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2#~r AL. Publication No. 39 Analysis of Storm Water Seepage Basins In Peninsular Florida By Hillel Rubin, John P. Glass and Anthony A. Hunt Department of Civil Engineering University of Florida Gainesville * ~ WIWI 9*1w EWA 77 TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . iii ABSTRACT . . . . . . iv CHAPTER 1: :E!E: pL INTRODUCTION . . . . 1 S' .: of the Study . . . . . 1 Objectives . . ... . . . 2 Methodology . . . . . . 2 Research Program . . . . . 3 CHAPTER 2: FACTORS AFFECTING THE DESIGN OF URBAN DRAINAGE SYSTEMS IN PENINSULAR FLORIDA . . . . 5 Introduction . . 7. .. .. .. 5 Legal Principles . . . . . 7 Federal, State and Local Regulations . . . 9 Project Feasibility . . . . 14 Preliminary and Detailed Design . . . 19 Conclusions . . . . . . 24 Appendix Flow Diagram for Urban Drainage Design . .. 26 References . . . . . . 32 CHV'TER 3: ANALYSIS OFTRANSIENT GROUNDWATER FLOW FROM SEEPAGE  i' . . . . . . . 33 Introduction . . . . . . 33 Calculation of Unsteady Seepage from a Pond . . 33 Calculation of the Response of the Groundwater Table . 39 Discussion and Conclusions . . . . 45 Notation . . . . . . 47 References . . . . . . 49 CHAPTER 4: THERMAL CONVECTION IN A CAVERNOUS AQUIFER . 50 Introduction . . . . . . 50 Basic Equations . . . . . 50 Linear Stability Analysis . . . . 55 Finite Amplitude Disturbances and Nonlinear Stability Analysis . . . . . . 58 Results and Discussion . . . . 65 Conclusions . . . . .. 71 Appendix Expressions for Coefficients of Heat Advection, Friction and Heat Dispersion Spectra . . . 72 Notation . . . . . . 73 References . . . . . . 77 CHAPTER 5: SEMINUMERICAL APPROACH FOR THE MATHEMATICAL MODELING OF SINGLY DISPERSIVE CONVECTION IN GROUNDWATERS . . 78 Introduction . . . . . . 78 Basic Equations . . . . . 79 The Flow Field Stability . . . . 82 Analysis of the Steady State Convection . . 83 Numerical Calculations . . . . 86 Conclusions . . . . . 89 Appendix I Expressions for the Coefficients of the Spectral Functions . . . . . 90 Notation . . . . . . 91 References . . . . . . 94 CHAPTER 6: NUMERICAL SIMULATION OF SINGLY DISPERSIVE CONVECTION IN GROUNDWATERS . . . . . . 96 Introduction . . . . . 96 Basic Equations . . . . . 96 The Flow Field Stability . . . . 100 Numerical Calculation of the Steady State Convection .. 102 Numerical Results and Discussion . . . 108 Conclusions . . . . . . 114 Notation . . . . . . 115 References . . . . . . 118 CHAPTER 7: SUMMARY AND CONCLUSIONS . . . 120 ACKNOWLEDGEMENTS This report summarizes the research results obtained in 1975 and 1976 in the Ai,,L .!lic Laboratory of the University of Florida. The research project was :,r.:v r",. by the Office of Water Research and Technology (OWRT) through the Florida Water Resources Research Center. The investigators are grateful to Dr. William H. Morgan, Director of the Florida Water Resources Research Center for his help through all phases of this investigation. Dr. Sylvester Petryk initiated the study and served as its principal investigator through October, 1975. His initiative and participation in this research are greatly appreciated. Chapters 2 and 3 of this report are mainly based on the Master of Science theses of A. A. Hunt and J. P. Glass. The investigators are very grateful to Dr. B. A. Christensen who served as the chairman of their respective graduate study committees. Dr. W. Huber served as coinvestigator in this research and partici pated in the graduate study committees. His activity is very much appreciated. ABSTRACT Rapid urbanization, currently proceeding in Florida, has resulted in significant problems with regard to both flood control and pollution abatement. The objective of the reported study was to search for the design procedures that may improve efficiency, safety and adequacy of drainage systems in peninsular Florida. Special attention was given to possibil ities of recharging groundwater aquifers with excess storm water. Through such a system, partial solution to problems of inadequate potable water supply in some areas can be achieved simply as a byproduct of flood control systems. In Chapter 2 of this report a conceptual design framework is presented. In developing this framework, a variety of disciplines involved in the solution of urban drainage systems in peninsular Florida are considered. Comparatively high levels of groundwater in many locations require consideration of two factors relating to the ability of seepage ponds to divert surface water to the aquifer. The engineer should study the ability of the pond to seep water and he should analyze the response of the groundwater table to the seeping water. With respect to these two factors, analyses and solutions adapted for engineering application are presented in Chapter 3 of this report. The continuing reduction in the availability of potable water in coastal zones of Florida has prompted several communities to design recharging systems utilizing treated storm water as well as effluents. Chapters 4 to 6 of this report concern flow conditions in the Floridan Aquifer that should be considered in connection with this subject. Methods by which the particular .T :. associated with flow conditions in the aquifer can be evaluated are presented. These methods cover several s.. tral ;...ion ,; .:aches as well as a complete numerical approach. CHAPTER 1 GENERAL INTRODUCTION SCOPE OF THE STUDY Since ancient times, people throughout the world have had to cope with periodic floods and inundations of lands and communities. Notably pragmatic solutions to flood control were developed in various locations. These solutions were dependent on human imagination and ingenuity applied through available resources to a broad variety of situations and conditions. Even for a relatively limited area such as the United States, we cannot imagine a singular methodology applicable in every situation. Effective flood control depends not only on resources and materials provided by 'Mother Nature' but, also on man's attitude toward the application of these resources. The very definition of the problem depends on the attitude of the people and their understanding. Peninsular Florida is a unique system in many respects. In the early part of this century, flood control projects were begun in earnest over an area frequently exposed to devastating hurricanes and extensive flooding. Since World War II, tremendous changes, inherent in the rapid development of the state, brought about improved techniques and better understanding of flood control. Presently, the state of Florida faces simultaneous problems of excessive storm water and limited water supplies particularly in coastal communities. Rapid growth and the influx of new people also resulted in reduced water quality, thus adding another factor to water management and flood control. This project has been initiated with the idea of simultaneously improving flood control techniques and enriching groundwater resources in peninsular Florida. UbULLiiVtc3 The investigators found it necessary to direct their efforts toward three principle objectives: (a) General management and development of conceptual design framework. (b) Development of a simplified analysis for the evaluation of seepage ponds in diverting stormwater to groundwater storage. (c) Development of approaches in the analysis of migration of contaminants in groundwater due to natural conditions as well as situations induced by artificial seepage. METHODOLOGY With respect to the listed objectives, Chapter 2 of this report concerns the development of the conceptual design framework as related to urban drainage systems in central and southern Florida. Chapter 3 of this report concerns management of water quantities. This effort in the investigation involved determining the ability of seepage ponds to divert collected stormwater to the groundwater aquifer. Emphasis was given to development of simplified methods that can easily be applied by drainage design personnel. Chapters 4 through 6 concern mathematical methods that can be used for the analysis of flow conditions in an aquifer similar to the Floridan Aquifer. Basic models are suggested and a variety of mathematical methods are checked from the point of view of applicability, efficiency and accuracy. These chapters form an introduction to analyses of water quality problems that have yet to be resolved in Florida. . 9r'i PROGRAM An initial research pr.:.'. was outlined by Dr. Sylvester Petryk in 1974 when he submitted the research proposal. The study, as envisioned in that f .::;.1, would consist of the following five tasks 1) Review of existing literature 2) Modeling of precipitation, ,..i,.;,7, and storm water seepage basins 3) Economic analysis and optimization of design 4) Design procedure 5) .>:. mental measurements. The research conducted during the past two years has been concerned with all of these areas. However, the program was modified with respect to the emphasis and the degree of effort devoted to each task. A review of the existing literature indicated that our efforts should be directed more toward improvement of the complete design framework and project .1.'.e':e.i, rather than involvement with particular design techniques or procedures. Chapter 2 of this report covers this part of our activity, which is partially related to each of tasks 3, 4, and 5 mentioned above. The design of seepage ponds in Florida often presents special problems because of high *.,..,, eter tables. These problems are the subject of r'".. 3 and are related to task 2 mentioned above. Problems of water quality have become increasingly important in Florida as well as in other parts of the country. One of our primary concerns in this regard was the effect that injection of impure water might have on the quality of Florida's groundwaters. Chapters 4 to 6 of this report are concerned with this problem, which is related to task 2 mentioned above. A series of field studies and measurements was conducted as suggested in task 5. Good results were obtained but the job was more or less routine in nature and we did not find it necessary to include them in this report. Our main objective in the field study program was to acquaint ourselves with local problems and to advise local people with respect to these subjects. '. i OTER 2 F.', AFFECT' THE DESIGN OF .Li DRAINAGE TEH. IN PENI[. AR FLORIDA demand for housing in Florida continues to climb as Americans seek a place in the sun. .: irement and tourism are rapidly displaci. agriculture in developing and utilizing an attractive natural environment. '.th, since the end of World War II, has been phenomenal. Land devel.r ment and home construction have become an integral part of the economy. These activities draw heavily on Florida's natural resources which are not limitless and, in many instances, are key elements in a sensitive environmental system. The demands of a progressive economy cannot be inn:?1, however, these demands can be adjusted to provide optimal use of a limited system of supply. Concerted efforts are being made to 7...ide both understanding of a complex and dynamic system, and a reason able balance of supply and demand. Water, one of Florida's most abundant resources, is a critical factor in this dynamic s. :tr Early management concentrated on drainage and fl;,. control to such an extent that damage to the system resulted. The past twenty years has seen a shift in management emphasis as water short .;. and environmental damage became more apparent. Contemporary manage ment practices are ...,.jing rapid change. The idea of water as a scarce commodity has ::. ,tEd basinwide regulation of consumptive use and natural flow. .:.'adation of water quality has led to more stringent pollution control laws and the development of improved abatement techni,.: :.o All of these efforts have recognized and addressed urban development as a leading cause of problems in maintaining water quality and managing flow. Urban storm runoff has been found to be a source of water pollu tion that is of equal or greater magnitude when compared with any other identifiable source. Home building and land developmen'L require drainage systems that accelerate runoff, consequently, creating or aggravating flood prone situations. Homes, streets, parking lots and commercial build ings retard or prevent infiltration of rainwater and necessary recharge of groundwater aquifers. In view of the current activity in water management and pollution control, the drainage engineer must be aware of the many facets of contem porary design and he must apply sound and systematic methods in development of his design. The natural environment of Florida presents a unique set of circumstances. Laws, regulations, procedures and techniques, although rooted in historic precedent and established standards, have all been developed with some consideration for these circumstances. The drainage engineer should have a thorough knowledge, not only of applied techniques, but of social, environmental and political impacts of development. He should understand recent changes in legal and governmental philosophies. The engineer needs to know sources of information, plan requirements, and the procedural aspects of government regulation and approval. Most importantly, systematic method in design synthesis permits the engineer to organize and evaluate his data, identify and augment weak points and effectively manage the design process. The objective of the present study is to review and suggest a conceptual design framework as related to the following topics: 1. Basic legal principles associated with drai..ay;., groundwater, land use and water courses. 2. Highlights and ..*, ..:' of legislative and regul =i. ," . on the federal, state and local level. 3. Elements of the feasibility study as the first *i.:,s in the design . . 4. Considerations and techniques applied to preliminary and detailed design of drainage systems. 5. Integration of the various design factors by means of a flow diagram. Finally, this discussion is oriented toward design factors particular to  .insular Florida. L5. PRINCIPLES runoff is of legal concern for a variety of reasons. 'n,:.., done concentrated ".'rc, either as a floodwave passing downstream or a backwater flooding of upstream lands, may result in injury to the :',ted parties. Alteration of flow directions, capacities or other drainage characteristics can and often does lead to environmental damage, e.g.,lowered water tables, water pollution, damage to vegetation, erosion and accretion. ,..*?ty damage from flooding of homes and businesses may be the direct result of poorly managed runoff and improper land use. Two basic principles of law concerning disposal of surface waters are 'the civil law rule' and 'the common enemy rule'. Under the civil law rule, the upland or dominant owner has an easement on the downstream owner for .;.: of surface runoff in its natural manner. The common enemy rule stipulates that the servient or lower owner may take measures neces 7 sary to keep these waters off of his land. Generally, Florida courts have followed the civil law rule modified by 'reasonable use'. For example, the general rule regarding drainage into a natural watercourse states that a riparian owner may discharge surface runoff without regard to either the 'civil law rule' or the 'common enemy rule'. However, this right is subject to three limitations which have been imposed by the courts in varying degrees. These limitations include: 1. Drainage must be reasonable. 2. Drainage must not come from outside the natural basin. 3. The natural capacity of the stream must not be exceeded. The concept of reasonable use has been applied in cases involving both land use and water rights. In several instances, the courts have ruled that land use, adversely affecting surface or groundwater flows, incurs no liability on the owner as long as his use is reasonable and legitimate [2, 3]. Rulings have also been made in light of the riparian doctrine, requiring reasonably unimpaired or undiminished flow. In conclusion, the engineer must consider possible liabilities and legal consequences in his design of drainage works. Stormwater systems designed to parallel natural discharges from a site are desirable from both an engineering and a legal viewpoint. Further, conflicts between water rights and legitimate land use have not been resolved in an entirely consistent manner. It is apparent that the body of law governing these activities and rights is not well defined and does not truly recognize the interrelation of land use and natural flow. p'":!' STATE AND LOCAL FF ,' ..ATIONS Florida's rapid :th has produced he.:,' and often conflicting demands on existing water resources. The geology of Florida '... .ides extensive groundwater sources that have been utilized in satisfying these demands. .. lesale acceleration and channelization of runoff and failure to provide te recharge to ,.s::rlying aquifers combined with concentrated with drawal from these aquifers has led, in several instances, to serious problems that cannot be resolved on a single situational basis. Total basin ' ,it has become the imperative solution. Comprehensive legislation at all levels of government has been enacted to provide a framework for ., rm t. Agencies with permitting and other regulatory powers have been created to affect solutions to apparent conflicts on a regional basis. The Federal Water Pollution Control Act of 1948 began what has become a massive effort to improve the quality of our national water resources. .e::;t'legislative efforts authorized federal assistance in research and development of methods of controlling pollution. Most recently, the F.W.P.C.A., Amendments of 1972 placed stronger fmirqpc,p is on storm runoff as a source of pollution. Section 208 of this act is directed toward areawide planning and management of both 'point source' and ': .r,'.cint source' discharges. Stormwater runoff, a nonpoint source, is to be evaluated on the basis of extensive data collection and monitor ing in the field. Methods for reducing pollutant loading are to be developdF at the community level and, subsequently integrated into a basinwide management plan [1]. Current technology in this area has not established a total 'cause and ( ':ct' relationship between runoff and stream quality. Loading rates are highly variable and depend on land use, storm duration and intensity, and seasonal weather patterns. This relationship or a reasonable approxi mation is singular in its importance with regard to effective modeling and comprehensive understanding of the problem. A second program of peaticular interest to the drainage engineer stems from the National Flood Insurance Act of 1968 and the Flood Disaster Protection Act of 1973. Through this program, federally subsidized flood insurance is available to homes and businesses located in floodprone communities. A prerequisite to qualification requires that designated communities adopt restrictive land use ordinances for areas subject to flooding on a frequency of once in onehundred years (equivalent to a one percent chance in any one year) [4]. The drainage engineer should recognize three significant points in these programs. First, the federal government has initiated comprehensive efforts to maintain and improve the quality of natural waters through strengthened programs at the state and local levels. Secondly, the flood insurance program is a substantial legislative effort directed toward better land use practices and flood control measures. Finally, these efforts actively involve the community and its citizens by promot ing improved local ordinances and related decision making. During 1975. Florida's environmental agencies were reorganized and consolidated into two major agencies, the Department of Environmental Regulation and the Department of Natural Resources. Additionally, the state's water management districts were redefined and were invested with broadened duties and powers. The Department of Environmental Regulation is responsible for enforcing pollution control laws and maintaining or i.: oving water quality. Permitting of 'i:iii;> treatment plants is a duty of this =.:/. In granting permits for point sources such as .. treatment plants, the D.E.R. will review and actively consider stormwater control, treatment and di,;::...,l as part of the permit request [7]. D~.', highlights of r 2 guidelines include [7]: 1. T.::..t of stormwater dis(cd'.:.e on the receiving waters will be viewed considering designated use of the waters, practical considerations and costeffectiveness. 2. Design of stormwater management systems should include a variety of techniques in reducing impact. Sod filters, vegetated buffers, sediment traps, primary considerations in design. 3. =... ,ion basins should be considered as essential in residential, business, industrial and highway devel..pa~erit.. Interim drainage systems, serving construction sites, should be anticipated and included. The Florida Water Resources Act of 1972 empowers the Department of Natural Resources to accomplish the conservation, protection, management and control of the waters of the state. In effect, this act makes all waters in the state subject to regulation [6]. This act further directed the environmental agencies to formulate a statewide water use plan. Elements of the plan are being prepared at the district level and will be comprehensive in scope and objectives. Other activities of the Department of ii. 1 e: uces through the water management districts include permitting for .:c., npive use by all users, constructing and maintaining lands and works incident to management activities, permitting and re, nation of wells, and management and storage of surface waters. It is readily apparent that the state is assuming the role of the riparian owner in many respects. Maintaining undiminished water quality and quantity is now a valid public concern as opposed to a strictly riparian right. In addition, the proposed water use plan will include recommendations as to flood plain zoning and protection from flood hazards. Thus, the state is assuming an ever increasing responsibility over land use management and water resources which, traditionally, have been inherent in private land ownership. Subdivision regulations and zoning ordinances form development regulations and standards applicable at the community level. Further, these regulations outline procedures in obtaining community approval for a proposed development. They provide an orderly exchange or basis of communication between the engineer and the community. Frequent discus sions during the approval process, enable the engineer and community planners and officials to effectively demonstrate their needs and desires. Individual citizens may express their opinions during public hearings before the local governing body as a matter of state law. Subdivision regulations vary considerably across the state. Commun ity resources and extent of development tend to impose practical limits on regulatory programs and requirements. Obviously, the most sophisticated regulations available are relatively ineffective without adequate staffing and community support. Federal and state efforts are aimed at supplement ing community programs and preventing abusive and costly development such as has occurred in the past. Development standards tend to fall into two broad categories. The first is concerned with conditions or programs unique to a community. Such tors as flood control . :..:Tfer recharge and salt water intrusion me,: ire regulations or procedures written specifically ', these conditions. For example, , ,.: County has recognized the need for con servi natural recha,: areas. Their .ulations outline :...ial measures maintaining natural highinfiltration in these areas. "Typical methods include the use of filtered rer c':,.; wells, bottomless inlets, :. :,. ted pipe, grading to retard .: artificial ::.:... basins, swales in street ri.,;':.ofway, and utilization of natural ,. ...,lation areas" [5]. The second ca' of regulations covers more detailed requirements. Minimum pipe size and material specification, street and lot i.'.ing, allowable overland flow distances, design methodology, allowable velocities in sales and ditches, drai.!..:e right. way i ,.irements, and storm water details are t ical items included in this category. Serviceability and ease maintenance are important considerations in these !..:ifications relatiJ :. ly to the 'hardware' of drainage. Routine maintenance e. dra" .. systems and streets is a ..jor cost item in municipal '...: ts. Flooded streets and overgrown ditches are common complaints .., tax ,.~' s and :2. .. is made to minimize such situations in writi ..; these regulations. In summary, legal principles ,... : .ing surface flow and .'....... ter are derived from ri ... associated with private ownership. Conflicts arising from alteration of natural flow patterns or from consumptive uses have been resolved considering reasonable use of land and water. It is :.rent that these principles are not adequate to resolve *.:.o iicts consider. cont. :y demands. As a result, regional or basinwide , :.:.:. of water resources is being affected t:..'