Muddy water seepage theory and its application

476Science in China Series E: Technological Sciences 2006 Vol.49 No.4 476- 484DOI: 10.1007/s11431-006-2011-4Muddy water seepage theoryand its applicationDANG Faning', LIU Yunhe', CHEN Junqiang' , ZHENG Zhongan2& WANG Zhouyu21. Institute of Water Resources and Hydro-electric Engineering, Xi'an University of Technology, Xi"an710048, China;2. City Water Matter Bureau, Shizuishan 753000, ChinaCorrespondence should be addressed to Dang Faning (dangfn@mail.xaut.edu.cn)Received July 21, 2004; accepted April 18, 2006AbstractOn condition that the seepage coefficients of sediment and stratum arenarrowly or widely different, seepage stratum is homogeneous or layer. The muddy waterseepage integral and differential equations based on Darcy's law are established, andthen their difference formula of the differential equations are derived. The methodpresented was used in the deep recharge test in Shizuishan, where power plant coolingtailing water was used to recharge and mine deep underground water. This is firstly usedin China. The law of sediment accumulation on the earth surface, and the rule of infiltrationcapacity changing with time are studied. The pattern of water table changing with timeaffected by the recharge irigation is predicted.Keywords: seepage, sediment accumulation, seepage coefficient, muddy water seepage differential equation.Common seepage calculation has a mature theory, including steady seepage andnon-steady seepage - , but they both are for clear water seepage. In refs. [9, 10] theinfluence of sedimentation of suspended matter on seepage was studied; the seepage lawof mud fluid in stratum was studied in refs. [11 - - 15]. But the influence of sedimentationof suspended matter on seepage has not been reported so far. The problem of muddywater seepage can often been touched in engineering practice, especially in the irrigatedarea of the Yellow River. Muddy water seepage which is different from variable headpermeability, is non-steady seepage. In the process of muddy water seepage, suspendedmatter will deposit unceasingly; seepage path in stratum and the total coefficient ofpermeability both change with time. This kind of engineering problems can be met asfollows: the influence of long-time recharge with myddv water gn coonare property; theinfluence of sediment accumulation in the upstre中国煤化工e through dam;the change of earth surface seepage property aYHCNMHGwater;motionstate of serous fluid in stratum during seepage proof; seepage analysis of fly ash hydr-www.scichina.com www. springerlink.comMuddy water seepage theory and its application477aulic fill dam, and so on.Shizuishan, an important industrial base, lies in the north of Ningxia Hui AutonomousRegion. From the 1980s on, the local industry has developed very fast, and ground wateris exploited more and more, thus water table of the city descended significantly. Atpresent, a big settlement crater, which covers almost half of the total basin, has beenformed in Shizuishan basin. Water table in the center of the settlement crater has fallenfrom about 50 to 80 m or so. The Yellow River Water Plant administrated by WaterResources Bureau of Shizuishan laid a scheme using power plant cooling tailing water torecharge and mine deep underground water in the vicinity of the new well field whichlies in the north of the basin. It is an unprecedented engineering project to recharge fromthe earth's surface to the depth of 70 m underground with muddy water. In order toinvestigate its feasibility, recharge test was first made in the recharging test site of theYellow River Water Plant. Because cooling water of the power plant is from the YellowRiver, it has a high temperature and sediment content. Thus recharge test is practically aproblem of muddy water infiltration. After seepage test with the sediment collected fromcooling water of the power plant, it is discovered that the seepage coefficient of thesediment is very low about 7.007x 10 "0 m/s, so the sediment can be used as impermeablematerial. The key problem in the recharge test