Research ArticleAnalytical Solution and Application for One-DimensionalConsolidation of Tailings Dam
Hai-ming Liu 12 Gan Nan1 Wei Guo1 Chun-he Yang2 and Chao Zhang2
1Faculty of Civil Engineering and Mechanics Kunming University of Science and Technology Kunming Yunnan 650224 China2State Key Laboratory of Geomechanics and Geotechnical Engineering Institute of Rock and Soil MechanicsChinese Academy of Sciences Wuhan 430071 China
Correspondence should be addressed to Hai-ming Liu haiming0871163com
Received 10 November 2017 Revised 16 January 2018 Accepted 12 February 2018 Published 19 March 2018
Academic Editor Shuo Wang
Copyright copy 2018 Hai-ming Liu 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 pore water pressure of tailings dam has a very great influence on the stability of tailings dam Based on the assumption ofone-dimensional consolidation and small strain the partial differential equation of pore water pressure is deduced The obtaineddifferential equation can be simplified based on the parameters which are constants According to the characteristics of the tailingsdam the pore water pressure of the tailings dam can be divided into the slope dam segment dry beach segment and artificial lakesegment The pore water pressure is obtained through solving the partial differential equation by separation variable method Onthis basis the dissipation and accumulation of pore water pressure of the upstream tailings dam are analyzedThe example of typicaltailings is introduced to elaborate the applicability of the analytic solution What is more the application of pore water pressure intailings dam is discussedThe research results have important scientific and engineering application value for the stability of tailingsdam
1 Introduction
Based on the assumption that the soil is isotropic and uni-form an external surface load is instantaneously applied andis held constant a classical one-dimensional (1D) consol-idation theory was proposed by Terzaghi [1] In order toanalyze time-dependent loading an analytical solution to thelayered consolidation problem for a general set of boundaryconditions and an arbitrary load history was presented bySchiffman and Stein [2] The 1D consolidation analyticalsolutions considering ramp loading were presented by Olson[3] A simple semianalytical method to solve the problem of1D consolidation by taking into account the varied compress-ibility of soil under cyclic loadings was brought up by Caiet al [4] A rigorous solution of the conventional Terzaghi1D consolidation under haversine cyclic loading with anyrest period was proposed by Muthing et al [5] which isachieved using Fourier harmonic analysis for the periodicfunction representing the rate of imposition of excess porewater pressure A semianalytical solution to 1D consolidationof viscoelastic unsaturated soils with a finite thickness under
oedometric conditions and subjected to a sudden loadingwas put forward by Qin et al [6] A semianalytical solutionto 1D consolidation equation of fractional derivative Kelvin-Voigt viscoelastic saturated soils subjected to different time-dependent loadings was presented by Wang et al [7] Underthe condition of the increasing weight of superincumbentmaterial and the length of the drainage path varies a solutionfor the 1D consolidation of a clay layer whose thicknessincreases with time was proposed by Gibson [8] An exactanalytical solution of the nonhomogeneous partial differen-tial equation governing the conventional 1D consolidationunder haversine repeated loading was derived and discussedby Razouki et al [9 10]
In order to analyze different boundary conditions sin-gle drainage solutions for several specific variations of thepermeability and shear modulus were given by Mahmoudand Deresiewicz [11] Several analytical solutions for the con-solidation analysis of a soil layer with fairly general lawsof variation of permeability and compressibility for botha single-drained condition and a double-drained conditionwere proposed by Zhu and Yin [12] Tang et al [13] propose
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 4573780 9 pageshttpsdoiorg10115520184573780
2 Mathematical Problems in Engineering
a closed-form solution for consolidation of three-layeredsoil with a vertical drain system and a method to solve aconvergence solution by enlarging the precision for the entirethickness of the foundation and adding controlling precisionfor the overall average degree of consolidation of each soillayerThe solution of Terzaghi 1D consolidation equationwithgeneral boundary conditions is proposed by Mei and Chen[14] whose solution is validated by comparing it to the clas-sical solution Hawlader et al [15] develop a new constitutivemodel for the compressibility behavior of soft clay sedimentsat low effective stress level and themodel is used to solve finitestrain 1D consolidation with pertinent initial and boundaryconditions A solution to the consolidation equation withboundary conditions that are cyclic with time is given byRahalt and Vuez [16] A nonlinear theory of consolidationhas been developed for an ideal normally consolidated soil byDavis and Raymond [17] A simple calculation procedure toanalyze the one-dimensional response of saturated soil layersto pore pressure variations at the boundary described by ageneral time-dependent function is developed by Conte andTroncone [18]
The tailings dam is an important geotechnical structure inmining engineering For a long time the theory of reservoirdam is applied to tailings dam without any modificationHowever there are many differences between tailings damand reservoir dam which lead to inaccurate calculationresults of the pore water pressure According to the classicalTerzaghi consolidation theory the analytical solution of thepore water pressure is discussed in this paper
2 The General Equation of 1D Consolidation
According to Darcyrsquos law the flow drag resistance of the 119911-direction is as follows
It is assumed that the water in porosity is incompressibleunder the consolidation process Thus 120574119908 is a constantDuring the process of soil consolidation because the porosityof soil particles is continuously compressed the permeabilitycoefficient of soil is decreased continuously and the perme-ability coefficient 119896 of soil is changed with depth that is119896 = 119891(119911) Taking the derivative of (3) one can have
According to fractional derivative rule (4) can be ex-panded as follows
12059721199011199081205971199112 + 119899120574119908 1119896120597V119911120597119911 minus 119899120574119908V119911 11198962
119889119896119889119911 = 0 (5)
Assume that the horizontal direction consolidation ofthe soil can be neglected that is the consolidation is underconfined compression condition According to the 1D con-solidation condition (6) can be deduced
Under the condition of 1D consolidation the axial strain120576V is equal to the volume strain 120576V of the soil According toPane and Schiffman [19] research (4) and (5) can be rewrittenas follows
12059721199011199081205971199112 + 120574119908119896120597120576119911120597119905 minus 119899120574119908V119911 11198962
119889119896119889119911 = 0 (8)
In the consolidation problem it is of great significanceto study the change of pore water pressure with time andposition under external load Therefore (8) can be expandedaccording to pore water pressure
Assuming that the soil profile is shown as Figure 1 thepore water pressure of a point located 119911 under load 1205900 andΔ1205900 is 119906 According to the Terzaghi effective stress principlethe effective stress 1205901015840 could be given as follows underconsolidation
1205901015840 = 120590 minus 119901119908 = 120590 minus [120574119908 (ℎ + 119911) + 119906] (9)
According to Figure 1 the total stress 120590 of a point located119911 is as follows120590 = (1205900 + Δ120590) + 120574119904119911 + 120574119908ℎ (10)
Mathematical Problems in Engineering 3
Initial dam
Tailings fine sand
Slope dam segment Dry beach segment Artificial lake segment
Tailings mud
Tailings silty sand amp tailings mealy sand
Initial topography
Tailings fill dam
1 m
Figure 2 Schematic diagram of the upstream tailings dam
Substituting (10) into (9) yields
1205901015840 = (1205900 + Δ120590) + 1205741015840119911 minus 119906 (11)
Take the derivative of (11) with respect to 119905 and considerthe relationship 120597119911120597119905 equals 120597119867120597119905
1205971205901015840120597119905 = 120597
120597119905 (Δ120590) + 1205741015840120597119867120597119905 minus 120597119906
120597119905 (12)
According to the compression curve of 1D consolidationtest the volume compressibility factor 119898V can be defined asfollows
Substituting (15) into (17) (17) can be expressed as porewater pressure Equation (18) can be obtained
12059721199061205971199112 +
119898V120574119908119896 [120597Δ120590120597119905 + 1205741015840 120597119867120597119905 minus 120597119906120597119905 ] +
1119896119889119896119889119911
120597119906120597119911 = 0 (18)
Equation (18) is a general equation which reflects the1D consolidation process It takes into account the situationwhere the applied load changes with time the soil thicknesschanges with time the soil permeability changes with depthand so on
3 Accumulation and Dissipation of PoreWater Pressure in Tailings Dam
Tailings areminewastes produced in themining engineeringwhich are sent to tailings reservoir by pipe or flume Thetailings dam is an important part in mining engineeringwhich is consisted by the initial dam and fill dam The initialdam is made by permeable rockfill generally and the fill damis formed by tailings In general the construction of a tailingdam takes many decades or even a century According toconstruction method the tailings dam can be divided intoupstream tailings dam downstream tailings dam midlinemethod tailings dam and so onThenumber of tailings pondsin China has reached more than 12000 by statistics Becausethe downstream of the tailings dam is residents living areaor mining production area generally the social and peoplersquosproperty damage a huge impact if the tailings dam is failed
Because the upstream tailings dam has advantages ofsimple operation low construction costs the less need forcoarse particles and so on according to statistics 95 of thetailings dams are adopting the construction of the upstreamtailings dam in China On the other hand the constructionprocess of the upstream tailings dam cannot precisely controlthe shape of the tailings dam The deposition structure ofthe tailings dam is very complex The upstream tailings haveshortcoming of long infiltration distant and poor stabilityIn view of the tailings slurry discharge and tailings particledeposition following the sediment mechanics the sectionprofile of the tailings dam has obvious regularity The coarserthe particle size is the shorter the average distance migratesand vice versa Generally the deposition order of the tailingsdam along the dry beach face is tailings fine sand tailings silttailings sand and tailings mud
If particle size distribution of the tailings and the lengthof the dry beach face are kept constant the interface of thetailingsmaterial should be substantially parallel to the surfaceof the dam slope which is shown in Figure 2 For most ofthe tailings dams because the tailings water contains moreheavy metals the initial topography and the side of tailingsdam are treated as an impermeable boundary On the otherhand the tailings dam extends very long in the direction ofthe reservoir areaTherefore the consolidation of the tailingsdam is a 1D problem
As can be seen from Figure 2 the closer to inside thereservoir area is the thicker the tailings mud is and the worse
4 Mathematical Problems in Engineering
the average consolidation is In other words the closer toinside the reservoir area is the greater pore water pressureof the tailings dam is and the lower shear strength of thetailings material is Generally speaking the construction ofa tailings dam needs several decades to reach the designelevation With the continuous production of mining theheight of the tailings dam is increasing During this processthe accumulation and dissipation of the excess pore waterpressure of the tailings material experience several differentstages of development
31 Tailings Thickness Increases with Time In this stage thethickness of the tailings increases with time The tailingsmaterial produces great compressive deformation under self-weight effect The pore water pressure produced by the self-weight pressure at the early stage is partially dissipatedNevertheless the pore water pressure caused by self-weightpressure is increased with the tailings thickness at the laterstage which leads to the increase of the pore water pressurewith time Therefore the net pore water pressure is accumu-lated during this process
