ELECTRSC MALOG MODEL STUDY OF THE HYDROLOGY OF THE SAGINAW FORMATION IN THE LANSING, MICHIGAN AREA Thesis for the Degree of M. S. MICHIGAN STATE UNIVERSITY MERLiiN L. WHEELER 1967 THESE WI 3 \\\\\1\\fl\j\\\|\1\\fiw\l\\MAW" PLACE IN REIURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1’ r} 4 . Nov 1 4 2005 2/05 a/cm.mrs ‘ ' “**" yve'f-j‘rrvm i‘mmmz... 1 W ABSTRACT ELECTRIC ANALOG MODEL STUDY OF THE HYDROLOGY OF THE SAGINAW FORMATION IN THE LANSING, MICHIGAN AREA by Merlin L. Wheeler The Saginaw Formation, a bedrock artesian aquifer, is the primary source of water for domestic and industrial use in the Lansing, Michigan area. Ground water with- drawal by the many centralized supply systems has resulted in a large cone of depression which is expanding rapidly. Community or city planning, desirable for many reasons, must include an analysis of the present and future ground water resources. Previous investigations, employing various types of theoretical analyses, have been hampered by the great complexity of the aquifer. The idealized assumptions made in these analyses have invalidated the results, and theoretical analysis without these assumptions is extremely difficult. To facilitate an analytical approach to the problem, an electric analog model was constructed. The initial design of the model utilized data presently available. The model was then modified until the analog simulation of the decline in the piezometric surface for several Merlin L. Wheeler periods agreed closely with the actual declines. The model is now considered to be an accurate analog of the aquifer. Analysis of the model has revealed several signifi— cant facts about the hydrology of the aquifer. The single most important source of recharge is from the leakage of water through the upper confining beds. Further, this leak— age has significantly lowered the Water levels in the saturated glacial material above the confining beds. The ' variation in the leakage rate throughout the area has been specified. The uniformity of the leakage rate results in a reduction of the horizontal movement of water through the aquifer. This means that the transmissibility of the aquifer is not as significant a factor as it was once considered to be. The artesian storage coefficient was found to have no effect upon the long term drawdowns in the aquifer. In areas where the aquifer has begun to dewater, the water table storage coefficient has only a minor effect. Recharge to the aquifer from the rivers flowing through the area has been determined, and was found to reach a maximum after the piezometric sur- face dropped below the bottom of the river channel. The electric analog model will be useful in the planning of future water resource development. ELECTRIC ANALOG MODEL STUDY OF THE HYDROLOGY OF THE SAGINAW FORMATION IN THE LANSING, MICHIGAN AREA By Merlin L; Wheeler A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Geology 1967 ACKNOWLEDGMENTS I would like to express my appreciation to everyone who has assited me in this study. I am indebted to Ken Vanlier, of the United States Geological Survey in Lansing, who suggested the project to me, and provided a great deal of extremely valuable criticism and encouragement as it proceeded. I would like to thank the United States Geologi— cal Survey in Lansing, which provided me with most of the data used in the study; also the municipal supply systems in the area, the Lansing Board of Water and Light, the City of East Lansing, Michigan State University, and the local townships, which were very cooperative in providing me with pumping information. Dr. William Hinze, my academic advisor, was very helpful throughout the study, and provided the equipment needed for the construction and analysis of the analog model, through the use of funds from the Water Resources Institute. In addition, I would like to thank all of the other people, too numerous to mention, who have helped me in so many ways. 11 TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES INTRODUCTION HISTORY OF INVESTIGATION THEORY OF ELECTRIC ANALOG MODELING. UTILIZATION OF THE ELECTRIC ANALOG MODEL TO OBSERVE OR PREDICT HEAD CHANGES. ERROR PRESENT IN STUDY. DESIGN OF THE ELECTRIC ANALOG MODEL OF THE SAGINAW FORMATION . . . . ANALYSIS OF THE MODEL Steady State Analysis Transient Analysis FINAL ANALYSIS OF LEAKAGE RATE CONCLUSIONS BIBLIOGRAPHY APPENDIX iii Page ii iv 16 2O 22 2A 2A 28 37 39 66 Table LIST OF TABLES Leakage Rates for Recharge Regions Shown in Figure 9, for Undeveloped Conditions of 1900. . . . . . . Flow from Aquifer to River in Undeveloped Condition of 1900 . . . . . . . River Recharge Rates Expected Ground Water Withdrawal by 1975. iv Page A2 LI3 AA 72. Figure l. 2. 10. ll. l2. 13. IA. 15. LIST OF FIGURES Aquifer Grid Electrical Network Block Diagram of Electronic Equipment Used in Analysis . . . Design and Analysis of Analog Model Transmissibility of the Saginaw Formation Piezometric Surface of the Saginaw Formation in 1900. Leakage to the Saginaw Formation from the Overlying Glacial Drift in 1900. Analog Simulation of the Piezometric Surface of the Saginaw Formation in 1900 Recharge Regions for Analog Model Based on Leakage Map Piezometric Surface of the Saginaw Formation in 1935. Piezometric Surface of the Saginaw Formation in 1945. Piezometric Surface of the Saginaw Formation in 1964. Decline in the Piezometric Surface of the Saginaw Formation from 1900 to 1935 Decline in the Piezometric Surface of the Saginaw Formation from 1935 to 1945 Decline in the Piezometric Surface of the Saginaw Formation from 1935 to 196A Page II 45 A6 47 AB “9 50 51 52 53 SA 55 56 57 Figure 16. Production Wells and Generalized Pumping Centers for Analog Model . . . . . 17. Average Daily Ground Water Withdrawal from Pumping Centers . . . . . . 18. Analog Simulation of the Decline in the Piezometric Surface of the Saginaw Formation from 1900 to 1935 19. Analog Simulation #1 of the Decline in the Piezometric Surface of the Saginaw Formation from 1935 to 1945 20. Analog Simulation #2 of the Decline in the Piezometric Surface of the Saginaw Formation from 1935 to 19u5 21. Analog Simulation of the Decline in the Piezometric Surface of the Saginaw Formation from 1935 to 196A 22. Potential Leakage to the Saginaw Formation from Overlying Glacial Material. 23. Centers of Ground Water Withdrawal from the Saginaw Formation . ,. . . . . 