Groundwater HydrologyConceptual and Computational Models

K.R. RushtonEmeritus Professor of Civil Engineering

University of Birmingham, UK

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Groundwater Hydrology

Groundwater HydrologyConceptual and Computational Models

K.R. RushtonEmeritus Professor of Civil Engineering

University of Birmingham, UK

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Preface xiii

1 Introduction 11.1 Groundwater Investigations – a Detective Story 11.2 Conceptual Models 21.3 Computational Models 21.4 Case Studies 31.5 The Contents of this Book 31.6 Units, Notation, Journals 5

Part I: Basic Principles 7

2 Background to Groundwater Flow 92.1 Introduction 92.2 Basic Principles of Groundwater Flow 9

2.2.1 Groundwater head 102.2.2 Direction of flow of groundwater 102.2.3 Darcy’s Law, hydraulic conductivity and permeability 112.2.4 Definition of storage coefficients 132.2.5 Differential equation describing three-dimensional time-variant groundwater flow 14

2.3 One-dimensional Cartesian Flow 142.3.1 Equation for one-dimensional flow 152.3.2 Aquifer with constant saturated depth and uniform recharge 162.3.3 Definition of transmissivity 172.3.4 Aquifer with constant saturated depth and linear variation in recharge 182.3.5 Aquifer with constant saturated depth and linear decrease in recharge towards lake 192.3.6 Confined aquifer with varying thickness 202.3.7 Unconfined aquifer with saturated depth a function of the unknown groundwater head 202.3.8 Time-variant one-dimensional flow 22

2.4 Radial Flow 232.4.1 Radial flow in a confined aquifer 232.4.2 Radial flow in an unconfined aquifer with recharge 242.4.3 Radial flow in an unconfined aquifer with varying saturated depth 262.4.4 Radial flow in a leaky aquifer 272.4.5 Time-variant radial flow 282.4.6 Time-variant radial flow including vertical components 28

2.5 Two-dimensional Vertical Section (Profile Model) 282.5.1 Steady-state conditions, rectangular dam 282.5.2 Time-variant moving water table 31

2.6 Regional Groundwater Flow 322.6.1 Analysis of Connorton (1985) 32

Contents

vi Contents

2.6.2 Illustrative numerical examples of one-dimensional flow 342.6.3 When vertical flows must be included explicitly 35

2.7 Numerical Analysis 362.7.1 One-dimensional flow in x-direction 362.7.2 Example 372.7.3 Radial flow 382.7.4 Vertical section 38

2.8 Monitoring and Additional Indirect Evidence 392.8.1 Introduction 392.8.2 Monitoring groundwater heads 402.8.3 Surface water monitoring 432.8.4 Monitoring borehole discharge 462.8.5 Monitoring groundwater quality 462.8.6 Data handling and presentation 47

2.9 Introduction to Quality Issues 492.9.1 Introduction 492.9.2 Fresh and saline groundwater 492.9.3 Conditions at the coast 512.9.4 Upconing 532.9.5 Monitoring the movement of a saline interface 542.9.6 Hydrodynamic dispersion and contaminant transport 56

2.10 Concluding Remarks 58

3 Recharge due to Precipitation or Irrigation 603.1 Introduction 60

3.1.1 Representative field situations 603.1.2 Symbols used in this chapter 61

3.2 Brief Review of Alternative Methods of Estimating Recharge 613.2.1 Methods based on field measurements 623.2.2 Estimates using properties of unsaturated soil 62

3.3 Conceptual models for the Soil Moisture Balance Technique 643.3.1 Introduction 643.3.2 Representation of moisture conditions in the soil 643.3.3 Bare soil evaporation 673.3.4 Crop transpiration 683.3.5 Nature of the soil 703.3.6 Runoff 703.3.7 Occurrence of recharge 703.3.8 Combining crops and bare soil 713.3.9 Water storage near the soil surface 71

3.4 Quantified Conceptual and Computational Models for the Soil Moisture Balance Technique 743.4.1 Crops and crop coefficients 743.4.2 Reduced transpiration when crops are under water stress 753.4.3 Reduced evaporation due to limited soil water availability 763.4.4 Runoff 773.4.5 Combining the influence of crop transpiration and bare soil evaporation 783.4.6 Soil moisture balances 783.4.7 Soil moisture balances with near surface soil storage 813.4.8 Algorithms for soil moisture balance 823.4.9 Annual soil moisture balance for a temperate climate 833.4.10 Estimating recharge in a semi-arid region of Nigeria 843.4.11 Estimating recharge due to precipitation and irrigation in a semi-arid region of India 87

3.4.12 Bypass recharge 903.4.13 Estimating catchment-wide recharge distributions 90

3.5 Estimating Recharge when Drift is Present 903.5.1 Introduction 903.5.2 Recharge factors 913.5.3 Recharge through Drift estimated using Darcy’s Law 923.5.4 Delay in recharge reaching the water table 93

3.6 Delayed Runoff and Runoff Recharge 953.6.1 Delayed runoff from minor aquifers in the Drift 953.6.2 Runoff recharge 973.6.3 Incorporating several recharge processes 98

3.7 Concluding Remarks 98

4 Interaction between Surface Water and Groundwater 1004.1 Introduction 1004.2 Canals 101

4.2.1 Introduction 1014.2.2 Classification of interaction of canals with groundwater 1014.2.3 More detailed consideration of boundary conditions and canal dimensions 1034.2.4 Effect of lining of canals 1064.2.5 Perched canals 1084.2.6 Estimation of losses from canals 1114.2.7 Artificial recharge using spreading techniques 111

4.3 Springs, Rivers, Lakes and Wetlands 1124.3.1 Introduction 1124.3.2 Spring–aquifer interaction 1134.3.3 River–aquifer interaction: basic theory and modelling 1154.3.4 River–aquifer interaction in practice 1164.3.5 Lakes and wetlands 1214.3.6 Impact of groundwater abstraction on springs, rivers, lakes and wetlands 125

4.4 Drains 1264.4.1 Introduction 1264.4.2 Field evidence 1264.4.3 Formulation of drainage problems 1274.4.4 Approximate methods of analysis for steady-state problems 1294.4.5 Numerical solutions 1314.4.6 Representative time-variant analysis 1344.4.7 Interceptor drains 135

4.5 Irrigated Ricefields 1384.5.1 Introduction 1384.5.2 Formulation of flow processes in ricefields 1394.5.3 Representative results 1404.5.4 Ricefields with different geometry and water levels 141

4.6 Concluding Remarks 142

Part II: Radial Flow 145

5 Radial Flow to Pumped Boreholes – Fundamental Issues 1475.1 Aquifer Response due to Pumping from a Borehole 147

5.1.1 Introduction 1475.1.2 Details of field study 147

Contents vii

viii Contents

5.1.3 Measurements in pumped borehole and observation piezometers 1485.1.4 Conceptual models of flows for different times 1495.1.5 Recovery phase 1515.1.6 Results for open borehole instead of individual piezometers 1525.1.7 Analysis of pumping test results 1535.1.8 Summary 154