is the interaction of sediment and seepage.This paper established seepage integral and differential equations of muddy waterbased on Darcy's law, and their difference formula of the differential equations arederived. The method presented above is utilized in the recharge test in Shizuishan, andthe rule of infiltration capacity changing with time is studied. Finally the pattern of watertable with time affected by the recharge is predicted.1 Law of muddy water seepageAccording to Darcy's law, seepage velocity is in direct proportion to hydraulic slope inlaminar flow state, that is .v=ki,(1)orq= kiA,(2)where v is seepage velocity; q is seepage quantity; i is hydraulic slope; A is sectional areavertical to the seepage path; and k is a proportional constant called seepage coefficient ofsoil.I.IWhen seepage coefficients of sediment and stratum are equivalentThe seepage question usually refers to unsaturated seepage or variable head seepage.When seepage coefficients of sediment and stratum are equivalent, the sediment willceaselessly cover on the surface of the stratum, with the result of extended seepage path,but water head is invariable. This kind of seep中国煤化工with variableseepage path, and seepage analysis of fly ash hy:YHCNMHGhis kind. Aftertime of T, the total seepage quantities per unit area Can be calculatea y78Science in China Series E: Technological SciencesQ(t)= |。 ki(t)dt,(3)where hydraulic slope i(t) is the function of time and decreases with the increase ofthickness.hi(t)=-(4)Lo+ L(t)’where h is the difference of total water head; Lo is seepage path at the beginning time; L()is increased seepage path because of sediment, and it can be expressed asL(t)= aQ(t),(5)where a is sediment concentration per unit volume of muddy water. Substituting eqs.(5) and (4) in eq. (3), the integral equation of muddy water seepage can be obtained asQ()=S%k-Lo + aQ(t)-dt.(6)The integral equation is not convenient for solution, so their difference formulas of thedifferential equations are derived. Muddy water seepage can be treated as steady seepage,over a small period of time, and seepage quantities of per unit area can be given by eq.(7):Q(t + Or)- Q(t)= ki()Ot.(7)After the first derivative by Ot，the differential equation of muddy water seepage canbe derived asQ(t + OI)- Q()= ki(t),(8)OtQ()= ki().(9)Substituting eqs. (5) and (4) in eq. (9), the other form of differential equation can begiven askhQ()=-Lo + aQ(1)(10)or(Lo + aQ())Q'()= kh.(11)This is a linear differential equation with one unknown. The numerical solution can beobtained with finite difference method, and its difference format can be written asQ(t + Ot)=-Lo + aQ()Ot + Q(t).(12)The initial condition of the equation can be expressed as Q(t)=0, and seepagequantities of later time can be derived from the seepage quantities of the previous one.1.2 When seepage coefficients of sediment and中国煤化工Muddy water seepage is apparently a kinTYHC N MH G with variable .seepage path. Because increment of the total seepage path IS much small, it has littleimpact on seepage. On the other hand, because sediment particle is very fine, seepageMuddy water seepage theory and its application479coefficient of the sediment is very small, and the sediment has a strong impact on thewhole stratum. Therefore muddy water seepage, in fact, is a kind of non-steady seepagewith variable seepage coefficient. Accordingly, after time of T, the total seepagequantities of per unit area can be given asQ(t)= I' k(t)idt.(13)(i) When the original pervious stratum is homogeneous. Assume that Lo is thicknessof stratum; ko is seepage coefficient of sediment; L(t) is thickness of sediment at t time.The seepage coefficient of stratum at t time can be expressed ask(t)=L(t)+ Lo(14)L(t) t LLg kwhere L(), thickness of sediment, can be given asL(t)= aQ(),(15)where a is sediment concentration per unit volume. Substituting eqs. (15) and (14) ineq. (13), integral equation of muddy water seepage can be derived asQ(1)= f' .aQ(t)+ Lo-idt.