2“. Predicted Piezometric Surface of the Saginaw Formation in 1975 . . . . . vi Page 58 59 61 62 63‘ 6A 65 73 7A INTRODUCTION The Saginaw Formation is the primary source of water for domestic and industrial use in the Lansing, Michigan area. Nearly all of the high-yield wells (greater than 200 gpm) tapping this bedrock aquifer are owned by municipal supply systems: The Lansing Board of Water and Light, the City of East Lansing, Michigan State University, and Lansing, Delta, Delhi, and Meridian Townships. The remaining high— yield wells are privately or commercially owned, and for the most part are located outside the present metropolitan area. Low—yield domestic wells are found in the many areas not presently served by a centralized water supply system. The average withdrawal of ground water from the aquifer in 196“ was in excess of 25 million gallons per day. The untreated water from the aquifer is quite consis- tent in chemical quality. The iron content, generally high, averages about 0.2 ppm, and the total carbonate hard— ness averages about 350 ppm. Other ions are present: chlorides, sulphates, etc., but none in any significant quantity. The water temperature is generally around 50° F. Domestic softening systems can lower the hardness, but in most cases will not affect the iron content. This has been a major reason for the expansion of municipal systems, which with superior treatment facilities can pro- duce water of a much higher quality. The Saginaw Formation varies radically in thickness and composition throughout the area, being composed mostly of interbedded sandstone and shale. The sandstone is quite consistent in permeability, about 100 gpd/ft2. The shale is relatively impermeable, therefore the transmissi- bility of the aquifer is primarily a function of the total thickness of sandstone. The bedrock is overlain by a mantle of Pleistocene glacial material, varying from out— wash sands and gravels to thick beds of clay. In most areas the sandstone beds are overlain by shale or clay, or both. These confining beds of shale and clay make the Saginaw Formation an artesian aquifer. That is, the water level in a well penetrating the aquifer will generally be above the bottom of the confining beds. The piezometric surface of an artesian aquifer is the imaginary surface de- fined by the elevation of the water levels in wells tapping the aquifer. The Saginaw Formation is in reality a leaky- artesian aquifer, because the confining beds are permeable enough to allow a significant flow of water between the aquifer and the overlying material. This movement occurs whenever there is a difference in elevation between the piezometric surface and the free water surface in the glacial material. The population of the Lansing, Michigan area is grow— ing at an increasing rate, and the resultant demand for high quality water even faster. Community or city planning, desirable for several reasons, must include an analysis of the present and future water supplies, both ground and sur- face water. To insure the most efficient and economical use of water resources, a thorough understanding of all hydro- logic systems in the area is necessary. The Saginaw Form- ation is one of these systems, and because of its size and complexity is perhaps one of the most difficult to under- stand. HISTORY OF INVESTIGATION The Saginaw Formation has long been the subject of intensive investigation in the Lansing area by Michigan State University, the Michigan Geological Survey, and the United States Geological Survey. In 1945, W. T. Stuart, then of the United States Geological Survey in Lansing, authored Ground Water Resources of the Lansing, Michigan Area (Stuart, 1945). This report utilized data collected over a period of many years. The study involved water level measurements, pumping tests, and other field investi- gations. The report served as a basis for further work, both in regard to the physical parameters of the aquifer and the general concept of the hydrologic processes in the area. The bulk of Stuart's work was aimed at determining the physical parameters of the aquifer (its transmissi- bility and storage coefficient),and the determination of the changes in the hydrologic conditions that had occurred up to that time. To do so, he compiled a great deal of data on water levels for both l9A5 and previous years. Extensive pumping tests were performed, using many of the large diameter, deep Lansing wells. Records were compiled of ground water usage by the Lansing Municipal System, as well as by East Lansing, Michigan State University, and by the various industrial and commercial establishments in the area. Stuart's report views the aquifer as a leaky—artesian system, with a transmissibility ranging from 3000 to 50,000 gpd/ft. The storage coefficient varied from 10"3 to 10-“, realistic values for an artesian aquifer. The study indi— cated that the aquifer was in equilibrium during the period from 1930 to l935. Pumping rates remained essentially con- stant, as did the position of the piezometric surface. Stuart felt that the aquifer was not in equilibrium in 19A5, and that a continuation of pumpage at the 1945 rate would cause a continued decline in the piezometric surface. For an aquifer to be in equilibrium, it must be re- charged at the same rate as water is discharged from it. Stuart listed three main sources of recharge to the aquifer; first, the hydrologic connection between the sandstone and the surface water bodies, especially the rivers and streams in the area; second, the direct recharge where the sand— stone is at or very near the ground surface; and third, the leakage into the sandstone from the saturated glacial material above the confining beds. However, Stuart was unable to quantify the total capabilities of recharge. He was able to determine the recharge rate in 1935, knowing that equilibrium conditions existed, but beyond this, analysis was very difficult. The dangers of developing the aquifer beyond its safe yield were pointed out. The Saginaw Formation is underlain by the Michigan Formation which is composed pre- dominately of shale, gypsum and limestone. This formation produces water which is high in sulphates and chlorides. If the water levels in the Saginaw are excessively low, water from the Michigan will begin to enter the aquifer, contaminating it. Later investigation in the area took primarily a reportative attitude. Two theses (Firouzian, 1962; Mencenberg, 1963) examined the aquifer in the light of further study, and reported the then existent hydrologic conditions. Neither report goes into any great detail on the recharge capabilities of the aquifer. All of the reports to the present have approached the problem on a theoretical level. Pumping tests were analyzed by theoretical techniques, involving certain ideal assumptions. Flow net analyses were performed, again using ideal assumptions. The hydrology in this area is, however, very far from idealized, and any attempt at predicting future hydrologic conditions must take into account all of the important aquifer and recharge characteristics. The complexity of