Seepage Flow Model and Deformation Properties of Coastal...
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Research Article Seepage Flow Model and Deformation Properties of Coastal Deep Foundation Pit under Tidal Influence Shu-chen Li, Can Xie , Yan-hong Liang, and Qin Yan Geotechnical and Structural Engineering Research Center, Shandong University, Shandong, China Correspondence should be addressed to Can Xie; [email protected] Received 6 January 2018; Revised 12 February 2018; Accepted 11 March 2018; Published 16 April 2018 Academic Editor: Qin Yuming Copyright © 2018 Shu-chen Li et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. As the coastal region is the most developed region in China, an increasing number of engineering projects are under construction in it in recent years. However, the quality of these projects is significantly affected by groundwater, which is influenced by tidal variations. erefore, the regional groundwater dynamic characteristics under tidal impact and the spatiotemporal evolution of the seepage field must be considered in the construction of the projects. en, Boussinesq function was introduced into the research to deduce the seepage equation under tidal influence for the coastal area. To determine the spatiotemporal evolution of the deep foundation pit seepage field and the coastal seepage field evolution model, numerical calculations based on changes in the tidal water level and seepage equation were performed using MATLAB. According to the developed model, the influence of the seepage field on the foundation pit supporting structure in the excavation process was analyzed through numerical simulations. e results of this research could be considered in design and engineering practice. 1. Introduction Groundwater seepage significantly impacts the stability of foundation pit engineering and the deformation of the foundation pit support structure and, hence, is a major factor in several foundation pit engineering accidents [1– 3]. Based on the generalized Darcy law, Atangana and Vermeulen [4] derived a new equation for groundwater flow and obtained an asymptotic analytical solution of the generalized groundwater flow equation by the Frobenius and Adomian decomposition method. And the feasibility of the solution was verified through comparisons with field test results. Finally, Atangana and Vermeulen further presented a proposition for reducing uncertainties in groundwater study. In order to investigate the influences of seepage field on mechanical property, Wang [5] developed a 3-Dimensional Stochastic Seepage finite element model and proposed a more comprehensive stochastic algorithm to analyze seepage field problems. Qiu et al. [6] established a statistical model and an artificial wavelet neural network model so as to improve predication accuracy. To explore the impact of changes in seawater tides on variations in groundwater levels, Jacob [7] first established a one-dimensional tidal seepage equation. e equation could be used to fit the movement of seawater by using the sine trigonometric function or cosine trigonometric function. Jeng et al. [8] established a new groundwater seepage model by considering the dynamic effects of the phreatic aquifer on the head fluctuations in the confined aquifer and accordingly derived a closed-form analytical solution. In contrast to the previous solutions, the newly developed solution could describe the interaction effect between tidal oscillations and semiconfined/phreatic coastal aquifers. Li et al. [9] constructed a two-dimensional per- meability model of coastal tides by using the boundary element method. ey demonstrated that the groundwater amplitude decreased compared with the tide amplitude and the vibration phase was also deferred. Zhang et al. [10] considered the mechanical properties and seepage charac- teristics of aquifers to explore the interaction between the water level and seepage flow and to express the relationship between confined groundwater and tides using a mathe- matical equation. Numerous theories exist to describe the seepage of water. Guo [11] investigated a multilayer aquifer system comprising an upper weak permeable layer, a lower Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 9714901, 10 pages https://doi.org/10.1155/2018/9714901
Transcript of Seepage Flow Model and Deformation Properties of Coastal...
Research ArticleSeepage Flow Model and Deformation Properties of CoastalDeep Foundation Pit under Tidal Influence
Shu-chen Li Can Xie Yan-hong Liang and Qin Yan
Geotechnical and Structural Engineering Research Center Shandong University Shandong China
Correspondence should be addressed to Can Xie xiecansdu163com
Received 6 January 2018 Revised 12 February 2018 Accepted 11 March 2018 Published 16 April 2018
Academic Editor Qin Yuming
Copyright copy 2018 Shu-chen Li 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
As the coastal region is the most developed region in China an increasing number of engineering projects are under constructionin it in recent years However the quality of these projects is significantly affected by groundwater which is influenced by tidalvariationsTherefore the regional groundwater dynamic characteristics under tidal impact and the spatiotemporal evolution of theseepage field must be considered in the construction of the projects Then Boussinesq function was introduced into the researchto deduce the seepage equation under tidal influence for the coastal area To determine the spatiotemporal evolution of the deepfoundation pit seepage field and the coastal seepage field evolution model numerical calculations based on changes in the tidalwater level and seepage equation were performed using MATLAB According to the developed model the influence of the seepagefield on the foundation pit supporting structure in the excavation process was analyzed through numerical simulationsThe resultsof this research could be considered in design and engineering practice
1 Introduction
Groundwater seepage significantly impacts the stability offoundation pit engineering and the deformation of thefoundation pit support structure and hence is a majorfactor in several foundation pit engineering accidents [1ndash3] Based on the generalized Darcy law Atangana andVermeulen [4] derived a new equation for groundwaterflow and obtained an asymptotic analytical solution of thegeneralized groundwater flow equation by the Frobenius andAdomian decomposition method And the feasibility of thesolution was verified through comparisons with field testresults Finally Atangana and Vermeulen further presented aproposition for reducing uncertainties in groundwater studyIn order to investigate the influences of seepage field onmechanical property Wang [5] developed a 3-DimensionalStochastic Seepage finite elementmodel and proposed amorecomprehensive stochastic algorithm to analyze seepage fieldproblems Qiu et al [6] established a statistical model andan artificial wavelet neural network model so as to improvepredication accuracy
To explore the impact of changes in seawater tides onvariations in groundwater levels Jacob [7] first established
a one-dimensional tidal seepage equation The equationcould be used to fit the movement of seawater by usingthe sine trigonometric function or cosine trigonometricfunction Jeng et al [8] established a new groundwaterseepage model by considering the dynamic effects of thephreatic aquifer on the head fluctuations in the confinedaquifer and accordingly derived a closed-form analyticalsolution In contrast to the previous solutions the newlydeveloped solution could describe the interaction effectbetween tidal oscillations and semiconfinedphreatic coastalaquifers Li et al [9] constructed a two-dimensional per-meability model of coastal tides by using the boundaryelement method They demonstrated that the groundwateramplitude decreased compared with the tide amplitude andthe vibration phase was also deferred Zhang et al [10]considered the mechanical properties and seepage charac-teristics of aquifers to explore the interaction between thewater level and seepage flow and to express the relationshipbetween confined groundwater and tides using a mathe-matical equation Numerous theories exist to describe theseepage of water Guo [11] investigated a multilayer aquifersystem comprising an upper weak permeable layer a lower
HindawiMathematical Problems in EngineeringVolume 2018 Article ID 9714901 10 pageshttpsdoiorg10115520189714901
2 Mathematical Problems in Engineering
weak permeable layer and a confined aquifer According tothe premise that the boundary between the sea and land isvertical amathematicmodel of groundwater level fluctuationwith tides was established and an analytical solution wasdetermined
Numerous theories exist to describe the seepage of waterIn particular the Boussinesq equation has been widelyemployed [12ndash17] Nielsen [12] analyzed the relationshipbetween the inland average water table and tidal amplitudeand solved the question of tidal dynamics on a slopingbeach on the basis of the Boussinesq equation and a fieldmonitoring test A new Boussinesq equation was proposedand a set of formulae for groundwater recession was derivedto use for groundwater flow in confined and unconfinedaquifers [14] Li et al [15] presented a newmethod to improvethe discontinuous boundary condition of the Boussinesqequation The method is useful for handling the movingboundary condition by replacing the Boussinesq equationwith an advectionndashdiffusion equation with an oscillationvelocity Teo et al [17] analyzed the effect of tidal fluctuationon the groundwater level and developed a new parameter toreplace the factor of tilted shores
It is suffice to note that foundation pit engineering incoastal areas and inland areas significantly differs in the influ-ence of groundwater levels In coastal areas the groundwaterlevel is directly affected by the tidal properties outside thepit Because seepage characteristics may be different from thesteady supply of the groundwater level it is crucial to explorethe seepage characteristics around a deep foundation pitunder a tidal dynamic cycleTherefore this research proposesa seepage equation under tidal influence according to theBoussinesq function To explore the deformation propertiesof the supporting structure in a deep foundation pit undertidal impact a spatiotemporal evolutionmodel of the seepagefield was established in accordance with the onsite waterlevel monitoring results of a deep foundation pit in a coastalarea In addition a finite difference simulation softwareprogram was used to simulate the excavation process of thepit
2 Seepage Flow Model
21 Seepage Equation A side of a deep foundation pit nearthe coast was chosen because of its significant tidal impactThe following assumptions were made (1) the seepage fieldhas a gradually varied horizontal flow (2) changes in thevertical seepage velocity are ignored because the verticalseepage velocity is much smaller than the horizontal seepagevelocity (3) the external precipitation recharge on the freesurface is ignored Because the tide level changes overtime the partial derivative with respect to time should bepreserved in the study to explore the timendashspace evolutioncharacteristics of the seepage field under tidal influence It issupposed that the bottom bedrock of the pit is impermeableand no water-resisting layer exists among the backfill layersabove the bedrock The normal direction of the pit is the 119909-axis the coastline is set as the origin of the 119909-axis and thedirection from the coastline to foundation pit is the positive
