《数学建模》美赛优秀论文：1999 C Locate the Pollution Source

Locate the pollution source 355 Locate the pollution source en quan Yang Zhenyu He xiaofei Zhejiang University Hangzhou, China Advisor: Zhang Chong Summary We develop a model for a strategy to detect new pollution. Three processes govern the movements of pollutants in groundwater: advection, dispersion, and retardation. Information from the wells is used to determine the rate and direction of groundwater movement, determine the horizontal and vertical extent of the pollutants, and analyze the underground structure and characteristics Regarding the diversity and complexity of the given data, we employ a two-step data selection to determine the pollutants most likely to cause new pollution <uring this period of time. We refine the data to choose those chemicals that est represent the variation during this period of time. Then, by using a grid search algorithm, we write a computer program to simulate the movement process and identify the location and start time of the pollution source. The program is written in C and runs on a PC. Four kinds of new pollution sources are located. The graph resulting from our model is in a good agreement with the given data. Finally, we test parameter sensitiv Assumptions All soil and aquifer properties are homogeneous and isotropic throughout both the saturated zone and the unsaturated zone The aquifer consists of sand and gravel The UMAP Journal 20(3)(1999)355-368. @Copyright 1999 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP

Locate the Pollution Source 355 Locate the Pollution Source Shen Quan Yang Zhenyu He Xiaofei Zhejiang University Hangzhou, China Advisor: Zhang Chong Summary We develop a model for a strategy to detect new pollution. Three processes govern the movements of pollutants in groundwater: advection, dispersion, and retardation. Information from the wells is used to • determine the rate and direction of groundwater movement, • determine the horizontal and vertical extent of the pollutants, and • analyze the underground structure and characteristics. Regarding the diversity and complexity of the given data, we employ a two-step data selection to determine the pollutants most likely to cause new pollution during this period of time. We refine the data to choose those chemicals that best represent the variation during this period of time. Then, by using a grid￾search algorithm, we write a computer program to simulate the movement process and identify the location and start time of the pollution source. The program is written in C and runs on a PC. Four kinds of new pollution sources are located. The graph resulting from our model is in a good agreement with the given data. Finally, we test parameter sensitivity. Assumptions • All soil and aquifer properties are homogeneous and isotropic throughout both the saturated zone and the unsaturated zone. • The aquifer consists of sand and gravel. The UMAP Journal 20 (3) (1999) 355–368.
c Copyright 1999 by COMAP, Inc. All rights reserved. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice. Abstracting with credit is permitted, but copyrights for components of this work owned by others than COMAP must be honored. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior permission from COMAP

356 The uMAP Journal 20.3(1999) Steady, uniform water flow occurs only in the vertical direction throughout the unsaturated zone, and only in the horizontal (longitudinal) plane in the saturated zone in the direction of groundwater velocity Physical processes play the greatest role, while the chemical processes are negligible All the parameters describing the characteristic of both zones are constant hroughout the monitoring period All the sources of the pollutants are point sources roblem one This problem is to estimate the location and start time of the source, so we consider the movement process of the pollution and the structure of the underground Data Analysis and Processing We assume that there is no interaction between pollutants so that we can process each pollutant separately. With the given data of the coordinate and water level of each well, we plot the water level map by using linear interpo- lation on the elevations of the monitoring wells, as in Figure 1. For simplicity of computation, we assume that all the underground water flows in the same direction o M4 MuD Mwz Direction of under ground water Figure 1. Water-level map

356 The UMAP Journal 20.3 (1999) • Steady, uniform water flow occurs only in the vertical direction throughout the unsaturated zone, and only in the horizontal (longitudinal) plane in the saturated zone in the direction of groundwater velocity. • Physical processes play the greatest role, while the chemical processes are negligible. • All the parameters describing the characteristic of both zones are constant throughout the monitoring period. • All the sources of the pollutants are point sources. Problem One This problem is to estimate the location and start time of the source, so we consider the movement process of the pollution and the structure of the underground. Data Analysis and Processing We assume that there is no interaction between pollutants so that we can process each pollutant separately. With the given data of the coordinate and water level of each well, we plot the water level map by using linear interpo￾lation on the elevations of the monitoring wells, as in Figure 1. For simplicity of computation, we assume that all the underground water flows in the same direction. Figure 1. Water-level map

