9 Ground pollution 9.1竞赛题 Background Several practically important but. theoretically difficult math matical problmes pertain to the assessment of pollution. One such problem consists in deriving accurate estimates of the location and amount of pollutants seeping inaccessibly underground, and the loca- tion of their source, on the basis of very few measurements taker only around, but not necessarily directly in, the suspected polluted regIony Example Adatasetislocatedathttp://www.comapcom/mcm/proc d Is The data set (an Excel file which can be downloaded into most spreadsheets )shows measurements of pollutants in underground wa ter from 10 monitoring wells (Mw)from 1990 to 1997. The units are micrograms per liter(ug/). The location and elevation for eight wells are known and given in Table 1. The first two numbers are the coordinates of the location of the well on a Cartesian grid on a map. The third number is the altitude in feet above mean Sea Level of the water level in the well ①1999年美国大学生数学建模竞赛C题。 229 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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Tahle I Locations for eight wells in Problen C Well Number x-Coordinate (ft) y-Coordinate( ft) Elevation (fu) Nw-1 4187.5 6375,0 」482.23 NW-3 9062.5 4375.0 1387.92 NW-7 7625.0 5812.5 1400.I9 nw-g 9125,g 4000.0 NW-12 9062.5 5187.5 1394.26 Nw-12 9062.5 4562.5 1388.9 NW-13 5000.0 1394.25 Nw-14 4750.0 2562.5 I412.00 The locations and elevations of the other two wells in the data set(MW-27 and MW-33)are not known. In the data set, you will also see the letter T, M, or B after the well number, indicating that the measurements were taken at the Top, Middle, or Bottom of the aquifer in the well. Thus, Mw-7B and Mw-7M are from the same well, but from the bottom and from the middLe. Also, other mea surements indicate that water tends to flow toward well mw-9 in this area Problme one Build a mathematical model to determine whether any new pol lution has begun during this time period in the area represented by the data set, I ify the new pollutants and the lo- cation anx time of their source Problme two Before the collection of any data, the question arises whether the intended type of data and model can yeild the desired assessment of the location and amount of pollutants. Liquid chemicals may have 230 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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leaked from one of the storage tanks amoung many simialr tanks in a storage facility built over a homogeneous soil. Because probing under the many large tanks would be prohibitively expensive and danger- OLS, measuring only near the periphery of the storage facility or on the surface of the terrain seems preferable. Determine what type and number of measurements, taken only outside the boundary or on the surface of the entire storage facility, can be used in a mathematical model lo determine whether a leak has occurred, when it occurred, where( from which tank) it occurred, and how much liquid has leaked 实题苘析 地下水污染测定问题是一个真实的实际问题,它的数据是从 许多井中测量得到的,没有任何修饰和改变。题目给出一组不同 测量井在一段时期内的地下水物质浓度的数据,要求分析这组数 据并求出污染物质的数量和定位污染源;同时要求给出能够监测 污染情况最少需要的测量井数,并给出测量井分布方案。 本问题是一个非常切合实际的问题,如何合理利用并管理好 地下水是目前一个热门的研究课题。本题有如下特点 1.数据量大。本题给出8个测量井、数百种物质的测量数 据,时间从1990年到1997年,数据多达数万条。 2.并非所有数据都可靠。由于测量时期长,有些数据有缺 失;有些数据采用不同的计量单位,但数据表上并未注明,导致数 千倍的变动;还有一些数据明显属于测量错误,比如地下水的强碱 性是不合理的 3、地下水的模型复杂,精确求解涉及解高次微分方程组,没 有闭式解,只能近似。对数据的不同分析和解释,以及对问题的不 同假设,将使解答产生很大的变化。 231 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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9.2参赛论文 Determination of groundwater pollution Hong Wei Liu Yanghua Tang Jianbin Advisor Dai Yonghong Summary In this mathemalical model, we develop an algorithm for the groundwater pollution deternination. First, we generate a model of pollutants transferring in groundwater. The distribution of pollutant density satisfies a partial differential equation called advection-dis- persion equation. If we use a step-change pollution source which we analyze is the most common and important type of source as the boundary condition, we can obtain the solution of the equation. This solution shows the relationship between the pollutants density and time at any position in this area. From the solution, we define a pol lution arriving time and the pollution arriving time of each well is different from others. We can obtain the pollution arriving time by regression analyzing the provided data so that we reduce the devia tion to minimum. Using the time difference among several wells and Darcy principle, we can get an equation system. Solving the equa- tion system, we can find where the pollution source locates and when it began to leak pollutant Using this model, we can also determine how to set monitoring wells to detect which storage tank leaks liquid. We evenly put the monitoring wells on a circle around the storage facility. Under the condition of unknowing the speed and direction of streamlines, we 232 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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give the minimum number of the wells. And we offer a method to determine the interval of sampling by significance test. Using the density-time relationship of the monitoring wells, we can also calcu- late the approximate amount of the pollutant by volume integral This model is simple to use but is accuratc enough in practice Assumption Because the factors that affect the environment are very com plex, we give the following assumptions to simplify our model In a certain small area, the geologic terrain and geologic structure is considerably stable The geologic materials dispersion property is isotropic and the hydraulic conductivity is also the same. Model Design We consider the pollutants are materials and energy that cause the natural water's quality to becme bad and unable to be used And if the pollutants infiltrate into the groundwater, which over the water self-purification ability, the waters quality will become lower. We call this phenomena groundwater pollution Model of pollutants transferring When the pollutants enter the groundwater, they will be trans ferring with the