《数学建模》美赛优秀论文：1999 C Pollution Detection Modeling an Underground Spill through Hydro-Chemical Analysis

Pollution detection 343 Pollution Detection: Modeling an Underground Spill through Hydro-Chemical analysis James r. garlick Savannah n. crites Earlham College Richmond. IN 47374 Advisors: Mic Jackson and Tekla Lewin Summary Data from ten monitoring wells in a region of suspected underground pol lution are used to assess the source, time, and amount of pollutant released into the d. The chemicals are sorted based on changes recorded in their concentrations over time to determine which were active pollutants during the data collection period and to account for discrepancy caused by an incomplete data set. Those chemicals found to be active during this time period change concentration simultaneously, indicating that each chemical is a component of a single leaking liquid involved in two major spills. The concentrations of selected active chemicals are combined to form a composite indicator whose concentration value is found at each well on each date. The composite indicator reveals that two spills occurred, the first between July 1991 and March 1993, and the second between January 1995 and April 1997, possibly continuing until the end of the data collection period. The primary chemical constituents of the leaking liquid are identified A Delaunay triangulation is used to interpolate a gradient of concentra- tion for the composite indicator at each date between the monitoring wells. Given that the general flow of groundwater in this region is directed toward well9, the time and location of the pollution source can be approximated based on changes in the concentration gradient over time. This spill is estimated to have originated in the region surrounding the point( 8000, 4500). Following the initial triangulation, Voronoi polygons are used to construct a convex hull he UMAP Journal20(3)(1999)343-354. @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

Pollution Detection 343 Pollution Detection: Modeling an Underground Spill through Hydro-Chemical Analysis James R. Garlick Savannah N. Crites Earlham College Richmond, IN 47374 Advisors: Mic Jackson and Tekla Lewin Summary Data from ten monitoring wells in a region of suspected underground pol￾lution are used to assess the source, time, and amount of pollutant released into the ground. The chemicals are sorted based on changes recorded in their concentrations over time to determine which were active pollutants during the data collection period and to account for discrepancy caused by an incomplete data set. Those chemicals found to be active during this time period change concentration simultaneously, indicating that each chemical is a component of a single leaking liquid involved in two major spills. The concentrations of selected active chemicals are combined to form a composite indicator whose concentration value is found at each well on each date. The composite indicator reveals that two spills occurred, the first between July 1991 and March 1993, and the second between January 1995 and April 1997, possibly continuing until the end of the data collection period. The primary chemical constituents of the leaking liquid are identified. A Delaunay triangulation is used to interpolate a gradient of concentra￾tion for the composite indicator at each date between the monitoring wells. Given that the general flow of groundwater in this region is directed toward well 9, the time and location of the pollution source can be approximated based on changes in the concentration gradient over time. This spill is estimated to have originated in the region surrounding the point (8000, 4500). Following the initial triangulation, Voronoi polygons are used to construct a convex hull The UMAP Journal 20 (3) (1999) 343–354.
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

344 The uMAP Journal 20.3(1999) representing the total volume and position of the spill (the volume of the con- aminated area). This polygon is comprised of smaller segments, each of a specific uniform concentration. The program Geomview is used to generate graphics of these polygons and convex hulls. A volume can be calculated at each concentration, and ultimately the total volume of polluting liquid can be found, if the concentration of the composite indicator in the original polluting liquid is known Finally, various testing and interpretation methods are explored and incor- information given in the data set to test the validity of the method o andso porated into a procedure for evaluating underground pollution. Each method is discussed in terms of its application to the scenario in Problem Tw Introduction Given the location and elevation of eight groundwater monitoring wells (two more wells exist at unknown locations), a complete chemical analy taken periodically at each well between 1990 and 1997, and the general di rection of groundwater flow, it is possible to accurately estimate the location, source, time of origin, and total volume of pollutants seeping underground. In the case of a suspected leak in a chemical storage facility built over homoge nous soil, cost and safety prohibited collection of analytical data directly below the suspected sight of the spill. Data from monitoring wells surrounding the periphery of but not necessarily directly in the suspected polluted region are In thematical model to determine whether a leak has time and location when the leak occurred and the amount of liquid that has during the data collection erIo Assumptions All monitoring wells are located below ground and are contained within an aquifer(a geological unit capable of storing and transmitting substantial volumes of water). This aquifer has an unobstructed constant flow rate which is inversely proportional to the porosity of the soil medium. The monitoring wells are permanent, allow free flow through their measuring devices, have no effect on the chemical or geological composition of the region, and provide an accurate reflection of the surrounding This ensures that the wells themselves do not contaminate or pollute the region to be assessed [ Soliman et al. 1997, 32] The volume of fluid is constant in each well, and all wells have the same volume. assuming a consistent volume between wells allows direct ratios to be assessed comparing concentrations of solutes in each well