5.2 Classification of Radial Flow Problems 1545.3 Steady Radial Flow in Confined Aquifers 1575.4 Analytical Solutions for Unsteady Radial Flow in Confined Aquifers 158

5.4.1 Analytical solutions 1585.4.2 Theis method using log-log graphs 1595.4.3 Cooper–Jacob technique 1605.4.4 Transmissivity and storage coefficient varying with radius 161

5.5 Numerical Solution for Unsteady Radial Flow 1625.5.1 Theoretical basis of the numerical model 1625.5.2 Comparison of analytical and numerical solutions 1635.5.3 Example of the use of the numerical model 1635.5.4 Representation of well losses 164

5.6 Analysis of the Recovery Phase for Unsteady Radial Flow in Confined Aquifers 1645.7 Comparison of Analytical and Numerical Techniques of Pumping Test Analysis 1655.8 Analysis of Leaky Aquifers Without Storage in the Aquitard 1675.9 Further Consideration of Single-Layer Aquifers 1685.10 Aquifer with Restricted Dimensions (Boundary Effects) 1695.11 Change in Transmissivity or Storage Coefficient with Radius 1705.12 Changing Saturated Depth and Changing Permeabilities in Unconfined Aquifers 1735.13 Varying Abstraction Rates 1745.14 Overflowing Artesian Boreholes 1745.15 Interfering Boreholes 176

5.15.1 Introduction to interference 1765.15.2 Theoretical analysis of interfering boreholes 1775.15.3 Practical example of the significance of interfering boreholes 178

5.16 Conditions Changing between Confined and Unconfined 1805.17 Delayed Yield 183

5.17.1 Background 1835.17.2 Inclusion of delayed yield in radial flow numerical model 1845.17.3 Comparison of analytical and numerical solutions of a field example

with delayed yield 1855.18 Concluding Remarks 185

6 Large Diameter Wells 1876.1 Introduction 1876.2 Description of Flow Processes for Large Diameter Wells 1876.3 Analytical Solutions for Large Diameter Wells 189

6.3.1 Conventional analyses 1896.3.2 Drawdown ratio method 1896.3.3 Alternative methods including Kernel function techniques 191

6.4 Numerical Analysis of Large Diameter Well Tests using Observation Well Data 1936.5 Analysis with Varying Abstraction Rates 1946.6 Use of Large Diameter Wells for Agriculture 195

6.6.1 Representative problem 1966.6.2 Agrowells in Sri Lanka 198

6.6.3 Case study of a Miliolite limestone aquifer 1996.7 Concluding Remarks 200

7 Radial Flow where Vertical Components of Flow are Significant 2017.1 Introduction 2017.2 Radial-Vertical Time-variant Flow [r, z, t] 203

7.2.1 Mathematical formulation 2037.2.2 Analytical solutions 2047.2.3 Numerical methods 2067.2.4 Examples of numerical solutions in [r, z, t] 206

7.3 Radial-Vertical Time-instant [r, z] 2097.3.1 Principles of the approach 2097.3.2 Case study: reduction of discharge due to partial penetration of borehole 2097.3.3 Case study: effectiveness of water table control using tubewells 210

7.4 Two-zone Approximation [r, t, vZ] 2137.4.1 Introduction 2137.4.2 Examples of formulation using the two-zone model 2137.4.3 Details of the two-zone model 2147.4.4 Discrete space–discrete time equations for the two-zone model 2167.4.5 Solution of simultaneous equations 2187.4.6 Examples of the use of the two-zone model 218

7.5 Inclusion of Storage in Aquitards [r, t: z, t] 2237.5.1 Introduction 2237.5.2 Analytical solutions 2237.5.3 Case study: influence of aquitard storage on an aquifer system; increase

and subsequent decrease in flows from the aquitard 2247.5.4 Influence on aquitard storage of pumping from sandstone aquifers 226

7.6 Concluding Remarks 227

8 Practical Issues of Interpreting and Assessing Resources 2288.1 Introduction 2288.2 Step Drawdown Tests and Well Losses 228

8.2.1 Introduction 2288.2.2 Confined, leaky or unconfined conditions 2308.2.3 Estimating the coefficients B and C 2308.2.4 Exploring well losses 2318.2.5 Causes of well loss 2318.2.6 Field examples of well losses 235

8.3 Packer Testing to Identify variations in Hydraulic Conductivity with Depth 2368.3.1 Conducting packer tests 2368.3.2 Interpretation of packer tests using analytical expressions 2378.3.3 Do the analytical solutions provide a reasonable approximation to hydraulic

conductivity variations? 2388.3.4 Effectiveness of fissures in collecting water from the aquifer 2408.3.5 Comparison of properties of sandstone aquifers based on cores, packer

testing and pumping tests 2418.3.6 Slug tests 241

8.4 Information about Groundwater Heads in the Vicinity of Production Boreholes 2428.4.1 Background 2428.4.2 Case Study: identifying the water table elevation 243

8.5 Realistic Yield from Aquifer Systems 2458.5.1 Introduction 245

Contents ix

x Contents

8.5.2 Weathered-fractured aquifers 2468.5.3 Alluvial aquifers with an uppermost layer of low hydraulic conductivity 2478.5.4 Response of an alluvium-sandstone aquifer system 252

8.6 Injection Wells and Well Clogging 2548.6.1 Introduction 2548.6.2 Alluvial aquifer in India 2548.6.3 Initial pumping test 2568.6.4 Artificial recharge results and interpretation 2578.6.5 North London Artificial Recharge Scheme 258

8.7 Variable Hydraulic Conductivity with Depth in Chalk and Limestone 2598.7.1 Introduction 2598.7.2 Case study in Berkshire Downs, the UK 2598.7.3 Consequences of variation in hydraulic conductivity 261

8.8 Horizontal Wells 2628.8.1 Collector wells 2628.8.2 Mathematical expressions for horizontal wells 2638.8.3 Horizontal well in a shallow coastal aquifer 264

8.9 Concluding Remarks 267

Part III: Regional Groundwater Flow 269

9 Regional Groundwater Studies in which Transmissivity is Effectively Constant 2719.1 Introduction 2719.2 Nottinghamshire Sherwood Sandstone Aquifer 271

9.2.1 Identifying the conceptual model, focus on recharge components 2719.2.2 Idealisations introduced in the regional groundwater model 2749.2.3 Quantifying the parameters of the conceptual model 2749.2.4 Numerical groundwater model 2769.2.5 Adequacy of model 2769.2.6 Flow balances 278

