Contents Page TABLE OF FIGURES... VII GLOSSARY...XI INTRODUCTION... 1

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3 Abstract Despite the long history of work on saltwater wedge intrusion in coastal aquifers, very little attention has been paid to the effect of basement irregularities on the wedge. This work combines physical and numerical modelling of a hypothetical aquifer in two dimensions to examine the effect of small scale basement heterogeneities. Physical modelling was carried out in a 2-D aquifer slice with a glass bead porous media. Two small walls at the base of the aquifer served as a small heterogeneity. Digital photography was used to record the position of a wedge as it progressed into the aquifer and then receded after a change in boundary conditions. The heterogeneity was found to have limited effect during the progression of the wedge, only affecting the movement of the interface near the toe as it moved over the walls of the heterogeneity. The heterogeneity had no effect on the steady-state position of the wedge. The heterogeneity, however, did have a significant effect on the wedge as it receded: the interface was deformed in the area immediately above the heterogeneity. As the interface receded past the walls of the heterogeneity, brine remained immediately upstream of the walls and in the cavity between them. The toe of the main wedge was attenuated downstream of the walls. The results of the physical modelling were used to calibrate a numerical model. Some adjustment of the hydraulic conductivity value observed in the tank was required to match the observed movements of the wedge. Good agreement was obtained between the numerical model and the tank during the progression and steady state stages, but not during the recession. Analysis indicated that the values of dispersivity used were too high, and that values were required one or more orders of magnitude lower than the minimum values that would not cause numerical instability. The calibrated model was used to more closely examine the flushing of cavity-like heterogeneities during wedge recession. Two flushing regimes were identified: advective and dispersive. A relationship was derived from the Baydon-Ghyben- Hertzberg principle uniting the effects of cavity geometry and aquifer boundary conditions which could effectively predict the relative importance of these two flushing modes. iii

4 Acknowledgements Firstly, for their assistance with the project itself, thankyou to David Reynolds, my supervisor, for his guidance, interest and sense of humour; Prabhakar Clement, who first taught me groundwater, originally devised the project and gave me many interesting ideas on which to build; Charles and later Bruce for their endless assistance and patience in the hydraulics laboratory during the experimental stage of the work; Matthew Simpson for many books, articles and sound advice; Gajan Sivandran, who did some important preliminary work which led to this project; and the rest of the staff and academics at the Centre for Water Research. This work was funded by a small ARC grant project titled "Experimental and Numerical Investigation of Unconfined Groundwater Flow Systems with Density Interfaces", which was awarded to Dr. Prabhakar Clement. This dissertation would have been a much more lonely and difficult task without the continual support of a large group of friends and supporters. Standouts include Georgia Pickering, my parents, Andrea, Alan and Garvey (my room-mates ), and the rest of the CWR crew. iv

5 Contents Page TABLE OF FIGURES... VII GLOSSARY...XI INTRODUCTION... 1 CHAPTER 1. LITERATURE REVIEW PRINCIPLES OF GROUNDWATER FLOW SALTWATER INTRUSION QUALITATIVE UNDERSTANDING AND ANALYTICAL APPROXIMATIONS OF THE INTERFACE POSITION PROBLEM THE FIELD STUDIES OF SALINE INTRUSION PHYSICAL MODELLING NUMERICAL MODELLING OF SALINE INTRUSION CONCLUSION CHAPTER 2. PHYSICAL MODELLING MATERIALS AND METHODS RESULTS DISCUSSION CONCLUSION CHAPTER 3. MODEL CALIBRATION CONCEPTUAL MODEL DISCRETISATION OF THE MODEL DOMAIN BOUNDARY CONDITIONS INITIAL CONDITIONS PARAMETERISATION DISCUSSION CHAPTER 4. SENSITIVITY ANALYSIS INTRODUCTION SETUP OF SENSITIVITY ANALYSIS RESULTS DISCUSSION DERIVATION OF AN ANALYTICAL DESCRIPTION OF FLUSHING MODES CONCLUSION - ADVECTIVE AND DISPERSIVE FLUSHING MODES CHAPTER 5. CONCLUSION REFERENCES APPENDIX A. EXPERIMENTAL DESIGN A.1 MODEL SETUP A.2 DENSITY AND HEAD GRADIENT A.3 OBSTRUCTION LOCATION v

6 APPENDIX B. COMPARISON OF MODEL OUTPUT WITH OBSERVED INTERFACE POSITION 117 B.1 HOMOGENOUS BASEMENT, PROGRESSION B.2 HOMOGENOUS BASEMENT, RECESSION B.3 HETEROGENOUS BASEMENT, PROGRESSION B.4 HETEROGENOUS BASEMENT, RECESSION APPENDIX C. MATLAB M-FILES C.1 M-FILES FOR CONVERTING SUTRA OUTPUT INTO MATLAB VARIABLES C.1.1. D3Q C.1.2. D3IN C.1.3. D4Q C.1.4. D4IN C.2 M-FILES FOR MANIPULATING AND PRESENTING MODEL DATA C.2.1. D3MMBGH C.2.2. CTOVC C.2.3. BGH C.3 M-FILES TO PROCESS DIGITAL IMAGES C.3.1. INTERFACE C.3.2. GETPOINTS APPENDIX D. FLUSHING FROM CAVITIES vi

7 Table of Figures FIGURE 1 : DEFINITION OF THE REPRESENTATIVE ELEMENTARY VOLUME (BEAR AND VERRUIJT 1987). HERE U V /U IF THE POROSITY, AND U 0 IS THE REV....5 FIGURE 2 : PORE-SCALE TRASPORT PROCESSES LEADING TO HYDRODYNAMIC DISPERSION (FROM FETTER 1994, P 456)...12 FIGURE 3 : RELATIONSHIP BETWEEN PECLET NUMBER AND DISPERSION. TAKEN FROM BEAR (1979)15 FIGURE 4 : SALTWATER INTRUSION. SALTWATER FROM THE OCEAN FLOWS IN UNDER THE COASTAL AQUIFER, WHILE FRESHWATER FLOWS OUT ABOVE IT. BETWEEN THEM A ZONE OF DIFFUSION EXISTS WHERE SALT-WATER MIXES WITH FRESH AND IS TRANSPORTED OUT OF THE AQUIFER (FROM COOPER 1964)...19 FIGURE 5 : THE BAYDON-GHYBEN HERTZBERG APPROXIMATION (NOTE HERE Γ S IS THE DENSITY OF SEAWATER (Ρ S ), Γ F OF THE DENSITY OF FRESHWATER (Ρ F ) AND H S =Z). (TAKEN FROM BEAR 1979)20 FIGURE 6: CROSS SECTION OF THE EXPERIMENTAL TANK (NOT TO SCALE). THE TANK BASICALLY A THREE DIMENSIONAL SLICE OF AN AQUIFER WITH A NARROW THIRD DIMENSION. THE HEAD AT EACH END IS CONTROLLED BY CHANGING THE ELEVATION OF THE DOWN-PIPES LOCATED IN EACH RESERVOIR...32 FIGURE 7 : CONSTRUCTION OF THE SIDE PANELS OF THE TANK. THE SLOTTED SCREEN (RIGHT) WAS WRAPPED IN GEOTEXTILE AND INSERTED INTO THE END OF THE TANK, FORMING A RESERVOIR. (LEFT) A PLAN VIEW OF THE DOWNSTREAM RESERVIOUR...33 FIGURE 8 : RESULTS OF FLOW-RATE TESTS TO DETERMINE HYDRAULIC CONDUCTIVITY (K). K IS THE GRADIENT OF THE LINE CONNECTING THE DATA POINTS, AS SHOWN IN EQUATION ( 2-1 )...42 FIGURE 9: POINT RELEASE TEST IN STAGE 1. THE PATHS OF THE DYE SLUGS INDICATE THE HOMOGENEITY OF THE TANK. THERE APPEARS TO BE AN AREA OF INCREASED CONDUCTIVITY ABOUT A THIRD OF THE WAY UP FROM THE BOTTOM OF THE TANK FIGURE 10 : RUN 0A AT 5 MINUTES. INTRUSION BEGINS FROM BOTTOM LEFT-HAND CORNER. BRINE MAY BE FLOWING DOWN THE BOUNDARY ON THE...44 FIGURE 11 : RUN 0A AT 20 MINUTES. EARLY GROWTH OF THE INTRUSION IS BOTH HORIZONTAL AND VERTICAL...44 FIGURE 12 : RUN 0A AT 85 MIN. THE WEDGE HAS CEASED TO INCREASE IN HEIGHT, WHILE THE TOE CONTINUES TO MOVE FORWARD FIGURE 13 : RUN 0A AT 365 MIN. THE WEDGE IS APPROACHING THE STEADY-STATE POSITION...45 FIGURE 14 : POINT RELEASE TEST FROM A FAILED RUN DURING THE FIRST STAGE. INITIAL POSITION OF THE RELEASED DYE IS SHOWN BY THE DASHED BLUE LINE FIGURE 15 : RUN 1A AT 45MIN. THE TOE OF THE WEDGE HAS REACHED THE FIRST WALL AND IS INCREASING IN HEIGHT. FURTHER BACK THE INCREASE IS LESS. MOVEMENT AT THE UPSTREAM BOUNDARY MAY NOT BE SIGNIFICANT FIGURE 16 : INTRUSION RUN 1A AT 75 MIN. BRINE FROM THE WEDGE BEGINS FILLING THE CAVITY. THE POSITION OF THE INTERFACE DOWNSTREAM OF THE WALL WAS CONSTANT DURING THE FILLING FIGURE 17 : RUN 1A AT 145 MIN. BRINE FLOWS OVER THE SECOND WALL AND INTO THE UPSTREAM AREA. NOTE THE DIFFUSED WEDGE IN THE UPSTREAM REGION AND THE LACK OF ANY CHANGE OF SLOPE ON THE INTERFACE IN THE VICINITY OF THE FIRST WALL FIGURE 18 : RUN 4A AT 0 MIN. THE WEDGE ALMOST AT STEADY STATE READY TO BEGIN A RECESSION TEST FIGURE 19 : RUN 0B : WEDGE MOVEMENT IMMEDIATELY FOLLOWING THE CHANGE IN BOUNDARY CONDITIONS. ARROWS INDICATE THE DIRECTION OF MOVEMENTS OF PARTS OF THE WEDGE..50 vii

8 FIGURE 20 : RUN 0B AT 110MIN. WEDGE RETRACTS VERTICALLY AND HORIZONTALLY TO NEW STEADY- STATE POSITION...50 FIGURE 21 : RUN 6B AT 20 MIN. ARROWS INDICATE THE MOVEMENT OF THE INTERFACE AS IT RECEDES. NOTE THE KINK ABOVE THE OBSTRUCTION...51 FIGURE 22 : RUN 6B AT 65 MIN. WEDGE CONTINUES TO RETRACT, LEAVING BEHIND BRINE IN A WEDGE UPSTREAM OF THE OBSTRUCTION AND WITHIN THE CAVITY. BRINE CONTINUES TO FLOW OVER THE WALLS FIGURE 23 : RUN 6B AT 160MIN. DOWNSTREAM WEDGE REMAINS CONNECTED TO OBSTRUCTION FOR SOME TIME. BRINE IN CAVITY AND UPSTREAM WEDGE IS REDUCED...53 FIGURE 24: RUN 3B AT 190MIN. AT STEADY STATE, THE INTERFACE ALMOST INTERSECTS THE CAVITY FIGURE 25 : INTERFACE POSITION OVER TIME FOR RUNS 0A AND 1A. RUN 0A IS PLOTTED IN RED, WHILE 1A IS BLUE. 1A INTRUDES FASTER THAN 0A (NOTE RUN 0A AT 75MIN OVERLAPS RUN 1A AT 45MIN)...54 FIGURE 26 : INTERFACE POSITION OVER TIME DURING RUNS 5B AND 0B. NOTE HOW THE PRESENCE OF THE LUMP APPEARS TO AFFECT THE HEIGHT OF THE INTERFACE JUST DOWNSTREAM OF THE OBSTRUCTION AT 11MIN...55 FIGURE 27 : INTERFACE POSITION AT 65MIN FOR RUNS 2B, 3B, 5B AND 6B. THESE RUNS HAD DOWNSTREAM HEAD BOUNDARY CONDITIONS OF 0.6M, 0.67M, 0.65M AND 0.63M OF SALTWATER HEAD RESPECTIVELY FIGURE 28: SEPARATION OF THE SEALANT FROM THE TANK. THE DYE MOVES OVER THE NEAR WALL AS IT SHOULD, BUT IS ALLOWED TO PASS THROUGH THE FAR WALL...62 FIGURE 29 : BOUNDARY CONDITIONS OF THE MODEL...67 FIGURE 30 : BASIC DISCRETISATION FOR FINE-SCALE MODELING...69 FIGURE 31 : DISCRETISATION INCLUDING OBSTRUCTION AT THE BASE OF THE DOMAIN...70 FIGURE 32 : OBSERVED AND MODELLED WEDGE POSITION DURING PROGRESSION INTO THE AQUIFER. THE MODEL DID NOT ACCURATELY REPRODUCE THE POINT OF INTERSECTION OF THE WEDGE INTERFACE AND DOWNSTREAM BOUNDARY...74 FIGURE 33 : OBSERVED AND MODELED WEDGE POSITION DURING RECESSION OUT OF THE AQUIFER PAST THE OBSTRUCTION. THE MODEL WAS UNABLE TO MATCH THE RATE OF FLUSHING AROUND THE OBSTRUCTION...75 FIGURE 34 : NORMALISED MASS REMAINING IN UPSTREAM WEDGE OVER TIME FOR THREE VALUES OF DISPERSIVITY AND THE OBSERVED DATA...76 FIGURE 35 : SPATIAL CONFIGURATION AND DISCRETISATION USED AS A BASE FOR THE SENSITIVITY ANALYSIS...79 FIGURE 36 : INITIAL CONDITION USED FOR BASE CASE IN SENSITIVITY ANALYSIS. ALL SENSITIVITY RUNS USED THE SAME INITIAL CONDITION, ALTHOUGH THE CAVITY WAS DIFFERENT TO THAT SHOWN HERE FOR TWO OF THE RUNS (SEE TEXT). THE UPSTREAM BOUNDARY IS ON THE LEFT AND THE DOWNSTREAM IS ON THE RIGHT...80 FIGURE 37 : CAVITY CONFIGURATION FOR THE 'WIDE' CASE IN THE SENSITIVITY ANALYSIS (RA= ¼)81 FIGURE 38 : CAVITY CONFIGURATION FOR THE 'DEEP' CASE IN THE SENSITIVITY ANALYSIS (RA= 4).81 FIGURE 39 : EXAMPLE OF THE COMBINED PLOTTING OF CONCENTRATION AND ADVECTIVE SOLUTE FLUX. CONCENTRATION IS SHOWN BY COLOUR, WHERE RED IS UNDILUTED BRINE AND BLUE IS FRESHWATER. AT THE RESOLUTION PRESENTED, IT MAY BE DIFFICULT TO DISCERN THE INDIVIDUAL ARROWS SHOWING SALT FLUX. (THE EXAMPLE IS FLUSHING OF THE WIDE CASE OF THE SENSITIVITY ANALYSIS AT 25 MINUTES [5 OUTPUT TIMESTEPS])...83 viii

9 FIGURE 40 : MASS AND ASPECT RATIO NORMALISED MASS OF SALT REMAINING IN CAVITY FOR EACH OF THE THREE ASPECT RATIOS EXAMINED IN THE SENSITIVITY ANALYSIS. MASS IS NORMALISED TO THE INITIAL MASS IN THE CAVITY...84 FIGURE 41 : MASS AND HEAD GRADIENT NORMALISED MASS OF SALT REMAINING IN CAVITY FOR EACH OF THE THREE HEAD GRADIENTS EXAMINED IN THE SENSITIVITY ANALYSIS. MASS IS NORMALISED TO THE INITIAL MASS IN THE CAVITY...85 FIGURE 42 : TIME RATE OF REMOVAL OF SALT FROM THE CAVITIES FOR EACH ASPECT RATIO IN THE SENSITIVITY ANALYSIS FIGURE 43 : TIME RATE OF REMOVAL OF SALT FROM THE CAVITIES FOR EACH HEAD GRADIENT IN THE SENSITIVITY ANALYSIS...87 FIGURE 44 : EXAMPLE OF FLOW THROUGH A CAVITY AT EARLY TIME. NOTE THE APPARENT LACK OF CONVECTIVE CELLS. (WIDE CASE AT 25 MINUTES, CLOSE-UP OF THE CAVITY)...89 FIGURE 45 : EXAMPLE OF FLOW THROUGH A CAVITY AFTER PEAK FLUX OUT. NOTE THE RELATIVE SHARPNESS OF THE INTERFACE AND THE PRESENCE OF A CONVECTIVE CELL WITHIN THE REMAINING BRINE. THE DOTTED LINE IS RELATED TO LATER DISCUSSION. (WIDE CASE AT 45 MINUTES, CLOSE-UP OF THE CAVITY)...89 FIGURE 46 : CONCENTRATION AND ADVECTIVE SOLUTE FLUX FIELD FOR THE BASE CASE AT 35 MIN. THE WHITE LINE REPRESENTS THE POSITION OF THE INTERFACE PREDICTED BY THE BAYDON GHYBEN HERTZBERG APPROXIMATION (WITH LINEAR HEAD GRADIENT ACROSS THE CAVITY). NOTE THE MODELED INTERFACE POSITION IS HIGHER THAN THE APPROXIMATION AT THE DOWNSTREAM END AND LOWER AT THE UPSTREAM END FIGURE 47 : CAVITY AT THE END OF THE ADVECTIVE FLUSHING PERIOD AT 30MIN FOR THE RUN WITH MORE HYDRAULIC GRADIENT (LEFT) AND AT 50MIN FOR THE CASE WITH LESS HYDRAULIC GRADIENT (RIGHT). NOT THE DIFFERENCE IN THE SLOPE OF THE INTERFACE FIGURE 48 : IDEALISED CAVITY SHOWING NOMENCLATURE USED IN THE DERIVATION...96 FIGURE 49 : THREE CASES USED TO DETERMINE THE VOLUME OF D AS A PROPORTION OF THE TOTAL CAVITY VOLUME...99 FIGURE 50 : D/V AS A FUNCTION OF Λ FIGURE 51 : FLUSHING REGIMES AS PREDICTED BY Λ AND MASS REMAINING IN CAVITY, M/MI FIGURE 52 : PREDICTED FLUSHING REGIMES AND DATA OBSERVED IN THE MODELLING FIGURE 53: THE ABOVE PLOT SHOWS THE POSITION OF THE TOE OF THE WEDGE FOR FOUR COMBINATIONS OF HEAD DIFFERENCE AND DENSITY DIFFERENCE ix

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11 Glossary Symbol Units Description A m 2 Area a ijkm m geometric dispersivity of the porous medium a L m Longitudinal dispersivity of porous media a T m Transverse dispersivity of porous media C kg/kg Mass fraction D Mechanical dispersion tensor D d m 2 /s co-efficient of molecular diffusion D * d m 2 /s Molecular diffusion Tensor in a porous medium D/V Proportion of cavity volume at transition point between advective and dispersive flushing regimes g m/s 2 Gravity h m Head h f head of freshwater h 2 m Saline head at downstream boundary J kg/m 2 s Mass flux k m 2 Intrinsic permeability k r relative permeability function K m/s Hydraulic Conductivity K m/s Hydraulic conductivity tensor L m Domain or cavity length M/M i Mass of salt divided by initial mass of salt in cavity P N/m 2 Pressure Pe Peclet number Pe m Mesh Peclet number q m/s Darcy flux q m 2 /s flow per unit width Q m 3 /s Flow Rate Q s m 3 /s Fluid mass source r a S w T * ij cavity aspect ratio (depth/width) Degree of saturation, θ/η Tortuosity tensor v m/s Mean pore velocity V m/s mean velocity α, n, m van Genuchten parameters δ characteristic length ratio of pore channel δ ij Kronecker delta x m Average mesh node spacing t s timestep η Porosity θ volumetric water content θ s saturated water content θ r residual saturation Θ effective saturation Λ Ratio of aspect ratio and interface gradient in cavity Ψ m Pressure head µ kg/ms viscosity ρ kg/m 3 Density ξ m depth from top of cavity to salt-fresh interface xi

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13 Introduction Saltwater occurs naturally in aquifers connected to the sea as a consequence of the greater density of the seawater relative to the water in the aquifer. The dense salt water tends to flow in under the freshwater and form a wedge. The wedge occupies a position where the density and pressure forces from the sea pushing the wedge inland are almost balanced by the forces of pressure within the aquifer. These forces are never quite balanced because of the effect of dispersion, eroding the wedge at the interface between the fresh and salt water. Salt is transported out of the aquifer with the freshwater. Simultaneously however, because the wedge is not quite at a position where the forces are balanced, saline water flows into the wedge from the sea. Once at this position of dynamic equilibrium, the position of the interface between the wedge and the freshwater remains more or less constant, as the flux of salt out equals the flux of salt in. The phenomenon is observable all around the world. The saltwater wedge becomes a significant problem if it begins to move inland. It can contaminate wells use for water supply or irrigation, rendering them useless. Ironically, the over-extraction of water from the coastal aquifer is usually the cause of the movement, as it changes the balance of pressure forces in the aquifer. In response the wedge moves inland until it reaches a new equilibrium position. The study of saltwater intrusion is not new. Work stretching back more than one hundred years has examined this groundwater phenomenon, beginning with the classic works of Badon-Ghyben (1888) and Hertzberg (1901). Over time, steadily more sophisticated techniques of describing the wedge, its position and its response to various changes have been developed. Today, finite element groundwater models are often used by resource managers for this purpose. The basement of the aquifer in most numerical models is assumed to be flat. At most, it may be approximated as curving or sloped in some way. Small scale heterogeneities in the basement are difficult to detect and are generally assumed to be of little consequence to the modelling. This work examines the effect of such small scale heterogeneities on saltwater intrusion in a hypothetical aquifer. The scale of these heterogeneities is small compared to the 1

