A numerical model of a fractured aquifer with dual porosity

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A numerical model of a fractured aquifer with dual porosity

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Title:
A numerical model of a fractured aquifer with dual porosity
Creator:
Morrison, Kevin E.
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Tampa, Florida
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University of South Florida
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English
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vii, 57 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Aquifers -- Mathematical models ( lcsh )
Groundwater flow -- Mathematical models ( lcsh )
Dissertations, Academic -- Geology -- Masters -- USF ( FTS )

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Thesis (M.S.)--University of South Florida, 1995. Includes bibliographical references (leaves 55-57).

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University of South Florida
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Universtity of South Florida
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021219287 ( ALEPH )
33884419 ( OCLC )
F51-00119 ( USFLDC DOI )
f51.119 ( USFLDC Handle )

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A NUMERICAL MODEL OF A FRACTURED AQUIFER WITH DUAL POROSITY by KEVIN E. MORRISON A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Geology University of south Florida May 1995 Major Professor: Mark T. Stewart, Ph.D.

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Graduate School University of South Florida Tampa, Florida CERTIFICATE OF APPROVAL Master's Thesis This is to certify that the Master's Thesis of KEVIN E. MORRISON with a major in Geology has been approved by the Examining Committee on April 10, 1995 as satisfactory for the thesis requirement for the Master of Science degree Examining Committee: Major Professor: Mark Stewart, Ph.D. Member: H. Len Ph.D. Member: !:'llf 4.

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ACKNOWLEDGEMENTS I would like to thank my major professor, Dr. Mark stewart for his guidance on this project, and the members of my committee, Dr. H.L. Vacher and Dr. Jeff Ryan, for their help. My sister Lisa and my parents have also been a great support during my time in graduate s chool.

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TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES . ABSTRACT . . . . . . . . . . . 1. INTRODUCTION General . Objectives 2. PREVIOUS STUDIES . . ..... Numerical Models . . . . . Field studies . . . . . . . Analytical Models . . . . . . 3. METHODS . . . . . Model Design and Analysis The Dual-Porosity Concept ..... Model Parameters . Analytical Solution . . . 4 RESULTS . . . . . . . . steady-state Model . . . . . Transient Case . . . . . . . Analytical Solution . . . . . . 5. DISCUSSION . . ..... Steady-state Model . . . Transient Model . . . . . . Analytical Solution . . . 6 CONCLUSIONS 7. LIST OF REFERENCES i ii iii v 1 1 5 6 6 7 8 10 10 12 13 15 16 16 33 44 47 47 49 52 53 55

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Table 1. Table 2. LIST OF TABLES Model Parameters Bourdet-Gringarten Analytical Solution Results 11 14 46

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LIST OF FIGURES Figure 1. Finite-difference Model Grid for the Steady-State Simulation. 4 Figure 2. Potentiometric Surface Contour Map of the Steady-State, Homogeneous Model. 17 Figure 3. Potentiometric Surface Contour Map of the Steady-State Model. 18 Figure 4. Potentiometric Surface Contour Map of the Steady-state Model. 19 Figure 5. Potentiometric Surface Contour Map of the steady-state Model. 20 Figure 6. Potentiometric surface Contour Map of the Steady-State Model. 22 Figure 7. Potentiometric surface Contour Map of the Steady-state Model. 23 Figure 8. Potentiometric Surface Contour Map of the Steady-state Model. 24 Figure 9. Potentiometric surface Contour Map of the steady-state Model. 25 Figure 10. Potentiometric surface Contour Map of the Steady-State Model. 26 Figure 11. Potentiometric Surface Contour Map of the Steady-state Model. 27 Figure 12. Potentiometric Surface Contour Map of the Steady-State Model. 28 Figure 13. Potentiometric Surface Contour Map of the steady-State Model. 30 Figure 14. Potentiometric Surface Contour Map of the Steady-State Model. 31 Figure 15. Potentiometric Surface Maps from Six steady-State Models. 32 iii

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Figure 16. Figure 17. Figure 18. Figure 19. Figure 20. Figure 21. Figure 22. Figure 23. Figure 24. Figure 25. Finite-Difference Model Grid for the Transient Simulation. Comparison of Time-Drawdown Data from the Homogeneous Case with the Theis (1935) Type Curve. Comparison of Time-Drawdown Data from the Floridan Aquifer with the Theis (1935) Type Curve. Comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type Curve. Comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type curve. comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type Curve. Comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type Curve. comparison of Time-Drawdown Data from the Floridan Aquifer to Model-Generated Data. Comparison of Time-Drawdown Data from the Floridan Aquifer to Model-Generated Data. Comparison of Time-Drawdown Data from the Floridan Aquifer to Model-Generated Data. iv 34 35 36 37 38 40 41 42 43 45

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A NUMERICAL MODEL OF A FRACTURED AQUIFER WITH DUAL POROSITY by KEVIN E. MORRISON An Abstract Of a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Geology University of South Florida May 1995 Major Professor: Mark T. Stewart, Ph.D. v

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Groundwater flow in a fractured aquifer with dual porosity can be simulated using a finite-difference numerical model. The fractured aquifer is discretized using variable grid spacing to define blocks and fractures. The fractures are vertical and orthogonal. Blocks have low permeability and high storage capacity, whereas fractures have high permeability and low storage capacity. A variety of fracture-network geometries is simulated. The steadystate models indicate that the configuration of the potentiometric surface is sensitive to the block-to-fracture hydraulic conductivity ratio, and to the particular configuration of the fracture network in relation to the outflow node. As the block-to-fracture conductivity ratio decreases, the system behaves more heterogeneously. The heads in the models decrease progressively as the fracture conductivity increases. Varying the geometry of the fracture network produces unpredictable results. Models with more fractures oriented parallel to the regional direction of flow have generally lower heads whereas systems with fractures oriented perpendicular to the regional flow direction have much higher heads. Systems with randomly generated fracture networks behave unpredictably. The discrete-fracture model was stressed with a pumping well to simulate transient aquifer response. Time-drawdown data from model runs with varying fracture density and vi

