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Groundwater HydrologyConceptual and Computational Models
K.R. RushtonEmeritus Professor of Civil Engineering
University of Birmingham, UK
Innodata0470871652.jpg
Groundwater Hydrology
Groundwater HydrologyConceptual and Computational Models
K.R. RushtonEmeritus Professor of Civil Engineering
University of Birmingham, UK
Copyright © 2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,West Sussex PO19 8SQ, EnglandTelephone (+44) 1243 779777
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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 threedimensional timevariant groundwater flow 14
2.3 Onedimensional Cartesian Flow 142.3.1 Equation for onedimensional 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 Timevariant onedimensional 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 Timevariant radial flow 282.4.6 Timevariant radial flow including vertical components 28
2.5 Twodimensional Vertical Section (Profile Model) 282.5.1 Steadystate conditions, rectangular dam 282.5.2 Timevariant 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 onedimensional flow 342.6.3 When vertical flows must be included explicitly 35
2.7 Numerical Analysis 362.7.1 Onedimensional flow in xdirection 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 semiarid region of Nigeria 843.4.11 Estimating recharge due to precipitation and irrigation in a semiarid region of India 87
3.4.12 Bypass recharge 903.4.13 Estimating catchmentwide 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 steadystate problems 1294.4.5 Numerical solutions 1314.4.6 Representative timevariant 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 loglog 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 SingleLayer 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 RadialVertical Timevariant 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 RadialVertical Timeinstant [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 Twozone Approximation [r, t, vZ] 2137.4.1 Introduction 2137.4.2 Examples of formulation using the twozone model 2137.4.3 Details of the twozone model 2147.4.4 Discrete space–discrete time equations for the twozone model 2167.4.5 Solution of simultaneous equations 2187.4.6 Examples of the use of the twozone 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 Weatheredfractured aquifers 2468.5.3 Alluvial aquifers with an uppermost layer of low hydraulic conductivity 2478.5.4 Response of an alluviumsandstone 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 MultiAquifer 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 runoffrecharge 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 Multilayered Aquifers 36412.4 TimeInstant Conditions 366
12.4.1 Introduction 36612.4.2 Basis of timeinstant approach 36712.4.3 Examples of sandstone and limestone aquifers 36712.4.4 Timeinstant 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 Twozone 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 groundwater investigation be planned?’ At that time I had difficulty in giving a convincing answer, but in the intervening 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 conceptual and computational models in the title of thisbook.
The number of groundwater investigations throughout the world continues to increase. The objectives ofthese investigations are varied, including meeting regulatory requirements, exploring the consequences ofgroundwater development and rectifying the results ofoverexploitation 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 understood 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 computational 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 introduced. 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 ofthreedimensional 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 important 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 investigations 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 coauthors 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 uncertainties 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 groundwater through aquifers were established by the early1970s. In addition, numerous analytical solutions wereavailable. At that time, numerical modelling of groundwater flow was also developing although limited computing 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 saturated 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, computational models are used to test whether the flow processesof the conceptual model are plausible and the parameter 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 pseudosteady 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 contents 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 iscrossreferencing to other similar studies. One advantage of using case studies based on the author’s experience 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: 0470850043
2 Groundwater hydrology
in the construction of the shallow piezometers. Consequently 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 weatheredfractured system wasoverlooked by moving straight to analysis and ignoring ‘inconvenient’ data.
1.3 COMPUTATIONAL MODELS
The phrase ‘computational models’ is used to emphasise that a wide variety of alternative models are available. Darcy’s Law is the fundamental computationalmodel; not only is it included in many of the governing equations for groundwater flow but it can also beused for preliminary calculations such as vertical flowsthrough low permeability strata. The diversity of computational models is illustrated by considering radialflow. There are many analytical models for radial flowto pumped boreholes including the Thiem equation forsteadystate flow in a confined aquifer, the timevariantTheis 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 differential 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 computer software is now widely available to simplify theprocedure. However, the ease of using computer software for matching field data to the analytical solutionsoften hides the inconsistencies between the assumptions of the analytical model and the real field conditions. Although less convenient to use, numericalradial flow models often represent more closely theactual behaviour of the aquifer system.
Real groundwater problems are threedimensional inspace and timevariant. Due to frequent increases inthe power of computers and userfriendly 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 investigations, 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 hypothesise 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 threedimensional diagrams. First attemptsat conceptual models are often based on simplified geological crosssections 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, specialised 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. Conceptual models are rarely prepared before embarkingon the analysis of pumping test data. One examplequoted in this book relates to a weatheredfracturedaquifer system with pumping from the underlying fractured 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 numerical models. Yet, if there is only a partial understandingof the hydrogeology, with limited data and informationabout the aquifer, any threedimensional timevariantanalysis 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 estimation based on a soil moisture balance technique,runoffrecharge simulation and groundwater–surfacewater interaction.