32 Load on the Tailings Layer Increases with Time The self-deposition of the tailings mud continues to extend to interiorof the tailings reservoir area in this stage The self-depositionof the tailings layer has ended where the end of the tailingsreservoir is near It overlapped with particles coarse tailingsand tailings mud The tailings mud has to bear a growingload with time The tailings mud continues to consolidateunder the combined action of self-weight and additionalload At this stage the pore pressure produced by the self-weight pressure dissipates a little part and the more porewater pressure produced by additional load accumulatesThenet pore water pressure tends to accumulate in this proc-ess
33TheAdditional Load of the TailingsMaterial Remains Con-stant For a specific part of the tailings dam the upperboundary of the tailings dam has reached the designedelevation at a certain moment Therefore the additional loadwill remain unchanged from this moment During this stagethe tailings mud will continue to consolidate under thecombined action of self-weight and constant additional loadThus the pore water pressure will dissipate within the tailingsdam Obviously the different parts of tailings mud havedifferent consolidation stages There is no additional loadat the thickest tailings mud in the tailings reservoir until itis closed Therefore the tailings mud is always in the firststage and will never enter the second and third stages Fromthe above qualitative analysis the pore water pressure whichis located at the slope dam segment reaches the maximumvalue when the additional load just stops growing The porewater pressure which is located at dry beach segment andartificial lake segment achieves the maximum value whenthe tailings dam reaches themaximumheightThe numericalvalues of the pore water pressure at different locations andmoments can be obtained by solving the partial differentialequations of consolidation problems
4 Analytical Solutions ofthe Pore Water Pressure
It is necessary to accurately calculate the accumulation anddissipation of the pore water pressure in tailings dam ofwhich a solution of 1D consolidation problem can be sim-plified During the consolidation process the soil parametersof tailings material (such as the change of bulk densitypermeability coefficient and consolidation coefficient) arechanged with the consolidation process The change law canbe determined through many experiments Under normalcircumstances it is difficult to obtain the analytical solutionwhen the change law of soil parameters is considered There-fore the exact solution of the problem can only be dependedon numerical calculation method
As the horizontal length of the tailings reservoir is usuallyfar greater than the vertical thickness (ie the horizontallength is 10 times more than the vertical thickness) thedrainage consolidation effect of the horizontal direction canbe ignored Therefore the vertical direction of drainage isneeded to consider Then it is only a 1D consolidation prob-lem Based on the assumption of small-strain and constantof soil parameters such as 119898] 119896 120597Δ120590120597119905 1205741015840 and 120597119867120597119905are constant (18) is degenerated into a constant coefficientnonhomogeneous parabolic partial differential equation
12059721199061205971199112 +
1119888] (120572 + 120574
1015840119876 minus 120597119906120597119905 ) = 0 (19)
where
120572 = 120597Δ120590120597119905
119876 = 120597119867120597119905
119888] = 119896119898]120574119908
(20)
The above three different stages are specifically discussedas follows
41 The Thickness Mud of Tailings Mud Increases with TimeThe overburden load of the tailings mud at this stage is zerothat is 120572 = 0 The consolidation equation yields
12059721199061205971199112 +
1119888] (1205741015840119876 minus 120597119906
120597119905 ) = 0 (21)
If the bottom of the tailings mud is impermeable thecoordinates origin is taken as the impermeable bottom Theboundary conditions of the problem can be given as follows
119911 = 0120597119906120597119911 = 0 (22)
119911 = 119867119906 = 0 (23)
Mathematical Problems in Engineering 5
The initial condition of the problem is given by
119905 = 0119906 = 0 (24)
According to Gibsonrsquos [20] study (25) is chosen as thesolution of (21)
It can be verified that (25) satisfies the boundary condi-tion (22) The choice of function 119892(120585) needs to satisfy (23)Substituting (25) into (23) gives
For any function ℎ(119905) the partial differential equation(26) has no numerical solution With regard to ℎ(119905) = 119898119905the following parameters can be introduced to transform theequation
It can be verified that (31) satisfies the initial condition of(24)
42 The Additional Load of Tailings Mud Increases with TimeDuring this stage the thickness of tailings mud not onlydoes not increase with time but also gradually decreases withthe increase of effective stress For the sake of simplicityassuming that the thickness of tailingsmud119867 is constant thebasic consolidation differential (19) can be simplified as
119888] 12059721199061205971199112 =
120597119906120597119905 minus
120597Δ120590120597119905 (32)
The initial condition of the problem is given by
119905 = 1199051119906 = 1199060 (119911) (33)
The boundary conditions of the problem can be given as
119911 = 0119906 = 0119911 = 119867
120597119906120597119911 = 0
(34)
The following three questions are called problem 119860problem 119861 and problem 119862 respectively119860
If the solution of problem119860 problem 119861 and problem119862 is119906119860 119906119861 and 119906119862 respectively it can be directly proved throughsubstitution method
119906119862 = 119906119860 + 119906119861 (38)
The solution of 119906119860 belongs to the Terzaghi classical1D consolidation problem The analytical solution can beobtained
119906119860 =infinsum119895=1
119860119899 sin 1198951205871199112119867 exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (39)
where
119860119899 = 1119867 int21198670
1199060 (119911) sin 1198951205871199112119867 119889119911 (40)
1199060(119911) is the solution of (31) Substituting (31) into (40) byMatlab software119860119899 can be obtained Substituting it into (39)the numerical solution 119906119860 can be given
For solution to the problem 119906119861 Schiffman [21] acquiresthe following series solutions
119906119861 = 1612057211986721205873119888]
infinsum135
11198953 sin
1198951205871199112119867
sdot 1 minus exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (41)
When the tailings mud is below the groundwater levelthe total stress 120590(119911) of the tailings mud at the depth 119911 can beexpressed as follows
120590 (119911) = 1205741015840119911 + 120597Δ120590120597119905 (119905 minus 1199051) (42)
The ratio of porewater pressure to total stress can be easilydetermined as follows
43 The Additional Load on the Tailings Mud Remains Con-stant In this stage the basic differential equation and thedefinite condition are problem119863119863
If the solution of problem 119860 is 119906119860 and the solution ofproblem 119864 is 119906119864 the solution of problem 119863 can be easilydemonstrated is 119906119863 as follows
119906119863 = 119906119860 + 119906119864 (46)
The definite condition of problem 119864 and problem 119860 isexactly the same in form except that the function 1199061(119911) isdifferent from the function 1199060(119911)
Suppose that
119906119864 =infinsum119895=1
119861119899 sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (47)
Then
119861119899 = 1119867 int21198670
1199061 (119911) sin 1198951205871199112119867 119889119911 (48)
where
1199061 (119911) = 1612057211986721205873119888V
sdot infinsum123
11198953 sin
1198951205871199112119867 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
(49)
Mathematical Problems in Engineering 7
Gravel
Tailings fine sand
Tailings sandTailings mud
1030
16 105 33 70 300
Impervious boundary
Slope dam segment Dry beach segment Artificial lake segment
Figure 3 Simplified section for calculation of pore water pressure in a tailings dam
150 100200 250 300 350
50
Figure 4 Distribution of pore water pressure at the moment when the dam rises to its maximum height
According to the orthogonal rule of the solution thefollowing relationship can be obtained
1119867 int21198670
sin 1198981205871199112119867 sin1198951205871199112119867 119889119911 =
0 119898 = 1198951 119898 = 119895 (50)
Equation (48) can be simplified through (50)
119861119899
= 1612057211986721205873119888V
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ] 119895 = 1 3 5
0 119895 = 2 4 6(51)
Substituting (51) into (47) yields
119906119864 = 1612057211986721205873119888V
infinsum135
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
sdot sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (52)
For the third stage the analytical solution of pore waterpressure is obtained
Similar to (42) the ratio of pore water pressure to totalstress at a certain depth of tailings mud at this stage is asfollows
During this stage 120590(119911) does not change with time and itcan take the following value
120590 (119911) = 1205741015840119911 + 120572 (1199052 minus 1199051) (54)
5 Example
The simplified calculation section of the representative sec-tion of a tailings dam is shown in Figure 3 The length ofthe tailings dam is 368m Therefore the problem can besimplified as 1D consolidationThe slime layer below the levelis assumed for impervious boundary The rising speed of the
dam height is 175my The saturation density of gravel is220 kNm3 the saturation density of the tailings fine sandis 196 kNm3 the saturation density of the tailings sand is191 kNm3 and the saturation density of the tailings mudis 188 kNm3 The permeability coefficient of gravel is 3 times10minus3ms the permeability coefficient of the tailings fine sandis 5times 10minus6ms the permeability coefficient of the tailings sandis 2 times 10minus6ms and the permeability coefficient of the tailingsmud is 15 times 10minus8ms The consolidation coefficient of gravelis 38 times 10minus1 cm2s the consolidation coefficient of the tailingsfine sand is 54 times 10minus3 cm2s the consolidation coefficient ofthe tailings sand is 28 times 10minus3 cm2s and the consolidationcoefficient of the tailings mud is 12 times 10minus5 cm2s Pleasedetermine the distribution of pore water pressure when thetailings dam reaches maximum height
The above problem can be solved through (31) (46)and (52) The calculation can be accomplished through theMATLAB software Firstly a coordinate system can be setup Then the corresponding coordinates are taken out ofthe interval 1m The region is determined according to thecoordinates Then the pore water pressure can be calculatedthrough the corresponding equation Finally the contourmap can be drawn through a set of array The results of thecalculation are shown in Figure 4
6 Discussion
Due to the similarity between the tailings dam and thereservoir dam in the geotechnical structure many scholarsdirectly introduce the calculation methods of the reservoirdam whose theory is relatively mature into the tailings damConsidering the difference on construction cycle construc-tion materials between the tailings dam and reservoir dam itmakes the calculation results inconsistent with the actual
Based on the Terzaghi consolidation theory of 1D thetailings dam is divided into the slope dam segment the drybeach segment and the artificial lake segment The solutionsof the pore water pressure are derived respectively Theanalysis shows that the additional load of the slope damsegment is unchanged which can be calculated using (46)The analysis indicates that tailings thickness of the artificial
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
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2 Mathematical Problems in Engineering
a closed-form solution for consolidation of three-layeredsoil with a vertical drain system and a method to solve aconvergence solution by enlarging the precision for the entirethickness of the foundation and adding controlling precisionfor the overall average degree of consolidation of each soillayerThe solution of Terzaghi 1D consolidation equationwithgeneral boundary conditions is proposed by Mei and Chen[14] whose solution is validated by comparing it to the clas-sical solution Hawlader et al [15] develop a new constitutivemodel for the compressibility behavior of soft clay sedimentsat low effective stress level and themodel is used to solve finitestrain 1D consolidation with pertinent initial and boundaryconditions A solution to the consolidation equation withboundary conditions that are cyclic with time is given byRahalt and Vuez [16] A nonlinear theory of consolidationhas been developed for an ideal normally consolidated soil byDavis and Raymond [17] A simple calculation procedure toanalyze the one-dimensional response of saturated soil layersto pore pressure variations at the boundary described by ageneral time-dependent function is developed by Conte andTroncone [18]
