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Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
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By THOMAS E. REILLY and DAVID W. POLLOCK
U.S . GEOLOGICAL SURVEY WATER-SUPPLY PAPER 2412
U.S . DEPARTMENT OF THE INTERIOR BRUCE BABBITT, Secretary
U.S . GEOLOGICAL SURVEY DALLAS L. PECK, Director
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Library of Congress Cataloging in Publication Data Reilly, Thomas E. Factors affecting areas contributing recharge to wells in shallow aquifers / by Thomas E. Reilly and David W. Pollock. p. cm. - (U.S. Geological Survey water-supply paper ; 2412) Includes bibliographical references. Supt. of Docs. no.: 119.13:2412 1 . Groundwater flow. 2. Aquifers. I. Pollock, David W. II. Title. 111 . Series. G131197.7.1345 1993 551 .49-dc20 93-7410 CIP
CONTENTS
Abstract ...... 1 Introduction ...... 1 Areas Contributing Recharge and Sources of Water to Wells ...... 2 Previous Related Work ...... 3 Factors Affecting Areas Contributing Recharge to Wells ...... 4 Method of Analysis ...... 4 Delineation of Areas Contributing Recharge to Wells in Hypothetical Ground-Water Systems ...... 5 Case 1-Homogeneous System ...... 5 Well Discharge of 2,000 Cubic Feet per Day from Three Different Screened Intervals ...... 6 Well Discharge of 10,000 Cubic Feet per Day from Three Different Screened Intervals ...... 7 Well Discharge of 30,000 Cubic Feet per Day ...... 8 Well Discharge of 30,000 Cubic Feet per Day in the Presence of an Additional Well Discharging 130,000 Cubic Feet per Day ...... 8 Case 2-Low-Hydraulic-Conductivity Local Confining Unit ...... 9 Case 3-High-Hydraulic-Conductivity Units ...... 12 Case 4-Random Distribution of Hydraulic Conductivity ...... 15 Discussion of Simulation Results ...... 19 Considerations for the Protection of Ground Water ...... 19 References Cited ...... 20
FIGURES 1 . Diagram showing the area contributing recharge to a single discharging well in a simplified hypothetical ground-water system ...... 2 2 . Diagram showing the area contributing recharge to a single discharging well in a simplified hypothetical ground-water system with a stream ...... 3 3 . Map view of hypothetical alluvial valley ...... 5 4 . Model grid for simplified alluvial-valley system showing stream and well location ...... 6 5 . Configuration of the water table for the simulated alluvial valley with no ground-water withdrawals ...... 7 6 . Simulated configuration of the water table and areas contributing recharge for a well discharging at a rate of 2,000 cubic feet per day screened at three different depths ...... 8 7 . Pathlines in the ground-water system with well 1 screened at 125 to 150 feet below the water table and discharging at a rate of 2,000 cubic feet per day ...... 9 8-10 . Simulated configuration of the water table and area contributing recharge for: 8 . A well discharging at a rate of 10,000 cubic feet per day screened at three different depths ...... 10 9 . A well discharging at a rate of 30,000 cubic feet per day ...... 11 10 . Two wells discharging simultaneously: well 1 discharges at a rate of 30,000 cubic feet per day, and well 2 discharges at a rate of 130,000 cubic feet per day ...... 12 11 . A, Cross section showing area of lateral inflow on finite-difference grid . B, Simulated configuration of the water table and area contributing recharge for a well discharging at a rate of 10,000 cubic feet per day, in the presence of a discontinuous confining unit ...... 13 12,13. Simulated configuration of the water table and area contributing recharge for: 12 . A well discharging at a rate of 10,000 cubic feet per day, with a layer of high hydraulic conductivity at the bottom of the aquifer...... 14 13 . A well discharging at a rate of 10,000 cubic feet per day, with a zone of high hydraulic conductivity (500 feet per day) 150 feet thick along the stream ...... 15
Contents III
14 . Map showing random two-dimensional hydraulic-conductivity distribution in a hypothetical alluvial-valley aquifer ...... 16 15 . Simulated configuration of the water table and areas contributing recharge for four cases ...... 17 16 . Map showing areas contributing recharge to a discharging well in an aquifer with two different randomly distributed hydraulic-conductivity fields and in an aquifer of uniform hydraulic conductivity ...... 18
METRIC CONVERSION FACTORS
Multiply By To obtain SI metric unit foot (ft) 0.3048 meter (m) foot per day (ft/d) 0.3048 meter per day (m/d) cubic feet per day (ft3/d) 0.0283 cubic meters per day (M3/d)
IV Contents Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers By Thomas E. Reilly and David W. Pollock
Abstract INTRODUCTION Ground water is the source of drinking water The source of water to wells is ultimately for about 50 percent of the population of the United the location where the water flowing to a well States (Solley and others, 1988). Ground water is enters the boundary surface of the ground-water generally obtained through wells and, because of the system. In ground-water systems that receive natural filtering properties of the rocks, commonly most of their water from areal recharge, the does not require treatment before use. Widespread location of the water entering the ground-water contamination of shallow ground-water supplies system is at the water table. The area contribut- from various land-use practices is affecting this ing recharge to a discharging well is the surface resource . To address this concern, the U.S . Envi- (USEPA), in response area that defines the location of the water enter- ronmental Protection Agency to the Safe Drinking Water Act Amendments of ing the ground-water system at the water table 1986, has encouraged the States to develop "Well- the well and is eventually dis- that flows to head Protection" strategies (U.S . Environmental charged from the well. Protection Agency, 1987). The objective of these where water carrying The calculation of areas contributing strategies is to define the area potential contaminants can enter the ground-water recharge to wells is complex because flow paths system and flow to a supply well and then to set change in response to in ground-water systems regulations to minimize the opportunity for contami- development, and the aquifer material in nation to occur in areas defined as sources of water ground-water systems is heterogeneous and is to wells. hidden from direct observation . Hypothetical Although the concept of "wellhead protection" experiments were undertaken to show the com- is straightforward and consistent with the ideals of plexities in the delineation of areas contributing protecting our water supplies, in many cases there in actually recharge to wells. Four different "cases" are are technical and conceptual difficulties amount of certainty, a examined to demonstrate the effect of different defining, with a reasonable fixed area that defines the recharge location of water conceptualized aquifer frameworks on determin- that flows to a particular well . These difficulties istically calculated areas contributing recharge . arise because of the inherent complexity of ground- from the experi- The main conclusion drawn water systems. Wellhead-protection strategies that ments is that, in order to understand the cause are based on overly simplified characterizations of and effect relations that affect the quality of the ground-water-flow system may needlessly pro- water derived from wells, the importance and tect areas that do not contribute water to a well and nature of the variability in the ground-water- instead fail to protect areas that do contribute water. flow system must be considered and accounted This report describes and illustrates some of for in any efforts to "protect" the water supply. the inherent difficulties in the determination of the
Introduction 1
areas contributing recharge to wells. Simulation Areal Recharge techniques are used to calculate the areas contribut- ing recharge under a variety of conditions defined Area for hypothetical aquifer systems . The areas contrib- Contributing Dischring ~ uting flow to wells as determined for the hypotheti- Recharge Land Surf cal systems are delineated in a series of figures. The use of simple hypothetical systems provides the mechanism to illustrate and compare the cause and Water Table effect relations among the many factors examined. Screened zone All the systems analyzed are unconfined valley-fill- Bounding, Flow aquifer systems that are undoubtedly much less Lines Not to scale complex than actual systems that would be encoun- A tered in nature .