For most of the case studies described in the following 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 substantial 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 investigator 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 semiarid, temperate and humidtropical. 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 imagination, 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 crossreferencing. 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 Principles (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 consideration of water quality issues. Chapter 3 considersrecharge estimation and includes detailed informationabout a soil moisture balance method of recharge estimation for both semiarid and temperate climates.Modifications to recharge processes due to the presenceof low permeability drift, including the estimation ofrunoff recharge, are described. Surface water–groundwater 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 determination 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, steppumping tests and packer tests arereviewed, and case studies concerning the reliable yieldof aquifer systems are introduced. Artificial rechargethrough injection wells and the hydraulics of horizontal wells are also topics in Chapter 8.
ically examining all the field information, developingconceptual models and then selecting suitable analytical 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 significant. 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 multiaquifer 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 developed 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 developed 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 conceptual 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 metreday. 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. Whenever 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 Hydrological Sciences, IAHS Press, as above.
J. Inst. Water & Env. Man.: Journal of Institution ofWater and Environmental Management, CharteredInstitution of Water and Environmental Management, 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 Institution 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 estimation 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 mathematical background to groundwater flow. OnedimensionalCartesian and radial flow are analysed under bothsteady state and timevariant conditions. Twodimensional vertical section formulations (profilemodels) are studied with an emphasis on boundary conditions. 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 considered 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 semiarid 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 runoffrecharge 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 provides 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 timevariantbehaviour is stressed. Interceptor drains intended tocollect canal seepage are used as an example of conceptual 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 practical problems, appropriate field information and monitoring 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 fundamental parameters of groundwater flow are introduced. Sections 2.3 to 2.6 consider groundwater flow using different coordinate systems, namely onedimensional Cartesian flow, radial flow, flow in a vertical 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 encouraged 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 packages 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 parameter 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 interaction. 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: 0470850043
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 relative to atmospheric pressure.
In Figure 2.1a a piezometer pipe enters the side ofan elemental volume of aquifer. However, an observation 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 groundwater head to identify the direction of groundwater
hpg
z= +r
multiplied by the crosssectional areas through which theflows occur, the quantities of water involved in groundwater 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 groundwater velocities within an aquifer. However, observation 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 identify 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. Consequently the direction of the flow is updip 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 piezometers 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 groundwater head in piezometer (iii) is below that of piezometer (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 experiments 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 discussed 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 crosssectionalarea of the cylinder A,
(2.2)vQA
=
Note that the calculation of the groundwater velocityignores the fact that the aquifer crosssection A contains 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; consequently 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 lefthand face is p, thepressure on the righthand 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= + ¥
= +ÊËˆ¯
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 resisting the flow of water. The water pressure is multiplied 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 flowing fluid and inversely proportional to a parametercalled the intrinsic permeability of the porous material. 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 conductivity 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 groundwater head which corresponds to an adjustment of thepressure within the aquifer system. For more detaileddiscussions on the physical basis of the storage coefficients, 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 corresponds 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 pressure 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 equation are Darcy’s Law and the principle of continuity.
• The magnitudes of the velocities change across theelement. Therefore, if the velocity on the lefthandface of the element is vx, on the righthand 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 lefthand side of the equation representsflows in the three coordinate directions, the righthand 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 coordinate 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 ONEDIMENSIONAL CARTESIAN FLOW
This section introduces a onedimensional mathematical formulation which can be used to understandsimple regional groundwater flow when the flow is predominantly in one horizontal direction. Initiallysteadystate 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 thickness 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 demonstrated in the pumping test case study in Section 5.1.
Storage properties are summarised in Table 2.1. Notethat the specific storage has dimensions L1 all the otherstorage coefficients are dimensionless. Typical rangesfor these three storage coefficients are
Although the confined response to a change in groundwater head is effectively instantaneous, the drainage ofwater as the water table falls can take some time. Hencethe concept of delayed yield is introduced for unconfined aquifers; for further information about delayedyield see Section 5.17.
2.2.5 Differential equation describing threedimensional timevariant groundwater flow
A mathematical derivation for threedimensionalregional 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 L1 confined pressureConfined storage SC area – confined pressureSpecific yield SY area – unconfined drainageUnconfined storage SY + SC area – unconfined drainage + pressure