The tailings dam is an important geotechnical structure inmining engineering For a long time the theory of reservoirdam is applied to tailings dam without any modificationHowever there are many differences between tailings damand reservoir dam which lead to inaccurate calculationresults of the pore water pressure According to the classicalTerzaghi consolidation theory the analytical solution of thepore water pressure is discussed in this paper
2 The General Equation of 1D Consolidation
According to Darcyrsquos law the flow drag resistance of the 119911-direction is as follows
It is assumed that the water in porosity is incompressibleunder the consolidation process Thus 120574119908 is a constantDuring the process of soil consolidation because the porosityof soil particles is continuously compressed the permeabilitycoefficient of soil is decreased continuously and the perme-ability coefficient 119896 of soil is changed with depth that is119896 = 119891(119911) Taking the derivative of (3) one can have
According to fractional derivative rule (4) can be ex-panded as follows
12059721199011199081205971199112 + 119899120574119908 1119896120597V119911120597119911 minus 119899120574119908V119911 11198962
119889119896119889119911 = 0 (5)
Assume that the horizontal direction consolidation ofthe soil can be neglected that is the consolidation is underconfined compression condition According to the 1D con-solidation condition (6) can be deduced
Under the condition of 1D consolidation the axial strain120576V is equal to the volume strain 120576V of the soil According toPane and Schiffman [19] research (4) and (5) can be rewrittenas follows
12059721199011199081205971199112 + 120574119908119896120597120576119911120597119905 minus 119899120574119908V119911 11198962
119889119896119889119911 = 0 (8)
In the consolidation problem it is of great significanceto study the change of pore water pressure with time andposition under external load Therefore (8) can be expandedaccording to pore water pressure
Assuming that the soil profile is shown as Figure 1 thepore water pressure of a point located 119911 under load 1205900 andΔ1205900 is 119906 According to the Terzaghi effective stress principlethe effective stress 1205901015840 could be given as follows underconsolidation
1205901015840 = 120590 minus 119901119908 = 120590 minus [120574119908 (ℎ + 119911) + 119906] (9)
According to Figure 1 the total stress 120590 of a point located119911 is as follows120590 = (1205900 + Δ120590) + 120574119904119911 + 120574119908ℎ (10)
Mathematical Problems in Engineering 3
Initial dam
Tailings fine sand
Slope dam segment Dry beach segment Artificial lake segment
Tailings mud
Tailings silty sand amp tailings mealy sand
Initial topography
Tailings fill dam
1 m
Figure 2 Schematic diagram of the upstream tailings dam
Substituting (10) into (9) yields
1205901015840 = (1205900 + Δ120590) + 1205741015840119911 minus 119906 (11)
Take the derivative of (11) with respect to 119905 and considerthe relationship 120597119911120597119905 equals 120597119867120597119905
1205971205901015840120597119905 = 120597
120597119905 (Δ120590) + 1205741015840120597119867120597119905 minus 120597119906
120597119905 (12)
According to the compression curve of 1D consolidationtest the volume compressibility factor 119898V can be defined asfollows
Substituting (15) into (17) (17) can be expressed as porewater pressure Equation (18) can be obtained
12059721199061205971199112 +
119898V120574119908119896 [120597Δ120590120597119905 + 1205741015840 120597119867120597119905 minus 120597119906120597119905 ] +
1119896119889119896119889119911
120597119906120597119911 = 0 (18)
Equation (18) is a general equation which reflects the1D consolidation process It takes into account the situationwhere the applied load changes with time the soil thicknesschanges with time the soil permeability changes with depthand so on
3 Accumulation and Dissipation of PoreWater Pressure in Tailings Dam
Tailings areminewastes produced in themining engineeringwhich are sent to tailings reservoir by pipe or flume Thetailings dam is an important part in mining engineeringwhich is consisted by the initial dam and fill dam The initialdam is made by permeable rockfill generally and the fill damis formed by tailings In general the construction of a tailingdam takes many decades or even a century According toconstruction method the tailings dam can be divided intoupstream tailings dam downstream tailings dam midlinemethod tailings dam and so onThenumber of tailings pondsin China has reached more than 12000 by statistics Becausethe downstream of the tailings dam is residents living areaor mining production area generally the social and peoplersquosproperty damage a huge impact if the tailings dam is failed
Because the upstream tailings dam has advantages ofsimple operation low construction costs the less need forcoarse particles and so on according to statistics 95 of thetailings dams are adopting the construction of the upstreamtailings dam in China On the other hand the constructionprocess of the upstream tailings dam cannot precisely controlthe shape of the tailings dam The deposition structure ofthe tailings dam is very complex The upstream tailings haveshortcoming of long infiltration distant and poor stabilityIn view of the tailings slurry discharge and tailings particledeposition following the sediment mechanics the sectionprofile of the tailings dam has obvious regularity The coarserthe particle size is the shorter the average distance migratesand vice versa Generally the deposition order of the tailingsdam along the dry beach face is tailings fine sand tailings silttailings sand and tailings mud
If particle size distribution of the tailings and the lengthof the dry beach face are kept constant the interface of thetailingsmaterial should be substantially parallel to the surfaceof the dam slope which is shown in Figure 2 For most ofthe tailings dams because the tailings water contains moreheavy metals the initial topography and the side of tailingsdam are treated as an impermeable boundary On the otherhand the tailings dam extends very long in the direction ofthe reservoir areaTherefore the consolidation of the tailingsdam is a 1D problem
As can be seen from Figure 2 the closer to inside thereservoir area is the thicker the tailings mud is and the worse
4 Mathematical Problems in Engineering
the average consolidation is In other words the closer toinside the reservoir area is the greater pore water pressureof the tailings dam is and the lower shear strength of thetailings material is Generally speaking the construction ofa tailings dam needs several decades to reach the designelevation With the continuous production of mining theheight of the tailings dam is increasing During this processthe accumulation and dissipation of the excess pore waterpressure of the tailings material experience several differentstages of development
31 Tailings Thickness Increases with Time In this stage thethickness of the tailings increases with time The tailingsmaterial produces great compressive deformation under self-weight effect The pore water pressure produced by the self-weight pressure at the early stage is partially dissipatedNevertheless the pore water pressure caused by self-weightpressure is increased with the tailings thickness at the laterstage which leads to the increase of the pore water pressurewith time Therefore the net pore water pressure is accumu-lated during this process
32 Load on the Tailings Layer Increases with Time The self-deposition of the tailings mud continues to extend to interiorof the tailings reservoir area in this stage The self-depositionof the tailings layer has ended where the end of the tailingsreservoir is near It overlapped with particles coarse tailingsand tailings mud The tailings mud has to bear a growingload with time The tailings mud continues to consolidateunder the combined action of self-weight and additionalload At this stage the pore pressure produced by the self-weight pressure dissipates a little part and the more porewater pressure produced by additional load accumulatesThenet pore water pressure tends to accumulate in this proc-ess
33TheAdditional Load of the TailingsMaterial Remains Con-stant For a specific part of the tailings dam the upperboundary of the tailings dam has reached the designedelevation at a certain moment Therefore the additional loadwill remain unchanged from this moment During this stagethe tailings mud will continue to consolidate under thecombined action of self-weight and constant additional loadThus the pore water pressure will dissipate within the tailingsdam Obviously the different parts of tailings mud havedifferent consolidation stages There is no additional loadat the thickest tailings mud in the tailings reservoir until itis closed Therefore the tailings mud is always in the firststage and will never enter the second and third stages Fromthe above qualitative analysis the pore water pressure whichis located at the slope dam segment reaches the maximumvalue when the additional load just stops growing The porewater pressure which is located at dry beach segment andartificial lake segment achieves the maximum value whenthe tailings dam reaches themaximumheightThe numericalvalues of the pore water pressure at different locations andmoments can be obtained by solving the partial differentialequations of consolidation problems
4 Analytical Solutions ofthe Pore Water Pressure
It is necessary to accurately calculate the accumulation anddissipation of the pore water pressure in tailings dam ofwhich a solution of 1D consolidation problem can be sim-plified During the consolidation process the soil parametersof tailings material (such as the change of bulk densitypermeability coefficient and consolidation coefficient) arechanged with the consolidation process The change law canbe determined through many experiments Under normalcircumstances it is difficult to obtain the analytical solutionwhen the change law of soil parameters is considered There-fore the exact solution of the problem can only be dependedon numerical calculation method
As the horizontal length of the tailings reservoir is usuallyfar greater than the vertical thickness (ie the horizontallength is 10 times more than the vertical thickness) thedrainage consolidation effect of the horizontal direction canbe ignored Therefore the vertical direction of drainage isneeded to consider Then it is only a 1D consolidation prob-lem Based on the assumption of small-strain and constantof soil parameters such as 119898] 119896 120597Δ120590120597119905 1205741015840 and 120597119867120597119905are constant (18) is degenerated into a constant coefficientnonhomogeneous parabolic partial differential equation
12059721199061205971199112 +
1119888] (120572 + 120574
1015840119876 minus 120597119906120597119905 ) = 0 (19)
where
120572 = 120597Δ120590120597119905
119876 = 120597119867120597119905
119888] = 119896119898]120574119908
(20)
The above three different stages are specifically discussedas follows
41 The Thickness Mud of Tailings Mud Increases with TimeThe overburden load of the tailings mud at this stage is zerothat is 120572 = 0 The consolidation equation yields
12059721199061205971199112 +
1119888] (1205741015840119876 minus 120597119906
120597119905 ) = 0 (21)
If the bottom of the tailings mud is impermeable thecoordinates origin is taken as the impermeable bottom Theboundary conditions of the problem can be given as follows
119911 = 0120597119906120597119911 = 0 (22)
119911 = 119867119906 = 0 (23)
Mathematical Problems in Engineering 5
The initial condition of the problem is given by
119905 = 0119906 = 0 (24)
According to Gibsonrsquos [20] study (25) is chosen as thesolution of (21)
It can be verified that (25) satisfies the boundary condi-tion (22) The choice of function 119892(120585) needs to satisfy (23)Substituting (25) into (23) gives
For any function ℎ(119905) the partial differential equation(26) has no numerical solution With regard to ℎ(119905) = 119898119905the following parameters can be introduced to transform theequation
It can be verified that (31) satisfies the initial condition of(24)
42 The Additional Load of Tailings Mud Increases with TimeDuring this stage the thickness of tailings mud not onlydoes not increase with time but also gradually decreases withthe increase of effective stress For the sake of simplicityassuming that the thickness of tailingsmud119867 is constant thebasic consolidation differential (19) can be simplified as