. combined 1.,:isla tive efforts on all levels of government. Preemption of private water rights appears to be justified in that the burden of flood damages and optimization of consumptive uses rests with the entire community. Finally, lawmakers have broad powers in preserving and protecting natural resources in the public interest. Thus, considerable legislation has been enacted to improve land use, relieve flood hazards and reduce pollution of natural waters. PROJECT FEASIBILITY Project initiation usually begins with a discussion including the client and key staff ,I.. ionals. Subsequent efforts depend to a great extent on several factors. Among these are the intended scope of the project, potential market, capital availability, anticipated scheduling of planning, engineering, construction and sales, legal considerations and immediate technical needs. Certainly, forty acres in an established residential area will have significantly different needs than tenthousand acres of undeveloped and agricultural lands. The experience of the client or developer will also influence project initiation. An experienced developer often will have established project economics, scope and approximate timing prior to discussion with design professionals. Project feasibility may, however, include multidisciplinary studies and analyses of the previously mentioned factors. One of the most fruitful approaches to evaluation and design is organization and coordination of all related efforts. Project planning draws each factor into the design process as it becomes pertinent. A systematic approach to project management helps to ensure optimization of technical i and ;:. :..tion and, that the quality of the : r.;c will be ':.. ....ily .. ...',ssional. i. ::. design as an i Yineering .: ., is well suited to an or ized ...  : ._.ign .. ,thesis and coordination of :i .. in drain Sineeri., is : .,!cally demonstrated in the flow diagram as shown in the '. ,,.x .J from Woodson [8]. Flow di c.:ams are an . . and proven :.... .t tool. Their inherent flexibility makes them adaptable to a variety of situations and design conditions. i. included flow di am,presented in the Appendix, is divided into three distinct phases allegingg the balance of this discussion. I,.' phases include ".'ect i ability, preliminary design and detailed design. Intensive data collection is the first step following project initi ation. Gathering information on existing drainage patterns and both current and future land'use follows several avenues. A thorough under standing of the existing drainage situation is essential not only in determining required improvements but, also in demonstrating the effect iveness of ,*'r,.~c~ improvements. Standard sources of information that will readily provide data on drainage, land use and flood p.;,stial should first be invest .... .*".L:g these sources are: 1. U. S. Geological Survey for topographic ',.:, flood studies, groundwater studies, geologic investigations, well :...., stream and lake gage records and general hydrologic information. 2. ";,.'icipal or County Engineering and Planning for subdivision regulations, zoning ordinances, aerial mapping, established drai .r. networks and other i '.<.on directly related to the site. 3. Soil Conservation Service for soil surveys, drainage techniques proven in the area, recommended best use of the site and extensive technical expertise in drainage and hydrology. 1. State age,: fk for re.,ia:y informationn and design guidelines in stormwater management. Specific informa tion as to local environmental considerations and current practices. 5. Federal agencies for flood insurance and flood plain information, possible federal permits, design standards for qualification under federal loan programs and many other technical services. The reconnaissance survey of the site of a proposed development is particularly important in that information resulting from visual inspec tion is of primary importance and is generally available from no other source. Initial data may be summarized and plotted or noted on a site plan to facilitate observations and comparisons during the survey. Veri fication of land use, drainage patterns and improvements, both on the site and in the surrounding catchment, is an important objective. Other factors to be considered during reconn' are: 1. Attractive or beneficial features on thesite that will contribute to aesthetics and environmental quality. 2. Hydraulic constrictions and control points, ponds, lakes, streams, springs and natural depressions collecting run off. 3. High water marks on trees and drai:.,., structures, water  elevations and groundwater or water table eleva tions. 4. Soil t . and area distribution. Soil .,. ..its such as 'c muck and that must be considered in design and construction. receiving waters ..bI': le quality of current .". and receive waters. 6. Identification of chronic drainage ::,.l':,.ems and existing i,.,....:,'A,.: , both on the site and in the immediate catch ment. '. i..: or following field investigation, the engineer should outline require ments , additional ;... ,aphic surveys, soils investigations, stream . ing, water quality evaluations, environmental surveys and similar e.'. . and techni :.: for field investigations are '.,er....:. on site conditions and the extent of proposed development. Detailed field notes and ;.rrnlementary "',,Co .. I are common to all surveys and provide documentation to :. :. subsequent j'.m.!i:., public presentations and technical discussions. Evaluation and incorporation of collected information involves simu lation of existing drainage conditions, potential for improvement, delin eation of drai....? systems and land use planning. 'r..!!';' calculations or :. ... imations should be made in evaluating the hydraulic and hydrologic characteristics of the existing system and the ..;.* improvements. drai .... engineer should work very closely with those involved in land planning. Land use, of course, will have multiple effects on any proposed drainage system. Compatibility of land use and soil characteristics is of particular importance in Florida. Planners must be aware of probable water table elevations and seasonal fluctuations. Much of Florida con sists of fine sandswith small percentages of silt in the surficial layers. Often, a semipermeable layer of consolidated fines and organic material underlies surface sands. Infiltration of stormwater is severely retarded and lateral or surficial flow may be very slow, particularly on the marine terraces and in the 'piney flat woods'. The drainage engineer should evaluate costeffective measures in reducing high water tables and flat gradients and, he should advise land planners of his findings. Land planning should be coordinated to reflect drainage methods and alternatives best suited for the site, required rightsofway, pond loca tions, flood elevations, fill sources and quantities, recharge sites and natural features to be preserved. Every effort should be made at this time to promote an aesthetically pleasing drainage network. An open ditch or floodway is patently obvious and prominent in appearance. The additional costs associated with a landscaped and meandered channel are more than justified considering heightened sales potential and consumer ideals. The preliminary engineering report or feasibility study reflects a combined effort in engineering, planning and project economics. Form and content for such a study are variable and, generally, depend on the scope of the project and desires of the client. It cannot be over emphasized that such a report is preliminary in nature and that subse quent efforts may contradict stated conclusions, cost projections and 1 .,ity estimates. Misunderstandings are frequent in this c.: t and the engineer should use care and discretion in making this point clear. [.: lations, ,,:,mitting and o.,... ei t coordination are often disegrJ..3 at this stage of the process. The feasibility study should include a Y'.L* :ji! discussion of all aspects of the permitting and J L_1 ,o Preliminary engineering design provides the client with valuable i '.fention .. project scheduling and funding. The feasibility report serves the developer in anticipating construction costs, cash flow, marketing and sales. Finally, the engineer and client can selectively evaluate alternatives and determine preliminary design objectives. PREL IV.i.Y AND.DETAILED DESIGN Preliminary design should be approached considering overall site i logy. Simulation techniques commonly used include the rational method, unit "ii ,;, synthetic hydrographs, regression equations and more complex computer modeling techniques such as those developed by the r*... of Engineers and the Soil Conservation Service. The object of all of these methods is to systematically quantify runoff and to route these T :;.ically, the outfall point is identified together with available records or some estimation of appropriate water surface elevation. The flat gradients, so common to much of Florida, necessitate a thorough investigation and a reasonable approximation of downstream water surfaces corresponding to the events being simulated. Collector systems are usually a combination of open channels and storage basins that depend on both physical ground slopes and hydraulic head in maintaining positive flow. Central Florida's highlands may require grade stabilization structures where slopes are relatively steep. Peak velocities in unlined channels generally should not exceed two feet per second in Florida's sandy soils. Less frequent storm events may be permitted to generate somewhat higher velocities. Storageelevation curves and stagedischarge relationships may be approximated in routing surface runoff through the system. Routing begins with generation of water quantities at the up stream end of the system and proceeds down the slope to the receiving waters. Peak discharges and water surface elevations, derived from the routing process, may then be used to calculate water surface profiles. Most systems generate peak flows over a relatively short period of time so that only peaks may be considered in calculating the profiles. Subcritical flow profiles must begin at the receiving waters and, based on the appropriate surface elevation, proceed upstream. Adjustments are made in channel geometry, storage, control structures and overland flow times until close agreement is achieved between the water surface profiles and the elevations generated by routing. This procedure begins in the preliminary design phase and is continually refined through the end of the design sequence. Many factors that are elemental to simulation will have considerable variation depending on the event being considered. Rain fall intensities, for example, will generate system responses markedly different for various events. It is advisable to simulate storms of several frequencies and durations even though regulations may specify design based on one or two standard events. Practical considerations, such as allowable design time, must be considered in managing this procedure. Internal drainage systems direct runoff to the collectors. The rational method is almost universally used to design local drainage such as swales, culverts, inlet size and location, street slopes and lot i'd. ing. Home building and related land development require simultaneous evaluation of earthwork and drainage. Generally, surface slopes may be adjusted to minimize drainage improvements. Cutting and filling required to achieve ideal slopes may, however, result in excessive costs in earth moving. Once again, several iterations of site engineering may be required to achieve optimal conditions in detailed design. Incorporation of detentionretention ponds in the drainage system is desirable to attenuate increased discharges resulting from development. Pondi .. is a recommended method in reducing pollutant loading. Common practice is to provide sufficient storage within the system for the volume of runoff resulting from a given storm event. These events or design rainfalls generally require enough storage so that runoff from high ,.*. .,ny events, such as afternoon thunderstorms, is totally retained or detained for an extended period. Silts and other suspended solids have a chance to settle, evaporation and infiltration effectively reduce the volume of stored runoff, biological action and dilution reduce the concentration of pollutants, filtering devices prevent dis charge of surface borne contaminants such as oil and grease, and trash and debris are prevented from entering the receiving waters. Natural storage areas, including cypress ponds and mangrove swamps, may be utilized in developing detention. These areas have the advantage of acting as filters for runoff, thus improving discharge quality. Care must be used to maintain minimum water levels in these areas and to prevent damaging pollutants from entering such a system that utilizes natural vegetation. Other considerations necessary to retentiondetention ponds are: 1. Storage capacity recovery through evaporation, infiltration, structural devices regulating outflow, modified underdrains and pumping. 2. Siltation or clogging in the pond. Twostage ponds should be provided where sediment loads will be high. Periodic cleanout of settling basins in the first stage will be required. A subsurface berm or equivalent structure should separate the settling basin from the second stage. 3. Maintenance, safety, vegetation, normal depth, fill needs, aesthetics, emergency overflow, and drawdown capability are among the additional considerations. Systems of interconnected lakes and ponds are frequently employed in Florida to provide fill material, aesthetic appeal and recreational benefits. In addition, extensive open systems tend to lower surrounding water tables in chronically wet areas. All drainage systems should be viewed considering the effects of the maximum probable storm. Surface drainage systems may be inundated in some areas causing streets and low lying areas to act as open channels or ponding areas. Utility services may be interrupted. High water in the outfall system may result in flood conditions for an extended period. In this regard, auxiliary floodways may be desirable, road elevations should be sufficiently high to allow limited continuous access, floor elevations near points of concentration should be higher than arbitrary minimums and offsite structures and embankments should be evaluated as flood hazards. In ? :.:t, preliminary design involves evaluation K. a selected development plan. . .ign efforts should reflect sufficient detail such that .' limited alternatives remain together with final details neces sary to construction. The flow diagram, .'.:' ted in the Appendix, outlines elements of this phase through client and agency approval. Presentation of the preliminary plan C.I...,, with supporting data, becomes part of a comprehensive review by the client and staff. These plans may then be submitted to local agencies for preliminary approval and required zoning *", .:,o Review and discussion with state and federal agencies should be conducted on an informal basis. Results of these interviews and evaluations should be summarized for incorporation into detailed design. :e third and final phase, detailed design, completes this sequence. Construction plans, specifications and detailed cost estimates are primary objectives. :.'isions to the preliminary design should be solicited from staff, client, agencies and potential construction contractors. The design r.:... must provide a thorough check of every detail and should coordinate with other members of the staff in locating and resolving possible conflicts. Almost invariably, insufficient clearances between sanitary sewerage, storm mains and water supply will become evident and necessitate : *e!...*. Experienced contractors will often point out plan discrepancies, ,*.,.ity errors and unfavorable site conditions during bid 0=.ip ation or contract negotiation. During this phase, the engineer should coordinate closely with permitting and approving agencies in ..iding ' ting data for selected design features. Unnecessary and costly changes, based on arbitrary interpretation or judgmental differ ences, may be avoided through foresight and preparation. All parties should be informed of required revisions and all plans should be noted accordingly. CONCLUSIONS Drainage design has become considerably more than estimating runoff and selecting culverts. The design professional is obliged to assume a broad viewpoint in developing his program and directing design efforts. In summary of this discussion, the following conclusions are presented: 1. The engineer should be familiar with basic legal principles related to drainage and how they have been applied in resolving conflicts and water rights litigation. 2. The engineer must be aware of rapid changes taking place with regard to water rights and resource management. Federal and state legislation in this area has shifted rights and responsibilities away from the landowner and toward the public entity. 3. Local regulations and ordinances are being rewritten or supplemented in order to effect better land use practices and flood protection measures. 4. Substantial efforts are being made to develop and implement improved pollution abatement techniques. Urban runoff, as a prime source, may be improved through a variety of techniques presently available to the designer. 5. Legislative mandates have been very direct. The engineering profession should take precautions in protecting their design pr*..H*;vves and, at the same time, align their efforts with these mandates. 6. The feasibility study is, possibly, the most if.o t..Iit phase in design sy.:': sis. This study develops a set of conditions influencing the entire project. 7. Florida's environment and .di['I! ..' on well */, f,2er water resources emphasizes the efforts of the drainage engineer. He must actively participate in several areas of project development. 8. ".i.itionretention systems have become a necessity in designing improvements. Ponding or storage improves water quality, reduces flood potential and provides increased aquifer recharge. 9. Management of the design process and attainment of technical efficiency may be aided through the use of a flow diagram presented in the Appendix. Considerable flexibility allows this technique to be adapted to most projects. Finally, the attitude of the engineer must be receptive to change and inno vation. Bending to accommodate is not the answer. Embracing and assertive leadership is a trademark of the professional. APPENDIX FLOW DIAGRAM FOR URBAN DRAINAGE DESIGN Phase I: Project Feasibility Client Desires Staff Discussion Project Scope Project Initiation  Project Economics Time Schedule Outline First Efforts Is there sufficient ? No information to proceed? Yes Existing Land Use Soil Survey Topographic Surveys Historical Records Interviews Data Collection   Future Land Use Studies Laws and Regulations Reconnaissance Survey  C Establish Preliminary DDesign Objectivesl o Are pre1imif"i.t design ? No objectives consistent and reasonable? Yes Phase I C o m p l e t e Phase II: Preliminary Design Development of Internal . inage System Development of Collector and Outfall System Identification of Critical Points Development of Hydrologic Model Development of Hydraulic Model Development of Preliminary Design  System Evaluation Through Models Identification of Limited Alternatives Water Quality Evaluation Prepare Preliminary Design Plans Have all sources  been tried? Delineation of Existing Flow Patterns Identification of Constraints Simulation of Existing Drainage Conditions Proposed Drainage Schemes Proposed Land Use Are proposed land use and drainage plans workable? Cost Projections Quantity Estimates Anticipated Permits and Approvals Staff Review and Coordination Owner Review and Approval Presentation of Feasibility Study Have preliminary design objectives been realized? Design Features Limited Alternatives Special Problems and Solutions Cost Estimate Additional Data Needs Presentation of Preliminary Design Obtain Client and Agency Approval Are final design features and objectives approved by client and agencies? Phase II Complete Phase III: Detailed Desiqn Incorporation of Testing Results and Survey Data Safety Considerations Cost Optimization Coordination of Design and Drafting Design and Specification of Special Features Preparation of Detailed Cost Estimate Preparation of General Specifications Checks and Revisions of Calculations and Drawings Project Plans, Estimates and Specifications Review Development of Detailed Design Issue Plans Reflecting Final Design E# Is the final design complete ? and acceptable to the client? Yes Permit Applications and Procedures Local Governmental Approval Contractor or Bidder Review Final Staff and Client Review Revision of Detailed Design Obtain Final Approval by All Parties i Are plans, specifications, estimates, and permits complete? Are Ep!.rovals in receipt? Phase III Complete Symbols Key Flow Direction: Specific Activity or Information: General Activity: Executive Action: Decision: .^^ ^ BBgffi.^ l l....DO. ,r. I REFERENCES 1. "Executive Summary of Section 208 Program for Designated Area  Federal Water Pollution Control Act Amendments of 1972", United States Environmental Protection Agency, Washington, D. C., 1974. 2. Linsley, R. and Franzini, J., Water Resources Engineering, Second Edition, MIcGawHill, New York, 1964. 3. Maloney, F., Plager, S., and Baldwin, F., Jr., Water Law and Administration, The Florida Experience, University of Florida Press, Gainesville, 1968. 4. "National Flood Insurance Program", United States Department of Housing and Urban Development, Washington, D. C., 1974. 5. "Orange County Subdivision Regulations", Orange County, Orlando, 1974. 6. Tebeau, C., "South Florida Water Management" and "Environments of South Florida: Past and Present", Miami Geological Society, Miami, 1974. 7. Thabaraj, G., "Regulatory Aspects of Storm Runoff Control", Proceedings of the StormWater Management Workshop, Florida Techno logical University, Orlando, February, 1975. 8. Woodson, T., Introduction to Engineering Design, McGrawHill, New York, 1966. CHAPTER 3 ANALYSIS OF TRANSIENT GROUNDWATER FLOW FROM SEEPAGE PONDS INTRODUCTION The ability of a seepage pond to divert storm runoff into the ground water aquifer depends on two processes: (a) the rate of seepage from the pond, and (b) the reaction of the groundwater table. Theacstenr ofa seepagepondindisposngofstormwatertisa transient phenomenon. The inflow to the pond, specified as a runoff hydro graph, causes variations in the depth of ponded water. Outflow is consid ered here to be entirely by infiltration, which varies not only because of changes in pond depth but also due to variations in such soil proper ties as hydraulic conductivity and storativity. The growth and decay of the groundwater mound in the underlying aquifer is also time dependent. These facts are well known among scientists but are usually not taken into consideration in the design of a seepage pond. The objectives of this study are to develop methods by which the designer will be able to estimate seepage pond effectiveness by considering both the rate of seepage from the pond and the variation in the groundwater table while taking into account the transient nature of the problem. CALCULATION OF UNSTEADY SEEPAGE FROM A POND The adequacy of a seepage pond is evaluated by a storage routing procedure, which is basically an account of the inflow, outflow, and change of stored volume over successive discrete time increments. The inflow is represented by the runoff hydrograph for the design storm on the area to be drained. For the purposes of the present exposition it will be considered a given function of time. The outflow, on the other hand, is dependent on time, depth of ponding, and the properties of the soil. Consider an initially dry seepage pond constructed in unsaturated soil. As it fills, the water begins to escape from it by vertical unsat urated flow, or infiltration. The most convenient way to describe this flow is by the formula of Green and Ampt [1911] as modified by Bouwer [1969]. It has been shown that this approach, which was long thought to be purely empirical, is soundly based on physical principles [MorelSeytoux & Khanji, 1974] and gives very good answers [Whistler and Bouwer, 1970]. Green and Ampt based their derivation on a simplified model of infil tration which treats the soil as a bundle of vertical capillary tubes. The vertical hydraulic conductivity and moisture content of the unsatur ated flow are considered constant, as is the capillary suction potential of the advancing wetting front. Applying Darcy's law to this idealized flow, the infiltration rate is described as H + L + 'n (1) w = K (1) where w = infiltration rate Kt = hyaraulic conductivity of the transmission zone H = depth of ponded water ln = capillary suction potential L = depth of penetration of the wetting front. .i, rate of advance of the wetting front is dL w (2) dt f where t = time f = the volumetric fraction of fillable pore space. The equation of continuity applied to the pond yields dv dH T f d (3) dt A = I f(AL) (3) where v = stored volume in the pond I = inflow A = area of pond surface Af = area involved in infiltration. Introducing equation 3 into 1 and 2 we obtain, after differentiation, the following nonlinear differential equation S2 dL dAf dt2(L2/2) + (fAf/Al) + fL/A dt I/A = 0 (4) dt which can be solved numerically. However, in most practical cases the rate of change of H is about an order of magnitude smaller than the rate of change of L. In such cases an integration of equation 2 between ti and ti+1 provides a simple discrete presentation of the problem, as follows i+1 i+1l f dt = w dL. ti L f The integration leads to ti+ t = f/Kt[Li+1Li(H +*n))n( H ++L++p )]. (5) In dimensionless form this can be expressed as AtK AL i' Li+i/Li + r. i/L fLi L. Li 1 + r /L(6) where rF = Hi + n L. = value of L at time t. L i+1 = value of L at time t.i+ At = ti+1 t AL = Li L. H = ()(H. + H i+), the mean value of H. A chart that can be applied for engineering design purposes is shown in Figure 1. The curves are based on the solution to equation 6. The application of the discrete approach of equation 6 was checked against experimental data reported by Weaver and Kuthy [1975]. In this experiment a seepage pond was constructed and filled with water at a controlled rate. The variation of the pond volume and area vs. depth is shown in Figure 2. The experimenters listed soil test data which led to the following values of soil parameters. Kt = 1.2 ft/hr f = 0.2 n = 0.5 ft. According to the reported measurements, the rate of change of H was 5 to 10 times smaller than the rate of change of L. This would indicate 0.5  _ _= L____ __ _ r=o.5 S 0.05 0.0 =0.1 0.01 r = 0.05 1 =.. 