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ratios of hydraulic conductivity were qualitatively compared to field data from pumping tests of the Floridan aquifer to determine the similarity between the model-derived data and field data. Three examples from the Floridan aquifer give good matches to time-drawdown data produced by a discrete-fracture model. The Floridan aquifer often yields time-drawdown data which match the Theis (1935) type curve from early to late time, indicating homogeneous behavior. In some cases, the response of the Floridan aquifer deviates from Theis behavior. In some of these cases, the Floridan aquifer response to pumping stresses can be successfully approximated using the discrete-fracture approach. Abstract Approved: Major Professor: Mark T. Stewart, Ph.D. Professor, Department of Geology Date Approved: vii

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1 1. INTRODUCTION General The purpose of this study is to use numerical models to simulate a double-porosity, fractured aquifer and investigate the factors influencing the nature of the flow regime within the aquifer. Dual-porosity systems consist of a porous medium with primary porosity which is a product of original petrogenesis and a secondary porosity resulting from subsequent processes (Streltsova, 1976). In this study, secondary porosity exists in fractures within the porous medium. Fractures are often enhanced by dissolution or weathering (Barenblatt, Zheltov, and Kochina, 1960), and are known to be major pathways for fluid flow as a result of what is known as the cubic law (Domenico and Schwartz, 1990). This law holds that the discharge through a fracture is proportional to the cube of the fracture aperture. The cubic law appears to be valid for a variety of geometries and stress conditions (Moench, 1984). In this study, the dual-porosity system will be conceptualized as a heterogeneous domain having lowpermeability blocks and higher-permeability fractures. In many studies of fractured systems, the porosity of the

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blocks has been assumed to be zero, and thus ignored. If the blocks have very low permeabilities this assumption may be reasonable. In many dual-porosity systems, however, the blocks may make a significant contribution to flow in the aquifer and so should not be ignored. In particular, storage in the blocks can be very significant. This study will consider a system in which blocks have some permeability and are the major source of storage in the aquifer. Fractures have higher permeability and smaller storage coefficients than the blocks. Fractured aquifers are common throughout the world. 2 Photolinears visible on aerial photographs have been shown to be zones of increased density of vertical or nearvertical fractures (Moore and Stewart, 1983). These zones are up to 300 m wide and have been shown to yield up to 100 times as much water as adjacent strata (Parizek, 1976). The high-yield zone is usually narrow, often a few tens of meters wide, and lies within the photolinear (Parizek, 1976). Moore and stewart (1983) found that fractures associated with photolinears are an important influence on the hydrogeology of the Floridan aquifer of west-central Florida. These fractures are oriented in four principal directions consistently throughout the world --WNW, NNW, NNE, and ENE --and are thought to be the result of tidal and rotational stresses (Blanchet, 1957). Analytical solutions exist for dual-porosity systems

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3 {McConnel, 1993; Moench, 1984; Streltsova, 1976; Warren and Root, 1963). Because analytical solutions are a general solution for the entire system, simplifying assumptions must be made. Many analytical solutions assume blocks to be impermeable {Witherspoon and others, 1981). Others require fractures to be horizontal and to have no storage capacity (Boulton and Streltsova, 1978). The geometry of blocks and fractures is usually not specified {Warren and Root, 1963). This study will use a numerical model instead of an analytical model. The advantages of the numerical model are numerous. Various geometries can be simulated, fractures can be vertical, and various relationships of block/fracture permeability can be simulated. The mathematics of the solution are greatly simplified by using a numerical model. Some assumptions are inherent to a numerical model, but they are much less restrictive. This study simulates a fractured aquifer with a onelayer model consisting of blocks and fractures. Fractures are vertical and orthogonal. A variable grid is used to create large blocks separated by narrow fractures {Figure 1). Blocks are assigned a low permeability and, in transient cases, high storage capacity. Fractures are assigned high permeability and low storage capacity. Recharge is applied uniformly to the model, and discharge is through a single spring node or well. Transient simulations are used to determine the aquifer's response to a pumping

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4 c:\groundwa\fract2.gmo (Cell Status) Layer:l 2283.105 feet Figure 1. Finite-Difference Model Grid for the Steady-State Simulation. Highlighted Cells are Constant Heads.

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well in the absence of recharge. steady-state simulations show the large-scale behavior of such a system discharging at a regional spring. Time-drawdown data from the transient model are qualitatively compared to field data from pumping tests of the Floridan aquifer. Analytical solutions are applied to both model-derived time-drawdown data and field data. 5 Objectives The specific objectives of this study are to: (1) develop a discrete-fracture model using variable discretization to define blocks and fractures, and simulate both transient and steady-state conditions for 1:10, 1:100, and 1:500 block-to-fracture hydraulic conductivity ratios for both fully interconnected systems and systems that have 25%, 50% and 75% active fractures; (2) observe the response of the model in each of the above scenarios; and (3) qualitatively compare model-derived time-drawdown data to field data from the Floridan aquifer.

