Environmental Isotopes in the Hydrological Cycle
Principles and Applications

VOLUME VI: MODELLING
co-ordinating editor Y. Yurtsever IAEA Vienna
Contributing Authors
A.Zuber, P.Maloszewski; M.E.Campana,
G.A.Harrington, L.Tezcan; L.F.Konikow 

Preface
CONTENTS
1	MODELLING INTRODUCTION		  1     

2	LUMPED PARAMETER MODELS		  5
	2.1	Introduction			  5
	2.2	Basic principles of the lumped-parameter approach for constant flow systems	  8
	2.3	Models				10
		2.3.1	The piston flow model		10
		2.3.2	The exponential model		11
		2.3.3	The combined exponential - piston flow model	12
		2.3.4	The dispersion model		13
	2.4	Cases of constant tracer input		14
	2.5	Cases of variable tracer input		15
		2.5.1	The tritium method		15
		2.5.2	The 3H-3He method		18
		2.5.3	The krypton-85 method		20
		2.5.4	The carbon-14 method		21
		2.5.5	The oxygen-18 and deuterium method	21
		2.5.6	Other potential methods		22
	2.6	Examples of 3H age determinations	23
	2.7	Determination of hydrogeologic parameters from tracer ages	24
	2.8	The lumped-parameter approach versus other approaches	29
	2.9	Concluding remarks			30
	List of references				31     

3	COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW
	SIMULATION				37
	3.1	Introduction			37
	3.2	A simple compartment model: theory and application to a regional
		groundwater flow system		38
		3.2.1	Theory			38
			3.2.1.1	Tracer mass balance	39
			3.2.1.2	Transient flow	39
			3.2.1.3	Age calculations	40
		3.2.2	Application to the Nevada Test Site flow system	41
			3.2.2.1	Introduction		42
			3.2.2.2	Hydrogeology	42
			3.2.2.3	Model development and calibration	44
			3.2.2.4	Results and discussion	46
			3.2.2.5	Concluding remarks	49
	3.3	Constraining regional groundwater flow models with environmental
		isotopes and a compartmental mixing-cell approach	50
		3.3.1	Introduction		50
		3.3.2	Governing equations		51
		3.3.3	Model design, input data and calibration procedure	52
	3.4	Mixing-cell model for the simulation of environmental isotope transport	56
		3.4.1	Introduction		56
		3.4.2	Mixing-cell model of flow and ytransport dynamics
			in karst aquifer systems		61
			3.4.2.1	Physical framework of the model	62
			3.4.2.2	Hydrologic model	63
			3.4.2.3	Transport model	65
		3.4.3	Conclusions		66
	3.5	Summary and conclusions		67
	List of references				68     

4	USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW
	AND TRANSPORT			75
	4.1	Introduction			75
	4.2	Models				76
	4.3	Flow and transport processes		77
	4.4	Governing equations			78
		4.4.1	Groundwater flow equation	79
		4.4.2	Seepage velocity		80
		4.4.3	Solute-transport equation	80
	4.5	Numerical methods to solve equations	84
		4.5.1	Basics of finite-difference methods	88
		4.5.2	Basics of finite-element methods	91
		4.5.3	Matrix solution techniques	92
		4.5.4	Boundary and initial conditions	93
	4.6	Model design, development and application	94
		4.6.1	Grid design		95
		4.6.2	Model calibration		96
		4.6.3	Model error		99
		4.6.4	Mass balance		101
		4.6.5	Sensitivity tests		102
		4.6.6	Calibration criteria		103
		4.6.7	Predictions and postaudits	103
		4.6.8	Model validation		104
	4.7	Case history: local-scale flow and transport in a shallow confined aquifer	105
	4.8	Available groundwater models		109
	List of references				110
     

LITERATURE					117     

IAEA PUBLICATIONS				119     

CONSTANTS					123     

SUBJECT INDEX				125