0.005 r=0.02  _ r=0.01 ;_: 0.001 r=o.005 : r=0.001 ___ __ 0.001 0.005 0.001 0.005 0.01 0.05 0.1 0.5 1.0 (AtK i)/(fLi) Figure 1. Solutions to Equation 6 fo: Li+ /L = 1.2 i+1 AVERA.E A PE (thousand square feet) 3 4 5 5 10 15 20 25 30 35 POND VOLUME (thousand cubic feet) Figure 2. Depth vs. Average Area and Volume for Test Pond that the method utilizing ..i',,tion 6 should give :J.oo results. The ,.ii." ison between the ..., : '.l data and the theoretical ," fiction is shown in Figure 3. The .,_ .n:en. is, indeed, quite good. CA,.. ACTION OF THE ,'.: ..;: OF THE GROUNDWATER TABLE The reaction of the water table aquifer to vertical infiltration a ... g pond is characterized by the growth of a groundwater mound. For analysis we consider a circular pond constructed in homogeneous and iso tropic soil. The aqui:.. is underlaid by a horizontal impervious layer and initially has a horizontal free surface. When the wetting front of vertical infiltration reaches this free surface the mound begins to form. From this time the infiltration is assumed to continue at a constant rate. 1 2 3 4 5 6 7 8 9 10 11 10 9 8 7 6 u 5  L LD A im 12 13 14 15 TIME (hours) Figure 3. Comparison of Experimental and Calculated Storaqe Routinq In a i'ndri cal coordinate L .:.centered on the ..e :.: ,. as .. in Fi ,.. 4 the axis :..'i, .c flow in the saturated mound is described h3 K K s h h. + + (7) at n 3r nr ar n where Ks = saturated J J.'..:.ic conductivity n = ., ific yield w = rate of vertical infiltration. w and n are ... .tions of radial distance, r. The value of specific yield is :t. tly reduced in the zone of infiltration under the pond. on the . i.:. mental evidence of Bodman and Coleman [1943] it seems reasonable to assign a s.i:.'fic yield to the zone of infiltration with a value of of the specific yield elsewhere. The rate of infiltration, w, is zero when r is greater than the pond radius, R . ing that h = S + a, equation 7 can be rearranged: 3S Ks +S Ks [S)2 s 9S w tS = S + a) + 2 + r S + a + (8) at n n ar rn ar n To make the ...tion dimensionless, introduce the following dimension less variables: t = tks a/R, r= r/R., S' = S/a Substitution of these variables along with some manipulation and omitting the .* 'mes we obtain: .a 2(S2/2) + 1 + r 2) + + 4 (S2/2) + 3 rr (S'1) + + W/%) (9) Figure 4. Groundwater Mound In this form, the nonlinear terms are distinctly separated from the linear terms. They are treated differently in the numerical analysis. S = 3 0 at r = 0 (10) and S = 0 at r = The initial condition is: S = 0 at t = 0 (11) To solve the problem by finite differences, we must set up an appropriate y;'"e time grid. The time increments are designated by i's and the space increments by j's. The time derivative of S is approximated by a ,.r,,:rd difference. a i Si, j/At (12) A i+lj 1,j The nonlinear spatial derivatives are represented by central difference r (S2/2 S iS2 /(4Ar) (13) 4 (S2/2) S2i,j+l 2S,j + S2,j+l/2Ar2 (14) The linear space derivatives are the most influential terms in the equation. If the solution were to become numerically unstable, it would probably be because of these terms. Therefore, they are approximated by the mean of the finite difference representations on the (i + 1)th and the (i)th time rows. This is the implicit method of solution developed by Crank and Nicolson [1947] which has good stability and convergence charac teristics: aS 1 ( )/(2r) r 2 I Si+l ,j+l i+l,j_ )/(2Ar) + (S i,+ Si,jl)/(2Ar)] (15) r4 I l Si+l,j+l 2S + Si+lj_l)/Ar2 + (Si,+l 2Sij + Si,jl)/Ar2 (16) When these finite difference approximations are substituted into equation 9, a tridiagonal system of equations is generated which can be solved by Gaussian elimination. The computer program developed from these finite difference operators was tested with several combinations of grid mesh ratio and grid size. It was found to give stable results with a ratio of At/Ar2 < 0.11. The mesh size chosen was Ar = 0.03. To simulate the boundary condition at infinity, the calculations were carried out to r = 25 at which point S was required to equal zero always. For comparison with the linearized analytical solution of Hantush [1967], the finite difference program was run with constant specific yield, n. The results of this run are shown by the dashed lines in Figure 5. They are practically identical to the results obtained by the method of Hantush. ,c n was allowed to vary as a function of r, the results were .. much different, as shown by the solid lines in Fi..L 5. D? .ION AND I.,LI:% ONS analysis of the hydraulic .,* .ation of a ::. .....d. should be conducted in two phases. The first phase is concerned with the rate at which water seeps out of the e.:Ci by vertical infiltration through its bottom. The equation for vertical unsaturated flow devel..:o. by Green andi Amt [1911] can be used to describe the outflow in a s:. .. routing :,,;.s. The storage routing is simple enough to be done by hand when the soil is assumed to be homogeneous. The Green and Ampt equation can also be adapted for use in soil where the hydraulic conductivity varies monotonically with depth [Bouwer, 1969] which is a very common occurrence. This calculation is about as simple as the s,:.:.e routing e...we but the combination of variable ponded dJei and variable !w!.aulic conductivity would be cumbersome enough to make the use of a ,o,,.i.r or a programmable calculator desirable. The second phase of the analysis deals with the response of the water table to recharge from the seepage pond. The determination of adequate S.:,_ity of the groundwater aquifer requires use of the design charts for each specific case. The design chart itself is based on the numerical solution of the nonlinear differential equation. The i ,~. ":'..: of the finite difference solution of the i.ev.,.,ter mound is that it makes possible the consideration of an axisymmetric variation r the specific yield. It would also be possible to include the effects of axisymmetric variation in other soil properties. 1.14 .. ... .3 P 0 .3 'J ... 0 8 ... ...... 1 12 ~~ ~.. . W 2. P = 0.15.... .... .... ... . 1.10 ... ...... ... S .02 .05 .. .2 .. 1 2 5 ca ... .. ... 1.02 __' ^' '.'" '_ _.*J ^ '. ___________*  Variable n .02 .05 .1 .2 .5 1 2 5 10 DIMENSIONLESS TIME, t' Figure 5. Mound Height at r' = 0, Comparison of Constant and Variable n: It should be noted that the saturated ,.aulic conductivities to in the two phases the anal ,is are, ... ...:., not the same. .' V1 tration is concerned with vertical flow while the ,l .,i..:.ter mound is mainly i enced horizontal flow. ,: difference between the saturated '. .... ic conductivities in the two directions usually stems ... hori zontal la "i the soil. NOTATION a initial thickness  ffer EL] A area FL 2 A f area of i IStration [L ] f volumetric ', tion of fillable .;. ; ... h thickness of aquifer EL] H : ". of ,. .. ' water EL] I i :ow to E&.,,. [L3] Ks . aulic conductivity of saturated soil [L/T] Kt '.K... aulic conductivity of the transmission zone in unsaturated flow [L/T] L .:th of .tration of the wetting .,. [L] n specific yield P dimensionless recharge intensity, wR /K a2 r radial distance EL] r '.. .ionless radial distance, r/R0 R0 radius of :.. pond [L] S rise of groundwater mound above water table [L] So dimensionless mound height, S/a t time [T] t dimensionless time, tK a/nR v stored volume in pond EL3] w infiltration rate [L/T] r On + H [L] In capillary suction potential at field capacity EL] ., G B and lean E A. ":",sture and .:,:.. Conditions . ,: .: .rd Entry Water into Soils", Soil Science Society of :__. ._ Vol. 8, 3, pp. 116122. , nfiltration Water into '.. .,;form Soil" '. I.=. .. .. .. :.., ^ .. L._,2 Irrigation and : .' 1. '** pp '.: . Di isr, H 'rank, J. and ,:colson, P., "A Practical Method for Numerical Heat Conduction T..L", . inqs Cambridge Phil(. ;i.cal Society, Vol. : .17, pp. 5067. Green, W. H. and R G. A., "Studies on Soil K.,ics. I. The Flow of Air and Water Th'.. Soils", Journal Agricultural Science, Vol. 4, I pp. 124. Hantush, M. S., r''th and Decay of GroundwaterMounds in Response to Uniform Percolation", Water *..L'.rces Research, Vol. 3, No. 1, 1967 '234. I:1 *toux, W. J. and .;r,;i, J., "Derivation of an Equation of Infiltration, Water ..i.rces :1.. ..rch, Vol. 10, No. 4, ''74, )p '. : Weaver, R. J. and Kuthy, R. A., Field Evaluation of a ae.. ge Basin, .:. York .'te Department of Tr,: .. ,ation, Engineering Research and Devel .'" Bureau, :.:.,.ch '.':. 26 1975. Whistler, F. D. and Bouwer, H., "Comparison of v;, : for Calculating Vertical :i,.:.. and Infiltration for Soils", Journal of Hydrol.._, Vol. 10, '!. 1, 1970, pp. 119. CHAPTER 4 THERMAL CONVECTION IN A CAVERNOUS AQUIFER INTRODUCTION In a previous article [Rubin, 1976], referred as I in this paper, the author analyzed instability criteria related to onset of thermohaline convection in an aquifer whose properties are similar to those of the Boulder Zone of the Floridan Aquifer. Such an aquifer is characterized by extremely large pore size and transmissivity leading to very intensive solute and heat dispersion as well as to invalidity of the laminar Darcy law even when flow velocities are extremely small. In I it was found that for moderate Reynolds numbers the doubly diffusive convection can be approximated by the singly diffusive convec tion, for in such cases mechanical dispersion is larger than molecular solute and heat diffusion. The objective of this study is to analyze transport phenomena in the cavernous aquifer subjected to singly diffusive convection. BASIC EQUATIONS The analysis is related to a flow field similar to the Boulder Zone (the deep regions) of the Floridan Aquifer. We assume that density gradients are induced by a single component only, referred to as temper ature. In the cavernous strata, turbulent effects, as well as mechanical heat dispersion, are induced by even extremely slow fluid motions. In I the basic equations related to the calculation and approximated by the Boussinesq approach were presented as follows: 0 x, + pgn + (1 + b)ui = 0 I _+ u T (E S axi. ij axj p = pO [1 a(T TO u. = velocity vector p = :" _.sure p = fluid density K = ,. i,,.,!,ility = porosity u = viscosity T = i.'.cient of thermal a'. ..i:;icient of thermal :h.n:. ion p oj = densi, and tmnrature of reference = a ..Y:,cient J.kined by pC + sCs(1 ) pC :. ,,4 a brief literature survey *:x.ined in I it was suggested that the friction function, b, can be approximated by: b = 0.014 _ In an isotropic medium the di:. .ion tensor, Ei.. can be ,. :... as a sum of the i ,I,:ic molecular diffusivity and the second order where (4) symmetric mechanical dispersion tensor, as in E = K6ij + Et4 (7) where E4. = E* 6.. + (E* E*) uiuj/U2 = (E K) 6.. 13 t _J k. t it + (E Et) ui u /U2 (8) Here subscripts t and k refer to transversal and longitudinal components respectively. An assumption of singly diffusive convection is justified for a multicomponent system for moderate Reynolds numbers. A model suitable for the description of the mechanical dispersion tensor in such cases was suggested by Saffman [1959]. According to this model t s + 2 Re (9) v 2(s + 1)(s + 3) ) (9) E* 2 S_ (s + 1) Re (10) v 2(1 s)(s + 2)(s + 3) () where s is a power coefficient describing the dependence between the velocity and the pressure drop (s varies between unity, for laminar flow, and half, for turbulent flow). The flow field model considered in this study while unperturbed conditions prevail, is described in Figure 1. This is a saturated porous layer of infinite horizontal extent bounded by two impermeable planes located at bottom and top of the aquifer. Temperatures on bottom and top of the porous layer are To and T AT, respectively. Through the o ~r 7/ / / 77 ^^^^JZ.J^^^^^^^^^Z TEMPERATUF2 *OF ILE E IMPERMEABLE BO'JDARI Fi,.,: 1. Schematical description of unperturbed conditions ' Uo . y porous layer the fluid flows uniformly in the longitudinal, x, direction. y and z are transversal and vertical coordinates respectively. The flow field variables can be nondimensionalized as follows: x = (x ut / )/d u1 = uid/E 1 i t t = tEt/d2 (p pogz)K xu0 vp E (1 + b ) Et T' = (T T )/ T U' = Ud/Et E = E /Et Ii ii t (11) where .i is a unit vector in the x direction. Substituting the dimension less variables of (11) in (1)(3) and omitting the primes we obtain P RTn. + (1 + B)u = 0 +t i ax. ax. ij ax. 1 1 J (12) (13) Here R is the Rayleigh number defined by R = ATKd v (Et(I + b) (14) The power s in (9) and (10) according to I can be calculated through s n(Re) Zn(1 + b) + kn(Re) (15) However, (15) can be utilized only when Re 2.77 if b = 0.014 Re. For ... turbed conditions (12) and (13) yield u = 0 (16) = 0 (17) T = z (18) p = P Rz2/2 (19) E = E t E E = 1 Eij(i j) = 0 (20) LT!C: STABILITY ANALYSIS Stability criteria of the flow field are determined through the linear stability analysis. The flow field is subjected to small distur bances in the velocity (u, v, w), temperature (6), dispersion tensor (e ij), friction function (W), friction power coefficient (c) and pressure. Disturbances are very small and through the linear stability analysis second order terms are negligible. For the linear stability analysis the module of the velocity vector in the perturbed flow field is approximated by U = [(u d/Et + u)2 + 2 + w2]0.5 u 0d/Et + u (21) Substituting (21) in (15), (9) and (10) we obtain expressions for the principal components of the dispersion tensor. By applying (8), all :,.nr..nits of the dispersion tensor in the perturbed flow field are obtained. It is convenient to express the velocity components by utilizing a scalar function Q as follows 2 22 u a v 2 w = 2 0 (22) where v= 2 + ~ (23) ax ay Introducing flow field perturbations in (12) and (13), neglecting second order terms and eliminating the pressure perturbation we obtain v (Re + v2 ) = 0 (24) y + V + 2 (25) t ax2 ay az 1 2 axaz2 2 2 2 2 tt 3x y z where v + 2 +  Dx2 @Y2 Z2 ax ay az E E E* S____ t c u d T = u d (26) 0 0 Assuming vanishing values of 0 and Q on top and bottom of the aquifer, these disturbances can be expanded by the following normal modes (e,a) = (oe,3,) sin (Trz)exp [i(a x + ayy) + (or + io)t] (27) where 0e and 2V, are constants, ax and ay are the wave number components. For point of stability (or = 0) substitution of (27) in (24) and (25) yields the following secular equation a + 7T a2 ia(cla2 Cr2) (E/Et )(a x/a V)2 + X = 2 (ax/a ) + 1 R 2 2 Xa + Tr + iyW= 0 The minimal value of R satisfying (28) is the critical Rayleigh r. yields the following criteria of point of inst ability X = 1 a ax 0 o  ay = a W = 0 (30) convection cells are two dimensional rolls whose parallel to the unperturbed velocity vector. Superposition ,.,,turbed velocity and the convection velocity leads to a field. axes are of the helical flow Convection currents conducted in two dimensional rolls were also obtained for the ordinary B6nard problem [Malkus and Veronis, 1958; :i.luter et al., 1965] as well as for free convection in a porous layer while mechanical dispersion effects are negligible [Straus, 1974]. However, in those cases only nonlinear stability analysis associated with stability analysis of the steady convection motion yield such a result, whereas, in our study the linear stability analysis indicated that phenomenon. In our case, calculations concerning the anisotropy of the di ':;;.'; on led to the conclusion that convection cells should be two dimensional rolls. (28) (29) numbe Convection sets out in planes where the effective Rayleigh number attains maximal values, namely, where the coefficients of hydrodynamic dispersion attain minimal values. Inertial effects associated with the invalidity of the laminar Darcy law introduce the friction function, b, in the expression f'r R but do not affect the linear stability analysis and predictions. FINITE AMPLITUDE DISTURBANCES AND NONLINEAR STABILITY ANALYSIS The effect of the convection motion on transport processes through the aquifer can be predicted through the solution of the nonlinear equations of motion and heat transport related to supercritical conditions. The convection motion is two dimensional, therefore, the velocity components can be expressed by the stream function as follows n w = (31) 3 y az y Substituting the finite amplitude disturbances in (12), (13) and elimin ating the pressure perturbation we obtain R + V2+ = B(i,8) (32) Y +e 2 a = H(p,O) + D(Eiz) + F(cij,e) (33) where B and H are the friction and the heat advection spectra, D and F are two parts of the heat dispersion spectrum, all of which are of the form B(ip,) ( i ) (34) 1 1x M 1 x H(Q,O) = ; ( 7 (35) ) (e ) (!) (iz x iz F j i x (37) i i jx Here g and cij are finite amplitude disturbances in the friction funtio ant thpe disptson tensor respectively. As 1.,.. as the convection velocity is smaller than the unperturbed velocity the absolute value of the velocity in the flow field can be .apco..imated by U [(u d/Et)2 + V2]0.5 = (od/Et)(1 + x) (38) where V2 v2 + w2 (39) E t 2 V t 2 V t 4 V2 x = ( ) [1 ( ) 4 + (_) 8 0 0 0 Et 6V6 5 )6 V + + ] (40) ,.L(:'lying (40), (6) and (15) we obtain = [b/(1 + b)]Ax (41) (1 + b)[zn(1 + b)] b[zn(Re)] (42) (1 + b)[En(Re)]rn[(1 + b)Re] Introducing (42) in (9), (10) and the dimensionless form of the dispersion tensor we obtain after minor approximations t 1 S. [A + s ( + 13 E s+2 d2 (Uod)2 E* + E (3  1 1 3 si + s + 3 ij 1)(1 + ) +E t t tE r 1 { s 1 s s)] 2s + 5 E s] (s + 2)(s + 3) (E* t,  1)} (1  As long as the Rayleigh number is not very high, x is very small and can be approximated by the first term of (40), leading to the following approximations B = ~1V2 = (1V2 1.. = & V2 13 1 (44) (45) (46) ii + 2 u u. l.) 2 1 j where b 1 2(1 + b) 2 E ts (1 + 1 2(u d)2 (1 + E*E & + 1 2(u 0d)2 1 E t 2 (u d) b)[Ekn(1 + b)] b[n Re)] b)[Tn(Re)],n[(1 + b)Re] 1 1 1 E s + 2 s + 1 s + 3 E E 9 t 1) t x)}u u (43) Et 2 (u 0 (47) If ( ") and ( .) are conditions ,e = 0 at z = 0,1 aect to the '"..lowi, ;.'.. : .,::.. boundary ( ,, then the system () and ( .) can be solved by means of a set of trun cated ei tions ': .si the finite ".*litude disturbances. as follows = 1 q si 'pay) sin(qrz) p 5 p 3q p=0 p,q q=1 cos(...y) sin(qmTz) 11 4 4) (~7) The calculation can be s .!lified by using the complex variable presentationn .. sin(:,.y) and cos(pay) leading to p (5)) ( ..) T ei 'Y sin(:.,,z) e sin(qrz) 1q Pq 1' ,,q 2 p1q (53) Z 00 a I ;ded that = 'V p,q 7,q The solution of the system (32) and (33) through the Fourier series expansion means the determination of the Fourier series coefficients which can be done by power series expansion. Such a method was first used by Kuo [1961] for the analysis of the ordinary Benard convection. Palm et al. [1972] and btobi, [1975] through different modifications applied this approach for the analysis of free convection with no dispersion and turbulence in porous media. Rubin and Christensen [1975] suggested some guidelines for the utilization of such an approach when analyzing insta bilities induced by salinity gradients in a saturated porous layer. The advantage of this method lies in its simplicity. According to (30) the inception of convection motion is of the marginal instability of exchange, namely, steady convection follows point of instability. Therefore, the first term in (33) vanishes when steady convection is attained. In such conditions the coefficients pq and e of the Fourier series expansion can be expressed through a power series expansion as follows: N (n) (n) n(55) (p,q p,q n=1 pq (55) where n is a small parameter defined by n = [(RR )/ ]0'5 (56) The Rayleigh number is also expanded by a finite power series as follows: R = R + Rs n2j (57) where = RS/(1 n S = N/2 (58) flow ri d perturbations like v, w, g, and .ij as well as ..tion spectra H, B, D, and F can be also . .. i in double ....'ier and series F ions for the coefficients H(n) Bn) D(n) and ,C o ,p,q' pq D p,q F(n) are ..ted in the ....dix. pq If the anal ..s is L:oi..;:., for N = 1, it reduces to the linear .I.> *.i l "' i c .. 1 ,. number as given in (30). There is a .,.: 'tive relationship between increases in '.l,,eigh numbers and wave numbers. H;v.vr, taking the assumption that under itical conditions the wave number remains constant does not signifi cantly a .*... Jictions of transport phenomena for quite a wide range of ".,,leigh numbers [Straus, 1974]. Such an assumption is not required f.. the method used here but considerably simplifies the analysis. Substitut' .,:i the series p.,,.ions in (32) and (33) we obtain nO (n2i) 2 2 2 (n) +(n) R pir e + Res pi e(n 2i)' + i (p +q )pn + B = 0 (59) pq os i=l pq ,q pq 2 2 2)(n) H(n) + D(n) + F(n) = 0 (60) pq p,q p,q p,q p,q The '*... ,ions q) and e are generated by superpositions of trigonometric 4.0...ons, For n = 1 the only nonvanishing coefficients are'T1j. Therefore, only .:. Tcients Y (n) and e(n) with even values of 1pj+q have nonvanishing p,q pq values. ... the values of the subscripts Ijp and q are smaller or equal to n. Coefficients with other subscripts vanish. According to (60). (n) (H(n) + D(n)+ ,F(n) /Ti22) (61) o,q o,q o,q o,q For p00, (59) and (60) yield (n) [(P2 2) 4] + Rs (n2i) + B()/(npR ) p,q 2 2 R i=1 p,q p,q o 4pr(p ( q) 0 (H(n) + D(n) + F,)/[pq2)] = 0 (62) p,q p,q p,q For p=q=l e(n) + E 0(n2i) + B(n+2)'/R 2(H(n+2) + D(n+2) + F(n+2))/R (63) 1,I i 1 1,1 l /(os)2 1 1 1,1 s (63) Through (61), (62) and (63) all the coefficients ,(n) and O(n) are p,q pq determined. According to (63) the coefficients T1 n) (n+l) and e0n+2) must be determined simultaneously. However, by simple arrangements e(n) can be expressed explicitly. Such a procedure avoids any trial process. Calculations of the mean horizontal temperature and the Nusselt number followed the determination of the series coefficients. These para meters are given by T = z +Z (n) [sin(qfTz)]nn (64) n=l ql1 o,q N n1 Nu N1 n r (n) + (il ) sq T(ni) Nu 1 no,q 1 q1= pq= q,s=1 pq ps n=l q=l1 =3 n1 i1 = qs (j) 'ij) (ni) *F .E Z E qs T T ea (65) 1 i= 2 j=i p,k= q,k,s=l p,q k, IP+kI,s Calculation of Nu determines the convergence of the method and the termination of the series expansion. Through the calculation, presented in the next section, we took N = 6, 8, 10, 12, 14 and 16 according to the variation in the Nusselt number. If Nu varied by less than 2% as N was increased from N to N + 2 then the expansion was terminated. F .! I 'J .ION .. rding to (44)(47) convection effects are not determined only 0L. the .leigh number but also ". the ', I:1ds (Re) and Prandtl (Pr) numbers, as well as :. the ratio between the *.'. .v: layer thickness and the charac teristi: ; K size (, .'.3.) whole .: 1 .:is .:,. a. is cable only when mechanical dirvs.sion is at least *..'..rable with the molecular dii ion, s relationship is determined by the magnitude of Re and It seems that if Pr ? 1, which is reasonable for practical pur..,..: of hydrol ., [Somerton, '"A], then dispersion effects are significant even f.a.. Re Y 3. However, if the Prandtl number is smaller than unity (if the solid fraction is a .' conductor) then dispersion effects become si '..ficant at hi.h.' Reynolds numbers. The : t of Re, Pr and d/d on the convection :w.n 1.j.:.. is intro duced in the analysis through S1, &1 and &2 (and &3 = a. + 2) which determine 2,.ts of turbulence and dispersion induced by the convection motion. Fi .,. 2 demonstrates :;,,.c. in these ,:c',: clients due to variations in *, Pr and d/d when the porosity is 0.4. The *: t; of Pr vanishes ".. high Oldss numbers as in such cases mechanical di *....ion _ .ts tr' , esses more than the molecular diffusion. .i,. the ': .ds number increases, according to model (9) and (10), the mechanical di ,..ion tensor becomes more and more isotropic. This phenomenon leads to a reduction in the value of Ia. Turbulent Jf5 :':. become more and more si ificant when the = .lds number increases, leading to a positive relationship between increases in the Reynolds number and the coefficient 51' According to (63) turbulent effects introduced through the friction spectrum B amplify dispersion effects. However as 2 decreases when Re increases, it was .