344 The UMAP Journal 20.3 (1999) representing the total volume and position of the spill (the volume of the con￾taminated area). This polygon is comprised of smaller segments, each of a specific uniform concentration. The program Geomview is used to generate graphics of these polygons and convex hulls. A volume can be calculated at each concentration, and ultimately the total volume of polluting liquid can be found, if the concentration of the composite indicator in the original polluting liquid is known. Finally, various testing and interpretation methods are explored and incor￾porated into a procedure for evaluating underground pollution. Each method is discussed in terms of its application to the scenario in Problem Two and uses information given in the data set to test the validity of the method. Introduction Given the location and elevation of eight groundwater monitoring wells (two more wells exist at unknown locations), a complete chemical analysis taken periodically at each well between 1990 and 1997, and the general di￾rection of groundwater flow, it is possible to accurately estimate the location, source, time of origin, and total volume of pollutants seeping underground. In the case of a suspected leak in a chemical storage facility built over homoge￾nous soil, cost and safety prohibited collection of analytical data directly below the suspected sight of the spill. Data from monitoring wells surrounding the periphery of but not necessarily directly in the suspected polluted region are used in a mathematical model to determine whether a leak has occurred, the time and location when the leak occurred, and the amount of liquid that has leaked during the data collection period. Assumptions • All monitoring wells are located below ground and are contained within an aquifer (a geological unit capable of storing and transmitting substantial volumes of water). This aquifer has an unobstructed constant flow rate which is inversely proportional to the porosity of the soil medium. The monitoring wells are permanent, allow free flow through their measuring devices, have no effect on the chemical or geological composition of the region, and provide an accurate reflection of the surrounding area. This ensures that the wells themselves do not contaminate or pollute the region to be assessed [Soliman et al. 1997, 32]. • The volume of fluid is constant in each well, and all wells have the same volume. Assuming a consistent volume between wells allows direct ratios to be assessed comparing concentrations of solutes in each well

Pollution detection 345 Different chemicals may travel through the aquifer at different rates. Chem ical substances have a constant and specific ability to move in aqueous solu tions depending on polarity of the molecules hydrophobicity, and the initial concentration of each compound present in the data set occur naturally in the ground water and are not products of pollution. Any chemical that exhibits no sig nificant change in concentration at any monitoring well over the course of the data collection period can be removed from consideration in the data. In addition, certain naturally appearing chemical components of groundwater can be expected to fluctuate between standard levels Concentrations of pollutants are highest near their source, and concentra tions decrease as time and distance from their source increases The given data set is incomplete. Some trends may be misrepresented or missed entirely due to lack of available data. Also, the values that are given must be appropriately evaluated so as not to treat the n/a values as zero Discrepancies in the data can be attributed to variations in the equipment used or in sampling and analyzing techniques over the course of the study and should not always be interpreted as changes in the environment, espe- cially those occurring on the same data in every sample tested Pollution is defined as a contaminant that is harmful to an organism, while contamination refers to a greater concentration of a substance than would occur naturally without necessarily causing harm [Blatt 1997, 76]. In this problem, we assume that both terms refer to the artificial contamination of an underground region, regardless of the effect that the contaminants may have on organisms Dealing with the Data To use or interpret such a large and varied data set effectively, specific criteria must be employed to organize and sort the known information. We converted the data from its original spreadsheet form into a database so that we could set up queries and selectively access any portion of the information everal components of the data were not chemical concentrations but other factors necessary for a thorough chemical analysis, such as specific conduc tivity and total dissolved solids. These were separated and stored in another spreadsheet. Although some methods of modeling pollution use these mea- ald not detect a sign in these values to indicate the presence of absence of pollution. Using line graphs mapping the concentration of a given chemical at all dates and at each well, we identified chemicals that exhibited a negligible change in concentration. These were removed and stored in a separate spreadsheet. This left 23 chemicals from an original set of 106 measurement categories