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6 2. PREVIOUS STUDIES Numerical Models A similar approach to this study was taken in a study by Powers (1992). In Powers' model, variable discretization was used to define blocks separated by fractures 50 feet wide. A grid consisting of approximately 50 by 50 cells was used. As a result of the size of the model, boundary effects were very pronounced and unreasonable values for transmissivity and storativity were obtained. The size of the model was restricted by limitations of computer memory. Because of the rapid advance of microcomputer technology, memory is no longer a limiting factor. Powers' model did yield some interesting results. Several simulations were run with different block-tofracture hydraulic conductivity ratios. At a 1:10 ratio the flow regime was affected very little, but the overall transmissivity of the system was larger as reflected by lower heads in the model compared to the homogeneous case. At lower block-to-fracture conductivity ratios the system behaved more like a heterogeneous aquifer. As the ratio

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reached 1:500, local flow systems beganto develop within each block. Powers' model forms the basis for this study. This study will use a much larger model to avoid boundary effects, and it will explore the role of interconnection in the overall behavior of the system. Field Studies 7 Fractured media have been extensively studied in relation to the underground storage of nuclear waste materials. Photolinears have also been studied to determine if they can be used to aid in the siting of water-supply wells (Lattman and Parizek, 1964). Photolinears are largescale features that are thought to be the surface expressions of zones of increased fracture density (Parizek, 1976). These zones are found to have permeabilities 10 to 1000 times larger than the surrounding rocks (Parizek, 1976). A widespread network of large-scale regional fracture zones is known to exist in west-central Florida (Faulkner, 1973). Moore and Stewart (1983) and stewart and Wood (1991) showed that fractures associated with photolinears are important features within the Floridan aquifer of Florida and that the fractures can be recognized

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8 by a characteristic geophysical signature. Analytical Models McConnell (1993) developed an analytical approach to determine aquifer characteristics of fractured carbonate rocks in the Ozark mountains of Arkansas. This method allows for storage in the blocks but assumes most of the water comes to the well through the fractures. McConnell used pumping-test data from municipal water wells completed in confined, fractured, carbonate rocks. McConnell's method yields smaller values for storativity than other analytical solutions, but it is often not possible to obtain a unique value for storativity, because it is interrelated with wellbore-storage and skin effects (McConnell, 1993). Moench (1984) developed an analytical solution for fractured aquifers that incorporates the fracture-skin concept. The fracture skin is a low-porosity. deposit between the fracture and the block. The skin creates most of the head gradient between the fracture and block and results in a relatively uniform distribution of head in the blocks. This effect makes it possible to neglect the divergence of flow in the blocks and so a pseudo-steadystate approach is theoretically justified (Moench, 1984).

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The assumption of pseudo-steady-state conditions greatly simplifies the mathematics of the analytical model. 9 Powers (1992) includes an extensive discussion of the application of analytical solutions to dual-porosity fractured aquifers. Generally, there has been little success in applying analytical solutions to these types of systems. Powers' attempt at using analytical solutions with fractured aquifers was unsuccessful and will not be repeated in this study.

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10 3. METHODS Model Design and Analysis This study utilizes a two-dimensional, finitedifference model with variable discretization to define discrete blocks and fractures. The model grid is 98 by 98 cells (Figure 1). The computer code used is Graphic Groundwater (Esling and Larson, 1992), a graphics-based version of MODFLOW, the u.s. Geological Survey's threedimensional finite-difference flow model (McDonald and Harbaugh, 1988). Graphic Groundwater runs in the Microsoft Windows environment and supports graphical construction and editing of model grids. Graphic Groundwater uses the Strongly Implicit Procedure to iterate to a solution. The model runs rather slowly, but the savings in time gained in editing the model more than makes up for the slowness of the program. Steady-state and transient cases are run in which 25% 50% 75% and 100% of the model fractures are active. To randomly determine which fracture segments are active, a grid of random numbers was generated with a spreadsheet.

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11 Each fracture segment is activated or deactivated according to its number. Recharge is applied uniformly to the steadystate models. In the Floridan aquifer of west-central Florida, groundwater discharge is primarily through springs (Faulkner, 1973). Two constant-head cells with zero head at the bottom edge of the model simulate a spring. All water in the model must discharge through the spring node. For steady-state cases, a potentiometric surface map is generated using head data from the model. Two constant-head cells were used to simulate the spring so that it would be in the center of the model boundary because there are an even number of cells on a side and there is no center cell. For transient cases, time drawdown data are plotted on a log-log plot so that the curves can be compared to field data and to type-curves for analytical solutions. Timedrawdown data from pumping tests of the Floridan aquifer are compared to the model data to determine if the model successfully approximates dual-porosity behavior in the Floridan aquifer. For transient cases, no spring node is used because a spring is a steady-state regional feature (Powers, 1992). Instead, a well is placed in one corner of the grid and is pumped at a constant rate. In Graphic Groundwater as in MODFLOW, a well is simulated by placing an extra flux term in the equation for the cell containing the well. The flux due to the well is averaged over the entire cell and is not applied at a point. The placement of

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12 the well in the corner is possible if one assumes the aquifer to be symmetrical in all directions and thus only one quadrant need be discretized. No recharge is applied to transient cases so that the response of the aquifer to pumping can be observed in the absence of recharge. Storativity in the fractures is assumed to be one order of magnitude smaller than that in the blocks. Fractures are shown as narrow rows and columns of cells separating wider blocks made of larger cells (Figure 1). The fracture-cell dimensions are 50 by 200 feet, a 1-to-4 ratio of length to width. In most cases cells should not exceed a 1-to-2 ratio. To ensure that the model was not affected by the fracture-cell dimensions a test model was run in which some fracture cells were divided in half so that they were 50 by 100 feet. The test model results are identical to those of the original model so, in this case, the 1-to-4 ratio of cell dimensions is acceptable. The Dual-Porosity Concept The dual-porosity concept was first suggested by Barenblatt, Zheltov, and Kochina (1960). Dual porosity in a porous medium consists of matrix blocks and fractures, each having their own properties. The fractures are

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characterized as having high permeability and low storage capacity, whereas blocks have low permeability and high storage capacity. If the fracture density is high enough, the fluid flow will occur only in the fractures, and the system will behave like an unconsolidated homogeneous aquifer (Kruseman and de Ridder, 1991). Model Parameters 13 Hydraulic conductivity, recharge, initial head and aquifer thickness are specified for the steady-state models. All inflow is through recharge and all outflow is through the constant-head cells. In transient models no recharge is applied so that drawdown response can be observed. All water in the transient model is derived from storage, and all discharge is through the well. Model parameters are given in table 1.