Z
J
z
DX
x
L
CoastlineFoundation pit
boundary A
Figure 1 Diagrammatic drawing of seepage flow model
direction of the 119909-axis An unstable flow has been previouslyintroduced [18ndash20]
120597120597119909 [119896119885 (119909 119905) 120597119885 (119909 119905)120597119905 ] = 120583120597119885 (119909 119905)120597119905 (1)
Here 119896 is the permeability coefficient 120583 is the gravita-tional specific yield and empirical value of 120583 is 023 119905 isthe seepage time 119909 is the seepage distance and 119885(119909 119905) is afunction of the seepage field free surface
According to assumptions that (1) the distance from thefoundation pit boundary to coastline is 119871 meters (2) thefunction 119891(119905) is introduced to calculate the tide accords and(3) 119885 = 0 on the impermeable plane as shown in Figure 1since there are water-stop curtains around the foundation pitand the support structure is impermeable the boundary con-dition at119909 = 119871 is obtainedThat is (120597119885120597119909)|119909=119871 = 0Then theboundary conditions of the seepage model can be expressedas follows
119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0 (2)
If the seepage flow is kept constant along the 119909-axis themean water levels of the seashore and foundation pit are 119863and 119869 respectively 119911 is the height of the phreatic line when119883 = 119909 as shown in Figure 1 The initial condition of themodel could be derived according to the gradually variedseepage flow equation with a flat slope as follows [21 22]
119902 = 1198962119871 (1198632 minus 1198692) = 1198962119909 (1198632 minus 1199112)= 1198962 (119871 minus 119909) (1199112 minus 1198692)
(3)
Here 119896 is the permeability coefficient 119902 represents theunit width flow and 119871 is the distance between the coastlineand foundation pit boundary
Mathematical Problems in Engineering 3
Table 1 Parameters of soil layer
Soil name BulkmodulusMp
ShearmodulusMp CohesionKpa Friction∘ Unit
weightKNm3Permeability
coefficient cmsSoil layer
thicknessmFilling soil 524 19 120 18 178 80 times 10minus5 3Mucky soil 216 089 196 15 184 20 times 10minus6 4Silty clay 198 103 207 25 202 60 times 10minus6 2Gravel sand 385 264 320 28 198 80 times 10minus2 2Strongly weathered granite 41000 31000 500 57 260 20 times 10minus10 30
Therefore the equation could be simplified to (4) and theinitial condition could be expressed as (5)
119911 = radic1198632 minus 1199091198632 minus 1198692119871 (4)
119885 (119909 0) = radic1198632 minus 1199091198632 minus 1198692119871 (5)
Based on the above parameters the seepage equation ofthe seepage field model could be given as follows
120597120597119909 [119896119885 (119909 119905) 120597119885 (119909 119905)120597119909 ] = 120583120597119885 (119909 119905)120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
119885 (119909 0) = radic1198632 minus 1199091198632 minus 1198692119871
(6)
22 Seepage Equation Solution Several researchers haveattempted to determine the solution of (120597120597119909)[119896119885(119909 119905)(120597119885(119909119905)120597119905)] = 120583(120597119885(119909 119905)120597119905) [23 24] Therefore the equation islinearized in this research The first 119885(119909 119905) on the left sideof the equation could be considered a constant and replacedwith 119885119898 (119885119898 = (119863 + 119869)2) Accordingly the linearized seep-age equation is as follows
11988511989811989612059721198851205971199092 = 120583120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0119885 (119909 0) = 119890 (119909)
in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871
(7)
Finally the solution of the equation is again (see theAppendixfor specific solutions)
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(8)
3 Application to a Real Field Case A CoastalFoundation Pit
31 General Situation of a Coastal Foundation Pit Thecoastalfoundation pit chosen in the research is at 90m from the bayThe field area mainly comprises Quaternary artificial fill andHolocenemarine depositional strataThe physicomechanicalparameters of every stratum are shown in Table 1 To supportthe foundation a bored pile and an anchor cable supportsystem is used in the engineeringThe diameters of the boredpile and thewaterproof pile are 800mmwhile the pile spacingis 1200mm The detailed profile of the support system is
shown in Figure 2 and the parameters of the three anchorcables are indicated in Table 2
According to the official hydrogeological data the meantide water level is 123m (119863 = 123m) and the average annualwater level on the side of the foundation pit is 104m (119869 =104m) By analyzing the official hydrogeological data thechange of the tidal water level satisfies the following law
119891 (119905) = 12 lowast sin(1205871199056 ) + 123 (9)
The timendashspace evolution equation of the seepage fieldis obtained based on (8) and (9) The relationship of the
4 Mathematical Problems in Engineering
Table 2 Calculated parameters of anchor
Cablenumber
Horizontalspacingm Angle∘
Total lengthof the
cablesm
Length of freesegmentm
Length ofanchoragem
Prestressinglocked
valueKNMG1 2400 30 19000 9000 10000 230MG2 2400 25 20000 10000 10000 290MG3 2400 25 15000 6000 9000 260
Cable
Waterproof pileSupporting pile
Top beam
Supporting pile
MG1
MG2
MG3Waterproof pile
20
Ｇ30
Ｇ30
Ｇ30
Ｇ
08 Ｇ08 Ｇ
Figure 2 Cross-section of foundation pit supporting structure
seepage field with time and infiltration distance is analyzedby MATLAB in two 24 h periods The spatiotemporal evolu-tionary characteristic surface of the seepage field is presentedin Figure 3
32 Numerical Simulation The FlAC3D was used to analyzethe stability of coastal deep foundation pitThe dimensions ofthe model established in the research are 80m times 50m times 40mwith an excavation depth of 11m and an excavation width of30m The model is divided into 80000 zones and includes8634 grid points The mesh sketch of the numerical modelis shown in Figure 4
The upper surface of the model is a free boundary andthe bottom surface of the model is a fixed boundary Thefront and back boundaries constrain the displacement in the119910 direction and the left and right boundaries constrain thedisplacement in the 119909 directionThe computation parametersof the soil and cable are given in Table 2 Based on the changein water level of the seepage path in a tidal cycle (as shownin Table 3) the pore water pressure was applied to the model
as dynamic external stress Then dynamic calculation wascarried out during excavation of foundation pit Figure 5shows the calculation flow chart of numerical simulation
4 Analysis of Numerical Results
Figure 6 shows that the change in the horizontal displacementof the model increases with the excavation depth Themaximum horizontal displacement occurs in the upper partof the foundation pit sidewall The 119883-displacement of themodel increases with the excavation depth of the foundationpitThe isosurface of the119883-displacement is diffused in an arcand gradually tends to zero
Figure 7 shows the surface subsidence around the foun-dation pit It can be seen that the surface subsidence aroundthe foundation pit gradually increases with the excavationdepth and the maximum settlement is 105mm The settle-ment mainly occurs near the top of the supporting structureThe maximum settlement emerges at the top of the pile witheach excavation step
Mathematical Problems in Engineering 5
9075
6045
3015
0 2418
126
0Time (h)The distance to foundation
pit boundary (Am)
9
10
11
12
13
14
Wat
er le
vel (
m)
Figure 3 Three-dimensional image of spatiotemporal evolution characteristics of seepage
X
Z
Foundation pit boundary A11 m
80 m
30m
40m
Silty clay Gravel sand Strongly weathered granite Foundation pit
Mucky soil Filling soil
Figure 4 Mesh sketch of numerical model
Table 3 Height of the water at different times in a cycle
Timeh The distance to foundation pit boundary Am0 10 20 30 40 50
0 1128401 1149439 11701 1190401 1210363 1231 1127748 1148821 1170548 1195825 1231686 12883172 1121904 1150565 1181959 1220792 1272075 13391273 1126592 1159036 1197209 1242218 1293792 13474784 1139505 1171119 1208309 1248299 1285621 13101225 115371 1180319 1209362 1235372 1250003 12415916 1162077 1181513 1198634 12075 1200285 11685517 1160111 1173016 117924 1174748 11554 11194198 1147615 1157178 1157891 1149244 1132484 11133129 1128705 1139417 1142241 1140428 1140198 115260610 1110159 1126067 1137873 1151354 1175339 122201211 1098743 112187 1146089 1177477 122408 129452612 1098548 1128111 1163385 1208405 1267334 1341934
6 Mathematical Problems in Engineering
Start calculation
Initial condition and boundary condition
Initial equilibrium
Apply the dynamic stress generatedby the tides
Set dynamical boundary condition
Dynamic calculation
Whether the excavation ends
End calculation
Yes
Seepage calculated by MATAB
No
excavation of foundation pitStatic calculation the sequential
Figure 5 Calculation flow chart of numerical simulation
Figures 8 and 9 show the results of horizontal and verticaldisplacements of the pile top at different excavation depthof deep foundation pit excavation The simulation resultsof the horizontal and vertical displacements of the pile topare basically consistent with the site monitoring resultsThe horizontal and vertical displacements of the pile topaffected by the tide are obviously increased comparedwith thedisplacements under anhydrous conditions The deviation ofthe displacements increases with the excavation depth Thehorizontal and vertical displacements of the pile top withouttidal influence are 105mm and 82mm during excavation atthe bottomof the foundation pit However the horizontal andvertical displacements of the pile top with tidal influence are141mm and 106mm respectively The increases are 343and 226 respectively It is shown that the bearing capacityand deformation of the supporting structure in the deepfoundation pit are adversely affected by tides Therefore thetidal effect should be considered in the design of the supportstructure for a coastal deep foundation pit
5 Conclusions
To solve the seepage mechanics problems of a foundationpit affected by tides the equation of the seepage flow wasdeduced and the timendashspace evolution model of a seepagefield for a deep foundation pit was established based on theBoussinesq functionThe effects of time and space on seepageproperties are considered sophisticated
According to the results of in situ monitoring thevariations in the tide and groundwater level of the foun-dation pit on the ocean side were analyzed Although the