the hydrologic conditions in the area invited some type of analytical approach. It was decided that an electric analog model would be the best method of investigation. This decision was prompted by several things. First, the size and complexity of the aquifer discouraged any form of mathematical approach, even with the utilization of electronic computers. Second, an electric analog model could be built for comparatively low cost, and further, could very easily be modified to simulate any hydrologic condition, past, present, or future. Third, once the model was built and verified using historical data, it could be used as a predictive agent, determining future safe yields, recharge capabili- ties, drawdown distributions, and other parameters of the water resource system. THEORY OF ELECTRIC ANALOG MODELING An electric analog model consists of an array of resistors and capacitors, interconnected in a particular manner. The resistors simulate the transmissibility or permeability of an aquifer, and the capacitors simulate the storage capacity of the aquifer. It is necessary to demonstrate the analogy between this electrical network and the aquifer, and to develop methods for building a specific model, given a particular aquifer (Walton and Prickett, 1963). The equation governing the two-dimensional flow of water in an infinite isotropic aquifer is: 82h 2 x2 3y2 0) h = hydraulic head at some point within the aquifer perpendicular horizontal axes within the aquifer >4 << II S = storage coefficient of the aquifer T = transmissibility of the aquifer t = time in appropriate units. Let some finite distance Ax = 3x, and by = By. En- vision a grid system placed within the aquifer, as depicted in Figure l, with spacing Ax and Ay, where Ax and Ay are small in relation to the aquifer size. *«u w M T fig 5 H- Ax-i <<———9 X-———§ Figure l.-—Aquifer Grid 8h Now, 3? is the change in head with distance in the x-direction. Using the points numbered in Figure l, gg may be approximated by: [3h] : hl - h” . [ii] a. Ei_:_ii (2) 3x ‘ Ax ’ ax Ax . 1 .2 an . . 32h . The change in 3; with distance in the x-direction, . is 8x4 then approximated by I ) [3h] hz - hl hl - h“ 2 3X 1 ~ AX AX AX AX 341% 32h 3x2 "2 OP 32h h2 + h1+ — 2h 3x2 (Ax)2 1 (3) III 10 In a similar manner, IR (A) ay2 (M)2 The area within any small rectangle of the grid is given by (Ax)(Ay). With Ax = Ay, the area, a2 is (Ax)2. Then, 32h + 32h g: h2 + h3 + h,+ + hS — Ah 3x2 ey2 (Ax)2 1 ~ - ' 3h 3?. <5) I-BIC/J Rearranging terms, and substituting a2 for (Ax)2, we obtain: T(h +h +11 +h_uh)=a233§. (6) 2 3 a 5 1 “ at We construct an electrical network of resistors such that the resistors connect a series of Junction, or nodes, as shown in Figure 2, page ll. A capacitor is connected from each node to a common ground. From Kirchoff's Laws for the flow of current through such a network: I + I + IL+ + I = C ——— . (7) Applying Ohm's Law, 2 1 3 1 u 1 5 1 3V (8) 11 where: = current flowing through a particular resistor; = voltage at some node; resistance of the resistor connecting two nodes; = capacitance of the capacitor connected to a node; cf (a :n 1< H n a time in appropriate units. Figure 2.--Electrical Network The aquifer is assumed to be isotropic and homogeneous. Similarly, let: R2 3 R3 = Ru = R5 = R. Substituting this into Equation (8) we obtain: 12 1 8V R_(v2 + v3 + V“ + V5 - 4V1) — —— . (9) I 0 Comparing this with Equation (6): 2 32 Sa at , T(h2 + h3 + ht+ + h5 - Ahl) we see that the two equations can be made similar if T is prOportional to < 3MP h is proportional to O aZS is proportional to The flow of water, (Q), through a section of the aquifer is given by: Q = T I L where: T = transmissibility of the aquifer I = hydraulic gradient L = width of the aquifer section. Returning to the aquifer grid, Figure l, we obtain: (hS " h1)AX AX Q = T ”5‘51““ (‘2‘ + ‘2‘) _ Ax Q - T(h5 - hl) Ky with Ax = Ay Q = T(h5 - hl). (10) 13 Referring to the electrical network, Figure 2, and applying Ohm's Law: If T is prOportional to and h is proportional to V, then :IJII—l Q is proportional to the current, I. A similar derivation is possible for non-homogeneous aquifers, which will yield the same proportionalities that were developed for the homogeneous case. For convenience the time units used in the analog model studies are much smaller than the real time units in— volved, years becoming milliseconds. To allow conversion from hydraulic units to electrical units, the following four proportionality constants have been defined: KA’ with units days/sec K3, with units gpd/amp K2, with units ft/Volt Kl’ with units gal/coulomb (Kl = K3 X KA)' Then sec = l/Ku (days) gpd = l/K3 (amperes) ft = l/K2 (volts) gal = l/Kl (coulombs). 14 These four scale factors can then be adjusted to fit the needs of a particular study, so as to best utilize the analytical equipment available, as well as to use readily available low cost electrical components. To build a specific model, it is necessary to relate T to R, and S to C. Using the four scale factors, and the proportionalities noted, we obtain: _ 1 (1) R - —— - T II N .B' 00 m N (,0 (2) C where: R = resistance in ohms C = capacitance in farads a = node spacing in feet S = storage coefficient T = transmissibility in gpd/ft. A leaky artesian aquifer, in which there is a significant amount of leakage of water into or out of the aquifer through one of the confining beds may be simulated by con— necting a resistor to ground from each node, scaled to the vertical permeability of the confining beds. The value of this resistor is given by: 15 K R = 3 g K2(P'/m')a2 where: P' = vertical permeability of the confining beds m' = saturated thickness of the confining beds a, K3, K2 as previously defined. These formulas allow us to build an electric analog model which duplicates the physical characteristics of a particu- lar aquifer. All aquifers are finite, and allowance must be made for this on our model. A negative boundary, such as might be caused by a fault or a rapid decrease in transmissibility, is simulated by simply terminating the electrical network. A negative boundary is typified by having no flow across it, and this method of simulation meets this criterion. A posi- tive boundary, such as a streamrmr a lake, is represented by a terminal strip connected to ground, connecting all of the nodes along this positive boundary. A positive boundary is typified by having no change in head across it, and again we see that our method of simulation is valid. For cases in which the boundaries are not perfect, such as the partial penetration of a stream into the aquifer, resistors may be used to control the amount of current entering or leaving the network along the boundary, just as the permeability of a river bottom