119888] 12059721199061205971199112 =
120597119906120597119905 minus
120597Δ120590120597119905 (32)
The initial condition of the problem is given by
119905 = 1199051119906 = 1199060 (119911) (33)
The boundary conditions of the problem can be given as
119911 = 0119906 = 0119911 = 119867
120597119906120597119911 = 0
(34)
The following three questions are called problem 119860problem 119861 and problem 119862 respectively119860
If the solution of problem119860 problem 119861 and problem119862 is119906119860 119906119861 and 119906119862 respectively it can be directly proved throughsubstitution method
119906119862 = 119906119860 + 119906119861 (38)
The solution of 119906119860 belongs to the Terzaghi classical1D consolidation problem The analytical solution can beobtained
119906119860 =infinsum119895=1
119860119899 sin 1198951205871199112119867 exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (39)
where
119860119899 = 1119867 int21198670
1199060 (119911) sin 1198951205871199112119867 119889119911 (40)
1199060(119911) is the solution of (31) Substituting (31) into (40) byMatlab software119860119899 can be obtained Substituting it into (39)the numerical solution 119906119860 can be given
For solution to the problem 119906119861 Schiffman [21] acquiresthe following series solutions
119906119861 = 1612057211986721205873119888]
infinsum135
11198953 sin
1198951205871199112119867
sdot 1 minus exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (41)
When the tailings mud is below the groundwater levelthe total stress 120590(119911) of the tailings mud at the depth 119911 can beexpressed as follows
120590 (119911) = 1205741015840119911 + 120597Δ120590120597119905 (119905 minus 1199051) (42)
The ratio of porewater pressure to total stress can be easilydetermined as follows
43 The Additional Load on the Tailings Mud Remains Con-stant In this stage the basic differential equation and thedefinite condition are problem119863119863
If the solution of problem 119860 is 119906119860 and the solution ofproblem 119864 is 119906119864 the solution of problem 119863 can be easilydemonstrated is 119906119863 as follows
119906119863 = 119906119860 + 119906119864 (46)
The definite condition of problem 119864 and problem 119860 isexactly the same in form except that the function 1199061(119911) isdifferent from the function 1199060(119911)
Suppose that
119906119864 =infinsum119895=1
119861119899 sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (47)
Then
119861119899 = 1119867 int21198670
1199061 (119911) sin 1198951205871199112119867 119889119911 (48)
where
1199061 (119911) = 1612057211986721205873119888V
sdot infinsum123
11198953 sin
1198951205871199112119867 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
(49)
Mathematical Problems in Engineering 7
Gravel
Tailings fine sand
Tailings sandTailings mud
1030
16 105 33 70 300
Impervious boundary
Slope dam segment Dry beach segment Artificial lake segment
Figure 3 Simplified section for calculation of pore water pressure in a tailings dam
150 100200 250 300 350
50
Figure 4 Distribution of pore water pressure at the moment when the dam rises to its maximum height
According to the orthogonal rule of the solution thefollowing relationship can be obtained
1119867 int21198670
sin 1198981205871199112119867 sin1198951205871199112119867 119889119911 =
0 119898 = 1198951 119898 = 119895 (50)
Equation (48) can be simplified through (50)
119861119899
= 1612057211986721205873119888V
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ] 119895 = 1 3 5
0 119895 = 2 4 6(51)
Substituting (51) into (47) yields
119906119864 = 1612057211986721205873119888V
infinsum135
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
sdot sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (52)
For the third stage the analytical solution of pore waterpressure is obtained
Similar to (42) the ratio of pore water pressure to totalstress at a certain depth of tailings mud at this stage is asfollows
During this stage 120590(119911) does not change with time and itcan take the following value
120590 (119911) = 1205741015840119911 + 120572 (1199052 minus 1199051) (54)
5 Example
The simplified calculation section of the representative sec-tion of a tailings dam is shown in Figure 3 The length ofthe tailings dam is 368m Therefore the problem can besimplified as 1D consolidationThe slime layer below the levelis assumed for impervious boundary The rising speed of the
dam height is 175my The saturation density of gravel is220 kNm3 the saturation density of the tailings fine sandis 196 kNm3 the saturation density of the tailings sand is191 kNm3 and the saturation density of the tailings mudis 188 kNm3 The permeability coefficient of gravel is 3 times10minus3ms the permeability coefficient of the tailings fine sandis 5times 10minus6ms the permeability coefficient of the tailings sandis 2 times 10minus6ms and the permeability coefficient of the tailingsmud is 15 times 10minus8ms The consolidation coefficient of gravelis 38 times 10minus1 cm2s the consolidation coefficient of the tailingsfine sand is 54 times 10minus3 cm2s the consolidation coefficient ofthe tailings sand is 28 times 10minus3 cm2s and the consolidationcoefficient of the tailings mud is 12 times 10minus5 cm2s Pleasedetermine the distribution of pore water pressure when thetailings dam reaches maximum height
The above problem can be solved through (31) (46)and (52) The calculation can be accomplished through theMATLAB software Firstly a coordinate system can be setup Then the corresponding coordinates are taken out ofthe interval 1m The region is determined according to thecoordinates Then the pore water pressure can be calculatedthrough the corresponding equation Finally the contourmap can be drawn through a set of array The results of thecalculation are shown in Figure 4
6 Discussion
Due to the similarity between the tailings dam and thereservoir dam in the geotechnical structure many scholarsdirectly introduce the calculation methods of the reservoirdam whose theory is relatively mature into the tailings damConsidering the difference on construction cycle construc-tion materials between the tailings dam and reservoir dam itmakes the calculation results inconsistent with the actual
Based on the Terzaghi consolidation theory of 1D thetailings dam is divided into the slope dam segment the drybeach segment and the artificial lake segment The solutionsof the pore water pressure are derived respectively Theanalysis shows that the additional load of the slope damsegment is unchanged which can be calculated using (46)The analysis indicates that tailings thickness of the artificial
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Substituting (15) into (17) (17) can be expressed as porewater pressure Equation (18) can be obtained
12059721199061205971199112 +
119898V120574119908119896 [120597Δ120590120597119905 + 1205741015840 120597119867120597119905 minus 120597119906120597119905 ] +
1119896119889119896119889119911
120597119906120597119911 = 0 (18)
Equation (18) is a general equation which reflects the1D consolidation process It takes into account the situationwhere the applied load changes with time the soil thicknesschanges with time the soil permeability changes with depthand so on
3 Accumulation and Dissipation of PoreWater Pressure in Tailings Dam
Tailings areminewastes produced in themining engineeringwhich are sent to tailings reservoir by pipe or flume Thetailings dam is an important part in mining engineeringwhich is consisted by the initial dam and fill dam The initialdam is made by permeable rockfill generally and the fill damis formed by tailings In general the construction of a tailingdam takes many decades or even a century According toconstruction method the tailings dam can be divided intoupstream tailings dam downstream tailings dam midlinemethod tailings dam and so onThenumber of tailings pondsin China has reached more than 12000 by statistics Becausethe downstream of the tailings dam is residents living areaor mining production area generally the social and peoplersquosproperty damage a huge impact if the tailings dam is failed
Because the upstream tailings dam has advantages ofsimple operation low construction costs the less need forcoarse particles and so on according to statistics 95 of thetailings dams are adopting the construction of the upstreamtailings dam in China On the other hand the constructionprocess of the upstream tailings dam cannot precisely controlthe shape of the tailings dam The deposition structure ofthe tailings dam is very complex The upstream tailings haveshortcoming of long infiltration distant and poor stabilityIn view of the tailings slurry discharge and tailings particledeposition following the sediment mechanics the sectionprofile of the tailings dam has obvious regularity The coarserthe particle size is the shorter the average distance migratesand vice versa Generally the deposition order of the tailingsdam along the dry beach face is tailings fine sand tailings silttailings sand and tailings mud
If particle size distribution of the tailings and the lengthof the dry beach face are kept constant the interface of thetailingsmaterial should be substantially parallel to the surfaceof the dam slope which is shown in Figure 2 For most ofthe tailings dams because the tailings water contains moreheavy metals the initial topography and the side of tailingsdam are treated as an impermeable boundary On the otherhand the tailings dam extends very long in the direction ofthe reservoir areaTherefore the consolidation of the tailingsdam is a 1D problem
As can be seen from Figure 2 the closer to inside thereservoir area is the thicker the tailings mud is and the worse
4 Mathematical Problems in Engineering
the average consolidation is In other words the closer toinside the reservoir area is the greater pore water pressureof the tailings dam is and the lower shear strength of thetailings material is Generally speaking the construction ofa tailings dam needs several decades to reach the designelevation With the continuous production of mining theheight of the tailings dam is increasing During this processthe accumulation and dissipation of the excess pore waterpressure of the tailings material experience several differentstages of development
31 Tailings Thickness Increases with Time In this stage thethickness of the tailings increases with time The tailingsmaterial produces great compressive deformation under self-weight effect The pore water pressure produced by the self-weight pressure at the early stage is partially dissipatedNevertheless the pore water pressure caused by self-weightpressure is increased with the tailings thickness at the laterstage which leads to the increase of the pore water pressurewith time Therefore the net pore water pressure is accumu-lated during this process
32 Load on the Tailings Layer Increases with Time The self-deposition of the tailings mud continues to extend to interiorof the tailings reservoir area in this stage The self-depositionof the tailings layer has ended where the end of the tailingsreservoir is near It overlapped with particles coarse tailingsand tailings mud The tailings mud has to bear a growingload with time The tailings mud continues to consolidateunder the combined action of self-weight and additionalload At this stage the pore pressure produced by the self-weight pressure dissipates a little part and the more porewater pressure produced by additional load accumulatesThenet pore water pressure tends to accumulate in this proc-ess
33TheAdditional Load of the TailingsMaterial Remains Con-stant For a specific part of the tailings dam the upperboundary of the tailings dam has reached the designedelevation at a certain moment Therefore the additional loadwill remain unchanged from this moment During this stagethe tailings mud will continue to consolidate under thecombined action of self-weight and constant additional loadThus the pore water pressure will dissipate within the tailingsdam Obviously the different parts of tailings mud havedifferent consolidation stages There is no additional loadat the thickest tailings mud in the tailings reservoir until itis closed Therefore the tailings mud is always in the firststage and will never enter the second and third stages Fromthe above qualitative analysis the pore water pressure whichis located at the slope dam segment reaches the maximumvalue when the additional load just stops growing The porewater pressure which is located at dry beach segment andartificial lake segment achieves the maximum value whenthe tailings dam reaches themaximumheightThe numericalvalues of the pore water pressure at different locations andmoments can be obtained by solving the partial differentialequations of consolidation problems