AREAS CONTRIBUTING RECHARGE AND SOURCES OF WATER TO WELLS
The withdrawal of water from a well in a ground-water system creates drawdown throughout the aquifer. The only limit to the areal and vertical extent of drawdown is the physical boundaries of the ground-water system (Brown, 1963) . Drawdown occurs in three dimensions and decreases with dis- tance away from the point of withdrawal. The change in head caused by the withdrawal of water causes flow to the well . The location of pathlines that define the flow paths to the well depend on the Notto scale hydrogeologic characteristics of the flow system, the B well location and discharge rate, and the boundary Figure 1 . Area contributing recharge to a single discharg- conditions of the flow system. ing well in a simplified hypothetical ground-water system . A, Cross-sectional view . B, Map view. For three-dimensional systems, the area con- tributing recharge to a discharging well is defined in this report as the surface area that delineates the location of the water entering the ground-water sys- The location of this area depends on many factors tem at the water table that eventually flows to the that describe the ground-water system and the well. well and discharges. For example, figure 1 illus- Depending on factors that describe the three- trates the flow paths to a well in a simplified aquifer dimensional flow system and the placement of the system with areal recharge. The ultimate source of well in the three-dimensional system, the area con- water being discharged at a constant rate from a par- tributing recharge to a well does not necessarily tially penetrating well is recharge to the water table. have to include the location of the well (fig. 1) . The area of this source of water must provide an The area contributing recharge to a well in the amount of recharge that balances the amount of vicinity of a stream is shown in figure 2 . In this water being discharged from the well . Thus, for this case, the well is capturing water that was flowing in simple case, the stream, so the contributing area is not a function of the areal recharge rate only Q=WA, but is also a function of the amount of water obtained from the stream. where Any contamination entering the river valley Q=discharge rate of well (L3IT), upstream from the well could affect the quality of W=areal recharge rate (L17), and the water in the stream and ultimately affect the A=area contributing recharge (L2). water discharged from the well.
Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
Areal Recharge PREVIOUS RELATED WORK The quantitative estimation of flow paths to and from wells has been the subject of investigation Dischargin since the work of Slichter (1899, p. 368) at the turn Stream Well Land Surface of the century. The description of flow lines near an Water Table --~ - injection or withdrawal well in a two-dimensional Bounding infinite aquifer is available in works by Slichter Flow Lines Flow Lines (1899), Jacob (1950, p. 344), Milne-Thomson Screened zone (1955, p. 199), Ogata (1963), Bear and Jacobs (1965), and others . The description of flow lines Not to scale around a three-dimensional point source or sink is available by Streeter (1948, p. 53), Milne-Thomson (1955, p. 436), Bear and Jacobs (1965), and others . These descriptions all deal with idealized, uniform, and infinite systems that have no local sources of water. Brown (1963) laid the foundation for explain- ing the source of water to wells in shallow aquifers subject to areal recharge in bounded two-dimen sional systems . This work, although for a simplified two-dimensional system, defined the basic problem for shallow systems subject to areal recharge . Brown illustrated the distinction between the cone of depression (which he called the area of influence) and the area that contributes water to the well (which he called the area of diversion) . Not to scale B Transport to wells in the immediate vicinity of the well was investigated by Mundorff and others Figure 2. Area contributing recharge to a single discharg- Kirkham and Sotres (1978), Phillips and ing well in a simplified hypothetical ground-water system (1972), with a stream. A, Cross-sectional view. 8, Map view. Gelhar (1978), Reilly (1978), and others. These studies examined the effects of partial penetration and the vertical movement of water from the water boundaries on the traveltime of the water An obvious point is that the quality of the table or to a single well. These investigations did not water withdrawn from a well is a function of (1) the account for the effects of regional boundaries on the quality of the water entering the area that contrib- flow to a well, but they did account for the impor- utes recharge to the well and (2) any chemical or tant local effects in determining the flow paths and transformations that take place during its biological traveltimes to a single, partially penetrating, dis- movement of water is usu- transit to the well. The charging well. ally relatively slow, and traveltimes are commonly The estimation of surface areas that are the long in most ground-water systems. During the long source of water to discharging wells received con- residence time, chemical and biological activity siderable renewed interest with the enactment of the often changes the quality of the water. In some USEPA's Wellhead Protection Program in the 1986 instances, the quality of the water can improve with Amendments to the Safe Drinking Water Act. time as the water flows through the system . Another USEPA provided "Guidelines for the Delineation of process that can affect the chemical quality of the Wellhead Protection Areas" (U .S . Environmental water is mixing, both natural, as a result of the het- Protection Agency, 1987) to enable the States to erogeneity of ground-water systems, and human begin their wellhead protection efforts. Extensions induced, as a result of the installation of wells and of the basic analytical solutions have been made by other construction. some investigators ; for example, Javandal and Tsang