controls the amount of water moving between it and the aquifer. UTILIZATION OF THE ELECTRIC ANALOG MODEL TO OBSERVE OR PREDICT HEAD CHANGES In an aquifer, the hydraulic head at any particular a point will change with time, in response to seasonal fluctuations in precipitation, evapotranspiration, ground water withdrawal, etc. Of these factors, ground water withdrawal is our major concern. The pumpage of water from the aquifer at a certain rate for a certain length of time will produce a decline in the water level in ob— servation wells tapping the aquifer. For each well, a time-drawdown curve results. In the analog model, a specified amount of current (determined by the pumping rate of the wells) is withdrawn for a given length of time (determined by the length of time the wells are pumped). Current is withdrawn by pro- ducing a negative voltage drop across a resistor connected between some voltage source and a node on the analog model. As a result, the voltages at the various nodes of the model change with time, producing a time-voltage curve for each node. As we have shown, this time-voltage curve is analogous to, and with application of the various scale factors can be read directly as, a time-drawdown curve. In most model studies we are concerned with the changes that occur in the 16 17 various hydrologic parameters, not with the absolute magnitude of the parameters. The time scale factors used in analog studies result in a time-voltage curve of very short duration. To ob— serve the curve, we utilize three instruments. First, a pulse generator, which produces a voltage pulse for a certain length of time; second, a waveform generator which triggers these pulses at some specified frequency; and third, an oscilloscope which projects the time-voltage curve on a screen for observation. By triggering the voltage pulses at a high enough repetition rate, the time- voltage curve will appear to remain stationary on the oscilloscope, allowing us to read the values of, or re— cord, this curve. Figure 3 is a block diagram of the equip- ment used for the analysis of the model. The resistor Rp, controls the rate at which current is withdrawn from a particular node, so that the current is proportional to the pumping rate of a well located at that point in the aquifer. If a number of wells were operated under the same pumping schedule, resistors in parallel are used to control the withdrawal rate from the respective nodes. If the withdrawal rate from a particular well changes significantly one or more times during the period being analyzed, a separate pulse generator is used for each pumping rate. The pulse generators are then triggered sequentially, each one producing a voltage pulse whose length is proportional 18 to the length of time that the well was pumped at a particular rate. The analog model also may be used to investigate the hydrology of the aquifer under equilibrium conditions. A particular period is selected when the aquifer was in hydro- logic equilibrium, and the inflow and outflow of water, both natural and artificial is determined. Utilizing a direct current constant voltage supply, electric current is made to flow through the model, proportional to the quantity of water flowing through the aquifer. The voltages observed at the nodes of the model will be analogous to the piezo- metric surface in the aquifer. The storage ability of the aquifer is not included in this steady-state analysis. To recapitulate, an electric analog model is con- structed, utilizing the known hydrologic data, and the apprOpriate scale factors. For transient analysis, the model is activated by the use of various electronic equip- ment. The resultant time-voltage curves are recorded, and compared with the known or expected time-drawdown curves. For steady-state analysis, electric current is made to flow through the model, and the resultant voltage distri- bution is compared with the known or expected piezometric surface. It_is at this point that the strong point of the analog model enters. If, within the limits of error, the model simulation agrees with the observed or expected 19 conditions in the aquifer, then we may assume that the model is truly representative of the aquifer, and further, that the original conceptualization of the aquifer was correct. If, however, some deviation occurs in the model simulation, we know that our original concept was wrong, and must be modified. Quite frequently, experimentation with the model can give us insight into what form of errors we have made; whether recharge areas were neglected, transmissibility or storage coefficients were incorrectly determined, or some other hydrologic phenomena were not considered. The ease with which the model can be altered, and the form of pre- sentation of the information obtained from the model further recommends its application to ground water problems. ERROR PRESENT IN STUDY The error inherent in the electric analog model stems from several sources; the accuracy with which time and voltage can be measured, the precision of the electrical components, and other errors inherent to the electrical network. The finite difference equation, Equation 5, is an approximation, and error is introduced here principally as a function of the ratio of node spacing to aquifer size. All of these errors, for most models, are less than ten per- cent, and theoretical studies indicate far less than this for most cases (Stallman, 1963). The most significant source of error is the accuracy of the basic data; water levels, transmissibilities, stor— age coefficients, pumpage rates, and the other hydrologic parameters. The effect of the error in the physical parameters of the aquifer will vary, and can only be deter- mined through experimentation with the model. The accuracy of water levels is usually given in feet, rather than as a percent. If two water level maps are each considered accurate to f 5 feet, as is the case in this study, the water level change map obtained by subtracting* them will *"Subtracting" is the process of determining the difference in the water levels at each point on the two maps, and contouring the result. 