial period of distance there is quickincrease of the sediment concentration In contrast the largerthe120572 value is the slower the increasing degree of the sedimentconcentration is in the initial distance

The groundwater seeps along the path and brings awaythe static sediment changing it into flowing sediment whilethe pore ratio at the same position increases at the same

moment sediment along the path enters into the watergradually which makes it difficult for sedimentrsquos enteringlater on thus the pore ratio of the following distance issmall showing a gradual decrease of the pore ratio along thedistance at the same time

Along the path sediment enters into the water andbecomes flowing sediment so the sediment concentration ofthe water increases gradually

At the same position as some static sediments havebecome flowing sediments in the water the total amount ofsediments of such type is decreasing as the soil containsmanylarge and small particles small particles are most likely to betaken away under the same flow and large particles will bewashed away when all possible small ones are taken Sincethe remaining large particles could not be brought away thesediment concentration is close to the minimum and tendsto be 0 and the potential sediment concentration of thisposition also tends to 0

7 Conclusions

Flushing of underground sediment begins with small parti-cles and large particles will be taken away later which is moredifficult the underground sediment flushing contains erosionand siltation and the macroperformance between the two istheir differences When groundwater movement brings awayunderground sediment particles away pore ratio of the soilincreases gradually When all possible particles are washedaway with large particles left the flow becomes direct waterseepage ultimately and the sediment concentration at thesame position in the water decreases over time The longerthe path the more the static sediment entering the waterand this reduces the increase of sediment concentrationHigh sediment concentration in the water will weaken staticsedimentrsquos diffusion into the water so the increase of thesediment concentration reduces with a part of sedimententering into the water the remaining part becomes less sothe concentration of maximal flushing potential sedimentdecreases along the path

From the underground sediment flushing model theabove underground sediment flushing law is obtained whichreflects the flushing mechanism of underground sediment

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (nos 51179177 and 50709015) theFundamental Research Funds for the Central Universities(no 2011YXL053) the Yellow River Key Laboratory of theMinistry of Water Resources of Sediment (no 2010005) andChina Postdoctoral Science Foundation (nos 200801081 and20070410531)

References

[1] W Y Lim A Z Aris and M P Zakaria ldquoSpatial variabilityof metals in surface water and sediment in the langat riverand geochemical factors that influence their water-sediment


interactionsrdquo Scientific World Journal vol 2012 Article ID652150 14 pages 2012

[2] W Guangqian S Jichao F Xudong W Jiahua and WBaosheng ldquoAn underground sand erosion model experimentalapparatus and methodrdquo China Patent no 2009100922499 2011

[3] W Guangqian S Jichao F Xudong W Jiahua and WBaosheng ldquoUnderground sand erosion model experimentalapparatusrdquo China Patent no 2009202226149 2010

[4] W Guangqian ldquoAdvances in river sediment researchrdquo Journalof Sediment Research vol 2 no 1 pp 64ndash81 2007 (Chinese)

[5] S Jichao and W Guangqian ldquoResearch on underground waterpollution caused by geological fault through radioactive stra-tumrdquo Journal of Radioanalytical andNuclear Chemistry vol 297no 1 pp 27ndash32 2013

[6] R Sandor and N Fodor ldquoSimulation of soil temperaturedynamics with models using different conceptsrdquo ScientificWorld Journal vol 2012 Article ID 590287 8 pages 2012

[7] J Sun Q Gao H Wang and Y Li ldquoNumerical simulationof coupled rainfall and temperature of unsaturated soilsrdquo KeyEngineering Materials vol 306ndash308 pp 1433ndash1438 2006

[8] J Sun GWang andQ Sun ldquoCrack spacing of unsaturated soilsin the critical staterdquo Chinese Science Bulletin vol 54 no 12 pp2008ndash2012 2009

[9] E J Gabet and P Sternberg ldquoThe effects of vegetative ash oninfiltration capacity sediment transport and the generation ofprogressively bulked debris flowsrdquo Geomorphology vol 101 no4 pp 666ndash673 2008

[10] H Haynes E Vignaga and W M Holmes ldquoUsing magneticresonance imaging for experimental analysis of fine-sedimentinfiltration into gravel bedsrdquo Sedimentology vol 56 no 7 pp1961ndash1975 2009

[11] P R Roberge ldquoPolymericmaterials for underground piping andrelated systemsrdquo Materials Performance vol 49 no 1 p 802010

[12] D G Fredlund andH Rahardjo Soil Mechanics for UnsaturatedSoils Wiley-Interscience 1993

[13] K Terzaghi ldquoDer grundbruch an stauwerken und seine verhue-tungrdquo Die Wasserkraft vol 17 pp 445ndash449 1922