groundwater level of the foundation pit on the ocean sideperiodically changes similar to the tide the amplitude ofthe groundwater level is less than that of the tide Basedon the developed timendashspace seepage model the dynamicwater level of the coastal foundation pit seepage path wascalculated by MATLAB and used to analyze the deformationof the support structure of the deep foundation pit undertidal influence It is obviously seen that the seepage affectedby tide has a significant influence on the deformation ofthe deep foundation pit and increases the security risk ofengineering Therefore in order to ensure the safety of thefoundation pit and improve the construction environmentthe adverse effects of tides must be considered in the designand construction of deep foundation pits in coastal regions
The dynamic and fluid coupling calculation is a rathercomplex question We turned a complex problem into tworelatively simple problems One was simulated by MATLABbased on the seepage model presented in our manuscript andthe other was simulated by dynamic calculation of FLAC3DThe study can provide a new idea for the stability analysis ofdeep foundation pit under the influence of tides
Appendix
For
11988511989811989612059721198851205971199092 = 119906120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
Mathematical Problems in Engineering 7
00000E + 00
Contour of X-displacement
minus60000E minus 04minus90000E minus 04minus12000E minus 03
minus30000E minus 04
minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03minus30000E minus 03minus32183E minus 03
(a)
00000E + 00
Contour of X-displacement
minus12000E minus 03minus60000E minus 04
minus18000E minus 03minus24000E minus 03minus30000E minus 03minus36000E minus 03minus42000E minus 03minus48000E minus 03minus54000E minus 03
minus66000E minus 03minus72000E minus 03minus78000E minus 03minus78211E minus 03
minus60000E minus 03
(b)
Contour of X-displacement
minus10000E minus 03minus20000E minus 03minus30000E minus 03minus40000E minus 03minus50000E minus 03minus60000E minus 03minus70000E minus 03minus80000E minus 03
minus10000E minus 02minus90000E minus 03
minus11000E minus 02minus11113E minus 02
00000E + 00
(c)
Contour of X-displacement
minus20000E minus 03minus40000E minus 03minus60000E minus 03minus80000E minus 03minus10000E minus 02minus12000E minus 02minus14000E minus 02
00000E + 00
minus16000E minus 02minus16120E minus 02
(d)
Figure 6 Contour of119883-displacement
119885 (119909 0) = 119890 (119909) in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871 (A1)
assume that 119881(119909 119905) = 119860(119905) lowast 119909 + 119861(119905) satisfies (A1) then119881 (119909 119905) = 119891 (119905) (A2)
Assume that 119885 (119909 119905) = 119881 (119909 119905) + 119882 (119909 119905) (A3)
The following equation will be obtained according to (A1)and (A2)
119882119905 minus 119901119882119909119909 = minus1198911015840 (119905)119882 (0 119905) = 0120597119882120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0119882 (119909 0) = 119873 (119909)
(A4)
where 119901 = 119906119885119898119896 and119873(119909) = 119890(119909) minus 119891(0)
Assume that119882(119909 119905) = 119882I (119909 119905) + 119882II (119909 119905) (A5)
Equation (A5) is equivalent to the following equations
119882I119905 minus 119901119882I
119909119909 = 0119882I (0 119905) = 0120597119882I
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882I (119909 0) = 119873 (119909)
(A6)
119882II119905 minus 119901119882II
119909119909 = minus1198911015840 (119905)119882II (0 119905) = 0120597119882II
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882II (119909 0) = 0
(A7)
8 Mathematical Problems in Engineering
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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2 Mathematical Problems in Engineering
weak permeable layer and a confined aquifer According tothe premise that the boundary between the sea and land isvertical amathematicmodel of groundwater level fluctuationwith tides was established and an analytical solution wasdetermined
Numerous theories exist to describe the seepage of waterIn particular the Boussinesq equation has been widelyemployed [12ndash17] Nielsen [12] analyzed the relationshipbetween the inland average water table and tidal amplitudeand solved the question of tidal dynamics on a slopingbeach on the basis of the Boussinesq equation and a fieldmonitoring test A new Boussinesq equation was proposedand a set of formulae for groundwater recession was derivedto use for groundwater flow in confined and unconfinedaquifers [14] Li et al [15] presented a newmethod to improvethe discontinuous boundary condition of the Boussinesqequation The method is useful for handling the movingboundary condition by replacing the Boussinesq equationwith an advectionndashdiffusion equation with an oscillationvelocity Teo et al [17] analyzed the effect of tidal fluctuationon the groundwater level and developed a new parameter toreplace the factor of tilted shores
It is suffice to note that foundation pit engineering incoastal areas and inland areas significantly differs in the influ-ence of groundwater levels In coastal areas the groundwaterlevel is directly affected by the tidal properties outside thepit Because seepage characteristics may be different from thesteady supply of the groundwater level it is crucial to explorethe seepage characteristics around a deep foundation pitunder a tidal dynamic cycleTherefore this research proposesa seepage equation under tidal influence according to theBoussinesq function To explore the deformation propertiesof the supporting structure in a deep foundation pit undertidal impact a spatiotemporal evolutionmodel of the seepagefield was established in accordance with the onsite waterlevel monitoring results of a deep foundation pit in a coastalarea In addition a finite difference simulation softwareprogram was used to simulate the excavation process of thepit
2 Seepage Flow Model
21 Seepage Equation A side of a deep foundation pit nearthe coast was chosen because of its significant tidal impactThe following assumptions were made (1) the seepage fieldhas a gradually varied horizontal flow (2) changes in thevertical seepage velocity are ignored because the verticalseepage velocity is much smaller than the horizontal seepagevelocity (3) the external precipitation recharge on the freesurface is ignored Because the tide level changes overtime the partial derivative with respect to time should bepreserved in the study to explore the timendashspace evolutioncharacteristics of the seepage field under tidal influence It issupposed that the bottom bedrock of the pit is impermeableand no water-resisting layer exists among the backfill layersabove the bedrock The normal direction of the pit is the 119909-axis the coastline is set as the origin of the 119909-axis and thedirection from the coastline to foundation pit is the positive
Z
J
z
DX
x
L
CoastlineFoundation pit
boundary A
Figure 1 Diagrammatic drawing of seepage flow model
direction of the 119909-axis An unstable flow has been previouslyintroduced [18ndash20]
120597120597119909 [119896119885 (119909 119905) 120597119885 (119909 119905)120597119905 ] = 120583120597119885 (119909 119905)120597119905 (1)
Here 119896 is the permeability coefficient 120583 is the gravita-tional specific yield and empirical value of 120583 is 023 119905 isthe seepage time 119909 is the seepage distance and 119885(119909 119905) is afunction of the seepage field free surface
According to assumptions that (1) the distance from thefoundation pit boundary to coastline is 119871 meters (2) thefunction 119891(119905) is introduced to calculate the tide accords and(3) 119885 = 0 on the impermeable plane as shown in Figure 1since there are water-stop curtains around the foundation pitand the support structure is impermeable the boundary con-dition at119909 = 119871 is obtainedThat is (120597119885120597119909)|119909=119871 = 0Then theboundary conditions of the seepage model can be expressedas follows
119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0 (2)
If the seepage flow is kept constant along the 119909-axis themean water levels of the seashore and foundation pit are 119863and 119869 respectively 119911 is the height of the phreatic line when119883 = 119909 as shown in Figure 1 The initial condition of themodel could be derived according to the gradually variedseepage flow equation with a flat slope as follows [21 22]
119902 = 1198962119871 (1198632 minus 1198692) = 1198962119909 (1198632 minus 1199112)= 1198962 (119871 minus 119909) (1199112 minus 1198692)
(3)
Here 119896 is the permeability coefficient 119902 represents theunit width flow and 119871 is the distance between the coastlineand foundation pit boundary
Mathematical Problems in Engineering 3
Table 1 Parameters of soil layer
Soil name BulkmodulusMp
ShearmodulusMp CohesionKpa Friction∘ Unit
weightKNm3Permeability
coefficient cmsSoil layer
thicknessmFilling soil 524 19 120 18 178 80 times 10minus5 3Mucky soil 216 089 196 15 184 20 times 10minus6 4Silty clay 198 103 207 25 202 60 times 10minus6 2Gravel sand 385 264 320 28 198 80 times 10minus2 2Strongly weathered granite 41000 31000 500 57 260 20 times 10minus10 30
Therefore the equation could be simplified to (4) and theinitial condition could be expressed as (5)
119911 = radic1198632 minus 1199091198632 minus 1198692119871 (4)
119885 (119909 0) = radic1198632 minus 1199091198632 minus 1198692119871 (5)
Based on the above parameters the seepage equation ofthe seepage field model could be given as follows
120597120597119909 [119896119885 (119909 119905) 120597119885 (119909 119905)120597119909 ] = 120583120597119885 (119909 119905)120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
119885 (119909 0) = radic1198632 minus 1199091198632 minus 1198692119871
(6)
22 Seepage Equation Solution Several researchers haveattempted to determine the solution of (120597120597119909)[119896119885(119909 119905)(120597119885(119909119905)120597119905)] = 120583(120597119885(119909 119905)120597119905) [23 24] Therefore the equation islinearized in this research The first 119885(119909 119905) on the left sideof the equation could be considered a constant and replacedwith 119885119898 (119885119898 = (119863 + 119869)2) Accordingly the linearized seep-age equation is as follows
11988511989811989612059721198851205971199092 = 120583120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0119885 (119909 0) = 119890 (119909)
in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871
(7)
Finally the solution of the equation is again (see theAppendixfor specific solutions)
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(8)
3 Application to a Real Field Case A CoastalFoundation Pit
31 General Situation of a Coastal Foundation Pit Thecoastalfoundation pit chosen in the research is at 90m from the bayThe field area mainly comprises Quaternary artificial fill andHolocenemarine depositional strataThe physicomechanicalparameters of every stratum are shown in Table 1 To supportthe foundation a bored pile and an anchor cable supportsystem is used in the engineeringThe diameters of the boredpile and thewaterproof pile are 800mmwhile the pile spacingis 1200mm The detailed profile of the support system is
shown in Figure 2 and the parameters of the three anchorcables are indicated in Table 2