Previous Related Work 3 (1986) used complex potential theory to analyze the top boundary (the water table) of the three- capture zones for multiple wells . In addition, work dimensional ground-water system is where most (or with numerical models to calculate areas contribut- all) of the water flowing to the system will origi- ing flow to wells has recently increased; for exam- nate. The system's transmitting properties, in con- ple, see Morrissey (1989), Barlow (1989), and Bair junction with the boundary conditions and frame- and Roadcap (1992) . work, determine the rates at which water flows Morrissey (1989) uses analytical, two- through the system. It is important to visualize dimensional, and three-dimensional flow models to ground-water systems as three-dimensional and com- determine the area that contributes flow to a dis posed of materials of different transmitting proper- charging well. His careful definition of terms and ties. Water that enters at the top boundary can flow discussion of the various methods forms a founda- deep into the system before eventually returning to tion for future work. Barlow (1989) demonstrated the surface as discharge . the use of flow simulation and particle tracking Because ground-water systems are dynamic, (Pollock, 1989) to determine contributing areas. human-induced stresses cause the flow of water in the system Barlow's (1989) work introduces the variability in to adjust in response to these stresses. Drawdown caused by the areas that is due to factors such as the geologic well discharges changes the environment and the dimensionality of the method head and flow patterns . Thus, as water withdrawals used. This report expands on this theme and change, so too do the sources of water to the wells. attempts to address the uncertainties in determining This is shown quantitatively in following sections . the areas that contribute flow to discharging wells . And just as human-induced changes affect the paths of water, so too do natural variations in boundary conditions, such as recharge. Changes in rates of FACTORS AFFECTING AREAS recharge over time affect the paths of flow and, ulti- CONTRIBUTING RECHARGE TO WELLS mately, the source of water to wells. The factors that influence the location of areas contributing recharge to wells can be categorized as METHOD OF ANALYSIS dependent either on the ground-water system or the well. The ground-water factors that affect the paths Numerical simulation is used to show the of water movement in three-dimensional ground- cause and effect relations among the various factors water systems are (1) the hydrogeologic framework that affect the location, shape, and extent of areas of the system, (2) system boundary conditions, (3) contributing recharge to wells. Simulations of system transmitting and storage properties, (4) selected hypothetical systems are controlled experi- stresses and change in stresses (water withdrawals), ments that are conducted to determine areas contrib- and (5) other transient effects . The well factors are uting recharge; these areas can then be compared to the location of the well and the depth of the one another. screened zone or open hole section of a well. In The U.S. Geological Survey's ground-water- addition, the rate at which the well discharges deter- flow model called MODFLOW (McDonald and Har- mines the size of the area contributing recharge, as baugh, 1988) is used for these experiments . The discussed previously, and also determines the extent results of the flow simulation are simulated heads to which flow paths in the ground-water-flow sys- and flows. A postprocessing model called MOD- tem are altered to supply water to the well. PATH (Pollock, 1989) is then used to calculate The extents and thicknesses of the various lay- steady-state pathlines in the simulated three- ers in the system of interest define its hydrogeologic dimensional ground-water system. The computer framework. Once the extents and thicknesses are program MODPATH-PLOT (Pollock, 1990) is used to defined, the description of the boundary conditions plot the computed pathlines . The methodologies explains how water enters and leaves the ground- used for MODFLOW, MODPATH, and MODPATH- water system. For example, if areal recharge is the PLOT are described in their separate documentation primary source of water to the ground-water system, as referenced.
4 Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
DELINEATION OF AREAS Lateral Inflow ofGround Water CONTRIBUTING RECHARGE TO WELLS IN HYPOTHETICAL GROUND-WATER SYSTEMS
The setting for the hypothetical ground-water systems used for the calculation of areas contribut- ing recharge is a simplified alluvial valley (fig. 3) . Permeable alluvial deposits are underlain and later- ally bounded by impermeable bedrock, and a mean- dering stream flows through the valley. The simu- lated valley segment is 8,625 feet long, and the permeable deposits are about 150 ft thick and extend 6,750 ft across the valley. The areal recharge rate is 0 .005 ft/d from infiltration of precipitation directly on the valley, and runoff from the impermeable bed- rock valley walls accounts for inflow at the lateral boundary of the permeable deposits of 2 (ft3/d)/ft. Lateral Inflow of Ground Water The stream is not deeply incised into the deposits . Figure 3. Map view of hypothetical alluvial valley. The numerical model used to represent this system consists of a three-dimensional array of cells with 54 rows, 69 columns, and 6 layers (fig. 4). Each model layer is 25 ft thick, and each grid cell is 291,090 ft3/d and a lateral inflow of 34,500 ft3/d. 125 ft by 125 ft. The lateral inflow is simulated as The areal recharge comprises 89.4 percent of the entering the top layer . The stream is treated as a water entering the system and is the most significant nonpenetrating stream with a depth of 2 ft, a part of the water budget for the system for the con- streambed conductance of 15,000 ft2/d, and a stream dition of no discharging wells . All the water enter- stage that varies from 152 .75 to 148.85 ft above an ing the system (325,590 ft3/d) discharges to the arbitrary datum from left to right (column 1 to col- stream under such an unstressed condition . umn 69). Well 1 is screened at different depths at node Several hydrologic and (or) hydraulic condi- 30,44 (row 30, column 44) in the following varia- tions that could exist in alluvial valleys are simu- tions of the simulations for Case 1 . The areas con lated as individual cases . Each simulated "case" has tributing recharge are determined for discharge rates a different hydraulic-conductivity distribution. The of 2,000, 10,000, and 30,000 ft3/d. The source area results of these simulations as they pertain to areas (area contributing recharge) for the water discharged contributing to wells are described in the remainder is calculated by using MODPATH and MODPATH- of this paper. PLOT (Pollock, 1989,1990) . The pathline calculation method is used to determine the paths of the water Case 1-Homogeneous System through the modeled system, thereby allowing iden- tification of the paths that eventually discharge to For Case 1, the alluvial deposits are assumed the well . The location where the paths entered the to be homogeneous and vertically anisotropic. The ground-water system are then determined and plot- hydraulic conductivity is 100 ft/d laterally and 25 ted for each case . ft/d vertically, and the deposits have a uniform The "areas contributing recharge" delineated porosity of 0.3 . To examine areas contributing on maps for each simulation indicate the area at the recharge to discharging wells, the system is assumed water table where water flowing to the well origi to be in equilibrium (steady state), and the wells are nates. A component of flow to the well can also pumped continuously at specified discharge rates. originate at the lateral inflow boundary or at the The simulated water-table contours for the stream; these components are not shown on most of system when no wells are being pumped is shown in the figures but are discussed as water-budget com- figure 5 . The system has a net areal recharge of ponents to the well.