[14] Z Bazant ldquoMeasuring soil deformation caused by the pressureof the seepagerdquo in Proceedings of the 17th International Naviga-tion Congress Lisbon Portugal 1949

[15] B Indraratna and S Radampola ldquoAnalysis of critical hydraulicgradient for particlemovement in filtrationrdquo Journal of Geotech-nical andGeoenvironmental Engineering vol 128 no 4 pp 347ndash350 2002

[16] N Qian and Z WanMechanics of Sediment Transport SciencePress Beijing China 2003

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Fixed soils

Potential flowing

sediment

Flow waterFlow sediment

Flus

hing

chan

ge

Flow waterFixed sedimentFlow sediment

Fixed water

Figure 1 Schematics of particle exchange and transport of under-ground sediment

papers have not been involved in underground sediment[2 3]

Studies of permeation fluid mechanics in this area focuson the seepage of flow in saturated [5 6] and unsaturatedsoils [7 8] Accompanied by the study of sediment trans-port defining of the state point is conducted based on theexperimental and numerical computation [9ndash12] There areonly a few studies on related theories of sediment transport ingroundwater especially in terms of the quantitative transportprocess

The two processes of subsurface erosion and volumeuplift [13] have been observed in the experiment of thetwo processes with the conclusion that the irreversibledisplacement of each soil particle corresponds to a criticalhydraulic gradient [14]These results of the experiments havesome practical values but they lack necessary and sufficientproof of relevant theory Therefore they are not so rationaland applicable in practical application

Force analysis of particles in the pore channels is con-ducted in the literature [15] obtaining the critical flowvelocities formula when sediment particles start to moveHowever sediment particles are nonsticky sediment with thesame diameter which might be inconsistent with the actualsituation as actual sediment Thus its application has certainlimitations

3 Underground Sediment TransportEquations with Stable Saturated Seepage

The assumption in the paper is that the diameter of under-ground sediment particles which can be transported is muchsmaller than the diameter of the rest of the soil particles

A small unit along the path of the underground sedimenttransport is studied which is shown in Figure 2

dy

swSdtdy

dx

nSdxdy

nSdxdy +120597(nS)

120597tdtdxdy

swSdt dy +120597(swS)

120597xdxdtdy

qsdt dx dy

Figure 2 Unit body of underground sediment transport

31 The Sediment Transport Equation Within unit body119889119909119889119910 and time 119889119905 the sediment volume carried by the flow

120597 (V119904119908119878)

120597119909119889119909119889119905119889119910 (1)

Within unit body 119889119909119889119910 and time 119889119905 the sediment volumein water increases as follows

120597 (119899119878)

120597119905119889119905119889119909119889119910 (2)

Sediment in the119910direction has no concentration gradientnamely there is no sediment diffusion in this directionthe sediment volume in water supplied by flush potentialsediment is

119902119904119889119905119889119909119889119910 (3)

wherein 119902119904is the volume of sediment in the soil entering the

water per unit time per unit volumeTherefore within the unit body 119889119909119889119910 and time 119889119905 the

increased sediment equation is

120597 (119899119878)

120597119905119889119905119889119909119889119910 = minus

120597 (V119904119908119878)

120597119909119889119909119889119905119889119910 + 119902

119904119889119905119889119909119889119910 (4)

The following equation is obtained after simplification

120597 (119899119878)

120597119905= minus120597 (V119904119908119878)

120597119909+ 119902119904 (5)

32 Relations between the Flush Potential Sediment andWaterSediment The sediment in water supplied by flush potentialsediment is shown in Figure 2 and the volume is

119902119904119889119905119889119909119889119910 (6)

wherein when 119902119904is positive it indicates the sediment volume

in soil entering in the water per unit time per unit volumewhen it is negative it indicates the speed of the sedimentin cementing to the soil which can be called the solventingspeed

The sediment will start to flow into the seepage water Asthey are saturated underground sediments the flow rate isconstant But at the same time a part of the fixed sedimentflows into the seepagewater with the end result that exchangehappens between this part of sediment and flow water of thesame volume namely the fixed sediment changes into flow


x

Diss

olut

ion

Prec

ipita

tes

qs

S lt Smaxn lt nmax

S gt Smaxn gt nmax





1205971198991

120597119905= 119902119904 (7)



1max Consider

1198991max = 1198991 (119905 997888rarr +infin) (8)









119902119904= 119870(119878max minus 119878)

120572

(9)









119878max = 119870119904V3

119904119908 (10)


119878max = 119870119904V119904119908 (11)





1198991max = 1198701198991V119904119908 (12)



1205971198991

120597119905= 119870(119870

119904V119904119908minus 119878)120572

(13)