Pollution Detection 345 • Different chemicals may travel through the aquifer at different rates. Chem￾ical substances have a constant and specific ability to move in aqueous solu￾tions depending on polarity of the molecules, hydrophobicity, and the initial concentration of each compound. • Some chemicals found present in the data set occur naturally in the ground￾water and are not products of pollution. Any chemical that exhibits no sig￾nificant change in concentration at any monitoring well over the course of the data collection period can be removed from consideration in the data. In addition, certain naturally appearing chemical components of groundwater can be expected to fluctuate between standard levels. • Concentrations of pollutants are highest near their source, and concentra￾tions decrease as time and distance from their source increases. • The given data set is incomplete. Some trends may be misrepresented or missed entirely due to lack of available data. Also, the values that are given must be appropriately evaluated so as not to treat the N/A values as zero. • Discrepancies in the data can be attributed to variations in the equipment used or in sampling and analyzing techniques over the course of the study and should not always be interpreted as changes in the environment, espe￾cially those occurring on the same data in every sample tested. • Pollution is defined as a contaminant that is harmful to an organism, while contamination refers to a greater concentration of a substance than would occur naturally without necessarily causing harm [Blatt 1997, 76]. In this problem, we assume that both terms refer to the artificial contamination of an underground region, regardless of the effect that the contaminants may have on organisms. Dealing with the Data To use orinterpret such alarge and varied data set effectively, specific criteria must be employed to organize and sort the known information. We converted the data from its original spreadsheet form into a database so that we could set up queries and selectively access any portion of the information. Several components of the data were not chemical concentrations but other factors necessary for a thorough chemical analysis, such as specific conduc￾tivity and total dissolved solids. These were separated and stored in another spreadsheet. Although some methods of modeling pollution use these mea￾surements, our models do not, because we could not detect a significant pattern in these values to indicate the presence of absence of pollution. Using line graphs mapping the concentration of a given chemical at all dates and at each well, we identified chemicals that exhibited a negligible change in concentration. These were removed and stored in a separate spreadsheet. This left 23 chemicals from an original set of 106 measurement categories

346 The uMAP Journal 20.3 (1999) Determining the Presence of pollution From the rapid increases shown in the line graphs of chemical concentra- tions over time, it was apparent that new pollution had occurred in this region over the testing period. Those chemicals detected as new pollutants include acetone, ammonia, arsenic, barium, bicarbonate, calcium, chloride iron, lead magnesium,manganese, nickel, nitrate/nitrite, potassium, sodium, TDS, sul- fate, vanadium, and zinc The concentration of the majority of the chemicals in the active data set rise and fall together, indicating that each is a constituent of a single liquid involved in the spill. Although the concentrations of all emicals in the data set follow obvious trends, the changes in concentration are much more amplified for some than for others. We chose these amplified chemicals as indicator chemicals to track the movement of the spill. To further simplify spill detection, we added the concentrations of these indicator chemicals(chloride sulfate, and nitrate/nitrite)together to form a composite indicator chemical, the concentration of which indicates the presence of pollution at each test site n a given date. We chose these chemicals also because they are common components of pollutants and are often used to monitor pollution[B C Ministry of Environment, Land, and Parks 1999] In choosing chemicals to serve as indicators for a spill, it is essential to find chemicals that were measured consistently on the same dates and at all wells throughout the data collection period. Three chemicals in this data set that fit this criterion are chloride sulfide and nitrate /nitrate and we used those in the composite indicator. Because the data set is not complete and the measurements were not taken consistently for all chemicals at all points or on all dates, it is important to ensure that the concentration of this composite indicator does not misrepresent trends in the movement of the spill due to a lack or abundance of data for a given well or on a given date. We went through the data set and eliminated dates that were recorded twice(taking an average of the concentrations listed at each well)and corrected other abnormalities in the data until each of the three chemicals had exactly one value at each test location on all dates needed. Exceptions to this include those wells for which values are not available at the beginning of the testing period; these are added as data from these wells became available The Time of the Spill A series of line graphs showing the concentration of the composite indicator plotted together so that each line represents a monitoring well thspill.When at a given well over time can be used to estimate the time of th hese graphs of concentration over time show when concentrations first start to increase and at which well(s) this increase is first recorded. This record of which wells show the first rise in concentration provides a rough estimation of the location of the

346 The UMAP Journal 20.3 (1999) Determining the Presence of Pollution From the rapid increases shown in the line graphs of chemical concentra￾tions over time, it was apparent that new pollution had occurred in this region over the testing period. Those chemicals detected as new pollutants include: acetone, ammonia, arsenic, barium, bicarbonate, calcium, chloride, iron, lead, magnesium, manganese, nickel, nitrate/nitrite, potassium, sodium, TDS, sul￾fate, vanadium, and zinc. The concentration of the majority of the chemicals in the active data set rise and fall together, indicating that each is a constituent of a single liquid involved in the spill. Although the concentrations of all active chemicals in the data set follow obvious trends, the changes in concentration are much more amplified for some than for others. We chose these amplified chemicals as indicator chemicals to track the movement of the spill. To further simplify spill detection, we added the concentrations of these indicator chemicals (chloride, sulfate, and nitrate/nitrite) together to form a composite indicator chemical, the concentration of which indicates the presence of pollution at each test site on a given date. We chose these chemicals also because they are common components of pollutants and are often used tomonitor pollution [B.C.Ministry of Environment, Land, and Parks 1999]. In choosing chemicals to serve as indicators for a spill, it is essential to find chemicals that were measured consistently on the same dates and at all wells throughout the data collection period. Three chemicals in this data set that fit this criterion are chloride, sulfide, and nitrate/nitrate, and we used those in the composite indicator. Because the data set is not complete and the measurements were not taken consistently for all chemicals at all points or on all dates, it is important to ensure that the concentration of this composite indicator does not misrepresent trends in the movement of the spill due to a lack or abundance of data for a given well or on a given date. We went through the data set and eliminated dates that were recorded twice (taking an average of the concentrations listed at each well) and corrected other abnormalities in the data until each of the three chemicals had exactly one value at each test location on all dates needed. Exceptions to this include those wells for which values are not available at the beginning of the testing period; these are added as data from these wells became available. The Time of the Spill A series of line graphs showing the concentration of the composite indicator at a given well over time can be used to estimate the time of the spill. When plotted together so that each line represents a monitoring well, these graphs of concentration over time show when concentrations first start to increase and at which well(s) this increase is first recorded. This record of which wells show the first rise in concentration provides a rough estimation of the location of the