9.3 Northern Extension of Nottinghamshire Sherwood Sandstone Aquifer 2799.3.1 Brief description of groundwater catchment 2809.3.2 Conceptual models 2819.3.3 Numerical groundwater model and flow balances 282

9.4 Lower Mersey Sandstone Aquifer 2849.4.1 Conceptual model 2849.4.2 Recharge through drift 2869.4.3 Saline water 2879.4.4 Numerical groundwater model 2899.4.5 Flow balances and predictions 291

9.5 Barind Aquifer in Bangladesh 2929.5.1 Background 2939.5.2 Development of conceptual models 2939.5.3 Can this rate of abstraction be maintained? 2969.5.4 Possible provision of a regional groundwater model 297

9.6 Concluding Remarks 298

10 Regional Groundwater Flow in Multi-Aquifer Systems 29910.1 Introduction 29910.2 Mehsana Alluvial Aquifer, India 299

10.2.1 Introduction 300

10.2.2 Description of the aquifer system 30110.2.3 Field records of groundwater head 30110.2.4 Flow processes in aquifer system 30410.2.5 Mathematical model of a vertical section 30510.2.6 Origin of flows as determined from vertical section model 30710.2.7 More detailed study of smaller area 30810.2.8 Concluding discussion 310

10.3 Vanathavillu Aquifer System, Sri Lanka 31110.3.1 Introduction 31110.3.2 Aquifer parameters 31210.3.3 Groundwater head variations and estimates of aquifer resources 313

10.4 San Luis Potosi Aquifer System, Mexico 31510.4.1 Shallow aquifer system 31610.4.2 Deeper aquifer system 31610.4.3 Input of deep thermal water: 31810.4.4 Further considerations 319

10.5 Bromsgrove Sandstone Aquifer, UK 31910.5.1 Summary of field information 32010.5.2 Conceptual model 32210.5.3 Mathematical model 32210.5.4 Presentation of model outputs 32310.5.5 Management issues 325

10.6 Further examples where vertical components of flow are significant 32610.6.1 Madras aquifer 32610.6.2 Waterlogging in Riyadh, Saudi Arabia 32710.6.3 SCARPS: Saline Control and Reclamation Projects in Pakistan 32810.6.4 Fylde aquifer, UK 329

10.7 Concluding Remarks 331

11 Regional Groundwater Flow with Hydraulic Conductivity Varying with Saturated Thickness 33211.1 Introduction 33211.2 Chalk Aquifer of the Berkshire Downs 333

11.2.1 Changing estimates of yields of the Lambourn Valley catchment 33411.2.2 Conceptual and mathematical modelling 335

11.3 Southern Lincolnshire Limestone 33711.3.1 General description of Southern Lincolnshire Limestone catchment 33711.3.2 Recharge including runoff-recharge 33811.3.3 Surface water–groundwater interaction 34111.3.4 Wild boreholes 34311.3.5 Variable hydraulic conductivity with depth 34311.3.6 Résumé of conceptual models 34511.3.7 Brief description of the total catchment models 34611.3.8 Selected results and insights from the numerical model 347

11.4 Miliolite Limestone Aquifer in Western India 35011.5 Gipping Chalk Catchment, Eastern England 351

11.5.1 Conceptual model 35111.5.2 Quantified conceptual model 353

11.6 Further Examples 35411.6.1 South Humberside Chalk 35411.6.2 Candover Augmentation Scheme 35411.6.3 Hesbaye aquifer, Belgium 357

11.7 Concluding Remarks 358

Contents xi

xii Contents

12 Numerical Modelling Insights 36012.1 Introduction 36012.2 Representation of Rivers 360

12.2.1 Intermittent rivers 360361

12.3 Representing Boreholes Pumping Water from Multi-layered Aquifers 36412.4 Time-Instant Conditions 366

12.4.1 Introduction 36612.4.2 Basis of time-instant approach 36712.4.3 Examples of sandstone and limestone aquifers 36712.4.4 Time-instant solutions 368

12.5 Initial Conditions 36812.5.1 Specified heads or specified flows 36812.5.2 Initial conditions for Sandstone and other high storage aquifers 36912.5.3 Initial conditions for aquifers with seasonal changes in transmissivity 370

12.6 Dimensions and Detail of Numerical Models 37212.6.1 Identification of the area to be represented by a numerical model 37212.6.2 Identification of the duration of a numerical model simulation 37412.6.3 External boundary conditions 37612.6.4 Estimating parameter values for a numerical model 37712.6.5 Refinement of numerical groundwater models 37812.6.6 Sensitivity analysis 37912.6.7 Preparing exploratory groundwater models with limited field information 379

12.7 Predictive Simulations 38112.7.1 Issues to be considered 38112.7.2 Representative example 382

12.8 Evaluation of Conceptual and Computational Models 38412.8.1 Approach to groundwater modelling 38412.8.2 Monitoring 38412.8.3 Recharge 38512.8.4 Model calibration and refinement 38512.8.5 Sustainability, legislation and social implications 38612.8.6 Climate change 38712.8.7 Substantive issues requiring further investigation 387

Appendix: Computer Program for Two-zone Model 389

List of Symbols 397

References 399

Index 408

12.2.2 Rivers with significant seasonal changes in stage and wetted perimeter

It was more than twenty years ago when visiting Indiathat I was asked the question, ‘How should a ground-water investigation be planned?’ At that time I had difficulty in giving a convincing answer, but in the inter-vening years, with involvement in many challengingpractical groundwater studies, the important issueshave become clearer. The key is understanding beforeanalysis; this is reflected by the use of the words con-ceptual and computational models in the title of thisbook.

The number of groundwater investigations through-out the world continues to increase. The objectives ofthese investigations are varied, including meeting regu-latory requirements, exploring the consequences ofgroundwater development and rectifying the results ofover-exploitation of groundwater resources. Althoughhydrogeologists and water resource engineers workingon these projects may not carry out the analytical andnumerical analysis themselves, it is vital that theyunderstand how to develop comprehensive quantifiedconceptual models and also appreciate the basis ofanalytical solutions or numerical methods of modellinggroundwater flow. The presentation of the results ofgroundwater investigations in a form that can be under-stood by decision makers is another important task.This book is designed to address these issues. The firsttask in every investigation is to develop conceptualmodels to explain how water enters, passes through andleaves the aquifer system. These conceptual models arebased on the interpretation of field data and otherinformation. Second, techniques and methodologiesare required to analyse the variety of flow processesidentified during the conceptual model development. Aconsiderable number of computational models areavailable.