Delineation of Areas Contributing Recharge to Wells In Hypothetical Ground-Water Systems 5
Column
10 20 30 40 50 60 I . I I I I I
10 HNNUMMENOMMMMMUN NUHHHHHHUM
...... ------, ...... 20 I..uu..HHHHHH. ..u .. .u......
3 ...... 0 0 NONE 94 Well 1 30 ...... VUHHHH HHHU ...... OEM on
MHURN 40
50
0 1000 FEET
Figure 4. Model grid for simplified alluvial-valley system showing stream and well location .
Well Discharge of 2,000 Cubic Feet per Day from well screened at this shallow depth is derived from Three Different Screened Intervals areal recharge at the water table; the contributing The vertical placement of the well screen area, therefore, equals the discharge rate divided by affects the location of the area contributing the areal recharge rate-4.0x 105 ft2. The maximum recharge. Figure 6 shows the calculated areas con time required for a particle of water to travel from tributing recharge to a well discharging 2,000 ft3/d its origin at the water table to the well is 41 .8 years, and located in the same place areally but with three and the minimum time is 0 .0 years . different vertical screen intervals. All three screened With the screen at an intermediate depth intervals are 25 ft. The areas contributing recharge (50-75 ft below the water table), the area contribut- for all three intervals are on the same side of the ing recharge to the well does not directly overlie the stream as the well. well but extends from near the well to the valley With the screen at the shallowest position walls . With this intermediate screen depth, the well (0-25 ft below the water table), the area contribut- derives 97 percent of its flow from areal recharge at ing recharge to the well lies directly above the well. the water table and 3 percent from the lateral inflow The area contributing recharge does not extend to at the valley wall. The maximum time required for a the valley walls. All of the water flowing to the particle of water to travel from its origin at the
6 Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
travel from the water table to the well is 58.9 years, and the minimum time is 18 .8 years . This series of figures (figs . 6 and 7) for a well having its screened interval at different depths high- lights the importance of recognizing the three dimensional aspect of a flow system in determining paths of ground-water flow to a well.
Well Discharge of 10,000 Cubic Feet per Day from Three Different Screened Intervals
The area contributing recharge to well 1 dis- charging at a rate of 10,000 ft3/d with a 25-ft-long screened interval located at depth intervals of 0 to 25 ft, 50 to 75 ft, and 125 to 150 ft below the water o loon FEET table is shown in figure 8 . The area contributing EXPLANATION recharge is not contiguous at this discharge rate for [m Stream all the screened intervals, and water that flows to -154- SIMULATED LINE OF EQUAL . Most WATER-TABLE ELEVATION. the well originates on both sides of the stream Interval 1 foot Arbitrary datum. of the contributing area is on the same side of the stream as the well, but a part is on the opposite side Figure 5. Configuration of the water table for the simu- of the stream . Because the stream is only partially lated alluvial valley with no ground-water withdrawals . penetrating, water can flow beneath the stream to the well . The water-table contours shown on figure 8 do not reflect these deep flow patterns for any of water table to the well is 54.2 years, and the mini- the screened intervals. No water is derived directly Water derived from lateral mum time is 4.0 years. from the stream for any of the screened intervals, the inflow takes longer to flow to the well than but the total discharge to the stream from the maximum time required for water derived from the ground-water system has decreased by 10,000 ft3/d, water table. the amount discharged (captured) by the well. The With the deep screen (125-150 ft below the three-dimensional aspect of the water movement in water table), the area contributing recharge to the ground-water systems is evidenced by the flow of well lies at some distance from the well and extends some of the water beneath the stream to the well to the valley wall . The area contributing recharge is discharging at a rate of 10,000 ft3/d. distant from the well because, for the specified The vertical position of the well screen in the screen depth and discharge rate, the well intercepts three-dimensional domain affects where the water deep pathlines that originate far upgradient in the flowing to the well originates, as shown previously system. These pathlines are shown in cross section for a well discharging at 2,000 ft3/d. Examination of in figure 7 . Well 1 is capturing some of the lateral the effect of the position of the screened interval on inflow, which represents flow from the valley walls, a well with a discharge greater than 2,000 ft3/d pro- and some of the areal recharge occurring over the vides some additional insight into understanding the entire valley . In this case, the area contributing location of areas contributing recharge. Although the recharge is located on the same side of the river as location of the areas contributing recharge are simi- the well . No water is derived directly from the lar for the different screened intervals, there are stream, but the total discharge to the stream from some obvious differences . In particular, the location the ground-water system is decreased by the amount of the area contributing recharge to the well discharged (captured) by the well, which is 2,000 screened at 0 to 25 ft below the water table overlies ft3/d. With this screen interval, the well derives 68 the well, and the range in time for water to travel to percent of its flow from areal recharge at the water the well from its entrance at the water table is 0 .0 to table and 32 percent from lateral inflow at the valley 81 years. For the deepest well screen, at a depth wall. The maximum time for a particle of water to interval of 125 to 150 ft below the water table, the
Delineation of Areas Contributing Recharge to Wells in Hypothetical Ground-Water Systems 7
EXPLANATION
Area contributing recharge to well Stream SIMULATED LINE OF EQUAL WATER-TABLE ELEVATION. Interval 1 ft. Arbitrary datum. Discharging well
C x,1000FEET
Figure 6. Simulated configuration of the water table and areas contributing recharge for a well discharging at a rate of 2,000 cubic feet per day screened at three different depths . A, Screened interval=0-25 ft below water table. 8, Screened interval=50-75 ft below water table. C, Screened interval= 125-150 ft below water table.