20 21 only be accurate to within : 10 feet. This, unavoidably, is the major source of error in this particular investi- gation. DESIGN OF THE ELECTRIC ANALOG MODEL OF THE SAGINAW FORMATION The design and analysis of the analog model is de- picted in the flow chart, Figure A. To construct the analog model, it was necessary to develop a conceptual model of the aquifer. The transmissibility distribution was Specified as shown in Figure 5. This iso-transmissi- bility map was produced by utilization of the information obtained by W. T. Stuart (Stuart, 19A5) and by the analysis of succeeding pumping tests. Where these were not avail— able, specific capacity information was utilized. Although this is not a completely reliable source of information, it, can provide general guidelines. Where no field measurements of any kind were available, isopachs of the sandstone, in conjunction with an average permeability were used. It is felt that the values obtained are reasonably representative of the aquifer in all parts of the area studied. A The low transmissibility zones surrounding the area allow us to define a discrete aquifer, and provide us with a natural boundary at which to terminate the model. Be- cause there is negligible horizontal flow across this boundary, we know that all of the water discharged from the aquifer must have entered it within the area studied. 22 23 The storage coefficient was assumed to be 10—“, a reasonable value for an artesian aquifer. A leaky—artesian system was assumed, with the rate of flow into or out of the aquifer proportional to the head differential between the piezometric surface and the free water surface. Water (”I7 was assumed to flow between the aquifer and the rivers in a similar manner. Further, it was assumed that there was no significant depletion in stream flow, and that there was no significant change in the water levels in the glacial material. It was not possible to give any specific values to the leakage rates. The choice of values for the four analog scale factors was based on: one, the range in the values of the trans— missibility and the value of the storage coefficient; two, the electrical components and analysis equipment available; and three, the accuracy required in the study. The values are 2 6.A X 10“ days/sec Ku= K3 = 2 X 1010 gallons per day/amp K2 = 10 ft/Volt K = K X KA = 1.28 X 105 gallons/coulomb. l 3 ANALYSIS OF THE MODEL Steady State Analysis In order to determine the leakage rate, a steady— state, or equilibrium analysis of the model was initiated, thus eliminating all time—dependent variable from the hydrologic system. A piezometric map for the equilibrium conditions of 1900 was drawn using data obtained from the United States Geological Survey water level records. How- ever, for much of the area studied, reliable records were not available for this period. In these cases, water levels for later years were examined. It was discovered that at many locations there had been very little change in the piezometric surface for a number of years. At one point, for instance, the water levels in l9A5 were the same as they had been in 1929. Therefore, it was reasonable to assume that they would have been the same in 1900. This principle was only applicable at locations that had not been significantly effected by ground water withdrawal after 1900. In 1900 the aquifer was undeveloped, that is, there was no significant pumpage from the aquifer. The shape of the piezometric surface was determined by the natural dis- charge from, and the recharge to, the aquifer. The 2A 25 piezometric map, Figure 6, indicates that the primary area of discharge from the Saginaw Formation was along the rivers flowing through the area. A flow net analysis was made, obtaining a value for the total rate at which water was being discharged to the rivers at that time. All of the water discharged from the aquifer must have entered it within the area studied, which enabled us to make a pre- liminary estimate of the recharge rates. A map was drawn showing the differences in head between the piezometric surface of 1900 and the water table in the glacial material (based on tOpographic maps of 1910). The areas in which the water table was above the piezometric surface were delineated. For convenience, these positive head differ— ences were divided into two groups, 0-10 feet, and 10-20 feet, as shown in Figure 7. All of the water entering the aquifer must do so within the delineated areas, and at a rate proportional to the determined head differences. Assuming that this prOportion was the same throughout the area studied, the following relation was then solved: Al(w) + A2(2w) = Q where: A1 = area within 0-10 foot contour A2 = area within 10-20 foot contour w = leakage in gpd/mi2 Q = total discharge or total leakage. 26 Similarly, the flow into each portion of the rivers and the head differences between the rivers and the piezo- metric surface allowed the determination of the extent of the hydraulic connection between the aquifer and the river for each river segment shown in Figure 6. With the model constructed along these guidelines, the analog simulation of the piezometric surface for 1900 was measured. It was apparent that an initial error had been made, because the observed piezometric surface was a great deal higher than the real piezometric surface. Cursory examination showed that the extent of the difference could only be eXplained by an error in the recharge rate, the discharge rate, or both. The transmissibility of the aquifer could not be changed enough (within the limits of feasibility) to produce the desired results. Because the most unknown factor involved was the leakage rate, it was decided to change this first. The rate was reduced, and in some areas the discharge rate to the rivers was adjusted slightly, until the analog simu— lation of the piezometric surface agreed closely with the eXpected. This is shown in Figure 8. Comparison of this with the contour map of the real piezometric surface, Figure 6, revealsa significant difference in one area, the vicinity of downtown Lansing. It was found impossible to reproduce these.contours on the model, even by making very drastic changes in the original assumptions. Examination of the original data revealed that the water level 27 measurements used to contour this area were made in shallow wells, penetrating only the upper portion of the aquifer. In this upper portion, the piezometric surface reflected very closely the water levels in the glacial material, rather than the true piezometric surface of the aquifer. This is borne out in other wells drilled at later times. The modification in the piezometric surface produced by correcting this error greatly reduced the total discharge from the formation, accounting for the necessity of reduc- ing the leakage rate to match the analog simulation with the real piezometric surface. For convenience in the analysis, the areas in which leakage was occurring were divided into eleven regions. These are shown in Figure 9, and the total leakage deter- mined from the model for each region is given in Table l. The leakage per square mile, per foot of head was deter- mined for each region. This was done by using the area of a particular region, the total leakage in that region, and an "average head" for the region, so that the leakage