119904119908remains


119899120597119878

120597119905= minusV119904119908

120597119878

120597119909+ 119870(119870

119904V119904119908minus 119878)120572

(14)


V119904119908(119899 + 1)120597119878

120597119909= 119870(119870

119904V119904119908minus 119878)120572

(15)



V119904119908(119899 + 1)119889119878

119889119909= 119870(119870

119904V119904119908minus 119878)120572

(16)


4 Equation Solving


when 120572 = 1

119878 = minus(V119904119908(119899 + 1)

119870 (120572 minus 1) (119909 + 1198621))

1(120572minus1)

+ 119870119904V119904119908

(17)

when 120572 = 1

119878 = 119870119904V119904119908+ 119890minus(119870119909V

119904119908(119899+1))

1198621 (18)





1max is

1198991max = 1198991 (119905 997888rarr + prop)

= int

+prop

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(19)


119899119905= 1198990+ 1198991(119905)

= 1198990+ int

119905

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(20)


119899119905(119894+1)= 119899119905(119894)+ 1198991119905(119894+1)

= 119899119905(119894)+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908minus 119878)120572

119889119905

(21)


is 119870119904V119904119908

When 120572 = 1

1198621 =(119899 + 1)119870

119904V2119904119908

119870119890120572 ln(119870119904V119904119908) (120572 minus 1) (22)


1198621 =(119899 + 1)119870

119904V2119904119908

119870(119870119904V119904119908)120572

(120572 minus 1) (23)

When 120572 = 1

1198621 = minus119870119904V119904119908 (24)



3






119878max =119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) (25)




119904119908= 005ms 119870 = 1 119899 = 02 119909 = 0 rarr 1




119904V119904119908




10

9

8

7

6

5

4

3

2

1

00 01 02 03 04 05 06 07 08 09 1

x distance (m)

yse

dim

ent c

once

ntra

tion

()

120572 = 10

120572 = 15

120572 = 20

120572 = 25120572 = 30



119899119905(119894+1)= 119899119905(119894)

+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

119889119905

= 119899119905(119894)+ [119905(119894+1)minus 119905(119894)]

times 119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

(26)









065

06

055

05

045

04

035

03

025

020 01 02 03 04 05 06 07 08 09 1

x distance (m)

ypo

re ra

tio

t = 200i (i = 1 2 3 10) unit s


0 01 02 03 04 05 06 07 08 09 1x distance (m)

10

9

8

7

6

5

4

3

2

1

0

yse

dim

ent c

once

ntra

tion

()

t = 200i (i = 1 2 3 10) unit s

SSmax



















7 Conclusions



Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










x

Diss

olut

ion

Prec

ipita

tes

qs

S lt Smaxn lt nmax

S gt Smaxn gt nmax





1205971198991

120597119905= 119902119904 (7)



1max Consider

1198991max = 1198991 (119905 997888rarr +infin) (8)









119902119904= 119870(119878max minus 119878)

120572

(9)









119878max = 119870119904V3

119904119908 (10)


119878max = 119870119904V119904119908 (11)





1198991max = 1198701198991V119904119908 (12)



1205971198991

120597119905= 119870(119870

119904V119904119908minus 119878)120572

(13)



119904119908remains


119899120597119878

120597119905= minusV119904119908

120597119878

120597119909+ 119870(119870

119904V119904119908minus 119878)120572

(14)


V119904119908(119899 + 1)120597119878

120597119909= 119870(119870

119904V119904119908minus 119878)120572

(15)



V119904119908(119899 + 1)119889119878

119889119909= 119870(119870

119904V119904119908minus 119878)120572

(16)


4 Equation Solving


when 120572 = 1

119878 = minus(V119904119908(119899 + 1)

119870 (120572 minus 1) (119909 + 1198621))

1(120572minus1)

+ 119870119904V119904119908

(17)

when 120572 = 1

119878 = 119870119904V119904119908+ 119890minus(119870119909V

119904119908(119899+1))

1198621 (18)





1max is

1198991max = 1198991 (119905 997888rarr + prop)

= int

+prop

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(19)


119899119905= 1198990+ 1198991(119905)

= 1198990+ int

119905

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(20)


119899119905(119894+1)= 119899119905(119894)+ 1198991119905(119894+1)

= 119899119905(119894)+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908minus 119878)120572

119889119905

(21)


is 119870119904V119904119908

When 120572 = 1

1198621 =(119899 + 1)119870

119904V2119904119908

119870119890120572 ln(119870119904V119904119908) (120572 minus 1) (22)


1198621 =(119899 + 1)119870

119904V2119904119908

119870(119870119904V119904119908)120572

(120572 minus 1) (23)