area contributing recharge is not located at the well some of the water must travel vertically downward bore, and the time of travel of water flowing to the from the water table to the screened interval . The well from its point of recharge at the water table range in traveltimes from the water table to the ranges from 3 .5 to 83 years. Thus, the area contrib- well for the area contributing recharge is 1 .5 to 87 uting recharge and three-dimensional flow paths are years . The area is in two parts, one on each side of again affected by the vertical location of the well the stream . The area is adjacent to the stream, but screen. no water is derived directly from the stream at this discharge rate. Well Discharge of 30,000 Cubic Feet per Day The area contributing recharge becomes larger Well Discharge of 30,000 Cubic Feet per Day in the if the well is screened at the bottom of the aquifer Presence of an Additional Well Discharging 130,000 Cubic Feet per Day (125- to 150-ft depth interval) and discharges 30,000 ft3/d (fig. 9) . At this discharge rate, part of Ground-water systems are dynamic, and flow the contributing area directly overlies the well, and patterns within them change in response to changes
Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
Area overlying flow Area contributing paths to well 1 that recharge to well Stream discharge to stream
;u `~w`~ v`~/l_ Location of lateral inflow MAP to well 100 MIR MR, 4,
------_ 1
0 1000 FEET EXPLANATION VERTICAL SCALE GREATLY EXAGGERATED Finite-difference cell representing well 1 Water that originates in the INNINNIN area contributing recharge to well 1 Water that originates as lateral inflow that flows to well 1 Ground-water pathline
Figure 7. Pathlines in the ground-water system with well 1 screened at 125 to 150 feet below the water table and discharging at a rate of 2,000 cubic feet per day.
in stress . This is illustrated by simulating the same quality of the water eventually discharged from hypothetical system as the one just discussed but well 1 . with the addition of a second discharging well. The area contributing recharge to well 2 is The original well (designated well 1) contin- very complex (even for this simplified hypothetical ues to discharge at a rate of 30,000 ft3/d. However, system) . The water-table contours reflect the well's a second well (well 2, located at row 22, column effect on the flow system on the same side of the 49) discharges at a rate of 130,000 ft3/d from the stream, but the parts of the area contributing bottom of the aquifer at a depth interval of 125 to recharge to well 2 that surround the area contribut- 150 ft below the water table. The heads throughout ing recharge to well 1 on the opposite side of the the entire aquifer system respond to this new stress, stream cannot be inferred readily from the water- and the shape and location of the area contributing table contours. recharge change . The area contributing recharge to well 1 changes, as shown in figure 10, even though Case 2-Low-Hydraulic-Conductivity Local the discharge rate of well 1 has not changed . Confining Unit The area contributing recharge to well 1 under these conditions is contiguous, but most of it is on Case 2 uses a hypothetical alluvial aquifer sys- the opposite side of the stream from the well loca tem similar to that in Case 1, except for the inclu- tion . Also, some of the water flowing to well 1 sion of a low-hydraulic-conductivity layer at 100 ft comes directly out of the stream. This means that beneath the stream. The low-hydraulic-conductivity the quality of the water in the stream may affect the layer, which acts as a confining unit, does not
Delineation of Areas Contributing Recharge to Wells in Hypothetical Ground-Water Systems 9
A B
EXPLANATION
Area contributing recharge to well ® Stream
"'155- SIMULATED LINE OF EQUAL WATER-TABLE ELEVATION. Interval 1 ft. Arbitrary datum. Discharging well
Figure 8. Simulated configuration of the water interval=0-25 ft below water table. B, Screened table and areas contributing recharge for a well dis- interval=50-75 ft below water table. C, Screened charging at a rate of 10,000 cubic feet per day interval= 125-150 ft below water table. screened at three different depths . A, Screened
extend over the entire basin but is present only in tions is that the water flows under the confining unit the vicinity of the stream. The confining unit is sim- and then cannot discharge until it reaches the well. ulated as a 1-ft-thick layer with a vertical hydraulic An examination of specific pathlines in Case 2 conductivity of 0 .0001 ft/d. helps in visualizing the details of the flow system If a well is simulated as being screened at the that cause the area contributing recharge to be two bottom of the aquifer and discharging at a rate of separate areas on the same side of the stream. Path- 10,000 ft3/d, the resultant area contributing recharge lines that represent the path taken by the water orig- is as shown in figure 11B. This contributing area inating at the water table along the top boundary of differs from that for a well discharging at the same the aquifer system are shown in figure 11B. Pathline rate from a homogeneous aquifer (fig . 8). Of partic- A extends from the water table deep into the system ular interest is that, although the area contributing beneath the confining unit and then emerges on the recharge remains in two parts, both parts are on the opposite side of the stream, where the water dis- same side of the stream, in contrast to a well in a charges into the stream. Pathline B extends from the homogeneous aquifer. The reason that water from water table deep into the system beneath the confin- the area contributing recharge on the left of figure ing unit, where the flow path is affected by the 11B (up valley) flows to the well under these condi- discharging well, and the water flows downgradient
10 Factors Affecting Areas Contributing Recharge to Wells In Shallow Aquifers
"area liiu~ .eiaiilalle .0l1MUM111, Millullor- ~°~leleolr
0 1000 FEET EXPLANATION a Area contributing recharge All to well - 155- SIMULATED LINE OF EQUAL WATER-TABLE ELEVATION. Interval 1 ft. Arbitrary datum Stream Discharging well
Figure 9. Simulated configuration of the water table and area contributing recharge for a well discharging at a rate of 30,000 cubic feet per day.
beneath the confining unit to the well, where the In this case, most (71 percent) of the water pathline ends. The water movement represented by flowing to the well originates as lateral inflow along pathline C is from the water table to the section of the valley walls. Areal recharge accounts for 29 per stream that is near the edge of the confining unit, so cent of the water discharged by the well . The mini- that the flow path never gets under the confining mum traveltime from the area that contributes unit to be "captured" by the well. It is important to recharge to the well is 41 years-a significant note that, although no water from the water table in increase from the 3 .5 years for the homogeneous the area of pathline C flows to the well, some of the system of Case 1 . lateral inflow from the north (top of the figure) The presence of the 1-ft-thick confining unit is boundary does flow under the confining unit and not indicated by the water-table configuration. In flows to the well, as shown by the area of lateral fact, the water-table configuration is similar to that inflow on the cross section in figure 11A . Pathline for the case of a homogeneous aquifer (fig . 8C). D extends from the water table deep into the system However, even with similar water-table configura- beneath the clay layer and ends at the well. tions, the localized confining unit significantly
Delineation of Areas Contributing Recharge to Wells in Hypothetical Ground-Water Systems
0 1000 FEET EXPLANATION l_I I I I J Area contributing recharge to well 1 Area contributing recharge El to well 2 -155 .- SMULATED LINE OF EQUAL WATER-TABLE ELEVATION. Interval 1 ft. Arbitrary datum. ® Stream Discharging well
Figure 10. Simulated configuration of the water table and areas contributing recharge for two wells discharging simultaneously : well 1 discharges at a rate of 30,000 cubic feet per day, and well 2 discharges at a rate of 130,000 cubic feet per day .