rate is given by: Total leakage area X "average head'T ' Leakage = However, the exact shape of the head distribution was not determined. Thus,.the average head value used is a very rough approximation. The values obtained for the leakage per foot of head should be used qualitatively only. The 28 leakage will be taken up at a later time in this paper, and more reliable values will be given. The total flow from the aquifer into each river segment is given in Table 2. Transient Analysis In the steady-state analysis the actual conditions were modeled. However, in the transient analysis, it is necessary to measure the changes that occur in the various time-dependent variables. We are concerned not with the total leakage, but with the increase or decrease in the leakage per foot of head change. We are concerned not with the piezometric surface, but with the change in the piezo- metric surface over a period of time. To use the model for transient analysis it must be modified somewhat. The capacitors must be added to the network. The value of the resistors controlling the leakage rates must be recalculated using the change in leakage per foot of head decline. The head gradients in the rivers, necessary for steady—state analysis, are eliminated. An original equilibrium condition is selected, and the changes in the piezometric surface from this condition are determined. The 1935 piezometric surface, as shown in Figure 10, was in equilibrium, and it was selected as a base from which future changes in the piezometric surface would be determined and modeled. In addition, piezometric maps for l9A5, Figure 11, and for 196A, Figure 12, were 29 drawn. The changes in the piezometric surface from 1900 to 1935, Figure 13, from 1935 to 19A5, Figure 1A, and from 1935 to 196A, Figure 15, were obtained by subtracting the respective piezometric maps. To carry the analysis further, it was necessary to make a detailed examination of the pumpage from the aquifer. Whereas most previous reports were concerned only with the total pumpage from the aquifer, it was necessary to impose the various pumping rates on the model in the proper area. A tabulation was prepared of the total pumpage from each individual well, by years, from all of the high-yield wells in the area studied. Because it would be difficult to im- pose the pumpage from each individual well on the model, wells were arbitrarily grouped together into "pumping centers," all of the wells in a particular pumping center being close to a common center. These are shown in Figure 16. The total pumpage from each of the resultant sixteen pumping centers was then determined. Ideally the various withdrawal rates from each of the pumping centers should be imposed on the model as a continuous function. However, this is impossible to do without very sophisticated equip- ment. Instead, the pumping rates were divided into a series of step functions, each step representing an average pumping rate from a particular pumping center for a certain length of time. To keep the analysis reasonably simple, a total of five steps was employed, covering the period from 1935 to 196A; l935-19A0, 19Al-19A5, 19A6-l952, 1953-1960, 30 1961—196A. In some cases the actual pumpage rates from various pumping centers did not fall conveniently into these five steps, and in these cases some generalizations were necessary. However, for the most part, as shown in Figure 17, the actual pumpage rates fitted these five steps quite well. With the model modified for transient analysis, one final check was made. Under leaky-artesian conditions, the aquifer would reach a given equilibrium condition very rapidly. Thus, if the pumpage rates from 1930-1935 were imposed on the model, the total drawdowns from the unde- veloped condition of 1900 to 1935 should result. However, when this was done, the drawdowns observed were much less than expected. Investigation revealed that there was too great a flow from the rivers into the aquifer. It-was originally assumed that the flow of water between the aquifer and the surface water bodies was proportional to the head differential. However, the pumpage in 1930-1935, and in subsequent years, lowered the piezometric surface below the bottom of the river channel. The high perme- ability of the channel material, and the relative imperme- ability of the adjoining glacial material resulted in the flow into the aquifer being a function of the distance from the river surface to the bottom of the channel, essentially a constant. A method was developed, employ- ing a transistorized switching circuit, for placing a 31 maximum value on the flow of current, in the model, from the river to the aquifer. To facilitate this, the rivers were divided into a series of segments, as shown in Figure 10, similar to those used in the original flow—net analysis. A maximum value was then placed on the current flowing from any one segment into the aquifer. .This involved a certain amount of generalization, but it is believed to be justified. The channel depth was substantially less than the head difference between the river surface and the piezometric surface, resulting in a substantial reduction in the flow of current into the aquifer. This resulted in greater draw- downs on the model. With this modification and some adjust- ment in the amount of water flowing from each river segment into the aquifer, the change in the piezometric surface from 1900 to 1935 was duplicated quite closely, as shown in Figures 13 and 18. The adjustment in the amount of water flowing from each river segment into the aquifer was felt justified for several reasons. First, the original values placed upon the maximum flow from each segment were based on the dis- charge in the undeveloped condition of 1900. The total recharge from the river consisted of intercepted discharge (when the piezometric surface was above the river level), and induced recharge (when the piezometric surface dropped below the river level). In most instances the two were assumed to be equal, but for river segments where the head differential had increased significantly, the value of the 32 induced recharge was assumed to be greater. Second, the values used were obtained by adding together the flows from each portion of the segment. The generalization in- volved would inevitably produce some error on the model. Third, and partially counterbalancing the previous two factors, would be a reduction of the flow caused by a silting up of the river bottom. It is reasonable to ex- pect a net change in the flow rate, and the model indicates that it increased. Table 3 compares the maximum value placed upon the flow from each river segment with the dis- charge observed in the undeveloped condition. The increase in the