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Table 1. Model Parameters. Steady-State Model Block Hydraulic Conductivity Fracture Hydraulic Conductivity Recharge Aquifer Thickness Conductivity ratio 1:1 1:10 1:100 1:500 Transient Model Block Hydraulic Conductivity Fracture Hydraulic Conductivity Block Storativity Fracture Storativity Well discharge Aquifer Thickness Conductivity ratio 1:1 1:10 1:100 1:500 14 100 ftjd 100 ft/d 1000 ft/d 10000 ft/d 50000 ft/d 5E-5 ft/d 100 ft 100 ft/d 100 ft/d 1000 ft/d 10000 ft/d 50000 ft/d 0.1 0.01 1E+6 ft3 /d 100 ft

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15 Analytical Solution The Bourdet-Gringarten curve-matching method as described in Kruseman and DeRidder {1991) is used to determine values of transmissivity and storativity for both transient-model data and Floridan aquifer data. This method uses the Theis (1935) type curve to obtain curve matches between the data and type curve at late time and early time. The method yields values for transmissivity and storativity of the fractures at early time and transmissivity of the combined block-fracture system at late time. Storativity at late time is given as a combination of fracture storativity and block storativity which cannot be resolved. Block transmissivity also cannot be resolved with this method. This method does, however, give a general idea of the overall characteristics of the system.

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16 4. RESULTS Steady-State Model The simulation of a homogeneous aquifer is used for comparison with fracture models for the steady-state case. The pattern of equipotentials in the homogeneous case indicates radial flow to the spring node (Figure 2). The flow pattern for the 1:10 block-to-fracture conductivity ratio (Figure 3) is similar to that for the homogeneous case. Heads in the fractured aquifer are lower, reflecting an overall increase in transmissivity. At a block-tofracture conductivity ratio of 1:100 (Figure 4), nonhomogeneous effects are apparent as equipotential contours refract across fractures. Heads in the 1:100-ratio case are significantly lower than in the homogeneous case. At a 1:500 block-to-fracture ratio (Figure 5), local flow systems develop within each block, and flow is radial from the center of each block toward the fractures. Heads in the 1:500-ratio case are more than 90 percent lower than those for the homogeneous case. In all cases, equipotential contours become closer together near the spring node

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17 1 600 1 400 1200 1000 2000 4000 6000 800 0 1 0000 1 2000 14000 1 6000 f e e t Figure 2. Potentiometric Surface Contour Map of the Steady state, Homogeneous Model. Contour Interval is 10 Feet.

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18 1800 0 I I I I I I I 1600 o1--1400 IQ-1200 10-f-1000 10---150 1-800 10-1---1-------1---------. l.------r---------tr--1-10-1------v / 600 v I / --------------1 0-1-...-----r------.. 400 I / v "'-.... "'---v 10-1-200 I f I f f "' \ \ I I I I I I I 2000 4000 6000 8000 10000 12000 1 4000 16000 f e el Figure 3. Potentiometri c Surface Contour Map of the Steady state Model. All Fractures are A ctive. Block-to-Fracture Conductivity Ratio is 1 :10. Fractures are Shown a s Heavy Double Lines. Contour Interval i s 10 Feet.

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19 800(} I I I I I I I l-/ u I"-6 0 0(}-!-----"" "--------"'--I-1---------400(}-_./ r-------1--I-1 ---------I-n r----200 (}---1--I-(-t------f-------1--f--6J t--()()()(). -t--I-1 ------<1J 1-, _____ <1J 1----1------t-t:::= 800(}6 ___ --r---..... 1-------1---------I---I--1-l----I---r----6 r-----. -t--, __________ r::: --t: t=:= ..... 600(}-1-------------p..c--I---1----=::-----r---------------b, 1----"" "---.... ..... 400(}-f.---/ // h -200(}-------2 600 40 00 .. 14000 16000 feel Figure 4. Potentiometric Surface Contour Map of the SteadyState Model. All Fractures are Active. Block-to-Fracture Conductivity Ratio is 1:100. Fractures are Shown as Heavy Double Lines. Contour Interval is 1 Foot.

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20 feet Figure 5. Potentiometric Surface Contour Map of the Steady State Model. All Fractures are Active. Block-to-Fracture Conductivity Ratio is 1:500. Fractures are Shown as Heavy Double Lines. Contour Interval is 0.1 Feet.

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21 indicating an increase in hydraulic gradient and flow velocity. Models with 25 percent of the fractures active were run at the same three block-to-fracture conductivity ratios. At a 1:10 conductivity ratio with 25 percent of the fractures active (Figure 6) the results are similar to the homogeneous case, but heads are about 50 feet lower. At lower block-to-fracture ratios the equipotential lines refract sharply around individual fractures. Heads in the 1:100-and 1:500-ratio models are progressively lower reflecting the higher transmissivity of the system as the fracture conductivity increases (Figures 7 and 8). For the case with 50 percent of the fractures active, the 1:10 block-to-fracture ratio gives results similar to those of the homogeneous case, but heads are about 100 feet lower. Equipotential lines refract slightly around fractures (Figure 9). The case with a 1:100 block-tofracture conductivity ratio and with 50 percent of the fractures active gives results in which heads are lower than the 1:10-ratio case, reflecting a higher transmissivity (Figure 10). The 1:500-ratio case (Figure 11) results in still lower heads and highly heterogeneous flow patterns. The case with 75 percent of the fractures active and a 1:10 conductivity ratio gives similar results to the homogeneous case with heads about 80 feet lower. Equipotential lines refract slightly around fractures (Figure 12). The 1:100-ratio case gives results in which

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22 1!0 1====----------110 I 140 100 2000 4000 6000 16000 Figure 6. Potentiometric Surface Contour Map of the SteadyState Model. Twenty-Five Percent of Fractures are Active. Block-to-Fracture Ratio is 1:10. Fractures are Shown as Heavy Solid Lines. contour Interval is 5 Feet.