Locate the pollution source 357 Data selection Because we have thousands of data points about concentration of various pollutants, we must select data carefully. We do this in the following steps Because pollutants are strongly influenced by layers of different permeabil- ity, measurements of critical parameters and pollutant concentrations need to be done at intervals over the depth of the aquifer. We need a method for sampling at different depths in an aquifer. By analyzing the data set, we find that almost every pollutant affects only one part of a well( top, middle, or bottom). Thus, for each pollutant we need to consider only the effect on one layer of the well. Furthermore, the data from the bottom of each well (if any)remain constant or nearly so, hence we can neglect such data. We delete the data for some pollutants, such as tetrachloroethane, acrolein benzene bromomethane chlorobenzene cobalt, and so on because there are hardly any changes in concentrations of these pollutants in each well. We think that the pulse fluctuation about the pollutant concentration dur- ing a relatively stable period, such as for manganese, is caused by random factors. Thus, we eliminate these pollutants from the data set There is a particular constituent, the CarbonTotalOrganic, whose concen tration value decreases significantly, from more than 1000 to less than 1.5 hus, we eliminate it. Now only four pollutants remain: calcium, chloride, magnesium, and TDS Reselection F maining pollutant, to accurately reflect the tendency for the centration to change we reselect its data as follows For each well, we choose two concentration values for each year, one from the first half of the year and the other from the second half Because we do not know the locations of mw-27 and MW-33 and. moreover the concentration changes in these two wells are small, we do not consider their data According to the groundwater flow direction, the average concentration value of MW-9 should not be higher than that of MW-3 and MW-12, which contradicts the given data for calcium, chloride, and so on. This is also true for barium.(In 1997, concentrations in MW3M and MW12M vary from 50 to 85, whereas they vary from 80 to 95 in MW9M )Therefore, we think that MW-9 is a pumping well(Figure 2). Thus, we do not use the data from MW-9 in our analysis Finally, we list in Table 1 the data for calcium that we use to calculate the source location

Locate the Pollution Source 357 Data Selection Because we have thousands of data points about concentration of various pollutants, we must select data carefully. We do this in the following steps: • Because pollutants are strongly influenced by layers of different permeabil￾ity, measurements of critical parameters and pollutant concentrations need to be done at intervals over the depth of the aquifer. We need a method for sampling at different depths in an aquifer. By analyzing the data set, we find that almost every pollutant affects only one part of a well (top, middle, or bottom). Thus, for each pollutant we need to consider only the effect on one layer of the well. Furthermore, the data from the bottom of each well (if any) remain constant or nearly so, hence we can neglect such data. • We delete the data for some pollutants, such as tetrachloroethane, acrolein, benzene, bromomethane, chlorobenzene, cobalt, and so on, because there are hardly any changes in concentrations of these pollutants in each well. • We think that the pulse fluctuation about the pollutant concentration dur￾ing a relatively stable period, such as for manganese, is caused by random factors. Thus, we eliminate these pollutants from the data set. • There is a particular constituent, the CarbonTotalOrganic, whose concen￾tration value decreases significantly, from more than 1000 to less than 1.5. Thus, we eliminate it. • Now only four pollutants remain: calcium, chloride, magnesium, and TDS. Reselection For each remaining pollutant, to accurately reflect the tendency for the con￾centration to change, we reselect its data as follows: • For each well, we choose two concentration values for each year, one from the first half of the year and the other from the second half. • Because we do not know the locations of MW-27 and MW-33 and, moreover, the concentration changes in these two wells are small, we do not consider their data. • According to the groundwater flow direction, the average concentration value of MW-9 should not be higher than that of MW-3 and MW-12, which contradicts the given data for calcium, chloride, and so on. This is also true for barium. (In 1997, concentrations in MW3M and MW12M vary from 50 to 85, whereas they vary from 80 to 95 in MW9M.) Therefore, we think that MW-9 is a pumping well (Figure 2). Thus, we do not use the data from MW-9 in our analysis. Finally, we list in Table 1 the data for calcium that we use to calculate the source location

358 The uMAP Journal 20.3(1999) the direction of the water direction of undergroyndw ater water velocity directs toward ping well the pumping well when near it ure 2. Groundwater movement near a pumping well Table 1 Data for calcium used in the model Date MW-3M MW-7M MW-1IT MW-12M 12/7/93 41 3/7/94 9/19/94 7/10/95 36.5 54.3 59.5 10/10/9519 43.2 3/6/96 507 824 10/9/ 619 3/18/97 125 12/15/97614 115 63.8 884 According to the data, there is some pollutant detected in an early year as 1990; we name it the background concentration(Cb). We think that the later pollutants concentrations consist of background concentration plus new injected concentration. According to Figure 1, MW-1 must be at the headwater level. Moreover, the data from its bottom hardly change during this period according to the data set. Thus, we estimate Cb using data from MW-IB as follows Cb= arithmetic mean of the concentration value from MW-1B during this period for a certain pollutant n Table 2 we collect the symbols used in this paper and their definitions