affects the location of the area contributing recharge zone of high hydraulic conductivity beneath the and the time of travel to well 1 . stream only in layer 6 (the 25-ft-thick screened interval at the bottom of the aquifer) and the other with the zone of high hydraulic conductivity extend- Case 3-High-Hydraulic-Conductivity Units ing throughout the total thickness of the aquifer (approximately 150 ft) beneath the stream. In both In Case 3, a hypothetical alluvial system simi- cases, the background horizontal hydraulic conduc- lar to that in Case 1 is simulated, but the system tivity is simulated as 100 ft/d, and the deposits with now includes a zone of high hydraulic conductivity a high hydraulic conductivity have a horizontal beneath and (or) adjacent to the stream. The well hydraulic conductivity of 500 ft/d. The vertical discharges at a rate of 10,000 ft3/d. Two different hydraulic conductivity is one-fourth the horizontal systems are evaluated for Case 3-one with the conductivity everywhere.
12 Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
IMP '"""""""" IIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIilllllillllllllllllllllllillllllllllllllllllllllllllllllllllllllll IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII IIIIIIIIIIIIIIIIIIIIIIIIIIIIIilllllllllllllllllllllllllllllllllllllll III IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIINIIIIIIIIIIIIIIIIIIIIIIIIII A VERTICAL EXAGGERATION x 20.0
B 0 1000 FEET
EXPLANATION Location of low permeability -155 - SIMULATED LINE OF confining unit 100 feet deep EQUAL WATER-TABLE Area contributing lateral ELEVATION. Interval 1 ft. . inflow to well Arbitrary datum Area contributing recharge Pathline ® Stream Discharging well
Figure 11 . A, Cross section showing area of lateral inflow on finite-difference grid . B, Simulated configuration of the water table and area contributing recharge for a well discharging at a rate of 10,000 cubic feet per day, in the presence of a discontinuous confining unit . See text for discussion of pathlines A-D .
Delineation of Areas Contributing Recharge to Wells in Hypothetical Ground-Water Systems 13
0 1000 FEET EXPLANAT10N Location of high hydraulic conductivity lower layer N8 Area contributing recharge tm Stream ~' 155 -- SIMULATED LINE OF EQUAL WATER-TABLE ELEVATION. Interval l ft. Arbitrary datum. Discharging well
Figure 12. Simulated configuration of the water table and area contributing recharge for a well discharging at a rate of 10,000 cubic feet per day, with a layer of high hydrau- lic conductivity at the bottom of the aquifer.
Figure 12 shows the calculated area contribut- tributing recharge is quite different, as shown in ing recharge for the case of a layer of high hydrau- figure 13 . The area contributing recharge now com- lic conductivity in the screened zone at the bottom prises three distinct parts. Recharge water from the of the aquifer in the vicinity of the stream. This part in the upper left corner now flows toward the contributing area and the water-table configuration stream along the left boundary of the valley, and, are almost identical to those for the homogeneous once under the stream, it flows in the high- system as shown in figure 8C. As simulated, the hydraulic-conductivity section of aquifer until it dis- 25-ft-thick layer of high hydraulic conductivity has charges at the well. The water-table configuration very little effect on the location of the area contrib- differs from that shown in figures 8C and 12, but uting recharge to the well. However, if the entire the difference is not great, even though the differ- thickness of aquifer adjacent to the stream has a ence in the locations of the areas contributing high hydraulic conductivity, the resultant area con- recharge is significant .
14 Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
0 1000 FEET EXPLANATION Zone of high hydraulic conductivity E-83 Area contributing recharge Stream ~' 155 SIMULATED LINE OF EQUAL WATER-TABLE ELEVATION. Interval 1 ft. Arbitrary datum. Discharging well
Figure 13. Simulated configuration of the water table and area contributing recharge for a well discharging at a rate of 10,000 cubic feet per day, with a zone of high hydrau- lic conductivity (500 feet per day) 150 feet thick along the stream.
Case 4-Random Distribution of Hydraulic hydraulic conductivity is simulated in Case 4. A Conductivity correlated random field assumes that the parameter of interest is a random variable but that the parame- Properties of natural systems can vary signifi- ter values at adjacent points are related. This rela- cantly in space. The manner in which these proper- tion is specified such that, as the distance between ties vary in aquifers can be very complex, depend increases, they become less related (or ing on the depositional environment of the rock or two points unconsoli- unconsolidated materials that constitute the aquifer. correlated). The deposition processes of One conceptualization that accounts for the variabil- dated porous media would produce a hydraulic- ity in hydraulic conductivity treats hydraulic conduc- conductivity distribution that would probably be cor- tivity as a random variable. To obtain a sense of the related spatially. The simulation of a correlated effect of this randomness on the location of the area random distribution of hydraulic conductivity is a contributing recharge, a correlated random field of simplified means to test the significance of variabil-
Delineation of Areas Contributing Recharge to Wells in Hypothetical Ground-Water Systems 15
0 EXPLANATION Distribution of Hydraulic Conductivity Less than 25 feet per day . 25 to 75 feet per day 75 to 125 feet per day H125 to 175 feet per day Greater than 175 feet per day Arithmetic mean equals 100 feet per day
Figure 14. Random two-dimensional hydraulic-conductivity distribution in a hypothetical alluvial-valley aquifer.