According to the official hydrogeological data the meantide water level is 123m (119863 = 123m) and the average annualwater level on the side of the foundation pit is 104m (119869 =104m) By analyzing the official hydrogeological data thechange of the tidal water level satisfies the following law
119891 (119905) = 12 lowast sin(1205871199056 ) + 123 (9)
The timendashspace evolution equation of the seepage fieldis obtained based on (8) and (9) The relationship of the
4 Mathematical Problems in Engineering
Table 2 Calculated parameters of anchor
Cablenumber
Horizontalspacingm Angle∘
Total lengthof the
cablesm
Length of freesegmentm
Length ofanchoragem
Prestressinglocked
valueKNMG1 2400 30 19000 9000 10000 230MG2 2400 25 20000 10000 10000 290MG3 2400 25 15000 6000 9000 260
Cable
Waterproof pileSupporting pile
Top beam
Supporting pile
MG1
MG2
MG3Waterproof pile
20
Ｇ30
Ｇ30
Ｇ30
Ｇ
08 Ｇ08 Ｇ
Figure 2 Cross-section of foundation pit supporting structure
seepage field with time and infiltration distance is analyzedby MATLAB in two 24 h periods The spatiotemporal evolu-tionary characteristic surface of the seepage field is presentedin Figure 3
32 Numerical Simulation The FlAC3D was used to analyzethe stability of coastal deep foundation pitThe dimensions ofthe model established in the research are 80m times 50m times 40mwith an excavation depth of 11m and an excavation width of30m The model is divided into 80000 zones and includes8634 grid points The mesh sketch of the numerical modelis shown in Figure 4
The upper surface of the model is a free boundary andthe bottom surface of the model is a fixed boundary Thefront and back boundaries constrain the displacement in the119910 direction and the left and right boundaries constrain thedisplacement in the 119909 directionThe computation parametersof the soil and cable are given in Table 2 Based on the changein water level of the seepage path in a tidal cycle (as shownin Table 3) the pore water pressure was applied to the model
as dynamic external stress Then dynamic calculation wascarried out during excavation of foundation pit Figure 5shows the calculation flow chart of numerical simulation
4 Analysis of Numerical Results
Figure 6 shows that the change in the horizontal displacementof the model increases with the excavation depth Themaximum horizontal displacement occurs in the upper partof the foundation pit sidewall The 119883-displacement of themodel increases with the excavation depth of the foundationpitThe isosurface of the119883-displacement is diffused in an arcand gradually tends to zero
Figure 7 shows the surface subsidence around the foun-dation pit It can be seen that the surface subsidence aroundthe foundation pit gradually increases with the excavationdepth and the maximum settlement is 105mm The settle-ment mainly occurs near the top of the supporting structureThe maximum settlement emerges at the top of the pile witheach excavation step
Mathematical Problems in Engineering 5
9075
6045
3015
0 2418
126
0Time (h)The distance to foundation
pit boundary (Am)
9
10
11
12
13
14
Wat
er le
vel (
m)
Figure 3 Three-dimensional image of spatiotemporal evolution characteristics of seepage
X
Z
Foundation pit boundary A11 m
80 m
30m
40m
Silty clay Gravel sand Strongly weathered granite Foundation pit
Mucky soil Filling soil
Figure 4 Mesh sketch of numerical model
Table 3 Height of the water at different times in a cycle
Timeh The distance to foundation pit boundary Am0 10 20 30 40 50
0 1128401 1149439 11701 1190401 1210363 1231 1127748 1148821 1170548 1195825 1231686 12883172 1121904 1150565 1181959 1220792 1272075 13391273 1126592 1159036 1197209 1242218 1293792 13474784 1139505 1171119 1208309 1248299 1285621 13101225 115371 1180319 1209362 1235372 1250003 12415916 1162077 1181513 1198634 12075 1200285 11685517 1160111 1173016 117924 1174748 11554 11194198 1147615 1157178 1157891 1149244 1132484 11133129 1128705 1139417 1142241 1140428 1140198 115260610 1110159 1126067 1137873 1151354 1175339 122201211 1098743 112187 1146089 1177477 122408 129452612 1098548 1128111 1163385 1208405 1267334 1341934
6 Mathematical Problems in Engineering
Start calculation
Initial condition and boundary condition
Initial equilibrium
Apply the dynamic stress generatedby the tides
Set dynamical boundary condition
Dynamic calculation
Whether the excavation ends
End calculation
Yes
Seepage calculated by MATAB
No
excavation of foundation pitStatic calculation the sequential
Figure 5 Calculation flow chart of numerical simulation
Figures 8 and 9 show the results of horizontal and verticaldisplacements of the pile top at different excavation depthof deep foundation pit excavation The simulation resultsof the horizontal and vertical displacements of the pile topare basically consistent with the site monitoring resultsThe horizontal and vertical displacements of the pile topaffected by the tide are obviously increased comparedwith thedisplacements under anhydrous conditions The deviation ofthe displacements increases with the excavation depth Thehorizontal and vertical displacements of the pile top withouttidal influence are 105mm and 82mm during excavation atthe bottomof the foundation pit However the horizontal andvertical displacements of the pile top with tidal influence are141mm and 106mm respectively The increases are 343and 226 respectively It is shown that the bearing capacityand deformation of the supporting structure in the deepfoundation pit are adversely affected by tides Therefore thetidal effect should be considered in the design of the supportstructure for a coastal deep foundation pit
5 Conclusions
To solve the seepage mechanics problems of a foundationpit affected by tides the equation of the seepage flow wasdeduced and the timendashspace evolution model of a seepagefield for a deep foundation pit was established based on theBoussinesq functionThe effects of time and space on seepageproperties are considered sophisticated
According to the results of in situ monitoring thevariations in the tide and groundwater level of the foun-dation pit on the ocean side were analyzed Although the
groundwater level of the foundation pit on the ocean sideperiodically changes similar to the tide the amplitude ofthe groundwater level is less than that of the tide Basedon the developed timendashspace seepage model the dynamicwater level of the coastal foundation pit seepage path wascalculated by MATLAB and used to analyze the deformationof the support structure of the deep foundation pit undertidal influence It is obviously seen that the seepage affectedby tide has a significant influence on the deformation ofthe deep foundation pit and increases the security risk ofengineering Therefore in order to ensure the safety of thefoundation pit and improve the construction environmentthe adverse effects of tides must be considered in the designand construction of deep foundation pits in coastal regions
The dynamic and fluid coupling calculation is a rathercomplex question We turned a complex problem into tworelatively simple problems One was simulated by MATLABbased on the seepage model presented in our manuscript andthe other was simulated by dynamic calculation of FLAC3DThe study can provide a new idea for the stability analysis ofdeep foundation pit under the influence of tides
Appendix
For
11988511989811989612059721198851205971199092 = 119906120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
Mathematical Problems in Engineering 7
00000E + 00
Contour of X-displacement
minus60000E minus 04minus90000E minus 04minus12000E minus 03
minus30000E minus 04
minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03minus30000E minus 03minus32183E minus 03
(a)
00000E + 00
Contour of X-displacement
minus12000E minus 03minus60000E minus 04
minus18000E minus 03minus24000E minus 03minus30000E minus 03minus36000E minus 03minus42000E minus 03minus48000E minus 03minus54000E minus 03
minus66000E minus 03minus72000E minus 03minus78000E minus 03minus78211E minus 03
minus60000E minus 03
(b)
Contour of X-displacement
minus10000E minus 03minus20000E minus 03minus30000E minus 03minus40000E minus 03minus50000E minus 03minus60000E minus 03minus70000E minus 03minus80000E minus 03
minus10000E minus 02minus90000E minus 03
minus11000E minus 02minus11113E minus 02
00000E + 00
(c)
Contour of X-displacement
minus20000E minus 03minus40000E minus 03minus60000E minus 03minus80000E minus 03minus10000E minus 02minus12000E minus 02minus14000E minus 02
00000E + 00
minus16000E minus 02minus16120E minus 02
(d)
Figure 6 Contour of119883-displacement
119885 (119909 0) = 119890 (119909) in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871 (A1)
assume that 119881(119909 119905) = 119860(119905) lowast 119909 + 119861(119905) satisfies (A1) then119881 (119909 119905) = 119891 (119905) (A2)
Assume that 119885 (119909 119905) = 119881 (119909 119905) + 119882 (119909 119905) (A3)
The following equation will be obtained according to (A1)and (A2)
119882119905 minus 119901119882119909119909 = minus1198911015840 (119905)119882 (0 119905) = 0120597119882120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0119882 (119909 0) = 119873 (119909)
(A4)
where 119901 = 119906119885119898119896 and119873(119909) = 119890(119909) minus 119891(0)
Assume that119882(119909 119905) = 119882I (119909 119905) + 119882II (119909 119905) (A5)
Equation (A5) is equivalent to the following equations
119882I119905 minus 119901119882I
119909119909 = 0119882I (0 119905) = 0120597119882I
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882I (119909 0) = 119873 (119909)
(A6)
119882II119905 minus 119901119882II
119909119909 = minus1198911015840 (119905)119882II (0 119905) = 0120597119882II
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882II (119909 0) = 0
(A7)
8 Mathematical Problems in Engineering
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
Hindawiwwwhindawicom Volume 2018
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Mathematical Problems in Engineering
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Mathematical Problems in Engineering 3
Table 1 Parameters of soil layer
Soil name BulkmodulusMp
ShearmodulusMp CohesionKpa Friction∘ Unit
weightKNm3Permeability
coefficient cmsSoil layer
thicknessmFilling soil 524 19 120 18 178 80 times 10minus5 3Mucky soil 216 089 196 15 184 20 times 10minus6 4Silty clay 198 103 207 25 202 60 times 10minus6 2Gravel sand 385 264 320 28 198 80 times 10minus2 2Strongly weathered granite 41000 31000 500 57 260 20 times 10minus10 30
Therefore the equation could be simplified to (4) and theinitial condition could be expressed as (5)
119911 = radic1198632 minus 1199091198632 minus 1198692119871 (4)
119885 (119909 0) = radic1198632 minus 1199091198632 minus 1198692119871 (5)
Based on the above parameters the seepage equation ofthe seepage field model could be given as follows
120597120597119909 [119896119885 (119909 119905) 120597119885 (119909 119905)120597119909 ] = 120583120597119885 (119909 119905)120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
119885 (119909 0) = radic1198632 minus 1199091198632 minus 1198692119871
(6)