Developments of both conceptual and com-putational models for groundwater hydrology havecontinued from early in the twentieth century to thepresent day. Initially, computational models relied onanalytical methods but there is now a greater use of numerical models. Computational models foranalysing groundwater problems are the subject of

many articles in the literature. However, conceptualmodels are not discussed as widely. Real groundwaterproblems are frequently so complex that they can onlybe analysed when simplifying assumptions are intro-duced. Imagination and experience are required toidentify the key processes which must be included inconceptual and computational models. Furthermore,the selection of appropriate aquifer parameters is notstraightforward. A computational model does not need to be complex, despite the availability of modelcodes which include both saturated and unsaturatedflow conditions with the option of a large number ofthree-dimensional mesh subdivisions and numeroustime steps. Simpler, more flexible analytical or numerical models are often suitable for the early stages of modelling; more complex models can beintroduced when there is confidence that the importantfeatures of the aquifer system have been recognised. Inan attempt to indicate how the key processes are identified and aquifer parameters are selected, morethan fifty major case studies are included in this book.

Case studies illustrate how crucial insights are gainedwhich lead to a breakthrough in identifying the impor-tant flow processes which must be incorporated in theconceptual models. Comparing and contrasting anaquifer system with other field problems has proved tobe of immense benefit. A further advantage of casestudies is that they can indicate appropriate aquiferparameter values. For example, selecting suitable valuesfor the effective vertical hydraulic conductivities of lowpermeability strata is notoriously difficult. However,experience gained from other investigations in similarsituations often provides suitable first estimates. Due todifferences in climate, geology and the way in whichgroundwater is utilised, the reader may find it difficultto appreciate the significance of some of the casestudies. Yet much can be learnt from groundwaterstudies in other countries. Although the reader maynever be involved in studying losses from lined canalsor irrigated ricefields, insights gained from these inves-tigations can be transferred to other projects involving

Preface

xiv Preface

of a wide diversity of problems. Some of my colleaguescan be identified from more than sixty co-authors listedin papers in the References, but there are many moreindividuals and organisations with whom I have beenfortunate to collaborate.

surface water–groundwater interaction and rechargeestimation.

Groundwater studies involve teamwork, respectinginsights from different disciplines and sharing in uncer-tainties and ambiguities. I am most grateful to themany colleagues with whom I have shared the challenge

1.1 GROUNDWATER INVESTIGATIONS – A DETECTIVE STORY

Many of the basic principles of the flow of ground-water through aquifers were established by the early1970s. In addition, numerous analytical solutions wereavailable. At that time, numerical modelling of ground-water flow was also developing although limited com-puting power restricted applications. Since the 1970sthere have been significant advances in:

• broadening the understanding of important flowprocesses in a wide variety of aquifer systems,

• developing and applying analytical and numericaltechniques to represent these flow processes.

There are many different flow processes in aquifersystems. Inflows occur due to recharge which can resultfrom rainfall, runoff or groundwater–surface waterinteraction. Outflows occur from natural features suchas springs and rivers and through wells and boreholes.The actual flow processes may be complex when, forexample, a borehole takes different proportions of thedischarge from a number of layers within the aquifersystem. Flow paths within an aquifer are frequentlycomplicated with horizontal and vertical components;these flows may be strongly influenced by layering inthe aquifer system. Furthermore, the response of anaquifer with substantial seasonal variations in satu-rated thickness is different from one in which the seasonal water table fluctuations are small compared to the total saturated depth. Additional uncertaintiesoccur when deciding how large an area needs to bestudied. Although the purpose of the investigation mayrelate to the impact of changed abstraction in an areaof a few square kilometres, the aquifer response maybe influenced by hundreds, if not thousands, of squarekilometres of the aquifer system. Recognising and

resolving these issues is the key to conceptual modeldevelopment. The conceptual model must represent allthese flow processes; estimates of the magnitudes of allthe relevant parameters are also required.

Having developed a conceptual model, computa-tional models are used to test whether the flow processesof the conceptual model are plausible and the para-meter values are appropriate so that the computationalmodel replicates observed field responses. As a firststep, it is often helpful to simplify the physical setting,perhaps eliminating one or two coordinate dimensionsor considering a pseudo-steady state. These idealisedproblems can be studied using analytical models orsimple numerical models. Initial studies can be followedby the development of complex regional groundwatermodels. Insights from computational models often leadto modifications of conceptual models.

Although each groundwater study has its own specialfeatures, there has been a growing consensus about theimportance of certain flow processes. For example, inmany early studies, vertical flows through strata oflower permeability were considered to be sufficientlysmall to be neglected. Then, a number of investigationsin very different hydrogeological settings identified theimportance of vertical flow components through lowpermeability strata. Failure to include these verticalflow components explains why certain earlier resourcestudies provided unreliable predictions.

The purpose of this book is to describe advances inconceptual and computational modelling. The con-tents of the book are selected to introduce a wide rangeof issues which are important in practical groundwaterstudies. Many of the examples are taken from studiesin which the author was involved, although there iscross-referencing to other similar studies. One advan-tage of using case studies based on the author’s expe-rience is that information can be presented about initialmisunderstandings and invalid concepts.

1

Introduction

Groundwater Hydrology: Conceptual and Computational Models. K.R. Rushton© 2003 John Wiley & Sons Ltd ISBN: 0-470-85004-3

2 Groundwater hydrology

in the construction of the shallow piezometers. Conse-quently the original analysis was carried out assumingthat the fractured zone acts as a confined aquifer.However, preparation of a conceptual sketch of flowsthrough the aquifer system indicates that substantialflows occur from the weathered to the fractured zone.This key feature of the weathered-fractured system wasoverlooked by moving straight to analysis and ignor-ing ‘inconvenient’ data.

1.3 COMPUTATIONAL MODELS

The phrase ‘computational models’ is used to empha-sise that a wide variety of alternative models are avail-able. Darcy’s Law is the fundamental computationalmodel; not only is it included in many of the govern-ing equations for groundwater flow but it can also beused for preliminary calculations such as vertical flowsthrough low permeability strata. The diversity of com-putational models is illustrated by considering radialflow. There are many analytical models for radial flowto pumped boreholes including the Thiem equation forsteady-state flow in a confined aquifer, the time-variantTheis equation, classical leaky aquifer theory and manymore complex analyses which may include vertical flowcomponents. However, in all of these analytical models,assumptions are introduced so that mathematicalexpressions can be derived. These assumptions areoften discounted or disregarded by those carrying outthe analysis. As an alternative, radial flow through anaquifer system can be represented by numerical modelsbased on the finite difference approximation to the dif-ferential equations; fewer assumptions are required inderiving the numerical models. On what basis should acomputational model be selected? If the objective ofthe study is to estimate aquifer parameters from apumping test, the analytical solution which mostclosely represents the field situation can be comparedwith field data. Originally type curves were preparedfor graphical matching with field information but com-puter software is now widely available to simplify theprocedure. However, the ease of using computer soft-ware for matching field data to the analytical solutionsoften hides the inconsistencies between the assump-tions of the analytical model and the real field con-ditions. Although less convenient to use, numericalradial flow models often represent more closely theactual behaviour of the aquifer system.