(16)aQ(). LoLkThe integral equation is not convenient for solution, and then its difference formula ofthe differential equations is derived. Muddy water seepage can be treated as steadyseepage over a small period of time, and then seepage quantities of per unit area can begiven asQ(t + Ot)- Q(t)= k(t)iSt .(17)ThereforeQ(t + Ot)- Q(t)= k(t)i.(18)\tAfter the first derivative by△t，the differential equation of muddy water seepage canbe obtained asQ()= k(t)i.(19)Substituting eqs. (15) and (14) in eq. (19), the other form of differential equation ofmuddy water seepage can be derived asQ(t)=-i，(20)aQ(t). LLoor中国煤化工(aQ()+上|Q'()=aiqMHCNMHG(21)kok,This is a linear differential equation with one unknown, and numerical solution can be480Science in China Series E: Technological Sciencesobtained with finite difference method, that isQ(t + Ot)=aQ(t)+ LoiOt + Q(t).(22)aQ(t). . Lk(The initial condition of the equation can be expressed as Q(t)=0, and seepagequantities of later time can be derived from the seepage quantities of the previous one.(i) When the original pervious stratum is a layered structure. Assume that n is the .layer number of stratum; L; is the thickness of i layer; k; is seepage coefficient of i layer;L() is thickness of sediment at t time. Therefore the seepage coefficient of stratum can beexpressed asL(t)+ Lok()=-L(t)」4+L2+L +(23)kokk2nThe total thickness of stratum Lo=L+L+..Ln; the thickness of sediment L() can begiven asL(1)= aQ(t).(24)Substituting eqs. (24) and (23) in eq. (13), the integral equation of muddy water seepagecan be derived asaQ()+ LQ()= |aQ(t)+4+乡+L + L,-idt.(25)kok k2k,In a similar way, within a small period of time,Q(t +△r)- Q()= k()iOt .(26)The two sides of the equation are divided by Ot ，then differential equation of muddywater seepage can be given asQ'(t)= k(t)i.(27)Substituting eqs. (24) and (23) in eq. (27), the other form of differential equation ofmuddy water seepage can be expressed asQ'(t)=-i.(28)aQ(t). L ↓LThis is a linear differential equation with one unknown, and numerical solution can beaQ()+ LoQ(t+ Ot)=aQ(t)+ 4.5 +iOt + Q(t) .(29)kokky中国煤化工The initial condition of the equation canMHCNMHG,andseepageMuddy water seepage theory and its application4812 Seepage analysis of muddy water in the refilling test site in ShizuishanRecharge test site lies in about 1000 m east of the Yellow River Water Plant. Thestratum and the topographical features of the area are similar to those of the recharge testsite, which belongs to the alluvial fan territory of the Helan Mountains. The recharge testsite is mainly composed of a rectangle recharge pond dug by manpower and an irregularrecharge pit formed by half manpower and half natural power. The calculated depth ofseepage is 75 m. It is assumed that the free surface of water table in the recharge test siteis horizontal. The recharge pond is 27 m in width and 200 m in length. The width of therecharge pit varies from 60 to 70 m, and its measured depth is about 110 m.Fig. 1 shows the geologic section plane of the recharge pit (pond) per unit length. Thestratum is made up with silt seam, sand and gravel layer, clay blanket and aquiferouslayer, and the corresponding seepage cofficients are 7.007x 10-I0, 2.315x10-, 3.98x10 6and 2.397x10 4 m/s(1.678x10+ m/s) respectively.Equation of the drawdown curve on the left and right sides of the recharge pit (pond)can be expressed asπK,Kx=m. KHe‘(0≤y<∞).2When y- >∞，both branches of the drawdown curves have their own verticalasymptote, which can be given as xo =m-7 ; ，and thus the maximal seepage quantities2K’can be expressed as L=2x。 =二According to these equations, the maximal seepageK’quantities in the recharge pit L=-= 68 m，and the maximal seepage width in theKrecharge pond L=I=31.8 m .850403020.10010203040506070X (m)中国煤化工Fig. 1. Simplified stratigraphilMYHCNMHGAccording to the theory above, the initial average seepage intensity in the recharge pit482Science in China Series E: Technological Sciencesand pond is 2.01x10 5 m /s; the average seepage intensity after one week, half month andone month is 1 .6805x10-，1.368x10- and 9.300x10。