22 Seepage Equation Solution Several researchers haveattempted to determine the solution of (120597120597119909)[119896119885(119909 119905)(120597119885(119909119905)120597119905)] = 120583(120597119885(119909 119905)120597119905) [23 24] Therefore the equation islinearized in this research The first 119885(119909 119905) on the left sideof the equation could be considered a constant and replacedwith 119885119898 (119885119898 = (119863 + 119869)2) Accordingly the linearized seep-age equation is as follows
11988511989811989612059721198851205971199092 = 120583120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0119885 (119909 0) = 119890 (119909)
in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871
(7)
Finally the solution of the equation is again (see theAppendixfor specific solutions)
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(8)
3 Application to a Real Field Case A CoastalFoundation Pit
31 General Situation of a Coastal Foundation Pit Thecoastalfoundation pit chosen in the research is at 90m from the bayThe field area mainly comprises Quaternary artificial fill andHolocenemarine depositional strataThe physicomechanicalparameters of every stratum are shown in Table 1 To supportthe foundation a bored pile and an anchor cable supportsystem is used in the engineeringThe diameters of the boredpile and thewaterproof pile are 800mmwhile the pile spacingis 1200mm The detailed profile of the support system is
shown in Figure 2 and the parameters of the three anchorcables are indicated in Table 2
According to the official hydrogeological data the meantide water level is 123m (119863 = 123m) and the average annualwater level on the side of the foundation pit is 104m (119869 =104m) By analyzing the official hydrogeological data thechange of the tidal water level satisfies the following law
119891 (119905) = 12 lowast sin(1205871199056 ) + 123 (9)
The timendashspace evolution equation of the seepage fieldis obtained based on (8) and (9) The relationship of the
4 Mathematical Problems in Engineering
Table 2 Calculated parameters of anchor
Cablenumber
Horizontalspacingm Angle∘
Total lengthof the
cablesm
Length of freesegmentm
Length ofanchoragem
Prestressinglocked
valueKNMG1 2400 30 19000 9000 10000 230MG2 2400 25 20000 10000 10000 290MG3 2400 25 15000 6000 9000 260
Cable
Waterproof pileSupporting pile
Top beam
Supporting pile
MG1
MG2
MG3Waterproof pile
20
Ｇ30
Ｇ30
Ｇ30
Ｇ
08 Ｇ08 Ｇ
Figure 2 Cross-section of foundation pit supporting structure
seepage field with time and infiltration distance is analyzedby MATLAB in two 24 h periods The spatiotemporal evolu-tionary characteristic surface of the seepage field is presentedin Figure 3
32 Numerical Simulation The FlAC3D was used to analyzethe stability of coastal deep foundation pitThe dimensions ofthe model established in the research are 80m times 50m times 40mwith an excavation depth of 11m and an excavation width of30m The model is divided into 80000 zones and includes8634 grid points The mesh sketch of the numerical modelis shown in Figure 4
The upper surface of the model is a free boundary andthe bottom surface of the model is a fixed boundary Thefront and back boundaries constrain the displacement in the119910 direction and the left and right boundaries constrain thedisplacement in the 119909 directionThe computation parametersof the soil and cable are given in Table 2 Based on the changein water level of the seepage path in a tidal cycle (as shownin Table 3) the pore water pressure was applied to the model
as dynamic external stress Then dynamic calculation wascarried out during excavation of foundation pit Figure 5shows the calculation flow chart of numerical simulation
4 Analysis of Numerical Results
Figure 6 shows that the change in the horizontal displacementof the model increases with the excavation depth Themaximum horizontal displacement occurs in the upper partof the foundation pit sidewall The 119883-displacement of themodel increases with the excavation depth of the foundationpitThe isosurface of the119883-displacement is diffused in an arcand gradually tends to zero
Figure 7 shows the surface subsidence around the foun-dation pit It can be seen that the surface subsidence aroundthe foundation pit gradually increases with the excavationdepth and the maximum settlement is 105mm The settle-ment mainly occurs near the top of the supporting structureThe maximum settlement emerges at the top of the pile witheach excavation step
Mathematical Problems in Engineering 5
9075
6045
3015
0 2418
126
0Time (h)The distance to foundation
pit boundary (Am)
9
10
11
12
13
14
Wat
er le
vel (
m)
Figure 3 Three-dimensional image of spatiotemporal evolution characteristics of seepage
X
Z
Foundation pit boundary A11 m
80 m
30m
40m
Silty clay Gravel sand Strongly weathered granite Foundation pit
Mucky soil Filling soil
Figure 4 Mesh sketch of numerical model
Table 3 Height of the water at different times in a cycle
Timeh The distance to foundation pit boundary Am0 10 20 30 40 50
0 1128401 1149439 11701 1190401 1210363 1231 1127748 1148821 1170548 1195825 1231686 12883172 1121904 1150565 1181959 1220792 1272075 13391273 1126592 1159036 1197209 1242218 1293792 13474784 1139505 1171119 1208309 1248299 1285621 13101225 115371 1180319 1209362 1235372 1250003 12415916 1162077 1181513 1198634 12075 1200285 11685517 1160111 1173016 117924 1174748 11554 11194198 1147615 1157178 1157891 1149244 1132484 11133129 1128705 1139417 1142241 1140428 1140198 115260610 1110159 1126067 1137873 1151354 1175339 122201211 1098743 112187 1146089 1177477 122408 129452612 1098548 1128111 1163385 1208405 1267334 1341934
6 Mathematical Problems in Engineering
Start calculation
Initial condition and boundary condition
Initial equilibrium
Apply the dynamic stress generatedby the tides
Set dynamical boundary condition
Dynamic calculation
Whether the excavation ends
End calculation
Yes
Seepage calculated by MATAB
No
excavation of foundation pitStatic calculation the sequential
Figure 5 Calculation flow chart of numerical simulation
Figures 8 and 9 show the results of horizontal and verticaldisplacements of the pile top at different excavation depthof deep foundation pit excavation The simulation resultsof the horizontal and vertical displacements of the pile topare basically consistent with the site monitoring resultsThe horizontal and vertical displacements of the pile topaffected by the tide are obviously increased comparedwith thedisplacements under anhydrous conditions The deviation ofthe displacements increases with the excavation depth Thehorizontal and vertical displacements of the pile top withouttidal influence are 105mm and 82mm during excavation atthe bottomof the foundation pit However the horizontal andvertical displacements of the pile top with tidal influence are141mm and 106mm respectively The increases are 343and 226 respectively It is shown that the bearing capacityand deformation of the supporting structure in the deepfoundation pit are adversely affected by tides Therefore thetidal effect should be considered in the design of the supportstructure for a coastal deep foundation pit
5 Conclusions
To solve the seepage mechanics problems of a foundationpit affected by tides the equation of the seepage flow wasdeduced and the timendashspace evolution model of a seepagefield for a deep foundation pit was established based on theBoussinesq functionThe effects of time and space on seepageproperties are considered sophisticated
According to the results of in situ monitoring thevariations in the tide and groundwater level of the foun-dation pit on the ocean side were analyzed Although the
groundwater level of the foundation pit on the ocean sideperiodically changes similar to the tide the amplitude ofthe groundwater level is less than that of the tide Basedon the developed timendashspace seepage model the dynamicwater level of the coastal foundation pit seepage path wascalculated by MATLAB and used to analyze the deformationof the support structure of the deep foundation pit undertidal influence It is obviously seen that the seepage affectedby tide has a significant influence on the deformation ofthe deep foundation pit and increases the security risk ofengineering Therefore in order to ensure the safety of thefoundation pit and improve the construction environmentthe adverse effects of tides must be considered in the designand construction of deep foundation pits in coastal regions
The dynamic and fluid coupling calculation is a rathercomplex question We turned a complex problem into tworelatively simple problems One was simulated by MATLABbased on the seepage model presented in our manuscript andthe other was simulated by dynamic calculation of FLAC3DThe study can provide a new idea for the stability analysis ofdeep foundation pit under the influence of tides
Appendix
For
11988511989811989612059721198851205971199092 = 119906120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
Mathematical Problems in Engineering 7
00000E + 00
Contour of X-displacement
minus60000E minus 04minus90000E minus 04minus12000E minus 03
minus30000E minus 04
minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03minus30000E minus 03minus32183E minus 03
(a)
00000E + 00
Contour of X-displacement
minus12000E minus 03minus60000E minus 04
minus18000E minus 03minus24000E minus 03minus30000E minus 03minus36000E minus 03minus42000E minus 03minus48000E minus 03minus54000E minus 03
minus66000E minus 03minus72000E minus 03minus78000E minus 03minus78211E minus 03
minus60000E minus 03
(b)
Contour of X-displacement
minus10000E minus 03minus20000E minus 03minus30000E minus 03minus40000E minus 03minus50000E minus 03minus60000E minus 03minus70000E minus 03minus80000E minus 03
minus10000E minus 02minus90000E minus 03
minus11000E minus 02minus11113E minus 02
00000E + 00
(c)
Contour of X-displacement
minus20000E minus 03minus40000E minus 03minus60000E minus 03minus80000E minus 03minus10000E minus 02minus12000E minus 02minus14000E minus 02
00000E + 00
minus16000E minus 02minus16120E minus 02
(d)
Figure 6 Contour of119883-displacement
119885 (119909 0) = 119890 (119909) in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871 (A1)
assume that 119881(119909 119905) = 119860(119905) lowast 119909 + 119861(119905) satisfies (A1) then119881 (119909 119905) = 119891 (119905) (A2)
Assume that 119885 (119909 119905) = 119881 (119909 119905) + 119882 (119909 119905) (A3)
The following equation will be obtained according to (A1)and (A2)
119882119905 minus 119901119882119909119909 = minus1198911015840 (119905)119882 (0 119905) = 0120597119882120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0119882 (119909 0) = 119873 (119909)
(A4)
where 119901 = 119906119885119898119896 and119873(119909) = 119890(119909) minus 119891(0)
Assume that119882(119909 119905) = 119882I (119909 119905) + 119882II (119909 119905) (A5)
Equation (A5) is equivalent to the following equations
119882I119905 minus 119901119882I
119909119909 = 0119882I (0 119905) = 0120597119882I
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882I (119909 0) = 119873 (119909)
(A6)
119882II119905 minus 119901119882II
119909119909 = minus1198911015840 (119905)119882II (0 119905) = 0120597119882II