14 scale of the aquifer and the wedge. They may intrude into the aquifer or form a cavity in its basement. A thorough literature review was conducted on groundwater flow in general, saltwater intrusion and physical and numerical modelling. No work was found that examined the effect of such small scale basement phenomena. This literature review is found in Chapter 1 of this thesis. The effect of the heterogeneities was first examined through physical modelling of an aquifer cross-section was conducted in a thin slab sand-box with homogenous properties and carefully controlled boundary conditions. Because no work was previously found on the phenomena, the experimentation was designed to be initially as general as possible, before identifying interesting areas for closer examination. Therefore the aims of this experimentation were To examine the effect of basement heterogeneities on a salt water wedge To identify phenomenon for further work To obtain observed data with which to calibrate a numerical model to further investigate this phenomenon. The methods, results, discussion and conclusions of this experimental work can be found in Chapter 2. The results of the tank experiments were used to calibrate a numerical model. This model was intended as a tool to further examine the areas of interest identified by the physical work. The model used was SUTRA (Saturated Unsaturated Transport, Voss 1984): a finite element code capable of modelling density dependant solute transport in saturated and unsaturated porous media. This calibration process and its results are discussed in Chapter 3. The model was used to further examine the effect of the basement heterogeneities. In particular, flushing was examined from cavities in the aquifer basement as a saltwater wedge retreats from above it. This examination involved analysing the effect of cavity aspect ratio and aquifer head gradient on the rate of flushing. This analysis is described in Chapter 4. The analysis led to the development of an equation describing the relative dominance of advective and dispersive flushing regimes in the cavities. The derivation of this equation is also given in Chapter 4. 2

15 Chapter 1. Literature Review As the population of the world increases and with it demand for fresh water, it is expected that groundwater will become increasingly important as a source of potable water (Essink 2001). Of the available (not ice) freshwater on earth, surface water makes up only 1%. The rest is almost entirely groundwater 1 (Essink 2001). Groundwater in aquifers has been used for agriculture and potable water since the first well was dug millennia ago. Where rain is scarce, civilisations have come to rely on groundwater for survival. Roughly half the world's population live within 60km of the shoreline (Essink 2001). The cities and communities living on the coast often rely on the coastal aquifers beneath them. These groundwater resources require careful management to ensure they are not degraded. In some coastal aquifers, such as those in Israel (Stern et al. 1956), Holland (Volker 1961) and Florida (Kohout 1960), excessive pumping of the aquifer has led to intrusion of the wedge of salt water that extends inland from the seabed. This leads to salinisation of coastal wells and reduced water quality and crop yields (Essink 2001). An understanding of the processes and patterns of moving and mixing in the saline wedge is necessary for the management of these resources. The economic importance of coastal groundwater aquifers has led to a significant amount of research into salt water intrusion since the late 19 th Century (Reilly and Goodman 1985). However, questions still remain, especially with regard to the effect of heterogeneity in the aquifer on the wedge. There have been many recent studies examining the effect of heterogenous aquifers on variable density systems, including Schincariol and Schwartz (1990), Oostrom et al. (1992), Swartz and Schwartz (1998) Simmons et al. (2001), and Silliman et al. (2002). However, little work appears to have covered the question of the effects of basement morphology. The base of the aquifer is generally idealised as smooth and flat. In reality, paleochannels, fault-lines and intrusions may create impermeable boundaries within the aquifer. Abarca et al. (2002) has conducted some three-dimensional modelling on the effect of large-scale heterogenaity in basement formation, and shown 1 Freshwater also occurs in the atmosphere and in biota. 3

16 that its effect may be significant. The effects of small scale heterogeneity may be very different To date, no experimental work has been published on the effect of basement heterogeneities, and no work exists at all on the effects of small scale basement heterogeneities. This paper will report on experimental work, modelling and analysis involving small-scale heterogeneities. The following literature review will discuss the published work relevant to this research. It will begin by outlining the basic principles of groundwater flow. The important works on saltwater intrusion will be reviewed. It will then discuss the principles and practice of physically modelling groundwater systems. Finally it will discuss the numerical modelling techniques, focussing on the numerical model used in this work. 1.1 Principles of Groundwater Flow Saltwater intrusion is a groundwater phenomenon with a long history of study. In order to fully describe the mathematical models of saltwater intrusion both the early analytical approximations and the later numerical solutions it is necessary to lay out the basic principles of groundwater flow. This section will begin by describing the classical approach to groundwater flow in saturated soil, and expand on to more recent treatments that include unsaturated zone flow and transport. The principles of solute transport in porous media will then be outlined, including the relevant Advection- Dispersion equations. Following this treatment of the basic principles, the work done on saltwater intrusion is discussed, beginning with the analytical approximations of the position of the interface. This is followed by the more recent work on numerical solutions to the Groundwater Flow and Advection-Dispersion Equations discussed in the first two sections Unconfined Groundwater Flow Groundwater flow is the movement of fluid through the pore space between soil particles. This is a microscopic process and fluid velocities and pathlines may vary over very small distances; even within a particular pore space. On the other hand, the aquifers through which the water moves may be several hundred kilometres in size. 4

17 This discrepancy between microscopic and macroscopic scales is overcome in groundwater modelling through the use of a Representative Elementary Volume (REV) and the Continuum Approximation as described in Bear (1979). The Continuum Approximation allows the variations at the microscopic level to be ignored. Instead, each point within the domain of the aquifer is part of a field, and subject to the same relationships, irrespective of whether the point is part of the void space or the solid matrix. The relationships implicitly assume the presence of both. These relationships require the properties of the porous matrix, such as the porosity, to be specified. In the continuum approach, the properties of a point in the domain are the average properties of the solid matrix in a certain volume surrounding the point. However, if this volume is too small, the properties will vary depending on whether the point intersects with the pore-space or the solid matrix. If it is too large, it will intersect variations in the type of porous media. In between, an REV is assumed to exist. This is shown in Figure 1. The properties of a point can be defined as being the average properties of the REV that surrounds it. Figure 1 : Definition of the Representative Elementary Volume (Bear and Verruijt 1987). Here U v /U if the porosity, and U 0 is the REV. These ideas are the fundamental framework on which all the theories of groundwater flow utilised in this work are based. Darcy's Law The first systematic study of the flow of water through porous media was by Henry Darcy (Fetter 1994), who showed that the rate at which water flows through a given 5

18 porous media is proportional to the area through which it flows, and the hydraulic gradient, such that 2 : Q = KA( h h ) 2 1 L ( 1-1 ) where Q is the flow rate [m 3 /s], A is the area of flow [m 2 ], h 1 and h 2 [m] are head of water above an arbitrary datum at each end of a flow domain of length L [m], and K [m/s] is the hydraulic conductivity (discussed below). The equation above is known as Darcy's Law and may be defined in differential form as: dh Q = KA dx ( 1-2 ) or in three dimensional vector form as: q = K h ( 1-3 ) where q is known as Darcy flux [m 3 m -2 s -1 ] and is flow per unit area per time, and K is the saturated hydraulic conductivity tensor. It is important to note that while q has units of velocity, this is misleading. The actual average pore velocity, v [m/s], is equal to the Darcy flux divided by the porosity (η): q v = η ( 1-4 ) where η [unitless] is the proportion of a volume of aquifer that is void space. The head term, h, in Darcy's law is the relative energy of a particle of water per unit weight. It is also (in fresh water) the height to which water in a piezometer installed at that point will rise to. The head is composed of two parts: the pressure head (Ψ) and the elevation head (z) as measured from an arbitrary datum, such that: h = Ψ + z ( 1-5 ) Pressure head (Ψ) can be expressed in terms of pressure thus: 2 The information in this section is found in all textbooks on groundwater flow. In particular, (Fetter 1994)has been used as a references for this section. 6

19 Ψ = P ρg ( 1-6 ) Where P is the pressure [N/m 2 ], ρ is the density [kg/m 3 ] and g is gravity [m/s 2 ]. In saturated flow, hydraulic conductivity is a property of the porous media and the fluid flowing through it. The effects of the fluid properties may be expressed separately from the solid matrix properties as: K = ρg k µ ( 1-7 ) where k [m 2 ] is the intrinsic permeability of the porous media and µ is the absolute viscosity [kg/ms]. The value of K must usually be observed directly in the field using pump-tests. However, Some work has been done trying to determine a relationship between K and grain-size distribution. Fetter (1994) describes the work of Hazen (1911), which concluded that hydraulic conductivity was proportional to the square of the effective grain size. Shepherd (1989) surveyed the literature and found a slightly more sophisticated relationship. He found that K was related to the mean grain size: j K = Cd 50 ( 1-8 ) where K is the hydraulic conductivity [ft/day], d 50 is the mean grain size [mm], C is a shape factor and j is an exponent. C and j are higher for texturally mature, well sorted samples. For glass spheres, C is 40,000 and j is 2, while for consolidated sediments, C is 100 and j is 1.5. In any real (non-ideal) porous medium, hydraulic conductivity varies in space. Such systems are said to be heterogenous. Heterogeneity is the result of spatial variations in the material type, porosity, grain-size distribution or arrangement of grains in the porous media. Only carefully constructed laboratory porous media can be assumed to be homogenous, and even then, this can only be an assumption. In addition, the hydraulic conductivity of a particular porous medium may vary depending on the angle with which the fluid is flowing through it. Such porous media are said to be anisotropic. Those without this property are isotropic. Anisotropy results from grains that are not symmetrical, such as layered sediments. Pore spaces may tend to be more 7

20 likely to be oriented in a particular direction, promoting flow in that direction and inhibiting flow perpendicular to it. Unsaturated Flow In unsaturated flow, Darcy's Law still holds but K is not a constant, rather a function of the volumetric water content of the soil (θ), such that K r = Kk r (θ ) ( 1-9 ) Here k r is called the relative permeability function and varies between 1 for θ=θ s =η, where θ s is the saturated water content, and zero or almost zero for θ=θ r, the residual saturation after the soil drains under gravity. The exact form of this relationship must be determined empirically, as it a soil-specific property. In turn, θ is a function of the pressure head Ψ, which in the unsaturated zone is negative. Again this is an empirical relationship, known as the soil water retention function. These two functions together are used to describe flow in unsaturated soils. Equations modelling the relative permeability function and soil water retention function have been developed by van Genuchten (1980). If Θ is the effective saturation given by: θ θ r Θ = θ θ s r ( 1-10 ) where θ s is the saturation water content and θ r is the residual saturation, then: Θ = 1+ 1 n ( α Ψ ) m ( 1-11 ) where α, n and m=1-1/n are parameters that depend on soil properties. A relationship is also given by van Genuchten (1980) for the relative permeability function: kr = Θ 1 2 [ ( ) ] 1/ m m 1 1 Θ 2 ( 1-12 ) where m is the same as in Equation ( 1-11 ). 8

21 Groundwater Flow Equations A general form of Darcy's law can be created by combining equations ( 1-3 ) to ( 1-9 ) that will be used later by the SUTRA numerical model (Voss 1984): kk r v = ηs µ w ( P ρg) ( 1-13 ) Where S w is the degree of saturation given by S w = θ /η ( 1-14 ) The flow equations can be combined with the continuity equation, or conservation of mass to give the groundwater flow equation. Voss (1984) gives a derivation of the groundwater flow equations as used by the SUTRA model. This derivation is described in part below. Conceptually, conservation of mass is simply the idea that within a control volume, the change in mass over time must equal the mass entering the volume minus the mass leaving the volume plus the sum of sources and sinks within the volume. If the mass of water per unit aquifer volume is the porosity (η) times the degree of saturation (S w ) times the density (ρ) then this concept can be described mathematically for groundwater flow as: ( ηs w t ρ) = ( ηs w ρv) + Q s ( 1-15) where Q s is the fluid mass source [m 3 /s] (including pure water mass plus solute mass dissolved in the source water). The term on the left hand side is the total change in fluid mass with respect to time. The first term on the right hand side accounts for the advection of fluid. The fluid mass source term accounts for sources and sinks of fluid. The left hand side of this equation can be expanded to account for the possible changes in mass within a given volume (VOL) of aquifer. If we confine these possibilities to a change in pressure P [N/m 2 ] or a change in solute concentration, C [kg/kg], the change in total fluid mass in VOL can be expressed as follows: 9

22 ( ηs wρ) ( ηs wρ) VOL d( η S wρ) = VOL dp + dc P C ( 1-16 ) The differentials on the right hand side can be expanded using the product rule and simplified by the fact that saturation, S w, depends only on pressure, and porosity (η) is independent of concentration, to give: ( ηρ) S w ρ VOL d( ηs wρ) = VOL S w + ηρ dp + ηs w dc P P C ( 1-17 ) The three differentials on the right hand side can be interpreted in the following way. The first is related to the compressibility of the aquifer through fluid and solid matrix compressibility. Although Voss (1984) gives a relationship for these properties and includes them in the SUTRA model, it is not necessary to do so here, as it will be assumed that the fluid and solid matrix are incompressible, and this term is zero. The second differential describes the change in the degree of saturation for a change in pressure. This is simply the derivative of ( 1-11 ) taking into account the relationship of pressure head to pressure ( 1-6 ), and effective saturation to degree of saturation, ( 1-10 ) and ( 1-14 ). The third differential describes the change in density for a change in solute concentration. Voss (1984) suggests this value is generally considered to be a constant. For seawater at 20 C, when C [kg/kg] is the mass fraction of total dissolved solids it is approximately 700kg/m3. The relationship in ( 1-17 ) can be incorporated into ( 1-15 ) by cancelling VOL from each side and dividing by dt, to give: ( η S wρ) ( ηρ) S w P = S w + ηρ + ηs t P P t w ρ C C t ( 1-18 ) Finally, Darcy's Law ( 1-13 ) can be substituted into the velocity term on the right hand side of the conservation of mass equation ( 1-15 ) to give the final form of the groundwater flow equation: S 10 w ( ηρ) S w P + ηρ + ηs P P t w ρ C C t kk r ρ = µ ( P ρg) + Qs ( 1-19 )

23 Advection and Dispersion Salt water intrusion is a phenomenon involving the transport of a solute through a porous medium. The study of this phenomenon therefore requires a solid understanding of the way solutes move through porous media, and the range of transport mechanisms at work. The section above discussed in some detail the equations of groundwater flow. Those equations describe the flow field within a groundwater system. If a slug of an ideal tracer (ie one that does not affect the properties of the water, or has identical properties, and is not retarded by adsorption onto the solid matrix) were added to a particular flow field, the flow equations would predict that that tracer would move along the flowlines set by the equation. This is known as advection. However, the tracer would simultaneously tend to 'spread out' both longitudinally (in the direction of mean velocity) and transversally (perpendicular to the direction of mean velocity). This phenomenon is known as hydrodynamic dispersion. The following sections will describe the phenomenon of dispersion and the way it is usually modelled. First, a qualitative description of the processes of dispersion will be given. This will be followed by a discussion of the way dispersion has been parameterised using coefficients of dispersivity and diffusion. A brief treatment will be given to the problem of dispersion in the field and the on-going work in understanding this phenomenon. Finally the Advection-Dispersion equation will be given in the form used in the SUTRA model Hydrodynamic Dispersion Hydrodynamic dispersion is a combination of two interlinked but distinct phenomenon: mechanical dispersion and molecular diffusion. Each of these contributes to the spreading phenomenon described above. Molecular diffusion will be described first, as it not only occurs in the absence of flow, but also plays a role in mechanical dispersion. Mechanical dispersion only occurs in moving water. Molecular diffusion is the net transport of molecules down a concentration gradient. Molecules move randomly, and so will tend to move from areas of high concentration to low concentration at a greater rate than they move back. This creates a net flux. The rate of flux is generally assumed to be linearly proportional to the concentration gradient. Molecular diffusion acts even in the absence of flow to spread out a tracer. 11

24 Mechanical dispersion occurs at the microscopic scale. Groundwater flow, despite the large-scale regions in which it occurs, is based on a small-scale phenomenon: water moving between grains of porous media. Although at the macroscopic level the flow lines of this water may appear uniform, this is in fact only a representation of the average flow. At the microscopic level, water flows haphazardly around particles. Water close to a particle may flow slower than water near the centre of a pore space. Water in some pore spaces may flow faster than others. These processes are illustrated in Figure 2. Figure 2 : Pore-scale trasport processes leading to hydrodynamic dispersion (from Fetter 1994, p 456). If the water flowing through the porous medium contains a tracer, this heterogeneity of flow will lead to a general 'spreading out' of the tracer and close contact between areas of low and high concentration. The resultant concentration gradients promote molecular diffusion. Mechanical dispersion therefore always involves some diffusion. This complication will be discussed in more detail in the next section. The combined effect of pure molecular diffusion and mechanical dispersion (which may itself imply some diffusion) is known as hydrodynamic dispersion. This is the macroscopic end-product of these processes. For various systems and flows, hydrodynamic dispersion may be attributable mainly to diffusion or to dispersion. In most groundwater systems, both are present but hydrodynamic dispersion generally dominates. 12

25 Parameterisation of dispersion Theories on dispersion have been developing since the 1950's (Bear 1979). This work has focussed on: incorporating the phenomenon into equations of transport that may be coupled with the groundwater flow equations understanding the nature of the coefficients that arise in these equations and their relationships to the properties of the porous media and flow It is generally assumed that molecular diffusion can be expressed by Fick's Law which states that flux is proportional to the gradient of the concentration: J = D d C ( 1-20 ) where D d [m 2 /s] is the coefficient of molecular diffusion of a particular tracer in a particular fluid (here C is the volumetric concentration [kg/m 3 ]). In groundwater flow, the tracer must diffuse through the pore spaces and around the solid matrix. This results in the net flux of tracer relative to the porous media being different to that predicted by the equation above. Therefore a different coefficient is used, D * d, which is defined: * * ( D d ) ij = D d T ij ( 1-21 ) where T * ij is the tortuosity of the porous medium (Bear 1972, 1979), a second rank symmetric tensor. Therefore molecular diffusion in groundwater may be expressed as: * J = Dd C ( 1-22 ) Mechanical dispersion is usually also described as a Fickian process, but the coefficient of mechanical dispersion, D, is much more complex. It depends on the structure of the porous medium, the velocity and even on molecular diffusion, as mentioned above. Bear (1979) draws on previous work to suggest the relationship between D and these forces is: D ij = a ijkm V k V V m f ( Pe, δ ) ( 1-23 ) 13

26 where a ijkm [m]is the geometrical dispersivity of the porous medium, V is the magnitude of the mean velocity [m/s]and f(pe,δ) is a function which "introduces the effect of transfer by molecular diffusion between adjacent streamlines at the microscopic level" (Bear 1979). Pe is the Peclet number and δ is a characteristic length ratio of a pore channel to its hydraulic ratio. These parameters are described in more detail below. The geometrical dispersivity, a ijkm is a fourth rank tensor which describes the effect of the orientation and distribution of channels within the porous medium. In an isotropic porous medium it can expressed as: a ijkm al at = atδ ijδ km + ( δ ikδ jm + δ 2 im δ jk ) ( 1-24 ) where a L and a T [m]are the longitudinal (in the direction of average flow) and transverse (perpendicular to the direction of average flow) dispersivities of the porous media respectively. The δ ij is the Kronecker delta (not to be confused with the δ length ratio). The Kronecker delta equals 1 when i=j and 0 otherwise. The function f(pe,δ) describes the non-linear effect of molecular diffusion on the dispersion. As described above, mechanical dispersion tends to bring regions of low and high concentration in close proximity, thus promoting molecular diffusion between them. The Peclet number, Pe, represents the ratio of advective and diffusive timescales within the porespace. It is defined as: Pe = lv D d ( 1-25 ) where l [m]is the characteristic length of the pore space. The form of this relationship is proposed as (Bear and Bachmat 1967, Bear 1979): f ( Pe, δ ) = Pe /(2 + Pe + 4δ 2 ) ( 1-26 ) For most purposes this is approximated to be unity. This allows us to combine ( 1-23 ) and ( 1-24 ) to give D ij = at Vδ ij + ( al at ) ViV j / V ( 1-27 ) 14

27 For the special case where one of the axes of a two dimensional Cartesian set of coordinates is parallel to the direction of average flow at a point, this reduces to: D ij alv = 0 0 at V ( 1-28 ) In this case coefficient D 11 is known as the coefficient of longitudinal dispersion, D L, and D 22 and D 33 are the coefficient of transversal dispersion, D T. Hydrodynamic dispersion, as mentioned above, is the sum of mechanical dispersion and molecular diffusion. That is: D = D + * h D d ( 1-29 ) If the longitudinal component of D h is divided by D d the result is a non-dimensional number which is a function of the Peclet number. Bear (1979) presents a graph showing the results of a large number of experiments and appears to verify the existence of such a function (Figure 3). Figure 3 : Relationship between Peclet Number and Dispersion. Taken from Bear (1979) At the lower end of this function, for small Peclet numbers (Pe<<1), molecular diffusion dominates hydrodynamic dispersion and is independent of Peclet number, as would be expected. Molecular diffusion happens at the same rate even in the absence of flow. For Pe=1 mechanical dispersion and molecular diffusion are of approximately 15

28 the same order of magnitude. As Peclet number increases, mechanical dispersion increases relative to molecular diffusion, The relationship may be approximated as D hl / D d = α( Pe) m ; ( 1-30 ) with α 0.5 and 1<m<1.2. For even greater Peclet numbers, a linear approximation of α 1.8 and m=1 may be more appropriate. Problems with Dispersion The above approach to dispersion appears most often in the literature, but it is obvious that it has serious shortcomings. This should be expected, as it is a simple approximation of a process that is too complex to be modelled deterministically. Shortcomings revolve around the assumption that the longitudinal and transverse dispersivities, a L and a T, are constant properties of the porous medium. In general the function f(pe,δ) is assumed to be unity and the coefficients of dispersion of a particular porous medium are calculated by determining a L and a T as defined above. It has been noted that field measurements of dispersivity are often several orders of magnitude greater than laboratory tests of the same material. This is assumed to be a consequence of large scale heterogeneity in the hydraulic conductivity field, which creates areas of lesser or greater flow, and therefore more overall dispersion. Gelhar et al. (1992) present a critical review of 59 field measurements of dispersivity, and found an increasing trend of longitudinal dispersivity with scale of measurement. However, they did not suggest an empirical relationship for this. No relationship was found when only examining data of high reliability; the relationship was only apparent when all the data was included. Additionally, at any particular scale, the longitudinal dispersivity ranges over 2-3 orders of magnitude (Gelhar et al. 1992). However, a common approximation of the relationship is that longitudinal dispersivity is one tenth the scale of the system, and that transverse dispersivity is one hundredth. Of greater interest is the more recently discovered phenomenon of the density dependence of longitudinal dispersivity (Hassanizadeh and Leijnse 1995, Schotting et al. 1999). Experimental work showed that longitudinal dispersion coefficients were exponentially lower than expected for the case of a high concentration fluid displacing a low concentration fluid. It was hypothesised that at very high dispersive fluxes, the 16