358 The UMAP Journal 20.3 (1999) Figure 2. Groundwater movement near a pumping well. Table 1. Data for calcium used in the model. Date MW-3M MW-7M MW-11T MW-12M 12/7/93 41 50 39 42 3/7/94 42 50 43 47 9/19/94 42 45 41 41 7/10/95 36.5 54.3 44.7 59.5 10/10/95 19.2 53 43.2 54.7 3/6/96 62.4 65.1 50.7 82.4 10/9/96 60.2 61.9 53.3 87.6 3/18/97 63.8 125 53.2 87.6 12/15/97 61.4 115 63.8 88.4 • According to the data, there is some pollutant detected in an early year such as 1990; we name it the background concentration (Cb). We think that the later pollutants’ concentrations consist of background concentration plus new injected concentration. According to Figure 1, MW-1 must be at the headwater level. Moreover, the data from its bottom hardly change during this period according to the data set. Thus, we estimate Cb using data from MW-1B as follows: Cb = arithmetic mean of the concentration value from MW-1B during this period for a certain pollutant In Table 2 we collect the symbols used in this paper and their definitions

Locate the pollution source 359 Table 2 Symbols used horizontal dispersion coefficient(m) cal dispersion coefficient(m) pollutant concentration (mg/liter) background concentration(described above) Co concentration in the pollutant source(mg/liter pervasion coefficient(m2/s) H water level (ft) hydraulic conductivity(gal/day/ft2) L horizontal distance in the direction of water flow(ft) discharge rate of the pollutant(mg/day) effective porosity discharge rate of the pollutant(liter/day) retardation factor so comp und parameter start time of the pollution (yr) angle between the direction of underground water and the x-axis Va groundwater velocity(ft/day) hantush function pollution source coordinate Model design Model formulation The movement of pollutants consists of advection, dispersion, and retarda- tion. Furthermore, regarding the large scale of the area, the vertical movement is negligible. Thus, movement of pollutant in the soil(saturated and unsatu rated) can be described by the following two-dimensional equation: ac Vaal 0.z2+ ao a2C 8C a2C Rd Model explanation The model equation applies to steady uniform flow. An analytical solu- tion to the equation can be developed for both continuous(step-function) and pulsed inputs of pollutants as boundary conditions. A step function implie the input of a constant concentration pollutant for an infinite amount of time, while a pulse load is a constant concentration input for a finite amount of time The terms"infinite"and"finite"are relative to the time frame of the analysis Ne assume that the pollution source is applied as a step function(continu- ously) with the following boundary conditions (x,y,0)=0,(x,y)≠(0,0) C(0,0,t)=C t)=C(x,土∞,t)=0

Locate the Pollution Source 359 Table 2. Symbols used. αL horizontal dispersion coefficient (m) αT vertical dispersion coefficient (m) C pollutant concentration (mg/liter) Cb background concentration (described above) C0 concentration in the pollutant source (mg/liter) D pervasion coefficient (m2/s) H water level (ft) I hydraulic gradient K hydraulic conductivity (gal/day/ft2) L horizontal distance in the direction of water flow (ft) m discharge rate of the pollutant (mg/day) n effective porosity q discharge rate of the pollutant (liter/day) Rd retardation factor S compound parameter t0 start time of the pollution (yr) θ angle between the direction of underground water and the x-axis Vd groundwater velocity (ft/day) W hantush function (x0, y0) pollution source coordinate Model Design Model Formulation The movement of pollutants consists of advection, dispersion, and retarda￾tion. Furthermore, regarding the large scale of the area, the vertical movement is negligible. Thus, movement of pollutant in the soil (saturated and unsatu￾rated) can be described by the following two-dimensional equation: Rd ∂C ∂t = VdαL ∂2C ∂x2 + Vdαt ∂2C ∂y2 − Vd ∂C ∂x . (1) Model Explanation The model equation applies to steady uniform flow. An analytical solu￾tion to the equation can be developed for both continuous (step-function) and pulsed inputs of pollutants as boundary conditions. A step function implies the input of a constant concentration pollutant for an infinite amount of time, while a pulse load is a constant concentration input for a finite amount of time. The terms “infinite” and “finite” are relative to the time frame of the analysis. We assume that the pollution source is applied as a step function (continu￾ously) with the following boundary conditions: C(x, y, 0) = 0, (x, y)
= (0, 0); C(0, 0, t) = C0; C(±∞, y, t) = C(x, ±∞, t)=0, t ≥ 0