pumpage from 1935 to 19A5 was imposed on the model. Several things became apparent. First, the artesian storage coefficient which had been in- cluded in the model had no effect on the total drawdowns observed for this 10-year period. Its only effect was to change the shape of the very first portion of the time— drawdown curve. Second, and more importantly, the total drawdowns observed on the model were significantly less than those actually occurring during this period. Several factors could have caused this. A reduction in the trans- missibility might produce the additional drawdown. However, the analysis of the 1935 condition eliminated this possi- bility. Also, the amount of water entering the aquifer from the rivers might have been significantly less in 19A5 than in 1935. This could be caused by an additional silt- ing up of the river bottom. It did not appear, however, 33 that this would cause the magnitude of effect required. One important factor had not been considered. It was originally assumed that no changes occurred in the water levels in the glacial material. However, examination of water level records revealed that with the increase in the flow of water from the glacial material to the aquifer, the water levels in the glacial material had begun to drop, in effect draining the glacial material. This meant that the storage capability of the glacial material must be con- sidered. When this drainage effect was imposed on the model, the resulting piezometric change map was more similar to the expected. However, significant differences still existed between the actual piezometric change and that observed on the model. Examination of the l9A5 piezometric map revealed that one area in the northeast part of Lansing was contributing a great deal more water to the aquifer than had been realized. Examination of well logs in the vicinity indicated the possibility of very permeable saturated sands and gravels lying directly on the sandstone. In addition, the sands and gravels in this area appeared to be connected hydro- logically with other larger bodies of sand and gravel to the north and west, composing a portion of the Mason Esker. It seemed reasonable then, to assume an extensively re- charged body of gravel, connected directly to the aquifer. The model was modified to fit this condition, producing the desired modification in the piezometric change map, 3A shown in Figure 19. This area of recharge was not signifi- cant in 1935 because the drawdowns in the area were quite small. However, with increased drawdown, it became more important. Comparison of the observed and expected piezometric change maps for both 1900—1935, Figures 13 and 18, and 1935-19A5, Figures 1A and 19, reveals that in the southern portion of the area, observed drawdowns were greater than expected in both cases. This additional drawdown was not considered significant in the 1935 analysis, because it was within the expected error of the model. However, the appearance of a similar error in both analyses indicated that an error in the original conceptual model might have been made. Reducing the transmissibility to half its original value had little effect on the observed drawdowns. Similarly, changing the leakage rate within reasonable limits had only a minor effect. Investigation of well logs revealed that in the center of this area the Mason Esker rested directly on the sandstone, and appeared to be re- charged by the Sycamore Creek. It seemed reasonable that a large quantity of water was entering the aquifer in this region. Experimentation with various values on the model revealed that approximately five times as much water was entering the aquifer in this area as had been originally specified. Figure 20 shows the piezometric change map ob- served on the model when this additional recharge was added. Comparison with the original, Figure 19, and with 35 the expected, Figure 1A, reveals the importance of this recharge area. The modified model was then accepted as being representative of the aquifer in l9A5. In l9A5 the aquifer was artesian in all areas. How- ever, continued pumpage lowered the piezometric surface and began to dewater a portion of the aquifer. This area of water table conditions continued to expand, reaching by 196A the extent shown in Figure 12. The inclusion of this dewatering in the model analysis was a difficult problem. It was felt that it should be included because dewatering would produce a reduction in the drawdowns occurring in the aquifer, but to do so in a precise manner would involve a great deal of electronic circuitry. To determine the impor- tance of this dewatered area, it was first assumed that the aquifer remained a leaky-artesian system. The increased pumpage from 1935 to 196A was imposed on the model, and the resultant drawdowns recorded. Then, capacitors that would simulate the water table conditions were added to the model, and it was assumed that this condition existed from 1935 to 196A. This would permit the evaluation of the maximum ef- fect that the water table conditions exhibited, because they actually existed for a shorter time. The increase in pumpage from 1935 to 196A was again imposed on the model, and the resultant drawdowns compared with those obtained under assumed leaky-artesian conditions. Very little difference was observed, generally less than 10 feet over the 30-year period. This error is well within the limits 36 of accuracy of the data, and it was felt that the effect of the water table conditions was not significant. The lack of significance of the water table storage coefficient was attributed to several factors. First, the total dewatered area is very small in relation to the total area of the aquifer, less than two percent. Second, the total amount of water released from water table storage is less than two percent of the total amount of water with- drawn from the aquifer over this 30-year period. Third, it has already been shown that the most significant factor af- fecting the drawdown distribution is the leakage to the aquifer from the overlying material, and this would not be significantly affected by the dewatering. However, as the aquifer is developed further, the areas of water table conditions will become increasingly important. It was felt that to avoid excessive drawdowns in this portion of the model during future analyses, the water table storage coefficients should be included in the model, under the assumption that this portion of the aquifer was water table in 1935. The observed drawdowns were then compared with the expected drawdowns, Figure 15, and found to agree quite closely. A few minor changes were made in the resistors controlling the leakage from the glacial material, and the final observed drawdowns determined, as shown in Figure 21. The model was then accepted as being representative of the aquifer in 196A. FINAL ANALYSIS OF LEAKAGE RATE It was desired to determine the leakage that was occurring from the glacial drift, in units of gallons per day per square mile, per foot of head difference. To do so, the values of the resistors controlling the leakage were recorded. Each resistor controlled the leakage over one square mile. The leakage was then determined by: q=%XK3 where: = leakage in gpd/mi2 V = total voltage drop across resistor R = measured value of resistor K3 = scale factor (2 x 105 gpd/amp). If the voltage equivalent to a one—foot head differ- ential is used, namely, 0.1 volts, the expression becomes: _ 0.1 q - —E— X K3 where "q" is now in gpd/mi2/ft. 