29 linear assumption of dispersion breaks down. A new non-linear approximation has been postulated which reduces to the linear Fickian equation for small fluxes. This discovery confirms the intuitive understanding that mixing between high concentration difference fluids should be different to the case when the fluids are similar concentrations, especially when the concentration difference leads to a significant density difference. More energy is required to mix stratified fluids in a surface water body, and therefore the mixing process is dampened. It is reasonable to suspect a similar dampening of dispersion in porous media. However the traditional linear theory of dispersion does not accommodate this idea. The Advection Dispersion Equation The coefficients of dispersion mentioned above are used to formulate a generalised expression for the advection and dispersion of a tracer in a porous medium. This equation is derived from a mass balance for the tracer. Bear (1979) and Voss (1984) give identical expressions for this, albeit in slightly different forms. Neglecting adsorption, production and sources/sinks of the tracer, the mass balance equation is: (η S ρc) w t = ( CηS ρv ηs ρd C) w w h ( 1-31 ) where C is the mass of tracer per mass of fluid. The left hand side of this equation represents the time rate of change of tracer per unit volume of the aquifer. The right hand side represents the two mechanisms causing this change, advection and diffusion. It takes into account both the advective flux of the mean flow (the first term in the brackets) and the dispersive flux (the second term). 1.2 Saltwater Intrusion Qualitative Understanding and Analytical Approximations of the Interface Position Problem Where a fresh groundwater body flows into the ocean, the density difference between the bodies will tend to cause saline water to flow into the base of the aquifer, forcing the freshwater flow to converge upwards and over this 'wedge'. If increased pressure on the water resource or reduced recharge reduces the head gradient across the 17

30 aquifer, this wedge can intrude into the aquifer and reduce the water quality. This problem has been observed since the 19 th century, and is a significant problem in some areas of the world (eg Israel, Holland, Florida). In order to manage the groundwater resources for the future, it is necessary to understand this phenomenon and model it effectively. The literature on groundwater intrusion is extensive. However, a number of works stand out for their contribution to understanding. This section will discuss these works and the way the current understanding of saline intrusion has developed. Two reviews of the work on analytical solutions to the saline intrusion problem have been very useful for this section. They are Reilly and Goodman (1985) and Cheng and Ouazar (1999). Work on saltwater intrusion has typically involved trying to predict the position, and later the movement, of the interface between the saline wedge and freshwater aquifer. As originally discussed by Cooper (1964) this interface is in fact a 'zone of diffusion' where dispersive forces mix the salt and fresh water and lead to the transport of saline water along the interface and out of the aquifer. This flux out is compensated by a flux of salt water into the aquifer from the ocean boundary. However, the head losses associated with this flux and the dispersion of the interface prevent the saline wedge from intruding as far inland as it otherwise might (Cooper 1964). 18

31 Figure 4 : Saltwater intrusion. Saltwater from the ocean flows in under the coastal aquifer, while freshwater flows out above it. Between them a zone of diffusion exists where salt-water mixes with fresh and is transported out of the aquifer (from Cooper 1964). Analytical solutions of the problem of saltwater intrusion are usually steady state and generally make a number of simplifying assumptions. Commonly, the most important of these is that the saline and fresh waters do not mix. Since dispersion of the interface is the driving force behind transport in the wedge, the first assumption implies a second: that the saline wedge is stagnant. Analytical solutions also generally assume the aquifer is homogenous. Approaches that do not assume a sharp interface will be discussed later. As well as being the original thrust of research in this area, analytical solutions are important to understand as they provide a 'first approximation' for engineers. They also provide insight into the forces driving salt-water intrusion that might not be apparent in generalised groundwater flow and advection-dispersion equations. The most well-known analytical approximation of the interface position problem was independently developed by Badon-Ghyben (1888) and Hertzberg (1901). It is essentially a balance of the pressure at a point on th interface, overlain by fresh water, and the pressure at the same elevation in the adjoining ocean. This implies that the depth of the interface is proportional to the hight of the freshwater head above sea 19

32 level, and inversely proportional to the relative density difference of the saline and fresh water. Expressed mathematically: z = ρ s ρ f ρ f h ( 1-32 ) where z is the depth of the interface below sea level, h is the height of freshwater head above sea level and ρ f and ρ s are the fresh and saltwater head respectively. Figure 5 : The Baydon-Ghyben Hertzberg approximation (note here γ s is the density of seawater (ρ s ), γ f of the density of freshwater (ρ f ) and h s =z). (Taken from Bear 1979) This formulation is simple, and is useful as a first approximation. Indeed from it comes the rule-of-thumb that the saline wedge is at a depth of about 40 times the freshwater head above sea level. The assumption that the formula rests upon is that there are no vertical head gradients in the aquifer. That is, the head at the surface is the same as the head at the interface, is the same as the head at the base of the aquifer. This assumption is also a foundation of the Dupuit assumptions (Fetter 1994). An obvious consequence of the assumption is that the thickness of the freshwater outlet is zero. In reality the area of freshwater flux out of the aquifer may be many meters wide, even without considering the presence of a seepage face. Hubbert (1940) brought the understanding of the forces involved in saltwater intrusion forward by observing that the pressure in the saltwater and freshwater flow fields must be equal at the interface. The interface is therefore a boundary surface that 20

33 couples the two separate flow fields together. If P is the pressure in the aquifer at a point, then: h P = ρ g f + f z ( 1-33 ) and h P = ρ g s + s z ( 1-34 ) where h f and h s are the fresh and salt water heads above sea level. According to Hubbert (1940), at the interface, these equations can be solved simultaneously by eliminating P, yielding: z = ρ s ρ f ρ f h f ρ s ρ ρ s f h s ( 1-35 ) For the case of no horizontal head gradient across the wedge, the saltwater head is zero everywhere, and the equation collapses to the Gyben-Hertzberg relationship. Otherwise, if the assumption is made that h f and h s do not significantly vary vertically within their respective layers (a variant of the Dupuit assumption) the equation provides a better approximation of the interface position. However, this requires two observation bores be drilled: one in the fresh zone and one in the salt, both quite close to the interface position (Cheng and Ouazar 1999). It is questionable whether ( 1-35 ) represents a significant practical improvement over ( 1-32 ) given that to have direct usefulness, the Dupuit assumption still needs to be made (although not to the same extent) and two, rather then one observation bores need to be drilled. However, the relationship does have significance as a boundary condition coupling the two flow fields. Glover (1959) used this property to develop an equation which uses discharge, density and hydraulic conductivity to predict the interface position: z 2 2 2Q Q x 2 γk γ K 2 = 0 ( 1-36 ) 21

34 where x is the distance inland from the point at which h f equals zero, and γ is the specific gravity. The assumption is again made here that the saline wedge is stagnant. This formulation is a significant improvement over previous ones as it does allow for an outflow face, although not a seepage face. The work of Henry (1964) attempted to account for dispersion in a saltwater wedge. Henry attempted to derive an analytical solution for a steady-state wedge in idealised conditions. This solution has been used as a benchmark problem for numerical models for some time, but none have been able to replicate Henry s solution (Segol 1994). Part of the problem has been the use of an incorrect dispersion co-efficient due to confusion over the meaning of the value originally used by Henry. However, even after this error was corrected, the analytical solution has not been replicated. Henry s problem is simplified by the assumption that dispersion is not velocity dependant, but is simply a constant. The problem domain is a confined rectangular aquifer with a constant flux of freshwater in from the upstream end and hydrostatic saltwater at the downstream end. Despite the inability of any model to replicate the analytical solution, Henry s problem is still used as bench-mark problem. This is not a futile exercise, as it provides a standard salt-water intrusion problem with which to compare different models, even if they can t be compared to an analytical solution. Some work has been done building on Hubbert's theory to model the position of the toe from an areal 2-D perspective (Strack 1976). This is useful for modelling the effect of a pumping well near the coast on toe position. However, it is only applicable at the regional scale for vertically homogenous aquifer systems that have small vertical head gradients and relatively thin zones of diffusion (Reilly and Goodman 1985). It is not known how common this situation is. 1.3 The Field Studies of Saline Intrusion Observations regarding the encroachment of saltwater wedges into coastal aquifers also extend back over a hundred years. However, systematic, scientific studies have only been common since the 1940s. 22

35 According to Reilly and Goodman (1985) studies were being done in Europe around the turn of the last century, including Badon-Ghyben (1888), Hertzberg (1901), and D'Andrimont (1902), and later in the US, such as Brown (1925). From the 1940s to the 1960s extensive field investigations were carried out on saltwater intrusion. These included Kohout (1960) on the Biscayne Aquifer in Florida, Lusczynski and Swarenski (1966) on the Long-Island aquifer, Stern et al. (1956) on the coastal aquifer in Israel and Volker (1961) in the Netherlands. Reilly and Goodman (1985) point out that many of these studies indicated the importance of the geological structures and formations on the saltwater wedge. The effect of heterogeneity in the formation was first discussed by Lusczynski and Swarenski (1966) who documented the staggered interface of the Long Island aquifer caused by a multi-layered groundwater system and variations in the permeability characteristics of the system. More recently, numerical modelling has had to grapple with this heterogeneity when modelling the systems. For instance, Stakelbeek (1999) took into account the presence of clay-loam layers in a coastal aquifer when examining the effect of deep-well infiltration systems on brackish water migration. The presence of this heterogeneity had a significant effect on the transport of brackish water in the aquifer. Commonly, the heterogeneity of the basement is assumed to be insignificant and is simply modelled as a flat or linear boundary. For instance, the basement of the Nile delta aquifer slopes from 720m deep at the coast to almost nothing near Cairo. To model it Sherif (1999) assumed it could be represented as a smooth, linear, sloping boundary. Rarely is mention made of the possible effects of this assumption. 1.4 Physical Modelling Physical modelling in a laboratory is generally used where the parameters and boundary conditions of the system to be modelled can be accurately replicated, and where little is known about the process being studied. Numerical modelling is preferred where it is assumed that all the relevant processes are captured by the equations of the model. Moreover, numerical models are closed and controllable, (usually) less time intensive and produce detailed information. Physical models are difficult to close, potentially subject to experimental error, require a greater investment 23

36 of time and capital, and information is only available to the degree that it can be sampled. For this reason, physical modelling of groundwater, especially in homogenous porous media, is quite rarely published today. There are many numerical models available that can adequately reproduce many of the systems required. Most of the physical modelling of groundwater today is concerned with contaminant plume transport, especially in unstable and heterogenous systems (see for example Schincariol and Schwartz 1990, Oostrom et al. 1992, Oostrom and Hayworth 1992, Swartz and Schwartz 1998, Simmons et al. 2001, Silliman et al. 2002). Before the advent of high-powered computing, physical modelling was more common. Important studies on saline intrusion include Mualem and Bear (1974) and Collins et al. (1972) which used Hele-Shaw cells to re-create the systems. Hele-Shaw cells use the viscous properties of water sandwiched between two clear plates to simulate the aquifer in cross section. In such models there is no dispersion as long as flow rates are low enough to avoid turbulence. Molecular diffusion still applies. Such models are not useful where the effects of dispersion are significant. More recently, Demetriou et al. (1993) have used a two dimensional thin slab sand tank to examine the velocity of salt-water interfaces between two forcing regimes. The sand tank is essentially a box of sand or similar porous medium that is very narrow in one horizontal direction. It is usually constructed of Perspex to allow visual observation of the porous medium, and dyes are used to differentiate fluids. This approach is common among the contaminant transport experiments cited above. Recently more sophisticated methods of data collection in sand tanks have become common. Conductivity meters embedded in the tank wall were used by Demetriou et al. (1993). Schincariol et al. (1992) have discussed the use of image analysis to map concentration distributions based on dye reflectance. Image analysis requires calibration to match salt concentration with dye intensity with image values. Software capable of such image analysis is also required. The greatest difficulty in using the thin slab approach is ensuring the porous medium has the parameters required, and that these parameters are reproduced with successive packings. As Olivera et al. (1996) demonstrated, the packing technique employed can have significant effects on the properties of the porous medium. They recommended 24

37 the use of a medium with uniform grain-size distribution, and packing by distribution in thin layers, compacted with a heavy pestle Relationship to Surface Flushing Experiments The effect of basement morphology on density affected flow has also been studied experimentally in surface water. Debler and Imberger (1996) have examined flushing of denser saltwater from cavities in the base of estuaries in the presence of surface flows. They found through experimental work that these cavities could be flushed in a single shot if a wave of sufficient magnitude passes over them. If not, the brine in the cavity was more slowly eroded away by turbulent diffusion. It was found that the first type of flushing, known as one-shot, occurred for systems with large densiometric Froud numbers, while smaller valued systems experienced the second type. The shape and geometry of the cavity also affected which type of flushing would occur. This surface water phenomenon is interesting, but the results of this experiment are not applicable to similar groundwater phenomenon. The turbulent, inertial processes of surface water are very different to the slow, laminar processes in groundwater. However, this work does raise some interesting aspects of flushing from cavities that are worth examining, such as the effect of aspect ratio and cavity configuration on flushing rates. It also suggests that there may be more than one kind of flushing occurring in a given system. 1.5 Numerical Modelling of Saline Intrusion It is common practice today to attempt to create a numerical model of the groundwater in areas where saline intrusion is anticipated as a problem. The models can be used to provide insight into the processes occurring within the aquifer, as well as provide predictions as to the future movements of the wedge under various management scenarios. Numerical models of saltwater intrusion can be divided into two types: sharp-interface and advection-dispersion. The sharp interface approach assumes that the zone of dispersion is small compared to the thickness of the aquifer. They generally assume no dispersion or approximate flux across the interface as constant. Advection-dispersion approaches solve the groundwater-flow and advection-dispersion equations over a set 25

38 domain. While these are usually able to model the most general class of problems, they can require long run-times and may require more parameterisation, which introduces more sources of uncertainty in the model process. Shortening run times through coarser discretisation can lead to problems with stability and numerical dispersion. Sharp-interface models are often simpler and require less discretisation, but may not always be appropriate. These two types of models will be briefly discussed here. The numerical model SUTRA (Saturated Unsaturated TRAnsport) will be described in detail. This advectiondispersion based model is used in this thesis for modelling saltwater intrusion Sharp-Interface Approach The sharp-interface approach generally assumes the aquifer contains two fluids saltwater and freshwater and these fluids are immiscible. This assumption may be adequate where the zone of diffusion between the two fluids is small (Essaid 1999). Sharp-interface models assume that the two fluids can be represented as two flow domains coupled by flux and pressure continuity boundary conditions. The pressure boundary condition used by the USGS SHARP model (Essaid 1999) is simply equation ( 1-35 ) above. These boundary conditions are highly non-linear but can be simplified. A only a few sharp-interface models exist, including SHARP (Essaid 1999), 3DSIM and 2DSIM (Bear 1999). Some rely on sophisticated numerical techniques such as the boundary element method (Bear and Verruijt 1987) Coupled Advection-Dispersion-Groundwater Flow equation Approach The vast majority of models are based on solutions to both the advection-dispersion and density dependant groundwater flow equations. These equations must be solved over a domain with certain properties, boundary and initial conditions. Two of the most common methods for solving these problems are the finite difference method and the finite element method. Finite difference is the older and simpler method. It simply divides each of the dimensions of the domain (spatial x-y and temporal t) into a series of segments small enough that the solution can be assumed to vary linearly between them. The finite element method is similar to finite difference in some respects, but is more sophisticated in others. The method involves defining a network 26

39 of nodes connected by elements, over which the governing equations are solved. The finite element assumes the solution can be approximated by a (usally linear) shape function between the nodes. The elements of this betwork may in theory be any shape but are generally triangular or quadrilateral. This network need not be regular, and may trace around arbitrary boundaries and be coarser or finer as need dictates. This is one of the most powerful aspects of the method. Two important considerations when using numerical models are stability and accuracy (Bear and Verruijt 1987). In models that are unstable, small numerical errors such as round-off or truncation error compound and increase in magnitude over time. This leads to a condition where the numerical solution may contain extreme oscillations and break down completely. Accuracy requires that errors do not build up over time, even if they do not create instability. This occurs, for instance where mesh size is too large to represent smaller scale processes. An example in solute transport is numerical dispersion of the solute caused by approximation of the first-order derivatives of the concentration field (Bear and Verruijt 1987). Both these errors can be minimised by increasing the spatial and temporal discretisation. This increases the computation time required. However, it is possible to determine criteria for the discretisation that minimises these errors. The Mesh Peclet number ensures that numerical dispersion is smaller than the actual dispersion of the porous medium: Pe m V x = or D h Pe m x = since D α L V α L ( 1-37 ) where Pe m is the Mesh Peclet number, x is the distance between nodes, V is the velocity, D h is the co-efficient of hydrodynamic dispersion and α L is the coefficient of longitudinal dispersivity. Pe m should be set to below 2, but may be set as high as 4 for some applications (Voss 1984). It is also necessary to set the temporal discretisation such that the time step is smaller than the time taken for a particle of fluid to cover the distance between two nodes. Failure to do so results in numerical instability. This can be expressed as: t < x V ( 1-38 ) 27

40 These two conditions permit the estimation of maximum spatial and temporal discretisations with acceptable levels of stability and accuracy. There are a wide range of numerical models that solve the groundwater flow and advection-dispersion equations. These models vary in their numerical solution techniques, computational efficiency and ability to model other processes such as unsaturated flow, multi-phase flow, adsorption and chemical reactions. Examples of these types of models used in saltwater intrusion include HST3d and SWIP. The particular model used in this work (SUTRA) is discussed in more detail in the following section SUTRA The US Geological Survey s SUTRA code (Saturated Unsaturated TRAnsport) was developed by Clifford Voss and has been available since Its author claims it is the most widely used simulator in the world for seawater intrusion (Voss 1999). SUTRA is a two dimensional hybrid finite element and integrated finite difference model of density dependant saturated or unsaturated groundwater flow. It can be used to model either solute transport with adsorption to the porous medium or first or zero order production or decay; or transport of thermal energy (Voss 1984). SUTRA solves the density dependant groundwater equation and the advection dispersion equation simultaneously on a finite element mesh. A hybridised finite difference time discretisation is used to increase computational efficiency. Picard iterations are used for resolving non-linearities. The mesh is composed of bilinear quadrilateral elements. Flux terms are defined element-wise, while non-flux terms are defined node-wise. The SUTRA model, as stated previously, is widely used for the simulation of saltwater intrusion. It is robust and reliable, and there is a great deal of literature that discusses both the code and its applications (see for example Voss 1984, Sherif 1999, Voss 1999). The code includes a consistent velocity calculation that counteracts the spurious vertical velocities that can occur in saltwater intrusion simulations where concentration fields undergo rapid spatial changes (such as at the interface, see Voss 1984). The code was written for numerical robustness, and so may lack efficiency and speed compared to other codes. Other drawbacks include the fact that the code for the model 28

41 must be obtained and re-compiled to change unsaturated properties or set timevarying boundary conditions. The model requires specification of pressure and concentration boundary conditions. Because the model simulates density-dependant flow, head cannot be used to define boundary conditions. Parameters such as dispersivity, hydraulic conductivity and porosity must be specified for each element. 1.6 Conclusion Despite the long history of work on saline intrusion, very little work has been done on the effect of basement heterogeneity. Abarca et al. (2002) examined the effect of sloping and concave basements using the SUTRA numerical model and found that heterogeneity can have significant effects on the shape of the interface and flow within the saline zone. However, no work is available on the effect of small-scale heterogeneities which might have the ability to trap or conduct dense saline flows. Most modelling of salt-water intrusion assumes the basement of the aquifer is perfectly flat and, by implication, that any heterogeneity in the basement has little or no effect on the shape of the wedge. 29

42 Chapter 2. Physical Modelling Physical modelling of the hypothetical aquifer was conducted with three aims: To examine the effect of basement heterogeneities on a salt water wedge To identify phenomenon for further work To obtain observed data with which to calibrate a numerical model to further investigate this phenomenon. As discussed in the literature review, no published work was found of experimental work on the effect of basement morphology on saline wedges. Most aquifer basements are conceptually modelled as flat, or at most sloped in some way. This experimental work was conducted to examine, from as general perspective as possible, the effect of small heterogeneities on saltwater wedges. Small heterogeneities refers to structures in the basement at a scale smaller than the wedge. These structures might be intruding into the aquifer formation or cavity-like. If these structures were composed of impermeable material, or even material of lower permeability than the aquifer, they might have a significant effect on the intrusion, recession, and steady-state position of the wedge. The experimental work was conducted in a sand box that replicated a cross-section of an aquifer. The sand-box was constructed of Perspex and dyes were used examine the flow within the hypothetical aquifer. As little work had been done in this area, it was necessary to start with a broad approach, identify areas that might be of particular interest and then explore them. A broad approach was taken by examining a hypothetical aquifer with idealised properties. The porous medium of the tank was packed to be as homogenous as possible and the boundary conditions were carefully controlled. The work attempted to examine both intruding and cavity-like heterogeneities at the same time, by constructing a heterogeneity that contained both intruding and cavity-like parts. This amounted to two small walls mounted at the base of the tank a short distance apart. The walls created an intruding obstruction, while between them was a cavity. Careful attention was paid to the form and movement of the salt-water wedge as it moved in, reached steady state, and was pushed out again as the boundary conditions of the tank were changed. Additionally, digital photography was used to record the 30

43 wedge position over time. This allowed qualitative analysis of the movement of the wedge and speculation of the processes at work, as well as quantitative results which could be used to calibrate a numerical model. This calibration is described in the next chapter. This chapter is divided into four sections. The first section on Materials and Methods describes the equipment used for the experimental work and the way the experiments were carried out. The second section presents the Results of the experiments in the form of brief descriptions of the wedge progression and recession, both with and without the obstruction in place. It also presents plots of the interface position over time, obtained by transforming the recorded digital images into x-y co-ordinates. Not all the data is presented; only what is necessary to show the general processes and patterns of the results. These processes and patterns are examined in the third section of the chapter, Discussion. Finally, the fourth section, Conclusions, draws together the discussion and relates the findings to the aims of the experimental work described above. 2.1 Materials and Methods The experimental portion of this work was carried out predominantly in the Hydraulics Laboratory at the Centre for Water Research at the University of Western Australia. Materials used were restricted to those available at the Hydraulics Laboratory. The exception to this was the porous media, which was purchased specifically for this work. The experimental work centred on physically modelling a two dimensional vertical cross-section of a hypothetical aquifer. To achieve this, a tank with a narrow third dimension was filled with a porous media. The upstream and downstream heads in the tank were fixed in reservoirs at each end to create a hydraulic gradient and thus horizontal flow. To create the saline downstream boundary, the downstream reservoir was filled and constantly replenished with saline water of known density and constant colouration. Experimental runs were conducted first with a flat basement. This first stage, sometimes referred to as the homogenous case was essentially used as a control case. In the second heterogeneous stage, an obstruction was added to the basement. All runs were digitally photographed and their parameters recorded. 31

44 Tank The tank was not specifically constructed for this work and has been used for a number of different purposes in the past. A cross section of the tank is shown in Figure 6 below. Figure 6: Cross section of the experimental tank (not to scale). The tank basically a three dimensional slice of an aquifer with a narrow third dimension. The head at each end is controlled by changing the elevation of the down-pipes located in each reservoir. The walls of the tank were constructed of 1.5mm thick Perspex, reinforced by one horizontal and three vertical steel bars. Perspex plates were bolted on to each end to allow the reservoirs to be seen. The base was also Perspex. The whole assembly was held off the ground on a steel frame. Silicon sealant was used extensively to ensure the tank is water tight. The reservoirs were separated from the porous media by vertical slotted Perspex panels and a geotextile of unknown specifications. The slotted panels had 1cmx13cm slots in 5 staggered groups, though which water could pass. 32