Real groundwater problems are three-dimensional inspace and time-variant. Due to frequent increases inthe power of computers and user-friendly input andoutput routines, regional groundwater flow problems of

There are many similarities between detective storiesand groundwater investigations. Important activities ofa good detective include thorough investigations at thescene of the crime, imaginative interpretation of all theavailable information, and identifying the crucial issuesfrom a wealth of information. In groundwater investi-gations, this corresponds to the need to become veryfamiliar with all aspects of the study area, to examineimaginatively all field information (not just the ‘good’data sets) and to use all this information to hypothe-sise on the groundwater flow processes. Inevitably falseleads will be followed yet information, which appearsnot to fit, is often a key to identifying crucial aspectsof the flow processes. In a detective story there isusually a famous detective who succeeds in identifyingvital clues which lead to the naming of the culprit.However, groundwater studies do not rely on a singleindividual: instead, contributions are made by anumber of specialists. Each member of the team mustlisten to others, respecting their skills and insights; thisleads to the breakthroughs that are required duringmost groundwater investigations.

1.2 CONCEPTUAL MODELS

Conceptual models describe how water enters anaquifer system, flows through the aquifer system andleaves the aquifer system. Conceptual models start withsimple sketches although in their final form they maybe detailed three-dimensional diagrams. First attemptsat conceptual models are often based on simplified geo-logical cross-sections with sketches of flow directions.Then diagrams are prepared for various seasons; moredetailed drawings for certain parts of the aquifersystem are also helpful. Conceptual models allow othermembers of the team to assess critically the currentthinking and to provide further insights. In addition toconceptual models for the whole aquifer system, spe-cialised models are required for smaller scale processessuch as recharge estimation, the performance ofpumped boreholes and river–aquifer interaction.

To demonstrate the importance of conceptualmodels, consider the analysis of pumping tests. Con-ceptual models are rarely prepared before embarkingon the analysis of pumping test data. One examplequoted in this book relates to a weathered-fracturedaquifer system with pumping from the underlying frac-tured zone and monitoring in both the weathered andfractured zones (Section 7.4.6). When pumping fromthe fractured zone stopped, drawdowns continued toincrease in shallow piezometers in the weathered zone.This continuing fall was explained away as being a fault

considerable complexity can be represented by numeri-cal models. Yet, if there is only a partial understandingof the hydrogeology, with limited data and informationabout the aquifer, any three-dimensional time-variantanalysis is likely to be tentative. In the early stages ofan investigation, it is preferable to simplify or idealisethe conceptual models so that preliminary analyses canbe achieved using analytical or simpler numericalmodels.

Computational models are required for other aspectsof groundwater investigations such as recharge estima-tion based on a soil moisture balance technique,runoff-recharge simulation and groundwater–surfacewater interaction.

For most of the case studies described in the follow-ing chapters, detailed information is not included aboutthe actual numerical technique or software used in theanalysis. Some of the problems were originally studiedusing analogue computers; this was necessary due tothe limited power and accuracy of digital computers atthe time when the analysis was carried out. Todaydigital computer software is usually available, whichprovides effectively the same results. Changes have alsooccurred in the numerical techniques used to solve thefinite difference equations; this is partly due to sub-stantial increases in computer memory and speed ofoperation. Developments in software and hardware arecertain to continue. Nevertheless, it is essential to checkthat the software does perform what the investigatorrequires. With graphical interfaces separating the inves-tigator from the actual code, there are instances wherethe software does not perform the task required (see,for example, Section 12.2.2).

1.4 CASE STUDIES

So that the reader can gain insights into both the development of conceptual models and the selectionand application of computational models, there aredetailed descriptions of many case studies. Case studiesare taken from eleven different countries with climatesincluding semi-arid, temperate and humid-tropical. Awide variety of geological formations are considered.The objectives of the studies are also varied. Eachchapter contains a number of case studies. For eachcase study one or two aspects are highlighted; furtherissues are raised in subsequent examples. To obtain fullvalue from individual case studies, the reader needs toappreciate the actual situation; this requires imagina-tion, especially when the case study refers to climaticconditions, hydrogeology or aquifer exploitation whichare outside the experience of the reader. Focusing on

case studies permits explanations of misconceptionsand inadequate assumptions which often occur in theearly stages of a study. The locations of many of thecase studies in England and Wales are indicated inFigure 1.1.

Another advantage of case studies is that conceptualand computational models can be examined in thecontext of their actual use, rather than in hypotheticalsituations. The approach and parameter values from anexisting case study can often be adapted and modifiedas the basis for the development of conceptual andcomputational models for new projects.

The wide use of case studies means that guidance is required to identify where information can be found about the alternative concepts and techniques.Summary tables are provided in Chapters 5 to 11 toindicate where different topics are considered. For eachof the major case studies in regional groundwater flow,objectives of the study and key issues are highlighted.In addition, there is extensive cross-referencing. Theindex is also helpful in finding where specific topics areconsidered.

No problem is totally solved; no study is complete.Nevertheless, individual case studies do make positivecontributions because they move understandingforward. It is for those who revisit a case study to takethe current understanding and advance it further.

1.5 THE CONTENTS OF THIS BOOK

This book is divided into three parts. Part I: Basic Prin-ciples (Chapters 2 to 4) considers three fundamentaltopics in groundwater hydrology. The principles ofgroundwater flow are presented concisely in Chapter 2,where reference is also made to numerical methods,collecting and presenting field data plus a brief con-sideration of water quality issues. Chapter 3 considersrecharge estimation and includes detailed informationabout a soil moisture balance method of recharge esti-mation for both semi-arid and temperate climates.Modifications to recharge processes due to the presenceof low permeability drift, including the estimation ofrunoff recharge, are described. Surface water–ground-water interaction is considered in Chapter 4; thesurface water bodies include canals, springs, rivers andlakes, drains and flooded ricefields.

Radial flow towards wells and boreholes is thesubject of Part II: Radial Flow (Chapters 5 to 8). Toooften, radial flow analysis is restricted to the determi-nation of aquifer parameters using analytical solutions.However, far more can be learnt about the aquifer andthe ability of a well or borehole to collect water by crit-

Introduction 3

4 Groundwater hydrology

components are important are reviewed in Chapter 7.