'This calculation was made for each region of similar leak- age, and the results are shown in Figure 22. The term 37 38 "Potential Leakage" is used, because this is the leakage rate that can be expected to exist when the piezometric surface in an area is below the water table in the glacial material. In many areas, this condition does not presently exist. It is felt that the values given in Figure 22 are more representative than those given in Table l, for several reasons. First, the data used in their determination, namely the piezometric decline over many years, are more accurate than the information used in the study of the 1900 piezometric surface. Second, the total leakage is much greater, thus an error in the leakage rate in a given area is more likely to be detected. Third, the area in which leakage is occurring is better known than it was in the study of the 1900 condition. This is due to better knowledge of both the position of the water table and of the elevation of the piezometric surface. CONCLUSIONS During the analysis of the electric analog model, several significant facts were revealed about the hydro- logy of the Saginaw Formation. The single most important source of recharge to the aquifer is from the leakage through the upper confining beds. The variation in the leakage rate, as determined from the transient analysis of the model, is shown in Figure 22. This leakage has lowered the water levels in the saturated material over— lying the confining beds. Because the leakage is quite uniform, the horizontal movement of water through the aquifer is greatly reduced. This means that the trans— missibility of the aquifer is not as significant a factor as it was originally considered to be. Often, in an attempt- to make the drawdowns observed on the model agree with the expected results, the transmissibility of the model was changed by as much as 100 per cent. This had very little effect on the drawdown configuration. The degree to which the transmissibility of the aquifer becomes unimportant is related more to the leakage rate than to the pumpage rate. The rivers also were considered to be an important source of recharge to the aquifer. It is now realized that the natural recharge from this source is limited, 39 A0 not by the water available, but by the nature of the re- charge. On the model, the total recharge from each river segment had to have a maximum value placed on it. This maximum had already been reached for a large portion of the river courses in 19A5. The rivers thus serve as a constant source of natural recharge, but cannot be counted on for further development without a concentrated program of artificial recharge. The artesian storage coefficient of the aquifer was found to have no effect upon the total drawdowns, and the water table storage coefficients of the dewatered portion only a small effect. The storage coefficient of the glacial material is of great significance. However, the order of magnitude of the storage coefficient is much more important than minor variations. That is, we know that the storage coefficient must be in the order of 10—1, but whether it 2 or 1.5 x 10"1 is 5 X 10- does not appear significant. This statement is based on experimentation with different values on the model, and the observation of the resultant variations in the piezometric change maps for the various periods. As has been mentioned, the aquifer was in equilibrium in 1935. In addition, the model analysis indicates that the aquifer was in near equilibrium in l9A5. That is, had pumping rates continued at the l9A5 level, a new equilibrium would have been reached. Al, In summary, an electric analog model of the Saginaw Formation was constructed on the basis of a conceptual model of the aquifer. The analog model was analyzed, and modified within the limits of hydrologic feasibility to make the observed changes in the piezometric surface agree with the actual changes which occurred. The final analog model, and the resulting conceptual model, is then con- sidered to be an accurate representation of the aquifer, and should be useful as a predictive tool in the planning of future water resource development in the Lansing area. By imposing expected future withdrawal rates and other changing hydrologic conditions on the model, an estimate of future drawdowns can be obtained. In addition, other schemes, such as the feasibility of artificial recharge to the aquifer may be investigated. As more knowledge is gained about the aquifer through future field investié gations, the model will be updated. A2 TABLE l.--Leakage rates for recharge regions shown in Figure 9, for undeveloped conditions of 1900. Leakage Leakage Rate Region gpd/mi2 gpd/mi2/ft 1 25,000 A000 2 A0,000 3300 3 A0,000 5200 A 50,000 5000 5 60,000 7000 6 20,000 2000 7 15,000 1500 8 A0,000 5000 9 32,000 A000 10 25,000 A000 11 25,000 A500 A3 TABLE 2.——F1ow from aquifer to river in undeveloped condition of 1900. River Segment Flow Into From Segment Figure 6 mgd 1—2 ........ 0.28 2—3 0.12. 3-A 0.58 A—5 0.05 5—6 ........ 0.30 6-7 0.58, 7-8 0.29 8-9 0.27 9-10 0.15 10-11 ........ 0.26 11-12 0.17 l3-lA 0.05 lA-15 0.05 15-16 0.16 16-17 ........ 0.1A 17-18 0.20 18-6 0.2A 19—20 0.08 20-21 0.05 .21-22 ........ 0.19 22-23 0.19- 23-2A 0.2A 2A—25 0.1A 25—18 0.10 total A.88 mgd AA TABLE 3.--River recharge rates. River Segment ,Flow from Aquifer to Maximum Recharge from River in Undeveloped Rate Specified Figure 10 Condition of 1900 on Analog Model mgd mgd l 0.lA 0.28 2 0.30 0.60 3 0.31 0.62 A 0.17 0.3A 5 0.0A 0.08 6 1.13, 3.60 7 0.36 0.72 8 0.18 0.36 9 0.18 0.36 10 0.22 0.AA 11 0.07 0.1A 12 0.25 0.50 13 0.19 0.38 1A 0.AA 0.88 15 0.08 0.16 16 ‘ 0.1A 0.56 17 ' 0.23 0.56 18 0.A5 0.90 Total A.88 mgd ll.A8 mgd A5 Flgure 3 Block DIagram of Electronlc Equipment used In AnalysIs “var-"om /\ Trigger GENER- to ATOR Pulse Gener- PULSE PULSE PULSE ator GENER- GENER- GENER- # ATOR ATOR ATOR Sync. 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