45 Figure 7 : Construction of the side panels of the tank. The slotted screen (right) was wrapped in geotextile and inserted into the end of the tank, forming a reservoir. (left) A plan view of the downstream reserviour. The slotted panels were not sealed onto the sides of the tank. Instead they were fitted with side pieces that sat almost flush with the sides of the tank and the geotextile was inserted in the gap (Figure 7). The right hand side panel was not quite vertical and was further into the tank than the left hand side. For this reason the right hand side was later designated as the upstream boundary, where these imperfections would have minimal effect. The internal dimensions of the tank are shown in Table 1. Table 1: Dimensions of the tank Dimension Value [m] Total internal length 1.8 Internal width Total depth 1.04m Depth of porous media 0.8m Length of left (downstream) reservoir Length of right (upstream reservoir) (top) (bottom) Average width of porous media Porous Media Glass beads were chosen for the porous media. The tank had previously been packed with filtered sand (size range 0.5-1mm). The sand had proved difficult and laborious to 33

46 pack. Three days had been spent packing the sand for a previous trial, and was still known to contain heterogeneities (Sivandran, pers comm). Glass beads were investigated as a better option for achieving an homogenous medium in a minimum of time. The choice of packing material was affected largely by the ability of a material to produce a consistent, homogenous hydraulic conductivity and by the particle size distribution. It is easier to obtain a uniform packing if a high bulk density (low porosity) is achieved, either through vibration or compaction (Yaron et al 1966, in Olivera et al 1996). Both of these methods can lead to lateral particle segregation, either through the free fall of particles during deposition or through the bounce of particles during the vibration or compaction. It is therefore preferable to obtain a granular medium of as narrow a grain size distribution as possible. The sand that had previously been available had a grain size of microns. Glass beads can be obtained with very narrow grain sizes, such as those of technical quality used in some of the research cited above (eg Olivera et al (1996) uses microns). Technical glass beads could not be obtained quickly and at low cost. However, glass beads commonly used for sand-blasting were available relatively cheaply. These lack the technical quality, but are of sufficiently narrow particle size distribution to be advantageous. The chosen beads were GB1(A) Sandblasting beads. These beads have a nominal diameter of microns. Supplier information indicated at least 65% round. They contain no free silica. The specific gravity of the glass is 2.5g/cm 3. The tank was packed in narrow layers under a small amount (<2cm) of water. Glass beads were sprinkled from a height of usually around 10cm and no greater than 20cm. A specially constructed shovel exactly 20cm wide was used to sprinkle the beads. The layers of glass beads were 2-4cm thick. Each layer was compacted using a 20cmx20cm metal plate attached to a broom handle. The plate was rammed against the glass beads in a regular pattern. This process did cause some of the beads to splash up. It also entrained some air, but this was dissolved by running water through the tank for 24 hours. Once the layer was compacted the water level was raised to approximately 2cm above the top of the beads, and the next layer was added. The total depth of porous medium was 0.8m. 34

47 Camera and Lighting Initial runs without the obstruction were recorded using time-lapse photography on a Hewlett Packard Photosmart 618 digital camera. The camera was set to automatically take one shot every five minutes. The frame of the shot was set to the boundaries of the tank to maximise resolution. Later runs were recorded similarly using a Ricoh Caplio RR1 digital camera. An Inaebeam tungsten lamp was used to illuminate the tank during the first set of runs. The light was set to a position off to one side and illuminated the tank obliquely. Before beginning the second set of runs with the obstruction, it was noted that the supporting frame of the tank cast shadows across the area around the obstruction. Therefore the lighting was supplemented by a second tungsten light on the opposite side Dye Aeroplane brand food dye was used in the point-release and intrusion tests. For cases where the dye was injected into the tank using a syringe, the dye was used either undiluted or mixed with brine to give it the density of the fluid it was entering as closely as possible. This proved quite difficult. The very slow rates of movement in the brine meant that even very slow buoyancy driven flows resulting from small differences in density cause significant error. During the intrusion experiments the dye was mixed with saline water at a ratio of 50ml per 10L of saline water Hydraulic Conductivity Tests The hydraulic conductivity was initially tested using a simple flow-rate test. This test was based on the Dupuit Equation (Fetter 1994): q = 1 2 h K 2 1 h L 2 2 ( 2-1 ) where q is the flow per unit width, K is the hydraulic conductivity, h 1 and h 2 are the head at each boundary and L is the distance between the boundaries. For the tests, the head at each boundary was set and the flow rate, Q, measured. By dividing Q by the width of the tank, q was obtained. 35

48 The observed valued of q were plotted against the Dupuit gradient : Dupuit gradient = 1 2 h 2 1 h L 2 2 ( 2-2 ) A line was fitted to the points on this graph using a least-squares method. The slope of the line was interpreted as the value of K. This technique was used for the first and second stage of experiments. Results are presented in Section Point Release Tests Point release tests were conducted with two aims in mind: To determine if heterogeneities existed in the porous medium To examine flow patterns during intrusion tests Initial point release tests consisted of setting a particular head difference across the tank and releasing dye at points within the porous medium close to the upstream end. Dye was inserted via a syringe connected to thin metal tubing (external diameter 2mm). The tank was then photographed at 5 minute increments. Three head differentials were set up. For each configuration, three sets of at least four slugs of dye were released, giving 12 replicates in total. Each set was created by inserting a long thin metal syringe into the porous medium and releasing dye first close to the bottom, and then progressively higher. Later point release tests were used to examine flow during intrusion experiments. Dye was injected into both the freshwater flow-field and the wedge in a similar fashion to the above. However, it became apparent that the high density difference between the dye and the saline waterand the very slow flow regimes in the wedge caused the dye to inaccurately represent the flow. To remedy this, the dye was mixed with brine before injection into the salt wedge. This technique had mixed results Saltwater Intrusion Tests The tests were run in two stages. In the first stage the base of the tank was flat. In the second stage two small walls were installed to make an intrusive obstruction with a cavity. 36

49 Boundary conditions were established in both stages the same way. The right hand side was designated the upstream boundary. A hose connected to the mains continuously added water to the upstream reservoir, while a downflow pipe within the reservoir maintained the head of water at 0.8 meters. The (left-hand) downstream boundary of the tank replicated the high-salinity, low head downstream boundary of a coastal aquifer. The head of this boundary was varied in each simulation to reproduce the desired head gradient across the tank. It was also necessary to maintain the downstream boundary at a set salinity at all times. This salinity was set to a concentration of approximately 0.21 kg/kg salt, giving a density of 1150kg/m 3 for all experimental runs (see 0). Maintaining this proved to be one of the most difficult parts of the experimentation. The salt was mixed with warm water to dissolve it until a density of 1150kg/m 3 was obtained. Density was measured using a hydrometer. Saline water was mixed in batches up to 90L at a time to maintain consistent density during experiments. Dye was added to these batches in a constant proportion. The saltwater was transferred to a feeder bucket on the top of the tank. This bucket was topped up regularly. At the base of the bucket a valve controlled the flow out of the bucket and into the reservoir. This valve had to be set manually and adjusted to maintain the correct flow rate. A hose was connected to this valve and ran into the bottom of the reservoir, which was initially filled with freshwater. To initiate the experiment, the valve in the feeder bucket was opened and saline water ran out of the bucket and displaced the freshwater, which ran into a down pipe in the centre of the downstream reservoir. Once all the freshwater was displaced, the valve was adjusted to reduce the flow of saltwater. Freshwater continuously entered the downstream reservoir from the porous medium. This water either entered from the seepage face and floated on the surface of the saltwater before flowing into the down-pipe, or flowed from below the waterline, and diluted the saltwater. The valve on the bottom of the saltwater bucket had to be set high enough to avoid significant dilution of the downstream reservoir. This meant sacrificing significant amounts of partially diluted saltwater down the down-pipe. Maintaining the correct flow rate from the feeder bucked required trial and error and constant attention. Almost all the runs of the first stage of experimentation suffered 37

50 from inadequate attention to the degree of dilution in the downstream reservoir. The reservoir became diluted and consequently the pressure along the boundary dropped. If there was a wedge in the tank at this time, the loss of pressure at the downstream boundary caused the wedge to retreat. Consequently these runs could not be used for calibration purposes with the model. Only on the last run of the homogenous stage, performed on the 14 th of July, was the boundary condition maintained with sufficient accuracy to be used for calibration purposes. This run consisted of an upstream freshwater head of 0.8m and an initial downstream saline head of 0.7m (datum is the base of the tank). The wedge was allowed to progress into the aquifer for 225 minutes, during which time 46 photographs were taken (1 per 5 min). After this time the tank appeared to have reached approximately steady state. It was maintained at this position for 140 minutes, and 28 images were collected. The downstream head was then reduced to 0.65m. The wedge was allowed to recede for 115 minutes, and 23 images were taken. The first set of boundary conditions used in this run (upstream boundary 0.8m freshwater, downstream boundary 0.7m saline) are important and will be referred to as the maximum-wedge boundary conditions. These conditions result in the wedge achieving steady state with the toe of the wedge just touching the upstream end of the tank. For more discussion see 0. It is recognised that data from a single experimental run is insufficient for model calibration where a significant degree of confidence required. However it was decided that since the homogenous case, as stated above, was only to be used to ensure that the model could reproduce the significant features of the observed tank, this was adequate. For the second stage of experimentation, the glass beads were removed and two small walls extending across the thickness of the tank were inserted at the base of the tank (for discussion how the size and position of these was decided see 0). The walls were 103mm high, 14mm thick and set 110mm apart. The centre point between the walls was 900mm from the downstream boundary. The walls were constructed of Perspex and held in place by two Perspex braces running between them along the base of the tank. The walls were positioned over an outlet on the bottom of the tank to allow the cavity inside the walls to be drained. Silicone sealant was used to prevent water moving between the walls and the base or sides of the tank. However, the silicone was 38

51 only applied to the outer edges of the walls. As will be discussed later, this sealing was found to be insufficient. The glass beads were spread out on material and allowed to dry for several days. This ensured they could be packed into the tank in the same way as in the previous stage. Packing was complicated be the presence of the obstruction, which prevented the porous medium within the cavity of the obstruction being packed with the same regime of compressions that the rest of the tank was. This may have resulted in different hydraulic properties within the cavity of the obstruction. For the second stage a more careful approach was taken to the experiments, and in all but one run the boundary conditions were maintained satisfactorily. A total of 6 runs were conducted. The first run was a progression of the wedge into the aquifer from completely freshwater initial conditions and maximum-wedge boundary conditions, as discussed in the homogenous case presented above. The rest began with the wedge at steady state with maximum wedge boundary conditions, which were then altered to drive the wedge out. The initial condition of the wedge at steady state was obtained by raising the downstream head to almost 0.8m, allowing the wedge to intrude quickly, and then reducing it to 0.7m, allowing it to reach steady state. The following table summarises all the runs that were conducted in the second stage of the experimental work, and the only run from the first stage to be used for quantitative analyses. For each run a separate record has been entered each time the boundary conditions change (labelled 2a, 2b etc): 39

52 Table 2 : Summary of the second stage experimental runs Run Initial Downstream Length of Condition boundary condition run [min] h 2 [m of saline head, * indicates error] (number of images) Comments 0a Completely fresh (73) Initial progression for homogenous case. Used for calibration. 0b Wedge SS (22) Recession for homogenous case. Used for calibration. 1a Completely fresh (47) Initial Intrusion for second stage with obstruction. Used for calibration. 2a (2) To check steady state 2b Wedge SS (24) 2c 0.6* 110 (23) BC not maintained 4 3a (8) To check steady state 3b Wedge SS (38) 4a (9) To check steady state 4b Wedge SS (18) h 2 incorrectly set (meant to be 0.65) 4c (15) h 2 corrected 4d 0.65* 445 (89) BC not maintained 5a (6) To check steady state 5b Wedge SS (20) Repeat of run 4 with correct BC 6a (6) To check steady state 6b Wedge SS (55) Used for calibration. 6c 0.63* 40 (9) BC not maintained For each run a number of digital images were taken. A large ruler was placed in front of the tank to indicate length scales that would later be used when digitising the data Data collection The digital images collected were analysed to obtain the interface position over time. At the early stages of experimentation, this was done crudely be measuring the lengths off printouts of the images. Later a short program was written using MATLAB to input the position of the interface into a form that could be used in the calibration process. Scaling factors were obtained by measuring features of the tank and comparing these with their size on the images. For the first stage of the experimentation every fifth image (that is, every 25 minutes) was analysed manually. The data obtained was used for some initial calibration of the model. The distance of the interface from the downstream boundary was measured at heights of 10cm, 20cm, 30cm and 40cm from the base of the tank. The position of the 3 Signifies wedge at steady state with previously specified boundary conditions 4 Indicates downstream reservoir was allowed to become diluted with freshwater. 40

53 toe of the interface at the base of the tank had to be obtained by extrapolating the shape of the interface. The bottom 7cm of the tank is obscured by a metal brace. For the later calibration process, a MATLAB scrip was used to obtain x-y co-ordinates of the interface within the tank. The script read the image files and displayed them on screen. The user then indicated the length of 1 metre, using the ruler that was positioned at the base of the tank in the later photographs, or from the features on the tank. The position of the origin was also inputted, in this case the bottom left hand corner of the tank. Then by clicking on the interface, the important points are recorded as horizontal (x) and vertical (y) distances from the origin in meters. The code for this is given in Appendix C. 2.2 Results The results of the experimental work have been analysed a number of ways, and data relevant to each of these analyses is presented here. First, results of the initial analysis to determine the hydraulic properties of the tank are presented. Second, the sequence of phenomenon that occurred during the intrusion and recession of the wedge is described. This analysis, though qualitative, was one of the most valuable aspects of the experimental process. Third, the movement of the interface is described quantitatively through the interface data collected from the digital images. This information was useful in the model calibration process described in the next chapter. Finally, results showing the effect of the various head gradients on the flushing from behind and within the obstruction are presented. This analysis is related to the sensitivity analysis discussed in Chapter Properties of the tank The flow-rate tests showed very strong correlations between Dupuit gradient and flow-rate (Figure 8). This relationship broke down for large gradients, as unsaturated effects and the presence of a seepage face became important. The relationship was strong in both stages and produced very similar values of K. In stage 1, hydraulic conductivity was 225m/day, while in stage 2 it was 226m/day. This similarity tentatively validates the choice of the glass beads as a porous medium. In this particular case they have been shown to provide a consistent hydraulic conductivity in two separate packings. 41

54 Dupuit gradient vs q' q' [m2/day] Stage 1 Stage 2 y = 224.6x R 2 = y = x R 2 = Dupuit Gradient [m] Figure 8 : Results of flow-rate tests to determine hydraulic conductivity (K). K is the gradient of the line connecting the data points, as shown in equation ( 2-1 ). The point release tests also indicated the homogeneity of the porous medium. An example of these tests is shown in Figure 9. In the first stage, no significant areas of low hydraulic conductivity were observed, though an area of slightly higher hydraulic conductivity was observed about a third of the way up the tank and extending all the way across. This may indicate a layer where the packing regime for the glass beads was not followed exactly. 42

55 Figure 9: Point release test in stage 1. The paths of the dye slugs indicate the homogeneity of the tank. There appears to be an area of increased conductivity about a third of the way up from the bottom of the tank. It is also interesting to note in this image the degree of dispersion of the dye slugs. No quantitative estimation of dispersivity has been made from these tests. However, the dye slugs, which were initially small and round, have doubled in height and become much longer. This may indicate that the common rule of thumb which sets transverse dispersivity at one tenth of longitudinal dispersivity may be a reasonable approximation Qualitative description of the wedge progression for homogenous and heterogenous cases The observations of wedge intrusion phenomenon are described here as they were observed for both the flat, homogenous tank in the first stage of experiments and the second stage of experiments with the obstruction fitted. The events are described chronologically. Attention is paid to the position of the interface at each time, without speculation as to the processes at work. Discussion of the processes will be given later. The homogenous case is described first, using images from Run 0a. 43

56 At the initiation of each experiment, the tank was fresh and clear of salt. On initiation, the left-hand downstream tank was filled with brine and this brine began to intrude into the tank. As shown in Figure 10 the intrusion began in the bottom left-hand corner of the tank. The interface is slightly curved convexly and there is evidence of brine flowing down the boundary above the intrusion. Figure 10 : Run 0a at 5 minutes. Intrusion begins from bottom left-hand corner. Brine may be flowing down the boundary on the As the intrusion grew in the initial stages, the height of the intrusion increased at approximately the same rate as the toe moved forward. The wedge at this stage was equally long as tall (Figure 11). Figure 11 : Run 0a at 20 minutes. Early growth of the intrusion is both horizontal and vertical. At some point the top of the wedge ceased to move up the boundary, and reached an equilibrium position. For the case shown this was at about 30cm from the top of the porous medium (Figure 12). The toe of the wedge continued to move upstream, though its pace slowed. 44

57 The interface at this point took on the shape that it maintained at steady state. Previously the interface was roughly linear or at most slightly concave. Once the height of the wedge became fixed, the interface became much more linear towards the toe and more concave towards the upstream boundary. Figure 12 : Run 0a at 85 min. The wedge has ceased to increase in height, while the toe continues to move forward. As the toe continued to intrude, the linear portion of the interface was attenuated, while the concave portion remained roughly the same length. The pace at which the toe intruded became extremely slow as it approached the steady state position. Figure 13 : Run 0a at 365 min. The wedge is approaching the steady-state position. Point release tests were conducted during some of the failed first stage trials to examine the rate of water movement in the brine, in the freshwater and at the interface. One of these is shown in Figure

58 Figure 14 : Point release test from a failed run during the first stage. Initial position of the released dye is shown by the dashed blue line. This image was taken at a time at which the wedge was moving forward and was not at steady state. The dye was initially injected as a vertical streak where the blue dotted line is shown in the Figure 14. The dye in the freshwater section at the top flowed downstream, converging towards the seepage face at the downstream boundary. Dye within the wedge moved upstream. The dye appeared to have moved vertically also, but this was likely to be a consequence of the relative buoyancy of the dye compared to the brine. It appeared to have been moving horizontally with the wedge at approximately the same rate as the toe. Dye released at the interface became attenuated along the interface, such that a continuous streak connected the dye in the brine with the dye in the freshwater. In the second stage of the experimental process, the obstruction appeared to have little or no effect of the progresion of the wedge until the toe of the wedge reached it. The wedge became higher and flatter until the upstream end of the wedge was able to flow over the first wall and fill the cavity in the obstruction. Once this was filled, the brine flowed over the second wall. The brine formed a wedge on the upstream side of this wall, initially with a large zone of diffusion. As this wedge was filled, it became continuous with the downstream part of the wedge. It then moved towards a steadystate position identical to the position of the steady-state wedge in the first stage of the experiments. This process will now be described in more detail. The initial phase of the wedge progression for the second stage was identical to that for the first stage. The wedge began to intrude from the bottom left hand corner of the tank and increased in size vertically and horizontally until the height of the wedge at the 46

59 boundary reached some equilibrium position. The toe moved upstream until it reached the first wall of the obstruction. Once the toe reached the first wall the interface moved vertically as the wedge filled behind the first wall. The interface closest to the wall moved up the most, while further back the movement was less. The top of the wedge at the boundary may have moved vertically, but it may be too small to be significant (Figure 15). Figure 15 : Run 1a at 45min. The toe of the wedge has reached the first wall and is increasing in height. Further back the increase is less. Movement at the upstream boundary may not be significant. When the end of the wedge reached the top of the first wall, it flowed over the wall and began filling the cavity. The position of the interface just downstream of the wall continued to increase slightly, but most of the interface remained stationary while the cavity filled. Brine within the cavity was diffused in some parts. Figure 16 : Intrusion Run 1a at 75 min. Brine from the wedge begins filling the cavity. The position of the interface downstream of the wall was constant during the filling. 47

60 Once the cavity was filled, the brine within it became continuous with the rest of the wedge. The interface became smooth without any obvious change in slope in the vicinity of the first wall. The brine flowed over the second wall and into the upstream region past the obstruction. Here there was no further impediment to the movement of the brine and it flowed out across the base of the tank. Its progression at the toe appeared to slow as it moved upstream, and a wedge began to form behind the toe. This wedge was initially highly diffuse. As the brine continued to flow over the obstruction and into the upstream area, the wedge became less diffuse. Figure 17 : Run 1a at 145 min. Brine flows over the second wall and into the upstream area. Note the diffused wedge in the upstream region and the lack of any change of slope on the interface in the vicinity of the first wall. The interface of the upstream wedge became continuous with the interface of the rest of the wedge. The interface again did not show any change in slope in the region of the second wall. Eventually the wedge moved forwards towards the steady state position, which was identical to the steady state position in the first stage of experiments. 48

61 Figure 18 : Run 4a at 0 min. The wedge almost at steady state ready to begin a recession test. This steady state position was the initial condition for all recession tests, which are described in the next section Qualitative description of the wedge recession for homogenous and heterogenous cases The recession of the wedges occurred when the head at the downstream end of the tank was reduced. In all cases the wedge that was previously at steady state in the tank moved back downstream towards a new steady state position. In the first stage of experiments, no obstruction prevented the wedge from moving back to a new steadystate position. In the second sage, the presence of the obstruction has a significant effect on the recession. The description of the movement in the first stage will be presented first. Beginning with the wedge at steady state, the head was changed in Run 0b from 0.7m to 0.65m of saltwater head. Immediately, the wedge began to recede. The recession was the most dramatic near the toe in the early stages. The entire linear portion of the wedge, as described above, moved backwards. In contrast, the concave section near the top of the wedge increased marginally in height immediately after the change in boundary conditions. 49

62 Figure 19 : Run 0b : Wedge movement immediately following the change in boundary conditions. Arrows indicate the direction of movements of parts of the wedge. After this initial rise, the top of the wedge fell steadily. The toe of the wedge moved downstream and the linear section of the wedge retracted. Figure 20 : Run 0b at 110min. Wedge retracts vertically and horizontally to new steady-state position. Run 0b was completed before the wedge reached steady state with the new boundary conditions. The presence of the obstruction in the second stage of the experiments had a significant effect on the recession of the wedge in all the second stage recession runs. Images will be presented here from Run 6b (which is used later for calibration purposes) but the sequence of phenomenon was similar to the other runs. The effect of the differences in head gradient in the other runs is presented later. Beginning with steady state, the head boundary in the second stage experiments was dropped, and the wedge was allowed to move towards equilibrium with the new set of boundary conditions. In Run 6b the boundary was dropped from 0.7m to 0.63m of salt- 50