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23 9\:1'-----feet Figure 7. Potentiometric surface Contour Map of the Steady State Model. Twenty-Five Percent of Fractures are Active. Block-to-Fracture Conductivity Ratio is 1:100. Fractures are Shown as Heavy Solid Lines. Contour Interval is 3 Feet.

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24 1 800 1600 1400 1200 1000 ...., C1J 800 600 S3 0 2000 4000 6000 8000 16000 feet Figure 8 Potentiometric Surface Contour Map of the SteadyState Model. Twenty-Five Percent of Fractures are Active. Block-to-Fracture Conductivity Ratio is 1:500. Fractures are Shown as Heavy Solid Lines. Contour Interval is 2 Feet.

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25 1------t--6 ----------1---------63----4-2000 4000 6000 8000 10000 12000 14000 16000 feet Figure 9. Potentiometric Surface Contour Map of the steady State Model. Fifty Percent of Fractures are Active. Blockto-Fracture Conductivity Ratio is 1:10. Fractures are Shown as Heavy Solid Lines. Contour Interval is 3 Feet.

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26 1 600 1400 1200 42 1000 Q) 800 600 400 200 0 2000 4000 6000 8000 10000 12000 1 6000 feet Figure 10. Pote ntiom e tric Surface Contour Map of the SteadyState Model. Fifty Percent of Fractures are Active. Blockto-Fracture Conductivity Ratio is 1 :100. Fractures are Shown as Heavy Solid Lines. Contour Interval is 2 Feet.

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27 1600 1400 1200 1000 800 600 400 200 0 2000 4000 6000 8000 10000 12000 1 4000 16000 fee t Figure 11. Potentiometric Surface Contour Map of the Steady State Model. Fifty Percent of Fractures are Active. Blockto-Fracture Conductivity Ratio is 1:500. Fractures are Shown a s Heavy Solid Lines. Contour Interval is 2 Feet.

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1800 IU' II II I I I I II 1600 IQ= = 1400 1()-p. 94 94--9< 1200 lo-1-lo--=--1------1---t---1000 800 lo881-----ss-----]...---r--.. 10-v=== 1-------.............. .... 1-v-at?---......._ v / v .. 1(}: / --r---. -10:: ---a, \ 1:: 0 1\\ \ \ II II I I I I II 600 400 200 0 2000 4000 6000 8000 10000 12000 14000 16000 feet 28 Figure 12. Potentiometric Surface Contour Map of the steady State Model. Seventy-Five Percent of Fractures are Active. Block-to-Fracture Conductivity Ratio is 1:10. Fractures are Shown as Heavy Solid Lines. Contour Interval is 3 Feet.

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29 heads are about 5 feet lower. Equipotentials in the 1:100ratio case refract sharply around fractures {Figure 13). The 1:500-ratio case displays heads about 50 feet lower than the 1:100-ratio case, reflecting the higher transmissivity. The system appears more heterogeneous than the higher ratios, and equipotential lines refract sharply around fractures (Figure 14). When the models with fractures which are 25% 50% and 75% active are compared, no trend is found with respect to heads. It would be expected that heads would decline as more fractures become active. To investigate this unexpected result, six models were run with 50% of the fractures active and the potentiometric surfaces were compared (Figure 15). Two of the mod els have random fracture geometries (Figures 15a and 15b). Two of the models were constructed so that one has more fractures oriented parallel to the regional gradient and one has more fractures oriented perpendicular to the regional gradient (Figures 15c and 15d). Figure 15e has fractures only perpendicular to the gradient and figure 15f 'has fractures only parallel to the gradient. The six models indicate a trend from lower heads in models with parallel-to-gradient fractures to higher heads in models with more perpendicularto-gradient fractures. Models with random fracture orientations have unpredictable head distributions.

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30 1 OUUlF II II I I I I II 1 6000= = 1 400()----__..... 1----200o-1 "------1 oooof-e1 t'---......---sooof-1'---Bi -cu f'\nnn. ---f.v-/ 4000= f" '\ / -2000:: / I= {\. ( (; \ 2o'oo 4o'oo 6doo ac:oo 10000 12000 14000 16000 feet Figure 13. Potentiometric Surface Contour Map of the SteadyState Model. Seventy-Five Percent of Fractures are Active. Block-to-Fracture Conductivity Ratio is 1:100. Fractures are Shown as Heavy Solid Lines. Contour Interval is 3 Feet.

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31 feel Figure 14. Potentiometric surface Contour Map of the steadyState Model. Seventy-Five Percent of Fractures are Active. Block-to-Fracture Conductivity Ratio is 1:500. Fractures are Shown as Heavy Solid Lines. Contour Interval is 1 Foot.