Analytical and numerical techniques of analysis aredescribed and the significance of aquitard storage in assessing aquifer resources is also discussed. InChapter 8, step-pumping tests and packer tests arereviewed, and case studies concerning the reliable yieldof aquifer systems are introduced. Artificial rechargethrough injection wells and the hydraulics of horizon-tal wells are also topics in Chapter 8.

ically examining all the field information, developingconceptual models and then selecting suitable ana-lytical or numerical techniques to analyse the data.Chapter 5 is concerned with radial flow problems inwhich vertical flow components are not significant.Large diameter wells are reviewed in Chapter 6; theseare widely used in developing countries. Practicalexamples include analysing pumping tests in largediameter wells and the use of large diameter wells overextended time periods. Situations where vertical flow

Figure 1.1 Locations of studies in England and Wales reported in this book

The first three chapters of Part III: RegionalGroundwater Flow are concerned with different typesof aquifer systems; the discussion is based on twelvemajor case studies. Chapters 9 refers to aquifer systemswhere the transmissivity remains effectively constant.In Chapter 10, case studies include situations wherevertical flows through lower permeability strata are sig-nificant. Chapter 11 focuses on aquifer systems wherethe hydraulic conductivity varies over the saturateddepth. In Chapter 12 a number of techniques andissues related to modelling are examined, includingboreholes in multi-aquifer systems, the lateral extentand time period for regional groundwater studies,model refinement and sensitivity analyses together withthe use of groundwater models for predications. In thefinal section of Chapter 12 a brief evaluation is madeof conceptual and computational models with a reviewof achievements and limitations and an indication ofthe possible direction of future work.

In each part of this book, concepts and techniquesare introduced in different ways. In Part I, each chapteris concerned with a specific topic; as ideas are devel-oped they are illustrated by numerical examples. InPart II, the major derivations and techniques aredescribed in Chapter 5 which considers radial flow, andin Chapter 7 where vertical flow components are alsoincluded. Chapters 6 and 8 are concerned mainly withapplications of these techniques to practical problems.In Part III, the concepts and procedures involved inregional groundwater studies are introduced and devel-oped within the case studies. Each case study is used to present new material. In Chapter 12 many of theinsights of the previous three chapters are drawntogether. For those with limited experience of con-ceptual or computational modelling in groundwaterhydrology, it will be helpful to quickly read through thematerial in this book, paying particular attention to thefigures. This will provide an overall appreciation of theapproach to the wide range of situations which arise ingroundwater hydrology. More detailed examination ofspecific topics can then follow.

1.6 UNITS, NOTATION, JOURNALS

Metric units are used throughout this book, usually interms of metre-day. These units are convenient forgroundwater resource problems because response timesdue to pumping, recharge, etc. are typically in days ortens of days, with annual groundwater fluctuations inthe range of one to fifteen metres. For rainfall andrecharge it is preferable to use mm/d. Pumping ratesfrom major boreholes often exceed 1000m3/d, hence itis advantageous to use units of Ml/d (million litres perday or 1000m3/d). Hydraulic conductivity for sands liein the range 1.0 to 10m/d, while for clays the range isusually 0.0001 to 0.01m/d.

Since this book relates to several disciplines, itincludes a number of conventional notations. When-ever possible, standard symbols are used and they aredefined in the text. Occasionally there is duplication,for example KC is the hydraulic conductivity of clay andalso the crop coefficient in recharge estimation.However, the meaning of the symbol is clear from thecontext. A List of Symbols can be found after theAppendix and before the References.

Some of the journals quoted in the references maynot be familiar to all readers.

BHS: British Hydrological Society, contact throughInstitution of Civil Engineers, 1 Great George Street,Westminster, London SW1P 3AA, UK.

Hydrol. Sci. J.: Hydrological Sciences Journal, IAHSPress, CEH, Wallingford, Oxon OX10 8BB, UK.

IAHS Publ. No.: International Association of Hydro-logical Sciences, IAHS Press, as above.

J. Inst. Water & Env. Man.: Journal of Institution ofWater and Environmental Management, CharteredInstitution of Water and Environmental Manage-ment, 15 John Street, London WC1N 2EB, UK.

Quart. J. Eng. Geol.: Quarterly Journal of EngineeringGeology, Geological Society Publishing House, Unit7, Brassmill Enterprise Centre, Bath BA1 3JN, UK.

Proc. Inst. Civ. Eng. (London): Proceedings of Institu-tion of Civil Engineering, as above.

Introduction 5

PART I: BASIC PRINCIPLES

Part I considers three basic issues in groundwaterhydrology. Chapter 2 introduces the principles ofgroundwater flow, Chapter 3 considers recharge esti-mation and Chapter 4 examines various forms ofgroundwater–surface water interaction. In each chapterconceptual models of the flow processes are developed.Specific problems are then analysed using a variety ofcomputational techniques.

Chapter 2 summarises the physical and mathemati-cal background to groundwater flow. One-dimensionalCartesian and radial flow are analysed under bothsteady state and time-variant conditions. Two-dimensional vertical section formulations (profilemodels) are studied with an emphasis on boundary con-ditions. An examination of regional groundwater flowin the x–y plane demonstrates that this formulationimplicitly contains information about vertical flowcomponents. A brief introduction to numerical modelsserves as a preparation for their use in later chapters.The importance of monitoring both groundwater headsand river flows is stressed; case studies are included toillustrate suitable techniques. The chapter concludeswith a brief discussion of quality issues with a focus onthe analysis of the fresh–saline water interface.

The estimation of groundwater recharge is consid-ered in Chapter 3. Detailed information is provided for potential recharge estimation using soil moisture

balance techniques, and the physical basis of thisapproach is highlighted. Examples include rechargeestimates for temperate climates and for semi-arid areasunder both rainfed and irrigated regimes. Sufficientinformation is provided for readers to use this approachfor their own setting. Modifications to recharge occurdue to the unsaturated zone between the soil and thewater table; delays occur before the recharge reachesthe water table. Several examples of runoff-recharge aredescribed where runoff subsequently enters the aquiferin more permeable locations.

Chapter 4 considers different forms of interactionbetween groundwater and surface water. For canals, theimportance of the regional groundwater setting and theeffect of lining are examined. The study of canals pro-vides useful insights into river–aquifer interaction; thedimensions of the river and any deposits in the riverbedmay not be the most important factors in quantifyingthe interaction. Springs and lakes are also considered.Next, a reappraisal of the theory of horizontal drainsshows that it is incorrect to assume that the water tableintersects the drain. The importance of time-variantbehaviour is stressed. Interceptor drains intended tocollect canal seepage are used as an example of con-ceptual and numerical modelling. Finally, water lossesfrom flooded ricefields are studied; the major cause oflosses is water flowing through the bunds.

2.1 INTRODUCTION

The purpose of this chapter is to introduce both thebasic principles of groundwater flow and the associatedanalytical and numerical methods of solution whichare used in later chapters. These principles andmethods of analysis are the basis of conceptual andcomputational models. To apply them to specific prac-tical problems, appropriate field information and mon-itoring is required; this is the second topic consideredin this chapter. The chapter concludes with a briefintroduction of water quality issues.