ity in hydraulic conductivity on the areas contribut- correlation length, an arithmetic mean of 100 ft/d, ing flow to a well in aquifer systems . and a standard deviation of 30 ft/d are calculated The hydraulic conductivity used in Case 4 var- and used to illustrate potential effects of hydraulic- ies areally but is the same for each layer. This two- conductivity variability on areas contributing dimensional hydraulic-conductivity field was gener recharge . Again, the well is screened over the bot- ated using the program TUBA (Mantoglou and tom 25 ft of the aquifer and discharges at a rate of Wilson, 1982 ; Zimmerman and Wilson, 1989), 10,000 ft3/d. which generates two-dimensional spatially correlated The first realization of the hydraulic conduc- random fields by the turning-bands method. For the tivity distribution used is shown in figure 14 . The purposes of this report, three realizations (or three hydraulic conductivity ranged from approximately specific distributions out of the infinity of possible 10 ft/d to 190 ft/d. The resultant area contributing distributions) of a random hydraulic-conductivity recharge for this first realization is shown in figure field that has a normal distribution with a 1,000-ft 15B. The area contributing recharge consists of two
16 Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
A P_' 1000 FEET
x,1000 FEET C 01 1000 FEET
EXPLANATION Area contributing recharge to well
~' 155- SIMULATED LINE OF EQUAL WATER-TABLE ELEVATION. Interval 1 ft. Arbitrary datum. ® Stream Discharging well
Figure 15. Simulated configuration of the water table and areas contributing recharge for four cases. A, Homogeneous hydraulic conductivity of 100 feet per day, B-D, Three realizations of a random hydraulic- conductivity distribution with an arithmetic mean of 100 feet per day and the same variance .
parts, one on each side of the stream. The location it differs slightly at some locations. The water-table is similar to the area calculated for the homogeneous configurations in figures 15A and 15B have the case (fig. 8C, which is reproduced as fig. 15A), but same general shape that reflects the stream gradient
Delineation of Areas Contributing Recharge to Wells in Hypothetical Ground-Water Systems 1 7
0 1000 FEET
EXPLANATION Area contributing recharge for aquifer with uniform hydraulic conductivity Outline of area contributing recharge for one random hydraulic-conductivity distribution ^Outline of area contributing recharge for a second random hydraulic-conductivity distribution ® Stream o Discharging well
Figure 16. Areas contributing recharge to a discharging well in an aquifer with two different randomly distributed hydraulic-conductivity fields and in an aquifer of uniform hydraulic conductivity .
and geometry; any differences probably could not be indistinguishable, from a field perspective . Although detected by field measurements . subtle, differences are evident in the location of the Two other realizations of the random areas contributing recharge (fig . 16) . hydraulic-conductivity field were analyzed. The The standard deviation of the hydraulic con- areas contributing recharge and the water-table con ductivity used in the three realizations is probably figurations for the homogeneous case (also shown as smaller than that for most actual ground-water sys fig . 8C) and for the three realizations of the random tems. However, even this small variation in hydrau- hydraulic-conductivity field are shown in figure 15 . lic conductivity about the mean causes differences in All of the water-table surfaces for the four realiza- the location of the area contributing recharge (fig . tions (one homogeneous and three random) are 16) . Because hydraulic properties of natural systems
1 8 Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
vary spatially and cannot be quantified exactly, the uate the flow paths on a systemwide basis instead of areas contributing recharge to discharging wells can- on an individual well basis in order to delineate not be delineated without some degree of uncer- these areas correctly . Once areas contributing tainty. This uncertainty needs to be accounted for recharge to a well are delineated for a particular when attempting to define areas contributing site, they must be reevaluated as new wells are recharge under field conditions. added to the ground-water system . Uncertainties in the definition of the hydroge- ologic framework and transmitting properties of the DISCUSSION OF SIMULATION RESULTS aquifer system can have a major effect on the shape The variation in the size and location of the and location of areas contributing recharge . The calculated areas contributing recharge to a well in specified heterogeneities in Cases 2 and 3 affected the hypothetical alluvial-aquifer systems represented the shape and location of the area contributing in Cases 1-4 illustrates the complexity of ground- recharge compared to the location and area delin- water systems . Although the cases examined are not eated for the homogeneous system in Case 1 (fig. exhaustive, they do provide explicit examples, under 8C) . The relatively mild random distribution (a nor- specified conditions, of the cause and effect rela- mal distribution with a small variance) of hydraulic tions that must be considered in attempting to deter- conductivity in Case 4 also affected the location of mine the sources of water (areas contributing the area contributing recharge. recharge) to wells. The experimental results for Cases 1-4 show As illustrated in Case 1 (figs . 6-10), the loca- that the boundary conditions of the system and the tion and size of the area contributing recharge location of the well screen (particularly if the well is depend on the overall geometry and boundary condi partially penetrating) are very important in determin- tions of the system, as well as the location, screened ing the area contributing recharge. In calculating interval, and discharge rate of the well. If the and evaluating areas contributing recharge (or the boundary conditions, such as recharge rates or the contributing area and zone of contribution), it is stream stage, change, then the area contributing important to consider the entire three-dimensional recharge to a well will change. The location and ground-water system . The volume of earth material size of the area contributing recharge to the well and fluid that is associated with flow to a discharg- also will change as a function of the discharge rate, ing well should be envisioned as a "bundle" of flow well location, and position of the screened interval. tubes that changes shape as it traverses from its Results of the simulation for all 4 of the cases starting point in the flow system to its exit point show that the area contributing recharge to a single from the system at the discharging well . When well is not necessarily one contiguous area and does using terms to describe this complex "bundle" of not have to surround or be contiguous with the well. flow tubes, care needs to be taken to qualify the Even for the relatively simple systems considered assumptions used and the three-dimensional volumes here, flow paths to a well were shown to be a com- defined . plex reflection of the subtle interactions among boundary conditions and hydraulic properties that CONSIDERATIONS FOR THE take place in three-dimensional systems . PROTECTION OF GROUND WATER A discharging well is not an independent force on flow paths in a ground-water system. Flow Many State agencies are currently (1993) within a ground-water system reflects all the stresses developing wellhead-protection programs. The thrust imposed on that system . Flow paths change in of some of these programs is to protect water sup response to new stresses on the system, even if they plies by determining the areas contributing recharge are imposed at locations other than at the well of to water-supply wells and by specifying regulations interest. Figure 10, for Case 1, illustrated that even to minimize the opportunity for contamination of the though the discharge rate for well 1 did not change, recharge water by activities at the land surface. The the area contributing recharge to it changed in areas contributing recharge to wells delineated under response to the addition of another well. Any calcu- specified hypothetical conditions illustrate that, at lations to determine the contributing area must eval- least under some conditions, the determination of