*. i through the calculation that the net effect of an increase in Re led to a minor reduction in the influence of the spectra B, D and F. According to Figure 2 the main parameter that determines the effect of the spectra B, D and F on the convection phenomenon is d/d The coefficients &a' &2 and I are aii,,:.. inversely proportional to the square of this parameter. It should be mentioned that the absolute value of the coefficient r, is usually much smaller than al. Hence changes in s due to the convection motion are almost negligible. We may usually neglect terms d:e! ,ing on this coefficient in expressions (47). As an example we present in Figures 3 and 4 descriptions of mean hori zontal temperature and Nusselt numbers for an aquifer when p = 0.4, Re = 3 and Pr = 1. Figure 3 demonstrates profiles of mean horizontal temperature for various Rayleigh numbers and values of d/d As explained before, a decrease in d/d increases the effects of dispersion and turbulence induced by the convection motion. The convection motion increases the value of the effective dispersion coefficient. However, this effect leads to a reduction in values of temperature gradients at top and bottom of the porous layer as demonstrated in Figure 3. 01  SFigure 2. .cription of ,^L /0.5 S B, .5, and a2 vs. Re 5 various values of T. and d/d (1=0.4). 0.5 ...  .   / _ A r 4 3 . I 1.0 0.8 0.6 1.0 0.8 0.6 0.4 0.2 0.4 0.2 0.0 T Figure 3. Mean horizontal temperature profiles for various values of R/R and d/d (=O.4, Pr=l, Re=3). At high Rayleigh numbers a thick region in the center of the r.... layer should achieve a nearly isothermal state in the mean. However, as .. ted in Fi,,. 3 such conditions lead to a positive temperature gradient or reversal of temperatures. Such a phenomenon was also ident ified in the ordinary Benard problem [Kuo, 1961; Veronis, 1966]. Veronis [1966] tried to explain the origin of such a strange phenomenon. However, better choice of wave numbers diminishes this effect. Figure 3 also demonstrates the creation of boundary layers on top and bottom of the aquifer leading to invalidity of the continuum approach even for moderate 7.'leigh numbers if d/d is not very large. Figure 4 presents variations of Nusselt number with Rayleigh number for various values of d/d According to this presentation, the net effect of the mechanical dispersion induced by the convection motion leads to a reduction in the heat transport through the aquifer. Disper sion and turbulence induced by the convection motion act as a stabilizing mechanism in the flow field (ironical interpretation). However, this mechanism is more complicated than just a stabilization. In the calculation it was found that the phenomenon was associated with increased values of the higher modes of the series expansion, leading to a reduction in the convergence of the series expansions. Through the calculation, the maximal values of X and V were continu ously checked in order to examine the validity of approximations (44) (47) and to follow .!.. ,c in the Reynolds and the Peclet numbers. It Nu 6 5 Description of Nu vs. R/R for various values of d/d (4=0.4, Pr=1, Re=3). 3 2 I pO  p /P=1 ) 2 4 6 8 ) 2 4 6 8 Figure 4. 10 R/Ro was .. :,. that i .' the ., R/R!0 10, T_, was much smaller T,:,; unity which u;.. ifies utilization of (44)(47) and yields only minor in the "_. Ids number. 1 IONS The anisotropic character of the dispersion tensor leads to convec tion motions conducted in two dimensional rolls whose axes are parallel to the ..,.bed velocity vector. By choosing a new definition :, the Ray1l..'( number, turbulence and dispersion effects can be introduced in the linear stability analysis with no substantial complications in the calculations and results. Finite amplitude analysis for homogeneous boundary conditions can be conducted by Fourier series and power series expansions. The significance of mechanical dispersion and turbulence induced by the convection motion .:ps ic, on Pr, Re and mainly on d/d P An increase in Re diminishes the effect of Pr and reduces effects of dispersion and turbulence associated with the convection motion. An increase in '1d reduces significantly e~.': s of dispersion and turbulence due to the convection motion. For d/d 102 these effects practically vanish. '. r, for smaller values of d/d these effects lead to a reduction in The si. ly diffusive analysis presented in this article can be .'.,lied when density gradients are induced by a single component or when ..,!. conditions and effective dispersion coefficients are identical f. all components in a multi,,ri.one,! t system. The latter conditions *....: ... .'! ly satisfied when mechanical dispersion e:: s are 1,.. : than : molecular diffusion in the aquifer. Expressions for coefficients of heat advection, friction and heat dispersion spectra T (i) p (ni) (v=1,2;3)] + k[s (ni) (v=2;1,3)]} k, jpkks v V pk, , (66) Z(i) 2[ 2 (n i) k, l +3)k(pk)&2(pk)2+2&i 2 (ni) (v=1,2;3)]} S(67) (67) s3= q (68) (69) Olpk ,sv (v=2;1,3)= O pkj,s2e pki,sl pk1,s3 " (j) (ij) k,m=m k,k m,h t,h=1 {zh[4k(pkm)+(pkm)2 +3 s 2](ni) (v=l +[3k2hSvY2m(p=km)][(ni) (v=3,4,5;1 (n) .4 nl i1 c, pi,q 16 i=2 j=1 k,m=cc 9,h=1 (j) k,9 ,2,3,4;5,6,7)] ,2,6,7)]} (ij) m,h {th[(1 + &3)k(pkm)+&3(pkm) 2+&s 2][eni) ,s(v=5,6,7;1,2,3,4)] +m[2&1k22 2)(pkm)+&1k(pkm)2+&3ks2][O(ni) (v=3,4,5;1,2,6,7)] +h[&2k2+2 2k(pkm)212(& +& 3)mk][sE e(ni) (v=1,3,5,7;2,4,6)]} In (70) and (71) the subscript sv may obtain seven different values as follows: s3=qz+h s4=q+zh s2=q+(+h s7=Ahq APPENDIX 2 n1 2 1=1 3 n1 7 i=1 H(n) Ip ,q where where CO k  9,=1 k=co s =qP, B(n) p,q 4 ni iI 1 4 i=2 j=1 (70) (71) s5= +hq s2=q+k s, =qyh s6=hkq (72) NOTATION a ax a y a 0 b B B (n) B B ( psq pq ClC2 C Cs d d p D D D(n) pq p,q Et,E. I3 i.j E*, E E*,E EtEt F F F (n) p,q p,q g H H ,H(n) p,q p,q K ni wave number (o' ./i,!ts of wave number critical wave number friction function friction spectrum c.L:f:.ients in seriesexpansions forB coefficients defined in (26) specific heat of fluid specific heat of solid porous layer thickness characteristic pore size part of heat dispersion spectrum coefficients in series expansions for D heat dispersion tensors (mechanical and hydrodynamical respec tively) longitudinal heat dispersion coefficients transversal heat dispersion coefficients part of heat dispersion spectrum coefficients in series expansions for F gravity acceleration heat advection spectrum coefficients in series expansions for H permeability unit vector in the longitudinal direction unit vector in the vertical direction N total number of terms in the series expansion Nu Nusselt number p pressure pO pressure at the coordinates origin Pr Prandtl number (v/K) R Rayleigh number defined in (14) R critical Rayleigh number Ros parameter defined in (58) Re Reynolds number (pUd /v) s power coefficient S = N/2 t time T temperature T temperature at z = 0 u longitudinal velocity perturbation ui velocity vector u unperturbed velocity U module of velocity vector v lateral velocity perturbation V module of velocity perturbation w vertical velocity perturbation xi coordinate x,y,z coordinate system coefficient of volumetric thermal expansion 1' 2' 3 coefficients defined in (47), (c3 = a1 + a2). 6 perturbation in the friction function ,1 coefficient defined in (47) y parameter defined in (5) 6.. Kronecker's delta 13 AT difference in temperature between bottom and top of the porous Eij dispersion tensor perturbation n small parameter defined in (56) 6 ..:mre. ture perturbation 06 constant A. .ined in (27) (n) 0e e e( ; i>*:,'cients in series expansions for e K thermal diffusivity of saturated porous medium x parameter defined in (40) viscosity v kinematic viscosity C perturbation in s C1i ,.:.ficient defined in (47) p fluid density p density at z = 0 ps solid density or parameter expressing amplification of small disturbances Sporosity x parameter defined in (29) 9 stream :iuc.tion T p ?T(n) coefficients in series expansions for i pq, p,q' p,q om parameter expressing oscillations scalar function defined in (22) 1 constant defined in (27) REFERENCES Kuo, H. L., "Solution of the Nonlinear Equations of Cellular Convection and Heat Transport", J. Fluid Mech., 10, 1961, pp. 611 634. ' Ikus, W. V. R., and Veronis, G., "Finite Amplitude Cellular Convection", J. Fluid Mech., 4, 1958, pp. 225260. Palm E., Weber, J. E., and Kevernold, 0., "On Steady Convection in a Porous Medium",J. Fluid Mech., 54(1), 1972, pp. 153161. .'r n,, H., ',;. the Analysi o*f Ce' ,ular Convection in Porous Media", Int. J. Heat & :t.,Trans., 18, 1975, pp. 14831486. Rubin, H., "Onset of Thermohaline Convection in a Cavernous Aquifer", to be published in Water Resourc. Res., 1976. Rubin, H, and Christensen, B. A., "Convection Currents Associated With Hydrodynamic Dispersion in a Porous Medium", 16th Congress of ., Sao Paul, Brazil, 1975. .*f'', P. G., "A Theory of Dispersion in a Porous Medium", J. Fluid *!_., 6(3), 1959, pp. 321349. SchlUter, A., Lortz, D., and Busse, F., "On the Stability of St.dv Finite Amplitude Convection", J. Fluid Mech., 23(1), 1965, pp. 129 144. Somerton, H. W., "Some Thermal Characteristics of Porous Rocks", J. Petrol] Tech., 10, Note 2008, 1958, pp. 6165. Straus, J. M., "Large Amplitude Convection in Porous Media", J. Fluid Mech., 64(1), 1974, pp. 5163. Veronis, G., "Large Amplitude Benard Convection", J. Fluid Mech., 26(1), 1966 pp. 4968. CHAPTER 5 SEMINUMERICAL APPROACH FOR THE MATHEMATICAL MODELING OF SINGLY DISPERSIVE CONVECTION IN GROUNDWATERS INTRODUCTION Considerable interest is now focused in the State of Florida, as well as in other locations in the United States, for the possible utilization of deep saline aquifers for waste disposal. Vernon (16) delineates the properties of the deep zones of the Floridan aquifer which makes this stratum available for such application. Henry and Kohout (2) mention the fact that in a thick system, like the Floridan aquifer, the effect of geothermal activity should be considered, too. One of these authors has postulated in previous articles (3,4) that geothermal activity induces groundwater circulation in the Floridan aquifer. Singly diffusive convection is the convection motion induced by a single dissolved component (e.g. temperature or salinity) in a fluid layer. The hydrodynamics of this phenomenon in a saturated porous media has been studied while, in most instances, assuming that the fluid is initially at rest (7,8). However, groundwaters are generally subject to hydraulic gradients leading to slow, effectively horizontal flow of the subsurface water. This movement through a formation, similar to the Boulder Zone in the deep saline region of the Floridan aquifer (1,4), leads to intensive mechanical dispersion of heat and soluted materials in the aquifer, as well as inertial effects, as demonstrated by invalidity of the laminar Darcy law. In such a system molecular diffusion effects are usually less significant than mechanical dispersion. Convection phenomenon under such conditions may therefore, be called dispersive convection. A system is subject to singly dispersive convec tion if one of the following criteria is satisfied: a) density gradients are introduced by a si: le component; or b) all molecular diffusivities of the dissolved components are much smaller than the mechanical dispersion, and all of them have the same boundary conditions. The objective of this article is to present a rather simple method by which flow conditions in such an aquifer can be simulated. The basic ,:....:,tions applied for the analysis are the equations of Sitnuit. motion, and ,'.persion, subject to the Boussinesq a.::.ima.i,.:. you = 0 . . . . . . . (1) vp + pg + (1+b)u = 0 . . . . . (2) y + + O vT = v7 (E VT) . . . . . (3) The ..,rfcient y appearing in Equation 3 is defined by: Y = [pCwV + PsC )] . . . . . (4) In a brief literature survey, presented in a previous study(10), it was s5'..,= ted that the friction function, b, appearing in Equation 2 can be approximated by: b = 0.014 Re . . . . . . (5) It is assumed that the fluid density depends linearly on the dissolved .... t, which is the temperature, as follows: p = po[I (TT )] . . . . . . (6) zf LmA /T To AT/////////// ZZ// T///////T TEMPERATURE PROFILE \ IMPERMEABLE BOUNDARIES I 7777 x Schematical description of the aquifer with no convection motion. 0 Figure 1. 9 9 0 y Sow .eld model considered prior to the inception of the convection motion, as presented in Figure 1, is a cavernous aquifer consisting of a porous layer of infinite horizontal extent. It is bounded by two impermeable planes on which the ,, ..ture is constant. Through the porous layer the fluid flows uni.. ly in the 1 ... tudinal x direction. The transversal and vertical coordinates are y, and z, respectively. A movi.; coordinate system is applied with the velocity u /y in the x direction, and .' ider d, AT, e et/d, d2/et, et(1+b)/K . . . . (7) as characteristic length, temperature, dispersion coefficient, velocity, time, and :, respectively. In such a manner the following dimensionless basic equations are obtained (in .. :. :,ions all variables are dimensionless): v u = 0 . . . . . . . . (8) vp RTn + (1+3)u = 0 . . . . . . (9) @T 4 y + u VT = V7 (!ovT) . . . . . .(10) the Rayleigh number, R, and the variable a, are defined as follows: v t (ATKd In an isotropic medium the dimensionless dispersion tensor can be E = E t + (E Et)u u/U . . . . . (12) As long as there is no convection motion Equations 810 yield: u = 0 = 0 T = z p = P Rz /2 S =E =E =1 E =x E = Eyz =0 . . .(13) xx E yy zz xy xz yz .(.. ) THE FLOW FIELD STABILITY The stability of flow conditions presented by Equations 13 can be determined by analyzing the growth of small disturbances in the aquifer. These are disturbances in the velocity, u, temperature, o, dispersion tensor, E and pressure, P. Second order terms depending on these disturbances are negligible. It is convenient to express the velocity vector through the scalar vari able o as follows: U = where = nx vx . . . . (14) By substituting the small disturbances in the basic equations, and applying the boundary conditions o'v20 = 0 at z = 0, 1 . . . . . .. .(15) (where v2 = we obtain: ylV2 RV2 2 V2d R[(E 1)/u ][V(V )]*k + R(aEt/au )[V(Q.1)] . . . (16) where 2 2 2 7v = vV 7d = (E 1)P: VV + V . . . .. (17) By expandi in the follwinq normal mode, [sin(7rz)] exp[i(axx ayy) + (ol+ia2)t]. . . . (18) we obtain for the point of instability (a1 = 0, and minimum value of R) ax = 0 ay = a G2 = 0 ao = R = 4T2 (19) Hence the instability is of the marginal of exchange type. Convection cells are two dimensional rolls whose axes are parallel to the unperturbed velocity vector. Superposition of the unperturbed motion and the convection motion yields a helical flow field. _ .IS OF THE STEADY C,i ..TCTION In the . ous section it was , .;" that convection cells are two dimensional rolls. It is convenient to :.'; .: the velocity by the stream .,,tion > 4 u = x 7 = [ n, vs] . . . (20) where the square brackets s ,, lize the box product. Substituting Equation 20 in the basic qur.*tions, and eliminating the pressure perturbation, we obtain: R[~ ?, VT] + V2 = B( . . . . . (21) DO 2 Y1t + ve [n, v ] = H( 9,e) + D(c*) + F(, ). . . (22) where T is the finite amplitude disturbance in the dispersion tensor. .iables B, H, D and F are nonlinear terms defined as follows: !, 8) = v.(ev ) . . . . . . (23) H( 0, e) = [ v v ] . . . . . (24) D(to ) = v(Tn) . . . . . . (25) F(e, e) = v ( voe) . . . . . . (26) As long as convection velocity is smaller than the unperturbed velocity the absolute value of the flow field velocity can be approximated by: U = (1 + A 1/2X2 + 1/2x3 . . . . (27) where S = V2/(2u ) V2 = u . . . . (28) If x << 1 terms depending on high orders of A can be neglected. In such conditions, the expressions for g and the dispersion tensor can be . :. .. i mated by: S= 2 . . . . . . (29) e = alV I + e2uu + a3(ut + zu) + a4V 2t. . . . ( ) where 1 = b/[2(l+b)u ] . . . . . . (31) aI = (1/2Uq)(aE t/au) a2 = (E,l)/u2 a3 = (E)/o 4 = (E)/u + (1/2u )[(E Et )/u ] . (32) The system of the differential equations 21 and 22 is subject to the following boundary conditions: 4, e = 0 at z = 0,1 . . . . . . (33) Such boundary conditions are simple according to the definitions presented by Orszag (9). In such cases, accurate simulation of incompressible flows can be obtained by spectral methods. Assuming that + and e are periodic in the horizontal direction, these variables can be presented by sets of truncated eigenfunctions as suggested by Veronis in similar studies (17,18). 0 = ? ^ sin(pay)sin(qrz) . . . . (34) p,q=l p,q e = o e cos(pay)sin(qrz). . . . . .. (35) p=0 pq q=1 The calculation can be simplified by using the complex variable presentation of sin(pay) and cos(pay) leading to = i i p [sin(qfrz)] exp(ipay). . . . (36) 0=W p,q o = 2 e [sin(qrz)] exp(ipay) . . . . (37) p=o q=1 provided that Pages Missing or Unavailable The convergence of the expression for the Nusselt number also determines the truncation of the Fourier series expansion (the value of N). NUMERICAL CPACLJ!ATIQNS Experiments concerning heat transfer characteristics of porous rocks (e.g. 5, 6) showed that the phenomenon of heat dispersion due to the fluid movement in the stratum is very similar in nature to the characteristics of solute dispersion in porous media. We may adopt, then, models of mechanical dispersion, which are available in the literature, for the quantitative evaluation of the effect of the convection phenomenon on the intensity of transport processes in the aquifer. Saffman (11) suggested the following expressions for the coefficients of dispersion. et 1 s + 2 Re e 2Is + 2 Re) . . . . (42) ej + (s + I)2 Re v Pr 2(ls)(s+2)(s+3) ( 43 where s is a power coefficient describing the dependence between the flow velocity and the pressure gradient. In a previous article (10) it was suggested that this coefficient can be calculated through the following expression s = in(Re)/{tn[(1+b)Re]} . . . . . (44) Equation 4+ :an R= armlied 'r T > 2.77. Through the calculation, it was assume that Pr = 1, which seems to be a reasonable value for limestone and dolomite aquifers (13). The convergence of the numerical integration was very moderate. Several approaches were applied to speed convergence of the calculations. These were: (a) Each calculation was conducted in two steps, in the first step we took d (which leads to :; '=0), and, ..:' obtaining results :' such conditions the actual value d/d was introduced into the ~n ram; (b) . Its obtained with lower values N 1 ., !/or R were used as initial 7..;ities higher values : these parameters. As a criterion for *.' state, Straus (14l5) ted. in such calculations, to take { (' /dt)/e }max < 104 .() in our calculation, it was "::.j that such a criterion is ... conservative. 11y when ,1 a criterion of one hundred times less conservative, vari. ,ons less than 1% were obtained in the Nusselt Iw ., T. :.;, the ., .. process, deca :ng oscillations were detected in the value .. the 7. t number. Fi ,. 2 presents variations of the J;:;.lt number with :.leigh number various values :, d/d P These are the maximal values of Nu obtained when S i the value of the wave number (12). The wave number increased with increase : values of '/R o nationss in the wave number induced minor ,h ... in Nu ,:.' a ..' ,. ylei number S... to Fi .. *. 2, mechanical di ..ion, due to the convection motion, leads to a reduction in the intensity of transport processes through the aqui.'r. calculation indicated that in small values of d/d the magnitude :" the di .ion cents increases, when convection occurs, leading to a reduction in the : .itude of the ...., :, gradients on top and bottom '! the .q .., . This reduction in the i**..'... ture gradients affects transport ,.' :: more than the increase in the value of the dispersion .,. icients. It was also found that lower values of d/d give rise to the higher modes of the Fourier series expansions and reduce the *:: .,. :, of the calculation. For very 6 4  d/d // 220 / d/dp= 20  d/dp= 10 0I  I I 0 2 4 6 8 10 R/Ro Figure 2. Description of Nusselt number vs. R/R0 for various values of d/dp (=0.4; Pr=1; Re=3). small values of d/d the analysis fails to follow the physical ...r .,. even at moderate Rayleigh numbers, as the thickness of the boundary layers developed on top and bottom of the aquifer is of the same order of magnitude as the characteristic pore size. In such cases the continuum ;n,,.:;. applied th ..' the analysis is not satisfied. Through the calculation, the maximal value of A was continuously checked in order to examine the validity of the ,, ....,tions presented in Equations 2732. It was found that for ,I. <10, the maximal val.,: XH x was much smaller than unity. CONCLUSI'. Singly dispersive convection in aquifers can be analyzed by expanding the flow field disturbances in eigenfunctions. The ani:.;, ..,..ic character of the mechanical dispersion determines the plane in which convection motions are conducted. According to the analysis, convection cells are two dimensional rolls whose axes are parallel to the .' i:. :ed velocity vector. Analysis of steady state convection can be conducted by transforming the equations of motion and continuity to a set of general first order dif,.:realal equations that can be integrated through available subroutines. The effect of mechanical dispersion and turbulence, induced by the convection motion, depends mainly on the ratio between the porous layer thickness and the characteristic pore size; Prandtl and Reynolds numbers have less significant effects on the physical phenomenon. The analysis of the singly dispersive convection presented in this study can be :ilied when density gradients are induced by a single component, or when boundary conditionL and effective dispersion coefficients are identical for all components in a mul'icomponent system. APPENDIX I EXPRESSIONS FOR THE COEFFICIENTS OF THE SPECTRAL FUNCTIONS N1 NIk H = H_ pq = (a/2) H z Pq Pq k=1N s=1 + k[w v pkl, w (v=2;1,3)]} . . .. . ....... (46) D q = D ,q psq pAq 2 N1 NIkI = (Tra /2) Z E k=1N s=1 sT ks{[ (ctl +cx3)k(pk) + 2ua (72/a2)w2 u2(pk)2] pk,w (v=1,2;3)]} . . ... (47) where wj = qs w2 = q+s w3 = sq eIpklWv (v=2;1,3) = e pki' pklw2e p kl 3 *.. ...... (48) Bpq = B Pq PAq 2 2 N1 = (2a2/4) N k,m=1 N NjIk NIm E h= s=1 h=1 {sh[4k(pkm)+(pkm)2 + 3(2 /a2)w2 ][ kmw (v=1,2,3,4;5,6,7)] v pkm,w , V p,q p,q 2 2 N1 NIkl NIml = (ra /4) z zE k,m=1N s=l h=l {sh[(al+a3)k(pkm)+a3(pkm)2+al(T 2/a2)W2][eipkm l,wv(v=5,6,7;1,2,3,4)] + m[(pkm)(2 k2a22 2s2) + 2k(a2 s2)(pk )2 + 23kw 2][ p km2,Wy (v=3,4,5;1,2,6,7)] + h[a2k2 +2a2k(pkm)2as2 (72/a2) (al+a3)mk] V=' wI = qsh w5 = s+hq .,6)]} . . . . (50) w2 = q+s+h W6 = hsq w3 = qs+h w4 = q+sh W7 = shq . . . (51) k,s m,h Tk,s Tm,h where T ks fs(pk)[E) lpki, w (v=1,2;3)] v a = wave number; ao = critical wave number; ax, a = wave number components; b = friction function defined in Equation 5; B = friction spectrum defined in Equation 23; B =coefficients in the Fourier series expanded for B q,q Cs, C = specific heats of soil and water, respectively; d = porous layer thickness; d = characteristic pore size; D = part of heat dispersion spectrum defined in Equation 25; D = coefficients in the Fourier series expanded for D; p,q e et = longitudinal and lateral dispersion coefficients, respectively; E Et = dimensionless dispersion coefficients (dispersion coefficients divided by et existing prior to convection conditions); E = dispersion tensor; F = part of heat dispersion spectrum defined in Equation 26; F = coefficients in the Fourier series expanded for F; p,q g = gravity acceleration; H = heat advection spectrum defined in Equation 24; H = coefficients in the Fourier series expanded for H; P,q I = unit matrix; K = permeability; 9 = unit vector in the longitudinal direction; n = unit vector in the vertical direction; N = truncation .*ii:r; Nu = Nusselt number; p = pressure; '^ ",! t h..I , Po = pressure at z = 0; Pr = Prandtl number (=v/K); R = Rayleigh number defined in Equation 11; R = critical Rayleigh number; Re = Reynolds number (=pUd /v); s = power coefficient; t = time; T = temperature u = velocity vector; u = longitudinal velocity existing prior to convection conditions; U = module of velocity vector; V = module of convection velocity; x, y, z = coordinates; a = coefficient of volumetric thermal expansion; i(i=1, ..,4) = coefficients defined in Equation 32; 8 = perturbation in friction function; a1 = coefficient defined in Equation 32; y = parameter defined in Equation 4; = operator defined in Equation 14; AT = temperature difference between bottom and top of the aquifer; e = dispersion tensor perturbation; e = 'mpe. !Ae p, LcuroL en; ^q,, pq = coefficients in the Fourier series expanded for o; K = thermal diffusivity of saturated porous media; x = parameter defined in Equation 28; p = viscosity; = kinematic viscosity; = fluid density; = fluid density at z = 0; = solid density = parameters expressing growth and oscillation of disturbances; p,q, Tp,q porosity; stream function; coefficients in the Fourier series expanded for {; scalar variable defining velocity perturbations; constant defined in Equation 18; P p ps 01, C02 REFERENCES 1. Burke, R. G., "Sun Whips Florida's Boulder Zone", Oil and Gas Journal, Vol. 65, No. 7, 1967, pp. 126127. 2. Henry, H. R., and Kohout, F. A., "Circulation Pattern of Saline Ground water Affected by Geothermal Heating as Related to Waste Disposal", Proceedings of the 1st Symposium on Underground Waste Management and Environmental Implications, T. D. Cook (ed.), American Association of Petroleum Geologists, Tulsa, Oklahoma, 1972, pp. 202221. 3. Kohout, F. A., "A Hypothesis Concerning Cyclic Flow of Salt Water Related to Geothermal Heating in the Floridan Aquifer, New York Academy of Science Transactions, Vol. 28, No. 2, 1965, pp. 249271. 4. Kohout, F. A., "Groundwater Flow and the Geothermal Regime of the Floridan Plateau", Symposium Geological History of the Gulf of Mexico Caribbean Antillen Basin, Gulf Coast Association of Geological Societies Transactions, Vol. 17, 1967, pp. 339354. 5. Kunii, D., and Smith, J. M., "Heat Transfer Characteristics of Porous Rock", American Institute of Chemical Engineering Journal, Vol. 6, No. 1, 1960, pp. 7178. 6. Kunii, D., and Smith, J. M., "Heat Transfer Characteristics of Porous Rocks: II. Thermal Conductivities of Unconsolidated Particles with Flowing Fluids", American Institute of Chemical Engineering Journal, Vol. 7, No. 1, 1961, pp. 2934. 7. Lapwood, E. R., "Convection of a Fluid in Porous Media", Proceedings of Cambridge Philosophical Society, Vol. 44, 1948, pp. 508521T 8. Nield, D. A., "Onset of Thermohaline Convection in a Porous Medium", Water Resources Research, Vol. 4, No. 3, 1968, pp. 553560. 9. Orszag, S. A., "Numerical Simulation of Incompressible Flows Within Simple Boundaries; Accuracy", Journal of Fluid Mechanics, Vol. 49, Pt. 1, 1971, pp. 75112. 10. Rubin, H., "Onset of Thermohaline Convection in a Cavernous Aquifer", Water Resources :search, Vol. 1?, No. 2, 1976, pp. 141147. 11. >i. ,*, P. G., "A Theory of ;ispersion in a Porous Medium", Journal of Fluid Mechanics, Vol. 6, Pt. 3, 1959, pp. 321349. 12. Schluter, A., Lortz, D., and Busse, F., "On the Stability of Steady Finite Amplitude Convection", Journal of Fluid Mechanics, Vol. 23, Pt. 1, 1965, pp. 129144. 13. Somerton, H. W., "Some Thermal Characteristics of Porous Rocks", Journal of Petroleum Technology, Vol. 10, Note 2008, 1958, pp. 6165. 