63 water head. As the wedge retracted, brine remained in the cavity and in a wedge upstream of the obstruction. The wedge retracted to a steady-state position some distance downstream of the obstruction. The wedge upstream of the obstruction and the brine in the cavity were slowly reduced over time. These phenomenon will now be discussed in more detail. The wedge began to retract immediately after the boundary conditions were changed. The toe of the wedge moved back and the top of the wedge moved downwards. The rise in the wedge at the boundary that was noticeable in the homogenous case was absent. A kink was noticeable in the interface above the obstruction very soon after recession began. This kink remained above the wedge as the interface receded (Figure 21). Figure 21 : Run 6b at 20 min. Arrows indicate the movement of the interface as it recedes. Note the kink above the obstruction As the interface approached the obstruction, the toe of the wedge began to slow its retreat. Between the toe and the obstruction, the brine remained as a wedge. Downstream, the interface continued to retract and move downwards. As it reached the obstruction, the end of the downstream wedge dropped below the height of the first wall. Brine continued to flow over this wall out of the cavity. Brine also flowed from the upstream wedge over the second wall and into the cavity (Figure 22). 51

64 Figure 22 : Run 6b at 65 min. Wedge continues to retract, leaving behind brine in a wedge upstream of the obstruction and within the cavity. Brine continues to flow over the walls. Eventually the downstream wedge moved away from the obstruction altogether. Inspection through the base of the tank, which is also Perspex, revealed that the final separation of the of the toe of the downstream wedge and the obstruction was somewhat later than might be assumed form extrapolation of the interface that can be seen in the images. The flow out of the cavity and out of the upstream wedge became less obvious over time. However, both the brine in the cavity and the upstream wedge reduced over time. The shape and angle of the upstream wedge did not appear to change much as it reduced. Within the cavity, the rate of reduction was slower, but the brine took on distinct shape. It sloped downwards from the downstream wall, much like the wedge, but sloped upwards again near at the upstream wall (Figure 23). 52

65 Figure 23 : Run 6b at 160min. Downstream wedge remains connected to obstruction for some time. Brine in cavity and upstream wedge is reduced. Eventually the downstream wedge reached a steady state position. In the case above, this position is some way past the obstruction, but this was not always the case. For Run 3b for instance, the steady state position of the wedge placed the interface almost intersecting the cavity in the obstruction (Figure 24). Figure 24: Run 3b at 190min. At steady state, the interface almost intersects the cavity. This qualitative description of the observed phenomenon has been presented to show the process of intrusion and recession and the effect of the obstruction on that process. These effects will be discussed in detail in the discussion section of this chapter. 53

66 Quantitative description of interface position over time The position of the interface over time was analysed using the digital images taken every 5 minutes during each run and converted into x-y co-ordinates using MATLAB, as described above. A total of approximately 322 images were taken during runs 0 to 6 and most of these were processed into x-y co-ordinates. Presenting so much data here would be excessive and unnecessary. Instead, only that data which shows relevant trends and patterns, and is used is used the discussion, is presented. Data on the interface position which was used during the calibration process discussed in the next chapter, is identical to the data presented here. However, whereas the position at only a handful of time-steps is presented here, all the time steps were considered during the calibration process. The intrusion of the interface over time was analysed in runs 0a and 1a. Results show that run 1a, which included the presence of the obstruction, intruded faster than 0a (Figure 25). Figure 25 : Interface position over time for Runs 0a and 1a. Run 0a is plotted in red, while 1a is blue. 1a intrudes faster than 0a (Note Run 0a at 75min overlaps Run 1a at 45min). Possible reasons for this different rate of intrusion, including experimental error and different hydraulic conductivities, are discussed later. Interface position during the recession was also converted to x-y data for all recession runs. However, only 110minutes of the recession in 0b was recorded and it does not 54

67 appear to have reached steady state at that time. The position of the interface at two (for 0a) or three (for 5b) times is shown in Figure 26. Figure 26 : Interface Position over time during Runs 5b and 0b. Note how the presence of the lump appears to affect the height of the interface just downstream of the obstruction at 11min Data relevant to showing effect of varying head gradients on flushing The effect of head gradient on the recession was examined by plotting the interface for Runs 2b, 3b, 5b and 6b at a single point in time. Recall that these runs have downstream head boundary conditions of 0.6m, 0.67m, 0.65m and 0.63m of saltwater head respectively. The interface is shown in Figure

68 Figure 27 : Interface position at 65min for runs 2b, 3b, 5b and 6b. These runs had downstream head boundary conditions of 0.6m, 0.67m, 0.65m and 0.63m of saltwater head respectively. The wedge downstream of the obstruction appears to be steeper when the head gradient across the whole system is steeper. 2.3 Discussion Properties of the tank The aim of the analysis hydraulic conductivity was to determine the hydraulic conductivity of the porous medium and confirm the consistency of the hydraulic conductivity between the two stages of the experiments. However, the results of the flow-rate tests and the intrusion interface position shown in Figure 25 offer conflicting information about the consistency of the hydraulic conductivity of the tank between the two stages. Recall that the glass beads were removed between these two stages to allow the obstruction to be installed. It had been hoped that the use of the glass beads, as opposed to sand or other less homogenous porous medium, would lead to strong consistency of packing (and therefore hydraulic properties). The flow-rate tests indicated a miniscule variance in K between the two cases (Figure 8). The line-of-best-fit connecting all but the last set of data-points shows very high correlation, indicating normal experimental error was small in both cases. This high correlation both within each test and between the two tests allows good confidence in the veracity of the results. Some analysis of possible sources of error is warranted though. 56

69 The Dupuit assumptions, which underlie the equation used to form the graph, break down with large unsaturated flow regions near the outflow, especially in the presence of a seepage face. This breakdown can be seen in the graph divergence from the trend of the last data-point. However, since those data-points were excluded from the analysis, and the rest of the data-points lie on a straight line, it is reasonable to assume that there is no significant break-down in the Dupuit assumption over the ranges included in the analysis. It is possible that the slotted panels at the boundaries of the tank had a significant effect on the observed flow-rates. Recall that fluid exiting the tank must flow through a limited number of narrow slots in these panels. It may be that this is an impediment to flow and the effect of reducing the observed flow rate in the experiments. However, this source of error would apply equally to both stages. Therefore, while it might be assumed the observed K represents an underestimation of the actual properties of the tank (assuming there were no other source of error), it may still be concluded that the K is similar between the two stages. The results of the intrusion runs shown in Figure 25 paint a slightly different picture. Despite the presence of the obstruction, the interface at the second stage appears to move faster than the interface in the first stage. Assuming no experimental error, this may be ascribable to differences in K. A lower value of K would naturally lead to slower intrusion of the wedge. There may be another interpretation of this data other than different hydraulic conductivities. The last interface position presented in Figure 25 appears to show that the wedges (Stage 1 and Stage 2) may be reaching steady state at slightly different positions. This difference in positions may be due to a difference in boundary conditions. Therefore it may be suspected that experimental error, specifically in the height of the boundary head, is the cause of the apparent discrepancy in K. Given that the Stage 2 wedge appears to be reaching the steady-state position at a point close to that predicted by the initial modelling (see 0), it may be assumed that the majority of the error lies in the boundary conditions of the Run 0a. 57

70 Wedge progression and the effect of the obstruction The observations of the wedge progression steady-state and recession recorded earlier in this chapter will be discussed in this and the next section of the discussion. Particular attention is paid to the effects of the obstruction on these phenomena. We begin with some general comments about the shape and movements of the interface. Two areas are worthy of comment. Firstly, the point at which the wedge intersects the boundary varies significantly with a change in the boundary conditions. Immediately after the change in Run 0b, the head increased before declining significantly towards a new steady-state position. In Run 1b, there was no initial rise but the drop was more rapid. The drop in the top of the wedge is of course explainable by the drop in the head at that point. However, the increased flow rate through the seepage face that coincides with a drop in the downstream boundary conditions would also be expected to affect the position of the wedge near the seepage face, through increased vertical head gradients, convergent flow and the presence of unsaturated effects. The second point relates to the movement of fluid in the vicinity of the interface. The movement of fluid along the interface is shown by the dye in Figure 14. Here it can be seen that there is horizontal flow within the body of the wedge during progression and horizontal flow in the freshwater, converging and accelerating towards the seepage face. At the interface, a streak links the dye in the freshwater with that in the wedge. This streak does not appear to become diluted over time, indicating that it is not simply the same particles of dye that were initially released at the interface diluted along the interface. Rather, it is dye that was initially released at and below the interface, that is being drawn towards the interface by dispersion and entrained into the external flowfield of the freshwater. This possibility supports the conclusions of Cooper (1964), that dispersion along the interface leads to entrainment of the brine and its recirculation to the boundary. The obstruction appears to have some significant effects on the wedge, but only as the toe passes over it, and there are no apparent effects on the position of the interface at steady state. The obstruction impedes the movement of the wedge only slightly. As the brine builds up behind the wall, there must be an increase in the total head at the upstream end of 58

71 the wedge next to the wall. This would reduce the head gradient driving the wedge forward, but the reduction is apparently slight enough not to affect the flow of brine in from the boundary significantly. The dispersion that is apparent as the brine flows over both walls is likely to be caused by the relatively high velocity of the brine as it flows down the face. Within the cavity, this dispersed brine, being less dense, is displaced out of the cavity and appears to flow back towards the boundary. The dispersed wedge that forms upstream of the obstruction is a similar phenomenon, but the dispersed brine is trapped by the diverging flow lines around the obstruction of the freshwater flow field. The presence of the obstruction has no effect on the shape or position of the steadystate wedge. Indeed, the interface shows no evidence of the first wall s presence once the cavity is filled and the brine is flowing over the second wall. Similarly, once the upstream wedge has become contiguous with the main portion of the wedge, no evidence of the second wall can be seen in the interface either Recession and the effect of the obstruction The presence of the obstruction had a far more significant and obvious effect on the recession of the wedge than on the progression. The effect was obvious almost immediately on the shape of the interface. The obstruction trapped brine both within the cavity and upstream of the second wall. Where the steady-sate position of the wedge was downstream of the cavity, the toe of the wedge took some time to detach from the base of the first wall. These effects contrast with the recession of the wedge in the first stage of the experiments. There, the wedge moved quickly at first, and then slowed towards the new steady state position. The movement is caused by the net head gradient within the wedge that is created when the downstream head is dropped. Previously, at steadystate head gradients within the wedge would be small; created by the erosion of the interface through dispersion and creating only enough flow from the boundary to compensate this erosion. Once the downstream head is changed, a net head gradient across the wedge is created and the brine flows out. As it does so, a point any particular distance from the boundary but within the wedge experiences less and less pressure as the interface moves downward above it. At some point the head at a point within the wedge becomes lower than the head at the boundary. At that point in time, wedge stops receding. This is the new steady state position. 59

72 The presence of the obstruction complicates this process. Immediately after the change in head gradient, the interface in the Stage 2 experiments became kinked above the change in head. This phenomenon is difficult to explain from the results presented thus far. The obstruction has some effect on the toe of the wedge downstream of it. The extended toe observed after the wedge had moved downstream linking the wedge to the first wall is very likely related to vertical head gradients in the vicinity of the wall. Again however, it is difficult to show this quantitatively. The brine within the cavity flows out and over the first wall even after the wedge has receded past it. This indicates some flushing process at work. The flushing is greatest in the first stages, immediately after the wedge has passed over, and reduces later. At some point the interface within the cavity takes on a particular shape. In slopes down from the first wall and turns up again slightly at the second. As it is slowly flushed out, it maintains this shape. The flushing at this point is purely through dispersion and therefore velocity dependant. The cavity forces flowlines to diverge and therefore slow as they enter it. Thus as the brine is flushed out, the rate at which it is flushed would be expected to reduce also. Without quantitative analysis of this in the data it is impossible to say if there is conclusive evidence of this. However, observation appeared to support this assertion. Upstream of the obstruction, the wedge that was left by the retreating brine is trapped by the second wall. This phenomenon might be explained by observing that at the wall, the brine (initially) reaches the top of the wall, but in the absence of a vertical head gradient, cannot flow over it. Further back, the interface of the wedge is lower, but (almost) stationary, as the pressure created by the head of brine at the wall is matched by the increasing freshwater head away from the wall. This argument is tantamount to saying that the wedge behind the second wall is acting much like the main wedge at steady-state. The difference is that while both are slowly eroded by dispersion along the interface, the main wedge is refreshed by inflow from the boundary, while this smaller wedge is not. It therefore slowly reduces in size. This argument is given further credence by effect of the different head gradients on the shape of the upstream wedge, shown in Figure 27. Just as for a greater head gradient the interface position for a normal wedge recedes, this figure shows that for greater head gradients, the angle of the downstream wedge also recedes. 60

73 Sources of error in the experimental procedure Before drawing conclusions from this experimental work it is necessary to speak at the very least in qualitative terms about possible sources of error in the work and flaws in its general set-up. This experimental work was conceived without reference to previous work, and most of the methods were devised from scratch. As the work was conducted, a number of sources of error were observed that might have been avoided had they been addressed from the outset. These included flaws in the equipment and set-up, flaws in the experimental design and experimental error. By describing these sources here, they can hopefully be avoided in the future by any who wish to carry on this work or work like it. There were three significant flaws in the experimental setup which may have affected the results. Firstly, the boundaries between the porous medium and the reservoirs were slotted Perspex. The non-uniform distribution of the slots has the potential to create deviations in the flow-field close to those boundaries. This could have and effect on the movement of water and solutes across those boundaries, and thus affect the results. However, the effects would be limited to the immediate vicinity of the wall. Secondly, the toe of the wedge was obscured by the reinforcing bars along the base of the tank. Toe position had to be estimated by extrapolating the shape of the wedge down to the base of the tank. This prevented quantification of the attenuation of the wedge toe during the second stage recession. Thirdly, the sealant between the walls of the obstruction and the sides of the tank was insufficient and leaked. The sealant was put on while the tank was empty. Once filled, the sides of the tank bowed outward slightly, allowing glass beads to move between the wall ends and the sides of the tank, and eventually to separation of the silicone sealant from the tank. This is shown graphically in Figure

74 Figure 28: Separation of the sealant from the tank. The dye moves over the near wall as it should, but is allowed to pass through the far wall. Although this may appear to be a catastrophic flaw in the experimental work, it must be remembered that its effects are isolated to one end of one wall in the tank. Only small amounts can leak across this wall at any one time. As well as these flaws in the physical setup of the experiments, there were also flaws in the experimental design. These flaws were avoidable in hindsight. Firstly, there is a serious lack of replicate runs for each set of boundary and initial conditions. Each run was only conducted once. This prevents any analysis of errorbounds in the rate of progression and recession of the interface. This is especially significant since the results of this experimental work are used to calibrate a numerical model (described in the next chapter). Additionally this lack of knowledge about the range of error over-shadows the discussion about the differing rates of wedge progression observed between the first and second stages see (section 2.3.1). It is indeterminable whether this can be attributable to real differences between the properties of the tank or flow phenomenon or is simply experimental error. The second flaw in the experimental design concerns the decision to keep the upstream boundary constant and vary the downstream boundary. In the real world it is more often the upstream boundary the head in the coastal aquifer that is varied, while the sea level stays constant. By holding the downstream boundary constant and varying the upstream boundary condition, a more realistic situation would be simulated. 62

75 Additionally, holding the downstream boundary constant would have allowed easier interpretation of the movement of the wedge top along this boundary. The movement up and down of the top of the wedge in response to flow-rates and pressures would be distinguishable from movement resulting from simple changes in the height of that boundary. Thirdly, the design of the obstruction did not truly separate the effects of the different types of basement heterogeneities. Recall that the obstruction was designed such that intruding heterogeneities could be simulated by the presence of the two walls, while cavity-like heterogeneities were simulated by the cavity between them. However, these two phenomena were not truly independent, as brine from the upstream wedge could be transported into the cavity, where its presence may have changed the concentration gradients across the interface and thus the dispersive transport. In addition, due to the flaw in the sealant between the upstream wall and the side of the tank, brine from the cavity may have replenished brine within the upstream wedge, thus slowing the removal of that wedge. However, the rate of movement across the flaw was too small to have been significant. Finally, there are sources of normal experimental error. These generally relate to the boundary conditions. Firstly, the height of these may be set incorrectly. It was found during the preliminary modelling that small changes in the boundary conditions had large effects on the steady-state positions of the wedge. Thus error in the height of the boundary may have led to inaccurate runs. Secondly, the flow of freshwater in to the downstream reservoir diluted the brine there, reducing the pressure. This was compensated by having high rates of brine replenishment from the feeder tank. However, this does not rule out the possibility that the rate of replenishment was insufficient. Finally, there may have been slight differences in the concentration of each batch of brine. Large batches were made to try to avoid this, but the possibility remains. To avoid all these flaws and errors in the future, the following recommendations are made. The slotted screens should be replaced with stainless-steel mesh screens, fitted with a geotextile of known properties More effective and flexible sealants should be used 63

76 A panel should be fitted to the base of the tank, effectively raising the base above the reinforcing bar. At least three replicates of each run should be conducted, between which boundary and initial conditions are re-set, and new batches of brine are used. The downstream boundary should be kept constant, while the upstream boundary is varied The obstruction should be re-designed to avoid interference between phenomena. Ideally, only one phenomenon should be observed for each obstruction. The increased number of replicates should help to quantify experimental error, which could be minimised through care and improved methods of maintaining the boundary conditions. 2.4 Conclusion In the introduction to this chapter, the aims of the experimental work were laid out. They were: To examine the effect basement heterogeneities on a salt water wedge To identify phenomenon for further work To obtain observed data with which to calibrate a numerical model to further investigate this phenomenon. The effect of basement heterogeneity has been examined by creating a laboratory scale model of a hypothetical aquifer in cross section with idealised properties. Wedge progression has been examined both with and without basement heterogeneity. From this examination, the following tentative conclusions have been drawn: The presence of small scale heterogeneities in basement morphology may have little effect on the intrusion of saline wedge, except in the vicinity of the toe of the wedge as it passes over the heterogeneity The presence of small scale heterogeneities within the wedge may have little or no effect on the position of the wedge or shape of the interface at steady state. However, heterogeneities upstream of the wedge may have an effect on toe 64 position at steady state.

77 The presence of small scale heterogeneities may have a significant effect on the shape and position of the wedge during recession. These effects may include changing the shape of the interface above an intruding structure during the early stages of recession; attenuating the toe of the wedge once it has moved downstream of an intruding structure ; and creating pockets of brine both within a cavity and upstream of an intruding structure that are slowly flushed out through dispersion. These conclusions point to the recession of the wedge as an area of possible further study. Examining this area may be questionable given that saltwater wedges are generally a concern when they encroach into an aquifer. This encroachment is usually a result of over-extraction from the coastal aquifer, which is difficult to slow or stop, let alone reverse. Despite this, it was decided that the recession of the wedge would be the object of further study in this report. For this purpose the results of this experimental work were used to calibrate a numerical model of the hypothetical aquifer. This calibration process will be examined in the next chapter. 65

78 Chapter 3. Model Calibration The numerical modelling of the experimental data was undertaken with three aims To determine if the system can be modelled by the numerical model To better understand the processes occurring in the system To create a tool with which to further examine specific processes The numerical model chosen, SUTRA, was descried in detail in Chapter 1. A conceptual model of the sand-box model was used to create a model domain with boundary and initial conditions. The model parameters were initially based on experimental data or best guesses. These were refined during the calibration process that is discussed later in this section. There were in fact two stages to the modelling process. Initially, preliminary modelling with a coarse mesh was conducted in order to determine some parameters used in the experimental work. The aims and outcomes of this preliminary modelling are described in 0. In this section, the selection of discretisation, boundary conditions and parameters in this preliminary modelling will be mentioned concurrently with the discussion concerning the final detailed modelling. The STURA numerical model described in the literature review earlier was used to create a model of the tank. The model input files were created using the Graphical User interface ARGUS-ONE and the SUTRA-PIE add-in. This platform made the discretisation, mesh optimisation, and input file creation very easy. It was also simple to modify boundary conditions and create new scenarios. 3.1 Conceptual model The conceptual model in this case is quite simple (Figure 29). It consists of a rectangular domain of homogenous, isotropic porous media in which Darcy s Law holds. This porous media is 0.8m high and 1.68m wide. The width of the porous media is constant and unity. Its properties (hydraulic conductivity, dispersivity, porosity) are yet to be determined, and will be discussed later in the parameterisation section. The porous media is unconfined and unsaturated flow may occur. 66

79 Figure 29 : Boundary conditions of the model For part of the modelling the base of the domain is flat and homogenous. Later, an obstruction consisting of two walls 112mm high, 112mm apart and 16mm thick are positioned 704mm and 832mm from the left-hand boundary. These walls were intended be in the same position as the ones in the experimental setup in the previous section. All flow is essentially two-dimensional. There is no flow other than in the plane of the model. The base of the domain is impervious and no flow may occur across it. The top boundary is unconfined, but also designated as no-flow. The right hand boundary has a constant freshwater head of 0.8m from the base of the domain. All flow in from this boundary has a solute concentration of zero and a density of 1000kg/m 3. The left hand boundary has a variable saltwater hydrostatic head of up to 0.7m from the base of the domain. The solute concentration of the fluid entering from this boundary is 0.214kg/kg, giving it a density of 1150kg/m 3. Initially, the domain does not contain any solute. The following sections will describe the way a numerical model was created around this conceptual model. Discretisation of the model domain will be described, as will boundary and initial conditions and the calibration and selection of parameters. 67

80 3.2 Discretisation of the Model Domain The model domain was discretised into regular bilinear quadrilaterals. To reduce computation time, the elements were all orthogonal and there were no pinch nodes defined. Pinch nodes increase the bandwidth of the solution matrix. The time steps used were 30 seconds for the preliminary modelling and 3 seconds for the fine-scale modelling. The results were outputted every 5 minutes Preliminary modelling The preliminary modelling was conducted over a coarse mesh. The domain was a rectangle 0.8m high and 1.68m wide. This was divided into 63 elements (64 nodes) horizontally and 30 elements (31 nodes) vertically. The nodes were equally spaced m apart in both directions. A number of obstruction sizes and positions were trialled on the coarse grid. These were created simply by removing columnar sections of elements at the base of domain, and defining no-flow boundaries around where those elements were Fine-scale modelling The main set of modelling was conducted over a domain of identical size to the preliminary modelling: a rectangle 0.8m high and 1.68m wide. This was divided into 105 elements (106 nodes) horizontally and 50 elements (51 nodes) vertically. The nodes were equally spaced 0.016m apart in both directions. This basic setup is shown in Figure