Pollution detection 347 source as well. [EDITOR'S NOTE: We cannot effectively reproduce the authors hs here in black and white Two spills probably occurred, the first between uly 1991 and March 1993. dur- ing these times, the concentrations in wells believed to be closest to the spill increased dramatically, then receded back toward normal levels. The second probably began in January 1995 and continued at least until January 1997. At this time, concentrations were starting to descend, but this could result from a de- crease in the rate of the spill and may not indicate that the leak stopped Locating the Source The line graphs generated by queries from the database are extreme ful in determining the presence of a spill, the time at which it occurred, and the chemicals involved. However, finding the source of the spill is more effec tively accomplished with a visual interpolation showing the concentration of the composite indicator at each well over time. This way, we can determine where the concentrations rose first and the general direction the spill moved in. Knowing the general direction of the spill, we can develop bounds within which the source of the spill must lie. This can be done in three dimensions by creating a Voronoi polygon. This method of interpolation organizes data points into a triangles with their natural neighbors and partitions areas around each known point into polygons such that an arbitrary point placed in the polygon is closer to that data point than any other. The triangulation of a map is unique and effectively weights the value of any point in the region as a function of its distance from three natural neighbors While the line graphs show approximate dates when a spill might have occurred and at which wells the changes in concentration were detected the Voronoi polygon method interpolates between the known data points to show more precisely the location of the spill source. From a series of diagrams of the concentration of the composite indicator chemical at each well over a selection of dates, the progress of the spill is very apparent, and the location of the source can be found by following the flow patterns in the underground system backward from the point where the spill first occurred. [ EDITOR'S NOTE: We do not reproduce the authors' maps I A Procedure for Evaluating Underground Contamination The problem of detecting the presence of underground liquids is an old one, and due to its applications in locating water sources, petroleum reserves, and mineral deposits, an abundance of information about techniques and methods is available. Drilling sampling or monitoring wells is clearly necessary at some

Pollution Detection 347 source as well. [EDITOR’S NOTE: We cannot effectively reproduce the authors’ graphs here in black and white.] Two spills probably occurred, the first between July 1991 and March 1993. Dur￾ing these times, the concentrations in wells believed to be closest to the spill increased dramatically, then receded back toward normal levels. The second probably began in January 1995 and continued at least until January 1997. At this time, concentrations were starting to descend, but this could result from a de￾crease in the rate of the spill and may not indicate that the leak stopped. Locating the Source The line graphs generated by queries from the database are extremely use￾ful in determining the presence of a spill, the time at which it occurred, and the chemicals involved. However, finding the source of the spill is more effec￾tively accomplished with a visual interpolation showing the concentration of the composite indicator at each well over time. This way, we can determine where the concentrations rose first and the general direction the spill moved in. Knowing the general direction of the spill, we can develop bounds within which the source of the spill must lie. This can be done in three dimensions by creating a Voronoi polygon. This method of interpolation organizes data points into a triangles with their natural neighbors and partitions areas around each known point into polygons such that an arbitrary point placed in the polygon is closer to that data point than any other. The triangulation of a map is unique and effectively weights the value of any point in the region as a function of its distance from three natural neighbors. While the line graphs show approximate dates when a spill might have occurred and at which wells the changes in concentration were detected, the Voronoi polygon method interpolates between the known data points to show more precisely the location of the spill source. From a series of diagrams of the concentration of the composite indicator chemical at each well over a selection of dates, the progress of the spill is very apparent, and the location of the source can be found by following the flow patterns in the underground system backward from the point where the spill first occurred. [EDITOR’S NOTE: We do not reproduce the authors’ maps.] A Procedure for Evaluating Underground Contamination The problem of detecting the presence of underground liquids is an old one, and due to its applications in locating water sources, petroleum reserves, and mineral deposits, an abundance of information about techniques and methods is available. Drilling sampling or monitoring wells is clearly necessary at some