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PAGE 1 WATER IiRESOURCES researc center Publication No. 39 Analysis of Storm Water Seepage Basins In Peninsular Florida By Hillel Rubin, John P. Glass and Anthony A. Hunt Department of Civil Engineering University of Florida Gainesville UNIVERSITY OF FLORIDA PAGE 2 ACKNOWLEDGEMENTS. ABSTRACT .... TABLE OF CONTENTS CHAPTER 1: GENERAL INTRODUCTION Scope of the Study Objectives Methodology ........ Research Program . CHAPTER 2: FACTORS AFFECTING THE DESIGN OF URBAN DRAINAGE SYSTEMS IN PENINSULAR FLORIDA . Introduction ........... Legal Principles ......... Federal, State and Local Regulations Project Feasibility ...... Preliminary and Detailed Design. Conclusions. . . .. .... Appendix Flow Diagram for Urban Drainage Design References .... . CHAPTER 3: ANALYSIS OF TRANSIENT GROUNDWATER FLOW FROM SEEPAGE PONDS ......................... Introduction ................... Calculation of Unsteady Seepage from a Pond .... Calculation of the Response of the Groundwater Table Discussion and Conclusions Notation ................... References . . . . iii iv 1 1 2 2 3 5 5 7 9 14 19 24 26 32 33 33 33 39 45 47 49 CHAPTER 4: THERMAL CONVECTION IN A CAVERNOUS AQUIFER 50 Introduction . . . . 50 Basic Equations . . . . 50 Linear Stability Analysis. . . . 55 Finite AmplitudeDisturbances and Nonlinear Stability Analysis . . . . 58 Results and Discussion . . . 65 Conclusions. . . . . . .. 71 Appendix Expressions for Coefficients of Heat Advection, Friction and Heat Dispersion Spectra ............ 72 Notation . . . . .. . .. 73 References .. . . . . . .. 77 CHAPTER 5: SEMINUMERICAL APPROACH FOR THE MATHEMATICAL MODELING OF SINGLY DISPERSIVE CONVECTION IN GROUNDWATERS 78 Introduction . . . 78 Basic Equations . . . 79 The Flow Field Stability . 82 Analysis of the Steady State Convection 83 Numerical Calculations .. . 86 i PAGE 3 Conclusions ................ Appendix I Expressions for the Coefficients Spectral Functions Notation ................. References . . . '.' of the CHAPTER 6: NUMERICAL SIMULATION OF SINGLY DISPERSIVE CONVECTION 89 90 91 94 IN GROUNDWATERS . . . . .. 96 _'_, _. '._' ,_" __ '! _____ ,_._,_ 96 Basic Equattons_ .. _. __ ... ... _.. .. __ The Flow Field Stability. . . . 100 Numerical Calculation of the Steady State Convection 102 Numerical Results and Discussion 108 Conclusions .. 114 Notation . . . 115 References . 118 CHAPTER 7: SUMMARY AND CONCLUSIONS 120 i i PAGE 4 ACKNOWLEDGEMENTS This report summarizes the research results obtained in 1975 and 1976 in the Hydraulic Laboratory of the University of Florida. The research project was sponsored by the Office of Water Research and Technology (OWRT) through the Florida Water Resources Research Center. The investigators are grateful to Dr. William H. Morgan, Director of the Florida Water Resources Research Center for his help through all phases of this investigation. Dr. Sylvester Petryk initiated the study and served as its principal investigator through October, 1975. His initiative and participation in this research are greatly appreciated. Chapters 2 and 3 of this report are mainly based on the Master of Science theses of A. A. Hunt and J. P. Glass. The investigators are very grateful to Dr. B. A. Christensen who served as the chairman of their respective graduate study committees. Dr. W. Huber served as coinvestigator in this research and participated in the graduate study committees. His activity is very much appreciated. ii i PAGE 5 ABSTRACT Rapid urbanization, currently proceeding in Florida, has resulted in significant problems with regard to both flood control and pollution abatement. The objective of the reported study was to search for the design procedures that may improve efficiency, safety and adequacy of drainage systems in peninsular Florida. Special attention was given to possibilities of recharging groundwater aquifers with excess storm water. Through such a system, partial solution to problems of inadequate potable water supply in some areas can be achieved simply as a byproduct of flood control systems. In Chapter 2 of this report a conceptual design framework is presented. In developing this framework, a variety of disciplines involved in the solution of urban drainage systems in peninsular Florida are considered. Comparatively high levels of groundwater in many locations require consideration of two factors relating to the ability of seepage ponds to divert surface water to the aquifer. The engineer should study the ability of,the pond to seep water and he, should analyze the response of the groundwater table to the seepi,ng water. With respect to these two factors, analyses and solutions adapted for engineering application are presented in 3 of this report. The continuing reduction in the availability of potable water in coastal zones of Florida has prompted several communities to design recharging systems utilizing treated storm water as well as effluents. Chapters 4 to 6 of this report concern flow conditions in the Floridan iv PAGE 6 Aquifer that should be considered in connection with this subject. Methods by which the particular phenomena associated with flow conditions in the aquifer can be evaluated are presented. These methods cover several spectral expansion approaches as well as a complete numerical approach. v PAGE 7 SCOPE OF THE STUDY CHAPTER 1 GENERAL INTRODUCTION Since ancient times, people throughout the world have had to cope with periodic floods and inundations of lands and communities. Notably pragmatic solutions to flood control were developed in various locations. so utions were dependent on human imagination and ingenuity applied through available resources to a broad variety of situations and conditions. Even for a relatively limited area such as the United States, we cannot imagine a singular methodology applicable in every situation. Effective flood control depends not only on resources and materials provided by 'Mother Nature but, also on man's attitude toward the application of these resources. The very definition of the problem depends on the attitude of the people and their understanding. Peninsular Florida is a unique system in many respects. In the early part of this century, flood control projects were begun in earnest over an area frequently exposed to devastating hurricanes and extensive flooding. Since World War II, tremendous changes, inherent in the rapid development of the state, brought about improved techniques and better understanding of flood control. Presently, the state of Florida faces simultaneous problems of excessive storm water and limited water supplies particularly in coastal communities. Rapid growth and the influx of new people also resulted in reduced water quality, thus adding another factor to water management and flood control. This project has been initiated with the idea of simultaneously improving flood control techniques and enriching groundwater resources in peninsular Florida. 1 PAGE 8 iJb0t.L. 11 V t.:) The investigators found it necessary to direct their efforts toward three principle objectives: (a) General management and development of conceptual design framework. (b) Development of a simplified analysis for the evaluation of seepage ponds in diverting stormwater to groundwater storage. (c) Development of approaches in the analysis of migration of contaminants in groundwater due to natural conditions as well as situations induced by artificial seepage. METHODOLOGY With respect to the listed bbjectives, Chapter 2 of this report concerns the development of the conceptual design framework as related to urban drainage systems in central and southern Florida. Chapter 3 of this report concerns management of water quantities. This effort in the investigation involved determining the ability of seepage ponds to divert collected stormwater to the groundwater aquifer. Emphasis was given to development of simplified methods that can easily be applied by drainage design personnel. Chapters 4 through 6 concern mathematical methods that can be used for the analysis flow conditions in an aquifer similar to the Floridan Aquifer. Basic models are suggested and a variety of mathematical methods are checked from the point of view of applicability, efficiency and accuracy. These chapters form an introduction to analyses of water quality problems that have yet to be resolved in Florida. PAGE 9 RESEARCH PROGRAM An initial research program was outlined by Dr. Sylvester Petryk in 1974 when he submitted the research proposal. The study, as envisioned in that proposal, would consist of the following five tasks 1) Review of existing literature 2) Modeling of precipitation, runoff, and storm water 3) Economic analysis and optimization of design 4) Design procedure 5) Experimental measurements. The research conducted during the past two years has been concerned with all of these areas. However, the program was modified with respect to the emphasis and the degree of effort devoted to each task. A review of the existing literature indicated that our efforts should be directed more toward improvement of the complete design framework and project management rather than involvement with particular design techniques or procedures. Chapter 2 of this report covers this part of our activity, which is partially related to each of tasks 3, 4, and 5 mentioned above. The design of seepage ponds in Florida often presents special problems because of high groundwater tables. These problems are the subject of Chapter 3 and are related to task 2 mentioned above. Problems of water quality have become increasingly important in Florida as well as in other parts of the country. One of our primary concerns in this regard was the effect that injection of impure water might have on the quality of Florida's groundwaters. Chapters 4 to 6 of this report are concerned with this problem. which is related to task 2 mentioned above. 3 PAGE 10 A series of field studies and measurements was conducted as suggested in task 5. Good results were obtained but the job was more or less routine in nature and we did not find it necessary to include them in this report. Our main objective in the field study program was to acquaint oursel ves with local protrlems and toadvise local peoplewith respect to these subjects. 4 PAGE 11 CHAPTER 2 FACTORS AFFECTING THE DESIGN OF URBAN DRAINAGE SYSTEMS IN PENINSULAR FLORIDA INTRODUCTION The demand for housing in Florida continues to climb as Americans seek a place in the sun. Retirement and tourism are rapidly displacing agriculture in developing and utilizing an attractive natural environment. Growth, since the end of World War II, has been phenomenal. Land development and home construction have become an integral part of the economy. These activities draw heavily on Florida's natural resources which are not limitless and, in many instances, are key elements in a sensitive environmental system. The demands of a progressive economy cannot be ignored, however, these demands can be adjusted to provide optimal use of a limited of supply. Concerted efforts are being made to provide both of a complex and dynamic system, and a reasonable balance of supply and demand. Water, one of Florida's most abundant resources, is a critical factor in this dynamic system. Early management concentrated on drainage and flood control to such an extent that damage to the system resulted. The past twenty years has seen a shift in management emphasis as water shortages and environmental damage became more apparent. Contemporary management practices are undergoing rapid change. The idea of water as a scarce commodity has prompted basinwide regulation of consumptive use and natural flow. Degradation of water quality has led to more stringent pollution control laws and the development of improved abatement techniques. 5 PAGE 12 All of these efforts have recognized and addressed urban development as a leading cause of problems in maintaining water quality and managing flow. Urban storm runoff has been found to be a source of water pollu tion that is of equal or greater magnitude when compared with any other identifiable source. Home building and_tanddevelopment require drainage systems that accelerate runoff, consequently, creating or aggravating flood prone situations. Homes, streets, parking lots and commercial build ings retard or prevent infiltration of rainwater and necessary recharge of groundwater aquifers. In view of the current activity in water management and pollution control, the drainage engineer must be aware of the many facets of con temporary design and he must apply sound and systematic methods in development of his design. The natural environment of Florida presents a unique set of circumstances. Laws, regulations, procedures and techniques, although rooted in historic precedent and established standards, have all been developed with some consideration for these circumstances. The drainage engineer should have a thorough knowledge, not only of applied techniques, but of social, environmental and political impacts of development. He should understand recent changes in legal and governmental philosophies. The engineer needs to know sources of information, plan requirements, and the procedural aspects of government regulation and approval. Most importantly, systematic method in design synthesis permits the engineer to organize and evaluate his data, identify and augment weak points and effectively manage the design process. The objective of the present study is to review and suggest a conceptual design framework as related to the following topics: 6 PAGE 13 1. Basic legal principles associated with drainage, groundwater, land use and water courses. 2. Highlights and purposes of legislative and regulatory efforts on the federal, state and local level. 3. Elements of the feasibility study as the first phase in the design process. 4. Considerations and techniques applied to preliminary and detailed design of drainage systems. 5. Integration of the various design factors by means of a flow diagram. Finally, this discussion is oriented toward design factors particular to peninsular Florida. LEGAL PRINCIPLES Surface runoff is of legal concern for a variety of reasons. Damage done by concentrated runoff, either as a floodwave passing downstream or a backwater flooding of upstream lands, may result in injury to the affected parties. Alteration of flow directions, capacities or other drainage characteristics can and often does lead to environmental damage, e.g., lowered water tables, water pollution, damage to vegetation, erosion and accretion. Property damage from flooding of homes and businesses may be the direct result of poorly managed runoff and improper land use. Two basic principles of law concerning disposal of surface waters are 'the civil law rule and 'the common enemy rule' Under the civil law rule, the upland or dominant owner has an easement on the downstream owner for passage of surface runoff in its natural manner. The common enemy rule stipulates that the servient or lower owner may take measures neces7 PAGE 14 sary to keep these waters off of hi s 1 and. Generally, Flori da courts have followed the civil law rule modified by 'reasonable use'. For example, the general rule regarding drainage into a natural watercourse states that a riparian owner may discharge surface runoff without regard to either the the_ rule'. right is subject to three limitations which have been imposed by the courts in varying degrees. These limitations include: 1. Drainage must be reasonable. 2. Drainage must not come from outside the natural basin. 3. The natural capacity of the stream must not be exceeded. The concept of reasonable use has been applied in cases involving both land use and water rights. In several instances) the courts have ruled that land use, adversely affecting surface or groundwater flows, incurs no liability on the owner as long as his use is reasonable and legitimate [2, 3J. Rulings have also been made in light of the riparian doctrine, requiring reasonably unimpaired or undiminished flow. In conclusion, the engineer must consider possible liabilities and legal consequences in his design of drainage works. Stormwatersystems designed to parallel natural discharges from a site are desirable from both an engineering and a legal viewpoint. Further. conflicts between water rights and legitimate land use have not been resolved in an entirely consistent manner. It is apparent that the body of law governing these activities and rights is not well defined and does not truly recognize the interrelation of land use and natural flow. 8 PAGE 15 FEDERAL. STATE AND LOCAL REGULATIONS Florida's rapid growth has produced heavy and often conflicting demands on existing water resources. The geology of Florida provides extensive groundwater sources that have been utilized in satisfying these demands. Wholesale acceleration and channelization of runoff and failure to provide adequate recharge to supplying aquifers combined with concentrated withdrawal from these aquifers has led, in several instances, to serious problems that cannot be resolved on a single situational basis. Total basin management has become the imperative solution. Comprehensive legislation at all levels of government has been enacted to provide a framework for management. Agencies with permitting and other regulatory powers have been created to affect solutions to apparent conflicts on a regional basis. The Federal Water Pollution Control Act of 1948 began what has become a massive effort to improve the quality of our national water resources. Subsequent'legislative efforts authorized federal assistance in research and development of methods of controlling pollution. Most recently, the F.W.P.C.A., Amendments of 1972 placed stronger emphasis on storm runoff as a source of pollution. Section 208 of this act is directed toward areawide planning and management of both 'point source' and 'nonpoint source' discharges. Stormwater runoff, a nonpoint source, is to be evaluated on the basis of extensive data collection and monitoring in the field. Methods for reducing pollutant loading are to be developed at the community level and, subsequently integrated into a basinwide management plan [lJ. Current technology in this area has not established a total 'cause and effect' relationship between runoff and stream quality. Loading rates 9 PAGE 16 are highly variable and depend on land use, storm duration and intensity, and seasonal weather patterns. This relationship or a reasonable approxi mation is singular in its importance with regard to effective modeling and comprehensive understanding of the problem. A second ar intel"est tothe drainage engineer stems from the National Flood Insurance Act of 1968 and the Flood Disaster Protection Act of 1973. Through this program, federally subsidized flood insurance is available to homes and businesses located in floodprone communities. A prerequisite to qualification requires that designated communities adopt restrictive land use ordinances for areas subject to flooding on a frequency of once in onehundred years (equivalent to a one percent chance in anyone year) [4J. The drainage engineer should recognize three significant points in these programs. First, the federal government has initiated comprehensive efforts to maintain and improve the quality of natural waters through strengthened programs at the state and local levels. Secondly, the flood insurance program is a substantial legislative effort directed toward better land use practices and flood control measures. Finally, these efforts actively involve the community and its citizens by promot ing improved local ordinances and related decision During 1975: Florida1s environmental agencies were reorganized and consolidated into two major agencies, the Department of Environmental Regulation and the Department of Natural Resources. Additionally, the state1s water management districts were redefined and were invested with broadened duties and powers. The Department of Environmental Regulation is responsible for enforcing pollution control laws and maintaining or 10 PAGE 17 improving water quality. Permitting of sewage treatment plants is a duty of this agency. In granting permits for point sources such as sewage treatment plants, the D.E.R. will review and actively consider stormwater control, treatment and disposal as part of the permit request [7J. Other highlights of agency guidelines include [7J: 1. Impact of stormwater discharges on the receiving waters will be viewed considering designated use of the waters, practical considerations and costeffectiveness. 2. Design of stormwater management systems should include a variety of techniques in reducing impact. Sod filters, vegetated buffers, sediment traps, primary considerations in design. 3. Detention basins should be considered as essential in residential, business, industrial and highway development. Interim drainage systems, serving construction sites, should be anticipated and included. The Florida Water Resources Act of 1972 empowers the Department of Natural Resources to accomplish the conservation, protection, management and control of the waters of the state. In effect, this act makes all waters in the state subject to regulation [6J. This act further directed the environmental agencies to formulate a statewide water use plan. Elements of the plan are being prepared at the district level and will be comprehensive in scope and objectives. Other activities of the Department of Natural Resources through the water management districts include permitting for consumptive use by all users, constructing and maintaining lands and works incident to management activities, permitting and regulation of wells. and management and storage of surface waters. 11 PAGE 18 It is readily apparent that the state is assuming the role of the riparian owner in many respects. Maintaining undiminished water quality and quantity is now a valid public concern as opposed to a strictly riparian right. In addition, the proposed water use plan will include recomrnendat fonsast6 flood pTain z6nfng and proleeli on lromfl 60ahazards. Thus, the state is assuming an ever increasing responsibility over land use management and water resources which, traditionally, have been inherent in private land ownership. Subdivision regulations and zoning ordinances form development regulations and standards applicable at the community level. Further, these regulations outline procedures in obtaining community approval for a proposed development. They provide an orderly exchange or basis of communication between the engineer and the community. Frequent discus sions during the approval process, enable the engineer and community planners and officials to effectively demonstrate their needs and desires. Individual citizens may express their opinions during public hearings before the local governing body as a matter of state law. Subdivision regulations vary considerably across the state. Community resources and extent of development tend to impose practical limits on regulatory programs and requirements. Obviously, the most sophisticated regulations available relatively ineffective without adequate staffing and community support. Federal and state efforts are aimed at supplement ing community programs and preventing abusive and costly development such as has occurred in the past. Development standards tend to fall into two broad categories. The first is concerned with conditions or programs unique to a community. 