4 Analytical Solutions ofthe Pore Water Pressure
It is necessary to accurately calculate the accumulation anddissipation of the pore water pressure in tailings dam ofwhich a solution of 1D consolidation problem can be sim-plified During the consolidation process the soil parametersof tailings material (such as the change of bulk densitypermeability coefficient and consolidation coefficient) arechanged with the consolidation process The change law canbe determined through many experiments Under normalcircumstances it is difficult to obtain the analytical solutionwhen the change law of soil parameters is considered There-fore the exact solution of the problem can only be dependedon numerical calculation method
As the horizontal length of the tailings reservoir is usuallyfar greater than the vertical thickness (ie the horizontallength is 10 times more than the vertical thickness) thedrainage consolidation effect of the horizontal direction canbe ignored Therefore the vertical direction of drainage isneeded to consider Then it is only a 1D consolidation prob-lem Based on the assumption of small-strain and constantof soil parameters such as 119898] 119896 120597Δ120590120597119905 1205741015840 and 120597119867120597119905are constant (18) is degenerated into a constant coefficientnonhomogeneous parabolic partial differential equation
12059721199061205971199112 +
1119888] (120572 + 120574
1015840119876 minus 120597119906120597119905 ) = 0 (19)
where
120572 = 120597Δ120590120597119905
119876 = 120597119867120597119905
119888] = 119896119898]120574119908
(20)
The above three different stages are specifically discussedas follows
41 The Thickness Mud of Tailings Mud Increases with TimeThe overburden load of the tailings mud at this stage is zerothat is 120572 = 0 The consolidation equation yields
12059721199061205971199112 +
1119888] (1205741015840119876 minus 120597119906
120597119905 ) = 0 (21)
If the bottom of the tailings mud is impermeable thecoordinates origin is taken as the impermeable bottom Theboundary conditions of the problem can be given as follows
119911 = 0120597119906120597119911 = 0 (22)
119911 = 119867119906 = 0 (23)
Mathematical Problems in Engineering 5
The initial condition of the problem is given by
119905 = 0119906 = 0 (24)
According to Gibsonrsquos [20] study (25) is chosen as thesolution of (21)
It can be verified that (25) satisfies the boundary condi-tion (22) The choice of function 119892(120585) needs to satisfy (23)Substituting (25) into (23) gives
For any function ℎ(119905) the partial differential equation(26) has no numerical solution With regard to ℎ(119905) = 119898119905the following parameters can be introduced to transform theequation
It can be verified that (31) satisfies the initial condition of(24)
42 The Additional Load of Tailings Mud Increases with TimeDuring this stage the thickness of tailings mud not onlydoes not increase with time but also gradually decreases withthe increase of effective stress For the sake of simplicityassuming that the thickness of tailingsmud119867 is constant thebasic consolidation differential (19) can be simplified as
119888] 12059721199061205971199112 =
120597119906120597119905 minus
120597Δ120590120597119905 (32)
The initial condition of the problem is given by
119905 = 1199051119906 = 1199060 (119911) (33)
The boundary conditions of the problem can be given as
119911 = 0119906 = 0119911 = 119867
120597119906120597119911 = 0
(34)
The following three questions are called problem 119860problem 119861 and problem 119862 respectively119860
If the solution of problem119860 problem 119861 and problem119862 is119906119860 119906119861 and 119906119862 respectively it can be directly proved throughsubstitution method
119906119862 = 119906119860 + 119906119861 (38)
The solution of 119906119860 belongs to the Terzaghi classical1D consolidation problem The analytical solution can beobtained
119906119860 =infinsum119895=1
119860119899 sin 1198951205871199112119867 exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (39)
where
119860119899 = 1119867 int21198670
1199060 (119911) sin 1198951205871199112119867 119889119911 (40)
1199060(119911) is the solution of (31) Substituting (31) into (40) byMatlab software119860119899 can be obtained Substituting it into (39)the numerical solution 119906119860 can be given
For solution to the problem 119906119861 Schiffman [21] acquiresthe following series solutions
119906119861 = 1612057211986721205873119888]
infinsum135
11198953 sin
1198951205871199112119867
sdot 1 minus exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (41)
When the tailings mud is below the groundwater levelthe total stress 120590(119911) of the tailings mud at the depth 119911 can beexpressed as follows
120590 (119911) = 1205741015840119911 + 120597Δ120590120597119905 (119905 minus 1199051) (42)
The ratio of porewater pressure to total stress can be easilydetermined as follows
43 The Additional Load on the Tailings Mud Remains Con-stant In this stage the basic differential equation and thedefinite condition are problem119863119863
If the solution of problem 119860 is 119906119860 and the solution ofproblem 119864 is 119906119864 the solution of problem 119863 can be easilydemonstrated is 119906119863 as follows
119906119863 = 119906119860 + 119906119864 (46)
The definite condition of problem 119864 and problem 119860 isexactly the same in form except that the function 1199061(119911) isdifferent from the function 1199060(119911)
Suppose that
119906119864 =infinsum119895=1
119861119899 sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (47)
Then
119861119899 = 1119867 int21198670
1199061 (119911) sin 1198951205871199112119867 119889119911 (48)
where
1199061 (119911) = 1612057211986721205873119888V
sdot infinsum123
11198953 sin
1198951205871199112119867 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
(49)
Mathematical Problems in Engineering 7
Gravel
Tailings fine sand
Tailings sandTailings mud
1030
16 105 33 70 300
Impervious boundary
Slope dam segment Dry beach segment Artificial lake segment
Figure 3 Simplified section for calculation of pore water pressure in a tailings dam
150 100200 250 300 350
50
Figure 4 Distribution of pore water pressure at the moment when the dam rises to its maximum height
According to the orthogonal rule of the solution thefollowing relationship can be obtained
1119867 int21198670
sin 1198981205871199112119867 sin1198951205871199112119867 119889119911 =
0 119898 = 1198951 119898 = 119895 (50)
Equation (48) can be simplified through (50)
119861119899
= 1612057211986721205873119888V
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ] 119895 = 1 3 5
0 119895 = 2 4 6(51)
Substituting (51) into (47) yields
119906119864 = 1612057211986721205873119888V
infinsum135
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
sdot sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (52)
For the third stage the analytical solution of pore waterpressure is obtained
Similar to (42) the ratio of pore water pressure to totalstress at a certain depth of tailings mud at this stage is asfollows
During this stage 120590(119911) does not change with time and itcan take the following value
120590 (119911) = 1205741015840119911 + 120572 (1199052 minus 1199051) (54)
5 Example
The simplified calculation section of the representative sec-tion of a tailings dam is shown in Figure 3 The length ofthe tailings dam is 368m Therefore the problem can besimplified as 1D consolidationThe slime layer below the levelis assumed for impervious boundary The rising speed of the
dam height is 175my The saturation density of gravel is220 kNm3 the saturation density of the tailings fine sandis 196 kNm3 the saturation density of the tailings sand is191 kNm3 and the saturation density of the tailings mudis 188 kNm3 The permeability coefficient of gravel is 3 times10minus3ms the permeability coefficient of the tailings fine sandis 5times 10minus6ms the permeability coefficient of the tailings sandis 2 times 10minus6ms and the permeability coefficient of the tailingsmud is 15 times 10minus8ms The consolidation coefficient of gravelis 38 times 10minus1 cm2s the consolidation coefficient of the tailingsfine sand is 54 times 10minus3 cm2s the consolidation coefficient ofthe tailings sand is 28 times 10minus3 cm2s and the consolidationcoefficient of the tailings mud is 12 times 10minus5 cm2s Pleasedetermine the distribution of pore water pressure when thetailings dam reaches maximum height
The above problem can be solved through (31) (46)and (52) The calculation can be accomplished through theMATLAB software Firstly a coordinate system can be setup Then the corresponding coordinates are taken out ofthe interval 1m The region is determined according to thecoordinates Then the pore water pressure can be calculatedthrough the corresponding equation Finally the contourmap can be drawn through a set of array The results of thecalculation are shown in Figure 4
6 Discussion
Due to the similarity between the tailings dam and thereservoir dam in the geotechnical structure many scholarsdirectly introduce the calculation methods of the reservoirdam whose theory is relatively mature into the tailings damConsidering the difference on construction cycle construc-tion materials between the tailings dam and reservoir dam itmakes the calculation results inconsistent with the actual
Based on the Terzaghi consolidation theory of 1D thetailings dam is divided into the slope dam segment the drybeach segment and the artificial lake segment The solutionsof the pore water pressure are derived respectively Theanalysis shows that the additional load of the slope damsegment is unchanged which can be calculated using (46)The analysis indicates that tailings thickness of the artificial
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
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4 Mathematical Problems in Engineering
the average consolidation is In other words the closer toinside the reservoir area is the greater pore water pressureof the tailings dam is and the lower shear strength of thetailings material is Generally speaking the construction ofa tailings dam needs several decades to reach the designelevation With the continuous production of mining theheight of the tailings dam is increasing During this processthe accumulation and dissipation of the excess pore waterpressure of the tailings material experience several differentstages of development
31 Tailings Thickness Increases with Time In this stage thethickness of the tailings increases with time The tailingsmaterial produces great compressive deformation under self-weight effect The pore water pressure produced by the self-weight pressure at the early stage is partially dissipatedNevertheless the pore water pressure caused by self-weightpressure is increased with the tailings thickness at the laterstage which leads to the increase of the pore water pressurewith time Therefore the net pore water pressure is accumu-lated during this process
32 Load on the Tailings Layer Increases with Time The self-deposition of the tailings mud continues to extend to interiorof the tailings reservoir area in this stage The self-depositionof the tailings layer has ended where the end of the tailingsreservoir is near It overlapped with particles coarse tailingsand tailings mud The tailings mud has to bear a growingload with time The tailings mud continues to consolidateunder the combined action of self-weight and additionalload At this stage the pore pressure produced by the self-weight pressure dissipates a little part and the more porewater pressure produced by additional load accumulatesThenet pore water pressure tends to accumulate in this proc-ess
33TheAdditional Load of the TailingsMaterial Remains Con-stant For a specific part of the tailings dam the upperboundary of the tailings dam has reached the designedelevation at a certain moment Therefore the additional loadwill remain unchanged from this moment During this stagethe tailings mud will continue to consolidate under thecombined action of self-weight and constant additional loadThus the pore water pressure will dissipate within the tailingsdam Obviously the different parts of tailings mud havedifferent consolidation stages There is no additional loadat the thickest tailings mud in the tailings reservoir until itis closed Therefore the tailings mud is always in the firststage and will never enter the second and third stages Fromthe above qualitative analysis the pore water pressure whichis located at the slope dam segment reaches the maximumvalue when the additional load just stops growing The porewater pressure which is located at dry beach segment andartificial lake segment achieves the maximum value whenthe tailings dam reaches themaximumheightThe numericalvalues of the pore water pressure at different locations andmoments can be obtained by solving the partial differentialequations of consolidation problems