A clear understanding of fundamental concepts suchas groundwater head, groundwater velocity and storagecoefficients are a prerequisite for the interpretation offield data, the development of quantified conceptualmodels and the selection of appropriate analytical ornumerical methods of analysis. In Section 2.2 the fun-damental parameters of groundwater flow are intro-duced. Sections 2.3 to 2.6 consider groundwater flow using different co-ordinate systems, namely one-dimensional Cartesian flow, radial flow, flow in a verti-cal section and regional groundwater flow. Whereappropriate, analytical solutions to representativeproblems are introduced. This is followed by a briefintroduction to numerical methods of analysis. Thefinite difference method is selected because it is thebasis of many of the commonly used packages.

The study of groundwater flow requires the use ofthe language of mathematics; readers who are notfamiliar with the mathematical approach are encour-aged to examine the steps in obtaining the solutionseven though the mathematical manipulation is not fully understood. The derivations in this chapter are straightforward; for more rigorous derivations see Bear (1979) and Zheng and Bennett (2002). Awarenessof the basic steps in obtaining the mathematical

solution, and appreciating the assumptions made,will enable readers to judge whether a particularapproach is appropriate for use in practical problems.Furthermore, the same basic steps are followed inobtaining numerical solutions. With computer pack-ages designed to simplify the inputting of data, there is a vital need to appreciate the basic approach to studying groundwater flow, otherwise there is a riskthat the physical problem will not be specified adequately, thereby leading to incorrect or misleadingconclusions.

A further requirement of a successful groundwaterstudy is the collection and collation of aquifer para-meter values and the field monitoring of groundwaterflows and heads. The estimation of aquifer parametervalues is considered elsewhere in the book, especially inPart II; monitoring aquifer responses is considered inSection 2.8. The fundamental importance of collectingand interpreting field data and other information isstressed with examples of good practice.

The final section of this chapter contains a briefintroduction to the study of groundwater quality. Someof the principles are summarised with introductoryexamples related to saline water–fresh water interac-tion. A thorough treatment of contaminant transportissues with associated numerical models and casestudies is available in Applied Contaminant TransportModeling by Zheng and Bennett (2002).

2.2 BASIC PRINCIPLES OF GROUNDWATER FLOW

This book is concerned with groundwater flow in a widevariety of situations. Although groundwater travelsvery slowly (one hundred metres per year is a typicalaverage horizontal velocity and one metre per year is a typical vertical velocity), when these velocities are

2

Background to Groundwater Flow

Groundwater Hydrology: Conceptual and Computational Models. K.R. Rushton© 2003 John Wiley & Sons Ltd ISBN: 0-470-85004-3

10 Groundwater hydrology

(2.1)

where p is the pressure head, r is the density of the fluidand z is the height above a datum. All pressures are rel-ative to atmospheric pressure.

In Figure 2.1a a piezometer pipe enters the side ofan elemental volume of aquifer. However, an observa-tion well usually consists of a vertical drilled hole witha solid casing in the upper portion of the hole and anopen hole or a screened section with slotted casing atthe bottom of the solid casing. So that the groundwaterhead can be identified at a specific location in anaquifer, the open section of the piezometer shouldextend for no more than one metre. When the boreholehas a long open section, unreliable results may occur,as described in Section 5.1.6.

2.2.2 Direction of flow of groundwater

Groundwater flows from a higher to a lower head (orpotential). Typical examples of the use of the ground-water head to identify the direction of groundwater

hpg

z= +r

multiplied by the cross-sectional areas through which theflows occur, the quantities of water involved in ground-water flow are substantial. Consequently, the essentialfeature of an aquifer system is the balance between theinflows, outflows and the quantity of water stored.

Generally it is not possible to measure the ground-water velocities within an aquifer. However, observa-tion piezometers (boreholes) can be constructed todetermine the elevation of the water level in thepiezometer. This water level provides information aboutthe groundwater head at the open section of thepiezometer. Groundwater head gradients can be used toestimate the magnitude and direction of groundwatervelocities. A thorough understanding of the concept ofgroundwater heads is essential to identify and quantifythe flow processes within an aquifer system.

2.2.1 Groundwater head

The groundwater head for an elemental volume in anaquifer is the height to which water will rise in apiezometer (or observation well) relative to a consistentdatum. Figure 2.1a shows that the groundwater head

Figure 2.1 Groundwater head definition and determination of flow directions from groundwater head gradients

flow are shown in Figure 2.1b. In this confined aquiferthere are two piezometers which can be used to iden-tify the direction of flow. In the upper diagram the flowis from left to right because the lower groundwaterhead is in the piezometer to the right; this is in thedirection of the dip of the strata. For the lowerdiagram, the groundwater flow is to the left since thewater level in the left hand piezometer is lower. Conse-quently the direction of the flow is up-dip in theaquifer. This flow could be caused by the presence of apumped well or a spring to the left of the section.

Figure 2.1c refers to an unconfined aquifer in whichthe elevation of the water table falls to the right. Thereare three piezometers; their open ends are positionedto identify both horizontal and vertical components offlow. Piezometer (ii) (the numbering of the piezometerscan be found at the top of the figure) penetrates justbelow the water table, hence it provides informationabout the water table elevation. The other two piezome-ters have a solid casing within the aquifer with a shortopen section at the bottom; they respond to thegroundwater head at the base of the piezometer. Sincethe open sections of piezometers (i) and (ii) are at thesame elevation they provide information about the horizontal velocity component (values of the horizontalhydraulic conductivity are required to quantify thevelocity); the horizontal flow is clearly from left toright. Piezometer (iii) is positioned almost directlybelow piezometer (ii), therefore it provides informationabout the vertical velocity component. Since the ground-water head in piezometer (iii) is below that of piezom-eter (ii) there is a vertically downward component offlow. The horizontal and vertical velocity componentscan be combined vectorially to give the magnitude anddirection of the flow.

2.2.3 Darcy’s Law, hydraulic conductivity and permeability

It was in 1856 that Henry Darcy carried out experi-ments which led to what we now call Darcy’s Law. Anunderstanding of assumptions inherent in Darcy’s Lawis important; significant steps in the derivation are dis-cussed below.

Groundwater (Darcy) velocity and seepage velocity

Figure 2.2 illustrates flow through a cylinder of aquifer;the groundwater velocity or Darcy velocity is definedas the discharge Q divided by the total cross-sectionalarea of the cylinder A,

(2.2)vQA

=

Note that the calculation of the groundwater velocityignores the fact that the aquifer cross-section A con-tains both solid material and pores; consequently thegroundwater velocity is an artificial velocity which hasno direct physical meaning. Nevertheless, because of itsconvenience mathematically, the groundwater velocityis frequently used.

An approximation to the actual seepage velocity vscan be obtained by dividing the Darcy velocity by theeffective porosity N.

(2.3)

Derivation of Darcy’s Law

A mathematical derivation of Darcy’s Law is presentedin Figure 2.3, and some of the steps are summarisedbelow.