12 PAGE 19 Such factors as flood control, aquifer recharge and salt water intrusion may require regulations or procedures written specifically for these conditions. For example, Orange County has recognized the need for con serving natural recharge areas. Their regulations outline special measures for maintaining natural highinfiltration in these areas. "Typical methods include the use of recharge wells, bottomless inlets, perforated pipe, grading to retard runoff, artificial seepage basins, swales in street rightsofway, and utilization of natural percolation areas" [5J. The second category of regulations covers more detailed requirements. Minimum pipe size and material specification, street and lot grading, allowable overland flow distances, design methodology, allowable velocities in swales and ditches, drainage rightofway requirements, and storm water details are typical items included in this category. Serviceability and ease of maintenance are important considerations in these specifications relating primarily to the 'hardware' of drainage. Routine maintenance of drainage systems and streets is a major cost item in municipal budgets. Flooded streets and overgrown ditches are common complaints from tax payers and every effort is made to minimize such situations in writing these regulations. In summary, legal principles governing surface flow and groundwater are derived from rights associated with private ownership. Conflicts arising from alteration of natural flow patterns or from consumptive uses have been resolved reasonable use of land and water. It is apparent that these principles are not adequate to resolve conflicts considering contemporary demands. As a result, regional or basinwide management of water resources is being affected through combined legisla13 PAGE 20 tive efforts on all levels of government. Preemption of private water rights appears to be justified in that the burden of flood damages and optimization of consumptive uses rests with the entire community. Finally, lawmakers have broad powers in preserving and protecting natural resources i nthellubltc s icrtibn l1a s been enacte a to improve land use, relieve flood hazards and reduce pollution of natural waters. PROJECT FEASIBILITY Project initiation usually begins with a discussion including the client and key staff professionals. Subsequent efforts depend to a great extent on several factors. Among these are the intended scope of the project, potential market, capital availability, anticipated scheduling of planning, engineering, construction and sales, legal considerations and immediate technical needs. Certainly, forty acres in an established residential area will have significantly different needs than tenthousand acres of undeveloped and agricultural lands. The experience of the client or developer will also influence project initiation. An experienced developer often will have established project economics, scope and approximate timing prior to discussion with design professionals. Project feasibility may, however, include multidisciplinary studies and analyses of the previ mentioned factors. One of the most fruitful approaches to evaluation and design is organization and coordination of all related efforts. Project planning draws each factor into the design process as it becomes pertinent. A systematic approach to project management helps to ensure optimization of 14 PAGE 21 technical effort and production and, that the quality of the effort will be thoroughly professional. Drainage design as an engineering process, is well suited to an organized approach. Design synthesis and coordination of effort in drainage engineering is graphically demonstrated in the flow diagram as shown in the Appendix adapted from Woodson [8J. Flow diagrams are an accepted and proven management tool. Their inherent flexibility makes them adaptable to a variety of situations and design conditions. The included flow diagram, presented in the Appendix, is divided into three distinct phases paralleling the balance of this discussion. These phases include project feasibility, preliminary des"ign and detailed design. Intensive data collection is the first step following project initiation. Gathering information on existing drainage patterns and both current and future land' use follows several avenues. A thorough under standing of the existing drainage situation is essential not only in determining required improvements but, also in demonstrating the effectiveness of proposed improvements. Standard sources of information that will readily provide data on drainage, land use and flood potential should first be investigated. Among these sources are: 1. U. S. Geological Survey for topographic maps, flood studies, groundwater studies, geologic investigations, well surveys, stream and lake gage records and general hydrologic information. 2. Municipal or County Engineering and Planning for subdivision regulations, zoning ordinances, aerial mapping, established "drainage networks and other information directly related to the site. 15 PAGE 22 3. Soil Service for soil surveys, drainage techniques proven in the area, recommended best use of the site and extensive technical expertise in drainage and hydrology. guidelines in stormwater management. Specific information as to local environmental considerations and current practices. 5. Federal agencies for flood insurance and floodplain information, possible federal permits, design standards for qualification under federal loan programs and many other technical services. The reconnaissance survey of the site of a proposed development is particularly important in that information resulting from visual inspec tion is of primary importance and is generally available from no other source. Initial data may be summarized and plotted or noted on a site pl an to facil itate observati ons and compari sons duri ng the survey. Verification of land use, drainage patterns and improvements, both on the site and in the surrounding catchment, is an important objective. Other factors to be considered during I recon I are: 1. Attrar.tive or beneficial features on the*site that will contribute to aesthetics and environmental quality. 2. Hydraulic constrictions and control points, ponds, lakes, streams, springs and natural depressions collecting runoff. 16 PAGE 23 3. High water marks on trees and drainage structures, water surface elevations a groundwater or water table elevations. 4. Soil and areal distribution. Soil deposits such as ic muck and peat that must be considered in design and construction. receiving waters. Probable quality of current runoff and recei waters. 6. Identi cation of chronic drainage problems and existing improvements both on the site and in the immediate catch ment. During or following field investigation. the engineer should outline requirements for additional topographic surveys, soils investigations. stream gag ing, water quality environmental surveys and similar efforts. Methods and techniques for field investigations are dependent on site condi ons and the extent of proposed development. Detailed field notes and supplementary photographs are common to all surveys and provide documentation to support subsequent judgments, public presentations and technical discussions. Evaluation and incorporation of collected information involves simulation of existing drainage conditions, potential for delineation of drainage systems and land use planning. 'Rough' calculations or approximations should be made in evaluating the hydraulic and hydrologic characteristics of the sting system and the proposed improvements. The drai engineer should work very closely with those involved in land 17 PAGE 24 planning. land use, of course, will have multiple effects on any proposed dra i nage system. Compati bil ity of 1 and use and soil character; sti cs is of particular importance in Florida. Planners must be aware of probable water table elevations and seasonal fluctuations. Much of Florida cons;sts of fine sands withsmall percentagesof silt in the surficial layers. Often, a semipermeable layer of consolidated fines and organic material underlies surface sands. Infiltration of stormwater is severly retarded and 1 atera 1 or surfi ci a 1 flow may be very slow, parti cul arly on the marine terraces and in the 'piney flat woods' The drainage engineer should evaluate costeffective measures in reducing high water tables and flat gradients and, he should advise land planners of his findings. Land planning should be coordinated to reflect drainage methods and alternatives best suited for the site, required rightsofway, pond locations, flood elevations, fill sources and quantities, recharge sites and natural features to be preserved. Every effort should be made at this time to promote an aesthetically pleasing drainage network. An open ditch or floodway is patently obvious and prominent in appearance. The additional costs associated a landscaped and meandered channel are more than justified considering heightened sales potential and consumer idea 1 s. The preliminary engineering report or feasibility study reflects a combined effort in engineering, planning and project economics. Form and content for such a study are variable and. generally, depend on the scope of the project and desires of the client. It cannot be over emphasized that such a report is preliminary in nature and that subse quent efforts may contradict stated conclusions, cost projections and 18 PAGE 25 quantity estimates. Misunderstandings are frequent in this respect and the engineer should use care and discretion in making this point clear. Regulations, permitting and government coordination are often disregarded at this stage of the process. The feasibility study should include a thorough discussion of all aspects of the permitting and approval process. Preliminary engineering design provides the client with valuable information for project scheduling and funding. The feasibility report serves the developer in anticipating construction costs, cash flow, marketing and sales. F1nally, the engineer and client can selectively evaluate alternatives and determine preliminary design objectives. PRELIMINARY AND. DETAILED DESIGN Preliminary design should be approached considering overall site hydrology. Simulation techniques commonly used include the rational method, unit hydrographs, synthetic hydrographs, regression equations and more complex computer modeling techniques such as those developed by the Corps of Engineers and the Soil Conservation Service. The object of all of these methods is to systematically quantify runoff and to route these quantities through the system. Typically, the outfall point is identified together with available gage records or some estimation of appropriate water surface elevation. The flat gradients, so common to much of Florida, necessitate a thorough investigation and a reasonable approximation of downstream water surfaces corresponding to the events being simulated. Collector systems are usually a combination of open channels and storage basins that depend on 19 PAGE 26 both physical ground slopes and hydraulic head in maintaining positive flow. Central Floridais highlands may require grade stabilization structures where slopes are relatively steep. Peak velocities in unlined channels generally should not exceed two feet per second in Florida's sandy soils. Less frequent storm events maybe perm; tted to generate somewhat higher velocities. Storageelevation curves and stagedischarge relationships may be approximated in routing surface runoff through the system. Routing begins with generation of water quantities at the upstream end of the system and proceeds down the slope to the receiving waters. Peak discharges and water surface elevations, derived from the routing process, may then be used to calculate water surface profiles. Most systems generate peak flows over a relatively short period of time so that only peaks may be considered in calculating the profiles. Subcritical flow profiles must begin at the receiving waters and, based on the appropriate surface elevation, proceed Adjustments are made in channel geometry, storage, control structures and overland flow times until close agreement is achieved between the water surface profiles and the elevations generated by routing. This procedure begins in the preliminary design phase and is continually refined through the end of the design sequence. Many factors that are elemental to simulation will have considerable variation depending on the event being considered. Rainfall intensities, for example, will generate system responses markedly different for various events. It is advisable to simulate storms of several frequencies and durations even though regulations may specify design based on one or two standard events. Practical considerations, such as allowable design time. must be considered in managing this procedure. 20 PAGE 27 Internal drainage systems direct runoff to the collectors. The rational method is almost universally used to design local drainage such as swales, culverts, inlet size and location, street slopes and lot grad ing. Home building and related land development require simultaneous evaluation of earthwork and drainage. Generally, surface slopes may be adjusted to minimize drainage improvements. Cutting and filling required to achieve ideal slopes may, however, result in excessive costs in earth moving. Once again, several iterations of site engineering may be required to achieve optimal conditions in detailed design. Incorporation of detentionretention ponds in the drainage system is desirable to attenuate increased discharges resulting from development. Ponding is a recommended method in reducing pollutant loading. Common practice is to provide sufficient storage within the system for the volume of runoff resulting from a given storm event. These events or design rainfalls generally require enough storage so that runoff from high frequency events, such as afternoon thunderstorms, is totally retained or detained for an extended period. Silts and other suspended solids have a chance to settle, evaporation and infiltration effectively reduce the volume of stored runoff, biological action and dilution reduce the concentration of pollutants, filtering devices prevent discharge of surface borne contaminants such as oil and grease, and trash and debris are prevented from entering the receiving waters. Natural storage areas, including cypress ponds and mangrove swamps, may be utilized in developing detention. These areas have the advantage of acting as filters for runoff, thus improving discharge quality. Care must be used to maintain minimum water levels in these areas and to prevent 21 PAGE 28 damaging pollutants from entering such a system that utilizes natural vegetation. Other considerations necessary to retentiondetention ponds are: 1. Storage capacity recovery through evaporation; infiltration, outfJow,modifiedundeY'drainsand pumping. 2. Siltation or clogging in the pond. Twostage ponds should be provided where sediment loads will be high. Periodic cleanout of settling basins in the first stage will be required. A subsurface berm or equivalent structure should separate the settling basin from the second stage. 3. Maintenance, safety, vegetation, normal depth, fill needs, aesthetics, emergency overflow, and drawdown capability are among the additional considerations. Systems of interconnected lakes and ponds are frequently employed in Florida to provide fill material. aesthetic appeal and recreational benefits. In addition, extensive open systems tend to lower surrounding water tables in chronically wet areas. All drainage systems should be viewed considering the effects of the maximum probable storm. Surface drainage systems may be inundated in some areas cat1sing streets and low lying areas to act as open channels or ponding areas. Utility services may be interrupted. High water in the outfall system may result in flood conditions for an extended period. In this regard, auxiliary floodways may be desirable. road elevations should be sufficiently high to allow limited continuous access, floor elevations near points of concentration should be higher than arbitrary 22 PAGE 29 minimums and offsite structures and embankments should be evaluated as flood hazards. In effect, preliminary design involves evaluation of a selected development plan. Design efforts should reflect sufficient detail such that very limited alternatives remain together with final details neces sary to construction. The flow diagram,presented in the Appendix,outlines elements of this phase through client and agency approval. Presentation of the preliminary plan together with supporting data, becomes part of a comprehensive review by the client and staff. These plans may then be submitted to local agencies for preliminary approval and required zoning changes. Review and discussion with state and federal agencies should be conducted on an informal basis. Results of these interviews and evaluations should be summarized for incorporation into detailed design. The third and final phase, detailed design, completes this sequence. Construction plans, specifications and detailed cost estimates are primary objectives. Revisions tothe preliminary design should be solicited from staff, client, agencies and potential construction contractors. The design manager must provide a thorough check of every detail and should coordinate with other members of the staff in locating and resolving possible conflicts. Almost invariably, insufficient clearances between sanitary sewerage, storm mains and water supply will become evident and necessitate change. Experienced contractors will often point out plan discrepancies, quantity errors and unfavorable site conditions during bid preparation or contract negotiation. During this phase, the engineer should coordinate closely with permitting and approving agencies in providing supporting data for selected design features. Unnecessary and 23 PAGE 30 costly changes, based on arbitrary interpretation or judgmental differences, may be avoided through foresight and preparation. All parties should be informed of required revisions and all plans should be noted accordingly. CONCLUSIONS Drainage design has become considerably more than estimating runoff and selecting culverts. The design professional is obliged to assume a broad viewpoint in developing his program and directing design efforts. In summary of this discussion, the following conclusions are presented: 1. The engineer should be familiar with basic legal principles related to drainage and how they have been applied in resolving conflicts and water rights litigation. 2. The engineer must be. aware of rapid changes taking place with regard to water rights and resQurcemanagement. Federal and state legislation in this area has shifted rights and responsibilities away from the landowner and toward the public entity. 3. Local regulations and ordinances are being rewrittenor supplemented in order to effect better land use. practices and flood protection measures. 4. efforts are being made to develop and implement improved pollution abatement techniques. Urban runoff, as a prime source, may be improved through a variety of techniques presently available to the designer. 5. Legislative mandates have been very direct. The engineering profession should take precautions in protecting their design 24 PAGE 31 prerogatives and, at the same time, align their efforts with these mandates. 6. The feasibility study is, possibly, the most important phase in design synthesis. This study develops a set of conditions influencing the entire project. 7. Florida's environment and dependence on well managed water resources emphasizes the efforts of the drainage engineer. He must actively participate in several areas of project development. 8. Detentionretention systems have become a necessity in designing improvements. Ponding or storage improves water quality, reduces flood potential and provides increased aquifer recharge. 9. Management of the design process and attainment of technical efficiency may be aided through the use of a flow diagram presented in the Appendix. Considerable flexibility allows this technique to be adapted to most projects. Finally, the attitude of the engineer must be receptive to change and inno vation. Bending to accomodate is not the answer. Embracing and assertive leadership is a trademark of the professional. 25 PAGE 32 APPENDIX FLOW DIAGRAM FOR URBAN DRAINAGE DESIGN Phase I: Project Feasibility Client Desires Staff Discussion Project Scope Project Economics Time Schedule Is there sufficient information to proceed? Existing Land Use Soil Survey Topographic Surveys Historical Records Interviews Future Land Use Studies Laws and Regulations Reconnaissance Survey Project Initiation Outline First Efforts Yes Data Collection Summarize and Catalog Data 26 No PAGE 33 Are preliminary design objectives consistent and reasonable? Development of Internal Drainage System Development of Collector and Outfall System Identification of Critical Points Development of Hydrologic Model Development of Hydraulic Model System Evaluation Through Models Identification of Limited Alternatives Water Quality Evaluation 27 Establish Preliminary Design Objectives __ No Yes Phase I Phase II: Preliminary Design Development of Preliminary Design Prepare Preliminary Design Plans PAGE 34 Have all sources been tried? Delineation of Existing Flow Patterns Identification of Constraints Simulation of Existing Drainage Conditions Proposed Drainage Schemes Proposed Land Use Are proposed land use and drainage plans workable? Cost Projections Quantity Estimates Anticipated Permits and Approvals Staff Review and Coordination Owner Review and Approval t >..... No Yes Evaluation and Incorporation Develop Alternative Site Plans 28 >...... No Yes Presentation of Feasibility Study PAGE 35 Have preliminary design objectives been realized? Design Features Limited Alternatives Special Problems and Solutions Cost Estimate Ad9itional Data Needs Are final design features and objectives approved by client and agencies? Incorporation of Testing Results and Survey Data Safety Considerations Cost Optimization Coordination of Design and Drafting ">... No Yes Presentation of Preliminary Design Obtain Client and Agency Approval >1 ..... No Yes t Phase II Complete Phase III: Detailed Design 29 PAGE 36 Design and Specification of Special Features Preparation of Detailed Cost Estimate Preparation of General Spec ifi cations . Checks and Revisions of Calculations and Drawings Project Plans, Estimates and Specifications Review Development of Detailed Design Issue Plans Reflecting Final Design I s the fi na 1 des i gn complete ..... and acceptable to the client? Permit Applications and Procedures Local Governmental Approval Contractor or Bidder Review Final Staff and Client Review 30 Yes Revision of Detailed Design Obtain Final Approval by All Parties i I I PAGE 37 Are plans, specifications, estimates, and permits complete? Are approvals in receipt? >No Yes End Design Sequence Phase III Complete SymbOl s Key Flow Direction: Specific Activity or Information: General Activity: Executive Action: Decision: 31 PAGE 38 REFERENCES 1. "Executive Summary of Section 208 Program for Designated Area Federal Water Pollution Control Act Amendments of 197211, United States Environmental Protection Agency, Washington, D. C., 1974. 2. Linsley. R. and Franzin;, J., Water Resources Engineering, Second Edition,nMcGrawHilL, New York, 1964. _. _._. 3. Maloney, F., Plager, S., and Baldwin, F., Jr., Water Law and Administration, The Florida Experience, University of Florida Press, Gainesville, 1968, 4. IINational Flood Insurance Program", United States Department of Housing and Urban Development, Washington, D. C., 1974. 5. "Orange County Subdivision Regulations", Orange County, Orlando, 1974. 6. Tebeau, C., IISouth Florida Water Management" and IIEnvironments of South Florida: Past and Presentll, Miami Geological Society, Miami, 1974. 7. Thabaraj, G., "Regulatory Aspects of Storm Runoff Control", Proceedings of the StormWater Management Workshop, Florida Technological University, Orlando, February, 1975. 8. Woodson, T., Introduction to Engineering Design, McGrawHill, New York, 1966. 