4 Analytical Solutions ofthe Pore Water Pressure
It is necessary to accurately calculate the accumulation anddissipation of the pore water pressure in tailings dam ofwhich a solution of 1D consolidation problem can be sim-plified During the consolidation process the soil parametersof tailings material (such as the change of bulk densitypermeability coefficient and consolidation coefficient) arechanged with the consolidation process The change law canbe determined through many experiments Under normalcircumstances it is difficult to obtain the analytical solutionwhen the change law of soil parameters is considered There-fore the exact solution of the problem can only be dependedon numerical calculation method
As the horizontal length of the tailings reservoir is usuallyfar greater than the vertical thickness (ie the horizontallength is 10 times more than the vertical thickness) thedrainage consolidation effect of the horizontal direction canbe ignored Therefore the vertical direction of drainage isneeded to consider Then it is only a 1D consolidation prob-lem Based on the assumption of small-strain and constantof soil parameters such as 119898] 119896 120597Δ120590120597119905 1205741015840 and 120597119867120597119905are constant (18) is degenerated into a constant coefficientnonhomogeneous parabolic partial differential equation
12059721199061205971199112 +
1119888] (120572 + 120574
1015840119876 minus 120597119906120597119905 ) = 0 (19)
where
120572 = 120597Δ120590120597119905
119876 = 120597119867120597119905
119888] = 119896119898]120574119908
(20)
The above three different stages are specifically discussedas follows
41 The Thickness Mud of Tailings Mud Increases with TimeThe overburden load of the tailings mud at this stage is zerothat is 120572 = 0 The consolidation equation yields
12059721199061205971199112 +
1119888] (1205741015840119876 minus 120597119906
120597119905 ) = 0 (21)
If the bottom of the tailings mud is impermeable thecoordinates origin is taken as the impermeable bottom Theboundary conditions of the problem can be given as follows
119911 = 0120597119906120597119911 = 0 (22)
119911 = 119867119906 = 0 (23)
Mathematical Problems in Engineering 5
The initial condition of the problem is given by
119905 = 0119906 = 0 (24)
According to Gibsonrsquos [20] study (25) is chosen as thesolution of (21)
It can be verified that (25) satisfies the boundary condi-tion (22) The choice of function 119892(120585) needs to satisfy (23)Substituting (25) into (23) gives
For any function ℎ(119905) the partial differential equation(26) has no numerical solution With regard to ℎ(119905) = 119898119905the following parameters can be introduced to transform theequation
It can be verified that (31) satisfies the initial condition of(24)
42 The Additional Load of Tailings Mud Increases with TimeDuring this stage the thickness of tailings mud not onlydoes not increase with time but also gradually decreases withthe increase of effective stress For the sake of simplicityassuming that the thickness of tailingsmud119867 is constant thebasic consolidation differential (19) can be simplified as
119888] 12059721199061205971199112 =
120597119906120597119905 minus
120597Δ120590120597119905 (32)
The initial condition of the problem is given by
119905 = 1199051119906 = 1199060 (119911) (33)
The boundary conditions of the problem can be given as
119911 = 0119906 = 0119911 = 119867
120597119906120597119911 = 0
(34)
The following three questions are called problem 119860problem 119861 and problem 119862 respectively119860
If the solution of problem119860 problem 119861 and problem119862 is119906119860 119906119861 and 119906119862 respectively it can be directly proved throughsubstitution method
119906119862 = 119906119860 + 119906119861 (38)
The solution of 119906119860 belongs to the Terzaghi classical1D consolidation problem The analytical solution can beobtained
119906119860 =infinsum119895=1
119860119899 sin 1198951205871199112119867 exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (39)
where
119860119899 = 1119867 int21198670
1199060 (119911) sin 1198951205871199112119867 119889119911 (40)
1199060(119911) is the solution of (31) Substituting (31) into (40) byMatlab software119860119899 can be obtained Substituting it into (39)the numerical solution 119906119860 can be given
For solution to the problem 119906119861 Schiffman [21] acquiresthe following series solutions
119906119861 = 1612057211986721205873119888]
infinsum135
11198953 sin
1198951205871199112119867
sdot 1 minus exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (41)
When the tailings mud is below the groundwater levelthe total stress 120590(119911) of the tailings mud at the depth 119911 can beexpressed as follows
120590 (119911) = 1205741015840119911 + 120597Δ120590120597119905 (119905 minus 1199051) (42)
The ratio of porewater pressure to total stress can be easilydetermined as follows
43 The Additional Load on the Tailings Mud Remains Con-stant In this stage the basic differential equation and thedefinite condition are problem119863119863
If the solution of problem 119860 is 119906119860 and the solution ofproblem 119864 is 119906119864 the solution of problem 119863 can be easilydemonstrated is 119906119863 as follows
119906119863 = 119906119860 + 119906119864 (46)
The definite condition of problem 119864 and problem 119860 isexactly the same in form except that the function 1199061(119911) isdifferent from the function 1199060(119911)
Suppose that
119906119864 =infinsum119895=1
119861119899 sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (47)
Then
119861119899 = 1119867 int21198670
1199061 (119911) sin 1198951205871199112119867 119889119911 (48)
where
1199061 (119911) = 1612057211986721205873119888V
sdot infinsum123
11198953 sin
1198951205871199112119867 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
(49)
Mathematical Problems in Engineering 7
Gravel
Tailings fine sand
Tailings sandTailings mud
1030
16 105 33 70 300
Impervious boundary
Slope dam segment Dry beach segment Artificial lake segment
Figure 3 Simplified section for calculation of pore water pressure in a tailings dam
150 100200 250 300 350
50
Figure 4 Distribution of pore water pressure at the moment when the dam rises to its maximum height
According to the orthogonal rule of the solution thefollowing relationship can be obtained
1119867 int21198670
sin 1198981205871199112119867 sin1198951205871199112119867 119889119911 =
0 119898 = 1198951 119898 = 119895 (50)
Equation (48) can be simplified through (50)
119861119899
= 1612057211986721205873119888V
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ] 119895 = 1 3 5
0 119895 = 2 4 6(51)
Substituting (51) into (47) yields
119906119864 = 1612057211986721205873119888V
infinsum135
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
sdot sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (52)
For the third stage the analytical solution of pore waterpressure is obtained
Similar to (42) the ratio of pore water pressure to totalstress at a certain depth of tailings mud at this stage is asfollows
During this stage 120590(119911) does not change with time and itcan take the following value
120590 (119911) = 1205741015840119911 + 120572 (1199052 minus 1199051) (54)
5 Example
The simplified calculation section of the representative sec-tion of a tailings dam is shown in Figure 3 The length ofthe tailings dam is 368m Therefore the problem can besimplified as 1D consolidationThe slime layer below the levelis assumed for impervious boundary The rising speed of the
dam height is 175my The saturation density of gravel is220 kNm3 the saturation density of the tailings fine sandis 196 kNm3 the saturation density of the tailings sand is191 kNm3 and the saturation density of the tailings mudis 188 kNm3 The permeability coefficient of gravel is 3 times10minus3ms the permeability coefficient of the tailings fine sandis 5times 10minus6ms the permeability coefficient of the tailings sandis 2 times 10minus6ms and the permeability coefficient of the tailingsmud is 15 times 10minus8ms The consolidation coefficient of gravelis 38 times 10minus1 cm2s the consolidation coefficient of the tailingsfine sand is 54 times 10minus3 cm2s the consolidation coefficient ofthe tailings sand is 28 times 10minus3 cm2s and the consolidationcoefficient of the tailings mud is 12 times 10minus5 cm2s Pleasedetermine the distribution of pore water pressure when thetailings dam reaches maximum height
The above problem can be solved through (31) (46)and (52) The calculation can be accomplished through theMATLAB software Firstly a coordinate system can be setup Then the corresponding coordinates are taken out ofthe interval 1m The region is determined according to thecoordinates Then the pore water pressure can be calculatedthrough the corresponding equation Finally the contourmap can be drawn through a set of array The results of thecalculation are shown in Figure 4
6 Discussion
Due to the similarity between the tailings dam and thereservoir dam in the geotechnical structure many scholarsdirectly introduce the calculation methods of the reservoirdam whose theory is relatively mature into the tailings damConsidering the difference on construction cycle construc-tion materials between the tailings dam and reservoir dam itmakes the calculation results inconsistent with the actual
Based on the Terzaghi consolidation theory of 1D thetailings dam is divided into the slope dam segment the drybeach segment and the artificial lake segment The solutionsof the pore water pressure are derived respectively Theanalysis shows that the additional load of the slope damsegment is unchanged which can be calculated using (46)The analysis indicates that tailings thickness of the artificial
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
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Mathematical Problems in Engineering
Applied MathematicsJournal of
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
It can be verified that (25) satisfies the boundary condi-tion (22) The choice of function 119892(120585) needs to satisfy (23)Substituting (25) into (23) gives
For any function ℎ(119905) the partial differential equation(26) has no numerical solution With regard to ℎ(119905) = 119898119905the following parameters can be introduced to transform theequation
It can be verified that (31) satisfies the initial condition of(24)
42 The Additional Load of Tailings Mud Increases with TimeDuring this stage the thickness of tailings mud not onlydoes not increase with time but also gradually decreases withthe increase of effective stress For the sake of simplicityassuming that the thickness of tailingsmud119867 is constant thebasic consolidation differential (19) can be simplified as
119888] 12059721199061205971199112 =
120597119906120597119905 minus
120597Δ120590120597119905 (32)
The initial condition of the problem is given by
119905 = 1199051119906 = 1199060 (119911) (33)
The boundary conditions of the problem can be given as
119911 = 0119906 = 0119911 = 119867
120597119906120597119911 = 0
(34)
The following three questions are called problem 119860problem 119861 and problem 119862 respectively119860
If the solution of problem119860 problem 119861 and problem119862 is119906119860 119906119861 and 119906119862 respectively it can be directly proved throughsubstitution method
119906119862 = 119906119860 + 119906119861 (38)
The solution of 119906119860 belongs to the Terzaghi classical1D consolidation problem The analytical solution can beobtained
119906119860 =infinsum119895=1
119860119899 sin 1198951205871199112119867 exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (39)
where
119860119899 = 1119867 int21198670
1199060 (119911) sin 1198951205871199112119867 119889119911 (40)
1199060(119911) is the solution of (31) Substituting (31) into (40) byMatlab software119860119899 can be obtained Substituting it into (39)the numerical solution 119906119860 can be given
For solution to the problem 119906119861 Schiffman [21] acquiresthe following series solutions
119906119861 = 1612057211986721205873119888]
infinsum135
11198953 sin
1198951205871199112119867
sdot 1 minus exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (41)
When the tailings mud is below the groundwater levelthe total stress 120590(119911) of the tailings mud at the depth 119911 can beexpressed as follows