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882II (119909 0) = 0
(A7)
8 Mathematical Problems in Engineering
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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
4 Mathematical Problems in Engineering
Table 2 Calculated parameters of anchor
Cablenumber
Horizontalspacingm Angle∘
Total lengthof the
cablesm
Length of freesegmentm
Length ofanchoragem
Prestressinglocked
valueKNMG1 2400 30 19000 9000 10000 230MG2 2400 25 20000 10000 10000 290MG3 2400 25 15000 6000 9000 260
Cable
Waterproof pileSupporting pile
Top beam
Supporting pile
MG1
MG2
MG3Waterproof pile
20
Ｇ30
Ｇ30
Ｇ30
Ｇ
08 Ｇ08 Ｇ
Figure 2 Cross-section of foundation pit supporting structure
seepage field with time and infiltration distance is analyzedby MATLAB in two 24 h periods The spatiotemporal evolu-tionary characteristic surface of the seepage field is presentedin Figure 3
32 Numerical Simulation The FlAC3D was used to analyzethe stability of coastal deep foundation pitThe dimensions ofthe model established in the research are 80m times 50m times 40mwith an excavation depth of 11m and an excavation width of30m The model is divided into 80000 zones and includes8634 grid points The mesh sketch of the numerical modelis shown in Figure 4
The upper surface of the model is a free boundary andthe bottom surface of the model is a fixed boundary Thefront and back boundaries constrain the displacement in the119910 direction and the left and right boundaries constrain thedisplacement in the 119909 directionThe computation parametersof the soil and cable are given in Table 2 Based on the changein water level of the seepage path in a tidal cycle (as shownin Table 3) the pore water pressure was applied to the model
as dynamic external stress Then dynamic calculation wascarried out during excavation of foundation pit Figure 5shows the calculation flow chart of numerical simulation
4 Analysis of Numerical Results
Figure 6 shows that the change in the horizontal displacementof the model increases with the excavation depth Themaximum horizontal displacement occurs in the upper partof the foundation pit sidewall The 119883-displacement of themodel increases with the excavation depth of the foundationpitThe isosurface of the119883-displacement is diffused in an arcand gradually tends to zero
Figure 7 shows the surface subsidence around the foun-dation pit It can be seen that the surface subsidence aroundthe foundation pit gradually increases with the excavationdepth and the maximum settlement is 105mm The settle-ment mainly occurs near the top of the supporting structureThe maximum settlement emerges at the top of the pile witheach excavation step
Mathematical Problems in Engineering 5
9075
6045
3015
0 2418
126
0Time (h)The distance to foundation
pit boundary (Am)
9
10
11
12
13
14
Wat
er le
vel (
m)
Figure 3 Three-dimensional image of spatiotemporal evolution characteristics of seepage
X
Z
Foundation pit boundary A11 m
80 m
30m
40m
Silty clay Gravel sand Strongly weathered granite Foundation pit
Mucky soil Filling soil
Figure 4 Mesh sketch of numerical model
Table 3 Height of the water at different times in a cycle
Timeh The distance to foundation pit boundary Am0 10 20 30 40 50
0 1128401 1149439 11701 1190401 1210363 1231 1127748 1148821 1170548 1195825 1231686 12883172 1121904 1150565 1181959 1220792 1272075 13391273 1126592 1159036 1197209 1242218 1293792 13474784 1139505 1171119 1208309 1248299 1285621 13101225 115371 1180319 1209362 1235372 1250003 12415916 1162077 1181513 1198634 12075 1200285 11685517 1160111 1173016 117924 1174748 11554 11194198 1147615 1157178 1157891 1149244 1132484 11133129 1128705 1139417 1142241 1140428 1140198 115260610 1110159 1126067 1137873 1151354 1175339 122201211 1098743 112187 1146089 1177477 122408 129452612 1098548 1128111 1163385 1208405 1267334 1341934
6 Mathematical Problems in Engineering
Start calculation
Initial condition and boundary condition
Initial equilibrium
Apply the dynamic stress generatedby the tides
Set dynamical boundary condition
Dynamic calculation
Whether the excavation ends
End calculation
Yes
Seepage calculated by MATAB
No
excavation of foundation pitStatic calculation the sequential
Figure 5 Calculation flow chart of numerical simulation
Figures 8 and 9 show the results of horizontal and verticaldisplacements of the pile top at different excavation depthof deep foundation pit excavation The simulation resultsof the horizontal and vertical displacements of the pile topare basically consistent with the site monitoring resultsThe horizontal and vertical displacements of the pile topaffected by the tide are obviously increased comparedwith thedisplacements under anhydrous conditions The deviation ofthe displacements increases with the excavation depth Thehorizontal and vertical displacements of the pile top withouttidal influence are 105mm and 82mm during excavation atthe bottomof the foundation pit However the horizontal andvertical displacements of the pile top with tidal influence are141mm and 106mm respectively The increases are 343and 226 respectively It is shown that the bearing capacityand deformation of the supporting structure in the deepfoundation pit are adversely affected by tides Therefore thetidal effect should be considered in the design of the supportstructure for a coastal deep foundation pit
5 Conclusions
To solve the seepage mechanics problems of a foundationpit affected by tides the equation of the seepage flow wasdeduced and the timendashspace evolution model of a seepagefield for a deep foundation pit was established based on theBoussinesq functionThe effects of time and space on seepageproperties are considered sophisticated
According to the results of in situ monitoring thevariations in the tide and groundwater level of the foun-dation pit on the ocean side were analyzed Although the
groundwater level of the foundation pit on the ocean sideperiodically changes similar to the tide the amplitude ofthe groundwater level is less than that of the tide Basedon the developed timendashspace seepage model the dynamicwater level of the coastal foundation pit seepage path wascalculated by MATLAB and used to analyze the deformationof the support structure of the deep foundation pit undertidal influence It is obviously seen that the seepage affectedby tide has a significant influence on the deformation ofthe deep foundation pit and increases the security risk ofengineering Therefore in order to ensure the safety of thefoundation pit and improve the construction environmentthe adverse effects of tides must be considered in the designand construction of deep foundation pits in coastal regions
The dynamic and fluid coupling calculation is a rathercomplex question We turned a complex problem into tworelatively simple problems One was simulated by MATLABbased on the seepage model presented in our manuscript andthe other was simulated by dynamic calculation of FLAC3DThe study can provide a new idea for the stability analysis ofdeep foundation pit under the influence of tides
Appendix
For
11988511989811989612059721198851205971199092 = 119906120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
Mathematical Problems in Engineering 7
00000E + 00
Contour of X-displacement
minus60000E minus 04minus90000E minus 04minus12000E minus 03
minus30000E minus 04
minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03minus30000E minus 03minus32183E minus 03
(a)
00000E + 00
Contour of X-displacement
minus12000E minus 03minus60000E minus 04
minus18000E minus 03minus24000E minus 03minus30000E minus 03minus36000E minus 03minus42000E minus 03minus48000E minus 03minus54000E minus 03
minus66000E minus 03minus72000E minus 03minus78000E minus 03minus78211E minus 03
minus60000E minus 03
(b)
Contour of X-displacement
minus10000E minus 03minus20000E minus 03minus30000E minus 03minus40000E minus 03minus50000E minus 03minus60000E minus 03minus70000E minus 03minus80000E minus 03
minus10000E minus 02minus90000E minus 03
minus11000E minus 02minus11113E minus 02
00000E + 00
(c)
Contour of X-displacement
minus20000E minus 03minus40000E minus 03minus60000E minus 03minus80000E minus 03minus10000E minus 02minus12000E minus 02minus14000E minus 02
00000E + 00
minus16000E minus 02minus16120E minus 02
(d)
Figure 6 Contour of119883-displacement
119885 (119909 0) = 119890 (119909) in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871 (A1)
assume that 119881(119909 119905) = 119860(119905) lowast 119909 + 119861(119905) satisfies (A1) then119881 (119909 119905) = 119891 (119905) (A2)
Assume that 119885 (119909 119905) = 119881 (119909 119905) + 119882 (119909 119905) (A3)
The following equation will be obtained according to (A1)and (A2)
119882119905 minus 119901119882119909119909 = minus1198911015840 (119905)119882 (0 119905) = 0120597119882120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0119882 (119909 0) = 119873 (119909)
(A4)
where 119901 = 119906119885119898119896 and119873(119909) = 119890(119909) minus 119891(0)
Assume that119882(119909 119905) = 119882I (119909 119905) + 119882II (119909 119905) (A5)
Equation (A5) is equivalent to the following equations
119882I119905 minus 119901119882I
119909119909 = 0119882I (0 119905) = 0120597119882I
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882I (119909 0) = 119873 (119909)
(A6)
119882II119905 minus 119901119882II
119909119909 = minus1198911015840 (119905)119882II (0 119905) = 0120597119882II
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882II (119909 0) = 0
(A7)
8 Mathematical Problems in Engineering
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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 5
9075
6045
3015
0 2418
126
0Time (h)The distance to foundation
pit boundary (Am)
9
10
11
12
13
14
Wat
er le
vel (
m)
Figure 3 Three-dimensional image of spatiotemporal evolution characteristics of seepage
X
Z
Foundation pit boundary A11 m
80 m
30m
40m
Silty clay Gravel sand Strongly weathered granite Foundation pit
Mucky soil Filling soil
Figure 4 Mesh sketch of numerical model
Table 3 Height of the water at different times in a cycle
Timeh The distance to foundation pit boundary Am0 10 20 30 40 50
0 1128401 1149439 11701 1190401 1210363 1231 1127748 1148821 1170548 1195825 1231686 12883172 1121904 1150565 1181959 1220792 1272075 13391273 1126592 1159036 1197209 1242218 1293792 13474784 1139505 1171119 1208309 1248299 1285621 13101225 115371 1180319 1209362 1235372 1250003 12415916 1162077 1181513 1198634 12075 1200285 11685517 1160111 1173016 117924 1174748 11554 11194198 1147615 1157178 1157891 1149244 1132484 11133129 1128705 1139417 1142241 1140428 1140198 115260610 1110159 1126067 1137873 1151354 1175339 122201211 1098743 112187 1146089 1177477 122408 129452612 1098548 1128111 1163385 1208405 1267334 1341934
6 Mathematical Problems in Engineering
Start calculation
Initial condition and boundary condition
Initial equilibrium
Apply the dynamic stress generatedby the tides
Set dynamical boundary condition
Dynamic calculation
Whether the excavation ends
End calculation
Yes
Seepage calculated by MATAB
No
excavation of foundation pitStatic calculation the sequential
Figure 5 Calculation flow chart of numerical simulation