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32 ( A) (B) (C) (D) -66 1 --60 -S. .
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33 Transient Case For the transient cases, models were run at the same three conductivity ratios with fractures that are 2 5 % 50%, 75%, and 100% active. A discharge node representing a well is placed in the upper right corner cell. In order to locate the well in a fracture the eight columns on the right and the upper eight rows are deactivated (Figure 16). In the cases in which the well is in the center of a block four columns on the right and the upper four rows are deactivated. Drawdown data are taken from the cell containing the well. Time-drawdown data from the homogeneous case are plotted on a log-log scale and compared to the Theis (1935) type curve (Figure 17). The model-generated data match the Theis curve from early to late time. Time-drawdown data from some pumping tests of the Floridan aquifer match the Theis type curve from early to late time (Figure 18). In cases in which the data from the Floridan aquifer deviate from the Theis type curve, the behavior can be compared to that of the discrete-fracture model. Time-drawdown data from the transient models in which the well is located in a fracture display a characteristic three-segment curve which can be matched to the Theis (1935) type curve at early and late time (Figures 19 and 20).

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c :\groundwa\transl. gmo (Cell Status) Layer: l Figure 16. Finite-Difference Model Grid for the Transient Simulation. Highlighted Cells are Inactive. 34 2283.105 feet

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I 1 W ( u I -------Theis curve I Figure 17. Comparison of Time-Drawdown Data from the Homogeneous Case with the Theis (1935) Type curve. 35

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36 s I I T h e I S curve 1 /u Figure 18. Comparison of Time-Drawdown Data from the Floridan Aquifer with the Theis (1935) Type Curve. Data are from Plant City Well Field, Hillsborough Co., Florida (Wolansky and Corral, 1985).

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37 W ( u ) _! _______ t i The i s curve 1 I I I I I '--'---'-'--'-'--'--'-'-'-'-'--'-'-..LLJ.l.J 1 / u Figure 19. Comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type Curve. Block-to-Fracture Conductivity Ratio is 1:500. Simulated Well is Located in a Fracture. Match is at Late Time.

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I I I ---------r-1 \ f----The is cu r ve 1 /u 38 Figure 20. Comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type Curve. Block-to-Fracture Conductivity Ratio is 1:500. Simulated Well is Located in a Fracture. Match is at Early Time.

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39 In cases in which the well is located within a block, the time-drawdown plot lacks the three distinct segments but still matches the Theis (1935) type curve at early and late time (Figures 21 and 22). The model data fall above the Theis curve at early time when the match is at late time, and late-time data fall above the Theis curve when the match is at early time. Time-drawdown data from the Floridan aquifer in westcentral Florida can be compared to model-derived data for both fracture-pumped and block-pumped models. The shape of the curves is compared by overlaying log-log plots of timedrawdown data from the Floridan aquifer and from the models to obtain the best fit between the two curves. A nearly perfect match is obtained in comparing data from the blockpumped, fully active model with a 1:500 block-to-fracture conductivity ratio to field data from the Section 21 well field in Hillsborough County, Florida. The model data and field data match from early to late time (Figure 23). A good match is obtained when data from the fracturepumped model with 25 percent of the fractures active and a 1:500 block-to-fracture conductivity ratio are compared to field data from the Verna well field in Sarasota County, Florida. The data match well for late-time, but a lack of field data at early time prevents a complete match from early to late time (Figure 24).

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40 s The i s curve 1 /u Figure 21. Comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type Curve. Block-to-Fracture Conductivity Ratio is 1:500. Simulated Well is Located in a Block. Match is at Late Time.

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s W ( u ) I i ----f-41 curve 1 /u Figure 22. Comparison of Time-Drawdown Data from the Model of a Heterogeneous Aquifer with the Theis (1935) Type Curve. Block-to-Fracture Conductivity Ratio is 1:500. Simulated Well is Located in a Block. Match is at Early Time.

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D Model data 1 /u e Floridan aquifer data Figure 23. Comparison of Time-Drawdown Data from the Floridan Aquifer to Model-Generated Data. Model Block-toFracture Conductivity Ratio is 1:500. Simulated Well is Located in a Block. Field Data are From Section 21 Well Field, Hillsborough, Co., Florida (Wolansky and Corral, 1985). 42

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43 i ------W ( u ) Theis curve D Model data 1 /u Floridan aquifer dolo Figure 24. Comparison of Time-Drawdown Data from the Floridan Aquifer to Model-Generated Data. Model Block-toFracture Conductivity Ratio is 1:500. Twenty-Five Percent of Fractures are Active. Simulated Well is Located in a Fracture. Field Data are from Verna Well Field, Sarasota, Co., Florida {Wolansky and Corral, 1985).

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44 The fracture-pumped model with 75 percent of the fractures active and a 1:100 block-to-fracture conductivity ratio gives a good match at early time when compared to field data from Sixmile Creek test site of Hillsborough County (Figure 25). This test displays the three segments of the time-drawdown curve which are characteristic of the fracture-pumped models. Analytical Solution Results Values determined by the Bourdet-Gringarten curvematching method are given in Table 2. The values are similar to published values from the Floridan aquifer.

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W ( u ) Theis trve D Model data Floridan aquifer data 1 1 /u Figure 25. Comparison of Time-Drawdown Data from the Floridan Aquifer to Model-Generated Data. Model Block-toFracture Conductivity Ratio is 1:100. Seventy-Five Percent of Fractures are Active. Simulated Well is Located in a Fracture. Field Data are from sixmile Creek Test Site, Hillsborough, Co., Florida (Wolansky and Corral, 1985). 45

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Table 2. Bourdet-Gringarten Analytical Solution Results (Kruseman and deRidder, 1991) A. Model Data Blk/Frac. Ratio 1:100 1:10 1:100 1:500 1:10 1:100 1:500 1:10 1:100 1:500 B. Field Data Verna Well Field, Sarasota Co. Fla. %Active 100 25 25 25 50 50 50 75 75 75 Section 21 Well Field, Hillsborough, Co., Fla. T (early) ft2/d 40000 10000 40000 200000 20000 60000 70000 7000 40000 80000 20000 70000 Sixmile creek Test Site, Northeast Hillsborough Co., Fla. 400000 T (late) ft2/d N/A 4000 7000 6000 5000 8000 80000 N/A 100000 40000 0.5 0.01 0.05 0.05 0.01 0.1 0.5 0.3 0.04 0.2 0.01 2E-5 0.2 46