120590 (119911) = 1205741015840119911 + 120597Δ120590120597119905 (119905 minus 1199051) (42)
The ratio of porewater pressure to total stress can be easilydetermined as follows
43 The Additional Load on the Tailings Mud Remains Con-stant In this stage the basic differential equation and thedefinite condition are problem119863119863
If the solution of problem 119860 is 119906119860 and the solution ofproblem 119864 is 119906119864 the solution of problem 119863 can be easilydemonstrated is 119906119863 as follows
119906119863 = 119906119860 + 119906119864 (46)
The definite condition of problem 119864 and problem 119860 isexactly the same in form except that the function 1199061(119911) isdifferent from the function 1199060(119911)
Suppose that
119906119864 =infinsum119895=1
119861119899 sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (47)
Then
119861119899 = 1119867 int21198670
1199061 (119911) sin 1198951205871199112119867 119889119911 (48)
where
1199061 (119911) = 1612057211986721205873119888V
sdot infinsum123
11198953 sin
1198951205871199112119867 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
(49)
Mathematical Problems in Engineering 7
Gravel
Tailings fine sand
Tailings sandTailings mud
1030
16 105 33 70 300
Impervious boundary
Slope dam segment Dry beach segment Artificial lake segment
Figure 3 Simplified section for calculation of pore water pressure in a tailings dam
150 100200 250 300 350
50
Figure 4 Distribution of pore water pressure at the moment when the dam rises to its maximum height
According to the orthogonal rule of the solution thefollowing relationship can be obtained
1119867 int21198670
sin 1198981205871199112119867 sin1198951205871199112119867 119889119911 =
0 119898 = 1198951 119898 = 119895 (50)
Equation (48) can be simplified through (50)
119861119899
= 1612057211986721205873119888V
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ] 119895 = 1 3 5
0 119895 = 2 4 6(51)
Substituting (51) into (47) yields
119906119864 = 1612057211986721205873119888V
infinsum135
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
sdot sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (52)
For the third stage the analytical solution of pore waterpressure is obtained
Similar to (42) the ratio of pore water pressure to totalstress at a certain depth of tailings mud at this stage is asfollows
During this stage 120590(119911) does not change with time and itcan take the following value
120590 (119911) = 1205741015840119911 + 120572 (1199052 minus 1199051) (54)
5 Example
The simplified calculation section of the representative sec-tion of a tailings dam is shown in Figure 3 The length ofthe tailings dam is 368m Therefore the problem can besimplified as 1D consolidationThe slime layer below the levelis assumed for impervious boundary The rising speed of the
dam height is 175my The saturation density of gravel is220 kNm3 the saturation density of the tailings fine sandis 196 kNm3 the saturation density of the tailings sand is191 kNm3 and the saturation density of the tailings mudis 188 kNm3 The permeability coefficient of gravel is 3 times10minus3ms the permeability coefficient of the tailings fine sandis 5times 10minus6ms the permeability coefficient of the tailings sandis 2 times 10minus6ms and the permeability coefficient of the tailingsmud is 15 times 10minus8ms The consolidation coefficient of gravelis 38 times 10minus1 cm2s the consolidation coefficient of the tailingsfine sand is 54 times 10minus3 cm2s the consolidation coefficient ofthe tailings sand is 28 times 10minus3 cm2s and the consolidationcoefficient of the tailings mud is 12 times 10minus5 cm2s Pleasedetermine the distribution of pore water pressure when thetailings dam reaches maximum height
The above problem can be solved through (31) (46)and (52) The calculation can be accomplished through theMATLAB software Firstly a coordinate system can be setup Then the corresponding coordinates are taken out ofthe interval 1m The region is determined according to thecoordinates Then the pore water pressure can be calculatedthrough the corresponding equation Finally the contourmap can be drawn through a set of array The results of thecalculation are shown in Figure 4
6 Discussion
Due to the similarity between the tailings dam and thereservoir dam in the geotechnical structure many scholarsdirectly introduce the calculation methods of the reservoirdam whose theory is relatively mature into the tailings damConsidering the difference on construction cycle construc-tion materials between the tailings dam and reservoir dam itmakes the calculation results inconsistent with the actual
Based on the Terzaghi consolidation theory of 1D thetailings dam is divided into the slope dam segment the drybeach segment and the artificial lake segment The solutionsof the pore water pressure are derived respectively Theanalysis shows that the additional load of the slope damsegment is unchanged which can be calculated using (46)The analysis indicates that tailings thickness of the artificial
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
If the solution of problem119860 problem 119861 and problem119862 is119906119860 119906119861 and 119906119862 respectively it can be directly proved throughsubstitution method
119906119862 = 119906119860 + 119906119861 (38)
The solution of 119906119860 belongs to the Terzaghi classical1D consolidation problem The analytical solution can beobtained
119906119860 =infinsum119895=1
119860119899 sin 1198951205871199112119867 exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (39)
where
119860119899 = 1119867 int21198670
1199060 (119911) sin 1198951205871199112119867 119889119911 (40)
1199060(119911) is the solution of (31) Substituting (31) into (40) byMatlab software119860119899 can be obtained Substituting it into (39)the numerical solution 119906119860 can be given
For solution to the problem 119906119861 Schiffman [21] acquiresthe following series solutions
119906119861 = 1612057211986721205873119888]
infinsum135
11198953 sin
1198951205871199112119867
sdot 1 minus exp[minus11989521205872119888] (119905 minus 1199051)41198672 ] (41)
When the tailings mud is below the groundwater levelthe total stress 120590(119911) of the tailings mud at the depth 119911 can beexpressed as follows
120590 (119911) = 1205741015840119911 + 120597Δ120590120597119905 (119905 minus 1199051) (42)
The ratio of porewater pressure to total stress can be easilydetermined as follows
43 The Additional Load on the Tailings Mud Remains Con-stant In this stage the basic differential equation and thedefinite condition are problem119863119863
If the solution of problem 119860 is 119906119860 and the solution ofproblem 119864 is 119906119864 the solution of problem 119863 can be easilydemonstrated is 119906119863 as follows
119906119863 = 119906119860 + 119906119864 (46)
The definite condition of problem 119864 and problem 119860 isexactly the same in form except that the function 1199061(119911) isdifferent from the function 1199060(119911)
Suppose that
119906119864 =infinsum119895=1
119861119899 sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (47)
Then
119861119899 = 1119867 int21198670
1199061 (119911) sin 1198951205871199112119867 119889119911 (48)
where
1199061 (119911) = 1612057211986721205873119888V
sdot infinsum123
11198953 sin
1198951205871199112119867 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
(49)
Mathematical Problems in Engineering 7
Gravel
Tailings fine sand
Tailings sandTailings mud
1030
16 105 33 70 300
Impervious boundary
Slope dam segment Dry beach segment Artificial lake segment
Figure 3 Simplified section for calculation of pore water pressure in a tailings dam
150 100200 250 300 350
50
Figure 4 Distribution of pore water pressure at the moment when the dam rises to its maximum height
According to the orthogonal rule of the solution thefollowing relationship can be obtained
1119867 int21198670
sin 1198981205871199112119867 sin1198951205871199112119867 119889119911 =
0 119898 = 1198951 119898 = 119895 (50)
Equation (48) can be simplified through (50)
119861119899
= 1612057211986721205873119888V
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ] 119895 = 1 3 5
0 119895 = 2 4 6(51)
Substituting (51) into (47) yields
119906119864 = 1612057211986721205873119888V
infinsum135
11198953 1 minus exp[minus11989521205872119888V (1199052 minus 1199051)41198672 ]
sdot sin 1198951205871199112119867 exp[minus11989521205872119888V (119905 minus 1199052)41198672 ] (52)
For the third stage the analytical solution of pore waterpressure is obtained
Similar to (42) the ratio of pore water pressure to totalstress at a certain depth of tailings mud at this stage is asfollows
During this stage 120590(119911) does not change with time and itcan take the following value
120590 (119911) = 1205741015840119911 + 120572 (1199052 minus 1199051) (54)
5 Example
The simplified calculation section of the representative sec-tion of a tailings dam is shown in Figure 3 The length ofthe tailings dam is 368m Therefore the problem can besimplified as 1D consolidationThe slime layer below the levelis assumed for impervious boundary The rising speed of the
dam height is 175my The saturation density of gravel is220 kNm3 the saturation density of the tailings fine sandis 196 kNm3 the saturation density of the tailings sand is191 kNm3 and the saturation density of the tailings mudis 188 kNm3 The permeability coefficient of gravel is 3 times10minus3ms the permeability coefficient of the tailings fine sandis 5times 10minus6ms the permeability coefficient of the tailings sandis 2 times 10minus6ms and the permeability coefficient of the tailingsmud is 15 times 10minus8ms The consolidation coefficient of gravelis 38 times 10minus1 cm2s the consolidation coefficient of the tailingsfine sand is 54 times 10minus3 cm2s the consolidation coefficient ofthe tailings sand is 28 times 10minus3 cm2s and the consolidationcoefficient of the tailings mud is 12 times 10minus5 cm2s Pleasedetermine the distribution of pore water pressure when thetailings dam reaches maximum height
The above problem can be solved through (31) (46)and (52) The calculation can be accomplished through theMATLAB software Firstly a coordinate system can be setup Then the corresponding coordinates are taken out ofthe interval 1m The region is determined according to thecoordinates Then the pore water pressure can be calculatedthrough the corresponding equation Finally the contourmap can be drawn through a set of array The results of thecalculation are shown in Figure 4
6 Discussion
Due to the similarity between the tailings dam and thereservoir dam in the geotechnical structure many scholarsdirectly introduce the calculation methods of the reservoirdam whose theory is relatively mature into the tailings damConsidering the difference on construction cycle construc-tion materials between the tailings dam and reservoir dam itmakes the calculation results inconsistent with the actual
Based on the Terzaghi consolidation theory of 1D thetailings dam is divided into the slope dam segment the drybeach segment and the artificial lake segment The solutionsof the pore water pressure are derived respectively Theanalysis shows that the additional load of the slope damsegment is unchanged which can be calculated using (46)The analysis indicates that tailings thickness of the artificial
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
During this stage 120590(119911) does not change with time and itcan take the following value
120590 (119911) = 1205741015840119911 + 120572 (1199052 minus 1199051) (54)
5 Example
The simplified calculation section of the representative sec-tion of a tailings dam is shown in Figure 3 The length ofthe tailings dam is 368m Therefore the problem can besimplified as 1D consolidationThe slime layer below the levelis assumed for impervious boundary The rising speed of the
dam height is 175my The saturation density of gravel is220 kNm3 the saturation density of the tailings fine sandis 196 kNm3 the saturation density of the tailings sand is191 kNm3 and the saturation density of the tailings mudis 188 kNm3 The permeability coefficient of gravel is 3 times10minus3ms the permeability coefficient of the tailings fine sandis 5times 10minus6ms the permeability coefficient of the tailings sandis 2 times 10minus6ms and the permeability coefficient of the tailingsmud is 15 times 10minus8ms The consolidation coefficient of gravelis 38 times 10minus1 cm2s the consolidation coefficient of the tailingsfine sand is 54 times 10minus3 cm2s the consolidation coefficient ofthe tailings sand is 28 times 10minus3 cm2s and the consolidationcoefficient of the tailings mud is 12 times 10minus5 cm2s Pleasedetermine the distribution of pore water pressure when thetailings dam reaches maximum height