81 Figure 30 : Basic discretisation for fine-scale modeling To model the second stage of the physical experiments, the original mesh was altered to account for the presence of a small obstruction at the base of the model domain. This obstruction was created by simply removing the bottom seven elements of column 45 and 53 in the mesh. This did not reduce the number of nodes in the model domain, but simply created four new vertical no-flow boundaries seven elements high on the sides of where those elements used to be and two new vertical no-flow boundaries one element wide each at the top. The cavity space created by these walls contained 49 elements and 64 nodes. The discretised domain is shown in Figure

82 Figure 31 : Discretisation including obstruction at the base of the domain 3.3 Boundary conditions The top and bottom boundaries of the model were designated no-flow boundaries. This meant that the component of the velocity vector, the gradient of the concentration field and the gradient of the pressure field perpendicular to them were also zero. These boundary conditions applied to the top and bottom boundaries in both the preliminary and fine-scale modelling. They also applied to the walls of the obstruction. The upstream boundary was defined as hydrostatic freshwater head of 0.8m. In SUTRA, boundary conditions must be defined in term of pressure. The pressure for each node was calculated as density times gravity times the depth below 0.8m defined from the base of the domain. This placed the water table at the upstream boundary at the top of the domain. The concentration of the boundary was set to 0. The downstream boundary was similarly hydrostatic with a constant head of brine (referred to as h 2 ), which varied for each model run. This head was converted to pressure by multiplying the density of the brine (1150kg/m 3 ) by gravity and the depth below h 2, again defined from the base of the domain. A value of h 2 =0.7 was used to obtain a wedge at steady state that extended across the full width of the model domain (how this value was obtained is described in Error! Reference source not found.0). Because of its importance as the steady-state boundary 70

83 condition, this is referred to as the base boundary condition. The value of h 2 was also set to 0.67m, 0.65m 0.63m, 0.60m or 0.57m to replicate the boundary condition of the physical tank. 3.4 Initial Conditions The initial conditions of the model run were either a zero concentration field or the wedge at steady state. The zero concentration field is the default initial condition of SUTRA. This was also the initial condition of the tank in the physical modelling. Steady state was obtained by running the model with the base boundary conditions for 7200 time steps (6 hours) in the fine-scale modelling and 3000 time steps (25 hours) in the preliminary modelling. This simulated the progression of the wedge into the tank from initially fresh conditions. The final concentration and pressure conditions from these runs were saved and converted into initial conditions. To model the wedge recession these new initial conditions were used with an altered downstream boundary condition. 3.5 Parameterisation The two main parameters for calibration in the model were the permeability and dispersivity. A value of hydraulic conductivity was obtained from the tank in the physical modelling, but this value was inferred from flow-rate tests, and may not have been an accurate representation of the hydraulic conductivity of the tank. Dispersivity had not been directly measured in the tank, and would have to be calibrated Permeability Permeability was initially set using the value of hydraulic conductivity obtained from the flow-rate test performed on the experimental tank (see sections and 2.2.1). This value was 2.6x10-3 m/s. This had to be converted into intrinsic permeability (k) [m 2 ] for input into SUTRA. Using the following relationship. k = Kµ ρg ( 3-1 ) 71

84 where K is the hydraulic conductivity [m/s], µ is the viscosity [kg/m/s], g is gravity [m/s 2 ] and ρ is density [kg/m 3 ]. Values for these constant were the same as those inputted into the model: µ = 10-6 kg/ms g = 9.81m/s 2 ρ = 1000kg/m 3 This yielded an intrinsic permeability of 2.65x10-10 m 2. The model was run with this value of permeability and the interface position over time was compared to the observed interface position. Both the interface progression and recession were examined. Observations from both the first and second stage of the experimental work were compared separately. The model interface position over time was found to move faster than the observed interface. This was observed both during progression and recession and in both stages of the experimental work. Lower values of permeabilities were tried. For the first stage homogenous case, permeabilities of 50%, 60%, 70% 80% and 90% of the original value were tried. It was found that a permeability of 60% of the original value produced the closest match to the observed data for the first stage. For the second stage heterogenous case, permeabilities of 65%, 75% and 85% of the original value were tried. It was found that a permeability of 85% of the original value obtained the best match with the observed data Dispersivity No value of dispersivity was directly observed from the tank. Initial values for dispersivity were obtained from literature. Voss (1984), states that for a homogenous medium under laboratory conditions, longitudinal dispersivity is in the order of grainsize. This implies that the dispersivity in the experimental tank would be less than one millimetre. For numerical stability, the mesh Peclet number should be less than two and must be less than four. To create a model that was stable and able to model dispersivity so small, more than 1.3 million elements would be required over the model domain. This is prohibitive. 72

85 Taking into account the size of the mesh discussed previously, longitudinal dispersivity was set at 7.5x10-3 m. This gives a mesh Peclet number of Transverse dispersivity is typically set an order of magnitude lower than longitudinal (Voss 1984). This convention was maintained throughout this work. The results of the point-release test given in Section appear to support this approximation. 3.6 Discussion The calibrated model output and observed interface positions can be found in Appendix B. The model calibration process was able, overall, to obtain a good match to the observed data. However, there are a number of issues that warrant discussion. The hydraulic conductivity for both the homogenous and heterogenous stage had to be altered significantly to match the observed data. The original value was obtained from flow-rate tests conducted on the experimental tank. The hydraulic conductivity for both stages was found to be very similar in this test, despite the porous media being completely removed and repacked between the stages. However, to match the observed data, the hydraulic conductivity had to be reduced significantly, and by different amounts, from this value. That it had to be reduced at all may suggest that the original experimental observations were methodologically incorrect. There were sufficient replicates to rule out experimental error. However, the tests provide an averaged hydraulic conductivity, which may not be that experienced by the brine in the wedge. On the other hand, results from point release appeared to indicate no such heterogeneity. Another explanation may be offered by the work of Hassanizadeh and Leijnse (1995). They found during experimental work that intrinsic permeability appeared to be a linear function of mass fraction. In fact they found that permeability reduced by 15% in solutions with a mass fraction 0f 0.24kg/kg. In The work described here, the mass fraction of the brine is approximately 0.21kg/kg, and a reduction in permeability of about 15% was also found. The explanation offered by Hassanizadeh and Leijnse (1995) is that the resistivity of the porous medium is a function of the fluid properties, as well as the matrix. They argue that the assumption that permeability is independent of fluid properties is, in fact, 73

86 incorrect. The contrasts with the commonly accepted formulation of permeability. Hydraulic conductivity, as the empirical constant in Darcy s Law, is usually said to be proportional to gravity and inversely proportional to the kinematic viscosity of the fluid. The constant of proportionality which is the permeability is assumed to be solely a function of the porous medium and is related to grain-size distribution, grainshape, tortuosity, specific surface, and porosity (Bear and Verruijt 1987). This traditional formulation cannot explain the apparent reduction in permeability with increasing concentration. More work is certainly required before any conclusion can be reached on this subject. Even if this helps to explain the need to reduce the permeability, it does not explain the different permeabilities between the two stages. Experimental error is the most likely cause of this discrepancy. The lack of replicates of the observed data reduces their credibility. Despite great care, boundary conditions may not have been set accurately, influencing the rate at which the brine has moved in and out of the cavity. The adjustment of the hydraulic conductivity improved the agreement between the observed and modelled data, but there remain at least two particular areas where the model was not able to replicate the observed results. The first relates to the progression of the wedge into the aquifer. The point at which the wedge intersects the downstream boundary was observed to be lower in the model. This was observed in both the homogenous and heterogenous cases. An example of this is shown in Figure 32. Figure 32 : Observed and modelled wedge position during progression into the aquifer. The model did not accurately reproduce the point of intersection of the wedge interface and downstream boundary. 74

87 The most likely cause of this discrepancy is the effect of the unsaturated flow near the outlet zone. No attempt was made to calibrate the unsaturated properties in the model to the glass bead porous medium used in the experimental work. Doing so would require that measurements be made to determine those properties and that the code of the model be altered and re-compiled. Given more time to do so, this would hopefully improve the agreement between the model and the observed interface. The second area of disagreement between the model and the observed data occurs during the recession of the wedge over the heterogenous basement. An example of this is shown in Figure 33. Figure 33 : Observed and modeled wedge position during recession out of the aquifer past the obstruction. The model was unable to match the rate of flushing around the obstruction. The model predicted much faster rates of flushing than were observed from around the obstruction. In some cases, the model predicted that the upstream wedge would be completely flushed when it was in fact still quite large. This discrepancy is the result of the inability to model very low dispersivity with the mesh size used in the model. As stated earlier, the best guess available for dispersivity in a homogenous laboratory-scale porous medium is in the order of grain size. In this case that is around 7.5x10-4, an order of magnitude less than the dispersivity used in the model. To examine this further, the results of the numerical model and the observed data were analysed in more detail. The area of the observed upstream wedge was estimated for each time step and used to estimate the mass of salt remaining there. The model was run two more times with greater values of dispersivity (0.0375m and 0.075m). In all of these model runs, transversal dispersivity was an order of magnitude less than 75

88 longitudinal dispersivity. The output of concentration from the model was used to determine the mass of salt remaining in the upstream wedge for each time step in these two new models, as well as the original. The results of this are shown in Figure 34. Figure 34 : Normalised mass remaining in upstream wedge over time for three values of dispersivity and the observed data. There is a clear relationship here between dispersivity and flushing time. Higher values of dispersivity led to very rapid reductions in flushing time. The effect was most pronounced between the m and m values in the model. The difference between the observed flushing rates and the modelled was quite large and indicated a significant decrease in dispersivity was required to adequately model the data. But how much of a decrease? It is difficult to extrapolate the results of the above sensitivity analysis to determine an appropriate value. A value in the order of grainsize may be appropriate, but it too may be too large. The work of Hassanizadeh and Leijnse (1995) and Schotting et al. (1999) has indicated that for very high concentration gradients, Fickian transport breaks down and that values of the dispersion co-efficients may vary. Their experimental work dealt with brine of about half the concentration of the brine in this work, but if their results are extrapolated, it indicates the longitudinal 76

89 dispersion co-efficient may need to be one and a half orders of magnitude less than it would be with freshwater Schotting et al. (1999). It is therefore impossible to say what the exact value of dispersivity should be. More work is required to examine the effect of density-dependant dispersivity and whether those effects are significant in the area of saltwater wedges. Finally, an error in the modelling was noted late in the work. In the numerical model domain the walls of the obstruction were placed 704mm and 832mm from the downstream boundary. In fact they should have been placed 845mm and 955mm from the boundary. This error had been overlooked during the calibration process as a scaling factor had been introduced to correct an unrelated error in the data collection program. The error was corrected but the scaling factor was not removed. The error is unfortunate and warrants correction. However it was discovered too late in the process to be acted upon. Its effects are insignificant to the remainder of this thesis. Not only is the scale of the error small, but the rest of the work presented here does not rely on the model being accurately calibrated. 77

90 Chapter 4. Sensitivity Analysis 4.1 Introduction A sensitivity analysis was performed to determine the effect of varying important parameters on the flushing of a cavity. The configuration of the cavity in this analysis is somewhat different than the configuration of the obstruction in the previous two sections. This is a deliberate attempt to overcome some of the methodological difficulties of the previous section, such as the transfer of solute from the upstream wedge into the cavity. The aim of the sensitivity analysis is to examine in greater detail the processes of flushing in the cavity as the salt-water wedge recedes past it, and the effects of some of the governing parameters on these processes. This is accomplished by running a number of simulations with variations in cavity geometry and head gradient across the system. A base configuration of geometry and head gradient was run as a link between these two types of variation. Two models were run that varied from the base case in the aspect ratio (depth/length) of the cavity, but with identical cavity volume and head gradient across the system. Two others were run with identical cavities, but with varying head gradients. The results are shown as the mass of solute remaining in the cavity as a function of time, the rate of change of solute mass in the cavity and the derivative of this rate of change. Solute concentration and advective flux fields around the cavities are also given to show the flushing modes that have been identified. These flushing modes are examined in detail in the discussion section, where a distinction will be drawn between advective and dispersive flushing modes. This discussion will also examine some of the notable features of the results, and attempt to place these in the context of the flushing processes presented. This is followed by derivation of an equation to analytically estimate the point (in mass fraction, not temporal, terms) at which the transition from one flushing mode to another occurs and the relative importance of the flushing modes on a particular configuration of head gradient and aspect ratio. This technique is based on the Baydon- Ghyben-Hertzberg approximation of salt-water interface position. 78

91 4.2 Setup of Sensitivity Analysis Configurations of model space for sensitivity analysis The sensitivity analysis was run with a base configuration consisting of the same domain as the homogenous case of the above analysis, but with a square cavity in the base. The domain consisted of 106 nodes horizontally, spaced 0.016m apart to give a total width of 1.68m. The main part of the domain was 51 nodes high, also spaced 0.016m apart to give a height of 0.8m. The cavity was 11 nodes by 11 nodes. This gives it an area of 100 elements exactly, or 25.6cm 2. This configuration is shown in Figure 35. Figure 35 : Spatial configuration and discretisation used as a base for the sensitivity analysis All runs were initialised to set up a wedge of salt-water at steady state extending from the downstream boundary almost to the upstream boundary. The cavity was filled by this wedge with salt water. This steady-state condition was accomplished with an upstream head of 0.8m and a downstream head of 0.7m. This wedge was used as the initial condition for all the runs (Figure 36). 79

92 Figure 36 : Initial condition used for base case in sensitivity analysis. All sensitivity runs used the same initial condition, although the cavity was different to that shown here for two of the runs (see text). The upstream boundary is on the left and the downstream is on the right. The base case model run consisted of lowering the downstream head to push the wedge out and past the cavity, thus exposing the cavity to fresh-water. This caused flushing of the cavity to occur. The downstream head in the base case was lowered to 0.6m, which pushed the wedge to a steady-state position about 40-50cm from the downstream end. Two types of variation of the model were performed for the sensitivity analysis: variations in aspect ratio and variations in the downstream head. The variations in aspect ratio (referred to as r a = depth/length) maintained the volume of the cavity constant but varied the ratio of the depth of the cavity to its length. In one variation, the depth was set at 5 elements, 6 nodes and the width at 20 elements, 21 nodes, giving a wide cavity. This cavity has an identical volume to the base case (100 equal elements) but an aspect ratio of r a =¼. In the other the cavity was set at 5 elements, 6 nodes wide and 20 elements, 21 nodes deep, (r a = 4), giving a deep cavity. These configurations are shown in Figure 37 and Figure

93 Figure 37 : Cavity configuration for the 'wide' case in the sensitivity analysis (ra= ¼) Figure 38 : Cavity configuration for the 'deep' case in the sensitivity analysis (ra= 4) The second set of sensitivity analyses consisted of a geometrical configuration identical to the one for the base case but with variations in the downstream head (referred to as 81

94 h 2 ). The same initial conditions were used, but rather than changing the downstream head to 0.6m, it was set at 0.63m and 0.57m. This changed the final steady state position of the wedge, but the toe was still downstream of the cavity in both cases. The permeability in these cases was kept constant at 2.65x10-10 m 2. Similarly, dispersivity was set constant at the smallest value possible for this discretisation: 7.5x10-3 m. Time steps were set at 3sec, with output produced every 5 minutes. All other parameters were identical to those used in the previous chapter. 4.3 Results Results were analysed by reading the output files into MATLAB. The results were examined two ways. Plots were made of the concentration field and advective solute flux field. The concentration field is a pseudo-colour plot of the concentration at each node for a particular output time-step. The concentration field is an output of the model. The advective solute flux field is obtained by multiplying the concentration field with the fluid velocity field, which is also an output of the model. However, because the velocity is calculated for each element, and the concentration for each node, the concentration was interpolated for each element from the four nodes surrounding it, weighted equally. Multiplying this by the velocity at each element gives an approximation of the rate at which solute is moving through that element. It should be noted that this ignores the dispersive part of solute flux. Dispersive flux is calculated in the model as a function of the concentration gradient and the velocity field, but is not outputted. The advective flux was plotted as a quiver graph, with arrows representing the direction and magnitude of the flux vector at each point. This was plotted over the concentration flux graph to give a combined plot of the concentration field and advective flux field for each time step. An example of this type of plotting is shown below. 82

95 Figure 39 : Example of the combined plotting of concentration and advective solute flux. Concentration is shown by colour, where red is undiluted brine and blue is freshwater. At the resolution presented, it may be difficult to discern the individual arrows showing salt flux. (The example is flushing of the wide case of the sensitivity analysis at 25 minutes [5 output timesteps]) The results were also examined by calculating the mass of salt in the cavity as it is flushed. This information was calculated from the concentration field within the cavity. The concentration of salt is given by the model at each node in dimensionless form as mass of salt per mass of solution. To convert this to mass per volume it must be multiplied by density. The density is a function of concentration, given by: ρ = C ( 4-1 ) This an empirical relation, used in the model (see section 1.1.1). Once the mass-pervolume concentration was obtained it had to be interpolated for each element from the four surrounding nodes. This was done giving equal weighting for each node in all elements. It may have been more accurate to give less weighting to the boundary nodes, but this was not considered at the time, and is unlikely to be a significant source of error in the results. 83

96 This concentration was multiplied by the area of each element (2.54x10-4 m 2 ) to give the mass of salt per unit length and the result summed over all the elements in the cavity 5. This gives the total mass of salt in the cavity at each time step Mass of Salt with Time The following two plots show the mass of salt in the cavities over time for each of the sensitivity analyses. The first (Figure 40) shows the mass over time for each aspect ratio. The second (Figure 41) shows the mass over time for each head gradient. Note that the central blue line in each plot is the same model run (r a = 1, Downstream head = 0.6). This is the base configuration mentioned earlier. 1 Normalised Mass in Cavity vs Time Deep M/Mi 0.5 Base Wide Time [min] Figure 40 : MASS and ASPECT RATIO Normalised mass of salt remaining in cavity for each of the three aspect ratios examined in the sensitivity analysis. Mass is normalised to the initial mass in the cavity. 5 The per unit length is a consequence of the 2-dimensional nature of the model. Length is perpendicular to the field of the model. 84

97 1 Normalised Mass in Cavity vs Time M/Mi 0.5 More Gradient Less Gradient Base Time [min] Figure 41 : MASS and HEAD GRADIENT Normalised mass of salt remaining in cavity for each of the three head gradients examined in the sensitivity analysis. Mass is normalised to the initial mass in the cavity. Figure 40 shows significant differences in the mass of salt over time for each of the aspect ratios. The wide cavity, with an aspect ratio of ¼, flushes very quickly and is 90% flushed after just 75 minutes (15 x 5 minute output timesteps). At the other end of the scale, the deep cavity, with an aspect ratio of 4, is barely 20% flushed after 6 hours. The square cavity of the base case, with aspect ratio 1, sits in the middle of these extremes and is about 60% flushed after 6 hours of simulation. In all cases the most flushing appears to occur in the early part of the simulation, after a small lag period. This lag period is attributable to the time during which the salt wedge is moving back over the cavity. Figure 41 shows less extreme variation than Figure 40 but there is still a clear trend. The case of h2=0.57m, which represents a greater head gradient across the cavity, flushes faster than the others. On the other hand, the downstream head of 0.63, which represents a lower head gradient, flushes slower. Again, most of this flushing appears 85

98 to occur in the early stages. In this case, the later rate of flushing is similar in all three cases Flushing Rate with Time An approximation of the rate of flushing can be obtained by differentiating the above graphs. This gives the rate of change of mass in the cavity with respect to time. Given that once the salt wedge moves past the cavity there is no salt entering the cavity from upstream, the rate of change of mass represents the rate of flux of salt leaving the cavity. The following two graphs show this derivative with respect to time. Mass Flux from Cavity vs Time -1 Rate of flushing [5min ] 0.1 Wide Base Deep Time [min] Figure 42 : Time rate of removal of salt from the cavities for each aspect ratio in the sensitivity analysis. 86

99 Mass Flux from Cavity vs Time -1 Rate of flushing [5min ] 0.1 More gradient Base Less gradient Time [min] Figure 43 : Time rate of removal of salt from the cavities for each head gradient in the sensitivity analysis The first plot (Figure 42) shows the rate of removal of salt from the cavity as a function of time for each of the aspect ratios. This plot is consistent with the impression given by Figure 40: that the majority of flushing occurs in the initial stages, after some lag. The rate of flushing appears to increase quickly in every case to a maximum between time step 5 and 6. The maximum rate of flushing occurs at exactly the same time for each aspect ratio. After this initial peak, the rate of flushing drops quickly with an exponential type decay. There is a strong trend in the peak rate of flushing for the various aspect ratios. The peak rate of flushing for the wide case (r a = ¼) is the greatest, while the peak rate for the deep case is quite small. When compared to the later rates of flushing, this difference between the wide and deep cases is even greater. The flux out of the wide case drops very sharply after the initial peak, and actually reduces below the flux rate for the base and deep cases. There is less flux out of the wide cavity than there is out of the square cavity after about 75 minutes, and less than out of the deep cavity after 165 minutes. Figure 43 shows the flux out of the square cavity for each of the three various head gradients. This plot shows a clear difference to Figure 42, in that the peak rate of flux 87

100 does not occur at the same time in each case. The peak rate for the case of h 2 =0.57m occurs earlier than for h 2 =0.60m. The case of h 2 =0.63m experiences a peak flushing the latest. This timing difference is attributable to the rate at which the wedge moves out past the cavity. Under a smaller head gradient the salt wedge will move out slower and thus expose the cavity to fresh water later. The magnitude of the peak rate of flushing also shows a clear trend, increasing with increasing head gradient. However, the variation around the base case for the higher and lower head gradient s flushing rates is not as extreme as for the variations of aspect ratio Flushing Modes Concentration and advective flux plots for the cavities were obtained for a number of time steps and are shown in Appendix D. In each case, a plot is shown of the early flushing, either as or just after the wedge passes over the cavity. Another plot is also presented for each case some time after the peak flushing time. Some of these are presented here, but all may be found in the Error! Reference source not found.appendix D. A number of common patterns can be discerned in all of these plots. At the early stages, the cavities contain undiluted brine. As the wedge moves over the cavities, flowlines through the cavity appear to be continuous with those upstream and downstream. That is, there does not appear to be any separate convective cells within the cavities. Fluid at the base of the wide cavity is moving, while in the square and deep cavities it appears to be almost stagnant. An example of this can be seen in Figure

101 Figure 44 : Example of flow through a cavity at early time. Note the apparent lack of convective cells. (Wide case at 25 minutes, close-up of the cavity) In contrast, at a later time, flowlines in the cavity change significantly. The interface between parts of the cavity that are fresh and parts that are saline sharpens as the brine either forms a wedge at the downstream end (in the case of the wide cavity), or slopes down distinctly from the downstream end to the upstream end (in all other cases). This interface becomes the boundary of a convective cell within the bottom of the cavities. An example of this is shown in Figure 45. Figure 45 : Example of flow through a cavity after peak flux out. Note the relative sharpness of the interface and the presence of a convective cell within the remaining brine. The dotted line is related to later discussion. (Wide case at 45 minutes, close-up of the cavity) 89