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47 5. DISCUSSION steady-state Model The simulation of a heterogeneous aquifer with dual porosity reveals that the system is sensitive to the blockto-fracture hydraulic-conductivity ratio. At a 1:10 conductivity ratio the system behaves similar to the homogeneous case on a regional scale. The flow regime is radial around the outflow node in both cases. The heterogeneous case is characterized by significantly lower heads caused by the increased transmissivity of the system due to the presence of the fracture zones (Figure 3). Systems which have 25%, 50%, and 75% of the fractures active also behave in a manner similar to the homogeneous case when the block-to-fracture conductivity ratio is 1:10 (Figures 6, 9, and 12). At a 1:10 conductivity ratio the calculated potentiometric surface looks like the homogeneous case. Consequently, the presence of fractures would be impossible to detect by analyzing a regional potentiometric map. At lower conductivity ratios, the system becomes more heterogeneous. Potentiometric contours refract sharply across fractures. The effect is most pronounced at some

PAGE 58

48 distance from the outflow node where the hydraulic gradient is relatively low. At a conductivity ratio of 1:500 local flow systems develop within each block in the case with 100 percent of the fractures active (Figure 5). Flow is radial toward the fractures from the center of each block. Near the outflow node the hydraulic gradient is sufficiently high to prevent local flow systems from developing, and flow is still radial toward the outflow node in this region of the model. Heads in this case are extremely low compared to the homogeneous case, reflecting a very high transmissivity. Local flow systems do not develop in systems in which less than 100 percent of the fractures are active. systems that have less than 100 percent of the fractures active are strongly affected by the geometry of the fracture network, which is usually unknown and unpredictable. When the block-to-fracture conductivity ratio is low (1:100 or 1:500) the flow regime is no longer symmetrical around the outflow node but is distorted by the fractures. The average head in the system is strongly affected by the particular geometry of the fracture network. If fractures are preferentially oriented parallel to the regional direction of flow, lower heads result (Figures 15c,15f). The opposite occurs when fractures are oriented perpendicular to the regional flow direction (Figures 15d,15e). When the geometry of the fracture network is randomly generated, the effect is unpredictable (Figures

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49 15a, 15b). Thus, systems which have higher percentages of active fractures (and supposedly higher regional transmissivity) can have higher heads than systems with fewer fractures, indicating a lower regional transmissivity. The head distribution is therefore affected by the geometry of the fracture network and by the location of the discharge node within the fracture network. This effect would make the interpretation of the potentiometric surface produced from field data much more difficult as a result of the unknown nature of the fracture configuration. Modeling of a specific region is made more difficult by this effect due to the difficulty of determining the particular geometry of the fracture network. Enormous amounts of head data and geophysical data would be needed to give a sufficiently detailed description of the area to allow a unique discretefracture model to be constructed. A stochastic approach would probably be more effective when simulating a specific region with this type of model. Transient Model Transient simulations were run to determine the response of the discrete-fracture system to a pumping well in the absence of recharge. The main objective was to

PAGE 60

50 determine if the discrete-fracture approach can be applied to the Floridan aquifer of west-central Florida. Timedrawdown curves for field data and model-generated data were compared to see if the model results are similar to the observed response of the Floridan aquifer. No attempt was made to choose model parameters to directly approximate the Floridan aquifer. However, comparison of time-drawdown curves is possible on the basis of shape because a change in transmissivity will simply shift a time-drawdown plot up or down the y-axis, and changes in storativity will shift the plot along the x-axis. The overall shape of the plot will not change. Many time-drawdown curves from the Floridan aquifer match a Theis (1935) type curve from early to late time (Figure 18). In cases where they do not, however, the discrete-fracture models do behave similarly to the Floridan aquifer. When data from the block-pumped model with 100 percent of the fractures active and a 1:500 block-tofracture conductivity ratio are compared to data from a pumping test from Section 21 well field in Hillsborough county, Florida, the field data and model data match from early to late time (Figure 23). Both field data and model data deviate from the Theis curve. The plot of the data from the fracture-pumped model with 25 percent of the fractures active and a 1:500 blockto-fracture conductivity ratio displays a three-segmented time-drawdown curve. This curve is typical for a dual-

PAGE 61

51 porosity system. The drawdown at early time represents water derived from storage in the fractures. At late time the water is derived from storage in the blocks, which constitutes most of the storage in the system. When compared to field data from the Verna well field in Sarasota county, Florida, the data match well from the end of the transition from fracture storage to block storage through late time (Figure 24). A lack of early-time data in the field data prevent a complete match, but the systems do show similar behavior at late time. The plot of the data from the fracture-pumped model with 75 percent of the fractures active and a 1:100 blockto-fracture conductivity ratio displays a three-segmented time-drawdown curve. When this case is compared to field data from the Sixmile Creek test site in Hillsborough county, Florida, the two data sets match from early time through the transition portion of the curve and into the steeper portion of the curve which continues into late time (Figure 25). A lack of field data at late time prevents a complete match, but all three segments of the model curve are represented in the field data. It is not possible to uniquely determine the block-tofracture conductivity ratio and the geometry of the fracture network by matching model-generated data to pumping-test data because of the non-unique nature of the head distribution for any given combination of fracture geometry

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52 and conductivity ratio (as illustrated in Figure 15). Analytical Solution The Bourdet-Gringarten method of curve matching was used to obtain transmissivity and storativity values from model and field data. This method yields values for transmissivity of the fractures at early time, transmissivity of the combined block-fracture system at late-time, storativity in fractures at early time and a combined storativity for blocks and fractures at late time. A more extensive discussion of the application of analytical solutions to dual-porosity systems can be found in Powers (1992). The results of the Bourdet-Gringarten solutions demonstrate that the system becomes more transmissive as the block-to-fracture conductivity ratio decreases, and is also affected by the level of interconnection of the fracture network. The values obtained from the analytical solutions are generally well within the range of published values for the Floridan aquifer of west-central Florida. The difference between transmissivity values obtained from early-time and late-time matches increases as the block-tofracture conductivity ratio decreases, indicating that the system is behaving more heterogeneously.