When 120572 = 1

1198621 = minus119870119904V119904119908 (24)



3






119878max =119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) (25)




119904119908= 005ms 119870 = 1 119899 = 02 119909 = 0 rarr 1




119904V119904119908




10

9

8

7

6

5

4

3

2

1

00 01 02 03 04 05 06 07 08 09 1

x distance (m)

yse

dim

ent c

once

ntra

tion

()

120572 = 10

120572 = 15

120572 = 20

120572 = 25120572 = 30



119899119905(119894+1)= 119899119905(119894)

+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

119889119905

= 119899119905(119894)+ [119905(119894+1)minus 119905(119894)]

times 119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

(26)









065

06

055

05

045

04

035

03

025

020 01 02 03 04 05 06 07 08 09 1

x distance (m)

ypo

re ra

tio

t = 200i (i = 1 2 3 10) unit s


0 01 02 03 04 05 06 07 08 09 1x distance (m)

10

9

8

7

6

5

4

3

2

1

0

yse

dim

ent c

once

ntra

tion

()

t = 200i (i = 1 2 3 10) unit s

SSmax



















7 Conclusions



Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











V119904119908(119899 + 1)119889119878

119889119909= 119870(119870

119904V119904119908minus 119878)120572

(16)


4 Equation Solving


when 120572 = 1

119878 = minus(V119904119908(119899 + 1)

119870 (120572 minus 1) (119909 + 1198621))

1(120572minus1)

+ 119870119904V119904119908

(17)

when 120572 = 1

119878 = 119870119904V119904119908+ 119890minus(119870119909V

119904119908(119899+1))

1198621 (18)





1max is

1198991max = 1198991 (119905 997888rarr + prop)

= int

+prop

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(19)


119899119905= 1198990+ 1198991(119905)

= 1198990+ int

119905

0

119870(119870119904V119904119908minus 119878)120572

119889119905

(20)


119899119905(119894+1)= 119899119905(119894)+ 1198991119905(119894+1)

= 119899119905(119894)+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908minus 119878)120572

119889119905

(21)


is 119870119904V119904119908

When 120572 = 1

1198621 =(119899 + 1)119870

119904V2119904119908

119870119890120572 ln(119870119904V119904119908) (120572 minus 1) (22)


1198621 =(119899 + 1)119870

119904V2119904119908

119870(119870119904V119904119908)120572

(120572 minus 1) (23)

When 120572 = 1

1198621 = minus119870119904V119904119908 (24)



3






119878max =119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) (25)




119904119908= 005ms 119870 = 1 119899 = 02 119909 = 0 rarr 1




119904V119904119908




10

9

8

7

6

5

4

3

2

1

00 01 02 03 04 05 06 07 08 09 1

x distance (m)

yse

dim

ent c

once

ntra

tion

()

120572 = 10

120572 = 15

120572 = 20

120572 = 25120572 = 30



119899119905(119894+1)= 119899119905(119894)

+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

119889119905

= 119899119905(119894)+ [119905(119894+1)minus 119905(119894)]

times 119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

(26)









065

06

055

05

045

04

035

03

025

020 01 02 03 04 05 06 07 08 09 1

x distance (m)

ypo

re ra

tio

t = 200i (i = 1 2 3 10) unit s


0 01 02 03 04 05 06 07 08 09 1x distance (m)

10

9

8

7

6

5

4

3

2

1

0

yse

dim

ent c

once

ntra

tion

()

t = 200i (i = 1 2 3 10) unit s

SSmax



















7 Conclusions



Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










10

9

8

7

6

5

4

3

2

1

00 01 02 03 04 05 06 07 08 09 1

x distance (m)

yse

dim

ent c

once

ntra

tion

()

120572 = 10

120572 = 15

120572 = 20

120572 = 25120572 = 30



119899119905(119894+1)= 119899119905(119894)

+ int

119905(119894+1)

119905(119894)

119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

119889119905

= 119899119905(119894)+ [119905(119894+1)minus 119905(119894)]

times 119870(119870119904V119904119908

1198991max minus 1198990

(1198991max minus 119899119908) minus 119878)

120572

(26)









065

06

055

05

045

04

035

03

025

020 01 02 03 04 05 06 07 08 09 1

x distance (m)

ypo

re ra

tio

t = 200i (i = 1 2 3 10) unit s


0 01 02 03 04 05 06 07 08 09 1x distance (m)

10

9

8

7

6

5

4

3

2

1

0

yse

dim

ent c

once

ntra

tion

()

t = 200i (i = 1 2 3 10) unit s

SSmax



















7 Conclusions



Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation






















7 Conclusions



Acknowledgments


References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Transport Model of Underground Sediment