102 The convective cell is strongest in the wide cavity and weakest in the deep cavity. It also appears stronger in the case of h 2 =0.57m (greater hydraulic gradient) than in h 2 =0.63m (lesser hydraulic gradient). Flowlines along the interface in the convective cell and in the external flow are parallel to the interface and in the same direction. As time progresses the brine is removed from the cavity and the interface moves downward. However, its slope appears to remain roughly constant. In all cases except the wide cavity, the interface reaches from one vertical boundary to the other. It is roughly linear in the downstream half, but flattens out towards the upstream end. 4.4 Discussion Flushing in the cavities presented shows similar patterns and trends for all the variations examined. From the results presented there appears to be two stages to the flushing process. One occurs in early time and is characterised by rapid rates of flushing and penetration of flow through the full depth of the cavity (or near stagnation at the bottom). The other is characterised by low flushing rates and a sloping interface between freshwater and brine that forms a boundary between the outer flow field and a convection cell within the brine. For reasons that will become clear, these two modes of flushing will be referred to as advective and dispersive flushing. The following discussion will describe these two types of flushing in greater detail, including hypotheses about their causes and the effects of the variation in aspect ratio and head gradient upon them Advection The initial stages of flushing are described as advective because it is proposed that they are a result of brine flowing directly out of the cavity as a consequence of the head gradient across the cavity. The saltwater wedge that has been modelled in every simulation of this work moves in and out of the aquifer as a direct response to changes in the head gradients within the aquifer. For each combination of head boundary conditions, there is a different steady state position of the wedge at which the pressure forces at the downstream boundary are balanced by the pressure within the aquifer. This is the classical description of saltwater wedges, and can be traced back to the work of Badon-Ghyben (1888) and Hertzberg (1901). As previously discussed, the Baydon-Ghyben-Hertzberg is a simple 90

103 formula for the steady state depth of the wedge below the downstream watertable ξ(x), given the height of the freshwater head above it h f (x) ρ f ξ ( x) = h f ( x) ρ ρ s f ( 4-2 ) where ρ f and ρ s are the fresh and saltwater densities respectively. This formulation makes a number of assumptions including the lack of vertical head gradients, zero dispersion and stagnant saltwater within the wedge (that is, zero horizontal head gradients within the wedge). Using the same argument used by Baydon-Ghyben and Hertzberg, and using the same assumptions, it can be argued that as the wedge recedes past the cavity, the brine within the cavity must reach a position where the density-driven forces match the external head gradient. Prior to reaching this position, there is a horizontal head gradient within the brine in the cavity. This gradient causes advection and consequently brine advects out the downstream end and freshwater flows in the upstream end, as shown in Figure 44 above. The early part of the flushing is therefore advective and is a consequence of horizontal pressure gradients within the brine. This part of the flushing ends when the sloping interface balances these pressure gradients. This argument, as stated above is based on several assumptions, two of which are zero dispersion and zero vertical head gradients. The effect of dispersion will be discussed in the following section. The assumption of zero vertical head gradients is important as it allows the assumption that the head at the water table is the same as the head experienced by the interface. Where the head at the interface is lower than the head at the water table, the interface may occupy a position higher than it otherwise would. Figure 46 shows the effect of vertical head gradients on the interface relative to the Baydon Ghyben Hertzberg theory. At the upstream end, flowlines entering the cavity must diverge, creating lower pressure at the interface than hydrostatic (assuming hydrostatic conditions upstream a reasonable assumption given the boundary is hydrostatic). Note how the interface at the upstream end does not follow the trend of the rest of the interface. 91

104 Figure 46 : Concentration and advective solute flux field for the base case at 35 min. The white line represents the position of the interface predicted by the Baydon Ghyben Hertzberg approximation (with linear head gradient across the cavity). Note the modeled interface position is higher than the approximation at the downstream end and lower at the upstream end. At the downstream end however, these flowlines must converge again, creating head gradients in the opposite direction and higher than hydrostatic pressure at the interface. Note how the interface at the downstream end is lower than that predicted by Baydon Ghyben Hertzberg. Advective flux can be examined in the various scenarios modelled with different head gradients across the system and different cavity aspect ratios. The case with the greater head gradient appears to have a relatively steeper interface at the end of the advective flux phase and therefore an increased the proportion of the volume removed by advective flux (Figure 47). However, the time taken to remove that brine was less than for the case with the lower head gradient. Although the volume to be removed is increased, the rate at which it is removed is increased also. The case of lower hydraulic gradient required almost twice as long to reach the end of the advective flushing mode than the greater hydraulic gradient. 92

105 Figure 47 : Cavity at the end of the advective flushing period at 30min for the run with more hydraulic gradient (left) and at 50min for the case with less hydraulic gradient (right). Not the difference in the slope of the interface. The cavity geometry of the runs with different aspect ratio affects the flushing in significant ways. The slope of the interface was affected only a little by the different cavity aspect ratios, perhaps as a result of greater or lesser vertical head gradients within the cavity. However, this meant that because eof the cavity geometry, a much greater proportion of the cavity was flushed during the advective stage for the base case cavity than for the deep case, while the wide case was flushed more still Dispersion After the interface reaches the relatively stable position at the end of the advective stage, and pressure forces have (mostly) been balanced within the brine, the cavity contains two distinct flow fields. One, the external field, has a limiting flow-line that travels along the base of the model domain, into the cavity, along the interface and out again. The other is a convection cell within the brine in the cavity. The limiting flowline of this cell runs along the inside of the cavity and along the interface parallel to the limiting flow-line of the outer flow field. The only way solute in the brine can exit the convective cell is through transversal dispersion between these two flow-fields. For this reason, the flushing at this time is known as dispersive. That is not to say that dispersion is not an important process during the advective flushing phase. In fact, dispersive flux may be quite large during the advective phase. While the brine is being advected it is subject to both longitudinal and transversal dispersion. During the dispersive flushing phase, only transversal flushing occurs, which is usually much smaller than longitudinal dispersion. The coefficient of transversal dispersivity used in this modelling is an order of magnitude smaller than the coefficient of longitudinal dispersivity. 93

106 The dispersion along the interface and transport of dense water out of the cavity creates a small net pressure gradient across the brine. It is this gradient that generates the circulation within the brine. This process is identical to the circulation of salt within the greater saline wedge at steady state. However, where the greater wedge is connected to the source of saline water and can be thus replenished, the brine in the cavity is not. The rate at which the solute is removed from the cavity during dispersive flushing presumably depends on a number of factors including the dispersivity, the length of the interface and the velocity of the fluid moving along the interface. Greater dispersivity and fluid velocity implies more net dispersive flux per unit area of interface. A longer interface implies greater total mass removed. How then can the changes in flux shown in Figure 42 and Figure 43 be explained? Figure 43 shows the flux out of the cavity over time for the three head gradients. The flux out of the cavities for all three appears to converge. This may be explained by the increased depth of the interface, which leads to greater flow divergence as fluid enters the cavity and reduced the flow-rates at the interface, despite the greater flow rate in the main section of the aquifer. These two forces appear to have roughly balanced each other out in the cases shown, leading to similar rates of dispersive flux out of the cavities. Figure 42 is slightly more puzzling at first. It shows the rate of flux for the three aspect ratios. The base case rate of flux is always greater than the deep case, and although they do appear to be converging, it is unclear if they will reach the same value. However, the rate of flux for the wide case reduces much faster and intercepts and reduces below the rate of flushing for first the base case and then the wide case. This dramatic reduction in the rate of flushing can be understood when it is observed that at the point at which the flushing rates for the base and wide case cross, the amount of brine remaining in the wide cavity is only about 10% of the original mass. The length of the interface is much smaller and the interface is near the bottom of the cavity. In other words, very little is being flushed out simply because there is very little there to flush. On the other hand, the rate of flushing from the deep cavity changes only a little. The rate of flushing is always small, and therefore the factors that affect the rate of flushing flow-rate past the interface and length of the interface change little also. 94

107 These observations highlight the non-linear nature of the rate of dispersive flushing from the cavity. Over time, as brine is flushed out, the length of the interface decreases and the velocity past the interface slows due to the greater divergence of flow into the cavity. The rate of flushing is reduced, therefore so is the rate at which these factors change. This analysis assumes that the two flushing modes are distinct, and that advective flushing is much faster than dispersive flushing. There is no reason to believe this is always the case. In fact, it is entirely possible that the recession of saline wedges is slow enough that dispersive flushing in fact always dominates. More work is required to determine when exactly this will be the case. 4.5 Derivation of an analytical description of flushing modes The following is a derivation of a relationship which describes the relative dominance of the flushing modes described above on a particular cavity. We begin by deriving an expression for the interface position at steady state within the aquifer, under the assumption of zero dispersion and negligible vertical head gradients. This is achieved using the Baydon Ghyben Hertzberg approximation. Using this approximation, and the geometrical properties of the cavity (assuming it is roughly rectangular in section), an equation is derived that gives the proportion of the volume in the cavity that is flushed through advection, and the portion that must be flushed through dispersion. This derivation gives rise to a dimensionless parameter Λ, which is the ratio of the aspect ratio of the cavity and the gradient of the interface within the cavity. Using this equation, it is shown that the flushing mode dominant at any point can be determined if the mass of salt remaining in the cavity is known Baydon-Ghyben-Hertzberg approximation of steady state solute field The first step of this derivation is to determine the steady-state interface position within the cavity. To do so the Baydon Ghyben Hertzberg approximation is used. This well known formula is based on a number of assumptions. Three of these are: Zero dispersion of the interface 95

108 No vertical head gradients No horizontal head gradients within the wedge. We will use these assumptions to derive the required relationship, and then comment on their appropriateness. The formula is usually derived by balancing the pressure at an arbitrary point within the wedge with the pressure in the ocean. In this case the area of brine is not connected with a purely salt-water boundary, so the formulation must be derived again based on these particular circumstances. Consider the following diagram: Figure 48 : Idealised cavity showing nomenclature used in the derivation. ρ f ξ(h f ) ρ s In this diagram, the following nomenclature is used: x, z : are the direction of the x and z axes [m] d : depth of the cavity [m] L : length of the cavity [m] D : volume (per unit width) of brine that is not flushed out by advection [m 3 /m] h f : head of fresh water above the top of the cavity [m] 96 h f0 : head of fresh water above the top of the cavity at x=0 [m]

109 ξ : depth of the interface from the top of the cavity [m] ρ s : density of brine [kg/m 3 ] ρ f : density of freshwater [kg/m 3 ] We begin by assuming that the depth of brine at the downstream (left hand) end of the cavity is equal to the depth d. That is, the brine reaches right to the top of the cavity at this point. Based on the modelling and laboratory results shown above, this is a reasonable assumption. We can then use this point as a known head, in lieu of a fully saline boundary, such as the ocean, used in the original derivation of the Baydon Ghyben Hertzberg equation. As state above, this known head must be balanced with the head at an arbitrary point within the brine. With the assumption that there are no vertical or horizontal head gradients within the brine, any arbitrary point can be used. For the sake of simplicity, the point shall be the base of the brine. If the (hydrostatic) pressure at the base of the cavity on the downstream boundary is equated to some point also at the base, as distance x from the downstream boundary, the result is: ( d ξ ( h )) + ρ g( h ( ξ ) ρ gd + ρ gh = ρ g ) + s f f 0 s f f f x ( 4-3 ) Where the variables are as they are shown in Figure 48. Solving for ξ(h f ), this becomes: ξ ( h f ) ( h ( x) h ) = f f 0 ρ s ρ f ρ f ( 4-4 ) This gives a relationship for the position of the interface as a function of the freshwater head it. Recall that this is based on a number of assumptions, namely that: there is no dispersion, there are no vertical head gradients anywhere and there are no horizontal head gradients within the brine. These assumptions will now be examined in turn. The assumption that there is no dispersion is necessary to distinguish between advective and dispersive flushing of the cavity. No sensitivity analysis has been done on this parameter (other than that discussed in Chapter 3). However, it is assumed that for small values of dispersivity, and relatively high rates of advective flushing, these two flushing modes are distinguishable and dispersive flushing is negligible over the 97

110 timescales of advective flushing. If the timescales of advective flushing were similar to the timescales of dispersive flushing, the two flushing modes may not be distinguishable, and therefore the relationship derived above would be meaningless. The assumption of no vertical head gradients is required to ensure the piezometric freshwater head at the top of the cavity is equal to the freshwater head at the interface. If this were not the case, equation ( 4-3 ) would not be possible because the head at a point in the brine would not be a linear function of the head of water above it. Looking at the modelling presented earlier in this chapter and in Error! Reference source not found.appendix D, at the end of the advective flushing period, there are vertical head gradients at the upstream end of the cavity which reduce the head at the interface and deflect the interface upwards. However, the error this creates is small in the cases presented. It has not been determined under what circumstanced the error would be significant. The assumption of zero horizontal head gradient is required for the two sides of equation ( 4-3 ) to be equal. If it were not the case, the head gradient across the cavity would cause flow within the brine. As can be seen in the modelling, during the transition phase from advective to dispersive flushing modes, there is no (or very little) flow within the brine. However, as dispersion becomes dominant, a convective cell forms within the brine. This cell is caused by the dispersive erosion of the interface, which creates a head gradient across the brine. Therefore, the equation described above only holds for the transition point between advective and dispersive flushing regimes, and in fact it could be argued that it describes the position of the interface at which this transition occurs Steady-state solute field volume as a function of dimensionless parameter Λ Now that an equation for the position of the interface at steady state under conditions of zero dispersion has been derived, it will in turn be used to derive an expression for the volume of brine that must be removed from the cavity under the action of dispersion. Once a certain volume of brine has advected out, the volume remaining (call this D) can be calculated as the volume under the interface given by( 4-4 ) above. To calculate D simply, the assumption is made that the freshwater head varies linearly across the cavity. This assumption is reasonable for small cavities. This allows an equation for ξ to be derived as a function of x only: 98

111 ξ ( x) = h f ρ f x ρ ( 4-5 ) where ρ [kg/m 3 ] is the difference between the brine and freshwater densities, and [m/m] is the (constant) head gradient. D/V is defined as the volume of D normalised to the volume of the cavity. To calculate D/V three cases must be considered (Figure 49). h f D D D Figure 49 : Three cases used to determine the volume of D as a proportion of the total cavity volume In the first case, the interface intersects with the base of the cavity. In this case D is a triangle whose area is sought. At the point of intersection ξ(x)=d. Equation ( 4-5 ) thus gives the length of the base and D/V can be calculated as: D / V = d ρ h ρ f f dl ( 4-6 ) At this point, let us define Λ as a dimensionless parameter describing the ratio of the aspect ratio of the cavity r a to the gradient of the interface: ra d ρ Λ = = ξ ( x) Lh ρ ' f f ( 4-7) 99

112 Equation ( 4-6 ) above simplifies to D/V=½Λ. Moreover, each of the three cases in Figure 49 represent cases of Λ<1, Λ=1 and Λ>1 respectively. It can be shown in fact that D/V is a piecewise continuous function of Λ, such that: 1 2 Λ 1 D / V ( Λ) = Λ Λ < 1 Λ = 1 Λ > 1 ( 4-8 ) This equation gives the proportion of the cavity s volume that must be flushed out by dispersion, assuming the assumptions stated above are adequate. The function is shown in the following diagram, plotted on a semi-logarithmic axis D/V as a function of Deep Base Less Gradient More Gradient D/V Wide (log scale) Figure 50 : D/V as a function of Λ. A number of points can be made about this function. For small Λ, the flushing is mostly advective, and only a small amount is left for dispersion. In fact for Λ=0.1, 95% of the volume of the cavity is flushed by advection. At the other end of the scale, for large Λ flushing is mostly dispersive, with only 5% removed by advection for Λ=

113 As Λ goes to zero, D/V also goes to zero. This is expected since for Λ=0, either d=0, indicating that there is no cavity, or ρ=0, indicating that there is no difference between brine and freshwater concentration, and therefore advection is the only means necessary to flush out the cavity. On the other hand, as Λ approaches infinity, D/V asymptotically approaches 1, indicating that the effects of advection approach zero for deep cavities (large d) dense brines (large ρ) or small head gradients (small h f ) Flushing modes as a function of dimensionless parameter Λ and mass remaining in cavity Using equation ( 4-8 ), predictions can be made about the relative importance of the flushing modes at particular point in time given the mass of salt remaining in the cavity. At the point of transition between flushing modes D/V is approximately equal to the mass of salt remaining in the cavity divided by the initial mass of salt in the cavity before it was flushed. Defining this ratio as M/M i the following statements can be made: If M/M i >>D/V, advective flushing will be dominating the cavity If M/M i D/V, the cavity will be in transition between advective and dispersive flushing and the position of the interface will be predicted by ( 4-5 ) If M/M i <<D/V, dispersive flushing will be dominating the cavity. These statements can be summarised by the following diagram: 101

114 Figure 51 : Flushing regimes as predicted by Λ and mass remaining in cavity, M/Mi The predictions made by the theory can be easily tested against the model data. Testing against the observed experimental data would give greater credence to the results, but this has not been done at this stage. Results from each of the five model runs used in the sensitivity analysis were analysed. The velocity field was examined to determine the time-step at which the transition from advective to dispersive flushing occurred. The presence of significant downward velocities at the downstream wall, indicating the presence of a convection cell, was used as the indicator that the transition had occurred. Values of Λ for each run were calculated using pressure data outputted from the model and the aspect ratio and ( 4-8 ). The table below shows the values used in this calculation. 102

115 Table 3 : Values used for the calculation of Λ for each model run Run Head Gradient Aspect ratio Λ Base Wide Deep More gradient Less gradient The values of M/Mi for each model run were plotted versus the values of Λ. Advective flushing at a time-step was indicated by a circle, while dispersive flushing was indicated by a cross. The result is shown in Figure 52. Figure 52 : Predicted flushing regimes and data observed in the modelling. This result supports the hypothesis that this approach can predict flushing regimes, at least within the range of the data modeled. The line indicating the predicted transition zone between the two flushing types coincides almost exactly with the point that this transition was observed in the modeled data. If there is any discrepancy between the Λ relationship and the observed data, the Λ relationship does tend to predict the onset of dispersive flushing slightly later than 103

116 observed. More work is required to understand the reason for this error, and the circumstances under which it is exacerbated or reduced. 4.6 Conclusion - Advective and dispersive flushing modes This chapter has presented an analysis of flushing from cavities in the basement of a hypothetical aquifer as a salt-water wedge recedes past it. The analysis was carried out using a numerical model calibrated to replicate a hypothetical aquifer. The analysis involved creating an aquifer with a cavity in the basement. The salt-water wedge was initially completely covering the cavity, and filling it with brine. A change in the head gradient across the system caused the wedge to recede past the cavity and allow it to be flushed. The analysis focussed on the processes of brine transport within the cavity once this flushing process began. Two distinct modes of flushing were identified advective and dispersive. Advective flushing is caused by a net pressure gradient across the brine. The brine flows out under the action of this pressure gradient The peak rate of flushing for the wide case compared to the later flushing rate is many times greater than peak flushing rate of the deep case. This indicates that if there are two types of flushing occurring at early and late time, the importance of the early flushing to the wide case is many times greater than it is to the deep case. However, after 75 minutes, here is very little salt left in the wide case to flow out. 104

117 Chapter 5. Conclusion The study of saltwater intrusion has a long and detailed history. This work has looked at a small set of phenomenon that has not yet been examined in this field: the effect of small-scale basement heterogeneities on the movement and position of salt-water wedges. Small scale heterogeneities do not appear to inhibit the progression of the wedge into the aquifer significantly if the ultimate steady-state position of the wedge completely envelops the heterogeneity. It is possible that an intruding heterogeneity may inhibit the movement if it is large enough, but more work is required to examine this further. The most significant effects appear if the wedge is forced out of the aquifer through a change in the boundary conditions. An intruding heterogeneity affects the shape of the interface during the recession even before it is close to the interface. It can also trap brine in a wedge upstream. Cavities can trap brine which may persist for a long time where dispersion and head gradients are low. The behaviour of brine around the heterogeneities is primarily driven be density and horizontal head gradients, but vertical head gradients have also been identified as important for understanding this behaviour. Vertical head gradients caused by the divergence of flow lines close to the heterogeneity were observed to cause attenuation of the toe of the wedge downstream of the heterogeneity. This has important implications for the modelling of salt-water wedges as not accounting for these vertical head gradients may result in under-estimation of the extent of salt-water encroachment by aquifer managers. A theory describing the flushing processes of brines in basement cavities has been presented which has the potential to be useful in a number of areas. Cavities may act as a reservoir of salt within an aquifer for a long time after remediation efforts reverse the encroachment of a salt-water wedge. This work may aid the understanding of how the brine may behave in those cavities if they can be identified. However, physical evidence supporting the theory is currently lacking, and could be the subject of future study. Work in the future could also examine the effect of intruding obstructions, for which similar theories might be developed and tested. More work is required expanding the scope of these findings before this work has applicability to non-idealised systems. Much more analysis of different types of 105

118 intruding and cavity-like heterogeneities is necessary. More important still is analysis of flushing through dispersive systems. The hypothetical aquifer studied had extremely low dispersivity, leading to a very sharp interface. It is possible that the findings would have little relevance where the width of the interface was of a scale similar to the heterogeneity. Eventually it will be necessary to examine the phenomena in a three-dimensional model. This work also touched upon other areas that may provide research opportunities in the future. In particular there is an opportunity to relate the findings of Schotting et al. (1999) on dispersion across high concentration gradients to brines in aquifers. In the high concentration gradient field used in the work presented here, an exact value of dispersivity could not be obtained. A non-linear breakdown in Fick s Law suggested by Schotting et al. (1999) may partially explain the very low dispersivity observed in the experimental work. Also, the permeability of the porous media may have been affected by the high concentration of the brine an observation also made by Schotting et al. (1999). However, a far more careful study is required to rule out experimental error as a source of this observation. 106