Figures 8 and 9 show the results of horizontal and verticaldisplacements of the pile top at different excavation depthof deep foundation pit excavation The simulation resultsof the horizontal and vertical displacements of the pile topare basically consistent with the site monitoring resultsThe horizontal and vertical displacements of the pile topaffected by the tide are obviously increased comparedwith thedisplacements under anhydrous conditions The deviation ofthe displacements increases with the excavation depth Thehorizontal and vertical displacements of the pile top withouttidal influence are 105mm and 82mm during excavation atthe bottomof the foundation pit However the horizontal andvertical displacements of the pile top with tidal influence are141mm and 106mm respectively The increases are 343and 226 respectively It is shown that the bearing capacityand deformation of the supporting structure in the deepfoundation pit are adversely affected by tides Therefore thetidal effect should be considered in the design of the supportstructure for a coastal deep foundation pit
5 Conclusions
To solve the seepage mechanics problems of a foundationpit affected by tides the equation of the seepage flow wasdeduced and the timendashspace evolution model of a seepagefield for a deep foundation pit was established based on theBoussinesq functionThe effects of time and space on seepageproperties are considered sophisticated
According to the results of in situ monitoring thevariations in the tide and groundwater level of the foun-dation pit on the ocean side were analyzed Although the
groundwater level of the foundation pit on the ocean sideperiodically changes similar to the tide the amplitude ofthe groundwater level is less than that of the tide Basedon the developed timendashspace seepage model the dynamicwater level of the coastal foundation pit seepage path wascalculated by MATLAB and used to analyze the deformationof the support structure of the deep foundation pit undertidal influence It is obviously seen that the seepage affectedby tide has a significant influence on the deformation ofthe deep foundation pit and increases the security risk ofengineering Therefore in order to ensure the safety of thefoundation pit and improve the construction environmentthe adverse effects of tides must be considered in the designand construction of deep foundation pits in coastal regions
The dynamic and fluid coupling calculation is a rathercomplex question We turned a complex problem into tworelatively simple problems One was simulated by MATLABbased on the seepage model presented in our manuscript andthe other was simulated by dynamic calculation of FLAC3DThe study can provide a new idea for the stability analysis ofdeep foundation pit under the influence of tides
Appendix
For
11988511989811989612059721198851205971199092 = 119906120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
Mathematical Problems in Engineering 7
00000E + 00
Contour of X-displacement
minus60000E minus 04minus90000E minus 04minus12000E minus 03
minus30000E minus 04
minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03minus30000E minus 03minus32183E minus 03
(a)
00000E + 00
Contour of X-displacement
minus12000E minus 03minus60000E minus 04
minus18000E minus 03minus24000E minus 03minus30000E minus 03minus36000E minus 03minus42000E minus 03minus48000E minus 03minus54000E minus 03
minus66000E minus 03minus72000E minus 03minus78000E minus 03minus78211E minus 03
minus60000E minus 03
(b)
Contour of X-displacement
minus10000E minus 03minus20000E minus 03minus30000E minus 03minus40000E minus 03minus50000E minus 03minus60000E minus 03minus70000E minus 03minus80000E minus 03
minus10000E minus 02minus90000E minus 03
minus11000E minus 02minus11113E minus 02
00000E + 00
(c)
Contour of X-displacement
minus20000E minus 03minus40000E minus 03minus60000E minus 03minus80000E minus 03minus10000E minus 02minus12000E minus 02minus14000E minus 02
00000E + 00
minus16000E minus 02minus16120E minus 02
(d)
Figure 6 Contour of119883-displacement
119885 (119909 0) = 119890 (119909) in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871 (A1)
assume that 119881(119909 119905) = 119860(119905) lowast 119909 + 119861(119905) satisfies (A1) then119881 (119909 119905) = 119891 (119905) (A2)
Assume that 119885 (119909 119905) = 119881 (119909 119905) + 119882 (119909 119905) (A3)
The following equation will be obtained according to (A1)and (A2)
119882119905 minus 119901119882119909119909 = minus1198911015840 (119905)119882 (0 119905) = 0120597119882120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0119882 (119909 0) = 119873 (119909)
(A4)
where 119901 = 119906119885119898119896 and119873(119909) = 119890(119909) minus 119891(0)
Assume that119882(119909 119905) = 119882I (119909 119905) + 119882II (119909 119905) (A5)
Equation (A5) is equivalent to the following equations
119882I119905 minus 119901119882I
119909119909 = 0119882I (0 119905) = 0120597119882I
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882I (119909 0) = 119873 (119909)
(A6)
119882II119905 minus 119901119882II
119909119909 = minus1198911015840 (119905)119882II (0 119905) = 0120597119882II
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882II (119909 0) = 0
(A7)
8 Mathematical Problems in Engineering
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
Hindawiwwwhindawicom Volume 2018
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Mathematical Problems in Engineering
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Submit your manuscripts atwwwhindawicom
6 Mathematical Problems in Engineering
Start calculation
Initial condition and boundary condition
Initial equilibrium
Apply the dynamic stress generatedby the tides
Set dynamical boundary condition
Dynamic calculation
Whether the excavation ends
End calculation
Yes
Seepage calculated by MATAB
No
excavation of foundation pitStatic calculation the sequential
Figure 5 Calculation flow chart of numerical simulation
Figures 8 and 9 show the results of horizontal and verticaldisplacements of the pile top at different excavation depthof deep foundation pit excavation The simulation resultsof the horizontal and vertical displacements of the pile topare basically consistent with the site monitoring resultsThe horizontal and vertical displacements of the pile topaffected by the tide are obviously increased comparedwith thedisplacements under anhydrous conditions The deviation ofthe displacements increases with the excavation depth Thehorizontal and vertical displacements of the pile top withouttidal influence are 105mm and 82mm during excavation atthe bottomof the foundation pit However the horizontal andvertical displacements of the pile top with tidal influence are141mm and 106mm respectively The increases are 343and 226 respectively It is shown that the bearing capacityand deformation of the supporting structure in the deepfoundation pit are adversely affected by tides Therefore thetidal effect should be considered in the design of the supportstructure for a coastal deep foundation pit
5 Conclusions
To solve the seepage mechanics problems of a foundationpit affected by tides the equation of the seepage flow wasdeduced and the timendashspace evolution model of a seepagefield for a deep foundation pit was established based on theBoussinesq functionThe effects of time and space on seepageproperties are considered sophisticated
According to the results of in situ monitoring thevariations in the tide and groundwater level of the foun-dation pit on the ocean side were analyzed Although the
groundwater level of the foundation pit on the ocean sideperiodically changes similar to the tide the amplitude ofthe groundwater level is less than that of the tide Basedon the developed timendashspace seepage model the dynamicwater level of the coastal foundation pit seepage path wascalculated by MATLAB and used to analyze the deformationof the support structure of the deep foundation pit undertidal influence It is obviously seen that the seepage affectedby tide has a significant influence on the deformation ofthe deep foundation pit and increases the security risk ofengineering Therefore in order to ensure the safety of thefoundation pit and improve the construction environmentthe adverse effects of tides must be considered in the designand construction of deep foundation pits in coastal regions
The dynamic and fluid coupling calculation is a rathercomplex question We turned a complex problem into tworelatively simple problems One was simulated by MATLABbased on the seepage model presented in our manuscript andthe other was simulated by dynamic calculation of FLAC3DThe study can provide a new idea for the stability analysis ofdeep foundation pit under the influence of tides
Appendix
For
11988511989811989612059721198851205971199092 = 119906120597119885120597119905119885 (0 119905) = 119891 (119905)120597119885120597119909
10038161003816100381610038161003816100381610038161003816119909=119897 = 0
Mathematical Problems in Engineering 7
00000E + 00
Contour of X-displacement
minus60000E minus 04minus90000E minus 04minus12000E minus 03
minus30000E minus 04
minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03minus30000E minus 03minus32183E minus 03
(a)
00000E + 00
Contour of X-displacement
minus12000E minus 03minus60000E minus 04
minus18000E minus 03minus24000E minus 03minus30000E minus 03minus36000E minus 03minus42000E minus 03minus48000E minus 03minus54000E minus 03
minus66000E minus 03minus72000E minus 03minus78000E minus 03minus78211E minus 03
minus60000E minus 03
(b)
Contour of X-displacement
minus10000E minus 03minus20000E minus 03minus30000E minus 03minus40000E minus 03minus50000E minus 03minus60000E minus 03minus70000E minus 03minus80000E minus 03
minus10000E minus 02minus90000E minus 03
minus11000E minus 02minus11113E minus 02
00000E + 00
(c)
Contour of X-displacement
minus20000E minus 03minus40000E minus 03minus60000E minus 03minus80000E minus 03minus10000E minus 02minus12000E minus 02minus14000E minus 02
00000E + 00
minus16000E minus 02minus16120E minus 02
(d)
Figure 6 Contour of119883-displacement
119885 (119909 0) = 119890 (119909) in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871 (A1)
assume that 119881(119909 119905) = 119860(119905) lowast 119909 + 119861(119905) satisfies (A1) then119881 (119909 119905) = 119891 (119905) (A2)
Assume that 119885 (119909 119905) = 119881 (119909 119905) + 119882 (119909 119905) (A3)
The following equation will be obtained according to (A1)and (A2)
119882119905 minus 119901119882119909119909 = minus1198911015840 (119905)119882 (0 119905) = 0120597119882120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0119882 (119909 0) = 119873 (119909)
(A4)
where 119901 = 119906119885119898119896 and119873(119909) = 119890(119909) minus 119891(0)
Assume that119882(119909 119905) = 119882I (119909 119905) + 119882II (119909 119905) (A5)
Equation (A5) is equivalent to the following equations
119882I119905 minus 119901119882I
119909119909 = 0119882I (0 119905) = 0120597119882I
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882I (119909 0) = 119873 (119909)
(A6)
119882II119905 minus 119901119882II