Considerations for the Protection of Ground Water 19 areas contributing recharge is complex, because all water. The overriding conclusion is that, for any ground-water systems are three-dimensional, and the management approach to be successful, the uncer- transmitting properties, hydrogeologic framework, tainties associated with ground-water flow paths and boundary conditions are very difficult to deter- need to be considered. mine with certainty. Thus, any determination of the areas contributing recharge needs to include an esti- mation of the uncertainties involved and the poten- REFERENCES CITED tial changes in stress on the system and needs to Bair, E .S., and Roadcap, G .S., 1992, Comparison of identify an area that comprises the combined areas flow models used to delineate capture zones of under the suite of possible conditions for effective wells: 1, Leaky-confined fractured-carbonate aquifer: protection of a well's recharge area. The combined Ground Water, v. 30, no. 2, p. 199-211 . area, where recharge might occur under different Barlow, P.M., 1989, Delineation of contributing areas to conditions, could be considerably larger than the public supply wells in stratified glacial-drift aquifers, area contributing recharge for any one condition, but in Protecting ground-water from the bottom up: Pro- the combined area would account for the range of ceedings, U.S . Environmental Protection Agency, uncertainty in the estimation of a contributing area. Region I, Oct. 2-3, Danvers, Mass ., p . 145-164. Even though the area contributing recharge to Bear, J., and Jacobs, M., 1965, On the movement of a well is distant from the location of the well, a water bodies injected into aquifers: Journal of potential exists for contamination outside of the con Hydrology, v. 3, p. 37-57 . Brown, tributing area to affect the quality of the water flow- R.H ., 1963, The cone of depression and the area of diversion around a discharging well in an infinite ing to the well. For example, in the situation shown strip aquifer subject to uniform recharge: U.S . Geo- in figure 7, an area exists above the flow paths to logical Survey Water-Supply Paper 1545-C, p. the well where water entering the system discharges C69-C85 . to the stream. Water entering this area does not flow Jacob, C .E., 1950, Flow of ground water, in Rouse, to the well. However, if other wells, even nondis- Hunter, ed., Engineering hydraulics: New York, charging wells, have screens or are open holes that N.Y., John Wiley & Sons, p . 321-386. penetrate both the upper and lower zones, water can Javandal, I., and Tsang, C., 1986, Capture-zone type move downward through the well and "short circuit" curves: A tool for aquifer cleanup: Ground Water, v. the flow paths, allowing the mixing of the shallow 24, no. 5, p. 616-625. and deep waters. In addition, unknown heterogene- Kirkham, D., and Sotres, M.O., 1978, Case depths and ities in the hydraulic-conductivity distribution can solute travel times to wells: Water Resources cause mixing in the vertical direction, or dense, Research, v. 14, no . 2, p . 237-243 . immiscible contaminants may "sink" through the Mantoglou, A., and Wilson, J .L., 1982, The turning overlying water and affect the quality of the water bands method for the simulation of random fields flowing to the well. Additional stresses and varia- using line generation by a spectral method: Water tions in recharge over time can also induce mixing . Resources Research, v. 18, no . 5, p. 1379-1394. Thus, the area of concern perhaps is not only the McDonald, M.G., and Harbaugh, A.W., 1988, A modu- area where the water enters the ground-water system lar three-dimensional finite-difference ground-water and flows to the well but also is the entire area flow model: Techniques of Water-Resources Investi- overlying any flow paths to the well . gations of the U.S. Geological Survey, book 6, An implication of this discussion is that very chap. Al, 586 p . large areas may have to be considered for protec- Milne-Thomson, L .M., 1955, Theoretical hydrodynamics tion, if protection is deemed the most appropriate (3d ed.) : New York, N.Y ., The Macmillan Co., 632 means of ensuring high-quality ground water. Other P. Morrissey, D options to ensure a high-quality ground-water supply .J ., 1989, Estimation of the recharge area contributing water to a pumped well in a glacial- could include relocation of wells as problems arise, drift, river-valley aquifer: U.S . Geological Survey a combination of regulating some aspects of the Water-Supply Paper 2338, 41 p. areas contributing recharge in conjunction with Mundorff, M .J ., Bennett, G.D ., and Ahmad, M., 1972, chemically treating (cleaning) the water discharged Electric analog studies of flow to wells in the Punjab from wells before distribution, and complete reli- aquifer of West Pakistan: U.S . Geological Survey ance on water treatment to provide high-quality Water-Supply Paper 1608-N, 28 p.
20 Factors Affecting Areas Contributing Recharge to Wells in Shallow Aquifers
Ogata, Akio, 1963, Effects of the injection scheme on the American Society of Civil Engineers, v. 104, no. spread of tracers in ground-water reservoirs, in Short HY12, p. 1565-1575 . papers in geology and hydrology: U.S . Geological Slichter, C .S ., 1899, Theoretical investigation of the Survey Professional Paper 475-B, p . 13199-13202. motion of ground waters, in U.S Geological Sur- Phillips, K.J., and Gelhar, L.W., 1978, Contaminant vey's 19th Annual Report, 1897-1898, Part II: p . transport to deep wells: Journal of the Hydraulics 297-384. Division, American Society of Civil Engineers, v. Solley, W.B ., Merk, C.F., and Pierce, R.R., 1988, Esti- 104, no. HY6, p . 807-819. mated use of water in the United States in 1985: Pollock, D.W., 1989, Documentation of computer pro- U.S . Geological Survey Circular 1004, 82 p. grams to compute and display pathlines using results Streeter, V.L., 1948, Fluid dynamics : New York, N.Y., from the U.S . Geological Survey modular three McGraw-Hill Book Co., 263 p. dimensional finite-difference ground-water flow U.S . Environmental Protection Agency, 1987, Guidelines model: U.S. Geological Survey Open-File Report for delineation of wellhead protection areas: Office 89-381, 188 p. of Ground-Water Protection, U .S . Environmental 1990, A graphical kernel system (GKS) version of Protection Agency, June 22, 1987, 131 p . computer program MODPATH-PLOT for displaying Zimmerman, D.A., and Wilson, J .L., 1989, Description pathlines generated from the U .S. Geological Survey of and user's manual for TUBA, A computer code three-dimensional ground-water flow model: U .S . for generating two-dimensional random fields via the Geological Survey Open-File Report 89-622, 49 p. Turning Bands Method: Socorro, N .M., Hydrology Reilly, T.E., 1978, Convective contaminant transport to Program, New Mexico Institute of Mining and Tech- pumping well: Journal of the Hydraulics Division, nology, 76 p .
References Cited 21