119 References Abarca, E., J. Carrera, C. I. Voss, and X. Sánchez-Vila Effect of aquifer bottom morphology on seawater intrusion. in 17th Salt Water Intrusion Meeting (SWIM), Delft. Badon-Ghyben, W Nota in verband met de voorgenomen putboring nabij Amsterdam (notes on the probable results of well drilling near Amsterdam). Tijdschrift van het Koninklijk Instituut van Ingenieurs, The Hague 9:8-22. Bear, J Dynamics of fluids in porous media. American Elsevier Publishing Company, New York. Bear, J Hydraulics of Groundwater. McGraw Hill Inc. Bear, J Mathematical Modeling. Pages in J. Bear, A. H.-D. Cheng, S. Sorek, D. Ouazar, and I. Herrera, editors. Seawater intrusion in coastal aquifers - Concepts, Methods and Practices. Kluwer Academic Publishers. Bear, J., and Y. Bachmat A generalised theory on hydrodynamic dispersion in porous media. Pages 7-16 in I. A. S. H. symposium on artificial recharge and management of aquifers. I. A. S. H., Haifa, Israel. Bear, J., and A. Verruijt Modeling groundwater flow and pollution. D. Reidel Publishing Company, Dordrecht, Holland. Brown, J. S A study of coastal groundwater with special reference to Connecticut. US Geological Survey, Water Supply Paper 537:101. Cheng, A. H.-D., and D. Ouazar Analytical solutions. Pages in J. Bear, A. H.-D. Cheng, S. Sorek, D. Ouazar, and I. Herrera, editors. Seawater intrusion in coastal aquifers - Concepts, Methods and Practices. Kluwer Academic Publishers. Collins, M. A., L. W. Gelhar, and J. L. Wilson Hele-Shaw model of Long-island aquifer system. Journal of the Hydraulics Division : proceedings of the American Society of Civil Engineers 98: Cooper, H A hypothesis concerning the dynamic balance of freshwater and saltwater in a coastal aquifer. Geological Survey Water-Supply Paper 1613-C. 107

120 D'Andrimont, R Notes sur l'hydrologie du littoral belge (notes on the hydrology of the Belgian coast). Ann Soc Geog Belg 29:M129-M144. Debler, W., and J. Imberger Flushing criteria in Estuarine and Laboratory Experiements. Journal of Hydraulic Engineering 122: Demetriou, C., R. E. Volker, and H. De Silva Rates of movement of density affected contaminant plumes in Unconfined aquifers. in Hydrology and Water Resources Symposium, Newcastle. Essaid, H. I USGS SHARP Model. Pages in J. Bear, A. H.-D. Cheng, S. Sorek, D. Ouazar, and I. Herrera, editors. Seawater intrusion in coastal aquifers - Concepts, Methods and Practices. Kluwer Academic Publishers. Essink, G. O Improving fresh groundwater supply - problems and solutions. Ocean and Coastal Management 44: Fetter, C. W Applied Hydrogeology, 3 edition. Prentice-Hall Inc., New Jersey. Gelhar, L. W., C. Welty, and K. R. Rehfeldt A critical reveiw of data on field-scale dispersion in aquifers. Water Resources Research 28: Glover, R. E The pattern of fresh-water flow in a coastal aquifer. Journal of Geophysical Research 64: Hassanizadeh, S. M., and A. Leijnse A Non-Linear Theory of High- Concentration-Gradient Dispersion in Porous-Media. Advances in Water Resources 18: Hazen, A Discussion: Dams on Sand Foundations. Transactions, American Society of Civil Engineers 73:199. Henry, H. R Effects of Dispersion on Salt Encroachment in Coastal Aquifers. Pages C70-C84 in US Geological Survey Water Supply Paper 1613-C, Seawater In Coastal Aquifers. Hertzberg, A Die Wasserversorgung einiger Nordseebder (The water supply of parts of the North Sea coast in Germany. Z. Gasbeleucht. Wasserversorg 44: Hubbert, M. K The theory of groundwater motion. Journal of Geology 48:

121 Kohout, F. A Flow pattern of fresh and salt water in the Biscayne Aquifer of the Miami area, Florida. Int. Assoc. Sci. Sci. Hydrol., Commission of Subterrainian Waters: Lusczynski, N. J., and W. V. Swarenski Salt-Water Encroachment in southern nassau and southeastern Queens counties Long Island, N. Y. US Geological Survey, Water Supply Paper 1613-F:76. Mualem, Y., and J. Bear The shape of the interface in steady flow in a stratified aquifer. Water Resources Research 6: Olivera, I. B., A. H. Demond, and A. Salehzadeh Packing Sands for the production of Homogeneous Porous Media. Soil Science Society of America Journal 60: Oostrom, M., J. S. Hayworth, J. H. Dane, and O. Güven Experimental Investigation of Dense Solute Plumes in an unconfined Aquifer Model. Water Resources Research 28: Oostrom, M., and J. S. D. Hayworth, J. H.Güven, O Behaviour of Dense Aqueous phase Leachate plumes in Homogeneous porous media. Water Resources Research 28: Reilly, T. E., and A. S. Goodman Quantitative analysis of saltwater-freshwater relationships in groundwater systems - a historical perspective. Journal of Hydrology 80: Schincariol, R. A., E. E. Herderick, and F. W. Schwartz On the application of image analysis to determine concentration distributions in laboratory experiments. Journal of Contaminant Hydrology 12: Schincariol, R. A., and F. W. Schwartz An experimental Investigation of Variable density flow and mixing in Homogeneous and Heterogeneous Media. Water Resources Research 26: Schotting, R. J., H. Moser, and S. M. Hassanizadeh High-concentration-gradient dispersion in porous media: experiments, analysis and approximations. Advances in Water Resources 22: Segol, G Classic groundwater simulations : proving and improving numerical models / Genevieve Segol. PTR Prentice Hall,, Englewood Cliffs, N.J. :. 109

122 Shepherd, R. G Correlations of permeability and grain size. Ground Water 27: Sherif, M Nile Delta Aquifer in Egypt. Pages in J. Bear, A. H.-D. Cheng, S. Sorek, D. Ouazar, and I. Herrera, editors. Seawater intrusion in coastal aquifers - Concepts, Methods and Practices. Kluwer Academic Publishers. Silliman, E. S., B. Berkowitz, J. Simunek, and M. T. van Genuchten Fluid flow and solute migration within the capillary fringe. Groundwater 40: Simmons, C. T., T. R. Fenstemaker, and J. M. Sharp Variable-density groundwater flow and solute transport in heterogeneous porous media: approaches, resolutions and future challenges. Journal of Contaminant Hydrology 52: Stakelbeek, A Movement of Brackish Groundwater Near a Deep-Well Infiltration System in the Netherlands. Pages in J. Bear, A. H.-D. Cheng, S. Sorek, D. Ouazar, and I. Herrera, editors. Seawater intrusion in coastal aquifers - Concepts, Methods and Practices. Kluwer Academic Publishers. Stern, W., M. Jacobs, and S. Schmorak Hydrogeological investigations in the southern coastal plain of Israel. Hydrol. Serv., State of Israel,. Strack, O. D. L A single potential solution for regional interface problems in coastal aquifers. Water Resources Research 12: Swartz, C. H., and F. W. Schwartz An experimental study of mixing and instability development in variable density systems. Journal of Contaminant Hydrology 34: van Genuchten, M. T A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal 44: Volker, A Source of the brackish groundwater in Pleistocene formations beneath the Dutch ponderland. Econ. Geol. 56: Voss, C. I A finite element simulation model for saturated-unsaturated, fluiddensity-dependant groundwater flow with energy transport or chemically reactive single-species solute transport. Water Resources Investigations Report , US Geological Survey. 110

123 Voss, C. I UGS SUTRA Code - History, Practical Use, and Application in Hawaii. Pages in J. Bear, A. H.-D. Cheng, S. Sorek, D. Ouazar, and I. Herrera, editors. Seawater intrusion in coastal aquifers - Concepts, Methods and Practices. Kluwer Academic Publishers. 111

124 112

125 Appendix A. Experimental Design The first stage of the work involved conducting experiments in a sand-box to simulate a 2-dimensional slice of a hypothetical aquifer. The sand-box was already constructed and had previously been used for other experiments. This reduced the flexibility of the experimental design. However it was still necessary to make some decisions about certain aspects of the experimentation. To do this, a rough draft numerical model was developed. A.1 Model Setup The model used for this rough draft was the density dependant groundwater flow and transport model SUTRA. This model and the model setup are described in detail in Chapter 3. The setup of the model will be briefly summarised below. A coarse grid was constructed of 64 by 31 nodes equally spaced within a 1.68m wide by 0.8m high 2-D domain. The right hand boundary was set as the upstream boundary with 0.8m of freshwater head (measured from the base). The left-hand boundary was set as the downstream boundary. It s density and head were varied. The intrinsic permeability used was 2.56x10-10, which had been determined after the initial packing of the tank (see section Hydraulic Conductivity Tests). The value of longitudinal dispersivity used was 0.01m close to the minimum possible that would not create numerical instability. Transverse dispersivity was an order of magnitude smaller. The model was run with time steps of 30sec, outputted every 5min. A.2 Density and head gradient The movement of water through a porous medium is generally a very slow process, even with highly permeable media such as glass beads. The experimental setup had to be designed such that timescales were reasonable. It was assumed that the rate at which the salt wedge moved in and out of the aquifer was related to the density of the brine and the head gradient across the system, as well as the hydraulic conductivity. According to the Baydon-Ghyben-Hertzberg 113

126 approximation, head gradient and brine density would also affect the steady-state position of the wedge. The aim of the first set of modelling was therefore to determine a combination of brine density and head gradient that brought the wedge to a position of steady state quickly and at an appropriate position in the tank. A.1.1. Method To determine the best combination of density and head gradient, trial and error was used. Models were run at 1x, 2x, 3x and 6x the concentration of seawater (assumed for simplicity to be kg/kg or 1025km/m 3 ). The downstream head was adjusted and the models run repeatedly until the wedge reached steady state at approximately 1.5m from the downstream end of the tank. The time at which the toe of the wedge reached 1.45m was recorded. A.1.2. Results The results of the preliminary modelling are shown in the following table: Table 4 : Results of the first set of preliminary modeling, showing the effect of head gradients and brine densities on time taken to reach steady state. Concentration [kg/kg] Density [kg/m 3 ] Downstream head [m brine] 1xseawater hr 2xseawater hr 3xseawater hr 6xseawater hr Time to reach 1.45m The toe position as a function of time is shown for each of these cases in Figure

127 Figure 53: The above plot shows the position of the toe of the wedge for four combinations of head difference and density difference. A.1.3. Discussion These results indicate that very high brine concentrations are required for it to take less than 5 hours to reach steady state. To achieve a concentration of 6 times seawater requires about a fifth of the mass of brine to be salt. There was a concern that it would be difficult to produce brine in the laboratory of this concentration quickly and easily. In fact in cold water it was found to be almost impossible. However, if pure salt was stirred continuously while hot water was added to it, brine of sufficient density could be obtained. This brine needed to be cooled before it could be used. A.3 Obstruction location The second aim of the preliminary modelling was to choose a height and position of the obstruction at the base of the tank that was large enough to be observable, but would be completely enclosed in brine at steady state for one set of boundary conditions and completely exposed at another. The obstruction was to consist of two walls bounding a cavity that was roughly square in the plane of the aquifer cross-section. The position and height of these walls had to be selected such that the wedge could progress into the aquifer from one side, move 115

128 over the obstruction and become disassociated from it before the toe of the wedge reached the upstream boundary. This was necessary in order to examine the effect of the obstruction while within the wedge at steady state. However, the wedge had to be some way forward from the downstream boundary to allow the wedge to retreat past it completely following a change in boundary conditions. A number of obstruction positions were trialled, and it was found that an obstruction in roughly the middle of the tank, about 10-12cm high would be optimum. However the exact location and height could be varied without significant effect on the wedge, as long as it was small enough that the progressing wedge could move over it. 116

129 Appendix B. Comparison of model output with observed interface position The following plots show the concentration field calculated by the SUTRA model and the observed interface position. The interface is shown by the light grey line. These plots include the erroneous scaling factor discussed in the model calibration section. B.1 Homogenous basement, progression 117

130 B.2 Homogenous basement, recession 118

131 * No observed data at this time step B.3 Heterogenous basement, progression 119

132 120

133 121

134 * No observed data at this time step B.4 Heterogenous basement, recession 122

135 123

136 Appendix C. Matlab M-Files C.1 M-Files for converting SUTRA output into MATLAB variables C.1.1. d3q % Function for opening a SUTRA *.d3 output file with filename 'file' (string), % and returning: % steps - the number of timesteps % tstep - a vector of the timesteps % nodes - the number of nodes % fid - the file handle %This initialised file is to be read with the d3in function function [steps, tstep, nodes,fid]=d3q(file) fid=fopen(strcat(file,'.d3')); frewind(fid); fgetl(fid); fgetl(fid); fgetl(fid); fscanf(fid,'%c',6) nodes=fscanf(fid,'%i',1); fgetl(fid); fscanf(fid,'%c',4); steps=fscanf(fid,'%i',3); fgetl(fid); fgetl(fid); fgetl(fid); A=fscanf(fid,'%s %e %s %e %e %e',[10 steps]); A=A'; tstep=a(1:steps,10); fgetl(fid); fgetl(fid); fgetl(fid); C.1.2. d3in % Function for reading data from a *.d3 SUTRA output file % opened using the d3q function. This function takes imputs % of the file handle 'fid' and the number of nodes 'nodes' and outputs: % P - a matrix containing the pressure at each node [N/m2] % S - a matrix containing the saturation at each node [] % C - a matrix containing the concentration at each node [kg/kg] % the code is currently set up to accomodate a rectangular domain % of rectangular elements, which may have a cavity at the base function [P,S,C]=d3in(fid,nodes); S=1; A=ones(nodes,6).*NaN; for i=1:nodes s=fgetl(fid); 124

137 A(i,1:6)=str2num(s); end fgetl(fid); fgetl(fid); fgetl(fid); fgetl(fid); X=A(:,2); Y=A(:,3); press=a(:,4); conc=a(:,5); conc(find(conc==1))=0; conc(find(conc==2))=0; conc(find(conc==3))=0; ny=length(find(x==1.68)); nx=length(find(y==0.8)); nsx=max(x)/(nx-1); nsy=max(y)/(ny-1); cx=length(find(y==min(y))); cy=-min(min(y))./nsy; C(1:(ny+cy),1:nx)=NaN; S(1:(ny+cy),1:nx)=NaN; P(1:(ny+cy),1:nx)=NaN; for i=1:length(x) C(round(Y(i)/nsy)+1+cy, round(x(i)/nsx)+1)=conc(i); S(round(Y(i)/nsy)+1+cy, round(x(i)/nsx)+1)=sat(i); P(round(Y(i)/nsy)+1+cy, round(x(i)/nsx)+1)=press(i); end C.1.3. d4q % Function for opening a SUTRA *.d4 output file with filename 'file' (string), % and returning: % steps - the number of timesteps % tstep - a vector of the timesteps % nodes - the number of nodes % fid - the file handle %This initialised file is to be read with the d4in function function [steps, tstep, nodes,fid]=d4q(file) fid=fopen(strcat(file,'.d4')); frewind(fid) fgetl(fid); fgetl(fid); fgetl(fid); fscanf(fid,'%c',6); nodes=fscanf(fid,'%i',1); fgetl(fid); fscanf(fid,'%c',4); steps=fscanf(fid,'%i',3); fgetl(fid); fgetl(fid); fgetl(fid); A=fscanf(fid,'%s %e %s %e %e %e',[10 steps]); A=A'; tstep=a(1:steps,10); fgetl(fid); fgetl(fid); fgetl(fid); C.1.4. d4in % Function for reading data from a *.d4 SUTRA output file % opened using the d4q function. This function takes imputs 125

138 % of the file handle 'fid' and the number of nodes 'nodes' and outputs: % VX - a matrix containing the velocity in the X direction [m/s] % VX - a matrix containing the velocity in the Y direction [m/s] % the code is currently set up to accomodate a rectangular domain % of rectangular elements, which may have a cavity at the base function [VX,VY]=d4in(fid,nodes) A=ones(nodes,5).*NaN; for i=1:nodes s=fgetl(fid); A(i,1:5)=str2num(s); end X=A(:,2); Y=A(:,3); vx=a(:,4); vy=a(:,5); fgetl(fid); fgetl(fid); fgetl(fid); fgetl(fid); ny=length(find(x==max(x))); nx=length(find(y==max(y))); nsx=max(x)/(nx-0.5); nsy=max(y)/(ny-0.5); cx=length(find(y==min(y))); cy=-min(min(y))./nsy+0.5; VX(1:(ny+cy),1:nx)=NaN; VY(1:(ny+cy),1:nx)=NaN; for i=1:length(x) VX(round((Y(i)-nsy/2)/nsy)+1+cy, round((x(i)-nsy/2)/nsx)+1)=vx(i); VY(round((Y(i)-nsy/2)/nsy)+1+cy, round((x(i)-nsy/2)/nsx)+1)=vy(i); en C.2 M-Files for manipulating and presenting model data C.2.1. d3mmbgh % A number of m-files were used to plot the output of the SUTRA model. % The following is just one example that happens to contain many of the % subroutines written to manipulate the data in different ways % Function to output the concentration field and advective salt flux % Inputs are: % file - string containing the name of the simulation % t - the length of the timestep % cmax - the maximum concentration % get - a vector listing the timesteps to be outputted % Output is a vector, m, containing the mass of salt in a cavity at the base of % the model domain function m=d3mmbgh(file,t,cmax,get) % Initialise the STURA output files for reading [steps, tstep, nodes,fid]=d3q(file); [steps4, tstep4, nodes4,fid4]=d4q(file); 126

139 % Loop over timesteps for i=1:steps % read output from SUTRA files [H S C]=d3in(fid,nodes); [VX,VY]=d4in(fid4,nodes4); % if the output is to be outputted if ~isempty(find(i==(get/t+1))) %% create a mirror image of the data (not required) C=fliplr(C); H=fliplr(H); VY=fliplr(VY); VX=-fliplr(VX); % caluculate the advective salt flux [CX,CY]=CtoVC(C,VX,VY); % convert the concentration from mass-fraction to kg/m3 density=ones(size(c)).*1000+c.*700; c=c.*density; % determine the size and position of the cavity d=max(find(isnan(c(:,1)))); % depth s=min(find(isfinite(c(1,:)))); % first column of nodes fi=max(find(isfinite(c(1,:)))); % last column of nodes % Calculate the mass of salt in the cavity [ce a]=ctovc(c(1:(d+1),s:fi),ones(size(c(1:(d),s:(fi-1)))),1); m(i)=sum(sum(ce))*(0.016*0.016); % Calculate the Baydon-Ghyben-Hertzberg approximation % of the interface in the cavity [bgh,hgrad]=bgh(h,s,fi,d,0); % plot the data f=figure; % First, concentration pcolor(linspace(0,1.68,size(c,2)),linspace(- d*0.016,0.8,size(c,1)),c); hold on % then, advective salt flux q=quiver(linspace(0.008,1.672,size(cx,2)),linspace(- d* , ,size(cx,1)),cx,cy,6,'k'); shading interp; caxis([0 cmax]); title(strcat(file,' salt concentration and flux at t=',num2str((i-1)*t),'min')); axis image; axis([0,1.68,-d*0.016,0.8]); set(gcf,'position',[ ]) pause; % save as a.jpg %saveas(gcf,strcat(file,'t',num2str((i-1)*t),'.jpg')); % save as a black and white jpg colormap(flipud(gray)); set(gca,'color',[0.3,0.3,0.3]); set(q,'color',[0.65,0.65,0.65]) %saveas(gcf,strcat(file,'t',num2str((i-1)*t),'bw','.jpg')); % plot the BGH line p=plot(linspace((s-1)*0.016,(fi-1)*0.016,fi-s+1),bgh,'--w'); set(p,'linewidth',2); set(p,'color',[0.2,0.2,0.2]) 127

140 end % save as a *.fig file %saveas(gcf,strcat(file,'t',num2str((i-1)*t),'bw','.fig')); pause; close(f); end if i==(max(get)/t+1) break end C.2.2. CtoVC % Function to convert concentration from node-wise data into % element-wise data through evenly weighted interpolation. The % concentration is multiplied by the components of the velocity % vector to yeild an advective salt flux function [CX,CY]=CtoVC(C,VX,VY) [nyc,nxc]=size(c); [nyv,nxv]=size(vx); for i=1:nxv for j=1:nyv CN(j,i)=mean(mean(C([j,j+1],[i,i+1]))); end end CX=CN.*VX; CY=CN.*VY; C.2.3. BGH % Function to calculate the depth of the interface as predicted % by the Baydon Ghyben Hertzberg approximation. % Inputs are % P - The pressure feild % s - first column of nodes in the cavity % fi - last column of nodes in the cavity % d - depth of the cavity % hgrad - the head gradient across the cavity % outputs are % bgh - depth of BGH interface below basement at each node in the cavity % hgrad - as above % note hgrad can either be inputted or calculated function [bgh,hgrad]=bgh(p,s,fi,d,hgrad) if hgrad==0 hgrad=(p(d+1,fi)-p(d+1,s))/(9810*(fi-s)*0.016); end bgh=-(0:(fi-s))*0.016*hgrad*1000/150; C.3 M-files to process digital images C.3.1. interface % Function that handles the input of the interface position. % Current directory must contain all the image files. Files 128

141 % should have a common prefix and a sequentially numbered suffix. % Inputs are % filestart - string of file suffix % init - suffix integer of first file % final - suffix integer of last file % outputs are % points - 3 dimensional matrix of x and y co-ordinates of interface % position for each image % note that a file IFpoints is created which saves the matrix points function points=interface(filestart,init,final) c=1; points=ones(10,2,final-init).*nan; origin=0; x0=0; y0=0; s=0; for i=init:final if i>999 bit=''; else if i>99 bit='0'; else if i>9 bit='00'; else bit='000'; end end end [a x0 y0 s]=getpoints(strcat(filestart,bit,num2str(i),'.jpg'),i,origin,x0,y0,s) ; origin=1; points(1:size(a,1),1:size(a,2),c)=a; points(find(points==0))=nan; save IFpoints points; end C.3.2. getpoints % Function that displays a picture file and then accepts inputs % of base lengths unless a scaling factor has been specified, % an origin if one is not already specified and finally the % location of mouse clicks on the picture. Right mouse button % clicks are recorded as NaNs. function [points, x0, y0, s]=getpoints(file,i,origin,x0,y0,s) A=imread(file,'jpg'); h=figure; set(gcf,'position',[ 1 image(a); ]) axis image; axis manual; title(strcat('image ',num2str(i))); hold on; if origin==0 [x00,y00]=ginput(1); [x11,y11]=ginput(1); [x0,y0]=ginput(1); s=(x11-x00)*1; plot(x00,y00,'o'); plot(x11,y11,'o'); plot(x0,y0,'o'); y0=ones(size(y0))*size(a,1)-y0; 129

142 end [x y b]=ginput; r=find(b==3); x(r,:)=nan; y(r,:)=nan; y=ones(size(y))*size(a,1)-y; x=x-ones(size(x0))*x0; y=y-ones(size(y0))*y0; x=x./s; y=y./s; close(h); points=[x y]; 130

143 Appendix D. Flushing from cavities Deep Cavity at 20 min and 30 min More gradient at 15 min and 30 min Base case at 20 min and 35 min 131

144 Less gradient at 25 min and 50 min Wide case at 25 min and 45min 132