groundwater in the aquifer. The thickness of aquifer is small, so the transport will be modeled as a two-dimen sional-flow problem. In the homogenic aquifer, the density of pollu tants will satisfy this advection-dispersion equation ac aC In this equation, i is the curvilinear coordinate direction of thc streamline; v is the mean linear velocity of the groundwater stream and C is the density of the pollutants PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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D is the hydrodynamical dispersion factor, and it can be repre sented by two items. (2) The al is called dynamical dispersion factor, and D'is molecule dis persion factor. Since the D"is very small, we can neglect it For convenience, we consider the density of the pollutants source is o before the new pollution occurring, and become Co just after that time timc e Figure 1 step-change source This model of pollution source has pracical value because the pollution that lasts for a long time does more harm to people than that lasts for a short period, and we must locate it and manage to eliminate it. So in this model, we put the emphasis on the step change source and try to locate it Use the step-change source to represent the boundary cond tions C(l,0)=0 >=0 (3) C(0,l)=C (4) (+∞,t)=0 >=0 (5) Then use these boundary conditions, we can get the analytic solution of the equation (1) C_1 Croc( tl 0 2√D)(6) 234 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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Here I is the distance along the path of the groundwater stream. In the practical condition, Dt is very large, so the second item of the formuia could be ignored This formula can be used to calculate the density curves of the section plane C 1 erfc( t Determine the pollutants arrival time At any position in the polluted area, use the solution of the e- quation(1), we can get the density-time curve. It is a erfc func- (C/C)0.5 Lime t Figure 2 density-time curve We may think the pollutants come to this position at t.We can determine the t, by the solution At ts time, C/C0=0.5 Because erfc(0)=1, so we can get 0 u could be calculated by the Darcy Principle, it shows that dh (8) where k is the hydraulic conductivity In this small area, we think the streamline should be approxi- 235 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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mately linear. So we can simplify the formula to Ah (9) So we can get k=t =L (10) That is also 大△ht= (11) More generally speaking, we let the monitoring position is (x, y, 2)in the Cartesian coordinate. the source point is (20, yo, zo and begin to give off pollutants at to. The time the pollutants come to the monitoring point is t kAht =(x-xo)+(y-y 2+(之-z0 )2(12) In this equation, we have five unknown variable To, yo,2o to,k. So we must have at least five groups of(r, y, 2, t)to solve this equation. They are(x1,v1, 21, 11),(x2, y2, 22, t2),(x3, y3 ts, y5, zs, t kMt=(x1-x0)2+(y1-y)2+(x1-z0)2 kMht=(x2-x0)2+(y2-y)2+(x2-:0)2 kAnt = 0)2+(y3-y 222 )2(13) 0 (z4 kAnt=(xs- y5-y0 The locations of each monitoring point(monitoring well )have been given in the problen. Then, we must determine the pollutant arrival time t. We can obtain this information from the data set provided by the prob lem. But there are too many data, we must analyze them and try to cut down the quantity of usable data, before using them Analyzing the data set In the data set, there are a few items that are prominent from the 236 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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adjoining data occasionally. Although the item changes abnormally it turns back at the next time. We believe that it declares that the pollutant is in the range of the water self-purification and it will bring little harm to the groundwater. Moreover there may be ran dom errors that cause the changes. Considering the practicability of the mathematical model, we don t think it reflects pollution only according to these data. the Alkalinity column's data of the data set must have some problem. Bccause they are the same as the Bicarbonate'column and if we get pH value according to the‘ Alkalinity’ column’ s data, it is quite different from the'pH'column's data. But in fact, they should be coordinate. So it must have some problems Example On 95-3-21 data of MW1IB, the Alkalinity is 110mg/ Then Ig 11.8 (14) Obviously, the groundwatcr appeared considerably high alka line. It is not acord with the practical circumstance In the Carbon Total Organic' column of the data set, since 96 10-9, the unit of the data is 10 times of the preceding data and it is not ug/ but mg/ Example he consistency of carbon total organic on 97-3-18 in MW1lB is mgA, but not 1. 5ug/ In the Temperaturc column of the data set some data obvious- ly bigger than ordinary data. Through analyzing, we find that the unusual data may be Fahrenheit but not centigrade degree E 237 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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On 97-3-18, the temperature of Mw1iB is 57F,so 5*(57-32)/9=13.89℃ To the data such as <a, we think that it might be lower than the accuracy of the sensor, or it is under the standard and has little effect on the groundwater, so we use the upper limit to take the place of the original data We have modified all the obviously mistaken data. Then we must to determine the pollution from the data set Determine the pollutants According to the given data set, determine whe ether any new pollution has begun during this time pcriod in the area representee by it. Then determine the new pollutants and estimate the location and time of their source We draw all the density-time figure of each matter and each well. For example 60 500}--÷-- 400}---- l600180020002200240026002800 Figure 3 MWllM TDS density-time figure In this picture the x-axis is the time and y-axis is density. The unit is day and we define 91-7-1 is the day o In order to decrease the effect of occasional factors and random 238 PDF文件使用" pdffactory"试用版本创建w, fineprint,com,cn
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