119909119909 = minus1198911015840 (119905)119882II (0 119905) = 0120597119882II
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882II (119909 0) = 0
(A7)
8 Mathematical Problems in Engineering
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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 7
00000E + 00
Contour of X-displacement
minus60000E minus 04minus90000E minus 04minus12000E minus 03
minus30000E minus 04
minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03minus30000E minus 03minus32183E minus 03
(a)
00000E + 00
Contour of X-displacement
minus12000E minus 03minus60000E minus 04
minus18000E minus 03minus24000E minus 03minus30000E minus 03minus36000E minus 03minus42000E minus 03minus48000E minus 03minus54000E minus 03
minus66000E minus 03minus72000E minus 03minus78000E minus 03minus78211E minus 03
minus60000E minus 03
(b)
Contour of X-displacement
minus10000E minus 03minus20000E minus 03minus30000E minus 03minus40000E minus 03minus50000E minus 03minus60000E minus 03minus70000E minus 03minus80000E minus 03
minus10000E minus 02minus90000E minus 03
minus11000E minus 02minus11113E minus 02
00000E + 00
(c)
Contour of X-displacement
minus20000E minus 03minus40000E minus 03minus60000E minus 03minus80000E minus 03minus10000E minus 02minus12000E minus 02minus14000E minus 02
00000E + 00
minus16000E minus 02minus16120E minus 02
(d)
Figure 6 Contour of119883-displacement
119885 (119909 0) = 119890 (119909) in which 119890 (119909) = radic1198632 minus 1199091198632 minus 1198692119871 (A1)
assume that 119881(119909 119905) = 119860(119905) lowast 119909 + 119861(119905) satisfies (A1) then119881 (119909 119905) = 119891 (119905) (A2)
Assume that 119885 (119909 119905) = 119881 (119909 119905) + 119882 (119909 119905) (A3)
The following equation will be obtained according to (A1)and (A2)
119882119905 minus 119901119882119909119909 = minus1198911015840 (119905)119882 (0 119905) = 0120597119882120597119909
10038161003816100381610038161003816100381610038161003816119909=119871 = 0119882 (119909 0) = 119873 (119909)
(A4)
where 119901 = 119906119885119898119896 and119873(119909) = 119890(119909) minus 119891(0)
Assume that119882(119909 119905) = 119882I (119909 119905) + 119882II (119909 119905) (A5)
Equation (A5) is equivalent to the following equations
119882I119905 minus 119901119882I
119909119909 = 0119882I (0 119905) = 0120597119882I
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882I (119909 0) = 119873 (119909)
(A6)
119882II119905 minus 119901119882II
119909119909 = minus1198911015840 (119905)119882II (0 119905) = 0120597119882II
120597119909100381610038161003816100381610038161003816100381610038161003816119909=119871 = 0
119882II (119909 0) = 0
(A7)
8 Mathematical Problems in Engineering
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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
00000E + 00
Contour ofZ-displacement
minus10000E minus 04minus15000E minus 04minus20000E minus 04
minus50000E minus 05
minus25000E minus 04minus30000E minus 04minus35000E minus 04minus40000E minus 04minus45000E minus 04minus50000E minus 04minus55000E minus 04minus56290E minus 04
(a)
00000E + 00
Contour of Z-displacement
minus60000E minus 03minus30000E minus 04
minus90000E minus 03minus12000E minus 03minus15000E minus 03minus18000E minus 03minus21000E minus 03minus24000E minus 03minus27000E minus 03
minus33000E minus 03minus36000E minus 03minus37642E minus 03
minus30000E minus 03
(b)
Contour of Z-displacement
minus80000E minus 04minus16000E minus 03minus24000E minus 03minus32000E minus 03minus40000E minus 03minus48000E minus 03minus56000E minus 03minus64000E minus 03
minus80000E minus 03minus72000E minus 03
minus88000E minus 03
00000E + 00
minus88487E minus 03
(c)
Contour of Z-displacement
minus90000E minus 04minus18000E minus 03minus29000E minus 03minus36000E minus 03minus45000E minus 03minus54000E minus 03minus63000E minus 03
00000E + 00
minus72000E minus 03minus81000E minus 03
00000E + 00
minus90000E minus 03minus99000E minus 03minus10544E minus 02
(d)
Figure 7 Surface subsidence around the foundation pit
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
50
100
150
200
X-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 8 Horizontal displacement of pile top
The solution of (A6) was obtained by the method of separa-tion of variables
119882I (119909 119905) = infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 (A8)
Simulation results under influence of tideSimulation results under anhydrous conditionsSite monitoring results
00
30
60
90
120
150
Z-d
is at
the t
op o
f pile
(mm
)
30 60 90 12000The depth of excavation (m)
Figure 9 Vertical displacement of pile top
Here 119862119899 = (2119871) int1198710119873(119909) sdot sin((119899 + 05)120587119909119871)119889119909 119873(119909) =119890(119909) minus 119891(0)
For (A7) based on theorem of impulse the solution wasadopted
Mathematical Problems in Engineering 9
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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
Let119882II (119909 119905) = int1199050119878 (119909 119905 120580) 119889120580 (A9)
The following equations were obtained by combining (A7)and (A9)
119878119905 minus 119901119878119909119909 = 0119878 (0 119905) = 0
12059711987812059711990910038161003816100381610038161003816100381610038161003816119909=119871 = 0
119878 (119909 120591) = minus1198911015840 (120591) (A10)
The solution of (A10) was also obtained by the method ofseparation of variables
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580)
In which 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A11)
Above all
119885 (119909 119905) = 119891 (119905) + infinsum119899=1
119862119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)119905 + int1199050119878 (119909 119905 120580) 119889120580
in which 119901 = 119885119898119896119906 119862119899 = 2119871 int1198710119873(119909) sdot sin (119899 + 05) 120587119909119871 119889119909 119873 (119909) = 119890 (119909) minus 119891 (0)
119878 (119909 119905 120580) = infinsum119899=1
119860119899 sin (119899 + 05) 120587119909119871 sdot 119890((119899+05)212058721198712119901)(119905minus120580) 119860119899 = 2119871 int1198710119872(119909 120580) sdot sin (119899 + 05) 120587119909119871 119889119909 119872 (119909 120580) = minus1198911015840 (120591)
(A12)
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The research work presented in this paper was supported bythe National Natural Science Foundation of China (Grant no51379113) and the Specialized Research Fund of the NationalKey Research andDevelopment Program of China (Grant no2016YFC0600803)
References
[1] J Wang X Liu Y Wu et al ldquoField experiment and numericalsimulation of coupling non-Darcy flow caused by curtainand pumping well in foundation pit dewateringrdquo Journal ofHydrology vol 549 pp 277ndash293 2017
[2] A Cong ldquoDiscussion on several issues of seepage stability ofdeep foundation pit inmultilayered formationrdquoChinese Journal
of Rock Mechanics and Engineering vol 28 no 10 pp 2018ndash2023 2009
[3] N Zhou P A Vermeer R Lou Y Tang and S JiangldquoNumerical simulation of deep foundation pit dewateringand optimization of controlling land subsidencerdquo EngineeringGeology vol 114 no 3-4 pp 251ndash260 2010
[4] A Atangana and P D Vermeulen ldquoAnalytical solutions of aspace-time fractional derivative of groundwater flow equationrdquoAbstract and Applied Analysis Article ID 381753 Art ID 38175311 pages 2014
[5] Y Wang ldquo3-Dimensional Stochastic Seepage Analysis of aYangtze River Embankmentrdquo Mathematical Problems in Engi-neering vol 2015 Article ID 420264 2015
[6] JQiuD Zheng andKZhu ldquoSeepagemonitoringmodels studyof earth-rock dams influenced by rainstormsrdquo MathematicalProblems in Engineering vol 2016 Article ID 1656738 11 pages2016
[7] C E Jacob Flow of ground water John Wiley amp Son Inc NewYork NY USA 1950
[8] D S Jeng L Li and D A Barry ldquoAnalytical solution fortidal propagation in a coupled semi-confinedphreatic coastalaquiferrdquo Advances in Water Resources vol 25 no 5 pp 577ndash584 2002
[9] L Li D A Barry and C B Pattiaratchi ldquoNumerical modellingof tide-induced beach water table fluctuationsrdquo Coastal Engi-neering Journal vol 30 no 1-2 pp 105ndash123 1997
[10] D S Zhang H J Zheng and J Geng ldquoPhysical mechanism andunitary mathematical equation for tidal phenomena of ground
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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
10 Mathematical Problems in Engineering
waterrdquo Chinese Journal of Seismology and Geology vol 24 no 2pp 208ndash214 2002
[11] Q N Guo ldquoTide-induced groundwater fluctuation in a beachaquiferrdquo Chinese Journal of Geotechnical Investigation amp Survey-ing no 5 pp 36ndash39 2010
[12] P Nielsen ldquoTidal dynamics of the water table in beachesrdquoWaterResources Research vol 26 no 9 pp 2127ndash2134 1990
[13] D Liang H Gotoh and A Khayyer ldquoBoussinesq modellingof solitary wave and N-wave runup on coastrdquo Applied OceanResearch vol 42 pp 144ndash154 2013
[14] N Su ldquoThe fractional Boussinesq equation of groundwater flowand its applicationsrdquo Journal of Hydrology vol 547 pp 403ndash4122017
[15] L Li D A Barry F Stagnitti J-Y Parlange and D-S JengldquoBeach water table fluctuations due to spring-neap tides Mov-ing boundary effectsrdquo Advances in Water Resources vol 23 no8 pp 817ndash824 2000
[16] H Jiang F Liu I Turner and K Burrage ldquoAnalytical solu-tions for the multi-term time-space Caputo-Riesz fractionaladvection-diffusion equations on a finite domainrdquo Journal ofMathematical Analysis and Applications vol 389 no 2 pp 1117ndash1127 2012
[17] H T Teo D S Jeng B R Seymour D A Barry and L Li ldquoAnew analytical solution for water table fluctuations in coastalaquiferswith sloping beachesrdquoAdvances inWater Resources vol26 no 12 pp 1239ndash1247 2003
[18] J Boussinesq ldquoRecherches theoriques sur lrsquoecoulement desnappes drsquoeau infiltrees dans le sol et sur debit de sourcesrdquo JournalDe Mathematiques Pures Et Appliquees vol 10 no 5 pp 5ndash781904
[19] P W Werner ldquoSome problems in non-artesian ground-waterflowrdquo Eos Transactions American Geophysical Union vol 38no 4 pp 511ndash518 1957
[20] B H GildingMathematical modelling of saturated and unsatu-rated groundwater flow World Scientific Singapore 1992
[21] K Sato and Y Iwasa Groundwater Hydraulics Springer Japan2000
[22] A M Muir Wood Coastal Hydraulics Macmillan EducationLondon UK 1969
[23] A S Telyakovskiy and M B Allen ldquoPolynomial approximatesolutions to the Boussinesq equationrdquo Advances in WaterResources vol 29 no 12 pp 1767ndash1779 2006
[24] A S Telyakovskiy S Kurita and M B Allen ldquoPolynomial-based approximate solutions to the Boussinesq equation neara wellrdquo Advances in Water Resources vol 96 pp 68ndash73 2016
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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
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
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
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
![[11] Seepage [Rev2]](https://static.fdocuments.net/doc/165x107/55cf9210550346f57b932a1e/11-seepage-rev2.jpg)