32 PAGE 39 CHAPTER 3 ANALYSIS OF TRANSIENT GROUNDWATER FLOW FROM SEEPAGE PONDS INTRODUCTION The ability of a seepage pond to divert storm runoff into the ground water aquifer depends on two processes: (a) the rate of seepage from the pond, and (b) the reaction of the groundwater table. transient phenomenon. The inflow to the pond. specified as a runoff hydro graph, causes variations in the depth of ponded water. Outflow is consid ered here to be entirely by infiltration, which varies not only because of changes in pond depth but also due to variations in such soil properties as hydraulic conductivity and storativity. The growth and decay of the groundwater mound in the underlying aquifer is also time dependent. These facts are well known among scientists but are usually not taken into consideration in the design of a seepage pond. The objectives of this study are to develop methods by which the designer will be able to estimate seepage pond effectiveness by considering both the rate of seepage from the pond and the variation in the groundwater table while taking into account the transient nature of the problem. CALCULATION OF UNSTEADY SEEPAGE FROM A POND The adequacy of a seepage pond is evaluated by a storage routing procedure, which is basically an account of the inflow. outflow, and change of stored volume over successive discrete time increments. The inflow is represented by the runoff hydrograph for the design storm on the area to be drained. For the purposes of the present exposition it 33 PAGE 40 will be considered a given function of time. The outflow, on the other hand, is dependent on time, depth of ponding, and the properties of the soi 1. Consider an initially dry seepage pond constructed in unsaturated soi1. As it fills, the water begins to escape from it by vertical unsaturated flow, or infiltration. The most convenient way to describe this flow is by the formula of Green and Ampt [1911J as modified by Bouwer [1969J. It has been shown that this which was long thought to be purely empirical, is soundly based on physical principles [MorelSeytoux & Khanji, 1974J and gives very good answers [Whistler and Bouwer, 1970J. Green and Ampt based their derivation on a simplified model of infiltration which treats the soil as a bundle of vertical capillary tubes. The vertical hydraulic conductivity and moisture content of the unsatur ated flow are considered constant, as is the capillary suction potential of the advanc i ng wett i ng front. App lyi ng Darcy I slaw to th is idea 1 i zed I flow, the infiltration rate is described as where W ::: K H + L + 1/In t L w = infiltration rate K t = hyarau1ic conductivity of the transmission zone H = depth of ponded water 1/In = capillary suction potential l = depth of penetration of the wetting front. 34 ( 1) PAGE 41 The rate of advance of the wetting front is (2) where t = time f = the volumetric fraction of fillable pore space. The equation of continuity applied to the pond yields dv = AdH = I fddt{AfL) dt dt (3) where v = stored volume in the pond I = inflow A = area of pond surface Af = area involved in infiltration. Introducing equation 3 into 1 and 2 we obtain, after differentiation, the following nonlinear differential equation d 2 2 dL dAf dt2(L /2) + (fAf/Al)dt + fL/A dt I/A = 0 (4) which can be solved numerically. However, in most practical cases the rate of change of H is about an order of magnitude smaller than the rate of change of L. In such cases an integration of equation 2 between ti and ti+l provides a simple discrete presentation of the problem, as follows t'+l J 1 t, 1 35 PAGE 42 The integration leads to (5) In dimensionless form this can be expressed as f1tKt AL r'+l L'+l/L + r+1/L. __ = _0. w( 1 1 1 "2 1). fL L l 1 + r. +1 / L 1 1 1 1 "2 1 (6) where r '+1 = 1 "2 H "+1 + 1 "2 l/In L. = value of l at time t. 1 1 li+l = value of L at time ti+l f1t = ti+l t. 1 f1l = li+l l. 1 H '+1 = 1 "2 (H. 1 + Hi+l)' the mean value of H. A chart that can be applied for engineering design purposes is shown in Figure 1. The curves are based on the solution to equation 6. The application of the approach of equation 6 was checked against experimental data reported by Weaver and Kuthy [1975J. In this experiment a seepage pond was constructed and filled with water at a controlled rate. The variation of the pond volume and area vs. depth is shown in Figure 2. The experimenters listed soil test data which led to the follow;n9 values of soil parameters. K t = 1. 2 ft/hr f = 0.2 l/In = 0.5 ft. According to the reported measurements, the rate of change of H was 5 to 10 times smaller than the rate of change of l. This would indicate 36 PAGE 43 L'lL c:1 w ....J 0.5 0.1 0.05 T 1 I 1 11111 I I I T F III I I I J I [Il ++ I l r 11l I11I1 I I11I1 I I III TIV J I II I iIII I 1 I I 111111 j= I 1111111 1 1 J#f118 I I I II E r=0.5 _L, I r=0.2 LJ r=O.l Milli' :1 /eWf1JW' I l..k:=::::' l,_ A/17" +HI I I /1 d' I rtD III' l I R=R iii I I fffii I III111 ld 0.01 Ir=o.o) A' IA.a2' .1 1L1N +r=O.02 I' f 0.005 I I I I iHF::fzbi711 + ill I I III H 0.001 I r =0. 0 1 I r=0.005 0.001 0.005 0.001 0.005 0.01 (L'ltK; )/(fL;) 0.05 'I I I I I q.l Figure 1. Solutions to Equation 6 for li+l/l; ::: [1.2 : I ttl I II I r 0.5 1.0 PAGE 44 w co 6 5 +> 4 4:r: le... 3 a z: o e... 2 1 o AV13RAGE..AREA (thousand square feet) o 1 2 3 4 5 6 7 I I I \ II I I I I I I I I / VI I I ./' I I I I i I It/ /1 i I i I I I i I I I I I I I I i I I 1/ I I i / Volume I I I I I I i IYi / I I I I \/ I I I I I I / V i Ifi Area /! I / t I V/ i I I I 1 /1 I i I I V I I I VI I I I I I i I IIV'! i I I I I I r' I I I J l I I i I o 5 10 15 20 25 30 35 POND VOLUME (thousand cubic feet) I Figure 2. Depth vs. Average Area and Volume for Test Pond 8 Y I : I I. I I I I I I i I [ : I I I 40 PAGE 45 that the method utilizing equation 6 should give good results. The comparison between the experimental data and the theoretical prediction is shown in Figure 3. The agreement is, indeed, quite good. CALCULATION OF THE RESPONSE OF THE GROUNDWATER TABLE The reaction of the water table aquifer to vertical infiltration from a seepage pond is characterized by the growth of a groundwater mound. For analysis we consider a circular pond constructed in homogeneous and isotropic soil. The aquifer is underlaid by a horizontal impervious layer and initially has a horizontal free surface. When the wetting front of vertical infiltration reaches this free surface the mound begins to form. From this time the infiltratiori is assumed to continue at a constant rate. 39 PAGE 46 100 90 ,:t80 +l ClJ ClJ 70 4U ..... 60 ..c ::s u c 50 C rtI III ::s 0 40 ..s::: +l .j:::> 3 30 a a .....J I.L. z: 20 ...... 10 0 V Ot ::el'ved Je ptr, 0 / C31 cul a :.eJ nfl 0 .... 1 ..... V / .. i v tJ ( ( V 0.3/ '" ,.J / 0'( ;V/ V D r V 0 1', 1 9 10 11 12 13 14 15 2 4 5 6 7 8 3 10 9 8 7 ... 6 +l ClJ ClJ 4........ 5 :t: Ia.. I.J.J 4 0 3 2 1 0 TIME (hours) Figure 3. Comparison of Experimental and Calculated Storage Routing I PAGE 47 In a cylindrical coordinate system centered on the seepage pond as shown in Figure 4 the axisymmetric flow in the saturated mound is described by where nr ar + w n Ks = saturated hydraulic conductivity n = specific yield w = rate of vertical infiltration. (7) Both wand n are functions of radial distance, r. The value of specific yield is greatly reduced in the zone of infiltration under the pond. Based on the experimental evidence of Bodman and Coleman [1943] it seems reasonable to assign a specific yield to the'zone of infiltration with a value of 20% of the specific yield elsewhere. The rate of infiltration, w, is zero when r is greater than the pond radius, Ro' Noting that h = S + a, equation 7 can be rearranged: .' '" Ks (s + a) a2S + Ks (aSJ 2 + f.s + a) as + w at n arzn ar] rn l ar n (8) To make the equation dimensionless, introduce the following dimension less variables: Substitution of these variables along with some manipulation and omitting the primes we obtain: as a2 (2 ) n"IT arzS /2 41 (9) PAGE 48 s ,t h a r Figure 4. Groundwater Mound PAGE 49 In this form, the nonlinear terms are distinctly separated from the linear terms. They are treated differently in the numerical analysis. 2 = = 0 at r = 0 ar ar 2 (10) and S = 0 at r = 00 The initial condition is: S = 0 at t = 0 (11) To solve the problem by finite differences, we must set up an appropriate space time grid. The time increments are designated by ;IS and the space increments by j1s. The time derivative of S is approximated by a forward difference. (12 ) The nonl inear spatial derivatives are represented by central difference operators: a ar (S2/2) '" [S2,. J.+l 2S2 .. + S2. +11/2!J.r2 ,J J (13) (14 ) The linear space derivatives are the most influential terms in the equation. If the solution were to become numerically unstable, it would 43 PAGE 50 probably be because of these terms. Therefore, they are approximated by the mean of the finite difference representations on the (1 + l)th and the (i)th time rows. This;s the implicit method of solution developed by Crank and Nicolson [1947J which has good stability and convergence characteristics: '" ((Si+l,j+l Si+l,j_l)/(2Llr) + (S;,j+l S;,j_,)/(2Llr)] a2S [ 3rT '" 2 S;+l,j+l 2S;+1,j + S;+1,j_l)/Llr2 + (S. '+1 2S .. + S .. 1 )/Llr2 ) 1 ,.1 1 ,J 1 ,JWhen these finite difference approximations are substituted into equation 9, a tridiagonal system of equations is generated which can be solved by Gaussian elimination. (15 ) (16 ) The computer program developed from these finite difference operators was tested with several combinations of grid mesh ratio and grid size. 2 It was found to give stable results with a ratio of Llt/Llr 0.11. The mesh size chosen was Llr = 0.03. To simulate the boundary condition at infinity, the calculations were carried out to r = 25 at which point S was required to equal zero always. For comparison with the linearized analytical solution of Hantush [1967], the finite difference program was run with constant specific yield, n. The results of this run are shown by the dashed lines in 44 PAGE 51 Figure 5. They are practically identical to the results obtained by the method of Hantush. When n was allowed to vary as a function of r, the results were very much different, as shown by the solid lines in Figure 5. DISCUSSION AND CONCLUSIONS An analysis of the hydraulic operation of a seepage pond should be conducted in two phases. The first phase is concerned with the rate at which water seeps out of the pond by vertical infiltration through its bottom. The equation for vertical unsaturated flow developed by Green and Ampt [1911J can be used to describe the outflow in a storage routing process. The storage routing is simple enough to be done by hand when the soil is assumed to be homogeneous .. The Green and Ampt equation can also be adapted for use in soil where the hydraulic conductivity varies monotonically with depth [Bouwer, 1969J which is a very common occurrence. This calculation is about as simple as the storage routing procedure but the combination of variable ponded depth and variable hydraulic conductivity would be cumbersome enough to make the use of a computer or a programmable calculator desirable. The second phase of the analysis deals with the response of the water table to recharge from the seepage pond. The determination of adequate capacity of the aquifer requires use of the design charts for each specific case. The design chart itself is based on the numerical solution of the nonlinear differential equation. The importance of the finite difference solution of the groundwater mound is that it makes possible the consideration of an axisymmetric variation of the specific yield. It would also be possible to include the effects of axisymmetric variation in other soil properties. 45 PAGE 52 + Vl 1 14 P 0 3 I II ,p 0 3 j" .. ,. l,T"II .. ij"I''I"'I,"lTl .. .. I ............. ......... ........ :::: :: : .:.. =. :::'r::r: . ....... '"T''' "l",r::I":r"T"r:r:1 < >:: :<:::::' ::>< :: :::::::::::::: '.: :: :::: :::: : :::: :: :: .. ::. III .. .... .. ...... ... 1 .......... ................ .. 1.1211, ;Vl,.,> ... !iP .. .. t ...... .. .... l.. vJ R . ... .j .... ./V.. .. ::j:. .... ", ........... .. :::+::: :::: :::: ::::,:::. p = ;: j' :::;'::.::<::" .::. :::.. .. : 1: V:: ::: :;::1:::: :::: :::: :::: :::::::. .... .......... I .. Ka .... I .. j./ ..... ............... .. ...... .. :::: :::: :::: :::: :::::::: s : .. ::::i ,':: :::: :::: ::.: ......... j ..................... .. .. ++ ++1 I 1 1 1 1 1 1 1 1, : 1 .... ( ,, .. tI,II, ...... .... ...... .. .. 1;, .. 1 ......... 1 = C3 ...... L.U ::I: ....... z ::::> en 0 ::E: Vl Vl L.U ....J Z 0 .... Vl Z UJ ::E: ...... ... /. :41"ll:::::n p= 1. 08 : : .!:: : ",A. J .. 1.:, I.. .. ". ..................... 1 .. 1 ...... .......... .. .j ............ .... ......... .. 1 ......... ......... 0... I ,.j.' .... I .... t ... li ... ........ "I .... '.... .. .. 1 :: .... :: i" ... \ .......... .. I .. 'j .. .. I .. ,. r! ... .". .. ...................... .. ......... .: .... :. : :::: To. :::: :::: ::J::: :". :::: :::: :::::.. :1,<: : ':l::: ::':tj : :::: : .. :,::,.':. :H:" re :::: ::.: :J::.f:: 1.04 ... : ;: ::::1.1': :::1:': j ,. ,I. M' .... I" ..... t .... I ..... i . Cpn$tant n 1 02 .. Vbdable n .02 .05 .1 .2 .5 2 I I 5 10 DIMENSIONLESS TIME, tl Figure 5. Mound Height at rl = O,CQmparison of Constant .and Variable: ni PAGE 53 It should be noted that the saturated hydraulic conductivities referred to in the two phases of the analysis are, very often, not the same. Infiltration is concerned with vertical flow while the groundwater mound is mainly influenced by horizontal flow. The difference between the saturated hydraulic conductivities in the two directions usually stems from horizontal layering of the soil. NOTATION a h H I L n p r r' S' t initial thickness of aquifer [LJ area [L 2 J area of infiltration [L2J volumetric fraction of fillable pore space thickness of aquifer [LJ depth of ponded water [LJ inflow to pond [L3J hydraulic conductivity of saturated soil [L/TJ hydraulic conductivity of the transmission zone in unsaturated flow [LIT] depth of penetration of the wetting front [LJ specific yield .dimensionless recharge intensity, radial distance [LJ dimensionless radial distance, r/Ro radius of seepage pond [LJ rise of groundwater mound above water table [LJ dimensionless mound height, S/a time [TJ 47 PAGE 54 t' v w r dimensionless time, stored volume in pond [L3 ] infiltration rate [LIT] 1/In + H [L] cap;ll QDpoten{iiiliifflelcfcaJ1acity[L] 48 PAGE 55 REFERENCES Bodman, G. B. and Coleman, E. A., IIMoisture and Energy Conditions During Downward Entry of Water into Soilsll, Soil Science Society of America Proceedings, Vol. 8, 1943, pp. 116122. Bouwer, H., IIInfiltration of Water into Nonuniform Soil II Proceedings of the American Society of Civil Engineers, Irrigation and Drainage Division, Vol. 95, No. IR4, pp. 451462, 1969. Crank, J. and Nicolson, P., IIA Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type II Proceedings Cambridge Philosophical Society, Vol. 43, 1947, pp. 5067 Green, W. H. and Ampt, G. A., "Studies on Soil PhYSics. I. The Flow of Air and Water Through Soils", Journal Agricultural Science, Vol. 4, 1911 pp. 124. Hantush, M. S., I'Growth and Decay of GroundwaterMounds in Response to Uniform Perco1ationll, Water Resources Research, Vol. 3, No.1, 1967 pp. 227234. MorelSeytoux, W. J. and Khanji, J., IIDerivation of an Equation of Infiltration, Water Resources Research, Vol. 10, No.4, 1974, pp. 795800. Weaver, R. J. and Kuthy, R. A., Field Evaluation of a Recharge Basin, New York State Department of Transportation, Engineering Research and Development Bureau, Research Report 26, 1975. Whistler, F. D. and Bouwer, H., "Comparison of Methods for Calculating Vertical Drainage and Infiltration for Soils", Journal of Hydrology, Vol. 10, No.1, 1970, pp. 119. 49 PAGE 56 CHAPTER 4 THERMAL CONVECTION IN A CAVERNOUS AQUIFER INTRODUCTION the author ana lyzed i nstabil itycri terla related to onset of thermoha 1 i ne convection in an aquifer whose properties are similar to those of the Boulder Zone of the Floridan Aquifer. Such an aquifer is characterized by extremely large pore size and transmissivity leading to very intensive solute and heat dispersion as well as to invalidity of the laminar Darcy law even when flow velocities are extremely small. In I it was found that for moderate Reynolds numbers the doubly diffusive convection can be approximated by the singly diffusive convection, for in such cases mechanical dispersion is larger than molecular solute and heat diffusion. The objective of this study is to analyze transport phenomena in the cavernous aquifer subjected to singly diffusive convection. BASIC EQUATIONS The analysis is related to a flow field similar to the Boulder Zone (the deep regions) of the Floridan Aquifer. We assume that density gradients are induced by a single component only, referred to as temper ature. In the cavernous strata, turbulent effects, as well as mechanical heat dispersion, are induced by even extremely slow fluid motions. In I the basic equations related to the calculation and approximated by the Boussinesq approach were presentedas follows: 50 PAGE 57 where au. 1 0 ax. , + pgni + (1 + b)ui = 0 1 aT lL.. = _a_ (E. lL..) + u,. a a. aX. J a J Ui = velocity vector p = pressure p = fluid density K = permeability = porosity Jl = vi scos ity T = temperature a = coefficient of thermal expansion po,TO = density and temperature of reference Y = a coefficient defined by pC + p s C s(l (1) (2) (3) (4) (5) Through a brief literature survey presented in I it was suggested that the friction function, b, can be approximated by: b = 0.014 Re (6) In an isotropic medium the dispersion tensor, E;j' can be expressed as a sum of the isotropic molecular diffusivity and the second order 51 PAGE 58 symmetric mechanical dispersion tensor, as in E = KO + lJ lJ lJ where Here subscriptst and refer to transversal and longitudinal components respectively. (7) (8) An assumption of singly diffusive convection is justified for a mUlticomponent system for moderate Reynolds numbers. A model suitable for the description of the mechanical dispersion tensor in such cases was suggested by Saffman [1959J. According to this model Ei = \I E* \I s + 2 (Re) 2(s + 1) (s + 3) $ (s + 1)2 Re 2(1 s)(s + 2)(s + 3) ($) (9) (10) where s is a power coefficient describing the dependence between the velocity and the pressure drop (s varies between unity, for laminar flow, and half, for turbulent flow). The flow field model considered in this study while unperturbed conditions prevail, is described in Figure This is a saturated porous layer of infinite horizontal extent bounded by two impermeable planes located at bottom and top of the aquifer. Temperatures on bottom and top of the porous layer are To and respectively. Through the 52 PAGE 59 A '7 7/ // Uo TEMPERATURE PROFILE IMPERMEABLE Figure 1. Schematical description of unperturbed conditions 53 PAGE 60 porous layer the fluid flows uniformly in the longitudinal, x, direction. y and z are transversal and vertical coordinates respectively. The flow field variables can be nondimensionalized as follows: TI= (T Ta)/ T_ UI = Ud/E t E!. = E .jEt 1 J 1 J (p p gz)K xu PI 0 + 0 (11) + b) where ti is a unit vector in the x direction. Substituting the dimensionless variables of (11) in (1)(3) and omitting the primes we obtain RTn. + (1 + s)u. = 0 aXi 1 1 3.T + _a_ (E rat ui ax. ax. ij ax. 1 1 J Here R is the Rayleigh number defined by The power s in (9) and (10) according to I can be calculated through (12) (13) .( 14) s = tn(Re) ( ) tn(1 + b) + tn(Re) 15 However, (15) can be utilized only when Re > 2.77 if b = 0.014 Re. 54 PAGE 61 For unperturbed conditions (12) and (13) yield u. = 0 1 8 = 0 T = z 2 P = P Rz /2 o LINEAR STABILITY ANALYSIS E.;(i;' j} = 0 lJ (16 ) (I7) (18) (19) (20) Stability criteria of the flow field are determined through the linear stability analysis. The flow field is subjected to small disturbances in the velocity (u, v, w), temperature (6), dispersion tensor (E .. ), friction function (8), friction power coefficient (s) and lJ pressure. Disturbances are very small and through the linear stability analysis second order terms are negligible. For the linear stability analysis the module of the velocity vector in the perturbed flow field is approximated by (21) Substituting (21) in (15), (9) and (10) we obtain expressions for the principal components of the dispersion tensor. By applying (8), all components of the dispersion tensor in the perturbed flow field are obtained. 55 PAGE 62 It is convenient to express the velocity components by utilizing a scalar function Q as follows (22) where (23) Introducing flow field perturbations in (12) and (13), neglecting second order terms and eliminating the pressure perturbation we obtain where 02 02 02 02 v =2+2+2 ax ay az E* t c2 = Ud o (24) (25 ) (26) Assuming vanishing values of 6 and Q on top and bottom of the aquifer, these disturbances can be expanded by the following normal modes (6,Q) = (61,Q1) sin (nz)exp [i(a x + a.y) + (0 + iw)tJ (27) x yr where 61 and Ql' are constants, ax and ay are the wave number components. For point of stability (or = 0) substitution of (27) in (24) and (25) yields the following secular equation 56 PAGE 63 2 + 2 R I a 1T 2 ( 2 2 2 + 1T2 + a lax cIa C21T ) xa iYWI= a (28) where x = + 1 (29) (a/ay)2 + 1 The minimal value of R satisfying (28) is the critical Rayleigh number. Therefore, (28) yields the following criteria of point of instabil ity x = 1 a = 1T o a = a a = a x y w = a R = 4i o (30) Hence convection cells are two dimensional rolls whose axes are parallel to the unperturbed velocity vector. Superposition of the unperturbed velocity and the convection velocity leads to a helical flow field. Convection currents conducted in two dimensional rolls were also obtained for the ordinary Benard problem [Malkus and Veronis, 1958; Schluter et a1., 1965J as well as for free convection in a porous layer while mechanical dispersion effects are negligible [Straus, 1974J. However, in those cases only nonlinear stability analysis associated with stability analysis of the steady convection motion yield such a result,. whereas, in our study the linear stability analysis indicated that phenomenon. In our case, calculations concerning the anisotropy of the dispersion led to the conclusion that convection cells should be two dimensional rolls. 57 PAGE 64 Convection sets out in planes where the effective Rayleigh number attains maximal values, namely. where the coefficients of hydrodynamic dispersion attain minimal values. Inertial effects associated with the invalidity of the laminar Darcy law introduce the friction function, D. in the expression for R but do not affect the linear stability analysis and predictions. FINITE AMPLITUDE DISTURBANCES AND NONLINEAR STABILITY ANALYSIS The effect of the convection motion on transport processes through the aquifer can be predicted through the solution of the nonlinear equations of motion and heat transport related to supercritical conditions. The convection motion is two dimensional, therefore, the velocity components can be expressed by the stream function as follows v = 8z w 8y (31) Substituting the finite amplitude disturbances in (12). (13) and eliminating the pressure perturbation we obtain (32) (33) where Band H are the friction and the heat advection spectra, D and F are two parts of the heat dispersion spectrum, all of which are of the form = (6 1 1 (34) 58 PAGE 65 = a ( ae F(E .. ,e) = ,,E .. ,,) lJ aX. lJ aX. 1 J (35) (36) (37) Here sand Eij are finite amplitude disturbances in the friction functi on and the di spers i on tensor' r'espec ti vely. As long as the convection velocity is smaller than the unperturbed velocity the absolute value of the velocity in the flow field can be approximated by (38) where (39) (40) Applying (40), (6) and (15) we obtain s = [b/(1 + b)JA (41) = (1 + + b)J S (1 + + b)ReJ SA (42) Introducing (42) in (9), (10) and the dimensionless form of the dispersion tensor we obtain after minor approximations 59 PAGE 66 E2 EQ, E* 1 + t {(1)(1 + A) { [1 5 (UOd)2 Et Et 12 5 E* 2s + 5 E* + E! (3 5)J Q, I)} (1 A)} u u (5 + 2)(5 + 3) (E* t t J (43) As long as the Rayleigh number is not very high, A is very small and can be approximated by the first term of (40), leading to the following approximations (44) (45) (46 ) where b E (31 = ( t )2 2(1 + b) ua 0 2 sl = Et 5 2(u od)2 A EtEt 1 1 1 E:t 0,1 = .. + r; (s + 2 s + 1 s + 3) Et 2(u od)2 1 ( 47) E 2 ERA t 0,2 = 2 (1) (uod) Et 60 PAGE 67 If (32) and (33) are subject to the following homogeneous boundary conditions = 0 at z = 0,1 (48) then the system (32) and (33) can be solved by means of a set of truncated eigenfunctions expressing the finite amplitude disturbances. Double Fourier series expansions may conveniently be applied for such purposes as follows 00 A ,1, I: 'l' 'f' p,q=l p,q sin(pay) e 0p ,q cos(pay) q=l (49) (50) The calculation can be simplified by using the complex variable presentation of sin(pay) and cos(pay) leading to = i 'l'p.,q e ipay p=_oo q=l 00 e = I: p=_oo 8 e ipay p,q provided that 'l' p,q 1 0/ P.q = 2' p,q 8 =8 =18 p,q p,q 2 p,q 61 (51) (52) (53) (54) PAGE 68 The solution of the system (32) and (33) through the Fourier series expansion means the determination of the Fourier series coefficients which can be done by power series expansion. Such a method was first used by Kuo [1961J for the analysis of the ordinary Benard convection. Palm et al. RUbiO[1975 j through approach for the analysis of free convection with no dispersion and turbulence in porous media. Rubin and Christensen [1975J suggested some guidelines for the utilization of such an approach when analyzing instabilities induced by salinity gradients in a saturated porous layer. The advantage of this method lies in its simplicity. According to (30) the inception of convection motion is of the marginal instability of exchange, namely, steady convection follows point of instability. Therefore, the first term in (33) vanishes when steady convection is attained. In such conditions the toefficients 'p,q and 8 p ,q of the Fourier series expansion can be expressed through a power series expansion as follows: N (, ,8 )= p,q p,q n=l (,(n) 8(n)) n p,q' p,q n where n is a small parameter defined by (55) (56) The Rayleigh number is also expanded by a finite power series as follows: S 2' R = R + R n J o os J= (57) 62 PAGE 69 where S = N/2 (58) Other flow field perturbations like v, w, a, andcEij as well as perturbation spectra H, B, 0, and F can be also expanded in double Fourier (n) (n) (n) and power series. Expressions for the coefflclents Hp,q' Bp,q' Dp,q and F(n) are presented in the Appendix. p,q If the analysis is conducted for N = 1, it reduces to the linear stability analysis, yielding critical values of wave number and Rayleigh number as given in (30). There is a pos i ti ve rel a ti onshi p between increases in Rayl ei gh numbers and wave numbers. However, taking the assumption that under supercritical conditions the wave number remai ns constant does not significantly affect predictions of transport phenomena for quite a wide range of Rayleigh numbers [Straus, 1974]. Such an assumption is not required by the method used here but considerably simplifies the analysis. Substituting the series expansions in (32) and (33) we obtain R e(n) + R e(n2i) + + B(n) = 0 O P1f p.,q os II m p q Tp q P q i =1 (59) (60) The functions and e are generated by superpositions of trigonometric functions. For n = 1 the only nonvanishing coefficients Therefore, only coefficients and with even values of lpl+q nonvanishing values. Moreover the values of the subscripts Ipi and q are smaller or equal to n. Coefficients with other subscripts vanish. According to (60). 