The above problem can be solved through (31) (46)and (52) The calculation can be accomplished through theMATLAB software Firstly a coordinate system can be setup Then the corresponding coordinates are taken out ofthe interval 1m The region is determined according to thecoordinates Then the pore water pressure can be calculatedthrough the corresponding equation Finally the contourmap can be drawn through a set of array The results of thecalculation are shown in Figure 4
6 Discussion
Due to the similarity between the tailings dam and thereservoir dam in the geotechnical structure many scholarsdirectly introduce the calculation methods of the reservoirdam whose theory is relatively mature into the tailings damConsidering the difference on construction cycle construc-tion materials between the tailings dam and reservoir dam itmakes the calculation results inconsistent with the actual
Based on the Terzaghi consolidation theory of 1D thetailings dam is divided into the slope dam segment the drybeach segment and the artificial lake segment The solutionsof the pore water pressure are derived respectively Theanalysis shows that the additional load of the slope damsegment is unchanged which can be calculated using (46)The analysis indicates that tailings thickness of the artificial
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Mathematical Problems in Engineering
lake segment increases with time during construction periodwhich can be calculated by (31) The situation of the drybeach section is slightly complicated The additional loadof the dry beach section which is under the slope damsegment is basically the samewhich can be calculated by (52)The additional load on the other sections of the dry beachsegment increases with time which can be calculated using(31)
The theoretical derivation is based on 1D consolidationtheory It is only considering consolidation in the verticaldirection Since the horizontal scale of most tailings dams ismuch larger than the vertical direction it has little influenceto ignore the drainage of horizontal From the point ofengineering view it is conservative to the stability of thetailings damConsidering drainage of the horizontal the porewater pressure will be lessened and the safety factor of thetailings dam will be greater
It is assumed that the deformation of the tailings is smalldeformation during the consolidation process If the actualtailings are loose relatively the deformation of the tailingsis large deformation He et al [22] studied the pore waterpressure of saturated soils using the updated Lagrangianformulation of large strain method (ULM) total Lagrangianformulation of large strain method (TLM) and small strainmethod (SSM) The results point out that the pore waterpressure of the above three methods is exactly the samewhen the strain is less than 24 When the strain is greaterthan 24 the pore water pressure of the ULM and SSMstill keeps the same and the pore water pressure of theTLM is obviously less than the ULM and SSM When thestrain reaches 75 the pore water pressure of the ULMand SMM began to bifurcate When the deformation of thetailings dam is examined for a relatively short time theconsolidation of prophase tailings dam has been completedunder previous load The consolidation deformation of thewhole tailings dam is relatively small and its deformationconforms to the small deformation assumption On the otherhand it is difficult to calculate and solve partial differentialequations using large deformation theory The differencesbetween large deformation and small deformation need to befurther studied
It is supposed that the mechanical parameters such aspermeability coefficient and consolidation coefficient areconstant during the consolidation process Previous studies[23] have shown that the permeability coefficient of rock andsoil decreases with the increase of deformation during theconsolidation process It is believed that the value of 119862119888119862119896decides whether it is necessary to take under considerationthe effect of nonlinear property by Zhuang et al [24] Theresults show that pore water pressure calculated depends onthe ratio of loading intensity to the initial effective verticalstress subjected to time-dependent loading by Conte andTroncone [25] How the permeability coefficient of tailingsmaterial changes during the consolidation process is lessstudied by relevant scholars How the permeability coefficientof tailings material changes during the consolidation processis less studied by relevant scholars From the aspect ofengineering view the larger the pore water pressure of actualis the smaller the safety factor of tailings dam is Therefore
it is dangerous for the tailings dam The law of permeabilitycoefficient of tailing material during consolidation processneeds to be further studied
7 Conclusion
Tailings dam is a very important geotechnical structure ofmine engineeringThe calculation of pore water pressure hasa great impact on the safety factor of tailings dam slope Howto accurately estimate pore water pressure is very difficultBased on the assumption of 1D consolidation and smallstrain of tailings material a general equation of the porewater pressure is proposed According to dissipation andaccumulation characteristics of the pore water pressure in thetailings dam the tailings dam can be divided into the slopedam segment the dry beach segment and the artificial lakesegmentThe analytic solutions of the corresponding segmentare obtained through solving the partial differential equationwhich has great significance to the stability of the tailingsdam
Notations
120591 Shear strength of the tailings material120590(119911) Maximum increases in vertical total stressas a function of depth 119911120585 A parameter introduced to transform anequation120572 The rate of the additional load on thetailings mud 120572 = 120597Δ120590120597119905119909 Coordinate of 119909 direction119901 A parameter introduced to transform anequation119899 Porosity of the tailings material119898 119895 Counters 1 2 3 119892(120591) A function of variable 120591119892(120585) A function introduced into solving partialdifferential equation119876 The rate of the thickness of the tailingsmud 119876 = 120597119867120597119905119865(120591) A function introduced to transform anequation120590 Stress of the tailings material119911 Coordinate of 119911 direction119906 The pore water pressure119896 Coefficient of coefficientℎ The height of static water level119867 Thickness of tailings mud119890 Porosity ratioΔ120590 Increment of stress1199061(119911) Termination pore water pressure as afunction of depth 1199111199060(119911) Initial pore water pressure as a function ofdepth 1199111199052 Termination time1199051 Initial time119888] Coefficient of consolidation119880119911 Saturation119861119899 Coefficients to be determined
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 9
119860119899 Coefficients to be determined1205900 Initial total stress1205901015840 Effective vertical stress120576119911 Strain in the 119911 direction120574119908 Bulk density of water120574119904 Saturated bulk density of tailings dam1205741015840 Buoyancy unit weight of tailings materialV119911 The actual velocity along the flow
direction in the tailings dam119901119908 Water pressure119898V Coefficient of volume compressibility119894119911 Hydraulic gradient in the 119911 direction1198671 The total height of tailings mud plus staticwater level119865119911 The drag resistance force on the pore wallof a unit volume in the 119911 direction
Conflicts of Interest
The authors declare that they have no conflicts of interest
Acknowledgments
This paper is supported by the Open Research Fund ofState Key Laboratory of Geomechanics and GeotechnicalEngineering Institute of Rock and Soil Mechanics ChineseAcademy of Sciences (Grant no Z013009) the National KeyResearch and Development Program of China (Project no2017YFC0804601) the National Natural Science Foundationof China (Grant nos 51764020 51741410 51234004) andthe Natural Science Foundation of Yunnan Province (Grantno 2015FB130) The authors would like to thank them forproviding the financial support for conducting this research
References
[1] K TerzaghiTheoretical SoilMechanics JohnWileyamp SonsNewYork NY USA 1943
[2] R L Schiffman and J R Stein ldquoOne-dimensional consolidationof layered systemsrdquo Journal of the Soil Mechanics and Founda-tions Division vol 96 no 4 pp 1499ndash1504 1970
[3] R E Olson ldquoConsolidation under time-dependent loadingrdquoJournal of the Geotechnical Engineering Division vol 103 no 1pp 55ndash60 1977
[4] Y-Q Cai X-Y Geng and C-J Xu ldquoSolution of one-dimen-sional finite-strain consolidation of soil with variable compress-ibility under cyclic loadingsrdquo Computers amp Geosciences vol 34no 1 pp 31ndash40 2007
[5] N Muthing S S Razouki M Datcheva and T Schanz ldquoRig-orous solution for 1-D consolidation of a clay layer underhaversine cyclic loading with rest periodrdquo SpringerPlus vol 5no 1 article no 1987 2016
[6] A Qin D Sun and J Zhang ldquoSemi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soilsrdquoComputers amp Geosciences vol 62 pp 110ndash117 2014
[7] L Wang D Sun P Li and Y Xie ldquoSemi-analytical solutionfor one-dimensional consolidation of fractional derivative vis-coelastic saturated soilsrdquo Computers amp Geosciences vol 83 pp30ndash39 2017
[8] R E Gibson ldquoThe Progress of Consolidation in a Clay LayerIncreasing in Thickness with Timerdquo Geotechnique vol 8 no 4pp 171ndash182 1958
[9] S S Razouki and T Schanz ldquoOne-dimensional consolida-tion under haversine repeated loading with rest periodrdquo ActaGeotechnica vol 6 no 1 pp 13ndash20 2011
[10] S S Razouki P Bonnier M Datcheva and T Schanz ldquoAna-lytical solution for 1D consolidation under haversine cyclicloadingrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 37 no 14 pp 2367ndash2372 2013
[11] M S Mahmoud and H Deresiewicz ldquoSettlement of inhomo-geneous consolidating soilsmdashI The single-drained layer underconfined compressionrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 4 no 1 pp 57ndash721980
[12] G Zhu and J-H Yin ldquoAnalysis and mathematical solutions forconsolidation of a soil layer with depth-dependent parametersunder confined compressionrdquo International Journal of Geome-chanics vol 12 no 4 pp 451ndash461 2012
[13] X W Tang B Niu G C Cheng and H Shen ldquoClosed-formsolution for consolidation of three-layer soil with a verticaldrain systemrdquo Geotextiles and Geomembranes vol 36 pp 81ndash91 2013
[14] G-X Mei and Q-M Chen ldquoSolution of Terzaghi one-dimen-sional consolidation equation with general boundary condi-tionsrdquo Journal of Central South University vol 20 no 8 pp2239ndash2244 2013
[15] B C Hawlader B Muhunthan and G Imai ldquoState-dependentconstitutive model and numerical solution of self-weight con-solidationrdquo Geotechnique vol 58 no 2 pp 133ndash141 2008
[16] M A Rahalt and A R Vuez ldquoAnalysis of settlement andpore pressure induced by cyclic loading of silordquo Journal ofGeotechnical andGeoenvironmental Engineering vol 124 no 12pp 1208ndash1210 1998
[17] E H Davis and G P Raymond ldquoA non-linear theory of consol-idationrdquo Geotechnique vol 15 no 2 pp 161ndash173 1965
[18] E Conte and A Troncone ldquoSoil layer response to pore pressurevariations at the boundaryrdquo Geotechnique vol 58 no 1 pp 37ndash44 2008
[19] V Pane and R L Schiffman ldquoA note on sedimentation andconsolidationrdquo Geotechnique vol 35 no 1 pp 69ndash72 1985
[20] R E Gibson ldquoA heat conduction problem involving a specifiedmoving boundaryrdquo Quarterly of Applied Mathematics vol 16no 4 pp 426ndash430 1959
[21] R L Schiffman ldquoConsolidation of soil under time-dependentloading and varying permeabilityrdquo in Proceedings of the Thirty-Seventh Annual Meeting of the Highway Research Board vol 37pp 584ndash617 1958
[22] K S He Z J Shen and X X Peng ldquoThe comparison oflarge strain method using total and updated Lagrangian finiteelement formulation and small strain methodrdquo Chinese Journalof Geotechnical Engineering vol 22 no 1 pp 30ndash34 2000
[23] G Mesri and R E Olson ldquoMechanisms controlling the perme-ability of claysrdquo Clays and Clay Minerals vol 19 no 3 pp 151ndash158 1971
[24] Y-C Zhuang K-H Xie and X-B Li ldquoNonlinear analysis ofconsolidation with variable compressibility and permeabilityrdquoJournal of Zhejiang University (Engineering Science) vol 6 no3 pp 181ndash187 2005
[25] E Conte andA Troncone ldquoNonlinear consolidation of thin lay-ers subjected to time-dependent loadingrdquo Canadian Geotechni-cal Journal vol 44 no 6 pp 717ndash725 2007
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
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