1. In Figure 2.3, the elemental volume of water isinclined at an angle a to the horizontal; conse-quently the analysis which leads to Darcy’s Lawapplies to flow in any direction and is not restrictedto horizontal or vertical flow.

2. If the water pressure on the left-hand face is p, thepressure on the right-hand face, which is a distancedl to the right, will equal

3. There are four forces acting on the element. Twoforces are due to water pressures on the ends of the

pressure rate of change of distance= + ¥

= +Êˈ¯

p p

pdpdl

ld

vvN

QANs

= =

Background to groundwater flow 11

Figure 2.2 Darcy velocity

12 Groundwater hydrology

Figure 2.3 Derivation of Darcy’s Law

element, and there is also a force due to the weightof water in the element and a frictional force resist-ing the flow of water. The water pressure is multi-plied by NdA, where N is the effective porosity.

4. In Eq. C in Figure 2.3, the resisting force per unitvolume of water in the element is selected to be proportional to the dynamic viscosity of the flow-ing fluid and inversely proportional to a parametercalled the intrinsic permeability of the porous mate-rial. This is consistent with field experiments.

5. Darcy’s Law from Eq. (E) is

(2.4)

where the hydraulic conductivity K is a function of the dynamic viscosity of water and the intrinsicpermeability of the porous material.

6. The minus sign in Darcy’s Law emphasises thatwater flows from a higher to a lower groundwaterhead.

7. Strictly the phrase hydraulic conductivity shouldalways be used but in groundwater hydraulics andsoil mechanics the word permeability is often usedas an alternative; in the remainder of this text thesewords are considered to be interchangeable.

Various authors provide tables of the hydraulic con-ductivity and storage properties of unconsolidated andconsolidated materials. See, for example, McWhorterand Sunada (1977), Bouwer (1978), Todd (1980),Driscoll (1986), Walton (1991) and Fetter (2001).

2.2.4 Definition of storage coefficients

The change in groundwater head with time results inthe release of water from storage or water taken intostorage. Two mechanisms can apply; the release ofwater at the water table due to a change in the watertable elevation and/or the release of water throughoutthe full depth of the aquifer due to a change in ground-water head which corresponds to an adjustment of thepressure within the aquifer system. For more detaileddiscussions on the physical basis of the storage coeffi-cients, see Bear (1979) and Fetter (2001).

Specific yield: to appreciate the meaning of specificyield, consider the decline in water surface elevation ina surface reservoir. This fall in water surface corre-sponds to a reduction in the volume of water storeddue to outflows to a river, withdrawal for water supply,or evaporation. The effect of a fall in the water tableelevation in an aquifer is based on similar concepts; see

v Kdhdl

= -

Figure 2.4a. However, account must be taken of thefact that the aquifer contains both solid material andwater, hence it is necessary to multiply the decrease involume due to the water table fall by the specific yield,SY. The specific yield is defined as follows:

The volume of water drained from a unit plan area ofaquifer for a unit fall in head is SY

Specific storage: a fall in groundwater head within asaturated aquifer also causes a reduction in pressurewithin the aquifer system (Eq. (2.1)). This fall in press-ure means that less water can be contained within aspecified volume of the aquifer, and the volume of thesoil or rock matrix also changes. Consequently, wateris released from storage. The specific storage SS relatesto a unit volume of aquifer; see the lower part ofFigure 4b:

The volume of water released from a unit volume ofaquifer for a unit fall in head is SS

Confined storage: the specific yield refers to a unitplan area of aquifer whereas the specific storage relatesto a unit volume. Therefore, it is helpful to define a storage coefficient for pressure effects related to a unit plan area. The confined storage coefficient equals

Background to groundwater flow 13

Figure 2.4 Diagrammatic explanation of the three storagecoefficients

14 Groundwater hydrology

certain aspects of the derivation are discussed below.

• The two fundamental principles of the governing equa-tion are Darcy’s Law and the principle of continuity.

• The magnitudes of the velocities change across theelement. Therefore, if the velocity on the left-handface of the element is vx, on the right-hand face at

distance dx the velocity will become

• In the flow balance there are four components, netflows in the x, y and z directions and a further component due to the compressibility of the aquifersystem which equals the specific storage SS multipliedby the rate of change of groundwater head with time.These four components must sum to zero leading toEq. (B).

• The final part of the derivation is to substituteDarcy’s Law into the continuity equation; this leadsto Eq. (C), the governing equation which is repeatedbelow. The left-hand side of the equation representsflows in the three co-ordinate directions, the right-hand side represents the storage contribution due tothe change in groundwater head with time.

(2.5)

In most practical situations, there is no need to workin the three co-ordinate dimensions plus the timedimension. In the next four sections of this chapter,analyses are described in which one or more of thedimensions are excluded.

2.3 ONE-DIMENSIONAL CARTESIAN FLOW

This section introduces a one-dimensional mathemati-cal formulation which can be used to understandsimple regional groundwater flow when the flow is predominantly in one horizontal direction. Initiallysteady-state problems are considered, subsequently a

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂x

Khx y

Khy z

Khy

Shtx y z S

ÊË

ˆ¯ +

ÊË

ˆ¯ +

ÊË

ˆ¯ =

vvx

dxxx+

∂∂

.

the specific storage multiplied by the saturated thick-ness m,

Accordingly, the compressive properties of a confinedaquifer are quantified as follows:

The volume of water released per unit plan area ofaquifer for a unit fall in head is SC

Note that, as illustrated in Figure 2.4b, the saturatedthickness extends from the impermeable base of theaquifer to the upper surface which may be a confininglayer or a water table. Therefore, unconfined aquifersalso have a confined storage response. The importanceof considering both the unconfined and confinedstorage properties of a water table aquifer is demon-strated in the pumping test case study in Section 5.1.

Storage properties are summarised in Table 2.1. Notethat the specific storage has dimensions L-1 all the otherstorage coefficients are dimensionless. Typical rangesfor these three storage coefficients are

Although the confined response to a change in ground-water head is effectively instantaneous, the drainage ofwater as the water table falls can take some time. Hencethe concept of delayed yield is introduced for uncon-fined aquifers; for further information about delayedyield see Section 5.17.

2.2.5 Differential equation describing three-dimensional time-variant groundwater flow

A mathematical derivation for three-dimensionalregional groundwater flow is outlined in Figure 2.5;

10 10

10 3 10

10 0 25

6 4 1

5 3

3

- - -

- -

-

£ £

£ £ ¥

£ £

S

S

S

S

C

Y

m

.

S S mC S= ¥

Table 2.1 Summary of storage properties

Type Symbol Unit Dimension Name Pressure/drainage

Specific storage SS cube L-1 confined pressureConfined storage SC area – confined pressureSpecific yield SY area – unconfined drainageUnconfined storage SY + SC area – unconfined drainage + pressure