63 PAGE 70 (61) For PFO, (59) and (60) yield 222 2 R [(p +q ) 4p ] + E p,q Ro i=1 (62) For p=q=l e(n) + E e(n2i) + )_2(H(n+2) + D(n+2) + F(n+2/R (63) 1,1 i=ll,l 1,1 os 1,1 1,1 1,1 os Through (61), (62) and (63) all the coefficients and e(n) are p,q p,q detennined. According to (63) the coefficients and e(n+2) 1,1' 2,20,2 must pe determined simultaneously. However, by simple arrangements can be expressed explicitly. Such a procedure avoids any trial process. Calculations of the mean hori zontal temperature and the Nussel t number followed the determination of the series coefficients. These para meters are given by N T = z (64) N () nl (i) (n i) Nu=lqen +al''Ii"'!!l"L 'n=l q=l O,q 1,p,q p,S Calculation of Nu determines the convergence of the method and the termination of the series expansion. Through the calculation, presented in the next section, we took N = 6, 8, 10, 12, 14 and 16 according to the variation in the Nusselt number. If Nu varied by less than 2% as N was increased from N to N + 2 then the expansion was terminated. 64 PAGE 71 RESULTS AND DISCUSSION According to (44)(47) convection effects are not determined only by the Rayleigh number but also by the Reynolds (Re) and Prandtl (Pr) numbers, as well as by the ratio between the porous layer thickness and the characteristi cfJore ( d/dp whole analys i cableonly when mechanical dispersion is at least comparable with the molecular diffusion. Th;s relationship is determined by the magnitude of Re and Pro It seems that if Pr 1, which is reasonable for practical purposes of hydrology [Somerton, 1958J, then dispersion effects are significant even for Re 3. However, if the Prandtl number is smaller than unity (if the solid fraction is a good conductor) then dispersion effects become significant at higher Reynolds numbers. The effect of Re, Pr and d/dp on the convection phenomenon is introduced in the analysis through 6 1 and (and = + which determine effects of turbulence and dispersion induced by the convection motion. Figure 2 demonstrates changes in these coefficients due to variations in Re, Pr and d/dp when the porosity is 0.4. The effect of Pr vanishes for high Reynolds numbers as in such cases mechanical dispersion affects transport processes more than the molecular diffusion. When the Reynolds number increases, according to model (9) and (10), the mechanical dispersion tensor becomes more and more isotropic. This phenomenon leads to a reduction in the value of Turbulent effects become more and more significant when the Reynolds number increases, leading to a positive relationship between increases in the Reynolds number and the coefficient 65 PAGE 73 1\ a2 1\ al (31 01 "J 101 3 102 3 103 3 104 3 105 3 106 3 ===== Pr '!!!!!I!!II__ ;:;.;z;ft 0.5 iil O did =103 _ p .=________ + ________ __________ __ 3 10 30 100 Re Figure 2. Description of 8, ,a, and &2 vs. Re for various values of Pr and d/d p(cp=O.4). PAGE 74 R/Ro:5..1 dId =20 p dId:::: 10 p R/Ro= 10 R/Ro=4 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6 0.4 0.2 0.0 T Z Figure 3. Mean horizontal temperature profiles for various values of R/R and did Pr=l. o p Re=3) 68 PAGE 75 At high Rayleigh numbers a thick region in the center of the porous layer should achieve a nearly isothermal state in the mean. However, as presented in Figure 3 such conditions lead to a positive temperature gradient or reversal of temperatures. Such a phenomenon was also identified in the ordinary Benard problem [Kuo, 1961; Veronis, 1966J. Veronis [1966J tried to explain the origin of such a strange phenomenon. However, better choice of wave numbers diminishes this effect. Figure 3 also demonstrates the creation of boundary layers on top and bottom of the aquifer leading to invalidity of the continuum approach even for moderate Rayleigh numbers if d/d p is not very large. Figure 4 presents variations of Nusselt number with Rayleigh number for various values of d/d p According to this presentation, the net effect of the mechanical dispersion induced by the convection motion leads to a reduction in the heat transport through the aquifer. Disper sion and turbulence induced by the convection motion act as a stabilizing mechanism in the flow field (ironical interpretation). However, this mechanism is more complicated than just a stabilization. In the calculation it was found that the phenomenon was associated with increased values of the higher modes of the series expansion, leading to a reduction in the convergence of the series expansions. Through the calculation, the maximal values of A and V were continu ously checked in order to examine the validity of approximations (44)(47) and to follow changes in the Reynolds and the Peclet numbers. It 69 PAGE 76 " o Figure 4. Description of Nu vs. R/Ro for various values of d/dp(=Oo4, Pr=l, Re=3). Nu 7 6 5 4 3 2 o 2 .. ."..:d/dp= 00 ,d/dp= 20 d/d = 10 I P 4 6 8 10 R/RO PAGE 77 was found that for the range R/Ro :: 10, >max was much sma 11 er than unity whi justi es utili on of (44)(47) and yields only minor changes in the lds number. CONCLUS IONS The anisotropic character of the dispersion tensor leads to convection motions conducted in two dimensional rolls whose axes are paranel the ullpedurbedve lodty vector. By choos i ng a new defi ni ti on for the Rayleigh number, rbulence and dispersion effects can be introduced in the linear stability analysis no substantial complications in the calculations and results. Finite amplitude analysis for homogeneous boundary conditions can be by Fourier series and power series expansions. The significance of mechanical dispersion and tlwbulence induced by the convection motion depends on PI", Re and mainly on d/dp An increase in Re diminishes the effect of PI" and reduces effects of dispersion and turbulence associated with the convection motion. An increase in d/d p reduces significantly effects of dispersion and turbulence due to the convection motion. For d/dp 102 these effects practically vanish. However. for smaller values of d/d p these effects lead to a reduction in Nuo The singly diffusive analysis presented in this article can be applied when density gradients are induced by a single component or IlIIhen boundary condit; ons and effecti ve di spers; on coeffi ci ents are identical for all components in a system. The latter canditionis approximately satisfied when mechanical dispersion effects are larger than any molecular diffusion in the aquifer. 71 PAGE 78 APPENDIX where sl=qt Expressions for coefficients of heat advection, friction and heat dispersion spectra e!pk!,S2e !pk! ( ) 4 n1 i 1 00 ( .) (. ) B n 1T L L J'J p,q 8 1 "4 j=l k ,m=oo If'k,t If' m,h t,h=l {th[4k(pkm)+(pkm)2+ 3 s (v=1,2,3,4;S,6,7)] v +[3k2hs t2m(pkrn)][,(nki) (v=3,4,5;1,2,6,7)]} v pm,sv ( ) 4 n1 i 1 ( ) (.' ) F n 1T L L r If' J If' lJ !p! ,q16 i=2 j=l k,moo k,t m,h t,h=l {th[(&l + ,sv(V=S,6,7;1,2,3,4)] +m[2&1 k 2 &2t2 ) [e f I,s/ v=3,4,S; 1 ,2,6,7)] (68) (69) (70) ,sv(v=1,3,S,7;2,4,6)]} (71) In (70) and (7l) the subscript Sv may obtain seven different values as follows: sl=qth s6=h:..tq s2=q+t+h s7=thq '72 (72) PAGE 79 NOTATION a wave number components of wave number critical wave number b friction function B friction spectrum Bp;,q. hmts insari OflSFOr B c1,c2 C C s d d p o D ,O(n) p,q P.q E'!': E lJ lJ q,Et F F F(n) p,q p,q 9 H H ,H (n) p,q P.q K JI.,. 1 n 1 coefficients defined in (26) specific heat of fluid specH; c heat of solid porous layer thickness characteristic pore size part of heat dispersion spectrum coefficients in series expansions for D heat dispersion tensors (mechanical and hydrodynamical respect i ve 1 y ) longitudinal heat dispersion coefficients transversal heat dispersion coefficients part of heat dispersion spectrum coefficients in series expansions for F gravity acceleration heat advection spectrum coefficients in series expansions for H permeabil i ty unit vector in the longitudinal direction unit vector in the vertical direction 73 PAGE 80 N total number of terms in the series expansion Nu Nusselt number p pressure Po pressure at the coordinates origin Pr Prandtl number ( \lId R Rayleigh number defined in (14) Ro critical Rayleigh number Ros parameter defined in (58) Re Reynolds number s power coefficient s = N/2 t time T temperature To temperature at z = a u longitudinal velocity perturbation Ui velocity vector Uo unperturbed velocity U module of velocity vector v lateral velocity perturbation V module of velocity perturbation w vertical velocity perturbation x coordinate 1 x.y,z coordinate system a coefficient of volumetric thermal expansion a 1,a2,a3 coefficients defined in (47), (a3 = a1 + a2)' 8 perturbation in the friction function 74 PAGE 81 6 1 coefficient defined in (47) y parameter defined in (5) 0.. Kronecker1s delta lJ 6T difference in temperature between bottom and top of the porous layer Eij dispersion tensor perturbation n small parameter defined in (56) 8 temperature perturbation 81 constant defined in (27) A (n) ff"' t" f 8p,q8p,q8p,q coe lClen s ln serles expanslons or e K thermal diffusivity of saturated porous medium parameter defined in (40) v viscosity v kinematic viscosity perturbation in s coefficient defined in (47) P fluid density Po density at z = 0 P s solid density Gr parameter expressing amplification of small disturbances porosity x parameter defined in (29) stream funttion (n) ff" t" f coe lClen s ln serles expanslons or w parameter expressing oscillations 75 PAGE 82 Q scalar function defined in (22) Q1 constant defined in (27) 76 PAGE 83 REFERENCES Kuo, H. L., IISo 1 u tion of the Nonlinear Equations of Cellular Convection and Heat Transportll, J. Fluid Mech., 1.Q, 1961, pp. 611634. Malkus, W. V. R., and Veronis, G., IIFinite Amplitude Cellular Convection", J. Fluid Mech., 1, 1958, pp. 225260. Palm E., Weber, J. E., and Kevernold, 0., liOn Steady Convection in a Porous Medium", J. Fluid Mech., 54(1), 1972, pp. 153161. lHr.=,,,lIftQrrn 1ti"tbfl=lertAnrr.a:rtl ys ; 5 of Ce 11 u 1 a r Co nvec t lOn n Porou s Med i a II Int. J. Heat & Mass Trans., ]&, 1975, pp. 14831486. Rubin, H., "Onset of Thermohaline Convection in a Cavernous Aquifer", to be published in Water Resourc. Res., 1976. Rubin, H. and Christensen, B. A., IIConvection Currents Associated With Hydrodynamic Dispersion in a Porous Mediumll, 16th Congress of IAHR, Sao Paul, Brazil, 1975. Saffman, P. G., "A Theory of Dispersion in a Porous Medium", Fluid Mech., 6(3), 1959, pp. 321349. SchlUter, A., Lortz, D., and Busse, F., liOn the Stability of Steady Finite Ampl itude Convectiontl, J. Fluid Mech., 23(1), 1965, pp. 129144. Somerton, H. W., "Some Thermal Characteristics of Porous Rocks", J. Petrolr Tech., Note 2008, 1958, pp. 6165. Straus, J. M., "Large Amplitude Convection in Porous Media", Fluid Mech., 64(1), 1974, pp. 5163. Veronis, G., "Large Amplitude Benard Convection", J. Fluid Mech., 26(1),1966, pp. 4968. 77 PAGE 84 INTRODUCTION CHAPTER 5 SEMINUMERICAL APPROACH FOR THE MATHEMATICAL MODELING OF SINGLY DISPERSIVE CONVECTION IN GROUNDWATERS Considerable interest ;s now focused in the State of ElDrida, as well as in other locations in the United States, for the possible utilization of deep saline aquifers for waste disposal. Vernon (16) delineates the properties of the deep zones of the Floridan aquifer which makes this stratum available for such application. Henry and Kohout (2) mention the fact that in a thick system. like the Floridan aquifer, the effect of geothermal activity should be considered, too. One of these authors has postulated in previous articles (3,4) that geothermal activity induces groundwater circulation in the Floridan aquifer. Singly diffusive convection is the convection motion induced by a single dissolved component (e.g. temperature or salinity) in a fluid layer. The hydrodynamics of this phenomenon in a saturated porous media has been studied while, in most instances, assuming that the fluid is initially at rest (7,8). However, groundwaters are generally subject to hydraulic gradients leading to slow, effectively horizontal flow of the subsurface water. This movement through a formation, similar to the Boulder Zone in the deep saline region of the Floridan aquifer (1,4), leads to intensive mechanical dispersion of heat and soluted in the aquifer, as well as inertial effects, as demonstrated by invalidity of the laminar Darcy law. In such a system molecular diffusion effects are usually less significant than mechanical dispersion. Convection phenomenon under such conditions may therefore, be called dispersive convection. A system is subject to singly dispersive convectionif one of the following criteria ;s satisfied: a) density gradients are 78 PAGE 85 in a single component; or b) all molecular diffusivities of the dissolved components are much smaller than the mechanical dispersion, and all of them have the same boundary conditions. The objective of this art"icle is to present a rather simple method by which flow conditions in such an aquifer can simulated. BASIC EQUATIONS The basic equations applied for the analysis are the equations of __ cQnti nuitY3_ mot "ion ,_i:ul(l di sperstolJ, sublect to the_Boussi + \7U = 0 0 \7p + pgh + k (l+b)u = 0 aT ;(=) Y3f + u'\7T \7E\7T .. The coefficient y appearing in Equation 3 is defined by: (1) (2 ) (3) . ( 4 ) In a brief literature survey, presented in a previous study(lO), it was suggested that the friction function, b, appearing in Equation 2 can be approximated by: . . ( 5 ) It is assumed that the fluid density depends linearly on the dissolved component, which is the temperature, as follows: ...... ....... (6) 79 PAGE 86 Uo .. .. TEMPERATURE .. .. PROFILE Figure 1. Schematical description of the aquifer with no convection motion. 80 PAGE 87 The flow field model considered prior to the inception of the convection motion, as presented in Figure 1, is a cavernous aquifer consisting of a porous layer of infinite horizontal It is bounded by two impermeable planes on which the temperature is constant. Through the porous layer the fluid flows uniformly in the longitudinal x direction. The transversal and vertical coordinates are y, and z, respectively. A moving coordinate system is applied with the velocity uo/y in the x direction, and consider (7) as characteristic length, temperature, dispersion coefficient, velocity, time, and pressure, respectively. In such a manner the following dimensionless basic equations are obtained (in further expressions all variables are dimensionless): + vu = 0 . (8) vp RTh + = 0 (9) vT = V.(r'VT) (10) Here the Rayleigh R agll TKd number, R, and the variable B, are defined as follows: B = b[(U/u o ) l]/(l+b) (11) In an isotropic medium the dimensionless dispersion tensor can be expressed by: . (12) As long as there is no convection motion Equations 810 yield: + 0 B = 0 T = Rz2/2 u = z p = p 0 Exx = E Eyy = Ezz = 1 Exy = Exz = Eyz = 0 (13) JI, . . 81 PAGE 88 THE FLOW FIELD STABILITY The stability of flow conditions presented by Equations 13 can be determined by analyzing the growth of small disturbances in the aquifer. These are disturbances in the velocity, u, temperature, e, dispersion tensor, E and pressure,P. Second order terms depending on these disturbances are negligible. It is convenient to express the velocity vector through the scalar variable Q asf611ows: where t :: rtx'i,7X'i,7 . . . . (14) By substituting the small disturbances in the basic equations, and applying the boundary conditions 2 e,\71Q = 0 at z = 0,1 ........................ (15) (where :: we obtain: R'i,7iQ = + . . . (16) 2 ++ 2 'i,7d = 'i,7\7 + 'i,7 ................. (17) By expand PAGE 89 E STEADY STATE CONVECTION In the prey; ous sect; on it was plAoved that convect; on cell s two dimens'iona"J rolls. It is convenient to express the velocity by the stream function . . ( 20 ) where the square brackets symbolize the box product. Substituting Equation 20 in the basic equations, and eliminating the pressure perturbatiQn, we_obtain: R[rt, 1. VT] + = S) .. + v 2 e crt, =: + + F(, e). where is the finite amplitude disturbance in the dispersion tensor. Variables B, H. 0 and F are nonlinear terms defined as follows: S) = e) = [1. ve] = e) = v(E:ve). (21) (22) (23) (24) (25) (26) As long as convection velocity is smaller than the unperturbed velocity the absolute value of the flow field velocity can be approximated by: (27) where 2 ++ V ::: UU If 1 terms depending on high orders of can be neglected. In such conditions. the expressions for B and the dispersion tensor can be by: 8 =: 8 1 V 2 83 (28) (29) (30) PAGE 90 where 2 6 1 = b/[2(1+b)uoJ .. a 1 = (1/2u o)(aEt/au o ) a = 3 .. . . (31) 2 a2 = (E9,l)/u o + (1/2u o)[a(E9,Et)/au oJ . (32) The system of the differential equations 21 and 22_ ;s s_ubject to the foll ow; ng boundary cond it i on s : e = 0 at z = 0,1 ........................ (33) Such boundary conditions are simple according to the definitions presented by Orszag (9). In such cases, accurate simulation of incompressible flows can be obtained by spectral methods. Assuming that and e are periodic in the horizontal direction, these variables can be presented by sets of truncated eigenfunctions as suggested by Veronis in similar studies (17,18). = ; sin(pay)sin(qwz) .................. (34) p.q=l p,q e = e cos(pay)sin(qwz). .................. (35) p=o p,q q=l The calculation can be simplified by using the complex variable presentation of sin(pay) and cos (pay) leading to = i E_ exp(ipay) ................ (36) Doo 00 e = E p=oo e [sin(qwz)] exp(ipay) ................. (37) p,q q=l provided that 84 PAGE 91 The convergence of the expression for the Nusselt number also determines the truncation of the Fourier series expansion (the value of N). NUMERICAL CALCULATIQNS Experiments concerning heat transfer characteristics of porous rocks (e.g. 5,6) showed that the phenomenon ofheatdjspersion_duetothe fluid movement in the stratum is very similar in nature to the characteristics of solute dispersion in porous media. We may adopt, then, models of mechanical dispersion, which are available in the literature, for the quantitative evaluation of the effect of the convection phenomenon on the intensity of transport processes in the aquifer. Saffman (11) suggested the following expressions for the coefficients of dispersion. et 1 + s + 2 \) Pr 2 ( s + 1 )(s + 3 ) ( :e ) . . . . (42 ) (s + 1)2 + 2(1s)(s+2)(s+3) (:e). . . (43) where s is a power coefficient describing the dependence between the flow velocity and the pressure gradient. In a previous article (10) it was suggested that this coefficient can be calculated through the following expression s = In(Re)/{ln[(l+b)Re]} (44) Equation 41+ ;an h", "'d' ,":,; ;2.77. Through the calculation, it was assumea that PI" = 1, which seems to be a reasonable value for limestone and dolomite aquifers (13). The convergence of the numerical integration was very moderate. Several approaches were applied to speed convergence of the calculations. These were: (a) Each calculation was conducted in two steps, in the first step we took 86 PAGE 92 and. after ining resul for was introduced into the program; (b) Results lower values Nand/or R were used as initial quanti es hi parameters. As a criterion for steady calculations, to ta { I ( ., 04 < Ii 0 (45) In our calcula ion, it was that such a criterion is very conservative. a criterion of one hundred times less conservative, vari 1% were obtained in the selt number. Through the ng oscillations were detected in the value the 2 ations of the Nusselt number with Rayleigh number va 0(15 values These are the maximal values of filL! obtained when va value of the wave number (12). The wave number increased with values of ions in the wave number induced minor changes in Nu for a constant ei to Fi 2, mechanical dispersion, due to the convection motion, leads to a ion in the intensity of transport processes through the aquifer. The calculation 1 icated that in small values of d/d p the magnitude of the dis in This s es ion cients increases, when convection occurs, leading to a reduction the tempet'ature gradients on top and bottom of the on in temperature gradients affects transport processes more increase in ue of the dispersion coefficients. It was also t "IO'IrJer va 1 ues d/d p give rise to the higher modes of the Fourier the convergence of the calculation, For very ions a 87 PAGE 93 co co u 7 6 5 4 3 2 o 2 4 6 d/dp:: 00 d/dp:: 20 d/dp:: 10 8 10 R/RO Figure 2. Description of Nusselt number vs. R/Ro for various values of Pr=l; Re=3). PAGE 94 small values of did the analysis fails to follow the physical phenomenon, p even at moderate Rayleigh numbers, as the thickness of the boundary layers developed on top and bottom of the aquifer is of the same order of magnitude as the characteristic pore size. In such cases the continuum approach applied through the analysis is not satisfied. Through the calculation, the maximal value of was continuously checked in order to examine the validity of the approximations presented in Equations 2732. It was found that fOt' R,LR<"IO, the maximalvalue of .7. was mtfchsmallerthanunity. o CONCLUSIONS Singly dispersive convection in aquifers can be analyzed by expanding the flow field disturbances in eigenfunctions. The anisotropic character of the mechanical dispersion determines the plane in convection motions are conducted. According to the analysis. convection cells are two dimensional rolls whose axes are parallel to the unperturbed velocity vector. Analysis of steady state convection can be conducted by transforming the equations of motion and continuity to a set of general first order differential equations that can be integrated through available subroutines. The effect of mechanical dispersion and turbulence, induced by the convection motion, depends mainly on the ratio between the porous layer thickness and the characteristic pore size; Prandtl and Reynolds numbers have less significant effects on the physical phenomenon. The analysis of the singly dispersive convection presented in this study can be appl ied when density grad"ients are induced by a single or when bOL!ndar'y condition_ and effective dispersion coefficients are identical for all components in a mul{icomponent system. 89 PAGE 95 APPENDIX I EXPRESSIONS FOR THE COEFFICIENTS OF THE SPECTRAL FUNCTIONS Nl NI k I Hp,q = H_p,q = (wa/2) 'k,S{S(pk}[Slp_kJ, W y (Y=1,2;3}J + W y (Y=2;1,3}J} .................... (46) .... ___ 2 __ _!! ... .... .Dp,q =D."p:q= 222 2 + 2cxl(n /a }wy cx2(pk) J['p_k W (Y=1,2;3}J} ............ (47) y where w1 = qs Slp_kl ,wy(y=2;1 ,3} = Slp_kl 'W1 Slp_kl ,w2 Slp_kl 'W3 ............ (48) 2 2 N1 NIkl NIml B = B = (w a /4) L L L p,q p,q k,m=lN s=l h=l k,s m,h 2 2 2 2J ( . ) J {sh[4k(pkm)+(pkm) + 3(w /a }wy y1,2,3,4,S,6,7 + [3(a2/n2}k2hWys2m(pkm}J['p_k_m,Wy(y=3,4,5;1,2,6,7}J} ...... (49) where W 1 = qsh ,w5 = s+hq W 2 = q+s+h W6 = hsq W3 = qs+h W 7 = shq 90 W 4 = q+sh . (51) PAGE 96 NOTATION a = wave number; a o = critical wave number; ax' a = y wave number components; b = friction function defined in Equation 5; B = friction spectrum defined in Equation 23 ; B coefficients in the Fourier series q.q C s C w = specific heats of soil and water, respectively; d = porous layer thickness; d = characteristic pore size; p D = part of heat dispersion spectrum defined in Equation 25; D p,q = coefficients in the Fourier series expanded for D; eQ,' et = longitudinal and lateral dispersion coefficients. respectively; F H = dimensionless dispersion coefficients (dispersion coefficients divided by e t existing prior to convection conditions); E = dispersion tensor; F = part of heat dispersion spectrum defined in Equation 26; = coefficients in the Fourier series expanded for F; P.q g = gravity acceleration; H = heat advection spectrum defined in Equation 24; = coefficients in the Fourier series expanded for H; P.q I = unit matrix; K = permeability; + Q, = unit vector in the longitudinal direction; + n = unit vector in the vertical direction; N = truncation parameter; Nu = Nusselt number; p = pressure; 91 PAGE 97 Po = pressure at z = 0; Pr = Prandtl number (=V/K); R = Rayleigh number defined in Equation 11, Ro = critical Rayleigh number; Re = Reynolds number (= PAGE 98 \! kinematic viscosity; P = fluid dens ity; Po = fluid dens ity at z = 0; P s = solid density 0", 0"2 = parameters expressing growth and oscillation of disturbances;