m'/s, respectively. Fig. 2 showsthe distribution of average seepage intensity with time and the average seepage intensityhere refers to seepage quantities per unit area per unit time.The initial average seepage intensity in the recharge pit and pond is 2.01x10 5 m'/s,when sediments on the earth's surface is 0.1 m in depth, the average seepage intensity is5.121x107 7 m'/s, and when sediments on the earth's surface is 0.2, 0.4 and 0.6 m in depth,the corresponding average seepage intensity is 2.594x10-, 1.305x10-7, and 8.721x10 8m /s, respectively. Fig. 3 shows the relationship of the average seepage intensity with thedepth of the sedimentation.0.0872.0x 10-6 :宜1.6x10*0.06-1.2x10-目0.04-8.0x10-7勇0.024.0x10~70300600900“12001500.020406080100120140160Time (h)Depth of sediment (cm)Fig. 2. Distribution of average seepage intensity Fig. 3. Distribution of average seepage intensity with thewith time.depth of sediment.According to laboratory experiment, unit weight of the sediment is 1.2 g/cm' and theaverage sand-carry capacity of the Yellow River in the latest three years is 3.62 kg/m',which is based on the statistical data of Shizuishan hydrological station. After calculation,the depth of earth's surface sediment with time in the recharge test is ilustrated in Fig. 4.The initial total water recharge capacity per unit time in the recharge pit is 477.576m/h, and 390.744 m/h in the recharge pond. Taking the influence of sediment onseepage coefficient into account, after 24 h, the total water recharge capacity per unittime in the pit is 466.439 m'/h, and 381.632 m'/h in the pond; and after 720 h, 221.105m'/h in the pit and 180.904 m'/h in the pond. The total water recharge capacity per unittime in the pit and pond is shown in Table 1 and Fig. 5.Table 1 Total water recharge capacity with timeTime22448963607201440Total water rechargecapacity in the pit (m/h)477.576 472.481 466.439 454.586 431.775 399.687 325.302 221.105 102. 146capacity in the pond (m/h)390.744 386.576 381.632 371.934 353.270 327.017 266.156 180.904 83.574Fig. 6 shows the distribution of water table wi中国煤化工g the sediment,from which it can be concluded that water table \HCNMHGVhenthedepthof sediment rises to some extent, the recharge capacity will decrease, and water table willbegin to decline. Therefore, the right desilting time can be determined, for theMuddy water seepage theory and its application4830.20]会6目500一x一Pit0.15]4000.101旨3000.05旨200后1000.00-0300600900 1200 1500800 1200 1600Time (h)Fig. 4. Distribution of the depth of sediment with time.Fig. 5. Distribution of total water recharge capacitywith time.。10d20 -30 d←60d|15 .90 d--120 d10-0十-200-100100200Center distence between pit and pool (m)Fig. 6. Distribution of water table with time.convenience of construction.3 ConclusionsThe muddy water seepage integral and differential equations based on Darcy's law areestablished in this paper, and their difference formulas of the differential equations arederived. The theory presented above is utilized in the recharge test in Shizuishan, and therule of recharge capacity changing with time is studied. Finally the patterm of water tablechanging with time affected by the recharge test is predicted. The model proposed in thepaper is simple in theory, meaningful in physics, valuable in practice, and has universalguiding significance in the projects such as muddy water seepage, grouting, and so on.Acknowledgements This work was supported by the National Natural Science Foundation ofChina (Grant No.10372078).中国煤化工ReferencesMHCNMHG.1 YuanLJ, Li ZQ, Wu S Z, et al. Engineering Seepage Mechanics and Its Application (in Chinese). Bejing:China Building Industry Press, 2001484Science in China Series E: Technological Sciences2 Malkawi A I H, Al-Sheriadeh M. Evaluation and rehabilition of dam seepage problems. Eng Geol, 2000, 56:335 - 3453 Bathe K T, Khoshgoftaar M R. Finite element free surface analysis without mesh iteration. Intern J NumberMethods Engrg, 1979, 4(1): 13-224 Geo-Slope Intermational Ltd. SEEPW Engineering Book. Version 5, User's Guide. Calgary: Geo-SlopeIntermational Ltd, 20015 OdenJ T, Kikuchi N. 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