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53 6. CONCLUSIONS The steady-state models demonstrate that a heterogeneous aquifer with dual porosity is sensitive to the block-to-fracture hydraulic-conductivity ratio and to the geometry of the fracture network. The recognition of such a system in the field is problematic because of the quantity of field data that would be necessary to describe the system in sufficient detail. At lower block-to-fracture conductivity ratios, the overall configuration of the potentiometric surface is often similar to the homogeneous case with radial flow towards the outflow node. The configuration of the system is highly dependent on the particular geometry of the fracture network, which is rarely, if ever, known. Fractures oriented parallel to the regional direction of flow create systems with lower heads compared to systems with fractures which are perpendicular to the regional direction of flow. The time-drawdown data from the transient simulation of the heterogeneous aquifer displays the three curve segments expected in a dual-porosity system. Transient behavior is affected by the proximity of the well to a fracture, the distribution and orientation of active fractures, the

PAGE 64

percentage of active or open fractures, and the block-tofracture conductivity ratio. 54 Time-drawdown data from pumping tests of the Floridan aquifer often match the Theis type curve, which suggests that the Floridan aquifer usually behaves like a homogeneous aquifer. The behavior of the Floridan aquifer sometimes deviates from Theis behavior. Non-Theis behavior in the Floridan aquifer can be successfully approximated by the discrete-fracture approach. Pumping-test data from the Floridan aquifer that deviate from Theis behavior are possibly the result of wells located in or near fracture zones.

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55 7. LIST OF REFERENCES Barenblatt, G.I., I.P. Zheltov, and N. Kochina, 1960, Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. Journal of Applied Mathematics and Mechanics, v. 24, p. 1286-1303. Blanchet, P.H., 1957, Development of fracture analysis as an exploration method. Am. Assoc. Pet. Geol. Bull., v. 41, p.1748-1759. Boulton, N.S. and T.D. Streltsova, 1978b, Unsteady flow to a pumped well in an unconfined fissured aquifer. Journal of Hydrology, v. 37, p. 349-363. Domenico, P.A. and F.W. Schwartz, 1990, Physical and Chemical Hydrogeology. John Wiley and Sons, Inc., 824 p. Esling, S.P. and T.A. Larson, 1993, Graphic Groundwater. Micro-innovations, Inc. Faulkner, G.L., 1973b, Geohydrology of the Cross-Florida Barge canal area with special reference to the Ocala vicinity: u.s. Geological Survey Water Resources Investigations Report 1-73. Kruseman, G.P. and N.A. deRidder, 1991, Analysis and Evaluation of Pumping Test Data, 2nd edition. Pub. 47, International Institute for Land Reclamation and Improvement. Wageningen, The Netherlands, 377 p. Lattman, L.H. and R.R. Parizek, 1964, Relationship between fracture traces and the occurrence of ground water in carbonate rocks. Journal of Hydrology, v. 2, p. 73-91.

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56 McConnel, C.L., 1993, Double porosity well testing in the fractured carbonate rocks of the ozarks. Groundwater, v. 31, p. 75-83. McDonald, M.G. and A.W Harbaugh, 1988, A Modular ThreeDimensional Finite Difference Ground-water Flow Model: u.s. Geological Survey Open File Report 83-875. Moench, A.F., 1984, Double-porosity models for a fissured ground-water reservoir with fracture skin. water Resources Research, v. 20, p. 831-846. Moore, D.L. and M. stewart, 1983, fracture zones in a karst aquifer. v. 61, p. 325-340. Geophysical signatures of Journal of Hydrology, Parizek, R.R., 1976, On the significance of fracture traces and lineaments in carbonate and other terranes: in Karst Hydrology and Water Resources, proceedings U.S.-Yugoslav Symp.l, Dubrovnik, 1975, Water Resources Publications. Powers, J.A., 1992, Numerical and Analytical Modeling of Double-Porosity Aquifers. Master's thesis: University of South Florida, Tampa, Florida. 65 p. Stewart, M. and J. Wood, 1990, Geologic and geophysical character of fracture zones in a tertiary carbonate aquifer, Florida. In: Geotechnical and Environmental Geophysics, S.T. Ward (ed.), Society of Exploration Geophysics, Investigations in Geophysics, v. 2, pp. 235-244. Streltsova, T.D. 1976. Hydrodynamics of ground-water flow in a fractured formation. Water Resources Research, 12 (3), pp. 405-414. Theis, c.v., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage., Trans. Am. Geophys. Union. Annual Meeting, 16th, pp. 519-524. W ren J E and P J Root, 1963. The behavior of naturally ar . fractured reservoirs. Soc. Eng. Jour., vol. 3, pp. 245-255

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57 Witherspoon, P.A., Y.W. Tsang, J.C.S. Long, and J. Noorishad, 1981. New approaches to problems of fluid flow in fractured rock masses. Proc. 22nd u.s. symp. Rock Mechs., Boston, Mass., pp. 1-20. Wolansky, R.M. and M.A. Corral, Jr., 1985. Aquifer tests in west-central Florida, 1952-76: u.s. Geological survey WaterResources Investigations Report 84-4044.


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